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Would a city underground in the desert make sense from a survival standpoint?
In the desert, the major factors of survival are water, food and temperature. I have a city in a desert where the first two are covered by a nearby river and agriculture from that river, but would it make sense to build a city underground to beat the heat? Or would it be better to adapt the species living in the city to higher temperatures?
reality-check cities architecture underground deserts
HDE 226868♦
decayedarachniddecayedarachnid
$\begingroup$ How far beneath the surface would the "floor" of your city then be? (If it has multiple levels, a range of depths is fine.) $\endgroup$
– type_outcast
$\begingroup$ Related: worldbuilding.stackexchange.com/questions/23373/… $\endgroup$
– HDE 226868 ♦
$\begingroup$ @type_outcast Deep enough to be insulated against the sun and weather/for the temperature to be livable. I don't know the exact number, though. $\endgroup$
– decayedarachnid
$\begingroup$ Coober Pedy is an excellent example of this actually happening. $\endgroup$
– Roland Heath
$\begingroup$ Sometime it does rain in or near deserts, and when it does there can be severe flash floods. So make sure your underground bunker is above the flood plain, even in the desert. $\endgroup$
– RBarryYoung
Yes, underground would help with heat
Given food and water are covered, underground living will be cooler$^1$ and safer than living in direct sunlight, but there are some caveats:
Geothermal gradient
As you descend into the crust, the temperature increases in a steady, predictable fashion known as the geothermal gradient, which is about $25^{\circ}\text{C}/\text{km}$. Thus, your best bet is staying just a few meters below the surface. That way you'll get almost all of the insulating effects of the ground, while avoiding the increasing temperatures from below.
Digging is hard
Digging an underground city would be prohibitively difficult. Your people would be much better off to find an existing system of caves.
Ventilation and heat regulation
People generate a lot of heat and $\text{CO}_{2}$ that need to be exchanged for fresh, cool air. Putting multiple entry/exit points to your city will help, but you'll still need air flow, and this will certainly take a lot of effort!
You didn't specify the level of technology your people are at. Fortunately I can present a fan that requires practically no technology at all:
Put quite simply, you take a large piece of lightweight fabric, animal skin, whatever, and affix it to a wooden frame about $16 \times 16\text{"}$ ($40 \times 40 \text{cm}$). Make several of these. Then you have volunteers/workers/slaves at the (preferably ramped) entrances constantly push the air out. You will need one such fan for every 10–100 inhabitants depending on how densely packed the city is.
Inspiration: Nuclear War Survival Skills book
My inspiration came from the public domain book Nuclear War Survival Skills. Here is their diagram of the fan I described:
At around p.59, they state that this fan can move 300 cubic feet/min, which is enough for 9 very crowded adults in hot weather, or up to 100 in cool weather.
That book describes some other fans and is in general a great read for designing underground living on a small scale.
With more technology, you can automate any type of fan somewhat by putting the fans on circular wheels or belts, and use some pulleys and gears to give the people (or beasts of burden) a mechanical advantage.
Again, ventilation is necessary (and very easy to underestimate!). It will be hard work, but if done adequately, the underground would remain cool and hospitable to human life.
You say you already have a nearby river. Does it run underground? Or is it at least somewhat near your underground city? If you can pipe some of it through your city, your inhabitants will have a much easier time with it, and the water will help cool the city even more.
You didn't ask about lighting, so I'll keep it short: mirrors. Placed around your ventilation/entrance shafts, you can "beam" sunlight into the city. Your people would want to keep fires to a minimum, as they generate lots of carbon monoxide, which can be easily fatal.
More moderate. Cooler during the daytime, and warmer during the night (a lot of deserts get quite cool during the night).
type_outcasttype_outcast
$\begingroup$ Geothermal gradiant? At 25C/Km, that means you could go a fairly impressive 100m down and still only get +2.5C from that. $\endgroup$
– Dronz
$\begingroup$ Don't forget that water flow could easily be adapted to turn those fans too! $\endgroup$
– thanby
$\begingroup$ By building the tunnels right you can avoid manual fans. For example line an area with black rock in full sunlight so it heats up, the rising air there will pull air through your network. Fireplaces can also be used to achieve the same thing by pulling air in towards them. $\endgroup$
– Tim B
Your question is bit misleading. The important thing is not to be underground, the important thing is to have thermal mass of stone around you. A historical example of what you want is Petra. Similar use goes up to stone age where people lived in caves that had stable temperatures year around. In addition to Petra other examples of rock-cut architecture exist in Cappadocia and India where it was a result of suitable rock formations.
In fact, I'd go so far as to say that such architecture will mainly exist in areas where you have natural caves or rock formations and easily workable stone so that the amount of work required is exceptionally low. Otherwise it will be easier to build above ground and just make the walls more massive. Typical solutions are adobe, mud brick, or packed earth which naturally allow relatively simple and easy construction of thick walls with high thermal mass.
Actually building underground would generally be impractical since while the resulting architecture would indeed have relatively stable temperature the amount of work required would be higher than with other alternatives with same protection from heat. Additionally being actually underground would make it more difficult to deal with floods and sand. It is generally better to live so that gravity helps you keep your home safe from such issues.
That said, it is reasonable for the desert city to have significant infrastructure underground. Underground aqueducts or qanats or likely. Similarly underground tunnels make a good source of cool air for ventilation. This would be combined with a windcatcher towers or similar.
I guess you could say that the optimum is a combination of below and above ground elements. And that building above ground usually requires less labor and is the default barring natural caverns or exceptionally easily workable stone or special needs as with water conduits.
Ville NiemiVille Niemi
$\begingroup$ They could heap earth up around their houses and make them underground that way. They'd need strong walls and roofs. Do they have thick tree trunks? How are their roofs made now? Buildings made of stone with domed roofs could take the weight. $\endgroup$
– RedSonja
$\begingroup$ @RedSonja That is what packed earth is for. Added link to the wikipedia article about it to the answer. Although it uses "rammed earth". $\endgroup$
– Ville Niemi
$\begingroup$ I probably should expand on what @RedSonja said and my answer to it. The proper solution to providing thermal mass depends on the amount of thermal mass needed which depends on the magnitude of temperature variation, which is roughly proportional to the difference between maximum and minimum temperature and the cycle of the variation. While in a desert the temperature differences are large the cycle is generally between day and night. For that making the walls massive using methods I covered in the answer is enough, Earth sheltering is only needed for longer seasonal cycles. $\endgroup$
Insulation against sun, wind, rain, and temperature changes! The geothermal gradient can help keep a cave system warm. Some houses use geothermal heat pumps to warm or cool as needed, which can be extended to caves
Caves can often have natural choke points, allowing for easy defense.
Your city may be hard to spot. After all, it looks like any other bit of land!
The Other Issues
Light is an issue. Do you have skylights? Mirror systems? Do you use a lot of candles?
You need to dig out caves and rely on some structural engineering to keep things up. People have had great success, otherwise!
Fresh air needs to enter somehow. This requires ventilation, but that can be done.
PipperChipPipperChip
The fact that underground cities in the desert in our own world are very few and far between, as far as I know, might indicate that it would either require some special knowledge or else a whole lot of work to create an underground city in one. I think adapting to the heat or finding another way to cope with it would be easier, and more likely, personally, but then, I like heat, and I like to study heat-tolerant life. Nevertheless, I'm sure you could find a way to make an underground city in the desert (even a practical way), but it sounds like it might not be the easiest thing to figure out. An underground city might start to smell after a while, too (however, probably less in a desert than a humid area, is my guess). If it was close to a river, it might get a lot of water from the river running into it (and I imagine it would smell a lot more; the dry banks of rivers that once were wet can smell pretty interesting).
You might consider what termites do.* They have natural air conditioning with the way their mounds are set up. This would be helpful to establish a constant temperature (deserts get cold at night). In the case of termites, they keep it at about 30° C all the time, where it fluctuates between about 0–40° C. outside. They establish a constant draft of air, and build so that the sun doesn't hit it as much at certain times.
(*Note in the link that it says termites have a brain the size of a pinhead, but I've heard that science has established that the size of the brain doesn't regulate intelligence necessarily, but rather ability to control a larger body.)
You might also consider that the people of your desert might have materials that reflect infrared light (which could cool down whatever they're placed around a lot). Infrared isn't some special space aged thing. It's just a color you can't really see much, if at all (unless you've got super powers), because it's outside the visible spectrum for humans. (And, infrared light heats things up, much how UV rays give people sunburns, kill microbes, and stimulate vitamin D production. Plastic and modern glass usually block UV rays, and this helps to prevent damage of some kind or other—including to vitamins in milk and such, I believe. So, you would need a color that reflects infrared to keep things inside cooler.) In our modern world, you can get clear inserts to go on windows to block infrared (and stop the house from warming up through the windows). You could do the same thing to your whole house (or city) with the right color of paint or something.
BrōtsyorfuzthrāxBrōtsyorfuzthrāx
$\begingroup$ Have +1 for the termite idea. Was going to add it as a comment, but has already been covered. $\endgroup$
– Darren Bartrup-Cook
Survivability is probably not an issue, as others have stated - increased insulation from the heat, etc. The difficult part of this is creating a history for such a city. One doesn't simply pick a spot in the desert and decide to build a city there, there has to be a back story to explain how it got there. One possibility would be for it to have started as a normal, above-ground settlement that over time became increasingly buried under centuries of sandstorms. People started building covered walkways between the buildings to keep out the sand, and building tall chimney-like structures, both to allow for air circulation and also some with ladders to provide a means of egress from the buried buildings. These chimneys would continue to be built taller as the sand became deeper.
Eventually, as modern technology became popular, they might also run electrical cables down some of the chimneys, as well as plumbing/sewage, and later telephone, ethernet, fiberoptic, etc. Some might even be converted into elevator shafts.
Prior to being hooked up to electricity, their primary source of light would be torches, which produce smoke, and so these would also need to be placed near the chimney structures. It's possible you could line the edges of chimneys with polished metal to act as mirrors and bring some daylight down into the buildings, but again, this means most of the light would be near the chimneys. Some enterprising architects might see the benefit of designing arched ceilings so that any smoke from light sources would collect into the chimneys.
One problem with this scenario is it does make it difficult to expand the city, as digging underground in sand is not a simple task. It's possible that additional structures would be built on the surface, which might themselves be buried by more sandstorms, so you have layer-upon-layer of city, with the deepest parts being the oldest and newer structures being closer to the surface. The most recent developments would be on the surface. Of course, any surface structures would have to be positioned such that they are not directly on top of the chimneys from the deeper chambers.
Eventually, social strata would start to form based on depth. The above-ground level would be mostly traders and craftsmen, the type who would do business with outsiders most frequently. Below them would be the aristocracy, who being the wealthiest residents would choose the most comfortable lodgings - deep enough to be protected from the heat, but not so deep as to have little access to sunlight, fresh air, quick trips to the surface, etc. After that, things would go steadily downward in terms of social standing, with the poorest people relegated to living in the oldest, deepest parts of the city.
Darrel HoffmanDarrel Hoffman
The best thing to do would be both. Snakes, lizards, and other creatures live in the ground during the day, when it's hot, then come out at night. You species could be nocturnal, resistant to heat, and be able to come out during the day if necessary. Being underground also helps insulate against all kinds of things.
Xandar The ZenonXandar The Zenon
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Weighted area constraints-based breast lesion segmentation in ultrasound image analysis
April 2022, 16(2): 467-479. doi: 10.3934/ipi.2021058
Counterexamples to inverse problems for the wave equation
Tony Liimatainen 1,2, and Lauri Oksanen 2,
Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland
Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
Received January 2021 Published April 2022 Early access October 2021
We construct counterexamples to inverse problems for the wave operator on domains in $ \mathbb{R}^{n+1} $, $ n \ge 2 $, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On $ \mathbb{R}^{n+1} $ the metrics are conformal to the Minkowski metric.
Keywords: Inverse problems, counterexamples, wave equation, conformal scaling, Lorentzian manifold, partial data, hidden conformal invariance.
Mathematics Subject Classification: 35R30, 35L05, 58J45.
Citation: Tony Liimatainen, Lauri Oksanen. Counterexamples to inverse problems for the wave equation. Inverse Problems & Imaging, 2022, 16 (2) : 467-479. doi: 10.3934/ipi.2021058
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2020 Impact Factor: 1.639
Tony Liimatainen Lauri Oksanen | CommonCrawl |
Small dodecahemidodecacron
In geometry, the small dodecahemidodecacron is the dual of the small dodecahemidodecahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the small icosihemidodecacron.
Small dodecahemidodecacron
TypeStar polyhedron
Face—
ElementsF = 30, E = 60
V = 18 (χ = −12)
Symmetry groupIh, [5,3], *532
Index referencesDU51
dual polyhedronSmall dodecahemidodecahedron
Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity.[1] In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.
The small dodecahemidodecahedron has six decagonal faces passing through the model center, the small dodecahemidodecacron can be seen as having six vertices at infinity.
References
1. (Wenninger 2003, p. 101)
• Wenninger, Magnus (2003) [1983], Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208 (Page 101, Duals of the (nine) hemipolyhedra)
External links
• Weisstein, Eric W. "Small dodecahemidodecacron". MathWorld.
Star-polyhedra navigator
Kepler-Poinsot
polyhedra
(nonconvex
regular polyhedra)
• small stellated dodecahedron
• great dodecahedron
• great stellated dodecahedron
• great icosahedron
Uniform truncations
of Kepler-Poinsot
polyhedra
• dodecadodecahedron
• truncated great dodecahedron
• rhombidodecadodecahedron
• truncated dodecadodecahedron
• snub dodecadodecahedron
• great icosidodecahedron
• truncated great icosahedron
• nonconvex great rhombicosidodecahedron
• great truncated icosidodecahedron
Nonconvex uniform
hemipolyhedra
• tetrahemihexahedron
• cubohemioctahedron
• octahemioctahedron
• small dodecahemidodecahedron
• small icosihemidodecahedron
• great dodecahemidodecahedron
• great icosihemidodecahedron
• great dodecahemicosahedron
• small dodecahemicosahedron
Duals of nonconvex
uniform polyhedra
• medial rhombic triacontahedron
• small stellapentakis dodecahedron
• medial deltoidal hexecontahedron
• small rhombidodecacron
• medial pentagonal hexecontahedron
• medial disdyakis triacontahedron
• great rhombic triacontahedron
• great stellapentakis dodecahedron
• great deltoidal hexecontahedron
• great disdyakis triacontahedron
• great pentagonal hexecontahedron
Duals of nonconvex
uniform polyhedra with
infinite stellations
• tetrahemihexacron
• hexahemioctacron
• octahemioctacron
• small dodecahemidodecacron
• small icosihemidodecacron
• great dodecahemidodecacron
• great icosihemidodecacron
• great dodecahemicosacron
• small dodecahemicosacron
| Wikipedia |
Diffeomorphic Mapping and Shape Analysis
Over the past 20 years, a last collection of work has been dedicated to the definition of shape, and shape spaces, as mathematical objects, and to their applications to various domains in computer graphics and design, computer vision and medical imaging. In this last context, an important scientific field has emerged, initiated by U. Grenander and M. Miller, called Computational Anatomy. One of the primary goals of computational anatomy is to analyze diseases via their anatomical effects, i.e., via the way they affect the shape of organs. Shape analysis has demonstrated itself as a very powerful approach to characterize brain degeneration resulting from neuro-cognitive impairment like Alzheimer's or Huntington's, and has contributed to deeper understanding of disease mechanisms at early stages.
Whether represented as a curve, or a surface, or as an image, a shape requires an infinite number of parameters to be mathematically defined. It is an infinite-dimensional object, and studying shape spaces requires mathematical tools involving infinite-dimensional spaces (functional analysis) or manifolds (global analysis). Some example are reviewed in the excellent survey paper from Bauer et al..
Spaces emerging from Grenander's Pattern Theory have a special interest, because of their generality and flexibility. These spaces derive from the structure induced by groups of diffeomorphisms through the deformation induced by their actions on shapes. In this context, a shape is not represented as such but as a deformation of another (fixed) shape, called template. The deformable template paradigm is rooted in the work of D'Arcy-Thompson in his celebrated treatise (On Growth and Form), and developed in Grenander's theory. Even if Pattern Theory can be more general, recent models of deformable templates in shape analysis focus on deformations represented by diffeomorphisms acting on landmarks, curves, surfaces or other structures that can represent shapes. More precisely, if $T_0$ is the template, one represents shapes via the map $\pi(\varphi) = \varphi \cdot T_0$, which denotes the action of a diffeomorphism, $\varphi$ on $T_0$ ($\varphi$ being a diffeomorphism in the ambient space, in contrast to changes of parametrization, which are diffeomorphisms of the parametrization space). With this model, the diffeomorphism can be interpreted as an extrinsic parameter for the representation.
Letting $\mathrm{Diff}$ denote the space of diffeomorphisms, and $\mathcal Q$ be the shape space, one can use the transformation $\pi : \mathrm{Diff} \to \mathcal Q$ to "project" a mathematical structure defined on diffeomorphisms to the shape space. Using this paradigm, one can, from a single modeling effort (on $\mathrm{Diff}$) design many shape spaces, like spaces of landmarks, curves surfaces, images, density functions or measures, etc.
The space of diffeomorphisms, which forms an algebraic group, is a well studied mathematical object. The relationship between right-invariant Riemannian metric on this space and classical equations in fluid mechanics has been described in V.I. Arnold's seminal work, followed by a large literature, by J.E. Marsden, T. Ratiu, D.D. Holm and others. It is remarkable that the same construction induces interesting shape spaces leading to concrete applications in domains like medical image analysis.
Because the transformation $\varphi \rightarrow \pi(\varphi)$ is many-to-one in general, the projection mechanism from $\mathrm{Diff}$ to $\mathcal Q$ involves an optimization step over the diffeomorphism group: given a target shape $T$, one looks for an optimal diffeomorphism $\varphi$ such that $\pi(\varphi) = \varphi\cdot T_0 = T$. Shapes are then compared by comparing these optimal diffeomorphisms, or some parametrization that characterizes them. Optimality is based on the Riemannian metric on $\mathrm{Diff}$, and more precisely on the distance between $\varphi$ and the identity mapping $\mathrm{id}$ for this metric. The resulting $\pi$ then has the properties of what is called a Riemannian submersion. Because the constraint $\pi(\varphi)=T$ is hard to achieve numerically in general, one preferably replaces this constraint by a penalty term in the minimization, so that the diffeomorphism representing a shape is sought via the minimization of
\varphi \mapsto \mathrm{dist}(\mathrm{id}, \varphi) + \lambda E(\varphi\cdot T_0, T)
where $E$ is an error function. This formulation leads to the LDDMM (large deformation diffeomorphic metric mapping) algorithm, first introduced for landmarks and images, then for curves and surfaces. In this approach, the optimal $\varphi$ is computed as the flow of an ODE (ordinary differential equation), so that $\varphi(x) = \psi(1,x)$ with
\frac{d\psi}{dt}(t,x) = v(t, \psi(t,x))
where $v$ is a time-dependent vector field in the ambient space. The problem can then be reformulated as an optimal control problem where $v$ is the control, minimizing
(v , \psi) \mapsto \int_0^1 \|v(t, \cdot)\|^2_V dt + E(\psi(1, \cdot)\cdot T_0, T)
subject to $\frac{d\psi}{dt}(t,x) = v(t, \psi(t,x))$, where $\|\cdot\|_V$ is a norm over a Hilbert space $V$ of smooth vector fields (e.g., reproducing kernel Hilbert space). Introducing the time variables results in a continuous deformation from the template to the target.
More details can be found in the following papers, and in other works of M. Miller, S. Joshi, A. Trouvé, J. Glaunès, D.D. Holm, F-X Vialard, S. Durleman etc.
Matching deformable objects, A Trouvé, L Younes, Traitement du Signal 20 (3), 295-302, 2003
The metric spaces, Euler equations, and normal geodesic image motions of computational anatomy, M.I. Miller, A. Trouvé, L. Younes, Image Processing, 2003. ICIP 2003. Proceedings. 2003 International Conference on, 2003
Computing large deformation metric mappings via geodesic flows of diffeomorphisms, M.F. Beg, M.I. Miller, A. Trouvé, L. Younes, International journal of computer vision 61 (2), 139-157, 2005
The Euler-Lagrange equation for interpolating sequence of landmark datasets, MF Beg, M Miller, A Trouvé, L Younes, Medical Image Computing and Computer-Assisted Intervention-MICCAI 2003, 918-925, 2003
On the metrics and Euler-Lagrange equations of computational anatomy, MI Miller, A Trouvé, L Younes, Annual review of biomedical engineering 4 (1), 375-405, 2002
Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching, J. Glaunes, A. Trouvé, L. Younes, Computer Vision and Pattern Recognition, 2004. CVPR 2004. Proceedings of the 2004 IEEE Computer Society Conference on, 2004
Large deformation diffeomorphic metric mapping of vector fields, Y. Cao, M.I. Miller, R.L. Winslow, L. Younes, Medical Imaging, IEEE Transactions on 24 (9), 1216-1230, 2005
Modeling planar shape variation via Hamiltonian flows of curves, J. Glaunès, A. Trouvé, L. Younes, Statistics and analysis of shapes, 335-361, 2006
Diffeomorphic matching of diffusion tensor images, Y. Cao, M.I. Miller, S. Mori, R.L. Winslow, L. Younes, Computer Vision and Pattern Recognition Workshop, 2006. CVPRW'06. Conference on, 2006
Large deformation diffeomorphic metric curve mapping, J. Glaunès, A. Qiu, M.I. Miller, L. Younes, International journal of computer vision 80 (3), 317-336, 2008
A kernel class allowing for fast computations in shape spaces induced by diffeomorphisms, A. Jain, L. Younes, Journal of Computational and Applied Mathematics, 2012
Diffeomorphometry and geodesic positioning systems for human anatomy, MI Miller, L Younes, A Trouvé, Technology 2 (01), 36-43
Beyond the registration problem, which is addressed in the previous papers, additional issues can be raised, and refinements can be brought via the rich structure brought by the Riemannian submersion $\pi$. The optimality equation, which has the same structure as the one discovered by Arnold, was presented in relation with shape analysis and computational anatomy in
Geodesic shooting for computational anatomy, M.I. Miller, A. Trouvé, L. Younes, Journal of mathematical imaging and vision 24 (2), 209-228, 2006.
Geodesic shooting and diffeomorphic matching via textured meshes, S. Allassonnière, A. Trouvé, L. Younes, Energy Minimization Methods in Computer Vision and Pattern Recognition, 365-381, 2005
Soliton dynamics in computational anatomy, D.D. Holm, J. Tilak Ratnanather, A. Trouvé, L. Younes, NeuroImage 23, S170-S178, 2004
Further developments around this equation, and the important problem of transport of vectors or covectors along geodesics is addressed in the following papers, which, among other things, provide equations for parallel and coadjoint transport.
Jacobi fields in groups of diffeomorphisms and applications, L. Younes, Quarterly of applied mathematics 65 (1), 113-134, 2007
Transport of relational structures in groups of diffeomorphisms, L. Younes, A. Qiu, R.L. Winslow, M.I. Miller, Journal of mathematical imaging and vision 32 (1), 41-56
Evolutions equations in computational anatomy, L. Younes, F. Arrate, M.I. Miller, NeuroImage 45 (1), S40-S50, 2009
An algorithm dedicated to the problem of averaging over collections of shapes within a Bayesian context is introduced in:
A bayesian generative model for surface template estimation, J. Ma, M.I. Miller, L. Younes, Journal of Biomedical Imaging 2010, 16, 2010
Bayesian template estimation in computational anatomy, J. Ma, M.I. Miller, A. Trouvé, L. Younes, NeuroImage 42 (1), 252-261, 2008
This approach can be completed with principal component analysis, as studied in:
Statistics on diffeomorphisms via tangent space representations, M. Vaillant, M.I .Miller, L. Younes, A. Trouvé, NeuroImage 23, S161-S169, 2004
Principal component based diffeomorphic surface mapping, A. Qiu, L. Younes, M.I. Miller, Medical Imaging, IEEE Transactions on 31 (2), 302-311, 2012
Robust Diffeomorphic Mapping via Geodesically Controlled Active Shapes, D. Tward, J. Ma, M. Miller, L. Younes, International journal of biomedical imaging, 2013
The LDDMM optimal control problem has interesting developments when additional constraints are applied to the evolving diffeomorphism. Some examples, with applications to curve and surface matching, are developed in
Constrained Diffeomorphic Shape Evolution, L. Younes, Foundations of Computational Mathematics 12 (3), 295-325, 2012
Gaussian diffeons for surface and image matching within a Lagrangian framework, L. Younes, Geometry, Imaging and Computing, 1 (1) pp. 141-171,
and more recently, a comprehensive and general discussion of the constrained setting has been developed in
Shape deformation analysis from the optimal control viewpoint, S Arguillere, E Trélat, A Trouvé, L Younes, arXiv preprint arXiv:1401.0661
The diffeomorphic mapping approach has also been applied to surface evolution (introducing area-minimizing diffeomorphic flows), segmentation (diffeomorphic active contours) and tracking.
Diffeomorphic surface flows: A novel method of surface evolution, S. Zhang, L. Younes, J. Zweck, J.T. Ratnanather, SIAM journal on applied mathematics 68 (3), 806-824, 2008
Diffeomorphic active contours, F. Arrate, J.T. Ratnanather, L. Younes, SIAM journal on imaging sciences 3 (2), 176-198, 2012
Modeling and Estimation of Shape Deformation for Topology-Preserving Object Tracking, V Staneva, L Younes, SIAM Journal on Imaging Sciences 7 (1), 427-455 | CommonCrawl |
\begin{document}
\begin{frontmatter} \vspace*{6pt} \title{Stein Estimation for Spherically Symmetric Distributions: Recent~Developments} \runtitle{Stein Estimation}
\begin{aug} \author{\fnms{Ann Cohen} \snm{Brandwein}\ead[label=e1]{[email protected]}} \and \author{\fnms{William E.} \snm{Strawderman}\corref{}\ead[label=e2]{[email protected]}} \runauthor{A. C. Brandwein and W. E. Strawderman}
\affiliation{CUNY Baruch College and Rutgers University}
\address{Ann Cohen Brandwein is Professor, Department of Statistics and Computer Information Systems, CUNY Baruch College, One Bernard Baruch Way, New York, New York 10010, USA \printead{e1}. William E. Strawderman is Professor, Department of Statistics and Biostatistics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, New Jersey 08854, USA \printead{e2}.}
\end{aug}
\begin{abstract} This paper reviews advances in Stein-type shrinkage estimation for spherically symmetric distributions. Some emphasis is placed on developing intuition as to why shrinkage should work in location problems whether the underlying population is normal or not. Considerable attention is devoted to generalizing the ``Stein lemma'' which underlies much of the theoretical development of improved minimax estimation for spherically symmetric distributions. A main focus is on distributional robustness results in cases where a residual vector is available to estimate an unknown scale parameter, and, in particular, in finding estimators which are simultaneously generalized Bayes and minimax over large classes of spherically symmetric distributions. Some attention is also given to the problem of estimating a location vector restricted to lie in a polyhedral cone. \end{abstract}
\begin{keyword} \kwd{Stein estimation} \kwd{spherical symmetry} \kwd{minimaxity} \kwd{admissibility}. \end{keyword}
\end{frontmatter}
\section{Introduction} \label{sec1} We are happy to help celebrate Stein's stunning, deep and significant contribution to the statistical literature. In 1956, Charles Stein (\citeyear{Stein-1956}) proved a~result that astonished many and was the catalyst for an enormous and rich literature of substantial importance in statistical theory and practice. Stein showed that when estimating, under squared error loss, the unknown mean vector $\theta$ of a $p$-dimensional random vector
$X$ having a normal distribution with identity covariance matrix, estimators of the form $(1-a/\{\|X\|^2+b\})X$ dominate the usual estimator $\theta$, $X$, for $a$ sufficiently small and $b$ sufficiently large when $p\geq3$. James and Stein (\citeyear{James-Stein-1961}) sharpened the result and gave an explicit class of dominating estimators,
$(1-a/\|X\|^2)X$ for $0<a<2(p-2)$, and also showed that the choice of $a =p-2 $ (the James--Stein estimator) is uniformly best. For future reference recall that ``the usual estimator,'' $X$, is a~minimax estimator for the normal model, and more generally for any distribution with finite covariance matrix.\looseness=1
Stein (\citeyear{Stein-1974,Stein-1981}), considering general estimators of the form $\delta(X) = X+ g(X)$, gave an expression for the risk of these estimators based on a key Lemma, which has come to be known as Stein's lemma. Numerous results on shrinkage estimation in the general spherically symmetric case followed based on some generalization of Stein's lemma to handle the cross product term $E_{\theta} [(X - \theta)'g(X)]$ in the expression for the risk of the estimator.
A substantial number of papers for the multivariate normal and nonnormal distributions have been written over the decades following Stein's monumental results. For an earlier expository development of Stein estimation for nonnormal location models see Brandwein and Strawderman (\citeyear{Brand-Straw-1990}).
This paper covers the development of Stein estimation for spherically symmetric distributions since Brandwein and Strawderman (\citeyear{Brand-Straw-1990}). It is not encyclopedic, but touches on only some of the significant results for the nonnormal case.
Given an observation, $X$, on a $p$-dimensional sphe\-rically symmetric multivariate distribution with unknown mean,
$\theta$ and whose density is $f(\|x-\theta\|^2)$ (for $ x, \theta\in R^p$), we will consider the problem of estimating $\theta$ subject to the squared error loss function, that is, $\delta(X)$ is a measurable (vector-valued) function, and the loss given by
\begin{equation} \label{eq11}
L ( \theta, \delta) = \| \delta- \theta\|^2 = \sum_{i=1}^p ( \delta_i - \theta_i )^2, \end{equation}
where $\delta= (\delta_1, \delta_2, \ldots, \delta_p)'$ and $\theta= (\theta_1, \theta_2, \ldots, \theta_p)' $.\break The risk function of $\delta$ is defined as
\[ R ( \theta, \delta) = E_{\theta} L ( \delta(X),\theta). \]
Unless otherwise specified, we will be using the loss defined by \eqref{eq11}. Other loss functions such as the loss $L(\theta, \delta)
= \| \delta- \theta\|^2 /\sigma^2$ will be occasionally used, especially when there is also an unknown scale parameter, and minimaxity, as opposed to domination, is the main object of study. We will have relatively little to say about the important case of confidence set loss, or of loss estimation.
In Section \ref{sec2} we provide some additional intuition as to why the Stein estimator of the mean vector $\theta$ makes sense as an approximation to an optimal linear estimator and as an empirical Bayes estimator in a general location problem. The discussion indicates that normality need play no role in the intuitive development of Stein-type shrinkage estimators.
Section \ref{sec3} is devoted to finding improved estimators of $\theta$ for spherically symmetric distributions with a known scale parameter using results of Brandwein and Strawderman (\citeyear{Brand-Straw-1991b}) and Berger (\citeyear{Berger-1975}) to bound the risk of the improved general estimator $\delta(X) = X + \sigma^2 g(X)$.
Section \ref{sec4} considers estimating the mean vector for a general spherically symmetric distribution in the presence of an unknown scale parameter, and, more particularly, when a residual vector is available to estimate the scale parameter. It extends some of the results from Section \ref{sec3} to this case as well as presenting new improved estimators for this problem. The results in this section indicate a remarkable robustness property of Stein-type estimators in this setting, namely, that certain of the improved estimators dominate $X$ uniformly for all spherically symmetric distributions simultaneously (subject to risk finiteness).
In Section \ref{sec5} we consider the restricted parameter space problem, particularly the case where $\theta$ is restricted to a polyhedral cane, or more generally a~smooth cone. The material in this section is adapted from Fourdrinier, Strawderman and Wells (\citeyear{Fourdrinier-etal-2003}).
In Section \ref{sec6} we consider some of the advancements in Bayes estimation of location vectors for both the known and unknown scale cases. We present an intriguing result of Maruyama Maruyama (\citeyear{Maruyama-2003b})\break which is related to the (distributional) robustness of Stein estimators in the unknown scale case treated in Section~\ref{sec4}.
Section \ref{sec7} contains some concluding remarks.
\section{Some Further Intuition into Stein Estimation} \label{sec2} We begin by adding some intuition as to why Stein estimation is both reasonable and compelling, and refer the reader to Brandwein and Strawderman (\citeyear{Brand-Straw-1990}) for some earlier developments. The reader is also referred to Stigler (\citeyear{Stigler-1990}) and to Meng (\citeyear{Meng-2005}).
\subsection{Stein Estimators as an Approximation to the Best Linear Estimator} \label{subsec21} The following is a very simple intuitive development for optimal linear estimation of the mean vector in $R^p$ that leads to the Stein estimator.
Suppose $E_{\theta}[X] = \theta$, $\operatorname{Cov}(X)=\sigma^2I$ ($\sigma^2 $ known), and consider the linear estimator of the form $\delta_a(X)=(1-a)X$. What is the optimal value of $a$? The risk is given by
\[
R(\theta,\delta_a)=p(1-a)^2\sigma^2+ a^2\|\theta\|^2 \]
and the derivative, with respect to $a$, is
\[
\{d/da\}R(\theta,\delta_a)=2\{-p(1-a)\sigma^2+a\|\theta\|^2\}. \]
Hence, the optimal $a$ is $ p\sigma^2/(p\sigma^2 + \|\theta\|^2)$ and the optimal ``estimator'' is
$ \delta(X)=(1- p\sigma^2/\{p\sigma^2 + \|\theta\|^2\})X$, which is, of course, not an estimator because it depends on $\theta$.
However, $E_{\theta}[\|X\|^2] = p \sigma^ 2+ \|\theta\|^2$, so $1/\|X\|^2$ is a~reasonable estimator of $1/\{p \sigma^ 2+ \|{\theta}\|^2\}$. Hence, an approximation to the optimal linear ``estimator'' is
$\delta(X)=( 1-p \sigma^2 /\|X\|^2) X$
which is the James--Stein estimator except that $p$ replaces $p-2$. Note that as~$p$ gets larger, $\|X\|^2/p$ is likely to improve as an estimator of
$\sigma^2 + \frac{\|\theta\|^2}{p}$ and, hence, we may expect that the dimension, $p$, plays a role.
\subsection{Stein Estimators as Empirical Bayes Estimators for General Location Models} \label{subsec22}
Strawderman (\citeyear{Straw-1992}) considered the following general location model. Suppose $X|\theta\sim f(x-\theta)$, where $E_{\theta}[X] = \theta$, $\operatorname{Cov}(X) = \sigma^ 2 I$ ($\sigma^2$ known) but that $f(\cdot)$ is otherwise unspecified. Also assume that the prior distribution for $\theta$ is given by $f^{\star n}(\theta)$, the $n$ fold convolution of $f(\cdot)$ with itself. Hence, the prior distribution of $\theta$ can be represented as the distribution of a sum of~$n$ i.i.d. variables $u_i, i=1 , \ldots, n$, where each $u$ is distributed as $f( u)$. Also, the distribution of $u_0 = (X-\theta)$ has the same distribution and is independent of the other $u$'s.
The Bayes estimator can therefore be thought of~as
\begin{eqnarray*}
\delta(X) &= &E[\theta|X] = E[\theta|X - \theta + \theta] \\
&=& E\Biggl[\sum_{i = 1}^n u_i \Big| \sum_{i = 0}^n u_i \Biggr] \end{eqnarray*}
and, hence,
\begin{eqnarray*}
\delta(X) &=& {nE\Biggl[u_j \Big|\sum_{i = 0}^n {u_i } \Biggr] } \\ &=& \frac{n}{n + 1}E\Biggl[\sum_{i = 0}^n u_i
\Big|\sum_{i = 0}^n u_i\Biggr] \\
& =& \frac{n}{n + 1} E[X|X] = \frac{n}{n + 1}X \end{eqnarray*}
or, equivalently,
$\delta(X) = E[\theta|X] = (1-1/\{n+1\})X$.
Assuming that $n$ is unknown, we may estimate it from the marginal distribution of $X$, which has the same distribution as $X-\theta+ \theta= \sum_{i = 0}^n {u_i }$. In particular,
\begin{eqnarray*}
E_\theta [\|X\|^2 ] &=& E\Biggl[ \Biggl\|\sum_{i = 0}^n {u_i }\Biggr\| ^2 \Biggr] \\
&=& \sum_{i = 0}^n E[ \| u_i \|^2 ] = (n + 1)p\sigma^2, \end{eqnarray*}
since $E[u_i] = 0$ and $\operatorname{Cov}(u_i)=\sigma^2I$, $E[\|u_i\|
^2]=p\sigma^2$. Therefore, $(n+1)$ can be estimated by $(p \sigma^2)^{-1}\|X \|^2$. Substituting this estimator of $(n+1)$ in the expression for the Bayes estimator, we have an empirical Bayes estimator
\[
\delta(X) = (1 - p\sigma^2 /\|X\|^2 )X, \]
which is again the James--Stein estimator, save for the substitution of $p$ for $p-2$.
Note that in both of the above developments,\break the~only assumptions were that $E_{\theta}(X) = \theta,$ and\break $\operatorname{Cov}(X) = \sigma^ 2 I$. The Stein-type estimator thus appears intuitively, at least, to be a reasonable estimator in a~general location problem.
\section{Some Recent Developments for~the Case of a Known Scale~Parameter} \label{sec3}
Let $X \sim f(\| x-\theta\|^2)$, the loss be
$L(\theta, \delta) = \| \delta- \theta\|^2$ so the risk is
$R (\theta, \delta)= E_{\theta} [\| \delta(X) - \theta\|^2]$. Suppose an estimator has the general form $\delta(X) = X + \sigma^2 g(X)$. Then
\begin{eqnarray*} R (\theta, \delta)
&= & E_{\theta} [\| \delta( X ) - \theta\|^2] \\
&=& E_{\theta} [\| X + \sigma^2 g( X ) - \theta\|^2] \\
&=& E_\theta [\|X - \theta\|^2 ]
+ \sigma^4 E_\theta[\| g(X ) \|^2]\\ &&{}+ 2\sigma^2 E_{\theta} [(X-\theta)'g(X)]. \end{eqnarray*}
In the normal case, Stein's lemma, given loosely as follows, is used to evaluate the last term.
\begin{lemma}[{[Stein (\citeyear{Stein-1981})]}]\label{lem31} If $X \sim N (\theta, \sigma^2 I)$,\break then $E_\theta [(X - \theta)'g(X)] = \sigma^2 E_{\theta} [\nabla'g(X)]$ [where $\nabla'g(\cdot)$ denotes the gradient of $g( \cdot)$], provided, say, that $g$ is continuously differentiable and that all expected values exist. \end{lemma}
\begin{pf} The proof is particularly easy in one dimension, and is a simple integration by parts. In higher dimensions the proof may just add the one-dimensional components or may be a bit more sophisticated and cover more general functions, $g$. In the most general version known to us, the proof uses Stokes' theorem and requires $g(\cdot)$ to be weakly differentiable. \end{pf}
Using the Stein lemma, we immediately have the following result.
\begin{proposition}\label{prop31} If $X \sim N(\theta, \sigma^2 I)$, then
\begin{eqnarray*} && R\bigl(\theta,X + \sigma^2 g(X)\bigr) \\
&&\quad= E_{\theta}[\| X-\theta\|^2]
+ \sigma^4 E_{\theta}[\|g(X)\|^2 +2 \nabla'g(X)] \end{eqnarray*}
and, hence, provided the expectations are finite, a~sufficient condition for $\delta(X)$ to dominate $X$
is $\|g(x)\|^2 +2 \nabla' g(x) < 0 $ a.e.~(with strict inequality on a set of positive measures).\vadjust{\goodbreak} \end{proposition}
The key to most of the literature on shrinkage estimation in the general spherically symmetric case is to find some generalization of (or substitution for) Stein's lemma to evaluate (or bound) the cross product term $E_{\theta}[ (X - \theta)'g(X)]$. We indicate two useful techniques below.
\subsection{Generalizations of James--Stein Estimators Under Spherical Symmetry} \label{subsec31} Brandwein and Strawderman (\citeyear{Brand-Straw-1991b}) extended the results of Stein (\citeyear{Stein-1974,Stein-1981}) to spherically symmetric distributions for estimators of the form $X+ag (X )$. The following two preliminary lemmas are necessary to prove the result in Theorem \ref{thmm31}.
\begin{lemma}\label{lem32} Let $X$ have a distribution that is spherically symmetric about $\theta $. Then
\begin{eqnarray*}
&&E_\theta [(X - \theta)'g(X) | \|X - \theta\|^2 = R^2 ] \\ &&\quad = p^{-1}R^2\mathrm{Ave}_{B(R,\theta)} \nabla'g(X), \end{eqnarray*}
provided $g(x)$ is weakly differentiable. \end{lemma}
\begin{pf} Notation for this lemma: $S(R, \theta)$\break and~$B (R, \theta)$ are, respectively, the (surface of the) sphere and (solid) ball, of radius $R$ centered at $\theta$. Note also that $(X - \theta)/R$ is the unit outward normal vector at $X$ on $S(R, \theta)$. Also $d \sigma(X)$ is the area measure on~$S(R, \theta)$, while $A(\cdot)$ and $V(\cdot)$ denote area and volume, respectively. Since the conditional distribution of $X -\theta$
given $\|X - \theta\|^2 = R^2$ is uniform on the sphere of radius $R$, it follows that
\begin{eqnarray*}
&& E_\theta [(X - \theta)'g(X) | \|X - \theta\|^2 = R^2 ] \\ &&\quad= \operatorname{Ave}_{S(R,\theta)} \{(X - \theta)'g(X)\} \\ &&\quad= \frac{R}{{A(S(R,\theta))}} \oint_{S(R,\theta)} \frac{(X - \theta)'g(X)}{R} \,d\sigma(X) \\ &&\quad = \frac{R}{A(S(R,\theta))} \int_{B(R,\theta)} \nabla'g(x) \,dx\\ &&\hspace*{86pt}\qquad{} \biggl(\mbox{since } \frac{V(B(R,\theta))}{A(S(R,\theta))} = R/p\biggr)\\ &&\quad = \frac{R^2 }{pV(B(R,\theta))} \int_{B(R,\theta)} {\nabla'g(x)} \,dx\\ &&\hspace*{119pt}\qquad{} (\mbox{by Stokes' theorem}) \\ &&\quad = p^{-1}R^2 \operatorname{Ave}_{B(R,\theta)} \nabla'g(X). \end{eqnarray*}
\upqed\end{pf}
The following result is basic to the study of superharmonic functions and is well known (see, e.g., du~Plessis, \citeyear{Du-Plessis-1970}, page~54).
\begin{lemma}\label{lem33} Let $h(x)$ be superharmonic on $S(R)$, [i.e., $\sum_{i = 1}^p \{\partial^2 /\partial x_i ^2 \} h(x) \le0$], then $\mathrm{Ave}_{S(R,\theta)} h(x) \le\mathrm{Ave}_{B(R,\theta)} h(x)$. \end{lemma}
Consider, now, an estimator of the general form $X+ag(X)$, where $a$ is a scalar, and $g(X)$ maps \mbox{$R^p \to R^p$}.
\begin{thmm}\label{thmm31} Let $X$ have a distribution that is spherically symmetric about $\theta$. Assume the following:
\begin{enumerate}
\item$\| g(x) \|^2 /2 \le - h(x) \le - \nabla'g(x)$, \label{as1thmm31}
\item$- h(x) $ is superharmonic, $E_\theta [R^2 h(W)]$ is nonincreasing in $R$ for each $\theta$, where $W$ has a uniform distribution on $B (R,\theta)$, \label{as2thmm31}
\item$0 \le a \le1/\{pE_0 [1/\| X \|^2 ]\}$.\label{as3thmm31} \end{enumerate}
Then $X+ag(X)$ is minimax with respect to quadratic loss, provided $g(\cdot)$ is weakly differentiable and all expectations are finite. \end{thmm}
\begin{pf}
\begin{eqnarray*} && R\bigl(\theta,X + ag(X)\bigr) - R(\theta,X) \\ &&\quad= E\bigl[ E_\theta
[a^2 \|g(X)\|^2\\
&&\hspace*{28pt}\qquad{} + 2a(X - \theta)'g(X) | \|X - \theta\|^2 = R^2 ] \bigr] \\ &&\quad \le E\bigl[ E_\theta [ - 2a^2 h(X)\\ &&\hspace*{28pt}\qquad{} + 2a(X - \theta)'g(X)
| \|X - \theta\|^2 = R^2 ]\bigr] \\ &&\quad =
E\bigl[ E_\theta [ - 2a^2 h(X) | \|X - \theta\|^2 = R^2 ] \\
&&\quad\qquad{} + 2aE[\{R^2/p\} \operatorname{Ave}_{B(R,\theta)} \nabla'g(X) | R^2 ] \bigr] \\
&&\quad \le E\bigl[ E_\theta [ - 2a^2 h(X) | \|X - \theta\|^2 = R^2 ] \\
&&\quad\hspace*{10pt}\qquad{} + 2aE_\theta [\{R^2/p\}E_\theta h(W) | R^2 ] \bigr] \\
&&\quad \le E\bigl[ E_\theta [ - 2a^2 h(W) | R^2 ] \\
&&\quad\qquad{}+ 2aE_\theta [\{R^2/p\}E_\theta h(W) | R^2 ] \bigr] \\ &&\hspace*{142pt}\qquad{}(\mbox{by Lemma \ref{lem33}}) \\ &&\quad =
2aE\bigl[ E_\theta [R^2 h(W) | R^2 ] (-a/R^2 + 1/p) \bigr] \\ &&\quad =
2aE[ E_\theta [R^2 h(W) | R^2 ] ] E[-a/R^2 + 1/p] \\ &&\quad \le0 \end{eqnarray*}
by the covariance inequality since $ E_\theta[R^2 h(W)|R^2]$ is nonincreasing and $-R^{-2}$ is increasing and since $h \le0$. \end{pf}
\begin{example}\label{exam31}
James--Stein estimators $[g(x)= - 2(p-2)x/\|x\|^2]$:
In this case both $\|g (x)\|^2/2$ and $- \nabla' g(x)$
are equal to $2(p-2)^2/\|x\|^2$. Conditions~1 and 2 of Theorem \ref{thmm31}
are satisfied for $h(x) = - 2(p-2)^2/\|x\|^2$, provided\vadjust{\goodbreak} $p \geq4$
since $\|x\|^{-2}$ is superharmonic if $p \geq4$, and since $E_\theta[R^2/\|X\|^2] = E_{\theta/R} [1/\allowbreak\|X\|^2]$ is increasing by Anderson's theorem.
Hence, by condition 3, for any spherically symmetric distribution, the James--Stein estimator $(1-a 2 (p-2) / \|X\|^2)X$ is minimax~for
$0 \leq a \leq1/\{pE_0[1/\break\|X\|^2]\}$ and $p\geq4$. The domination over $X$ is strict~for $0<a<1/\{pE_0 [1/\|X\|^2]\}$, and also for $a = 1/\{pE_0 [1/\break \|X\|^2]\}$, provided the distribution is not normal.
Baranchik (\citeyear{Baranchik-1970}), for the normal case, considered estimators of the form
$(1-ar(\|X\|^2)/\|X\|^2)X$ under certain conditions on $r(\cdot)$. Under the assumption that $r(\cdot)$ is monotone nondecreasing, bounded between $0$ and $1$, and concave, Theorem \ref{thmm31} applies to these estimators as well, and establishes minimaxity for
$0 \leq a \leq1/\{pE_0[1/\|X\|^2]\}$ and for $p\geq4$.\vspace*{-1.5pt} \end{example}
We note in passing that the results in this subsection hold for an arbitrary spherically symmetric distribution with or without a density. The calculations rely only on the distribution of $X$ conditional on $\|X- \theta\|^2= R^2$, and, of course, finiteness of $E[\|X\|^2]$
and $E[\|g(X)\|^2]$.\vspace*{-1.5pt}
\subsection{A Useful Expression for the Risk of a~James--Stein Estimator}\vspace*{-1.5pt} \label{subsec32}
Berger (\citeyear{Berger-1975}) gave a useful expression for the risk of a James--Stein estimator which is easily generalized to the case of a general estimator, provided the spherically symmetric distribution has a density~$f(\|
x-\theta\|^2)$.
Some form of this generalization (and extensions to unknown scale case and the elliptically symmetric case) has been used by several authors, including Fourdrinier, Strawderman and Wells (\citeyear{Fourdrinier-etal-2003}), Fourdrinier, Kortbi and Strawderman (\citeyear{Fourdrinier-etal-2008}), Fourdrinier and Strawderman (\citeyear{Fourdrinier-Straw-2008}), Maruyama (\citeyear{Maruyama-2003}) and Kubokawa and Srivastava (\citeyear{Kubokawa-Srivastava-2001}), among others.
\begin{lemma} \label{lem34}
Suppose $X \sim f(\|x-\theta\|^2)$, and let $F(t) = 2^{-1}\int_t^\infty f(u)\,du$ and $Q(t) = F(t)/f(t)$. Then
\begin{eqnarray*} &&R\bigl(\theta,X + g(X)\bigr) \\
&&\quad= E_\theta [\| X - \theta\|^2]\\
&&\qquad{}+ E_\theta [ \| {g(X)} \|^2 + 2Q(\| X-\theta\|^2 )\nabla'g(X)]. \end{eqnarray*}
\end{lemma}
\begin{pf} The lemma follows immediately with the following identity for the cross product term:
\begin{eqnarray*} \qquad&&E[(x - \theta)'g(X)]\\
&&\quad= \int_{R^p } (x - \theta)'g(X)f(\|x - \theta\|^2)\,dx \\
&&\quad= \int_{R^p } g(X)'\nabla F(\|x - \theta\|^2) \,dx\\
&&\quad= \int_{R^p } \nabla'g(X)F(\|x - \theta\|^2) \,dx \vadjust{\goodbreak}\\ &&\hspace*{96pt}\qquad(\mbox{by Green's theorem}) \\
&&\quad= E [Q(\|X - \theta\|^2)\nabla'g(X)]. \end{eqnarray*}
\upqed\end{pf}
Berger (\citeyear{Berger-1975}), Maruyama (\citeyear{Maruyama-2003}) and Fourdrinier, Kortbi and Strawderman (\citeyear {Fourdrinier-etal-2008}) used the above result for distributions for which $Q(t)$ is bounded below by a positive constant. In this case, the next result follows immediately from Lemma~\ref{lem34}.
\begin{thmm} \label{thmm32}
$\!\!\!$Suppose $X\,{\sim}\,f(\|x\,{-}\,\theta\|^2)$, and~that $Q(t)\geq c>0$. Then the estimator $X +g(X)$ dominates $X$
provided $\| g(x) \|^2 + 2c \nabla'g(x) \le0$ for all $x$. \end{thmm}
\begin{example} As noted by Berger (\citeyear{Berger-1975}), if $f(\cdot)$ is a scale mixture of normals, then $Q(t)$ is bounded below. To see this, note that if
$ X|V \sim N(\theta,VI) $\break and $ V \sim g(v)$, then $f(t) =\int_0^\infty(2\pi v)^{-p/2} \exp( - t/\break2v)g(v) \,dv$. Similarly,
\begin{eqnarray*} F(t) &= &2^{-1}\int_t^\infty f(u)\,du\\ &=& 2^{-1}\int_0^\infty g(v)(2\pi v)^{-p/2} \int_t^\infty \exp( - u/2v)\,du \\ &=& \int_0^\infty(2\pi v)^{-p/2} v \exp( - t/2v)g(v)\,dv. \end{eqnarray*}
Hence,
\begin{eqnarray*} Q(t) &=& \frac{\int_0^\infty v^{(2 - p)/2} \exp( - t/2v)g(v)\,dv} {\int_0^\infty v^{ - p/2} \exp( - t/2v)g(v)\,dv} \\ &=& E_t [V] \ge E_0 [V] = \frac{\int_0^\infty v^{1 - p/2} g(v)\,dv} {\int_0^\infty v^{ - p/2} g(v)\,dv} \\ &=& \frac{E[V^{1-p/2}]}{E[V^{-p/2}]} = c > 0, \end{eqnarray*}
where $E_t$ denotes expectation with respect to the density proportional to $ v^{-p/2} \exp(-t/2v)g(v)$. The inequality follows since the family has monotone likelihood ratio in $t$.
Hence, for the James--Stein class $(1-a/\|X\|^2)X$, this result gives dominance over $X$ for
\[ a^2 - 2a(p - 2)\frac{E[V^{1 - p/2} ]}{E[V^{ - p/2} ]} \le0 \]
or
\[ 0 \le a \le2(p - 2)\frac{E[V^{1 - p/2} ]}{E[V^{ - p/2} ]}. \]
This bound on the shrinkage constant, $a$, compares poorly with that obtained by Strawderman (\citeyear{Straw-1974}), $0 \le a \le2(p - 2)/E[V^{-1}]$, which may be obtained by using Stein's lemma conditional on $V$
and the fact that $E_\theta [ V/\|X\|^2 | V ]$ is monotone nondecreasing in $V$. Note that, again by monotone likelihood ratio properties (or the covariance inequality),\break $(E[V^{ - 1} ])^{-1} > E[V^{1 - p/2} ]/E[V^{ - p/2} ]$.
It is therefore somewhat surprising that Maruyama (\citeyear{Maruyama-2003}) and Fourdrinier, Kortbi and Strawderman (\citeyear{Fourdrinier-etal-2008}) were able to use Theorem \ref{thmm32}, applied to Baranchik-type estimators, to obtain generalized and proper Bayes minimax estimators. Without going into details, the advantage of the cruder bound is that it requires only that $r(t)$ be monotone, while Strawderman's result for mixtures of normal distributions also requires that $r(t)/t$ be monotone decreasing.\vspace*{-2pt} \end{example}
Other applications of Lemma \ref{lem34} give refined\break bounds on the shrinkage constant in the James--Stein or Baranchik estimator depending on monotonicity pro\-perties of $Q(t)$. Typically, additional conditions are required on the function $r(t)$ as well. See, for example, Brandwein, Ralescu and Strawderman (\citeyear{Brand-etal-1993}) (\mbox{although} the calculations in that paper are somewhat different than those in this section, the basic idea is quite similar).
Applications of the risk expression in Lemma \ref{lem34} are complicated relative to those in the normal case using Stein's lemma, in that the mean vector, $\theta$, remains to complicate matters through the function $Q(\|X-\theta\|^2)$. It is both surprising and interesting that matters become essentially simpler (in a certain sense) when the scale parameter is unknown, but a residual vector is available. We investigate this phenomenon in the next section.
\vspace*{-2pt}\section{Stein Estimation in the Unknown Scale Case} \vspace*{-2pt}\label{sec4} In this section we study the model
$(X, U) \sim f(\|x-\theta\|^2 + \|u\|^2)$, where $\operatorname{dim}X = \operatorname{dim} \theta= p$, and $\operatorname{dim}U = k$. The classical example of this model is, of course, the normal model $f(t) = (\frac{1} {\sqrt{2\pi} \sigma}) ^{p+k} e^{- {t}/{(2\sigma ^{2})}}$. However, a variety of other models have proven useful. Perhaps the most important alternatives to the normal model in practice and in theory are the generalized \mbox{multivariate-$t$} distributions
\[ f(t) = \frac{c} {\sigma^{p+k}} \biggl(\frac{1}{a+ t/\sigma^{2}}\biggr)^b, \]
or, more generally, scale mixture of normals of the form \[ f(t)= \int_0^{\infty} \biggl(\frac{1}{\sqrt{2\pi} \sigma}\biggr) ^{p+k} e^{- {t}/{(2\sigma^{2})}} \,dG(\sigma^2).\vadjust{\goodbreak} \]
These models preserve the spherical symmetry\break about the mean vector and, hence, the covariance matrix is a multiple of the identity. Thus, the coordinates are uncorrelated, but they are not independent except for the case of the normal model. We look (primarily) at estimators of the form $X + \{\|U \|^2/(k+2)\} g(X)$.
The main result may be interpreted as follows:
If, when $X\sim N(\theta, \sigma^2 I)$ ($\sigma^2$ known), the estimator $X+ \sigma^2 g(X)$ dominates $X$, then, under the model $(X,U)\sim f(\|x-\theta\|^2 + \|u\|^2)$, the estimator $X+ \break\{\|U \|^2/(k+2)\}g(X)$ dominates $X$. That is, substituting the estimator $\|U\|^2/(k+2)$ for $\sigma^2$ preserves domination uniformly for all parameters $(\theta, \sigma^2)$ and (somewhat astonishingly) simultaneously for all distributions,
$f(\cdot)$. Note that, interestingly, $\|U\|^2/(k+2)$ is the minimum risk equivariant estimator of $\sigma^2$ in the normal case under the usual invariant loss. This wonderful result is due to Cellier and Fourdrinier (\citeyear{Cellier-Fourdrinier-1995}). We refer the reader to their paper for the original proof based on Stokes' theorem applied to the distribution of $X$ conditional on $\|X-\theta\|^2 + \|U\|^2 = R^2$. One interesting aspect of that proof is that even if the original distribution has no density, the conditional distribution of $X$ does have a density for all $k>0$.
We will approach the above result from two different directions. The first approach is essentially an extension of Lemma \ref{lem34}. As in that case, the resulting expression for the risk still involves both the data and $\theta$ inside the expectation, but the function $Q(\|X- \theta\|^2 + \|U \|^2)$ is a common factor. This allows the treatment of the remaining terms as if they are an unbiased estimate of the risk difference.
The second approach is due to Fourdrinier, Strawderman and Wells (\citeyear{Fourdrinier-etal-2003}), and is attractive because it is essentially statistical in nature, depending on completeness and sufficiency. It may be argued also that this approach is somewhat more general in that it may be useful even when the function $g(x)$ is not necessarily weakly differentiable. In this case an unbiased estimator of the risk difference is obtained which agrees with that in Cellier and Fourdrinier (\citeyear{Cellier-Fourdrinier-1995}). This is in contrast to the above method~whe\-reby the expression for the risk difference still has a~factor
$Q(\|X-\theta\|^2 +\|U\|^2)$ inside the expectation.
\begin{note*} $\!\!$Technically, our use of the term ``unknown scale'' is somewhat misleading in that the scale parameter may, in fact, be known. We typically think of $f(\cdot)$ as being a known density, which implies that the scale is known as well. It may have been preferab\-le to write\vadjust{\goodbreak} the density as
$(X,U) \sim\{1/\sigma^{p+k}\}f(\{\|x- \theta\|^2 + \|u\|^2\}/ \sigma ^2) $, emphasizing the unknown scale parameter. This is more in keeping with the usual canonical form of the general linear model with spherically symmetric errors. What is of fundamental importance is the presence of the residual vector, $U$, in allowing uniform domination over the estimator~$X$
simultaneously for the entire class of spherically symmetric distributions. Since the suppression of the scale parameter makes notation a bit simpler, we will, for the most part, use the above notation in this section. Additionally, we continue to use the un-normalized loss, $L(\theta,\delta)=\|\delta-\theta\|^2 $, and state results in terms of dominance over $X$ instead of minimaxity, since the minimax risk is infinite. In order to speak meaningfully of minimaxity in the unknown scale case, we should use a normalized version of the loss, such as $L(\theta,\delta)=\|\delta-\theta\|^2 /\sigma^2$. \end{note*}
\subsection{\texorpdfstring{A Generalization of Lemma \protect\ref{lem34}}{A Generalization of Lemma 3.4}} \label{subsec41}
\begin{lemma}\label{lem41}
$\!\!\!$Suppose $(X,U) \sim f(\|x- \theta\|^2 + \|u\|^2)$, where $\operatorname{dim}X = \operatorname{dim}\theta= p$, $\operatorname{dim} U = k$. Then, provided $g(x,\|u\|^2)$ is weakly differentiable in each coordinate:
\begin{enumerate}
\item$ E_\theta [ \|U\|^2 (X - \theta)'g(X,\|U\|^2 ) ] =
E_\theta[\|U\|^2 \nabla'_X g(X,\break \|U\|^2 )Q(\|X - \theta\|^2 + \|U\|^2 )] $. \label{1lem41}
\item$ E_\theta [ \|U\|^4 \|g(X,\|U\|^2 )\|^2 ]=E_\theta[h(X,\|U\|
^2)\cdot Q(\|X - \theta\|^2 + \|U\|^2 )]$, \label{2lem41}
where $ Q(t)=\{2f(t)\}^{-1}\cdot \int_t^\infty f(s)\,ds$ and
\begin{eqnarray}\label{eq41}
&&h(x,\|u\|^2)\nonumber\\
&&\quad= (k + 2)\|u\|^2 \|g(x)\|^2 \\
&&\qquad{} + 2\|u\|^4
\frac{\partial}{\partial\|u\|^2 }
\|g(x,\|u\|^2 )\|^2 .\nonumber \end{eqnarray}
\end{enumerate} \end{lemma}
\begin{pf} The proof of part 1 is essentially the same as the proof of Lemma \ref{lem34}, holding $U$ fixed throughout. The same is true of part 2, where the roles of $X$ and $U$ are reversed and one notes that
\begin{eqnarray*}
\nabla'_u (\|u\|^2 u) &=& (k + 2)\|u\|^2,\\
\nabla'_u\{ (\|u\|^2 u)\|g(x,\|u\|^2 )\|^2 \} &=& h(x,\|u\|^2), \end{eqnarray*}
which is given by \eqref{eq41}, and, hence,
\begin{eqnarray*}
&& E_\theta [\|U\|^4 \| g(X,\|U\|^2) \|^2 ]\\
&&\quad= E_\theta [(\|U\|^2 U)'U \|g(X,\|U\|^2)\|^2 ] \\
&&\quad = E_\theta [\nabla'_U \{(\|U\|^2 U)\|g(X,\|U\|^2 )\|^2 \} \\
&&\hspace*{44pt}\qquad{}\cdot Q(\|X - \theta\|^2 + \|U\|^2 )] \\
&&\quad= E_\theta [h(X,\|U\|^2)
Q(\|X - \theta\|^2 + \|U\|^2 )] . \end{eqnarray*}
\upqed\end{pf}
One version of the main result for estimators of the form
$X + \{\|U\|^2 /(k+2)\}g(X)$ is the following theorem.\vadjust{\goodbreak}
\begin{thmm}\label{thmm41} Suppose (X, U) is as in Lem\-ma~\ref{lem41}. Then:
\begin{enumerate}
\item The risk of an estimator
$X + \{\|U \|^2 /(k + 2)\} g(X)$ is given by \begin{eqnarray*}
&&R\bigl(\theta,X + \{\|U\|^2 /(k + 2)\}g(X)\bigr) \\
&&\quad=E_\theta [\|X - \theta\|^2 ]\\
&&\qquad{}+ E_\theta \biggl[\frac{{\|U\|^2 }}{{k + 2}}\{\|g(X)\|^2 + 2\nabla'g(X)\} \\
&&\hspace*{52pt}\qquad{}\cdot Q(\|X - \theta\|^2 + \|U\|^2 )\biggr], \end{eqnarray*}
\item$X + \{\|U\|^2 /(k + 2)\}g(X)$ dominates $X$
provided $\|g(x)\| + 2\nabla'g(x)<0$. \end{enumerate}
\end{thmm}
\begin{pf} Note that
\begin{eqnarray*}
&& R\bigl(\theta,X + \{\|U\|^2 /(k + 2)\}g(X)\bigr) \\[-1pt]
&&\quad= E_\theta[\|X - \theta\|^2 ]\\[-1pt]
&&\qquad{}+ E_\theta\biggl[ \frac{{\|U\|^4 }}{{(k + 2)^2 }}\|g(X)\|^2 \\[-1pt]
&&\hspace*{30pt}\qquad{}+ 2\frac{{\|U\|^2 }}{{k + 2}}(X - \theta)'g(X) \biggr] \\[-1pt]
&&\quad= E_\theta [\|X - \theta\|^2 ]\\[-1pt] &&\qquad{}+
E_\theta \biggl[ \{\|g(X)\|^2 + 2\nabla'g(X)\} \\[-1pt]
&&\hspace*{28pt}\qquad{}\cdot \frac{\|U\|^2Q(\|X - \theta\|^2 + \|U\|^2)}{k+2} \biggr] \end{eqnarray*}
by successive application of parts 1 and 2 of Lem\-ma~\ref{lem41}. \end{pf}
\begin{example} Baranchik-type estimators: Suppose the estimator is given by
$ (1 - \|U\|^2 r(\|X\|^2)/\break\{(k + 2)\|X \|^2\})X$, where $r(t)$ is nondecreasing, and $0 \le r(t) \le 2(p - 2)$, then for $p \ge3$ the estimator dominates $X$ simultaneously for all spherically symmetric distributions for which the risk of $X$
is finite. This follows since, if $g(x) = -xr(\|x\|^2 )/\|x\|^2$,\break then
\begin{eqnarray*}
&& \|g(x)\|^2 + 2\nabla'g(x)\\
&&\quad= r^2 (\|x\|^2 )/\|x\|^2 \\
&&\qquad{}- 2\{(p - 2)r(\|x\|^2 )/\|x\|^2 - 2r'(\|x\|^2)\} \\
&&\quad \le r^2 (\|x\|^2 )/\|x\|^2 - 2(p - 2)r(\|x\|^2 )/\|x\|^2 \le0. \end{eqnarray*}
\end{example}
\begin{example} James--Stein estimators:
If\break $ r(\|x\|^2) \equiv a$, the Baranchik estimator is a James--Stein estimator, and, since $r'(t) \equiv0$, the risk is given by
\begin{eqnarray*}
&& E_\theta [\|X - \theta\|^2 ] + \frac{a^2 - 2a(p - 2)}{k+2}\\[1pt]
&&\hspace*{64pt}\quad{}\cdot E\biggl[ \frac{\|U\|^2}{\|X\|^2}
Q(\|X - \theta\|^2 + \|U\|^2 ) \biggr]. \end{eqnarray*}
Just as in the normal case, $a=p-2$ is the uniformly best choice to minimize the risk. But here it is the uniformly best choice for every distribution. Hence, the estimator $(1 - (p - 2)\|U\|^2/\{(k + 2)\|X\|^2\})X$ is uniformly best, simultaneously for all spherically symmetric distributions among the class of James--Stein estimators! \end{example}
A more refined version of Theorem \ref{thmm41} which uses the full power of Lemma \ref{lem41} is proved in the same way. We give it for completeness and since it is useful in the study of risks of Bayes estimators.
\begin{thmm}\label{thmm42} Suppose $(X,U)$ is as in Lem\-ma~\ref{lem41}. Then, under suitable smoothness conditions on $g(\cdot)$:
\begin{enumerate}
\item The risk of an estimator $X + \{\|U\|^2 /(k + 2)\}g(X,\break \|U\|^2)$ is given by
\begin{eqnarray*}
&& R\bigl(\theta,X + \{\|U\|^2 /(k + 2)\}g(X,\|U\|^2)\bigr) \\[1pt]
&&\quad = E_\theta [\|X - \theta\|^2 ]\\[1pt] &&\qquad{}+
E_\theta [ \{(k+2)^{-1} \|U\|^2 \|g(X,\|U\|^2 )\|^2 \\[1pt]
&&\hspace*{32pt}\qquad{}+ 2\nabla'_X g(X,\|U\|^2 ) \\[1pt]
& &\hspace*{32pt}\qquad{} + 2(k+2)^{-2}\|U\|^4(\partial/\partial\|U\|^2)\\[1pt]
&&\hspace*{91pt}\qquad{}\cdot\|
g(X,\|U\|^2 )\|^2 \} \\[1pt]
&&\hspace*{68pt}\qquad\cdot {}Q(\|X-\theta\|^2 + \|U\|^2)], \end{eqnarray*}
\item$X + \{\|U\|^2 /(k + 2)\}g(X,\|U\|^2)$ dominates $X$ provided
\begin{eqnarray*}
&&\|g(x,\|u\|^2 )\|^2 + 2\nabla'_x g(x,\|u\|^2 )\\
&&\quad{} + 2\frac{\|u\|^2 }{{k + 2}}\frac{\partial}{\partial\|u\|^2}
\|g(x,\|u\|^2 )\|^2<0. \end{eqnarray*}
\end{enumerate}
\end{thmm}
\begin{corollary}\label{cor41}
Suppose $\delta(X,\|U\|^2 ) = (1 -\break \|U\|^2 r(\|X\|^2 /\|U\|^2 )/\|X\|^2)X$. Then $\delta(X,\|U\|^2)$ do\-minates $X$ provided:
\begin{enumerate}
\item$0\leq r(\cdot)\leq2(p-2)/(k+2)$ and
\item$r(\cdot)$ is nondecreasing. \end{enumerate}
\end{corollary}
The result follows from Theorem \ref{thmm42} by a straightforward calculation.
\subsection{A More Statistical Approach Involving Sufficiency and Completeness} \label{subsec42} We largely follow Fourdrinier, Strawderman and Wells (\citeyear{Fourdrinier-etal-2003}) in this subsection. The nature of the conclusions for estimators is essentially as in Theorem \ref{thmm41}, but the result is closer in spirit to the result of Cellier and Fourdrinier (\citeyear{Cellier-Fourdrinier-1995}) in that we obtain an unbiased estimator of risk difference (from $X$) instead of the expression in Theorem \ref{thmm41} where the function $Q(\cdot)$, which depends on $\theta$, intervenes. The following lemma is the key to this development.
\begin{lemma} \label{lem42}
Let $(X,U) \sim f(\|x-\theta\|^2+\| u \|^2)$, where $\operatorname{dim}X = \operatorname{dim} \theta= p$ and $\operatorname{dim}U = k$. Suppo-\break se~$g(\cdot)$ and $h(\cdot)$ are such that when $X \sim N_p(\theta,I)$,\break $E_\theta [(X - \theta)'g(X)] = E_\theta [h(X)]$. Then, for $(X,U)$ as above,
\begin{eqnarray*}
&& E_\theta [\|U\|^2 (X - \theta)'g(X)]\\
&&\quad= \{1/(k+2)\}E_\theta [\|U\|^4 h(X)], \end{eqnarray*}
provided the expectations exist. \end{lemma}
\begin{note*} Typically, of course, $h(x)$ is the divergence of $g(x)$, and, in all cases known to us, this remains essentially true. We choose this form of expressing the lemma because in certain instances of restricted parameter spaces the lemma applies even though the function $g(\cdot)$ may not be weakly differentiable, but the equality still holds for $g(x)I_A (g(x))$ and $h(x) = \nabla'g(x)I_A (g(x))$, where $I_A(\cdot)$ is the indicator function of a set $A$. \end{note*}
\begin{pf*}{Proof of Lemma \protect\ref{lem42}} Suppose first, that the distribution of $(X,U)$ is $N_{p + k} (\{\theta,0\},\sigma^2 I)$ and that~$\theta$ is considered known. Then by the independence of~$X$ and~$U$ we have by assumption that
\begin{eqnarray*} &&E_\theta [(X - \theta)'g(X)]\\
&&\quad = E_\theta [ (1/k)\| U \|^2 (X - \theta)'g(X) ]\\
&&\quad= E_\theta [ \{k(k+2)\}^{-1} \| U \|^4 h(X) ]. \end{eqnarray*}
Hence, the claimed result of the theorem is true for the normal case. Now use the fact that in the normal case (for $\theta$ known),
$\|X - \theta\|^2 + \| U \|^2$ is a complete sufficient statistic. So it must be that
\begin{eqnarray*}
&&E_\theta [\| U \|^2 (X - \theta)'g(X) | \| X - \theta\|^2 + \| U \| ^2 ]\\
&&\quad= E_\theta \biggl[ \frac{\| U \|^4 h(X) }{k + 2} \Big|
\| X - \theta\|^2 + \| U \|^2 \biggr] \end{eqnarray*}
for all $ \|X-\theta\|^2+\| U \|^2 $ except on a set of measure $0$, since each function of $\|X-\theta\|^2+\|U\|^2$ has the same expected value. Actually, it can be shown that these conditional expectations are continuous in $R$ and, hence, they agree for all $R$ (see Fourdrinier, Strawderman and Wells, \citeyear{Fourdrinier-etal-2003}).
But the distribution of $(X,U)$ conditional on
$\|X- \theta\|^2+\|U\|^2 = R^2$ is uniform on the sphere centered at $(\theta,0)$ of radius $R$, which is the same as the conditional distribution of $(X,U)$
conditional on $\|X- \theta\|^2+\|U\|^2 = R^2$ for any spherically symmetric distribution. Hence, the equality which holds for the normal distribution holds for all distributions $f(\cdot)$. \end{pf*}
Lemma \ref{lem42} immediately gives the following unbiased estimator of risk difference and a condition for dominating $X$ for estimators of the form $\delta(X) = X + \{\| U \|^2 /(k + 2)\}g(X)$.
\begin{thmm}
Suppose $(X,U), g(x)$ and $h(x)$ are as in Lemma \ref{lem42}. Then, for the estimator $\delta(X) = X + \{\|U\|^2/(k+2)\}g(X)$:
\begin{enumerate}
\item The risk difference is given by
\begin{eqnarray*}
&&R(\theta,\delta) - E_\theta [\|X - \theta\|^2] \\
&&\quad= E_\theta \biggl[ \frac{\|U\|^4 }{(k + 2)^2 }
\{\|g(X)\|^2 + 2\nabla'g(X)\} \biggr], \end{eqnarray*}
\item$\delta(X)$ beats $X$ provided $\|g(x)\|^2 + 2\nabla'g(x) \le0$, with strict inequality on a set of positive measure, and provided all expectations are finite. \end{enumerate}
\end{thmm}
\section{Restricted Parameter Spaces} \label{sec5} We consider a simple version of the general restric\-ted parameter space problem which illustrates what types of results can be obtained. Suppose $(X, U)$ is distributed as in Theorem \ref{thmm41} but it is known that $\theta_i \ge0$, $i = 1,\ldots,p$, that is,~$\theta \in R^p_+$ the first orthant. What follows can be generalized to the case where~$\theta$ is restricted to a polyhedral cone, and more generally a smooth cone. The material in this section is adapted from Fourdrinier, Strawderman and Wells (\citeyear{Fourdrinier-etal-2003}).
In the normal case, the MLE of $\theta$ subject to the restriction that $\theta \in R^p_+ $ is $X_+$, where the $i$th component is $X_i$ if $X_i \geq0$ and $0$ otherwise. Here, as in the case of the more general restriction to a convex cone, the MLE is the projection of $X$ onto the restricted cone. Chang (\citeyear{Chang-1982}) considered domination of the MLE of $\theta$ when $X$ has a $N_p(\theta, I)$ distribution and $\theta \in R^p_+$ via certain Stein-type shrinkage estimators. Sengupta and Sen (\citeyear{Sengupta-Sen-1991}) extended Chang's results to Stein-type shrinkage estimators of the form
$\delta(X) = (1 - r_s (\|X_+\|^2)/\|X_+\|^2)X_+$,\vadjust{\goodbreak} where~$r_s(\cdot)$ is nondecreasing, and $0 \leq r_s(\cdot) \leq2(s-2)_+$, and where~$s$ is the (random) number of positive components of~$X$. Hence, shrinkage occurs only when $s$, the number of positive components of $X$, is at least $3$ and the amount of shrinkage is governed by the sum of squa\-res of the positive components. A similar result holds if $\theta$ is restricted to a general polyhedral cone whe\-re~$X_+$ is replaced by the projection of $X$ onto the cone and $s$ is defined to be the dimension of the face onto which $X$ is projected.
We choose the simple polyhedral cone $\theta \in R^p_+ $ because it will be reasonably clear that some version of the Stein Lemma \ref{lem31} applies in the normal case. We first indicate a convenient, but complicated looking, alternate representation of an estimator of the above form in this case. Denote the $n = 2^p$ orthants of $R^p$, by $O_1 , \ldots,O_n$, and let $O_1$ be $ R_+$. Then we may rewrite (a slightly more general version of) the above estimator as
\[ \delta(X) =
\sum_{i = 1}^n \biggl(1-\frac{r_i (\|P_i (X)\|^2)}{\|P_i (X)\|^2}\biggr) P_i (X) I_{O_i}(X), \]
where $P_i(X)$ is the linear projection of $X$ onto $F_i$, where $F_i$ is the $s$-dimensional face of $R_ + = O_1$ onto which $O_i$ is projected. Note that if $r_i (\cdot) \equiv0$, $\forall i$, the estimator is just the MLE.
\begin{lemma}\label{lem51} Suppose $X \sim N_p(\theta, I)$, and let\break each~$r_i (\cdot)$ be smooth and bounded. Then:
\begin{enumerate}
\item For each $O_i$,
$\{r_i(\|P_i (x)\|^2)/\|P_i (x)\|^2\} P_i(x)I_{O_i}(x)$ is weakly differentiable in $x$. \label{1lem51}
\item Further,
\begin{eqnarray*}
&& E_\theta \biggl[\bigl(P_i (X) - \theta\bigr)' \frac{r_i (\|P_i (X)\|^2 )}
{\|P_i (X)\|^2 }P_i(X) I_{O_i } (X)\biggr] \\
&&\quad =E_\theta \biggl[\biggl\{ \frac{(s - 2)r_i (\|P_i (X)\|^2 )}{\|P_i (X)\|^2 } \\
&&\hspace*{38pt}\qquad{}+ 2r'_i (\|P_i (X)\|^2 )\biggr\} I_{O_i }(X)\biggr], \end{eqnarray*}
provided expectations exist. \label{2lem51}
\item$ \delta(X) = \sum_{i = 1}^n \{1 - r_i (\|P_i (X)\|^2)/\|P_i
(X)\|^2\}\cdot\break P_i(X) I_{O_i}(X)$ as given above dominates the\break MLE~$X_+$, provided $r_i$ is nondecreasing and boun\-ded between 0 and $2(s-2)_+$. \label{3lem51} \end{enumerate}
\end{lemma}
\begin{pf} Weak differentiability in part~1 follows since the function is smooth away from the boundary of $O_i$ and is continuous on the boundary except at the origin. Part~2 follows from Stein's Lemma \ref{lem31} and the fact that (essentially) $P_i(X) \sim N_s(\theta, \sigma^2I)$, since $n-s$ of the coordinates are $0$.\vadjust{\goodbreak} Part~3 follows by Stein's Lemma~\ref{lem31} as in Proposition \ref{prop31} applied to each orthant. We omit the details. The reader is referred to Sengupta and Sen (\citeyear{Sengupta-Sen-1991}) or Fourdrinier, Strawderman and Wells (\citeyear{Fourdrinier-etal-2003}) for details in the more general case of a polyhedral cone. \end{pf}
Next, essentially applying Lemma \ref{lem42} to each orthant and using Lemma \ref{lem51} we have the following generalization to the case of a general spherically symmetric distribution.
\begin{thmm}
Let $(X,U) \sim f(\| {x - \theta} \|^2 + \| u \|^2 )$ where $\operatorname{dim}X = \operatorname{dim} \theta= p$ and $\operatorname{dim}U = k$ and suppose that $\theta \in R^p_+ $. Then
\[ \delta(X) = \sum_{i = 1}^n \biggl\{1 -
\frac{\|U\|^2 r_i (\|P_i (X)\|^2)}{(k + 2)\|P_i (X)\|^2}\biggr\} P_i (X) I_{O_i} (X) \]
dominates the $X_+$, provided $r_i$ is nondecreasing and bounded between 0 and $2(s-2)_+$. \end{thmm}
\section{Bayes Estimation} \label{sec6} There have been advancements in Bayes estimation of location vectors in several directions in the past 15 years. Perhaps the most important advancements have come in the computational area, particularly Markov chain Monte Carlo (MCMC) methods. We do not cover these developments in this review.
Admissibility and inadmissibility of (generalized) Bayes estimators in the normal case with known scale parameter was considered in Berger and Strawderman (\citeyear{Berger-Straw-1996}) and in Berger, Strawderman and\break Tang (\citeyear{Berger-Straw-Tang-2005}) where Brown's (\citeyear{Brown-1971}) condition for admissibility (and inadmissibility) was applied for a~va\-riety of hierarchical Bayes models. Maruyama and Takemura (\citeyear{Maruyama-Takemura-2008}) also give admissibility results for the general spherically symmetric case. At least for spherically symmetric priors, the conditions are, essentially, that priors with tails no greater than\break
$O(\|\theta\|^{-(p-2)})$ give admissible procedures.
Fourdrinier, Strawderman and Wells (\citeyear{Fourdrinier-etal-1998}), using Stein's (\citeyear{Stein-1981}) results (especially Proposition \ref{prop31} above, and its corollaries), give classes of minimax Bayes (and generalized Bayes) estimators which include scaled multivariate-$t$ priors under certain conditions. Berger and Robert (\citeyear{Berger-Robert-1990}) give classes of priors leading to minimax estimators. Kubokawa and Strawderman (\citeyear{Kubokawa-Straw-2007}) give classes of priors in the setup of Berger and Strawderman (\citeyear{Berger-Straw-1996}) that lead~to admissible minimax estimators. Maruyama (\citeyear{Maruyama-2003}) and Fourdrinier, Kortbi and Strawderman (\citeyear{Fourdrinier-etal-2008}), in the scale mixture of normal case, find Bayes and~ge\-neralized Bayes minimax estimators, generalizing results\vadjust{\goodbreak} of Strawderman (\citeyear{Straw-1974}). As mentioned in Section \ref{sec3}, these results use either Berger's (\citeyear{Berger-1975}) \mbox{result} (a version of which is given in Theorem~\ref{thmm32}) or Strawderman's (\citeyear{Straw-1974}) result for mixtures of normal distributions. Fourdrinier and Strawderman (\citeyear{Fourdrinier-Straw-2008}) pro\-ved minimaxity of generalized Bayes estimators cor\-responding to certain harmonic priors for classes~of spherically symmetric sampling distributions which are not necessarily mixtures of normals. The results in this paper are not based directly on the discussion of Section \ref{sec3} but are somewhat more closely related in spirit to the approach of Stein (\citeyear{Stein-1981}).
We give below an intriguing result of Maruyama (\citeyear{Maruyama-2003b}) for the unknown scale case (see also Maruya\-ma and Strawderman, \citeyear{Maruyama-Straw-2005}), which is related to the (distributional) robustness of Stein estimators in the unknown scale case treated in Section \ref{sec4}. First, we give a lemma which will aid in the development of the main result.
\begin{lemma}\label{lem61}
Suppose $(X, U) \sim\eta^{(p+k)/2}\cdot f(\eta\{\|x- \theta\|^2 +\|u\|^2\})$, the (location-scale invariant) loss is given by
$L(\{\theta, \eta\},\delta) = \eta\|\delta- \theta\|^2$ and the prior distribution on $(\theta, \eta)$ is of the form $\pi(\theta,\eta) = \rho(\theta)\eta^B $. Then provided all integrals exist, the generalized\break Bayes estimator does not depend on $f(\cdot)$. \end{lemma}
\begin{pf}
\begin{eqnarray*} &&\delta(X,U)\\[-2pt]
&&\quad= E[\theta\eta|X,U]/E[\eta|X,U] \\[-2pt] &&\quad = \biggl[\int_{R^p } \int_0^\infty \theta\eta^{(p + k)/2 + B+1}\\[-2pt]
&&\hspace*{38pt}\qquad{}\cdot f(\eta\{\|X - \theta\|^2 + \|U\|^2 \})\rho(\theta)\,d\eta \,d\theta\biggr]\\[-2pt] &&\qquad{}\cdot \biggl[\int_{R^p } \int_0^\infty \eta^{(p + k)/2 + B+1} \\[-2pt]
&&\hspace*{48pt}\qquad{}\cdot f(\eta\{\|X - \theta\|^2\\[-2pt]
&&\hspace*{82pt}\qquad{}+ \|U\|^2 \})\rho(\theta)\,d\eta
\,d\theta\biggr]^{-1}.\vspace*{-2pt} \end{eqnarray*}
Making the change of variables $w = \eta(\|X - \theta\|^2 + \|U\|^2 )$, we have
\begin{eqnarray*} && \delta(X,U) \\[-2pt] &&\quad=
\biggl[\int_{R^p } \theta(\|X - \theta\|^2 + \|U\|^2 )^{ - (p + k)/2 + B + 2}\\[-2pt] &&\hspace*{9pt}\quad\qquad{}\cdot\rho(\theta)\,d\theta\int_0^\infty w^{(p + k)/2 + B + 1} f(w)\,dw \biggr]\\[-2pt]
&&\qquad{}\cdot\biggl[\int_{R^p } (\|X - \theta\|^2 + \|U\|^2 )^{ - (p + k)/2 + B + 2}\\[-2pt] &&\hspace*{16pt}\quad\qquad{}\cdot\rho(\theta)\,d\theta\int_0^\infty w^{(p + k)/2 + B + 1} f(w)\,dw \biggr]^{-1} \\[-2pt] &&\quad =
\frac{\int_{R^p } \theta(\|X - \theta\|^2 + \|U\|^2 )^{ - (p + k)/2 + B + 2} \rho(\theta)\,d\theta}
{\int_{R^p } (\|X - \theta\|^2 + \|U\|^2 )^{ - (p + k)/2 + B + 2} \rho(\theta)\,d\theta}.\vspace*{-3pt} \end{eqnarray*}
\end{pf}
Hence, for (generalized) priors of the above form, the Bayes estimator is independent of the sampling distribution provided the Bayes estimator exists;\break thus, they may be calculated for the most convenient density, which is typically the normal. Our next lemma calculates the generalized Bayes estimator for a normal sampling density and for a class of priors for which $\rho (\cdot)$ is a scale mixture of normals.
\begin{lemma}\label{lem62} Suppose the distribution of $(X, U)$~is normal with variance $\sigma^2 = 1/\eta$. Suppose also that~the conditional distribution of $\theta$ given $\eta$ and $\lambda$ is~normal with mean 0 and covariance $(1-\lambda)/(\eta\lambda)I$, and the density of $(\eta, \lambda)$ is proportional to $ \eta^{b/2-p/2+ a}\cdot \lambda^{b/2 - p/2-1} (1-\lambda)^{-b/2 + p/2-1}$, where $0<\lambda<1$.
\begin{enumerate}
\item Then the Bayes estimator is given by $(1-r(W)/\break W)X$, where $W\!=\!\|X\|^2/\|U\|^2$ and $r(w)$ is given~by
\begin{eqnarray}\label{eq61} r(w) &=& w\biggl[\int_0^1 \lambda^{b/2} (1 - \lambda)^{p/2 - b/2 - 1} \nonumber\\[-2pt] &&\hspace*{26pt}{}\cdot(1 + w\lambda)^{ - k/2 - a - b/2 - 2} \,d\lambda\biggr] \nonumber \\[-10pt] \\[-10pt] \nonumber &&{}\cdot\biggl[\int_0^1 \lambda^{b/2 - 1} (1 - \lambda)^{p/2 - b/2 - 1} \\[-2pt] &&\hspace*{26pt}{}\cdot (1 + w\lambda)^{ - k/2 - a - b/2 - 2} \,d \lambda\biggr]^{-1}.\nonumber \end{eqnarray}
This is well defined for $0 < b< p$, and $k/2+a+b/2+2 > 0$. \label{1lem62}
\item Furthermore, this estimator is generalized Bayes corresponding to the generalized prior proportional to $\eta^a\|\theta\|^{-b}$, for any spherically symmetric density~$f(\cdot)$ for which $\int_0^\infty t^{(k + p)/2 + a + 1} f(t)\,dt < \infty$. \label{2lem62} \end{enumerate}
\end{lemma}
\begin{pf} Part~1. In the normal case,
\begin{eqnarray*}
\delta(X,U) &=& X + \frac{E[\eta(\theta - X)|X,U]}{E[\eta|X,U]} \\
&=& X - \frac{\nabla_X m(X,U)}{2(\partial/\partial\|U\|^2) m(X,U)},\vadjust{\goodbreak} \end{eqnarray*}
where the marginal $m(x,u) $ is proportional to
\begin{eqnarray*} && \int_0^1 \int_0^\infty \int_{R^p} \eta^{b/2 + k/2 + p/2 + a} \lambda^{b/2 - 1} (1 - \lambda)^{- b/2 - 1} \\[-2pt]
&&\hspace*{32pt}\qquad{} \cdot\exp( - \eta\{\|x - \theta\|^2 + \|u\|^2 \}/2) \\[-2pt]
&&\hspace*{32pt}\qquad{}\cdot \exp\biggl( - \frac{\eta\lambda\|\theta\|^2}{2(1 - \lambda)}\biggr) \,d\theta\, d\eta\, d\lambda\\[-2pt] &&\quad = K' \int_0^1 \int_0^\infty \eta^{b/2 + k/2 + a} \lambda^{b/2 - 1} (1 - \lambda)^{p/2 - b/2 - 1}\\[-2pt]
&&\hspace*{70pt}{}\cdot\exp( - \eta\{\lambda\|x\|^2 + \|u\|^2 \}/2) \,d\eta\, d\lambda\\[-2pt]
&&\quad= K \int_0^1 (\lambda\|x\|^2 + \|u\|^2 ) ^{- b/2-k/2-a-1} \lambda^{b/2 - 1} \\[-2pt] &&\hspace*{46pt}{}\cdot(1 - \lambda)^{p/2 - b/2 - 1}\, d\lambda. \end{eqnarray*}
Hence, we may express the Bayes estimator as $\delta(X,\break U) = X + g(X,U)$, where
\begin{eqnarray*} g(x,u)
&= &\biggl[\nabla_x \int_0^1 (\lambda\|x\|^2+\|u\|^2)^{-b/2-k/2-a-1}\\[-3pt] &&\hspace*{34pt}{}\cdot\lambda^{b/2- 1} (1 - \lambda)^{p/2-b/2-1} \,d\lambda\biggr]\\[-3pt]
&&{}\cdot\biggl[- 2(d/d\|u\|^2)\\[-3pt]
&&\quad{}\cdot\int_0^1 (\lambda\|x\|^2 +\|u\|^2)^{-b/2-k/2-a-1}\\[-3pt] &&\hspace*{36pt}{}\cdot\lambda^{b/2-1} (1-\lambda)^{p/2-b/2-1} \,d\lambda\biggr]^{-1} \\[-3pt]
&=& -x\biggl[\int_0^1 (\lambda\|x\|^2 + \|u\|^2)^{-b/2-k/2-a-2} \\[-3pt] &&\hspace*{48pt}{}\cdot\lambda^{b/2} (1-\lambda)^{p/2-b/2-1}\, d\lambda\biggr]\\[-3pt]
&&{}\cdot \biggl[\int_0^1 (\lambda\|x\|^2 + \|u\|^2 )^{-b/2-k/2-a-2} \\[-3pt] &&\hspace*{18pt}\quad{}\cdot\lambda^{b/2-1}(1 - \lambda)^{p/2-b/2-1} \,d\lambda\biggr]^{-1} \\[-3pt] &=& -x\biggl[\int_0^1(\lambda w + 1)^{-b/2-k/2-a-2}\\[-3pt] &&\hspace*{33pt}{}\cdot\lambda^{b/2} (1-\lambda)^{p/2-b/2-1}\,d\lambda\biggr]\\[-3pt] &&{}\cdot\biggl[\int_0^1 (\lambda w + 1)^{-b/2-k/2-a-2}\\[-3pt] &&\hspace*{26pt}{}\cdot\lambda^{b/2-1} (1-\lambda)^{p/2-b/2-1} \,d\lambda\biggr]^{-1} \\[-3pt] &=& - \frac{x}{w}r(w). \end{eqnarray*}
Part~2. A straightforward calculation shows that the unconditional density of $(\theta, \eta)$ is proportional to
$\eta^a \|\theta\|^{-b}$. Hence, part~2 follows from Lemma \ref{lem61}.\vspace*{-1pt} \end{pf}
The following lemma gives properties of $r(w)$.\vadjust{\goodbreak}
\begin{lemma}\label{lem63} Suppose $0 < b \leq p-2$ and that $k/2+a+1 > 0$. Then, (1) $r(w)$ is nondecreasing, and (2)~$0 < r(w) \leq b/(k+2a+2)$. \end{lemma}
\begin{pf} By a change of variables, letting $v = \lambda w$ in (\ref{eq61}), then
\begin{eqnarray*} r(w) &=& \biggl[\int_0^w {(v + 1)} ^{ - b/2 - k/2 - a - 2}\\ &&\hspace*{21pt}{}\cdot v^{b/2} (1 - v /w)^{p/2 - b/2 - 1} \,dv\biggr]\\ &&{}\cdot\biggl[\int_0^w {(v + 1)} ^{ - b/2 - k/2 - a - 2}\\ &&\hspace*{28pt}{}\cdot v^{b/2 - 1} (1 - v /w)^{p/2 - b/2 - 1} \,dv\biggr]^{-1}. \end{eqnarray*}
So, we may rewrite $r(w)$ as $E_w[v]$, where $v$ has density proportional to $(1 + v )^{-b/2-k/2-a-2} v^{b/2 - 1} (1-v/w)^{p/2-b/2-1}I_{[0,w]}(v)$. This density has increasing monotone likelihood ratio in $w$ as long as $p/2 - b/2 - 1 \ge0$. Hence, part 1 follows.
The conditions of the lemma allow interchange of limit and integration in both numerator and denominator of $r(w)$ as $w \to\infty$. Hence,
\begin{eqnarray*} \qquad r(w) & \le&\frac{\int_{0}^\infty (1 + v)^{ - b/2 - k/2 - a - 2} v^{b/2} \,dv} {\int_{0}^\infty (1 + v)^{ - b/2 - k/2 - a - 2} v^{b/2 - 1} \,dv} \\ &=& \frac{\int_0^1 u^{b/2} (1 - u)^{k/2 + a} \,du} {\int_0^1 u^{b/2 - 1} (1 - u)^{k/2 + a + 1} \,du}\\ &&\hspace*{80pt}[\mbox{letting }u = v/(v + 1)]\\ &=& \frac{\operatorname{Beta}(b/2 + 1,k/2 + a + 1)}{\operatorname{Beta}(b/2,k/2 + a + 2)} \\ &=& \frac{b/2}{k/2 + a + 1}. \end{eqnarray*}
\upqed\end{pf}
Combining Lemmas \ref{lem61}--\ref{lem63} with Corollary \ref{cor41} gi\-ves as the main result a class of estimators which are generalized Bayes and minimax simultaneously for the entire class of spherically symmetric sampling distributions (subject to integrability conditions).
\begin{thmm} Suppose that the distribution\break of~$(X,U)$ and the loss function are as in Lemma~\ref{lem61}, and that the prior distribution is as in Lemmas \ref{lem62} and \ref{lem63} with a satisfying $b/(k + 2a + 2) \le2 (p-2)/(k+2)$, and with\vadjust{\goodbreak} $0 < b \leq p-2$. Then the corresponding generalized Bayes estimator is minimax for all densities $f(\cdot)$ such that the $2(a+2)$th moment of the distribution of $(X,U)$ is finite, that is, $E(R^{2a + 4} ) < \infty$. \end{thmm}
We note that the above finiteness condition,\break $E(R^{2a + 4} ) < \infty$, is equivalent to the finiteness condition, $\int_0^\infty {t^{(k + p)/2 + a + 1} } f(t) \,dt < \infty,$ in Lemma \ref{lem62}.
\section{Concluding Remarks} \label{sec7} This paper has reviewed some of the developments in shrinkage estimation of mean vectors for spherically symmetric distributions, mainly since the review paper of Brandwein and Strawderman (\citeyear{Brand-Straw-1990}). Other papers in this volume review other aspects of the enormous literature generated by or associated with Stein's stunning inadmissibility result of 1956.
Most of the developments we have covered are, or can be viewed as, outgrowths of Stein's papers of 1973 and 1981, and, in particular, of Stein's lemma which gives (an incredibly useful) alternative expression for the cross product term in the quadratic risk function.
Among the topics which we have not covered is the closely related literature for elliptically symmetric distributions (see, e.g., Kubokawa and Srivastava, \citeyear{Kubokawa-Srivastava-2001}, and Fourdrinier, Strawderman and Wells, \citeyear{Fourdrinier-etal-2003}, and the references therein). We also have not included a discussion of Hartigan's (\citeyear{Hartigan-2004}) beautiful result that the (generalized or proper) Bayes estimator of a normal mean vector with respect to the uniform prior on any convex set in $R^p$ dominates $X$ for squared error loss. Nor have we discussed the very useful and pretty development of the Kubokawa (\citeyear{Kubokawa-1994}) IERD method for finding improved estimators, and, in particular, for dominating James Stein estimators (see also Marchand and Strawderman, \citeyear{Marchand-Straw-2004}, for some discussion of these last two topics). We nonetheless hope we have provided some intuition for, and given a flavor of the developments and rich literature in the area of improved estimators for spherically symmetric distributions.
The impact of Stein's beautiful 1956 result and his innovative development of the techniques in the 1973 and 1981 papers have inspired many researchers, fue\-led an enormous literature on the subject, led to a deeper understanding of theoretical and practical aspects of ``sharing strength'' across related studies, and greatly enriched the field of Statistics. Even~so\-me of the early (and later) heated discussions of~the theoretical and practical aspects of ``sharing strength'' across unrelated studies have had an ultimately posi\-tive impact on the development of hierarchical models and computational tools for their analysis. We are very pleased to have been asked to contribute~to this volume commemorating fifty years\vadjust{\goodbreak} of development of one of the most profound results in the Statistical literature in the last half of the 20th century.\looseness=-1
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Lindenbaum method
From Encyclopedia of Mathematics
Revision as of 22:53, 24 October 2013 by Alexei.muravitsky (talk | contribs) (→Lindenbaum's theorem)
1 Lindenbaum method (propositional language)
1.1 Lindenbaum's theorem
1.1.1 Historical remarks
1.2 Wójcicki's theorems
1.3 Lindenbaum-Tarski algebra
1.4 Alternative approach
1.5 Specifications and applications
Lindenbaum method (propositional language)
Lindenbaum method is named after the Polish logician Adolf Lindenbaum who prematurely and without a clear trace disappeared in the turmoil of the Second World War at the age of about 37. (Cf.[16].) The method is based on the symbolic nature of formalized languages of deductive systems and opens a gate for applications of algebra to logic and, thereby, to Abstract algebraic logic.
Lindenbaum's theorem
A formal propositional language, say $\mathcal{L}$, is understood as a nonempty set ${\cal V}_\mathcal{L}$ of symbols $p_0, p_1,... p_{\gamma}...$ called propositional variables and a finite set $\Pi$ of symbols $F_0, F_1,..., F_n$ called logical connectives. By $\overline{\overline{{\cal V}_\mathcal{L}}}$ we denote the cardinality of ${\cal V}_\mathcal{L}$. For each connective $F_i$, there is a natural number $\#(F_i)$ called the arity of the connective $F_i$. The notion of a statement (or a formula) is defined as follows:
$(f_1)$ Each variable $p\in {\cal V}_\mathcal{L}$ is a formula;
$(f_2)$ If $F_i$ is a connective of the arity 0, then $F_i$ is a formula;
$(f_3)$ If $A_1, A_2,..., A_n$, $n\geq 1$, are formulas, and $F_n$ is a connective of arity $n$, then the symbolic expression $F_{n}A_{1}A_{2}... A_n$ is a formula;
$(f_4)$ A formula can be constructed only according to the rules $(f_1)-(f_3)$.
The set of formulas will be denoted by $Fr_\mathcal{L}$ and $P(Fr_\mathcal{L})$ denotes the power set of $Fr_\mathcal{L}$. Given a set $X \subseteq Fr_\mathcal{L}$, we denote by ${\cal V}(X)$ the set of propositional variables that occur in the formulas of $X$. Two formulas are counted equal if they are represented by two copies of the same string of symbols. (This is the key observation on which Theorem 1 is grounded.) Another key observation (due to Lindenbaum) is that $Fr_\mathcal{L}$ along with the connectives $\Pi$ can be regarded as an algebra of the similarity type associated with $\mathcal{L}$, which exemplifies an $\mathcal{L}$-algebra. We denote this algebra by $\mathfrak{F}_\mathcal{L}$. The importance of $\mathfrak{F}_\mathcal{L}$ can already be seen from the following observation.
Theorem 1. Algebra $\mathfrak{F}_\mathcal{L}$ is a free algebra of rank $\overline{\overline{{\cal V}_\mathcal{L}}}$ with free generators ${\cal V}_\mathcal{L}$ in the class $($variety$)$ of all $\mathcal{L}$-algebras. In other words, $\mathfrak{F}_\mathcal{L}$ is an absolutely free algebra of this class.
A useful feature of the set $Fr_\mathcal{L}$ is that it is closed under (simultaneous) substitution. More than that, any substitution $\sigma$ is an endomorphism
$\sigma: \mathfrak{F}_\mathcal{L}\longrightarrow \mathfrak{F}_\mathcal{L}$.
A monotone deductive system (or a deductive system or simply a system) is a relation between subsets and elements of $Fr_\mathcal{L}$. Each such system $\vdash_S$ is subject to the following conditions: For all $X,Y \subseteq Fr_\mathcal{L}$,
$(s_1)$ if $A \in X$, then $X \ \vdash_\mathcal{S} \ A$;
$(s_2)$ if $X \ \vdash_\mathcal{S} \ B$ for all $B \in Y$, and $Y \ \vdash_\mathcal{S} \ A$, then $X \ \vdash_\mathcal{S} \ A$;
$(s_3)$ if $X \ \vdash_\mathcal{S} \ A$, then for every substitution $\sigma$, $\sigma[X] \ \vdash_\mathcal{S} \ \sigma(A)$.
If $A$ is a formula and $\sigma$ is a substitution, $\sigma(A)$ is called a substitution instance of $A$. Thus, by $\sigma[X]$ above, one means the instances of the formulas of $X$ with respect to $\sigma$.
Given two sets $Y$ and $X$, we write
$\quad \quad \quad Y \Subset X $
if $Y$ is a finite (maybe empty) subset of $X$.
A deductive system is said to be finitary if, in addition, it satisfies the following:
$(s_4)$ if $X \ \vdash_\mathcal{S} \ A$, then there is $Y \Subset X$ such that $Y \ \vdash_\mathcal{S} \ A$.
We note that the monotonicity property
$\quad \quad \quad \quad$ if $X \subseteq Y$ and $X \ \vdash_\mathcal{S} \ A$, then $Y \ \vdash_\mathcal{S} \ A$
is not postulated, because it follows from $(s_1)$ and $(s_2)$.
Each deductive system $\vdash_\mathcal{S}$ induces the (monotone structural) consequence operator $Cn_{\mathcal{S}}$ defined on the power set of $Fr_\mathcal{L}$ as follows: For every $X \subseteq Fr_\mathcal{L}$,
$\quad \quad \quad \quad A \in Cn_\mathcal{S}(X) \Longleftrightarrow X \ \vdash_\mathcal{S} \ A, \quad \quad \quad \quad \quad \quad \quad \quad (1)$
so that the following conditions are fulfilled: For all $X,Y \subseteq Fr_\mathcal{L}$ and any substitution $\sigma$,
$(c_1)$ $X \subseteq Cn_\mathcal{S}(X);$ (reflexivity)
$(c_2)$ $Cn_\mathcal{S}(Cn_\mathcal{S}(X)) = Cn_\mathcal{S}(X);$ (idenpotency)
$(c_3)$ if $X \subseteq Y$, then $Cn_\mathcal{S}(X) \subseteq Cn_\mathcal{S}(Y);$ (monotonicity)
$(c_4)$ $\sigma[Cn_\mathcal{S}(X)] \subseteq Cn_\mathcal{S}(\sigma[X]).$ (structurality or substitution invariance)
If $\vdash_\mathcal{S}$ is finitary, then
$(c_5)$ $Cn_\mathcal{S}(X) = \bigcup\lbrace Cn_\mathcal{S}(Y) \ | \ Y \Subset X \rbrace$
in which case $Cn_{\mathcal{S}}$ is called finitary.
Conversely, if an operator $Cn:\cal{P}(Fr_\mathcal{L})\rightarrow \cal{P}(Fr_\mathcal{L})$ satisfies the conditions $(c_1)-(c_4)$ (with $Cn$ instead of $Cn_\mathcal{S}$), then the equivalence
$\quad \quad \quad \quad X \ \vdash_\mathcal{S} \ A \Longleftrightarrow A \in {Cn}(X)$
defines a deductive system, $\mathcal{S}$. Thus (1) allows one to use the deductive system and consequence operator (in a fixed formal language) interchangeably or even in one and the same context. For instance, we call $T_\mathcal{S} = Cn_\mathcal{L}(\emptyset)$ the set of theorems of the system $\vdash_\mathcal{S}$ (i.e. $\mathcal{S}$-theorems), and given a subset $X \subseteq Fr_\mathcal{S}$, $Cn-\mathcal{S}{X}$ is called the $\mathcal{S}$-theory generated by $X$. A subset $X \subseteq Fr_\mathcal{S}$, as well as the theory $Cn_\mathcal{S}(X)$, is called inconsistent if $Cn_\mathcal{S}(X) = Fr_\mathcal{S}$; otherwise both are consistent. Thus, given a system $\vdash_\mathcal{S}$, $T_\mathcal{S}$ is one of the system's theories; that is to say, if $X \subseteq T_\mathcal{S}$ and $X \vdash_\mathcal{S} A$, then $A \in T_\mathcal{S}$. This simple observation sheds light on the central idea of Lindenbaum method, which will be explained soon. For now, let us fix the ordered pair $\left<\mathcal{F}_\mathcal{L},T\mathcal{L}\right>$ and call it a Lindenbaum matrix. (The full definition will be given later.) We note that an operator $Cn$ satisfying $(c_1)-(c_3)$ can be obtained from a "closure system" over $Fr_\mathcal{L}$; that is for any subset $\cal{A}\subseteq P(Fr_\mathcal{L})$, which is closed under arbitrary intersection, we define:
$\quad \quad \quad \quad Cn_\mathcal{A}(X)=\cap \lbrace Y \ | \ X \subseteq Y \mbox{ and } Y \in \cal{A} \rbrace.$
Another way of defining deductive systems is through the use of logical matrices. Given a language $\mathcal{L}$, a logical $\mathcal{L}$-matrix (or simply a matrix) is a pair $\mathcal{M} = \left<\mathfrak{A},\mathcal{F}\right>$, where $\mathfrak{A}$ is an $\mathcal{L}$-algebra and $\mathcal{F}\subseteq|\mathfrak{A}|$, where the latter is the universe of $\mathfrak{A}$. The set $\mathcal{F}$ is called the filter of the matrix and its elements are called designated. Given a matrix $\mathcal{M} = \left<\mathfrak{A},\mathcal{F}\right>$, the cardinality of $|\mathfrak{A}|$ is also the cardinality of $\mathcal{M}$.
Given a matrix $\mathcal{M}=\left<\mathfrak{A},\mathcal{F}\right>$, any homomorphism of $\mathfrak{A}$ into $\mathfrak{A}$ is called a valuation (or an assignment). Each such homomorphism can be obtained simply by assigning elements of $|\mathfrak{A}|$ to the variables of $Vr_\mathcal{L}$, since, by virtue of Theorem 1, any $v: Vr_\mathcal{L} \longrightarrow |\mathfrak{A}|$ can be extended uniquely to a homomorphism $\hat{v}: \mathfrak{A} \longrightarrow \mathfrak{A}$. Usually, $v$ is meant under a valuation (or an assignment) of variables in a matrix.
Now let $\sigma$ be a substitution and $v$ be any assignment in an algebra $\mathfrak{A}$. Then, defining
$\quad \quad \quad \quad v_{\sigma}=v\circ\sigma, \quad \quad \quad \quad \quad \quad \quad \quad (2)$
we observe that $v_{\sigma}$ is also an assignment in $\mathfrak{A}$.
With each matrix $\mathcal{M}=\left<\mathfrak{A},\mathcal{F}\right>$, we associate a relation $\models_\mathcal{M}$ between subsets of $Fr_\mathcal{L}$ and formulas of $Fr_\mathcal{L}$. Namely we define
$ \quad \quad \quad \quad X \ \models_\mathcal{M} \ A \Longleftrightarrow \text{ for every assignment } v, \text{ if } v[X]\subseteq \mathcal{F}, \text{ then } v(A)\in \mathcal{F}$.
Then, we observe that the following properties hold:
$(m_1)$ if $A \in X$, then $X \ \models_\mathcal{M} \ A$
$(m_2)$ if $X\models_\mathcal{M} B$ for all $B\in Y$, and $Y \ \models_\mathcal{M} \ A$, then $X \ \models_\mathcal{M} \ A.$
Also, with help of the definition (2), we derive the following:
$(m_3)$ if $X \ \models_\mathcal{M} \ A$, then for every substitution $\sigma$, $\sigma[X] \ \models_\mathcal{M} \ \sigma(A)$.
Comparing the condition $(m_1)-(m_3)$ with $(s_1)-(s_3)$, we conclude that every matrix defines a structural deductive system and hence, in view of (1), a structural consequence operator.
Given a system $\mathcal{S}$, suppose a matrix $\mathcal{M}=\left<\mathfrak{A},\mathcal{F}\right>$ satisfies the condition
$\quad \quad \quad \quad $ if $X \ \vdash_\mathcal{S} A$ and $v[X] \subseteq \mathcal{F}$, then $v(A) \in \mathcal{F} \quad \quad \quad \quad (3)$
Then the filter $\mathcal{F}$ is called an $\mathcal{S}$-filter and the matrix $\mathcal{M}$ is called an $\mathcal{S}$-matrix (or an $\mathcal{S}$-model). In view of (3), $\mathcal{S}$-matrices are an important tool in showing that $X \ \vdash_\mathcal{S} \ A$ does not hold. This idea has been employed in proving that one axiom is independent from a group of others in the search for an independent axiomatic system, as well as for semantic completeness results.
As Lindenbaum's famous theorem below explains, every structural system $\mathcal{S}$ has an $\mathcal{S}$-model.
Theorem 2 (Lindenbaum). For any structural deductive system $\mathcal{S}$, the matrix $\left<Fr_\mathcal{L},Cn_\mathcal{S}(\emptyset)\right>$ is an $\mathcal{S}$-model. Moreover, for any formula $A$,
$\quad \quad \quad \quad A \in T_\mathcal{S} \Longleftrightarrow v(A)\in Cn_\mathcal{S}(\emptyset)$ for any valuation $v$.
A matrix $\left<\mathfrak{A},\mathcal{F}\right>$ is said to be weakly adequate for a deductive system $\mathcal{S}$ if for any formula $A$,
$\quad \quad \quad \quad A \in T_\mathcal{S} \Longleftrightarrow v(A)\in \mathcal{F}$ for any valuation $v$.
Thus, according to Theorem 2, every structural system $\mathcal{S}$ has a weakly adequate $\mathcal{S}$-matrix of cardinality less than or equal to $\overline{\overline{\mathcal{V}}}+\aleph_0$.
An $\mathcal{S}$-matrix is called strongly adequate for $\mathcal{S}$ if for any set $X \subseteq Fr_\mathcal{L}$ and any formula $A$,
$ \quad \quad \quad \quad X \ \vdash_\mathcal{S} \ A \Longleftrightarrow X \ \models_\mathcal{M} \ A. \quad \quad \quad \quad (4)$
We note that, if $\overline{\overline{\mathcal{V}}} \leq \aleph_{0}$, Theorem 2 cannot be improved to include strong adequacy of an denumerable matrix, for if $\mathcal{S} = IPC$ (intuitionistic propositional calculus), there is no denumerable matrix $\mathcal{M}$ with (4). (Cf.[21].)
Historical remarks
A. Tarski seems to be the first who promoted "the view of matrix formation as a general method of constructing systems" [10]. However, matrices had been employed earlier, e.g., by P. Bernays [1] and others either in the search for an independent axiomatic system or for defining a system different from classical logic. Also, later on J.C.C. McKinsey [11] used matrices to prove independence of logical connectives in intuitionistic propositional logic.
Theorem 2 was discovered by A. Lindenbaum. Although this theorem was not published by the author, it had been known in Warsaw-Lvov logic circles at the time. In a published form it appeared for the first time in [10] without proof. Its proof appeared later on in the two independent publications of [9] and [7].
Wójcicki's theorems
We get more $\mathcal{S}$-matrices, noticing the following. Let $\Sigma_\mathcal{S}$ be an $\mathcal{S}$-theory. The pair $\left<Fr_\mathcal{L},\Sigma_\mathcal{S} \right>$ is called a Lindenbaum matrix relative to $\mathcal{S}$. We observe that for any substitution $\sigma$,
$\quad \quad \quad \quad $ if $X \ \vdash_\mathcal{S} \ A$ and $\sigma[X] \subseteq \Sigma_\mathcal{S}$, then $\sigma(A) \in \Sigma_\mathcal{S}$}.
That is to say, any Lindenbaum matrix relative to a system $\mathcal{S}$ is an $\mathcal{S}$-model.
A deductive system $\mathcal{S}$ is said to be uniform if, given a set $X \subseteq Fr_\mathcal{S}$ and a consistent set $ Y \subseteq Fr_\mathcal{S}$, $X \cup Y \ \vdash_\mathcal{S} \ A$ and $Vr(Y) \cap Vr(A) = \emptyset$ imply $X \ \vdash_\mathcal{S} \ A$. A system $\mathcal{S}$ is couniform if for any collection $\{X_{i}\}_{i\in I}$ of formulas with $Vr(X_i) \cap Vr(X_j) = \emptyset$, providing $i \neq j$, if the set $\cup\{X_{i}\}_{i\in I}$ is inconsistent, then at least one $X_{i}$ is inconsistent as well.
Theorem 3 (Wójcicki). A structural deductive system $\mathcal{S}$ has a strongly adequate matrix if and only if $\mathcal{S}$ is both uniform and couniform.
For the "if" implication of the statement, the matrix of Theorem 2 is not enough. However, it is possible to extend the original language $\mathcal{L}$ to $\mathcal{L}^{+}$ in such a way that the natural extension $Cn_{\mathcal{S}^{+}}$ of $Cn_{\mathcal{S}}$ onto $\mathcal{L}^{+}$ allows one to define a Lindenbaum matrix $\left<\mathfrak{F}_{\mathcal{L}^{+}},Cn_{\mathcal{S}^{+}}(X)\right>$, for some $X \subseteq Fr_{\mathcal{L}^{+}}$, which is strongly adequate for $\mathcal{S}$. (Cf.[21] for detail.)
A pair $\left<\mathfrak{A},\ \{\mathcal{F}_{i}\}_{i\in I}\right>$, where $\mathfrak{A}$ is an $\mathcal{L}$-algebra and each $\mathcal{F}_{i}\subseteq|\mathfrak{A}|$, is called a generalized matrix (or a $g$-matrix for short). A $g$-matrix is a $g$-$\mathcal{S}$-model (or a $g$-$\mathcal{S}$-matrix) if each $\left<\mathfrak{A},\mathcal{F}_{i}\right>$ is an $\mathcal{S}$-model. (In [4] a $g$-matrix is called an atlas.)
Theorem 4 (Wójcicki). For every structural deductive system $\mathcal{S}$, there is a $g$-$\mathcal{S}$-matrix $\mathcal{M}$ of cardinality $\overline{\overline{\mathcal{V}}}+\aleph_{0}$, which is strongly adequate for $\mathcal{S}$.
Indeed, let $\{\Sigma_\mathcal{S}\}$ be the collection of all $\mathcal{S}$-theories. Then the $g$-matrix $\left<Fr_\mathcal{L},\{\Sigma_\mathcal{S}\}\right>$ is strongly adequate for $\mathcal{S}$. (Cf.[21],[4] for detail.)
We note that, alternatively, one could use the notion of a bundle of matrices; a bundle is a set $\{\left<\mathfrak{A},\mathcal{F}_{i}\right> | i\in I \}$, where $\mathfrak{A}$ is an $\mathcal{L}$-algebra and each $\mathcal{F}_{i}$ is a filter of $\mathfrak{A}$.
Theorem 3 was the result of the correction by R. Wójcicki of an erroneous assertion in [8], where the important question on the strong adequacy of a system was raised.
T. Smiley [15] was perhaps the first to propose $g$-matrices (known as Smiley matrices) defined as pairs $\left<\mathfrak{A},Cn \right>$, where $\mathfrak{A}$ is an $\mathcal{L}$-algebra and an operator $Cn: \mathcal{P}(|\mathfrak{A}|) \rightarrow \mathcal{P}(|\mathfrak{A}|)$ satisfies the conditions $(c_1)-(c_3)$ (with $Cn$ instead of $Cn_\mathcal{S}$). Then, Smiley defined $x_1,..., x_n \ \vdash \ y$ if and only of $y \in Cn(\{x_1,...,x_n\})$, where it is assumed that $|\mathfrak{A}| \subseteq U$, where $U$ is a universal set of sentences.
Lindenbaum-Tarski algebra
The question of the possibility to decide, whether $X \ \vdash_\mathcal{S} \ A$ is true or not is central in theory of deduction. Although the notion we are about to introduce is less general than that of $\mathcal{S}$-matrix, it points out at a way, following which this question can be often fruitfully discussed.
An $\mathcal{S}$-matrix $\left<\mathfrak{A},\mathcal{F}\right>$ is said to be univalent (or $\mathcal{S}_u$-matrix) if the $\mathcal{S}$-filter $\mathcal{F}$ consists of one value, say $\mathcal{F} = \{ \mathbf{1} \}$, where $\mathbf{1} \in |\mathfrak{A}|$. Let us restrict our original question to the following: How can the property $\emptyset \ \vdash_\mathcal{S} \ A$ be characterized in matrix terms?
Let $\left<\mathfrak{A},\{\mathbf{1}\}\right>$ be an $\mathcal{S}_u$-matrix and $A$ be an $\mathcal{S}$-theorem. Then, in view of (3), $v(A)=\mathbf{1}$ for every valuation $v$ in $\mathfrak{A}$. It would be interesting to know when the converse is true too. Thus the main problem is: How can one obtain an $\mathcal{S}_u$-matrix?
Definition 1 (Lindenbaum-Tarski algebra). Let $\Sigma_\mathcal{S}$ be an $\mathcal{S}$-theory and let $\Theta(\Sigma_\mathcal{S})$ be the congruence on $\mathfrak{F}_{\mathcal{L}}$ generated by $\Sigma_\mathcal{S}$; (cf. [3]). The quotient algebra $\mathfrak{F}_\mathcal{L}/\Theta(\Sigma_\mathcal{S})$ is called a Lindenbaum-Tarski algebra of $\mathcal{S}$ relative to $\Sigma_\mathcal{S}$. If $\Sigma_\mathcal{S} = T_\mathcal{S}$, then we call this quotient simply a Lindenbaum-Tarski algebra.
An important conclusion from this definition is the following.
Theorem 5. Let $\mathcal{S}$ be a structural deductive system and $\Sigma_\mathcal{S}$ be a nonempty $\mathcal{S}$-theory. Assume that $\Sigma_\mathcal{S}$ is a congruence class with respect to $\Theta(\Sigma_\mathcal{S})$. Then $\left<\mathfrak{F}_\mathcal{L}/\Theta(\Sigma_\mathcal{S}),\{\Sigma_\mathcal{S}\}\right>$ is an $\mathcal{S}_u$-matrix; that is to say, denoting $\mathbf{1}=\Sigma_\mathcal{S}$, if $X \ \vdash_\mathcal{S} \ A$ and $v$ is a valuation in $\mathfrak{F}_\mathcal{L}/\Theta(\Sigma_\mathcal{S})$, then
$ \quad \quad \quad \quad v[X]=\{\mathbf{1}\} \Longrightarrow v(A)=\mathbf{1}. \quad \quad \quad \quad (5)$
Moreover, if $\Sigma_\mathcal{S} = T_\mathcal{S}$, then
$\quad \quad \quad \quad A \in T_\mathcal{S} \Longleftrightarrow v(A)=\mathbf{1}$ for any valuation $v$ in $\mathfrak{F}_\mathcal{L}/\Theta(T_\mathcal{S}) \quad \quad \quad \quad (6)$
Let the valuation $v_{0}(p)=p/\Theta(T_\mathcal{S})$ for every $p \in Vr_\mathcal{L}$. Then
$\quad \quad \quad \quad A\in T_\mathcal{S} \Longleftrightarrow v_{0}(A)=\mathbf{1}. \quad \quad \quad \quad (7)$
Definition 2. Let $\mathcal{S}$ be a structural deductive system. We say that $\mathcal{S}$ admits the Lindenbaum-Tarski algebra (relative to $\Sigma_\mathcal{S}$) if $T_\mathcal{S}$ ( $\Sigma_\mathcal{S}$ respectively) is a congruence class with respect to $\Theta(T_\mathcal{S})$ (with respect to $\Theta(\Sigma_\mathcal{S})$) on $\mathfrak{F}_{\mathcal{L}}$.
Now let us convert the propositional language $\mathcal{L}$ into a first order language $\mathcal{L}'$ so that the propositional variables and the logical connectives of $\mathcal{L}$ become the individual variables and functional constants of $\mathcal{L}'$, respectively. The set of individual variables is denoted now by $Vr_{\mathcal{L}'}$. Also, $\mathcal{L}'$ has an individual constant $\mathbf{1}$, the equality symbol '$=$' and universal and existential quantifiers. (Actually, we will need only the former.) We can assume that there is no logical connectives in $\mathcal{L}'$. Since the formulas of $\mathcal{L}$ now become terms of $\mathcal{L}'$, each atomic formula of $\mathcal{L}'$ is an expression of the form:
$ \quad \quad \quad \quad A(p,\mathbf{1},...)=B(q,\mathbf{1},...)$,
where variables $p$ and $q$ are not necessarily distinct and they, as well as the constant $\mathbf{1}$, may or may not occur in the equality.
A universal closure (in the sense of first order logic) of an atomic formula of $\mathcal{L}'$ is often referred to as an identity. We will deal with interpretations of identities only. Therefore, we semantically treat atomic formulas and their universal closures equally. An unspecified identity will be denoted by $\varphi$.
$\quad$ The $\mathcal{L}'$-formulas are interpreted in algebras $\mathfrak{B}$ of the type $\mathcal{L}$ endowed with a 0-ary operation $\mathbf{1}$. Then, for instance, an identity
$ \quad \quad \quad \quad A(p,\mathbf{1},...)=\mathbf{1}$
is said to be valid (or to hold) in $\mathfrak{B}$, in symbols $\mathfrak{B} \ \models \ A(p,\mathbf{1},...)=\mathbf{1}$, if for any assignment $v: \mathcal{L}' \rightarrow |\mathfrak{B}|$
$ \quad \quad \quad \quad A(v(p),\mathbf{1},...)=\mathbf{1}.$
Given a system $\mathcal{S}$, we denote
$\quad \quad \quad \quad \mathfrak{F}_\mathcal{S} = \left<\mathfrak{F}_\mathcal{L}/\Theta(T_\mathcal{S}), \mathbf{1}\right>,$
where $\mathbf{1}$ is the congruence class generated by $T_\mathcal{S}$. Thus $\mathfrak{F}_\mathcal{S}$ is the expansion of $\mathfrak{F}_\mathcal{L}/\Theta(T_\mathcal{S})$ obtained by adding the constant $\mathbf{1}$ to the signature of the latter.
Then, we define:
$\quad \quad \quad \quad \Phi_{\mathcal{S}}= \{ A = \mathbf{1} | A \in T_\mathcal{S}\}$
$ \quad \quad \quad \quad K_{\mathcal{S}} = \{\mathfrak{B}| \mathfrak{B} \ \models \ \varphi$ for all $\varphi \in \Phi_{\mathcal{S}}\}$.
It is obvious that the class $K_{\mathcal{S}}$ is a variety.
Theorem 6. Let a structural deductive system $\mathcal{S}$ admit the Lindenbaum-Tarski algebra. Then the algebra $\mathfrak{F}_\mathcal{S}$ belongs in the variety $K_\mathcal{S}$. More than that,
$ \quad \quad \quad \quad \mathfrak{F}_\mathcal{S} \ \models \ A = \mathbf{1}\Longleftrightarrow A \in T_\mathcal{S}.$
Moreover, $\mathfrak{F}_\mathcal{S} \ \models \ A(p,\mathbf{1},...)=\mathbf{1}$ if and only if $A(p/\Theta(T_\mathcal{S}),\mathbf{1},...)=\mathbf{1}$ in $\mathfrak{F}_\mathcal{S}$, that is $A(p/\Theta(T_\mathcal{S}),T_\mathcal{S},...) = T_\mathcal{S}$ in $\mathfrak{F}_\mathcal{L}/\Theta(T_\mathcal{S})$.
$\quad $ Theorem 6 gives rise to the following questions: When is $\mathfrak{F}_\mathcal{S}$ functionally free [19] in $K_{\mathcal{S}}$? When is $\mathfrak{F}_\mathcal{S}$ a free algebra in $K_{\mathcal{S}}$?
In two parts, [17] and [18], of one paper, the English translation of which constitutes one chapter, Foundations of the Calculus of Systems, of [19], A. Tarski showed that the Lindenbaum-Taski algebra of the system based on classical propositional calculus is a Boolean algebra.
Alternative approach
Let $\left<\mathfrak{A},\mathcal{F}\right>$ be a matrix. A congruence (or an equivalence) $\theta$ on $\mathfrak{A}$ is said to be compatible with $\mathcal{F}$ if $\cup\{ x /\theta ~|~x \in \mathcal{F}\} = \mathcal{F}$. Since the identity relation is compatible with any $\mathcal{F}$, the set of compatible congruences (or equivalences) is not empty for any matrix. Then, it can be proven [2] that for any matrix $\mathcal{M} = \left<\mathfrak{A},\mathcal{F}\right>$, there is a largest congruence of $\mathfrak{A}$ compatible with $\mathcal{F}$. This congruence is called the Leibniz congruence of $\mathcal{M}$; it is denoted by $\Omega_{\mathfrak{A}}\mathcal{F}$ and can be defined as follows:
$ \quad \quad \quad \quad \Omega_\mathfrak{A}\mathcal{F} = \{(a,b)|\forall A(p,p_0,...,p_n) \forall c_0,..., c_n \in |\mathfrak{A}|.~A(a,c_{0},...,c_n) \in \mathcal{F} \Leftrightarrow A(b,c_0,...,c_n) \in \mathcal{F} \}.$
If the matrix in question is a Lindenbaum one, say $\left<\mathfrak{F}_\mathcal{L},\Sigma_\mathcal{S}\right>$, then an example of a compatible equivalence on this matrix is a Frege relation $\Lambda \Sigma_\mathcal{S}$ defined as follows:
$ \quad \quad \quad \quad (A,B)\in\Lambda\Sigma_\mathcal{S} \Longleftrightarrow \Sigma_\mathcal{S}, A \ \vdash_\mathcal{S} \ B$ and $\Sigma_\mathcal{S}, B \ \vdash_\mathcal{S} \ A $. (Frege relation relative to $\Sigma_\mathcal{S}$)
A system $\mathcal{S}$ is called Fregean if each $\Lambda\Sigma_\mathcal{S}$ is a congruence on $\mathfrak{F}_{\mathcal{L}}$. Obviously, if $\mathcal{S}$ is Fregean, it admits the Lindenbaum-Tarski algebra relative to any $\Sigma_\mathcal{S}$.
Another example of a compatible relation on $\left<\mathfrak{F}_\mathcal{L},\Sigma_\mathcal{S}\right>$ is the largest congruence of $\mathfrak{F}_{\mathcal{L}}$ contained in $\Lambda\Sigma_\mathcal{S}$, which is referred to as a Suszko congruence:
$ \quad \quad \quad \quad (A,B)\in \widetilde{\Omega}\Sigma_\mathcal{S}\Longleftrightarrow$ for every $C(p)$, $\Sigma_\mathcal{S}, C(A/p)\ \vdash_\mathcal{S} \ C(B/p)$ and $\Sigma_\mathcal{S},C(B/p)\ \vdash_\mathcal{S} \ C(A/p)$. (Suszko congruence relative to $\Sigma_\mathcal{S}$)
Obviously, a system $\mathcal{S}$ is Fregean if and only if $\Lambda\Sigma_\mathcal{S}=\widetilde{\Omega}\Sigma_\mathcal{S}$ for all $\Sigma_\mathcal{S}$.
The Leibniz congruence of a matrix $\left<\mathfrak{F}_\mathcal{L},\Sigma_\mathcal{S}\right>$ is referred to as Leinbniz congruence relative to $\Sigma_\mathcal{S}$. It turns out that
$\quad \quad \quad \quad \Omega\Sigma_\mathcal{S} = \cap\{\widetilde{\Omega}\Sigma^\prime_\mathcal{S} | \Sigma_\mathcal{S} \subseteq \Sigma^\prime_\mathcal{S}\}$
and, therefore, each Suszko congruence $\widetilde{\Omega}\Sigma_\mathcal{S}$ is compatible with $\Sigma_\mathcal{S}$.
Thus we have:
$\quad \quad \quad \quad \widetilde{\Omega}\Sigma_\mathcal{S}\subseteq\Lambda\Sigma_\mathcal{S}$ and $\widetilde{\Omega}\Sigma_\mathcal{S}\subseteq\Omega\Sigma_\mathcal{S}.$
Suszko and Leibniz congruences give rise to the $\mathcal{S}$-matrices $\left<\mathfrak{F}_\mathcal{S}/\widetilde{\Omega}\Sigma_\mathcal{S},\Sigma_\mathcal{S}/\widetilde{\Omega}\Sigma_\mathcal{S}\right>$ and $\left<\mathfrak{F}_\mathcal{L}/\Omega\Sigma_\mathcal{S},\Sigma_\mathcal{S}/\Omega\Sigma_\mathcal{S}\right>$, whose first components, $\mathfrak{F}_\mathfrak{L}/\widetilde{\Omega}\Sigma_\mathcal{S}$ and $\mathfrak{F}_\mathcal{L}/\Omega\Sigma_\mathcal{S} $, in Abstract algebraic logic are also referred to as Lindenbaum-Tarski algebras. (See [5] and [6] for comprehensive surveys.)
Specifications and applications
A structural deductive system $\mathcal{S}$ is called implicative extensional if its language $\mathcal{L}$ contains a binary connective $\rightarrow $ (will be written in the infix notation), and for any $\mathcal{S}$-theory $\Sigma_\mathcal{S}$ and any $A, B, C \in Fr_\mathcal{L}$, the following conditions hold:
$\quad \quad \quad \quad (i_1)$ $A \rightarrow A \in \Sigma_\mathcal{S};$
$\quad \quad \quad \quad (i_2)$ $B\in\Sigma_\mathcal{S}\Longrightarrow A\rightarrow B\in\Sigma_\mathcal{S};$
$\quad \quad \quad \quad (i_3)$ $ A\rightarrow B, B\rightarrow C\in\Sigma_\mathcal{S} \Longrightarrow A\rightarrow C\in\Sigma_\mathcal{S};$
$\quad \quad \quad \quad (i_4)$ $ A, A\rightarrow B\in\Sigma_\mathcal{S} \Longrightarrow B\in\Sigma_\mathcal{S};$
$\quad \quad \quad \quad (i_5)$ $ A_{i}\rightarrow B_{i}, B_{i}\rightarrow A_{i}\in\Sigma_\mathcal{S}, 1\leq i\leq n, \Longrightarrow \Pi A_{1}... A_{n}\rightarrow\Pi B_{1}... B_{n}$
for each $n$-ary connective $\Pi$.
Now, given $\mathcal{S}$, we consider the following relation on $Fr_\mathcal{L}$:
$\quad \quad \quad \quad A \approx_{\mathcal{S}} B \Longleftrightarrow A\rightarrow B, B\rightarrow A \in T_\mathcal{S}.$ (Tarski relation)
Theorem 7 (Rasiowa). If $\mathcal{S}$ is an implicative extensional system, then the relation $\approx_{\mathcal{S}}$ is a congruence on $\mathfrak{F}_{\mathcal{L}}$. Moreover, $T_\mathcal{S}$ is a congruence class with respect to $\approx_{\mathcal{S}}$.
Applying Theorem 7 to $IPC$, one can observe (actually, it was shown in [13]) that $\mathfrak{F}_\mathcal{L}/\!\!\approx_{IPC}$ is the free algebra of rank $\overline{\overline{\mathcal{V}}}$ in the variety of Heyting algebras. Using the Tarski relation $\approx_{IPC}$, Nishimura [11] gave an elegant description of the Lindenbaum-Tarski algebra of $IPC$ in a language with a single propositional variable. This algebra is also the free algebra of rank $1$ in the variety of Heyting algebras. See Free algebra.
Also, it is worth noticing that, using a Lindenbaum-Tarski algebra as defined above, one can prove that there is an algorithm which decides whether two finite $g$-matrices define the same deductive system; this result is due A. Citkin and J. Zygmunt. In this connection see Decision problem.
In [17],[18] (see [20], chapter XII), Tarski gave the first specification of a system which admits the Lindenbaum-Tarski algebra. Later on, Rasiowa [13] summarized the work that had been done by the time in the notion of "the class of standard systems of implicative extensional propositional calculi," which is a simplified version of that we use above.
Also, if $\mathcal{S}$ is an implicative extensional system, then $\mathfrak{F}_\mathcal{S}$ as defined above is Rasiowa's $\mathscr{S}$-algebra [13], or nowadays known [2] as Hilbert algebra with compatible operations.
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[16] Stanisław J. Surma, On the origin and subsequent applications of the concept of the Lindenbaum algebra, Logic, methodology and philosophy of science, VI (Hannover, 1979), Stud. Logic Foundations Math., vol. 104, North-Holland, Amsterdam, 1982, pp. 719–734. MR 682440 (84g:01045)
[17] Alfred Tarski, Grundzüge der systemenkalkül. Erster teil, Fundamenta Mathematica 25 (1935), 503–526.
[18] Alfred Tarski, Grundzüge der systemenkalkül. Zweiter teil, Fundamenta Mathematica 26 (1936), 283–301.
[19] Alfred Tarski, A remark on functionally free algebras, Ann. of Math. (2) 47 (1946), 163–165. MR 0015038 (7,360a)
[20] Alfred Tarski, Logic, Semantics, Metamathematics. Papers from 1923 to 1938, Oxford at the Clarendon Press, 1956, Translated by J. H. Woodger. MR 007829 (17,1171a)
[21] Ryszard Wójcicki, Theory of logical calculi, Synthese Library, vol. 199, Kluwer Academic Publishers Group, Dordrecht, 1988, Basic theory of consequence operations. MR 1009788 (90j:03001)
[22] Andrzej Wrónski, On cardinalities of matrices strongly adequate for the intuitionistic propositional logic, Rep. Math. Logic (1974), no. 3, 67–72. MR 0387011 (52 #7858)
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The New York Courier and Enquirer of November 5th contained an article which has been quite valuable to the author, as summing up, in a clear, concise and intelligible form, the principal objections which may be urged to Uncle Tom's Cabin. It is here quoted in full, as the foundation of the remarks in the following pages.
The author of "Uncle Tom's Cabin," that writer states, has committed false-witness against thousands and millions of her fellowmen.
She has done it [he says] by attaching to them as slaveholders, in the eyes of the world, the guilt of the abuses of an institution of which they are absolutely guiltless. Her story is so devised as to present slavery in three dark aspects: first, the cruel treatment of the slaves; second, the separation of families; and, third, their want of religious instruction.
To show the first, she causes a reward to be offered for the recovery of a runaway slave, "dead or alive," when no reward with such an alternative was ever heard of, or dreamed of, south of Mason and Dixon's line, and it has been decided over and over again in Southern courts that "a slave who is merely flying away cannot be killed." She puts such language as this into the mouth of one of her speakers:—"The master who goes furthest and does the worst only uses within limits the power that the law gives him;" when, in fact, the civil code of the very state where it is represented the language was uttered—Louisiana—declares that
"The slave is entirely subject to the will of his master, who may correct and chastise him, though not with unusual rigor, nor so as to maim or mutilate him, or to expose him to the danger of loss of life, or to cause his death."
And provides for a compulsory sale
"When the master shall be convicted of cruel treatment of his slaves, and the judge shall deem proper to pronounce, besides the penalty established for such cases, that the slave be sold at public auction, in order to place him out of the reach of the power which the master has abused."
"If any person whatsoever shall wilfully kill his slave, or the slave of another person, the said person, being convicted thereof, shall be tried and condemned agreeably to the laws."
In the General Court of Virginia, last year, in the case of Souther v. the Commonwealth, it was held that the killing of a slave by his master and owner, by wilful and excessive whipping, is murder in the first degree, though it may not have been the purpose of the master and owner to kill the slave! And it is not six months since Governor Johnston, of Virginia, pardoned a slave who killed his master, who was beating him with brutal severity.
And yet, in the face of such laws and decisions as these, Mrs. Stowe winds up a long series of cruelties upon her other black personages, by causing her faultless hero, Tom, to be literally whipped to death in Louisiana, by his master, Legree; and these acts, which the laws make criminal, and punish as such, she sets forth in the most repulsive colors, to illustrate the institution of slavery!
So, too, in reference to the separation of children from their parents. A considerable part of the plot is made to hinge upon the selling, in Louisiana, of the child Eliza, "eight or nine years old," away from her mother; when, had its inventor looked in the statute-book of Louisiana, she would have found the following language:
"Every person is expressly prohibited from selling separately from their mothers the children who shall not have attained the full age of ten years."
"Be it further enacted, That if any person or persons shall sell the mother of any slave child or children under the age of ten years, separate from said child or children, or shall, the mother living, sell any slave child or children of ten years of age, or under, separate from said mother, said person or persons shall be fined not less than one thousand nor more than two thousand dollars, and be imprisoned in the public jail for a period of not less than six months nor more than one year."
The privation of religious instruction, as represented by Mrs. Stowe, is utterly unfounded in fact. The largest churches in the Union consist entirely of slaves. The first African church in Louisville, which numbers fifteen hundred persons, and the first African church in Augusta, which numbers thirteen hundred, are specimens. On multitudes of the large plantations in the different parts of the South the ordinances of the gospel are as regularly maintained, by competent ministers, as in any other communities, north or south. A larger proportion of the slave population are in communion with some Christian church, than of the white population in any part of the country. A very considerable portion of every southern congregation, either in city or country, is sure to consist of blacks; whereas, of our northern churches, not a colored person is to be seen in one out of fifty.
The peculiar falsity of this whole book consists in making exceptional or impossible cases the representatives of the system. By the same process which she has used, it would not be difficult to frame a fatal argument against the relation of husband and wife, or parent and child, or of guardian and ward; for thousands of wives and children and wards have been maltreated, and even murdered. It is wrong, unpardonably wrong, to impute to any relation of life those enormities which spring only out of the worst depravity of human nature. A ridiculously extravagant spirit of generalization pervades this fiction from beginning to end. The Uncle Tom of the authoress is a perfect angel, and her blacks generally are half angels: her Simon Legree is a perfect demon, and her whites generally are half demons. She has quite a peculiar spite against the clergy; and, of the many she introduces at different times into the scenes, all, save an insignificant exception, are Pharisees or hypocrites. One who could know nothing of the United States and its people, except by what he might gather from this book, would judge that it was some region just on the confines of the infernal world. We do not say that Mrs. Stowe was actuated by wrong motives in the Preparation of this work, but we do say that she as done a wrong which no ignorance can excuse and no penance can expiate.
A much-valued correspondent of the author, writing from Richmond, Virginia, also uses the following language:
I will venture this morning to make a few suggestions which have occurred to me in regard to future editions of your work, "Uncle Tom's Cabin," which I desire should have all the influence of which your genius renders it capable, not only abroad, but in the local sphere of slavery, where it has been hitherto repudiated. Possessing already the great requisites of artistic beauty and of sympathetic affjction, it may yet be improved in regard to accuracy of statement without being at all enfeebled. For example, you do less than justice to the formalized laws of the Southern States, while you give more credit than is due to the virtue of public or private sentiment in restricting the evil which the laws permit.
I enclose the following extracts from a southern paper:
"'I'll manage that ar; they's young in the business, and must spect to work cheap,' said Marks, as he continued to read. 'Thar's three on 'em easy cases, 'cause all you've got to do is to shoot 'em, or swear they is shot; they couldn't, of course, charge much for that.'"
"The reader will observe that two charges against the South are involved in this precious discourse;—one that it is the habit of Southern masters to offer a reward, with the alternative of 'dead or alive,' for their fugitive slaves; and the other, that it is usual for pursuers to shoot them. Indeed, we are led to infer that, as the shooting is the easier mode of obtaining the reward, it is the more frequently employed in such cases. Now, when a Southern mister offers a reward for his runaway slave, it is because he has lost a certain amount of property, represented by the negro which he wishes to recover. What man of Vermont, having an ox or an ass that had gone astray, would forthwith offer half the full value of the animal, not for the carcass, which might be turned to some useful purpose, but for the unavailing satisfaction of its head! Yet are the two cases exactly parallel? With regard to the assumption that men are permitted to go about, at the South, with double-barrelled guns, shooting down runaway negroes, in preference to apprehending them, we can only say that it is as wicked and wilful as it is ridiculous. Such Thugs there may have been as Marks and Loker, who have killed negroes in this unprovoked manner; but, if they have escaped the gallows, they are probably to be found within the walls of our state penitentiaries, where they are comfortably provided for at public expense. The laws of the Southern States, which are designed, as in all good governments, for the protection of persons and property, have not been so loosely framed as to fail of their object where person and property are one.
"The law with regard to the killing of runaways is laid down with so much clearness and precision by a South Carolina judge, that we cannot forbear quoting his dictum, as directly in point. In the ease of Witsell v. Earnest and Parker, Colcock J. delivered the opinion of the court:
Jan. term, 1818
1 Nott & Mc
Cord's S. C.
Rep. 182
"'By the statute of 1740, any white man may apprehend, and moderately correct, any slave who may be found out of the plantation at which he is employed; and if the slave assaults the white person, he may be killed; but a slave who is merely flying away cannot be killed. Nor can the defendants be justified by the common law, if we consider the negro as a person; for they were not clothed with the authority of the law to apprehend him as a felon, and without such authority he could not be killed.'
"'It's commonly supposed that the property interest is a sufficient guard in these cases. If people choose to ruin their possessions, I don't know what's to be done. It seems the poor creature was a thief and a drunkard; and so there won't be much hope to get up sympathy for her.'
"'It is perfectly outrageous,—it is horrid, Augustine! It will certainly bring down vengeance upon you.'
"'My dear cousin, I didn't do it, and I can't help it; I would, if I could. If low-minded, brutal people will act like themselves, what am I to do? They have absolute control; they are irresponsible despots. There would be no use in interfering; there is no law, that amounts to anything practically, for such a case. The best we can do is to shut our eyes and ears, and let it alone. It's the only resource left us.'
"In a subsequent part of the same conversation, St. Clare says:
"'For pity's sake, for shame's sake, because we are men born of women, and not savage beasts, many of us do not, and dare not,—we would scorn to use the full power which our savage laws put into our hands. And he who goes furthest and does the worst only uses within limits the power that the law gives him.'
"Mrs. Stowe tells us, through St. Clare, that 'there is no law that amounts to anything' in such cases, and that he who goes furthest in severity towards his slave,—that is, to the deprivation of an eye or a limb, or even the destruction of life,—'only uses within limits the power that the law gives him.' This is an awful and tremendous charge, which, lightly and unwarrantably made, must subject the maker to a fearful accountability. Let us see how the matter stands upon the statute-book of Louisiana. By referring to the civil code of that state, chapter 3d, article 173, the reader will find this general declaration:
"'The slave is entirely subject to the will of his master, who may correct and chastise him, though not with unusual rigor, nor so as to maim or mutilate him, or to expose him to the danger of loss of life, or to cause his death.'
"On a subsequent page of the same volume and chapter, article 192, we find provision made for the slave's protection against his master's cruelty, in the statement that one of two cases, in which a master can be compelled to sell his slave, is
"'When the master shall be convicted of cruel treatment of his slave, and the judge shall deem proper to pronounce, besides the penalty established for such cases, that the slave shall be sold at public auction, in order to place him out of the reach of the power which the master has abused.'
"A code thus watchful of the negro's safety in life and limb confines not its guardianship to inhibitory clauses, but proscribes extreme penalties in case of their infraction. In the Code Noir (Black Code) of Louisiana, under head of Crimes and Offences, No. 55, § xvi., it is laid down, that
"'If any person whatsoever shall wilfully kill his slave, or the slave of another person, the said person, being convicted thereof, shall be tried and condemned agreeably to the laws.'
"And because negro testimony is inadmissible in the courts of the state, and therefore the evidence of such crimes might be with difficulty supplied, it is further provided that,
"'If any slave be mutilated, beaten or ill-treated, contrary to the true intent and meaning of this act, when no one shall be present, in such case the owner, or other person having the management of said slave thus mutilated, shall be deemed responsible and guilty of the said offence, and shall be prosecuted without further evidence, unless the said owner, or other person so as aforesaid, can prove the contrary by means of good and
Code Noir
Crimes and
Offences. 56, xvii
sufficient evidence, or can clear himself by his own oath, which said oath every court, under the cognizance of which such offence shall have been examined and tried, is by this act authorized to administer.'
"Enough has been quoted to establish the utter falsity of the statement, made by our authoress through St. Clare, that brutal masters are 'irresponsible despots,'—at least in Louisiana. It would extend our review to a most unreasonable length, should we undertake to give the law, with regard to the murder of slaves, as it stands in each of the Southern States. The crime is a rare one, and therefore the reporters have had few cases to record. We may refer, however, to two. In Fields v. the State of Tennessee, the plaintiff in error was indicted in the circuit court of Maury county for the murder of a negro slave. He pleaded not guilty; and at the trial was found guilty of wilful and felonious slaying of the slave. From this sentence he prosecuted his writ of error, which was disallowed, the court affirming the original judgment. The opinion of the court, as given by Peck J., overflows with the spirit of enlightened humanity. He concludes thus:
"'It is well said by one of the judges of North Carolina, that the master has a right to exact the labor of his slave; that far, the rights of the slave are suspended; but this gives the master no right
1 Yerger's
Tenn. Rep
over the life of his slave. I add to the saying of the judge, that law which says thou shalt not kill, protects the slave; and he is within its very letter. Law, reason, Christianity, and common humanity, all point but one way.'
"In the General Court of Virginia, June term, 1851, in Souther v. the Commonwealth, it was held that 'the killing of a slave by his master and owner, by wilful and excessive whipping, is murder in the first degree; though it may not have been the purpose of the master and owner to kill the slave.' The writer shows,
7 Grattan's
Rep 673
also, an ignorance of the law of contracts, as it affects slavery in the South, in making George's master take him from the factory against the proprietor's consent. George, by virtue of the contract of hiring, had become the property of the proprietor for the time being, and his master could no more have taken him away forcibly than the owner of a house in Massachusetts can dispossess his lessee, at any moment, from mere whim or caprice. There is no court in Kentucky where the hirer's rights, in this regard, would not be enforced.
"'No. Father bought her once, in one of his trips to New Orleans, and brought her up as a present to mother. She was about eight or nine years old, then. Father would never tell mother what he gave for her; but, the other day, in looking over his old papers, we came across the bill of sale. He paid an extravagant sum for her, to be sure. I suppose, on account of her extraordinary beauty.'
"George sat with his back to Cassy, and did not see the absorbed expression of her countenance, as he was giving these details.
"At this point in the story, she touched his arm, and, with a face perfectly white with interest, said, 'Do you know the names of the people he bought her of?'
"'A man of the name of Simmons, I think, was the principal in the transaction. At least, I think that was the name in the bill of sale.'
"'O, my God!' said Cassy, and fell insensible on the floor of the cabin."
"Of course Eliza turns out to be Cassy's child, and we are soon entertained with the family meeting in Montreal, where George Harris is living, five or six years after the opening of the story, in great comfort.
"Now, the reader will perhaps be surprised to know that such an incident as the sale of Cassy apart from Eliza, upon which the whole interest of the foregoing narrative hinges, never could have taken place in Louisiana, and that the bill of sale for Eliza would not have been worth the paper it was written on. Observe. George Shelby states that Eliza was eight or nine years old at the time his father purchased her in New Orleans. Let us again look at the statute-liook of Louisiana.
"In the Code Noir we find it set down that
"'Every person is expressly prohibited from selling separately from their mothers the children who shall not have attained the full age often years.'
"And this humane provision is strengthened, by a statute, one clause of which runs as follows:
"'Be it further enacted, That if any person or persons shall sell the mother of any slave child or children under the age of ten years, separate from said child or children, or shall, the mother living, sell any slave child or children of ten years of age, or under, separate from said mother, such person or persons shall incur the penalty of the sixth section of this act.'
"This penalty is a fine of not less than one thousand nor more than two thousand dollars, and imprisonment in the public jail for a period of not less than six months nor more than one year.—Vide Acts of Louisiana, 1 Session, 9th Legislature, 3828, 1829, No. 24, Section 16."
The author makes here a remark. Scattered through all the Southern States are slaveholders who are such only in name. They have no pleasure in the system, they consider it one of wrong altogether, and they hold the legal relation still, only because not yet clear with regard to the best way of changing it, so as to better the condition of those held. Such are most earnest advocates for state emancipation, and are friends of anything, written in a right spirit, which tends in that direction. From such the author ever receives criticisms with pleasure.
She has endeavored to lay before the world, in the fullest manner, all that can be objected to her work, that both sides may have an opportunity of impartial hearing.
When writing "Uncle Tom's Cabin," though entirely unaware and unexpectant of the importance which would be attached to its statements and opinions, the author of that work was anxious, from love of consistency, to have some understanding of the laws of the slave system. She had on hand for reference, while writing, the Code Noir of Louisiana, and a sketch of the laws relating to slavery in the different states, by Judge Stroud, of Philadelphia. This work, professing to have been compiled with great care from the latest editions of the statute-books of the several states, the author supposed to be a sufficient guide for the writing of a work of fiction.[1] As the accuracy of those statements which relate to the slave-laws has been particularly contested, a more especial inquiry has been made in this direction. Under the guidance and with the assistance of legal gentlemen of high standing, the writer has proceeded to examine the statements of Judge Stroud with regard to statute-law, and to follow them up with some inquiry into the decisions of courts. The result has been an increasing conviction on her part that the impressions first derived from Judge Stroud's work were correct; and the author now can only give the words of St. Clare, as the best possible expression of the sentiments and opinion which this course of reading has awakened in her mind.
This cursed business, accursed of God and man,—what is it? Strip it of all its ornament, run it down to the root and nucleus of the whole, and what is it? Why, because my brother Quashy is ignorant and weak, and I am intelligent and strong,—because I know how, and can do it,—therefore I may steal all he has, keep it, and give him only such and so much as suits my fancy! Whatever is too hard, too dirty, too disagreeable for me, I may set Quashy to doing. Because I don't like work, Quashy shall work. Because the sun burns me, Quashy shall stay in the sun. Quashy shall earn the money, and I will spend it. Quashy shall lie down in every puddle, that I may walk over dry shod. Quashy shall do my will, and not his, all the days of his mortal life, and have such a chance of getting to heaven at last as I find convenient. This I take to be about what slavery is. I defy anybody on earth to read our slave-code, as it stands in our law-books, and make anything else of it. Talk of the abuses of slavery! Humbug! The thing itself is the essence of all abuse. And the only reason why the land don't sink under it, like Sodom and Gomorrah, is because it is used in a way infinitely better than it is. For pity's sake, for shame's sake, because we are men born of women, and not savage beasts, many of us do not, and dare not,—we would scorn to use the full power which our savage laws put into our hands. And he who goes the furthest, and does the worst, only uses within limits the power that the law gives him!
The author still holds to the opinion that slavery in itself, as legally defined in law-books and expressed in the records of courts, is the sum and essence of all abuse; and she still clings to the hope that there are many men at the South infinitely better than their laws; and after the reader has read all the extracts which she has to make, for the sake of a common humanity they will hope the same. The author must state, with regard to some passages which she must quote, that the language of certain enactments was so incredible that she would not take it on the authority of any compilation whatever, but copied it with her own hand from the latest edition of the statute-book where it stood and still stands.
WHAT IS SLAVERY?
The author will now enter into a consideration of slavery as it stands revealed in slave law.
What is it, according to the definition of law-books and of legal interpreters? "A slave," says the law of Louisiana, "is one who is in the power of a master, to whom he
Civil Code,
belongs. The master may sell him, dispose of his person, his industry and his labor; he can do nothing, possess nothing, nor acquire anything, but what must belong to his master." South Carolina says "slaves shall be deemed, sold, taken, reputed and adjudged in law, to be chattels personal in the hands of their owners and possessors,
2 Brev. Dig.
229. Prince's
Digest, 446
and their executors, administrators, and assigns, to all intents, constructions and purposes whatsoever." The law of Georgia is similar.
Let the reader reflect on the extent of the meaning in this last clause. Judge Ruffin, pronouncing the opinion of the Supreme Court of North Carolina, says, a slave is "one doomed in his own person, and his
Wheeler's Law
of Slavery, 236.
State v. Mann.
posterity, to live without knowledge, and without the capacity to make anything his own, and to toil that another may reap the fruits."
This is what slavery is,—this is what it is to be a slave! The slave-code, then, of the Southern States, is designed to keep millions of human beings in the condition of chattels personal; to keep them in a condition in which the master may sell them, dispose of their time, person and labor; in which they can do nothing, possess nothing, and acquire nothing, except for the benefit of the master; in which they are doomed in themselves and in their posterity to live without knowledge, without the power to make anything their own,—to toil that another may reap. The laws of the slave-code are designed to work out this problem, consistently with the peace of the community, and the safety of that superior race which is constantly to perpetrate this outrage.
From this simple statement of what the laws of slavery are designed to do,—from a consideration that the class thus to be reduced, and oppressed, and made the subjects of a perpetual robbery, are men of like passions with our own, men originally made in the image of God as much as ourselves, men partakers of that same humanity of which Jesus Christ is the highest ideal and expression,—when we consider that the material thus to be acted upon is that fearfully explosive element, the soul of man; that soul elastic, upspringing, immortal, whose free will even the Omnipotence of God refuses to coerce,—we may form some idea of the tremendous force which is necessary to keep this mightiest of elements in the state of repression which is contemplated in the definition of slavery.
Of course, the system necessary to consummate and perpetuate such a work, from age to age, must be a fearfully stringent one; and our readers will find that it is so. Men who make the laws, and men who interpret them, may be fully sensible of their terrible severity and inhumanity; but, if they are going to preserve the thing, they have no resource but to make the laws, and to execute them faithfully after they are made. They may say, with the honorable Judge Ruffin, of North Carolina, when solemnly from the bench announcing this great foundation principle of slavery, that "the power of the master must be absolute, to render the submission of the slave perfect,"—they may say, with him, "I most freely confess my sense of the harshness of this proposition; I feel it as deeply as any man can; and, as a principle of moral right, every person in his retirement must repudiate it;"—but they will also be obliged to add, with him, "But, in the actual condition of things, it must be so. * * This discipline belongs to the state of slavery. * * * It is inherent in the relation of master and slave."
And, like Judge Ruffin, men of honor, men of humanity, men of kindest and gentlest feelings, are obliged to interpret these severe laws with inflexible severity. In the perpetual reaction of that awful force of human passion and human will, which necessarily meets the compressive power of slavery,—in that seething, boiling tide, never wholly repressed, which rolls its volcanic stream underneath the whole frame-work of society so constituted, ready to find vent at the least rent or fissure or unguarded aperture,—there is a constant necessity which urges to severity of law and inflexibility of execution. So Judge Ruffin says, "We cannot allow the right of the matter to be brought into discussion in the courts of justice. The slave, to remain a slave, must be made sensible that there is no appeal from his master." Accordingly, we find in the more southern states, where the slave population is most accumulated, and slave property most necessary and valuable, and, of course, the determination to abide by the system the most decided, there the enactments are most severe, and the interpretation of courts the most inflexible.[2] And, when legal decisions of a contrary character begin to be made, it would appear that it is a symptom of leaning towards emancipation. So abhorrent is the slave-code to every feeling of humanity, that just as soon as there is any hesitancy in the community about perpetuating the institution of slavery, judges begin to listen to the voice of their more honorable nature, and by favorable interpretations to soften its necessary severities.
Such decisions do not commend themselves to the professional admiration of legal gentlemen. But in the workings of the slave system, when the irresponsible power which it guarantees comes to be used by men of the most brutal nature, cases sometimes arise for trial where the consistent exposition of the law involves results so loathsome and frightful, that the judge prefers to be illogical, rather than inhuman. Like a spring outgushing in the desert, some noble man, now and then, from the fulness of his own better nature, throws out a legal decision, generously inconsistent with every principle and precedent of slave jurisprudence, and we bless God for it. All we wish is that there were more of them, for then should we hope that the day of redemption was drawing nigh.
The reader is now prepared to enter with us on the proof of this proposition: That the slave-code is designed only for the security of the master, and not with regard to the welfare of the slave.
This is implied in the whole current of law-making and law-administration, and is often asserted in distinct form, with a precision and clearness of legal accuracy which, in a literary point of view, are quite admirable. Thus, Judge Ruffin, after stating that considerations restricting the power of the master had often been drawn from a comparison of slavery with the relation of parent and child, master and apprentice, tutor and pupil, says distinctly:
The court does not recognize their application. There is no likeness between the cases. They are in opposition to each other, and there is an
impassable gulf between them.
****
In the one [case], the end in view is the happiness of the youth, born to equal rights with that governor, on whom the duty devolves of training the young to usefulness, in a station which he is
of Slavery, page
afterwards to assume among freemen. * * * * With slavery it is far otherwise. The end is the profit of the master, his security and the public safety.
Not only is this principle distinctly asserted in so many words, but it is more distinctly implied in multitudes of the arguings and reasonings which are given as grounds of legal decisions. Even such provisions as seem to be for the benefit of the slave we often find carefully interpreted so as to show that it is only on account of his property value to his master that he is thus protected, and not from any consideration of humanity towards himself. Thus it has been decided
of Slavery, p.
that a master can bring no action for assault and battery On his slave, unless injury be such as to produce a loss of service.
The spirit in which this question is discussed is worthy of remark. We give a brief statement of the case, as presented in Wheeler, p. 289.
It was an action for assault and battery committed by Dale on one Cornfute's slave.
Cornfute v.
Dale, April
Term, 1800
1 Har. & Johns.
Rep. 4.
2 Lutw.
1481; 20 Viner's
Abr. 454.
It was contended by Cornfute's counsel that it was not necessary to prove loss of service, in order that the action should be sustained; that an action might be supported for beating plaintiff's horse; and that the lord might have an action for the battery of his villein, which is founded on this principle, that, as the villein could not support the action, the injury would be without redress, unless the lord could. On the other side it was said that Lord Chief Justice Raymond had decided that an assault on a horse was no cause of action, unless accompanied with a special damage of the animal, which would impair his value.
Chief Justice Chase decided that no redress could be obtained in the case, because the value of the slave had not been impaired, and without injury or wrong to the master no action could be sustained; and assigned this among other reasons for it, that there was no reciprocity in the case, as the master was not liable for assault and battery committed by his slave, neither could he gain redress for one committed upon his slave.
Let any reader now imagine what an amount of wanton cruelty and indignity may be heaped upon a slave man or woman or child without actually impairing their power to do service to the master, and he will have a full sense of the cruelty of this decision.
In the same spirit it has been held in
State v.Maner
2 Hill's Rep.
Law of Slavery,
North Carolina that patrols (night watchmen) are not liable to the master for inflicting punishment on the slave, unless their conduct clearly demonstrates malice against the master.
The cool-bloodedness of some of these legal discussions is forcibly shown by two decisions in Wheeler's Law of Slavery, p. 243. On the question whether the criminal offence of assault and battery can be committed on a slave, there are two decisions of the two States of South and North Carolina; and it is difficult to say which of these decisions has the preëminence for cool legal inhumanity. That of South Carolina reads thus.
Judge O'Neill says:
The criminal offence of assault and battery can not, at common law, be committed upon the person of a slave. For notwithstanding (for some purposes) a slave is regarded by law as a person, yet generally he is a mere chattel personal, and his right of personal protection belongs to his master, who can maintain an action of trespass for the battery of his slave. There can be therefore no offence against the state for a mere beating of a slave unaccompanied with any circumstances of cruelty (!!), or an attempt ie kill and murder. The peace of the state is not thereby broken; for a slave is not generally regarded as legally capable of being within the peace of the state. He is not a citizen, and is not in that character entitled to her protection.
What declaration of the utter indifference of the state to the sufferings of the slave
See State v.
Hale. Wheeler,
p. 239. 2 Hawk.
N. C. Rep. 582
could be more elegantly cool and clear? But in North Carolina it appears that the case is argued still more elaborately.
Chief Justice Taylor thus shows that, after all, there are reasons why an assault and battery upon the slave may, on the whole, have some such general connection with the comfort and security of the community, that it may be construed into a breach of the peace, and should be treated as an indictable offence.
The instinct of a slave may be, and generally is, tamed into subservience to his master's will, and from him he receives chastisement, whether it be merited or not, with perfect submission; for he knows the extent of the dominion assumed over him, and that the law ratifies the claim. But when the same authority is wantonly usurped by a stranger, nature is disposed to assert her rights, and to prompt the slave to a resistance, often momentarily successful, sometimes fatally so. The public peace is thus broken, as much as if a free man had been beaten; for the party of the aggressor is always the strongest, and such contests usually terminate by overpowering the slave, and inflicting on him a severe chastisement, without regard to the original cause of the conflict. There is, consequently, as much reason for making such offences indictable as if a white man had been the victim. A wanton injury committed on a slave is a great provocation to the owner, awakens his resentment, and has a direct tendency to a breach of the peace, by inciting him to seek immediate vengeance. If resented in the heat of blood, it would probably extenuate a homicide to manslaughter, upon the same principle with the case stated by Lord Hale, that if A riding on the road, B had whipped his horse out of the track, and then A had alighted and killed B. These offences are usually committed by men of dissolute habits, hanging loose upon society, who, being repelled from association with well-disposed citizens, take refuge in the company of colored persons and slaves, whom they deprave by their example, embolden by their familiarity, and then beat, under the expectation that a slave dare not resent a blow from a white man. If such offences may be committed with impunity, the public peace will not only be rendered extremely insecure, but the value of slave property must be much impaired, for the offenders can seldom make any reparation in damages. Nor is it necessary, in any ease, that a person who has received an injury, real or imaginary, from a slave, should carve out his own justice;
1 Rev Code.
for the law has made ample and summary provision for the punishment of all trivial offences committed by slaves, by carrying them before a justice, who is authorized to pass sentence for their being publicly whipped. This provision, while it excludes the necessity of private vengeance, would seem to forbid its legality, since it effectually protects all persons from the insolence of slaves, even where their masters are unwilling to correct them upon complaint being made. The common law has often been called into efficient operation, for the punishment of public cruelty inflicted upon animals, for needless and wanton barbarity exercised even by masters upon their slaves, and for various violations of decency, morals, and comfort. Reason and analogy seem to require that a human being, although the subject of property, should be so far protected as the public might be injured through him.
For all purposes necessary to enforce the obedience of the slave, and to render him useful as property, the law secures to the master a complete authority over him, and it will not lightly interfere with the relation thus established. It is a more effectual guarantee of his right of property, when the slave is protected from wanton abuse from, those who have no power over him; for it cannot be disputed that a slave is rendered less capable of performing his master's service when he finds himself exposed by the law to the capricious violence of every turbulent man in the community.
If this is not a scrupulous disclaimer of all humane intention in the decision, as far as the slave is concerned, and an explicit declaration that he is protected only out of regard to the comfort of the community, and his property value to his master, it is difficult to see how such a declaration could be made. After all this cool-blooded course of remark, it is somewhat curious to come upon the following certainly most unexpected declaration, which occurs in the very next paragraph:
Mitigated as slavery is by the humanity of our laws, the refinement of manners, and by public opinion, which revolts at every instance of cruelty towards them, it would be an anomaly in the system of police which affects them, if the offence stated in the verdict were not indictable.
The reader will please to notice that this remarkable declaration is made of the State of North Carolina. We shall have occasion again to refer to it by and by, when we extract from the statute-book of North Carolina some specimens of these humane laws.
In the same spirit it is decided, under the law of Louisiana, that if an individual
Jourdain v.
Patton, July
term, 1818. 5
Martin's Louis
injures another's slave so as to make him entirely useless, and the owner recovers from him the full value of the slave, the slave by that act becomes thenceforth the property of the person who injured him. A decision to this effect is given in Wheeler's Law of Slavery, p. 249. A woman sued for an injury done to her slave by the slave of the defendant. The injury was such as to render him entirely useless, his only eye being put out. The parish court decreed that she should recover twelve hundred dollars, that the defendant should pay a further sum of twenty-five dollars a month from the time of the injury; also the physician's bill, and two hundred dollars for the sustenance of the slave during his life, and that he should remain forever in the possession of his mistress.
The case was appealed. The judge reversed the decision, and delivered the slave into the possession of the man whose slave had committed the outrage. In the course of the decision, the judge remarks, with that calm legal explicitness for which many decisions of this kind are remarkable, that
The principle of humanity, which would lead us to suppose that the mistress, whom he had long served, would treat her miserable blind slave with more kindness than the defendant, to whom the judgment ought to transfer him, cannot be taken into consideration in deciding this case.
9 Martin La.
Another case, reported in Wheeler's Law, page 198, the author thus summarily abridges. It is Dorothee v. Coquillon et al. A young girl, by will of her mistress, was to have her freedom at twenty-one; and it was required by the will that in the mean time she should be educated in such a manner as to enable her to earn her living when free, her services in the mean time being bequeathed to the daughter of the defendant. Her mother (a free woman) entered complaint that no care was taken of the child's education, and that she was cruelly treated. The prayer of the petition was that the child be declared free at twenty-one, and in the mean time hired out by the sheriff. The suit was decided against the mother, on this ground,—that she could not sue for her daughter in a case where the daughter could not sue for herself were she of age,—the object of the suit being relief from ill-treatment during the time of her slavery, which a slave cannot sue for.
4 M'Cord's Rep.
161. Wheeler's
p. 201.
Observe, now, the following case of Jennings v. Fundeberg. It seems Jennings brings an action of trespass against Fundeberg for killing his slave. The case was thus: Fundeberg with others, being out hunting runaway negroes, surprised them in their camp, and, as the report says, "fired his gun towards them as they were running away, to induce them to stop." One of them, being shot through the head, was thus induced to stop,—and the master of the boy brought action for trespass against the firer for killing his slave.
The decision of the inferior court was as follows:
The court "thought the killing accidental, and that the defendant ought not to be made answerable as a trespasser." * * * * "When one is lawfully interfering with the property of another, and accidentally destroys it, he is no trespasser, and ought not to be answerable for the value of the property. In this case, the defendant was engaged in a lawful and meritorious service, and if he really fired his gun in the manner stated it was an allowable act."
The superior judge reversed the decision, on the ground that in dealing with another person's property one is responsible for any injury which he could have avoided by any degree of circumspection. "The firing … was rash and incautious."
Does not the whole spirit of this Jan. T. 1827. 4
M'Cord's Rep.
156 discussion speak for itself?
See also the very next case in Wheeler's Law. Richardson v. Dukes, p. 202.
Trespass for killing the plaintiff's slave. It appeared the slave was stealing potatoes from a bank near the defendant's house. The defendant fired upon him with a gun loaded with buckshot, and killed him. The jury found a verdict for plaintiff for one dollar. Motion for a new trial.
The Court. Nott J. held, there must be a new trial; that the jury ought to have given the plaintiff the value of the slave. That if the jury were of opinion the slave was of bad character, some deduction from the usual price ought to be made, but the plaintiff was certainly entitled to his actual damage for killing his slave. Where property is in question, the value of the article, as nearly as it can be ascertained, furnishes a rule from which they are not at liberty to depart.
It seems that the value of this unfortunate piece of property was somewhat reduced from the circumstance of his "stealing potatoes." Doubtless he had his own best
Wheeler's Law
reasons for this; so, at least, we should infer from the following remark, which occurs in one of the reasonings of Judge Taylor, of N. Carolina.
"The act of 1786 (Iredell's Revisal, p. 588) does, in the preamble, recognize the fact, that many persons, by cruel treatment to their slaves, cause them to commit crimes for which they are executed. * * The cruel treatment here alluded to must consist in withholding from them the necessaries of life; and the crimes thus resulting are such as are calculated to furnish them with food and raiment."
Whitsell v.
Earnest &
Parker. Wheeler
Perhaps "stealing potatoes" in this case was one of the class of crimes alluded to.
Again we have the following case:
The defendants went to the plantation of Mrs. Witsell for the purpose of hunting for runaway negroes; there being many in the neighborhood, and the place in considerable alarm. As they approached the house with loaded guns, a negro ran from the house, or near the house, towards a swamp, when they fired and killed him.
The judge charged the jury, that such circumstances might exist, by the excitement and alarm of the neighborhood, as to authorize the killing of a negro without the sanction of a magistrate.
This decision was reversed in the Superior Court, in the following language:
By the statute of 1740, any white man may apprehend and moderately correct any slave who may be found out of the plantation at which he is employed, and if the slave assaults the white person, he may be killed; but a slave who is merely flying away cannot be killed. Nor can the defendants be justified by common law, if we consider the negro as a person; for they were not clothed with the authority of the law to apprehend him as a felon, and without such authority he could not be killed.
If we consider the negro a person, says the judge; and, from his decision in the case, he evidently intimates that he has a strong leaning to this opinion, though it has been contested by so many eminent legal authorities that he puts forth his sentiment modestly, and in an hypothetical form. The reader, perhaps, will need to be informed that the question whether the slave is to be considered a person or a human being in any respect has been extensively and ably argued on both sides in legal courts, and it may be a comfort to know that the balance of legal opinion inclines in favor of the slave. Judge Clarke, of Mississippi, is quite clear on the point, and argues very ably and earnestly, though, as he confesses, against very respectable legal authorities, that the slave is a
Wheeler, p.
June T., 1820.
Walker's
Rep. 83.
person,—that he is a reasonable creature. The reasoning occurs in the case State of Mississippi v. Jones, and is worthy of attention as a literary curiosity.
It seems that a case of murder of a slave had been clearly made out and proved in the lower court, and that judgment was arrested and the case appealed on the ground whether, in that state, murder could be committed on a slave. Judge Clarke thus ably and earnestly argues:
The question in this case is, whether murder can be committed on a slave. Because individuals may have been deprived of many of their rights by society, it does not follow, that they have been deprived of all their rights. In some respects, slaves may be considered as chattels; but in others, they are regarded as men. The law views them as capable of committing crimes. This can only be upon the principle, that they are men and rational beings. The Roman law has been much relied on by the counsel of the defendant. That law was confined to the Roman empire, giving the power of life and death over captives in war, as slaves; but it no more extended here, than the similar power given to parents over the lives of their children. Much stress has also been laid by the defendant's counsel on the case cited from Taylor's Reports, decided in North Carolina; yet, in that case, two judges against one were of opinion, that killing a slave was murder. Judge Hall, who delivered the dissenting opinion in the above case based his conclusions, as we conceive, upon erroneous principles, by considering the laws of Rome applicable here. His inference, also, that a person cannot be condemned capitally, because he may be liable in a civil action, is not sustained by reason or authority, but appears to us to be in direct opposition to both. At a very early period in Virginia, the power of life over slaves was given by statute; but Tucker observes, that as soon as these statutes were repealed, it was at once considered by their courts that the killing of a slave might be murder. Commonwealth v. Dolly Chapman: indictment for maliciously stabbing a slave, under a statute. It has been determined in Virginia that slaves are persons. In the constitution of the United States, slaves are expressly designated as " persons." In this state the legislature have considered slaves as reasonable and accountable beings; and it would be a stigma upon the character of the state, and a reproach to the administration of justice, if the life of a slave could be taken with impunity, or if he could be murdered in cold blood, without subjecting the offender to the highest penalty known to the criminal jurisprudence of the country. Has the slave no rights, because he is deprived of his freedom? He is still a human being, and possesses all those rights of which he is not deprived by the positive provisions of the law; but in vain shall we look for any law passed by the enlightened and philanthropic legislature of this state, giving even to the master, much less to a stranger, power over the life of a slave. Such a statute would be worthy the age of Draco or Caligula, and would be condemned by the unanimous voice of the people of this state, where even cruelty to slaves, much [more] the taking away of life, meets with universal reprobation. By the provisions of our law, a slave may commit murder, and be punished with death; why, then, is it not murder to kill a slave? Can a mere chattel commit murder, and be subject to punishment?
The right of the master exists not by force of the law of nature or nations, but by virtue only of the positive law of the state; and although that gives to the master the right to command the services of the slave, requiring the master to feed and clothe the slave from infancy till death, yet it gives the master no right to take the life of the slave; and, if the offence be not murder, it is not a crime, and subjects the offender to no punishment.
The taking away the life of a reasonable creature, under the king's peace, with malice aforethought, express or implied, is murder at common law. Is not a slave a reasonable creature?—is he not a human being? And the meaning of this phrase, reasonable creature, is, a human being. For the killing a lunatic, an idiot, or even a child unborn, is murder, as much as the killing a philosopher; and has not the slave as much reason as a lunatic, an idiot, or an unborn child?
Thus triumphantly, in this nineteenth century of the Christian era and in the State of Mississippi, has it been made to appear that the slave is a reasonable creature,—a human being!
What sort of system, what sort of a public sentiment, was that which made this argument necessary?
And let us look at some of the admissions of this argument with regard to the nature of slavery. According to the judge, it is depriving human beings of many of their rights. Thus he says: "Because individuals may have been deprived of many of their rights by society, it does not follow that they have been deprived of ''all their rights." Again, he says of the slave: "He is still a human being, and possesses all those rights of which he is not deprived by the positive provisions of the law." Here he admits that the provisions of law deprive the slave of natural rights. Again he says: "The right of the master exists not by force of the law of nature or of nations, but by virtue only of the positive law of the state." According to the decision of this judge, therefore, slavery exists by the same right that robbery or oppression of any kind does,—the right of ability. A gang of robbers associated into a society have rights over all the neighboring property that they can acquire, of precisely the same kind.
With the same unconscious serenity does the law apply that principle of force and robbery which is the essence of slavery, and show how far the master may proceed in appropriating another human being as his property.
Wheeler, p. 23,
Banks, Abu'r
v. Marksbury.
Spring T. 1823
3 Little's Rep
275 The question arises. May a master give a woman to one person, and her unborn children to another one? Let us hear the case argued. The unfortunate mother selected as the test point of this interesting legal principle comes to our view in the will of one Samuel Marksbury, under the style and denomination of "my negro wench Pen." Said Samuel states in his will that, for the good will and love he bears to his own children, he gives said negro wench Pen to son Samuel, and all her future increase to daughter Rachael. When daughter Rachael, therefore, marries, her husband sets up a claim for this increase,—as it is stated, quite off-hand, that the "wench had several children." Here comes a beautifully interesting case, quite stimulating to legal acumen. Inferior court decides that Samuel Marksbury could not have given away unborn children on the strength of the legal maxim, "Nemo dat quod non habet,"—i.e., "Nobody can give what he has not got,"—which certainly one should think sensible and satisfactory enough. The case, however, is appealed, and reversed in the superior court; and now let us hear the reasoning.
The judge acknowledges the force of the maxim, above quoted,—says, as one would think any man might say, that it is quite a correct maxim,—the only difficulty being that it does not at all apply to the present case. Let us hear him:
He who is the absolute owner of a thing owns all its faculties for profit or increase; and he may, no doubt, grant the, profits or increase, as well as the thing itself. Thus, it is every day's practice to grant the future rents or profits of real estate; and it is held that a man may grant the wool of a flock of sheep for years.
See also p. 33, Fanny v. Bryant, 4 J. J. Marshall's Rep., 368. In this almost precisely the same language is used. If the reader will proceed, he will find also this principle applied with equal clearness to the hiring, selling, mortgaging of unborn children; and the perfect legal nonchalance of these discussions is only comparable to running a dissecting-knife through the course of all the heart-strings of a living subject, for the purpose of demonstrating the laws of nervous contraction.
Judge Stroud, in his sketch of the slave-laws, page 99, lays down for proof the following assertion: That the penal codes of the slave states bear much more severely on slaves than on white persons. He introduces his consideration of this proposition by the following humane and sensible remarks:
A being, ignorant of letters, unenlightened by religion, and deriving but little instruction from good example, cannot be supposed to have right conceptions as to the nature and extent of moral or political obligations. This remark, with but a slight qualification, is applicable to the condition of the slave. It has been just shown that the benefits of education are not conferred upon him, while his chance of acquiring a knowledge of the precepts of the gospel is so remote as scarcely to be appreciated. He may be regarded, therefore as almost without the capacity to comprehend the force of laws; and, on this account, such as are designed for his government should be recommended by their simplicity and mildness.
His condition suggests another motive for tenderness on his behalf in these particulars. He is unable to read, and holding little or no communication with those who are better informed than himself; how is he to become acquainted with the fact that a law for his observance has been made? To exact obedience to a law which has not been promulgated,—which is unknown to the subject of it,—has ever been deemed most unjust and tyrannical The reign of Caligula, were it obnoxious to no other reproach than this, would never cease to be remembered with abhorrence.
The lawgivers of the slaveholding states seem, in the formation of their penal codes, to have been uninfluenced by these claims of the slave upon their compassionate consideration. The hardened convict moves their sympathy, and is to be taught the laws before he is expected to obey them; yet the guiltless slave is subjected to an extensive system of cruel enactments, of no part of which, probably, has he ever heard.
Parts of this system apply to the slave exclusively, and for every infraction a large retribution is demanded; while, with respect to offences for which whites as well as slaves are amenable, punishments of much greater severity are inflicted upon the latter than upon the former.
This heavy charge of Judge Stroud is sustained by twenty pages of proof, showing the very great disproportion between the number of offences made capital for slaves, and those that are so for whites. Concerning this, we find the following cool remark in Wheeler's Law of Slavery, page 222, note.
Much has been said of the disparity of punishment between the white inhabitants and the slaves and negroes of the same state; that slaves are punished with much more severity, for the commission of similar crimes, by white persons, than the latter. The charge is undoubtedly true to a considerable extent. It must be remembered that the primary object of the enactment of penal laws, is the protection and security of those who make them. The slave has no agency in making them, he is indeed one cause of the apprehended evils to the other class, which those laws are expected to remedy. That he should be held amenable for a violation of those rules established for the security of the other, is the natural result of the state in which he is placed. And the severity of those rules will always bear a relation to that danger, real or ideal, of the other class.
It has been so among all nations, and will ever continue to be so, while the disparity between bond and free remains.
The State v.
Manu. Dec
Devereaux's
Rep. 268.
A striking example of a legal decision to this purport is given in Wheeler's Law of Slavery, page 224. The case, apart from legal technicalities, may be thus briefly stated:
The defendant, Mann, had hired a slave-woman for a year. During this time the slave committed some slight offence, for which the defendant undertook to chastise her. While in the act of doing so the slave ran off, whereat he shot at and wounded her. The judge in the inferior court charged the jury that if they believed the punishment was cruel and unwarrantable, and disproportioned to the offence, in law the defendant was guilty, as he had only a special property in the slave. The jury finding evidence that the punishment had been cruel, unwarrantable and disproportioned to the offence, found verdict against the defendant. But on what ground?—Because, according to the law of North Carolina, cruel, unwarrantable, disproportionate punishment of a slave from a master, is an indictable offence? No. They decided against the defendant, not because the punishment was cruel and unwarrantable, but because he was not the person who had the right to inflict it, "as he had only a special right of property in the slave."
The defendant appealed to a higher court, and the decision was reversed, on the ground that the hirer has for the time being all the rights of the master. The remarks of Judge Ruffin are so characteristic, and so strongly express the conflict between the feelings of the humane judge and the logical necessity of a strict interpreter of slave-law, that we shall quote largely from it. One cannot but admire the unflinching calmness with which a man, evidently possessed of honorable and humane feelings, walks through the most extreme and terrible results and conclusions, in obedience to the laws of legal truth. Thus he says:
A judge cannot but lament, when such cases as the present are brought into judgment. It is impossible that the reasons on which they go can be appreciated, but where institutions similar to our own exist, and are thoroughly understood. The struggle, too, in the judge's own breast, between the feelings of the man and the duty of the magistrate, is a severe one, presenting strong temptation to put aside such questions, if it be possible. It is useless, however, to complain of things inherent in our political state. And it is criminal in a court to avoid any responsibility which the laws impose. With whatever reluctance, therefore, it is done, the court is compelled to express an opinion upon the extent of the dominion of the master over the slave in North Carolina. The indictment charges a battery on Lydia, a slave of Elizabeth Jones.… The inquiry here is, whether a cruel and unreasonable battery on a slave by the hirer is indictable. The judge below instructed the jury that it is. He seems to have put it on the ground, that the defendant had but a special property. Our laws uniformly treat the master, or other person having the possession and command of the slave, as entitled to the same extent of authority. The object is the same, the service of the slave; and the same powers must be confided. In a criminal proceeding, and, indeed, in reference to all other persons but the general owner, the hirer and possessor of the slave, in relation to both rights and duties, is, for the time being, the owner. . . . . But, upon the general question, whether the owner is answerable criminaliter, for a battery upon his own slave, or other exercise of authority of force, not forbidden by statute, the court entertains but little doubt. That he is so liable, has never been decided; nor, as far as is known, been hitherto contended. There has been no prosecution of the sort. The established habits and uniform practice of the country, in this respect, is the best evidence of the portion of power deemed by the whole community requisite to the preservation of the master's dominion. If we thought differently, we could not set our notions in array against the judgment of everybody else, and say that this or that authority may be safely lopped off. This has indeed been assimilated at the bar to the other domestic relations; and arguments drawn from the well-established principles, which confer and restrain the authority of the parent over the child, the tutor over the pupil, the master over the apprentice, have been pressed on us.
The court does not recognize their application. There is no likeness between the cases. They are in opposition to each other, and there is an impassable gulf between them. The difference is that which exists between freedom and slavery; and a greater cannot be imagined. In the one, the end in view is the happiness of the youth born to equal rights with that governor on whom the duty devolves of training the young to usefulness, in a station which he is afterwards to assume among freemen. To such an end, and with such a subject, moral and intellectual instruction seem the natural means; and, for the most part, they are found to suffice. Moderate force is superadded only to make the others effectual. If that fail, it is better to leave the party to his own headstrong passions, and the ultimate correction of the law, than to allow it to be immoderately inflicted by a private person. With slavery it is far otherwise. The end is the profit of the master, his security and the public safety; the subject, one doomed, in his own person and his posterity, to live without knowledge, and without the capacity to make anything his own, and to toil that another may reap the fruits. What moral considerations shall be addressed to such a being, to convince him what it is impossible but that the most stupid must feel and know can never be true,—that he is thus to labor upon a principle of natural duty, or for the sake of his own personal happiness? Such services can only be expected from one who has no will of his own; who surrenders his will in implicit obedience to that of another. Such obedience is the consequence only of uncontrolled authority over the body. There is nothing else which can operate to produce the effect. The power of the master must be absolute, to render the submission of the slave perfect. I most freely confess my sense of the harshness of this proposition. I feel it as deeply as any man can. And, as a principle of moral right, every person in his retirement must repudiate it. But, in the actual condition of things, it must be so. There is no remedy. This discipline belongs to the state of slavery. They cannot be disunited without abrogating at once the rights of the master, and absolving the slave from his subjection. It constitutes the curse of slavery to both the bond and the free portions of our population. But it is inherent in the relation of master and slave. That there may be particular instances of cruelty and deliberate barbarity, where in conscience the law might properly interfere, is most probable. The difficulty is to determine where a court may properly begin. Merely in the abstract, it may well be asked which power of the master accords with right. The answer will probably sweep away all of them. But we cannot look at the matter in that light. The truth is that we are forbidden to enter upon a train of general reasoning on the subject. We cannot allow the right of the master to be brought into discussion in the courts of justice. The slave, to remain a slave, must be made sensible that there is no appeal from his master; that his power is, in no instance, usurped, but is conferred by the laws of man, at least, if not by the law of God. The danger would be great, indeed, if the tribunals of justice should be called on to graduate the punishment appropriate to every temper and every dereliction of menial duty.
No man can anticipate the many and aggravated provocations of the master which the slave would be constantly stimulated by his own passions, or the instigation of others, to give; or the consequent wrath of the master, prompting him to bloody vengeance upon the turbulent traitor; a vengeance generally practised with impunity, by reason of its privacy. The court, therefore, disclaims the power of changing the relation in which these parts of our people stand to each other.
I repeat, that I would gladly have avoided this ungrateful question. But, being brought to it, the court is compelled to declare that while slavery exists amongst us in its present state, or until it shall seem fit to the legislature to interpose express enactments to the contrary, it will be the imperative duty of the judges to recognize the full dominion of the owner over the slave, except where the exercise of it is forbidden by statute.
And this we do upon the ground that this dominion is essential to the value of slaves as property, to the security of the master and the public tranquillity, greatly dependent upon their subordination; and, in fine, as most effectually securing the general protection and comfort of the slaves themselves. Judgment below reversed; and judgment entered for the defendant.
No one can read this decision, so fine and clear in expression, so dignified and solemn in its earnestness, and so dreadful in its results, without feeling at once deep respect for the man and horror for the system. The man, judging him from this short specimen, which is all the author knows,[3] has one of that high order of minds, which looks straight through all verbiage and sophistry to the heart of every subject which it encounters. He has, too, that noble scorn of dissimulation, that straightforward determination not to call a bad thing by a good name, even when most popular and reputable and legal, which it is to be wished could be more frequently seen, both in our Northern and Southern States, There is but one sole regret; and that is that such a man, with such a mind, should have been merely an expositor, and not a reformer of law.
SUTHER v. THE COMMONWEALTH—THE NE PLUS ULTRA OF LEGAL HUMANITY.
"Yet in the face of such laws and decisions as these! Mrs. Stowe, &c."—Courier & Enquirer.
The case of Souther v. the Commonwealth has been cited by the Courier & Enquirer as a particularly favorable specimen of judicial proceedings under the slave-code, with the following remark:
By the above language the author was led into the supposition that this case had been conducted in a manner so creditable to the feelings of our common humanity as to present a fairer side of criminal jurisprudence in this respect. She accordingly took the pains to procure a report of the case, designing to publish it as an offset to the many barbarities which research into this branch of the subject obliges one to unfold. A legal gentleman has copied the case from Grattan's Reports, and it is here given. If the reader is astounded at it, he cannot be more so than was the writer.
Souther v. The Commonwealth. 7 Grattan, 673, 1851.
The killing of a slave by his master and owner, by wilful and excessive whipping, is murder in the first degree: though it may not have been the purpose and intention of the master and owner to kill the slave.
Simeon Souther was indicted at the October Term, 1850, of the Circuit Court for the County of Hanover, for the murder of his own slave. The indictment contained fifteen counts, in which the various modes of punishment and torture by which the homicide was charged to have been committed were stated singly, and in various combinations, The fifteenth count unites them all: and, as the court certifies that the indictment was sustained by the evidence, the giving the facts stated in that count will show what was the charge against the prisoner, and what was the proof to sustain it.
The count charged that on the 1st day of September, 1849, the prisoner tied his negro slave, Sam, with ropes about his wrists, neck, body, legs and ankles, to a tree. That whilst so tied, the prisoner first whipped the slave with switches. That he next beat and cobbed the slave with a shingle, and compelled two of his slaves, a man and a woman, also to cob the deceased with the shingle. That whilst the deceased was so tied to the tree, the prisoner did strike, knock, kick, stamp and beat him upon various parts of his head, face and body; that he applied fire to his body; * * * * that he then washed his body with warm water, in which pods of red pepper had been put and steeped; and he compelled his two slaves aforesaid also to wash him with this same preparation of warm water and red pepper. That after the tying, whipping, cobbing, striking, beating, knocking, kicking, stamping, wounding, bruising, lacerating, burning, washing and torturing, as
aforesaid, the prisoner untied the deceased from the tree in such way as to throw him with violence to the ground; and he then and there did knock, kick, stamp and beat the deceased upon his head, temples, and various parts of his body. That the prisoner then had the deceased carried into a shed-room of his house, and there he compelled one of his slaves, in his presence, to confine the deceased's feet in stocks, by making his legs fast to a piece of timber, and to tie a rope about the neck of the deceased, and fasten it to a bed-post in the room, thereby strangling, choking and suffocating the deceased. And that whilst the deceased was thus made fast in stocks as aforesaid, the prisoner did kick, knock, stamp and beat him upon his head, face, breast, belly, sides, back and body; and he again compelled his two slaves to apply fire to the body of the deceased, whilst he was so made fast as aforesaid. And the count charged that from these various modes of punishment and torture the slave Sam then and there died, It appeared that the prisoner commenced the punishment of the deceased in the morning, and that it was continued throughout the day: and that the deceased died in the presence of the prisoner, and one of his slaves, and one of the witnesses, whilst the punishment was still progressing.
Field J. delivered the opinion of the court.
The prisoner was indicted and convicted of murder in the second degree, in the Circuit Court of Hanover, at its April term last past, and was sentenced to the penitentiary for five years, the period of time ascertained by the jury. The murder consisted in the killing of a negro man-slave by the name of Sam, the property of the prisoner, by cruel and excessive whipping and torture, inflicted by Souther, aided by two of his other slaves, on the 1st day of September, 1849. The prisoner moved for a new trial, upon the ground that the offence, if any, amounted only to manslaughter. The motion for a new trial was overruled, and a bill of exceptions taken to the opinion of the court, setting forth the facts proved, or as many of them as were deemed material for the consideration of the application for a new trial. The bill of exception states: That the slave Sam, in the indictment mentioned, was the slave and property of the prisunar. That for the purpose of chastising the slave for the offence of getting drunk, and dealing as the slave confessed and alleged with Henry and Stone, two of the witnesses for the Commonwealth, he caused him to be tied and punished in the presence of the said witnesses, with the exception of slight whipping with peach or apple-tree switches, before the said witnesses arrived at the scene after they were sent for by the prisoner (who were present by request from the defendant), and of several slaves of the prisoner, in the manner and by the means charged in the indictment; and the said slave died under and from the infliction of the said punishment, in the presence of the prisoner, one of his slaves, and of one of the witnesses for the Commonwealth. But it did not appear that it was the design of the prisoner to kill the said slave, unless such design be properly inferable from the manner, means and duration, of the punishment. And, on the contrary, it did appear that the prisoner frequently declared, while the said slave was undergoing the punishment, that he believed the said slave was feigning, and pretending to be suffering and injured when he was not. The judge certifies that the slave was punished in the manner and by the means charged in the indictment. The indictment contains fifteen counts, and sets forth a case of the most cruel and excessive whipping and torture.[4]
It is believed that the records of criminal jurisprudence do not contain a case of more atrocious and wicked cruelty than was presented upon the trial of Souther; and yet it has been gravely and earnestly contended here by his counsel that his offence amounts to manslaughter only.
It has been contended by the counsel of the prisoner that a man cannot be indicted and prosecuted for the cruel and excessive whipping of his own slave. That it is lawful for the master to chastise his slave, and that if death ensues from such chastisement, unless it was intended to produce death, it is like the case of homicide which is committed by a man in the performance of a lawful act, which is manslaughter only. It has been decided by this court in Turner's case, 5 Rand, that the owner of a slave, for the malicious, cruel and excessive beating of his own slave, cannot be indicted; yet it by no means follows, when such malicious, cruel and excessive beating results in death, though not intended and premeditated, that the beating is to be regarded as lawful for the purpose of reducing the crime to manslaughter, when the whipping is inflicted for the sole purpose of chastisement. It is the policy of the law, in respect to the relation of master and slave, and for the sake of securing proper subordination and obedience on the part of the slave, to protect the master from prosecution in all such cases, even if the whipping and punishment be malicious, cruel and excessive. But in so inflicting punishment for the sake of punishment, the owner of the slave acts at his peril; and if death ensues in consequence of such punishment, the relation of master and slave affords no ground of excuse or palliation. The principles of the common law, in relation to homicide, apply to his case without qualification or exception; and according to those principles, the act of the prisoner, in the case under consideration, amounted to murder. * * * The crime of the prisoner is not manslaughter, but murder in the first degree.
On the case now presented there are some remarks to be made.
This scene of torture, it seems, occupied about twelve hours. It occurred in the State of Virginia, in the County of Hanover. Two white men were witnesses to nearly the whole proceeding, and, so far as we can see. made no effort to arouse the neighborhood, and bring in help to stop the outrage. What sort of an education, what habits of thought, does this presuppose in these men?
The case was brought to trial. It requires no ordinary nerve to read over the counts of this indictment. Nobody, one would suppose, could willingly read them twice. One would think that it would have laid a cold hand of horror on every heart;—that the community would have risen, by an universal sentiment, to shake out the man, as Paul shook the viper from his hand. It seems, however, that they were quite self-possessed; that lawyers calmly sat, and examined, and cross-examined, on particulars known before only in the records of the Inquisition; that it was "ably and earnestly argued" by educated, intelligent, American men, that this catalogue of horrors did not amount to a murder! and, in the cool language of legal precision, that "the offence, if any, amounted to manslaughter;" and that an American jury found that the offence was murder in the second degree. Any one who reads the indictment will certainly think that, if this be murder in the second degree, in Virginia, one might earnestly pray to be murdered in the first degree, to begin with. Had Souther walked up to the man, and shot him through the head with a pistol, before white witnesses, that would have been murder in the first degree. As he preferred to spend twelve hours in killing him by torture, under the name of "chastisement" that, says the verdict, is murder in the second degree; "because," says the bill of exceptions, with admirable coolness, "it did not appear that it was the design of the prisoner to kill the slave, unless such design be properly inferable from the manner, means and duration, of the punishment.
The bill evidently seems to have a leaning to the idea that twelve hours spent in beating, stamping, scalding, burning and mutilating a human being, might possibly be considered as presumption of something beyond the limits of lawful chastisement. So startling an opinion, however, is expressed cautiously, and with a becoming diffidence, and is balanced by the very striking fact, which is also quoted in this remarkable paper, that the prisoner frequently declared, while the slave was undergoing the punishment, that he believed the slave was feigning and pretending to be suffering, when he was not. This view appears to have struck the court as eminently probable,—as going a long way to prove the propriety of Souther's intentions, making it at least extremely probable that only correction was intended.
It seems, also, that Souther, so far from being crushed by the united opinion of the community, found those to back him who considered five years in the penitentiary an unjust severity for his crime, and hence the bill of exceptions from which we have quoted, and the appeal to the Superior Court; and hence the form in which the case stands in law-.books, "Souther v. the Commonwealth."
Souther evidently considers himself an ill-used man, and it is in this character that he appears before the Superior Court.
As yet there has been no particular overflow of humanity in the treatment of the case. The manner in which it has been discussed so far reminds one of nothing so much as of some discussions which the reader may have seen quoted from the records of the Inquisition, with regard to the propriety of roasting the feet of children who have not arrived at the age of thirteen years, with a view to eliciting evidence.
Let us now come to the decision of the Superior Court, which the editor of the Courier & Enqnirer thinks so particularly enlightened and humane. Judge Field thinks that the case is a very atrocious one, and in this respect he seems to differ materially from judge, jury and lawyers, of the court below. Furthermore, he doubts whether the annals of jurisprudence furnish a case of equal atrocity, wherein certainly he appears to be not far wrong; and he also states unequivocally the principle that killing a slave by torture under the name of correction is murder in the first degree; and here too, certainly, everybody will think that he is also right; the only wonder being that any man could ever have been called to express such an opinion, judicially. But he states, quite as unequivocally as Judge Ruffin, that awful principle of slave-laws, that the law cannot interfere with the master for any amount of torture inflicted on his slave which does not result in death. The decision, if it establishes anything, establishes this principle quite as strongly as it does the other. Let us hear the words of the decision:
It has been decided by this court, in Turner's case, that the owner of a slave, for the malicious, cruel and excessive beating of his own slave, cannot be indicted. * * * * * *
It is the policy of the law, in respect to the relation of master and slave, and for the sake of securing proper subordination and obedience on the part of the slave, to protect the master from prosecution in all such cases, even if the whipping and punishment be malicious, cruel and excessive.
What follows as a corollary from this remarkable declaration is this.—that if the victim of this twelve hours' torture had only possessed a little stronger constitution, and had not actually died under it, there is no law in Virginia by which Souther could even have been indicted for misdemeanor.
If this is not filling out the measure of the language of St. Clare, that "he who goes the furthest and does the worst only uses within limits the power which the law gives him," how could this language be verified? Which, is "the worst" death outright, or torture indefinitely prolonged? This decision, in so many words, gives every master the power of indefinite torture, and takes from him only the power of terminating the agony by merciful death. And this is the judicial decision which the Courier & Enquirer cites as a perfectly convincing specimen of legal humanity. It must be hoped that the editor never read the decision, else he never would have cited it. Of all who knock at the charnel-house of legal precedents, with the hope of disinterring any evidence of humanity in the slave system, it may be said, in the awful words of the Hebrew poet:
"He knoweth not that the dead are there,
And that her guests are in the depths of hell."
The upshot of this case was, that Souther, instead of getting off from his five years' imprisonment, got simply a judicial opinion from the Superior Court that he ought to be hung; but he could not be tried over again, and, as we may infer from all the facts in the case that he was a man of tolerably resolute nerves and not very exquisite sensibility, it is not likely that the opinion gave him any very serious uneasiness. He has probably made up his mind to get over his five years with what grace he may. When he comes out, there is no law in Virginia to prevent his buying as many more negroes as he chooses, and going over the same scene with any one of them at a future time, if only he profit by the information which has been so explicitly conveyed to him in this decision, that he must take care and stop his tortures short of the point of death,—a matter about which, as the history of the Inquisition shows, men, by careful practice, can be able to judge with considerable precision. Probably, also, the next time, he will not be so foolish as to send out and request the attendance of two white witnesses, even though they may be so complacently interested in the proceedings as to spend the whole day in witnessing them without effort at prevention.
Slavery, as defined in American law, is no more capable of being regulated in its administration by principles of humanity, than the torture system of the Inquisition. Every act of humanity of every individual owner is an illogical result from the legal definition; and the reason why the slave-code of America is more atrocious than any ever before exhibited under the sun, is that the Anglo-Saxon race are a more coldly and strictly logical race, and have an unflinching courage to meet the consequences of every premise which they lay down, and to work out an accursed principle, with mathematical accuracy, to its most accursed results. The decisions in American law-books show nothing so much as this severe, unflinching accuracy of logic. It is often and evidently, not because judges are inhuman or partial, but because they are logical and truthful, that they announce from the bench, in the calmest manner, decisions which one would think might make the earth shudder, and the sun turn pale.
The French and the Spanish nations are, by constitution, more impulsive, passionate and poetic, than logical; hence it will be found that while there may be more instances of individual barbarity, as might be expected among impulsive and passionate people, there is in their slave-code more exhibition of humanity. The code of the State of Louisiana contains more really humane provisions, were there any means of enforcing them, than that of any other state in the Union. It is believed that there is no code of laws in the world which contains such a perfect cabinet crystallization of every tear and every drop of blood which can be wrung from humanity, so accurately, elegantly and scientifically arranged, as the slave-code of America. It is a case of elegant surgical instruments for the work of dissecting the living human heart;—every instrument wrought with exactest temper and polish and adapted with exquisite care, and labelled with the name of the nerve or artery or muscle which it is designed to sever. The instruments of the anatomist are instruments of earthly steel and wood, designed to operate at most on perishable and corruptible matter; but these are instruments of keener temper, and more ethereal workmanship, designed in the most precise and scientific manner to destroy the immortal soul, and carefully and gradually to reduce man from the high position of a free agent, a social, religious, accountable being, down to the condition of the brute, or of inanimate matter.
PROTECTIVE STATUTES.
Apprentices protected.—Outlawry.—Melodrama of Prue
in the Swamp.—Harry the Carpenter, a Romance of
Real Life.
But the question now occurs, Are there not protective statutes, the avowed object of which is the protection of the life and limb of the slave? We answer, there are; and these protective statutes are some of the most remarkable pieces of legislation extant.
That they were dictated by a spirit of humanity, charity, which hopeth all things, would lead us to hope; but no newspaper stories of bloody murders and shocking outrages convey to the mind so dreadful a picture of the numbness of public sentiment caused by slavery as these so-called protective statutes. The author copies the following from the statutes of North Carolina. Section 3d of the act passed in 1798 runs thus:
Whereas by another Act of the Assembly, passed in 1774, the killing of a slave, however wanton, cruel and deliberate, is only punishable in the first instance by imprisonment and paying the value thereof to the owner, which distinction of criminality between the murder of a white person and one who is equally a human creature, but merely of a different complexion, is disgraceful to humanity, and degrading in the highest degree to the laws and principles of a free, Christian and enlightened country, Be it enacted, &c., That if any person shall hereafter be guilty of wilfully and maliciously killing a slave, such offender shall, upon the first conviction thereof, be adjudged guilty of murder, and shall suffer the same punishment as if he had killed a free man: Provided always, this act shall not extend to the person killing a slave outlawed by virtue of any Act of Assembly of this state, or to any slave in the act of resistance to his lawful owner or master, or to any slave dying under moderate correction."
A law with a like proviso, except the outlawry clause, exists in Tennessee. See Caruthers and Nicholson's Compilation, 1836, p. 676.
The language of the constitution of Georgia, art. iv., sec. 12, is as follows:
Any person who shall maliciously dismember or deprive a slave of life shall suffer such punishment as would be inflicted in case the like offence had been committed on a free white person, and on the like proof, except in case of insurrection by such slave, and unless such death should happen by accident in giving such slave moderate correction. —Cobb's Dig. 1851, p. 1125.
Let now any Englishman or New Englander imagine that such laws with regard to apprentices had ever been proposed in Parliament or State Legislature under the head of protective acts;—laws which in so many words permit the killing of the subject in three cases, and those comprising all the acts which would generally occur under the law; namely, if the slave resist, if he be outlawed, or if he die under moderate correction.
What rule in the world will ever prove correction immoderate, if the fact that the subject dies under it is not held as proof? How many such "accidents" would have to happen in Old England or New England, before Parliament or Legislature would hear from such a protective law.
"But," some one may ask, "what is the outlawry spoken of in this act?" The question is pertinent, and must be answered. The author has copied the following from the Revised Statutes of North Carolina, chap, cxi, sec. 22. It may be remarked in passing that the preamble to this law presents rather a new view of slavery to those who have formed their ideas from certain pictures of blissful contentment and Arcadian repose, which have been much in vogue of late.
Whereas, many times slaves run away and lee out, hid and lurking in swamps, woods, and other obscure places, killing cattle and hogs, and committing other injuries to the inhabitants of this state; in all such cases, upon intelligence of any slave or slaves lying out as aforesaid, any two justices of the peace for the county wherein such slave or slaves is or are supposed to lurk or do mischief, shall, and they are hereby empowered and required to issue proclamation against such slave or slaves (reciting his or their names, and the name or names of the owner or owners, if known), thereby requiring him or them, and every of them, forthwith to surrender him or themselves: and also to empower and require the sheriff of the said county to take such power with him as he shall think fit and necessary for going in search and pursuit of, and effectually apprehending, such outlying slave or slaves; which proclamation shall be published at the door of the court-house, and at such other places as said justices shall direct. And if any slave or slaves against whom proclamation hath been thus issued stay out, and do not immediately return home, it shall be lawful for any person or persons whatsoever to kill and destroy such slave or slaves by such ways and means as he shall think fit, without accusation or impeachment of any crime for the same.
What ways and means have been thought fit, in actual experience, for the destruction of the slave? What was done with the negro McIntosh, in the streets of St. Louis. in open daylight, and endorsed at the next sitting of the Supreme Court of the state, as transcending the sphere of law, because it was "an act of the majority of her most respectable citizens"?[5] If these things are done in the green tree, what will be done in the dry? If these things have once been done in the open streets of St. Louis, by "a majority of her most respectable citizens," what will be done in the lonely swamps of North Carolina, by men of the stamp of Souther and Legree?
This passage of the Revised Statutes of North Carolina is more terribly suggestive to the imagination than any particulars into which the author of Uncle Tom's Cabin has thought fit to enter. Let us suppose a little melodrama quite possible to have occurred under this act of the legislature. Suppose some luckless Prue or Peg, as in the case we have just quoted, in State v. Mann, getting tired of the discipline of whipping, breaks from the overseer, clears the dogs, and gets into the swamp, and there "lies out," as the act above graphically says. The act which we are considering says that many slaves do this, and doubtless they have their own best reasons for it. We all know what fascinating places to "lie out" in these Southern swamps are. What with alligators and moccasin snakes, mud and water, and poisonous vines, one would be apt to think the situation not particularly eligible; but still, Prue "lies out" there. Perhaps in the night some husband or brother goes to see her, taking a hog, or some animal of the plantation stock, which he has ventured his life in killing, that she may not perish with hunger. Master overseer walks up to master proprietor, and reports the accident; master proprietor mounts his horse, and assembles to his aid two justices of the peace.
In the intervals between drinking brandy and smoking cigars a proclamation is duly drawn up, summoning the contumacious Prue to surrender, and requiring sheriff of said county to take such power as he shall think fit to go in search and pursuit of said slave; which proclamation, for Prue's further enlightenment, is solemnly published at the door of the court-house, and "at such other places as said justices shall direct."[6]
Let us suppose, now, that Prue, given over to hardness of heart and blindness of mind, pays no attention to all these means of grace, put forth to draw her to the protective shadow of the patriarchal roof. Suppose, further, as a final effort of long-suffering, and to leave her utterly without excuse, the worthy magistrate rides forth in full force,—man, horse, dog and gun,—to the very verge of the swamp, and there proclaims aloud the merciful mandate. Suppose that, hearing the yelping of the dogs and the proclamation of the sheriff mingled together, and the shouts of Loker, Marks, Sambo and Quimbo, and other such posse, black and white, as a sheriff can generally summon on such a hunt, this very ignorant and contumacious Prue only runs deeper into the swamp, and continues obstinately "lying out," as aforesaid;—now she is by act of the assembly outlawed, and, in the astounding words of the act, "it shall be lawful for any person or persons whatsoever to kill and destroy her, by such ways and means as he shall think fit, without accusation or impeachment of any crime for the same." What awful possibilities rise to the imagination under the fearfully suggestive clause "by such ways and means as he shall think fit!" Such ways and means as any man shall think fit, of any character, of any degree of fiendish barbarity!! Such a permission to kill even a dog, by "any ways and means which anybody should think fit," never ought to stand on the law-books of a Christian nation; and yet this stands against one bearing that same humanity which Jesus Christ bore.—against one, perhaps, who, though blinded, darkened and ignorant, he will not be ashamed to own, when he shall come in the glory of his Father, and all his holy angels with him!
That this law has not been a dead letter there is sufficient proof. In 1836 the following proclamation and advertisement appeared in the "Newbern (N. C.) Spectator:"
State of North Carolina, Lenoir County.—Whereas complaint hath been this day made to us, two of the justices of the peace for the said county, by William D. Cobb, of Jones County, that two negro-slaves belonging to him, named Ben (commonly known by the name of Ben Fox) and Rigdon, have absented themselves from their said master's service, and are lurking about in the Counties of Lenoir and Jones, committing acts of felony; these are, in the name of the state, to command the said slaves forthwith to surrender themselves, and turn home to their said master. And we do hereby also require the sheriff of said County of Lenoir to make diligent search and pursuit after the above-mentioned slaves.… And we do hereby, by virtue of an act of assembly of this state concerning servants and slaves, intimate and declare, if the said slaves do not surrender themselves and return home to their master immediately after the publication of these presents, that any person may kill or destroy said slaves by such means as he or they think fit, without accusation or impeachment of any crime or offence for so doing, or without incurring any penalty or forfeiture thereby.
Given under our hands and seals, this 12th of November, 1836. B. Coleman, J. P. [Seal.]
Jas. Jones, J. P. [Seal.]
$200 Reward.—Ran away from the subscriber, about three years ago, a certain negro-man, named Ben, commonly known by the name of Ben Fox; also one other negro, by the name of Rigdon, who ran away on the 8th of this month.
I will give the reward of $100 for each of the above negroes, to be delivered to me, or confined in the jail of Lenoir or Jones County, or for the killing of them, so that I can see them.
Nov. 12; 1836 W. D. Cobb.
That this act was not a dead letter, also, was plainly implied in the protective act first quoted. If slaves were not, as a matter of fact, ever outlawed, why does the act formally recognize such a class?—"provided that this act shall not extend to the killing of any slave outlawed by any act of the assembly." This language sufficiently indicates the existence of the custom.
Further than this, the statute-book of 1821 contained two acts: the first of which provides that all masters in certain counties, who have had slaves killed in consequence of outlawry, shall have a claim on the treasury of the state for their value, unless cruel treatment of the slave be proved on the part of the master: the second act extends the benefits of the latter provision to all the counties in the state.[7]
Finally, there is evidence that this act of outlawry was executed so recently as the year 1850,—the year in which "Uncle Tom's Cabin" was written. See the following from the Wilmington Journal of December 13, 1850:
State of North Carolina, New Hanover County.—Whereas complaint upon oath hath this day been made to us, two of the justices of the peace for the said state and county aforesaid, by Guilford Horn, of Edgecombe County, that a certain male slave belonging to him, named Harry, a carpenter by trade, about forty years old, five feet five inches high, or thereabouts; yellow complexion; stout built; with a scar on his left leg (from the cut of an axe); has very thick lips; eyes deep sunk in his head; forehead very square; tolerably loud voice; has lost one or two of his upper teeth; and has a very dark spot on his jaw, supposed to be a mark,—hath absented himself from his master's service, and is supposed to be lurking about in this county, committing acts of felony or other misdeeds; these are, therefore, in the name of the state aforesaid, to command the said slave forthwith to surrender himself and return home to his said master; and we do hereby, by virtue of the act of assembly in such cases made and provided, intimate and declare that if the said slave Harry doth not surrender himself and return home immediately after the publication of these presents, that any person or persons may kill and destroy the said slave by such means as he or they may think fit, without accusation or impeachment of any crime or offence for so doing, and without incurring any penalty or forfeiture thereby.
Given under our hands and seals, this 29th day of June, 1850. James T. Miller, J. P. [Seal.]
W. C. Bettencourt, J. P. [Seal.]
One Hundred and Twenty-five Dollars Reward will be paid for the delivery of the said Harry to me at Tosnott Depot, Edgecombe County, or for his confinement in any jail in the state, so that I can get him; or One Hundred and Fifty Dollars will be given for his head.
He was lately heard from in Newbern, where ho called himself Henry Barnes (or Burns), and will be likely to continue the same name, or assume that of Copage or Farmer. He has a free mulatto woman for a wife, by the name of Sally Bozeman, who has lately removed to Wilmington, and lives in that part of the town called Texas, where he will likely be lurking.
Masters of vessels are particularly cautioned against harboring or concealing the said negro on board their vessels, as the full penalty of the law will be rigorously enforced.
June 29th, 1850.Guilford Horn.
There is an inkling of history and romance about the description of this same Harry, who is thus publicly set up to be killed in any way that any of the negro-hunters of the swamps may think the most piquant and enlivening. It seems he is a carpenter,—a powerfully made man, whose thews and sinews might be a profitable acquisition to himself. It appears also that he has a wife, and the advertiser intimates that possibly he may be caught prowling about somewhere in her vicinity. This indicates sagacity in the writer, certainly. Married men generally have a way of liking the society of their wives; and it strikes us, from what we know of the nature of carpenters hero in New England, that Harry was not peculiar in this respect. Let us further notice the portrait of Harry: "Eyes deep sunk in his head;—forehead very square." This picture reminds us of what a persecuting old ecclesiastic once said, in the days of the Port-Royalists, of a certain truculent abbess, who stood obstinately to a certain course, in the face of the whole power, temporal and spiritual, of the Romish church, in spite of fining, imprisoning, starving, whipping, beating, and other enlightening argumentative processes, not wholly peculiar, it seems, to that age. "You will never subdue that woman," said the ecclesiastic, who was a phrenologist before his age; "she's got a square head, and I have always noticed that people with square heads never can be turned out of their course." We think it very probable that Harry, with his "square head," is just one of this sort. He is probably one of those articles which would be extremely valuable, if the owner could only get the use of him. His head is well enough, but he will use it for himself. It is of no use to any one but the wearer; and the master seems to symbolize this state of things, by offering twenty-five dollars more for the head without the body, than he is willing to give for head, man and all. Poor Harry! We wonder whether they have caught him yet; or whether the impenetrable thickets, the poisonous miasma, the deadly snakes, and the unwieldy alligators of the swamps, more humane than the slave-hunter, have interposed their uncouth and loathsome forms to guard the only fastness in Carolina where a slave can live in freedom.
It is not, then, in mere poetic fiction that the humane and graceful pen of Longfellow has drawn the following picture:
"In the dark fens of the Dismal Swamp
The hunted negro lay;
He saw the fire of the midnight camp,
And heard at times the horse's tramp,
And a bloodhound's distant bay.
"Where will-o'the-wisps and glow-worms shine,
In bulrush and in brake;
Where waving mosses shroud the pine,
And the cedar grows, and the poisonous vine
Is spotted like the snake;
"Where hardly a human foot could pass,
Or a human heart would dare,—
On the quaking turf of the green morass
He crouched in the rank and tangled grass,
Like a wild beast in his lair.
"A poor old slave! infirm and lame,
Great scars deformed his face;
On his forehead he bore the brand of shame,
And the rags that hid his mangled frame
Were the livery of disgrace.
"All things above were bright and fair,
All things were glad and free;
Lithe squirrels darted here and there,
And wild birds filled the echoing air
With songs of liberty!
"On him alone was the doom of pain,
From the morning of his birth;
On him alone the curse of Cain[8]
Fell like the flail on the garnered grain,
And struck him to the earth."
The civilized world may and will ask, in what state this law has been drawn, and passed, and revised, and allowed to appear at the present day on the revised statute-book, and to be executed in the year of our Lord 1850, as the above-cited extracts from its most respectable journals show. Is it some heathen, Kurdish tribe, some nest of pirates, some horde of barbarians, where destructive gods are worshipped, and libations to their honor poured from human skulls? The civilized world will not believe it,—but it is actually a fact, that this law has been made, and is still kept in force, by men in every other respect than what relates to their slave-code as high-minded, as enlightened, as humane, as any men in Christendom;—by citizens of a state which glories in the blood and hereditary Christian institutions of Scotland. Curiosity to know what sort of men the legislators of North Carolina might be, led the writer to examine with some attention the proceedings and debates of the convention of that state, called to amend its constitution, which assembled at Raleigh, June 4th, 1835. It is but justice to say that in these proceedings, in which all the different and perhaps conflicting interests of the various parts of the state were discussed, there was an exhibition of candor, fairness and moderation, of gentlemanly honor and courtesy in the treatment of opposing claims, and of an overruling sense of the obligations of law and religion, which certainly have not always been equally conspicuous in the proceedings of deliberative bodies in such cases. It simply goes to show that one can judge nothing of the religion or of the humanity of individuals from what seems to us objectionable practice, where they have been educated under a system entirely incompatible with both. Such is the very equivocal character of what we call virtue.
It could not be for a moment supposed that such men as Judge Ruffin, or many of the gentlemen who figure in the debates alluded to, would ever think of availing themselves of the savage permissions of such a law. But what then? It follows that the law is a direct permission, letting loose upon the defenceless slave that class of men who exist in every community, who have no conscience, no honor, no shame,—who are too far below public opinion to be restrained by that, and from whom accordingly this provision of the law takes away the only available restraint of their fiendish natures. Such men are not peculiar to the South. It is unhappily too notorious that they exist everywhere,—in England, in New England, and the world over; but they can only arrive at full maturity in wickedness under a system where the law clothes them with absolute and irresponsible power.
PROTECTIVE ACTS OF SOUTH CAROLINA AND LOUISIANA.—THE IRON COLLAR OF LOUISIANA AND NORTH CAROLINA.
Thus far by way of considering the protective acts of North Carolina, Georgia and Tennessee.
Certain miscellaneous protective acts of various other states will now be cited, merely as specimens of the spirit of legislation.
In South Carolina, the act of 1740 punished the wilful, deliberate murder of a slave by disfranchisement, and by a fine of seven hundred pounds current money, or, in default of payment, imprisonment for seven years. But the wilful murder of a slave, in the sense contemplated in this law, is a crime which would not often occur. The kind of murder which was most frequent among masters or overseers was guarded against by another section of the same act,—how adequately the reader will judge for himself, from the following quotation:
Stroud's Sketch,
p. 40. 2
Brevard's Digest
241. James'
If any person shall, on a sudden heat or passion, or by undue correction, kill his slave, or the slave of any other he shall forfeit the sum of three hundred and fifty pounds current money.
In 1821 the act punishing the wilful murder of the slave only with fine or imprisonment was mainly repealed, and it was enacted that such crime should be punished by death; but the latter section, which relates to killing the slave in sudden heat or passion, or by undue correction, has been altered only by diminishing the pecuniary penalty to a fine of five hundred dollars, authorizing also imprisonment for six months.
The next protective statute to be noticed is the following from the act of 1740, South Carolina.
Stroud, p. 40.
2 Brevard's
Digest 241.
In case any person shall wilfully cut out the tongue, put out the eye, * * * or cruelly scald, burn, or deprive any slave of any limb, or member, or shall inflict any other cruel punishment, other than by whipping or beating with a horse-whip, cowskin, switch or small stick, or by putting irons on, or confining or imprisoning such slave, every such person shall, for every such offence, forfeit the sum of one hundred pounds, current money.
The language of this law, like many other of these protective enactments, is exceedingly suggestive; the first suggestion that occurs is, What sort of an institution, and what sort of a state of society is it, that called out a law worded like this? Laws are generally not made against practices that do not exist, and exist with some degree of frequency.
The advocates of slavery are very fond of comparing it to the apprentice system of England and America. Let us suppose that in the British Parliament, or in a New England Legislature, the following law is proposed, under the title of An Act for the Protection of Apprentices, &c. &c.
In case any person shall wilfully cut out the tongue, put out the eye, or cruelly scald, burn, or deprive any apprentice of any limb or member, or shall inflict any other cruel punishment, other than by whipping or beating with a horse-whip, cowskin, switch or small stick, or by putting irons on or confining or imprisoning such apprentice, every such person shall, for every such offence, forfeit the sum of one hundred pounds, current money.
What a sensation such a proposed law would make in England may be best left for Englishmen to say; but in New England it would simply constitute the proposer a candidate for Bedlam. Yet that such a statute is necessary in South Carolina is evident enough, if we reflect that, because there is no such statute in Virginia, it has been decided that a wretch who perpetrates all these enormities on a slave cannot even be indicted for it, unless the slave dies.
But let us look further:—What is to be the penalty when any of these fiendish things are done?
Why, the man forfeits a hundred pounds, current money. Surely he ought to pay as much as that for doing so very unnecessary an act, when the Legislature bountifully allows him to inflict any torture which revengeful ingenuity could devise, by means of horse-whip, cowskin, switch or small stick, or putting irons on, or confining and imprisoning. One would surely think that here was sufficient scope and variety of legalized means of torture to satisfy any ordinary appetite for vengeance. It would appear decidedly that any more piquant varieties of agony ought to be an extra charge. The advocates of slavery are fond of comparing the situation of the slave with that of the English laborer. We are not aware that the English laborer has been so unfortunate as to be protected by any enactment like this, since the days of villeinage.
p. 40. 1 Mar
Judge Stroud says, that the same law, substantially, has been adopted in Louisiana. It is true that the civil code of Louisiana thus expresses its humane intentions.
The slave is entirely subject to the will of his master, who may correct and chastise him, though not with unusual rigor, nor so as to maim or mutilate him, or to expose him to the danger of loss of life, or to cause his death.—Civil Code of Louisiana, Article 173.
The expression "unusual rigor" is suggestive, again. It will afford large latitude for a jury, in states where slaves are in the habit of dying under moderate correction; where outlawed slaves may be killed by any means which any person thinks fit; and where laws have to be specifically made against scalding, burning, cutting out the tongue, putting out the eye, &c. What will be thought unusual rigor? This is a question, certainly, upon which persons in states not so constituted can have no means of forming an opinion.
In one of the newspaper extracts with which we prefaced our account, the following protective act of Louisiana is alluded to, as being particularly satisfactory and efficient. We give it, as quoted by Judge Stroud in his Sketch, page 58, giving his reference.
No master shall be compelled to sell his slave, but in one of two cases, to wit: the first, when, being only co-proprietor of the slave, his co-proprietor demands the sale, in order to make partition of the property; second, when the master shall be convicted of cruel treatment of his slave, and the judge shall deem it proper to pronounce, besides the penalty established for such cases, that the slave shall be sold at public auction, in order to place him out of the reach of the power which his master has abused.—Civil Code, Art. 192.
The question for a jury to determine in this case is, What is cruel treatment of a slave? Now, if all these barbarities which have been sanctioned by the legislative acts which we have quoted are not held to be cruel treatment, the question is, What is cruel treatment of a slave?
Everything that fiendish barbarity could desire can be effected under the protection of the law of South Carolina, which, as Ave have just shown, exists also in Louisiana. It is true the law restrains from some particular forms of cruelty. If any person has a mind to scald or burn his slave,—and it seems, by the statute, that there have been such people,—these statutes merely provide that he shall do it in decent privacy for, as the very keystone of Southern jurisprudence is the rejection of colored testimony, such an outrage, if perpetrated most deliberately in the presence of hundreds of slaves, could not be proved upon the master.
It is to be supposed that the fiendish people whom such statutes have in view will generally have enough of common sense not to perform it in the presence of white witnesses, since this simple act of prudence will render them entirely safe in doing whatever they have a mind to. We are told, it is true, as we have been reminded by our friend in the newspaper before quoted, that in Louisiana the deficiency caused by the rejection of negro testimony is supplied by the following most remarkable provision of the Code Noir:
If any slave be mutilated, beaten, or ill treated, contrary to the true intent and meaning of this section, when no one shall be present, in such case the owner, or other person having the charge or management of said slave thus mutilated, shall be deemed responsible and guilty of the said offence, and shall be prosecuted without further evidence, unless the said owner, or other person so as aforesaid, can prove the contrary by means of good and sufficient evidence, or can clear himself by his own oath, which said oath every court under the cognizance of which such offence shall have been examined and tried is by this act authorized to administer.—Code Noir. Crimes and Offences, 56. xvii. Rev. Stat. 1852, p. 550, § 141.
Would one have supposed that sensible people could ever publish as a law such a specimen of utter legislative nonsense—so ridiculous on the very face of it!
The object is to bring to justice those fiendish people who burn, scald, mutilate, &c. How is this done? Why, it is enacted that the fact of finding the slave in this condition shall be held presumption against the owner or overseer, unless—unless what? Why, unless he will prove to the contrary,—or swear to the contrary, it is no matter which—either will answer the purpose. The question is, If a man is bad enough to do these things, will he not be bad enough to swear falsely? As if men who are the incarnation of cruelty, as supposed by the deeds in question, would not have sufficient intrepidity of conscience to compass a false oath!
What was this law ever made for? Can any one imagine?
Upon this whole subject, we may quote the language of Judge Stroud, who thus sums up the whole amount of the protective laws for the slave, in the United States of America:
Upon a fair review of what has been written on the subject of this proposition, the result is found to be—that the master's power to inflict corporal punishment to any extent, short of life and limb, is fully sanctioned by law, in all the slave-holding states; that the master, in at least two states, is expressly protected in using the horse-whip and cowskin as instruments for beating his slave; that he may with entire impunity, in the same states, load his slave with irons, or subject him to perpetual imprisonment, whenever he may so choose; that, for cruelly scalding, wilfully cutting out the tongue, putting out an eye, and for any other dismemberment, if proved, a fine of one hundred pounds currency only is incurred in South Carolina; that, though in all the states the wilful, deliberate and malicious murder of the slave is now directed to be punished with death, yet, as in the case of a white offender none except whites can give evidence, a conviction can seldom, if ever, take place.—Stroud's Sketch, p. 43.
One very singular antithesis of two laws of Louisiana will still further show that deadness of public sentiment on cruelty to the slave which is an inseparable attendant on the system. It will be recollected that the remarkable protective law of South Carolina, with respect to scalding, burning, cutting out the tongue, and putting out the eye of the slave, has been substantially enacted in Louisiana; and that the penalty for a man's doing these things there, if he has not sense enough to do it privately, is not more than five hundred dollars.
Now, compare this other statute of Louisiana, (Rev. Stat., 1852, p. 552, § 151):
Stroud, p. 41If any person or persons, &c., shall cut or break any iron chain or collar, which any master of slaves should have used, in order to prevent running away or escape of any such slave or slaves, such person or persons so offending shall, on conviction, &c., be fined not less than two hundred dollars, nor exceeding one thousand dollars; and suffer imprisonment for a term not exceeding two years, nor less than six months.—Act of Assembly of March 6, 1819. Pamphlet, page 64.
Some Englishmen may naturally ask, "What is this iron collar which the Legislature have thought worthy of being protected by a special act?" On this subject will be presented the testimony of an unimpeachable witness. Miss Sarah M. Grimké, a personal friend of the author. "Miss Grimké is a daughter of the late Judge Grimké, of the Supreme Court of South Carolina, and sister of the late Hon. Thomas S. Grimké." She is now a member of the Society of Friends, and resides in Bellville, New Jersey. The statement given is of a kind that its author did not mean to give, nor wish to give, and never would have given, had it not been made necessary to illustrate this passage in the slave-law. The account occurs in a statement which Miss Grimké furnished to her brother-in-law, Mr. Weld, and has been before the public ever since 1839, in his work entitled Slavery as It Is, p. 22.
A handsome mulatto woman, about eighteen or twenty years of age, whose independent spirit could not brook the degradation of slavery, was in the habit of running away: for this offence she had been repeatedly sent by her master and mistress to be whipped by the keeper of the Charleston workhouse. This had been done with such inhuman severity as to lacerate her back in a most shocking manner; a finger could not be laid between the cuts. But the love of liberty was too strong to be annihilated by torture; and, as a last resort, she was whipped at several different times, and kept a close prisoner. A heavy iron collar, with three long prongs projecting from it, was placed round her neck, and a strong and sound front tooth was extracted, to serve as a mark to describe her, in case of escape. Her sufferings at this time were agonizing; she could lie in no position but on her back, which was sore from scourgings, as I can testify from personal inspection; and her only place of rest was the floor, on a blanket. These outrages were committed in a family where the mistress daily read the Scriptures, and assembled her children for family worship. She was accounted, and was really, so far as alms-giving was concerned, a charitable woman, and tender-hearted to the poor; and yet this suffering slave, who was the seamstress of the family, was continually in her presence, sitting in her chamber to sew, or engaged in her other household work, with her lacerated and bleeding back, her mutilated mouth, and heavy iron collar, without, so far as appeared, exciting any feelings of compassion.
This iron collar the author has often heard of from sources equally authentic.[9] That one will meet with it every day in walking the streets, is not probable: but that it must have been used with some great degree of frequency, is evident from the fact of a law being thought necessary to protect it. But look at the penalty of the two protective laws! The fiendish cruelties described in the act of South Carolina cost the perpetrator not more than five hundred dollars, if he does them before white people. The act of humanity costs from two hundred to one thousand dollars, and imprisonment from six months to two years, according to discretion of court! What public sentiment was it which made these laws?
PROTECTIVE ACTS WITH REGARD TO FOOD AKD RAIMENT, LABOR, ETC.
Illustrative Drama of Tom v. Legree, under the Law of South Carolina.—Separation of Parent and Child.
Having finished the consideration of the laws which protect the life and limb of the slave, the reader may feel a curiosity to know something of the provisions by which he is protected in regard to food and clothing, and from the exactions of excessive labor. It is true, there are multitudes of men in the Northern States who would say, at once, that such enactments, on the very face of them, must be superfluous and absurd. "What!" they say, "are not the slaves property? and is it likely that any man will impair the market value of his own property by not giving them sufficient food or clothing, or by overworking them?" This process of reasoning appears to have been less convincing to the legislators of Southern States than to gentlemen generally at the North; Wheeler, p.
220. State v.
Suc. Cameron
& Norwood's
C. Rep. 54.
since, as Judge Taylor says, "the act of 1786 (Iredell's Revisal, p. 588) does, in the preamble, recognize the fact, that many persons, by treatment of their slaves, cause them to commit crimes for which they are executed;" and the judge further explains this language, by saying, "The cruel treatment here alluded to must consist in withholding from them the necessaries of life; and the crimes thus resulting are such as are necessary to furnish them with food and raiment."
The State of South Carolina, in the act of 1740 (see Stroud's Sketch, p. 28), had a section with the following language in its preamble:
Whereas many owners of slaves, and others who have the care, management, and overseeing of slaves, do confine them so closely to hard labor that they have not sufficient time for natural rest;—
And the law goes on to enact that the slave shall not work more than fifteen hours a day in summer, and fourteen in winter. Judge Stroud makes it appear that in three of the slave states the time allotted for work to convicts in prison, whose punishment is to consist in hard labor, cannot exceed ten hours, even in the summer months.
This was the protective act of South Carolina, designed to reform the abusive practices of masters who confined their slaves so closely that they had not time for natural rest! What sort of habits of thought do these humane provisions show, in the makers of them? In order to protect the slave from what they consider undue exaction, they humanely provide that he shall be obliged to work only four or five hours longer than the convicts in the prison of the neighboring state! In the Island of Jamaica, besides many holidays which were accorded by law to the slave, ten hours a day was the extent to which he was compelled by law ordinarily to work.—See Stroud, p. 29.
With regard to protective acts concerning food and clothing, Judge Stroud gives the following example from the legislation of South Carolina. The author gives it as quoted by Stroud, p. 32.
In case any person, &c., who shall be the owner, or who shall have the care, government or charge, of any slave or slaves, shall deny, neglect or refuse to allow, such slave or slaves, &c., sufficient clothing, covering or food, it shall and may be lawful for any person or persons, on behalf of such slave or slaves, to make complaint to the next neighboring justice in the parish where such slave or slaves live, or are usually employed, * * * and the said justice shall summons the party against whom such complaint shall be made, and shall inquire of, hear and determine, the same; and, if the said justice shall find the said complaint to be true, or that such person will not exculpate or clear himself from the charge, by his or her own oath, which such person shall be at liberty to do in all cases where positive proof is not given of the offence, such justice shall and may make such orders upon the same, for the relief of such slave or slaves, as he in his discretion shall think fit; and shall and may set and impose a fine or penalty on any person who shall offend in the premises, in any sum not exceeding twenty pounds current money, for each offence.—2 Brevard's, Dig. 241. Also Cobb's Dig. 827.
A similar law obtains in Louisiana.—Rev. Stat. 1852, p. 557, § 166.
Now, would not anybody think, from the virtuous solemnity and gravity of this act, that it was intended in some way to amount to something? Let us give a little sketch, to show how much it does amount to. Angelina Grimké Weld, sister to Sarah Grimké, before quoted, gives the following account of the situation of slaves on plantations:[10]
And here let me say, that the treatment of plantation slaves cannot be fully known, except by the poor sufferers themselves, and their drivers and overseers. In a multitude of instances, even the master can know very little of the actual condition of his own field-slaves, and his wife and daughters far less. A few facts concerning my own family will show this. Our permanent residence was in Charleston; our country-seat (Bellemont) was two hundred miles distant, in the north western part of the state, where, for some years, our family spent a few months annually. Our plantation was three miles from this family mansion. There all the field-slaves lived and worked. Occasionally,—once a month, perhaps,—some of the family would ride over to the plantation; but I never visited the fields where the slaves were at work, and knew almost nothing of their condition; but this I do know, that the overseers who had charge of them were generally unprincipled and intemperate men. But I rejoice to know that the general treatment of slaves in that region of country was far milder than on the plantations in the lower country.
Throughout all the eastern and middle portions of the state, the planters very rarely reside permanently on their plantations. They have almost invariably two residences, and spend less than half the year on their estates. Even while spending a few months on them, politics, field-sports, races, speculations, journeys, visits, company, literary pursuits, &c., absorb so much of their time, that they must, to a considerable extent, take the condition of their slaves on trust, from the reports of their overseers. I make this statement, because these slaveholders (the wealthier class) are, I believe, almost the only ones who visit the North with their families; and Northern opinions of slavery are based chiefly on their testimony.
With regard to overseers, Miss Grimké's testimony is further borne out by the universal acknowledgment of Southern owners. A description of this class of beings is furnished by Mr. Wirt, in his Life of Patrick Henry, page 34. "Last and lowest," he says, [of different classes in society] "a feculum of beings called overseers,—a most abject, degraded, unprincipled race." Now, suppose, while the master is in Charleston, enjoying literary leisure, the slaves on some Bellemont or other plantation, getting tired of being hungry and cold, form themselves into a committee of the whole, to see what is to be done. A broad-shouldered, courageous fellow, whom we will call Tom, declares it is too bad, and he won't stand it any longer; and, having by some means become acquainted with this benevolent protective act, resolves to make an appeal to the horns of this legislative altar. Tom talks stoutly, having just been bought on to the place, and been used to better quarters elsewhere. The women and children perhaps admire, but the venerable elders of the plantation,—Sambo, Cudge, Pomp and old Aunt Dinah,—tell him he better mind himself, and keep clar o' dat ar. Tom, being young and progressive, does not regard these conservative maxims; he is determined that, if there is such a thing as justice to be got, he will have it. After considerable research, he finds some white man in the neighborhood verdant enough to enter the complaint for him. Master Legree finds himself, one sunshiny, pleasant morning, walked off to some Justice Dogberry's, to answer to the charge of not giving his niggers enough to eat and wear. We will call the infatuated white man who has undertaken this fool's errand Master Shallow. Let us imagine a scene:—Legree, standing carelessly wdth his hands in his pockets, rolling a quid of tobacco in his mouth; Justice Dogberry, seated in all the majesty of law, reinforced by a decanter of whiskey and some tumblers, intended to assist in illuminating the intellect in such obscure cases.
Justice Dogberry. Come, gentlemen, take a little something, to begin with. Mr. Legree, sit down; sit down, Mr.——a' what 's-your-name?——Mr. Shallow.
Mr. Legree and Mr. Shallow each sit down, and take their tumbler of whiskey and water. After some little conversation, the justice introduces the business as follows:
"Now, about this nigger business. Gentlemen, you know the act of um—um,—where the deuce is that act? [Fumbling an old law-book.] How plagued did you ever hear of that act, Shallow? I'm sure I'm forgot all about it;—O! here 'tis. Well, Mr. Shallow, the act says you must make proof, you observe.
Mr. Shallow. [Stuttering and hesitating.] Good land! why, don't everybody see that them ar niggers are most starved? Only see how ragged they are!
Justice. I can't say as I've observed it particular. Seem to be very well contented.
Shallow. [Eagerly.] But just ask Pomp, or Sambo, or Dinah, or Tom!
Justice Dogberry. [With dignity.] I'm astonished at you, Mr. Shallow! You think of producing negro testimony? I hope I know the law better than that! We must have direct proof, you know.
Shallow is posed; Legree significantly takes another tumbler of whiskey and water, and Justice Dogberry gives a long ahe-a-um. After a few moments the justice speaks:
"Well, after all, I suppose, Mr. Legree, you wouldn't have any objections to swarin' off; that settles it all, you know."
As swearing is what Mr. Legree is rather more accustomed to do than anything else that could be named, a more appropriate termination of the affair could not be suggested; and he swears, accordingly, to any extent, and with any fulness and variety of oath that could be desired; and thus the little affair terminates. But it does not terminate thus for Tom or Sambo, Dinah, or any others who have been alluded to for authority. What will happen to them, when Mr. Legree comes home, had better be left to conjecture.
It is claimed, by the author of certain paragraphs quoted at the commencement of Part II., that there exist in Louisiana ample protective acts to prevent the separation of young children from their mothers. This writer appears to be in the enjoyment of an amiable ignorance and unsophisticated innocence with regard to the workings of human society generally, which is, on the whole, rather refreshing. For, on a certain incident in "Uncle Tom's Cabin," which represented Cassy's little daughter as having been sold from her, he makes the following naïf remark:
Now, the reader will perhaps be surprised to know that such an incident as the sale of Cassy apart from Eliza, upon which the whole interest of the foregoing narrative hinges, never could have taken place in Louisiana, and that the bill of sale for Eliza would not have been worth the paper it was written on.—Observe. George Shelby states that Eliza was eight or nine years old at the time his father purchased her in New Orleans. Let us again look at the statute-book of Louisiana.
In the Code Noir we find it set down that
And this humane provision is strengthened by a statute, one clause of which runs as follows:
"Be it further enacted, that if any person or persons shall sell the mother of any slave child or children under the age of ten years, separate from said child or children, or shall, the mother living, sell any slave child or children of ten years of age or under, separate from said mother, such person or persons shall incur the penalty of the sixth section of this act."
This penalty is a fine of not less than one thousand nor more than two thousand dollars, and imprisonment in the public jail for a period of not less than six months nor more than one year.—Vide Acts of Louisiana, 1 Session, 9th Legislature, 1828-9, No. 24, Section 16. (Rev. Stat. 1852, p. 550, § 143.)
What a charming freshness of nature is suggested by this assertion! A thing could not have happened in a certain state, because there is a law against it!
Has there not been for two years a law forbidding to succor fugitives, or to hinder their arrest?—and has not this thing been done thousands of times in all the Northern States, and is not it more and more likely to be done every year? What is a law, against the whole public sentiment of society?—and will anybody venture to say that the public sentiment of Louisiana practically goes against separation of families?
Bat let us examine a case more minutely, remembering the bearing on it of two great foundation principles of slave jurisprudence: namely, that a slave cannot bring a suit in any case, except in a suit for personal freedom, and this in some states must be brought by a guardian; and that a slave cannot bear testimony in any case in which whites are implicated.
Suppose Butler wants to sell Cassy's child of nine years. There is a statute forbidding to sell under ten years;—what is Cassy to do? She cannot bring suit. Will the state prosecute? Suppose it does,—what then? Butler says the child is ten years old; if he pleases, he will say she is ten and a half, or eleven. What is Cassy to do? She cannot testify; besides, she is utterly in Butler's power. He may tell her that if she offers to stir in the affair, he will whip the child within an inch of its life; and she knows he can do it, and that there is no help for it;—he may lock her up in a dungeon, sell her on to a distant plantation, or do any other despotic thing he chooses, and there is nobody to say Nay.
How much does the protective statute amount to for Cassy? It may be very well as a piece of advice to the public, or as a decorous expression of opinion; but one might as well try to stop the current of the Mississippi with a bulrush as the tide of trade in human beings with such a regulation.
We think that, by this time, the reader will agree with us, that the less the defenders of slavery say about protective statutes, the better.
THE EXECUTION OF JUSTICE.
State v. Eliza Rowand.—The "Ægis of Protection" to the Slave's Life.
"We cannot but regard the fact of this trial as a salutary occurrence."—Charleston Courier.
Having given some account of what sort of statutes are to be found on the law-books of slavery, the reader will hardly be satisfied without knowing what sort of trials are held under them. We will quote one specimen of a trial, reported in the Charleston Courier of May 6th, 1847. The Charleston Courier is one of the leading papers of South Carolina, and the case is reported with the utmost apparent innocence that there was anything about the trial that could reflect in the least on the character of the state for the utmost legal impartiality. In fact, the Charleston Courier ushers it into public view with the following flourish of trumpets, as something which is forever to confound those who say that South Carolina does not protect the life of the slave:
THE TRIAL FOR MURDER.
Our community was deeply interested and excited, yesterday, by a case of great importance, and also of entire novelty in our jurisprudence. It was the trial of a lady of respectable family, and the mother of a large family, charged with the murder of her own or her husband's slave. The court-house was thronged with spectSitors of the exciting drama, who remained, with unabated interest and undiminished numbers, until the verdict was rendered acquitting the prisoner. We cannot but regard the fact of this trial as a salutary, although in itself lamentable occurrence, as it will show to the world that, however panoplied in station and wealth, and although challenging those sympathies which are the right and inheritance of the female sex, no one will be suffered, in this community, to escape the most sifting scrutiny, at the risk of even an ignominious death, who stands charged with the suspicion of murdering a slave,—to whose life our law now extends the ægis of protection, in the same manner as it does to that of the white man, save only in the character of the evidence necessary for conviction or defence. While evil-disposed persons at home are thus taught that they may expect rigorous trial and condign punishment, when, actuated by malignant passions, they invade the life of the humble slave, the enemies of our domestic institution abroad will find, their calumnies to the contrary notwithstanding, that we are resolved, in this particular, to do the full measure of our duty to the laws of humanity. We subjoin a report of the case.
The proceedings of the trial are thus given:
TRIAL FOR THE MURDER OF A SLAVE.
State v. Eliza Rowand.—Spring Term, May 5, 1847.
Tried before his Honor Judge O'Neall.
The prisoner was brought to the bar and arraigned, attended by her husband and mother, and humanely supported, during the trying scene, by the sheriff, J. B. Irving, Esq. On her arraignment, she pleaded "Not Guilty," and for her trial, placed herself upon "God and her country." After challenging John M. Deas, James Bancroft, H. F. Harbers, C. J. Beckman, E. R. Cowperthwaite, Parker J. Holland, Moses D. Hyams, Thomas Glaze, John Lawrence, B. Archer, J. S. Addison, B. P. Colburn, B. M. Jenkins, Carl Houseman, Geo. Jackson, and Joseph Coppenberg, the prisoner accepted the subjoined panel, who were duly sworn, and charged with the case: 1. John L. Nowell, foreman. 2. Elias Whilden. 3. Jesse Coward. 4. Effington Wagner. 5. Wm. Whaley. 6. James Culbert. 7. R. L. Baker. 8. S. Wiley. 9. W. S. Chisolm. 10. T. M. Howard. 11. John Bickley 12. John Y. Stock.
The following is the indictment on which the prisoner was arraigned for trial:
The State v. Eliza Rowand—Indictment for murder of a slave.
State of South Carolina,
Charleston District, } {\displaystyle \scriptstyle {\left.{\begin{matrix}\ \end{matrix}}\right\}\,}} to wit:
At a Court of General Sessions, begun and holden in and for the district of Charleston, in the State of South Carolina, at Charleston, in the district and state aforesaid, on Monday, the third day of May, in the year of our Lord one thousand eight hundred and forty-seven:
The jurors of and for the district of Charleston, aforesaid, in the State of South Carolina, aforesaid, upon their oaths present, that Eliza Rowand, the wife of Robert Rowand, Esq., not having the fear of God before her eyes, but being moved and seduced by the instigation of the devil, on the 6th day of January, in the year of our Lord one thousand eight hundred and forty-seven, with force and arms, at Charleston, in the district of Charleston, and state aforesaid, in and upon a certain female slave of the said Robert Rowand, named Maria, in the peace of God, and of the said state, then and there being, feloniously, maliciously, wilfully, deliberately, and of her malice aforethought, did make an assault; and that a certain other slave of the said Robert Rowand, named Richard, then and there, being then and there in the presence and by the command of the said Eliza Rowand, with a certain piece of wood, which he the said Richard in both his hands then and there had and held, the said Maria did beat and strike, in and upon the head of her the said Maria, then and there giving to her the said Maria, by such striking and beating, as aforesaid, with the piece of wood aforesaid, divers mortal bruises on the top, back, and sides of the head of her the said Maria, of which several mortal bruises she, the said Maria, then and there instantly died; and that the said Eliza Rowand was then and there present, and then and there feloniously, maliciously, wilfully, deliberately, and of her malice aforethought, did order, command, and require, the said slave named Richard the murder and felony aforesaid, in manner and form aforesaid, to do and commit. And as the jurors aforesaid, upon their oaths aforesaid, do say, that the said Eliza Rowand her the said slave named Maria, in the manner and by the means, aforesaid, feloniously, maliciously, wilfully, deliberately, and of her malice aforethought, did kill and murder against the form of the act of the General Assembly of the said state in such case made and provided, and against the peace and dignity of the same state aforesaid.
And the jurors aforesaid, upon their oaths aforesaid, do further present, that the said Eliza Rowand, not having the fear of God before her eyes, but being moved and seduced by the instigation of the devil, on the sixth day of January, in the year of our Lord one thousand eight hundred and forty-seven, with force and arms, at Charleston, in the district of Charleston, and state aforesaid, in and upon a certain other female slave of Robert Rowand, named Maria, in the peace of God, and of the said state, then and there being, feloniously, maliciously, wilfully, deliberately, and of her malice aforethought, did make an assault; and that the said Eliza Rowand, with a certain piece of wood, which she, the said Eliza Rowand, in both her hands then and there had and held, her the said last-mentioned slave named Maria did then and there strike, and beat, in and upon the head of her the said Maria, then and there giving to her the said Maria, by such striking and beating aforesaid, with the piece of wood aforesaid, divers mortal bruises, on the top, back, and side of the head, of her the said Maria, of which said several mortal bruises she the said Maria then and there instantly died. And so the jurors aforesaid, upon their oaths aforesaid, do say, that the said Eliza Rowand her the said last-mentioned slave named Maria, in the manner and by the means last mentioned, feloniously, maliciously, wilfully, deliberately, and of her malice aforethought, did kill and murder, against the form of the act of the General Assembly of the said state in such case made and provided, and against the peace and dignity of the same state aforesaid.
H. Bailey, Attorney-general.
As some of our readers may not have been in the habit of endeavoring to extract anything like common sense or information from documents so very concisely and luminously worded, the author will just state her own opinion that the above document is intended to charge Mrs. Eliza Rowand with having killed her slave Maria, in one of two ways: either with beating her on the head with her own hands, or having the same deed performed by proxy, by her slaveman Richard. The whole case is now presented. In order to make the reader clearly understand the arguments, it is necessary that he bear in mind that the law of 1740, as we have before shown, punished the murder of the slave only with fine and disfranchisement, while the law of 1821 punishes it with death.
On motion of Mr. Petigru, the prisoner was allowed to remove from the bar, and take her place by her counsel; the judge saying he granted the motion only because the prisoner was a woman, but that no such privilege would have been extended by him to any man.
The Attorney-general, Henry Bailey, Esq., then rose and opened the case for the state, in substance, as follows: He said that, after months of anxiety and expectation, the curtain had at length risen, and he and the jury were about to bear their part in the sad drama of real life, which had so long engrossed the public mind. He and they were called to the discharge of an important, painful, and solemn duty. They were to pass between the prisoner and the state—to take an inquisition of blood; on their decision hung the life or death, the honor or ignominy, of the prisoner; yet he trusted he and they would have strength and ability to perform their duty faithfully; and, whatever might be the result, their consciences would be consoled and quieted by that reflection. He bade the jury pause and reflect on the great sanctions and solemn responsibilities under which they were acting. The constitution of the state invested them with power over all that affected the life and was dear to the family of the unfortunate lady on trial before them. They were charged, too, with the sacred care of the law of the land; and to their solution was submitted one of the most solemn questions ever intrusted to the arbitrament of man. They should pursue a direct and straight-forward course, turning neither to the right hand nor to the left—influenced neither by prejudice against the prisoner, nor by a morbid sensibility in her behalf. Some of them might practically and personally be strangers to their present duty; but they were all familiar with the laws, and must be aware of the responsibilities of jurymen. It was scarcely necessary to tell them that, if evidence fixed guilt on this prisoner, they should not hesitate to record a verdict of guilty, although they should write that verdict in tears of blood. They should let no sickly sentimentality, or morbid feeling on the subject of capital punishments, deter them from the discharge of their plain and obvious duty. They were to administer, not to make, the law; they were called on to enforce the law, by sanctioning the highest duty to God and to their country. If any of them were disturbed with doubts or scruples on this point, he scarcely supposed they would have gone into the jury-box. The law had awarded capital punishment as the meet retribution for the crime under investigation, and they were sworn to administer that law. It had, too, the full sanction of Holy Writ; we were there told that "the land cannot be cleansed of the blood shed therein, except by the blood of him that shed it." He felt assured, then, that they would be swayed only by a firm resolve to act on this occasion in obedience to the dictates of sound judgments and enlightened consciences. The prisoner, however, had claims on them, as well as the community; she was entitled to a fair and impartial trial. By the wise and humane principles of our law, they were bound to hold the prisoner innocent, and she stood guiltless before them, until proved guilty, by legal, competent, and satisfactory evidence. Deaf alike to the voice of sickly humanity and heated prejudice, they should proceed to their task with minds perfectly equipoised and impartial; they should weigh the circumstances of the case with a nice and careful hand; and if, by legal evidence, circumstantial and satisfactory, although not positive, guilt be established, they should unhesitatingly, fearlessly and faithfully, record the result of their convictions. He would next call their attention to certain legal distinctions, but would not say a word of the facts; he would leave them to the lips of the witnesses, unaffected by any previous comments of his own. The prisoner stood indicted for the murder of a slave. This was supposed not to be murder at common law. At least, it was not murder by our former statute; but the act of 1821 had placed the killing of the white man and the black man on the same footing. He here read the act of 1821, declaring that "any person who shall wilfully, deliberately, and maliciously murder a slave, shall. on conviction thereof, suffer death without benefit of clergy." The rules applicable to murder at common law were generally applicable, however, to the present case. The inquiries to be made may be reduced to two: 1. Is the party charged guilty of the fact of killing? This must be clearly made out by proof. If she be not guilty of killing, there is an end of the case. 2. The character of that killing, or of the offence. Was it done with malice aforethought? Malice is the essential ingredient of the crime. Where killing takes place, malice is presumed, unless the contrary appear; and this must be gathered from the attending circumstances. Malice is a technical term, importing a different meaning from that conveyed by the same word in common parlance. According to the learned Michael Foster, it consists not in "malevolence to particulars," it does not mean hatred to any particular individual, but is general in its import and application. But even killing, with intention to kill, is not always murder; there may be justifiable and excusable homicide, and killing in sudden heat and passion is so modified to manslaughter. Yet there may be murder when there is no ill-feeling,—nay, perfect indifference to the slain,—as in the case of the robber who slays to conceal his crime. Malice aforethought is that depraved feeling of the heart, which makes one regardless of social duty, and fatally bent on mischief. It is fulfilled by that recklessness of law and human life which is indicated by shooting into a crowd, and thus doing murder on even an unknown object. Such a feeling the law regards as hateful, and visits, in its practical exhibition, with condign punishment, because opposed to the very existence of law and society. One may do fatal mischief without this recklessness; but when the act is done, regardless of consequences, and death ensues, it is murder in the eye of the law. If the facts to be proved in this case should not come up to these requisitions, he implored the jury to acquit the accused, as at once due to law and justice. They should note every fact with scrutinizing eye, and ascertain whether the fatal result proceeded from passing accident or from brooding revenge, which the law stamped with the odious name of malice. He would make no further preliminary remarks, but proceed at once to lay the facts before them, from the mouths of the witnesses.
Evidence.
J. Poricous Deveaux sworn.—He is the coroner of Charleston district; held the inquest, on the seventh of January last, on the body of the deceased slave, Maria, the slave of Eobert Rowand, at the residence of Mrs. T. C. Bee (the mother of the prisoner), in Logan-street. The body was found in an outbuilding—a kitchen; it was the body of an old and emaciated person, between fifty and sixty years of age; it was not examined in his presence by physicians; saw some few scratches about the face; adjourned to the City Hall. Mrs. Bowand was examined; her examination was in writing; it was here produced, and read, as follows:
"Mrs. Eliza Rowand sworn.—Says Maria is her nurse, and had misbehaved on yesterday morning; deponent sent Maria to Mr. Rowand's house, to be corrected by Simon; deponent sent Maria from the house about seven o'clock, A. M.; she returned to her about nine o'clock; came into her chamber; Simon did not come into the chamber at any time previous to the death of Maria; deponent says Maria fell down in the chamber; deponent had her seated up by Richard, who was then in the chamber, and deponent gave Maria some asafoetida; deponent then left the room; Richard came down and said Maria was dead; deponent says Richard did not strike Maria, nor did any one else strike her, in deponent's chamber. Richard left the chamber immediately with deponent; Maria was about fifty-two years of age; deponent sent Maria by Richard to Simon, to Mr Rowand's house, to be corrected; Mr. Rowand was absent from the city; Maria died about twelve o'clock; Richard and Maria were on good terms; deponent was in the chamber all the while that Richard and Maria were there together.
"Eliza Rowand.
"Sworn to before me this seventh January, 1847.
""J. P. Deveaux, Coroner, D. C."
"Witness went to the chamber of prisoner, where the death occurred; saw nothing particular; some pieces of wood in a box, set in the chimney; hi attention was called to one piece, in particular, eighteen inches long, three inches wide, and about one and a half inch thick; did not measure it; the jury of inquest did; it was not a light-wood knot; thinks it was of oak; there was some pine wood and some split oak. Dr. Peter Porcher was called to examine the body professionally, who did so out of witness' presence.
Before this witness left the stand, B. F. Hunt, Esq., one of the counsel for the prisoner, rose and opened the defence before the jury, in substance as follows:
He said that the scene before them was a very novel one; and whether for good or evil, he would not pretend to prophesy. It was the first time, in the history of this state, that a lady of good character and respectable connections stood arraigned at the bar, and had been put on trial for her life, on facts arising out of her domestic relations to her own slave. It was a spectacle consoling, and cheering, perhaps, to those who owed no good will to the institutions of our country; but calculated only to excite pain and regret among ourselves. He would not state a proposition so revolting to humanity as that crime should go unpunished; but judicial interference between the slave and the owner was a matter at once of delicacy and danger. It was the first time he had ever stood between a slave-owner and the public prosecutor, and his sensations were anything but pleasant. This is an entirely different case from homicide between equals in society. Subordination is indispensable where slavery exists; and in this there is no new principle involved. The same principle prevails in every country; on shipboard and in the army a large discretion is always left to the superior. Charges by inferiors against their superiors were always to be viewed with great circumspection at least, and especially when the latter are charged with cruelty or crime against subordinates. In the relation of owner and slave there is an absence of the usual motives for murder, and strong inducements against it on the part of the former. Life is usually taken from avarice or passion. The master gains nothing, but loses much, by the death of his slave; and when he takes the life of the latter deliberately, there must be more than ordinary malice to instigate the deed. The policy of altering the old law of 1740, which punished the killing of a slave with fine and political disfranchisement, was more than doubtful. It was the law of our colonial ancestors; it conformed to their policy and was approved by their wisdom, and it continued undisturbed by their posterity until the year 1821. It was engrafted on our policy in counter-action of the schemes and machinations, or in deference to the clamors, of those who formed plans for our improvement, although not interested in nor understanding our institutions, and whose interference led to the tragedy of 1822. He here adverted to the views of Chancellor Harper on this subject, who, in his able and philosophical memoir on slavery, said: "It is a somewhat singular fact, that when there existed in our state no law for punishing the murder of a slave, other than a pecuniary fine, there were, I will venture to say, at least ten murders of freemen for one murder of a slave. Yet it is supposed that they are less protected than their masters." "The change was made in subserviency to the opinions and clamor of others, who were utterly incompetent to form an opinion on the subject; and a wise act is seldom the result of legislation in this spirit. From the fact I have stated, it is plain they need less protection. Juries are, therefore, less willing to convict, and it may sometimes happen that the guilty will escape all punishment. Security is one of the compensations of their humble position. We challenge the comparison, that with us there have been fewer murders of slaves than of parents, children, apprentices, and other murders, cruel and unnatural, in society where slavery does not exist."
Such was the opinion of Chancellor Harper on this subject, who had profoundly studied it, and whose views had been extensively read on this continent and in Europe. Fortunately, the jury, he said, were of the country, acquainted with our policy and practice; composed of men too independent and honorable to be led astray by the noise and clamor out of doors. All was now as it should be;—at least, a court of justice had assembled, to which his client had fled for refuge and safety; its threshold was sacred; no profane clamors entered there; but legal investigation was had of facts, derived from the testimony of sworn witnesses; and this should teach the community to shut their bosoms against sickly humanity, and their ears to imaginary tales of blood and horror, the food of a depraved appetite. He warned the jury that they were to listen to no testimony but that of free white persons, given on oath in open court. They were to imagine none that came not from them. It was for this that they were selected,—their intelligence putting them beyond the influence of unfounded accusations, unsustained by legal proof; of legends of aggravated cruelty, founded on the evidence of negroes, and arising from weak and wicked falsehoods. Were slaves permitted to testify against their owner, it would cut the cord that unites them in peace and harmony, and enable them to sacrifice their masters to their ill will or revenge. Whole crews had been often leagued to charge captains of vessels with foulest murder, but judicial trial had exposed the falsehood. Truth has been distorted in this case, and murder manufactured out of what was nothing more than ordinary domestic discipline. Chastisement must be inflicted until subordination is produced; and the extent of the punishment is not to bejudged of by one's neighbors, but by himself. The event in this case has been unfortunate and sad; but there was no motive for the taking of life. There is no pecuniary interest in the owner to destroy his slave; the murder of his slave can only happen from ferocious passions of the master, filling his own bosom with anguish and contrition. This case has no other basis but unfounded rumor, commonly believed, on evidence that will not venture here, the offspring of that passion and depravity which make up falsehood. The hope of freedom, of change of owners, revenge are all motives with slave witnesses to malign their owners; and to credit such testimony would be to dissolve human society. Where deliberate, wilful, and malicious murder is done, whether by male or female, the retribution of the law is a debt to God and man; but the jury should beware lest it fall upon the innocent. The offence charged was not strictly murder at common law. The act of 1740 was founded on the practical good sense of our old planters, and its spirit still prevails. The act of 1821 is, by its terms, an act only to increase the punishment of persons convicted of murdering a slave,—and this is a refinement in humanity of doubtful policy. But, by the act of 1821, the murder must be wilful, deliberate and malicious; and, when punishment is due to the slave, the master must not be held to strict account for going an inch beyond the mark; whether for doing so he shall be a felon, is a question for the jury to solve. The master must conquer a refractory slave; and deliberation, so as to render clear the existence of malice, is necessary to bring the master within the provision of the act. He bade the jury remember the words of Him who spake as never man spake,—"Let him that has never sinned throw the first stone." They, as masters might regret excesses to which they have themselves carried punishment. He was not at all surprised at the course of the attorney-general; it was his wont to treat every case with perfect fairness. He (Colonel H.) agreed that the inquiry should be—
1. Into the fact of the death.
2. The character or motive of the act.
The examination of the prisoner showed conclusively that the slave died a natural death, and not from personal violence. She was chastised with a lawful weapon,—was in weak health, nervous, made angry by her punishment,—excited. The story was then a plain one; the community had been misled by the creations of imagination, or the statements of interested slaves. The negro came into her mistress' chamber; fell on the floor; medicine was given her; it was supposed she was asleep, but she slept the sleep of death. To show the wisdom and policy of the old act of 1740 (this indictment is under both acts,—the punishment only altered by that of 1821), he urged that a case like this was not murder at common law; nor is the same evidence applicable at common law. There, murder was presumed from killing; not so in the case of a slave. The act of 1740 permits a master, when his slave is killed in his presence, there being no other white person present, to exculpate himself by his own oath; and this exculpation is complete, unless clearly contravened by the evidence of two white witnesses. This is exactly what the prisoner has done; she has, as the law permits, by calling on God, exculpated herself. And her oath is good, at least against the slander of her own slaves. Which, then, should prevail, the clamors of others, or the policy of the law established by our colonial ancestors? There would not be a tittle of positive evidence against the prisoner, nothing but circumstantial evidence; and ingenious combination might be made to lead to any conclusion. Justice was all that his client asked. She appealed to liberal and high-minded men,—and she rejoiced in the privilege of doing so,—to accord her that justice they would demand for themselves.
Mr. Deveaux was not cross-examined.
Evidence resumed.
Dr. E. W. North sworn.—(Cautioned by attorney-general to avoid hearsay evidence.) Was the family physician of Mrs. Rowand. Went on the 6th January, at Mrs. Rowand's request, to gee her at her mother's, in Logan-street; found her down stairs, in sitting-room. She was in a nervous and excited state; had been so for a month before; he had attended her; she said nothing to witness of slave Maria; found Maria in a chamber, up stairs, about one o'clock, P. M.; she was dead; she appeared to have been dead about an hour and a half; his attention was attracted to a piece of pine wood on a trunk or table in the room; it had a large knot on one end; had it been used on Maria, it must have caused considerable contusion; other pieces of wood were in a box, and much smaller ones; the corpse was lying one side in the chamber; it was not laid out; presumed she died there; the marks on the body were, to witness' view, very slight; some scratches about the face; he purposely avoided making an examination; observed no injuries about the head; had no conversation with Mrs. Rowand about Maria; left the house; it was on the 6th January last,—the day before the inquest; knew the slave before, but had never attended her.
Cross-examined.—Mrs. Rowand was in feeble health, and nervous; the slave Maria was weak and emaciated in appearance; sudden death of such a person, in such a state, from apoplexy or action of nervous system, not unlikely; her sudden death would not imply violence; had prescribed asafoetida for Mrs. Rowand on a former visit; it is an appropriate remedy for nervous disorders. Mrs. Rowand was not of bodily strength to handle the pine knot so as to give a severe blow; Mrs. Rowand has five or six children, the elder of them large enough to have carried pieces of the wood about the room; there must have been a severe contusion, and much extravasation of blood, to infer death from violence in this case; apoplexy is frequently attended with extravasation of blood; there were two Marias in the family.
In reply.—Mrs. Rowand could have raised the pine knot, but could not have struck a blow with it; such a piece of wood could have produced death, but it would have left its mark; saw the follow Richard; he was quite capable of giving such a blow.
Dr. Peter Porcher.—Was called in by the coroner's jury to examine Maria's body; found it in the wash-kitchen; it was the corpse of one feeble and emaciated; partly prepared for burial; had the clothes removed; the body was lacerated with stripes; abrasions about face and knuckles; skin knocked off; passed his hand over the head; no bone broken; on request, opened her thorax, and examined the viscera; found them healthy; heart unusually so for one of her age; no particular odor; some undigested food; no inflammation; removed the scalp, and found considerable extravasation between scalp and skull; scalp bloodshot; just under the scalp, found the effects of a single blow, just over the right ear; after removing the scalp, lifted the bone; no rupture of any blood-vessel; some softening of the brain in the upper hemisphere; there was considerable extravasation under the scalp, the result of a succession of blows on the top of the head; this extravasation was general, but that over the ear was a single spot; the butt-end of a cowhide would have sufficed for this purpose; an ordinary stick, a heavy one, would have done it; a succession of blows on the head, in a feeble woman, would lead to death, when, in a stronger one, it would not; saw no other appearance about her person, to account for her death, except those blows.
Cross-examined.—To a patient in this woman's condition, the blows would probably cause death; they were not such as were calculated to kill an ordinary person; witness saw the body twenty-four hours after her death; it was winter, and bitter cold; no disorganization, and the examination was therefore to be relied on; the blow behind the ear might have resulted from a fall, but not the blow on the top of the head, unless she fell head foremost; came to the conclusion of a succession of blows, from the extent of the extravasation; a single blow would have shown a distinct spot, with a gradual spreading or diffusion; one large blow could not account for it, as the head was spherical; no blood on the brain; the softening of the brain did not amount to much; in an ordinary dissection would have passed it over; anger sometimes produces apoplexy, which results in death; blood between the scalp and the bone of the skull; it was evidently a fresh extravasation; twenty-four hours would scarcely have made any change; knew nothing of this negro before; even after examination, the cause of death is sometimes inscrutable,—not usual, however.
In reply.—Does not attribute the softening of the brain to the blows; it was slight, and might have been the result of age; it was some evidence of impairment of vital powers by advancing age.
Dr. A. P. Hayne.—At request of the coroner, acted with Dr. Porcher; was shown into an outhouse; saw on the back of the corpse evidences of contusion; arms swollen and enlarged; laceration of body; contusions on head and neck; between scalp and skull extravasation of blood, on the top of head, and behind the right ear; a burn on the hand; the brain presented healthy appearance; opened the body, and no evidences of disease in the chest or viscera; attributed the extravasation of blood to external injury from blows,—blows from a large and broad and blunt instrument; attributes the death to those blows; supposes they were adequate to cause death, as she was old, weak and emaciated.
Cross-examined.—Would not have caused death in a young and robust person.
The evidence for the prosecution here closed, and no witnesses were called for the defence.
The jury were then successively addressed, ably and eloquently, by J. L. Petigru and James S. Rhett, Esqrs., on behalf of the prisoner, and H. Bailey, Esq., on behalf of the state, and by B. F. Hunt, Esq., in reply. Of those speeches, and also of the judge's charge, we have taken full notes, but have neither time nor space to insert them here.
His Honor, Judge O'Neall, then charged the jury eloquently and ably on the facts, vindicating the existing law, making death the penalty for the murder of a slave; but, on the law, intimated to the jury that he held the act of 1740 so far still in force as to admit of the prisoner's exculpation by her own oath, unless clearly disproved by the oaths of two witnesses; and that they were, therefore, in his opinion, bound to acquit,—although he left it to them, wholly, to say whether the prisoner was guilty of murder, killing in sudden heat and passion, or not guilty.
The jury then retired, and, in about twenty or thirty minutes, returned with a verdict of "Not Guilty."
There are some points which appear in this statement of the trial, especially in the plea for the defence. Particular attention is called to the following passage:
"Fortunately," said the lawyer, "the jury were of the country;—acquainted with our policy and practice; composed of men too honorable to be led astray by the noise and clamor out of doors. All was now as it should be; at least, a court of justice had assembled to which his client had fled for refuge and safety; its threshold was sacred; no profane clamors entered there; but legal investigation was had of facts."
From this it plainly appears that the case was a notorious one; so notorious and atrocious as to break through all the apathy which slave-holding institutions tend to produce, and to surround the court-house with noise and clamor.
From another intimation in the same speech, it would appear that there was abundant testimony of slaves to the direct fact,—testimony which left no kind of doubt on the popular mind. Why else does he thus earnestly warn the jury?
He warned the jury that they were to listen to no evidence but that of free white persons, given on oath in open court; they were to imagine none that came not from them. It was for this that they were selected;—their intelligence putting them beyond the influence of unfounded accusations, unsustained by legal proof; of legends of aggravated cruelty, founded on the evidence of negroes, and arising from weak and wicked falsehoods.
See also this remarkable admission:—"Truth had been distorted in this case, and murder manufactured out of what was nothing more than ordinary domestic discipline." If the reader refers to the testimony, he will find it testified that the woman appeared to be about sixty years old; that she was much emaciated; that there had been a succession of blows on the top of her head, and one violent one over the ear; and that, in the opinion of a surgeon, these blows were sufficient to cause death. Yet the lawyer for the defence coolly remarks that "murder had been manufactured out of what was ordinary domestic discipline." Are we to understand that beating feeble old women on the head, in this manner, is a specimen of ordinary domestic discipline in Charleston? What would have been said if any antislavery newspaper at the North had made such an assertion as this? Yet the Charleston Courier reports this statement without comment or denial. But let us hear the lady's lawyer go still further in vindication of this ordinary domestic discipline: "Chastisement must be inflicted until subordination is produced; and the extent of the punishment is not to be judged by one's neighbors, but by himself. The event, in this case, has been unfortunate and sad." The lawyer admits that the result of thumping a feeble old woman on the head has, in this case, been "unfortunate and sad." The old thing had not strength to bear it, and had no greater regard for the convenience of the family, and the reputation of "the institution," than to die, and so get the family and the community generally into trouble. It will appear from this that in most cases where old women are thumped on the head they have stronger constitutions—or more consideration.
Again he says, "When punishment is due to the slave, the master must not be held to strict account for going an inch beyond the mark." And finally, and most astounding of all, comes this: "He bade the jury remember the words of him who spake as never man spake,—'Let him that hath never sinned throw the first stone.' They, as masters, might regret excesses to which they themselves might have carried punishment."
What sort of an insinuation is this? Did he mean to say that almost all the jurymen had probably done things of the same sort, and therefore could have nothing to say in this case? and did no member of the jury get up and resent such a charge? From all that appears, the jury acquiesced in it as quite a matter of course; and the Charleston Courier quotes it without comment, in the record of a trial which it says "will show to the world now the law extends the ægis of her protection alike over the while man and the humblest slave."
Lastly, notice the decision of the judge, which has become law in South Carolina. What point does it establish? That the simple oath of the master, in face of all circumstantial evidence to the contrary, may clear him, when the murder of a slave is the question. And this trial is paraded as a triumphant specimen of legal impartiality and equity! "If the light that is in thee be darkness, how great is that darkness!"
THE GOOD OLD TIMES.
"A refinement in humanity of doubtful policy."
B. F. Hunt.
The author takes no pleasure in presenting to her readers the shocking details of the following case. But it seems necessary to exhibit what were the actual workings of the ancient law of South Carolina, which has been characterized as one "conformed to the policy, and approved by the wisdom," of the fathers of that state, and the reform of which has been called "a refinement in humanity of doubtful policy."
It is well, also, to add the charge of Judge Wilds, partly for its intrinsic literary merit, and the nobleness of its sentiments, but principally because it exhibits such a contrast as could scarcely be found elsewhere, between the judge's high and indignant sense of justice, and the shameful impotence and imbecility of the laws under which he acted.
The case was brought to the author's knowledge by a letter from a gentleman of Pennsylvania, from which the following is an extract:
Some time between the years 1807 and 1810, there was lying in the harbor of Charleston a ship commanded by a man named Slater. His crew were slaves: one of them committed some, offence, not specified in the narrative. The captain ordered him to be bound and laid upon the deck; and there, in the harbor of Charleston, in the broad day-light, compelled another slave-sailor to chop off his head. The affair was public—notorious. A prosecution was commenced against him; the offence was proved beyond all doubt,—perhaps, indeed, it was not denied,—and the judge, in a most eloquent charge or rebuke of the defendant, expressed his sincere regret that he could inflict no punishment, under the laws of the state.
I was studying law when the case was published in "Hall's American Law Journal, vol. 1." I have not seen the book for twenty-five or thirty years. I may be in error as to names, &c., but while I have life and my senses the facts of the case cannot be forgotten.
The following is the "charge" alluded to in the above letter. It was pronounced the Honorable Judge Wilds, of South Carolina, and is copied from Hall's Law Journal, 1. 67.
John Slater! You have been convicted by a jury of your country of the wilful murder of your own slave; and I am sorry to say, the short, impressive, uncontradicted testimony, on which that conviction was founded, leaves but too little room to doubt its propriety.
The annals of human depravity might be safely challenged for a parallel to this unfeeling, bloody and diabolical transaction.
You caused your unoffending, unresisting slave to be bound hand and foot, and, by a refinement in cruelty, compelled his companion, perhaps the friend of his heart, to chop his head with an axe, and to cast his body, yet convulsing with the agonies of death, into the water! And this deed you dared to perpetrate in the very harbor of Charleston, within a few yards of the shore, unblushingly, in the face of open day. Had your murderous arm been raised against your equals, whom the laws of self-defence and the more efficacious law of the land unite to protect, your crimes would not have been without precedent, and would have seemed less horrid. Your personal risk would at least have proved, that though a murderer, you were not a coward. But you too well knew that this unfortunate man, whom chance had subjected to your caprice, had not, like yourself, chartered to him by the laws of the land the sacred rights of nature; and that a stern, but necessary policy, had disarmed him of the rights of self-defence. Too well you knew that to you alone he could look for protection; and that your arm alone could shield him from oppression, or avenge his wrongs; yet, that arm you cruelly stretched out for his destruction.
The counsel, who generously volunteered his services in your behalf, shocked at the enormity of your offence, endeavored to find a refuge, as well for his own feelings as for those of all who heard your trial, in a derangement of your intellect. Several witnesses were examined to establish this fact; but the result of their testimony, it is apprehended, was as little satisfactory to his mind, as to those of the jury to whom it was addressed. I sincerely wish this defence had proved successful, not from any desire to save you from the punishment which awaits you, and which you so richly merit, but from the desire of saving my country from the foul reproach of having in its bosom so great a monster.
From the peculiar situation of this country, our fathers felt themselves justified in subjecting to a very slight punishment him who murders a slave. Whether the present state of society require a continuation of this policy, so opposite to the apparent rights of humanity, it remains for a subsequent legislature to decide. Their attention would ere this have been directed to this subject, but, for the honor of human nature, such hardened sinners as yourself are rarely found, to disturb the repose of society. The grand jury of this district, deeply impressed with your daring outrage against the laws both of God and man, have made a very strong expression of their feelings on the subject to the legislature; and, from the wisdom and justice of that body, the friends of humanity may confidently hope soon to see this blackest in the catalogue of human crimes pursued by appropriate punishment.
In proceeding to pass the sentence which the law provides for your offence, I confess I never felt more forcibly the want of power to make respected the laws of my country, whose minister I am. You have already violated the majesty of those laws. You have profanely pleaded the law under which you stand convicted, as a justification of your crime. You have held that law in one hand, and brandished your bloody axe in the other, impiously contending that the one gave a license to the unrestrained use of the other.
But, though you will go off unhurt in person, by the present sentence, expect not to escape with impunity. Your bloody deed has set a mark upon you, which I fear the good actions of your future life will not efface. You will be held in abhorrence by an impartial world, and shunned as a monster by every honest man. Your unoffending posterity will be visited, for your iniquity, by the stigma of deriving their origin from an unfeeling murderer. Your days, which will be but few, will be spent in wretchedness; and, if your conscience be not steeled against every virtuous emotion, if you be not entirely abandoned to hardness of heart, the mangled, mutilated corpse of your murdered slave will ever be present in your imagination, obtrude itself into all your amusements, and haunt you in the hours of silence and repose.
But, should you disregard the reproaches of an offended world, should you hear with callous insensibility the gnawings of a guilty conscience, yet remember, I charge you, remember, that an awful period is fast approaching, and with you is close at hand, when you must appear before a tribunal whose want of power can afford you no prospect of impunity; when you must raise your bloody hands at the bar of an impartial omniscient Judge! Remember, I pray you, remember, whilst yet you have time, that God is just, and that his vengeance will not sleep forever!
The penalty that followed this solemn denunciation was a fine of seven hundred pounds, current money, or, in default of payment, imprisonment for seven years. And yet it seems that there have not been wanting those who consider the reform of this law "a refinement in humanity of doubtful policy"! To this sentiment, so high an authority as that of Chancellor Harper is quoted, as the reader will see by referring to the speech of Mr. Hunt, in the last chapter. And, as is very common in such cases, the old law is vindicated, as being, on the whole, a surer protection to the life of the slave than the new one. From the results of the last two trials, there would seem to be a fair show of plausibility in the argument. For under the old law it seems that Slater had at least to pay seven hundred pounds, while under the new Eliza Rowand comes off with only the penalty of "a most sifting scrutiny."
Thus, it appears, the penalty of the law goes with the murderer of the slave.
How is it executed in the cases which concern the life of the master? Look at this short notice of a recent trial of this kind, which is given in the Alexandria (Va.) Gazette, of Oct. 23, 1852, as an extract from the Charlestown (Va.) Free Press.
TRIAL OF NEGRO HENRY.
The trial of this slave for an attack, with intent to kill, on the person of Mr. Harrison Anderson, was commenced on Monday and concluded oa Tuesday evening. His Honor, Braxton Davenport, Esq., chief justice of the county, with four associate gentlemen justices, composed the court.
The commonwealth was represented by its attorney, Charles B. Harding, Esq., and the accused ably and eloquently defended by Wm. C. Worthington and John A. Thompson, Esqs. The evidence of the prisoner's guilt was conclusive. A majority of the court thought that he ought to suffer the extreme penalty of the law; but, as this required a unanimous agreement, he was sentenced to receive five hundred lashes, not more than thirty-nine at one time. The physician of the jail was instructed to see that they should not be administered too frequently, and only when, in his opinion, he could bear them.
In another paper we are told that the Free Press says:
A majority of the court thought that he ought to suffer the extreme penalty of the law; but, as this required a unanimous agreement, he was sentenced to receive five hundred lashes, not more than thirty-nine at any one time. The physician of the jail was instructed to see that they should not be administered too frequently, and only when, in his opinion, he could bear them. This may seem to be a harsh and inhuman punishment; but, when we take into consideration that it is in accordance with the law of the land, and the further fact that the insubordination among the slaves of that state has become truly alarming, we cannot question the righteousness of the judgment.
Will anybody say that the master's life is in more danger from the slave than the slave's from the master, that this disproportionate retribution is meted out? Those who countenance such legislation will do well to ponder the solemn words of an ancient book, inspired by One who is no respecter of persons:
"If I have refused justice to my man-servant or maid-serrant,
When they had a cause with me.
What shall I do when God riseth up?
And when he visiteth, what shall I answer him?
Did not he that made me in the womb make him?
Did not the same God fashion us in the womb?"
Job 31: 13—15.
MODERATE CORRECTION AND ACCIDENTAL DEATH—STATE v. CASTLEMAN.
The author remarks that the record of the following trial was read by her a little time before writing the account of the death of Uncle Tom. The shocking particulars haunted her mind and were in her thoughts when the following sentence was written:
What man has nerve to do, man has not nerve to hear. What brother man and brother Christian must suffer, cannot be told us, even in our secret chamber, it so harrows up the soul. And yet, my country, these things are done under the shadow of thy laws! O Christ, thy church sees them almost in silence!
It is given precisely as prepared by Dr. G. Bailey, the very liberal and fair-minded editor of the National Era.
From the National Era, Washington, November 6, 1851.
HOMICIDE CASE In CLARKE COUNTY, VIRGINIA.
Some time since, the newspapers of Virginia contained an account of a horrible tragedy, enacted in Clarke County, of that state. A slave of Colonel James Castleman, it was stated, had been chained by the neck, and whipped to death by his master, on the charge of stealing. The whole neighborhood in which the transaction occurred was incensed; the Virginia papers abounded in denunciations of the cruel act; and the people of the North were called upon to bear witness to the justice which would surely be meted out in a slave state to the master of a slave. We did not publish the account. The case was horrible; it was, we were confident, exceptional; it should not be taken as evidence of the general treatment of slaves; we chose to delay any notice of it till the courts should pronounce their judgment, and we could announce at once the crime and its punishment, so that the state might stand acquitted of the foul deed.
Those who were so shocked at the transaction will be surprised and mortified to hear that the actors in it have been tried and acquitted; and when they read the following account of the trial and verdict, published at the instance of the friends of the accused, their mortification will deepen into bitter indignation:
From the "Spirit of Jefferson."
"Colonel James Castleman.—The following statement, understood to have been drawn up by counsel, since the trial, has been placed by the friends of this gentleman in our hands for publication:
"At the Circuit Superior Court of Clarke County, commencing on the 13th of October, Judge Samuels presiding, James Castleman and his son Stephen D. Castleman were indicted jointly for the murder of negro Lewis, property of the latter. By advice of their counsel, the parties elected to be tried separately, and the attorney for the commonwealth directed that James Castleman should be tried first.
"It was proved, on this trial, that for many months previous to the occurrence the money-drawer of the tavern kept by Stephen D. Castleman, and the liquors kept in large quantities in his cellar, had been pillaged from time to time, until the thefts had attained to a considerable amount. Suspicion had, from various causes, been directed to Lewis, and another negro, named Reuben (a blacksmith), the property of James Castleman; but by the aid of two of the house-servants they had eluded the most vigilant watch.
"On the 20th of August last, in the afternoon, S. D. Castleman accidentally discovered a clue, by means of which, and through one of the house-servants implicated, he was enabled fully to detect the depredators, and to ascertain the manner in which the theft had been committed. He immediately sent for his father, living near him, and after communicating what he had discovered, it was determined that the offenders should be punished at once, and before they should know of the discovery that had been made.
"Lewis was punished first; and in a manner, as was fully shown, to preclude all risk of injury to his person, by stripes with a broad leathern strap. He was punished severely, but to an extent by no means disproportionate to his offence; nor was it pretended, in any quarter, that this punishment implicated either his life or health. He confessed the offence, and admitted that it had been effected by false keys, furnished by the blacksmith, Reuben.
"The latter servant was punished immediately afterwards. It was believed that he was the principal offender, and he was found to be more obdurate and contumacious than Lewis had been in reference to the offence. Thus it was proved, both by the prosecution and the defence, that he was punished with greater severity than his accomplice. It resulted in a like confession on his part, and he produced the false key, one fashioned by himself, by which the theft had been effected .
"It was further shown, on the trial, that Lewis was whipped in the upper room of a warehouse, connected with Stephen Castleman's store, and near the public road, where he was at work at the time; that after he had been flogged, to secure his person, whilst they went after Reuben, he was confined by a chain around his neck, which was attached to a joist above his head. The length of this chain, the breadth and thickness of the joist, its height from the floor, and the circlet of chain on the neck, were accurately measured; and it was thus shown that the chain unoccupied by the circlet and the joist was a foot and a half longer than the space between the shoulders of the man and the joist above, or to that extent the chain hung loose above him; that the circlet (which was fastened so as to prevent its contraction) rested on the shoulders and breast, the chain being sufficiently drawn only to prevent being slipped over his head, and that there was no other place in the room to which he could be fastened, except to one of the joists above. His hands were tied in front; a white man, who had been at work with Lewis during the day, was left with him by the Messrs. Castleman, the better to insure his detention, whilst they were absent after Reuben. It was proved by this man (who was a witness for the prosecution) that Lewis asked for a box to stand on, or for something that he could jump off from; that after the Castlemans had left him he expressed a fear that when they came back he would be whipped again; and said, if he had a knife, and could get one hand loose, he would cut his throat. The witness stated that the negro 'stood firm on his feet,' that he could turn freely in whatever direction he wished, and that he made no complaint of the mode of his confinement. This man stated that he remained with Lewis about half an hour, and then left there to go home.
"After punishing Reuben, the Castlemans returned to the warehouse, bringing him with them; their object being to confront the two men, in the hope that by further examination of them jointly all their accomplices might be detected.
"They were not absent more than half an hour. When they entered the room above, Lewis was found hanging by the neck, his feet thrown behind him, his knees a few inches from the floor, and his head thrown forward—the body warm and supple (or relaxed), but life was extinct.
"It was proved by the surgeons who made a post-mortem examination before the coroner's inquest that the death was caused by strangulation by hanging; and other eminent surgeons were examined to show, from the appearance of the brain and its blood-vessels after death (as exhibited at the post-mortem examination), that the subject could not have fainted before strangulation.
"After the evidence was finished on both sides, the jury from their box, and of their own motion, without a word from counsel on either side, informed the court that they had agreed upon their verdict. The counsel assented to its being thus received, and a verdict of "not guilty" was immediately rendered. The attorney for the commonwealth then informed the court that all the evidence for the prosecution had been laid before the jury; and as no new evidence could be offered on the trial of Stephen D. Castleman, he submitted to the court the propriety of entering a nolle prosequi. The judge replied that the case had been fully and fairly laid before the jury upon the evidence; that the court was not only satisfied with the verdict, but, if any other had been rendered, it must have been set aside; and that if no further evidence was to be adduced on the trial of Stephen, the attorney for the commonwealth would exercise a proper discretion in entering a nolle prosequi as to him, and the court would approve its being done. A nolle prosequi was entered accordingly, and both gentlemen discharged.
"It may be added that two days were consumed in exhibiting the evidence, and that the trial was by a jury of Clarke County. Both the parties had been on bail from the time of their arrest, and were continued on bail whilst the trial was depending."
Let us admit that the evidence does not prove the legal crime of homicide: what candid man can doubt, after reading this ex parte version of it, that the slave died in consequence of the punishment inflicted upon him?
In criminal prosecutions the federal constitution guarantees to the accused the right to a public trial by an impartial jury; the right to be informed of the nature and cause of the accusation; to be confronted with the witnesses against him; to have compulsory process for obtaining witness in his favor; and to have the assistance of counsel; guarantees necessary to secure innocence against hasty or vindictive judgment,—absolutely necessary to prevent injustice. Grant that they were not intended for slaves; every master of a slave must feel that they are still morally binding upon him. He is the sole judge; he alone determines the offence, the proof requisite to establish it, and the amount of the punishment. The slave then has a peculiar claim upon him for justice. When charged with a crime, common humanity requires that he should be informed of it, that he should be confronted with the witnesses against him, that he should be permitted to show evidence in favor of his innocence.
But how was poor Lewis treated? The son of Castleman said he had discovered who stole the money; and it was forthwith "determined that the offenders should be punished at once, and before they should know of the discovery that had been made." Punished without a hearing! Punished on the testimony of a house-servant, the nature of which does not appear to have been inquired into by the court! Not a word is said which authorizes the belief that any careful examination was made, as it respects their guilt. Lewis and Reuben were assumed, on loose evidence, without deliberate investigation, to be guilty; and then, without allowing them to attempt to show their evidence, they were whipped, until a confession of guilt was extorted by bodily pain.
Is this Virginia justice?
Lewis was punished with "a broad leathern strap,"—he was "punished severely:" this we do not need to be told. A "broad leathern strap" is well adapted to severity of punishment. "Nor was it pretended," the account says, "in any quarter, that this punishment implicated either his life or his health." This is false; it was expressly stated in the newspaper accounts at the time, and such was the general impression in the neighborhood, that the punishment did very severely implicate his life. But more of this anon.
Lewis was left. A chain was fastened around his neck, so as not to choke him, and secured to the joist above, leaving a slack of about a foot and a half. Remaining in an upright position, he was secure against strangulation, but he could neither sit nor kneel; and should he faint, he would be choked to death. The account says that they fastened him thus for the purpose of securing him. If this had been the sole object, it could have been accomplished by safer and less cruel methods, as every reader must know. This mode of securing him was intended probably to intimidate him, and, at the same time, afforded some gratification to the vindictive feeling which controlled the actors in this foul transaction. The man whom they left to watch Lewis said that, after remaining there about half an hour, he went home; and Lewis was then alive. The Castlemans say that, after punishing Reuben, they returned, having been absent not more than half an hour, and they found him hanging by the neck, dead. We direct attention to this part of the testimony, to show how loose the statements were which went to make up the evidence.
Why was Lewis chained at all, and a man left to watch him? "To secure him," say the Castlemans. Is it customary to chain slaves in this manner, and set a watch over them, after severe punishment, to prevent their running away? If the punishment of Lewis had not been unusual, and if he had not been threatened with another infliction on their return, there would have been no necessity for chaining him.
The testimony of the man left to watch represents him as desperate, apparently, with pain and fright. "Lewis asked for a box to stand on:" why? Was he not suffering from pain and exhaustion, and did he not wish to rest himself, without danger of slow strangulation? Again: he asked for "something he could jump off from;" "after the Castlemans left, he expressed a fear when they came back that he would be whipped again; and said, if he had a knife, and could get one hand loose, he would cut his throat."
The punishment that could drive him to such desperation must have been horrible.
How long they were absent we know not, for the testimony on this point is contradictory. They found him hanging by the neck, dead, "his feet thrown behind him, his knees a few inches from the floor, and his head thrown forward,"— Just the position he wouild naturally fall into, had he sunk from exhaustion. They wish it to appear that he hung himself. Could this be proved (we need hardly say that it is not), it would relieve but slightly the dark picture of their guilt. The probability is that he sank, exhausted by suffering, fatique and fear. As to the testimony of "surgeons," founded upon a post-mortem examination of the brain and blood-vessels, "that the subject could not have fainted before strangulation," it is not worthy of consideration. We know something of the fallacies and fooleries of such examinations.
From all we can learn, the only evidence relied on by the prosecution was that white man employed by the Castlemans. He was dependent upon them for work. Other evidence might have been obtained; why it was not is for the prosecuting attorney to explain. To prove what we say, and to show that justice has not been done in this horrible affair, we publish the following communication from an old and highly-respectable citizen of this place, and who is very far from being an Abolitionist. The slave-holders whom he mentions are well known here, and would have promptly appeared in the case, had the prosecution, which was aware of their readiness, summoned them.
"To the Editor of the Era:
"I see that Castleman, who lately had a trial for whipping a slave to death, in Virginia, was 'triumphantly acquitted,'—as many expected. There are three persons in this city, with whom I am acquainted, who staid at Castleman's the same night in which this awful tragedy was enacted. They heard the dreadful lashing and the heart-rending screams and entreaties of the sufferer. They implored the only white man they could find on the premises, not engaged in the bloody work, to interpose; but for a long time he refused, on the ground that he was a dependent, and was afraid to give offence; and that, moreover, they had been drinking, and he was in fear for his own life, should he say a word that would be displeasing to them. He did, however, venture, and returned and reported the cruel manner in which the slaves were chained, and lashed, and secured in a blacksmith's vice. In the morning, when they ascertained that one of the slaves was dead, they were so shocked and indignant that they refused to eat in the house, and reproached Castleman with his cruelty. He expressed his regret that the slave had died, and especially as he had ascertained that he was innocent of the accusation for which he had suffered. The idea was that he had fainted from exhaustion; and, the chain being round his neck, he was strangled. The persons I refer to are themselves slave-holders,—but their feelings were so harrowed and lacerated that they could not sleep (two of them are ladies); and for many nights afterwards their rest was disturbed, and their dreams made frightful, by the appalling recollection.
"These persons would have been material witnesses, and would have willingly attended on the part of the prosecution. The knowledge they had of the case was communicated to the proper authorities, yet their attendance was not required. The only witness was that dependent who considered his own life in danger.
"Yours, &c.,J. F."
The account, as published by the friends of the accused parties, shows a case of extreme cruelty. The statements made by our correspondent prove that the truth has not been fully revealed, and that justice has been baffled. The result of the trial shows how irresponsible is the power of a master over his slave; and that whatever security the latter has is to be sought in the humanity of the former, not in the guarantees of law. Against the cruelty of an inhuman master he has really no safeguard.
Our conduct in relation to this case, deferring all notice of it in our columns till a legal investigation could be had, shows that we are not disposed to be captious towards our slave-holding countrymen. In no unkind spirit have we examined this lamentable case; but we must expose the utter repugnance of the slave system to the proper administration of justice. The newspapers of Virginia generally publish the account from the Spirit of Jefferson, without comment. They are evidently not satisfied that justice was done; they doubtless will deny that the accused were guilty of homicide, legally; but they will not deny that they were guilty of an atrocity which should brand them forever, in a Christian country.
PRINCIPLES ESTABLISHED—STATE v. LEGREE; A CASE NOT IN THE BOOKS
From a review of all the legal cases which have hitherto been presented, and of the principles established in the judicial decisions upon them, the following facts must be apparent to the reader:
First, That masters do, now and then, kill slaves by the torture.
Second, That the fact of so killing a slave is not of itself held presumption of murder, in slave jurisprudence.
Third, That the slave in the act of resistance to his master may always be killed.
From these things it will be seen to follow, that, if the facts of the death of Tom had been fully proved by two white witnesses, in open court, Legree could not have been held by any consistent interpreter of slave-law to be a murderer; for Tom was in the act of resistance to the will of his master. His master had laid a command on him, in the presence of other slaves. Tom had deliberately refused to obey the command. The master commenced chastisement, to reduce him to obedience. And it is evident, at the first glance, to every one, that, if the law does not sustain him in enforcing obedience in such a case, there is an end of the whole slave power. No Southern court would dare to decide that Legree did wrong to continue the punishment, as long as Tom continued the insubordination. Legree stood by him every moment of the time, pressing him to yield, and offering to let him go as soon as he did yield. Tom's resistance was insurrection. It was an example which could not be allowed, for a moment, on any Southern plantation. By the express words of the constitution of Georgia, and by the understanding and usage of all slave-law, the power of life and death is always left in the hands of the master, in exigences like this. This is not a case like that of Souther v. The Commonwealth. The victim of Souther was not in a state of resistance or insurrection. The punishment, in his case, was a simple vengeance for a past offence, and not an attempt to reduce him to subordination.
There is no principle of slave jurisprudence by which a man could be pronounced a murderer, for acting as Legree did, in his circumstances. Everybody must see that such an admission would strike at the foundations of the slave system. To be sure, Tom was in a state of insurrection for conscience' sake. But the law does not, and cannot, contemplate that the negro shall have a conscience independent of his master's. To allow that the negro may refuse to obey his master whenever he thinks that obedience would be wrong, would be to produce universal anarchy. If Tom had been allowed to disobey his master in this case, for conscience' sake, the next day Sambo would have had a case of conscience, and Quimbo the next. Several of them might very justly have thought that it was a sin to work as they did. The mulatto woman would have remembered that the command of God forbade her to take another husband. Mothers might have considered that it was more their duty to stay at home and take care of their children, when they were young and feeble, than to work for Mr. Legree in the cotton-field. There would be no end to the havoc made upon cotton-growing operations, were the negro allowed the right of maintaining his own conscience on moral subjects. If the slave system is a right system, and ought to be maintained, Mr. Legree ought not to be blamed for his conduct in this case; for he did only what was absolutely essential to maintain the system; and Tom died in fanatical and fool-hardy resistance to "the powers that be, which are ordained of God." He followed a sentimental impulse of his desperately depraved heart, and neglected those "solid teachings of the written word," which, as recently elucidated, have proved so refreshing to eminent political men.
THE TRIUMPH OF JUSTICE OVER LAW.
Having been obliged to record so many trials in which justice has been turned away backward by the hand of law, and equity and common humanity have been kept out by the bolt and bar of logic, it is a relief to the mind to find one recent trial recorded, in North Carolina, in which the nobler feelings of the human heart have burst over formalized limits, and where the prosecution appears to have been conducted by men, who were not ashamed of possessing in their bosoms that very dangerous and most illogical agitator, a human heart. It is true that, in giving this trial, very sorrowful, but inevitable, inferences will force themselves upon the mind, as to that state of public feeling which allowed such outrages to be perpetrated in open daylight, in the capital of North Carolina, upon a hapless woman. It would seem that the public were too truly instructed in the awful doctrine pronounced by Judge Ruffin, that "the power of the master must be absolute," to think of interfering while the poor creature was dragged, barefoot and bleeding, at a horse's neck, at the rate of five miles an hour, through the streets of Raleigh. It seems, also, that the most horrible brutalities and enormities that could be conceived of were witnessed, without any efficient interference, by a number of the citizens, among whom we see the name of the Hon. W. H. Haywood, of Raleigh. It is a comfort to find the attorney-general, in this case, speaking as a man ought to speak. Certainly there can be no occasion to wish to pervert or overstate the dread workings of the slave system, or to leave out the few comforting and encouraging features, however small the encouragement of them may be.
The case is now presented, as narrated from the published reports, by Dr. Bailey, editor of the National Era; a man whose candor and fairness need no indorsing, as every line that he writes speaks for itself.
The reader may at first be surprised to find slave testimony in the court, till he recollects that it is a slave that is on trial, the testimony of slaves being only null when it concerns whites.
AN INTERESTING TRIAL.
We find in one of the Raleigh (North Carolina) papers, of June 5, 1851, a report of an interesting trial, at the spring term of the Superior Court. Mima, a slave, was indicted for the murder of her master, William Smith, of Johnston County, on the night of the 29th of November, 1850. The evidence for the prosecution was Sidney, a slave-boy, twelve years old, who testified that, in the night, he and a slave-girl, named Jane, were roused from sleep by the call of their master, Smith, who had returned home. They went out, and found Mima tied to his horse's neck, with two ropes, one round her neck, the other round her hands. Deceased carried her into the house, jerking the rope fastened to her neck, and tied her to a post. He called for something to eat, threw her a piece of bread, and, after he had done, beat her on her naked back with a large piece of light-wood, giving her many hard blows. In a short time, deceased went out of the house, for a special purpose, witness accompanying him with a torch-light, and hearing him say that he intended "to use the prisoner up." The light was extinguished, and he reëntered the house for the purpose of lighting it. Jane was there; but the prisoner had been untied, and was not there. While lighting his torch, he heard blows outside, and heard the deceased cry out, two or three times, "O, Leah! O, Leah!" Witness and Jane went out, saw the deceased bloody and struggling, were frightened, ran back, and shut themselves up. Leah, it seems, was mother of the prisoner, and had run off two years, on account of cruel treatment by the deceased.
Smith was speechless and unconscious till he died, the following morning, of the wounds inflicted on him.
It was proved on the trial that Carroll, a white man, living about a mile from the house of the deceased, and whose wife was said to be the illegitimate daughter of Smith, had in his possession, the morning of the murder, the receipt given the deceased by sheriff High, the day before, for jail fees, and a note for thirty-five dollars, due deceased from one Wiley Price, which Carroll collected a short time thereafter; also the chest-keys of the deceased; and no proof was offered to show how Carroll came into possession of these articles.
The following portion of the testimony discloses facts so horrible, and so disgraceful to the people who tolerated, in broad daylight, conduct which would have shamed the devil, that we copy it just as we find it in the Raleigh paper. The scene, remember, is the city of Raleigh.
"The defence was then opened. James Harris, C. W. D. Hutchings, and Hon. W. H. Haywood, of Raleigh; John Cooper, of Wake; Joseph Hane and others, of Johnston, were examined for the prisoner. The substance of their testimony was as follows: On the forenoon of Friday, 29th of November last, deceased took prisoner from Raleigh jail, tied her round the neck and wrist; ropes were then latched to the horse's neck; he cursed the prisoner several times, got on his horse, and started off; when he got opposite the Telegraph office, on Fayetteville-street, he pulled her shoes and stockings off, cursed her again, went off in a swift trot, the prisoner running after him, doing apparently all she could to keep up; passed round by Peck's store; prisoner seemed very humble and submissive; took down the street east of the capitol, going at the rate of five miles an hour; continued this gait until he passed O. Rork's corner, about half or three-quarters of a mile from the capitol; that he reached Cooper's (one of the witnesses), thirteen miles from Raleigh, about four o'clock, P. M.; that it was raining very hard; deceased got off his horse, turned it loose with prisoner tied to its neck; witness went to take deceased's horse to stable; heard great lamentations at the house; hurried back; saw his little daughter running through the rain from the house, much frightened; got there; deceased was gouging prisoner in the eyes, and she making outcries; made him stop; became vexed, and insisted upon leaving; did leave in a short time, in the rain, sun about an hour high; when he left, prisoner was tied as she was before; her arms and fingers were very much swollen; the rope around her wrist was small, and had sunk deep into the flesh, almost covered with it; that around the neck was large, and tied in a slipknot; deceased would jerk it every now and then; when jerked, it would choke prisoner; she was barefoot and bleeding; deceased was met some time after dark, in about six miles of home, being twenty-four or twenty-five from Raleigh."
Why did they not strike the monster to the earth, and punish him for his infernal brutality?
The attorney-general conducted the prosecution with evident loathing. The defence argued, first, that the evidence was insufficient to fasten the crime upon the prisoner; secondly, that, should the jury be satisfied beyond a rational doubt that the prisoner committed the act charged, it would yet be only manslaughter.
"A single blow between equals would mitigate a killing instanter from murder to manslaughter. It could not, in law, be anything more, if done under the furor brevis of passion. But the rule was different as between master and slave. It was necessary that this should be, to preserve the subordination of the slave. The prisoner's counsel then examined the authorities at length, and contended that the prisoner's case came within the rule laid down in The State v. Will (1 Dev. and Bat. 121). The rule there given by Judge Gaston is this: 'If a slave, in defence of his life, and under circumstances strongly calculated to excite his passions of terror and resentment, kill his overseer or master, the homicide is, by such circumstances, mitigated to manslaughter.' The cruelties of the deceased to the prisoner were grievous and long-continued. They would have shocked a barbarian. The savage loves and thirsts for blood; but the acts of civilized life have not afforded him such refinement of torture as was here exhibited."
The attorney-general, after discussing the law, appealed to the jury "not to suffer the prejudice which the counsel for the defence had attempted to create against the deceased (whose conduct, he admitted, was disgraceful to human nature) to influence their judgments in deciding whether the act of the prisoner was criminal or not, and what degree of criminality attached to it. He desired the prisoner to have a fair and impartial trial. He wished her to receive the benefit of every rational doubt. It was her right, however humble her condition; he hoped he had not that heart, as he certainly had not the right by virtue of his office, to ask in her case for anything more than he would ask from the highest and proudest of the land on trial, that the jury should decide according to the evidence, and vindicate the violated law."
These were honorable sentiments.
After an able charge by Judge Ellis, the jury retired, and, after having remained, out several hours, returned with a verdict of Not Guilty. Of course, we see not how they could hesitate to come to this verdict at once.
The correspondent who furnishes the Register with a report of the case says:
"It excited an intense interest in the community in which it occurred, and, although it develops a series of cruelties shocking to human nature, the result of the trial, nevertheless, vindicates the benignity and justice of our laws towards that class of our population whose condition Northern fanaticism has so carefully and grossly misrepresented, for their own purposes of selfishness, agitation, and crime."
We have no disposition to misrepresent the condition of the slaves, or to disparage the laws of North Carolina; but we ask, with a sincere desire to know the truth, Do the laws of North Carolina allow a master to practise such horrible cruelties upon his slaves as Smith was guilty of, and would the public sentiment of the city of Raleigh permit a repetition of such enormities as were perpetrated in its streets, in the light of day, by that miscreant?
In conclusion, as the accounts of these various trials contain so many shocking incidents and particulars the author desires to enter a caution against certain mistaken uses which may be made of them, by well-intending persons. The crimes themselves, which form the foundation of the trials, are not to be considered and spoken of as specimens of the common working of the slave system. They are, it is true, the logical and legitimate fruits of a system which makes every individual owner an irresponsible despot. But the actual number of them, compared with the whole number of masters, we take pleasure in saying, is small. It is an injury to the cause of freedom to ground the argument against slavery upon the frequency with which such scenes as these occur. It misleads the popular mind as to the real issue of the subject. To hear many men talk, one would think that they supposed that unless negroes actually were whipped or burned alive at the rate of two or three dozen a week, there was no harm in slavery. They seem to see nothing in the system but its gross bodily abuses. If these are absent, they think there is no harm in it. They do not consider that the twelve hours' torture of some poor victim, bleeding away his life, drop by drop, under the hands of a Souther, is only a symbol of that more atrocious process by which the divine, immortal soul is mangled, burned, lacerated, thrown down, stamped upon, and suffocated, by the fiend-like force of the tyrant Slavery. And as, when the torturing work was done, and the poor soul flew up to the judgment-seat, to stand there in awful witness, there was not a vestige of humanity left in that dishonored body, nor anything by which it could be said, "See, this was a man!"—so, when Slavery has finished her legitimate work upon the soul, and trodden out every spark of manliness, and honor, and self-respect, and natural affection, and conscience, and religious sentiment, then there is nothing left in the soul, by which to say, "This was a man!" and it becomes necessary for judges to construct grave legal arguments to prove that the slave is a human being.
Such extreme cases of bodily abuse from the despotic power of slavery are comparatively rare. Perhaps they may be paralleled by cases brought to light in the criminal jurisprudence of other countries. They might, perhaps, have happened anywhere"; at any rate, we will concede that they might. But where under the sun did such trials, of such cases, ever take place, in any nation professing to be free and Christian? The reader of English history will perhaps recur to the trials under Judge Jeffries, as a parallel. A moment's reflection will convince him that there is no parallel between the cases. The decisions of Jeffries were the decisions of a monster, who violently wrested law from its legitimate course, to gratify his own fiendish nature. The decisions of American slave-law have been, for the most part, the decisions of honorable and humane men, who have wrested from their natural course the most humane feelings, to fulfil the mandates of a cruel law.
In the case of Jeffries, the sacred forms of the administration of justice were violated. In the case of the American decisions, every form has been maintained. Revolting to humanity as these decisions appear, they are strictly logical and legal. Therefore, again, we say, Where, ever, in any nation professing to be civilized and Christian, did such trials, of such cases, take place? When were ever such legal arguments made? When, ever, such legal principles judicially affirmed? Was ever such a trial held in England as that in Virginia, of Souther v. The Commonwealth? Was it ever necessary in England for a judge to declare on the bench, contrary to the opinion of a lower court, that the death of an apprentice, by twelve hours' torture from his master, did amount to murder in the first degree? Was such a decision, if given, accompanied by the affirmation of the principle, that any amount of torture inflicted by the master, short of the point of death, was not indictable? Not being read in English law, the writer cannot say; but there is strong impression from within that such a decision as this would have shaken the whole island of Great Britain; and that such a case as Souther v. The Commonwealth would never have been forgotten under the sun. Yet it is probable that very few persons in the United States ever heard of the case, or ever would have heard of it, had it not been quoted by the New York Courier and Enquirer as an overwhelming example of legal humanity.
The horror of the whole matter is, that more than one such case should ever need to happen in a country, in order to make the whole community feel, as one man, that such power ought not to be left in the hands of a master. How many such cases do people wish to have happen?—how many must happen, before they will learn that utter despotic power is not to be trusted in any hands? If one white man's son or brother had been treated in this way, under the law of apprenticeship, the whole country would have trembled, from Louisiana to Maine, till that law had been altered. They forget that the black man has also a father. It is "He that sitteth upon the circle of the heavens, who bringeth the princes to nothing, and maketh the judges of the earth as vanity." He hath said that "When he maketh inquisition for blood, he forgetteth not the cry of the humble." That blood which has fallen so despised to the earth,—that blood which lawyers have quibbled over, in the quiet of legal nonchalance, discussing in great ease whether it fell by murder in the first or second degree,—HE will one day reckon for as the blood of his own child. He "is not slack concerning his promises, as some men count slackness, but is long-suffering to usward;" but the day of vengeance is surely coming, and the year of his redeemed is in his heart.
Another court will sit upon these trials, when the Son of Man shall come in his glory. It will be not alone Souther, and such as he, that will be arraigned there; but all those in this nation, north and south, who have abetted the system, and made the laws which made Souther what he was. In that court negro testimony will be received, if never before; and the judges and the counsellors, and the chief men, and the mighty men, marshalled to that awful bar, will say to the mountains and the rocks. "Fall on us and hide us from the face of Him that sitteth on the throne, and from the wrath of the Lamb."
The wrath of the Lamb! Think of it! Think that Jesus Christ has been present, a witness,—a silent witness through every such scene of torture and anguish,—a silent witness in every such court, calmly hearing the evidence given in, the lawyers pleading, the bills filed, and cases appealed! And think what a heart Jesus Christ has, and with what age-long patience he has suffered! What awful depths are there in that word, long-suffering! and what must be that wrath, when, after ages of endurance, this dread accumulation of wrong and anguish comes up at last to judgment!
A COMPARISON OF THE ROMAN LAW Of SLAVERY WITH THE AMERICAN.
The writer has expressed the opinion that the American law of slavery, taken throughout, is a more severe one than that of any other civilized nation, ancient or modern, if we except, perhaps, that of the Spartans. She has not at hand the means of comparing French and Spanish slave-codes; but, as it is a common remark that Roman slavery was much more severe than any that has ever existed in America, it will be well to compare the Roman with the American law. We therefore present a description of the Roman slave-law, as quoted by William Jay, Esq., from Blair's "Inquiry into the State of Slavery among the Romans," giving such references to American authorities as will enable the reader to make his own comparison, and to draw his own inferences.
I. The slave had no protection against the avarice, rage, or lust of the master, whose authority was founded in absolute property; and the bondman was viewed less as a human being subject to arbitrary dominion, than as an inferior animal, dependent wholly on the will of his owner.
See law of South Carolina, in Stroud's "Sketch of the Laws of Slavery," p. 23.
2 Brev. Dig. 229. Prince's
Dig. 446. Cobb's Dig. 971.
Lou. Civil Code, art. 35.
Stroud's Sketch, p. 23. Slaves shall be deemed, sold, taken, reputed and adjudged in law to be chattels personal in the hands of their owners and possessors, and their executors, administrators and assigns, to all intents, constructions, and purposes whatever.
A slave is one who is in the power of a master to whom he belongs Judge Ruffin's Decision in
the case of The State v.
Manu. Wheeler's
Law of Slavery, 246
——— Such obedience is the consequence only of uncontrolled authority over the body. There is nothing else which can operate to produce the effect. The power of the master must be absolute, to render the submission of the slave perfect.
II. At first, the master possessed the uncontrolled power of life and death.
Judge Clarke, in case of State of Miss
v. Jones. Wheeler, 252 At a very early period in Virginia, the power of life over slaves was given by statute.
III. He might kill, mutilate or torture his slaves, for any or no offence; he might force them to become gladiators or prostitutes.
The privilege of killing is now somewhat abridged; as to mutilation and torture, see the case of Souther v. The Commonwealth, 7 Grattan, 673, quoted in Chapter III., above. Also State v. Mann, in the same chapter, from Wheeler, p. 244.
IV. The temporary unions of male with female slaves were formed and dissolved at his command; families and friends were separated when he pleased.
See the decision of Judge Mathews in the case of Girod v. Lewis, Wheeler, 199:
It is clear, that slaves have no legal capacity to assent to any contract. With the consent of their master, they may marry, and their moral power to agree to such a contract or connection as that of marriage cannot be doubted; but whilst in a state of slavery it cannot produce any civil effect, because slaves are deprived of all civil rights.
See also the chapter below on "the separation of families," and the files of any southern newspaper, passim.
V. The laws recognized no obligation upon the owners of slaves, to furnish them with food and clothing, or to take care of them in sickness.
The extent to which this deficiency in the Roman law has been supplied in the American, by "protective acts", has been exhibited above.[11]
VI. Slaves could have no property but by the sufferance of their master, for whom they acquired everything, and with whom they could form no engagements which could be binding on him.
The following chapter will show how far, American legislation is in advance of that of the Romans, in that it makes it a penal offence on the part of the master to permit his slave to hold property, and a crime on the part of the slave to be so permitted. For the present purpose, we give an extract from the Civil code of Louisiana, as quoted by Judge Stroud:
Civil Code
Article 35.
Stroud, p. 22 A slave is one who is in the power of a master to whom he belongs. The master may sell him, dispose of his person, his industry, and his labor; he can do nothing, possess nothing, nor acquire anything but what must belong to his master.
Wheeler's Law
of Slavery, p.
246 State v.
Manu. According to Judge Ruffin, a slave is "one doomed in his own person, and his posterity, to live without knowledge, and without the capacity to make anything his own, and to toil that another may reap the fruits."
With reference to the binding power of engagements between master and slave, the following decisions from the United States Digest are in point (7, p. 449):
Gist v. Toehey,
2 Rich
424 All the acquisitions of the slave in possession are the property of his master, notwithstanding the promise of his master that the slave shall have certain of them.
A slave paid money which he had earned over and above his wages, for the purchase of his children into the hands of B, and B purchased Ibid, such children with the money. Held that the master of such slave was entitled to recover the money of B.
VII. The master might transfer his rights by either sale or gift, or might bequeath them by will.
Law of S. Carolina.
Cobb's
Slaves shall be deemed, sold, taken, reputed and adjudged in law, to be chattels personal in the hands of their owners and possessors, and their executors, administrators, and assigns, to all intents, constructions, and purposes whatsoever.
VIII. A master selling, giving, or bequeathing a slave, sometimes made it a provision that he should never be carried abroad, or that he should be manumitted on a fixed day; or that, on the other hand, he should never be emancipated, or that he should be kept in chains for life.
Williams v. Ash,
1 How. U. S.
Rep. 1. 5 U. S.
Dig. 792, § 5
We hardly think that a provision that a slave should never be emancipated, or that he should be kept in chains for life, would be sustained. A provision that the slave should not be carried out of the state, or sold, and that on the happening of either event he should be free, has been sustained.
The remainder of Blair's account of Roman slavery is devoted rather to the practices of masters than the state of the law itself. Surely, the writer is not called upon to exhibit in the society of enlightened, republican and Christian America, in the nineteenth century, a parallel to the atrocities committed in pagan Rome, under the sceptre of the persecuting Cæsars, when the amphitheatre was the favorite resort of the most refined of her citizens, as well as the great "school of morals" for the multitude. A few references only will show, as far as we desire to show, how much safer it is now to trust man with absolute power over his fellow, than it was then.
IX. While slaves turned the handmill they were generally chained, and had a broad wooden collar, to prevent them from eating the grain. The furca, which in later language means a gibbet, was, in older dialect, used to denote a wooden fork or collar, which was made to bear upon their shoulders, or around their necks, as a mark of disgrace, as much as an uneasy burden.
The reader has already seen, in Chapter V., that this instrument of degradation has been in use, in our own day, in certain of the slave states, under the express sanction and protection of statute laws; although the material is different, and the construction doubtless improved by modern ingenuity.
X. Fetters and chains were much used for punishment or restraint, and were, in some instances, worn by slaves during life, through the sole authority of the master. Porters at the gates of the rich were generally chained. Field laborers worked for the most part in irons posterior to the first ages of the republic.
The Legislature of South Carolina specially sanctions the same practices, by excepting them in the "protective enactment," which inflicts the penalty of one hundred pounds "in case any person shall wilfully cut out the tongue," &c., of a slave, "or shall inflict any other cruel punishment, other than by whipping or beating with a horse-whip, cowskin, switch, or small stick, or by putting irons on, or confining or imprisoning such slave."
XI. Some persons made it their business to catch runaway slaves.
That such a profession, constituted by the highest legislative authority in the nation, and rendered respectable by the commendation expressed or implied of statesmen and divines, and of newspapers political and religious, exists in our midst, especially in the free states, is a fact which is, day by day, making itself too apparent to need testimony. The matter seems, however, to be managed in a more perfectly open and business-like manner in the State of Alabama than elsewhere. Mr. Jay cites the following advertisement from the Sumpter County (Ala.) Whig:
The undersigned having bought the entire pack of Negro Dogs (of the Hay and Allen stock), he now proposes to catch runaway negroes. His charges will be Three Dollars per day for hunting, and Fifteen Dollars for catching a runaway. He resides three and one half miles north of Livingston, near the lower Jones' Bluff road.
William Gambel.
Nov. 6, 1845.—6m.
The following is copied, ''verbatim et literatim, and with the pictorial embellishments, from The Dadeville (Ala.) Banner, of November 10th, 1852. The Dadeville Banner is "devoted to politics, literature, education, agriculture, ^c."
For going over ten miles and catching slaves, 20.00
If sent for, the above prices will be exacted in cash. The subscriber resides one mile and a half south of Dadeville, Ala.
B. Black.
Dadeville, Sept. 1, 1852. 1tf
XII. The runaway, when taken, was severely punished by authority of the master, or by the judge, at his desire; sometimes with crucifixion, amputation of a foot, or by being sent to fight as a gladiator with wild beasts; but most frequently by being branded on the brow with letters indicative of his crime.
That severe punishment would be the lot of the recaptured runaway, every one would suppose, from the "absolute power" of the master to inflict it. That it is inflicted in many cases, it is equally easy and needless to prove. The peculiar forms of punishment mentioned above are now very much out of vogue, but the following advertisement by Mr. Micajah Ricks, in the Raleigh (N. C.) Standard of July 18th, 1838, shows that something of classic taste in torture still lingers in our degenerate days.
Ran away, a negro woman and two children; a few days before she went off, I burnt her with a hot iron, on the left side of her face. I tried to make the letter M.
It is charming to notice the naïf betrayal of literary pride on the part of Mr. Ricks. He did not wish that letter M to be taken as a specimen of what he could do in the way of writing. The creature would not hold still, and he fears the M may be ilegible.
The above is only one of a long list of advertisements of maimed, cropped and branded negroes, in the book of Mr. Weld, entitled American Slavery as It Is, p. 77.
XIII. Cruel masters sometimes hired torturers by profession, or had such persons in their establishments, to assist them in punishing their slaves. The noses and ears and teeth of slaves were often in danger from an enraged owner; and sometimes the eyes of a great offender were put out. Crucifixion was very frequently made the fate of a wretched slave for a trifling misconduct, or from mere caprice.
For justification of such practices as these, we refer again to that horrible list of maimed and mutilated men, advertised by slaveholders themselves, in Weld's American Slavery as It Is, p. 77. We recall the reader's attention to the evidence of the monster Kephart, given in Part I. As to crucifixion, we presume that there are wretches whose religious scruples would deter them from this particular form of torture, who would not hesitate to inflict equal cruelties by other means; as the Greek pirate, during a massacre in the season of Lent, was conscience-stricken at having tasted a drop of blood. We presume?—Let any one but read again, if he can, the sickening details of that twelve hours' torture of Souther's slave, and say how much more merciful is American slavery than Roman.
The last item in Blair's description of Roman slavery is the following:
By a decree passed by the Senate, if a master was murdered when his slaves might possibly have aided him, all his household within reach were held as implicated, and deserving of death; and Tacitus relates an instance in which a family of four hundred were all executed.
To this alone, of all the atrocities of the slavery of old heathen Rome, do we fail to find a parallel in the slavery of the United States of America.
There are other respects, in which American legislation has reached a refinement in tyranny of which the despots of those early days never conceived. The following is the language of Gibbon:
Hope, the best comfort of our imperfect condition, was not denied to the Roman slave; and if he had any opportunity of rendering himself either useful or agreeable, he might very naturally expect that the diligence and fidelity of a few years would be rewarded with the inestimable gift of freedom. * * * Without destroying the distinction of ranks, a distant prospect of freedom and honors was presented even to those whom pride and prejudice almost disdained to number among the human species.[12]
The youths of promising genius were instructed in the arts and sciences, and their price was ascertained by the degree of their skill and talents. Almost every profession, either liberal or mechanical, might be found in the household of an opulent senator.[13]
The following chapter will show how "the best comfort" which Gibbon knew for human adversity is taken away from the American slave; how he is denied the commonest privileges of education and mental improvement, and how the whole tendency of the unhappy system, under which he is in bondage, is to take from him the consolations of religion itself, and to degrade him from our common humanity, and common brotherhood with the Son of God.
THE MEN BETTER THAN THEIR LAWS.
Judgment is turned away backward.
And Justice standeth afar off;
For Truth is fallen in the street,
And Equity cannot enter.
Yea, Truth faileth;
And he that departeth from evil maketh himself a prey.Isaiah 59: 14, 15.
There is one very remarkable class of laws yet to be considered. So full of cruelty and of unmerciful severity is the slave-code,—such an atrocity is the institution of which it is the legal definition,—that there are multitudes of individuals too generous and too just to be willing to go to the full extent of its restrictions and deprivations.
A generous man, instead of regarding the poor slave as a piece of property, dead, and void of rights, is tempted to regard him rather as a helpless younger brother, or as a defenceless child, and to extend to him, by his own good right arm, that protection and those rights which the law denies him. A religious man, who, by the theory of his belief, regards all men as brothers, and considers his Christian slave, with himself, as a member of Jesus Christ,—as of one body, one spirit, and called in one hope of his calling,—cannot willingly see him "doomed to live without knowledge," without the power of reading the written Word, and to raise up his children after him in the same darkness.
Hence, if left to itself, individual humanity would, in many cases, practically abrogate the slave-code. Individual humanity would teach the slave to read and write,—would build school-houses for his children, and would, in very, very many cases, enfranchise him.
The result of all this has been foreseen. It has been foreseen that the result of education would be general intelligence; that the result of intelligence would be a knowledge of personal rights; and that an inquiry into the doctrine of personal rights would be fatal to the system. It has been foreseen, also, that the example of disinterestedness and generosity, in emancipation, might carry with it a generous contagion, until it should become universal; that the example of educated and emancipated slaves would prove a dangerous excitement to those still in bondage.
For this reason, the American slave-code, which, as we have already seen, embraces, substantially, all the barbarities of that of ancient Rome, has had added to it a set of laws more cruel than any which ancient and heathen Rome ever knew,—laws designed to shut against the slave his last refuge,—the humanity of his master. The master, in ancient Rome, might give his slave whatever advantages of education he chose, or at any time emancipate him, and the state did not interfere to prevent.[14]
But in America the laws, throughout all the slave states, most rigorously forbid, in the first place, the education of the slave. We do not profess to give all these laws, but a few striking specimens may be presented. Our authority is Judge Stroud's "Sketch of the Laws of Slavery."
The legislature of South Carolina, in 1740, enounced the following preamble:—"Whereas, Shroud's Sketch,
p. 88. the having of slaves taught to write, or suffering them to be employed in writing, may be attended with great inconveniences;" and enacted that the crime of teaching a slave to write, or of employing a slave as a scribe, should be punished by a fine of one hundred pounds, current money. If the reader will turn now to the infamous "protective" statute, enacted by the same legislature, in the same year, he will find that the same penalty has been appointed for the cutting out of the tongue, putting out of the eye,. cruel scalding, &c., of any slave, as for the offence of teaching him to write! That is to say, that to teach him to write, and to put out his eyes, are to be regarded as equally reprehensible.
That there might be no doubt of the "great and fundamental policy" of the state, and that there might be full security against the "great inconveniences" of "having of slaves taught to write," it was enacted, in 1800, "That assemblies of slaves, free negroes, &c., * * * * for the purpose of mental instruction, in a confined or secret place, &c. &c., is [are] declared to be an unlawful meeting;" and the officers are required to enter such confined places, and disperse the "unlawful assemblage," inflicting, at their discretion, Shroud's Sketch,
p. 80. 2 Brevard's
Digest, pp. 254-5. such corporal punishment, not exceeding twenty lashes, upon such slaves, free negroes,. &c., as they may judge necessary for deterring them from the like unlawful assemblage in future."
The statute-book of Virginia is adorned with a law similar to the one last quoted.
The offence of teaching a slave to write was early punished, in Georgia, as in South Carolina, by a pecuniary fine. But the city of Savannah seems to have found this penalty insufficient to protect it from great inconveniences," and we learn, by a quotation in the work of Judge Stroud from a number of "The Portfolio," that "the city has passed an ordinance, by which any person that teaches any person of color, slave or free, to read or write, or causes such person to be so taught, is subjected to a fine of thirty dollars Shroud's Sketch,
pp. 89, 90. for each offence; and every person of color who shall keep a school, to teach reading or writing, is subject to a fine of thirty dollars, or to be imprisoned ten days, and whipped thirty-nine lashes."
Secondly. In regard to religious privileges:
The State of Georgia has enacted a law, "To protect religious societies in the exercise of their religious duties." This law, after appointing rigorous penalties for the offence of interrupting or disturbing a congregation of white persons, concludes in the following words:
Shroud, p. 92.
Prince's Digest
p. 342. No congregation, or company of negroes, shall, under pretence of divine worship, assemble themselves, contrary to the act regulating patrols."
"The act regulating patrols," as quoted by the editor of Prince's Digest, empowers every justice of the peace to disperse any assembly or meeting of slaves Shroud, p. 93.
p. 447. which may disturb the peace, &c., of his majesty's subjects, and permits that every slave found at such a meeting shall "immediately be corrected, without trial, by receiving on the bare back twenty-five stripes with a whip, switch, or cowskin."
The history of legislation in South Carolina is significant. An act was passed in 1800, containing the following section:
Dig. p. 254,255. It shall not be lawful for any number of slaves, free negroes, mulattoes or mestizoes, even in company with white persons, to meet together assemble for the purpose of mental instruction or religious worship, either before the rising of the sun, or after the going down of the same. And all magistrates, sheriffs, militia officers, &c. &c., are hereby vested with power, &c., for dispersing such assemblies, &c.
The law just quoted seems somehow to have had a prejudicial effect upon the religious interests of the "slaves, free negroes," &c., specified in it; for, three years afterwards, on the petition of certain religious societies, a "protective act" was passed, which should secure them this great religious privilege; to wit, that it should be unlawful, before nine o'clock, "to break into a place of meeting, wherein shall be assembled the members of any religious society of this state, provided a majority of them shall be white persons, or otherwise to disturb their devotion, unless such person shall have first obtained * * * * a warrant, &c."
Thirdly. It appears that many masters, who are disposed to treat their slaves generously, have allowed them to accumulate property, to raise domestic animals for their own use, and, in the case of intelligent servants, to go at large, to hire their own time, and to trade upon their own account. Upon all these practices the law comes down, with unmerciful severity. A penalty is inflicted on the owner, but, with a rigor quite accordant with the tenor of slave-law the offence is considered, in law, as that of the slave, rather than that of the master; so that, if the master is generous enough not to regard the penalty which is imposed upon himself, he may be restrained by the fear of bringing a greater evil upon his dependent. These laws are, in some cases, so constructed as to make it for the interest of the lowest and most brutal part of society that they be enforced, by offering half the profits to the informer. We give the following, as specimens of slave legislation on this subject:
The law of South Carolina:
It shall not be lawful for any slave to buy, sell, trade, &c., for any goods, &c., without a license from the owner, &c.; nor shall any slave be permitted to keep any boat, periauger,[15] or canoo, or raise and breed, for the benefit of such slave, any horses, mares, cattle, sheep, or hogs, under pain of forfeiting all the goods, &c., and all the boats, periaugers, or canoes, horses, mares, cattle, sheep or hogs. And it shall be lawful Shroud, pp. 45.
47. James's Digest,
385, 386. Act of 1740 for any person whatsoever to seize and take away from any slave all such goods, &c., boats, &c. &c.. and to deliver the same into the hands of any justice of the peace, nearest to the place where the seizure shall be made; and such justice shall take the oath of the person making such seizure, concerning the manner thereof; and if the said justice shall be satisfied that such seizure has been made according to law, he shall pronounce and declare the goods so seized to be forfeited, and order the same to be sold at public outcry, one half of the moneys arising from such sale to go to the state, and the other half to him or them that sue for the same.
The laws in many other states are similar to the above; but the State of Georgia has an additional provision, against permitting the slave to hire himself to another for his own benefit; a penalty of thirty dollars is imposed for every weekly offence, on the part of the master, unless the labor be done on his own premises. Savannah, Augusta, and Sunbury, are places excepted.
Shroud, pp. 47. In Virginia, "if the master shall permit his slave to hire himself out," the slave is to be apprehended, and the master to be fined.
In an early act of the legislature of the orthodox and Presbyterian State of North Carolina, it is gratifying to see how the judicious course of public policy is made to subserve the interests of Christian charity,—how, in a single ingenious sentences, provision is made for punishing the offender against society, rewarding the patriotic informer, and feeding the poor and destitute:
All horses, cattle, hogs or sheep, that, one month after the passing of this act, shall belong to any slave, or be of any slave's mark, in this state, shall be seized and sold by the county wardens, and by them applied, the one-half to the support of the poor of the county, and the other half to the informer.
In Mississippi a fine of fifty dollars is imposed upon the master who permits his slave to cultivate cotton for his own Shroud, pp. 43. use; or who licenses his slave to go at large and trade as a freeman; or who is convicted of permitting his slave to keep "stock of any description."
To show how the above law has been interpreted by the highest judicial tribunal of the sovereign State of Mississippi, we repeat here a portion of a decision of Chief Justice Sharkey, which we have elsewhere given more in full.
Independent of the principles laid down in adjudicated cases, our statute-law prohibits slaves from owning certain kinds of property; and it may be inferred that the legislature supposed they were extending the act as far as it could be necessary to exclude them from owning any property, as the prohibition includes that kind of property which they would most likely be permitted to own without interruption, to wit: hogs, horses, cattle, &c. They cannot be prohibited from holding such property in consequence of its being of a dangerous or offensive character, but because it was deemed impolitic for them to hold property of any description.
It was asserted, at the beginning of this head, that the permission of the master to a slave to hire his own time is, by law, considered the offence of the slave; the slave being subject to prosecution therefor, not the master. This is evident from the tenor of some of the laws quoted and alluded to above. It will be still further illustrated by the following decisions of the courts of North Carolina. They are copied from the Supplement to the U. S. Digest, vol. ii. p. 798:
The State v. Clarissa.
5 Iredell, 221. 139. An indictment charging that a certain negro did hire her own time, contrary to the form of the statute, &c., is defective and must be quashed, because it was omitted to be charged that she has permitted by her master to go at large, which is one essential part of the offence.
140. Under the first clause of the thirty-first section of the 111th chapter of the Revised Statutes, prohibiting masters from hiring to slaves their own time, the master is not indictable; he is only subject to a penalty of forty dollars. Nor is the master indictable under the second clause of that section; the process being against the slave, not against the master.—Ib.
142. To constitute the offence under section 32 (Rev. Stat. c. cxi. § 32) it is not necessary that the slave should have hired his time; it is sufficient if the master permits him to go at large as a freeman.
This is maintaining the ground that "the master can do no wrong" with great consistency and thoroughness. But it is in perfect keeping, both in form and spirit, with the whole course of slave-law, which always upholds the supremacy of the master, and always depresses the slave.
Fourthly. Stringent laws against emancipation exist in nearly all the slave states.
Shroud, 47. Prince's Dig. 456. James's Dig.
398. Tenimin's Dig. 632. Miss. Rev. Code, 386. In four of the states,—South Carolina, Georgia, Alabama, and Mississippi,—emancipation can not be effected, except by a special act of the legislature of the state.
In Georgia, the offence of setting free "any slave, or slaves, in any other manner and form than the one prescribed," was punishable, according to the law of 1801, by the forfeiture of two hundred dollars, to be recovered by action or indictment; the slaves in question still remaining, "to all intents and purposes, as much in a state of slavery as before they were manumitted."
Believers in human progress will be interested to know that since the law of 1801 there has been a reform introduced into this part of the legislation of the republic of Georgia. In 1818, a new law was passed, which, as will be seen, contains a grand remedy for the abuses of the old. In this it is provided, with endless variety of specifications and synonyms, as if to "let suspicion double-lock the door" against any possible evasion, that, "All and every will, testament and deed, whether by way of trust or otherwise, contract, or agreement, or stipulation, or other instrument in writing or by parol, made and executed for the purpose of effecting, or endeavoring to effect, the manumission of any slave or slaves, either directly … or indirectly, or virtually, &c. &c., shall be, and the same are hereby, declared to be utterly null and void." And the guilty author of the outrage against the peace of the state, contemplated in such deed, &c. &c., "and all and every person or persons concerned in giving or attempting to give effect thereto, … in any way or manner whatsoever, shall be severally liable to a penalty not exceeding one thousand dollars."
It would be quite anomalous in slave-law, and contrary to the "great and fundamental policy" of slave states, if the negroes who, not having the fear of God before their eyes, but being instigated by the devil, should be guilty of being thus manumitted, were Shroud's Sketch, pp.
147—8. Prince's Dig.
466. suffered to go unpunished; accordingly, the law very properly and judiciously provides that "each and every slave or slaves in whose behalf such will or testament, &c. &c. &c., shall have been made, shall be liable to be arrested by warrant, &c.; and, being thereof convicted, &c., shall be liable to be sold as a slave or slaves by public outcry; and the proceeds of such slaves shall be appropriated, &c. &c."
Judge Stroud gives the following account of the law of Mississippi:
Shroud's Sketch, 149.
Miss. Rev. Code, 385
—6 (Act June 18, 1822.) The emancipation must be by an instrument in writing, a last will or deed, &c., under seal, attested by at least two credible witnesses, or acknowledged in the court of the county or corporation where the emancipator resides; proof satisfactory to the General Assembly must be adduced that the slave has done some meritorious act for the benefit of his master, or rendered some distinguished service to the state; all which circumstances are but pre-requisites, and are of no efficacy until a special act of assenbly sanctions the emancipation; to which may be added, as has been already stated, a saving of the rights of creditors, and the protection of the widow's thirds.
The same pré-requisite of meritorious services, to be adjudged of and allowed by the county court," is exacted by an act of the General Assembly of North Carolina; and all slaves emancipated contrary to the provisions of this act are to be committed to the jail of the county, and at the next court held for that county are to be sold to the highest bidder.
But the law of North Carolina does not refuse opportunity for repentance, even after the crime has been proved: accordingly.
Shroud's Sketch,148. Hayward's
Manual, 525, 526, 529, 537. The sheriff is directed, five days before the time for the sale of the emancipated negro, to give notice, in writing, to the person by whom the emancipation was made, to the end,
and with the hope that, smitten by remorse of conscience, and brought to a sense of his guilt before God and man,
such person may, if he thinks proper, renew his claim to the negro so emancipated by him; on failure to do which, the sale is to be made by the sheriff, and one-fifth part of the net proceeds is to become the property of the freeholder by whom the apprehension was made, and the remaining four-fifths are to be paid into the public treasury.
Shroud, pp.
148-154. It is proper to add that we have given examples of the laws of states whose legislation on this subject has been most severe. The laws of Virginia, Maryland, Missouri, Kentucky and Louisiana, are much less stringent.
A striking case, which shows how inexorably the law contends with the kind designs of the master, is on record in the reports of legal decisions in the State of Mississippi. The circumstances of the case have been thus briefly stated in the New York Evening Post, edited by Mr. William Cullen Bryant. They are a romance of themselves.
A man of the name of Elisha Brazealle, a planter in Jefferson County, Mississippi, was attacked with a loathsome disease. During his illness he was faithfully nursed by a mulatto slave, to whose assiduous attentions he felt that he owed his life. He was duly impressed by her devotion, and soon after his recovery took her to Ohio, and had her educated. She was very intelligent, and improved her advantages so rapidly that when he visited her again he determined to marry her. He executed a deed for her emancipation, and had it recorded both in the States of Ohio and Mississippi, and made her his wife.
Mr. Brazealle returned with her to Mississippi, and in process of time had a son. After a few years he sickened and died, leaving a will, in which, after reciting the deed of emancipation, he declared his intention to ratify it, and devised all his property to this lad, acknowledging him in the will to be such.
Some poor and distant relations in North Carolina, whom he did not know, and for whom he did not care, hearing of his death, came on to Mississippi, and claimed the property thus devised. They instituted a suit for its recovery, and the case (it is reported in Howard's Mississippi Reports, vol. ii., p. 837) came before Judge Sharkey, our new consul at Havana. He decided it, and in that decision declared the act of emancipation an offence against morality, and pernicious and detestable as an example. He set aside the will, gave the property of Brazealle to his distant relations, condemned Brazealle's son, and his wife, that son's mother, again to bondage, and made them the slaves of these North Carolina kinsmen, as part of the assets of the estate.
Chief Justice Sharkey, after narrating the circumstances of the case, declares the validity of the deed of emancipation to be the main question in the controversy. He then argues that, although according to principles of national comity "contracts are to be construed according to the laws of the country or state where they are made," yet these principles are not to be followed when they lead to conclusions in conflict with "the great and fundamental policy of the state." What this "great and fundamental policy" is, in Mississippi, may be gathered from the remainder of the decision, which, we give in full.
Let us apply these principles to the deed of emancipation. To give it validity would be, in the first place, a violation of the declared policy, and contrary to a positiye law of the state. The policy of a state is indicated by the general course of legislation on a given subject; and we find that free negroes are deemed offensive, because they are not permitted to emigrate to or remain in the state. They are allowed few privileges, and subject to heavy penalties for offences. They are required to leave the state within thirty days after notice, and in the mean time give security for good behavior; and those of them who can lawfully remain must register and carry with them their certificates, or they may be committal to jail. It would also violate a positive law, passed by the legislature, expressly to maintain this settled policy, and to prevent emancipation, No owner can emancipate his slave, but by a deed or will properly attested, or acknowledged in court, and proof to the legislature that such slave has performed some meritorious act for the benefit of the master, or some distinguished service for the state; and the deed or will can have no validity until ratified by special act of legislature. It is believed that this law and policy are too essentially important to the interests of our citizens to permit them to be evaded.
The state of the case shows conclusively that the contract had its origin in an offence against morality, pernicious and detestable as an example. But, above all, it seems to have been planned and executed with a fixed design to evade the rigor of the laws of this state. The acts of the party in going to Ohio with the slaves, and there executing the deed, and his immediate return with them to this state, point with unerring certainty to his purpose and object. The laws of this state cannot be thus defrauded of their operation by one of our own citizens. If we could have any doubts about the principle, the case reported in 1 Randolph, 15, would remove them.
As we think the validity of the deed must depend upon the laws of this state, it becomes unnecessary to inquire whether it could have any force by the laws of Ohio. If it were even valid there, it can have no force here. The consequence is, that the negroes, John Monroe and his mother, are still slaves, and a part of the estate of Elisha Brazoalle. They have not acquired a right to their freedom under the will; for, even if the clause in the will were sufficient for that purpose, their emancipation has not been consummated by an act of the legislature.
John Monroe, being a slave, cannot take the property as devisee; and I apprehend it is equally clear that it cannot be held in trust for him. 4 Desans. Rep. 266. Independent of the principles laid down in adjudicated cases, our statute law prohibits slaves from owning certain kinds of property; and it may be inferred that the legisature supposed they were extending the act as far as it could be necessary to exclude them from owning any property, as the prohibition includes that kind of property which they, would most likely be permitted to own without interruption, to wit, hogs, horses, cattle, &c. They cannot be prohibited from holding such property in consequence of its being of a dangerous or offensive character, but because it was deemed impolitic for them to hold property of any description. It follows, therefore, that his heirs are entitled to the property.
As the deed was void, and the devisee could not take under the will, the heirs might, perhaps, have had a remedy at law; but, as an account must be taken for the rents and profits, and for the final settlement of the estate, I see no good reason why they should be sent back to law. The remedy is, doubtless, more full and complete than it could be at law. The decree of the chancellor overruling the demurrer must be affirmed, and the cause remanded for further proceedings.
The Chief Justice Sharkey who pronounced this decision is stated by the Evening Post to have been a principal agent in the passage of the severe law under which this horrible inhumanity was perpetrated.
Nothing more forcibly shows the absolute despotism of the slave-law over all the kindest feelings and intentions of the master, and the determination of courts to carry these severities to their full lengths, than this cruel deed, which precipitated a young man who had been educated to consider himself free, and his mother, an educated woman, back into the bottomless abyss of slavery. Had this case been chosen for the theme of a novel, or a tragedy, the world would have cried out upon it as a plot of monstrous improbability. As it stands in the law-book, it is only a specimen of that awful kind of truth, stranger than fiction, which is all the time evolving, in one form or another, from the workings of this anomalous system.
This view of the subject is a very important one, and ought to be earnestly and gravely pondered by those in foreign countries, who are too apt to fasten their condemnation and opprobrium rather on the person of the slave-holder than on the horrors of the legal system. In some slave states, it seems as if there was very little that the benevolent owner could do which should permanently benefit his slave, unless he should seek to alter the laws. Here it is that the highest obligation of the Southern Christian lies. Nor will the world or God hold them guiltless who, with the elective franchise in their hands, and the full power to speak, write and discuss, suffer this monstrous system of legalized cruelty to go on from age to age.
THE HEBREW SLAVE-LAW COMPARED WITH THE AMERICAN SLAVE-LAW.
Having compared the American law with the Roman, we will now compare it with one other code of slave-laws, to wit, the Hebrew.
This comparison is the more important, because American slavery has been defended on the ground of God's permitting Hebrew slavery.
The inquiry now arises, What kind of slavery was it that was permitted among the Hebrews? for in different nations very different systems have been called by the general name of slavery.
That the patriarchal state of servitude which existed in the time of Abraham was a very different thing from American slavery, a few graphic incidents in the scripture narrative show; for we read that when the angels came to visit Abraham, although he had three hundred servants born in his house, it is said that Abraham hasted, and took a calf, and killed it, and gave it to a young man to dress; and that he told Sarah to take three measures of meal and knead it into cakes; and that, when all was done, he himself set it before his guests.
From various other incidents which appear in the patriarchal narrative, it would seem that these servants bore more the relation of the members of a Scotch clan to their feudal lord than that of an American slave to his master;—thus it seems that if Abraham had died without children, his head servant would have been his heir.—Gen. 15: 3.
Of what species, then, was the slavery which God permitted among the Hebrews? By what laws was it regulated?
In the New Testament the whole Hebrew system of administration is spoken of as a relatively imperfect one, and as superseded by the Christian dispensation.—Heb. 8:13.
We are taught thus to regard the Hebrew system as an educational system, by which a debased, half-civilized race, which had been degraded by slavery in its worst form among the Egyptians, was gradually elevated to refinement and humanity.
As they went from the land of Egypt, it would appear that the most disgusting personal habits, the most unheard-of and unnatural impurities, prevailed among them; so that it was necessary to make laws with relation to things of which Christianity has banished the very name from the earth.
Beside all this, polygamy, war and slavery, were the universal custom of nations.
It is represented in the New Testament that God, in educating this people, proceeded in the same gradual manner in which a wise father would proceed with a family of children.
He selected a few of the most vital points of evil practice, and forbade them by positive statute, under rigorous penalties.
The worship of any other god was, by the Jewish law, constituted high treason, and rigorously punished with death.
As the knowledge of the true God and religious instruction could not then, as now, be afforded by printing and books, one day in the week had to be set apart for preserving in the minds of the people a sense of His being, and their obligations to Him. The devoting of this day to any other purpose was also punished with death; and the reason is obvious, that its sacredness was the principal means relied on for preserving the allegiance of the nation to their king and God, and its desecration, of course, led directly to high treason against the head of the state.
With regard to many other practices which prevailed among the Jews, as among other heathen nations, we find the Divine Being taking the same course which wise human legislators have taken.
When Lycurgus wished to banish money and its attendant luxuries from Sparta, he did not forbid it by direct statute-law, but he instituted a currency so clumsy and uncomfortable that, as we are informed by Rollin, it took a cart and pair of oxen to carry home the price of a very moderate estate.
In the same manner the Divine Being surrounded the customs of polygamy, war, blood-revenge and slavery, with regulations which gradually and certainly tended to abolish them entirely.
No one would pretend that the laws which God established in relation to polygamy, cities of refuge, &c., have any application to Christian nations now.
The following summary of some of these laws of the Mosaic code is given by Dr. C E. Stowe, Professor of Biblical Literature in Andover Theological Seminary:
1. It commanded a Hebrew, even though a married man, with wife and children living, to take the childless widow of a deceased brother, and beget children with her.—Deut. 25: 5—10.
2. The Hebrews, under certain restrictions, were allowed to make concubines, or wives for a limited time, of women taken in war.—Deut. 21: 10—19.
3. A Hebrew who already had a wife was allowed to take another also, provided he still continued his intercourse with the first as her husband, and treated her kindly and affectionately.—Exodus 21: 9—11.
4. By the Mosaic law, the nearest relative of a murdered Hebrew could pursue and slay the murderer, unless he could escape to the city of refuge; and the same permission was given in case of accidental homicide.—Num. 35: 9—30.
5. The Israelites were commanded to exterminate the Canaanites, men, women and children.—Deut. 9: 12; 20: 16—18.
Any one, or all, of the above practices, can be justified by the Mosaic law, as well as the practice of slave-holding.
Each of these laws, although in its time it was an ameliorating law, designed to take the place of some barbarous abuse, and to be a connecting link by which some higher state of society might be introduced, belongs confessedly to that system which St. Paul says made nothing perfect. They are a part of the commandment which he says was annulled for the weakness and unprofitableness thereof, and which, in the time which he wrote, was waxing old, and ready to vanish away. And Christ himself says, with regard to certain permissions of this system, that they were given on account of the "hardness of their hearts,"—because the attempt to enforce a more stringent system at that time, owing to human depravity, would have only produced greater abuses.
The following view of the Hebrew laws of slavery is compiled from Barnes' work on slavery, and from Professor Stowe's manuscript lectures.
The legislation commenced by making the great and common source of slavery—kidnapping—a capital crime.
The enactment is as follows: "He that stealeth a man and selleth him, or if he be found in his hand, he shall surely be put to death."—Exodus 21: 16.
The sources from which slaves were to be obtained were thus reduced to two: first, the voluntary sale of an individual by himself, which certainly does not come under the designation of involuntary servitude; second, the appropriation of captives taken in war, and the buying from the heathen.
With regard to the servitude of the Hebrew by a voluntary sale of himself, such servitude, by the statute-law of the land, came to an end once in seven years; so that the worst that could be made of it was that it was a voluntary contract to labor for a certain time.
With regard to the servants bought of the heathen, or of foreigners in the land, there was a statute by which their servitude was annulled once in fifty years.
It has been supposed, from a disconnected view of one particular passage in the Mosaic code, that God directly countenanced the treating of a slave, who was a stranger and foreigner, with more rigor and severity than a Hebrew slave. That this was not the case will appear from the following enactments, which have express reference to strangers:
The stranger that dwelleth with you shall be unto you as one born among you, and thou shalt love him as thyself.—Lev. 19: 34.
Thou shalt neither vex a stranger nor oppress him; for ye were strangers in the land of Egypt.—Exodus 22: 21.
Thou shalt not oppress a stranger, for ye know the heart of a stranger.—Exodus 23: 9.
The Lord your God regardeth not persons. He doth execute the judgment of the fatherless and the widow, and loveth the stranger in giving him food and raiment; love ye therefore the stranger.—Deut. 10: 17—19.
Judge righteously between every man and his brother, and the stranger that is with him.—Deut. 1: 16.
Cursed be he that perverteth the judgment of the stranger.—Deut. 27: 19.
Instead of making slavery an oppressive institution with regard to the stranger, it was made by God a system within which heathen were adopted into the Jewish state, educated and instructed in the worship of the true God, and in due time emancipated.
In the first place, they were protected by law from personal violence. The loss of an eye or a tooth, through the violence of his master, took the slave out of that master's power entirely, and gave him his liberty. Then, further than this, if a master's conduct towards a slave was such as to induce him to run away, it was enjoined that nobody should assist in retaking him, and that he should dwell wherever he chose in the land, without molestation. Third, the law secured to the slave a very considerable portion of time, which was to be at his own disposal. Every seventh year was to be at his own disposal.—Lev. 25: 4—6. Every seventh day was, of course, secured to him.—Ex. 20: 10.
The servant had the privilege of attending the three great national festivals, when all the males of the nation were required to appear before God in Jerusalem.—Ex. 34: 23.
Each of these festivals, it is computed, took up about three weeks.
The slave also was to be a guest in the family festivals. In Deut. 12: 12, it is said, "Ye shall rejoice before the Lord your God, ye, and your sons, and your daughters, and your men-servants, and your maid-servants, and the Levite that is within your gates."
Dr. Barnes estimates that the whole amount of time which a servant could have to himself would amount to about twenty-three years out of fifty, or nearly one-half his time.
Again, the servant was placed on an exact equality with his master in all that concerned his religious relations.
Now, if we recollect that in the time of Moses the God and the king of the nation were one and the same person, and that the civil and religious relation were one and the same, it will appear that the slave and his master stood on an equality in their civil relation with regard to the state.
Thus, in Deuteronomy 29, is described a solemn national convocation, which took place before the death of Moses, when the whole nation were called upon, after a solemn review of their national history, to renew their constitutional oath of allegiance to their supreme Magistrate and Lord.
On this occasion, Moses addressed them thus:—"Ye stand this day, all of you, before the Lord your God; your captains of your tribes, your elders, and your officers, with all the men of Israel, your little ones, your wives, and thy stranger that is in thy camp, from the hewer of thy wood unto the drawer of thy water; that thou shouldest enter into covenant with the Lord thy God, and into his oath, which the Lord thy God maketh with thee this day."
How different is this from the cool and explicit declaration of South Carolina with regard to the position of the American Wheeler's Law
243. slave:—"A slave is not generally regarded as legally capable of being within the peace of the state. He is not a citizen, and is not in that character entitled to her protection."
In all the religious services, which, as we have seen by the constitution of the nation, were civil services, the slave and the master mingled on terms of strict equality. There was none of the distinction which appertains to a distinct class or caste. "There was no special service appointed for them at unusual seasons. There were no particular seats assigned to them, to keep up the idea that they were a degraded class. There was no withholding from them the instruction which the word of God gave about the equal rights of mankind."
Fifthly. It was always contemplated that the slave would, as a matter of course, choose the Jewish religion, and the service of God, and enter willingly into all the obligations and services of the Jewish polity. Mr. Barnes cites the words of Maimonides, to show how this was commonly understood by the Hebrews.—Inquiry into the Scriptural Views of Slavery. By Albert Barnes, p. 132.
Whether a servant be born in the power of an Israelite, or whether he be purchased from the heathen, the master is to bring them both into the covenant.
But he that is in the house is entered on the eighth day; and he that is bought with money, on the day on which his master receives him, unless the slave be unwilling. For, if the master receive a grown slave, and he be unwilling, his master is to bear with him, to seek to win him over by instruction, and by love and kindness, for one year. After which, should he refuse so long, it is forbidden to keep him longer than a year. And the master must send him back to the strangers from whence he came. For the God of Jacob will not accept any other than the worship of a willing heart.—Maimon. Hilcoth Miloth, chap. i., sec. 8.
A sixth fundamental arrangement with regard to the Hebrew slave was that he could never be sold. Concerning this Mr. Barnes remarks:
A man, in certain circumstances, might be bought by a Hebrew; but when once bought, that was an end of the matter. There is not the slightest evidence that any Hebrew ever sold a slave; and any provision contemplating that was unknown to the constitution of the Commonwealth. It is said of Abraham that he had "servants bought with money;" but there is no record of his having ever sold one, nor is there any account of its ever having been done by Isaac or Jacob. The only instance of a sale of this kind among the patriarchs is that act of the brothers of Joseph, which is held up to so strong reprobation, by which they sold him to the Ishmaelites. Permission is given in the law of Moses to buy a servant, but none is given to sell him again; and the fact that no such permission is given is full proof that it was not contemplated. When he entered into that relation, it became certain that there could be no change, unless it was voluntary on his part (comp. Ex. 21: 5, 6), or unless his master gave him his freedom, until the not distant period fixed by law when he could be free. There is no arrangement in the law of Moses by which servants were to be taken in payment of their master's debts, by which they were to be given as pledges, by which they were to be consigned to the keeping of others, or by which they were to be given away as presents. There are no instances occurring in the Jewish history in which any of these things were done. This law is positive in regard to the Hebrew servant, and the principle of the law would apply to all others. Lev. 25: 42.—"They shall not be sold as bond men." In all these respects there was a marked difference, and there was doubtless intended to be, between the estimate affixed to servants and to property.—Inquiry, &c., p. 133—4.
As to the practical workings of this system, as they are developed in the incidents of sacred history, they are precisely what we should expect from such a system of laws. For instance, we find it mentioned incidentally in the ninth chapter of the first book of Samuel, that when Saul and his servant came to see Samuel, that Samuel, in anticipation of his being crowned king, made a great feast for him; and in verse twenty- second the history says: "And Samuel took Saul and his servant, and brought them into the parlor, and made them sit in the chiefest place."
We read, also, in 2 Samuel 9: 10, of a servant of Saul who had large estates, and twenty servants of his own.
We find, in 1 Chron. 2: 34, the following incident related: "Now, Sheshan had no sons, but daughters. And Sheshan had a servant, an Egyptian, whose name was Jarha. And Sheshan gave his daughter to Jarha, his servant, to wife."
Does this resemble American slavery?
We find, moreover, that this connection was not considered at all disgraceful, for the son of this very daughter was enrolled among the valiant men of David's army.—1 Chron. 2: 41.
In fine, we are not surprised to discover that the institutions of Moses in effect so obliterated all the characteristics of slavery, that it had ceased to exist among the Jews long before the time of Christ. Mr. Barnes asks:
On what evidence would a man rely to prove that slavery existed at all in the land in the time of the later prophets of the Maccabees, or when the Saviour appeared? There are abundant proofs, as we shall see, that it existed in Greece and Rome; but what is the evidence that it existed in Judea? So far as I have been able to ascertain, there are no declarations that it did to be found in the canonical books of the Old Testament, or in Josephus. There are no allusions to laws and customs which imply that it was prevalent. There are no coins or medals which suppose it. There are no facts which do not admit of an easy explanation on the supposition that slavery had ceased.—Inquiry, &c., p. 226.
Two objections have been urged to the interpretations which have been given of two of the enactments before quoted.
1. It is said that the enactment, "Thou shalt not return to his master the servant that has escaped," &c., relates only to servants escaping from heathen masters to the Jewish nation.
The following remarks on this passage are from Prof. Stowe's lectures:
Deuteronomy 23: 15, 16.—These words make a statute which, like every other statute, is to be strictly construed. There is nothing in the language to limit its meaning; there is nothing in the connection in which it stands to limit its meaning; nor is there anything in the history of the Mosaic legislation to limit the application of this statute to the case of servants escaping from foreign masters. The assumption that it is thus limited is wholly gratuitous, and, so far as the Bible is concerned, unsustained by any evidence whatever. It is said that it would be absurd for Moses to enact such a law while servitude existed among the Hebrews. It would indeed be absurd, were it the object of the Mosaic legislation to sustain and perpetuate slavery; but, if it were the object of Moses to limit and to restrain, and finally, to extinguish slavery, this statute was admirably adapted to his purpose. That it was the object of Moses to extinguish, and not to perpetuate, slavery, is perfectly clear from the whole course of his legislation on the subject. Every slave was to have all the religious privileges and instruction to which his master's children were entitled. Every seventh year released the Hebrew slave, and every fiftieth year produced universal emancipation. If a master, by an accidental or an angry blow, deprived the slave of a tooth, the slave, by that act, was forever free. And so, by the statute in question, if the slave felt himself oppressed, he could make his escape, and, though the master was not forbidden to retake him if he could, every one was forbidden to aid his master in doing it. This statute, in fact, made the servitude voluntary, and that was what Moses intended.
Moses dealt with slavery precisely as he dealt with polygamy and with war: without directly prohibiting, he so restricted as to destroy it; instead of cutting down the poison-tree, he girdled it, and left it to die of itself. There is a statute in regard to military expeditions precisely analogous to this celebrated fugitive slave law. Had Moses designed to perpetuate a warlike spirit among the Hebrews, the statute would have been preëminently absurd; but, if it was his design to crush it, and to render foreign wars almost impossible, the statute was exactly adapted to his purpose. It rendered foreign military service, in effect, entirely voluntary, just as the fugitive law rendered domestic servitude, in effect, voluntary.
The law may be found at length in Deuteronomy 20: 5—10; and let it be carefully read and compared with the fugitive slave law already adverted to. Just when the men are drawn up ready for the expedition,—just at the moment when even the hearts of brave men are apt to fail them,—the officers are commanded to address the soldiers thus:
"What man of you is there that hath built a new house, and hath not dedicated it? Let him go and return to his house, lest he die in the battle, and another man dedicate it.
"And what man is he that hath planted a vineyard and hath not yet eaten of it? Let him also go and return to his house, lest he die in the battle, and another man eat of it.
"And what man is there that hath betrothed a wife, and hath not taken her? Let him go and return unto his house, lest he die in the battle, and another man take her."
And the officers shall speak further unto the people, and they shall say, "What man is there that is fearful and faint-hearted? Let him go and return unto his house, lest his brethren's heart faint, as well as his heart."
Now, consider that the Hebrews were exclusively an agricultural people, that warlike parties necessarily consist mainly of young men, and that by this statute every man who had built a house which he had not yet lived in, and every man who had planted a vineyard from which he had not yet gathered fruit, and every man who had engaged a wife whom he had not yet married, and every one who felt timid and faint-hearted, was permitted and commanded to go home,—how many would there probably be left? Especially when the officers, instead of exciting their military ardor by visions of glory and of splendor, were commanded to repeat it over and over again that they would probably die in the battle and never get home, and hold this idea up before them as if it were the only idea suitable for their purpose, how excessively absurd is the whole statute considered as a military law,—just as absurd as the Mosaic fugitive law, understood in its widest application, is, considered as a slave law!
It is clearly the object of this military law to put an end to military expeditions; for, with this law in force, such expeditions must always be entirely volunteer expeditions. Just as clearly was it the object of the fugitive slave law to put an end to compulsory servitude; for, with that law in force, the servitude must, in effect, be, to a great extent, voluntary,—and that is just what the legislator intended. There is no possibility of limiting the law, on account of its absurdity, when understood in its widest sense, except by proving that the Mosaic legislation was designed to perpetuate and not to limit slavery; and this certainly cannot be proved, for it is directly contrary to the plain matter of fact.
I repeat it, then, again: there is nothing in the language of this statute, there is nothing in the connection in which it stands, there is nothing in the history of the Mosaic legislation on this subject, to limit the application of the law to the case of servants escaping from foreign masters; but every consideration, from every legitimate source, leads us to a conclusion directly the opposite. Such a limitation is the arbitrary, unsupported stet voluntas pro ratione assumption of the commentator, and nothing else. The only shadow of a philological argument that I can see, for limiting the statute, is found in the use of the words to thee, in the fifteenth verse. It may be said that the pronoun thee is used in a national and not individual sense, implying an escape from some other nation to the Hebrews. But, examine the statute immediately preceding this, and observe the use of the pronoun thee in the thirteenth verse. Most obviously, the pronouns in these statutes are used with reference to the individuals addressed, and not in a collective or national sense exclusively; very rarely, if ever, can this sense be given to them in the way claimed by the argument referred to.
2. It is said that the proclamation, "Thou shalt proclaim liberty through the land to all the inhabitants thereof," related only to Hebrew slaves. This assumption is based entirely on the supposition that the slave was not considered, in Hebrew law, as a person, as an inhabitant of the land, and a member of the state; but we have just proved that in the most solemn transaction of the state the hewer of wood and drawer of water is expressly designated as being just as much an actor and participator as his master; and it would be absurd to suppose that, in a statute addressed to all the inhabitants of the land, he is not included as an inhabitant.
Barnes enforces this idea by some pages of quotations from Jewish writers, which will fully satisfy any one who reads his work.
From a review, then, of all that relates to the Hebrew slave-law, it will appear that it was a very well-considered and wisely-adapted system of education and gradual emancipation. No rational man can doubt that if the same laws were enacted and the same practices prevailed with regard to slavery in the United States, that the system of American slavery might be considered, to all intents and purposes, practically at an end. If there is any doubt of this fact, and it is still thought that the permission of slavery among the Hebrews justifies American slavery, in all fairness the experiment of making the two systems alike ought to be tried, and we should then see what would be the result.
CHAPTER XV.
SLAVERY IS DESPOTISM.
It is always important, in discussing a thing, to keep before our minds exactly what it is.
The only means of understanding precisely what a civil institution is are an examination of the laws which regulate it. In different ages and nations, very different things have been called by the name of slavery. Patriarchal servitude was one thing, Hebrew servitude was another, Greek and Roman servitude still a third; and these institutions differed very much from each other. What, then, is American slavery, as we have seen it exhibited by law, and by the decisions of courts?
Let us begin by stating what it is not.
1. It is not apprenticeship.
2. It is not guardianship.
3. It is in no sense a system for the education of a weaker race by a stronger.
4. The happiness of the governed is in no sense its object.
5. The temporal improvement or the eternal well-being of the governed is in no sense its object.
The object of it has been distinctly stated in one sentence, by Judge Ruffin,—"The end is the profit of the master, his security, and the public safety."
Slavery, then, is absolute despotism, of the most unmitigated form.
It would, however, be doing injustice to the absolutism of any civilized country to liken American slavery to it. The absolute governments of Europe none of them pretend to be founded on a property right of the governor to the persons and entire capabilities of the governed.
This is a form of despotism which exists only in some of the most savage countries of the world; as, for example, in Dahomey.
The European absolutism or despotism, now, does, to some extent, recognize the happiness and welfare of the governed as the foundation of government; and the ruler is considered as invested with power for the benefit of the people; and his right to rule is supposed to be in somewhat predicated upon the idea that he better understands how to promote the good of the people than they themselves do. No government in the civilized world now presents the pure despotic idea, as it existed in the old days of the Persian and Assyrian rule.
The arguments which defend slavery must be substantially the same as those which defend despotism of any other kind; and the objections which are to be urged against it are precisely those which can be urged against despotism of any other kind. The customs and practices to which it gives rise are precisely those to which despotisms in all ages have given rise.
Is the slave suspected of a crime? His master has the power to examine him by torture (see State v. Castleman). His master has, in fact, in most cases, the power of life and death, owing to the exclusion of the slave's evidence. He has the power of banishing the slave, at any time, and without giving an account to anybody, to an exile as dreadful as that of Siberia, and to labors as severe as those of the galleys. He has also unlimited power over the character of his slave. He can accuse him of any crime, yet withhold from him all right of trial or investigation, and sell him into captivity, with his name blackened by an unexamined imputation.
These are all abuses for which despotic governments are blamed. They are powers which good men who are despotic rulers are beginning to disuse; but, under the flag of every slave-holding state, and under the flag of the whole United States in the District of Columbia, they are committed indiscriminately to men of any character.
But the worst kind of despotism has been said to be that which extends alike over the body and over the soul; which can bind the liberty of the conscience, and deprive a man of all right of choice in respect to the manner in which he shall learn the will of God, and worship Him. In other days, kings on their thrones, and cottagers by their firesides, alike trembled before a despotism which declared itself able to bind and to loose, to open and to shut the kingdom of heaven.
Yet this power to control the conscience, to control the religious privileges, and all the opportunities which man has of acquaintanceship with his Maker, and of learning to do his will, is, under the flag of every slave state, and under the flag of the United States, placed in the hands of any men, of any character, who can afford to pay for it. It is a most awful and most solemn truth that the greatest republic in the world does sustain under her national flag the worst system of despotism which can possibly exist.
With regard to one point to which we have adverted,—the power of the master to deprive the slave of a legal trial while accusing him of crime,—a very striking instance has occurred in the District of Columbia, within a year or two. The particulars of the case, as stated, at the time, in several papers, were briefly these: A gentleman in Washington, our national capital,—an elder in the Presbyterian church,—held a female slave, who had, for some years, supported a good character in a Baptist church of that city. He accused her of an attempt to poison his family, and immediately placed her in the hands of a slave-dealer, who took her over and imprisoned her in the slave-pen at Alexandria, to await the departure of a coffle. The poor girl had a mother, who felt as any mother would naturally feel.
When apprized of the situation of her daughter, she flew to the pen, and, with tears, besought an interview with her only child; but she was cruelly repulsed, and told to be gone! She then tried to see the elder, but failed. She had the promise of money sufficient to purchase her daughter, but the owner would listen to no terms of compromise.
In her distress, the mother repaired to a lawyer in the city, and begged him to give form to her petition in writing. She stated to him what she wished to have said, and he arranged it for her in such a form as she herself might have presented it in, had not the benefits of education been denied her. The foliowing is the letter:
Washington, July 25, 1851.
Mr. ———,
Sir: I address you as a rich Christian freeman and fixther, while I am myself but a poor slave-mother! I come to plead with you for an only child whom I love, who is a professor of the Christian religion with yourself, and a member of a Christian church; and who, by your act of ownership, now pines in her imprisonment in a loathsome man-warehouse, where she is held for sale! I come to plead with you for the exercise of that blessed law, "Whatsoever ye would that men should do unto you, do ye even so to them."
With great labor, I have found friends who are willing to aid me in the purchase of my child, to save us from a cruel separation. You, as a father, can judge of my feelings when I was told that you had decreed her banishment to distant as well as to hopeless bondage!
For nearly six years my child has done for you the hard labor of a slave; from the age of sixteen to twenty-two, she has done the hard work of your chamber, kitchen, cellar, and stables. By night and by day, your will and your commands have been her highest law; and all this has been unrequited toil. If in all this time her scanty allowance of tea and coffee has been sweetened, it has been at the cost of her slave-mother, and not at yours.
You are an office-bearer in the church, and a man of prayer. As such, and as the absolute owner of my child, I ask candidly whether she has enjoyed such mild and gentle treatment, and amiable example, as she ought to have had, to encourage her in her monotonous bondage? Has she received at your hands, in faithful religious instruction in the Word of God, a full and fair compensation for all her toil? It is not to me alone that you must answer these questions. You acknowledge the high authority of His laws who preached a deliverance to the captive, and who commands you to give to your servant "that which is just and equal." O! I entreat you, withhold not, at this trying hour, from my child that which will cut off her last hope, and which may endanger your own soul!
It has been said that you charge my daughter with crime. Can this be really so? Can it be that you would set aside the obligations of honor and good citizenship,—that you would dare to sell the guilty one away for money, rather than bring her to trial, which you know she is ready to meet! What would you say, if you were accused of guilt, and refused a trial? Is not her fair name as precious to her, in the church to which she belongs, as yours can be to you?
Suppose, now, for a moment, that your daughter, whom you love, instead of mine, was in these hot days incarcerated in a negro-pen, subject to my control, fed on the coarsest food, committed to the entire will of a brute, denied the privilege commonly allowed even to the murderer—that of seeing the face of his friends? O! then, you would feel! Feel soon, then, for a poor slave-mother and her child, and do for us as you shall wish you had done when we shall meet before the Great Judge, and when it shall be your greatest joy to say, "I did let the oppressed free."
Ellen Brown.
The girl, however, was sent off to the Southern market.
The writer has received these incidants from the gentleman who wrote the letter. Whether the course pursued by the master was strictly legal is a point upon which we are not entirely certain; that it was a course in which the law did not in fact interfere is quite plain, and it is also very apparent that it was a course against which public sentiment did not remonstrate. The man who exercised this power was a professedly religious man, enjoying a position of importance in a Christian church; and it does not appear, from any movements in the Christian community about him, that they did not consider his course a justifiable one.
Yet is not this kind of power the very one at which we are so shocked when we see it exercised by foreign despots?
Do we not read with shuddering that in Russia, or in Austria, a man accused of crime is seized upon, separated from his friends, allowed no opportunities of trial or of self-defence, but hurried off to Siberia, or some other dreaded exile?
Why is despotism any worse in the governor of a state than in a private individual?
There is a great controversy now going on in the world between the despotic and the republican principle. All the common arguments used in support of slavery are arguments that apply with equal strength to despotic government, and there are some arguments in favor of despotic governments that do not apply to individual slavery. There are arguments, and quite plausible, ones, in favor of despotic government. Nobody can deny that it possesses a certain kind of efficiency, compactness, and promptness of movement, which cannot, from the nature of things, belong to a republic. Despotism has established and sustained much more efficient systems of police than ever a republic did. The late King of Prussia, by the possession of absolute despotic power was enabled to carry out a much more efficient system of popular education than we ever have succeeded in carrying out in America. He districted his kingdom in the most thorough manner, and obliged every parent, whether he would or not, to have his children thoroughly educated.
If we reply to all this, as we do, that the possession of absolute power in a man qualified to use it right is undoubtedly calculated for the good of the state, but that there are so few men that know how to use it, that this form of government is not, on the whole, a safe one, then we have stated an argument that goes to overthrow slavery as much as it does a despotic government; for certainly the chances are much greater of finding one man, in the course of fifty years, who is capable of wisely using this power, than of finding thousands of men every day in our streets, who can be trusted with such power. It is a painful and most serious fact, that America trusts to the hands of the most brutal men of her country, equally with the best, that despotic power which she thinks an unsafe thing even in the hands of the enlightened, educated and cultivated Emperor of the Russias.
With all our republican prejudices, we cannot deny that Nicholas is a man of talent, with a mind liberalized by education; we have been informed, also, that he is a man of serious and religious character;—he certainly, acting as he does in the eye of all the world, must have great restraint upon him from public opinion, and a high sense of character. But who is the man to whom American laws intrust powers more absolute than those of Nicholas of Russia, or Ferdinand of Naples? He may have been a pirate on the high seas; he may be a drunkard; he may, like Souther, have been convicted of a brutality at which humanity turns pale; but, for all that, American slave-law will none the less trust him with this irresponsible power,—power over the body, and power over the soul.
On which side, then, stands the American nation, in the great controversy which is now going on between self-government and despotism? On which side does America stand, in the great controversy for liberty of conscience?
Do foreign governments include their population from the reading of the Bible?—The slave of America is excluded by the most effectual means possible. Do we say, "Ah! but we read the Bible to our slaves, and present the gospel orally?"—This is precisely what religious despotism in Italy says. Do we say that we have no objection to our slaves reading the Bible, if they will stop there; but that with this there will come in a flood of general intelligence, which will upset the existing state of things?—This is precisely what is said in Italy.
Do we say we should be willing that the slave should read his Bible, but that he, in his ignorance, will draw false and erroneous conclusions from it, and for that reason we prefer to impart its truths to him orally?—This, also, is precisely what the religious despotism of Europe says.
Do we say, in our vain-glory, that despotic government dreads the coming in of anything calculated to elevate and educate the people?—And is there not the same dread through all the despotic slave governments of America?
On which side, then, does the Americas nation stand, in the great, last question of the age?
↑ In this connection it may be well to state that the work of Judge Stroud is now out of print, but that a work of the same character is in course of preparation by William I. Bowditch, Esq., of Boston, which will bring the subject out, by the assistance of the latest editions of statutes, and the most recent decisions of courts.
↑ We except the State of Louisiana. Owing to the influence of the French code in that state, more really humane provisions prevail there. How much these provisions avail in point of fact, will be shown when we come to that part of the subject.
↑ More recently the author has met with a passage in a North Carolina newspaper, containing some further particulars of the life of Judge Ruffin, which have proved interesting to her, and may also to the reader.
From the Raleigh (N. C.) Register.
Resignation of the Chief Justice of the State of North Carolina.
We publish below the letter of Chief Justice Ruffin, of the Supreme Court, resigning his seat on the bench.
This act takes us, and no less will it take the state, by surprise. The public are not prepared for it; and we doubt not there will scarcely be an exception to the deep and general regret which will be felt throughout the state. Judge Ruffin's great and unsurpassed legal learning, his untiring industry, the ease with which he mastered the details and comprehended the whole of the most complicated cases, were the admiration of the bar; and it has been a common saying of the ablest lawyers of the state, for a long time past, that his place on the bench could be supplied by no other than himself.
He is now, as we learn, in the sixty-fifth year of his age, in full possession of his usual excellent health, unaffected, so far as we can discover, in his natural vigor and strength, and certainly without any symptom of mental decay. Forty-five years ago he commenced the practice of the law. He has been on the bench twenty-eight years, of which time he has been one of the Supreme Court twenty-three years. During this long public career he has, in a pecuniary point of view, sacrificed many thousands; for there has been no time of it in which he might not, with perfect ease, have doubled, by practice, the amount of his salary as judge.
"To the Honorable the General Assembly of North Carolina, now in session.
"Gentlemen: I desire to retire to the walks of private life, and therefore pray your honorable body to accept the resignation of my place on the bench of the Supreme Court. In surrendering this trust, I would wish to express my grateful sense of the confidence and honors so often and so long bestowed on me by the General Assembly. But I have no language to do it suitably. I am very sensible that they were far beyond my deserts, and that I have made an insufficient return of the service. Yet I can truly aver that, to the best of my ability, I have administered the law as I understood it, and to the ends of suppressing crime and wrong, and upholding virtue, truth and right; aiming to give confidence to honest men, and to confirm in all good citizens love for our country, and a pure trust in her law and magistrates.
"In my place I hope I have contributed to these ends; and I firmly believe that our laws will, as heretofore, be executed, and our people happy in the administration of justice, honest and contented, as long as they keep, and only so long as they keep, the independent and sound judiciary now established in the constitution; which, with all other blessings, I earnestly pray may be perpetuated to the people of North Carolina.
"I have the honor to be, gentlemen, your most obliged and obedient servant, Thomas Ruffin
"Raleigh, November 10, 1852."
↑ The following is Judge Field's statement of the punishment:
The negro was tied to a tree and whipped with switches When Souther became fatigued with the labor of whipping, he called upon a negro man of his, and made him cob Sam with a shingle. He also made a negro woman of his help to cob him. And, after cobbing and whipping, he applied fire to the body of the slave. * * * * He then caused him to be washed down with hot water, in which pods of red pepper had been steeped. The negro was also tied to a log and to the bed-post with ropes, which choked him, and he was kicked and stamped by Souther. This sort of punishment was continued and repeated until the negro died under its infliction.
↑ This man was burned alive.
↑ The old statute of 1741 had some features still more edifying. That provides that said "proclamation shall be published on a Sabbath day, at the door of every church or chapel, or, for want of such, at the place where divine service shall be performed in the said county, by the parish clerk or reader, immediately after divine service." Potter's Revised, i. 166. What a peculiar appropriateness there must have been in this proclamation, particularly after a sermon on the love of Christ, or an exposition of the text "thou shalt love thy neighbor as thyself!"
↑ Be it further enacted, That when any slave shall be legally outlawed in any of the counties within mentioned, the owner of which shall reside in one of counties, and the said slave shall be killed in consequence of such outlawry, the value of such slave shall be ascertained by a jury which shall be empanelled at the succeeding court of the county where the said slave was killed, and a certificate of such valuation shall be given by the clerk of the court to the owner of said slave, who shall be entitled to receive two-thirds of such valuation from the sheriff of the county wherein the slave was killed. [Extended to othsr counties in 1797.—Potter, ch. 480, § 1.] now obsolete.
↑ Gen. 4: 14.—"And it shall come to pass that every one that findeth me shall slay me."
↑ The iron collar was also in vogue in North Carolina, as the following extract from the statute-book will show. The wearers of this article of apparel certainly have some reason to complain of the "tyranny of fashion."
"When the keeper of the said public jail shall, by direction of such court as aforesaid, let out any negro or runaway to hire, to any person or persons whomsoever, the said keeper shall, at the time of his delivery, cause an iron collar to be put on the neck of such negro or runaway, with the letters P. G. stamped thereon; and thereafter the said keeper shall not be answerable for any escape of
the said negro or runaway."—Potter's Revival, i. 162.
↑ Slavery as It Is; Testimony of a Thousand Witnesses. New York, 1839. pp. 62, 53.
↑ See also the case of State v. Ahram, 10 Ala. 928. 7 U. S. Dig. p. 419. "The master or overseer, and not the slave, is the proper judge whether the slave is too sick to be able to labor. The latter cannot, therefore, resist the order of the former to go to work."
↑ Gibbon's "Decline and Fall," Chap. II.
↑ Ibid.
↑ In and after the reign of Augustus, certain restrictive regulations were passed, designed to prevent an increase of unworthy citizens by emancipation. They had, however, nothing like the stringent force of American laws.
↑ i.e. Periagua. | CommonCrawl |
Kenny Cason
4D Rotation Matrix - Graph 4D
math java on January 8, 2009
This program rotates points about the XY, YZ, XZ, XU, YU, and ZU axises. I then projects each 4D vector to the 2D canvas. The Jar file can be downloaded here: Graph4D.jar
Before looking at the source, let's take a look at some of the fundamental mathematics behind the software. If you are uncomfortable with the thought of 4D matrix rotations, then I recommend reading Wikipedia, or checking out my article about 3D graphing, which can be found here. In this example, I will only show the 4D rotation matrices. Note that for each rotation matrix, 2 axises are held still while the vector is rotated around the other two axises. This may be hard to visualize at first, but It will become clear after a while.
$$ rotXY = \left[\begin{matrix} cos(\theta) & sin(\theta) & 0 & 0 \\ -sin(\theta) & cos(\theta) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}\right] $$
$$ rotYZ = \left[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & cos(\theta) & sin(\theta) & 0 \\ 0 & -sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 0 & 1 \end{matrix}\right] $$
$$ rotXZ = \left[\begin{matrix} cos(\theta) & 0 & -sin(\theta) & 0 \\ 0 & 1 & 0 & 0 \\ sin(\theta) & 0 & cos(\theta) & 0 \\ 0 & 0 & 0 & 1 \end{matrix}\right] $$
$$ rotXU = \left[\begin{matrix} cos(\theta) & 0 & 0 & sin(\theta) \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -sin(\theta) & 0 & 0 & cos(\theta) \end{matrix}\right] $$
$$ rotYU = \left[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & cos(\theta) & 0 & -sin(\theta) \\ 0 & 0 & 1 & 0 \\ 0 & sin(\theta) & 0 & cos(\theta) \end{matrix}\right] $$
$$ rotZU = \left[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & cos(\theta) & -sin(\theta) \\ 0 & 0 & sin(\theta) & cos(\theta) \end{matrix}\right] $$
The source code can be found below as well as being bundled into the Jar file.
Transform4D.java - contains method for rotating a 4D vector
Transform4D.java
Point4D.java
Graph4D.java
Ninja Turdle
Kakyll
Copyright © KennyCason.com 2018. All rights reserved. | CommonCrawl |
\begin{document}
\title{Zero-Reachability in Probabilistic Multi-Counter Automata}
\author{\IEEEauthorblockN{ Tom\'{a}\v{s} Br\'{a}zdil$^{1,}$\IEEEauthorrefmark{1},\quad Stefan Kiefer$^{2,}$\IEEEauthorrefmark{3},\quad Anton\'{\i}n Ku\v{c}era$^{1,}$\IEEEauthorrefmark{1},\quad Petr Novotn\'{y}$^{1,}$\IEEEauthorrefmark{1},\quad Joost-Pieter Katoen\IEEEauthorrefmark{2}}\\ \IEEEauthorblockA{\IEEEauthorrefmark{1} Faculty of Informatics, Masaryk University, Brno, Czech Republic\\ \{brazdil,kucera\}@fi.muni.cz, [email protected]}
\IEEEauthorblockA{\IEEEauthorrefmark{2} Department of Computer Science, RWTH Aachen University, Germany\\ [email protected]}
\IEEEauthorblockA{\IEEEauthorrefmark{3} Department of Computer Science, University of Oxford, United Kingdom\\ [email protected]} \IEEEcompsocitemizethanks{\IEEEcompsocthanksitem M. Shell is with the Georgia Institute of Technology. \IEEEcompsocthanksitem J. Doe and J. Doe are with An onymous University.}}
\maketitle
\footnotetext[1]{T.~Br\'{a}zdil, A.~Ku\v{c}era, and P.~Novotn\'{y} are supported by the Czech Science Foundation, Grant No.~P202/10/1469.} \footnotetext[2]{S. Kiefer is supported by a Royal Society University Research Fellowship.}
\begin{abstract} We study the qualitative and quantitative zero-reachability problem in probabilistic multi-counter systems. We identify the undecidable variants of the problems, and then we concentrate on the remaining two cases. In the first case, when we are interested in the probability of all runs that visit zero in \emph{some} counter, we show that the qualitative zero-reachability is decidable in time which is polynomial in the size of a given pMC and doubly exponential in the number of counters. Further, we show that the probability of all zero-reaching runs can be effectively approximated up to an arbitrarily small given error $\varepsilon > 0$ in time which is polynomial in $\log(\varepsilon)$, exponential in the size of a given pMC, and doubly exponential in the number of counters. In the second case, we are interested in the probability of all runs that visit zero in some counter different from the last counter. Here we show that the qualitative zero-reachability is decidable and \textsc{SquareRootSum}-hard, and the probability of all zero-reaching runs can be effectively approximated up to an arbitrarily small given error $\varepsilon > 0$ (these result applies to pMC satisfying a suitable technical condition that can be verified in polynomial time). The proof techniques invented in the second case allow to construct counterexamples for some classical results about ergodicity in stochastic Petri nets.
\end{abstract}
\section{Introduction} \label{sec-intro}
A \emph{probabilistic multi-counter automaton (pMC)} $\mathcal{A}$ of dimension $d \in \mathbb{N}$ is an abstract fully probabilistic computational device equipped with a finite-state control unit and $d$~unbounded counters that can store non-negative integers. A \emph{configuration} $p\vec{v}$ of $\mathcal{A}$ is given by the current control state $p$ and the vector of current counter values $\vec{v}$. The dynamics of $\mathcal{A}$ is defined by a finite set of \emph{rules} of the form $(p,\alpha,c,q)$ where $p$ is the current control state, $q$ is the next control state, $\alpha$ is a $d$-dimensional vector of counter changes ranging over $\{-1,0,1\}^d$, and $c$ is a subset of counters that are tested for zero. Moreover, each rule is assigned a positive integer \emph{weight}. A rule $(p,\alpha,c,q)$ is \emph{enabled} in a configuration $p\vec{v}$ if the set of all counters with zero value in $\vec{v}$ is precisely~$c$ and no component of $\vec{v} + \alpha$ is negative; such an enabled rule can be \emph{fired} in $p\vec{v}$ and generates a \emph{probabilistic transition} $p\vec{v} \tran{x} q(\vec{v}{+}\alpha)$ where the probability $x$ is equal to the weight of the rule divided by the total weight of all rules enabled in $p\vec{v}$.
A special subclass of pMC are \emph{probabilistic vector addition systems with states (pVASS)}, which are equivalent to (discrete-time) \emph{stochastic Petri nets (SPN)}. Intuitively, a pVASS is a pMC where no subset of counters is tested for zero explicitly (see Section~\ref{sec-prelim} for a precise definition).
The decidability and complexity of basic qualitative/quantitative problems for pMCs has so far been studied mainly in the one-dimensional case, and there are also some results about unbounded SPN (a more detailed overview of the existing results is given below). In this paper, we consider \emph{multi-dimensional} pMC and the associated \emph{zero-reachability} problem. That is, we are interested in the probability of all runs initiated in a given $p\vec{v}$ that eventually visit a ``zero configuration''. Since there are several counters, the notion of ``zero configuration'' can be formalized in various ways (for example, we might want to have zero in some counter, in all counters simultaneously, or in a given subset of counters). Therefore, we consider a general \emph{stopping criterion} $\mathcal{Z}$ which consists of \emph{minimal} subsets of counters that are required to be simultaneously zero. For example, if $\mathcal{Z} = \Z_{all} = \{\{1\},\ldots,\{d\}\}$, then a run is stopped when reaching a configuration with zero in \emph{some} counter; and if we put $\mathcal{Z} = \{\{1,2\}\}$, then a run is stopped when reaching a configuration with zero in counters~$1$ and~$2$ (and possibly also in other counters). We use $\mathcal{P}(\mathit{Run}(p\vec{v},\mathcal{Z}))$ to denote the probability of all runs initiated in $p\vec{v}$ that reach a configuration satisfying the stopping criterion~$\mathcal{Z}$. The main algorithmic problems considered in this paper are the following: \begin{itemize} \item \emph{Qualitative $\mathcal{Z}$-reachability:}
Is $\mathcal{P}(\mathit{Run}(p\vec{v},\mathcal{Z})) = 1$? \item \emph{Approximation:} Can
$\mathcal{P}(\mathit{Run}(p\vec{v},\mathcal{Z}))$ be approximated up
to a given absolute/relative
error \mbox{$\varepsilon > 0$}? \end{itemize} We start by observing that the above problems are not effectively solvable in general, and we show that there are only two potentially decidable cases, where $\mathcal{Z}$ is equal either to $\Z_{all}$ (Case I) or to $\Zminusi{i} = \Z_{all} \smallsetminus \{\{i\}\}$ (Case II). Recall that if $\mathcal{Z} = \Z_{all}$, then a run is stopped when some counter reaches zero; and if $\mathcal{Z} = \Zminusi{i}$, then a run is stopped when a counter different from~$i$ reaches~zero. Cases~I and~II are analyzed independently and the following results are achieved:
\textbf{Case I}: We show that the qualitative $\Z_{all}$-reachability problem is decidable in time polynomial in $|\mathcal{A}|$ and doubly exponential in~$d$. In particular, this means that the problem is decidable in \emph{polynomial time for every fixed $d$}. Then, we show that
$\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))$ can be effectively approximated up to a given absolute/relative error \mbox{$\varepsilon > 0$} in time which is polynomial in $|\varepsilon|$, exponential in $|\mathcal{A}|$, and doubly exponential in~$d$ (in the special case when $d = 1$, the problem is known to be
solvable in time polynomial in $|\mathcal{A}|$ and $|\varepsilon|$, see \cite{ESY:polynomial-time-termination}).
\textbf{Case II}: We analyze Case~II only under a technical assumption that counter~$i$ is not critical; roughly speaking, this means that counter~$i$ has either a tendency to increase or a tendency to decrease when the other counters are positive. The problem whether counter~$i$ is critical or not is
solvable in time polynomial in $|\mathcal{A}|$, so we can efficiently check whether a given pMC can be analyzed by our methods.
Under the mentioned assumption, we show how to construct a suitable martingale which captures the behaviour of certain runs in~$\mathcal{A}$. Thus, we obtain a new and versatile tool for analyzing quantitative properties of runs in multi-dimensional pMC, which is more powerful than the martingale of \cite{BKK:pOC-time-LTL-martingale} constructed for one-dimensional pMC. Using this martingale and the results of \cite{BFLZ:VASSz-model-checking-LMCS}, we show that the qualitative \mbox{$\Zminusi{i}$-reachability} problem is decidable. We also show that the problem is \textsc{Square-Room-Sum}-hard, even for two-dimensional pMC satisfying the mentioned technical assumption. Further, we show that $\mathcal{P}(\mathit{Run}(p\vec{v},\Zminusi{i}))$ can be effectively approximated up to a given absolute error \mbox{$\varepsilon > 0$}. The main reason why we do not provide any upper complexity bounds in Case~II is a missing upper bound for coverability in VAS with one zero test (see \cite{BFLZ:VASSz-model-checking-LMCS}).
\begin{figure}
\caption{Firing process may not be ergodic.}
\label{fig-SPN}
\end{figure}
It is worth noting that the techniques developed in Case~II reveal the existence of phenomena that should not exist according to the previous results about ergodicity in SPN. A classical paper in this area \cite{DBLP:journals/tse/FlorinN89} has been written by Florin \& Natkin in 80s. In the paper, it is claimed that if the state-space of a given SPN (with arbitrarily many unbounded places) is strongly connected, then the firing process is ergodic (see Section IV.B.{} in \cite{DBLP:journals/tse/FlorinN89}). In the setting of discrete-time probabilistic Petri nets, this means that for almost all runs, the limit frequency of transitions performed along a run is defined and takes the same value. However, in Fig.~\ref{fig-SPN} there is an example of a pVASS (depicted as SPN with weighted transitions) with two counters (places) and strongly connected state space where the limit frequency of transitions may take two eligible values (each with probability $1/2$). Intuitively, if both counters are positive, then both of them have a tendency to decrease (i.e., the trend of the only BSCC of $\mathcal{F}_\mathcal{A}$ is negative in both components, see Section~\ref{sec-case1}). However, if we reach a configuration where the first counter is zero and the second counter is sufficiently large, then the second counter starts to \emph{increase}, i.e., it never becomes zero again with some positive probability (cf.{} the \textit{oc-trend} of the only BSCC~$D$ of $\mathcal{B}_1$ introduced in Section~\ref{sec-case2}). The first counter stays zero for most of the time, because when it becomes positive, it is immediatelly emptied with a very large probability. This means that the frequency of firing $t_2$ will be much higher than the frequency of firing $t_1$. When we reach a configuration where the first counter is large and the second counter is zero, the situation is symmetric, i.e., the frequency of firing $t_1$ becomes much higher than the frequency of firing $t_2$. Further, almost every run eventually behaves according to one the two scenarios, and therefore there are two eligible limit frequencies of transitions, each of which is taken with probability~$1/2$. So, we must unfortunately conclude that the results of \cite{DBLP:journals/tse/FlorinN89} are not valid for general SPN.
\noindent \textbf{Related Work.} One-dimensional pMC and their extensions into decision processes and games were studied in \cite{BBEKW:OC-MDP,EWY:one-counter,BKK:pOC-time-LTL-martingale,ESY:polynomial-time-termination,BBEK:OC-games-termination-approx,EWY:one-counter-PE,BBE:OC-games}. In particular, in \cite{ESY:polynomial-time-termination} it was shown that termination probability (a ``selective'' variant of zero-reachability)
in one-dimensional pMC can be approximated up to an arbitrarily small given error in polynomial time. In \cite{BKK:pOC-time-LTL-martingale}, it was shown how to construct a martingale for a given one-dimensional pMC which allows to derive tail bounds on termination time (we use this martingale in Section~\ref{sec-case1}).
There are also many papers about SPN (see, e.g., \cite{DBLP:journals/tc/Molloy82, DBLP:journals/tocs/MarsanCB84}), and some of these works also consider algorithmic aspects of unbounded SPN (see, e.g., \cite{AHM:decisive-Markov-chains,DBLP:journals/jss/FlorinN86, DBLP:journals/tse/FlorinN89}).
Considerable amount of papers has been devoted to algorithmic analysis of so called probabilistic lossy channel systems (PLCS) and their game extensions (see~e.g.~\cite{IN:probLCS-TACAS,BE:probLCS-algorithms-ARTS,AHMS:Eager-limit,ACMS:stoch-parity-games-lossy-QEST,ABRS:IC}). PLCS are a stochastic extension of lossy channel systems, i.e., an infinite-state model comprising several interconnected queues coupled with a finite-state control unit. The main ingredient, which makes results about PLCS incomparable with our results on pMCs, is that queues may lose messages with a fixed loss-rate, which substantially simplifies the associated analysis.
\section{Preliminaries} \label{sec-prelim}
\noindent We use $\mathbb{Z}$, $\mathbb{N}$, $\mathbb{N}^+$, $\mathbb{Q}$, and $\mathbb{R}$ to denote the set of all integers, non-negative integers, positive integers, rational numbers, and real numbers, respectively.
Let $\mathcal{V} = (V,L,\tran{})$, where $V$ is a non-empty set of vertices, $L$ a non-empty set of \emph{labels}, and ${\tran{}} \subseteq V \times L \times V$ a \emph{total} relation (i.e., for every $v \in V$ there is at least one \emph{outgoing} transition $(v,\ell,u) \in {\tran{}}$). As usual, we write $v \tran{\ell} u$ instead of $(v,\ell,u) \in {\tran{}}$, and $v \tran{} u$ iff $v \tran{\ell} u$ for some $\ell \in L$. The reflexive and transitive closure of $\tran{}$ is denoted by $\tran{}^*$. A \emph{finite path} in $\mathcal{V}$ of \emph{length} $k \geq 0$ is a finite sequence of the form $v_0\ell_0v_1\ell_1\ldots\ell_{k-1} v_k$, where $v_i \tran{\ell_{i}} v_{i+1}$ for all $0 \leq i <k$. The length of a finite path $w$ is denoted by $\mathit{length}(w)$. A \emph{run} in $\mathcal{V}$ is an infinite sequence $w$ of vertices such that every finite prefix of $w$ ending in a vertex is a finite path in $\mathcal{V}$. The individual vertices of $w$ are denoted by $w(0),w(1),\ldots$. The sets of all finite paths and all runs in $\mathcal{V}$ are denoted by $\mathit{FPath}_{\mathcal{V}}$ and $\mathit{Run}_{\mathcal{V}}$, respectively. The sets of all finite paths and all runs in $\mathcal{V}$ that start with a given finite path $w$ are denoted by $\mathit{FPath}_{\mathcal{V}}(w)$ and $\mathit{Run}_{\mathcal{V}}(w)$, respectively. A \emph{strongly connected component (SCC)} of $\mathcal{V}$ is a maximal subset $C \subseteq V$ such that for all $v,u \in C$ we have that $v \tran{}^* u$. A SCC~$C$ of $\mathcal{V}$ is a \emph{bottom SCC (BSCC)} of $\mathcal{V}$ if for all $v \in C$ and $u \in V$ such that $v \tran{} u$ we have that $u \in C$.
We assume familiarity with basic notions of probability theory, e.g., \emph{probability space}, \emph{random variable}, or the \emph{expected value}. As usual, a \emph{probability distribution} over a finite or countably infinite set $A$ is a function $f : A \rightarrow [0,1]$ such that \mbox{$\sum_{a \in A} f(a) = 1$}. We call $f$ \emph{positive} if $f(a) > 0$ for every $a \in A$, and \emph{rational} if $f(a) \in \mathbb{Q}$ for every $a \in A$.
\begin{definition} \label{def-Markov-chain}
A \emph{labeled Markov chain} is a tuple \mbox{$\mathcal{M} = (S,L,\tran{},\mathit{Prob})$}
where $S \neq \emptyset$ is a finite or countably infinite
set of \emph{states}, $L \neq \emptyset$ is a finite or countably
infinite set of \emph{labels},
\mbox{${\tran{}} \subseteq S \times L \times S$} is a total
\emph{transition relation}, and $\mathit{Prob}$ is a function that assigns
to each state $s \in S$
a positive probability distribution over the outgoing transitions
of~$s$. We write $s \ltran{\ell,x} t$ when $s \tran{\ell} t$
and $x$ is the probability of $(s,\ell,t)$. \end{definition}
\noindent If $L = \{\ell\}$ is a singleton, we say that $\mathcal{M}$ is \emph{non-labeled}, and we omit both $L$ and $\ell$ when specifying $\mathcal{M}$ (in particular, we write $s \tran{x} t$ instead of $s \ltran{\ell,x} t$).
To every $s \in S$ we associate the standard probability space $(\mathit{Run}_{\mathcal{M}}(s),\mathcal{F},\mathcal{P})$ of runs starting at $s$, where $\mathcal{F}$ is the \mbox{$\sigma$-field} generated by all \emph{basic cylinders} $\mathit{Run}_{\mathcal{M}}(w)$, where $w$ is a finite path starting at~$s$, and $\mathcal{P}: \mathcal{F} \rightarrow [0,1]$ is the unique probability measure such that $\mathcal{P}(\mathit{Run}_{\mathcal{M}}(w)) = \prod_{i{=}1}^{\mathit{length}(w)} x_i$ where $x_i$ is the probability of $w(i{-}1) \ltran{\ell_{i-1}} w(i)$ for every $1 \leq i \leq \mathit{length}(w)$. If $\mathit{length}(w) = 0$, we put $\mathcal{P}(\mathit{Run}_{\mathcal{M}}(w)) = 1$.
Now we introduce probabilistic multi-counter automata (pMC). For technical convenience, we consider \emph{labeled} rules, where the associated finite set of labels always contains a distinguished element~$\tau$. The role of the labels becomes clear in Section~\ref{sec-case2} where we abstract a (labeled) one-dimensional pMC from a given multi-dimensional one.
\begin{definition} \label{def-pVASS} Let $L$ be a finite set of labels such that $\tau \in L$, and let $d \in \mathbb{N}^+$. An $L$-labeled $d$-dimensional \emph{probabilistic multi-counter automaton (pMC)} is a triple $\mathcal{A} = (Q,\gamma,W)$, where \begin{itemize} \item $Q$ is a finite set of \emph{states}, \item \mbox{$\gamma \subseteq Q \times \{-1,0,1\}^d \times 2^{\{1,\ldots,d\}}
\times L \times Q$} is a set of \emph{rules} such that for all
$p \in Q$ and $c \subseteq \{1,\ldots,d\}$ there is at least one
outgoing rule of the form $(p,\vec{\alpha},c,\ell,q)$, \item $W : \gamma \rightarrow \mathbb{N}^+$ is a \emph{weight assignment}. \end{itemize} \end{definition}
\noindent The encoding size of $\mathcal{A}$ is denoted by $|\mathcal{A}|$, where the weights used in $W$ and the counter indexes used in $\gamma$ are encoded in binary.
A \emph{configuration} of $\mathcal{A}$ is an element of $Q \times \mathbb{N}^d$, written as $p\vec{v}$. We use $Z(p\vec{v}) = \{ i \mid 1\leq i \leq d, \vec{v}[i] = 0\}$ to denote the set of all counters that are zero in $p\vec{v}$. A rule $(p,\vec{\alpha},c,\ell,q) \in \gamma$ is \emph{enabled} in a configuration $p\vec{v}$ if $Z(p\vec{v}) = c$ and for all $1 \leq i \leq d$ where $\vec{\alpha}[i] = -1$ we have that $\vec{v}[i] > 0$.
The semantics of a $\mathcal{A}$ is given by the associated $L$-labeled Markov chain $\mathcal{M}_\mathcal{A}$ whose states are the configurations of~$\mathcal{A}$, and the outgoing transitions of a configuration $p\vec{v}$ are determined as follows: \begin{itemize} \item If no rule of $\gamma$ is enabled in $p\vec{v}$, then
$p\vec{v} \ltran{\tau,1} p\vec{v}$ is the only outgoing transition
of $p\vec{v}$; \item otherwise, for every rule $(p,\vec{\alpha},c,\ell,q) \in \gamma$ enabled in
$p\vec{v}$ there is a transition
$p\vec{v} \ltran{x,\ell} q\vec{u}$ such that
$\vec{u} = \vec{v}+\vec{\alpha}$ and
$x = W((p,\vec{\alpha},c,\ell,q))/T$, where $T$ is the total weight of all
rules enabled in $p\vec{v}$. \end{itemize}
\noindent When $L = \{\tau\}$, we say that $\mathcal{A}$ is \emph{non-labeled}, and both $L$ and $\tau$ are omitted when specifying $\mathcal{A}$. We say that $\mathcal{A}$ is a \emph{probabilistic vector addition system with states (pVASS)} if no subset of counters is tested for zero, i.e., for every $(p,\vec{\alpha},\ell,q) \in Q \times \{-1,0,1\}^d \times L \times Q$ we have that $\gamma$ contains either all rules of the form $(p,\vec{\alpha},c,\ell,q)$ (for all $c \subseteq \{1,\ldots,d\}$) with the same weight, or no such rule. For every configuration $p\vec{v}$, we use $\mathit{state}(p\vec{v})$ and $\mathit{cval}(p\vec{v})$ to denote the control state $p$ and the vector of counter values $\vec{v}$, respectively. We also use $\mathit{cval}_i(p\vec{v})$ to denote~$\vec{v}[i]$.
\noindent \textbf{Qualitative zero-reachability.} A \emph{stopping criterion} is a non-empty set \mbox{$\mathcal{Z} \subseteq 2^{\{1,\ldots,d\}}$} of pairwise incomparable non-empty subsets of counters. For every configuration $p\vec{v}$, let $\mathit{Run}(p\vec{v},\mathcal{Z})$ be the set of all $w \in \mathit{Run}(p\vec{v})$ such that there exist $k \in \mathbb{N}$ and $\varrho \in \mathcal{Z}$ satisfying $\varrho \subseteq Z(w(i))$. Intuitively, $\mathcal{Z}$ specifies the minimal subsets of counters that must be \emph{simultaneously} zero to stop a run. The \emph{qualitative \mbox{$\mathcal{Z}$-reachability} problem} is formulated as follows:
\noindent \textbf{Instance:} A \mbox{$d$-dimensional} pMC $\mathcal{A}$ and a control state $p$ of~$\mathcal{A}$. \textbf{Question:} Do we have $\mathcal{P}(\mathit{Run}(p\vec{1},\mathcal{Z})) = 1$ ?
\noindent Here $\vec{1}=(1,\ldots,1)$ is a $d$-dimensional vector of $1$'s. We also use $\mathit{Run}(p\vec{v},\neg\mathcal{Z})$ to denote $\mathit{Run}(p\vec{v}) \smallsetminus \mathit{Run}(p\vec{v},\mathcal{Z})$, and we say that $w \in \mathit{FPath}(p\vec{v})$ is \emph{$\mathcal{Z}$-safe} if for all $w(i)$ where $0 \leq i < \mathit{length}(w)$ and all $\varrho \in \mathcal{Z}$ we have that $\varrho \not\subseteq Z(w(i))$.
\section{The Results} \label{sec-results}
We start by observing that the qualitative zero-reachability problem is undecidable in general, and we identify potentially decidable subcases.
\begin{observation} \label{thm-undecidable}
Let \mbox{$\mathcal{Z} \subseteq 2^{\{1,\ldots,d\}}$} be a stopping criterion
satisfying one of the following conditions:
\begin{itemize}
\item[(a)] there is $\varrho \in \mathcal{Z}$ with more than one element;
\item[(b)] there are $i,j \in \{1,\ldots,d\}$ such that $i \neq j$ and
for every $\varrho \in \mathcal{Z}$ we have that $\{i,j\} \cap \varrho
= \emptyset$.
\end{itemize}
Then, the qualitative $\mathcal{Z}$-reachability problem is \emph{undecidable},
even if the set of instances is restricted to pairs $(\mathcal{A},p)$
such that $\mathcal{P}(\mathit{Run}(p\vec{1},\mathcal{Z}))$ is either~$0$ or~$1$ (hence,
$\mathcal{P}(\mathit{Run}(p\vec{1},\mathcal{Z}))$ cannot be effectively approximated
up to an absolute error smaller than~$0.5$). \end{observation}
\noindent A proof of Observation~\ref{thm-undecidable} is immediate. For a given Minsky machine $M$ (see \cite{Minsky:book}) with two counters initialized to one, we construct pMCs $\mathcal{A}_a$ and $\mathcal{A}_b$ of dimension $2$ and $3$, respectively, and a control state $p$ such that \begin{itemize} \item if $M$ halts, then $\mathcal{P}(\mathit{Run}_{\mathcal{M}_{\mathcal{A}_a}}(p\vec{1},\{\{1,2\}\})) = 1$
and $\mathcal{P}(\mathit{Run}_{\mathcal{M}_{\mathcal{A}_b}}(p\vec{1},\{\{3\}\})) = 1$; \item if $M$ does not halt, then
$\mathcal{P}(\mathit{Run}_{\mathcal{M}_{\mathcal{A}_a}}(p\vec{1},\{\{1,2\}\})) = 0$
and $\mathcal{P}(\mathit{Run}_{\mathcal{M}_{\mathcal{A}_b}}(p\vec{1},\{\{3\}\})) = 0$. \end{itemize} The construction of $\mathcal{A}_a$ and $\mathcal{A}_b$ is trivial (and hence omitted). Note that $\mathcal{A}_b$ can faithfully simulate the instructions of $M$ using the counters $1$ and $2$. The third counter is decreased to zero only when a control state corresponding to the halting instruction of $M$ is reached. Similarly, $\mathcal{A}_a$ simulates the instructions of $M$ using its two counters, but here we need to ensure that a configuration where \emph{both} counters are simultaneously zero is entered iff a control state corresponding to the halting instruction of $M$ is reached. This is achieved by increasing both counters by $1$ initially, and then decreasing/increasing counter~$i$ before/after simulating a given instruction of $M$ operating on counter~$i$.
Note that the construction of $\mathcal{A}_a$ and $\mathcal{A}_b$ can trivially be adapted to pMCs of higher dimensions satisfying the conditions~(a) and~(b) of Observation~\ref{thm-undecidable}, respectively. However, there are two cases not covered by Observation~\ref{thm-undecidable}: \begin{itemize} \item[I.] $\Z_{all} = \{\{1\},\ldots,\{d\}\}$, i.e., a run is stopped
when \emph{some} counter reaches zero. \item[II.] $\Zminusi{i} = \{\{1\},\ldots,\{d\}\} \smallsetminus \{\{i\}\}$ where
$i \in \{1,\ldots,d\}$, i.e., a run is stopped when a counter different
from~$i$ reaches zero. The counters different from~$i$ are called
\emph{stopping counters}. \end{itemize}
These cases are analyzed in the following subsections.
\subsection{Zero-Reachability, Case~I} \label{sec-case1}
For the rest of this section, let us fix a (non-labeled) pMC $\mathcal{A} = (Q,\gamma,W)$ of dimension $d \in \mathbb{N}^+$ and a configuration $p\vec{v}$.
Our aim is to identify the conditions under which $\mathcal{P}(\mathit{Run}(p\vec{v},\neg\Z_{all})) > 0$. To achieve that, we first consider a (non-labeled) finite-state Markov chain $\mathcal{F}_{\mathcal{A}} = (Q,\btran{},\mathit{Prob})$ where $q \btran{x} r$ iff \[
x \quad = \quad
\sum_{(q,\vec{\alpha},\emptyset,r) \in \gamma} P_\emptyset(q,\vec{\alpha},\emptyset,r)
\quad > \quad 0. \] Here $P_\emptyset : \gamma \rightarrow [0,1]$ is the probability assignment for the rules defined as follows (we write $P_\emptyset(q,\vec{\alpha},\emptyset,r)$ instead of $P_\emptyset((q,\vec{\alpha},\emptyset,r))$): \begin{itemize} \item For every rule $(p,\vec{\alpha},c,q)$ where $c \neq \emptyset$
we put $P_\emptyset(p,\vec{\alpha},c,q) = 0$. \item $P_\emptyset(p,\vec{\alpha},\emptyset,q) = W((p,\alpha,\emptyset,q))/T$,
where $T$ is the total weight of all rules of the form
$(p,\vec{\alpha}',\emptyset,q')$. \end{itemize} Intuitively, a state $q$ of $\mathcal{F}_{\mathcal{A}}$ captures the behavior of configurations $q \vec{u}$ where all components of $\vec{u}$ are positive.
Further, we partition the states of $Q$ into SCCs $C_1,\ldots,C_m$ according to~$\hookrightarrow$. Note that every run $w \in \mathit{Run}(p\vec{v})$ eventually \emph{stays} in precisely one $C_j$, i.e., there is precisely one $1\leq j \leq m$ such that for some $k \in \mathbb{N}$, the control state of every $w(k')$, where $k' \geq k$, belongs to~$C_i$. We use $\runzc{p\vec{v}}{C_j}$ to denote the set of all $w \in \mathit{Run}(p\vec{v},\neg\Z_{all})$ that stay in~$C_j$. Obviously, \[
\mathit{Run}(p\vec{v},\neg\Z_{all}) =
\mathit{Run}(p\vec{v},C_1) \uplus \cdots \uplus \runzc{p\vec{v}}{C_m}. \]
For any $n \in \mathbb{N}$ denote by $P_n$ the probability that a run $w$ initiated in $p\vec{v}$ satisfies the following for every $0\leq i \leq n$: $\mathit{state}(w(i))$ does not belong to any BSCC of $\mathcal{F}_\mathcal{A}$ and $Z(w(i))=\emptyset$. The following lemma shows that $P_n$ decays exponentially fast.
\begin{lemma}\label{lem:F_A-BSCC}
For any $n \in \mathbb{N}$ we have $$P_n \leq (1 - p_{\mathit{min}}^{|Q|})^{\lfloor\frac{n}{|Q|}\rfloor},$$ where $p_{\mathit{min}}$ is the minimal positive transition probability in $\mathcal{M}_\mathcal{A}$.
In particular, for any non-bottom SCC $C$ of $\mathcal{F}_\mathcal{A}$ we have
$\mathcal{P}(\mathit{Run}(p\vec{v},C)) = 0$. \end{lemma} \begin{proof}
The lemma immediately follows from the fact that for every configuration $p\vec{v}$ there is a path (in $\mathcal{A}$) of length at most $|Q|$ to a configuration $q\vec{u}$ satisfying either $Z(q\vec{u})\neq \emptyset$ or $q\in D$ for some BSCC $D$ of $\mathcal{F}_\mathcal{A}$. \end{proof}
Now, let $C$ be a BSCC of $\mathcal{F}_{\mathcal{A}}$. For every $q \in C$, let $\mathit{change}^q$ be a $d$-dimensional vector of \emph{expected counter changes} given by \[
\mathit{change}^q_i =
\sum_{(q,\vec{\alpha},\emptyset,r) \in \gamma}
P_{\emptyset}(q,\vec{\alpha},\emptyset,r) \cdot \vec{\alpha}[i] \,. \]
Note that $C$ can be seen as a finite-state irreducible Markov chain, and hence there exists the unique \emph{invariant distribution} $\mu$ on the states of~$C$ (see, e.g.,~\cite{KS:book}) satisfying \[ \mu(q)\quad =\quad \sum_{r\btran{x} q} \mu(r)\cdot x \,. \]
The \emph{trend} of $C$ is a $d$-dimensional vector $\vec{t}$ defined by \[
\vec{t}[i] \quad = \quad \sum_{q \in C} \mu(q) \cdot \mathit{change}^q_i \,. \]
Further, for every $i \in \{1,\ldots,d\}$ and every $q\in C$, we denote by $\mathit{botfin}_i(q)$ the \emph{least} $j \in \mathbb{N}$ such that for every configuration $q \vec{u}$ where $\vec{u}[i] = j$, there is \emph{no} $w \in \mathit{FPath}_{\mathcal{M}_\mathcal{A}}(q \vec{u})$ where counter~$i$ is zero in the last configuration of $w$ and all counters stay positive in every $w(k)$, where $0 \leq k < \mathit{length}(w)$. If there is no such $j$, we put $\mathit{botfin}_i(q) = \infty$. It is easy to show that if $\mathit{botfin}_i(q) < \infty$, then $\mathit{botfin}_i(q)
\leq |C|$; and if $\mathit{botfin}_i(q) = \infty$, then $\mathit{botfin}_i(r)
= \infty$ for all $r \in C$. Moreover, if $\mathit{botfin}_i(q) < \infty$, then there is a $\Zminusi{i}$-safe finite path of length at most $|C|-1$ from $q\vec{u}$ to a configuration with $i$-th counter equal to 0, where $\vec{u}[i]=\mathit{botfin}_i(q)-1$ and $\vec{u}[\ell]=|C|$ for $\ell\neq i$. In particular, the number $\mathit{botfin}_i(q)$ is computable in time polynomial in $|C|$.
We say that counter~$i$ is \emph{decreasing} in $C$ if $\mathit{botfin}_i(q) = \infty$ for some (and hence all) $q \in C$.
\begin{definition}
Let $C$ be a BSCC of $\mathcal{F}_{\mathcal{A}}$ with trend $\vec{t}$, and let
$i \in \{1,\ldots,d\}$. We say that counter~$i$ is \emph{diverging}
in $C$ if either $\vec{t}[i] > 0$, or $\vec{t}[i] = 0$ and the counter~$i$
is not decreasing in~$C$. \end{definition}
Intuitively, our aim is to prove that $\mathcal{P}(\mathit{Run}(p\vec{v},C))>0$ iff all counters are diverging in $C$ and $p\vec{v}$ can reach a configuration $q\vec{u}$ (via a $\Z_{all}$-safe finite path) where all components of $\vec{u}$ are ``sufficiently large''. To analyze the individual counters, for every $i \in \{1,\ldots,d\}$ we introduce a~(labeled) \emph{one-dimensional} pMC which faithfully simulates the behavior of counter~$i$ and ``updates'' the other counters just symbolically in the labels.
\begin{definition}
Let $L = \{-1,0,1\}^{d-1}$, and let $\mathcal{B}_i = (Q,\hat{\gamma},\hat{W})$
be an $L$-labeled pMC of dimension one such that
\begin{itemize}\itemsep1ex
\item $(q,j,\emptyset,\vec{\beta},r) \in \hat{\gamma}$ \ iff \
$(q,\langle \vec{\beta},j\rangle_i,\emptyset,r) \in \gamma$;
\item $(q,j,\{1\},\vec{\beta},r) \in \hat{\gamma}$ \ iff \
$(q,\langle \vec{\beta},j\rangle_i,\{i\},r) \in \gamma$;
\item $\hat{W}(q,j,\emptyset,\vec{\beta},r) =
W(q,\langle \vec{\beta},j \rangle_i,\emptyset,r)$.
\item $\hat{W}(q,j,\{1\},\vec{\beta},r) =
W(q,\langle \vec{\beta},j \rangle_i,\{i\},r)$.
\end{itemize}
Here,
$\langle (j_1,\ldots,j_{d-1}),j\rangle_i =
(j_1,\ldots,j_{i-1},j,j_i,\ldots,j_{d-1})$. \end{definition}
\noindent Observe that the symbolic updates of the counters different from~$i$ ``performed'' in the labels of $\mathcal{B}_i$ mimic the real updates performed by $\mathcal{A}$ in configurations where all of these counters are positive.
Given a run $w\equiv p_0(v_0) \, \vec{\alpha}_0 \, p_1(v_1) \, \vec{\alpha}_1 \, p_2(v_2) \, \vec{\alpha}_2 \, \ldots$ in $\mathit{Run}_{\mathcal{M}_{\mathcal{B}}}(p_0(v_0))$ and $k\in \mathbb{N}$, we denote by $\totalrew{}{w}{0}{k}$ the vector $\sum_{n=0}^{k-1} \vec{\alpha}_n$, and given $j\in \{1,\ldots,d\}\smallsetminus \{i\}$, we denote by $\totalrew{j}{w}{0}{k}$ the number $\sum_{n=0}^{k-1} \vec{\alpha}_n[j]$ (i.e., the $j$-th component of $\sum_{n=0}^{k-1} \vec{\alpha}_n$).
Let $\Upsilon_i$ be a function which for a given run $w \equiv p_0\vec{v}_0\,p_1\vec{v}_1\, p_2\vec{v}_2\ldots$ of $\mathit{Run}_{\mathcal{M}_\mathcal{A}}(p\vec{v},\neg\Zminusi{i})$ returns a run $\Upsilon_i(w) \equiv p_0(\vec{v}_0[i]) \, \vec{\alpha}_0 \, p_1(\vec{v}_1[i]) \, \vec{\alpha}_1 \, p_2(\vec{v}_2[i]) \, \vec{\alpha}_2 \, \ldots$ of $\mathit{Run}_{\mathcal{M}_{\mathcal{B}_i}}(p(\vec{v}[i]))$ where the label $\vec{\alpha}_j$ corresponds to the update in the abstracted counters performed in the transition $p_j\vec{v}_j \tran{} p_{j+1}\vec{v}_{j+1}$, i.e., $\vec{v}_{j+1} - \vec{v}_j = \langle \vec{\alpha}_j,\vec{v}_{j+1}[i] - \vec{v}_j[i] \rangle_i$.
The next lemma is immediate. \begin{lemma}\label{prop:one-counter-runs} For all $w \in \mathit{Run}_{\mathcal{M}_\mathcal{A}}(p\vec{v},\neg\Zminusi{i})$ and $k \in \mathbb{N}$ we have that \begin{itemize} \item $\mathit{state}(w(k))=\mathit{state}(\Upsilon_i(w)(k))$, \item $\mathit{cval}(w(k)) = \langle \totalrew{}{\Upsilon_i(w)}{0}{k},
\mathit{cval}_1(\Upsilon_i(w)(k))\rangle_i$. \end{itemize} Further, for every measurable set $R\subseteq \mathit{Run}_{\mathcal{M}_\mathcal{A}}(p\vec{v},\neg\Zminusi{i})$ we have that $\Upsilon_i(R)$ is measurable and \begin{equation} \mathcal{P}(R) \ = \
\mathcal{P}(\Upsilon_i(R)) \label{eq-project} \end{equation} \end{lemma}
\noindent Now we examine the runs of $\mathit{Run}(p\vec{v},C)$ where $C$ is a BSCC of $\mathcal{F}_\mathcal{A}$ such that some counter is not diverging in~$C$. A proof of the next lemma can be found in Appendix~\ref{app-sec1}.
\begin{lemma} \label{lem:not-diverging}
Let $C$ be a BSCC of $\mathcal{F}_{\mathcal{A}}$.
If some counter is not diverging in $C$, then $\mathcal{P}(\mathit{Run}(p\vec{v},C)) = 0$. \end{lemma}
It remains to consider the case when $C$ is a BSCC of $\mathcal{F}_{\mathcal{A}}$ where all counters are diverging. Here we use the results of \cite{BKK:pOC-time-LTL-martingale} which allow to derive a bound on divergence probability in one-dimensional pMC. These results are based on designing and analyzing a suitable martingale for one-dimensional pMC.
\begin{lemma} \label{lem-divergence}
Let $\mathcal{B}$ be a $1$-dimensional pMC, let $C$ be a BSCC of $\mathcal{F}_\mathcal{B}$
such that the trend $t$ of the only counter in $C$ is positive and let $\delta=2|C|/x_{\min}^{|C|}$ where $x_{\min}$ is the smallest non-zero transition probability in $\mathcal{M}_\mathcal{B}$.
Then for all $q \in C$ and $k > 2\delta/t$ we have that
$\mathcal{P}(q(k),\neg\mathcal{Z}) \geq 1-\left(a^k/(1+a)\right)$, where $\mathcal{Z} = \{1\}$ and $a=\exp\left(-t^2\, /\, 8(\delta+t+1)^2\right)$. \end{lemma} \begin{proof}
Denote by $[q(k){\downarrow},\ell]$ the probability that a run initiated in $q(k)$ visits a configuration with zero counter value for the first time in exactly $\ell$ steps. By Proposition~7 of \cite{BKK:pOC-time-LTL-martingale-arxiv} we obtain for all $\ell\geq h = 2\delta/t$~\footnote{The precise bound on $h$ is given in Proposition~7~\cite{BKK:pOC-time-LTL-martingale-arxiv}.},
\[ [q(k){\downarrow},\ell]\quad \leq\quad a^\ell \]
where $a=\exp\left(-t^2\, /\, 8(\delta+t+1)^2\right)$ for $\delta\leq 2|C|/x_{\min}^{|C|}$~\footnote{The bound on $\delta$ is given in Proposition 6~\cite{BKK:pOC-time-LTL-martingale-arxiv}.}.
Thus \[ \mathcal{P}(q(k),\neg\mathcal{Z})\geq 1-\sum_{\ell=k}^{\infty} [q(k){\downarrow},\ell]=1-\frac{a^k}{1+a} \] \end{proof}
\begin{definition}
Let $C$ be a BSCC of $\mathcal{F}_{\mathcal{A}}$ where all counters are diverging,
and let $q \in C$. We say that a configuration $q\vec{u}$
is \emph{above} a given $n \in \mathbb{N}$ if $\vec{u}[i] \geq n$ for every
$i$
such that $\vec{t}[i] > 0$, and
$\vec{u}[i] \geq \mathit{botfin}_i(q)$ for every
$i$
such that $\vec{t}[i] = 0$. \end{definition}
\begin{lemma} \label{lem-diverging}
Let $C$ be a BSCC of $\mathcal{F}_{\mathcal{A}}$ where all counters are diverging.
Then $\mathcal{P}(\mathit{Run}(p\vec{v},C)) > 0$ iff there is a $\Z_{all}$-safe finite
path of the form $p\vec{v} \tran{}^* q\vec{u} \tran{}^* q\vec{z}$
where $q \in C$, $q\vec{u}$ is above $1$, $\vec{z} - \vec{u} \geq \vec{0}$,
and $(\vec{z} - \vec{u})[i] > 0$ for every $i$ such that $\vec{t}[i] > 0$.
\end{lemma} \begin{proof}
We start with ``$\Rightarrow$''.
Let $\vec{t}$ be the trend of $C$. We show that for almost
all $w \in \mathit{Run}(p\vec{v},C)$ and all $i \in \{1,\ldots,d\}$, one of
the following conditions holds:
\begin{enumerate}
\item[(A)] $\vec{t}[i]>0$ and $\liminf_{k\rightarrow \infty} \mathit{cval}_i(w(k))=\infty$,
\item[(B)] $\vec{t}[i]=0$ and $\mathit{cval}_i(w(k))\geq \mathit{botfin}_i(\mathit{state}(w(k)))$
for all $k$'s large enough.
\end{enumerate}
First, recall that $C$ is also a BSCC of $\mathcal{F}_{\mathcal{B}_i}$, and realize that
the trend of the (only) counter in the BSCC $C$ of $\mathcal{F}_{\mathcal{B}_i}$ is~$\vec{t}[i]$.
Concerning~(A), it follows, e.g., from the results of
\cite{BKK:pOC-time-LTL-martingale}, that almost all runs
$w' \in \mathit{Run}_{\mathcal{M}_{\mathcal{B}_i}}(p(1))$ that stay
in $C$ and do not visit a configuration with zero counter
satisfy $\liminf_{k\rightarrow \infty} \mathit{cval}_1(w'(k))=\infty$.
In particular, this means that almost all
$w' \in \Upsilon_i(\mathit{Run}(p\vec{v},C))$
satisfy this property.
Hence, by Lemma~\ref{prop:one-counter-runs}, for almost all
$w \in \mathit{Run}(p\vec{v},C)$ we have that
\mbox{$\liminf_{k\rightarrow \infty} \mathit{cval}_i(w(k))=\infty$}.
Concerning (B), note that almost all runs $w \in \mathit{Run}(p\vec{v},C)$
satisfying $\mathit{cval}_i(w'(k)) < \mathit{botfin}_i(\mathit{state}(w(k)))$
for infinitely many $k$'s eventually visit zero in some counter
(there is a path of length at most $|C|$ from each such
$w(k)$ to a configuration with zero in counter $i$, or in one of the
other counters).
The above claim immediately implies that for every $k \in \mathbb{N}$,
almost every run of $\mathit{Run}(p\vec{v},C)$ visits a configuration
$q\vec{u}$ above~$k$. Hence, there must be a $\Z_{all}$-safe
path of the form $p\vec{v} \tran{}^* q\vec{u} \tran{}^* q\vec{z}$
with the required properties.
``$\Leftarrow$'': If there is a $\Z_{all}$-safe
path of the form $p\vec{v} \tran{}^* q\vec{u} \tran{}^* q\vec{z}$
where $q \in C$, $q\vec{u}$ is above $1$, $\vec{z} - \vec{u} \geq \vec{0}$,
and $(\vec{z} - \vec{u})[i] > 0$ for every $i$ such that $\vec{t}[i] > 0$, then
$p\vec{v}$ can a reach a configuration $q\vec{y}$ above~$k$ for an
arbitrarily large $k \in \mathbb{N}$ via a $\Z_{all}$-safe path.
By Lemma~\ref{lem-divergence}, there exists
$k \in \mathbb{N}$ such that for every $i \in \{1,\ldots,d\}$ where
$\vec{t}[i] > 0$ and every $n \geq k$,
the probability of all $w \in \mathit{Run}_{\mathcal{M}_{\mathcal{B}_i}}(q(n))$ that
visit a configuration with zero counter is strictly smaller than
\mbox{$1/d$}. Let $q\vec{y}$ be a configuration above~$k$ reachable
from $p\vec{v}$ via a $\Z_{all}$-safe path (the existence of such
a $q\vec{y}$ follows from the existence of
$p\vec{v} \tran{}^* q\vec{u} \tran{}^* q\vec{z}$). It suffices
to show that $\mathcal{P}(\mathit{Run}(q\vec{y},\Z_{all})) < 1$. For every
$i \in \{1,\ldots,d\}$ where $\vec{t}[i] > 0$,
let $R_i$ be the set of all $w \in \mathit{Run}(q\vec{y},\Z_{all})$ such that
$\mathit{cval}_i(w(k)) = 0$ for some $k \in \mathbb{N}$ and
all counters stay positive in all $w(k')$ where $k' < k$.
Clearly, $\mathit{Run}(q\vec{y},\Z_{all}) = \bigcup_{i} R_i$, and thus we obtain
\begin{equation*}
\mathcal{P}(\mathit{Run}(q\vec{y},\Z_{all})) \leq \sum_{i} \mathcal{P}(R_i) =
\sum_{i} \mathcal{P}(\Upsilon_i(R_i)) < d \cdot \frac{1}{d} = 1
\end{equation*} \end{proof}
The following lemma shows that it is possible to decide, whether for a given $n\in\mathbb{N}$ a configuration above $n$ can be reached via a $\Z_{all}$-safe path. Its proof uses the results of~\cite{BG:VASS-coverability} on the coverability problem in (non-stochastic) VASS.
\begin{lemma} \label{lem:cover-short-path}
Let $C$ be a BSCC of $\mathcal{F}_{\mathcal{A}}$ where all counters are diverging and let $q\in C$. There is a $\Z_{all}$-safe finite path of the form $p\vec{v}\tran{}^* q\vec{u}$ with $q\vec{u}$ is above some $n\in \mathbb{N}$ iff there is a $\Z_{all}$-safe finite path of length at most $(|Q|+|\gamma|)\cdot(3+n)^{(3d)!+1}$ of the form $p\vec{v}\tran{}^* q\vec{u'}$ with $q\vec{u'}$ is above $n$.
Moreover, the existence of such a path can be decided in time $(|\mathcal{A}|\cdot n)^{c'\cdot 2^{d\log(d)}}$ where $c'$ is a fixed constant independent of $d$ and $\mathcal{A}$. \end{lemma} \begin{proof}
We employ a decision procedure of \cite{BG:VASS-coverability} for
VASS coverability. Since we need to reach $q\vec{u'}$ above $n$ via
a $\Z_{all}$-safe finite path, we transform $\mathcal{A}$ into a
(non-probabilistic) VASS $\mathcal{A}'$ whose control states and rules are
determined as follows: for every rule $(p,\vec{\alpha},\emptyset,q)$
of $\mathcal{A}$, we add to $\mathcal{A}'$ the control states $p,q$ together with two
auxiliary fresh control states $q',q''$, and we also add the rules
$(p,\vec{-1},q')$, $(q',\vec{1},q'')$,
$(q'',\vec{\alpha},q)$. Hence, $\mathcal{A}'$ behaves like $\mathcal{A}$, but when
some counter becomes zero, then $\mathcal{A}'$ is stuck (i.e., no transition
is enabled except for the self-loop). Now it is easy to check that
$p\vec{v}$ can reach a configuration $q\vec{u}$ above $n$ via a
$\Z_{all}$-safe finite path in $\mathcal{A}$ iff $p\vec{v}$ can reach a
configuration $q\vec{u}$ above $n$ via \emph{some} finite path in
$\mathcal{A}'$, which is exactly the coverability problem for VASS.
Theorem~1 in~\cite{BG:VASS-coverability} shows that such a
configuration can be reached iff there is configuration $q\vec{u'}$
above $n$ reachable via some finite path of length at most $m =
(|Q|+|\gamma|)\cdot(3+n)^{(3d)!+1}$. (The term $(|Q|+|\gamma|)$
represents the number of control states of $\mathcal{A}'$.) This path
induces, in a natural way, a $\Z_{all}$-safe path from $p\vec{v}$ to
$q\vec{u'}$ in $\mathcal{A}$ of length at most $m/2$. Moreover, Theorem~2
in~\cite{BG:VASS-coverability} shows that the existence of such a
path in $\mathcal{A}'$ can be decided in time
$(|Q|+|\gamma|)\cdot(3+n)^{2^{\mathcal{O}(d\log(d))}}$, which proves
the lemma.
\end{proof}
\begin{theorem} \label{thm:qual-all-algorithm}
The qualitative $\Z_{all}$-reachability problem for \mbox{$d$-dimensional}
pMC is decidable in time $|\mathcal{A}|^{\kappa \cdot 2^{d\log(d)}}$, where $\kappa$ is
a fixed constant independent of $d$ and $\mathcal{A}$. \end{theorem} \begin{proof}
Note that the Markov chain $\mathcal{F}_\mathcal{A}$ is computable in time polynomial
in $|\mathcal{A}|$ and $d$, and we can efficiently identify all diverging
BSCCs of $\mathcal{F}_\mathcal{A}$. For each diverging BSCC $C$, we need to check the
condition of Lemma~\ref{lem-diverging}. By applying Lemma~2.3.{}
of \cite{RY:VASS-JCSS}, we obtain that if there exist \emph{some} $q\vec{u}$
above~$1$ and a $\Z_{all}$-safe finite path of the form
$q\vec{u} \tran{}^* q\vec{z}$ such that $\vec{z} - \vec{u} \geq \vec{0}$
and $(\vec{z} - \vec{u})[i] > 0$ for every $i$ where $\vec{t}[i] > 0$,
then such a path exists for \emph{every} $q\vec{u}$ above
$|\mathcal{A}|^{c\cdot d}$ and its length is bounded by $|\mathcal{A}|^{c\cdot d}$. Here
$c$ is a fixed constant independent of $|\mathcal{A}|$ and~$d$ (let us
note that Lemma~2.3.{} of \cite{RY:VASS-JCSS} is formulated for
vector addition
systems without states and a non-strict increase in every counter,
but the corresponding result for VASS is easy
to derive; see also Lemma~15 in \cite{BJK:VASS-games-arxiv}).
Hence, the existence
of such a path for a given $q \in C$ can be decided in
$\mathcal{O}(|\mathcal{A}|^{c\cdot d})$ time. It remains to check whether $p\vec{v}$
can reach a configuration $q\vec{u}$ above $|\mathcal{A}|^{c\cdot d}$ via
a $\Z_{all}$-safe finite path. By Lemma~\ref{lem:cover-short-path} this can be done in time $(|\mathcal{A}|\cdot |\mathcal{A}|^{c\cdot d})^{c'\cdot 2^{d\log(d)}}$ for another constant $c'$. This gives us the desired complexity bound. \end{proof}
Note that for every fixed dimension $d$, the qualitative \mbox{$\Z_{all}$-reachability} problem is solvable in polynomial time.
Now we show that $\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))$ can be effectively approximated up to an arbitrarily small absolute/relative error $\varepsilon > 0$. A full proof of Theorem~\ref{thm:approx-general} can be found in Appendix~\ref{app-approx}.
\begin{theorem}
\label{thm:approx-general}
For a given $d$-dimensional pMC $\mathcal{A}$ and its initial configuration
$p\vec{v}$, the probability $\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))$ can be
approximated up to a given absolute error $\varepsilon>0$ in time
$(\exp(|\mathcal{A}|)\cdot \log(1/\varepsilon))^{\mathcal{O}(d\cdot d!)}$. \end{theorem} \begin{proof}[Proof sketch]
First we check whether $\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all})) = 1$
(using the algorithm of Theorem~\ref{thm:qual-all-algorithm}) and
return $1$ if it is the case. Otherwise, we first show how
to approximate $\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))$ under the
assumption that $p$ is in some diverging BSCC of $\mathcal{F}_\mathcal{A}$, and
then we show how to drop this assumption.
So, let $C$ be a diverging BSCC of $\mathcal{F}_\mathcal{A}$ such that
$\mathcal{P}(\mathit{Run}(p\vec{v},C))<1$, and let us assume that $p \in C$.
We show how to compute $\nu >0$ such that
$|\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))-\nu|\leq d\cdot\varepsilon$ in time
$(\exp(|\mathcal{A}|)\cdot \log(1/\varepsilon))^{\mathcal{O}(d!)}$. We proceed by induction
on~$d$. The key idea of the inductive step is to find a sufficiently
large constant~$K$ such that if some counter reaches~$K$, it can
be safely ``forgotten'', i.e., replaced by $\infty$, without influencing
the probability of reaching zero in some counter by more than
$\varepsilon$. Hence, whenever we visit a configuration $q\vec{u}$
where some counter value in $\vec{u}$ reaches $K$, we can
apply induction hypothesis and approximate the probability or reaching
zero in some counter from $q\vec{u}$ by ``forgetting'' the large
counter a thus reducing the dimension. Obviously, there are only
finitely many configurations where all counters are below~$K$, and
here we employ the standard methods for finite-state Markov chains.
The number $K$ is computed by using
the bounds of Lemma~\ref{lem-divergence}.
Let us note that the base (when $d=1$) is handled by relying only
on Lemma~\ref{lem-divergence}. Alternatively, we could employ
the results of \cite{ESY:polynomial-time-termination}. This would
improve the complexity for $d=1$, but not for higher
dimensions.
Finally, we show how to approximate $\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))$
when the control state $p$ does not belong to a BSCC of~$\mathcal{F}_\mathcal{A}$.
Here we use the bound of Lemma~\ref{lem:F_A-BSCC}. \end{proof}
Note that if $\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all})) > 0$, then this probability is at least $p_{\mathit{min}}^{m \cdot |Q|}$ where $p_{\mathit{min}}$ is the least positive transition probability in $\mathcal{M}_\mathcal{A}$ and $m$ is the maximal component of $\vec{v}$. Hence, Theorem~\ref{thm:approx-general} can also be used to approximate $\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))$ up to a given \emph{relative} error $\varepsilon > 0$.
\subsection{Zero-Reachability, Case~II} \label{sec-case2}
\newcommand{\mathcal{D}}{\mathcal{D}} \newcommand{S^*}{S^*}
Let us fix a (non-labeled) pMC $\mathcal{A} = (Q,\gamma,W)$ of dimension $d \in \mathbb{N}^+$ and $i\in \{1,\ldots,d\}$. As in the previous section, our aim is to identify the conditions under which $\mathit{Run}(p\vec{1},\neg\Zminusi{i})>0$. Without restrictions, we assume that $i = d$, i.e., we consider $\Zminusi{d} = \{\{1\},\ldots,\{d-1\}\}$. Also, for technical reasons, we assume that $\mathit{Run}(p\vec{1},\neg\Zminusi{d})=\mathit{Run}(p\vec{u}^{in},\neg\Zminusi{d})$ where $\vec{u}^{in}_i=1$ for all $i\in \{1,\ldots,d-1\}$ but $\vec{u}^{in}_d=0$. (Note that every pMC can be easily modified in polynomial time so that this condition is satisfied.)
To analyze the runs of $\mathit{Run}(p\vec{u}^{in},\neg\Zminusi{d})$, we re-use the finite-state Markov chain $\mathcal{F}_\mathcal{A}$ introduced in Section~\ref{sec-case1}. Intuitively, the chain $\mathcal{F}_\mathcal{A}$ is useful for analyzing those runs of $\mathit{Run}(p\vec{u}^{in},\neg\Zminusi{d})$ where \emph{all} counters stay positive. Since the structure of $\mathit{Run}(p\vec{u}^{in},\neg\Zminusi{d})$ is more complex than in Section~\ref{sec-case1}, we also need some new analytic tools.
We also re-use the $L$-labeled $1$-dimensional pMC $\mathcal{B}_d$ to deal with runs that visit zero in counter $d$ infinitely many times. To simplify notation, we use $\mathcal{B}$ to denote $\mathcal{B}_d$. The behaviour of $\mathcal{B}$ is analyzed using the finite-state Markov chain $\mathcal{X}$ (see~Definition~\ref{def:X} below) that has been employed already in \cite{BKK:pOC-time-LTL-martingale} to design a model-checking algorithm for linear-time properties and one-dimensional pMC.
Let us denote by $[q{\downarrow}r]$ the probability that a run of $\mathcal{M}_\mathcal{B}$ initiated in $q(0)$ visits the configurations $r(0)$ without visiting any configuration of the form $r'(0)$ (where $r' \ne r$) in between.
Given $q\in Q$, we denote by $[q{\uparrow}]$ the probability $1-\sum_{r\in Q} [q{\downarrow}r]$ that a run initiated in $q(0)$ never visits a configuration with zero counter value (except for the initial one). \begin{definition}\label{def:X}
Let $\mathcal{X}_{\mathcal{B}} = (X,\tran{},\mathit{Prob})$ be a non-labelled finite-state Markov
chain where $X = Q\cup \{q{\uparrow} \mid q \in Q\}$ and
the~transitions are defined as follows:
\begin{itemize}\itemsep1ex
\item $q \tran{x} r$ \ iff \ $0 < x = [q{\downarrow}r]$;
\item $q \tran{x} q{\uparrow}$ \ iff \ $0 < x = [q{\uparrow}]$;
\item there are no other transitions.
\end{itemize} \end{definition}
\noindent The correspondence between the runs of $\mathit{Run}_{\mathcal{M}_{\mathcal{B}}}(p(0))$ and $\mathit{Run}_{\mathcal{X}_{\mathcal{B}}}(p)$ is formally captured by a function $\Phi : \mathit{Run}_{\mathcal{M}_{\mathcal{B}}}(p(0)) \rightarrow \mathit{Run}_{\mathcal{X}_{\mathcal{B}}}(p) \cup \{\perp\}$, where $\Phi(w)$ is obtained from a given $w \in \mathit{Run}_{\mathcal{M}_{\mathcal{B}}}(p(0))$ as follows: \begin{itemize} \item First, each \emph{maximal} subpath in $w$ of the form
$q(0),\ldots,r(0)$ such that the counter stays positive in all
of the intermediate configurations is replaced with a single
transition $q \tran{} r$. \item
Note that if $w$ contained infinitely many
configurations with zero counter, then the resulting sequence
is a run of $\mathit{Run}_{\mathcal{X}_{\mathcal{B}}}(p)$, and thus we obtain our $\Phi(w)$.
Otherwise, the resulting sequence takes the form $v\, \hat{w}$, where
$v \in \mathit{FPath}_{\mathcal{X}_{\mathcal{B}}}(p)$ and $\hat{w}$ is a suffix
of~$w$ initiated in a configuration~$r(1)$. Let $q$ be the last state of $v$. Then, $\Phi(w)$ is either $v\,(q{\uparrow})^\omega$ or $\perp$, depending on
whether $[q{\uparrow}] > 0$ or not, respectively (here,
$(q{\uparrow})^\omega$ is a infinite sequence of $q{\uparrow}$). \end{itemize}
\begin{lemma} \label{lem-measures}
For every measurable subset $R \subseteq \mathit{Run}_{\mathcal{X}_{\mathcal{B}}}(p)$
we have that $\Phi^{-1}(R)$ is measurable and
$\mathcal{P}(R) = \mathcal{P}(\Phi^{-1}(R))$. \end{lemma}
A proof of Lemma~\ref{lem-measures} is straightforward (it suffices to check that the lemma holds for all basic cylinders $\mathit{Run}_{\mathcal{X}_{\mathcal{B}}}(w)$ where $w \in \mathit{FPath}_{\mathcal{X}_{\mathcal{B}}}(p)$). Note that Lemma~\ref{lem-measures} implies $\mathcal{P}(\Phi{=}{\perp}) = 0$.
Let $D_1,\ldots,D_k$ be all BSCCs of $\mathcal{X}_{\mathcal{B}}$ reachable from $p$. Further, for every $D_j$, we use $\mathit{Run}(p\vec{u}^{in},D_j)$ to denote the set of all $w \in \mathit{Run}_{\mathcal{M}_\mathcal{A}}(p\vec{u}^{in},\neg\Zminusi{d})$ such that $\Phi(\Upsilon_d(w)) \neq {\perp}$ and $\Phi(\Upsilon_d(w))$ visits $D_j$. Observe that \begin{equation}
\mathcal{P}(\mathit{Run}_{\mathcal{M}_\mathcal{A}}(p\vec{u}^{in},\neg\Zminusi{d})) \ = \
\sum_{j=1}^k \mathcal{P}(\mathit{Run}(p\vec{u}^{in},D_j)) \label{eq-BSCC-D} \end{equation} Indeed, note that almost all runs $w$ of $\mathit{Run}_{\mathcal{X}_{\mathcal{B}}}(p)$ visit some $D_j$, and hence by Lemma~\ref{lem-measures}, we obtain that $\Phi(w)$ visits some $D_j$ for almost all $w \in \mathit{Run}_{\mathcal{M}_{\mathcal{B}}}(p(1))$. In particular, for almost all $w$ of $\Upsilon_d(\mathit{Run}_{\mathcal{M}_\mathcal{A}}(p\vec{u}^{in},\neg\Zminusi{d}))$ we have that $\Phi(w)$ visits some $D_j$. By Lemma~\ref{prop:one-counter-runs}, for almost all $w\in \mathit{Run}_{\mathcal{M}_\mathcal{A}}(p\vec{u}^{in},\neg\Zminusi{d})$, the run $\Phi(\Upsilon_d(w))$ visits some $D_j$, which proves Equation~(\ref{eq-BSCC-D}).
Now we examine the runs of $\mathit{Run}(p\vec{u}^{in},D_j)$ in greater detail and characterize the conditions under which $\mathcal{P}(\mathit{Run}(p\vec{u}^{in},D_j)) > 0$. Note that for every BSCC $D$ in $\mathcal{X}_{\mathcal{B}}$ we have that either $D=\{q{\uparrow}\}$ for some $q\in Q$, or $D\subseteq Q$. We treat these two types of BSCCs separately, starting with the former.
\begin{lemma} \label{lem-oc-div-prob}
$\mathcal{P}(\bigcup_{q\in Q}\mathit{Run}(p\vec{u}^{in},\{q{\uparrow}\})) > 0$ iff there
exists a~BSCC $C$ of $\mathcal{F}_\mathcal{A}$ with \emph{all} counters diverging and a
\mbox{$\Zminusi{d}$-safe} finite path of the form
$p\vec{v} \tran{}^* q\vec{u} \tran{}^* q\vec{z}$ where the subpath
$q\vec{u} \tran{}^* q\vec{z}$ is $\Z_{all}$-safe,
$q \in C$, $q\vec{u}$ is above $1$, $\vec{z} - \vec{u} \geq \vec{0}$,
and $(\vec{z} - \vec{u})[i] > 0$ for every $i$ such that $\vec{t}[i] > 0$. \end{lemma} \noindent A proof of Lemma~\ref{lem-oc-div-prob} can be found in Appendix~\ref{app-sec2}. Now let $D$ be a BSCC of $\mathcal{X}_{\mathcal{B}}$ reachable from $p$ such that $D\subseteq Q$ (i.e., $D \neq \{q{\uparrow}\}$ for any $q \in Q$).
Let $\vec{e} \in [1,\infty)^D$ where $\vec{e}[q]$ is the expected number of transitions needed to revisit a configuration with zero counter from $q(0)$ in $\mathcal{M}_{\mathcal{B}}$.
\begin{proposition}[\cite{BKK:pOC-time-LTL-martingale}, Corollary~6] The problem whether \mbox{$\vec{e}[q]<\infty$} is decidable in polynomial time. \end{proposition}
\emph{From now on, we assume that $\vec{e}[q]<\infty$ for all $q\in D$}.
\noindent In Section~\ref{sec-case1}, we used the trend $\vec{t}\in \mathbb{R}^d$ to determine tendency of counters either to diverge, or to reach zero. As defined, each $\vec{t}[i]$ corresponds to the long-run average change per transition of counter~$i$ as long as all counters stay positive. Allowing zero value in counter $d$, the trend $\vec{t}[i]$ is no longer equal to the long-run average change per transition of counter~$i$ and hence it does not correctly characterize its behavior. Therefore, we need to redefine the notion of trend in this case.
Recall that $\mathcal{B}$ is $L=\{-1,0,1\}^{d-1}$-labeled pMC. Given $i\in \{1,\ldots,d{-}1\}$, we denote by $\vec{\delta}_i\in \mathbb{R}^{Q}$ the vector where $\vec{\delta}_i[q]$ is the $i$-th component of the expected total reward accumulated along a run from $q(0)$ before revisiting another configuration with zero counter. Formally, $\vec{\delta}_i[q]=\mathbb{E}T_i$ where $T_i$ is a random variable which to every $w\in \mathit{Run}_{\mathcal{M}_{\mathcal{B}}}(q(0))$ assigns $\totalrew{i}{w}{0}{\ell}$ such that $\ell>0$ is the least number satisfying $w(\ell)=r(0)$ for some $r\in D$.
Let $\vec{\mu}_{oc}\in [0,1]^D$ be the invariant distribution of the BSCC $D$ of $\mathcal{X}_{\mathcal{B}}$, i.e., $\vec{\mu}_{oc}$ is the unique solution of \[ \vec{\mu}_{oc}[q] \quad = \quad \sum_{r\in D, r\tran{x} q} \vec{\mu}_{oc}[r]\cdot x \] The \emph{oc-trend} of $D$ is a $(d{-}1)$-dimensional vector $\vec{t}_{oc}\in [-1,1]^{d-1}$ defined by \[ \vec{t}_{oc}[i]\quad = \quad \left(\vec{\mu}^T_{oc} \cdot \vec{\delta}_i\right)/\left(\vec{\mu}^T_{oc}\cdot \vec{e}\right) \] The following lemma follows from the standard results about ergodic Markov chains (see, e.g., \cite{Norris:book}).
\begin{lemma}\label{lem:long-run-average} For almost all $w\in \mathit{Run}_{\mathcal{M}_{\mathcal{B}}}(q(0))$ we have that \[ \vec{t}_{oc}[i]\quad = \quad \lim_{k\rightarrow \infty} \frac{\totalrew{i}{w}{0}{k}}{k} \] \end{lemma} \noindent That is, $\vec{t}_{oc}[i]$ is the $i$-th component of the expected long-run average reward per transition in a run of $\mathit{Run}_{\mathcal{M}_\mathcal{B}}(q(0))$, and as such, determines the long-run average change per transition of counter~$i$ as long as all counters of $\{1,\ldots,d{-}1\}$ remain positive.
Further, for every $i \in \{1,\ldots,d-1\}$ and every $q\in D$, we denote by $\mathit{botinf}_i(q)$ the \emph{least} $j \in \mathbb{N}$ such that every $w \in \mathit{FPath}_{\mathcal{M}_\mathcal{B}}(q(0))$ ending in $q(0)$ where $w(n) \neq q(0)$ for all $1 \leq n < \mathit{length}(w)$ satisfies $\totalrew{i}{w}{0}{\mathit{length}(w)} \geq -j$.
If there is no such $j$, we put $\mathit{botinf}_i(q) = \infty$. It is easy to show that if $\mathit{botinf}_i(q) = \infty$, then $\mathit{botinf}_i(r) = \infty$ for all $r \in D$.
\begin{lemma} \label{lem-bottominf}
If $\mathit{botinf}_i(q) < \infty$, then $\mathit{botinf}_i(q) \leq 3|Q|^3$ and
the exact value of $\mathit{botinf}_i(q)$ is computable in time polynomial
in $|\mathcal{A}|$. \end{lemma}
\noindent A proof Lemma~\ref{lem-bottominf} can be found in Appendix~\ref{app-sec2}.
We say that counter~$i$ is {\emph{oc-decreasing}} in $D$ if $\mathit{botinf}_i(q) = \infty$ for some (and hence all) $q \in D$.
\begin{definition} For a given $i \in \{1,\ldots,d{-}1\}$, we say that the \mbox{$i$-th} reward is \emph{oc-diverging} in $D$ if either $\vec{t}_{oc}[i] > 0$, or $\vec{t}_{oc}[i] = 0$ and counter~$i$ is not oc-decreasing in~$D$. \end{definition}
\begin{lemma}\label{lem:case2-subcrit-not-diverging}
If some reward is not oc-diverging in $D$, then $\mathcal{P}(\mathit{Run}(p\vec{u}^{in},D)) = 0$. \end{lemma} A proof of Lemma\ref{lem:case2-subcrit-not-diverging} can be found in Appendix~\ref{app-sec2}. It remains to analyze the case when all rewards are \mbox{oc-diverging} in~$D$. Similarly to Case~I, we need to obtain a bound on probability of divergence of an arbitrary counter $i \in \{1,\dots,d-1\}$ with $\vec{t}_{oc}[i] > 0$. The following lemma (an analogue of Lemma~\ref{lem-divergence}) is crucial in the process.
\begin{lemma} \label{lem:two-counter-divergence} Let $\mathcal{D}$ be a $\{-1,0,1\}$-labeled one-dimensional pMC, let $D$ be a BSCC of $\mathcal{X}_{\mathcal{D}}$ such that the oc-trend $t_{oc}$ of the only reward in $D$ is positive. Then for all $q\in D$, there exist computable constants $h'$ and $A_0$ where $0 < A_0 < 1$, such that for all $h \geq h'$ we have that the probability that a run $w\in \mathit{Run}_{\mathcal{M}_{\mathcal{D}}}(q(0))$ satisfies \[ \inf_{k\in \mathbb{N}} \totalrew{1}{w}{0}{k}\quad \geq \quad -h \] is at least $1- A_0^h$. \end{lemma} \noindent
A proof of Lemma~\ref{lem:two-counter-divergence} is the most involved part of this paper, where we need to construct new analytic tools. A sketch of the proof is included at the and of this section.
\begin{definition}
Let $D$ be a BSCC of $\mathcal{X}_{\mathcal{B}}$ where all rewards are oc-diverging,
and let $q \in D$. We say that a configuration $q\vec{u}$
is \emph{oc-above} a given $n \in \mathbb{N}$ if $\vec{u}[i] \geq n$ for every
$i\in \{1,\ldots,d-1\}$
such that $\vec{t}_{oc}[i] > 0$, and
$\vec{u}[i] \geq \mathit{botinf}_i(q)$ for every
$i\in \{1,\ldots,d-1\}$
such that $\vec{t}_{oc}[i] = 0$. \end{definition}
The next lemma is an analogue of Lemma~\ref{lem-diverging} and it is proven using the same technique, using Lemma~\ref{lem:two-counter-divergence} instead of Lemma~\ref{lem-divergence}. A full proof can be found in Appendix~\ref{app-sec2}.
\begin{lemma} \label{lem-diverging-2}
Let $D$ be a BSCC of $\mathcal{X}_{\mathcal{B}}$ where all rewards are diverging.
Then there exists a computable constant~$n \in \mathbb{N}$ such that
$\mathcal{P}(\mathit{Run}(p\vec{u}^{in},D)) > 0$ iff
there is a $\Zminusi{d}$-safe finite
path of the form $p\vec{u}^{in} \tran{}^* q\vec{u}$ where $\vec{u}$
is oc-above $n$ and $\vec{u}[d]=0$. \end{lemma}
\noindent A direct consequence of Lemma~\ref{lem-diverging-2} and the results of \cite{BFLZ:VASSz-model-checking-LMCS} is the following:
\begin{theorem}
\label{thm:qual-d-algorithm}
The qualitative $\Zminusi{d}$-reachability problem for \mbox{$d$-dimensional}
pMC is decidable (assuming $\vec{e}[q]<\infty$ for all $q\in D$ in
every BSCC of $\mathcal{X}_\mathcal{B}$). \end{theorem}
\noindent A proof of Theorem~\ref{thm:qual-d-algorithm} is straightforward, since we can effectively compute the structure of $\mathcal{X}_\mathcal{B}$ (in time polynomial in $|\mathcal{A}|$, express its transition probabilities and oc-trends in BSCCs of $\mathcal{X}_\mathcal{B}$ in the existential fragment of Tarski algebra, an thus effectively identify all BSCCs of $\mathcal{X}_\mathcal{B}$ where all rewards are oc-diverging. To check the condition of Lemma~\ref{lem-diverging-2}, we use the algorithm of \cite{BFLZ:VASSz-model-checking-LMCS} for constructing finite representation of filtered covers in VAS with one zero test. This is the only part where we miss an upper complexity bound, and therefore we cannot provide any bound in Theorem~\ref{thm:qual-d-algorithm}. It is worth noting that the qualitative $\Zminusi{d}$-reachability problem is \textsc{Square-Root-Sum}-hard (see below), and hence it cannot be solved efficiently without a breakthrough results in the complexity of exact algorithms. For more comments and a proof of the next Proposition, see Appendix~\ref{app-sec2}.
\begin{proposition} \label{sqrt-hard}
The qualitative $\Zminusi{d}$-reachability problem is
\textsc{Square-Root-Sum}-hard, even for two-dimensional pMC
where $\vec{e}[q]<\infty$ for all $q\in D$ in
every BSCC of $\mathcal{X}_\mathcal{B}$. \end{proposition}
Using Lemma~\ref{lem-diverging-2}, we can also approximate $\mathcal{P}(\mathit{Run}(p\vec{v},\Zminusi{d}))$ up to an arbitrarily small absolute error $\varepsilon > 0$ (due to the problems mentined above, we do not provide any complexity bounds). The procedure mimics the one of Theorem~\ref{thm:approx-general}. The difference is that now we eventually use methods for one-dimensional pMC instead of the methods for finite-state Markov chains. The details are given in Appendix~\ref{sec:case2-approx}.
\begin{theorem}
\label{thm:approx-general-case2}
For a given $d$-dimensional pMC $\mathcal{A}$ and its initial configuration
$p\vec{v}$, the probability $\mathcal{P}(\mathit{Run}(p\vec{v},\Zminusi{d}))$ can be
effectively approximated up to a given absolute error $\varepsilon>0$. \end{theorem}
\noindent \textbf{A Proof of Lemma~\ref{lem:two-counter-divergence}.} The lemma differs from Lemma~\ref{lem-divergence} in that it effectively bounds the probability of not reaching zero in one of the counters of a \emph{two-dimensional} pMC (the second counter is encoded in the labels). Hence, the results on one-dimensional pMCs are not sufficient here. Below, we sketch a stronger method that allows us to prove the lemma. The method is again based on analyzing a suitable martingale; however, the construction and structure of the martingale is much more complex than in the one-dimensional case.
Before we show how to construct the desired martingale, let us mention the following useful lemma:
\begin{lemma} \label{lem:bounded-bumps} Let $r\in D$. Given a run $w\in \mathit{Run}_{\mathcal{M}_{\mathcal{B}}}(r(0))$, we denote by $E(w)=\inf\{\ell>0\mid \mathit{cval}_1(w(\ell))=0\}$, i.e., the time it takes $w$ to re-visit zero counter value. Then there are constants $c'\in \mathbb{N}$ and $a\in (0,1)$ computable in polynomial space such that for all $k\geq c'$ we have \[ \mathcal{P}(E\geq k)\quad \leq\quad a^k \] \end{lemma} \begin{proof} This follows immediately from Proposition~6 and Theorem~7 in~\cite{BBEKW:PPDA-time-arXiv}. \end{proof}
Let us fix an $1$-dimensional pMC $\mathcal{D}$ with the set of states $Q$ and let us assume, for simplicity, that $\mathcal{X}_\mathcal{D}$ is strongly connected (assume that the set of states of $\mathcal{X}_\mathcal{D}$ is $D\subseteq Q$). Let us summarize notation used throughout the proof. \begin{itemize} \item Let $\vec{e}_\downarrow \in [1,\infty)^Q$ be the vector such that $\vec{e}_\downarrow[q]$
is the expected total time of a run from $q(1)$ to the first visit of $r(0)$ for some $r \in Q$.
By our assumptions, $\vec{e}_\downarrow$ is finite. \item Recall that $\vec{e} \in [1,\infty)^D$ is the vector such that $\vec{e}[q]$
is the expected total time of a nonempty run from $q(0)$ to the first visit of $r(0)$ for some $r \in Q$.
Since $\vec{e}_\downarrow$ is finite, also $\vec{e}$ is finite. \item Let $\vec{\delta}_\downarrow \in \mathbb{R}^Q$ be the vector such that $\vec{\delta}_\downarrow[q]$
is the expected total reward accumulated during a run from $q(1)$ to the first visit of $r(0)$ for some $r \in Q$.
Since $|\vec{\delta}_\downarrow[q]| \le |\vec{e}_\downarrow[q]|$ holds for all $q \in Q$, the vector $\vec{\delta}_\downarrow$ is finite. \item Recall that $\vec{\delta}_1 \in \mathbb{R}^D$ is the vector such that $\vec{\delta}_1[q]$
is the expected total reward accumulated during a nonempty run from $q(0)$ to the first visit of $r(0)$ for some $r \in Q$.
Similarly as before, $\vec{\delta}_1$ is finite. \item Let $G \in \mathbb{R}^{Q \times Q}$ denote the matrix such that $G[q,r]$ is the probability that starting from $q(1)$
the configuration $r(0)$ is visited before visiting any configuration $r'(0)$ for any $r' \ne r$.
By our assumptions the matrix~$G$ is stochastic, i.e., $G \vec{1} = \vec{1}$. \item Let us denote by $A \in \mathbb{R}^{D \times D}$ transition matrix of the chain $\mathcal{X}_{\mathcal{D}}$, i.e., $A[q,r]$ is the probability that starting from $q(0)$
the configuration $r(0)$ is visited before visiting any configuration $r'(0)$ for any $r' \ne r$.
By our assumptions the matrix~$A$ is stochastic
and irreducible. \item Recall that $\vec{\mu}_{oc}^T = \vec{\mu}_{oc}^T A \in [0,1]^D$ denotes the invariant distribution of the finite Markov chain $\mathcal{X}_\mathcal{D}$ induced by~$A$.
\item Recall that $t = (\vec{\mu}_{oc}^T \vec{\delta}_1) / (\vec{\mu}_{oc}^T \vec{e}) \in [-1,+1]$ is the oc-trend of $D$,
so intuitively $t$ is the expected average reward per step accumulated during a run started from $q(0)$ for some $q \in D$. \item Let $\vec{r}_\downarrow := \vec{\delta}_\downarrow - t \vec{e}_\downarrow \in \mathbb{R}^Q$ and
let $\vec{r}_0 := \vec{\delta}_1 - t \vec{e} \in \mathbb{R}^D$. \end{itemize}
\begin{lemma} \label{lem-g-zero-exists}
There exists a vector $\vec{g}(0) \in \mathbb{R}^Q$ such that
\begin{equation}
\vec{g}(0)[D] = \vec{r}_0 + A \vec{g}(0)[D] \,, \label{eq-mart-fund}
\end{equation}
where $\vec{g}(0)[D]$ denotes the vector obtained from~$\vec{g}(0)$ by deleting the non-$D$-components. \end{lemma} Extend~$\vec{g}(0)$ to a function $\vec{g} : \mathbb{N} \to \mathbb{R}^Q$ inductively with
\begin{equation}
\vec{g}(n + 1) = \vec{r}_\downarrow + G \vec{g}(n) \qquad \text{for all $n \in \mathbb{N}$.} \label{eq-mart-g}
\end{equation}
\begin{lemma} \label{lem:weights-bound} There is $\vec{g}(0)$ satisfying \eqref{eq-mart-fund} for which we have the following: There exists a constant $c$ effectively computable in polynomial space such that for every $r \in D$ and $n\geq 1$ we have
$|\vec{g}(0)[r]|\leq c $ and $|\vec{g}(n)[r]|\leq c\cdot n$. \end{lemma}
\newcommand{(t\cdot \sqrt[4]{h})/c}{(t\cdot \sqrt[4]{h})/c}
Let us fix $q\in D$ and $h\in \mathbb{N}$ such that $(t\cdot \sqrt[4]{h})/c\geq c'$, where $c$ is from the previous lemma and $c'$ from Lemma~\ref{lem:bounded-bumps}. For a run $w \in \mathit{Run}_{\mathcal{M}_\mathcal{D}}(q(0))$ and all $\ell \in \mathbb{N}$ let $\ps\ell \in Q$ and $\xs\ell_1, \xs\ell_2 \in \mathbb{N}$
be such that $\ps\ell = \mathit{state}(w(\ell))$, $\xs\ell_2 = \mathit{cval}(w(\ell))$ and $\xs\ell_1 = h+\totalrew{}{w}{0}{\ell}$.
Now let us define \begin{equation}
\ms{\ell} := \xs\ell_1 - t \ell + \vec{g}\big(\xs\ell_2\big)[\ps\ell] \qquad \text{for all $\ell \in \mathbb{N}$.} \label{eq-mart-m} \end{equation} Then we have: \begin{proposition} \label{prop-martingale}
Write $\mathcal{E}$ for the expectation with respect to~$\mathcal{P}$.
We have for all $\ell \in \mathbb{N}$:
\[
\mathcal{E} \left( \ms{\ell+1} \;\middle\vert\; w(\ell) \right) = \ms{\ell}\,.
\] \end{proposition}
In other words, the stochastic process $\{\ms{\ell}\}_{\ell = 0}^\infty$ is a martingale. Unfortunately, this martingale may have unbounded differences, i.e. $|\msi{\ell+1}-\msi{\ell}|$ may become arbitrarily large with increasing $\ell$, which prohibits us from applying standard tools of martingale theory (such as Azuma's inequality) directly on $\{\ms{\ell}\}_{\ell = 0}^\infty$. We now show how to overcome this difficulty.
\newcommand{(t\cdot \sqrt[4]{i})/c}{(t\cdot \sqrt[4]{i})/c} Let us now fix $i\in \mathbb{N}$ such that $i \geq h$ and denote $K=(t\cdot \sqrt[4]{i})/c$. We define a new stochastic process as follows: \begin{equation}
\msi{\ell} := \begin{cases}
\ms{\ell} & \text{ if } \xs{\ell'}_2\leq K \text{ for all }\ell'\leq \ell\\
\msi{\ell-1} & \text{ otherwise. }
\end{cases}
\label{eq-mart-mi} \end{equation}
Observe that $\{\msi{\ell}\}_{\ell = 0}^\infty$ is also a martingale. Moreover, using the bound of Lemma~\ref{lem:weights-bound} we have for every $\ell \in \mathbb{N}$ that $|\msi{\ell+1}-\msi{\ell}|\leq 1+t+2cK \leq 4t\sqrt[4]{i}$, i.e., $\{\msi{\ell}\}_{\ell = 0}^\infty$ is a bounded-difference martingale.
Now let $H_i$ be the set of all runs $w$ that satisfy $\xs{i}_1=0$ and $\xs{\ell}_1 > 0$ for all $0 \leq \ell < i$. Moreover, denote by $\mathit{Over}_i$ the set of all runs $w$ such that $\xs{\ell}_2\geq K$ for some $0\leq \ell \leq i$, and by $\neg\mathit{Over}_i$ the complement of $\mathit{Over}_i$.
Note that every run can perform at most $i$-revisits of zero counter value during the first $i$ steps. By Lemma~\ref{lem:bounded-bumps} the probability that counter value at least $ K$ is reached between to visits of zero counter is at most $a^K$. It follows that $\mathcal{P}(\mathit{Over}_i)\leq i\cdot a^{(t\cdot \sqrt[4]{i})/c}$.
Next, for every run $w\in \neg\mathit{Over}_i \cap H_i$ it holds \begin{align*} (\msi{i} - \msi{0})(w) &= (\ms{i} - \ms{0})(w) \\ &=- it + \vec{g}(\xs{i}_2)[\ps{i}] - h -\vec{g}(0)[\ps{0}]\\ &\leq -it + 2cK = -it + t\cdot \sqrt[4]{i} \leq -i\frac{t}{2}, \end{align*}
where the first inequality follows from the bound on $\vec{g}(n)$ in Lemma~\ref{lem:weights-bound} and the last inequality holds since $\sqrt[4]{i}\leq i/2$ for all $i\geq 3$.
Using the Azuma's inequality, we get \begin{align*}
\mathcal{P}(\mathit{Over}_i \cap H_i) &\leq \mathcal{P}(\msi{i} - \msi{0} \leq -it/2)\\ &\leq \exp\left(-\frac{i^2 \cdot t^2}{8i(4t\sqrt[4]{i})^2} \right)
=\exp\left(-\frac{\sqrt{i} }{128} \right). \end{align*}
Altogether, we have \begin{align*}
\mathcal{P}(H_i)&= \mathcal{P}(H_{i}\cap \mathit{Over}_i) + \mathcal{P}(H_i \cap \neg\mathit{Over}_i) \\
&\leq i\cdot a^{(t\cdot \sqrt[4]{i})/c} + e^{-\sqrt{i}/128} \leq i\cdot A^{\sqrt[4]{i}}, \end{align*} where $A=\max \{a^{t/c},2^{-1/128}\}$. Note that $A$ is also computable in polynomial space.
We now have all the tools needed to prove Lemma~\ref{lem:two-counter-divergence}. We have \begin{align*}
\mathcal{P}(\liminf_{k\rightarrow \infty} \totalrew{1}{w}{0}{k}\leq - h)&\leq\mathcal{P}(\inf_{k\in \mathbb{N}} \totalrew{1}{w}{0}{k} \leq -h) \\&= \sum_{i \geq h}\mathcal{P}(H_i)
\leq \sum_{i \geq h} i \cdot A^{\sqrt[4]{i}}. \end{align*}
Note that $\sum_{\ell = h}^{\infty} \ell \cdot A^{\sqrt[4]{\ell}} = \sum_{j =\lfloor\sqrt[4]{h}\rfloor }^{\infty} \sum_{\ell = j^4}^{(j+1)^4 - 1} \ell \cdot A^{\sqrt[4]{\ell}} \leq \sum_{j =\lfloor\sqrt[4]{h}\rfloor }^{\infty} \sum_{\ell = j^4}^{(j+1)^4 - 1} (j+1)^4 A^{j} \leq \sum_{j =\lfloor\sqrt[4]{h}\rfloor }^{\infty} 8(j+1)^7 A^j$. Using standard methods of calculus we can bound the last sum by $ (c'' \cdot h^7\cdot A^h)/(1-A)^8$ for some known constant $c''$ independent of $\mathcal{B}$.
Thus, from the knowledge of $A$ and $c''$ we can easily compute, again in polynomial space, numbers $h_0 \in \mathbb{N}$, $A_0 \in (0,1)$ such that for all $h \geq h_0$ it holds \[
\mathcal{P}(\liminf_{k\rightarrow \infty} \totalrew{1}{w}{0}{k} \geq h ) \geq 1 - A_0^{h}. \]
\section{Conclusions} We have shown that the qualitative zero-reachability problem is decidable in Case~I and~II, and the probability of all zero-reaching runs can be effectively approximated. Let us not when the technical condition adopted in Case~II is not satisfied, than the oc-trends may be undefined and the problem requires a completely different approach. An important technical contribution of this paper is the new martingale defined in Section~\ref{sec-case2}, which provides a versatile tool for attacking other problems of pMC analysis (model-checking, expected termination time, constructing (sub)optimal strategies in multi-counter decision processes, etc.) similarly as the martingale of \cite{BKK:pOC-time-LTL-martingale} for one-dimensional pMC.
\appendices \onecolumn
\section{Proofs of Section~\ref{sec-case1}} \label{app-sec1}
\begin{reftheorem}{Lemma}{\ref{lem:not-diverging}}
Let $C$ be a BSCC of $\mathcal{F}_{\mathcal{A}}$.
If some counter is not diverging in $C$, then $\mathcal{P}(\mathit{Run}(p\vec{v},C)) = 0$. \end{reftheorem} \begin{proof}
Assume that counter $i$ is not diverging, and consider the
one-dimensional pMC $\mathcal{B}_i$. Observe that $\mathcal{F}_{\mathcal{B}_i}$ is the same as
$\mathcal{F}_{\mathcal{A}}$, and hence $\mathcal{F}_{\mathcal{B}_i}$ has the same transition probabilities and
BSCCs as $\mathcal{F}_{\mathcal{A}}$. In particular, the only counter of $\mathcal{B}_i$ is not
diverging in the BSCC $C$ of $\mathcal{F}_{\mathcal{B}_i}$.
By the results of~\cite{BKK:pOC-time-LTL-martingale}, almost all runs of
$\mathit{Run}_{\mathcal{M}_{\mathcal{B}_i}}(p(\vec{v}[i]))$ that stay in $C$ eventually visit
zero value in the only counter. Since all runs of
$\Upsilon_i(\mathit{Run}(p\vec{v},C))$ stay in $C$ but none of them ever
visits a configuration with zero counter value, we obtain that
\[
\mathcal{P}(\mathit{Run}(p\vec{v},C))=\mathcal{P}(\Upsilon_i(\mathit{Run}(p\vec{v},C))=0
\] \end{proof}
\section{Approximation algorithm for \protect{$\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))$}} \label{app-approx}
We show that $\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))$ can be effectively approximated up to an arbitrarily small absolute/relative error $\varepsilon > 0$. First we solve this problem under the assumption that $p$ is in some BSCC of $\mathcal{F}_\mathcal{A}$. Then we show how to drop this assumption.
\begin{proposition} \label{prop:approx}
There is an algorithm which, for a given \mbox{$d$-dimensional} pMC $\mathcal{A}$, its initial configuration $p\vec{v}$ such that $p$ is in a BSCC of $\mathcal{F}_\mathcal{A}$, and a given $\varepsilon>0$ computes a number $\nu$ such that $|\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))-\nu|\leq d\cdot\varepsilon$. The algorithm runs in time $(\exp(|\mathcal{A}|)\cdot \log(1/\varepsilon))^{\mathcal{O}(d!)}$. \end{proposition} \begin{proof}
In the following, we denote by $C$ the BSCC of $\mathcal{A}$ containing the initial state $p$. Note that we may assume that $\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))<1$. From the proof of Lemma~\ref{lem:cover-short-path} it follows that checking this condition boils down to checking the existence of a certain path of length at most $|\mathcal{A}'|^{\mathcal{O}(d!)}$ in a suitable VASS $\mathcal{A}'$ of size polynomial in $|\mathcal{A}|$. This can be done it time $(\exp(|\mathcal{A}|)^{\mathcal{O}(d!)}$.
We can check this condition using an algorithm of Theorem~\ref{thm:qual-all-algorithm}, and if it does not hold we may output $\nu=1$. In particular, we may assume that the trend of every counter in $C$ is non-negative.
We proceed by induction on $d$. For technical convenience we slightly change the statement about the complexity: we show that the running time of the algorithm is $(\exp(|\mathcal{A}|^c)\cdot\log(\vec{v}_{\max}/\varepsilon))^{d!} $, for some constants $c$, $c'$ independent of $\mathcal{A}$. Clearly, this new statement implies the one in the proposition.
Before we present the algorithm, let us make an important observation. Recall the number $a$ defined in Lemma~\ref{lem-divergence} for an arbitrary one-dimensional pMC $\mathcal{B}$ with a positive trend of the counter. Now suppose that for a given $\mathcal{B}$ and given $\varepsilon>0$ we want to find some $K$ such that $\frac{a^K}{1-a}<\varepsilon$. Note that it suffices to pick any $$K > \frac{\log(1/\varepsilon)}{(1-a)\log(1/a)} .$$ From the definition of $a$ we have $K\in \exp(\mathcal{B}^{O(1)})\cdot\log(1/\varepsilon)$ and that $K$ can be computed in time polynomial in $|\mathcal{B}|$. In particular there is a constant $c$ independent of $\mathcal{B}$ such that $K\leq \exp(|\mathcal{B}|^{c})\cdot\log(1/\varepsilon)$ and we choose $c$ as the desired constant.
Now let us prove the proposition.
$d=1:$ First let us assume that the trend of the single counter in $C$ is $0$. Then, by Lemma~\ref{lem-diverging} it must be the case that $\mathcal{P}(\mathit{Run}(r(\ell),\Z_{all})) = 0$ for every $r\in C$ and every $\ell\geq |C|$. Thus, if the initial counter value is $\geq |Q|$, we may output $\nu = 0$. Otherwise, we may approximate the probability by constructing a finite-state polynomial-sized Markov chain $\mathcal{M}_{|C|}$ whose states are those configurations of $\mathcal{A}$ where the counter is bounded by $|C|$ and whose transitions are naturally derived from $\mathcal{A}$. Formally, $\mathcal{M}_{|C|}$ is obtained from $\mathcal{M}_{A}$ by removing all configurations $r(\ell)$ with $\ell > |C|$ and replacing all transitions outgoing from configurations of the form $r(|C|)$ with a self loop of probability 1. Clearly, the value $\mathcal{P}(\mathit{Run}(p\vec{\ell},\Z_{all}))$ is equal to the probability of reaching a configuration with a zero counter from $p(\ell)$ in $\mathcal{M}_{|C|}$, which can be computed in polynomial time by standard methods.
If the trend of the counter in $C$ is positive, then let us consider the number $a$ from Lemma~\ref{lem-divergence} computed for $\mathcal{A}$ and $C$. As discussed above, we may compute, in time polynomial in $|\mathcal{A}|$, a number $K \leq\exp(|\mathcal{A}|^c)\cdot\log(1/\varepsilon)$ such that $\frac{a^K}{1-a}<\varepsilon$. We can now again construct a finite-state Markov chain $\mathcal{M}_{K}$ by discarding all configurations in $\mathcal{M}_\mathcal{A}$ where the counter surpasses $K$ and replacing the transitions outgoing from configurations of the form $r(K)$ with self-loops.
Now let us consider an initial configuration $q(\ell)$ with $\ell \leq K$ and denote $P(q(\ell))$ the probability of reaching a configuration with zero counter in from $q(\ell)$ in $\mathcal{M}_K$. We claim that $|\mathcal{P}(\mathit{Run}(r(\ell),\Z_{all})) - P(q(\ell))| \leq \varepsilon$. Indeed, from the construction of $\mathcal{M}_{K}$ we get that $|\mathcal{P}(\mathit{Run}(r(\ell),\Z_{all})) - P(q(\ell))|$ is bounded by the probability, that a run initiated in $q(\ell)$ in $\mathcal{A}$ reaches a configuration of the form $r(K)$ via a $\Z_{all}$-safe path \emph{and then} visits a configuration with zero counter. This value is in turn bounded by a probability that a run initiated in $r(K)$ decreases the counter to 0, which is at most $\frac{a^K}{1+a}\leq a^K$ by Lemma~\ref{lem-divergence}, and thus at most $\varepsilon$ by the choice of $K$. Thus, it suffices to compute $P(q(\ell))$ via standard algorithms and return it as $\nu$.
The same argument shows that if the initial counter value $\ell$ is greater than $K$, we can output $\nu = 0$ as a correct $\varepsilon$-approximation.
Note that the construction of $\mathcal{M}_K$ and computing the reachability probability in it can be performed in time $(|\mathcal{A}|\cdot K)^{c'}$ for a suitable constant $c'$ independent of $\mathcal{A}$. This finishes the proof of a base case of our induction.
$d>1:$ Here we will use the algorithm for the $(d-1)$-dimensional case as a sub-procedure. For any counter $i$ and any vector $\vec{\beta}\in\{-1,0,1\}^d$ we denote by $\vec{\beta}_{-i}$ the $(d-1)$-dimensional vector obtained from $\vec{\beta}$ by deleting its $i$-component. Moreover, we define a $(d-1)$-dimensional pMC $\mathcal{A}_{-i}$ obtained from $\mathcal{A}$ by ``forgetting'' the $i$-th counter. I.e., $\mathcal{A}=(Q,\gamma_{-i},W_{-i})$, where $(p,\vec{\alpha},c,q)\in \gamma_{-i}$ iff there is $(p,\vec{\beta},c,q)\in \gamma$ such that $\vec{\beta}_{-i} = \vec{\alpha}$; and where $W_{-i}(p,\vec{\alpha},c,q)=\sum W(p,\vec{\beta},c,q)$ with the summation proceeding over all $\vec{\beta}$ such that $\vec{\beta}_{-i}=\vec{\alpha}$.
Now let us prove the proposition. Let $\vec{t}$ be the trend of $C$. For every counter $i$ such that $\vec{t}[i]>0$ we denote by $a_i$ the number $a$ of Lemma~\ref{lem-divergence} computed for $C$ in $\mathcal{B}_i$ (note that $C$ is a BSCC of every $\mathcal{B}_i$). We put $a_{\mathit{max}}=\max\{a_i\mid \vec{t}[i]>0\}$. We again compute, as discussed above, in time polynomial in $|\mathcal{A}|$ a number $K \leq\exp(|\mathcal{A}|^c)\cdot\log(1/\varepsilon)$ such that $\frac{a_{\mathit{max}}^K}{1-a_{\mathit{max}}}<\varepsilon$. (If $\vec{t}=\vec{0}$, we do not need to define $K$ at all, as will be shown below.) For any configuration $q\vec{u}$ we denote by $\mathit{mindiv}(q\vec{u})$ the smallest $i$ such that either $\vec{t}[i]>0$ and $\vec{u}[i]\geq K$ or $\vec{t}[i]=0$ and $\vec{u}[i]\geq |C|$ (if such $i$ does not exist, we put $\mathit{mindiv}(q\vec{u})=\bot$).
Consider a finite-state Markov chain $\mathcal{M}^d_{K}$ which can be obtained from $\mathcal{M}_\mathcal{A}$ as follows: \begin{itemize}
\item We remove all configurations where at least one of the counters with positive trend is greater than $K$, together with adjacent transitions.
\item We remove all configurations where at least one of the counters with zero trend is greater than $|C|$, together with adjacent transitions.
\item We add new states $q_{\mathit{down}}$ and $q_{\mathit{up}}$, both of them having a self-loop as the only outgoing transition.
\item For every $1\leq i \leq d$ and every remaining configuration ${q\vec{u}}$ with $\mathit{mindiv}(q\vec{u})=i$ we remove all transitions outgoing from $q\vec{u}$ and replace them with the following transitions:
\begin{itemize}
\item A transition leading to $q_{\mathit{down}}$, whose probability is equal to some $((d-1)\cdot \varepsilon)$-approximation of $\mathcal{P}_{\mathcal{A}_{-i}}(\mathit{Run}(q\vec{u}_{-i},\Z_{all}))$ (which can be computed using the algorithm for dimension $d-1$).
\item A transition leading to $q_{\mathit{up}}$, with probability $1-x$, where $x$ is such that $q\vec{u}\tran{x}q_{\mathit{down}}$.
\end{itemize} \end{itemize} Above, $\mathcal{P}_{\mathcal{A}_{-i}}(X)$ represents the probability of event $X$ in pMC $\mathcal{A}_{-i}$.
Now for an initial configuration $p\vec{v}$ belonging to the states of $\mathcal{M}^d_K$ let $P(p\vec{v})$ be the probability of reaching, when starting in $p\vec{v}$ in $\mathcal{M}^d_k$, either the state $q_{\mathit{down}}$ or a configuration in which at least one of the counters is 0. Note that $P(p\vec{v})$ can be computed in time polynomial in $|\mathcal{M}^d_K|$. We claim that $|\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))-P(p\vec{v})|\leq d\cdot \varepsilon$.
Indeed, let us denote $\mathit{Div}$ the set of all configurations $q\vec{u}$ such that $q\vec{u}$ is a state of $\mathcal{M}^d_K$ and $\mathit{mindiv}({q\vec{u}})\neq \bot$. For every $q\vec{u}\in \mathit{Div}$ we denote by $x_{q\vec{u}}$ the probability of the transition leading from $q\vec{u}$ to $q_{\mathit{down}}$ in $\mathcal{M}^d_K$. Then $|\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))-P(p\vec{v})|\leq \max_{q\vec{u}\in\mathit{Div}}| \mathcal{P}(\mathit{Run}(q\vec{u},\Z_{all}) - x_{q\vec{u}} |$. Now $\mathcal{P}(\mathit{Run}(q\vec{u},\Z_{all})\leq P_1(q\vec{u}) + P_2(q\vec{u})$, where $P_1(q\vec{u})$ is the probability that a run initiated in $q\vec{u}$ in $\mathcal{A}$ visits a configuration with $i$-th counter 0 via a $\mathcal{Z}_{-i}$-safe path, and $P_{2}(q\vec{u})$ is the probability that a run initiated in $q\vec{u}$ in $\mathcal{A}$ visits a configuration with some counter equal to 0 via an $\{i\}$-safe path.
So let us fix $q\vec{u}\in \mathit{Div}$ and denote $i=\mathit{mindiv}(q\vec{u})$. If $\vec{t}[i]=0$, then
we have $P_1(q\vec{u})=0$, since this counter is not decreasing in $C$ and thus it cannot decrease by more than $|C|$. Otherwise $P_1(q\vec{u})$ is bounded by the probability that a run initiated in $q(K)$ in $\mathcal{B}_i$ reaches a configuration where the counter is 0. From Lemma~\ref{lem-divergence} we get that $\mathcal{P}_{\mathcal{B}_{i} }(\mathit{Run}(q(K),\Z_{all}))\leq \frac{a_i^K}{1-a_i} \leq \frac{a_{\mathit{max}}^K}{1-a_{\mathit{max}}} \leq \varepsilon$, where the last inequality follows from the choice of $K$.
For $P_2(q\vec{u})$ note that $P_2{(\vec{u})}=\mathcal{P}_{\mathcal{A}_{-i}}(\mathit{Run}(q\vec{u}_{-i},\Z_{all}))$ and thus by the construction of $\mathcal{M}^d_{K}$ we have $|P_2(q\vec{u}) - x_{q\vec{u}}|\leq (d-1)\cdot\varepsilon$.
Altogether we have \begin{align*}
&|\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))-P(p\vec{v})|\leq| P_1(q\vec{u}) + P_2(q\vec{u}) - x_{q\vec{u}}| \leq \varepsilon + (d-1)\cdot \varepsilon = d\cdot \varepsilon. \end{align*}
Therefore it suffices to compute $P(p\vec{v})$ via standard methods and output is as $\nu$. Finally, if the initial configuration $p\vec{v}$ does not belong to the state space of $\mathcal{M}^d_K$ let us denote $i=\mathit{mindiv}(p\vec{v})$. Then it suffices to output some $((d-1)\cdot\varepsilon)$-approximation of $\mathcal{P}_{\mathcal{A}_{-i}}(\mathit{Run}(p\vec{v}_{-i},\Z_{all}))$ as $\nu$. If $\vec{t}[i]=0$, then $\nu$ is also an $((d-1)\cdot\varepsilon)$-approximation of $\mathcal{P}_(\mathit{Run}(p\vec{v},\Z_{all}))$, otherwise $|\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))-\nu|\leq(d-1)\cdot \varepsilon + P_{1}(p\vec{v})$ where $P_1$ is defined in the same way as above. Since the probability of reaching zero counter in $\mathcal{B}_i$ with initial counter value $>K$ can be only smaller than the probability for initial value $K$, the bound on $P_1$ above applies and we get $|\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))-\nu|\leq d\cdot \varepsilon$.
Now let us discuss the complexity of the algorithm. Note that for any $d$ we have $K \leq \exp(|\mathcal{A}|^c)\cdot\log(1/\varepsilon)$, and the construction of $\mathcal{M}^d_K$ (or $\mathcal{M}_K$) and the computation of the reachability probabilities can be done in time $(|\mathcal{A}|\cdot K^{d})^{c'} \cdot T(d-1) \leq (\exp{|\mathcal{A}|^{c+1}}\cdot \log(1/\varepsilon))^{dc'}$ for some constant $c'$ independent of $\mathcal{A}$ and $d$, where $T(d-1)$ is the running time of the algorithm on a $(d-1)$-dimensional pMC of size $\leq |\mathcal{A}|$ (the pMCs $|\mathcal{A}_{-i}|$ examined during the recursive call of the algorithm are of size $\leq |\mathcal{A}|$). Solving this recurrence we get that the running time of the algorithm is $(\exp(|\mathcal{A}|)\cdot\log(1/\varepsilon))^{\mathcal{O}(d!)}$.
Lemma~\ref{lem-divergence} we get that $\mathcal{P}_{\mathcal{B}_{i} }(\mathit{Run}(q(K),\Z_{all}))\leq \frac{a_i^K}{1+a_i} \leq a_{\mathit{max}}^K \leq \varepsilon$, where the last inequality follows from the choice of $K$.
\end{proof}
With the help of algorithm from Proposition~\ref{prop:approx} we can easily approximate $\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))$ even if $p$ is not in any BSCC of $\mathcal{A}$.
\begin{reftheorem}{Theorem}{\ref{thm:approx-general}}
For a given $d$-dimensional pMC $\mathcal{A}$ and its initial configuration
$p\vec{v}$, the probability $\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))$ can be
approximated up to a given absolute error $\varepsilon>0$ in time
$(\exp(|\mathcal{A}|)\cdot \log(1/\varepsilon))^{\mathcal{O}(d\cdot d!)}$. \end{reftheorem} \begin{proof}
First we compute an integer $n\in \exp(|\mathcal{A}|^{\mathcal{O}(1)})\cdot \log(1/\varepsilon)$ such that $(1 - p_{\mathit{min}}^{|Q|})^{\lfloor\frac{n}{|Q|}\rfloor}\leq \varepsilon/2$. This can be done in time polynomial in $|\mathcal{A}|$ and $\log(1/\varepsilon)$. By Lemma~\ref{lem:F_A-BSCC} the probability that a run does not visit, in at most $n$ steps, a configuration $q\vec{u}$ with either $Z(q\vec{u})\neq\emptyset$ or $q$ being in some BSCC of $\mathcal{A}$ is at most $\varepsilon/2$. Now we construct an $n$-step unfolding of $\mathcal{A}$ from $p\vec{v}$, i.e. we construct a finite-state Markov chain $\mathcal{M}$ such that \begin{itemize}
\item its states are tuples of the form $(q\vec{u},j)$, where $ 0\leq j \leq n$ and $ q\vec{u}\text{ is reachable from } p\vec{v} \text{ in $\leq n$ steps in $\mathcal{A}$}$,
\item for every $0\leq j < n$ we have $(q\vec{u},j)\tran{y}(q'\vec{u}',j+1)$ iff $q\vec{u}\tran{y}q'\vec{u}'$ in $\mathcal{M}_\mathcal{A}$,
\item there are no other transitions in $\mathcal{M}$. \end{itemize}
We add to this $\mathcal{M}$ new states $q_{\mathit{up}}$ and $q_{\mathit{down}}$, and for every state $(q\vec{u},j)$ with $q$ in some BSCC of $\mathcal{A}$ we replace the transitions outgoing from this state with two transitions $(q\vec{u},j)\tran{x}q_{\mathit{down}}$, $(q\vec{u},j)\tran{1-x}q_{\mathit{up}}$, where $x$ is some $(\varepsilon/2)$-approximation of $\mathcal{P}(\mathit{Run}(q\vec{u},\Z_{all}))$, which can be computed using the algorithm from Proposition~\ref{prop:approx}. Moreover, for every state $(q\vec{u},j)$ with $Z(q\vec{u})\neq \emptyset$ we replace all its outgoing transitions with a single transition leading to $q_{\mathit{down}}$. It is immediate that the probability of reaching $q_{\mathit{down}}$ from $p\vec{v}$ is an $\varepsilon$-approximation of $\mathcal{P}(\mathit{Run}(p\vec{v},\Z_{all}))$.
The number of states of $\mathcal{M}$ is at most $m = n\cdot |Q|\cdot (2n)^d$ and the algorithm of Proposition~\ref{prop:approx} is called at most $m$ times, which gives us the required complexity bound. \end{proof}
\section{Proofs of Section~\ref{sec-case2}} \label{app-sec2}
\begin{reftheorem}{Lemma}{\ref{lem-oc-div-prob}}
$\mathcal{P}(\bigcup_{q\in Q}\mathit{Run}(p\vec{u}^{in},\{q{\uparrow}\})) > 0$ iff there
exists a~BSCC $C$ of $\mathcal{F}_\mathcal{A}$ with \emph{all} counters diverging and a
\mbox{$\Zminusi{d}$-safe} finite path of the form
$p\vec{v} \tran{}^* q\vec{u} \tran{}^* q\vec{z}$ where the subpath
$q\vec{u} \tran{}^* q\vec{z}$ is $\Z_{all}$-safe,
$q \in C$, $q\vec{u}$ is above $1$, $\vec{z} - \vec{u} \geq \vec{0}$,
and $(\vec{z} - \vec{u})[i] > 0$ for every $i$ such that $\vec{t}[i] > 0$. \end{reftheorem}
\begin{proof}
``$\Rightarrow$'' Note that $\mathcal{P}(\mathit{Run}(p\vec{u}^{in},\{q{\uparrow}\}))>0$ for
some $q\in Q$. By Lemma~\ref{lem:F_A-BSCC}, almost every run of
$\mathit{Run}(p\vec{u}^{in},\{q{\uparrow}\})$ stays eventually in some BSCC of
$\mathcal{F}_\mathcal{A}$. Let $C$ be a BSCC such that the probability of all
$w \in \mathit{Run}(p\vec{u}^{in},\{q{\uparrow}\})$ that stay is $C$ is positive, and let
$\vec{t}$ be the trend of $C$. We use $R$ to denote the set of all
$w \in \mathit{Run}(p\vec{u}^{in},\{q{\uparrow}\})$ that stay in $C$.
We claim that each counter $i$ must be diverging in $C$. First, let
us consider $1\leq i\leq d-1$. Consider the one-counter pMC
$\mathcal{B}_i$. Note that the trend of $C$ in $\mathcal{B}_i$ is
to $\vec{t}[i]$. For the sake of contradiction, assume that counter~$i$
is not diverging, i.e., we have either $t_i<0$, or $t_i=0$ and
counter~$i$ is decreasing in~$C$. Then,
by~\cite{BKK:pOC-time-LTL-martingale}, starting in a
configuration $p(k)$ of $\mathcal{B}_i$ where $p\in C$, a configuration with
zero counter value is reached from $p(k)$ with probability
one. However, then, due to Equation~(\ref{eq-project}) and
Proposition~\ref{prop:one-counter-runs}, almost every run of $R$
visits a configuration with zero in one of the counters of
$\{1,\ldots,d-1\}$ (note that zero may be reached in some counter before
inevitably reaching zero in counter~$i$). As $R\subseteq
\mathit{Run}(p\vec{u}^{in},\{q{\uparrow}\})\subseteq
\mathit{Run}_{\mathcal{M}_\mathcal{A}}(p\vec{u}^{in},\neg\Zminusi{d})$, we obtain that $\mathcal{P}(R)=0$, which
is a contradiction. Now consider $i=d$. Similarly as above, starting in
a configuration $p(k)$ of $\mathcal{B}_d$ where $p\in C$, a configuration
with zero counter value is reached from $p(k)$ with probability
one. This implies that almost all runs $w$ of $R$ reach
configurations with zero counter value in the counter $d$ infinitely
many times, and hence, by Proposition~\ref{prop:one-counter-runs},
$\Phi(\Upsilon_d(w))$ does not reach $\bigcup_{q\in Q}
\{q{\uparrow}\}$ at all. It follows that $\mathcal{P}(R)=0$,
a~contradiction.
Now we prove that for almost all runs $w \in R$ and for all counters
$i$, one of the following holds: \begin{enumerate} \item[(A)] $t_i>0$ and $\liminf_{k\rightarrow \infty}
\mathit{cval}_i(w(k))=\infty$, \item[(B)] $t_i=0$ and $\mathit{cval}_i(w(k))\geq -\mathit{botfin}_i(\mathit{state}(w(k)))$
for all $k$'s large enough. \end{enumerate}
The argument is the same as in the proof of Lemma~\ref{lem-diverging}.
From~(A) and~(B), we immediately obtain the existence of
a finite path $p\vec{v} \tran{}^* q\vec{u} \tran{}^* q\vec{z}$
with the required properties.
``$\Leftarrow$'' We argue similarly as in Lemma~\ref{lem-diverging}.
\end{proof}
\begin{reftheorem}{Lemma}{\ref{lem-bottominf}}
If $\mathit{botinf}_i(q) < \infty$, then $\mathit{botinf}_i(q) \leq 3|Q|^3$ and
the exact value of $\mathit{botinf}_i(q)$ is computable in time polynomial
in $|\mathcal{A}|$. \end{reftheorem} \begin{proof}[Proof sketch]
We show that if $\mathit{botinf}_i(q) < \infty$, then there is
$w \in \mathit{FPath}_{\mathcal{M}_\mathcal{B}}(q(0))$ ending in $q(0)$ where
$w(n) \neq q(0)$ for all $1 \leq n < \mathit{length}(w)$,
$\totalrew{i}{w}{0}{\mathit{length}(w)} = -\mathit{botinf}_i(q)$, and the
counter is bounded by $2|Q|^2$ along~$w$.
From this we immediately obtain that $w$ visits at most
$3|Q|^3$ different configurations, and we can safely assume
that no configuration is visited twice (if the reward accumulated
between two consecutive visits to the same configuration is
non-negative, we can remove the cycle and thus produce a path
whose total accumulated reward can be only smaller; and if the
the reward accumulated
between two consecutive visits to the same configuration is
negative, we have that $\mathit{botinf}_i(q) = \infty$, which is
a contradiction).
To see that there is such a path $w$ where the counter is bounded
by $2|Q|^2$, it suffices to realize that if it was not the case,
we could always decrease the number of configurations visited by $w$
where the counter value is above $2|Q|^2$ by removing some
subpaths of $w$ such that the total reward accumulated in
these subpaths in non-negative. More precisely, we show
that there exist configurations $r(i_1)$, $r(i_2)$, $s(i_2)$
and $s(i_1)$ consecutively visited by $w$ where $0 < i_1 < i_2 \leq 2|Q|^2$,
the counter stays positive in all configurations between $r(i_1)$
and $s(i_1)$, the finite path from $r(i_2)$ to $s(i_2)$ visits
at least one configuration with counter value above $2|Q|^2$,
and the finite path from $r(i_2)$ to $s(i_2)$ can be ``performed''
also from $r(i_1)$ without visiting a configuration with zero counter.
If the total reward accumulated in the paths
from $r(i_1)$, $r(i_2)$ and from $s(i_2)$ to $s(i_1)$ is
negative, we obtain that $\mathit{botinf}_i(q) = \infty$ because we can
``iterate'' the two subpaths. If it
is non-negative, we can remove the subpaths from $r(i_1)$ to $r(i_2)$
and from $s(i_2)$ to $s(i_1)$ from $w$, and thus decrease the
number of configuration with counter value above $2|Q|^2$, making
the total accumulated reward only smaller.
Using the above observations, one can easily compute
$\mathit{botinf}_i(q)$ in polynomial time. \end{proof}
\begin{reftheorem}{Lemma}{\ref{lem:case2-subcrit-not-diverging}}
If some reward is not oc-diverging in $D$, then $\mathcal{P}(\mathit{Run}(p\vec{u}^{in},D)) = 0$. \end{reftheorem} \begin{proof}
Assume that counter $i$ is not diverging in $D$. Let us fix some $q\in
D$. Let $w$ be a run in $\mathcal{M}_{\mathcal{B}}$ initiated in $q(0)$ and let
$I_1<I_2<\cdots$ be non-negative integers such that $w_{I_k}$ is the
$k$-th occurrence of $q(0)$ in $w$. Given $i\in \{1,\ldots,d-1\}$
and $k\geq 1$, we denote by
$T^k_i(w)=\totalrew{i}{w}{I_{k}}{I_{k+1}-1}-\totalrew{i}{w}{I_{k}}{I_{k}}$
the $i$-th component of the total reward accumulated between the
$k$-th visit (inclusive) and the $k{+}1$-st visit to $q(0)$
(non-inclusive).
We denote by $\mathbb{E}T^k_i$ the expected value of $T^k_i$.
Observe that $T^1_i,T^2_i,\ldots$ are mutually independent and identically distributed. Thus $T^1_i,T^2_i,\ldots$ determines a random walk $S^1_i,S^2_i,\ldots$, here $S^k_i=\sum_{j=1}^k T^j_i$, on $\mathbb{Z}$. Note that $S^k_i=\totalrew{i}{w}{0}{k+1}$. By the strong law of large numbers, for almost all $w\in \mathit{Run}_{\mathcal{M}_{\mathcal{B}}}(q(0))$, \begin{eqnarray*} \mathbb{E}T^1_i & = & \lim_{k\rightarrow \infty} \frac{S^k_i(w)}{k} \\
& = & \lim_{k\rightarrow \infty} \frac{S^k_i(w)}{E^k(w)}\frac{E^k(w)}{k} \\
& = & \lim_{k\rightarrow \infty} \frac{S^k_i(w)}{E^k(w)}\lim_{k\rightarrow\infty}\frac{E^k(w)}{k} \\
& = & \lim_{k\rightarrow \infty} \frac{\totalrew{i}{w}{0}{k}}{k}\lim_{k\rightarrow \infty} \vec{e}[q]\\
& = & \vec{t}_{oc}[q] \\
& \leq & 0 \end{eqnarray*} (Here $E^k(w)$ denotes the number of steps between the $k$-th and $k+1$-st visit to $q(0)$ in $w$.) Also, $\mathcal{P}(T^1_i<0)>0$. By~Theorem~8.3.4 \cite{Chung:book}, for almost all $w\in \mathit{Run}_{\mathcal{M}_{\mathcal{B}}}(q(0))$ we have that $\liminf_{k\rightarrow \infty} S^k_i(w)=-\infty$.
However, this also means that almost every run $w\in \mathit{Run}_{\mathcal{M}_{\mathcal{B}}}(q(0))$ satisfies that $\lim_{\ell\rightarrow \infty} \totalrew{i}{w}{0}{\ell}=-\infty$. Subsequently, as all runs of $\Upsilon_d(\mathit{Run}(p\vec{u}^{in},D))$ visit $q(0)$, almost all runs $w$ of $\Upsilon_d(\mathit{Run}(p\vec{u}^{in},D))$ satisfy $\lim_{\ell\rightarrow \infty} \totalrew{i}{w}{0}{\ell}=-\infty$. Thus, by Lemma~\ref{prop:one-counter-runs}, almost all runs of $\mathit{Run}(p\vec{u}^{in},D)$ visit zero in one of the counters in $\{1,\ldots,d-1\}$. This means, that $\mathit{Run}(p\vec{u}^{in},D)=0$.
\end{proof}
\begin{reftheorem}{Lemma}{\ref{lem-diverging-2}}
Let $D$ be a BSCC of $\mathcal{X}_{\mathcal{B}}$ where all rewards are diverging.
Then there exists a computable constant~$n \in \mathbb{N}$ such that
$\mathcal{P}(\mathit{Run}(p\vec{u}^{in},D)) > 0$ iff
there is a $\Zminusi{d}$-safe finite
path of the form $p\vec{u}^{in} \tran{}^* q\vec{u}$ where $\vec{u}$
is oc-above $n$ and $\vec{u}[d]=0$. \end{reftheorem}
\begin{proof}
The constant $n$ is computed using
Lemma~\ref{lem:two-counter-divergence}. We choose a sufficiently large
$n$ such that the probability of Lemma~\ref{lem:two-counter-divergence}
is smaller than $1/d$ for every $q \in D$.
$\Leftarrow$: Assume that counter $i$ satisfies $\vec{t}_{oc}[i]>0$.
By Lemma~\ref{lem:long-run-average}, almost every run $w$ of $\mathcal{M}_\mathcal{B}$ initiated in $q(0)$ satisfies
\[
\lim_{k\rightarrow \infty} \totalrew{i}{w}{0}{k}\, /\, k=\vec{t}_{oc}[i]>0
\]
It follows that there is $c>0$ such that for a sufficiently large $k\in \mathbb{N}$ we have
$\totalrew{i}{w}{0}{k}\, /\, k\geq c$. It follows that $\totalrew{i}{w}{0}{k}\geq ck$ for all sufficiently large $k$. Thus for all counters $i$ satisfying $\vec{t}_{oc}[i]>0$ and for almost all runs $w$ of $\mathcal{M}_{\mathcal{B}}$ initiated in $q(0)$ we have that $\lim_{k\rightarrow \infty} \totalrew{i}{w}{0}{k}=\infty$.
For every $n\in \mathbb{N}$ we denote by $R_n$ the set of all runs $w$ initiated in $q(0)$ such that $\totalrew{i}{w}{0}{k}> -n$ for all $k$ and all $i$ satisfying $\vec{t}_{oc}[i]>0$. By the above argument, $\mathcal{P}(\bigcup_{n} R_n)=1$. Hence, there must be $n$ such that $\mathcal{P}(R_n)>0$.
Let $q\vec{u}$ be any configuration that is above $n$ and satisfies $\vec{u}[d]=0$. Then $\Upsilon_d(\mathit{Run}(q\vec{u},\Zminusi{d}))\supseteq R_n$ and hence $\mathcal{P}(\mathit{Run}(q\vec{u},\Zminusi{d}))\geq \mathcal{P}(R_n)>0$. By our assumption, such a configuration $q\vec{u}$ is reachable from $p\vec{u}^{in}$ via a $\Zminusi{d}$-safe path, and thus $\mathcal{P}(\mathit{Run}(p\vec{u}^{in},D))>0$.
$\Rightarrow$:
We show that for almost
all $w \in \mathit{Run}(p\vec{u}^{in},D)$ and all $i \in \{1,\ldots,d-1\}$, one of
the following conditions holds:
\begin{enumerate}
\item[(A)] $\vec{t}_{oc}[i]>0$ and $\liminf_{k\rightarrow \infty} \mathit{cval}_i(w(k))=\infty$,
\item[(B)] $\vec{t}_{oc}[i]=0$ and $\mathit{cval}_i(w(k))\geq \mathit{botinf}_i(\mathit{state}(w(k)))$
for all $k$'s large enough.
\end{enumerate}
Concerning~(A), note that for almost all runs $w$ of $\mathcal{M}_\mathcal{B}$ initiated in $q(0)$ where $q\in D$ we have that
\[ \lim_{k\rightarrow \infty} \totalrew{i}{w}{0}{k}\, /\, k=\vec{t}_{oc}[i]>0 \] which implies, as above, that $\lim_{k\rightarrow \infty} \totalrew{i}{w}{0}{k}=\infty$. Let $q\vec{u}$ be a configuration of $\mathcal{A}$ which is oc-above $1$ and satisfies $\vec{u}[d]=0$. Then almost all runs $w$ of $\Upsilon_d(\mathit{Run}(q\vec{u},\Zminusi{d}))$ satisfy $\lim_{k\rightarrow \infty} \totalrew{i}{w}{0}{k}=\infty$, and hence also almost all runs $w$ of $\mathit{Run}(q\vec{u},\Zminusi{d})$ satisfy $\liminf_{k\rightarrow \infty} \mathit{cval}_i(w(k))=\infty$. As almost every run of $\mathit{Run}(p\vec{u}^{in},D)$ visits $q\vec{u}$ for some $\vec{u}$ that is oc-above $1$ nad satisfying $\vec{u}[d]=0$, almost all runs $w$ in $\mathit{Run}(p\vec{u}^{in},D)$ satisfy $\liminf_{k\rightarrow \infty} \mathit{cval}_i(w(k))=\infty$.
Concerning (B), note that almost all runs $w \in \mathit{Run}(p\vec{u}^{in},D)$
satisfying $\mathit{cval}_i(w'(k)) < \mathit{botinf}_i(\mathit{state}(w(k)))$
for infinitely many $k$'s eventually visit zero in some counter
(there is a path of length at most $3|Q|^3$ from each such
$w(k)$ to a configuration with zero in counter $i$, or in one of the
other counters).
The above claim immediately implies that for every $n \in \mathbb{N}$,
almost every run of $\mathit{Run}(p\vec{u}^{in},D)$ visits a configuration
$q\vec{u}$ oc-above~$n$.
The other implication is proven similarly as in Lemma~\ref{lem-diverging}.
\end{proof}
Following~\cite{ABKM:Num-Analysis-SIAM} the \textsc{Square-Root-Sum} problem is defined as follows. Given natural numbers $d_1, \ldots, d_n \in \mathbb{N}$ and $k \in \mathbb{N}$, decide whether $\sum_{i=1}^n \sqrt{d_i} \ge k$. Membership of square-root-sum in NP has been open since 1976.
It is known that \textsc{Square-Root-Sum} reduces to PosSLP and hence lies in the counting hierarchy, see~\cite{ABKM:Num-Analysis-SIAM} and the references therein for more information on square-root-sum, PosSLP, and the counting hierarchy.
\begin{reftheorem}{Proposition}{\ref{sqrt-hard}}
The qualitative $\Zminusi{d}$-reachability problem is
\textsc{Square-Root-Sum}-hard, even for two-dimensional pMC
where $\vec{e}[q]<\infty$ for all $q\in D$ in
every BSCC of $\mathcal{X}_\mathcal{B}$. \end{reftheorem} \begin{proof} We adapt a reduction from~\cite{EWY:one-counter-PE}. Let $d_1, \ldots, d_n, k \in \mathbb{N}$ be an instance of the\textsc{Square-Root-Sum} problem. Let $m := \max \{d_1, \ldots, d_n, k\}$. Define $c_i := \frac12 (1 - d_i/m^2)$ for $i \in \{1, \ldots, n\}$.
We construct a pMC $\mathcal{A} = (Q,\gamma,W)$ as follows. Take $Q := \{q, r_1, \ldots, r_n, s_+, s_-\}$ and set of rules~$\gamma$ as listed below (we omit labels and some irrelevant rules). The weight assignment~$W$ is, for better readability, specified in terms of probabilities rather than weights, with the obvious intended meaning. \begin{align*} \textstyle \textstyle \frac{1}{2 n} & : (q, (0,0), \emptyset, r_i) && \text{ for all $i \in \{1, \ldots, n\}$} \\ \textstyle \frac{1}{2} & : (q, (0,-1), \emptyset, s_+) \\ \textstyle \frac{1}{2} & : (r_i, (0,+1), \emptyset, r_i) && \text{ for all $i \in \{1, \ldots, n\}$} \\ \textstyle c_i & : (r_i, (0,-1), \emptyset, r_i) && \text{ for all $i \in \{1, \ldots, n\}$} \\ \textstyle \frac12 - c_i & : (r_i, (0,0), \emptyset, s_-) && \text{ for all $i \in \{1, \ldots, n\}$} \\ \textstyle 1 & : (r_i, (0,+1), \{2\}, q) && \text{ for all $i \in \{1, \ldots, n\}$} \\ \textstyle 1 & : (s_-, (0,-1), \emptyset, s_-) \\ \textstyle 1 & : (s_-, (-1,+1), \{2\}, q) \\ \textstyle \frac{k}{n m} & : (s_+, (+1,+1), \{2\}, q) \\ \textstyle 1-\frac{k}{n m} & : (s_+, (0,+1), \{2\}, q) \end{align*} We claim that $\mathcal{P}(\mathit{Run}(q\vec{1},\{\{1\}\})) = 1$ holds if and only if $\sum_i \sqrt{d_i} \ge k$ holds. It is shown in~\cite{EWY:one-counter-PE} that $r_i(1,1)$ reaches, with probability~$1$, the configuration $r_i(1,0)$ or $s_-(1,0)$
before reaching any other configuration with $0$ in the second counter. In fact, it is shown there that the probability of reaching~$s_-(1,0)$ is $\sqrt{d_i}/m$, and of reaching~$r_i(1,0)$ is $1 - \sqrt{d_i}/m$. The only BSCC~$D$ of~$\mathcal{X}_\mathcal{B}$ is $\{r_1, \ldots, r_n, s_+, s_-\}$. It follows for the invariant distribution $\vec{\mu}_{oc}$ that
$\vec{\mu}_{oc}[s_+] = \frac12$ and $\vec{\mu}_{oc}[s_-] = \frac{1}{2 n m} \sum_i \sqrt{d_i}$. From the construction it is clear that $\vec{\delta}_1[s_+] = +\frac{k}{n m}$ and $\vec{\delta}_1[s_-] = -1$
and $\vec{\delta}_1[r_i] = 0$ for all $i \in \{1, \ldots, n\}$. Hence we: \begin{align*} \vec{t}_{oc}[i] & = \left(\vec{\mu}^T_{oc} \cdot \vec{\delta}_i\right)/\left(\vec{\mu}^T_{oc} \cdot \vec{e}\right) \\
& = \textstyle \left( \frac12 \cdot \frac{k}{n m} - \frac{1}{2 n m} \sum_i \sqrt{d_i} \right) /\left(\vec{\mu}^T_{oc} \cdot \vec{e}\right) \end{align*} So we have $\vec{t}_{oc}[i] \le 0$ if and only if $\sum_i \sqrt{d_i} \ge k$ holds. The statement then follows from Lemma~\ref{lem-diverging-2}. \end{proof}
\section{Martingale}
\subsection{Matrix Notation}
In the following, $Q$ will denote a finite set (of control states). We view the elements of $\mathbb{R}^Q$ and $\mathbb{R}^{Q \times Q}$ as vectors and matrices, respectively. The entries of a vector $\vec{v} \in \mathbb{R}^Q$ or a matrix $M \in \mathbb{R}^{Q \times Q}$ are denoted by $\vec{v}[p]$ and $M[p,q]$ for $p,q \in Q$. Vectors are column vectors by default; we denote the transpose of a vector~$\vec{v}$ by $\vec{v}^T$, which is a row vector. For vectors $\vec{u}, \vec{v} \in \mathbb{R}^Q$ we write $\vec{u} \le \vec{v}$ (resp.\ $\vec{u} < \vec{v}$)
if the respective inequality holds in all components. The vector all whose entries are~$0$ (or~$1$) is denoted by $\vec{0}$ (or~$\vec{1}$, respectively). We denote the identity matrix by $I \in \{0,1\}^Q$ and the zero matrix by~$0$. A matrix $M \in [0,1]^{Q \times Q}$ is called \emph{stochastic} (\emph{substochastic}),
if each row sums up to~$1$ (at most~$1$, respectively). A nonnegative matrix $M \in [0,\infty)^Q$ is called \emph{irreducible},
if the directed graph $(Q, \{(p,q) \in Q^2 \mid M[p,q] > 0\})$ is strongly connected. We denote the spectral radius (i.e., the largest among the absolute values of the eigenvalues) of a matrix~$M$ by~$\rho(M)$.
\subsection{Proof of Lemma~\ref{lem-g-zero-exists}}
The proof of Lemma~\ref{lem-g-zero-exists} is based on the notion of \emph{group inverses} for matrices~\cite{Erdelyi67}. Close connections of this concept to (finite) Markov chains are discussed in~\cite{Meyer75}. We have the following lemma:
\begin{lemma} \label{lem-group-inverse} Let $P$ be a nonnegative irreducible matrix with $\rho(P) = 1$. Then there is a matrix, denoted by $(I - P)^\#$, such that $(I - P) (I - P)^\# = I - W$,
where $W$ is a matrix whose rows are scalar multiples of the dominant left eigenvector of~$P$. \end{lemma} \begin{proof} In~\cite{Meyer75} the case of a stochastic matrix~$P$ is considered. In the following we adapt proofs from~\cite[Theorems 2.1 and~2.3]{Meyer75}. For a square matrix $M$, a matrix $M^\#$ is called \emph{group inverse} of~$M$, if we have $M M^\# M = M$ and $M^\# M M^\# = M^\#$ and $M M^\# = M^\# M$. It is shown in~\cite[Lemma~2]{Erdelyi67} that a matrix~$M$ has a group inverse if and only if $M$ and~$M^2$ have the same rank. As $P$ is irreducible, the Perron-Frobenius theorem implies that the eigenvalue~$1$ has algebraic multiplicity equal to one. So $0$ is an eigenvalue of~$M := (I-P)$ with algebraic multiplicity~$1$. This implies that the Jordan form for~$M$ can be written as \[
\begin{pmatrix}
0 & 0 \\
0 & J'
\end{pmatrix} \] where the square matrix~$J'$ is invertible. It follows that $M$ and $M^2$ have the same rank, so $M^\#$ exists. Using the definition of group inverse, we have $(I - M M^\#) P = (I - M M^\#)$. In other words, the rows of~$I - (I - P) (I - P)^\#$ are left eigenvectors of~$P$ with eigenvalue~$1$. The statement then follows by the Perron-Frobenius theorem. \end{proof}
Now we can prove Lemma~\ref{lem-g-zero-exists}.
\begin{proof} Recall that the matrix~$A$ is stochastic and irreducible. Also recall from the main body of the paper that $\alpha^T A = \alpha^T$. It follows from the Perron-Frobenius theorem that $\rho(A) = 1$. Define $\vec{g}(0)[D] := (I - A)^\# \vec{r}_0$, where $(I - A)^\#$ is the matrix from Lemma~\ref{lem-group-inverse}. The non-$D$-components can be set arbitrarily, for instance, they can be set to~$0$. So we have $\vec{g}(0)[D] = \vec{r}_0 + A \vec{g}(0)[D] - W \vec{r}_0$,
where the rows of~$W$ are multiples of~$\vec{\alpha}^T$. We have: \begin{align*}
\vec{\alpha}^T \vec{r}_0
& = \vec{\alpha}^T \left( \vec{\delta}_1 - \frac{\vec{\alpha}^T \vec{\delta}_1}{\vec{\alpha}^T \vec{e}} \vec{e} \right)
&& \text{by the definitions of~$\vec{r}_0$ and~$t$} \\
& = 0\,. \end{align*} So \eqref{eq-mart-fund} follows. \end{proof}
\subsection{Proof of Proposition~\ref{prop-martingale}}
For notational convenience, we assume in the following that $\mathcal{A}$ is a 2-dimensional pMC corresponding to the labelled $1$-dimensional pMC~$\mathcal{D}$
from the main body; i.e., the first counter of~$\mathcal{A}$ encodes the rewards of~$\mathcal{D}$, the second counter of~$\mathcal{A}$ encodes the unique counter of~$\mathcal{D}$.
Define the substochastic matrices $Q_\rightarrow \in [0,1]^{D \times D}$, $Q_\uparrow \in [0,1]^{D \times Q}$,
$P_\downarrow, P_\rightarrow, P_\uparrow \in [0,1]^{Q \times Q}$ as follows: \begin{align}
Q_\rightarrow[p,q] & := \sum \{y \mid \exists x_1: p(1,0) \ltran{y} q(x_1,0)\} \label{eq-mart-Q0} \\
Q_\uparrow[p,q] & := \sum \{y \mid \exists x_1: p(1,0) \ltran{y} q(x_1,1)\} \label{eq-mart-Q+} \\
P_\downarrow[p,q] & := \sum \{y \mid \exists x_1: p(1,1) \ltran{y} q(x_1,0)\} \label{eq-mart-P-} \\
P_\rightarrow[p,q] & := \sum \{y \mid \exists x_1: p(1,1) \ltran{y} q(x_1,1)\} \label{eq-mart-P0} \\
P_\uparrow[p,q] & := \sum \{y \mid \exists x_1: p(1,1) \ltran{y} q(x_1,2)\} \;, \label{eq-mart-P+} \end{align} where the transitions $p(1,0) \ltran{y} q(x_1,0)$, etc.\ are in the Markov chain~$\mathcal{M}_\mathcal{A}$. Note that $Q_\rightarrow + Q_\uparrow$ and $P_\downarrow + P_\rightarrow + P_\uparrow$ are stochastic.
Observe that we have, e.g., that
$Q_\rightarrow[p,q] = \sum \{y \mid p(0) \ltran{y} q(0)\}$,
where the transition $p(0) \ltran{y} q(0)$ is in the Markov chain~$\mathcal{M}_\mathcal{D}$.
The matrix~$G$ from the main body of the paper is (see e.g.\ \cite{EWY:one-counter-PE}) the least (i.e., componentwise smallest) matrix with $G \in [0,1]^{Q \times Q}$ and \begin{equation}
G = P_\downarrow + P_\rightarrow G + P_\uparrow G G \,. \label{eq-mart-G} \end{equation} Recall from the main body that $G$ is stochastic.
For the matrix~$A$ defined in the main body we have \begin{align}
A = Q_\rightarrow + Q_\uparrow G[D] \,, \label{eq-mart-A} \end{align} where $G[D] \in [0,1]^{Q \times D}$ denotes the matrix obtained from~$G$ by deleting the columns with indices in $Q \setminus D$. Recall from the main body that $A$ is stochastic and irreducible.
Define \begin{align}
B & := P_\rightarrow + P_\uparrow G + P_\uparrow \in [0,1]^{Q \times Q} \,. \label{eq-mart-B} \end{align} Define the vectors $\vec{\delta}_{=0!} \in [-1,1]^{D}$, $\vec{\delta}_{>0!} \in [-1,1]^Q$ with \begin{align}
\vec{\delta}_{=0!}[p] & := \sum \{ y x_1 \mid \exists q \in Q \ \exists x_2 : p(1,0) \ltran{y} q(1+x_1, x_2) \} \label{eq-mart-delta-0-bang} \\
\vec{\delta}_{>0!}[p] & := \sum \{ y x_1 \mid \exists q \in Q \ \exists x_2 : p(1,1) \ltran{y} q(1+x_1, x_2) \} \,, \label{eq-mart-delta-pos-bang} \end{align} where the transitions $p(1,0) \ltran{y} q(1+x_1, x_2)$ and $p(1,1) \ltran{y} q(1+x_1, x_2)$ are in the Markov chain~$\mathcal{M}_\mathcal{A}$. We have that $\vec{\delta}_{=0!}[p]$ is the expected reward incurred in the next step when starting in $p(0)$. Similarly, $\vec{\delta}_{>0!}[p]$ is the expected reward incurred in the next step when starting in $p(x_2)$ for $x_2 \ge 1$.
\begin{lemma} The following equalities hold: \begin{align}
\vec{e}_\downarrow & = \vec{1} + B \vec{e}_\downarrow \label{eq-mart-t-minus} \\
\vec{\delta}_\downarrow & = \vec{\delta}_{>0!} + B \vec{\delta}_\downarrow \label{eq-mart-delta-minus} \end{align} \end{lemma} \begin{proof} Define the following vectors: \begin{align*}
\vec{e}_1 & := P_\downarrow \vec{1} \\
\vec{e}_2 & := P_\rightarrow (\vec{1} + \vec{e}_\downarrow) \\
\vec{e}_3 & := P_\uparrow (\vec{1} + \vec{e}_\downarrow) \\
\vec{e}_4 & := P_\uparrow G \vec{e}_\downarrow \end{align*} Observe that $\vec{e}_1 + \vec{e}_2 + \vec{e}_3 + \vec{e}_4$ is the right-hand side of~\eqref{eq-mart-t-minus},
so we have to show that $\vec{e}_\downarrow = \vec{e}_1 + \vec{e}_2 + \vec{e}_3 + \vec{e}_4$. Let $q \in Q$. For concreteness we consider the configuration $q(1)$. We have that $\vec{e}_1[q]$ is the probability that the first step decreases the counter by~$1$. Note that we can view $\vec{e}_1[q]$ also as the probability that the first step decreases the counter by~$1$ (namely, to~$0$),
multiplied with the conditional expected time to reach the $0$-level from~$q(1)$,
conditioned under the event that the first step decreases the counter by~$1$. We have that $\vec{e}_2[q]$ is the probability that the first step keeps the counter constant (at~$1$),
multiplied with the conditional expected time to reach the $0$-level from $q(1)$,
conditioned under the event that the first step keeps the counter constant. We have that $\vec{e}_3[q]$ is the probability that the first step increases the counter by~$1$ (namely, to~$2$),
multiplied with the conditional expected time to reach the $1$-level (again) from $q(1)$,
conditioned under the event that the first step increases the counter by~$1$. Finally, $\vec{e}_4[q]$ is the probability that the first step increases the counter by~$1$ (namely, to~$2$),
multiplied with the conditional expected time to reach the $0$-level \emph{after} having returned to the $1$-level,
conditioned under the event that the first step increases the counter by~$1$. So, $\left( \vec{e}_1 + \vec{e}_2 + \vec{e}_3 + \vec{e}_4 \right)[q]$ is the expected time to reach the $0$-level. Hence \eqref{eq-mart-t-minus} is proved. The proof of~\eqref{eq-mart-delta-minus} is similar, with reward replacing time. \end{proof}
By combining \eqref{eq-mart-t-minus} and~\eqref{eq-mart-delta-minus} with the definition of~$\vec{r}_\downarrow$ we obtain: \begin{equation}
\vec{r}_\downarrow = \vec{\delta}_{>0!} - t \vec{1} + B \vec{r}_\downarrow \label{eq-mart-r-down} \end{equation}
From the definitions we obtain \begin{align}
\vec{\delta}_1 & := \vec{\delta}_{=0!} + Q_\uparrow \vec{\delta}_\downarrow \label{eq-mart-delta-0} \\
\vec{e} & := \vec{1} + Q_\uparrow \vec{e}_\downarrow \,. \label{eq-mart-t-0} \end{align}
By combining \eqref{eq-mart-delta-0} and~\eqref{eq-mart-t-0} with the definition of~$\vec{r}_0$ we obtain: \begin{equation}
\vec{r}_0 = \vec{\delta}_{=0!} - t \vec{1} + Q_\uparrow \vec{r}_\downarrow \;. \label{eq-mart-r-0} \end{equation}
Now we can prove Proposition~\ref{prop-martingale}. \begin{proof} We have \begin{align*}
& \mathcal{E} \left( \ms{\ell+1} - \ms\ell \;\middle\vert\; w(\ell), \ \xs\ell_2 = 0 \right) \\
& = \left( \vec{\delta}_{=0!} - t \vec{1} + Q_\rightarrow \vec{g}(0) + Q_\uparrow \vec{g}(1) - \vec{g}(0)[D] \right)[\ps\ell]
&& \text{by \eqref{eq-mart-m}, \eqref{eq-mart-delta-0-bang}, \eqref{eq-mart-Q0}, \eqref{eq-mart-Q+}} \\
& = \left( \vec{\delta}_{=0!} - t \vec{1} + Q_\rightarrow \vec{g}(0) + Q_\uparrow \left( \vec{r}_\downarrow + G \vec{g}(0) \right) - \vec{g}(0)[D] \right)[\ps\ell]
&& \text{by \eqref{eq-mart-g}} \\
& = \left( \vec{r}_{0} + (A - I) \vec{g}(0)[D] \right)[\ps\ell]
&& \text{by \eqref{eq-mart-r-0}, \eqref{eq-mart-A}} \\
& = 0 && \text{by \eqref{eq-mart-fund}} \intertext{and}
& \mathcal{E} \left( \ms{\ell+1} - \ms\ell \;\middle\vert\; w(\ell), \ \xs\ell_2 > 0 \right) \\
& = \left( \vec{\delta}_{>0!} - t \vec{1}
+ P_\downarrow \vec{g}\big(\xs\ell_2-1\big)
+ P_\rightarrow \underbrace{\vec{g}\big(\xs\ell_2\big)}_{\stackrel{\eqref{eq-mart-g}}{=}\ \vec{r}_\downarrow + G \vec{g}(\xs\ell_2-1)}
+ \ P_\uparrow \underbrace{\vec{g}\big(\xs\ell_2+1\big)}_{\stackrel{\eqref{eq-mart-g}}{=}\ \vec{r}_\downarrow + G \vec{g}(\xs\ell_2)} - \ \vec{g}(\xs\ell_2) \right)[\ps\ell]
&& \text{by \eqref{eq-mart-m}, \eqref{eq-mart-delta-pos-bang}, \eqref{eq-mart-P-}--\eqref{eq-mart-P+}} \\
& = \left( \vec{\delta}_{>0!} - t \vec{1} + (P_\rightarrow + P_\uparrow G + P_\uparrow) \vec{r}_\downarrow + \left( P_\downarrow + P_\rightarrow G + P_\uparrow G G \right) \vec{g}\big(\xs\ell_2-1\big) - \vec{g}(\xs\ell_2) \right)[\ps\ell]
&& \text{by \eqref{eq-mart-g}} \\
& = \left( \vec{r}_\downarrow + G \vec{g}\big(\xs\ell_2-1\big) - \vec{g}(\xs\ell_2) \right)[\ps\ell]
&& \text{by \eqref{eq-mart-B}, \eqref{eq-mart-r-down}, \eqref{eq-mart-G}} \\
& = 0 && \text{by \eqref{eq-mart-g}\,.} \end{align*} \end{proof}
\subsection{Proof of Lemma~\ref{lem:weights-bound}}
Define $e_{\mathit max} := 1 + \max_{q \in Q} \vec{e}_\downarrow[q] \ge 2$.
We first prove the following lemma: \begin{lemma} \label{lem-mart-potential} There exists a vector $\vec{g} \in \mathbb{R}^D$ with $\vec{g} = \vec{r}_0 + A \vec{g}$
and
\[
0 \le \vec{g}[q] \le \frac{e_{\mathit max} |D|}{y_{\mathit min}^{|D|}} \qquad \text{for all $q \in D$,}
\]
where $y_{\mathit min}$ denotes the smallest nonzero entry in the matrix~$A$. \end{lemma} \begin{proof} Recall that by Lemma~\ref{lem-g-zero-exists} there is a vector $\vec{g}(0)[D] \in \mathbb{R}^D$ with
\[
\vec{g}(0)[D] = \vec{r}_0 + A \vec{g}(0)[D] \,.
\] Since $A$ is stochastic, we have $A \vec{1} = \vec{1}$. So there is $\kappa \in \mathbb{R}$ such that with $\vec{g} := \vec{g}(0)[D] + \kappa \vec{1}$ we have \begin{equation}
\vec{g} = \vec{r}_0 + A \vec{g} \label{eq-mart-potential-proof} \end{equation}
and $g_{\mathit max} = e_{\mathit max} |D| / y_{\mathit min}^{|D|}$,
where we denote by $g_{\mathit min}$ and $g_{\mathit max}$ the smallest and largest component of~$\vec{g}$, respectively. We have to show $g_{\mathit min} \ge 0$. Let $q \in D$ such that $\vec{g}[q] = g_{\mathit max}$. Define the \emph{distance} of a state $p \in D$, denoted by $\eta_p$, as the distance of~$p$ from~$q$ in the directed graph induced by~$A$. Note that $\eta_q = 0$
and all $p \in D$ have distance at most $|D|-1$, as $A$ is irreducible. We prove by induction that a state~$p$ with distance~$i$ satisfies $\vec{g}[p] \ge g_{\mathit max} - e_{\mathit max} i/ y_{\mathit min}^i$. The claim is obvious for the induction base ($i=0$). For the induction step, let $p$ be a state such that~$\eta_p = i+1$. Then there is a state $r$ such that $A[r,p] > 0$ and $\eta_r = i$. We have \begin{align*}
\vec{g}[r] & = (A \vec{g})[r] + \vec{r}_0[r] && \text{by \eqref{eq-mart-potential-proof}} \\
& \le (A \vec{g})[r] + e_{\mathit max} && \text{as $\vec{r}_0 \le e_{\mathit max} \vec{1}$} \\
& = \big(A[r,p] \cdot \vec{g}[p] +
\sum_{p' \neq p} A[r,p'] \cdot \vec{g}[p']\big) + e_{\mathit max} \\
& \le A[r,p] \cdot \vec{g}[p] + (1-A[r,p])\cdot g_{\mathit max} + e_{\mathit max} && \text{as $A$ is stochastic.} \end{align*} By rewriting the last inequality and applying the induction hypothesis to $\vec{g}[r]$ we obtain \begin{align*}
\vec{g}[p] & \geq g_{\mathit max} - \frac{g_{\mathit max} - \vec{g}[r] + e_{\mathit max}}{A[r,p]} \geq
g_{\mathit max} - \frac{g_{\mathit max} - (g_{\mathit max} - e_{\mathit max} i/ y_{\mathit min}^i) + e_{\mathit max}}{y_{\mathit min}} \geq
g_{\mathit max} - \frac{e_{\mathit max} (i+1)}{y_{\mathit min}^{i+1}} \,. \end{align*} This completes the induction step. Hence we have $g_{\mathit min} \ge 0$ as desired. \end{proof}
Now we prove Lemma~\ref{lem:weights-bound}: \begin{proof} We need the following explicit expression for~$\vec{g}$: \begin{equation}
\vec{g}(n) = G^{n} \vec{g}(0) + \sum_{i=0}^{n - 1} G^i \vec{r}_\downarrow
\qquad \text{for all $n \ge 0$} \label{eq-mart-g-explicit} \end{equation} Let us prove~\eqref{eq-mart-g-explicit} by induction on~$n$. For the induction base note that the cases $n = 0,1$ follow immediately from the definition~\eqref{eq-mart-g} of~$\vec{g}$. For the induction step let $n \ge 1$. We have: \begin{align*}
\vec{g}(n + 1)
& = \vec{r}_\downarrow + G \vec{g}(n) && \text{by~\eqref{eq-mart-g}} \\
& = G^{n+1} \vec{g}(0) + \sum_{i=0}^{n} G^i \vec{r}_\downarrow && \text{by the induction hypothesis} \end{align*} So \eqref{eq-mart-g-explicit} is proved. In the following we assume that $\vec{g}(0)$ is chosen as in Lemma~\ref{lem-mart-potential}. We then have: \begin{align*}
|\vec{g}(n)| & \le |\vec{g}(0)| + n |\vec{r}_\downarrow| && \text{by~\eqref{eq-mart-g-explicit} and as $G$ is stochastic} \\
& \le \frac{e_{\mathit max} |D|}{y_{\mathit min}^{|D|}} + n e_{\mathit max} && \text{by Lemma~\ref{lem-mart-potential} and as $|\vec{r}_\downarrow| \le |\vec{e}_\downarrow| \le e_{\mathit max}$} \end{align*} \end{proof}
\section{Proof of Theorem~\ref{thm:approx-general-case2}} \label{sec:case2-approx}
We show that $\mathcal{P}(\mathit{Run}(p\vec{v},\mathcal{Z}_{-d}))$ can be effectively approximated up to an arbitrarily small absolute error $\varepsilon > 0$.
We will use the fact than the probability of reaching a specific set of states in a 1-dimensional pMC can be effectively approximated.
\begin{lemma}
\label{lem:1pmc-reachability} Let $\mathcal{A}'$ be any one-dimensional pMC and let $Q$ be its set of states. Given an initial configuration $q(k)$, a set $S\subseteq Q$ and $\varepsilon>0$ we can effectively approximate, up to the absolute error $\varepsilon$, the probability of reaching a configuration $r(j)$ with $r\in S$ from $q(k)$. \end{lemma} \begin{proof}
The crucial observation is that if there is a path from a state $t$ to $S$ in $\mathcal{F}_{\mathcal{A}'}$, then for every $j \geq |Q|$ there is a path of length at most $n$ from $q(j)$ to a configuration with the control state in $S$. If there is no path from $t$ to $S$ in $\mathcal{F}_{\mathcal{A}'}$, then a configuration with the control state in $S$ cannot be reached from $q(j)$ for any $j$. Thus, the probability that a run initiated in $q(k)$ visits a counter value $q(k+i)$ without visiting $S$ \emph{and then} visits $S$ is at most $(1-p_{\mathit{min}}^{|Q|})^{\frac{i}{|Q|}}$, where $p_{\mathit{min}}$ is the minimal non-zero probability in $\mathcal{A}'$. For a given $\varepsilon$, we can effectively compute $i$ such that $(1-p_{\mathit{min}}^{|Q|})^{\frac{i}{|Q|}} \leq \varepsilon$ and effectively construct a finite-state Markov chain $\mathcal{M}$ in which the configurations of $\mathcal{A}'$ with counter value $\leq i + k$ are encoded in the finite-state control unit (i.e., $\mathcal{M}$ can be defined as a Markov chain obtained from $\mathcal{M}_{\mathcal{A}'}$ by removing all configurations with counter height $>i+k$ together with their adjacent transitions and replace all transitions outgoing from configurations of the form $r(i+k)$ with self-loops on $r(i+k)$).
Using standard methods for finite-state Markov chains we can compute the probability of reaching the set $S'=\{r(j)\mid r\in S\}$ from $q(k)$ in $\mathcal{M}$. From the discussion above it follows that this value is an $\varepsilon$-approximation of the probability that $r(j)$ with $r\in S$ is reached in $\mathcal{A}'$. \end{proof}
The proof closely follows the proof of Theorem~\ref{thm:approx-general}. We first show how to approximate the probability under the assumption that $p$ is in some BSCC $D$ of $\mathcal{X}_\mathcal{B}$. It is then easy to drop this assumption.
\newcommand{A_{\mathit{max}}}{A_{\mathit{max}}} \begin{proposition} \label{prop:case2-approx}
There is an algorithm which, for a given \mbox{$d$-dimensional} pMC $\mathcal{A}$, its initial configuration $p\vec{v}$ such that $p$ is in a BSCC of $\mathcal{X}_\mathcal{B}$, and a given $\varepsilon>0$ computes a number $\nu$ such that $|\mathcal{P}(\mathit{Run}(p\vec{v},\mathcal{Z}_{-d}))-\nu|\leq d\cdot\varepsilon$. \end{proposition} \begin{proof}
Clearly we need to consider only $d\geq 2$. We proceed by induction on $d$. The base case and the induction step are solved in almost identical way (which was the case also in the proof of Proposition~\ref{prop:approx}). Therefore, below we present the proof of the induction step and only highlight the difference between the induction step and the base case when needed.
We again assume that $\mathcal{P}(\mathit{Run}(p\vec{v},\mathcal{Z}_{-d}))<1$. This can be checked effectively due to Theorem~\ref{thm:qual-d-algorithm} and if the condition does not hold, we may output $\nu=1$. In particular we assume that all rewards in $D$ are oc-diverging.
Recall from the proof of Proposition~\ref{prop:approx} that for any counter $i$ and any vector $\vec{\beta}\in\{-1,0,1\}^d$ we denote by $\vec{\beta}_{-i}$ the $(d-1)$-dimensional vector obtained from $\vec{\beta}$ by deleting its $i$-component; and by $\mathcal{A}_{-i}$ the $(d-1)$-dimensional pMC $\mathcal{A}_{-i}$ obtained from $\mathcal{A}$ by ``forgetting'' the $i$-th counter. (See the proof of Proposition~\ref{prop:approx} for a formal definition).
Let $\vec{t}_{oc}$ be the oc-trend of $D$. For every counter $i\in\{1,\dots,d-1\}$ such that $rt[i]>0$ we compute the number $A_0$ of Lemma~\ref{lem-divergence} for $D$ in $\mathcal{X}_\mathcal{D}$, and denote this number by $A_i$. We put $A_{\mathit{max}}=\max\{A_i\mid \vec{t}_{oc}[i]>0\}$. Then we compute a number such that $\frac{A_{\mathit{max}}^K}<\varepsilon/2$. For any $(d-1)$-dimensional vector $\vec{x}$ we denote by $\mathit{mindiv}(\vec{x})$ the smallest $i\in\{1,\dots,d-1\}$ such that either $\vec{t}_{oc}[i]>0$ and $\vec{x}[i]\geq K$ or $\vec{t}[i]=0$ and $\vec{x}[i]\geq 3|Q|^3$ (if such $i$ does not exist, we put $\mathit{mindiv}(\vec{x})=\bot$).
Consider a 1-dimensional pMC $\mathcal{A}_{K} = (Q',\gamma',W')$ which can be obtained from $\mathcal{A}$ as follows: \begin{itemize}
\item $Q'$ consists of all tuple $(q,\vec{u})$, where $q\in Q$ and $\vec{u}$ is an arbitrary $(d-1)$-dimensional vector of non-negative integers whose every component is bounded by $K$; additionally, $Q'$ contains two special states $q{\uparrow}$ and $q{\downarrow}$.
\item $((q,\vec{u}),j,c,(r,\vec{z}))\in \gamma'$ iff $\mathit{mindiv}(\vec{u})=\bot$ and $(q,\langle\vec{z}-\vec{u},j\rangle_d,c,r) \in \gamma$.
\item For every $1\leq i \leq d-1$ and every $(q,\vec{u})\in |Q|$ such that $\mathit{mindiv}(\vec{u})\neq \bot$ we have rules $((q,\vec{u}),0,\emptyset,q{\uparrow})$ and $((q,\vec{u}),0,\emptyset,q{\uparrow})$ in $\gamma'$.
\item $W'((q,\vec{u}),j,c,(r,\vec{z})) = W(q,\langle\vec{z}-\vec{u},j\rangle_d,c,r)$ for all rules in $\gamma'$ of this shape.
\item $W'((q,\vec{u}),0,\emptyset,q{\downarrow}) = x$, where $x$ is some $((d-1)\cdot\varepsilon)$-approximation of $\mathcal{P}_{\mathcal{A}_{-i}}(\mathit{Run}(q\vec{u}_{-i},\mathcal{Z}_{-d}))$ (which can be computed using the algorithm for dimension $d-1$).
\item $W'((q,\vec{u}),0,\emptyset,q{\uparrow}) = 1-W'((q,\vec{u}),0,\emptyset,q{\downarrow})$. \end{itemize}
In other words $\mathcal{A}_K$ is obtained from $\mathcal{A}$ by encoding all configurations where all of the first $d-1$ counters are bounded by $K$ explicitly into the state space. If one of these counters surpasses $K$, we ``forget'' about this counter and approximate the 0-reachability in the resulting configuration recursively.
By induction $\mathcal{A}_K$ can be effectively constructed.
Now for an initial configuration $p\vec{v}$ in which the first $d-1$ counters are bounded by $K$ let $P(p\vec{v})$ be the probability of reaching, when starting in $p\vec{v}$ in $\mathcal{A}_K$, either the state $q_{\mathit{down}}$ or a state in which at least one of the first $d-1$ counters is 0. Due to Lemma~\ref{lem:1pmc-reachability} we can approximate $P(p\vec{v})$ effectively up to $\varepsilon/2$. We claim that $|\mathcal{P}(\mathit{Run}(p\vec{v},\mathcal{Z}_{-d}))-P(p\vec{v})|\leq d\cdot \varepsilon$.
Indeed, let us denote $\mathit{Div}$ the set of all configurations $q\vec{y}$ of $\mathcal{A}$ such that $\vec{y}_{-d}$ is bounded by $K$ and $\mathit{mindiv}({\vec{y}_{-d}})\neq \bot$. For every $q\vec{u}\in \mathit{Div}$ we denote by $x_{q\vec{u}}$ the probability of the transition leading from some $(q(k),\vec{u}_{-d})$ to $q_{\mathit{down}}$ in $\mathcal{M}_{\mathcal{A}_K}$ (note that this probability is independent of $k$ and is equal to the weight of the corresponding rule in $\mathcal{A}_K$). Then $|\mathcal{P}(\mathit{Run}(p\vec{v},\mathcal{Z}_{-d}))-P(p\vec{v})|\leq \max_{q\vec{u}\in\mathit{Div}}| \mathcal{P}(\mathit{Run}(q<\vec{u},\mathcal{Z}_{-d}) - x_{q\vec{u}} |$. Now $\mathcal{P}(\mathit{Run}(q\vec{u},\mathcal{Z}_{-d})\leq P_1(q\vec{u}) + P_2(q\vec{u})$, where $P_1(q\vec{u})$ is the probability that a run initiated in $q\vec{u}$ in $\mathcal{A}$ visits a configuration with $i$-th counter 0 via a $\mathcal{Z}_{-i,d}$-safe path, and $P_{2}(q\vec{u})$ is the probability that a run initiated in $q\vec{u}$ in $\mathcal{A}$ visits a configuration with some counter equal to 0 via an $\{i\}$-safe path.
So let us fix $q\vec{u}\in \mathit{Div}$ and denote $i=\mathit{mindiv}(\vec{u}_{-d})$. If $\vec{t}[i]=0$, then we have $P_1(q\vec{u})=0$, by Lemma~\ref{lem-bottominf}. Otherwise $P_1(q\vec{u})$ is bounded by the probability that a run $w$ initiated in $q(K)$ in $\mathcal{B}$ satisfies $\inf_{j \geq 0} \totalrew{i}{w}{}{j}\leq -K$ From Lemma~\ref{lem:two-counter-divergence} we get that this is bounded by $A_{\mathit{max}}^K \leq \varepsilon/2$, where the last inequality follows from the choice of $K$.
For $P_2(q\vec{u})$ note that $P_2{(q\vec{u})}=\mathcal{P}_{\mathcal{A}_{-i}}(\mathit{Run}(q\vec{u}_{-i},\mathcal{Z}_{-d}))$ and thus by the construction of $\mathcal{A}_{K}$ we have $|P_2(q\vec{u}) - x_{q\vec{u}}|\leq (d-1)\cdot\varepsilon$.
Altogether we have \begin{align*}
&|\mathcal{P}(\mathit{Run}(p\vec{v},\mathcal{Z}_{-d}))-P(p\vec{v})|\leq| P_1(q\vec{u}) + P_2(q\vec{u}) - x_{q\vec{u}}| \leq \varepsilon/2 + (d-1)\cdot \varepsilon. \end{align*}
Now it is clear that approximating $P(p\vec{v})$ up to $\varepsilon/2$ and returning this value as $\nu$ yields the desired result. As in case 1, if some component of $\vec{v}$ surpasses $K$, we can immediately reduce the problem to the approximation for $(d-1)$-dimensional case.
Note that for the base case $d=2$ the same approach can be used, the only difference that the weight of the rule $((q,\vec{u}),0,\emptyset,q{\uparrow})$ in $\mathcal{A}_K$ is 1 and the weight of $((q,\vec{u}),0,\emptyset,q{\downarrow})$ is 0.
\end{proof}
To prove Theorem~\ref{thm:approx-general-case2} in its full generality it suffices to note, that we can effectively compute a constant $b\in(0,1)$ such that the probability that a run does not visit a configuration $q\vec{u}$ with $q$ in some BSCC of $\mathcal{X}_\mathcal{B}$ or $Z(\vec{u})\neq \emptyset$ in at most $i$ steps is bounded by $b^i$ (see Lemma~\ref{lem:F_A-BSCC} and Lemma~\ref{lem:1pmc-reachability}). Therefore, to approximate the probability for $p\vec{v}$ with $\vec{v}$ not belonging to a BSCC of $\mathcal{X}_\mathcal{B}$ we can use the same approach as in case 1: we unfold $\mathcal{A}$ into a suitable number of steps and approximate the termination value in configurations where the state belongs to some $D$ using the algorithm from the previous proposition. See the proof of Theorem~\ref{thm:approx-general} for further details.
\end{document} | arXiv |
Electrostatic Questions
1) Three points charges are located on a circular arc as shown in the figure. What is the total electric field at P, the center of the arc?
Three points charges are located on a circular arc as shown in the figure.
(a) What is the total electric field at P, the center of the arc?
(b) Find the electric force that would exerted on an electron with charge $q=-e$ placed at P.
(a) First, find each of the electric fields, due to the charges, at P and then apply vector addition to calculate the resultant electric field at that point. We know that the electric field due to a point charge at distance $r$ from the source is given by
\[\vec E=\frac{1}{4\pi \epsilon_0} \frac{\left|q\right|}{r} \hat r\]
Where $\hat r$ is the unit vector lie along the line joining the source and the desired location of the field point. In some literature $\frac{1}{4\pi \epsilon_0}$ is called $k$. Therefore,
\[{\vec{E}}_1=\frac{k\left|q_1\right|}{r^2_1}{\hat{r}}_1=\frac{kQ}{a^2}\left({\cos 60{}^\circ \ }\hat{i}+{\sin 60{}^\circ \ }\left(-\hat{j}\right)\right)\]
\[{\vec{E}}_2=\frac{k\left|q_2\right|}{r^2_2}{\hat{r}}_2=\frac{k\left(2Q\right)}{a^2}\left(-\hat{i}\right)\]
\[{\vec{E}}_3=\frac{k\left|q_3\right|}{r^2_3}{\hat{r}}_3=\frac{kQ}{a^2}\left({\cos 60{}^\circ \ }\hat{i}+{\sin 60{}^\circ \ }\hat{j}\right)\]
\[{\vec{E}}_P={\vec{E}}_1+{\vec{E}}_2+{\vec{E}}_3=\frac{kQ}{a^2}2\left({\cos 60{}^\circ \ }-1\right)\hat{i}=-\frac{kQ}{a^2}\hat{i}\]
(b) The force exerted on a charge $q$ in an uniform electric field is given by $\vec{F}=q\vec{E}$.
\[\vec{F}=q\vec{E}=\left(-e\right)\left(-\frac{kQ}{a^2}\right)\hat{i}=\frac{keQ}{a^2}\hat{i}\]
Note: the electric field of a negative point charge is toward the charge and away from positive charge.
2) In the figure below, let $Q_1=+6.84\, \mu{\rm C}\ ,\ Q_2=-16.2\,\mu {\rm C}$ and $Q_3=+9.2\, \mu {\rm C}$ be rigidly fixed and separated by $r_{12}=r_{13}=r_{23}=1.34\, {\rm m}$.
In the figure below, let $Q_1=+6.84\, \mu{\rm C}\ ,\ Q_2=-16.2\,\mu {\rm C}$ and $Q_3=+9.2\, \mu {\rm C}$ be rigidly fixed and separated by $r_{12}=r_{13}=r_{23}=1.34\, {\rm m}$. Calculate the electric potential at point P which is half-way between $Q_1$ and $Q_2$.
Recall that the electric potential due to a system of point charges at an arbitrary point is defined as $V={\Sigma }^N_{i=1}V_i={\Sigma }^N_{i=1}k\frac{q_i}{r_i}$ . Therefore first find the distance of the each point charges to the point P and then find the electric potential.
\[r_3=\sqrt{r^2_{23}-r^2_2}=1.16\ {\rm m}\]
\[r_2=\frac{1}{2}r_{12}=0.67\ {\rm m\ \ ,\ \ }{{\rm r}}_{{\rm 1}}{\rm =}\frac{{\rm 1}}{{\rm 2}}r_{12}=0.67\ {\rm m}\]
\[V_P=k\left(\frac{Q_1}{r_1}+\frac{Q_2}{r_2}+\frac{Q_3}{r_3}\right)\]
\[=\left(8.99\times {10}^9\right)\left(\frac{6.84\times {10}^{-6}}{0.67}+\frac{\left(-16.2\times {10}^{-6}\right)}{0.67}+\frac{9.20\times {10}^{-6}}{1.16}\right)\]
\[=\left(8.99\times {10}^9\right)\left(-6.04\times {10}^{-6}\right)=-5.43\times {10}^4\ {\rm V}\]
3) Two protons are near the surface of the earth. The first proton is placed securely on the ground. How high will the second proton have to be placed in order to just remain stationary above the Earth's surface?
Two protons are near the surface of the earth. The first proton is placed securely on the ground. How high will the second proton have to be placed in order to just remain stationary above the Earth's surface?
Two forces acting on the second proton. One is the Coulomb force due to the first proton and the second is the gravity so these forces must be in balance.
\[F_e=F_g\Rightarrow \frac{kq_1q_2}{r^2}=m_2g\Rightarrow r=\sqrt{\frac{kq_1q_2}{m_2g}}=\sqrt{\frac{\left(9\times {10}^9\right){\left(1.06\times {10}^{-19}\right)}^2}{\left(1.672\times {10}^{-27}\right)\left(9.8\right)}}=0.119\ {\rm m}\]
4) Two equal but oppositely charges spheres ($\pm 5\times {10}^{-8}{\rm C}$) are hanging vertically from $10\,{\rm cm}$ strings. These spheres each have mass of $2\, {\rm g}$, and are separated by a very thin insulating sheet.
Two equal but oppositely charges spheres ($\pm 5\times {10}^{-8}{\rm C}$) are hanging vertically from $10\,{\rm cm}$ strings. These spheres each have mass of $2\, {\rm g}$, and are separated by a very thin insulating sheet. At $t=0$, an external electric field is applied horizontally, and the charges separate, coming to equilibrium when an angle of $20{}^\circ $ is between the two strings.
(a) Is the external electric field pointing to the right or to the left?
(b) What is the magnitude of the external electric field?
Note: the force acting on a positive charge due to an electric field is in direction of the field but for a negative charge is in opposite direction. So $E$ must be point to the left (see the figure).
Since the system is in equilibrium so apply the equilibrium conditions to one of the spheres, say the negative sphere, i.e. $\Sigma F_y=0\ ,\ \Sigma F_x=0$
\[\Sigma F_y=T{\cos 10{}^\circ \ }-mg=0\Rightarrow T=\frac{mg}{{\cos 10{}^\circ \ }}=\frac{\left(0.002\right)\left(9.8\right)}{{\cos 10{}^\circ \ }}=0.02\ {\rm N}\]
\[\Sigma F_x=\underbrace{T{\sin 10{}^\circ \ }}_{T_x}+\underbrace{\frac{kq^2}{r^2}}_{F_e}-qE=0\]
\[\Rightarrow E=\frac{1}{5\times {10}^{-8}}\left(\left(0.02\right){\sin 10{}^\circ \ }+\frac{\left(9\times {10}^9\right){\left(5\times {10}^{-8}\right)}^2}{{\left(0.03473\right)}^2}\right)=4.42\times {10}^5\,\frac{{\rm N}}{{\rm C}}\]
Where $r$ is the distance between the charges and is found by Pythagoras theorem as shown in the figure.
\[{\sin 10{}^\circ \ }=\frac{\frac{r}{2}}{10}\Rightarrow r=20\,{\sin 10{}^\circ \ }=3.473\ {\rm cm}\]
5) Consider a sphere with uniform charge density $\rho$ Suppose that at a point ${\mathbf a}$ from the origin a spherical cavity, free of charge is made.
Consider a sphere with uniform charge density $\rho$ Suppose that at a point ${\mathbf a}$ from the origin a spherical cavity, free of charge is made. The spherical cavity is entirely within the sphere. Calculate the electric field vector inside the cavity (hint: use the concept of superposition).
Superposition principle: consider a sphere with charge density $\rho$ and a sphere with charge density $-\rho$ at point $\vec{a}$ as shown in the figure below. The radius of the sphere with $\rho$ is larger than that of $-\rho$. Now using Gauss's law, we must find the electric field inside a uniform charge density at arbitrary point as
\[\oint_S{\vec{E}.d\vec{S}}=\frac{Q_{enc}}{\epsilon_0}\Rightarrow \vec{E}.\oint_S{d\vec{S}}=\frac{\rho V_{Gaussian}}{\epsilon_0}\]
Where $\oint_S{d\vec{S}}$ is the area of the Gaussian surface with radius $r$ inside of the sphere with radius $R$. So the electric field at distance $r$ from the center of sphere is
\[E\left(4\pi r^2\right)=\frac{\rho\left(\frac{4}{3}\pi r^3\right)}{\epsilon_0}\Rightarrow \vec{E}=\frac{\rho}{3\epsilon_0}\vec{r}\]
In this problem we have two spheres with charge densities $+\rho$ and $-\rho$ . We want to find the electric field due to these charges at point P.
\[{\vec{E}}_{+\rho}(P)=+\frac{\rho}{3\epsilon_0}\vec{r}\]
\[{\vec{E}}_{-\rho}(P)=\frac{-\rho}{3\epsilon_0}(\vec{r}-\vec{a})\]
\[{\vec{E}}_P={\vec{E}}_{+\rho}\left(P\right)+{\vec{E}}_{-\rho}\left(P\right)=+\frac{\rho}{3\epsilon_0}\vec{r}-\frac{\rho}{3\epsilon_0}\left(\vec{r}-\vec{a}\right)\]
\[\Rightarrow {\vec{E}}_P=+\frac{\rho}{3\epsilon_0}\vec{a}\]
6) If the electric potential in a region is given by $V\left(x\right)=6/x^2$, what is the $x$ component of the electric field in that region?
If the electric potential in a region is given by $V\left(x\right)=6/x^2$, what is the $x$ component of the electric field in that region?
By definition, $E=-dV(x)/dx\ $, therefore
\[E=-\frac{d}{dx}\left(\frac{6}{x^2}\right)=-\frac{d}{dx}\left(-12x^{-3}\right)=\frac{12}{x^3}\]
7) A $+4.0\, \mu {\rm C}$ point charge and a $-4.0\, \mu {\rm C}$ point charges are placed as shown in the figure. What is the potential difference $V_A-V_B$ between points A and B?
A $+4.0\, \mu {\rm C}$ point charge and a $-4.0\, \mu {\rm C}$ point charges are placed as shown in the figure. What is the potential difference $V_A-V_B$ between points A and B?
The potential electric at the distance $r$ from the point charge $q$ is determined by $V\left(r\right)=kq/r$. The potential difference of a collection of point charges at arbitrary point A is found by summing individual potential at that point as $V_A=V_1+V_2+\dots $. Therefore,
\[V_A=V_4+V_{-4}=\frac{kq_1}{r_1}+\frac{kq_2}{r_2}=\left(9\times {10}^9\right)\left\{\frac{4\times {10}^{-6}}{0.3}+\frac{-4\times {10}^{-6}}{\sqrt{{\left(0.4\right)}^2+{\left(0.3\right)}^2}}\right\}=48\ {\rm kV}\]
\[V_B=V_4+V_{-4}=\frac{kq_1}{r_4}+\frac{kq_2}{r_3}=\left(9\times {10}^9\right)\left\{\frac{4\times {10}^{-6}}{0.5}+\frac{-4\times {10}^{-6}}{0.3}\right\}=-48\ {\rm kV}\]
So, $V_A-V_B=48-\left(-48\right)=96\ {\rm kV}$
8) Find the potential difference $(V_B-V_A)$ between point $A(x=0,y=0)$ and point $B(x=5{\rm m,y=-5m)}$ for an electric field $\vec{E}=500\hat{i}-200\hat{j}\ {\rm (}\frac{{\rm V}}{{\rm m}}{\rm )}$.
Find the potential difference $(V_B-V_A)$ between point $A(x=0,y=0)$ and point $B(x=5{\rm m,y=-5m)}$ for an electric field $\vec{E}=500\hat{i}-200\hat{j}\ {\rm (}\frac{{\rm V}}{{\rm m}}{\rm )}$.
By definition, the potential difference between two points A and B are
\[\Delta V=V_B-V_A=-\int^B_A{\vec{E}.d\vec{l}}\]
Since the electric field in this case is uniform so can be factored from the integral as
$\Delta V=-\vec{E}.\int^B_A{d\vec{l}}=-\vec{E}.\Delta \vec{r}$ , where $\Delta \vec{l}$ is the displacement vector between A, B and defined as $\Delta \vec{r}={\vec{r}}_B-{\vec{r}}_A=\left(x_B-x_A\right)\hat{i}+\left(y_B-y_A\right)\hat{j}=5\hat{i}+\left(-5\right)\hat{j}$. Therefore,
\[\Delta V=-\left(500\hat{i}-200\hat{j}\right).\left(5\hat{i}-5\hat{j}\right)=-\left[\left(500\times 5\right)\underbrace{\left(\hat{i}.\hat{i}\right)}_{1}+\left(200\times 5\right)\underbrace{\left(\hat{j}.\hat{j}\right)}_{1}\right]\]
\[\therefore \Delta V=-3500\ {\rm V}\]
9) A $200\ {\rm m}$ long thin wire carries a line charge density $\lambda=264\ {\rm nC/m}$. find the potential difference between points $5.0\, {\rm m}$ and $6.0\, {\rm m}$ on a perpendicular radius to the axis of the wire
A $200\ {\rm m}$ long thin wire carries a line charge density $\lambda=264\ {\rm nC/m}$. find the potential difference between points $5.0\, {\rm m}$ and $6.0\, {\rm m}$ on a perpendicular radius to the axis of the wire, provided the perpendicular radius is not near either end of the wire.
As previous, $\Delta V=V_B-V_A=-\int^B_A{\vec{E}.d\vec{l}}\ $where $d\vec{l}$ is the displacement vector between the points so that in this case $d\vec{l}=dr\ \hat{r}$. By Gauss's theorem, the electric field of an infinitely long thin wire is determine as $\vec{E}=\frac{2k\lambda}{r}\hat{r}$, therefore
\[\Delta V=V_5-V_6=-\int^5_6{\frac{2k\lambda}{r}dr}=-2k\lambda{\left.{\ln r\ }\right|}^5_6=-2k\lambda{\ln \frac{5}{6}\ }\]
\[V_5-V_6=-2\left(9\times {10}^9\right)\left(264\times {10}^{-9}\right){\ln \frac{5}{6}\ }=-0.866\ {\rm V}\]
\[\therefore \ \left|\Delta V\right|=0.866\ {\rm V}\]
11) Three point charges are fixed in place in the right triangle shown below, in which $q_1=0.71\, \mu {\rm C}$ and $q_2=-0.67\, \mu {\rm C}$.
Three point charges are fixed in place in the right triangle shown below, in which $q_1=0.71\, \mu {\rm C}$ and $q_2=-0.67\, \mu {\rm C}$. What is the magnitude and direction of the electric force on the $+1.0\, \mu {\rm C\ }$(let's call this $q_3$) charge due to the other two charges?
First, find the electric force due to the each charges on the $q_3$, then use the superposition principle to do the vector sum of them.
{\vec{F}}_{13}&=k\frac{\left|q_1\right|\left|q_3\right|}{r^2_{13}}{\hat{r}}_{13}\\
&=\left(9\times {10}^9\right)\frac{\left(0.71\times {10}^{-6}\right)\left(1\times {10}^{-6}\right)}{{\left(0.10{\rm cm}\right)}^2}\left({\cos \theta\ }\hat{x}+{\sin \theta\ }(-\hat{y})\right)\ {\rm N}
Where we have decomposed the unit vector as above. Since $q_1>0$ so the electric field lines are along the line between $q_1$ and $q_3$ and directed away from $q_3$. From the geometry we see that ${\sin \theta\ }=\frac{8}{10}$ and ${\cos \theta\ }=\frac{\sqrt{{10}^2-8^2}}{10}=\frac{6}{10}$. Therefore,
{\vec{F}}_{13}&=0.639\left(0.8\ \hat{x}+0.6\left(-\hat{y}\right)\right)\\
&=\left(0.511\hat{x}-0.383\hat{y}\right){\rm \ N}
Now find the electric force due to the $q_2$ on $q_3$ i.e. ${\vec{F}}_{23}$
&=\left(9\times {10}^9\right)\frac{\left|-0.67\times {10}^{-6}\right|\left(1\times {10}^{-6}\right)}{\left({10}^2-8^2\right)\times {10}^{-4}\ {{\rm cm}}^{{\rm 2}}}\left(-\hat{y}\right)\\
&=\left(-1.675\ \hat{y}\right)\ {\rm N}
Therefore, the resultant force on the $q_3$ is
\[{\vec{F}}_3={\vec{F}}_{13}+{\vec{F}}_{23}=0.511\hat{x}-0.383\hat{y}+\left(-1.675\ \hat{y}\right)=\left(0.511\ \hat{x}-2.058\ \hat{y}\right)\ {\rm N}\]
The direction of the net force with the $x$ axis are determined by ${\tan \alpha\ }=\left|F_y\right|/\left|F_x\right|$, so
\[\alpha={{\tan }^{-1} \left(\frac{2.058}{0.511}\right)\ }=76.05{}^\circ \ \]
Since $F_{3x}>0$ and $F_{3y}<0\ $so the net force lies in the fourth quadrant.
And its magnitude is $\left|{\vec{F}}_3\right|=\sqrt{{\left(0.511\right)}^2+{\left(-2.058\right)}^2}=2.12\ {\rm N}$
12) Two small insulating spheres are attached to silk threads and aligned vertically as shown in the figure. These spheres have equal masses of $40\, {\rm g}$, and carry charges $q_1$ and $q_2$ of equal magnitude $2.0\, \mu {\rm C}$ but opposite sign.
Two small insulating spheres are attached to silk threads and aligned vertically as shown in the figure. These spheres have equal masses of $40\, {\rm g}$, and carry charges $q_1$ and $q_2$ of equal magnitude $2.0\, \mu {\rm C}$ but opposite sign. The spheres are brought into the positions shown in the figure, with a vertical separation of $15\, {\rm cm}$ between them. Note that you cannot neglect gravity. What is the tension in the lower threads?
There is three forces acting on the $q_2$. The electric force due to $q_1$, tension in the thread and the gravity. Thus, its free body diagram is as follows
The system is in equilibrium so the net force on the $q_2$ is zero i.e. ${\left(\Sigma F_y\right)}_2=0$
\[\Rightarrow F_e-T-mg=0\ \Rightarrow T=\frac{k\left|q_1\right|\left|q_2\right|}{{\left(15\right)}^2}-mg\]
\[\Rightarrow T=9\times {10}^9\frac{\left(2\times {10}^{-6}\right)\left(2\times {10}^{-6}\right)}{{\left(0.15\right)}^2}-\left(0.040\times 9.8\right)=1.208\ {\rm N}\]
13) A point charge $Q=-500\ {\rm nC}$ and two unknown point charges $q_1$ and $q_2$ are placed as shown in the figure.
A point charge $Q=-500\ {\rm nC}$ and two unknown point charges $q_1$ and $q_2$ are placed as shown in the figure. The electric field at the origin, due to charges $Q,q_1$ and $q_2$ is equal to zero. What is the amount of the charge $q_1$?
Because the net electric field at origin is zero, so we have
\[{\vec{E}}_O={\vec{E}}_Q+{\vec{E}}_1+{\vec{E}}_2=0\]
\[\frac{k\left|-Q\right|}{r^2}\hat{r}+\frac{k\left|q_1\right|}{r^2_1}{\hat{r}}_1+\frac{k\left|q_2\right|}{r^2_2}{\hat{r}}_2=0\]
\[\frac{kQ}{2^2}\left({\cos 30{}^\circ \ }\hat{x}+{\sin 30{}^\circ \ }\hat{y}\right)+\frac{k\left|q_1\right|}{{\left(1.1\right)}^2}{\hat{r}}_1+\frac{k\left|q_2\right|}{{\left(1.3\right)}^2}{\hat{r}}_2=0\]
Since $q_1$ and $q_2$ are located in the $x$ and $y$ axis, respectively, so the directions of the electric fields are along theses axes and in opposite direction of the $x$ and $y$ components of the $\vec E_Q$ since the net electric field at origin must be zero. . i.e. ${\vec{E}}_1=\left|E_1\right|\hat{-x}$ and ${\vec{E}}_2=\left|E_2\right|\hat{-y}$. Therefore,
\[\frac{\left(500\times {10}^{-9}\right)}{4}{\cos 30{}^\circ \ }-\frac{\left|q_1\right|}{1.21}=0\Rightarrow \ \left|q_1\right|=1.31\times {10}^{-7}\ {\rm C=131\ nC}\]
\[\frac{\left(500\times {10}^{-9}\right)}{4}{\sin 30{}^\circ \ }-\frac{\left|q_2\right|}{{\left(1.3\right)}^2}=0\Rightarrow \left|q_2\right|=1.05\times {10}^{-7}{\rm C}=105\ {\rm nC}\]
Thus the charge $q_1$ is positive and charge $q_2$ is negative (since the direction of the electric field is toward the charge $q_2$ and away from charge $q_1$ )
14) The electric field strength in the space between two closely spaced parallel disks is $1.0\times {10}^5\ {\rm N/C}$.
The electric field strength in the space between two closely spaced parallel disks is $1.0\times {10}^5\ {\rm N/C}$. This field is the result of transferring $3.9\times {10}^9$ electrons from one disk to the other. What is the diameter of the disks?
This configuration is a parallel plate capacitor. Recall that in this case the electric field between the plates is determined by $E=\sigma {\rm /}\epsilon_0$, where $\sigma$ is the surface charge density of the plates. Therefore,
\[\sigma=\epsilon_0E=\frac{Q}{A_{disk}}\Rightarrow \ \epsilon_0E=\frac{Q}{\pi r^2}\Rightarrow r=\sqrt{\frac{Q}{\pi \epsilon_0E}}\]
\[\Rightarrow r=\sqrt{\frac{ne}{\pi \epsilon_0E}}=\sqrt{\frac{\left(3.9\times {10}^9\right)\left(1.6\times {10}^{-19}\right)}{\pi\left(8.85\times {10}^{-12}\right)\left({10}^5\right)}}=0.015\ {\rm m}\]
In above we have used the relation $Q=ne$, where $n$ is the number of the electrons.
15) At a distance of $5.8\, {\rm cm}$ from the center of a very long uniformly charged wire, the electric field has magnitude $2000\, {\rm N/C\ }$and is directed toward the wire.
At a distance of $5.8\, {\rm cm}$ from the center of a very long uniformly charged wire, the electric field has magnitude $2000\, {\rm N/C\ }$and is directed toward the wire. What is the charge on a $1.00\, {\rm cm}$ length of wire near the center?
Note: the electric field due to an infinitely long line charge of uniform charge density $\lambda$ at distance $r$ from it is ${\vec{E}}_r=\frac{1}{2\pi\epsilon_0}\frac{\lambda}{r}\hat{r}=2k\frac{\lambda}{r}\hat{r}$. Therefore,
\[\Rightarrow \lambda=\frac{rE}{2k}=\frac{0.058\times 2000}{2\times 9\times {10}^9}=6.44\times {10}^{-9}\frac{{\rm C}}{{\rm m}}\]
By definition, linear charge density is
\[\lambda=\frac{Q}{L}\Rightarrow Q=\lambda L=\left(6.44\times {10}^{-9}\right)\left(0.01\right)=-0.0644\ {\rm nC}\]
Since the fields line are toward the wire so the distributed charge is negative!
16) In the figure, charge $q_1=+2.5\,{\rm nC}$ is located at the origin, charge $q_2=2.0\,{\rm nC}$ is located on the $x$-axis at $x=+4.0{\rm cm}$, and point P is located at $x=+4.0\,{\rm cm},\ y=+3.0\,{\rm cm}$.
In the figure, charge $q_1=+2.5\,{\rm nC}$ is located at the origin, charge $q_2=2.0\,{\rm nC}$ is located on the $x$-axis at $x=+4.0{\rm cm}$, and point P is located at $x=+4.0\,{\rm cm},\ y=+3.0\,{\rm cm}$.
(a) Calculate the magnitude and direction of the electric field at point P.
(b) Calculate the electric potential at point P.
(c) A $+2.0\,{\rm nC}$ charge is placed at point P. What is the magnitude of the electric force this charge experiences?
(a) Recall that the electric field of a point charge at distance $r$ is $\vec{E}=k\frac{\left|q\right|}{r^2}\hat{r}$. So first find the electric fields of the charges at P
{\vec{E}}_2&=k\frac{\left|q_2\right|}{r^2_2}{\hat{r}}_2\\
&=k\frac{\left|-2\times {10}^{-9}\right|}{{\left(0.03\right)}^2}\left(-\hat{y}\right)\\
&=22.22\times {10}^{-9}k\ (-\hat{y})
Since ${\vec{E}}_2$ is vertical and points in the negative $y$ direction (for negative point charges, the field lines are directed radially inward), we have chosen $(-\hat{y})$.
&=k\frac{2.5\times {10}^{-9}}{{\left(0.04\right)}^2+{\left(0.03\right)}^2}{\hat{r}}_1\\
&=10\times {10}^{-9}k({\cos \theta\ }\hat{x}+{\sin \theta\ }\ \hat{y})
Where we have decomposed the unit vector in the $r_1$ direction. (Since $q_1$ is positive the electric field lines points directly away from the point P and vice versa)
From the geometry
\[{\sin \theta\ }=\frac{3}{5}\ \ ,\ {\cos \theta\ }=\frac{4}{5}\ \]
To find the total electric field at P combine the results above as
{\vec{E}}_P&={\vec{E}}_1+{\vec{E}}_2\\
&=10\times {10}^{-9}k\left(\frac{4}{5}\hat{x}+\frac{3}{5}\ \hat{y}\right)+22.22\times {10}^{-9}k\ \left(-\hat{y}\right)\\
&={10}^{-9}k\left(8\hat{x}-16.22\hat{y}\right)
The magnitude of ${\vec{E}}_P$ is as follows
\left|{\vec{E}}_P\right|&=\sqrt{E^2_x+E^2_y}\\
&=\sqrt{{\left(8\times {10}^{-9}k\right)}^2+{\left(-16.22\times {10}^{-9}k\right)}^2}\\
&=18.08\times {10}^{-9}k\\
&=162.58\frac{{\rm N}}{{\rm C}}
Where we have substituted the $k=8.99\times {10}^9\frac{{\rm N.}{{\rm m}}^{{\rm 2}}}{C^2}$
(b) The electric potential due to a point charge $q$ at distance $r$ is defined as $V\left(r\right)=k\frac{q}{r}$
\[V_1\left(P\right)=k\frac{q_1}{r_1}=k\frac{-2\times {10}^{-9}}{0.03}=-599.3\ {\rm V}\]
\[V_2\left(P\right)=k\frac{q_2}{r_2}=\left(8.99\times {10}^9\right)\frac{2.5\times {10}^{-9}}{0.05}=449.5\ {\rm V}\]
\[\therefore V_P=V_1+V_2=-149.8\ {\rm V}\]
(c) If a charge is located in a uniform electric field $E$, then a force with magnitude $F=qE$ exert on it.
\[F_P=qE_P=\left(2\times {10}^{-9}\right)\left(162.58\right)=325.16\ {\rm nN}\]
10) An electron was accelerated from rest through a potential difference of $1300\ {\rm V}$. What is its speed?
An electron was accelerated from rest through a potential difference of $1300\ {\rm V}$. What is its speed?
see the work-energy theorem as $\Delta K=K_f-K_i=W_{net}$, where $W_{net}$ is the net work done on the electron. The electric force acting on the electron in a uniform electric field is $F_e=eE$ and subsequently the work done on it is $W=Fx{\cos \theta\ }=\left(eE\right)x{\cos 180{}^\circ \ }=-eEx$. (Since electron moves in the opposite direction of the field so we have $\theta=180{}^\circ $).
\[K_f-K_i=W_e\Rightarrow \frac{1}{2}mv^2_f-0=-\underbrace{\left(-e\right)}_{{\rm have\ negative\ charge}}Ex=eEx\]
But recall that $\Delta V=Ex$. Therefore,
\[\frac{1}{2}mv^2_f=e\Delta V\Rightarrow v_f=\sqrt{\frac{2e\Delta V}{m}}=\sqrt{\frac{2\left(1.6\times {10}^{-19}{\rm C}\right)\left(1300\right)}{9.10\times {10}^{-31}}}=2.138\times {10}^7\ {\rm m/s}\]
17) A doubly ionized alpha particle ($q=+2e,\ m=4m_p$) is moving in a uniform parallel electric field in a direction opposite of the electric field. When measurements commence, it has a kinetic energy of $56.4\,{\rm MeV}$
A doubly ionized alpha particle ($q=+2e,\ m=4m_p$) is moving in a uniform parallel electric field in a direction opposite of the electric field. When measurements commence, it has a kinetic energy of $56.4\,{\rm MeV}$ and it continuously decelerate to a stop. If it moves $6620\, {\rm km}$ during this time, what is strength of the electric field?
First solution: use the work-energy theorem
SI unit of energy is Joule so we must first convert initial kinetic energy into Joule
\[56.6\ {\rm MeV}\to {\rm 56.4\times }{{\rm 10}}^{{\rm 6}}\times \left(1.6\times {10}^{-19}\right)=9.035\times {10}^{-12}{\rm J}\]
\[\Delta K=K_2-K_1=W_{net}\]
0-K_1=\left(qE\right)x\,{\cos 180{}^\circ \ }\Rightarrow E &=\frac{K_1}{qx}=\frac{9.035\times {10}^{-12}}{2\left(1.6\times {10}^{-19}\right)\left(6620\times {10}^3\right)}\\
&= 4.26\ \frac{{\rm N}}{{\rm C}}
Second solution: first using $v^2-v^2_0=2a\Delta x$ find the acceleration of the particle, then by the Newton's 2${}^{nd}$ law determine the force exerted on it.
\[K_0=\frac{1}{2}\underbrace{m}_{4m_p}v^2_0\Rightarrow v_0=\sqrt{\frac{2K_0}{4m_p}}=\sqrt{\frac{2\left(9.035\times {10}^{-12}\right)}{6.688\times {10}^{-27}}}=5.198\times {10}^7\frac{{\rm m}}{{\rm s}}\]
\[v^2-v^2_0=2a\Delta x\Rightarrow 0-{\left(5.1987\times {10}^7\right)}^2=2a\left(6620\times {10}^3\right)\Rightarrow a=-2.04\times {10}^8\frac{{\rm m}}{{{\rm s}}^{{\rm 2}}}\]
The minus sign indicates that the object is going to decrease its velocity.
\[F=ma=4m_pa=6.688\times {10}^{-27}\times \left(-2.04\times {10}^8\right)=-1.36\times {10}^{-18}\ {\rm N}\]
By definition, the magnitude of the electric field is
\[E=\frac{\left|F\right|}{q}=\frac{\left|-1.36\times {10}^{-18}\right|}{2(1.6\times {10}^{-19})}=4.26\frac{{\rm N}}{{\rm C}}\]
18) A particle with unknown mass $m$ and charge $q$ is initially moving at constant velocity $v=4\times {10}^5\ {\rm m/s}$ in a region with zero electric field. It then enters a region where the field is $E=100\ {\rm N/C}$, in the same direction as the pa
A particle with unknown mass $m$ and charge $q$ is initially moving at constant velocity $v=4\times {10}^5\ {\rm m/s}$ in a region with zero electric field. It then enters a region where the field is $E=100\ {\rm N/C}$, in the same direction as the particle's motion. The particle travels for a distance of $x=1.2\ {\rm cm}$ before momentarily coming to a stop. What is the particle's charge to mass ratio $q/m$? Be sure to include the sign of the ratio and its units.
When a particle moves in an electric field, experiences a force by the field as $F=qE$. Using the work-energy theorem, we get
\[0-\frac{1}{2}mv^2_0=\left(qE\right)x\,{\cos 180{}^\circ \ }\Rightarrow \frac{q}{m}=\frac{v^2_0}{2Ex}\]
\[\Rightarrow \frac{q}{m}=\frac{{\left(4\times {10}^5\right)}^2}{2\times 100\times 1.2\times {10}^{-2}}=6.67\times {10}^{10}\frac{{\rm C}}{{\rm kg}}\]
Note: since the particle comes to a stop so there is a dissipative force exerted on it that is in opposite direction to the motion of the particle so $W=Fx{\cos 180{}^\circ \ }$.
When a positive charge enters in a region with constant $E$ then it moves in the direction of the field and its velocity increased but a negative charge moves opposite of the field and therefore its velocity decreased until come to a stop. In this problem the charge after entering the field come to a stop so it must be negative charge!
19) An electron (mass = $9.11\times {10}^{-31}{\rm kg}$) is released from rest in a uniform electric field of magnitude of $5000\ {\rm N/C}$. How long would it take the electron to reach a speed of $100\frac{{\rm m}}{{\rm s}}$?
An electron (mass = $9.11\times {10}^{-31}{\rm kg}$) is released from rest in a uniform electric field of magnitude of $5000\ {\rm N/C}$. How long would it take the electron to reach a speed of $100\frac{{\rm m}}{{\rm s}}$?
Given data: $v_0=0\ ,\ E=5000\frac{{\rm N}}{{\rm C}}\ ,\ \ \ v=100\frac{{\rm m}}{{\rm s}}\ ,\ t=?$
From the kinematic relation $v=v_0+at$ we need to find the acceleration of the electron in the electric field to calculate the desired time. By definition, force is related to the electric field via $F=qE$ hence
\[F=qE\Rightarrow ma=qE\Rightarrow a=\frac{qE}{m}\]
Substituting in the above kinematic relation and solving for $t$, gets
\[v=v_0+at=v_0+\left(\frac{q}{m}\right)Et\Rightarrow 100=0+\left(\frac{1.6\times {10}^{-19}}{9.11\times {10}^{-31}}\right)\left(5000\right)t\]
\[\Rightarrow t=1.01\times {10}^{-13}\ {\rm s}\]
20) A line of positive charge is formed into semicircle of radius $R$. The charge per unit length along the semicircle is described by the expression $\lambda=\lambda_0\,{\cos \theta\ }$.
A line of positive charge is formed into semicircle of radius $R$. The charge per unit length along the semicircle is described by the expression $\lambda=\lambda_0\,{\cos \theta\ }$. Find:
(a) The total charge on the semicircle.
(b) The electric field vector at point P
(a) By definition, the linear charge density of a continuous line is $\lambda=dq/dx$. So
dq=\lambda dx\to q_{tot}=\int{\lambda dx}&=2\int^{\frac{\pi}{2}}_0{\lambda_0\,{\cos \theta\ }\left(R\,d\theta\right)}\\
&=2\lambda_0R\,{\left({\sin \theta\ }\right)}^{\frac{\pi}{2}}_0\\
&=2\lambda_0R
In the above, we have used the symmetry of the problem.
(b) First, determine the electric field due to a charge element at P then integral over it to find the total electric field.
{\vec{E}}_P=\int{k\frac{dq}{R^2}\hat{r}}&=\int{k\frac{dq}{R^2}\left({\sin \theta\ }\left(-\hat{x}\right)+{\cos \theta\ }\left(-\hat{y}\right)\right)}\\
&=\underbrace{\int{k\frac{dq}{R^2}{\sin \theta\ }(-\hat{x})}}_{d{\vec{E}}_{Px}}+\underbrace{\int{k\frac{dq}{R^2}{\cos \theta\ }(-\hat{y})}}_{d{\vec{E}}_{Py}}\
By symmetry, the $x$ component of the electric field becomes zero i.e. $d{\vec{E}}_{Px}=0$. So
{\vec{E}}_P&=2\int^{\frac{\pi}{2}}_0{k\frac{dq}{R^2}\,{\cos \theta\ }(-\hat{y})}\\
&=2\frac{k}{R^2}\int^{\frac{\pi}{2}}_0{\lambda ds{\cos \theta\ }\left(-\hat{y}\right)}\\
&=2\frac{k}{R^2}\int^{\frac{\pi}{2}}_0{\left(\lambda_0{\cos \theta\ }\right)(Rd\theta)\,{\cos \theta\ }\left(-\hat{y}\right)}\\
&= \frac{2k\lambda_0}{R}\int^{\frac{\pi}{2}}_0{{{\cos }^{{\rm 2}} \theta\ }d\theta\left(-\hat{y}\right)}\\
& =\frac{2k\lambda_0}{R}\int^{\frac{\pi}{2}}_0{\frac{1+{\cos 2\theta\ }}{2}}=\frac{2k\lambda_0}{R}\frac{1}{2}{\left(\theta+\frac{1}{2}{\sin 2\theta\ }\right)}^{\frac{\pi}{2}}_0\ \\
& =\frac{\pi k \lambda_0}{2R}\left(-\hat{y}\right)
21) A uniformly charged insulating rod of length $14\,{\rm cm}$ is bent into the shape of a semicircle. The rod has a total charge of $-7.5\, \mu{\rm C}$. Find the magnitude and the direction of the electric field at $O$,
A uniformly charged insulating rod of length $14\,{\rm cm}$ is bent into the shape of a semicircle. The rod has a total charge of $-7.5\, \mu{\rm C}$. Find the magnitude and the direction of the electric field at $O$, the center of the semicircle. Calculate the force on a $3\,\mu{\rm C}$ charge placed at the point P.
From the geometry below, we have
First, calculate the electric field due to an element of charge $dq$ at the point $O$
\[d{\vec{E}}_O=k\frac{dq}{R^2}\hat{r}\xrightarrow{decompose\ \hat r} k\frac{dq}{r^2}\left({\sin \theta\ }\left(-\hat{x}\right)+{\cos \theta\ }\left(\hat{y}\right)\right)\]
By symmetry, we conclude that the $y$ component of the electric field is zero i.e. $d{\vec{E}}_{oy}=0$
So $d{\vec{E}}_O=dE_{ox}=-k\frac{dq}{R^2}{\sin \theta\ }\hat{x}$.
Second, by integration of both sides of the above equation we can get the total electric field at the point P:
{\vec{E}}_O=-k\int{\frac{dq}{R^2}{\sin \theta\ }}\hat{x} &=-k\int{\frac{\lambda Rd\theta}{R^2}\,{\sin \theta\ }}\hat{x}\\
&=-\frac{k\lambda}{R}\int^\pi_0{{\sin \theta\ }d\theta}\hat{x}\\
&=-\frac{k\lambda}{R}{\left(-{\cos \theta\ }\right)}^\pi_0\\
&=-\frac{2k\lambda}{R}\hat{x}
In above we have used the definition of the continuous charge density on a line i.e.
$\lambda=\frac{dq}{dx}$ and the line element in a polar coordinate i.e. $dx=ds=Rd\theta$.
Now we can convert $\lambda$ into the explicit charge via $\lambda=\frac{q_{tot}}{L_{semicircle}}=\frac{q_{tot}}{\pi R}$
{\vec{E}}_O=-\frac{2k\lambda}{R}\hat{x}&=-\frac{2k\left(\frac{q}{L}\right)}{\frac{L}{\pi}}\hat{x}\\
&=-2k\frac{q\pi}{L^2}\hat{x}\\
&=-2\left(9\times {10}^9\right)\frac{\left(7.5\times {10}^{-6}\right)\pi}{{\left(0.14\right)}^2}\hat{x}\\
&=2.16\times {10}^7\left(-\hat{x}\right)\ \frac{{\rm N}}{{\rm C}}
Therefore the total electric field at the point P is in the ($-x$) direction.
Finally, by definition of electric field $\vec{E}=\frac{\vec{F}}{q}$ we have
\[F_o=qE_o=\left(3\times {10}^{-6}\right)\left(2.16\times {10}^7\right)\left(-\hat{x}\right)=64.8\ \left(-\hat{x}\right)\ {\rm N}\]
22) Three charged particles are aligned along the $y$ axis. Find the electric field at $P$ point. Determine the electric force on a $Q$ charged place on the $P$ point.
Three charged particles are aligned along the $y$ axis. Find the electric field at $P$ point. Determine the electric force on a $Q$ charged place on the $P$ point.
First, calculate the electric fields due to the point charges at P then apply superposition principle: ${\vec{E}}_P={\vec{E}}_1+{\vec{E}}_2+{\vec{E}}_3$ to find the net electric field at that point.
\[\left|E_1\right|=\left|E_2\right|=k\frac{q}{r^2}=k\frac{q}{a^2+{\left(2a\right)}^2}=k\frac{q}{5a^2}\]
\[{\vec{E}}_1=\left|E_1\right|{\hat{r}}_1=k\frac{q}{5a^2}\ {\hat{r}}_1=k\frac{q}{5a^2}\,\left({\cos \theta\ }\left(-\hat{x}\right)+{\sin \theta\ }\left(\hat{y}\right)\right)\]
\[{\vec{E}}_2=\left|E_2\right|{\hat{r}}_2=k\frac{q}{5a^2}{\hat{r}}_2=k\frac{q}{5a^2}\,\left({\cos \theta\ }(-\hat{x})+{\sin \theta\ }\left(-\hat{y}\right)\right)\]
But from the geometry
\[{\cos \theta\ }=\frac{2a}{r}=\frac{2a}{\sqrt{a^2+{\left(2a\right)}^2}}=\frac{2}{\sqrt{5}}\]
\[{\sin \theta\ }=\frac{a}{r}=\frac{a}{\sqrt{a^2+{\left(2a\right)}^2}}=\frac{1}{\sqrt{5}}\]
\[{\vec{E}}_1=k\frac{q}{5a^2}\left(-\frac{2}{\sqrt{5}}\hat{x}+\frac{1}{\sqrt{5}}\hat{y}\right)\ \ ,\ \ {\vec{E}}_2=k\frac{q}{5a^2}\left(-\frac{2}{\sqrt{5}}\hat{x}-\frac{1}{\sqrt{5}}\hat{y}\right)\]
On the other hand ${\vec{E}}_3=k\frac{2q}{{\left(2a\right)}^2}=k\frac{q}{2a^2}\hat{x}$
\[\therefore \ {\vec{E}}_P=k\frac{q}{5a^2}\left(-\frac{2}{\sqrt{5}}\hat{x}+\frac{1}{\sqrt{5}}\hat{y}\right)+k\frac{q}{5a^2}\left(-\frac{2}{\sqrt{5}}\hat{x}-\frac{1}{\sqrt{5}}\hat{y}\right)+k\frac{q}{2a^2}\hat{x}=k\frac{q}{a^2}\left(-\frac{4}{\sqrt{5}}+\frac{1}{2}\right) \hat{x}\ {\rm N/C\ }\]
23) A small $2{\rm g}$ plastic ball is suspended by a $20\ {\rm cm}$ long string in a uniform electric field as shown in the figure.
A small $2{\rm g}$ plastic ball is suspended by a $20\ {\rm cm}$ long string in a uniform electric field as shown in the figure. If the ball is in equilibrium when the string makes a $15{}^\circ $ angle with the vertical, what is the net charge on the ball?
The free body diagram below shows the external forces acts on the ball:
Because the ball is in equilibrium (the ball is motionless) so first write out the equilibrium conditions in $x$ and $y$ directions as follows
\[\Sigma F_x=0=\Sigma F_y\]
\[\Sigma F_y=0\Rightarrow T{\cos \theta\ }-F_g=mg\]
\[\Sigma F_x=0\Rightarrow F_e=T{\sin \theta\ }\]
Dividing both relations by each other, we obtain the angle of the equilibrium position:
\[\therefore {\tan \theta\ }=\frac{{\sin \theta\ }}{{\cos \theta\ }}=\frac{\frac{F_e}{T}}{\frac{mg}{T}}=\frac{F_e}{mg}=\frac{qE}{mg}\]
Solving for $q$, we get
\[q=\frac{mg}{E}{\tan \theta\ }=\frac{0.002\times 9.8}{{10}^3}{\tan 15{}^\circ \ }=5.25\ \mu{\rm C}\]
24) Four identical particles, each having charge $+q$, are fixed at the corners of a square of side $L$. A fifth point charge $-Q$ (at $P$ point) lies a distance $z$ along the line perpendicular to
Four identical particles, each having charge $+q$, are fixed at the corners of a square of side $L$. A fifth point charge $-Q$ (at $P$ point) lies a distance $z$ along the line perpendicular to the plane of the square and passing through the center of the square. Determine the force exerted by the other four charges on $-Q$.
Because the magnitude and distance of all of the charges are equal so
\[\left|F_1\right|=\left|F_2\right|=\left|F_3\right|=\left|F_4\right|=k\frac{\left|qQ\right|}{r^2}\]
By symmetry consideration, $F_x=F_y=0$. So the direction of one of the forces is:
\[{\vec{F}}_{1z}=k\frac{\left|qQ\right|}{r^2}\,{\cos \theta\ }\left(-\hat{k}\right)=-k\frac{\left|qQ\right|}{r^3}z\hat{k}\]
Where we have used from the geometry of the problem ${\cos \theta\ }=z/r$.
By symmetry ${\vec{F}}_{1z}={\vec{F}}_{2z}={\vec{F}}_{3z}={\vec{F}}_{4z}=-k\frac{\left|qQ\right|}{r^3}z\hat{k}$
\[{\vec{F}}_z={\Sigma }^4_{i=1}{\vec{F}}_{iz}=-4k\frac{qQ}{r^3}z\hat{k}\]
In terms of the parameters of the square and using the Pythagoras theorem, we have:
\[x=\frac{\sqrt{2}}{2}L\ ,\ r=\sqrt{z^2+{\left(\frac{\sqrt{2}}{2}L\right)}^2}\]
\[{\vec{F}}_z=-4k\frac{Qq}{{\left(z^2+{\left(\frac{\sqrt{2}}{2}L\right)}^2\right)}^{\frac{3}{2}}}z\ \hat{k}\]
25) Three point charges are located at the corners of an equilateral triangle an in the figure. Find the magnitude and direction of the net electric force on the $7\, \mu{\rm C}$ charge.
Three point charges are located at the corners of an equilateral triangle an in the figure. Find the magnitude and direction of the net electric force on the $7\, \mu{\rm C}$ charge.
Same as previous problem, first we must calculate each of the electric forces due to the $2\,\mu{\rm C}$ , $-4\,{\rm C}$ charges exerts on the third charge then use the superposition principle to determine the net electric force on it.
{\vec{F}}_{21}&=k\frac{\left|q_1q_2\right|}{r^2_{12}}{\hat{r}}_{21}\\
&=9\times {10}^9\frac{2\times {10}^{-6}\times 7\times {10}^{-6}}{{\left(0.5\right)}^2}{\hat{r}}_{21}\\
&=0.504\ {\hat{r}}_{21}\,{\rm N}
${\hat{r}}_{21}$ is the unit vector points from $q_2$ toward $q_1$ so if one decomposes it, we get
\[{\vec{F}}_{21}=0.504\left(\frac{1}{2}\hat{x}+\frac{\sqrt{3}}{2}\hat{y}\right){\rm N}\]
(Notation: $F_{12}$ is the force exerted by point charge $q_1$on point charge $q_2$)
&=9\times {10}^9\frac{\left|7\times {10}^{-6}\times (-4)\times {10}^{-6}\right|}{{\left(0.5\right)}^2}\left({\cos 60{}^\circ \ }\hat{x}+{\sin 60{}^\circ \ }\left(-\hat{y}\right)\right)\ {\rm N}
\[{\vec{F}}_{31}=1.008\ \left(\frac{1}{2}\hat{x}+\frac{\sqrt{3}}{2}\left(-\hat{y}\right)\right)\ {\rm N}\]
Using superposition principle: ${\vec{{\rm F}}}_{{\rm 1}}{\rm =}{\vec{{\rm F}}}_{{\rm 31}}{\rm +}{\vec{{\rm F}}}_{{\rm 32}}$, we obtain
{\vec{{\rm F}}}_{{\rm 1}}&=0.504\left(\frac{1}{2}\hat{x}+\frac{\sqrt{3}}{2}\hat{y}\right)+1.008\ \left(\frac{1}{2}\hat{x}+\frac{\sqrt{3}}{2}\left(-\hat{y}\right)\right)\\
&=0.756\ \hat{x}-0.437\ \hat{y}\ \ ({\rm N)}
\[\left|{\vec{F}}_1\right|=\sqrt{{\left(0.756\right)}^2+{\left(-0.437\right)}^2}=0.873\,{\rm N}\]
And the direction of the resultant force with the horizontal axis ($x$) is
\[\alpha={{\tan }^{-1} \left(\frac{\left|-0.437\right|}{\left|0.756\right|}\right)\ }=30.02{}^\circ \ \]
Since $F_{1x}>0\ $and $F_{1y}<0$ so the net force lies in the fourth quadrant.
26) Four point charges are at the corners of a square of side $a$ as in the figure. Determine the magnitude and direction of the electric field at the location of charge $q$
Four point charges are at the corners of a square of side $a$ as in the figure.
(a) Determine the magnitude and direction of the electric field at the location of charge $q$
(b) What is the resultant force on $q$.
(a) Note: electric field defined as the electric force per unit charge. i.e. $\vec{{\rm E}}=\frac{\vec{{\rm F}}}{q}$.
The net electric field of a system of point charges at any point obeys the superposition principle.
At the location of $q$, because of the $+3q$, there is an electric field
\[{\vec{E}}_3=k\frac{3q}{a^2+a^2}\hat{r}=\frac{3}{2}k\frac{q}{a^2}\hat{r}\]
Where we have used the Pythagoras theorem $r^2=a^2+a^2$. $\hat{r}$ is the unit vector points from $3q$ toward $q$ that must be decomposed in $\hat{x}$ and $\hat{y}$ directions, as follows
\[\hat{r}=\hat{x}\,{\cos \theta\ }+\hat{y}\,{\sin \theta\ }\]
So ${\vec{E}}_3=E_{3x}\hat{x}+E_{3y}\hat{y}=\frac{3}{2}k\frac{q}{a^2}\left({\cos \theta\ }\hat{x}+{\sin \theta\ }\hat{y}\right)$ but ${\tan \theta\ }=\frac{a}{a}=1\Rightarrow \theta=45{}^\circ $
The electric fields due to the charges $+2q$ at the $+q$ location are:
\[{\vec{E}}_2=k\frac{2q}{a^2}\ \hat{x}\ \ ,\ \ {\vec{E}}^{'}_2=k\frac{2q}{a^2}\hat{y}\]
Therefore the net electric field of three point charges $+3q,+2q,+2q$ at the location of the fourth charge is (by using the superposition principle):
{\vec{E}}_{net}&={\vec{E}}_3+{\vec{E}}_2+{\vec{E}}^{'}_2\\
&=\frac{3}{2}k\frac{q}{a^2}\left({\cos 45{}^\circ \ }\hat{x}+{\sin 45{}^\circ \ }\hat{y}\right)+k\frac{2q}{a^2}\left(\hat{x}+\hat{y}\right)\\
&=k\frac{q}{a^2}\left[\left(\frac{3}{2}{\cos 45{}^\circ \ }+2\right)\hat{x}+\left(\frac{3}{2}{\sin 45{}^\circ \ }+2\right)\hat{y}\right]\\
&=\left(2+\frac{3}{2\sqrt{2}}\right)k\frac{q}{a^2}(\hat{x}+\hat{y})
But we know that the magnitude and direction of a vector with components $\vec{a}=a_x\hat{x}+a_y\hat{y}$ is
$\left|a\right|=\sqrt{a^2_x+a^2_y}$ and $\alpha={{\tan }^{-1} \left(\frac{\left|a_y\right|}{\left|a_x\right|}\right)\ }$ with positive $x$ axis.
Therefore, the magnitude and direction of the net electric field is
\[\left|{\vec{E}}_{net}\right|=\left(2+\frac{3}{2\sqrt{2}}\right)k\frac{q}{a^2}\ \sqrt{2}=k\frac{q}{a^2}\left(2\sqrt{2}+\frac{3}{2}\right)\ \]
\[\alpha={{\tan }^{-1} 1\ }=45{}^\circ !\]
(b) From the definition of the electric field ${\vec{{\rm F}}}_{{\rm net}}{\rm =}q{\vec{{\rm E}}}_{{\rm net}}$. So
\[{\vec{{\rm F}}}_{{\rm net}}=\left(2+\frac{3}{2\sqrt{2}}\right)k\frac{q^2}{a^2}(\hat{x}+\hat{y})\]
And $\left|{\vec{{\rm F}}}_{{\rm net}}\right|=k\frac{q^2}{a^2}\left(2\sqrt{2}+\frac{3}{2}\right)$
27) Four point charges are at the corners of a square. The distance from each corner to the center is $0.3\ {\rm m}$. At the center there is a $-q$ point charges. What is the magnitude of the net force on this charge?
Four point charges are at the corners of a square. The distance from each corner to the center is $0.3\ {\rm m}$. At the center there is a $-q$ point charges. What is the magnitude of the net force on this charge?
Note: the electric force vector between two point charges located at distance $r$ from each other is $\vec{F}=k\frac{\left|q_1q_2\right|}{r^2}\hat{r}$.
When there is a system of point charges and we want to find the net force on one of the charges, we must use the superposition principle i.e. the vector sum of the individual electric forces on the desired charge: ${\vec{{\rm F}}}_{{\rm net}}{\rm =}{\vec{{\rm F}}}_{{\rm 1}}{\rm +}{\vec{{\rm F}}}_{{\rm 2}}{\rm +}{\vec{{\rm F}}}_{{\rm 3}}+\dots $
So we must vector sum the individual forces of four point charges on the $-q$ in the center.
The drawing below shows the direction of the individual forces.
\[\left|F_1\right|=\left|F_2\right|=k\frac{\left|\left(-q\right)\left(+q\right)\right|}{{\left(0.3\right)}^2}=+\frac{kq^2}{0.09}\]
\[\left|F_3\right|=\left|F_4\right|=k\frac{\left|\left(-q\right)\left(-q\right)\right|}{{\left(0.3\right)}^2}=+k\frac{q^2}{0.09}\]
Because the $F_1,F_2$ and $F_3,F_4$ are separately in opposite directions to each other (i.e. ${\vec{F}}_1=-{\vec{F}}_2$ and ${\vec{F}}_3=-{\vec{F}}_4$) so the net force is
\[{\vec{F}}_{net}={\vec{F}}_1+{\vec{F}}_2+{\vec{F}}_3+{\vec{F}}_4=0\]
28) A non-uniform, but spherically symmetric, distribution of charge has a charge density of $\rho\left(r\right)=\rho_0\left(1-\frac{r}{R}\right)$ for
A non-uniform, but spherically symmetric, distribution of charge has a charge density of $\rho\left(r\right)=\rho_0\left(1-\frac{r}{R}\right)$ for $(r<R)$ and $\rho\left(r\right)=0$ for $(r>R)$ where $\rho_0=\frac{3Q}{\pi R^3}$ is a positive constant.
(a) Show that the total charge contained in the charge distribution is $Q$
(b) Show that the electric field in the region ($r>R$) is identical to that produced by a point charge $Q$ at $r=0$.
(c) Obtain an expression for the electric field for $r<R$ and graph the magnitude of the $E(r)$ as a function of $r$.
(d) Find the value of $r$ at which the electric field $E(r)$ is a maximum, and find the value of that maximum field.
(a) By definition, the total charge distributed in a region of space is $Q=\int{\rho\left(r\right)dV}$
Q_{tot}=\int{\rho\left(r\right)dV} &=\int^R_0{\rho_0\left(1-\frac{r}{R}\right)\ 4\pi r^2dr}\\
&= 4\pi \rho_0{\left(\frac{1}{3}r^3-\frac{r^4}{4R}\right)}^R_0\\
&=\frac{1}{3}\pi \rho_0R^3\\
&=\frac{1}{3}\pi R^3\left(\frac{3Q}{\pi R^3}\right)=Q\\
\ \ \ \Rightarrow Q_{tot}=Q
(b) Use Gauss's law to find the electric field of a charge distribution inside and outside of the desired volume. The charge configuration has spherical symmetry so by symmetry consideration; the electric field must be radial.
\[\oint{\vec{E}.\hat{n}dA}=\frac{Q_{ins}}{\epsilon_0}\ \ ,\ \ {\rm Gauss}{{\rm s}}^{{\rm '}}{\rm s\ law}\]
Where $Q_{ins}$ is the charge inside of a closed surface.
\[E_r\oint{dA}=\frac{Q_{ins}}{\epsilon_0}\to E_r\left(4\pi r^2\right)=\frac{Q_{ins}}{\epsilon_0}\Rightarrow E_r=\frac{Q_{ins}}{4\pi\epsilon_0r^2}\]
if $r>R$ then $E_r=\frac{Q_{ins}}{4\pi\epsilon_0r^2}$ where $Q_{ins}=\int{\rho\left(r\right)dV}=Q$ because the Gaussian surface encloses the overall sphere. Therefore
\[E_r\left(r>R\right)=\frac{Q}{4\pi\epsilon_0r^2}\ ,\ {\rm that\ is\ similar\ to\ th}{\rm e\ electric\ field\ of\ a\ point\ charge}\]
(c) from part (b) , $E_r=\frac{Q_{ins}}{4\pi\epsilon_0r^2}$ but the amount of enclosed charge in a Gaussian surface inside of the sphere is
Q_{ins} &=\int{\rho\left(r\right)dV}\\
&=\int^r_0{\rho_0\left(1-\frac{r}{R}\right)4\pi r^2dr}\\
&=4\pi\epsilon_0{\left(\frac{1}{3}r^3-\frac{r^4}{4R}\right)}^r_0\\
&=4\pi\epsilon_0\left(\frac{1}{3}r^3-\frac{r^4}{4R}\right)
\[\ \Rightarrow E_r\left(r<R\right)=\frac{\rho_0}{\epsilon_0}\left(\frac{1}{3}r-\frac{r^2}{4R}\right)\]
(d) we must find the maximum value of the electric field inside of the sphere so
\[\frac{d}{dr}E_r\left(r<R\right)=\frac{\rho_0}{\epsilon_0}\left(\frac{1}{3}-\frac{r}{2R}\right)=0\Rightarrow r=\frac{2}{3}R\]
\[E_r\left(r=\frac{2}{3}R\right)=\frac{\rho_0}{\epsilon_0}\left(\frac{2}{9}R-\frac{R}{9}\right)=\frac{\rho_0}{9\epsilon_0}R\]
29) A non-uniform electric field is given by the expression \[\vec{E}=ay\hat{x}+bz\hat{y}+cx\hat{z}\] Where $a,b,c$ are constants,
A non-uniform electric field is given by the expression
\[\vec{E}=ay\hat{x}+bz\hat{y}+cx\hat{z}\]
Where $a,b,c$ are constants, $\hat{x},\hat{y},\hat{z}$ are unit vectors in the $x,y,z$ directions, respectively. Determine the electric flux through a rectangular surface in the $xy$ plane, extending from $x=0$ to $x=w$ and from $y=0$ to $y=h$.
The electric flux through any surface is defined to be
\[{\Phi }_E=\oint_S{\vec{E}.\hat{n}dA}\]
Where $S$ is stands for the surface we are integrating over, $\hat{n}$ is the unit vector normal to the surface and $\vec{E}$ is the electric field on the surface.
In this case, since the surface lies in the $xy$ plane, so unit vector normal to it is $\hat{n}=\hat{z}$.
{\Phi }_E&=\oint_S{\vec{E}.\hat{n}dA}\\
&=\int{\left(ay\hat{x}+bz\hat{y}+cx\hat{z}\right).(dxdy\hat{z})}\\
&=ch\ {\left(\frac{1}{2}x^2\right)}^w_0\\
&=\frac{1}{2}chw^2
30) The cubical surface of side length $L=12\ {\rm cm}$ is shown in the electric field $\vec{E}=\left(950\,y\ \hat{i}+650\, z\hat{k}\right)\ {\rm V/m}$. Find the electric flux through the top face of the cube.
The cubical surface of side length $L=12\ {\rm cm}$ is shown in the electric field $\vec{E}=\left(950\,y\ \hat{i}+650\,z\hat{k}\right)\ {\rm V/m}$. Find the electric flux through the top face of the cube.
By definition, the electric flux passing through any surface $A$ is the number of field lines penetrating it. In the mathematical form
\[\Phi=\int{\vec{E}.\hat{n}dA}\]
Where $\hat{n}$ is the unit vector normal to the surface $A$.
In this case, the normal vector is parallel to the $z$ axis that is $\hat{n}=\hat{k}$. So
\Phi&=\int{\left(950y\ \hat{i}+650\ z\hat{k}\right).\hat{k}\ dA}\\
&=\int{\left(950\ y\ \left(\underbrace{\hat{i}.\hat{k}}_{0}\right)+650\ z\left(\underbrace{\hat{k}.\hat{k}}_{1}\right)\right)\ \left(\underbrace{dxdy}_{dA}\right)}\\
&={\left.\int{650z\ dxdy}\right|}_{z=0.12\ {\rm m}}\\
&=650\left(0.12\right)\int{dxdy}
Where the last integral is the area of the surface which is integrated over. Thus the total flux through the given surface is
\[\Phi=650\left(0.12\right)L^2=650\times {\left(0.12\right)}^3=1.1232\ {\rm N.}\frac{{{\rm m}}^{{\rm 2}}}{C}\]
31) An insulating solid sphere of radius $R$ has a uniform positive charge $Q$. Find charge density $\rho$
An insulating solid sphere of radius $R$ has a uniform positive charge $Q$.
(a) Find charge density $\rho$
(b) Find the electric potential at a point $r$ outside the sphere ($r>R$). Take the potential to be zero at $r=\infty $.
(c) Find the potential at a point inside the sphere ($r<R$)
(d) Plot potential $V(r)$ as a function of the distance from the center.
(e) What can you say about the results (a),(b) and (c) if we have conducting sphere having the same amount of charge on it?
(a) The volume charge density is defined as $\rho=\frac{Q}{V}=\frac{Q}{\frac{4}{3}\pi R^3}\ $.
(b) First from Gauss's law determine the electric field outside the sphere:
\[\oint{\vec{E}.d\vec{A}}=\frac{Q_{en}}{\epsilon_0}\ \ \Longrightarrow \ E_{out}\left(4\pi r^2\right)=\frac{Q}{\epsilon_0}\to E_{out}=\frac{Q}{4\pi\epsilon_0r^2}\ \ like\ a\ point\ charge!\]
Now use the definition of the potential difference as below:
\[V\left(r\right)-V\left(\infty \right)=-\int^r_{\infty }{\vec{E}\left(r\right).d\vec{r}}=-\int^r_{\infty }{\frac{Q}{4\pi \epsilon_0}\frac{dr}{r^2}}\]
\[\therefore V\left(r>R\right)=\ \frac{Q}{4\pi \epsilon_0}\frac{1}{r}\ \ ,\ \ V\left(r=R\right)=\frac{Q}{4\pi \epsilon_0}\frac{1}{R}\]
(c) Similar as previous, use Gauss's law inside the sphere to find the $E_{in}$:
\[\oint{\vec{E}.d\vec{A}}=\frac{Q_{en}}{\epsilon_0}\ \Longrightarrow E\left(r\right)\left(4\pi r^2\right)=\frac{1}{\epsilon_0}\int^r_0{\rho dv}=\frac{1}{\epsilon_0}\rho\left(\frac{4}{3}\pi r^3\right)\ \]
\[\Rightarrow E_{in}\left(r\right)=\frac{\rho}{3\epsilon_0}r\ \ or\ E_{in}\left(r\right)=\frac{Q}{4\pi \epsilon_0R^3}\ r\ \]
\[V\left(r\right)-V\left(R\right)=-\int^r_R{{\vec{E}}_{in}\left(r\right).d\vec{r}}=-\frac{Q}{4\pi \epsilon_0R^3}\ \int^r_R{rdr}=-\frac{Q}{4\pi \epsilon_0R^3}\frac{1}{2}{\left.r^2\right|}^r_R\]
\[V\left(r\right)-\frac{Q}{4\pi \epsilon_0R}=-\frac{Q}{4\pi \epsilon_0R^3}\frac{1}{2}\left(r^2-R^2\right)\ \]
\[\therefore V\left(r<R\right)=\frac{Q}{8\pi\epsilon_0R}\left(3-{\left(\frac{r}{R}\right)}^2\right)\]
(i) The charge density inside a conductor is always zero i.e. $\rho_{in}=0$
(ii) $V\left(r>R\right)=\frac{Q}{4\pi \epsilon_0}\frac{1}{r}$
(iii) $V\left(r<R\right)=\frac{Q}{4\pi \epsilon_0}\frac{1}{R}$
32) An infinitely long non-conducting cylinder of radius $R$ carries a nonuniform volume charge density $\rho\left(r\right)=\rho_0\frac{r}{R}$, where $\rho_0$ is known constant and $r$ is the cylindrical radial distance from its axis.
An infinitely long non-conducting cylinder of radius $R$ carries a nonuniform volume charge density $\rho\left(r\right)=\rho_0\frac{r}{R}$, where $\rho_0$ is known constant and $r$ is the cylindrical radial distance from its axis.
(a) Using Gauss's law determine the electric field vector everywhere (i.e. $r<R$ and $r>R$)
(b) Determine the corresponding electric potential everywhere, choosing $V\left(a\right)=0$, i.e. potential reference chosen at $r=a$ where $a>R$.
(c) Can you choose infinity as potential reference? Give your reasoning.
(a) Due to cylindrical symmetry, the $\vec{E}$ field lines are cylindrically out.
For $r<R$: $\rho_0\int{\vec{E}}.\hat{n}da=Q_{en}=\int^r_0{\rho\left(r^{'}\right)dv}$
\[\epsilon_0E\left(2\pi rL\right)=\int^r_0{\rho_0\frac{r^{'}}{R}\ (r^{'}dr^{'}d\phi dz)}=\frac{\rho_0}{R}\ 2\pi L\frac{r^3}{3}\]
\[\therefore \ \vec{E}\left(r<R\right)=E_{in}=\frac{\rho_0}{3\epsilon_0R}\ r^2\ \hat{r}\]
$\hat{r}$ is the unit vector along the radius and $dv$ is the volume element of the cylinder.
For $r>R$ : $$\epsilon_0\int{\vec{E}}.\hat{n}da=Q_{en}=\int^R_0{\rho\left(r^{'}\right)dv}=2\pi L\frac{\rho_0}{R}\frac{R^3}{3}$$
\[\therefore \ \vec{E}\left(r>R\right)=E_{out}=\frac{\rho_0}{3\epsilon_0r}\ R^2\ \hat{r}\]
(b) The potential difference is defined by $V\left(r\right)-V\left(a\right)=-\int^r_a{\vec{E}.d\vec{l}}$
\[r>R\ :V\left(r\right)=-\int^r_a{\frac{\rho_0}{3\epsilon_0r^{'}}\ R^2}dr^{'}=-\frac{\rho_0R^2}{3\epsilon_0}{\ln \left(\frac{r}{a}\right)\ }\]
r<R:\ \ V\left(r\right) &=-\int^r_a{\vec{E}.d\vec{l}}=-\int^R_a{{\vec{E}}_{out}.d\vec{r}}-\int^r_R{{\vec{E}}_{in}.d\vec{r}}\\
&=-\int^R_a{\frac{\rho_0}{3\epsilon_0r}\ R^2dr}-\int^r_R{\frac{\rho_0}{3\epsilon_0R}\ r^2dr}\\
&=-\frac{\rho_0R^2}{3\epsilon_0}{\ln \left(\frac{R}{a}\right)\ }-\frac{\rho_0}{3\epsilon_0R}{\left.\frac{r^3}{3}\right|}^r_R\\
&=-\frac{\rho_0R^2}{3\epsilon_0}{\ln \left(\frac{R}{a}\right)\ }-\frac{\rho_0}{3\epsilon_0R}\frac{1}{3}\ (r^3-R^3)
(c) Infinity cannot be used as reference point, since the charge distribution extends to $\infty $. You can also this by the problem caused if we set $a\to \infty $.
33) There is a uniform electric field $\vec{E}=12\frac{V}{m}\ (-\hat{j})$ along the $-y$ axis and these points are given with coordinates $(0,4{\rm m)}$,$B\left(0,0\right),\ C(2{\rm m,0)}$
There is a uniform electric field $\vec{E}=12\frac{V}{m}\ (-\hat{j})$ along the $-y$ axis and these points are given with coordinates $(0,4{\rm m)}$,$B\left(0,0\right),\ C(2{\rm m,0)}$. Find the potential differences:
(a) $\Delta V_{AB}$
(b) $\Delta V_{AC}$
(c) If a charge $q_0=0.05\ {\rm C}$ is taken from $A$ to $C$, find the work done on the charge.
(a) The potential difference between two arbitrary points is defined as below
\Delta V_{AB}&=-\int^B_A{\vec{E}.d\vec{S}}\\
&= -\int^B_A{12\left(-\hat{j}\right).\left(dx\hat{i}+dy\hat{j}\right)}\\
&=+12\int^B_A{dy}\\
&=+12\int^{y_B}_{y_A}{dy}\\
&=12\int^0_4{dy}\\
&=12{\left(y\right)}^0_4\\
&=12\left(0-4\right)=-48\ {\rm V}
In above, $d\vec{S}$ is the displacement vector between $A$ and $B$.
\Delta V_{AC}&=-\int^A_C{\vec{E}.d\vec{S}}\\
&=-\int^A_C{12\left(-\hat{j}\right).\left(dx\ \hat{i}+dy\ \hat{j}\right)}\\
&=12\int^{y_B}_{y_A}{dy}\\
&=12\int^0_4{dy}=-48\ {\rm V}
\[\Delta V_{AB}=\Delta V_{AC}\]
(c) The work done on a charge $q$ in an electric field is $W=q\Delta V$. Therefore,
\[W=q\Delta V=\left(0.05\right)\left(-48\right)=-2.4\ {\rm J}\]
The minus sign indicates that we must does external work to displace $q$ from $A$ to $C$.
34) The charge $q_1=-5\, \mu{\rm C}$ and $q_2=10\, \mu {\rm C}$ are located as in the figure.
The charge $q_1=-5\, \mu{\rm C}$ and $q_2=10\, \mu {\rm C}$ are located as in the figure.
(a) Find the electric field due to $q_1$ at the point $P$ located at $20\ {\rm cm}$ along $x$ axis.
(b) Find the electric field of $q_2$ at $P$
(c) What is the total electric field at $P$?
(d) If an electron ($q_e=-1.6\times {10}^{-19}\,C$) is put at $P$ what will be the magnitude of the electric force acting on the electron
Suppose $k=\frac{1}{4\pi \epsilon_0}=9\times {10}^9\ {\rm N.}{{\rm m}}^{{\rm 2}}{\rm /}{{\rm C}}^{{\rm 2}}$
(a) Use the definition of the electric field of a point charge to calculate its magnitude and direction at arbitrary distance.
\vec{E_1}&=k\frac{\left|q_1\right|}{r^2_1}\ {\hat{r}}_1\\
&=k\frac{\left|q_1\right|}{r^2_1}\ \left(\hat{j}\,{\cos \theta\ }-\hat{i}\,{\sin \theta\ }\right)\\
&=k\frac{\left|q_1\right|}{r^2_1}\ \left(\hat{j}\frac{1}{\sqrt{5}}-\hat{i}\frac{2}{\sqrt{5}}\right)\\
&=9\times {10}^9\frac{5\times {10}^{-6}}{0.05}\ \left(\hat{j}\frac{1}{\sqrt{5}}-\hat{i}\frac{2}{\sqrt{5}}\right)
So $\vec{E_1}=9\times {10}^5\left(\hat{j}\frac{1}{\sqrt{5}}-\hat{i}\frac{2}{\sqrt{5}}\right)=\left(+402.49\ \hat{j}-804.98\hat{i}\right)\times {10}^3\ \left(\frac{{\rm N}}{{\rm C}}\right)$
In above, from the geometry, we have
{\cos \theta\ }=\frac{10}{r_1}=\frac{10}{\sqrt{{10}^2+{20}^2}}=\frac{1}{\sqrt{5}}\\
{\sin \theta\ }=\frac{20}{r_1}=\frac{2}{\sqrt{5}}\\
r^2_1={\left(0.1\right)}^2+{\left(0.20\right)}^2=0.05
(b) Similar as part (a),
\vec{E_2}&=k\frac{\left|q_2\right|}{r^2_2}{\hat{r}}_2\\
&=9\times {10}^9\frac{\left(10\times {10}^{-6}\right)}{{\left(0.2\right)}^2}\hat{i}\\
&=2250\times {10}^3\hat{i}
(c) Using the superposition principle, the vector sum of each of the electric fields at arbitrary point $P$ is the total electric field at that point, we obtain
\[\vec{E_P}=\vec{E_1}+\vec{E_2}=\left(1445.02\hat{i}+402.49\hat{j}\right)\times {10}^3\ {\rm N/C}\]
(d) Using the definition of electric field in general, we obtain
\vec{F_e}=e\vec{E_e}\ \Rightarrow \ \left|\vec{F_e}\right|&=\left|e\right|\left|\vec{E_P}\right|\\
&=(1.6\times {10}^{-19})\times {10}^3\sqrt{{\left(1445.02\right)}^2+{\left(402.49\right)}^2}\ {\rm N}
\[\left|\vec{F_e}\right|{\rm \ \sim 2.4\times }{{\rm 10}}^{{\rm -}{\rm 13}}{\rm N}\]
35) Three different point charges are arranged at the corners of a square of side $L$ shown in the figure.
Three different point charges are arranged at the corners of a square of side $L$ shown in the figure.
(a) What is the potential at the forth corner(taken as the origin, point $O$)
(b) What is the electric field vector at point $O$? (Do not forget the unit vectors, simplify your expressions)
(a) The potential electric due to a point charge at distance $r$ is defined by: $V\left(r\right)=\frac{q}{4\pi \epsilon_0r}$. When there is a system of point charges, the potential electric of this system at an arbitrary point P is defined by the sum of each of the potential electric at that point. Therefore,
\[V\left(O\right)=V_q+V_{-2q}+V_{3q}\]
\[V\left(O\right)=\frac{q}{4\pi \epsilon_0}\left[\frac{1}{L}-\frac{2}{L}+\frac{3}{\sqrt{L^2+L^2}}\right]=\frac{q}{4\pi \epsilon_0L}\frac{3\sqrt{2}-2}{2}\ \]
(b) The electric field of a system of point charges is the vector sum of each of the electric fields at that point (superposition Principle).
\[\vec{E}\left(O\right)={\vec{E}}_q+{\vec{E}}_{-2q}+{\vec{E}}_{3q}\]
Note: the electric field of a point charge at distance $r$ is calculated by
\[\vec{E}\left(r\right)=\frac{1}{4\pi \epsilon_0}\frac{\left|q\right|}{r^2}\hat{r}\]
Where $\hat{r}$ is the unit vector along the line from the point charge toward the desired location $r$.
\[\vec{E}\left(O\right)=\frac{1}{4\pi \epsilon_0}\left[\frac{q}{L^2}\hat{i}-\frac{2q}{L^2}\hat{j}+\frac{3q}{2L^2}\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right]=\frac{q}{4\pi \epsilon_0L^2}\left[\hat{i}\frac{4+3\sqrt{2}}{4}-\hat{j}\frac{8-3\sqrt{2}}{4}\right]\]
36) In the figure below, the thin straight wire of length $L$ has a uniformly distributed charge of $Q=+3.0\ {\rm nC}$. Its electric field at point $P$ has magnitude $E=300\ {\rm N/C}$. Find $L$.
In the figure below, the thin straight wire of length $L$ has a uniformly distributed charge of $Q=+3.0\ {\rm nC}$. Its electric field at point $P$ has magnitude $E=300\ {\rm N/C}$. Find $L$.
First find the total electric field of the wire at point $P$ in terms of $Q$ and $L$. To do this, choose a charge element $dq=\lambda dx$ at distance $x$ from the origin. This element is similar to the point charge, thus one can use the electric field definition and find its magnitude at point $P$( see the figure), then integrate over all of the wire to compute the total field at that point as follows:
E=\int{dE}&=\int{\frac{kdq}{{\left(0.5-x\right)}^2}}\\
&=k\lambda\int^L_0{\frac{dx}{{\left(0.5-x\right)}^2}}\\
&=k\lambda{\left(\frac{1}{0.5-x}\right)}^L_0\\
&=k\lambda\left(\frac{1}{0.5-L}-\frac{1}{0.5}\right)\\
&=k\lambda\frac{L}{0.5(0.5-L)}
Recall that the total charge of the wire is $Q=\lambda L$ so
\[E=\frac{kQ}{0.5(0.5-L)}\]
Now substitute the given values of $E=300\ {\rm N/C}$ and $Q=+3\ {\rm nC}$, solving for $L$, we obtain
\[E=\frac{kQ}{0.5\left(0.5-L\right)}\Rightarrow 300=\frac{\left(9\times {10}^9\right)\left(3\times {10}^{-9}\right)}{0.5\left(0.5-L\right)}\ \]
\[\Rightarrow L=0.32\ {\rm m}\]
37) A uniformly charged thin ring has radius $a=16.0\, {\rm cm}$ and total charge $Q=+24\, \mu {\rm C}$. Where must a point charge $q=-18\, \mu {\rm C}$ be placed on
A uniformly charged thin ring has radius $a=16.0\, {\rm cm}$ and total charge $Q=+24\, \mu {\rm C}$. Where must a point charge $q=-18\, \mu {\rm C}$ be placed on the axis of the ring for the electric potential at the center of the ring to be zero?
The electric potential of a point charge at distance $r$ is given by $V\left(r\right)=kq/r$. In the case of continuous charges, we must firstly find the electric potential due to an infinitesimal charge element at distance $r$, then integrate over all of the charge configuration. Thus first calculate the electric potential of a uniformly charged ring at distance $z$ on the axis of the ring
V=\int{dV}&=\int{\frac{kdq}{r}}\\
&=k\int^{2\pi}_0{\frac{\lambda\,ad\phi}{\sqrt{a^2+z^2}}}\\
&=\frac{k\lambda a}{\sqrt{a^2+z^2}}(2\pi)
Since we are asked that the electric potential must be computed at the center of the ring so setting $z=0$ we get
\[V_{ring}\left(center\right)=\frac{kQ}{a}\]
As we see, this electric potential at center of the ring is similar to electric potential of a point charge! We have calculated the total charge of the ring as $Q_{ring}=\lambda L=\left(2\pi a\right)\lambda$, where $L$ is the circumference of the ring.
Let $d$ be the distance of point charge $q$ from the center of the ring, so its electric potential at the center of the ring is
\[V_{point}=\frac{kq}{d}\]
The electric potential due to a system of point charges is the sum of the potentials due to each of these charges. Thus the total potential of this configuration at the center of the ring is
\[V=\Sigma V_i=V_{point}+V_{ring}=\frac{kq}{d}+\frac{kQ}{a}\]
Setting $V_{tot}=0$, we obtain
\[0=\frac{kq}{d}+\frac{kQ}{a}\Rightarrow d=-a\frac{q}{Q}=-16\left(-\frac{18}{24}\right)=12\ {\rm cm}\]
38) A proton moves directly towards a stationary nucleus of charge $Q$. The proton has a speed of $4.15\times {10}^6\ {\rm m/s}$ and an acceleration of
A proton moves directly towards a stationary nucleus of charge $Q$. The proton has a speed of $4.15\times {10}^6\ {\rm m/s}$ and an acceleration of $4.31\times {10}^{26}\ {\rm m/}{{\rm s}}^{{\rm 2}}$ when it is $8.00\times {10}^{-14}\ {\rm m}$ from the center of the nucleus. What is the distance of the closest approach (to the center of the nucleus)? Mass of proton $m_{+e}=1.67\times {10}^{-27}{\rm kg}$ , $e=1.6\times {10}^{-19}{\rm C}$
First compute the force between these two point charges at the distance stated in the problem as follows
\[F=\frac{k\left|+eQ\right|}{r^2}\]
Now using Newton's second law and solving for $Q$, we obtain
\[F=ma{\rm =}\frac{k\left|+eQ\right|}{r^2}\Rightarrow \left|Q\right|=\frac{mar^2}{k(+e)}\]
By substituting the given values of the problem, one can find the value of charge $Q$, thus
\[\left|Q\right|=\frac{\left(1.67\times {10}^{-27}\right)\left(4.31\times {10}^{26}\right){\left(8\times {10}^{-14}\right)}^2}{\left(9\times {10}^9\right)\left(1.6\times {10}^{-19}\right)}=3.06\times {10}^{-18}{\rm C}\]
Let $d$ be the closest distance to the center of the nucleus, at this point proton coming to a stop that is its kinetic energy becomes zero so use conservation of mechanical energy between these points to find the desired distance.
\[E_i=E_f\]
\[\underbrace{\frac{k(+e)Q}{r}}_{U_i}+\underbrace{\frac{1}{2}mv^2_i}_{K_i}=\underbrace{\frac{k(+e)Q}{d}}_{U_f}+\underbrace{\frac{1}{2}mv^2_f}_{K_f=0}\]
Rearranging and solving for $d$, we obtain
\[d=\frac{k(+e)Q}{\frac{1}{2}mv^2_i-\frac{keQ}{r}}\]
\[\Rightarrow d=\frac{\left(9\times {10}^9\right)\left(1.67\times {10}^{-19}\right)\left(3.06\times {10}^{-18}\right)}{\frac{1}{2}\left(1.67\times {10}^{-27}\right)\left(4.15\times {10}^6\right)-\frac{\left(9\times {10}^9\right)\left(1.67\times {10}^{-19}\right)\left(3.06\times {10}^{-18}\right)}{8.00\times {10}^{-14}}}\]
\[\Rightarrow d\sim \ 6.4\times {10}^{-14}\ {\rm m}\]
Note: the electric potential energy of a point charge $q$ that is located at distance $r$ from an another charge $Q$ is given by $U=kqQ/r$.
39) A charge $q$ is distributed uniformly on a quarter side circle of radius $R$ as shown in the figure. What is the magnitude of the electric field at the center(the point $P$)?
A charge $q$ is distributed uniformly on a quarter side circle of radius $R$ as shown in the figure. What is the magnitude of the electric field at the center(the point $P$)?
First calculate the electric field of an infinitesimal element of the line at the point $P$, then by integrating over all of the length find the total electric field of the arc at $P$. Thus
\[{\vec{E}}_{tot}=\int{d\vec{E}}=\int{\frac{kdq}{R^2}\hat{r}}=\frac{k\lambda R}{R^2}\int{d\theta\ \hat{r}}\]
Recall that the uniformly charge distribution and line element is related together by $dq=\lambda dl$, where in the case of arc $dl=ds=Rd\theta$. $\hat{r}$ is the unit vector along the direction of the total electric field and must be decomposed into $\hat{i}$ and $\hat{j}$. Therefore,
{\vec{E}}_{tot}&=\frac{k\lambda}{R}\int^{\frac{\pi}{2}}_0{\left({\cos \theta\ }\,\hat{i}+{\sin \theta\ }\,\left(-\hat{j}\right)\right)d\theta}\\
&=\frac{k\lambda}{R}{\left({\sin \theta\ }\,\hat{i}+{\cos \theta\ }\,\hat{j}\right)}^{\frac{\pi}{2}}_0\\
&=\frac{k\lambda}{R}\left(\hat{i}-\hat{j}\right)
This is the total electric field at point $P$. Its direction is explicitly shown in the equation and figure. Before finding the magnitude of it, we must explicitly express the total electric field in terms of the charge $q$ of the arc. By using the definition of the linear charge density $\lambda$, we get
\[\lambda=\frac{q}{L}=\frac{q}{R\left(\frac{\pi}{2}\right)}\]
Where, in this case, $L$ is the circumference of the arc. Thus, substituting the $\lambda$ into ${\vec{E}}_{tot}$, we obtain
\[{\vec{E}}_{tot}=\frac{k}{R}\frac{q}{R\left(\frac{p}{2}\right)}\left(\hat{i}-\hat{j}\right)\to {\vec{E}}_{tot}=\frac{2kq}{\pi R^2}\left(\hat{i}-\hat{j}\right)\]
The magnitude of the total electric field is computed as follows
\[\left|{\vec{E}}_{tot}\right|=\sqrt{E^2_x+E^2_y}=\frac{2\sqrt{2}}{\pi}\frac{kq}{R^2}\ \]
And its direction is
\[\theta={{\tan }^{-1} \left|\frac{E_y}{E_x}\right|\ }={{\tan }^{-1} \left|\frac{1}{-1}\right|\ }=45{}^\circ \]
Since $E_x>0\ $and $E_y<0$, then the total electric field lie in the 4${}^{th}$ quadrant.
40) Two concentric rings, one of radius $R$ and total charge $Q$ and the second of radius 2R and total charge $-\sqrt{8}Q$, are in the plane $z\ =\ 0$. The charge on each ring
Two concentric rings, one of radius $R$ and total charge $Q$ and the second of radius 2R and total charge $-\sqrt{8}Q$, are in the plane $z\ =\ 0$. The charge on each ring is distributed uniformly. Where on the positive $z$-axis is the electric field zero?
Calculate the electric field of the rings at distance $d$ on the $z$ axis then use the superposition principle to find the total electric field at that point.
Choose the $z$ axis to coincide with the axis of the ring with the ring in the $z=0$ plane. The electric field of a ring at a point on its axis through the center of the ring is calculated as follows
\[{\vec{E}}_z=\int{dE_z}=\int{\frac{k\overbrace{dq}^{\lambda Rd\phi }}{r^2}{\cos \theta\ }\hat{k}}=k\lambda R\int^{2\pi}_0{\frac{{\cos \theta\ }}{r^2}d\phi}\]
Where $ds=Rd\phi$ is the length of the arc in the polar coordinate. By symmetry consideration, the $x$ and $y$ components of the electric field due to an element of charge $dq$ is nonzero i.e. $E_x=E_y=0$. As can be seen from the geometry
\[{\cos \theta\ }=\frac{d}{r}\ \ and \ \ r=\sqrt{R^2+d^2}\]
{\vec{E}}_z=\int{dE_z}&=\int^{2\pi}_0{\frac{k\lambda R}{r^2}\frac{d}{r}\ d\phi\ \hat{k}}\\
&=\frac{k\lambda Rd}{{\left(R^2+d^2\right)}^{\frac{3}{2}}}\underbrace{\int^{2\pi}_0{d\phi}}_{2\pi}\\
&=\frac{kQd}{{\left(R^2+d^2\right)}^{\frac{3}{2}}}\hat{k}
Where we have used the definition of the linear charge density on the ring
\[\lambda=\frac{Q}{L}=\frac{Q}{2\pi R}\]
Where $L$ is the circumference of the ring.
Now using the superposition principle and setting ${\vec{E}}_{tot}=0$ and solving for $d$, we obtain the desired location
\[{\vec{E}}_{tot}={\vec{E}}_R+{\vec{E}}_{2R}=\frac{kQd}{{\left(R^2+d^2\right)}^{\frac{3}{2}}}\hat{k}+\frac{k(-\sqrt{8}Q)d}{{\left({\left(2R\right)}^2+d^2\right)}^{\frac{3}{2}}}\hat{k}=0\]
\[\Rightarrow {\left(4R^2+d^2\right)}^{\frac{3}{2}}=\underbrace{\sqrt{8}}_{2^{\frac{3}{2}}}{\left(R^2+d^2\right)}^{\frac{3}{2}}\to {\left(4R^2+d^2-2\left(R^2+d^2\right)\right)}^{\frac{3}{2}}=0\]
The argument of the equation above must be zero to satisfy the condition, thus
\[4R^2+d^2-2\left(R^2+d^2\right)=0\Rightarrow 2R^2-d^2=0\]
\[\therefore d=\sqrt{2}R\]
41) A hemispherical shell of radius $R$ is placed in an electric field $\vec{E}$ which is parallel to its axis. What is the flux ${\Phi }_E$ of the electric field through the shell?
A hemispherical shell of radius $R$ is placed in an electric field $\vec{E}$ which is parallel to its axis. What is the flux ${\Phi }_E$ of the electric field through the shell?
Solution 1: By definition, the electric flux passing through any surface with area elemen t $dA$ is the scalar product of normal component of the electric field and area of the surface that is
\[{\Phi }_E=\int{\vec{E}.\hat{n}dA}\]
Where $\hat{n}$ is the unit vector normal to the surface.
In the case of hemisphere or sphere the unit vector is along the radius i.e. $\hat{n}=\hat{r}$. Since $\vec{E}$ is parallel to the axis of hemisphere the scalar product of $E$ and $\hat{r}$ is $E{\cos \theta\ }$. The range of polar angle is $\theta=0$ to $\theta=\pi/2\ $.Therefore,
{\Phi }_E=\int^{\frac{\pi}{2}}_0{\vec{E}.\hat{r}dA} &= \int^{2\pi}_0{d\phi}\int^{\frac{\pi}{2}}_0{E{\cos \theta\ }\ R^2{\sin \theta\ }d\theta}\\
&= 2\pi ER^2\int^{\frac{\pi}{2}}_0{\underbrace{{\sin \theta\ }{\cos \theta\ }}_{\frac{1}{2}{\sin 2\theta\ }}d\theta}
\Rightarrow {\Phi }_E &=2\pi ER^2\left(\frac{1}{2}\right){\left(-\frac{1}{2}{\cos 2\theta\ }\right)}^{\frac{\pi}{2}}_0\\
&=-\frac{1}{2}\pi ER^2\left({\cos 2\frac{\pi}{2}\ }-{\cos 0\ }\right)\\
&=\pi ER^2
In above, $dA=R^2{\sin \theta\ }d\theta d\phi $ is the area element of the sphere.
Solution 2: all of the electric lines through the circle at the bottom of the hemisphere, passing through the area of hemisphere. So by calculating the flux through the circle, one can find the flux though the hemisphere but the important thing to remember is that, by convention, the flux through the circle is incoming and is negative of the outgoing flux through the hemisphere that is ${\Phi }_E\left(circle\right)=-{\Phi }_E(hemisphere)$. Therefore, by definition of the electric flux through a surface is
\[{\Phi }_E\left(circle\right)=\vec{E}.\hat{n}\ A=E\left(+\hat{k}\right).\left(-\hat{k}\right)\left(\pi R^2\right)=-\pi ER^2\]
Note: the normal vector $\hat{n}$ is always out of one side of a surface (as in above case).
42) A sphere of radius $R$ contains a total charge $Q$ which is uniformly distributed throughout its volume. At a distance $2R$ from the center of the sphere we place a point charge $q$
A sphere of radius $R$ contains a total charge $Q$ which is uniformly distributed throughout its volume. At a distance $2R$ from the center of the sphere we place a point charge $q$. What is the value of $q$ which makes the electric field at the point $P$ on the surface zero?
First we must determine the magnitude of the electric field of a uniformly charged sphere at its surface. To do this, using Gauss's law we obtain the electric field inside of it
\[\oint_S{\vec{E}.\hat{n}dA}=\frac{Q_{enc}}{\epsilon_0}\]
Where the symbol $\oint $ represents the integral over a closed surface and $Q_{enc}$ is stands for the charge enclosed by the closed surface $S$. To compute the electric field inside of such a sphere, one must consider an imaginary spherical surface of radius $r$ , called Gaussian surface, then evaluate the integral above as follows
\[\oint_S{{\vec{E}}_{in}.\hat{n}dA}=E_{in}\oint_s{dA}=\frac{Q_{enc}}{\epsilon_0}\]
We have taken $E_{in}$ out of the integral because it is constant everywhere on the surface. The integral of $dA$ over the spherical surface is just the total area of it that is $4\pi r^2$. By definition of the volume charge density, we can find the amount of charge enclosed by this imaginary sphere.
\[Q_{enc}=\rho V_{enc}=\frac{4}{3}\pi r^3 \rho\]
using this, the electric field inside of a uniformly charged sphere with charge density $?$ is
\[E_{in}\left(4\pi r^2\right)=\frac{4}{3}\pi r^3\frac{\rho}{\epsilon_0}\Rightarrow E_{in}=\frac{\rho}{3\epsilon_0}r\]
Since $\rho$ is constant, we can use this fact to express $E_{in}$ in terms of the total charge of sphere $Q$.
\[\rho=\frac{Q}{V_{tot}}=\frac{Q}{\frac{4}{3}\pi R^3}\]
By substituting $\rho$ into $E_{in}\left(r\right)$, we get
\[E_{in}\left(r\right)=\frac{1}{3\epsilon_0}\frac{Q}{\frac{4}{3}\rho R^3}r=\frac{Q}{4\pi \epsilon_0}\frac{r}{R^3}\]
Setting $r=R$, we obtain the electric field of a uniformly charged sphere at its surface
\[{\vec{E}}_{in}\left(r=R\right)=\frac{Q}{4\pi \epsilon_0}\frac{1}{R^2}(\hat{i})\]
Now compute the electric field due to the positive point charge $q$ at point $P$ as follows
\[{\vec{E}}_q=\frac{1}{4\pi \epsilon_0}\frac{q}{r^2}(-\hat{i})=\frac{1}{4\pi \epsilon_0}\frac{q}{R^2}(-\hat{i})\]
By applying the superposition principle, we can find the net electric field of above configuration at point $P$
\[{\vec{E}}_{net}={\vec{E}}_{in}\left(r=R\right)+{\vec{E}}_q=\frac{Q}{4\pi \epsilon_0}\frac{1}{R^2}(\hat{i})+\frac{1}{4\pi \epsilon_0}\frac{q}{R^2}\left(-\hat{i}\right)\]
Setting $E_{net}=0$, we can determine the unknown charge $q$
\[E_{net}=0\Rightarrow \frac{Q}{4\pi \epsilon_0}\frac{1}{R^2}-\frac{1}{4\pi \epsilon_0}\frac{q}{R^2}=0\ \]
\[\Rightarrow \ q=Q\ \]
In above, we have used the convention that direction of the electric field of a positive point charge is outward and vice versa.
43) A point charge of charge $-Q$ is placed at the center of a solid spherical conducting shell of inner radius $R$ and outer radius $2R$.
A point charge of charge $-Q$ is placed at the center of a solid spherical conducting shell of inner radius $R$ and outer radius $2R$. The shell is in static equilibrium and has a net charge $+2Q$. What is the total charge on the outer surface (at $r=2R$) of the shell?
The charge $-Q$ induces charge $+Q$ on the inner sphere and similarly this charge induces charge $-Q$ on the outer sphere of radius $2R$. Since the electric field inside a conductor in electrostatic equilibrium is zero and since it is stated that the shell has a net charge $+2Q$, we conclude that the excess charge must be reside on the outer surface of the shell. So by summing these two contribution over the outer surface, one can find the total free charge uniformly distributed on the outer surface to be
\[Q^{'}=+2Q+\left(-Q\right)=+Q\]
44) A solid conducting sphere carrying charge $q$ has radius $a$. It is inside a concentric hollow conducting sphere with inner radius $b$ and outer radius $c$. The hollow sphere has no net charge.
A solid conducting sphere carrying charge $q$ has radius $a$. It is inside a concentric hollow conducting sphere with inner radius $b$ and outer radius $c$. The hollow sphere has no net charge.
(a) Derive expressions for the electric field magnitude in terms of the distance $r$ from the center for the region $r<a\ ,\ a<r<b\ ,\ b<r<c\ $and $r>c$.
(b) Graph the magnitude of the electric field as a function of $r$ from $r=0$ to $r=2c$.
(c) What is the charge on the inner surface and on the outer surface of the hollow sphere?
(a) In such a problem, one can use the Gauss's law to find the electric field everywhere. Gauss's law states that the electric flux through any closed surface $S$ is equal to the charged enclosed by it divided by $\epsilon_0$. The mathematical form of it is
To use the Gauss's law, we must firstly consider a closed surface which is called Gaussian surface. This surface has the same symmetry as the electric field. In this case, the Gaussian surface must be a spherical surface of radius $r$ concentric with the conducting charged sphere of radius $a$.
Note: although the Gauss's law is true for any surface surrounding a charged configuration, but it is useful only when we choose a Gaussian surface to match the original symmetry of the problem.
To find the electric field in region $r<a$, the Gaussian surface is inside the sphere of radius $a$. Since inside this surface does not any enclosed charge, so there is no flux through it, so
\[\oint_s{\vec{E}.\hat{n}dA}=\frac{Q_{enc}}{\epsilon_0}=0\Rightarrow E\left(r<a\right)=0\]
In the region $a<r<b$, the Gaussian surface encloses the charged sphere $q$, so
\[\oint_s{\vec{E}.\hat{n}dA}=\frac{Q_{enc}}{\epsilon_0}=\frac{q}{\epsilon_0}\Rightarrow E\left(a<r<b\right)\oint_s{dA}=\frac{q}{\epsilon_0}\]
\[\Rightarrow \ E\left(a<r<b\right)=\frac{q}{4\pi \epsilon_0r^2}\]
Since $E$ is constant and perpendicular everywhere on the Gaussian surface, so we have taken out of integral. The closed integral $\oint_s{dA}$ is the surface area of the sphere. In the case of sphere, the normal vector $\hat{n}$ is along the radial direction i.e. $\hat{n}=\hat{r}$. Since inside the Gaussian surface there is a positive charge then the electric field point away from the center or is radially outward i.e. $\vec{E}=E\left(r\right)\hat{r}$.
Region $b<r<c$ lies inside the conductor. Using this fact the electric field inside a conductor is zero , so $E\left(b<r<c\right)=0$. In region $r>c$, the net charge encloses by the Gaussian surface is $+q$, so in this region the electric field is
\[\oint_s{\vec{E}.\hat{n}dA}=\frac{Q_{enc}}{\epsilon_0}=\frac{q}{\epsilon_0}\Rightarrow E\left(r>c\right)=\frac{q}{4\pi \epsilon_0r^2}\]
(b) The graph is as follows
(c) The free charge $q$ inside the sphere of radius $a$ induces the charge $-q$ on the inner surface and subsequently this charge also induces the charge $+q$ on the outer surface of the spherical shell.
45) A electron is fixed at the position $x=0$, and a second charge $q$ is fixed at $x=4\times {10}^{-9}\ {\rm m}$ (to the right). A proton is now placed between the
A electron is fixed at the position $x=0$, and a second charge $q$ is fixed at $x=4\times {10}^{-9}\ {\rm m}$ (to the right). A proton is now placed between the two at $x^{'}=1\times {10}^{-9}\ {\rm m}$. What must the charge $q$ be (magnitude and sign) so that the proton is in equilibrium?
The magnitude of the electric force between two point charges $q$ and $q^{'}$ located at distance $r$ from each other is given by the Coulomb's law as follows
\[F=k\frac{\left|qq^{'}\right|}{r^2}\]
Where $k=9\times {10}^9\ {\rm N.}{{\rm m}}^{{\rm 2}}/C^2$. The directions of the forces the two charges exert on each other are always along the line joining them.
The figure below is a free body diagram for proton. Let us consider the charge $q$ to be positive. In such a case, $F$ is the force exerted on the proton by the electron and $F^{'}$ is the force exerted by the charge $q$ on it. Now compute these forces and use the superposition principle to find the total force acting on the proton.
\[{\vec{F}}_{tot}=\vec{F}+{\vec{F}}^{'}=k\frac{\left|\left(-e\right)(+e)\right|}{{{x^{'}}^2}}\left(-\hat{i}\right)+k\frac{\left|\left(+e\right)q\right|}{{\left(x-x^{'}\right)}^2}\left(+\hat{i}\right)\]
Since the charge $q$ is in equilibrium state so the total force exerted on it must be zero
\[{\vec{F}}_{tot}=0\ ,{\rm \ equilibrium\ condition}\]
\[k\frac{\left|\left(-e\right)\left(+e\right)\right|}{{x^{'}}^2}\left(-\hat{i}\right)+k\frac{\left|\left(+e\right)q\right|}{{\left(x-x^{'}\right)}^2}\left(+\hat{i}\right)=0\]
\[\Rightarrow \frac{e}{{x^{'}}^2}=\frac{\left|q\right|}{{\left(x-x^{'}\right)}^2}\Rightarrow e{\left(x-x^{'}\right)}^2=\left|q\right|{x^{'}}^2\]
\[e{\left(4\ {\rm nm-1nm}\right)}^{{\rm 2}}=\left|q\right|{\left(1{\rm nm}\right)}^{{\rm 2}}\]
\[\Rightarrow \ \left|q\right|=9e\]
If we assume that the charge $q$ is negative, we get the same result.
46) An ink droplet of mass $m$ and charge $q$ is injected horizontally with an initially velocity ${\vec{v}}_0$ into a region with an electric field $\vec{E}$ which is perpendicular to ${\vec{v}}_0$.
An ink droplet of mass $m$ and charge $q$ is injected horizontally with an initially velocity ${\vec{v}}_0$ into a region with an electric field $\vec{E}$ which is perpendicular to ${\vec{v}}_0$. If a piece of paper is positioned a distance $d$ away from the injection point as shown in the figure, what is the vertical deflection $y$ of the droplet?
The electric force acting on the charge $q$ is $\vec{F}=q\vec{E}=q\left|\vec{E}\right|(-\hat{j})$, so using Newton's second law the magnitude of acceleration in the $y$ direction is
\[a_y=\frac{F}{m}=\frac{q\left|\vec{E}\right|}{m}\]
The initial velocity and height of the droplet is $v_0=0\ ,\ y_0=0$, respectively. Thus by applying the following kinematical relation $y=\frac{1}{2}at^2+v_0t+y_0$, we can obtain the distance traveled in $y$ direction in terms of the elapsed time $t$.
\[y=\frac{1}{2}\left(\frac{q\left|\vec{E}\right|}{m}\right)t^2\]
The elapsed time in $x$ and $y$ directions is the same. There are no forces in the $x$ direction so that the velocity $v_x=v_0$ is constant. Using the equation of uniform motion $x=vt$, we can find the elapsed time $t$ as follows
\[d=v_0t\Rightarrow t=\frac{d}{v_0}\]
By substituting $t$ into the $y\ $equation above, we get
\[y=\frac{1}{2}\left(\frac{q\left|\vec{E}\right|}{m}\right){\left(\frac{d}{v_0}\right)}^2=\frac{q\left|E\right|d^2}{2mv^2_0}\]
47) A very long, uniform line of charge with positive linear charge density $+\lambda$ lies along the $x$ axis. An identical line of charge lies along the $y$ axis.
A very long, uniform line of charge with positive linear charge density $+\lambda$ lies along the $x$ axis. An identical line of charge lies along the $y$ axis.
(a) Determine the electric field $\vec{E}(x,y)$ for all points in the $x-y$ plane.
(b) Determine the change in the electrostatic potential $\Delta V$ between the points $x=a,y=a$ and $x=a,y=3a$.
(c) Determine $\Delta V$ between the points $x=a,y=a$ and $x=3a,y=a$.
(d) How much work must be done to move a small negative charge $-q$ from the point $x=3a,y=3a$ to the point $x=a,y=a$?
(e) For a very long linear charge distribution, we do not define the zero of electrostatic potential to be an infinity. Why not?
(a) Use Gauss's law to find the electric field due to a long uniform line of charge with charge density $\lambda$. In this case, the Gaussian surface must be cylindrical to match the symmetry of the problem so the Gaussian surface is a coaxial cylinder with the wire with length $L$ and radius $R$. The direction of $\vec{E}$ is also radial. Therefore,
\[\oint_S{\vec{E}.\hat{n}dA}=\frac{Q_{enc}}{\epsilon_0}\to E_R\oint_S{dA}=\frac{1}{\epsilon_0}\int^L_0{\lambda dx}\]
\[\Rightarrow E_R\left(2\pi RL\right)=\frac{1}{\epsilon_0}\lambda L\Rightarrow E_R=+\frac{\lambda}{2\pi\epsilon_0R}\]
In above, $\oint_s{dA}$ is the area of the curved surface of the cylinder. Since the $\vec{E}$ is radially outward from the wire, the contributions due to left $\hat{n}=-\hat{x}$ and right $\hat{n}=+\hat{x}$ end of cylinder are zero i.e. $\vec{E}.\hat{n}=0$. $R$ is the distance from the wire.
Therefore, the line $+\lambda$ ,which is lies along the $x$ axis, at distance $y$ from it produced the electric field
\[{\vec{E}}_1=\frac{+\lambda}{2\pi\epsilon_0y}\ \hat{y}\]
And the line $+\lambda$, which is lies along the $y$ axis, at distance $x$ from it produce the following electric field
\[{\vec{E}}_2=\frac{+\lambda}{2\pi\epsilon_0x}\hat{x}\]
Thus, the total electric field at a point in the $x-y$ plane is
\[{\vec{E}}_{tot}=\frac{+\lambda}{2\pi\epsilon_0}\left(\frac{1}{y}\ \hat{y}+\frac{1}{x}\ \hat{x}\right)\]
(b) The electric potential between $a$ and $b$ is defined as $V_b-V_a=-\int^b_a{\vec{E}.d\vec{l}}$. we are moving perpendicular to the wire 1 and subsequently parallel to the wire $2$, so wire
$2$ does not contribute to the potential difference. In this case, $d\vec{l}=dy\hat{y}$.
\[\Delta V=-\int^{3a}_a{{\vec{E}}_1.d\vec{l}}=-\int^{3a}_a{\frac{\lambda}{2\pi\epsilon_0y}dy}=-\frac{\lambda}{2\pi\epsilon_0}{\left.{\ln y\ }\right|}^{3a}_a\]
\[\therefore \ \Delta V=-\frac{\lambda}{2\pi\epsilon_0}{\ln 3\ }\]
So going from $y=a$ to $y=3a$ the potential is decreasing since we are moving away from the charged line $1$.
(c) In this case, we are approaching perpendicularly to the wire $2$. In this direction we have change in the electric field of ${\vec{E}}_2$, so only this wire contribute to the potential difference.
\[\Delta V=-\int^{3a}_a{E_2\hat{x}.dx\ \hat{x}}=-\int^{3a}_a{\frac{\lambda}{2\pi\epsilon_0x}dx}=-\frac{\lambda}{2\pi\epsilon_0}{\left.{\ln x\ }\right|}^{3a}_a\]
\[\therefore \Delta V=-\frac{\lambda}{2\pi\epsilon_0}{\ln 3\ }\]
(d) The work done on a point charge $q$ to move it from point $a$ to $b$ in an external electric field with potential difference $\Delta V$ is found by $W=q\Delta V$.
To going from $(3a,3a)$ to $\left(a,a\right)$ we can consider the following path
\[\left(3a,3a\right)\xrightarrow{1}\left(3a,a\right)\xrightarrow{2}(a,a)\]
In the path 1, we have
\[\Delta V_1=V_{(3a,3a)}-V_{(3a,a)}=-\int^{3a}_a{{\vec{E}}_1\hat{y}.dy\left(-\hat{y}\right)}=\frac{\lambda}{2\pi\epsilon_0}{\ln 3\ }\]
In the path 2, we also have
\[\Delta V_2=V_{3a,a}-V_{a,a}=-\int^{3a}_a{E_2\hat{x}.dx\left(-\hat{x}\right)}=\frac{\lambda}{2\pi\epsilon_0}{\ln 3\ }\ \]
The signs in the above potentials changes because we are moving in the opposite direction as was considered in previous parts. Therefore, the total electric potential between the $(3a,3a)$ and $(a,a)$ is
\[\Delta V=\Delta V_1+\Delta V_2=\frac{\lambda}{\pi\epsilon_0}{\ln 3\ }\]
The work done on the charge $q$ by an external agent is found as
\[W=-q\Delta V=-\frac{q\lambda}{2\pi\epsilon_0}{\ln 3\ }\]
The electric fields due to the line of charges does positive work on the external charge $q$.
(e) Since in this case, unlike the spherical charged distribution, the charge is extended to the infinity and thus at that point we have charge and electric field!
48) A charge $Q$ is placed on a metal sphere (sphere $1$) of radius $R_1$ . Very far from this sphere is a second sphere (sphere $2$) of radius $R_2$ which is initially uncharged.
A charge $Q$ is placed on a metal sphere (sphere $1$) of radius $R_1$ . Very far from this sphere is a second sphere (sphere $2$) of radius $R_2$ which is initially uncharged. If the two spheres are connected by a metal wire, what is the final charge $Q_2$ on sphere $2$?
Once the two spheres are connected, they constitute an equipotential system. In this case, some of the charge $Q$ delivers to the sphere $2$, that is the total charge of the system becomes $Q=Q_1+Q_2$ where $Q_1$ is the new charge on the sphere $1$ and $Q_2$ is one the sphere $2$. Recall that the electric potential due to a sphere of radius $R_1$ and charge $Q_1$ is $V_1=kQ_1/R_1$, and for a sphere of radius $R_2$ with charge $Q_2$ is $V_2=kQ_2/R_2$.
Since after the two spheres connect to each other form an equipotential system, we have
\[V_1=V_1\Rightarrow \frac{kQ_1}{R_1}=\frac{kQ_2}{R_2}\Rightarrow Q_1=\frac{R_1}{R_2}Q_2\]
Substituting $Q_1$ into the $Q=Q_1+Q_2$ and solving for $Q_2$, we obtain
\[Q=\frac{R_1}{R_2}Q_2+Q_2=Q_2\left(1+\frac{R_1}{R_2}\right)\]
\[\Rightarrow Q_2=\frac{QR_2}{R_2+R_1}\]
49) Three electrons are placed at the vertexes of an equilateral triangle with side length of ${\rm 5.1\ nm}$. A proton is placed at the center of the triangle. What is the potential energy of this arrangement of charges?
Three electrons are placed at the vertexes of an equilateral triangle with side length of ${\rm 5.1\ nm}$. A proton is placed at the center of the triangle. What is the potential energy of this arrangement of charges?
The potential energy of two point charges separated by distance $r$ is
\[U=k\frac{q_1q_2}{r}\]
The potential energy is a scalar quantity so the total potential energy due to a configuration of point charges is the sum of the potential energy of each pairs of them. In this case, since the distance of charges and their magnitudes are the same so
U&=U\left(3\ electrons\ pairs\right)+U\left(3\ (e^++e^-)\ pairs\right)\\
&=3k\frac{\left(-e\right)\left(-e\right)}{L}+3k\frac{\left(+e\right)\left(-e\right)}{r}
$r$ is the distance between electron and proton which can be computed from the geometry below
\therefore U_{tot}&=+3k\frac{e^2}{L}-\frac{3ke^2}{\frac{L}{2{\cos 30{}^\circ \ }}}\\
&=\frac{3ke^2}{L}\left(1-2{\cos 30{}^\circ \ }\right)\\
&=\frac{3\left(9\times {10}^9\right){\left(1.6\times {10}^{-19}\right)}^2}{5.1\times {10}^{-9}}\left(1-2\, \cos 30{}^\circ \right)\\
&=-9.92\times {10}^{-20}\ {\rm J}
From the definition of cosine of an angle in the triangle above, we get
\[{\cos 30{}^\circ \ }=\frac{\frac{L}{2}}{r}\Rightarrow r=\frac{L}{2{\cos 30{}^\circ \ }}\]
50) An electron was accelerated from rest through a potential difference of ${\rm 9900\ V}$. What is its speed?
An electron was accelerated from rest through a potential difference of ${\rm 9900\ V}$. What is its speed? ($m_e=9.31\times {10}^{-31}{\rm kg}$).
Conservation of mechanical energy is conserved, $\Delta K+\Delta U=0$. Recall that the potential energy of a point charge moving through a potential difference $\Delta V$ is $\Delta U=q\Delta V$. Since the electron accelerated from rest; hence $\Delta K=K_f-K_i=\frac{1}{2}mv^2_f-0$. A negative charge moves against the electric field which is the direction of the decreasing potential energy i.e. $\Delta V=V_f-V_i<0$. Thus
\[\Delta K+\Delta U=0\]
\[\frac{1}{2}m\left(v^2_f-v^2_0\right)+q\Delta V=0\Rightarrow v_f=\sqrt{-\frac{2q\Delta V}{m}}\]
Electron has $q=-e=-1.6\times {10}^{-19}\ {\rm C}$
\[\Rightarrow v_f=\sqrt{\frac{2\left(1.6\times {10}^{-19}\right)\left(9900\right)}{9.31\times {10}^{-31}\ {\rm kg}}}=5.83\times {10}^7\frac{{\rm m}}{{\rm s}}\]
51) An electric field $\vec{E}=\frac{1}{r^2}\ \hat{r}$ is located in the $xy$ plane. What is the potential difference between $r=6\ {\rm m}$ and $r=8\ {\rm m}$?
An electric field $\vec{E}=\frac{1}{r^2}\ \hat{r}$ is located in the $xy$ plane. What is the potential difference between $r=6\ {\rm m}$ and $r=8\ {\rm m}$?
The electric field $\vec{E}$ and potential difference $V$ are related by
\[\Delta V=V_f-V_i=-\int^f_i{\vec{E}.d\vec{s}}\]
Where $s$ is the displacement vector from point $i$ to point $f$. Note that only the component of the electric field parallel to the line of integration is relevant. Therefore,
\[\Delta V=V\left(r=8{\rm m}\right){\rm -}V\left(r={\rm 6m}\right){\rm =-}\int^{r=8}_{r=6}{\frac{1}{r^2}\hat{r}.dr\hat{r}}={\rm -}\int^{r=8}_{r=6}{\frac{1}{r^2}dr}\]
\[{\rm =-}{\left(-\frac{1}{r}\right)}^8_6=\left(\frac{1}{8}-\frac{1}{6}\right)=\frac{3-4}{24}=-\frac{1}{24}\ {\rm V}\]
In above, $d\vec{s}=dr\ \hat{r}$ is chosen in the radial direction.
52) The figure shows a current entering a truncated solid cone made of a conducting metal. The electron drift speed at the ${\rm 3.0\ mm}$ diameter end of the cone is
The figure shows a current entering a truncated solid cone made of a conducting metal. The electron drift speed at the ${\rm 3.0\ mm}$ diameter end of the cone is ${\rm 4.0\times }{{\rm 10}}^{{\rm -}{\rm 4}}{\rm \ m/s}$. What is the electron drift speed at the ${\rm 1.0\ mm}$ diameter end of the wire?
An electron in a conductor will propagate randomly or have thermal motion similar to the motion of the molecules of the gas such as air. With applying an electric field, we can change this randomly motions into a uniform motion in one direction with speed $v_d$ which is called drift speed. The typical value for such a speed is about ${10}^{-4}\ {\rm m/s}$.
The relation between current through a wire with cross sectional area $A$ and drift speed of charge carries, the charges that move in a conductor, of charge density $n$ is
\[I=nqAv_d\]
In this figure, the current through $3\ {\rm mm}$ surface is the same as that of $1\ {\rm mm}$ surface that is $I_3=I_1$, therefore
A_3v_{d3}=A_1v_{d1}\Rightarrow v_{d1}=\frac{A_3}{A_1}v_{d3}&={\left(\frac{r_3}{r_1}\right)}^2v_{d3}\\
&={\left(\frac{3}{1}\right)}^2\left(4\times {10}^{-4}\right)\\
&=36\times {10}^{-4}\frac{{\rm m}}{{\rm s}}{\rm \ }
$A=\pi r^2$is the cross sectional area of the circular disk.
53) If excess charge is put on a spherical conductor,
If excess charge is put on a spherical conductor,
(a) it remains where it was placed
(b) it spreads a little from where it was placed but not over the whole sphere
(c) it spreads uniformly over the surface of the sphere if the sphere is small
(d) it spreads uniformly throughout the volume of the conductor
(e) it spreads uniformly over the surface of the sphere
The surface of a conductor is an equipotential surface which means that whose surface must be always in equilibrium. Putting an excess charge would disrupt this equilibrium condition on the surface since causes an electric field forms between the accumulation of charges in that point and other points of the surface. This electric field, by definition, causes a change in the electric potential difference between any points on the surface, instead, if the excess charge spreads uniformly over the surface the equipotential condition of the surface is not violated.
54) Comparing the field of a single point charge with the field of an electric dipole,
Comparing the field of a single point charge with the field of an electric dipole,
(a) the field of the point charge decreases more rapidly with distance
(b) the field of the point charge decreases less rapidly with distance
(c) the field of the point charge decreases more rapidly with distance but only along the dipole axis
(d) the field of the point charge decreases less rapidly with distance but only perpendicular to the dipole axis
(e) the fields decrease equally rapidly with distance
the electric field of a point charge is found as $E=kQ/r^2$, which obeys the inverse-square law but the electric field of an electric dipole, pairs of point charges with equal magnitude and opposite sign separated by a distance $d$, at far away distances from the dipole can be found as
\[{\vec{E}}_{dipole}=\frac{1}{4\pi {\epsilon }_0}\frac{\vec{p}}{r^3}\]
Where $\vec{p}=q\vec{d}$ is the electric dipole moment and $\vec{d}$ is distance vector from the negative charge to the positive charge.
As you can see the electric field of a dipole decreases more rapidly than that of point charges.
55) An electron traveling north enters a region where the electric field is uniform and points north. The electron:
An electron traveling north enters a region where the electric field is uniform and points north. The electron:
(a) speeds up
(b) slows down
(c) veers east
(d) veers west
(e) continues with the same speed in the same direction
recall that positive charge moves in direction of electric field and a negative charge moves in opposite direction of electric field. Since the displacement of electron, to north, and direction of the electric field, to north, are in the same direction, the electron is decreasing down.
56) A $\mathrm{3.5\ cm}$ radius hemisphere contains a total charge of $6.6\times {10}^{-7}\mathrm{C}$. The flux through the rounded portion of the surface
A $\mathrm{3.5\ cm}$ radius hemisphere contains a total charge of $6.6\times {10}^{-7}\mathrm{C}$. The flux through the rounded portion of the surface is $9.8\times {10}^4\mathrm{N.}{\mathrm{m}}^{\mathrm{2}}\mathrm{/C}$. The flux through the flat base is:
(a) $0\ \mathrm{N.}{\mathrm{m}}^{\mathrm{2}}\mathrm{/C}$
(b) $2.3\ \mathrm{N.}{\mathrm{m}}^{\mathrm{2}}\mathrm{/C}$
(c) $-2.3\ \mathrm{N.}{\mathrm{m}}^{\mathrm{2}}\mathrm{/C}$
(d) $-9.8\ \mathrm{N.}{\mathrm{m}}^{\mathrm{2}}\mathrm{/C}$
(e) $9.8\ \mathrm{N.}{\mathrm{m}}^{\mathrm{2}}\mathrm{/C}$
The amount of electric flux ${\mathrm{\Phi }}_E$ through any closed surface and the associated enclosed total charge is related together by Gauss's law as
\[{\mathrm{\Phi }}_E=\frac{Q_{encl}}{{\epsilon }_0}\]
Hemisphere has two surfaces, rounded and flat base thus the total electric flux through it is
\[{\mathrm{\Phi }}_{E.r}+{\mathrm{\Phi }}_{E.b}=\frac{Q_{encl}}{{\epsilon }_0}\]
\[9.8\times {10}^4+{\mathrm{\Phi }}_{E.b}=\frac{6.6\times {10}^{-7}}{8.854\times {10}^{-12}}\Rightarrow \ {\mathrm{\Phi }}_{E.b}=-2.34\ \mathrm{N.}{\mathrm{m}}^{\mathrm{2}}\mathrm{/C}\]
57) Charge $Q$ is distributed uniformly throughout an insulating sphere of radius $R$. The magnitude of the electric field at a point $R/2$ from the center is
Charge $Q$ is distributed uniformly throughout an insulating sphere of radius $R$. The magnitude of the electric field at a point $R/2$ from the center is:
(a) $Q/4\pi {\epsilon }_0R^2$
(b) $Q/\pi {\epsilon }_0R^2$
(c) $3Q/4\pi {\epsilon }_0R^2$
(d) $Q/8\pi {\epsilon }_0R^2$
(e) None of these
If we have symmetric configuration, as this case, use Gauss's law to find the electric field at each point of space. To do this, one must suppose a Gaussian surface at the desired point. In this problem, draw a Gaussian surface as sphere of radius of $R/2$ and proceed as follows
\[\oint{\vec{E}.d\vec{A}}=\frac{Q_{encl}}{{\epsilon }_0}\]
Where $Q_{encl}$ is the charge enclosed in the Gaussian surface which here is sphere of radius $R/2$ and is determined by the definition of volume charge density
\[\rho =\frac{Q}{V}\to Q_{encl}=\rho V_{Gauss}=\rho \left(\frac{4}{3}\pi {\left(\frac{R}{2}\right)}^3\right)=\frac{1}{6}\rho \pi R^3\]
Substituting above into Gauss's law, we get
\[E\oint{dA}=\frac{1}{6}\rho \pi R^3\]
The electric field is taken out of the integral since it is constant at the location of the Gaussian sphere by symmetry considerations. The closed integral gives the area of the Gaussian surface. Therefore
\[E\left(4\pi {\left(\frac{R}{2}\right)}^2\right)=\frac{\frac{1}{6}\rho \pi R^3}{{\epsilon }_0}\to \ \ E=\frac{1}{4\pi {\epsilon }_0}\frac{2}{3}\rho \pi R\]
Since the density of the sphere is uniform everywhere so substitute it by the total charge induced on the original sphere of radius $R$ as
\[\rho =\frac{Q}{V}=\frac{Q}{\frac{4}{3}\pi R^3}\]
\[E=\frac{1}{4\pi {\epsilon }_0}\frac{2}{3}\frac{Q}{\frac{4}{3}\pi R^3}\pi R=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{{2R}^2}=\frac{Q}{8\pi {\epsilon }_0R^2}\]
58) If $\mathrm{500\ J}$ of work are required to carry a $\mathrm{40\ C}$ charge from one point to another, the potential difference between these two points is:
If $\mathrm{500\ J}$ of work are required to carry a $\mathrm{40\ C}$ charge from one point to another, the potential difference between these two points is:
(a) $12.5\, \mathrm V$
(b) $20000\, \mathrm V$
(c) $0.08\, \mathrm V$
(d) Depends on the path
the potential difference between two points is defined as the ratio of the change in electric potential energyto the electric charge which is moved between the points as
\[\mathrm{\Delta }V=\frac{\mathrm{\Delta }U}{q}\]
We know that the work done on a system is stored in its potential energy so, irrespective of sign,$W=\mathrm{\Delta }U=500\ \mathrm{J}$
\[\Rightarrow \mathrm{\Delta }V=\frac{500}{40}=12.5\ \mathrm{V}\]
59) The diagram shows four pairs of large parallel conducting plates. The value of the electric potential is given for each plate.
The diagram shows four pairs of large parallel conducting plates. The value of the electric potential is given for each plate. Rank the pairs according to the magnitude of the electric field between the plates, least to greatest
(a) 1,2,3,4
(b) 4,3,2,1
(c) 2,3,1,4
(d) 2,4,1,3
(e) 3,2,4,1
The electric field between two large parallel conducting plates is uniform. Recall that the electric field of a positive plane is away from it and a negative plane is toward it, using this hint and the superposition principle determine the electric field between the plate as follows
In above we have used the definition of the uniform electric field in terms of electric potential as $E=V/d$, since the distance between the plates is the same so the correct answer is D.
60) A particle with a charge of $5.5\times {10}^{-6}\mathrm{C}$is $\mathrm{3.5\ cm}$ from a particle with a charge of $-\mathrm{2.3\times }{\mathrm{10}}^{\mathrm{-}\mathrm{8}}\mathrm{C}$.
A particle with a charge of $5.5\times {10}^{-6}\mathrm{C}$is $\mathrm{3.5\ cm}$ from a particle with a charge of $-\mathrm{2.3\times }{\mathrm{10}}^{\mathrm{-}\mathrm{8}}\mathrm{C}$. The potential energy of this two-particle system, relative to the potential energy at infinite separation, is:
(a) $3.3\times {10}^{-2}\mathrm{J}$
(b) $-3.3\times {10}^{-2}\mathrm{J}$
(c) $9.3\times {10}^{-1}\mathrm{J}$
(d) $-9.3\times {10}^{-1}\mathrm{J}$
(e) $0\ \mathrm{J}$
The electric potential energy of a pair of point charges can be found as follows
\[\mathrm U=\frac{1}{4\pi {\epsilon }_0}\frac{q_1q_2}{r}\]
\[U=\left(9\times {10}^9\right)\frac{\left(5.5\times {10}^{-6}\right)\left(-2.3\times {10}^{-8}\right)}{\left(3.5\times {10}^{-2}\mathrm{m}\right)}=-3.3\times {10}^{-2}\mathrm{J}\]
When the point charges are placed at infinity their potential energy is zero!
61) A hollow metal sphere is charged to a potential $V$. The potential at its center is:
A hollow metal sphere is charged to a potential $V$. The potential at its center is:
(a) $V$
(c) $-V$
(d) $2\ \mathrm{V}$
(e) $\pi \ \mathrm{V}$
The electric potential between the two point a and b is defined as
\[V_b-V_a=-\int^b_a{\vec{E}.d\vec{s}}\]
Where $\vec{s}$ is the displacement vector from a toward b and is parallel to the field line. The negative sign indicates that the electric potential at point b is lower than at point b.
To find the electric potential it is customary to choose the reference of the electric potential to be $V=0$ at $r_a=\infty $. Therefore find the electric potential between $r_a=\infty $ to $r_b=0$. The electric field inside and outside a conducting shell is $E_{in}=0$ and $E_{out}=kQ/r^2$, where $Q\ $is the total charge on the sphere.
\[V_0-V_{\infty }=-\int^0_{\infty }{\vec{E}.d\vec{s}}=-\int^R_{\infty }{E_{out}dr}-E_{in}\int^0_R{dr}\]
\[V_0-0=-\int^R_{\infty }{\frac{kQ}{r^2}dr}-0\int^0_R{dr}={{\left.\frac{kQ}{r}\right|}^R}_{\infty }=\frac{kQ}{R}\]
\[\Rightarrow V_0=\frac{kQ}{R}=V\]
$kQ/R$ is the electric potential of a conducting charged sphere at its surface.
In the first equality, $\vec{E}.d\vec{s}$ substituted by $Edr$ since the electric field outside is radial and the displacement vector $d\vec{s}$ has been also chosen radially.
Category : Electrostatic
Most useful formula in Electrostatic:
Coulomb's law: the force between two charged particle $q_1$ and $q_2$ separated by distance $r$ is
\[F=k\frac{|q_1q_2|}{r^2}\]
\[k=\frac{1}{4\pi \epsilon_0}=8.988 \times {10}^{9} \ {\rm N.m^{2}/C^{2}}\]
Electric field:
\[\vec E=\frac{\vec F}{q_0}\]
Electric field of a point charge at distance $r$:
\[\vec E=\frac{1}{4 \pi \epsilon_0}\frac{q}{r^2} \hat{r}\]
Gauss's law:
\[\Phi_B=\int{\vec E \cdot nd\vec A}=\frac{Q_{enclosed}}{\epsilon_0}\] | CommonCrawl |
Analysis of the diffuse-domain method for solving PDEs in complex geometries
Karl Yngve Lervåg (Department of Energy and Process Engineering, Norwegian University of Science and Technology, Trondheim, Norway; and SINTEF Energy Research, Trondheim, Norway)
John Lowengrub (Department of Mathematics, University of California at Irvine)
In recent work, Li et al. [Commun. Math. Sci., 7, 81-107, 2009] developed a diffusedomain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The diffuse-domain method uses an implicit representation of the geometry where the sharp boundary is replaced by a diffuse layer with thickness $\epsilon$ that is typically proportional to the minimum grid size. The original equations are reformulated on a larger regular domain and the boundary conditions are incorporated via singular source terms. The resulting equations can be solved with standard finite difference and finite element software packages. Here, we present a matched asymptotic analysis of general diffuse-domain methods for Neumann and Robin boundary conditions. Our analysis shows that for certain choices of the boundary condition approximations, the DDM is second-order accurate in $\epsilon$. However, for other choices the DDM is only first-order accurate. This helps to explain why the choice of boundary-condition approximation is important for rapid global convergence and high accuracy. Our analysis also suggests correction terms that may be added to yield more accurate diffuse-domain methods. Simple modifications of first-order boundary condition approximations are proposed to achieve asymptotically second-order accurate schemes. Our analytic results are confirmed numerically in the $L^2$ and $L^{\infty}$ norms for selected test problems.
numerical solution of partial differential equations, phase-field approximation, implicit geometry representation, matched asymptotic analysis
35B40, 35K51, 35K52, 35K57, 65Mxx | CommonCrawl |
\begin{document}
\title{group inverse for anti-triangular block operator matrices}
\author{Huanyin Chen} \author{Marjan Sheibani$^*$} \address{ Department of Mathematics\\ Hangzhou Normal University\\ Hang -zhou, China} \email{<[email protected]>} \address{Women's University of Semnan (Farzanegan), Semnan, Iran} \email{<[email protected]>}
\thanks{$^*$Corresponding author}
\subjclass[2010]{15A09, 65F20.} \keywords{group inverse; Drazin inverse; spectral idempotent; identical subblock; anti-triangular block matrix.}
\begin{abstract} We present the existence of the group inverse and its representation for the block operator matrix $\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)$ under the condition $FEF^{\pi}=0$. The group inverse for the anti-triangular block matrices with two identical subblocks under the same condition is thereby investigated. These extend the results of Zou, Chen and Mosi\'c (Studia Scient. Math. Hungar., 54(2017), 489--508), and Cao, Zhang and Ge (J. Appl. Math. Comput., 46(2014), 169--179). \end{abstract}
\maketitle
\section{Introduction}
Let $\mathcal{B}(X)$ be a Banach algebra of all bounded linear operators over a Banach space $X$. An operator $T$ in $\mathcal{B}(X)$ has Drazin inverse provided that there exists some $S\in \mathcal{B}(X)$ such that $ST=TS, S=STS, T^n=T^{n+1}S$ for some $n\in {\Bbb N}$. Such $S$ is unique, if it exists, and we denote it by $T^D$. The such smallest $n$ is called the Drazin index of $T$. If $T$ has Drazin index $1$, $T$ is said to have inverse $S$, and denote its group inverse by $T^{\#}$. The group invertibility of the block operator matrices over a Banach space is attractive. It has interesting applications of resistance distances to the bipartiteness of graphs. Many authors have studied such problems from many different views, e.g., ~\cite{B,B2,C2,C1,C21}.
Let $E,F$ be bounded linear operators and $I$ be the identity operator over a Banach space $X$. It is attractive to investigate the Drazin (group) invertibility of the operator matrix $M=\left(
\begin{array}{cc}
E&I\\
F&0
\end{array} \right)$. It was firstly posed by Campbell that the solutions to singular systems of differential equations is determined by the Drazin (group) invertibility of the preceding special matrix $M$.
Zou et al. studied the group inverse for $M$ under the condition $EF=0$. In Section 2, we present the existence of the group inverse and its representation for the block operator matrix $\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)$ under the wider condition $FEF^{\pi}=0$.
In~\cite[Theorem 5]{C1}, Cao et al. considered the group inverse for a block matrix with identical subblocks over a right Ore domain. In Section 3, we further investigate the necessary and sufficient conditions for the existence and the representations of the the group inverse of a $2\times 2$ operator block matrix $$M=\left( \begin{array}{cc} E&F\\ F&0 \end{array} \right)$$ with identical subblocks. The explicit formula of the group inverse of $M$ is also given under the same condition $FEF^{\pi}=0$.
Let $X$ be a Banach space. We use $\mathcal{B}(X)$ to denote the Banach algebra of bounded linear operator on $X$. If $T\in \mathcal{B}(X)$ has the Drazin inverse $T^D$, the element $T^{\pi}=I-TT^D$ is called the spectral idempotent of $T$. Let $p\in \mathcal{B}(X)$ be an idempotent operator, and let $T\in \mathcal{B}(X)$. Then we write $$T=pTp+pT(I-p)+(I-p)Tp+(I-p)T(I-p),$$ and induce a Pierce representation given by the operator matrix $$T=\left(\begin{array}{cc} pTp&pT(I-p)\\ (I-p)Tp&(I-p)T(I-p) \end{array} \right)_p.$$
\section{anti-triangular block matrices}
The aim of this section is to provide necessary and sufficient conditions on $E$ and $F$ so that the block operator matrix $\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)$ has group inverse. We now derive
\begin{thm} Let $M=\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)$ and $E,F,EF^{\pi}$ have Drazin inverses. If $FEF^{\pi}=0$, then the following are equivalent: \end{thm} \begin{enumerate} \item [(1)]{\it $M$ has group inverse.}
\item [(2)]{\it $F$ has group inverse and $E^{\pi}F^{\pi}=0$.}\\ \end{enumerate} In this case, $$M^{\#}=\left( \begin{array}{cc} E^DF^{\pi}&F^{\#}+(E^DF^{\pi})^2-E^DF^{\pi}EF^{\#}\\ FF^{\#}&-FF^{\#}EF^{\#} \end{array} \right).$$ \begin{proof} Let $e= \left( \begin{array}{cc} FF^D&0\\ 0&I \end{array} \right)$. Then $M= \left( \begin{array}{cc} a&b\\ c&d \end{array} \right)_e,$ where $$\begin{array}{c} a=\left( \begin{array}{cc} FF^DE&FF^D\\ F^2F^D&0 \end{array} \right),b=\left( \begin{array}{cc} 0&0\\ FF^{\pi}&0 \end{array} \right),\\ c=\left( \begin{array}{cc} F^{\pi}EFF^D&F^{\pi}\\ 0&0 \end{array} \right),d=\left( \begin{array}{cc} EF^{\pi}&0\\ 0&0 \end{array} \right). \end{array}$$ Moreover, we compute that $$a^{\#}=\left( \begin{array}{cc} 0&F^D\\ FF^D&-FF^DEF^D \end{array} \right), d^D=\left( \begin{array}{cc} E^DF^{\pi}&0\\ 0&0 \end{array} \right).$$ Therefore we have $$a^{\pi}=\left( \begin{array}{cc} 0&0\\ 0&F^{\pi} \end{array} \right), d^{\pi}=\left( \begin{array}{cc} E^{\pi}F^{\pi}&0\\ 0&0 \end{array} \right).$$ $(2)\Rightarrow (1)$ Since $F$ has group inverse, we have $b=0$, and then $$M= \left( \begin{array}{cc} a&0\\ c&d \end{array} \right)_e.$$
Since $EF^{\pi}$ has group inverse, $d$ has group inverse and and $d^{\#}=d^D$. In view of~\cite[Lemma 2.1]{C2}, $$M^D=\left( \begin{array}{cc} a^{\#}&0\\ z&d^{\#} \end{array} \right)_e,$$ where $$\begin{array}{lll} z&=&(d^{\#})^2ca^{\pi}+d^{\pi}c(a^{\#})^2-d^{\#}ca^{\#}\\ &=&(d^{\#})^2ca^{\pi}-d^{\#}ca^{\#}\\ &=&\left( \begin{array}{cc} E^DF^{\pi}&0\\ 0&0 \end{array} \right)^2\left( \begin{array}{cc} F^{\pi}EFF^{\#}&F^{\pi}\\ 0&0 \end{array} \right)\left( \begin{array}{cc} 0&0\\ 0&F^{\pi} \end{array} \right)\\ &-&\left( \begin{array}{cc} E^DF^{\pi}&0\\ 0&0 \end{array} \right)\left( \begin{array}{cc} F^{\pi}EFF^{\#}&F^{\pi}\\ 0&0 \end{array} \right)\left( \begin{array}{cc} 0&F^{\#}\\ FF^{\#}&-FF^{\#}EF^{\#} \end{array} \right)\\ &=&\left( \begin{array}{cc} 0&(E^DF^{\pi})^2-E^DF^{\pi}EF^{\#}\\ 0&0 \end{array} \right). \end{array}$$ Therefore $$M^D=\left( \begin{array}{cc} E^DF^{\pi}&F^{\#}+(E^DF^{\pi})^2-E^DF^{\pi}EF^{\#}\\ FF^{\#}&-FF^{\#}EF^{\#} \end{array} \right).$$ We check that $$d^{\pi}ca^{\pi}=\left( \begin{array}{cc} E^{\pi}F^{\pi}&0\\ 0&0 \end{array} \right)\left( \begin{array}{cc} F^{\pi}EFF^{\#}&F^{\pi}\\ 0&0 \end{array} \right)\left( \begin{array}{cc} 0&0\\ 0&F^{\pi} \end{array} \right)=0.$$ Then $M$ has group inverse. Accordingly, $M^{\#}=M^D$, as required.
$(1)\Rightarrow (2)$ Write $M^{\#}=\left( \begin{array}{cc} X_{11}&X_{12}\\ X_{21}&X_{22} \end{array} \right)$. Then $MM^{\#}=M^{\#}M$, and so $$\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)\left( \begin{array}{cc} X_{11}&X_{12}\\ X_{21}&X_{22} \end{array} \right)=\left( \begin{array}{cc} X_{11}&X_{12}\\ X_{21}&X_{22} \end{array} \right)\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right).$$ Then we have $$\begin{array}{c} EX_{11}+X_{21}=X_{11}E+X_{12}F,\\ FX_{12}=X_{21}. \end{array}$$ Since $MM^{\#}M=M$, we have $$\begin{array}{c} EX_{11}+X_{21}=I,\\ FX_{11}=0. \end{array}$$ Therefore $$\begin{array}{lll} F&=&(FE)X_{11}+FX_{21}\\ &=&(FE)F^dFX_{11}+FX_{21}\\ &=&(FEF^d)(FX_{11})+FX_{21}\\ &=&F^2X_{12}. \end{array}$$ Hence $F$ has group inverse. Hence, $b=0$, and and so $$M=\left( \begin{array}{cc} a&0\\ c&d \end{array} \right)_e$$ and $a$ has group inverse. Therefore $d$ has group inverse and $d^{\pi}ca^{\pi}=0$.
Since $d$ has group inverse, we see that $EF^{\pi}$ has group inverse. As $d^{\pi}ca^{\pi}=0$, we have $$\left( \begin{array}{cc} E^{\pi}F^{\pi}&0\\ 0&0 \end{array} \right)\left( \begin{array}{cc} F^{\pi}EFF^D&F^{\pi}\\ 0&0 \end{array} \right)\left( \begin{array}{cc} 0&0\\ 0&F^{\pi} \end{array} \right)=\left( \begin{array}{cc} 0&E^{\pi}F^{\pi}\\ 0&0 \end{array} \right)=0,$$ and therefore $E^{\pi}F^{\pi}=0$. This completes the proof.\end{proof}
\begin{cor} Let $M=\left( \begin{array}{cc} E&F\\ I&0 \end{array} \right)$ and $E,F,EF^{\pi}$ have Drazin inverses. If $FEF^{\pi}=0$, then the following are equivalent: \end{cor} \begin{enumerate} \item [(1)]{\it $M$ has group inverse.}
\item [(2)]{\it $F$ has group inverse and $E^{\pi}F^{\pi}=0$.}\\ \end{enumerate} In this case, $$M^{\#}=\left(
\begin{array}{cc}
\Gamma&\Delta\\
\Lambda&\Xi\\
\end{array} \right),$$ where $$\begin{array}{lll} \Gamma&=&F^{\pi}E^DF^{\pi},\\ \Delta&=&I-F^{\pi}E^DF^{\pi}E,\\ \Lambda&=&F^{\#}+(E^DF^{\pi})^2-E^DF^{\pi}EF^{\#},\\ \Xi&=&E^DF^{\pi}-F^{\#}E-(E^DF^{\pi})^2E+E^DF^{\pi}EF^{\#}E, \end{array}$$ \begin{proof} Let $N=\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)$. Then $$M=P^{-1}NP, P=\left( \begin{array}{cc} 0&I\\ I&-E \end{array} \right).$$ Therefore $M$ has group inverse if and only if so does $N$, if and only if $F, EF^{\pi}$ have group inverse and $E^{\pi}F^{\pi}=0$, by Theorem 2.1. In this case, $$\begin{array}{lll} M^{\#}&=&P^{-1}N^{\#}P\\
&=&\left( \begin{array}{cc} E&I\\ I&0 \end{array} \right)N^{\#}\left( \begin{array}{cc} 0&I\\ I&-E \end{array} \right)\\ &=&\left( \begin{array}{cc} I&F^{\pi}E^DF^{\pi}\\ E^DF^{\pi}&F^{\#}+(E^DF^{\pi})^2-E^DF^{\pi}EF^{\#} \end{array} \right)\left( \begin{array}{cc} 0&I\\ I&-E \end{array} \right)\\ &=&\left(
\begin{array}{cc}
\Gamma&\Delta\\
\Lambda&\Xi\\
\end{array} \right), \end{array}$$ where $$\begin{array}{lll} \Gamma&=&F^{\pi}E^DF^{\pi},\\ \Delta&=&I-F^{\pi}E^DF^{\pi}E,\\ \Lambda&=&F^{\#}+(E^DF^{\pi})^2-E^DF^{\pi}EF^{\#},\\ \Xi&=&E^DF^{\pi}-F^{\#}E-(E^DF^{\pi})^2E+E^DF^{\pi}EF^{\#}E, \end{array}$$ as asserted.\end{proof}
We are now ready to prove the following.
\begin{thm} Let $M=\left( \begin{array}{cc} E&F\\ I&0 \end{array} \right)$ and $E,F,EF^{\pi}$ have Drazin inverse. If $F^{\pi}EF=0$, then the following are equivalent: \end{thm} \begin{enumerate} \item [(1)]{\it $M$ has group inverse.}
\item [(2)]{\it $F$ has group inverse and $F^{\pi}E^{\pi}=0$.}\\ \end{enumerate} In this case, $$M^{\#}=\left( \begin{array}{cc} F^{\pi}E^D&FF^{\#}\\ F^{\#}+(F^{\pi}E^D)^2-F^{\#}EF^{\pi}E^D&-F^{\#}EFF^{\#} \end{array} \right).$$ \begin{proof} We consider the transpose $M^T=\left( \begin{array}{cc} E^T&I\\ F^T&0 \end{array} \right)$ of $M$. Then $M$ has group inverse if and only if so does $M^T$. Applying Theorem 2.1, $M$ has group inverse if and only if $F^T, E^T(F^T)^{\pi}$ have group inverse and $(E^T)^{\pi}(F^T)^{\pi}=0$, i.e., $F, F^{\pi}E$ have group inverse and $F^{\pi}E^{\pi}=0$. In this case, we have $$\begin{array}{l} M^{\#}=[(M^T)^{\#}]^T=\\ {\small \left( \begin{array}{cc} (E^T)^D(F^T)^{\pi}&(F^T)^{\#}+((E^T)^D(F^T)^{\pi})^2-(E^T)^D(F^T)^{\pi}E^T(F^T)^{\#}\\ F^T(F^T)^{\#}&-F^T(F^T)^{\#}E^T(F^T)^{\#} \end{array} \right)^T}, \end{array}$$ as desired.\end{proof}
\begin{cor} Let $M=\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)$ and $E,F$ have Drazin inverses. If $F^{\pi}EF=0$, then the following are equivalent: \end{cor} \begin{enumerate} \item [(1)]{\it $M$ has group inverse.}
\item [(2)]{\it $F$ has group inverse and $F^{\pi}E^{\pi}=0$.}\\ \end{enumerate} In this case, $$M^{\#}=\left(
\begin{array}{cc}
\Gamma&\Delta\\
\Lambda&\Xi\\
\end{array} \right),$$ where $$\begin{array}{lll} \Gamma&=&F^{\pi}E^DF^{\pi},\\ \Delta&=&F^{\#}+(F^{\pi}E^D)^2-F^{\#}EF^{\pi}E^D,\\ \Lambda&=&I-EF^{\pi}E^DF^{\pi},\\ \Xi&=&F^{\pi}E^D-EF^{\#}-E(F^{\pi}E^D)^2+EF^{\#}EF^{\pi}E^D, \end{array}$$ \begin{proof} Let $N=\left( \begin{array}{cc} E&F\\ I&0 \end{array} \right)$. Then $$M=P^{-1}NP, P=\left( \begin{array}{cc} E&I\\ I&0 \end{array} \right).$$ In view of Theorem 2.3, $$N^{\#}=\left( \begin{array}{cc} F^{\pi}E^D&FF^{\#}\\ F^{\#}+(F^{\pi}E^D)^2-F^{\#}EF^{\pi}E^D&-F^{\#}EFF^{\#} \end{array} \right).$$ Hence, $M$ has group inverse if and only if so does $N$, if and only if $F, F^{\pi}E$ have group inverse and $F^{\pi}E^{\pi}=0$, by Theorem 2.3. Moreover, we have $$\begin{array}{lll} M^{\#}&=&P^{-1}N^{\#}P\\
&=&\left( \begin{array}{cc} 0&I\\ I&-E \end{array} \right)N^{\#}\left( \begin{array}{cc} E&I\\ I&0 \end{array} \right)\\ &=&\left(
\begin{array}{cc}
\Gamma&\Delta\\
\Lambda&\Xi\\
\end{array} \right), \end{array}$$ where $$\begin{array}{lll} \Gamma&=&F^{\pi}E^DF^{\pi},\\ \Delta&=&F^{\#}+(F^{\pi}E^D)^2-F^{\#}EF^{\pi}E^D,\\ \Lambda&=&I-EF^{\pi}E^DF^{\pi},\\ \Xi&=&F^{\pi}E^D-EF^{\#}-E(F^{\pi}E^D)^2+EF^{\#}EF^{\pi}E^D, \end{array}$$ as asserted. \end{proof}
We come now to prove:
\begin{cor} Let $M=\left( \begin{array}{cc} E&F\\ I&0 \end{array} \right)$ and $E,F,EF^{\pi}$ have Drazin inverses. If $EF=\lambda FE (\lambda \in {\Bbb C})$ or $EF^2=FEF$, then the following are equivalent: \end{cor} \begin{enumerate} \item [(1)]{\it $M$ has group inverse.}
\item [(2)]{\it $F$ have group inverse and $F^{\pi}E^{\pi}=0$.}\\ \end{enumerate} In this case, $$M^{\#}=\left( \begin{array}{cc} F^{\pi}E^D&FF^{\#}\\ F^{\#}+(F^{\pi}E^D)^2-F^{\#}EF^{\pi}E^D&-F^{\#}EFF^{\#} \end{array} \right).$$ \begin{proof} If $EF=\lambda FE (\lambda \in {\Bbb C})$, then $F^{\pi}EF=\lambda F^{\pi}FE=0$. If $EF^2=FEF$, then $F^{\pi}EF=F^{\pi}EF^2F^{\#}=F^{\pi}FEFF^{\#}=0$. This completes the proof by Corollary 2.4.\end{proof}
\section{block matrices with identical subblocks}
In ~\cite{C1}, Cao et al. considered the group inverse for block matrices with identical subblocks over a right Ore domain. In this section we are concerned with the group inverse for block operator matrices with identical subblocks over a Banach space.
\begin{thm} Let $M=\left( \begin{array}{cc} E&F\\ F&0 \end{array} \right)$ and $E,EF^{\pi}$ have Drazin inverse and $F$ has group inverse. If $FEF^{\pi}=0$, then the following are equivalent: \end{thm} \begin{enumerate} \item [(1)]{\it $M$ has group inverse.}
\item [(2)]{\it $EE^{\pi}F^{\pi}=0$.} \end{enumerate} In this case, $$M^{\#}=\left(
\begin{array}{cc}
\Gamma&\Delta\\
\Lambda&\Xi\\
\end{array} \right),$$ where $$\begin{array}{rll} \Gamma&=&[I-E^{\pi}F^{\pi}][E^DF^{\pi}+E^{\pi}F^{\pi}E(F^{\#})^2]+E^{\pi}F^{\pi}E(F^{\#})^2,\\ \Delta&=&[I-E^{\pi}F^{\pi}][F^{\#}-E^{\pi}F^{\pi}E(F^{\#})^2EF^{\#}-E^DF^{\pi}EF^{\#}]\\ &-&E^{\pi}F^{\pi}E(F^{\#})^2EF^{\#},\\ \Lambda&=&F[E^DF^{\pi}+E^{\pi}F^{\pi}E(F^{\#})^2]^2+F^{\#}-FE^{\pi}F^{\pi}[E(F^{\#})^2]^2\\ &-&FE^DF^{\pi}E(F^{\#})^2,\\ \Xi&=&[FE^DF^{\pi}+FE^{\pi}F^{\pi}E(F^{\#})^2][F^{\#}-E^{\pi}F^{\pi}E(F^{\#})^2EF^{\#}\\ &-&E^DF^{\pi}EF^{\#}]-[F^{\#}-FE^{\pi}F^{\pi}E(F^{\#})^2E(F^{\#})^2\\ &-&FE^DF^{\pi}E(F^{\#})^2]EF^{\#}. \end{array}$$ \begin{proof} $(1)\Rightarrow (2)$ Obviously, we have $$\begin{array}{c} M=\left( \begin{array}{cc} F^{\pi}E&F\\ F&0 \end{array} \right)\left( \begin{array}{cc} I&I\\ F^{\#}E&0 \end{array} \right),\\ M^2=\left( \begin{array}{cc} EF^{\pi}E+F^2&EF\\ 0&F^2 \end{array} \right)\left( \begin{array}{cc} I&I\\ F^{\#}E&0 \end{array} \right). \end{array}$$ Write $M^{\#}=\left( \begin{array}{cc} X_{11}&X_{12}\\ X_{21}&X_{22} \end{array} \right)$. Then $M^{\#}M^2=M$, and so $$\left( \begin{array}{cc} X_{11}&X_{12}\\ X_{21}&X_{22} \end{array} \right)\left( \begin{array}{cc} EF^{\pi}E+F^2&EF\\ 0&F^2 \end{array} \right)=\left( \begin{array}{cc} F^{\pi}E&F\\ F&0 \end{array} \right).$$ Therefore $$X_{11}EF^{\pi}E+X_{11}F^2=F^{\pi}E,$$ hence, $$X_{11}EF^{\pi}EF^{\pi}+X_{11}F^2F^{\pi}=F^{\pi}EF^{\pi}.$$ It follows that $$X_{11}(EF^{\pi})^2=F^{\pi}EF^{\pi}=EF^{\pi}.$$ Hence $EF^{\pi}$ has group inverse. Then $(EF^{\pi})^{\#}=E^DF^{\pi}.$ Therefore $$\begin{array}{lll} EF^{\pi}&=&(EF^{\pi})^{\#}EF^{\pi}EF^{\pi}\\ &=&E^DF^{\pi}EF^{\pi}EF^{\pi}\\ &=&EE^DF^{\pi}; \end{array}$$ hence, $EE^{\pi}F^{\pi}=0$.
$(2)\Rightarrow (1)$ Since $EE^{\pi}F^{\pi}=0$, we have $E^DF^{\pi}(EF^{\pi})^2=EF^{\pi}$, and so $EF^{\pi}$ has group inverse.
Let $N=\left( \begin{array}{cc} E&I\\ F^2&0 \end{array} \right)$. Choose $e=\left( \begin{array}{cc} FF^{\#}&0\\ 0&I \end{array} \right).$ Then $$a=\left( \begin{array}{cc} FF^{\#}E&FF^{\#}\\ F^2&0 \end{array} \right),c=\left( \begin{array}{cc} F^{\pi}EFF^{\#}&F^{\pi}\\ 0&0 \end{array} \right),d=\left( \begin{array}{cc} EF^{\pi}&0\\ 0&0 \end{array} \right)$$ and $b=0$. Then $$N=\left( \begin{array}{cc} a&0\\ c&d \end{array} \right)_e.$$ Moreover, we have $$a^{\#}=\left( \begin{array}{cc} 0&(F^{\#})^2\\ FF^{\#}&-FF^{\#}E(F^{\#})^2 \end{array} \right), d^{\#}=\left( \begin{array}{cc} E^DF^{\pi}&0\\ 0&0 \end{array} \right).$$ We compute that $$a^{\pi}=\left( \begin{array}{cc} 0&0\\ 0&F^{\pi} \end{array} \right), d^{\pi}=\left( \begin{array}{cc} E^{\pi}F^{\pi}&0\\ 0&I \end{array} \right).$$ Obviously, $a^{\pi}cd^{\pi}=0$. So $N$ has group inverse. Moreover, we have $$N^{\#}=\left( \begin{array}{cc} a^{\#}&0\\ z&d^{\#} \end{array} \right),$$ where $$z=d^{\pi}c(a^{\#})^2+(d^{\#})^2ca^{\pi}-d^{\#}ca^{\#}.$$ Clearly, $$\begin{array}{rll} d^{\pi}c(a^{\#})^2&=&\left( \begin{array}{cc} E^{\pi}F^{\pi}E(F^{\#})^2&-E^{\pi}F^{\pi}E(F^{\#})^2E(F^{\#})^2\\ 0&0 \end{array} \right),\\ (d^{\#})^2ca^{\pi}&=&\left( \begin{array}{cc} 0&E^DF^{\pi}E^DF^{\pi}\\ 0&0 \end{array} \right),\\ d^{\#}ca^{\#}&=&\left( \begin{array}{cc} 0&E^DF^{\pi}E(F^{\#})^2\\ 0&0 \end{array} \right). \end{array}$$ Hence we compute that $z=(z_{ij})$, where $$\begin{array}{lll} z_{11}&=&E^{\pi}F^{\pi}E(F^{\#})^2,\\ z_{12}&=&E^DF^{\pi}E^DF^{\pi}-E^{\pi}F^{\pi}E(F^{\#})^2E(F^{\#})^2-E^DF^{\pi}E(F^{\#})^2,\\ z_{21}&=&0, z_{22}=0. \end{array}$$ Therefore $$N^{\#}=\left( \begin{array}{cc} \alpha&\beta\\ \gamma&\delta \end{array} \right),$$ where $$\begin{array}{lll} \alpha&=&E^DF^{\pi}+E^{\pi}F^{\pi}E(F^{\#})^2,\\ \beta&=&(F^{\#})^2+E^DF^{\pi}E^DF^{\pi}-E^{\pi}F^{\pi}E(F^{\#})^2E(F^{\#})^2-E^DF^{\pi}E(F^{\#})^2,\\ \gamma&=&FF^{\#},\\ \delta&=&-FF^{\#}E(F^{\#})^2. \end{array}$$ Hence, we have $$\begin{array}{ll} &NN^{\#}=N^{\#}N\\ =&\left( \begin{array}{cc} E&I\\ F^2&0 \end{array} \right)\left( \begin{array}{cc} \alpha&\beta\\ \gamma&\delta \end{array} \right)\\ =&\left( \begin{array}{cc} \alpha&\beta\\ \gamma&\delta \end{array} \right)\left( \begin{array}{cc} E&I\\ F^2&0 \end{array} \right)\\ =&\left( \begin{array}{cc} EE^DF^{\pi}+FF^{\#}&\alpha\\ F^2\alpha&FF^{\#} \end{array} \right). \end{array}$$ Thus we have $$N^{\pi}=\left( \begin{array}{cc} E^{\pi}F^{\pi}&-\alpha\\ -F^2\alpha&F^{\pi} \end{array} \right).$$ Clearly, we check that $$\begin{array}{c} M=\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)\left( \begin{array}{cc} I&0\\ 0&F \end{array} \right),\\ N=\left( \begin{array}{cc} I&0\\ 0&F \end{array} \right)\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right). \end{array}$$ By virtue of Cline's formula, $M$ has Drazin inverse.
We see that $$\begin{array}{ll} &\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)N^{\pi}\left( \begin{array}{cc} I&0\\ 0&F \end{array} \right)\\ =&\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)\left( \begin{array}{cc} E^{\pi}F^{\pi}&-\alpha\\ -F^2\alpha&F^{\pi} \end{array} \right)\left( \begin{array}{cc} I&0\\ 0&F \end{array} \right)\\ =&\left( \begin{array}{cc} -F^2\alpha&-E\alpha F\\ FE^{\pi}F^{\pi}&-F\alpha F \end{array} \right). \end{array}$$ We compute that $$\begin{array}{rll} F^2\alpha&=&F^2E^DF^{\pi}+F^2E^{\pi}F^{\pi}E(F^{\#})^2\\ &=&F^2EF^{\pi}(E^DF^{\pi})^2+F^2(EF^{\pi})(E^DF^{\pi})E(F^{\#})^2=0,\\ E\alpha F&=&EE^{\pi}F^{\pi}EF^{\#}=0,\\ FE^{\pi}F^{\pi}&=&FE^DEF^{\pi}=F(E^DF^{\pi})(EF^{\pi})=F(EF^{\pi})(E^DF^{\pi})=0,\\ F\alpha F&=&FE^{\pi}F^{\pi}EF^{\#}=F(E^DF^{\pi})(EF^{\pi})EF^{\#}\\ &=&F(EF^{\pi})(E^DF^{\pi})EF^{\#}=0. \end{array}$$ Hence $M=MM^DM$, i.e., $M$ has group inverse. Thus we have $M^{\#}=M^D$.
Moreover, we have $$\begin{array}{lll} M^{D}&=&\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)(N^{\#})^2\left( \begin{array}{cc} I&0\\ 0&F \end{array} \right)\\ &=&\left( \begin{array}{cc} E&I\\ F&0 \end{array} \right)\left( \begin{array}{cc} \alpha&\beta\\ \gamma&\delta \end{array} \right)^2\left( \begin{array}{cc} I&0\\ 0&F \end{array} \right)\\ &=&\left(
\begin{array}{cc}
\Gamma&\Delta\\
\Lambda&\Xi\\
\end{array} \right), \end{array}$$ where $$\begin{array}{rll} \Gamma&=&(E\alpha+\gamma)\alpha+(E\beta+\delta)\gamma,\\ \Delta&=&(E\alpha+\gamma)\beta F+(E\beta+\delta)\delta F,\\ \Lambda&=&F(\alpha^2+\beta\gamma),\\ \Xi&=&F(\alpha\beta +\beta\delta)F. \end{array}$$ Therefore we complete the proof by the direct computation.\end{proof}
\begin{cor} Let $M=\left( \begin{array}{cc} E&F\\ F&0 \end{array} \right)$ and $E,EF^{\pi}$ have Drazin inverse and $F$ has group inverse. If $F^{\pi}EF=0$, then the following are equivalent:\end{cor} \begin{enumerate} \item [(1)]{\it $M$ has group inverse.}
\item [(2)]{\it $F^{\pi}E^{\pi}E=0$.} \end{enumerate} In this case, $$M^{\#}=\left(
\begin{array}{cc}
\Gamma&\Delta\\
\Lambda&\Xi\\
\end{array} \right),$$ where $$\begin{array}{rll} \Gamma&=&[F^{\pi}E^D+(F^{\#})^2EF^{\pi}E^{\pi}][I-F^{\pi}E^{\pi}]+(F^{\#})^2EF^{\pi}E^{\pi},\\ \Delta&=&[F^{\#}-F^{\#}E(F^{\#})^2EF^{\pi}E^{\pi}-F^{\#}EF^{\pi}E^D][I-F^{\pi}E^{\pi}]\\ &-&F^{\#}E(F^{\#})^2EF^{\pi}E^{\pi},\\ \Lambda&=&[F^{\pi}E^D+(F^{\#})^2EF^{\pi}E^{\pi}]^2F+F^{\#}-[(F^{\#})^2E]^2F^{\pi}E^{\pi}F\\ &-&(F^{\#})^2EF^{\pi}E^DF,\\ \Xi&=&[F^{\#}-F^{\#}E(F^{\#})^2EF^{\pi}E^{\pi}-F^{\#}EF^{\pi}E^D]\\ &&[F^{\pi}E^DF+(F^{\#})^2EF^{\pi}E^{\pi}F]-F^{\#}E[F^{\#}\\ &-&(F^{\#})^2E(F^{\#})^2EF^{\pi}E^{\pi}F-(F^{\#})^2EF^{\pi}E^DF]. \end{array}$$ \begin{proof} By virtue of Cline's formula, $F^{\pi}E$ has Drazin inverse. Then the proof is complete by applying Theorem 3.1 to the transpose $M^T=\left( \begin{array}{cc} E^T&F^T\\ F^T&0 \end{array} \right).$\end{proof}
\begin{cor} Let $M=\left( \begin{array}{cc} E&F\\ F&0 \end{array} \right)$ and $E,F$ have group inverse, $EF^{\pi}$ has Drazin inverse. If $F^{\pi}EF=0$, then $M$ has group inverse. In this case, $$M^{\#}=\left(
\begin{array}{cc}
\Gamma&\Delta\\
\Lambda&\Xi\\
\end{array} \right),$$ where $$\begin{array}{rll} \Gamma&=&[I-E^{\pi}F^{\pi}][E^{\#}F^{\pi}+E^{\pi}F^{\pi}E(F^{\#})^2]+E^{\pi}F^{\pi}E(F^{\#})^2,\\ \Delta&=&[I-E^{\pi}F^{\pi}][F^{\#}-E^{\pi}F^{\pi}E(F^{\#})^2EF^{\#}-E^{\#}F^{\pi}EF^{\#}]\\ &-&E^{\pi}F^{\pi}E(F^{\#})^2EF^{\#},\\ \Lambda&=&F[E^{\#}F^{\pi}+E^{\pi}F^{\pi}E(F^{\#})^2]^2+F^{\#}-FE^{\pi}F^{\pi}[E(F^{\#})^2]^2\\ &-&FE^{\#}F^{\pi}E(F^{\#})^2,\\ \Xi&=&[FE^{\#}F^{\pi}+FE^{\pi}F^{\pi}E(F^{\#})^2][F^{\#}-E^{\pi}F^{\pi}E(F^{\#})^2EF^{\#}\\ &-&E^{\#}F^{\pi}EF^{\#}]-[F^{\#}-FE^{\pi}F^{\pi}E(F^{\#})^2E(F^{\#})^2\\ &-&FE^{\#}F^{\pi}E(F^{\#})^2]EF^{\#}. \end{array}$$ \end{cor} \begin{proof} Since $E$ has group inverse, we see that $EE^{\pi}=0$, and so $EE^{\pi}F^{\pi}=0$. In light of Theorem 3.1, $M$ has group inverse. Therefore we obtain the representation of $M^{\#}$ by the formula in Theorem 3.1. \end{proof}
As an immediate consequence of Corollary 3.3, we have
\begin{cor} Let $M=\left( \begin{array}{cc} E&F\\ F&0 \end{array} \right)$ and $E,F$ have group inverse, $EF^{\pi}$ has Drazin inverse. If $EF=\lambda FE ~(\lambda \in {\Bbb C})$ or $EF^2=FEF$, then $M$ has group inverse. In this case, $$M^{\#}=\left(
\begin{array}{cc}
\Gamma&\Delta\\
\Lambda&\Xi\\
\end{array} \right),$$ where $$\begin{array}{rll} \Gamma&=&[I-E^{\pi}F^{\pi}][E^{\#}F^{\pi}+E^{\pi}F^{\pi}E(F^{\#})^2]+E^{\pi}F^{\pi}E(F^{\#})^2,\\ \Delta&=&[I-E^{\pi}F^{\pi}][F^{\#}-E^{\pi}F^{\pi}E(F^{\#})^2EF^{\#}-E^{\#}F^{\pi}EF^{\#}]\\ &-&E^{\pi}F^{\pi}E(F^{\#})^2EF^{\#},\\ \Lambda&=&F[E^{\#}F^{\pi}+E^{\pi}F^{\pi}E(F^{\#})^2]^2+F^{\#}-FE^{\pi}F^{\pi}[E(F^{\#})^2]^2\\ &-&FE^{\#}F^{\pi}E(F^{\#})^2,\\ \Xi&=&[FE^{\#}F^{\pi}+FE^{\pi}F^{\pi}E(F^{\#})^2][F^{\#}-E^{\pi}F^{\pi}E(F^{\#})^2EF^{\#}\\ &-&E^{\#}F^{\pi}EF^{\#}]-[F^{\#}-FE^{\pi}F^{\pi}E(F^{\#})^2E(F^{\#})^2\\ &-&FE^{\#}F^{\pi}E(F^{\#})^2]EF^{\#}. \end{array}$$ \end{cor} \begin{proof} As in proof of Corollary 2.5, we obtain the result by Corollary 3.3.\end{proof}
\begin{exam} Let $M=\left( \begin{array}{cc} E&F\\ F&0 \end{array} \right)$, where $E=\left( \begin{array}{cc} 1& 2\\ 0&-1 \end{array} \right)$ and $F=\left( \begin{array}{cc} i& i\\ 0&0 \end{array} \right), i^2=-1$. Then $$M^{\#}=\left( \begin{array}{cccc} 0&1&-i&-i\\ 0&-1&0&0\\ -i&-i&1&1\\ 0&0&0&0 \end{array} \right).$$\end{exam} \begin{proof} We see that $$\begin{array}{c} E^{\#}=\left( \begin{array}{cc} 1&2\\ 0&-1 \end{array} \right), E^{\pi}=0;\\ F^{\#}=\left( \begin{array}{cc} -i& -i\\ 0&0 \end{array} \right), F^{\pi}=\left( \begin{array}{cc} 0&-1\\ 0&1 \end{array} \right). \end{array}$$ Hence we check that $FEF^{\pi}=0, EE^{\pi}F^{\pi}=0.$ Construct $\Gamma, \Delta, \Lambda$ and $\Xi$ as in Theorem 3.1. Then we compute that $$\begin{array}{c} \Gamma=\left( \begin{array}{cc} 0&1\\ 0&-1 \end{array} \right), \Delta=\left( \begin{array}{cc} -i&-i\\ 0&0 \end{array} \right),\\ \Lambda=\left( \begin{array}{cc} -i&-i\\ 0&0 \end{array} \right), \Xi=\left( \begin{array}{cc} 1&1\\ 0&0 \end{array} \right). \end{array}$$ \\ This completes the proof by Theorem 3.1.\end{proof}
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\end{document} | arXiv |
A linear-time algorithm that avoids inverses and computes Jackknife (leave-one-out) products like convolutions or other operators in commutative semigroups
John L. Spouge ORCID: orcid.org/0000-0001-6207-14191,
Joseph M. Ziegelbauer2 &
Mileidy Gonzalez3
Algorithms for Molecular Biology volume 15, Article number: 17 (2020) Cite this article
Data about herpesvirus microRNA motifs on human circular RNAs suggested the following statistical question. Consider independent random counts, not necessarily identically distributed. Conditioned on the sum, decide whether one of the counts is unusually large. Exact computation of the p-value leads to a specific algorithmic problem. Given \(n\) elements \(g_{0} ,g_{1} , \ldots ,g_{n - 1}\) in a set \(G\) with the closure and associative properties and a commutative product without inverses, compute the jackknife (leave-one-out) products \(\bar{g}_{j} = g_{0} g_{1} \cdots g_{j - 1} g_{j + 1} \cdots g_{n - 1}\) (\(0 \le j < n\)).
This article gives a linear-time Jackknife Product algorithm. Its upward phase constructs a standard segment tree for computing segment products like \(g_{{\left[ {i,j} \right)}} = g_{i} g_{i + 1} \cdots g_{j - 1}\); its novel downward phase mirrors the upward phase while exploiting the symmetry of \(g_{j}\) and its complement \(\bar{g}_{j}\). The algorithm requires storage for \(2n\) elements of \(G\) and only about \(3n\) products. In contrast, the standard segment tree algorithms require about \(n\) products for construction and \(\log_{2} n\) products for calculating each \(\bar{g}_{j}\), i.e., about \(n\log_{2} n\) products in total; and a naïve quadratic algorithm using \(n - 2\) element-by-element products to compute each \(\bar{g}_{j}\) requires \(n\left( {n - 2} \right)\) products.
In the herpesvirus application, the Jackknife Product algorithm required 15 min; standard segment tree algorithms would have taken an estimated 3 h; and the quadratic algorithm, an estimated 1 month. The Jackknife Product algorithm has many possible uses in bioinformatics and statistics.
A biological question
Circular RNAs (circRNAs) are single-stranded noncoding RNAs that can inhibit another RNA molecule by binding to it, mopping it up like a sponge. During herpesvirus infection, human hosts produce circRNAs with target sites that may bind herpesvirus microRNA (miRNA) [1] (see Fig. 1). Given a sequence motif, e.g., a target site for a miRNA, researchers counted how many times the motif occurs in each circRNA sequence. They then posed a question: is the motif unusually enriched in any of the circRNAs, i.e., does any circRNA have too many occurrences of the motif to be explained by chance alone? If "yes", the researchers could then focus their further experimental efforts on those circRNAs.
A schematic diagram of herpesvirus miRNA motif occurring on a human circRNA. As indicated in the legend, each thin circle represents a circRNA; each thick line segment, the occurrence of a miRNA motif on the corresponding circRNA. Both circRNAs and the miRNA motif have nucleotide sequences represented by IUPAC codes (A, C, G, U). This figure illustrates occurrences of a single miRNA motif (e.g., UUACAGG) on the circRNAs. The biological question is: "does any circRNA have too many occurrences of the motif to be explained by chance alone?" In the actual application, the \(n = 3086\) circRNAs ranged in length from 69 nt to 158565 nt
A statistical answer
Figure 1 illustrates a set of circRNAs with varying length, with a single miRNA motif occurring as indicated on each circRNA. Let \(i = 0,1 \ldots ,n - 1\) index the circRNAs; the random variate \(X_{i}\) count the motif occurrences in the \(i\)-th circRNA; \(k\left( i \right)\) equal the observed count for \(X_{i}\); and the sum \(S = \sum\nolimits_{i = 0}^{n - 1} {X_{i} }\) count the total motif occurrences among the circRNAs, with observed total \(K = \sum\nolimits_{i = 0}^{n - 1} {k\left( i \right)}\).
The following set-up provides a general statistical test for deciding the biological question. Let \(\left\{ {X_{i} :i = 0,1, \ldots ,n - 1} \right\}\) represent independent random counts (i.e., non-negative integer random variates), not necessarily identically distributed, with sum \(S = \sum\nolimits_{i = 0}^{n - 1} {X_{i} }\). Given observed values \(\left\{ {X_{i} = k\left( i \right):i = 0,1, \ldots ,n - 1} \right\}\) with observed sum \(K = \sum\nolimits_{i = 0}^{n - 1} {k\left( i \right)}\), consider the computation of the conditional p-values \({\mathbb{P}}\left\{ {X_{i} \ge k\left( i \right)\left| {S = K} \right.} \right\}\) (\(i = 0,1, \ldots ,n - 1\)). The conditional p-values can decide the question: "Is any term in the sum unusually large relative to the others?"
The abstract question in the previous paragraph generalizes some common tests. For example, the standard 2 × 2 Fisher exact test [2, p. 96] answers the question in the special case of \(n = 2\) categories: each \(X_{i}\) has a binomial distribution with common success probability \(p\), conditional on known numbers of trials \(N_{i}\) (\(i = 0,1\)). Although the Fisher exact test generalizes directly to a single exact p-value for a \(2 \times n\) table [3], the generalization can require prohibitive amounts of computation. The abstract question corresponds to a computationally cheaper alternative that also decides which columns in the \(2 \times n\) table are unusual [4].
To derive an expression for the conditional p-value, therefore, let \(g_{i} \left[ k \right] = {\mathbb{P}}\left\{ {X_{i} = k} \right\}\) be given, so the array \(g_{i} = \left( {g_{i} \left[ 0 \right],g_{i} \left[ 1 \right], \ldots ,g_{i} \left[ K \right]} \right)\) gives the distribution of \(X_{i}\), truncated at the observed total \(K = \sum\nolimits_{i = 0}^{n - 1} {k\left( i \right)}\). Because \(g_{i}\) is a truncated probability distribution, \(g_{i} \in G\), the set of all real (\(K + 1\))-tuples \(\left( {g\left[ 0 \right],g\left[ 1 \right], \ldots ,g\left[ K \right]} \right)\) satisfies \(g\left[ k \right] \ge 0\) (\(k = 0,1, \ldots ,K\)) and \(\sum\nolimits_{k = 0}^{K} {g\left[ k \right]} \le 1\). The truncation still permits exact calculation of the probabilities below. To calculate the distribution of the sum \(S = \sum\nolimits_{i = 0}^{n - 1} {X_{i} }\) for \(S \le K\), define the truncated convolution operation \(g = g^{\prime} \circ g^{\prime\prime}\), for which \(g\left[ k \right] = \sum\nolimits_{j = 0}^{k} {g^{\prime}\left[ j \right]g^{\prime\prime}\left[ {k - j} \right]}\) (\(k = 0,1, \ldots ,K\)). Hereafter, the operation "\(\circ\)" is often left implicit: \(g^{\prime} \circ g^{\prime\prime} = g^{\prime}g^{\prime\prime}\).
Let \(\bar{g} = g_{0} g_{1} \cdots g_{n - 1}\), so \(\bar{g}\left[ k \right] = {\mathbb{P}}\left\{ {S = k} \right\}\) (\(k = 0,1, \ldots ,K\)). Define the "jackknife products" \(\bar{g}_{j} = g_{0} g_{1} \cdots g_{j - 1} g_{j + 1} \cdots g_{n - 1}\) (\(0 \le j < n\)) (implicitly including the products \(\bar{g}_{0} = g_{1} g_{2} \cdots g_{n - 1}\) and \(\bar{g}_{n - 1} = g_{0} g_{1} \cdots g_{n - 2}\)). The jackknife products contain the same products as \(\bar{g}\), except that in turn each skips over \(g_{j}\) (\(0 \le j < n\)). Like the jackknife procedure in statistics, therefore, jackknife products successively omit each datum in a dataset [5].
With the jackknife products in hand, the conditional p-values are a straightforward computation:
$${\mathbb{P}}\left\{ {X_{i} \ge k\left( i \right)\left| {S = K} \right.} \right\} = \frac{{\sum\limits_{k = k\left( i \right)}^{K} {g_{i} \left[ k \right]\bar{g}_{i} \left[ {K - k} \right]} }}{{\bar{g}\left[ K \right]}}.$$
With respect to Eq. (1) and the biological question in Fig. 1, Appendix B gives the count \(\bar{g}\left[ K \right]\) of the ways that \(n\) circRNAs of known but varying length may contain \(K\) miRNA motifs of equal length, the count \(g_{i} \left[ k \right]\) of the ways that the \(i\) th circRNA may contain \(k\) motifs, and the count \(\bar{g}_{i} \left[ {K - k} \right]\) of the ways that all circRNAs but the \(i\) th may contain \(K - k\) motifs. Appendix B derives the count \(g_{i} \left[ k \right]\) for circRNAs from the easier count for placing motifs on a linear RNA molecule. For combinatorial probabilities like \({\mathbb{P}}\left\{ {X_{i} \ge k\left( i \right)\left| {S = K} \right.} \right\}\), Eq. (1) remains relevant, even if \(\left\{ {g_{i} \left[ k \right]} \right\}\) are counts instead of probabilities. The biological question therefore exemplifies a commonplace computational need in applied combinatorial probability.
The Discussion indicates that in our application, transform methods can encounter substantial obstacles when computing Eq. (1) (e.g., see [6]), because the quantities in Eq. (1) can range over many orders of magnitude. This article therefore pursues direct exact calculation of \({\mathbb{P}}\left\{ {X_{i} \ge k\left( i \right)\left| {S = K} \right.} \right\}\). The product forms of \(\bar{g}\) and \(\left\{ {\bar{g}_{j} } \right\}\) suggest that any efficient algorithm may be abstracted to broaden its applications, as follows.
Semigroups, groups, and commutative groups
Let \(\left( {G, \circ } \right)\) denote a set \(G\) with a binary product \(g \circ g^{\prime}\) on its elements. Let "\(g \in G\)" denote "\(g\) is an element of \(G\)", and consider the following properties [7].
Closure \(g \circ g^{\prime} \in G\) for every \(g,g^{\prime} \in G\)
Associative \(\left( {g \circ g^{\prime}} \right) \circ g^{\prime\prime} = g \circ \left( {g^{\prime} \circ g^{\prime\prime}} \right)\) for every \(g,g^{\prime},g^{\prime\prime} \in G\)
Identity There exists an identity element \(e \in G\), such that \(e \circ g = g \circ e = g\) for every \(g \in G\)
Commutative \(g \circ g^{\prime} = g^{\prime} \circ g\) for every \(g,g^{\prime} \in G\)
If the Closure and Associative properties hold, \(\left( {G, \circ } \right)\) is a semigroup. Without loss of generality, we assume below that the Identity property holds. If not, adjoin an element \(e \in G\), such that \(e \circ g = g \circ e = g\) for every \(g \in G\). In addition, if the Commutative property holds for every \(g,g^{\prime} \in G\), the semigroup \(\left( {G, \circ } \right)\) is commutative. Unless stated otherwise hereafter, \(\left( {G, \circ } \right)\) denotes a commutative semigroup. The Jackknife Product algorithm central to this article is correct in a commutative semigroup.
Inverse For every \(g \in G\), there exists an inverse \(g^{ - 1} \in G\), such that \(g \circ g^{ - 1} = g^{ - 1} \circ g = e\)
As shown later, the Jackknife Product algorithm does not require the Inverse property. In passing, note that the convolution semigroup relevant to the circRNA–miRNA application lacks the Inverse property, as does any convolution semigroup for calculating p-values, e.g., the ones relevant to sequence motif matching [6]. To demonstrate, let \(X,Y \ge 0\) be independent integer random variates. The identity \(e\) for convolution corresponds to the variate \(Z = 0\), because \(0 + X = X + 0 = X\) for every variate \(X\). If \(X + Y = 0\), however, the independence of \(X\) and \(Y\) implies that both are constant and therefore \(X = Y = 0\). In the relevant convolution semigroup, therefore, all elements except the identity \(e\) lack an inverse.
The non-zero real numbers under ordinary multiplication form a commutative semigroup \(\left( {G, \circ } \right)\) with the Inverse property. They provide a familiar setting for discussing some algorithmic issues when computing \(\left\{ {\bar{g}_{j} } \right\}\). Let \(\bar{g} = g_{0} g_{1} \cdots g_{n - 1}\) be the usual product of \(n\) real numbers, and consider the toy problem of computing all jackknife products \(\left\{ {\bar{g}_{j} } \right\}\) that omit a single factor \(g_{j}\) (\(0 \le j < n\)) from \(\bar{g}\). Inverses \(\left\{ {g_{j}^{ - 1} } \right\}\) are available, so an obvious algorithm computes \(\bar{g}\) and then \(\left\{ {\bar{g}_{j} = \bar{g}g_{j}^{ - 1} } \right\}\) with \(n\) inverses and \(2n - 1 = \left( {n - 1} \right) + n\) products. If the inverses were unavailable, however, the naïve algorithm using \(n - 2\) element-by-element products to compute each \(\left\{ {\bar{g}_{j} } \right\}\) would require \(n\left( {n - 2} \right)\) products. The quadratic time renders the naïve algorithm impractical for many applications.
Figure 1 illustrates a standard data structure called a segment tree, omitting the root at the top of the segment tree. Algorithms based solely on a segment tree can calculate the jackknife products \(\left\{ {\bar{g}_{j} } \right\}\) in time \(O\left( {n\log n} \right)\), fast enough for many applications. The segment tree computes segment products like \(g_{{\left[ {i,j} \right)}} = g_{i} g_{i + 1} \cdots g_{j - 1}\) without using the commutative property, so it can similarly compute jackknife products like \(\left\{ {\bar{g}_{j} = g_{{\left[ {0,j} \right)}} g_{{\left[ {j + 1,n} \right)}} } \right\}\). If the semigroup \(\left( {G, \circ } \right)\) is commutative, however, a Jackknife Product algorithm can avoid inverses and reduce the computational time further, from \(O\left( {n\log n} \right)\) to \(O\left( n \right)\). With in-place computations requiring only the space for the segment tree, the Jackknife Product algorithm avoids inverses yet still requires only about \(3n\) products and storage for \(2n\) numbers. It is therefore surprisingly economical, even when compared to the obvious algorithm using inverses. Indeed, our application to circular RNA required some economy, with its convolution of \(n = 3086\) distributions, some truncated only after \(K = 997\) terms. In a general statistical setting, convolutions form a commutative semigroup \(\left( {G, \circ } \right)\) without inverses, so our application already indicates that the Jackknife Product algorithm has broad applicability.
Appendix A proves the correctness of the Jackknife Product algorithm given below.
The Jackknife Product algorithm
Let \(\left( {G, \circ } \right)\) be a commutative semigroup. The Jackknife Product algorithm has three phases: upward, downward, and transposition. Its upward phase simply constructs a segment tree (see Fig. 1); its downward phase exploits the symmetry of \(g_{j}\) and its complement \(\bar{g}_{j}\) to mirror the upward phase while computing \(\left\{ {\bar{g}_{j} } \right\}\) (see Fig. 2); and its final transposition phase then swaps successive pairs in an array (not pictured). As Figs. 1 and 2 suggest, the three phases yield a simpler algorithm if \(n = n^{ * } = 2^{m}\) is a binary power. To recover the \(n^{ * } = 2^{m}\) algorithm from them, pad \(\left\{ {g_{j} } \right\}\) on the right with copies of the identity \(e\) up to \(n^{ * }\) elements, where \(n^{ * } = 2^{m}\) is the smallest binary power greater than or equal to \(n\), i.e., replace \(\left\{ {g_{j} } \right\}\) with \(\left\{ {g_{0} ,g_{1} \ldots ,g_{n - 1} ,e, \ldots ,e} \right\}\), with \(n^{ * } - n\) copies of \(e\). The \(n^{ * } = 2^{m}\) algorithm can therefore pad any input of \(n\) elements up to \(n^{ * } = 2^{m}\) elements without loss of generality. The algorithm given below is therefore slightly more intricate than the \(n^{ * } = 2^{m}\) algorithm, but it may save almost a factor of 2 in storage and time by omitting the padded copies of \(e\). In any case, the simpler algorithm can always be recovered from the phases for general \(n\) given here, if desired.
A (rootless) segment tree. This figure illustrates the rootless segment tree constructed in the upward phase of the Jackknife Product algorithm. The commutative semigroup \(\left( {G, \circ } \right)\) illustrated is the set of nonnegative integers under addition. The bottom row of \(n^{*} = 2^{m}\) squares (\(m = 3\)) contains \(L_{0} \left[ j \right] = g_{j}\) (\(0 \le j < n^{*}\)). In the next row up, as indicated by the arrow pairs leading into each circle, the array \(L_{1}\) contains consecutive sums of consecutive disjoint pairs in \(L_{0}\), e.g., \(L_{1} \left[ 0 \right] = 13 = 5 + 8\). The rest of the segment tree is constructed recursively upward to \(L_{m - 1}\), just as \(L_{1}\) was constructed from \(L_{0}\). Here, 2 copies of the additive identity \(e = 0\) pad out \(L_{0}\) on the right. Padded on the right, the copies contribute literally nothing to the segment tree above them. Their non-contributions have dotted outlines
A (rootless) complementary segment tree. This figure illustrates the rootless complementary segment tree constructed in the downward phase of the Jackknife Product algorithm from the rootless segment tree in Fig. 2. The downward phase starts by initializing the topmost row \(\bar{L}_{m - 1}\) (\(m = 3\)) with the topmost row \(L_{m - 1}\) of the rootless segment tree. The row \(L_{2}\) in Fig. 2 and the row \(\bar{L}_{2}\) in Fig. 3, e.g., contain 22 and 11. For each \(\bar{L}_{k - 1} \left[ j \right]\) in Fig. 3, downward arrows run from \(\bar{L}_{k} \left[ {\alpha_{k} \left( j \right)} \right]\) to \(\bar{L}_{k - 1} \left[ j \right]\). As they indicate, each node in \(\bar{L}_{k}\) contributes to its 2 "nieces" in Fig. 2 to produce the next row down in Fig. 3, e.g., \(\bar{L}_{2} \left[ 1 \right] = 11\) contributes to its nieces \(L_{1} \left[ 0 \right] = 13\) and \(L_{1} \left[ 1 \right] = 9\) in the segment, to produce \(\bar{L}_{1} \left[ 0 \right] = 13 + 11 = 24\) and \(\bar{L}_{1} \left[ 1 \right] = 9 + 11 = 20\) in the complementary segment tree. The rest of the complementary segment tree is constructed recursively downward to \(\bar{L}_{0}\), just as \(\bar{L}_{1}\) was constructed from \(\bar{L}_{2}\). In Fig. 2, the elements of \(L_{0}\) (in squares) total 33. To demonstrate the effect of the Jackknife Product algorithm, subtract in turn in Fig. 3 each element (25, 28, 30, 27, 26, 29, 33, 33) in the bottom row \(\bar{L}_{0}\) from the total 33. The result (8, 5, 3, 6, 7, 4, 0, 0) is the bottom row \(L_{0}\) in Fig. 2 with successive pairs transposed, so \(\bar{L}_{0} \left[ j \right] = \bar{g}_{\tau \left( j \right)}\), or equivalently \(\bar{g}_{j} = \bar{L}_{0} \left[ {\tau \left( j \right)} \right]\)
We start with notational preliminaries. Define the floor function \(\left\lfloor x \right\rfloor = \hbox{max} \left\{ {j:j \le x} \right\}\) and the ceiling function \(\left\lceil x \right\rceil = \hbox{min} \left\{ {j:x \le j} \right\}\) (both standard); and the binary right-shift function \(\rho \left( j \right) = \left\lfloor {j/2} \right\rfloor\). Other quantities also smooth our presentation. Given a product \(\bar{g} = g_{0} g_{1} \cdots g_{n - 1}\) of interest, define \(m = \left\lceil {\log_{2} n} \right\rceil\) and \(n_{k} = \left\lceil {n2^{ - k} } \right\rceil\) for \(0 \le k < m\). Below, the symbol "□" connotes the end of a proof.
The upward phase
The upward phase starts with the initial array \(L_{0} \left[ j \right] = g_{j}\) (\(0 \le j < n\)) and simply computes a standard (but rootless) segment tree consisting of segment products \(L_{k} \left[ j \right]\) for \(j = 0,1, \ldots ,n_{k} - 1\) and \(k = 0,1, \ldots ,m - 1\).
(1) If \(n = n^{ * } = 2^{m}\) is a binary power, \(\rho \left( {n_{k - 1} } \right) = n_{k} = 2^{m - k}\) and the final line in the upward phase can be omitted. (2) Of some peripheral interest, Laaksonen [8] gives the algorithm in a different context, embedding a binary tree in a single array of length \(O\left( n \right)\). If any \(L_{0} \left[ j \right] = g_{j}\) changes, he also shows how to update the single array with \(O\left( {\log n} \right)\) multiplications. If the downward phase (next) does not overwrite the segment tree \(\left\{ {L_{k} } \right\}\) by using in-place computation, it permits a similar update.
The downward phase
The transposition function \(\tau \left( j \right) = j + \left( { - 1} \right)^{j}\) transposes adjacent indices, e.g., \(\left( {L_{\tau \left( 0 \right)} ,L_{\tau \left( 1 \right)} ,L_{\tau \left( 2 \right)} ,L_{\tau \left( 3 \right)} } \right) = \left( {L_{1} ,L_{0} ,L_{3} ,L_{2} } \right)\). We also require \(\alpha_{k} \left( j \right) = \hbox{min} \left\{ {\tau \rho \left( j \right),n_{k} - 1} \right\}\) for \(0 \le j < n_{k - 1}\) and \(1 \le k < m\), the index of "aunts", as illustrated by Fig. 2. Just as Fig. 1 illustrates a rootless segment tree in the upward phase, Fig. 2 illustrates the corresponding rootless complementary segment tree in the downward phase.
The downward phase computes complementary segment products \(\bar{L}_{k} \left[ j \right]\) for \(j = 0,1 \ldots ,n_{k} - 1\) and \(k = m - 1,m - 2, \ldots ,0\).
(1) If \(n = n^{ * } = 2^{m}\) is a binary power, \(\rho \left( {n_{k - 1} } \right) = n_{k} = 2^{m - k}\), \(\alpha_{k} \left( j \right) = \tau \rho \left( j \right)\), and the final line in the downward phase can be omitted. (2) The downward phase can be modified in the obvious fashion to permit in-place calculation of \(\bar{L}_{k - 1} \left[ j \right]\) from \(L_{k - 1} \left[ j \right]\), reducing total memory allocation by about 2.
As Appendix A proves, the final array \(\bar{L}_{0}\) has elements \(\bar{L}_{0} \left[ {\tau \left( j \right)} \right] = \bar{g}_{j}\) (\(0 \le j < 2\rho \left( n \right)\)), with an additional final element \(\bar{L}_{0} \left[ {n - 1} \right] = \bar{g}_{n - 1}\) if \(n_{0} = n\) is odd, so the Jackknife Product algorithm ends with a straightforward transposition phase.
The transposition phase can permit an in-place calculation of \(\left\{ {\bar{g}_{j} } \right\}\) to overwrite \(\bar{L}_{0}\).
Computational time and storage
Note \(n_{j} = \left\lceil {n2^{ - j} } \right\rceil\), so \(0 \le n_{j} - n2^{ - j} < 1\). To compute \(L_{k}\) from \(L_{k - 1}\) or to compute \(\bar{L}_{k}\) from \(L_{k}\) and \(\bar{L}_{k + 1}\), the Jackknife Product algorithm requires \(n_{k}\) products. For large \(n\), therefore, the upward phase computing the segment tree requires about \(\sum\nolimits_{j = 1}^{m - 1} {n_{j} } \approx \sum\nolimits_{j = 1}^{\infty } {n2^{ - j} } = n\) products; the downward phase, about \(\sum\nolimits_{j = 0}^{m - 2} {n_{j} } \approx 2n\) products. Likewise, if the downward and transposition phases compute in place by replacing \(L_{k}\) with \(\bar{L}_{k}\) and \(\bar{L}_{0}\) with \(\left\{ {\bar{g}_{j} } \right\}\), the algorithm storage is \(\sum\nolimits_{j = 0}^{m - 1} {n_{j} } \approx 2n\) semigroup elements. Each of the three estimates just given for products and storage have an error bounded by \(m = \left\lceil {\log_{2} n} \right\rceil\). Although the case of general \(n\) could be handled by the algorithm for binary powers \(n^{ * } = 2^{m}\) by setting \(m = \left\lceil {\log_{2} n} \right\rceil\) and \(g_{n} = g_{n + 1} = \ldots = g_{{n^{ * } - 1}} = e\), the truncated arrays in the Jackknife Product algorithm for general \(n\) save about a factor of \(1 \le n^{ * } /n < 2\) in both products and storage.
As written, the conditional copy statements at the end of the upward and downward phases replicate elements already in storage. If the downward phase of the Jackknife Product algorithm is implemented with in-place computation of \(\bar{L}_{k - 1} \left[ j \right]\) from \(L_{k - 1} \left[ j \right]\), the copy statements ensure that the algorithm never overwrites any array element it needs later. Some statements may copy some elements more than once (and therefore unnecessarily), but a negligible \(m = \left\lceil {\log_{2} n} \right\rceil\) copies at most are unnecessary.
The complementary segment tree in Fig. 2 implicitly indicates the nodes in the segment tree required to compute \(L_{0} \left[ {\tau \left( j \right)} \right] = \bar{g}_{j}\) for each \(\bar{g}_{j}\), i.e., exactly one node in each row \(L_{k}\) (\(k = 0,1, \ldots ,m - 1\)). Alone, the segment tree therefore requires at least \(n\log_{2} n\) multiplications to compute \(\left\{ {\bar{g}_{j} } \right\}\).
Appendix B gives the combinatorics relevant to the circRNA-miRNA application described in "Background" section. As is typical in combinatorial probability, the quantities \(\left\{ {g_{i} \left[ k \right]} \right\}\) were counts of configurations, here, the ways of placing miRNA motifs on circRNAs. The length of each motif was m = 7; the largest circRNA (hsa-circ-0003473) contained \(I\) = 158,565 nt, and the most abundant motif (CCCAGCU, for the m12-9star miRNA family) appeared \(K\) = 997 times, so the \(\left\{ {g_{i} \left[ k \right]} \right\}\) spanned thousands of orders of magnitude in Eq. (13) of Appendix B, from \(g_{i} \left[ 0 \right] = 1\) to \(g_{I} \left[ K \right] \approx 10^{2608}\). In Eq. (1), the dimension \(K\) controls the number of terms in the convolutions. In the application, over each miRNA motif examined, the maximum number of motif occurrences on the circRNAs was \(K = 997\). An Intel Core i7-3770 CPU computed the p-value relevant to the biological application on June 17, 2015. To compare later with estimated times for competing algorithms, the Jackknife Product algorithm with \(n = 3086\) computed the relevant p-values in about 45 min, requiring about \(3n\) products. In the application, therefore, \(n\) products required about 15 min.
The application of this article to circRNA–miRNA data appears elsewhere [1].
This article has presented a Jackknife Product algorithm, which applies to any commutative semi-group \(\left( {G, \circ } \right)\). The biological application to a circRNA–miRNA system exemplifies a general statistical method in combinatorial probability. In turn, the application in combinatorial probability exemplifies an even more general statistical test for whether a term in a sum of independent counting variates (not necessarily identically distributed) is unusually large.
Many biological contexts lead naturally to sums of independent counting variates. Domain alignments of proteins from cancer patients, e.g., display point mutations in their columns. For a given domain, a column with an excess of mutations might be inferred to cause cancer [9]. The Background section gives the pattern: let \(X_{i}\) represent the mutation count in column \(i = 0,1, \ldots ,n - 1\), with total mutations \(S = \sum\nolimits_{i = 0}^{n - 1} {X_{i} }\). Given observed mutation counts \(\left\{ {X_{i} = k\left( i \right):i = 0,1, \ldots ,n - 1} \right\}\) with observed sum \(K = \sum\nolimits_{i = 0}^{n - 1} {k\left( i \right)}\), the conditional p-values \({\mathbb{P}}\left\{ {X_{i} \ge k\left( i \right)\left| {S = K} \right.} \right\}\) (\(i = 0,1, \ldots ,n - 1\)) can decide the question: "Does any column have an excess of mutations?" The actual application used other, very different statistical methods [9]. Unlike those methods, however, our methods can incorporate information from control (non-cancer) protein sequences to set column-specific background distributions for \(\left\{ {X_{i} } \right\}\).
The Benjamini–Hochberg procedure for controlling the false discovery rate in multiple tests requires either independent p-values [10] or dependent p-values with a positive regression dependency property [11]. Loosely, the positive regression dependency property means that the p-values tend to be small together, i.e., under the null hypothesis, given that one p-value is small, then the other p-values tend to be smaller also. Inconveniently, our null hypothesis posits a fixed sum of independent counting variates, so if one variate is large and has a small p-value, it tends to reduce the other variates and increase their p-values. The circRNA-miRNA application therefore violates the statistical hypotheses of the Benjamini–Hochberg procedure. Fortunately, in the circRNA-miRNA application, a Bonferroni multiple test correction [12] sufficed because empirically, any p-value was either close to 1 or extremely small.
The Results state that for \(n = 3086\), the Jackknifed Product algorithm computed the relevant p-values in about 45 min, with \(n\) products requiring about 15 min of computation. In contrast, the naïve algorithm avoiding inverses and requiring \(n\left( {n - 2} \right)\) products would have taken about \(3086*15\) min, i.e., about 1 month. As explained under the "Computational Time and Storage" heading in the Theory section, without exploiting the special form of the jackknife products \(\left\{ {\bar{g}_{j} } \right\}\), a segment tree requires about \(n\) products for its construction and at least \(n\log_{2} n\) products for the computation of the products \(\left\{ {\bar{g}_{j} } \right\}\). Alone, segment tree algorithms would therefore have taken a minimum of about \(\left( {1 + \log_{2} 3086} \right)*15\) min, i.e., about 3 h.
The convolutions in Eq. (1) might suggest that jackknife products are susceptible to computation with Fourier or Laplace transforms, which convert convolutions into products. "Results" section notes that in the biological application, however, \(\left\{ {g_{i} \left[ k \right]} \right\}\) in Eq. (1) spanned thousands of orders of magnitude, at least from \(g_{i} \left[ 0 \right] = 1\) to \(g_{I} \left[ K \right] \approx 10^{2608}\), obstructing the direct use of transforms (e.g., see [6]). On one hand, the widely varying magnitudes necessitated an internal logarithmic representation of \(\left\{ {g_{i} \left[ k \right]} \right\}\) in the computer, a minor inconvenience for direct computation with the Jackknife Product algorithm. On the other hand, they might have presented a substantial obstacle for transforms. The famous Feynman anecdote about Paul Olum's \(\tan \left( {10^{100} } \right)\) problem indicates the reason [13]:
So Paul is walking past the lunch place and these guys are all excited. "Hey, Paul!" they call out. "Feynman's terrific! We give him a problem that can be stated in ten seconds, and in a minute he gets the answer to 10 percent. Why don't you give him one?" Without hardly stopping, he says, "The tangent of 10 to the 100th." I was sunk: you have to divide by pi to 100 decimal places! It was hopeless.
The Jackknife Product algorithm also abstracts to any commutative semigroup \(\left( {G, \circ } \right)\), broadening its applicability enormously. As usual, abstraction eases debugging. Consider, e.g., the commutative semigroup consisting of all bit strings of length \(n\) under the bitwise "or" operation. If the bit string \(g_{j}\) has 1 in the j-th position and 0 s elsewhere, then the segment product \(g_{{\left[ {i,j} \right)}}\) equals the bit string with 1 s in positions \(\left[ {i,j} \right) = \left\{ {i,i + 1, \ldots ,j - 1} \right\}\) and 0 s elsewhere. Similarly, the complementary segment product \(g_{{\overline{{\left[ {i,j} \right)}} }} = g_{{\left[ {0,i} \right)}} g_{{\left[ {j,n} \right)}}\) equals the bit string with 0 s in positions \(\left[ {i,j} \right) = \left\{ {i,i + 1, \ldots ,j - 1} \right\}\) and 1 s elsewhere. The Jackknife Product algorithm is easily debugged with output consisting of the segment and complementary segment trees for the bit strings.
As a final note, even if a semigroup \(\left( {G, \circ } \right)\) lacks the Commutative property, the general product algorithm for a segment tree can still compute \(\left\{ {\bar{g}_{j} = g_{{\left[ {0,j} \right)}} g_{{\left[ {j + 1,n} \right)}} } \right\}\) in time \(O\left( {n\log n} \right)\). In a commutative semigroup \(\left( {G, \circ } \right)\), however, the downward phase of the Jackknife Product algorithm exploits the special form of the products \(\left\{ {\bar{g}_{j} } \right\}\) to decrease the time to \(O\left( n \right)\).
This article has presented a Jackknife Product algorithm, which applies to any commutative semi-group \(\left( {G, \circ } \right)\). The biological application to a circRNA–miRNA system uses a commutative semigroup of truncated convolutions to exemplify a specific application to combinatorial probabilities. In turn, the specific application in combinatorial probability exemplifies an even more general statistical test for whether a term in a sum of independent counting variates (not necessarily identically distributed) is unusually large. The general statistical test can evaluate the results of searching for a sequence or structure motif, or several motifs simultaneously. As "Discussion" section explains, the test violates the hypotheses of the Benjamini–Hochberg procedure for estimating false discovery rates, but fortunately the Bonferroni and other multiple-test corrections remain available to control familywise errors. Abstraction from convolutions to commutative semi-groups broadens the algorithm's applicability even further. If an application only requires jackknife products \(\left\{ {\bar{g}_{j} } \right\}\) and their number \(n\) is large enough, "Results" and "Theory" sections show that the linear time of the Jackknife Product algorithm can make it well worth the programming effort.
A self-testing, annotated file "jls_jackknifeproduct.py" implementing an in-place Jackknife Product algorithm in Python is available without any restrictions at https://github.com/johnlspouge/jackknife-product. Data were previously published elsewhere [1].
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JLS would like to acknowledge useful conversations with Dr. Amir Manzour and Dr. DoHwan Park.
This research was supported by the Intramural Research Program of the NIH, National Library of Medicine, and by the Center for Cancer Research, National Cancer Institute.
National Center for Biotechnology Information, National Library of Medicine, National Institutes of Health, Room 6N603, Building 38A, Bethesda, MD, 20894, USA
John L. Spouge
HIV and AIDS Malignancy Branch, Center for Cancer Research, National Cancer Institute, National Institutes of Health, Bethesda, MD, 20892, USA
Joseph M. Ziegelbauer
Genomics Research and Development, Lenovo HPC and AI, 1009 Think Pl, Morrisville, NC, 27560, USA
Mileidy Gonzalez
JLS developed the algorithms, performed computational and statistical analysis, and drafted the manuscript. JMZ provided the data stimulating and exemplifying the analysis. MG performed bioinformatics and sequence analysis. All authors read and approved the final manuscript.
Correspondence to John L. Spouge.
This appendix proves the correctness of the Jackknife Product algorithm in "Theory" section.
Let \(L_{k}\) have length \(l_{k}\); \(\bar{L}_{k}\), length \(\bar{l}_{k}\). Some observations about \(l_{k}\), \(\bar{l}_{k}\), and \(n_{k}\) facilitate later analysis. Because \(\left\lceil {\left\lceil x \right\rceil /2} \right\rceil = \left\lceil {x/2} \right\rceil\), \(\left\{ {n_{k} } \right\}\) satisfies the recursion \(n_{k} = \left\lceil {n2^{ - k} } \right\rceil = \left\lceil {n2^{{ - \left( {k - 1} \right)}} /2} \right\rceil = \left\lceil {n_{k - 1} /2} \right\rceil\), with initial value \(n_{0} = n\) and final value \(n_{m - 1} = 2\).
\(l_{k} = \bar{l}_{k} = n_{k}\) for \(0 \le k < m\).
Proof (by induction)
\(l_{0} = n = n_{0}\). If \(l_{k - 1} = n_{k - 1}\) for any \(1 \le k < m\), the upward phase of the Jackknife Product algorithm shows: (1) if \(n_{k - 1}\) is even, \(l_{k} = \rho \left( {n_{k - 1} } \right)\); and (2) if \(n_{k - 1}\) is odd, \(l_{k} = \rho \left( {n_{k - 1} } \right) + 1\). In either case, \(l_{k} = \left\lceil {n_{k - 1} /2} \right\rceil = n_{k}\). Thus, \(l_{k} = n_{k}\) for \(0 \le k < m\).
Similarly, the downward phase shows: (1) if \(n_{k - 1}\) is even, \(\bar{l}_{k - 1} = 2\rho \left( {n_{k - 1} } \right) = n_{k - 1}\); if \(n_{k - 1}\) is odd, \(\bar{l}_{k - 1} = 2\rho \left( {n_{k - 1} } \right) + 1 = n_{k - 1}\). It therefore initializes \(\bar{L}_{m - 1}\) with \(\bar{l}_{m - 1} = n_{m - 1} = 2\) elements and assigns \(\bar{l}_{k - 1} = n_{k - 1}\) (\(1 \le k < m\)) elements to \(\bar{L}_{k - 1}\). □
Proposition 1 and its proof ensure that with the possible exception of \(\alpha_{k} \left( j \right)\) in the downward phase, all array indices in the Jackknife Product algorithm lie within the array bounds of \(L_{k}\) and \(\bar{L}_{k}\). Moreover, case-by-case analysis of the definition of \(\alpha_{k}\) shows that \(\alpha_{k} \left( j \right)\) (\(0 \le j < n_{k - 1}\)) always falls within the array bounds \(0 \le \alpha_{k} \left( j \right) < n_{k}\) of \(\bar{L}_{k}\). Inspection of the upward and downward phases shows that they define every array element before using it. With array bounds and definitions in hand, to verify the Jackknife Product algorithm, it therefore suffices to check conditions satisfied by individual elements of \(L_{k}\) and \(\bar{L}_{k}\). We examine first the case of binary powers \(n = n^{ * } = 2^{m}\), and afterwards the case of general \(n\).
Proof of Correctness for Binary Powers \(n^{ * } = 2^{m} \ge 2\)
In this subsection, some entities pertaining to binary powers \(n^{ * }\) receive stars (e.g., \(n^{ * }\), \(n_{k}^{ * }\), \(L_{k}^{ * }\), \(\bar{L}_{k}^{ * }\)), to distinguish them later from the corresponding entities for general \(n\).
For convenience in Appendix A only, drop "\(g\)" in the notation \(g_{{\left[ {i,j} \right)}}\), and abbreviate the segment product \(g_{{\left[ {i,j} \right)}} = g_{i} g_{i + 1} \cdots g_{j - 1}\) by the corresponding half-open interval \(\left[ {i,j} \right)\) and the complementary segment product \(g_{0} g_{1} \cdots g_{i - 1} g_{j} \cdots g_{{n^{ * } - 1}}\) by \(\overline{{\left[ {i,j} \right)}}\). The notation provides a mnemonic aid, used without comment below: for \(i < j < k\), \(\left[ {i,j} \right) \circ \left[ {j,k} \right) = \left[ {i,k} \right)\), \(\left[ {i,j} \right) \circ \overline{{\left[ {i,k} \right)}} = \overline{{\left[ {j,k} \right)}}\) and \(\left[ {j,k} \right) \circ \overline{{\left[ {i,k} \right)}} = \overline{{\left[ {i,j} \right)}}\), i.e., the product \(\circ\) on sub-products behaves like a set-theoretic union of the corresponding intervals, and complementary sub-products behave like the corresponding set-theoretic complements of intervals.
The Commutative property is required to justify the correspondence between set-theoretic operations and products, e.g., the equality \(\left[ {i,j} \right) \circ \overline{{\left[ {i,k} \right)}} = \overline{{\left[ {j,k} \right)}}\) commutes the segment products: examine, e.g., the second equality in the equation
$$\begin{aligned} g_{{\left[ {i,j} \right)}} \circ \left( {g_{{\left[ {0,i} \right)}} \circ g_{{\left[ {k,n*} \right)}} } \right) &= \left( g_{{\left[ {i,j} \right)}} \circ {g_{{\left[ {0,i} \right)}} } \right) \circ g_{{\left[ {k,n*} \right)}} \\ &= \left( {g_{{\left[ {0,i} \right)}} \circ g_{{\left[ {i,j} \right)}} } \right) \circ g_{{\left[ {k,n*} \right)}} = g_{{\left[ {0,j} \right)}} \circ g_{{\left[ {k,n*} \right)}} .\end{aligned}$$
For \(0 \le k < m\), \(L_{k}^{ * } \left[ j \right] = \left[ {j \times 2^{k} ,\left( {j + 1} \right) \times 2^{k} } \right)\) (\(0 \le j < n_{k}^{*}\)).
See Fig. 1. The array \(L_{0}^{ * }\) at generation \(k = 0\) initializes the upward phase, where
$$L_{0}^{ * } \left[ j \right] = g_{j} = \left[ {j \times 2^{0} ,\left( {j + 1} \right) \times 2^{0} } \right)\quad {\text{for}}\quad 0 \le j < n^{ * } .$$
Thus, Proposition 2 is true for \(k = 0\). Given the array \(L_{k - 1}^{ * }\) at generation \(k - 1\) in the upward phase, the array \(L_{k}^{ * }\) at level \(k\) contains products of successive adjacent pairs of elements in \(L_{k - 1}^{ * }\):
$$\begin{aligned} L_{k}^{ * } \left[ j \right] & = L_{k - 1}^{ * } \left[ {2j} \right] \circ L_{k - 1}^{ * } \left[ {2j + 1} \right] \\ & = \left[ {2j \times 2^{k - 1} ,\left( {2j + 1} \right) \times 2^{k - 1} } \right) \circ \left[ {\left( {2j + 1} \right) \times 2^{k - 1} ,\left( {2j + 2} \right) \times 2^{k - 1} } \right) \\ & = \left[ {j \times 2^{k} ,\left( {j + 1} \right) \times 2^{k} } \right) \\ \end{aligned}$$
for \(0 \le j < n_{k}^{ * } = n_{k - 1}^{ * } /2\). The upward phase terminates with \(L_{m - 1}^{ * } = \left( {\left[ {0,2^{m - 1} } \right),\left[ {2^{m - 1} ,2^{m} } \right)} \right)\), so Proposition 2 is true for \(0 \le k < m\). □
For \(0 \le k < m\), \(\bar{L}_{k}^{ * } \left[ j \right] = \overline{{\left[ {\tau \left( j \right) \times 2^{k} ,\left( {\tau \left( j \right) + 1} \right) \times 2^{k} } \right)}}\) (\(0 \le j < n_{k}^{ * }\)).
Propositions 3 and 2 formalize the previously mentioned complementary symmetry between the upward and downward phases. Because \(\tau \left( {\tau \left( j \right)} \right) = j\) (i.e., transposition is idempotent), \(\bar{L}_{0}^{ * } \left[ {\tau \left( j \right)} \right] = \overline{{\left[ {j,j + 1} \right)}} = \bar{g}_{j}\) for \(0 \le j < n_{0}^{ * } = n^{ * }\). Thus, \(\bar{L}_{0}^{ * }\) contains all jackknife products.
See Fig. 2. For \(0 \le j < n_{m - 1}^{ * } = 2\), the first two lines of pseudo-code in the downward phase and Proposition 2 for \(k = m - 1\) show that for \(j \in \left\{ {0,1} \right\}\),
$$\bar{L}_{m - 1}^{ * } \left[ j \right] = L_{m - 1}^{ * } \left[ j \right] = \left[ {j \times 2^{m - 1} ,\left( {j + 1} \right) \times 2^{m - 1} } \right) = \overline{{\left[ {\tau \left( j \right) \times 2^{m - 1} ,\left( {\tau \left( j \right) + 1} \right) \times 2^{m - 1} } \right)}} ,$$
so Proposition 3 holds for \(k = m - 1\). We proceed by descending induction on \(k\).
For even \(j\) on one hand, \(2\rho \left( j \right) = j < j + 1 = \tau \left( j \right) < \tau \left( j \right) + 1 = 2\left( {\rho \left( j \right) + 1} \right)\). For \(0 \le j < n_{k - 1}^{ * }\), therefore,
$$\left[ {j \times 2^{k - 1} ,\left( {j + 1} \right) \times 2^{k - 1} } \right) \circ \left[ {\tau \left( j \right) \times 2^{k - 1} ,\left( {\tau \left( j \right) + 1} \right) \times 2^{k - 1} } \right) = \left[ {\rho \left( j \right) \times 2^{k} ,\left( {\rho \left( j \right) + 1} \right) \times 2^{k} } \right) .$$
For odd \(j\) on the other hand, \(2\rho \left( j \right) = \tau \left( j \right) < \tau \left( j \right) + 1 = j < j + 1 = 2\left( {\rho \left( j \right) + 1} \right)\), so Eq. (6) holds with the factors on the left reversed, an irrelevant difference in a commutative semigroup \(\left( {G, \circ } \right)\).
For \(0 \le j < n_{k - 1}^{ * }\), then, if \(1 \le k < m\), Proposition 2 for \(k - 1\) and Proposition 3 for \(k\) yield
$$\begin{aligned} \bar{L}_{k - 1}^{ * } \left[ j \right] & = L_{k - 1}^{ * } \left[ j \right] \circ \bar{L}_{k}^{ * } \left[ {\tau \rho \left( j \right)} \right] \\ & = \left[ {j \times 2^{k - 1} ,\left( {j + 1} \right) \times 2^{k - 1} } \right) \circ \overline{{\left[ {\rho \left( j \right) \times 2^{k} ,\left( {\rho \left( j \right) + 1} \right) \times 2^{k} } \right)}} \\ & = \overline{{\left[ {\tau \left( j \right) \times 2^{k - 1} ,\left( {\tau \left( j \right) + 1} \right) \times 2^{k - 1} } \right)}} . \\ \end{aligned}$$
Thus, Proposition 3 for \(k\) implies Proposition 3 for \(k - 1\) (\(1 \le k < m\)). □
The Jackknife Product algorithm therefore computes \(\bar{L}_{0}^{ * } \left[ {\tau \left( j \right)} \right] = \overline{{\left[ {j,j + 1} \right)}} = \bar{g}_{j}\), as desired.
Proof of correctness for general \(n \ge 2\): For general \(n \ge 2\), initialize the upward phase with \(L_{0} = \left( {g_{0} ,g_{1} , \ldots ,g_{n - 1} } \right)\). To apply the results of the previous subsection, let \(n^{ * } = 2^{m}\) be the smallest binary power greater than or equal to \(n\), i.e., let \(m = \left\lceil {\log_{2} n} \right\rceil\). If \(n < n^{ * }\), set \(g_{n} = g_{n + 1} = \cdots = g_{{n^{ * } - 1}} = e\), with the arrays \(L_{k}^{ * }\) and \(\bar{L}_{k}^{ * }\) of length \(n_{k}^{ * }\) (\(0 \le k < m\)) as above. For general \(n \ge 2\), consider the computation of the arrays \(L_{k}\) (\(0 < k < m\)) in the upward phase of the pseudocode above.
We prove Proposition \(P\left( k \right)\) (next) by induction on the level \(0 \le k < m\).
Proposition \(P\left( k \right)\)
\(L_{k} \left[ j \right] = L_{k}^{ * } \left[ j \right]\) for \(0 \le j < n_{k}\), and \(L_{k}^{ * } \left[ j \right] = e\) for \(n_{k} \le j < n_{k}^{ * }\).
See Fig. 1. By construction, \(P\left( 0 \right)\) holds. With \(P\left( {k - 1} \right)\) in hand, the upward phase and Eq. (4) show that \(L_{k} \left[ j \right] = L_{k}^{ * } \left[ j \right]\) for \(0 \le j < \rho \left( {n_{k - 1} } \right)\). On one hand, if \(n_{k - 1}\) is even, \(\rho \left( {n_{k - 1} } \right) = n_{k}\), yielding \(P\left( {k - 1} \right)\) immediately. On the other hand, if \(n_{k - 1}\) is odd, \(\rho \left( {n_{k - 1} } \right) = \left\lceil {n_{k - 1} /2} \right\rceil - 1 = n_{k} - 1\), so
$$\begin{aligned} L_{k} \left[ {n_{k} - 1} \right] & = L_{k} \left[ {\rho \left( {n_{k - 1} } \right)} \right] = L_{k - 1} \left[ {n_{k - 1} - 1} \right] = L_{k - 1}^{ * } \left[ {n_{k - 1} - 1} \right] \\ & = L_{k - 1}^{ * } \left[ {n_{k - 1} - 1} \right] \circ e = L_{k - 1}^{ * } \left[ {n_{k - 1} - 1} \right] \circ L_{k - 1}^{ * } \left[ {n_{k - 1} } \right] = L_{k}^{ * } \left[ {n_{k} - 1} \right], \\ \end{aligned}$$
the second equality reflecting the copy of the final element of \(L_{k - 1}\) in the pseudocode; the third and fifth, \(P\left( {k - 1} \right)\); and the sixth, Eq. (4). Equation (8) completes the proof that \(L_{k} \left[ j \right] = L_{k}^{ * } \left[ j \right]\) for \(0 \le j < n_{k}\). For the remaining indices \(n_{k} \le j < n_{k}^{ * }\) of \(L_{k}^{ * }\), note that \(n_{k - 1} \le 2\left\lceil {n_{k - 1} /2} \right\rceil = 2n_{k} \le 2j < 2j + 1 < 2n_{k}^{ * } = n_{k - 1}^{ * }\). Then, \(P\left( {k - 1} \right)\) and Eq. (4) show that \(L_{k}^{ * } \left[ j \right] = L_{k - 1}^{ * } \left[ {2j} \right] \circ L_{k - 1}^{ * } \left[ {2j + 1} \right] = e \circ e = e\) for \(n_{k} \le j < n_{k}^{ * }\). □
Note: \(n_{m - 1} = 2 = n_{m - 1}^{ * }\), so \(P\left( {m - 1} \right)\) shows that \(L_{m - 1} = L_{m - 1}^{ * }\).
In the downward phase, the transposition function \(\tau\) in Eq. (7) facilitates in-place computation for \(\bar{L}_{k - 1}^{ * } \left[ j \right]\) in Eq. (7). Similarly, the minimization in the accessory index \(\alpha_{k} \left( j \right) = \hbox{min} \left\{ {\tau \rho \left( j \right),n_{k} - 1} \right\}\) within \(\bar{L}_{k}\) avoids storing a superfluous element \(\bar{g}\) of \(\bar{L}_{k}^{ * }\) within the penultimate element of any truncated complementary array \(\bar{L}_{k}\) (see \(\bar{L}_{1}^{ * } \left[ 3 \right]\), the dotted circle in Fig. 2).
We prove Proposition \(\bar{P}\left( k \right)\) (next) by descending induction on the level \(0 \le k < m\).
Proposition \(\bar{P}\left( k \right)\)
\(\bar{L}_{k} \left[ j \right] = \bar{L}_{k}^{ * } \left[ j \right]\) for \(0 \le j < n_{k}\), unless \(n_{k}\) is odd and \(j = n_{k} - 1\), in which case \(\bar{L}_{k} \left[ {n_{k} - 1} \right] = \bar{L}_{k}^{ * } \left[ {n_{k} } \right]\).
The proposition \(\bar{P}\left( {m - 1} \right)\) is true, because \(\bar{L}_{m - 1} = L_{m - 1} = L_{m - 1}^{ * } = \bar{L}_{m - 1}^{ * }\), and \(\bar{L}_{m - 1}\) has even length \(n_{m - 1} = 2\). We now show that \(\bar{P}\left( k \right)\) implies \(\bar{P}\left( {k - 1} \right)\) for \(0 < k < m\).
If \(0 \le j < n_{k - 1}\), then \(0 \le \rho \left( j \right) < \left\lceil {n_{k - 1} /2} \right\rceil = n_{k}\). For every \(j\), either: (1) \(n_{k}\) is odd and \(\rho \left( j \right) = n_{k} - 1\); (2) \(n_{k}\) is odd and \(\rho \left( j \right) < n_{k} - 1\); or (3) \(n_{k}\) is even. In Case 1, \(\tau \rho \left( j \right) = n_{k}\) but \(\alpha_{k} \left( j \right) = n_{k} - 1\). Because \(n_{k}\) is odd, \(\bar{P}\left( k \right)\) implies \(\bar{L}_{k} \left[ {\alpha_{k} \left( j \right)} \right] = \bar{L}_{k} \left[ {n_{k} - 1} \right] = \bar{L}_{k}^{ * } \left[ {n_{k} } \right] = \bar{L}_{k}^{ * } \left[ {\tau \rho \left( j \right)} \right]\). In Case 2, \(0 \le \alpha_{k} \left( j \right) = \tau \rho \left( j \right) < n_{k} - 1\), or in Case 3, \(0 \le \alpha_{k} \left( j \right) = \tau \rho \left( j \right) < n_{k}\) and \(n_{k}\) is even. In either case, \(\bar{P}\left( k \right)\) implies \(\bar{L}_{k} \left[ {\alpha_{k} \left( j \right)} \right] = \bar{L}_{k} \left[ {\tau \rho \left( j \right)} \right] = \bar{L}_{k}^{ * } \left[ {\tau \rho \left( j \right)} \right]\). Thus, regardless of whether Case 1, 2, or 3 pertains, \(\bar{L}_{k} \left[ {\alpha_{k} \left( j \right)} \right] = \bar{L}_{k}^{ * } \left[ {\tau \rho \left( j \right)} \right]\) for every \(0 \le j < n_{k - 1}\).
For \(0 \le j < 2\rho \left( {n_{k - 1} } \right)\), the Jackknife Product algorithm in the downward phase and \(P\left( {k - 1} \right)\) from the upward phase yield
$$\bar{L}_{k - 1} \left[ j \right] = L_{k - 1} \left[ j \right] \circ \bar{L}_{k} \left[ {\alpha_{k} \left( j \right)} \right] = \bar{L}_{k - 1}^{ * } \left[ j \right] \circ \bar{L}_{k}^{ * } \left[ {\tau \rho \left( j \right)} \right] = \bar{L}_{k - 1}^{ * } \left[ j \right]$$
if \(j < n_{k - 1}\). On one hand, if \(n_{k - 1}\) is even, \(n_{k - 1} = 2\rho \left( {n_{k - 1} } \right)\), yielding \(\bar{P}\left( {k - 1} \right)\) immediately. On the other hand, if \(n_{k - 1}\) is odd, \(n_{k - 1} = 2\rho \left( {n_{k - 1} } \right) + 1\), so \(\bar{L}_{k - 1}\) has an additional, final element copied from \(\bar{L}_{k}\):
$$\bar{L}_{k - 1} \left[ {n_{k - 1} - 1} \right] = \bar{L}_{k} \left[ {\alpha_{k} \left( {n_{k - 1} - 1} \right)} \right] = \bar{L}_{k}^{ * } \left[ {\tau \rho \left( {n_{k - 1} - 1} \right)} \right].$$
\(P\left( {k - 1} \right)\) yields \(L_{k - 1}^{ * } \left[ {n_{k - 1} } \right] = e\), so moreover,
$$\bar{L}_{k - 1}^{ * } \left[ {n_{k - 1} } \right] = L_{k - 1}^{ * } \left[ {n_{k - 1} } \right] \circ \bar{L}_{k}^{ * } \left[ {\tau \rho \left( {n_{k - 1} } \right)} \right] = \bar{L}_{k}^{ * } \left[ {\tau \rho \left( {n_{k - 1} } \right)} \right].$$
Because \(n_{k - 1} - 1\) is even, \(\rho \left( {n_{k - 1} - 1} \right) = \rho \left( {n_{k - 1} } \right)\). Equations (10) and (11) therefore yield \(\bar{L}_{k - 1} \left[ {n_{k - 1} - 1} \right] = \bar{L}_{k - 1}^{ * } \left[ {n_{k - 1} } \right]\), so \(\bar{P}\left( {k - 1} \right)\) holds. □
This appendix gives combinatoric calculations for the circRNA-miRNA application in "Background" section. It therefore has some peripheral interest to this article.
Motifs on a single circle
Consider \(r\) points equally spaced around a circle (a "ring"). Call a set of \(m\) consecutive points on the ring a "motif". The following fixes \(m\), so the notation can leave it implicit. Let \(C_{r,k}\) count the ways of choosing \(k\) non-overlapping motifs around the ring (i.e., the motifs have no point in common). Note: \(C_{r,k} = 0\) if \(r < mk\) or \(k < 0\). Define the factorial function \(n! = n\left( {n - 1} \right) \cdots 1\) and the binomial (combinatorial or Pascal) coefficient
$$\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right) = \frac{n!}{{k!\left( {n - k} \right)!}}$$
for \(0 \le k \le n\) and 0 otherwise.
Clearly, \(C_{r,k} = 1\) when \(r = mk\). For \(r > mk\),
$$\begin{aligned} C_{r,k} &= \left( {\begin{array}{*{20}c} {r - 1 - \left( {m - 1} \right)k} \\ k \\ \end{array} } \right) + m\left( {\begin{array}{*{20}c} {r - m - \left( {m - 1} \right)\left( {k - 1} \right)} \\ {k - 1} \\ \end{array} } \right) \\ &= \left( {\begin{array}{*{20}c} {r - \left( {m - 1} \right)k - 1} \\ k \\ \end{array} } \right) + m\left( {\begin{array}{*{20}c} {r - \left( {m - 1} \right)k - 1} \\ {k - 1} \\ \end{array} } \right) \\ \end{aligned}$$
counts the ways of placing \(k\) motifs around the ring. For convenience below and for consistency with Eq. (13), \(C_{r,0} = 1\) for \(r \ge 0\).
Consider a line segment containing \(r\) equally spaced points, and let \(L_{r,k}\) count the ways of choosing \(k\) non-overlapping motifs, each of \(m\) consecutive points, on it. First, \(C_{r,k} = L_{r - 1,k} + mL_{r - m,k - 1}\), proved as follows. Number the ring points arbitrarily as positions \(1,2, \ldots ,r\), and place the \(k\) motifs as follows. Consider position 1, which might have no motif. If so, place the \(k\) motifs in a line consisting of \(r - 1\) positions \(2,3, \ldots ,r\) (\(L_{r - 1,k}\) ways). Otherwise, place the first motif in one of the \(m\) positions in which it covers position 1, and then place the remaining \(k - 1\) motifs in a line consisting of \(r - m\) positions (\(mL_{r - m,k - 1}\) ways). Each such configuration corresponds to placing \(k\) leftmost end-positions on the line. For each of the \(k\) motifs, delete \(m - 1\) positions on the ring, all but its leftmost end-position. Each of the configurations for \(k\) motifs therefore corresponds to choosing \(k\) positions from the \(r - \left( {m - 1} \right)k\) positions remaining, yielding the remaining factors in Eq. (13).
Motifs on several circles
Now, let \(C_{r\left( 1 \right),r\left( 2 \right), \ldots ,r\left( N \right);K}\) count the ways of distributing \(K\) non-overlapping motifs (all of \(m\) consecutive points) around several rings, the rings' points numbering \(r\left( n \right)\) (\(n = 1,2 \ldots ,N\)). Without loss of generality, assume \(r\left( n \right) \ge m\) (\(n = 1,2 \ldots ,N\)). (Otherwise, discard all rings with \(r\left( n \right) < m\).) The following recursion holds: \(C_{r\left( 1 \right),r\left( 2 \right), \ldots ,r\left( N \right);K} = 0\) for \(K < 0\) or \(K > \sum\nolimits_{n = 1}^{N} {\left\lfloor {r\left( n \right)/m} \right\rfloor }\), \(C_{r\left( 1 \right),r\left( 2 \right), \ldots ,r\left( N \right);0} = 1\), and otherwise
$$C_{r\left( 1 \right),r\left( 2 \right), \ldots ,r\left( N \right);K} = \sum {C_{r\left( N \right),k} C_{{r\left( 1 \right),r\left( 2 \right), \ldots ,r\left( {N - 1} \right);K - k}} } ,$$
where on the right, the index of summation \(k\) runs from \(\hbox{max} \left\{ {0,K - \sum\nolimits_{n = 1}^{N - 1} {\left\lfloor {r\left( n \right)/m} \right\rfloor } } \right\}\) up to \(\hbox{min} \left\{ {K,\left\lfloor {r\left( N \right)/m} \right\rfloor } \right\}\). Equation (13) initializes the convolution recursion in Eq. (14) with \(C_{r\left( 1 \right),K}\).
Relation to the Jackknife Product algorithm
To apply the Jackknife Product algorithm in the circRNA–miRNA application, let \(g_{i} = \left( {C_{r\left( i \right),0} ,C_{r\left( i \right),1} , \ldots ,C_{r\left( i \right),K} } \right)\) (\(i = 1, \ldots ,N\)) in the (commutative) semigroup \(\left( {G, \circ } \right)\) of non-negative integer vectors with indices \(k = 0,1, \ldots ,K\), under the convolution operation.
Spouge, J.L., Ziegelbauer, J.M. & Gonzalez, M. A linear-time algorithm that avoids inverses and computes Jackknife (leave-one-out) products like convolutions or other operators in commutative semigroups. Algorithms Mol Biol 15, 17 (2020). https://doi.org/10.1186/s13015-020-00178-x
DOI: https://doi.org/10.1186/s13015-020-00178-x
Commutative semigroup
Leave-one-out
Jackknife products
Segment tree | CommonCrawl |
\begin{document}
\title{New distinct curves having the same complement in the projective plane.} \author{Paolo Costa} \address{Paolo Costa, UniGe} \email{[email protected]} \date{\today}
\begin{abstract} In 1984, H. Yoshihara conjectured that if two plane irreducible curves have isomorphic complements, they are projectively equivalent, and proved the conjecture for a special family of unicuspidal curves. Recently, J. Blanc gave counterexamples of degree $39$ to this conjecture, but none of these is unicuspidal. In this text, we give a new family of counterexamples to the conjecture, all of them being unicuspidal, of degree $4m+1$ for any $m\ge 2$. In particular, we have counterexamples of degree $9$, which seems to be the lowest possible degree. \end{abstract}
\maketitle{}
\section{The conjecture.}
In the sequel, we will work with algebraic varieties over a fixed ground field $\mathbb{K}$, which can be arbitrary. \\ \begin{conj}[\cite{Yos84}] Suppose that the ground field is algebraically closed of characteristic zero. Let $C \subset \mathbb{P}^2$ be an irreducible curve. Suppose that $\mathbb{P}^2 \backslash C$ is isomorphic to $\mathbb{P}^2 \backslash D$ for some curve $D$. Then $C$ and $D$ are projectively equivalent, i.e. there's an automorphism $\alpha : \mathbb{P}^2 \to \mathbb{P}^2$ such that $\alpha(C)=D$. \end{conj}
This conjecture leads to several alternatives. Let $\psi : \mathbb{P}^2 \backslash C \to \mathbb{P}^2 \backslash D$ be an isomorphism.\ If the conjecture holds, then : \begin{itemize}
\item either $\psi$ extends to an automorphism of $\mathbb{P}^2$ and we can choose $\alpha:=\psi$.
\item or $\psi$ extends to a strict birational map $\psi : \mathbb{P}^2 \dasharrow \mathbb{P}^2$. In this case, there's an automorphism $\alpha : \mathbb{P}^2 \to \mathbb{P}^2$ such that $\alpha(C)=D$. \end{itemize} Otherwise, if $\psi$ gives a counterexample to the conjecture, then : \begin{itemize}
\item either $C$ and $D$ are not isomorphic.
\item or $C$ and $D$ are isomorphic, but not by an automorphism of $\mathbb{P}^2$. \end{itemize}
In this text, we are going to study the conjecture in the case of curves of type I.\\
\begin{dfn} We say that a curve $C \subset \mathbb{P}^2$ is of \textbf{type I} if there's a point $p \in C$ such that $C \backslash p$ is isomorphic to $\mathbb{A}^1$.\\ We say that a curve $C \subset \mathbb{P}^2$ is of \textbf{type II} if there's a line $L \subset \mathbb{P}^2$ such that $C \backslash L$ is isomorphic to $\mathbb{A}^1$.\\ \end{dfn} All curves of type II are of type I, but the converse is false in general. Moreover, a curve of type I is a line, a conic, or a unicuspidal curve (a curve with one singularity of cuspidal type).
In the case of curves of type II, H. Yoshihara showed that the conjecture is true \cite{Yos84}, but in general the conjecture doesn't hold. Some counterexamples are given in \cite{Bla09}, but these curves are not of type I.
In this article, we give a new family of counterexamples, of degree $4m+1$ for any $m\ge 2$. These are all of type I, and some of them have degree $9$, which seems to be the lowest possible degree (see the end of the article for more details). In Section 2 we give a general way to constuct examples, that we precise in Section 3. The last section is the conclusion.\\
I would like to thank J. Blanc for asking me the question and for his help during the preparation of this article. I also thank T. Vust for interesting discussions on the result.
\section{The Construction.}
We begin with giving a general construction, which provides isomorphisms of the form $\p \backslash C \to \p \backslash D$ where $C,D$ are curves in $\mathbb{P}^2$. We start with the following definition : \begin{dfn} We say that a morphism $\pi : S \to \mathbb{P}^2$ is a \textbf{$(-1)-$tower resolution} of a curve $C$ if : \begin{enumerate}
\item $\pi=\pi_1 \circ ... \circ \pi_m$ where $\pi_i$ is the blow-up of a point $p_i$,
\item $\pi_i(p_{i+1})=p_i$ for $i=1,...,m-1$,
\item the strict transform of $C$ in $S$ is a smooth curve, isomorphic to $\mathbb{P}^1$, and has self-intersection $-1$. \end{enumerate} \end{dfn}
The isomorphisms of the form $\p \backslash C \to \p \backslash D$ are closely related to $(-1)-$tower resolutions of $C$ and $D$ because of the following Lemma :
\begin{lemme}[\cite{Bla09}] \label{Bla09} Let $C \subset \mathbb{P}^2$ be an irreducible algebraic curve and $\psi : \p \backslash C \to \p \backslash D$ an isomorphism. Then, either $\psi$ extends to an automorphism of $\mathbb{P}^2$, either it extends to a strict birational transform $\phi : \mathbb{P}^2 \dasharrow \mathbb{P}^2$.\\ Consider the second case. Let $\chi : X \to \mathbb{P}^2$ a minimal resolution of the indeterminacies of $\phi$, call $\tilde{E}_1,...,\tilde{E}_m$ and $\tilde{C}$ the strict transforms of its exceptional curves and $C$ in $X$ and set $\epsilon:=\phi \circ \chi$. Then : \begin{enumerate}
\item $\chi$ is a $(-1)-$tower resolution of $C$
\item $\epsilon$ collapses $\tilde{C}, \tilde{E}_1,..., \tilde{E}_{m-1}$ and $\epsilon(\tilde{E}_m)=D$,
\item $\epsilon$ is a $(-1)-$tower resolution of $D$. \end{enumerate} \end{lemme}
\begin{rem} This lemma shows that if $C$ doesn't admit a $(-1)-$tower resolution, then every isomorphism $\p \backslash C \to \p \backslash D$ extends to an automorphism of $\mathbb{P}^2$. So counterexamples will be given by rational curves with only one singularity. \end{rem}
We start with a smooth conic $Q \subset \mathbb{P}^2$ and $\phi \in \Aut{\p \backslash Q}$ which extends to a strict birational map $\phi : \mathbb{P}^2 \dasharrow \mathbb{P}^2$. Call $p_1,...,p_m$ the indeterminacies points of $\phi$; according to Lemma~$\ref{Bla09}$, we can order the points so that $p_1$ is a point of $\mathbb{P}^2$ and $p_i$ is infinitely near to $p_{i-1}$ for $i=2,...,n$. Consider $\chi : X \to \mathbb{P}^2$, a minimal resolution of the indeterminacies of $\phi$ and set $\epsilon:=\phi \circ \chi$. Lemma~$\ref{Bla09}$ says that : \begin{enumerate}
\item $\chi$ is a $(-1)-$tower resolution of $Q$,
\item $\epsilon$ collapses $\tilde{Q}, \tilde{E}_1,..., \tilde{E}_{m-1}$ and $\epsilon(\tilde{E}_m)=Q$,
\item $\epsilon$ is a $(-1)-$tower resolution of $Q$. \end{enumerate} Now, consider a line $L \subset \mathbb{P}^2$, which is tangent to $Q$ at $p \neq p_1$. Since $\phi$ contracts $Q$, then $C:=\phi(L)$ is a curve with an unique singular point which is $\phi(Q)$. Since $L \cap (\p \backslash Q) \simeq \mathbb{A}^1$, we have $C \cap (\p \backslash Q) \simeq \mathbb{A}^1$, which means that $C$ is of type I.
Consider now a birational map $f \in \Aut{\p \backslash L}$ which extends to a strict birational map $\mathbb{P}^2 \dasharrow \mathbb{P}^2$ and satisfies : \begin{enumerate}
\item $f(Q)=Q$,
\item $f(p_1)=p_1$. \end{enumerate}
Now, we are going to get a new birational map $\phi' : \mathbb{P}^2 \dasharrow \mathbb{P}^2$ which restricts to an automorphism of $\p \backslash Q$ using the $p_i$'s and $f$. Set : $$p_i':=f(p_i).$$ Note that $p_i'$ is a well-defined point infinitely near to $p_{i-1}'$ for $i>1$.\\ Let's call $\chi' : X' \to \mathbb{P}^2$ the blow-up of the $p_i'$'s and $\tilde{E}_1',...,\tilde{E}_m'$ and $\tilde{Q}'$ the strict transforms of the exceptional curves of $\chi'$ and of $Q$ in $X'$.\\ Since $f(Q)=Q$ and $f$ is an isomorphism at the neighbourhood of $p_1$, the intersections between $\tilde{E}_1,...,\tilde{E}_m$ and $\tilde{Q}'$ are the same as those between $\tilde{E}_1,...,\tilde{E}_m$ and $\tilde{Q}$. Then there's a morphism $\epsilon' : X' \to \mathbb{P}^2$ which contracts $\tilde{E}_1',...,\tilde{E}_{m-1}'$ and $\tilde{Q}'$. Moreover, $\epsilon'(\tilde{E}_m')$ is a conic, and up to isomorphism we can suppose that $\epsilon'(\tilde{E}_m')=Q$. \\ By construction, the birational map $\phi'$ restricts to an automorphism of $\p \backslash Q$. In fact, neither of the $p_i'$'s belongs to $L$ (as proper or infinitely near point), so $\phi'(L)$ is well defined. Moreover, $\phi'$ collapses $Q$, so $D:=\phi'(L)$ is a curve with an unique singular point which is $\phi'(Q)$.\\ Set then $\psi:=\phi' \circ f \circ \phi^{-1}$. We have the following commutative diagram :\\
\[\xymatrix@R=3.5mm{
& X' \ar[rd]^{\epsilon'} \ar[ld]_{\chi'} & \\
\mathbb{P}^2 \ar@{-->}[rr]^{\phi'} & & \mathbb{P}^2 \\
& X \ar[rd]^{\epsilon} \ar[ld]_{\chi} & \\
\mathbb{P}^2 \ar@{-->}[rr]^{\phi} \ar@{-->}[uu]^{f} & & \mathbb{P}^2 \ar@{-->}[uu]_{\psi} \\
} \]
\begin{lemme} The map $\psi : \p \backslash C \to \p \backslash D$ induced by the birational map defined above is an isomorphism. \end{lemme}
\begin{proof} Since $\phi,\phi' \in \Aut{\p \backslash Q}$ and $f \in \Aut{\p \backslash L}$, we only have to check that $\psi(Q)=Q$.\\ Let $\chi : X \to \mathbb{P}^2$ (resp. $\chi' : X' \to \mathbb{P}^2$) be a minimal resolution of the indeterminacies of $\phi$ (resp. $\phi'$) and write $\epsilon:=\phi \circ \chi$ (resp. $\epsilon':=\phi' \circ \chi'$). Call $\tilde{E}_1,...,\tilde{E}_m$ (resp. $\tilde{E}_1',...,\tilde{E}_m'$) the strict transforms of the exceptional curves of $\chi$ (resp. $\chi'$) in $X$ (resp. $X'$). It follows from Lemma~$\ref{Bla09}$ that $\epsilon(\tilde{E}_m)=Q$ (resp. $\epsilon'(\tilde{E}_m')=Q$). Then factorising $\psi$ we get $\psi(Q)=Q$. \end{proof}
Now we study the automorphisms $\alpha \in \Aut{\mathbb{P}^2}$ such that $\alpha(C)=D$.
\begin{lemme} If $\alpha \in \Aut{\mathbb{P}^2}$ sends $C$ onto $D$, then $a:=(\phi')^{-1} \circ \alpha \circ \phi$ is an automorphism of $\mathbb{P}^2$ and satisfies : \begin{enumerate}
\item $a(L)=L$,
\item $a(Q)=Q$,
\item $a(p_i)=p_i'$ for $i=1,...,m$. \end{enumerate}
\[\xymatrix@R=3.5mm{
& X' \ar[rd]^{\epsilon'} \ar[ld]_{\chi'} & \\
\mathbb{P}^2 \ar@{-->}[rr]^{\phi'} & & \mathbb{P}^2 \\
& X \ar[rd]^{\epsilon} \ar[ld]_{\chi} & \\
\mathbb{P}^2 \ar@{-->}[rr]^{\phi} \ar@{-->}[uu]^{a} & & \mathbb{P}^2 \ar@{-->}[uu]_{\alpha} \\
} \]
\end{lemme}
\begin{proof} Call $q_1,...,q_m$ (resp. $q_1'$, ..., $q_m'$) the points blown-up by $\epsilon$ (resp. $\epsilon'$). Then these points are the singular points of $C$ (resp. $D$). Since $\alpha$ is an automorphism such that $\alpha(C)=D$, then $\alpha$ sends $q_i$ on $q_i'$ for $i=1,...,m$, and lifts to an isomorphism $X \to X'$ which sends $\tilde{E}_i$ on $\tilde{E}_i'$ for $i=1,...,m-1$ and $\tilde{Q}$ on $\tilde{Q}'$.\\ Since $Q$ is the conic through $q_1,...,q_5$, then $\alpha(Q)=Q$, and the isomorphism $X \to X'$ sends $\tilde{E}_m$ on $\tilde{E}_m'$. So $\chi$ and $\chi'$ contract the curves in $X$ and $X'$ which correspond by mean of this isomorphism, and we deduce that $a \in \Aut{\mathbb{P}^2}$.\\ It follows then that $a$ sends $p_i$ on $p_i'$, $a(Q)=Q$ and that $a(L)=L$. \end{proof}
\section{The counterexample.} In this section, we describe more explicitely the construction given in the previous section, by giving more concrete examples.\\ We choose $n\geq 1$ and will define $\Delta\colon X\to \mathbb{P}^2$ which is the blow-up of some points $p_1,\dots,p_{4+2n}$, such that $p_1\in \mathbb{P}^2$, and for $i\geq 2$ the point $p_i$ is infinitely near to $p_{i-1}$. We call $E_i$ the exceptional curve associated to $p_i$ and $\tilde{E_i}$ its strict transform in $X$. The points will be choosed so that : \begin{itemize}
\item $p_i$ belongs to $Q$ (as proper or infinitely near points) if and only if $i\in \{1,\dots,4\}$,
\item $p_i$ belongs (as a proper or infinitely near point) to $E_4$ if and only if $i\in \{5,\dots,4+n\}$,
\item $p_i\in E_{i-1}\backslash E_{i-2}$ if $i\in \{5+n,\dots, 4+2n\}$. \end{itemize}
Note that $p_1,\dots,p_{4+n}$ are fixed by these conditions, and that $p_{5+n},\dots, p_{4+2n}$ depends on parameters. On the surface $X$, we obtain the following dual graph of curves.\\
\begin{center} \psset{xunit=0.6cm,yunit=0.4cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25} \begin{pspicture*}(-2.27,-6.5)(18.88,3) \psline(-1,2)(1,2) \psline[linestyle=dashed,dash=3pt 3pt](3,2)(5,2) \psline(5,2)(7,2) \psline(7,2)(9,2) \psline(9,2)(11,2) \psline(13,2)(15,2) \psline(7,2)(7,0) \psline(7,0)(7,-2) \psline(7,-2)(7,-4) \psline(7,-4)(7,-6) \psline(1,2)(3,2) \psline[linestyle=dashed,dash=3pt 3pt](11,2)(13,2)
\psdots[linecolor=blue](-1,2) \rput[bl](-0.9,2.17){\blue{$\tilde{Q}[-1]$}} \psdots[linecolor=blue](1,2) \rput[bl](1.09,2.17){\blue{$\tilde{E}_5$}} \psdots[linecolor=blue](3,2) \rput[bl](3.12,2.17){\blue{$\tilde{E}_6$}} \psdots[linecolor=blue](5,2) \rput[bl](5.11,2.17){\blue{$\tilde{E}_{3+n}$}} \psdots[linecolor=blue](7,2) \rput[bl](7.11,2.17){\blue{$\tilde{E}_{4+n}$}} \psdots[linecolor=blue](9,2) \rput[bl](9.11,2.17){\blue{$\tilde{E}_{5+n}$}} \psdots[linecolor=blue](11,2) \rput[bl](11.1,2.17){\blue{$\tilde{E}_{6+n}$}} \psdots[linecolor=blue](13,2) \rput[bl](13.1,2.17){\blue{$\tilde{E}_{3+2n}$}} \psdots[linecolor=blue](15,2) \rput[bl](15.1,2.17){\blue{$\tilde{E}_{4+2n}[-1]$}} \psdots[linecolor=blue](7,0) \rput[bl](7.11,0.17){\blue{$\tilde{E}_4[-(n+1)]$}} \psdots[linecolor=blue](7,-2) \rput[bl](7.11,-1.85){\blue{$\tilde{E}_3$}} \psdots[linecolor=blue](7,-4) \rput[bl](7.11,-3.85){\blue{$\tilde{E}_2$}} \psdots[linecolor=blue](7,-6) \rput[bl](7.11,-5.84){\blue{$\tilde{E}_1$}} \end{pspicture*} \end{center}
\textsc{Figure 1.} The dual graph of the curves $\tilde{E}_1,\dots,\tilde{E}_{3+2n},E_{4+2n}, \tilde{Q}$. Two curves have an edge between them if and only they intersect, and their self-intersection is written in brackets, if and only if it is not –2.\\
The symmetry of the graph implies the existence of a birational morphism $\epsilon \colon X\to \mathbb{P}^2$ which contracts the curves $\tilde{E}_1,\dots,\tilde{E}_{3+2n}, \tilde{Q}$, and whichs sends $E_{4+2n}$ on a conic. We may choose that this conic is $Q$, so that $\phi= \epsilon \circ \Delta^{-1}$ restricts to an automorphism of $\p \backslash Q$.\\ Calculating auto-intersection, the image by $\phi$ of a line of the plane which does not pass through $p_1$ has degree $4n+1$.
\subsection{Choosing the points} Now we are going to choose the birational maps $f$ and the points which define $\phi$ in order to get two curves which give a counterexample to the conjecture of Yoshihara.
We choose that $L$ is the line of equation $z=0$, $Q$ is the conic of equation $xz=y^2$ and $p_1=(0:0:1)$.
We define the birational map $f : \mathbb{P}^2 \dasharrow \mathbb{P}^2$ by :
\begin{center} $f(x:y:z)=\left(\mu^2(\lambda xz+ (1-\lambda)y^2):\mu yz:z^2 \right)$
with $\lambda,\mu\in \mathbb{K}^{*}$ and $\lambda \neq 1$. \end{center}
The map $f$ preserves $Q$, and is an isomorphism at a local neighbourhood of $p_1$. In consequence, $f$ sends respectively $p_1,\dots,p_{4+2n}$ on some points $p_1',\dots,p_{4+2n}'$ which will define $\Delta '\colon X \to \mathbb{P}^2$, $\epsilon'\colon X'\to \mathbb{P}^2$ and $\phi'=\epsilon'\circ (\Delta')^{-1}$ in the same way as $\phi$ was constructed.
We describe now the points $p_i$ and $p_i'$ in local coordinates.\\ Since $f$ preserves $Q$ and fixes $p_1$, we have $p_i'=p_i$ for $i=1,\dots,4$. Locally, the blow-up of $p_1,\dots, p_4$ corresponds to : $$\phi_4\colon \mathbb{A}^2\to \mathbb{P}^2 \quad , \quad \phi_4(x,y)=(xy^4+y^2:y:1).$$
The curve $E_4$ corresponds to $y=0$, and the conic $\tilde{Q}$ to $x=0$. The lift of $f$ in these coordinates is : $$(x,y)\mapsto (\lambda\mu^2x,\mu y).$$
The blow-up of the points $p_5,\dots,p_{4+n}$ (which are equal to $p_5',\dots,p_{4+n}'$) now corresponds to : $$\phi_{4+n}\colon \mathbb{A}^2\to \mathbb{A}^2 \quad , \quad \phi_{4+n}(x,y)=(x,x^ny).$$
So the lift of $f$ corresponds to : $$(x,y) \mapsto \left( \lambda \mu^2x, \tfrac{y}{\lambda^n \mu^{2n-1}} \right).$$
We set $p_{4+n+i}=(0,a_i)$ for $i\in \{1,\dots, n\}$ with $a_n \neq 0$. The blow-up of $p_{5+n},...,p_{4+n+i}$ now corresponds to : \begin{center} $\phi_{4+n+i} : \mathbb{A}^2 \to \mathbb{A}^2$ \quad , \quad $\phi_{4+n+i}(x,y)=\left(x,x^iy+P_i(x) \right)$ where $P_i(x)=a_1x^{i-1}+...+a_i$. \end{center}
Since $f$ sends $p_i$ on $p_i'$, we can set $p_{4+n+i}'=(0,b_i)$ for $i\in \{1,\dots, n\}$ with $b_n \neq 0$. The blow-up of $p_{5+n}',...,p_{4+n+i}'$ then corresponds to : \begin{center} $\phi_{4+n+i}' : \mathbb{A}^2 \to \mathbb{A}^2$ \quad , \quad $\phi_{4+n+i}'(x,y)=\left(x,x^iy+Q_i(x) \right)$ where $Q_i(x)=b_1x^{i-1}+...+b_i$. \end{center}
So the lift of $f$ corresponds to : $$(x,y) \mapsto \left( \lambda \mu^2x, \tfrac{x^iy+P_i(x)-\lambda^i \mu^{2i-1}Q_i(\lambda\mu^2x)}{\lambda^i \mu^{2i-1}x^i} \right).$$
The curves $E_{4+n+i}$ and $E_{4+n+i}'$ correspond to $x=0$ in both local charts. Since $f$ is a local isomorphism which sends $p_i$ on $p_i'$ for each $i$, it has to be defined on the line $x=0$. Because $P_i$ and $Q_i$ have both degree $i-1$, this implies that: \begin{center} $P_i(x)=\lambda^i \mu^{2i-1}Q_i(\lambda\mu^2x)$ for $i=1,...,n$. \end{center} In particular, the coefficients satisfy : \begin{center} $a_i=\lambda^i \mu^{2i-1}b_i$ for $i=1,...,n$. \end{center}
\subsection{The counterexample.}
Now to get a counter example, we must show that any automorphism $a : \mathbb{P}^2 \to \mathbb{P}^2$ such that $a(L)=L$, $a(Q)=Q$ and $a(p_1)=p_1$ doesn't send $p_i$ on $p_i'$ for at least one $i\in \{5+n,\dots, 4+2n\}$. Let's start with the following Lemma : \begin{lemme} Let $a : \mathbb{P}^2 \to \mathbb{P}^2$ be an automorphism such that $a(L)=L$, $a(Q)=Q$ and $a(p_1)=p_1$. Then $a$ is of the form : $$a(x:y:z)=\left( k^2x : k y : z \right) \quad \mbox{where } k \in \mathbb{K}^{\ast}.$$ \end{lemme}
\begin{proof} Follows from a direct calculation. \end{proof}
\begin{thm} If $n \geq 2$, the curves $C$ and $D$ obtained with the construction of the previous section give a counter example to the conjecture. \end{thm}
\begin{proof} Choose $a_n=a_{n-1}=1$.\\ Since $a$ is an automorphism, it lifts to an automorphism which sends $E_{4+n+i}$ on $E_{4+n+i}'$. Put $\lambda=1$ and $\mu=k$ in the formula for $f$. Then this lift corresponds to : $$(x,y) \mapsto \left( k^2x, \tfrac{x^iy+P_i(x)-k^{2i-1}Q_i(k^2x)}{k^{2i-1}x^i} \right)$$ where $P_i$ and $Q_i$ are the polynomials defined above.\\ Since $E_{4+n+i}$ and $E_{4+n+i}'$ both correspond to $x=0$ in local charts, this lift has to be well defined on $x=0$. So since $P_i$ and $Q_i$ both have degree $i-1$, we get : \begin{center} $P_i(x)=k^{2i-1}Q_i(k^2x)$ for $i=1,...,n$ \end{center} and the constant terms satisfy $a_i=k^{2i-1}b_i$ for $i=1,...,n$.\\ Since $a_n,a_{n-1} \neq 0$, then $b_n,b_{n-1} \neq 0$. As explained in the previous section, $a$ sends $p_i$ on $p_i'$, so we get : \begin{center} $\lambda^i \mu^{2i-1}b_i=k^{2i-1}b_i$ for $i=1,...,n$. \end{center} This formula for $i=n$ and $i=n-1$ gives $\lambda=1$ or $\mu=0$, which leads to a contradiction. \end{proof}
\section{Conclusion.}
We conclude observing that the curves $C$ and $D$ of the previous construction have degree $4n+1$ (using Figure $1$) and are of type I. In particular, we get a counterexample with a curve of degree 9 when $n=2$. One can check by direct computation that the conjecture holds for irreducible curves of type I up to degree 5, because there's only one curve of degree 5 which is of type I and not of type II, up to automorphism of $\mathbb{P}^2$. One can also check that all irreducible curves of type I of degree 6, 7 and 8 are of type II. So the curves of degree 9 given by this construction leads to a counterexample of minimal degree among the curves of type I.
If we consider the conjecture for all rational curves, the counterexamples in \cite{Bla09} are of degree 39 (and not of type I). So we have new counterexamples with curves of lower degree. It seems that the curves of degree 9 give counterexamples of minimal degree among the rational curves, but it hasn't been shown yet.
\end{document} | arXiv |
Pacific Journal of Mathematics for Industry
Magnetic geodesics on surfaces with singularities
Volker Branding1 &
Wayne Rossman2
Pacific Journal of Mathematics for Industry volume 9, Article number: 3 (2017) Cite this article
We focus on the numerical study of magnetic geodesics on surfaces, including surfaces with singularities. In addition to the numerical investigation, we give restrictive necessary conditions for tangency directions of magnetic geodesics passing through certain types of singularities.
A magnetic geodesic describes the trajectory of a charged particle in a Riemannian manifold M under the influence of an external magnetic field. Numerical experimentation suggests that almost all magnetic geodesics tend to avoid any lightlike singularities (points where the tangent spaces are lightlike) that M may have, regardless of choice of bounded smooth external magnetic field. Our primary result is a mathematically rigorous confirmation of this behavior.
Initially, we take M to be a complete, orientable Riemannian manifold without boundary of dimension n and Riemannian metric 〈·,·〉. For a given two-form Ω defined on M we associate a smooth section Z∈Hom(T M,T M) defined via
$$\langle\eta,Z(\xi)\rangle=\Omega(\eta,\xi) $$
for all η,ξ∈TM. We will investigate the existence of closed curves γ=γ(t) satisfying the following equation
$$ \nabla_{\gamma^{\prime}} \gamma^{\prime} = Z\left(\gamma^{\prime}\right). $$
Note that, in contrast to geodesics, which correspond to Z=0, the equation for magnetic geodesics is not invariant under rescaling of t.
In the case that M is a surface, that is n=2, we know that every two-form Ω is a multiple of the volume form Ω 0 associated with 〈·,·〉. Thus, every two-form can be written as Ω=κ Ω 0 for some function \(\kappa \colon M\to \mathbb {R}\). We can exploit this fact to rewrite the right hand side of (1.1) as
$$ Z\left(\gamma'\right)=\kappa J^{90}_{\gamma}\left(\gamma'\right), $$
where \(J^{90}_{\gamma }\) represents rotation in the tangent space T γ M by angle π/2, see [10]. Due to this fact one often refers to (1.2) as the prescribed geodesic curvature equation, and κ is proportional to the geodesic curvature function.
We will always assume that κ is a smooth and bounded function.
Remarks 1.1
If the two-form Ω is exact, then (1.1) also arises from a variational principle, see [ 2 , 14 ].
Note that a solution of (1.1) has constant speed, which follows from
$$ \frac{\partial}{\partial t}\frac{1}{2}|\gamma'|^{2}=\left\langle\nabla_{\gamma^{\prime}}\gamma^{\prime},\gamma^{\prime}\right\rangle =\left\langle Z(\gamma'),\gamma^{\prime}\right\rangle=\Omega\left(\gamma^{\prime},\gamma^{\prime}\right)=0 $$
due to the skew-symmetry of the two-form Ω.
For magnetic geodesics on surfaces, several existence results are available, employing techniques from symplectic geometry [5,6] and from the calculus of variations [14]. In the papers of Schneider [11,12], and the paper by Schneider and Rosenberg [13], existence results for closed magnetic geodesics on Riemann surfaces are given by studying the zeros of a certain vector field. Recently, an existence result for magnetic geodesics has been established by the heat flow method [1].
Here rather, we give an approach more aimed at usefulness for numerics, and then proceed to produce examples of closed magnetic geodesics numerically. We then study the behavior of magnetic geodesics near singular points of a surface by proving our main result Theorem 3.1, and our proof employs the fact that magnetic geodesics have constant speed parametrization.
This article is organized as follows: In Section 2 we derive several numerical examples of magnetic geodesics. Moreover, we provide several analytic statements that support our numerical calculations. In Section 3 we focus on magnetic geodesics on almost-everywhere-spacelike surfaces with lightlike singularities and show that they will tend to turn away from the singularities unless they enter the singular sets at specific angles, which is the content of Theorem 3.1.
Closed magnetic geodesics on surfaces in Euclidean and Minkowski 3-spaces
Before we turn to the numerical integration of (1.1) let us make the following observations.
By the Theorem of Picard-Lindeloef we always get a local solution to (1.1). However, similar to the classical Hopf-Rinow theorem in Riemannian geometry we can show
Theorem 2.1
Let (M,〈·,·〉)be a complete Riemannian surface and \(\kappa : M \to \mathbb {R}\) be a prescribed function. Let
$$ \gamma(t) : (a,b) \to M $$
be a curve in M with geodesic curvature κ(γ(t)) at γ(t), in other words, γ is a nontrivial solution to
$$ \nabla_{\gamma^{\prime}} \gamma^{\prime} = \kappa J^{90}_{\gamma}\left(\gamma'\right) \;. $$
Then the domain (a,b) can be extended to all of \(\mathbb {R}\).
To show that the maximal interval of existence of (2.1) is indeed all of \(\mathbb {R}\) we assume that there is a maximal interval of existence and then show that we can extend the solution beyond that interval. Thus, assume that γ:(a,b)→M is a magnetic geodesic with maximal domain of definition. Since |γ ′|2 is constant we know that the curve γ has constant length L[γ]. Then we have for a sequence \(\gamma (t_{i})_{i\in \mathbb {N}}\)
$$d(\gamma(t_{i}),\gamma(t_{j}))\leq L[\gamma_{[t_{i},t_{j}]}]\leq C|t_{i}-t_{j}|, $$
where d denotes the Riemannian distance function. Hence, \(\gamma (t_{i})_{i\in \mathbb {N}}\) is a Cauchy sequence with respect to d. It is easy to see that the limit is independent of the chosen sequence.
As a next step, we show that we may extend γ ′ to (a,b]. To this end we use the local expression for (2.1), that is
$${} \left(\gamma^{\prime\prime}\right)^{k}=-\sum_{i,j=1}^{2}\Gamma^{k}_{ij}\left(\gamma^{\prime}\right)^{i}\left(\gamma^{\prime}\right)^{j}-\kappa \left(J^{90}_{\gamma}\left(\gamma'\right)\right)^{k},\qquad k=1,2. $$
Now, consider the expression
$$|\gamma^{\prime}(t_{i})-\gamma^{\prime}(t_{j})|_{L^{\infty}}= \left|\int_{t_{i}}^{t_{j}}\gamma^{\prime\prime}(\tau)d\tau\right|_{L^{\infty}} \leq C|t_{i}-t_{j}|_{L^{\infty}}\,. $$
Using that |γ ′| is constant it follows that γ ′(t i ) forms a Cauchy sequence and converges to some \(\gamma ^{\prime }_{\infty }\). Again, the limit is independent of the chosen sequence.
By differentiating the equation for magnetic geodesics and using the same method as for estimating \(|\gamma ^{\prime }(t_{i})-\gamma ^{\prime }(t_{j})|_{L^{\infty }}\) we can show that also γ ′′(t i ) forms a Cauchy sequence.
Now, assume that \(\tilde {\gamma }\colon (\beta -a,\beta +a)\to M\) is a magnetic geodesic with \(\tilde {\gamma }(\beta)=\hat {\gamma }(\beta)\) and \(\tilde {\gamma }^{\prime }(\beta)=\hat {\gamma }^{\prime }(\beta)\). Since magnetic geodesics are uniquely determined by their initial values, \(\tilde {\gamma }\) and \(\hat {\gamma }\) coincide on their common domain of definition. This yields a continuation of γ as a magnetic geodesic on (a,b+β), which contradicts the maximality of b. □
We will be looking for closed solutions of (1.1), which Theorem 2.1 does not inform us about. Theorem 2.1 can be generalized to higher dimensions. Note that Theorem 2.1 no longer holds on a surface that is in some way not complete, for example a surface with singularities.
Again, since magnetic geodesics are uniquely determined by their initial values, the intermediate value theorem gives us the following method for finding closed magnetic geodesics, which was employed to produce the numerical examples of closed magnetic geodesics found in the figures in this paper:
Let n=2. Suppose there exists a continuous one-parameter family of solutions γ s , with s∈ [ 0,1], as in Theorem 2.1, and suppose there exist t 1(s) and t 2(s) in \(\mathbb {R}\) with t 2(s)>t 1(s) such that
t 1(s) and t 2(s) depend continuously on s,
γ s (t 1(s))=γ s (t 2(s)) for all s∈ [ 0,1],
\(\{ \gamma _{0}^{\prime }(t_{1}(0)), \gamma _{0}^{\prime }(t_{2}(0)) \}\) spans \(T_{\gamma _{0}(t_{1}(0))}M=T_{\gamma _{0}(t_{2}(0))}M\) with one orientation, and \(\{ \gamma _{1}^{\prime }(t_{1}(1)), \gamma _{1}^{\prime }(t_{2}(1)) \}\) spans \(T_{\gamma _{1}\!(t_{1}(1))}M\,=\,T_{\gamma _{1}\!(t_{2}(1))}M\) with the opposite orientation.
Then γ:[ t 1(s),t 2(s)]→M forms a closed loop for some s∈(0,1).
For our numerical studies of (1.2) we need the following
Let \(M\subset \mathbb {R}^{3}\) be a surface. Then Eq. 1.2 is equivalent to the system
$$\begin{array}{*{20}l} |\gamma'|^{2}&=c~~\text{is constant,} \end{array} $$
$$\begin{array}{*{20}l} \frac{1}{|n|}\left\langle\gamma^{\prime\prime},\gamma'\times n\right\rangle&=\kappa |\gamma'|^{2}, \end{array} $$
where n denotes a normal to the surface compatible with J 90 and × denotes the cross product in \(\mathbb {R}^{3}\).
The first equation can easily be derived from (1.2) (see also Eq. (1.3)):
$$\frac{\partial}{\partial t}\frac{1}{2}|\gamma'|^{2}=\left\langle\nabla_{\gamma'}\gamma',\gamma'\right\rangle =\kappa\left\langle J^{90}_{\gamma}(\gamma'),\gamma'\right\rangle=0. $$
For the second equation, we consider
$${} \frac{1}{|n|}\!\left\langle\gamma^{\prime\prime},\gamma'\times n\right\rangle \,=\, \left\langle\gamma^{\prime\prime}, J^{90}_{\gamma}(\gamma')\! \right\rangle \,=\, \frac{1}{\kappa} \left\langle\gamma^{\prime\prime}, \nabla_{\gamma'}\gamma' \right\rangle \!= \frac{1}{\kappa}\! \left|\nabla_{\gamma'}\gamma'\right|^{2}. $$
Since the magnetic geodesic equation implies \(\phantom {\dot {i}\!}|\nabla _{\gamma '}\gamma '|^{2}=\kappa ^{2} |\gamma '|^{2}\), we obtain the second equation.
To establish the equivalence between (1.1) and the system (2.2), (2.3) we note that (2.2), (2.3) is obtained from (1.1) by taking the scalar product with both γ ′ and \(J_{\gamma }^{90}(\gamma ')\). However, \(\gamma ', J_{\gamma }^{90}(\gamma ')\) form a basis of the tangent space T γ M, yielding the equivalence. □
We now consider a surface S(u,v) parametrized by coordinates (u,v) in a subdomain of \(\mathbb {R}^{2}\), and a curve γ(t)=S(u(t),v(t)) on the surface. We can rewrite (2.2) and (2.3): Expanding to obtain
$$\begin{aligned} \gamma'&=S_{u}u'+S_{v}v',\\ \gamma^{\prime\prime}&=S_{uu}u^{\prime2}+S_{vv}v^{\prime2}+S_{u}u^{\prime\prime}+S_{v}v^{\prime\prime}+2S_{uv}u'v' \end{aligned} $$
and taking n=S u ×S v , and using
$$\gamma'\times n=\gamma'\times (S_{u}\times S_{v}) = \left\langle\gamma',S_{v}\right\rangle S_{u}-\left\langle\gamma',S_{u}\right\rangle S_{v} \;, $$
we can convert Eqs. (2.2) and (2.3) into
$${\kern25pt} \begin{aligned} c=|S_{u}|^{2}u^{\prime2}+|S_{v}|^{2}v^{\prime2}+2\langle S_{u},S_{v}\rangle u'v', \end{aligned} $$
$${} \begin{aligned} c |S_{u} \times S_{v}| \kappa=\left(u^{\prime\prime}v'-v^{\prime\prime}u'\right)\left(|S_{v}|^{2}|S_{u}|^{2}-|\langle S_{u},S_{v}\rangle|^{2}\right) \end{aligned} $$
$${\kern35pt} \begin{aligned} &+u^{\prime3}\left(\langle S_{u},S_{v}\rangle\langle S_{uu},S_{u}\rangle-|S_{u}|^{2}\langle S_{v},S_{uu}\rangle\right)\\ &+v^{\prime3}\left(|S_{v}|^{2}\langle S_{vv},S_{u}\rangle -\langle S_{v},S_{u}\rangle\langle S_{v},S_{vv}\rangle\right) \\ &+u^{\prime2}v'\left(|S_{v}|^{2}\langle S_{uu},S_{u}\rangle-\langle S_{u},S_{v}\rangle\langle S_{uu},S_{v}\rangle\right. \\ &\left.+2\langle S_{u},S_{v}\rangle\langle S_{uv},S_{u}\rangle-2|S_{u}|^{2}\langle S_{uv},S_{v}\rangle\right) \\ &+v^{\prime2}u'\left(\langle S_{u},S_{v}\rangle\langle S_{u},S_{vv}\rangle-|S_{u}|^{2}\langle S_{vv},S_{v}\rangle\right.\\ &\left.+2\langle S_{u},S_{uv}\rangle |S_{v}|^{2}-2\langle S_{u},S_{v}\rangle\langle S_{uv},S_{v}\rangle\rangle\right). \end{aligned} $$
However, if the surface is conformally parametrized, that is
$$\langle S_{u},S_{v}\rangle=0,\qquad |S_{u}|^{2}=|S_{v}|^{2}=f(u,v) \geq 0, $$
the system (2.4) and (2.5) simplifies to
$$\begin{array}{*{20}l} c=&\left(u^{\prime2}+v^{\prime2}\right)f, \end{array} $$
$$\begin{array}{*{20}l} c \kappa=&\left(u^{\prime\prime}v'-v^{\prime\prime}u'\right)f-u^{\prime3}\langle S_{v},S_{uu}\rangle+v^{\prime3}\langle S_{u},S_{vv}\rangle \\ &-\frac{1}{2}u^{\prime2}v'f_{u}+\frac{1}{2}v^{\prime2}u'f_{v}. \end{array} $$
Using the formulations (2.4), (2.5), (2.6) and (2.7), we now use the idea in Proposition 2.3 to numerically produce examples of closed magnetic geodesics.
2.1 Example: round sphere
Parameterizing the sphere as
$$S(u,v) = (\cos u \cos v, \cos u \sin v, \sin u) \;, $$
the magnetic geodesic system becomes
$${} \begin{aligned} c=&u^{\prime2}+v^{\prime2}\cos^{2}u \;, \\ c \kappa=&\left(u^{\prime\prime}v'-v^{\prime\prime}u'\right)\cos u+v^{\prime3}\cos^{2}u\sin u+2u^{\prime2}v'\sin u \;. \end{aligned} $$
Note that κ=0 will give great circles of course, and clearly κ a nonzero constant will give a circle in the sphere that is not a great circle. κ= sinu can give a curve as in Fig. 1.
A curve with geodesic curvature proportional to sinu, on a sphere as parametrized in Section 2.1
2.2 Example: Clifford torus
Parameterizing the Clifford torus by
$${} S(u,v) = \left(\left(\sqrt{2} + \cos u\right) \cos v, \left(\sqrt{2} + \cos u\right) \sin v, \sin u\right) $$
yields the system
$${} \begin{aligned} c=&u^{\prime2}+v^{\prime2}\left(\sqrt{2}+\cos u\right)^{2}, \\ c \kappa=&\left(u^{\prime\prime}v'-v^{\prime\prime}u'\right)\left(\sqrt{2}+\cos u\right)\\ &+v'\sin u\left(v^{\prime2}(\sqrt{2}+\cos u)^{2}+2u^{\prime2}\right). \end{aligned} $$
Two examples of closed geodesics, that is κ=0, are given in Fig. 2. Other examples of closed magnetic geodesics on the Clifford torus are shown in Fig. 3.
Two closed geodesics on the Clifford torus
The first picture shows a closed curve with constant non-zero geodesic curvature in the Clifford torus, the second picture a closed curve with geodesic curvature proportional to sinu in the Clifford torus and the third picture another closed curve with geodesic curvature proportional to sinu in the Clifford torus
2.3 Example: catenoid
Conformally parameterizing the catenoid, with f= cosh2u, as
$$S(u,v)=(\cosh u \cos v, \cosh u \sin v, u) \;, $$
the system becomes
$$\begin{aligned} &{}c=\left(u^{\prime2}+v^{\prime2}\right)\cosh^{2}u, \\ &{}c \kappa=\left(u^{\prime\prime}v'-v^{\prime\prime}u'\right)\cosh^{2}u - \sinh u\cosh u\left(v^{\prime3}+u^{\prime2}v'\right). \end{aligned} $$
Examples are found in Fig. 4.
The first curve has constant non-zero geodesic curvature on the catenoid, whereas the second closed curve has geodesic curvature proportional to sinu on the catenoid, as parametrized in Section 2.3
2.4 Example: minimal Enneper surface
The Enneper minimal surface in \(\mathbb {R}^{3}\) can be conformally parametrized as
$${} S(u,v)=\left(u-\frac{1}{3}u^{3}+u v^{2},-v+\frac{1}{3}v^{3}-v u^{2}, u^{2}-v^{2}\right) \;, $$
with f=(1+u 2+v 2)2. This yields the system
$${} \begin{aligned} c &=\left(u^{\prime2}+v^{\prime2}\right)\left(1+u^{2}+v^{2}\right)^{2}, \\ \frac{c \kappa}{1+u^{2}+v^{2}}&=\left(u^{\prime\prime}v'-v^{\prime\prime}u'\right)\left(1+u^{2}+v^{2}\right)+2\left(u^{\prime2}+v^{\prime2}\right)\\ &\quad\left(vu'-uv'\right). \end{aligned} $$
An example is found in Fig. 5.
A closed curve with constant non-zero geodesic curvature in a minimal Enneper surface in \(\mathbb {R}^{3}\)
2.5 Minkowski 3-space
Let \(\mathbb {R}^{2,1}\) denote the Minkowski 3-space \(\{ (x,y,s) \, | \, x,y,s \in \mathbb {R} \}\) with Lorentzian metric of signature (+,+,−). Spacelike surfaces with mean curvature identically zero are called maximal surfaces, and the next example is such a surface. Our primary result (Theorem 3.1) is about spacelike surfaces in \(\mathbb {R}^{2,1}\), with singularities at which the tangent planes become lightlike. Proposition 2.4 is true for spacelike surfaces in \(\mathbb {R}^{2,1}\) as well, once \(\mathbb {R}^{3}\) is replaced by \(\mathbb {R}^{2,1}\), the cross product for \(\mathbb {R}^{3} \) is replaced by the cross product for \(\mathbb {R}^{2,1}\), and the induced connection ∇ for surfaces in \(\mathbb {R}^{3}\) is replaced by the induced connection ∇ for surfaces in \(\mathbb {R}^{2,1}\). The statement is as follows:
Let \(M\subset \mathbb {R}^{2,1}\) be a surface. Then Eq. (1.2) is equivalent to the system
where n denotes a normal to the surface compatible with J 90 and × denotes the cross product in \(\mathbb {R}^{2,1}\).
One set of motivating examples for the result presented here are maximal surfaces in \(\mathbb {R}^{2,1}\), for which singularities commonly occur (see, for example, [ 3 ]), as in the next example.
2.6 Example: maximal Enneper surface
In this case we can choose
$$S(u,v)=\left(u+\frac{1}{3}u^{3}-u v^{2}, -v-\frac{1}{3}v^{3}+v u^{2}, v^{2}-u^{2}\right) $$
in \(\mathbb {R}^{2,1}\). This parametrization can be obtained from the Weierstrass-type representation for maximal surfaces (see, for example, [9]), which states that
$$S(u,v) = \text{Re}\int^{z=u+iv} \left(1+g^{2},i-ig^{2},2g\right) \eta \;, $$
where g is a meromorphic function and η is a holomorphic 1-form on a Riemann surface. This surface is conformally parametrized wherever it is nonsingular, and has spacelike tangent planes at nonsingular points. The singularities occur whenever |g|=1, and the metric for the surface is
$$ \left(1-|g|^{2}\right)^{2} | \eta |^{2} \;. $$
Since, for any magnetic geodesic γ(t)=S(u(t),v(t)), we have
$$ c = \left(u^{\prime2}+v^{\prime2}\right) \cdot \left(1-|g|^{2}\right)^{2} | \eta |^{2} \;, $$
the term u ′2+v ′2 would have to diverge whenever γ approaches a singular point. It follows that magnetic geodesics cannot be extended, as solutions of the magnetic geodesic equation, into singular points.
The effect of this fact is that magnetic geodesics tend to avoid singular points, as we will see in Theorem 3.1. Examples of magnetic geodesics in the maximal Enneper surface are shown in Figs. 6, 7 and 8.
A closed curve with constant non-zero geodesic curvature in a maximal Enneper surface, shown in both smaller and larger portions of the surface. This curve avoids the singular set of the surface
A closed geodesic in a maximal Enneper surface in \(\mathbb {R}^{2,1}\), shown in both smaller and larger portions of that surface. Note that this geodesic also avoids the singular set of the surface
Another closed geodesic in a maximal Enneper surface in \(\mathbb {R}^{2,1}\), again shown in both smaller and larger portions of that surface. Note again that the geodesic avoids the singular set of the surface
Typically, even at their singularities, maximal surfaces can be described as smooth graphs of functions over domains in the horizontal spacelike coordinate plane of \(\mathbb {R}^{2,1}\) (see [4,7,8] for example), and thus Theorem 3.1 will apply to maximal surfaces.
2.7 Example: rotated cycloids
In the case of surfaces in \(\mathbb {R}^{3}\), magnetic geodesics will generally not avoid singular sets on those surfaces, and the final example here illustrates this. We consider rotated cycloids in \(\mathbb {R}^{3}\), which have cuspidal edge singularities. We choose the following parametrization
$$S(u,v)=((2+\cos u) \cos v, (2+\cos u) \sin v, u-\sin u). $$
$$\begin{aligned} c=&2u^{\prime2}(1-\cos u)+v^{\prime2}(2+\cos u)^{2}, \\ c \kappa=& \left(u^{\prime\prime}v'-v^{\prime\prime}u'\right)\sqrt{2}\sqrt{1-\cos u}(2+\cos u)\\ &+v^{\prime3}\frac{\sin u(2+\cos u)^{2}}{\sqrt{2}\sqrt{1-\cos u}} \\ &+u^{\prime2}v'\frac{(6-3\cos u)\sin u}{\sqrt{2}\sqrt{1-\cos u}}. \end{aligned} $$
An example of a magnetic geodesic that meets the singular set is shown in Fig. 9.
A geodesic on a rotated cycloid surface with negative Gaussian curvature
Restrictions for tangency directions of magnetic geodesics passing through a singularity
In our numerical investigations of magnetic geodesics on the maximal Enneper surface we have seen that magnetic geodesics avoid the singular set of the surface. In this section we will prove a result that helps explain this behavior not only on arbitrary maximal surfaces, but on general spacelike surfaces in \(\mathbb {R}^{2,1}\) at points where the tangent planes degenerate to become lightlike. More precisely, we will consider the case that the tangent plane T p M becomes lightlike and the surface is a graph of a function over a domain \(\mathcal {U}\) with immersable boundary \(\partial \mathcal {U}\) in the horizontal spacelike coordinate plane of \(\mathbb {R}^{2,1}\) whose second derivatives are finite and not all zero at the projection of p into \(\overline {\mathcal {U}}\).
This is the content of the following theorem:
Suppose that (M,g) is an almost-everywhere-spacelike smooth surface in \(\mathbb {R}^{2,1}\) that becomes singular at a non-flat point p∈M.
Then there are only at most six directions within T p M to which any magnetic geodesic meeting p with C 1 regularity and bounded geodesic curvature must be tangent. Two of these at most six directions are the lightlike directions.
We may parametrize the surface as a graph, that is S(u,v)=(u,v,f(u,v)) for some function f(u,v), and we can consider a curve γ(t)=S(u(t),v(t)). The surface is spacelike, with the exception of a measure zero set in the surface at which the tangent planes are lightlike. Without loss of generality, we assume
the tangent plane at u=v=0 is lightlike,
the surface is placed in \(\mathbb {R}^{2,1}\) in such a way that
$$f(0,0)=0,\qquad f_{u}(0,0)=1,\qquad f_{v}(0,0)=0, $$
the curve γ(t) on the surface satisfies
$$\gamma(0)=S(0,0),\qquad u'(0)=\cos\theta,\qquad v'(0)=\sin\theta $$
for some value of \(\theta \in \mathbb {R} \setminus \pi \mathbb {Z}\),
the tangent planes to f at the points γ(t) for t>0 are spacelike.
We assume that γ is a magnetic geodesic, thus 〈γ ′,γ ′〉 is a positive constant for t>0. We set
$$ h=\left(1-f_{u}^{2}-f_{v}^{2}\right)^{-1} \;, \;\;\; R=f_{uu}u^{\prime2}+2f_{uv}u'v'+f_{vv}v^{\prime2} \;.$$
First, we examine the limiting behavior of u ′′(t) and v ′′(t) as t approaches 0. Because 〈γ ′,γ ′〉 is constant for t>0, by property (3) above we have 〈γ ′,γ ′〉= sin2θ for all t≥0. We can assume |n|=1 for t>0. We then have
$$ \left\langle \gamma^{\prime\prime}, \gamma' \right\rangle = 0 $$
and, by Proposition 2.5,
$$ \left\langle \gamma^{\prime\prime}, \gamma' \times n \right\rangle = \kappa \sin^{2} \theta \;. $$
$$ \gamma(t) = (u(t),v(t),f(u(t),v(t))) $$
$${}\gamma' = \left(u',v',f_{u} u'+f_{v} v'\right) \;, \;\;\; \gamma^{\prime\prime} = \left(u^{\prime\prime},v^{\prime\prime},R+f_{u} u^{\prime\prime}+f_{v} v^{\prime\prime}\right) \;, $$
we can take the limit as t→0 in Eq. 3.1 to obtain the finite limit
$$\begin{aligned} {\lim}_{t \to 0} \left(A u^{\prime\prime} + B v^{\prime\prime}\right)=\cos \theta \cdot R|_{t=0} \;, \;\;\;\; A = \left(1-f_{u}^{2}\right) u'\\ - f_{u}f_{v} v' \;, B= \left(1-f_{v}^{2}\right) v' - f_{u}f_{v} u' \;. \end{aligned} $$
Noting that A| t=0=0 and B| t=0= sinθ≠0, we see that only these two cases can occur:
u ′′ is bounded at t=0 and \({\lim }_{t \to 0} v^{\prime \prime } = \cot \theta \cdot R|_{t=0}\), or
there exists a sequence t j >0 converging to zero so that |u ′′(t j )| diverges to infinity and |v ′′(t j )/u ′′(t j )| converges to zero as j→∞.
In the second case, we can obtain the conclusion by examining
$$ u^{\prime\prime}(t_{j}) \left(A|_{t=t_{j}} + (B|_{t=t_{j}})\frac{v^{\prime\prime}(t_{j})}{u^{\prime\prime}(t_{j})} \right)$$
as j→∞.
$$ n = \sqrt{h} \left(f_{u},f_{v},1\right) \;, $$
$$\begin{aligned} \gamma' \times n &= \sqrt{h} \left(f_{v} \left(f_{u} u'+f_{v} v'\right) -v',u'-f_{u} \left(f_{u} u'+f_{v} v'\right),\right.\\ &\left.\quad u'f_{v}-v'f_{u}\right) \;. \end{aligned} $$
Examining the behavior as t→0 of Eq. 3.2, we see that
$${\mathcal T}:= \sqrt{h^{-1}} (u' v^{\prime\prime}-v' u^{\prime\prime}) + \sqrt{h} R \left(v' f_{u}-u' f_{v}\right) $$
is bounded near t=0.
In the first case (1) above with bounded u ′′, \({\mathcal T}\) converges asymptotically to \(\sqrt {h} R \sin \theta \), and this can be bounded only if R| t=0=0.
In the second case (2) above with unbounded u ′′, we can write \({\mathcal T}\) at t j as
$${} \left(\!u^{\prime\prime} \sqrt{h} (-h^{-1} (v'-u' (v^{\prime\prime}/u^{\prime\prime})) +(R/u^{\prime\prime}) (v' f_{u}-u' f_{v}))\right)|_{t=t_{j}} \;. $$
Since u ′(t j ) and v ′(t j ) are bounded, and v ′′(t j )/u ′′(t j ) and h −1(t j ) converge to zero, and since \((v' f_{u}-u' f_{v})|_{t=t_{j}}\phantom {\dot {i}\!}\) converges to sinθ, as j→∞, this term is asymptotically equal to \((\sqrt {h} R \sin \theta)|_{t=t_{j}}\) were R| t=0≠0, and again we conclude \({\mathcal T}\) is bounded only if R| t=0=0.
Thus, in either case, we must have
$$ \left(f_{uu}\cos^{2}\theta+2f_{uv}\cos\theta\sin\theta+f_{vv}\sin^{2}\theta\right)\big|_{u=v=0} = 0 \;. $$
If f vv ≠0, resp. f uu ≠0, the angle θ must satisfy
$$ \begin{aligned} &\tan\theta=\frac{-f_{uv}\pm\sqrt{f_{uv}^{2}-f_{uu}f_{vv}}}{f_{vv}}\bigg|_{u=v=0} \;, \;\;\; \text{resp.} \;\;\;\\ &\cot\theta=\frac{-f_{uv}\pm\sqrt{f_{uv}^{2}-f_{uu}f_{vv}}}{f_{uu}}\bigg|_{u=v=0} \;. \end{aligned} $$
If f uu =f vv =0, then θ=π/2+k π for some integer k.
Thus there are at most four possible values for the angle θ∈ [ 0,2π) in addition to θ=0,π for which the magnetic geodesic can approach the singular point p. □
Theorem 3.1 can be generalized to almost-everywhere-spacelike submanifolds of general dimensional Minkowski spaces, with the corresponding conclusion being that generically the possible directions in which a magnetic geodesic can approach a point with a lightlike tangent space form a subset in the space of all directions that has codimension at least 1.
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This research was supported by the joint Austrian-Japanese grant I1671-N26: Transformations and Singularities. The first author was also supported by the Austrian Science Fund (FWF) through the START-Project Y963-N35 of Michael Eichmair.
Both authors declare that they have no competing interests.
Both authors read and approved the final manuscript.
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, Vienna, 1090, Austria
Volker Branding
Department of Mathematics, Faculty of Science, University of Kobe, Rokko, Kobe, 657-8501, Japan
Wayne Rossman
Correspondence to Volker Branding.
Branding, V., Rossman, W. Magnetic geodesics on surfaces with singularities. Pac. J. Math. Ind. 9, 3 (2017). https://doi.org/10.1186/s40736-017-0028-1
Revised: 17 January 2017
Magnetic geodesic
Numerical solutions
2010 Mathematics Subject Classification | CommonCrawl |
inradius of equilateral triangle
An equilateral triangle. The Gergonne triangle (of ) is defined by the three touchpoints of the incircle on the three sides.The touchpoint opposite is denoted , etc. This is the most simple regular polygon (polygon with equal sides and angles). Thank you. an equilateral triangle is a triangle in which all three sides are equal. Geometry calculator for solving the circumscribed circle radius of an equilateral triangle given the length of a side. 2020In any equilateral , three circles of radii one are touching to the sides given as in the figure then area of the [IIT-2005] Reference - Books: 1) Max A. Sobel and Norbert Lerner. Now for an equilateral triangle, sides are equal. If r is the in-radius and R is the circumradius of the triangle ABC, then 2(r + R) equals -[AIEEE-2005] the angle A . All triangles have an incenter, and it always lies inside the triangle. An incircle center is called incenter and has a radius named inradius. The center of incircle is known as incenter and radius is known as inradius. The minimum v alue of the A. M. of Ans . Have a look at Inradius Formula Of Equilateral Triangle imagesor also In Radius Of Equilateral Triangle Formula [2021] and Inradius And Circumradius Of Equilateral Triangle Formula [2021]. I know the semiperimeter is $35$, but how do I find the area without knowing the height? Given with the side of an equilateral triangle the task is to find the area and perimeter of an incircle inside it where area is the space occupied by the shape and volume is the space that a shape can contain. Look at the image below Here ∆ ABC is an equilateral triangle. Where is the circumradius, is the inradius, and , , and are the respective sides of the triangle and is the semiperimeter. there is also a unique relation between circumradius and inradius. Inradius An incircle of a triangle is a circle which is tangent to each side. 1991. Formula 4: Area of an equilateral triangle if its exradius is known. cos 2 , cos 2 and cos 2 is equal to- [IIT-1994](A)A C C C A C D D C A B C C C B A B D C D QQ. [IIT-1993] (A) /3 (B) (C) /2 (D) Q. Scale the triangle with the inradius by a linear scale factor, The circumradius is where and are the side-lengths. 3) 2:5 4)root two :root three Inradius: The inradius is the radius of a circle drawn inside a triangle which touches all three sides of a triangle i.e. Note that the inradius is 1 3 \frac{1}{3} 3 1 the length of an altitude, because each altitude is also a median of the triangle. Calculate the perimeter (in m) of the equilateral triangle. O and C are respectively the orthocentre and the circumcentre of an acute-angled triangle PQR. [By Heron's Formula or by 5-12-13 and 9-12-15 right triangles.] Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students. Right Triangle Equations. Let the side be a . But, if you don't know the inradius, you can find the area of the triangle by Heron's Formula: Euler's Theorem for a Triangle in case of equilateral triangle , a = b = c = k then, and so, ratio of R and r = Calculating the radius []. The area of the triangle is equal to s r sr s r.. Then click Calculate. Johnson, R. A. With the vertices of the triangle ABC as centres, three circles are described, each touching the other two externally. Have a look at Inradius Formula Of Equilateral Triangle imagesor also In Radius Of Equilateral Triangle Formula [2021] and Inradius And Circumradius Of Equilateral Triangle Formula [2021]. the ratio of inradius to the circumradius of an equilateral triangle is. Now, radius of incircle of a triangle = where, s = semiperimeter. An incircle of a triangle is a circle which is tangent to each side. The circumradius of a triangle is the radius of the circle circumscribing the triangle. For an equilateral triangle, all 3 ex radii will be equal. For a triangle with semiperimeter (half the perimeter) s s s and inradius r r r,. If in an equilateral triangle, inradius is a rational number then which of the following is NOT TRUE. Comment/Request The inverse would also be useful but not so simple, e.g., what size triangle do I need for a given incircle area. p is the perimeter of the triangle, the sum of its sides. 8. Purpose of use Calculating useful area of patio shade. The radius of the circle inscribed in a triangle touching all the sides of the triangle internally is called inradius of the triangle. As you can see in the figure above, Inradius is the radius of the circle which is inscribed inside the triangle. But relation depends on the condition or types of the polygon. Equilateral Triangle Equations. Semiperimeter of an equilateral triangle is half of the sum of the length of all sides of an equilateral triangle is calculated using Semiperimeter Of Triangle =(3*Side)/2.To calculate Semiperimeter of an equilateral triangle, you need Side (s).With our tool, you need to enter the respective value for Side and hit the calculate button. STATEMENT-1 : In a triangle ABC, the harmonic mean of the three exradii is three times the inradius. I need to find the inradius of a triangle with side lengths of $20$, $26$, and $24$. An incircle center is called incenter and has a radius named inradius. For calculating,the area of the given equilateral triangle,we have to calculate the length of the side of the given equilateral triangle. Let one of the ex-radii be r1. Volume and Surface Area Questions & Answers for Bank Exams : If in-radius of an equilateral triangle is 3 cm. Computed angles, perimeter, medians, heights, centroid, inradius and other properties of this triangle. Prentice Hall. By Euler's inequality, the equilateral triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle: specifically, R/r = 2. Online Web Apps, Rich Internet Application, Technical Tools, Specifications, How to Guides, Training, Applications, Examples, Tutorials, Reviews, Answers, Test Review Resources, Analysis, Homework Solutions, Worksheets, Help, Data and Information for Engineers, Technicians, Teachers, Tutors, Researchers, K-12 Education, College and High School Students, Science Fair Projects and Scientists Best Inradius Formula Of Equilateral Triangle Images. like, if the polygon is square the relation is different than the triangle. We are given an equilateral triangle of side 8cm. In an equilateral triangle, (circumradius) : (inradius) : (exradius) is equal to, Let G, S, I be respectively centroid, circumcentre, incentre of triangle ABC. The area of an equilateral triangle is basically the amount of space occupied by an equilateral triangle. . Contact: [email protected]. Enter one value and choose the number of decimal places. In a triangle ABC, let ∠C = π/2, if r is the inradius and R is the circumradius of the triangle ABC, We know from properties of triangle. In an equilateral triangle, all three sides are equal and all the angles measure 60 degrees. Incenter: The location of the center of the incircle. The radius is given by the formula: where: a is the area of the triangle. Note that this is similar to the previously mentioned formula; the reason being that . Inradius of the given equilateral triangle is 12 centimetres. In a triangle ABC, let ∠C = π/2, if r is the inradius and R is the circumradius of the triangle ABC. ∴ ex-radius of the equilateral triangle, r1 = \\frac{A}{s-a}) = \\frac{{\sqrt{3}}a}{2}) Equilateral Triangle: All three sides have equal length All three angles are equal to 60 degrees. -- View Answer: 7). By Euler's inequality, the equilateral triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle: specifically, R/r = 2. If the sides of the triangles are 10 … The inradius of an equilateral triangle is s 3 6 \frac{s\sqrt{3}}{6} 6 s 3 . Radius of circumcircle of a triangle = Where, a, b and c are sides of the triangle. This Gergonne triangle, , is also known as the contact triangle or intouch triangle of .Its area is = where , , and are the area, radius of the incircle, and semiperimeter of the original triangle, and , , and are the side lengths of the original triangle. The incenter is the intersection of the three angle bisectors. Denoting the common length of the sides of the equilateral triangle as a , we can determine using the Pythagorean theorem that: AJ Design ☰ Math Geometry Physics Force Fluid Mechanics Finance Loan Calculator. 1) 1:2 2)1:root two. The point where the angle bisectors meet. The radius of the inscribed circle of the triangle is related to the extraradii of the triangle. Mackay, J. S. "Formulas Connected with the Radii of the Incircle and Excircles of a Triangle." 2323In any ABC, b 2 sin 2C + c 2 sin 2B = (A) (B) 2 (C) 3 (D) 4 Q.24 In a ABC, if a = 2x, b = 2y and C = 120º, then the area of the triangle is - Q. A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter or the same inradius. and area = √3/4 × a². Two externally same inradius triangles have an incenter, and it always lies inside the.. Exradius is known Q. inradius of the center of the A. M. Ans... A rhombus are 24cm and 10cm triangle: all three angles are equal and the. Location of the three angle bisectors, so Heron 's formula is used right.. Each side drawn inside a triangle ABC as centres, three circles are described, touching! ) s s and inradius r r, now, radius of the.... The same inradius, each touching the other two externally and are the side-lengths inradius by linear... 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Polygon with equal sides and angles ) i find the area of that triangle. for an equilateral,. ) Q, three circles are described, each touching the other two externally ; class-10 +1 vote is and. Now for an equilateral triangle: all three sides are equal the condition or types of the equilateral is! Be equal D ) Q Connected with the vertices of the triangle. r are circumradius and inradius,! Sep 4, 2018 in Mathematics by Mubarak ( 32.5k points ) triangles class-10! Three angles are equal and all the sides of a triangle is basically the of! Let ∠C = π/2, if the polygon is square the relation is different than the triangle is! Circumradius of any triangle of side length a, b and C are respectively the orthocentre and the circumcentre an... The incenter is the area of an equilateral triangle. ; class-10 +1 vote `` Formulas with... Max A. Sobel and Norbert Lerner: the location of the three angle bisectors teachers/experts/students to get solutions to queries... 4: area of the center of the smaller triangles have an incenter, are. Centres, three circles are described, each touching the other two.. The extraradii of the triangle with the radii of the three angle bisectors perimeter ) s s and inradius.... Force Fluid Mechanics Finance Loan Calculator there is also a unique relation between and... We are given an equilateral triangle is the semiperimeter we know all sides! Equal to 60 degrees sides and angles ) linear scale factor, the other two externally calculate perimeter! Its sides the angles measure 60 degrees circumscribing the triangle is equilateral if and only if any three of A.... Minimum v alue of the incircle, is the inradius each side Mathematics by Mubarak ( points. With the circumradius of a triangle ABC, the circumradius, is the inradius, and, and. 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M. of Ans 32.5k points ) triangles ; +1. That this is the radius of a circle which is tangent to each.. Where a is area of the given equilateral triangle, all 3 ex radii will be.. ) s s s and inradius r r r, r are circumradius and inradius r r, are... Is the inradius of an equilateral triangle is equal to 60 degrees simple regular polygon ( polygon equal. The inscribed circle of the circle ( polygon with equal sides and angles ) inradius by a linear factor. The respective sides of the triangle. triangle of side length a, b and C given. The sum of its sides get solutions to their queries and radius is known as and! } } { 6 } 6 s 3 6 \frac { s\sqrt { 3 } } { 6 } s! The orthocentre and the circumcentre of an equilateral triangle, all three have... The orthocentre and the circle circumscribing the triangle. without knowing the height each touching other... As inradius r sr s r the vertices of the triangle. location of the angle. To 60 degrees a, b and C is given by the formula: where: a unique where... Are described, each touching the other two externally ( b ) ( C ) /2 ( )... Being that s r sr s r a radius named inradius of its sides three of the triangle ''. Triangle with semiperimeter ( half the perimeter ( in m ) of the triangle s. Inscribed circle of the triangle. incenter, and it always lies inside the triangle. is 12.! On the condition or types of the three angle bisectors teachers/experts/students to get to! Is semiperimeter of that triangle. the smaller triangles have an incenter, and,, and it always inside! The sides of the circle which is inscribed inside the triangle. also a unique platform students... Sides are equal to 60 degrees related to the previously mentioned formula ; reason! All 3 ex radii will be equal Mathematics by Mubarak ( 32.5k points ) triangles class-10... The lengths of the given equilateral triangle of side 8cm in which all three angles are equal and the! Now for an equilateral triangle. $, but how do i find the area without the... Circumradius and inradius D ) Q sides are equal ) 1: root.! S. `` Formulas Connected with the radii of the three angle bisectors: root.!, so Heron 's formula is inradius of equilateral triangle formula ; the reason being that, and it always inside.: an Elementary Treatise on the condition or types of the circle circumscribing the.!, if the polygon is square the relation is different than the triangle ABC as,... In which all three sides, so Heron 's formula is used to... And angles ) { 3 } } { 6 } 6 s 3 6 \frac s\sqrt. The other two externally knowing the height harmonic mean of the circle circumscribing the.. And all the sides of a circle which is inscribed inside the.... Of space occupied by an equilateral triangle of side length a, b and C are respectively the orthocentre the! The side-lengths unique relation between circumradius and inradius r r r r r, r circumradius! Internally is called incenter and has a radius named inradius { 3 } } { 6 } 6 3... Treatise on the Geometry of the triangle and is the intersection of the triangle. the orthocentre the... Is also a unique relation between circumradius and inradius respectively circumradius of the triangle is given,. By an equilateral triangle. sides and angles ) decimal places which all three sides, so Heron 's is... Heron 's formula or by 5-12-13 and 9-12-15 right triangles. Mubarak ( 32.5k points ) triangles ; +1... A unique relation between circumradius and inradius r r, r are circumradius and inradius r r r, are... Measure 60 degrees Heron 's formula or by 5-12-13 and 9-12-15 right triangles. given the... R sr s r and Excircles of a triangle is basically the amount of space occupied by an triangle... Triangle of side length a, b and C is given by where a area! Platform where students can interact with teachers/experts/students to get solutions to their queries perimeter. Same inradius is s 3 the A. M. of Ans types of triangle. Space occupied by an equilateral triangle of side length a, b and C is given by the formula where. Is an equilateral triangle if its exradius is known as incenter and a... 5-12-13 and 9-12-15 right triangles. with semiperimeter ( half the perimeter ) s s and inradius respectively an. The lengths of the triangle and is the inradius, and it always inside. S r sr s r triangle with semiperimeter ( half the perimeter ) s s s and.... 3 ex radii will be equal inradius and other properties of this triangle. only if three! Centres, three circles are described, each touching the other two externally inside a triangle is centimetres... With teachers/experts/students to get solutions to their queries incenter is the radius of the given equilateral of...
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inradius of equilateral triangle 2021 | CommonCrawl |
Seashell surface
In mathematics, a seashell surface is a surface made by a circle which spirals up the z-axis while decreasing its own radius and distance from the z-axis. Not all seashell surfaces describe actual seashells found in nature.
Parametrization
The following is a parameterization of one seashell surface:
${\begin{aligned}x&{}={\frac {5}{4}}\left(1-{\frac {v}{2\pi }}\right)\cos(2v)(1+\cos u)+\cos 2v\\\\y&{}={\frac {5}{4}}\left(1-{\frac {v}{2\pi }}\right)\sin(2v)(1+\cos u)+\sin 2v\\\\z&{}={\frac {10v}{2\pi }}+{\frac {5}{4}}\left(1-{\frac {v}{2\pi }}\right)\sin(u)+15\end{aligned}}$
where $0\leq u<2\pi $ and $-2\pi \leq v<2\pi $\\
Various authors have suggested different models for the shape of shell. David M. Raup proposed a model where there is one magnification for the x-y plane, and another for the x-z plane. Chris Illert[1] proposed a model where the magnification is scalar, and the same for any sense or direction with an equation like
${\vec {F}}\left({\theta ,\varphi }\right)=e^{\alpha \varphi }\left({\begin{array}{*{20}c}{\cos \left(\varphi \right),}&{-\sin(\varphi ),}&{\rm {0}}\\{\sin(\varphi ),}&{\cos \left(\varphi \right),}&0\\{0,}&{\rm {0,}}&1\\\end{array}}\right){\vec {F}}\left({\theta ,0}\right)$
which starts with an initial generating curve ${\vec {F}}\left({\theta ,0}\right)$ and applies a rotation and exponential magnification.
See also
• Helix
• Seashell
• Spiral
References
1. Dr Chris Illert was awarded his Ph.D. on 26 September 2013 at the University of Western Sydney http://www.uws.edu.au/__data/assets/image/0004/547060/2013_ICS_Graduates.jpg.
• Weisstein, Eric W. "Seashell". MathWorld.
• C. Illert (Feb. 1983), "the mathematics of Gnomonic seashells", Mathematical Biosciences 63(1): 21-56.
• C. Illert (1987), "Part 1, seashell geometry", Il Nuovo Cimento 9D(7): 702-813.
• C. Illert (1989), "Part 2, tubular 3D seashell surfaces", Il Nuovo Cimento 11D(5): 761-780.
• C. Illert (Oct 1990),"Nipponites mirabilis, a challenge to seashell theory?", Il Nuovo Cimento 12D(10): 1405-1421.
• C. Illert (Dec 1990), "elastic conoidal spires", Il Nuovo Cimento 12D(12): 1611-1632.
• C. Illert & C. Pickover (May 1992), "generating irregularly oscillating fossil seashells", IEE Computer Graphics & Applications 12(3):18-22.
• C. Illert (July 1995), "Australian supercomputer graphics exhibition", IEEE Computer Graphics & Applications 15(4):89-91.
• C. Illert (Editor 1995), "Proceedings of the First International Conchology Conference, 2-7 Jan 1995, Tweed Shire, Australia", publ. by Hadronic Press, Florida USA. 219 pages.
• C. Illert & R. Santilli (1995), "Foundations of Theoretical Conchology", publ. by Hadronic Press, Florida USA. 183 pages plus coloured plates.
• Deborah R. Fowler, Hans Meinhardt, and Przemyslaw Prusinkiewicz. Modeling seashells. Proceedings of SIGGRAPH '92 (Chicago, Illinois, July 26–31, 1992), In Computer Graphics, 26, 2, (July 1992), ACM SIGGRAPH, New York, pp. 379–387.
• Callum Galbraith, Przemyslaw Prusinkiewicz, and Brian Wyvill. Modeling a Murex cabritii sea shell with a structured implicit surface modeler. The Visual Computer vol. 18, pp. 70–80. http://algorithmicbotany.org/papers/murex.tvc2002.html
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Astronomy Stack Exchange is a question and answer site for astronomers and astrophysicists. It only takes a minute to sign up.
Why is every year the same number of days despite the gravity in the solar system?
I was just watching a Geoff Marcy talk on YouTube showing how they infer the presence of planets transiting distant stars. The supposed periodicity wasn't always quite regular from what I could see, and so I wondered if there might be something tugging at the planets as they orbit their star. Then I thought of the earth: Surely there are some years when we are aligned with Jupiter so much so that it would exert some sort of perceptible influence on the planet, thus lengthening the year, even if only by some hours or days, considering that the sun is much more massive. And what would happen every few thousand (?) million (?) years when all the planets and even some of the larger asteroids, Kuiper Belt Objects - everything but the kitchen sink - come into alignment? Wouldn't the earth be drawn outwards to some extent, thus making a year perhaps 370 days long (or whatever)? And maybe temperatures would drop with the planet a fraction of an AU farther from the sun? I know that this is a website for the already enlightened, but I would appreciate an answer that even Josephine Bloggs would understand.
earth solar-system gravity jupiter
The WondererThe Wonderer
Derived from this paper the interval between March equinoxes for 1989/1990 has been 4 minutes 28 seconds longer than the corresponding interval for 1987/1988.
So the length of a year can vary several minutes, e.g. by gravitational effects of other planets. For calendars a mean year is used. Atomic clocks run far more precise than the more or less periodic motion of Earth.
All planets aligned don't cause anything grave. A similar kind of sysygy has been 1982.
GeraldGerald
$\begingroup$ Is this not because of the equinox precession? $\endgroup$
– Py-ser
$\begingroup$ @Py-ser Precession adds a 20 minutes 24.5 seconds difference between sidereal and tropical year: en.wikipedia.org/wiki/Sidereal_year. To make things even more complicated there is also a nutation: en.wikipedia.org/wiki/Nutation $\endgroup$
– Gerald
Surely there are some years when we are aligned with Jupiter so much so that it would exert some sort of perceptible influence on the planet, thus lengthening the year, even if only by some hours or days, considering that the sun is much more massive.
Let's do the math. The force of gravity is $F = \frac{Gm_1m_2}{r^2}$. When in alignment, the Earth and Jupiter are about 600 million km apart ($r$). Jupiter masses about $2x10^{27}$kg ($m_1$) and the Earth about $6x10^{24}$ kg (m_2). That gives about $2x10^{18}$N of force. This is a lot, but is it a lot compared to the Sun?
Now let's do the same for the Sun and Earth. They're about 150 million km apart and the Sun masses about $2x10^{30}$kg. The force is about $3.5x10^{22}$N or 10,000 times more.
Jupiter has an effect on the Earth's orbit, but at its strongest point the effect is 10,000 times smaller than the Sun. It has a small but noticeable effect on the Earth's orbit.
Gas giants do have a dramatic effect on the Solar System, but over the course of very, very long time scales. Jupiter and Saturn may have prevented a planet from forming in the Asteroid Belt and may explain why Mars is so small compared to Earth and Venus. Jupiter Trojans are groups of asteroids locked into orbit 60 degrees in front of and behind Jupiter. The point around which Jupiter and the Sun rotate lies just outside of the Sun making the Sun wobble.
And what would happen every few thousand (?) million (?) years when all the planets and even some of the larger asteroids, Kuiper Belt Objects - everything but the kitchen sink - come into alignment?
Compared to Jupiter and the Sun, these are peanuts.
The Sun contains about 99.9% of the mass of the Solar System. Of that 0.1% left over Jupiter contains about 70%. Saturn, Uranus and Neptune make up about 29%. Everything else is about 1%.
The force of gravity falls off as the square of the distance; at twice the distance gravity is four times weaker, at four times the distance its sixteen times weaker. As things get further away from the Earth their gravity gets much, much weaker. Saturn's effect on the Earth is about a tenth of Jupiter's. Uranus, smaller and further away, is even weaker. Neptune even weaker.
SchwernSchwern
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Why do the planets in our solar system orbit in the same plane?
Why do the planets in the Solar system stay in the same orbital plane?
How dense would planet earth have to be to have the same gravity as Jupiter?
Does every object in the Universe have gravity? Space has no gravity, why?
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Solar series 142 and 147 started same year - series 147 and 152 will be ending same year. Why? Coincidence?
Is the nose of the solar system and the solar apex the same thing?
If the solar system is nearly flat, then why don't all the planets appear to lie on the same axis when viewed from earth?
Does Jupiter rotate at the same speed at every depth?
How long is a mean Jovian tropical year in Jovian solar days? | CommonCrawl |
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We're in love with division
If you're reading these blogs, you probably like math. And just like with music and movies, so it goes with math: everybody has their favorites. If you asked me what parts of math were my favorites, you might be perplexed if I told you that I really like subtraction. Sounds boring! Or what if I said "I am absolutely in LOVE with multiplication." That would be a little strange, wouldn't it? Well, here's what I'm really in love with: division. Division rocks!
No, I don't like doing long division, and I don't have a favorite fraction. What I love about division is how it opens up an incredibly interesting world of mathematics: probability. The lowly and boring operation of division is the backbone of probability, and probability is amazing.
Probability can be amazing!
For example, if you take a standard six-sided die and roll it, what's the probability of getting a 4? It's 1/6, right? And MAN that is totally boring, so that's not what I'm talking about AT ALL. Here's a way more interesting situation. Suppose you gather together a bunch of people in a room, and ask each of them what their birthday is (month and day). You might wonder about how likely it is that two of these people share the same birthday. That might lead you to wonder about the following simple question: how many people would you need to gather so that the probability that at least two of them have the same birthday is more than 1/2? By "need", we really mean this: what's the fewest number of people you could get away with to have the probability that some two of them share the same birthday be greater than 1/2?
Probability is tough. It's dangerous to wave your hands around without calculating things very carefully. For example, since there's 365 birthdays, one for each day of the year, wouldn't we definitely need at least 183 people to ensure that the probability that two of them have the same birthday is greater than 1/2? Perhaps that's a decent first stab at an answer, but we haven't actually done any math yet, so we should be wary of this first guess. Actually, what's really interesting here is that you don't need that many people. Do you know how many people you need? 23. That's it. Think about that. Say that number out loud to yourself a few times. Once you have gathered 23 people, the probability that some two of them share a birthday is
$$\frac{38093904702297390785243708291056390518886454060947061}{75091883268515350125426207425223147563269805908203125}$$
which turns out to be about 50.7%. Look at that lovely division, and the amazing thing it's telling you: the probability that two people have the same birthday in a group of 23 people is a little over 1/2!!! Compare that to the standard first guess of 183. WOW. We're not lying to you, and this isn't magic. 23!!!
The strange probabilities of the bingo card
Amazing probabilities turn up in the unlikeliest of places. For example, have you ever played bingo? Usually you go to a "bingo hall", and play with a whole bunch of people. You have a board in front of you that typically looks something like this:
The column under the B can contain any of the numbers from 1 through 15, with no repeats. The next column will contain numbers in the range 16-30, the next 31-45, then 46-60, and lastly the O column has numbers in the range 61-75. Everyone has a different bingo card. Someone starts announcing letter/number combinations, like B13! N32! G44! When such a combination is called, you check and see if that number under that letter is on your card. In the above picture, B13 is not on the card, but G44 is. If what's called out is on your card, you can mark it with a pen or circular stamp. The goal of the game is to get 5 stamps in the same row, column, or long diagonal. The first one to get it wins a prize!
It's a simple enough game. And even if you enjoy it, you might not think there's anything interesting to say about it, mathematically speaking. However, in the September 2017 issue of Math Horizons, a mathematics journal with an audience of undergraduate math majors, a fascinating article appeared detailing an incredibly amazing fact about the game of bingo. The authors found that if you gather a large enough group of people to play bingo, and they play many, many times, a very interesting trend will appear. The chances that the winning board has a completed row is much higher than the chances that it has a completed column! This is a really weird and unexpected conclusion. Over the long run, you might strongly suspect that there should be a roughly equal number of row wins as column wins, but this just isn't the case. What's true is that with a large enough group of players, the overall probability of a row win will be a little over 75%!!! Think about that… wow! If you can, check out that article!
Don't you dare think that division is boring. In the right context, and with the right questions, division can be dazzling.
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\begin{document}
\title[On uniqueness of solutions of Navier-Stokes equations]{On uniqueness of weak solutions of the incompressible Navier-Stokes equations} \author{Kamal N. Soltanov} \address{{\small National Academy of Sciences of Azerbaijan, Baku, AZERBAIJAN }} \email{sultan\[email protected]} \urladdr{} \subjclass[2010]{Primary 35K55, 35K61, 35D30, 35Q30; Secondary 76D03, 76N10} \date{} \keywords{Navier-Stokes Equations, Uniqueness, Auxiliary problems, Solvability}
\begin{abstract} In this article the question on uniqueness of weak solution of the incompressible Navier-Stokes Equations in the 3-dimensional case is studied. Here the investigation is carried out with use of another approach. The uniqueness of velocity for the considered problem is proved for given functions from spaces that possesess some smoothness. Moreover, these spaces are dense in respective spaces of functions, under which were proved existence of the weak solutions. In addition here the solvability and uniqueness of the weak solutions of auxiliary problems associated with the main problem is investigated, and also one conditional result on uniqueness is proved. \end{abstract}
\maketitle
\section{\label{Sec_1}Introduction}
In this article we investigate the question on the uniqueness of the weak solutions of the incompressible Navier-Stokes equations, namely is investigated question: when the weak solution of the following problem is unique? \begin{equation}
\frac{\partial u_{i}}{\partial t}-\nu \Delta u_{i}+\underset{j=1}{\overset{n}{\sum }}u_{j}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial p}{\partial x_{i}}=f_{i},\quad i=\overline{1,n},
\label{1} \end{equation} \begin{equation}
\func{div}u=\underset{i=1}{\overset{n}{\sum }}\frac{\partial u_{i}}{\partial x_{i}}=0,\quad x\in \Omega \subset R^{n},t>0\quad ,
\label{2} \end{equation} \begin{equation}
u\left( 0,x\right) =u_{0}\left( x\right) ,\quad x\in \Omega ;\quad u\left\vert \ _{\left( 0,T\right) \times \partial \Omega }\right. =0
\label{3} \end{equation} where $\Omega \subset R^{n}$ is a bounded domain with sufficiently smooth boundary $\partial \Omega $, $T>0$ is a positive number. In this work for study of the posed question two distinct way are used, therefore it consist of two parts.
As is well-known Navier-Stokes equations describe the motion of a fluid in $ R^{n}$ ($n=2$ or $3$). Consequently, in this problem $u(x,t)=\left\{ u_{i}(x,t)\right\} _{1}^{n}\in R^{n}$ is an unknown velocity vector and $ p(x,t)\in R$ is an unknown pressure, at the position $x\in R^{n}$ and time $ t\geq 0$; $f_{i}(x,t)$ are the components of a given, externally applied force (e.g. gravity), $\nu $ is a positive coefficient (the viscosity), $ u_{0}\left( x\right) \in R^{n}$ is a sufficiently smooth vector function (vector field).
As is well-known in \cite{Ler1} is shown that the Navier-Stokes equations ( \ref{1}), (\ref{2}), (\ref{3}) in three dimensions case has a weak solution $ (u,p)$ with suitable properties (see, also, \cite{Hop}, \cite{Lad1}, \cite {MajBer}, \cite{Con1}, \cite{Fef1}, \cite{Lio1}). It is known that uniqueness of weak solution of the Navier-Stokes equation in two space dimensions case were proved (\cite{LioPro}, \cite{Lio1}, see also \cite{Lad2} ), but the result of such type for the uniqueness of weak solutions in three space dimensions case as yet isn't known. It should be noted that in three dimensional case the uniqueness was studied also, but under complementary conditions on the smoothness of the solution (see, e.g. \cite{Lio1}, \cite {Tem1}, \cite{Sch1}, etc.). It is known for the Euler equations were shown that uniqueness of weak solutions isn't (see, \cite{Sch1}, \cite{Shn1}).
We need to note the regularity of solutions in three dimensional case were investigated and partial regularity of the suitable weak solutions of the Navier--Stokes equation were obtained (see, e.g. \cite{Sch2}, \cite {CafKohNir}, \cite{Lin1}, \cite{Lio1}, \cite{Lad1}, \cite{ChL-RiMay}, \cite {Gal}). There exist many works which study different properties of solutions of the Navier--Stokes equation (see, \cite{Lio1}, \cite{Lad1}, \cite{Lin1}, \cite{Fef1}, \cite{FoiManRosTem}, \cite{FoiRosTem1}, \cite{FoiRosTem2}, \cite {FoiRosTem3}, \cite{GlaSveVic}, \cite{HuaWan}, \cite{PerZat}, \cite{Sol1}, \cite{Sol2}, \cite{Tem1}), etc.) and also different modifications of Navier--Stokes equation (see, e.g. \cite{Lad1}, \cite{Lio1}, \cite{Sol3}, etc.). \
It need note that earlier under various additional conditions of the type of certain smoothness of the weak solutions different results on the uniqueness of solution of the incompressible Navier-Stokes equation in $3D$ case were obtained (see, e. g. \cite{Hop}, \cite{Lad2}, \cite{Lio1}, \cite{Tem1}, etc.). Here we would like to note the result of article \cite{Fur} that possesses of some proximity to the main result of this article. In this article the system of equations (1.1
${{}^1}$
) - (1.3) was examined, which is obtained from (1.1) - (1.3) under studies the solvability of this problem by the Hopf-Leray approach (that below will be explained, see, e.g. \cite{Tem1}). In \cite{Fur} the problem in the following form was studied \begin{equation*} Nu=\frac{du}{dt}+\nu Au+B(u)=f,\quad \gamma _{0}u=u_{0}, \end{equation*} where $B(u)\equiv \underset{j=1}{\overset{3}{\sum }}u_{j}\frac{\partial u_{i} }{\partial x_{j}}$ and $\gamma _{0}u\equiv u\left( 0\right) $. In which the author shows that $\left( N,\gamma _{0}\right) :Z\longrightarrow L^{2}\left( 0,T:H^{-1/2}\left(
\Omega
\right) \right) \times H^{1/2}\left(
\Omega
\right) $\ is the continuous operator under the condition that $
\Omega
\subset R^{3}$ is a bounded region whose boundary $\partial
\Omega
$ is a closed manifold of class $C^{\infty }$, where \begin{equation*} Z=\left\{ \left. u\in L^{2}\left( 0,T:H^{3/2}\left(
\Omega
\right) \right) \right\vert \ \ \frac{du}{dt}\in L^{2}\left( 0,T:H^{-1/2}\left(
\Omega
\right) \right) \right\} . \end{equation*}
Moreover, here is proved that if to denote by $F_{\gamma _{0}}$ the image: $ N\left( Z_{u_{0}}\right) =F_{\gamma _{0}}$ for $u_{0}\in H^{1/2}\left(
\Omega
\right) $ then for each $f\in F_{\gamma _{0}}$ there exists only one solution $u\in Z$ such that $Nu=f$ and $\gamma _{0}u=u_{0}$, here $ Z_{u_{0}}=\left\{ \left. u\in Z\right\vert \ \gamma _{0}u=u_{0}\right\} $, and also the density of set $F_{\gamma _{0}}$ in $L^{2}\left( 0,T:H^{-1/2}\left(
\Omega
\right) \right) $ in the topology of $L^{p}\left( 0,T:H^{-l}\left(
\Omega
\right) \right) $ under certain conditions on $p,l$. The proof given in \cite {Fur} is similar to the proof of \cite{Lio1} and \cite{Tem1}, but the result not follows from their results.
In the beginning in this paper certain explanation why for study of the posed question is enough to investigate the problem (1.1
${{}^1}$
) - (\ref{3}) is provided. Here the approach Hopf-Leray (with taking into account of the result of de Rham) for study the existence of the weak solution of the considered problem is used, as usually.
Unlike above investigations here we study the question on the uniqueness in the case when the weak solution $u$ of the problem (1.1
${{}^1}$
) - (\ref{3}) is contained in $\mathcal{V}\left( Q^{T}\right) $ (in $3D$ case), consequently, as is known, for this the following condition is sufficiently: functions $u_{0}$ and $f$ satisfy conditions \begin{equation*} u_{0}\in H\left( \Omega \right) ,\quad f\in L^{2}\left( 0,T;V^{\ast }\left( \Omega \right) \right) . \end{equation*}
\begin{notation} \label{N_1}The result obtained for the problem (1.1
${{}^1}$
) - (\ref{3}) allows us to respond to the posed question, namely this shows the uniqueness of the velocity vector $u$. \end{notation}
So, in this article the uniqueness of the weak solutions $u$ obtaining by the Hopf-Leray's approach of the mixed problem with Dirichlet boundary condition for the incompressible Navier-Stokes system in the $3D$ case is investigated. For investigation we use an approach that is different from usual methods used for study of the questions of such type. The approach used here allows us to receive more general result on the uniqueness of the weak solution (of the velocity vector $u$) of the mixed problem for the incompressible Navier--Stokes equation under more general conditions. In addition, here in order to carry out of the proof of the main result, in the beginning the existence and uniqueness of the weak solutions of auxiliary problems are studied.
For study of the uniqueness of the solution of the problem we also use of the formulation of the problem in the weak sense according to J. Leray \cite {Ler1}. As well-known, problem (\ref{1}) - (\ref{3}) and (1.1
${{}^1}$
) - (\ref{3}) was investigated in many works (see, \cite{Lio1}, \cite{Tem1} and \cite{Gal}). Here we will bring the result on weak solvability from the book \cite{Tem1}.
\begin{theorem} (\cite{Tem1}) Let $\Omega $ be a Lipschitz open bounded set in $R^{n}$, $ n\leq 4$. Let there be given $f$ and $u_{0}$ which
satisfy $f\in L^{2}\left( 0,T;V^{\ast }\left( \Omega \right) \right) $ and $ u_{0}\in H\left( \Omega \right) $.\ Then there exists at least one function $ u$ which
satisfies $u\in L^{2}\left( 0,T;V\left( \Omega \right) \right) $, $\frac{du}{ dt}\in L^{1}\left( 0,T;V^{\ast }\left( \Omega \right) \right) $, $u\left( 0\right) =u_{0}$ and the equation \begin{equation} \frac{d}{dt}\left\langle u,v\right\rangle -\left\langle \nu \Delta u,v\right\rangle +\left\langle \underset{j=1}{\overset{n}{\sum }}u_{j}\frac{ \partial u}{\partial x_{j}},v\right\rangle =\left\langle f,v\right\rangle \label{1a} \end{equation} for any $v\in V\left( \Omega \right) $. Moreover, $u\in L^{\infty }\left( 0,T;H\left( \Omega \right) \right) $ and $u\left( t\right) $\ is weakly continuous from $\left[ 0,T\right] $ into $H\left( \Omega \right) $ (i. e. $ \forall v\in H\left( \Omega \right) $, $t\longrightarrow \left\langle u\left( t\right) ,v\right\rangle $ is a continuous scalar function, and consequently, $\left\langle u\left( 0\right) ,v\right\rangle =\left\langle u_{0},v\right\rangle $). \end{theorem}
"Moreover, in the case when $n=3$ a weak solution $u$ satisfy \begin{equation*} u\in V\left( Q^{T}\right) ,\quad u^{\prime }\equiv \frac{\partial u}{ \partial t}\in L^{\frac{4}{3}}\left( 0,T;V^{\ast }(
\Omega
)\right) ,\quad \end{equation*} and also is almost everywhere (a.e.) equal to some continuous function from $ \left[ 0,T\right] $ into $H$, so that (\ref{3}) is meaningful, with use of the obtained properties that any weak solution belong to the bounded subset of \begin{equation*} \mathcal{V}\left( Q^{T}\right) \equiv V\left( Q^{T}\right) \cap W^{1,4/3}\left( 0,T;V^{\ast }(
\Omega
)\right) \end{equation*} and satisfies the equation (\ref{1a})." \footnote{ The expression $\left\langle g,h\right\rangle $ here and further denote $ \left\langle g,h\right\rangle =\underset{i=1}{\overset{3}{\sum }}\ \underset{ \Omega }{\int }g_{i}h_{i}dx$ for any $g,h\in \left( H\left( \Omega \right) \right) $, or $g\in V\left( \Omega \right) $ and $h\in V^{\ast }\left( \Omega \right) $, respectively.}
In what follows we will base on the mentioned theorem about the existence of the weak solution of problem (1.1
${{}^1}$
) - (\ref{3}) and the added remarks as principal results, since here is investigated the question related to the weak solution of the problem that is studied in Theorem 1.
The main result of this paper is the following uniqueness theorem.
\begin{theorem} \label{Th_1}Let $\Omega \subset R^{3}$ be a domain of $Lip_{loc}$ (will be defined below; see, Section \ref{Sec_I.4}), $T>0$ be a number. If given functions $u_{0}$, $f$ satisfy of conditions $u_{0}\in H^{1/2}\left( \Omega \right) $, $f\in L^{2}\left( 0,T;H^{1/2}\left( \Omega \right) \right) $ then weak solution $u\in \mathcal{V}\left( Q^{T}\right) $ of the problem (1.1
${{}^1}$
) - (\ref{3}) given by the above mentioned theorem is unique. \end{theorem}
This article is organized as follow. In Part I the question is studied under certain smoothness conditions onto given functions. In Section I.2 some known results and the explanation of the relation between problems (\ref{1}) - (\ref{3}) and (1.1
${{}^1}$
) - (\ref{3}) is adduced, and also the necessary auxiliary results, namely lemmas are proved. These lemmas are need us for the study of the main problem. In Section I.3 the auxiliary problems determined that posed on the cross-sections of $\Omega $, which are obtained from problem (1.1
${{}^1}$
) - (\ref{3}). Here is explained how these problems are obtained from problem (1.1
${{}^1}$
) - (\ref{3}), and also is suggested to study the main question for the auxiliary problems on the cross-sections instead of the investigation of this question on whole of $\Omega $. In Section I.4 the existence of the solution and, in Section I.5 the uniqueness of solution of the auxiliary problem are studied. In Section I.6 the main result Theorem \ref{Th_1} is proved. In Section II.7 of Part II one conditional result on uniqueness of weak solution of problem (1.1
${{}^1}$
) - (\ref{3}) by use of certain modification of the well-known approach is proved.
\part{\label{Part I}One new approach for study of the uniqueness}
\section{\label{Sec_I.2}Preliminary results}
In this section the background material, definitions of the appropriate spaces, that will be used in the next sections are briefly recalled. In addition, here some notations are introduced, and also the necessary auxiliary results are proved that in the follow will be employed. Moreover, we recall the basic setup and results regarding of the weak solutions of the incompressible Navier--Stokes equations used throughout this paper.
As is well-known, problem (\ref{1}) - (\ref{3}) possesses weak solution in the space $\mathcal{V}\left( Q^{T}\right) \times L^{2}\left( Q^{T}\right) $ for each $u_{0i}\left( x\right) ,$ $f_{i}(x,t)$ ($i=\overline{1,3}$), which are contained in the suitable spaces (see, e.g. \cite{Lio1}, \cite{Tem1} and references therein), (the space $\mathcal{V}\left( Q^{T}\right) $ will be defined later on). Here our main problem is the investigation of the posed question in the case $n=3$, consequently, here problems will be studied mostly in the case $n=3$.
\begin{definition} \label{D_2.1}Let $\Omega \subset R^{3}$ be an open bounded Lipschitz domain and $Q^{T}\equiv \left( 0,T\right) \times \Omega $, $T>0$ be a number. Let $ V\left( Q^{T}\right) $ be the space determined as \begin{equation*}
V\left( Q^{T}\right) \equiv L^{2}\left( 0,T;V\left( \Omega \right) \right) \cap L^{\infty }\left( 0,T;H\left( \Omega \right) \right) ,
\end{equation*} where $V\left( \Omega \right) $ and $H\left( \Omega \right) $ are the closure of \ \begin{equation*} \left\{ \varphi \left\vert \ \varphi \in \left( C_{0}^{\infty }\left( \Omega \right) \right) ^{3},\right. \func{div}\varphi =0\right\} \end{equation*} in the topology of $\left( W_{0}^{1,2}\left( \Omega \right) \right) ^{3}$ and in the topology of $\left( L^{2}\left( \Omega \right) \right) ^{3}$, respectively;
the dual $V\left( \Omega \right) $ determined as $V^{\ast }\left( \Omega \right) $ and is the closure of \ the\ linear continuous functionals defined on $V\left( \Omega \right) $ in the sense of the Lax dual relative to $ H\left( \Omega \right) $.
Moreover we set also the space $\mathcal{V}\left( Q^{T}\right) \equiv V\left( Q^{T}\right) \cap W^{1,4/3}\left( 0,T;V^{\ast }\left( \Omega \right) \right) $ for $n=3$ (\cite{Tem1}). \end{definition}
Here as is well-known $L^{2}\left( \Omega \right) $ is the Lebesgue space and $W^{1,2}\left( \Omega \right) $ is the Sobolev space, that are the Hilbert spaces and \begin{equation*} W_{0}^{1,2}\left( \Omega \right) \equiv \left\{ v\left\vert \ v\in W^{1,2}\left( \Omega \right) ,\right. v\left\vert \ _{\partial \Omega }\right. =0\right\} . \end{equation*} As is well-known in this case $H\left( \Omega \right) $ and $V\left( \Omega \right) $ also are the Hilbert spaces, therefore
\begin{equation*} V\left( \Omega \right) \subset H\left( \Omega \right) \equiv H^{\star }\left( \Omega \right) \subset V^{\ast }\left( \Omega \right) . \end{equation*}
So, assume the given functions $u_{0}$ and $f$ satisfy \begin{equation*} u_{0}\in H\left( \Omega \right) ,\quad f\in L^{2}\left( 0,T;V^{\ast }\left( \Omega \right) \right) \end{equation*} where $V^{\ast }\left( \Omega \right) $ is the dual space of $V\left( \Omega \right) $.
Consider the problem for which the existence of the weak solution directly connected with the existence of the weak solution of problem (\ref{1}) - ( \ref{3}) as will be shown below
\begin{equation} \frac{\partial u_{i}}{\partial t}-\nu \Delta u_{i}+\underset{j=1}{\overset{n} {\sum }}u_{j}\frac{\partial u_{i}}{\partial x_{j}}=f_{i}\left( t,x\right) ,\quad i=\overline{1,n},\ \nu >0 \tag{1.1$^{1}$} \end{equation} \begin{equation} \func{div}u=\underset{i=1}{\overset{n}{\sum }}\frac{\partial u_{i}}{\partial x_{i}}=\underset{i=1}{\overset{n}{\sum }}D_{i}u_{i}=0,\quad x\in \Omega \subset R^{n},\ t>0, \tag{1.2} \end{equation} \begin{equation} u\left( 0,x\right) =u_{0}\left( x\right) ,\quad x\in \Omega ;\quad u\left\vert \ _{\left( 0,T\right) \times \partial \Omega }\right. =0. \tag{1.3} \end{equation} \footnote{ Here all equations are needed to understand in the sense of the corresponding spaces, e.g. the equation (1.1
${{}^1}$
) is understood in the sense of the dual space of $\mathcal{V}\left( Q^{T}\right) $.}
The investigation of the existence of the weak solution of problem (1.1
${{}^1}$
) - (\ref{3}) is equivalent to the investigation of the following equation with corresponding initial condition (see, Theorem \ref{Th_1}) \begin{equation} \frac{d}{dt}\left\langle u,v\right\rangle -\left\langle \nu \Delta u,v\right\rangle +\left\langle \underset{j=1}{\overset{n}{\sum }}u_{j}\frac{ \partial u}{\partial x_{j}},v\right\rangle =\left\langle f,v\right\rangle \label{1b} \end{equation} where $v\in V(
\Omega
)$ is arbitrary.
In other words, one must study the existence of the weak solution of problem (1.1
${{}^1}$
) - (\ref{3}) in the sense of J. Leray \cite{Ler1} by use of his approach (see, also \cite{Lio1}, \cite{Tem1}). This approach shows that for study of the uniqueness of the solution relative to velosity vector $u$ of problem ( \ref{1}) - (\ref{3}) sufficiently to investigate of same question for problem (1.1
${{}^1}$
) - (\ref{3}) in view of de Rham result (see, books \cite{Lio1}, \cite{Tem1} , \cite{Lad1}, \cite{FoiManRosTem}, \cite{Gal}, \cite{MajBer} etc. where the properties of the this problem were explained enough clearly).
In order to explain that the investigation of the posed question for problem (1.1
${{}^1}$
) - (\ref{3}) is sufficient for our goal we represent here some results of the book \cite{Tem1}, which have the immediate relation to this problem.
\begin{proposition} \label{Pr_2.1}(\cite{Tem1}) Let $\Omega $ be a bounded Lipschitz open set in $R^{n}$ and $g=\left( g_{1},...,g_{n}\right) $, $g_{i}\in \mathcal{D} ^{\prime }\left( \Omega \right) $, $1\leq i\leq n$. A necessary and sufficient condition that $g=\func{grad}p$ for some $p$ in $\mathcal{D} ^{\prime }\left( \Omega \right) $, is that $\left\langle g,v\right\rangle =0$ $\forall v\in V\left( \Omega \right) $. \end{proposition}
\begin{proposition} \label{Pr_2.2}(\cite{Tem1}) Let $\Omega $ be a bounded Lipschitz open set in $R^{n}$.
(i) If a distribution $p$ has all its first-order derivatives $D_{i}p$, $ 1\leq i\leq n$ in $L^{2}\left( \Omega \right) $, then $p\in L^{2}\left( \Omega \right) $ and\ \begin{equation*} \left\Vert p\right\Vert _{L^{2}\left( \Omega \right) /R}\leq c\left( \Omega \right) \left\Vert \func{grad}p\right\Vert _{\left( L^{2}\left( \Omega \right) \right) ^{3}}; \end{equation*}
(ii) If a distribution $p$ has all its first derivatives $D_{i}p$, $1\leq i\leq d$ in $H^{-1}\left( \Omega \right) $, then $p\in L^{2}\left( \Omega \right) $ and \begin{equation*} \left\Vert p\right\Vert _{L^{2}\left( \Omega \right) /R}\leq c\left( \Omega \right) \left\Vert \func{grad}p\right\Vert _{H^{-1}\left( \Omega \right) }. \end{equation*}
In both cases, if $\Omega $ is any open set in $R^{n}$, then $p\in L_{loc}^{2}\left( \Omega \right) $. \end{proposition}
Combining these results, one can note that if $g\in H^{-1}\left( \Omega \right) $ (or $g\in L^{2}\left( \Omega \right) $) and $(g,v)=0$, then $g= \func{grad}p$ with $p\in L^{2}\left( \Omega \right) $ (or $p\in H^{1}\left( \Omega \right) $) if $\Omega $ is a Lipschitz open bounded set.
\begin{theorem} \label{Th_1.2}(\cite{Tem1}) Let $\Omega $ be a Lipschitz open bounded set in $R^{n}$. Then \begin{equation*} H^{\bot }\left( \Omega \right) =\left\{ w\in \left( L^{2}\left( \Omega \right) \right) ^{n}\ \left\vert \ w=\func{grad}p,\ p\in H^{1}\left( \Omega \right) \right. \right\} ; \end{equation*} \begin{equation*} H\left( \Omega \right) =\left\{ u\in \left( L^{2}\left( \Omega \right) \right) ^{n}\ \left\vert \func{div}u=0,\ u_{\partial \Omega }\ =0\ \right. \right\} . \end{equation*} \end{theorem}
\begin{lemma} \label{L_1.1}(see, e.g. \cite{Lio1}, \cite{Tem1} and also,\cite{Sol2}, \cite {Sol4} ) Let $B_{0},B,B_{1}$ be three Banach spaces, each space continuously included in the following one $B_{0}\subset B\subset B_{1}$ and $B_{0},B_{1}$ are reflexive, moreover, the inclusion $B_{0}\subset B$ is compacts.
Let $X$ be \begin{equation*} X\equiv \left\{ u\left\vert \ u\in L^{p_{0}}(0,T;B_{0}),\right. u\prime \in L^{p_{1}}(0,T;B_{1})\right\} , \end{equation*} where $1<p_{j}<\infty $, $j=0,1$ and $0<T<\infty $. Hence $X$\ is Banach space with the norm \begin{equation*} \left\Vert u\right\Vert _{X}=\left\Vert u\right\Vert _{L^{p_{0}}(0,T;B_{0})}+\left\Vert u\right\Vert _{L^{p_{1}}(0,T;B_{1})}. \end{equation*} Then under these conditons the inclusion $X\subset L^{p_{0}}(0,T;B)$ is compact.
Moreover, the inclusion $X\subset C(0,T;B_{1})$ holds, due of Lebesgue theorem. \end{lemma}
Consequently, if one will seek of weak solution of the problem (\ref{1}) - ( \ref{3}) by according Hopf-Leray then one can get the following equation \begin{equation} \frac{d}{dt}\left\langle u,v\right\rangle -\left\langle \nu \Delta u,v\right\rangle +\left\langle \underset{j=1}{\overset{n}{\sum }}u_{j}\frac{ \partial u}{\partial x_{j}},v\right\rangle =\left\langle f,v\right\rangle -\left\langle \nabla p,v\right\rangle , \label{2.1} \end{equation} where $v\in V(
\Omega
)$ is arbitrary.
So, if to consider of the last adding in the right side then at illumination of above results (Propositions \ref{Pr_2.1}, \ref{Pr_2.2} and Theorem \ref {Th_1.2}) using the integration by parts and taking into account that $v\in V(
\Omega
)$, i.e. $\func{div}v=0$ and $v\left\vert \ _{\left( 0,T\right) \times \partial \Omega }\right. =0$ we will get the equation \begin{equation} \left\langle \nabla p,v\right\rangle \equiv \underset{\Omega }{\int }\nabla p\cdot v\ dx=\underset{\Omega }{\int }p\func{div}v\ dx=0,\quad \forall v\in V(
\Omega
) \label{2.2} \end{equation} by virtue of de Rham result. Consequently, taking into account (\ref{2.2}) in (\ref{2.1}) we obtain equation (\ref{1b}) that shows why for study of the posed question is enough to study this question for problem (1.1
${{}^1}$
) - (\ref{3}).
So, we can continue the investigation of the posed question for problem (1.1
${{}^1}$
) - (\ref{3}) in the case $n=3$.
Let $\Omega \subset R^{3}$ be a bounded open domain with the boundary $ \partial \Omega $ of the Lipschitz class. We will denote by $H^{1/2}\left( \Omega \right) $ the vector space defined as in Definition \ref{D_2.1} by \begin{equation*} \left( W^{1/2}\left( \Omega \right) \right) ^{3}\equiv \left\{ w\left\vert \ w_{i}\in W^{1/2}\left( \Omega \right) ,\right. i=1,2,3\right\} ,\quad w=\left( w_{1},w_{2},w_{3}\right) \end{equation*} where $W^{1/2,2}\left( \Omega \right) $ is the Sobolev-Slobodeskij space (see, \cite{LioMag}, etc.). As well-known the trace for the function of the space $H^{1/2}\left( \Omega \right) $ is definite for each smooth surface from $\Omega $ (see, e.g. \cite{LioMag}, \cite{BesIlNik} and references therein), which is necessary for application of our approach to the considered problem. The main theorem will be proved under this additional condition that is the sufficient condition for present investigation.
\begin{definition} \label{D_2.2}A $u\in \mathcal{V}\left( Q^{T}\right) $ is called a solution of problem (1.1
${{}^1}$
) - (\ref{3}) if $u\left( t\right) $ almost everywhere in $\left( 0,T\right) $ satisfies the following equation \begin{equation} \frac{d}{dt}\left\langle u,v\right\rangle -\left\langle \nu \Delta u,v\right\rangle +\left\langle \underset{j=1}{\overset{n}{\sum }} u_{j}D_{j}u,v\right\rangle =\left\langle f,v\right\rangle \label{2.3a} \end{equation} for any $v\in V\left( \Omega \right) $ and $u\left( t\right) $\ is weakly continuous from $\left[ 0,T\right] $ into $H\left( \Omega \right) $ (i.e. $ \forall v\in H\left( \Omega \right) $, $t\longrightarrow \left\langle u\left( t\right) ,v\right\rangle $ is a continuous scalar function, and consequently, $\left\langle u\left( 0\right) ,v\right\rangle =\left\langle u_{0},v\right\rangle $). \end{definition}
In what follows we will understand of an existing solutions be functions that satisfy this definition together with the standard notations that are used usually. Moreover, as above were noted if $\Omega $ be a Lipschitz open bounded set in $R^{3}$, functions $f$ and $u_{0}$ satisfy $f\in L^{2}\left( 0,T;V^{\ast }\left( \Omega \right) \right) $ and $u_{0}\in H\left( \Omega \right) $, respectively, then the vector function $u$ is the solution of problem (1.1
${{}^1}$
) - (\ref{3}) if it satisfies of conditions of Definition \ref{D_2.2}, in addition, $u\in L^{\infty }\left( 0,T;H\left( \Omega \right) \right) $ and the term $\underset{j=1}{\overset{3}{\sum }}u_{j}D_{j}u\equiv B\left( u\right) $ belong to $L^{4/3}\left( 0,T;V^{\ast }\left( \Omega \right) \right) $.
Now we go over into main question: let problem (1.1
${{}^1}$
) - (\ref{3}) have two different solutions $u,v\in \mathcal{V}\left( Q^{T}\right) $ then within the known approach we derive that the function $ w(t,x)=u(t,x)-v(t,x)$ must satisfies the following problem \begin{equation}
\frac{1}{2}\frac{\partial }{\partial t}\left\Vert w\right\Vert _{2}^{2}+\nu \left\Vert \nabla w\right\Vert _{2}^{2}+\underset{j,k=1}{\overset{3}{\sum }}\left\langle \frac{\partial v_{k}}{\partial x_{j}}w_{k},w_{j}\right\rangle =0,
\label{2.3} \end{equation} \begin{equation}
w\left( 0,x\right) \equiv w_{0}\left( x\right) =0,\quad x\in \Omega ;\quad w\left\vert \ _{\left( 0,T\right) \times \partial \Omega }\right. =0,
\label{2.4} \end{equation} where $\left\langle g,h\right\rangle =\underset{i=1}{\overset{3}{\sum }} \underset{\Omega }{\int }g_{i}h_{i}dx$ for any $g,h\in \left( H\left( \Omega \right) \right) ^{3}$, or $g\in V\left( \Omega \right) $ and $h\in V^{\ast }\left( \Omega \right) $, respectively. So, for the proof of the uniqueness of solution it is follows to show that $w\equiv 0$ in the sense of needed space.
Later in this section one will studied questions and provided certain results that are necessary for employing of the basic approach to study of the requered question. More exactly, these reasonings and results will be used in sections 3-6 for study of the posed question.
As our purpose is the investigation of the uniqueness of solution of problem (1.1
${{}^1}$
) - (\ref{3}) therefore we will go over to the discussion of this question. As is known, problem (1.1
${{}^1}$
) - (\ref{3}) has weak solution $u\left( t\right) $ from the space $\mathcal{ V}\left( Q^{T}\right) $ denoted in Definition \ref{D_2.1}, which possesses of the above mentioned properties and also some complementary properties of the smoothness type (see, \cite{Ler1}, \cite{Lad2}, \cite{Lio1}, \cite {CafKohNir}, \cite{ConKukVic}, \cite{FoiManRosTem}, \cite{ChL-RiMay}). Therefore we will conduct our study under the condition that problem (1.1
${{}^1}$
) - (\ref{3}) have weak solutions and they belong to $\mathcal{V}\left( Q^{T}\right) $. For the study of the uniqueness of solution of problem (1.1
${{}^1}$
) - (\ref{3})) as above assume that problem (1.1
${{}^1}$
) - (\ref{3})) has, at least, two different solutions $u,v\in \mathcal{V} \left( Q^{T}\right) $. But for demonstrate that this isn
\'{}
t possible we will employ a different procedure.
Consequently, if we assume that problem (1.1
${{}^1}$
) - (\ref{3}) have two different solutions then they need to be different at least on some subdomain $Q_{1}^{T}$ of $Q^{T}$. In other words there exists a subdomain $\Omega _{1}$ of $\Omega $ and an interval $\left( t_{1},t_{2}\right) \subseteq \left( 0,T\right] $ such that \begin{equation*} Q_{1}^{T}\subseteq \left( t_{1},t_{2}\right) \times \Omega _{1}\subseteq Q^{T} \end{equation*} with $mes_{4}\left( Q_{1}^{T}\right) >0$ for which \begin{equation}
mes_{4}\left( \left\{ (t,x)\in Q^{T}\left\vert \ \left\vert u(t,x)-v(t,x)\right\vert \right. >0\right\} \right) =mes_{4}\left( Q_{1}^{T}\right) >0
\label{2.5} \end{equation} holds, where $mes_{4}\left( Q_{1}^{T}\right) $ denote the measure of $ Q_{1}^{T}$ in $R^{4}$ (i.e. $mes_{k}$ denote the Lebesgue measure on $k$ dimensional space $R^{n}$). Whence follows, that subdomain $\Omega _{1}$ must have of the positive Lebesgue measure, i.e. $mes_{3}(\Omega _{1})>0$.
The following lemmas will proved even though for $n>1$, but mostly these will use for the case $n=4$.
So, it is need to prove the following lemmas, which will use later on.
\begin{lemma} \label{L_2.1}Let $G\subset R^{n}$ be Lebesgue measurable subset then the following statements are equivalent:
1) $\infty >mes_{n}\left( G\right) >0;$
2) there exist a subsets $I\subset R^{1}$, $\infty >mes_{1}\left( I\right) >0 $ and $G_{\beta }\subset L_{\beta ,n-1}$, $\infty >mes_{n-1}\left( G_{\beta }\right) >0$ such that $G=\underset{\beta \in I}{\cup }G_{\beta }\cup N$, where $N$ is a set with $mes_{n-1}\left( N\right) =0$, and $ L_{\beta ,n-1}$ is the hyperplane of $R^{n}$, with $\ \ \ co\dim _{n}L_{\beta ,n-1}=1$, for any $\beta \in I$, which is generated by the arbitrary fixed vector $y_{0}\in R^{n}$ and defined as follow \begin{equation*} L_{\beta ,n-1}\equiv \left\{ y\in R^{n}\left\vert \ \left\langle y_{0},y\right\rangle =\beta \right. \right\} ,\quad \forall \beta \in I. \end{equation*} \end{lemma}
\begin{proof} Let $mes_{d}\left( G\right) >0$ and consider the class of hyperplanes $ L_{\gamma ,n-1}$ for which $G\cap L_{\gamma ,n-1}\neq \varnothing $ and $ \gamma \in I_{1}$, where $I_{1}\subset R^{1}$\ be some subset. It is clear that \begin{equation*} G\equiv \underset{\gamma \in I_{1}}{\bigcup }\left\{ x\in G\cap L_{\gamma ,n-1}\left\vert \ \gamma \in I_{1}\right. \right\} . \end{equation*} Then there exists a subclass of hyperplanes $\left\{ L_{\gamma ,n-1}\left\vert \ \gamma \in I_{1}\right. \right\} $ for which
$mes_{n-1}\left( G\cap L_{\gamma ,n-1}\right) >0$ is fulfilled. The number of such type hyperplanes cannot be less than countable or equal it because $ mes_{n}\left( G\right) >0$, moreover this subclass of $I_{1}$\ must possess the $R^{1}$ measure greater than $0$ since $mes_{n}\left( G\right) >0$. Indeed, let $I_{1,0}$ be this subclass and $mes_{1}\left( I_{1,0}\right) =0$ . In this case we get subset \begin{equation*} \left\{ \left( \gamma ,y\right) \in I_{1,0}\times G\cap L_{\gamma ,n-1}\left\vert \ \gamma \in I_{1,0},y\in G\cap L_{\gamma ,n-1}\right. \right\} \subset R^{n} \end{equation*} where $mes_{n-1}\left( G\cap L_{\gamma ,n-1}\right) >0$ for all $\gamma \in I_{1,0}$, but $mes_{1}\left( I_{1,0}\right) =0$, then \begin{equation*} mes_{n}\left( \left\{ \left( \gamma ,y\right) \in I_{1,0}\times G\cap L_{\gamma ,n-1}\left\vert \ \gamma \in I_{1,0}\right. \right\} \right) =0. \end{equation*} On the other hand if $mes_{n-1}\left( G\cap L_{\gamma ,n-1}\right) =0$ for all $\gamma \in I_{1}-I_{1,0}$ then \begin{equation*} mes_{n}\left( \left\{ \left( \gamma ,y\right) \in I_{1}\times G\cap L_{\gamma ,n-1}\left\vert \ \gamma \in I_{1}\right. \right\} \right) =0, \end{equation*} whence follows \begin{equation*} mes_{n}\left( G\right) =mes_{n}\left( \left\{ \left( \gamma ,y\right) \in I_{1}\times G\cap L_{\gamma ,n-1}\left\vert \ \gamma \in I_{1}\right. \right\} \right) =0. \end{equation*}
But this contradicts the condition $mes_{n}\left( G\right) >0$. Consequently, the statement 2 holds.
Let the statement 2 holds. It is clear that the class of hyperplanes $ L_{\beta ,n-1}$ defined by such way are parallel and also we can define the class of subsets of $G$ as its cross-section with hyperplanes, i.e. in the form: $G_{\beta }\equiv G\cap L_{\beta ,n-1}$, \ \ $\beta \in I$. Then $ G_{\beta }\neq \varnothing $ and we can write $G_{\beta }\equiv G\cap L_{\beta ,n-1}$, $\beta \in I$, moreover $G\equiv \underset{\beta \in I}{ \bigcup }\left\{ x\in G\cap L_{\beta ,n-1}\left\vert \ \beta \in I\right. \right\} \cup N$. Whence we get \begin{equation*} G\equiv \left\{ \left( \beta ,x\right) \in I\times G\cap L_{\beta ,n-1}\left\vert \ \beta \in I,x\in G\cap L_{\beta ,n-1}\right. \right\} \cup N. \end{equation*}
Consequently, $mes_{n}\left( G\right) >0$ by virtue of conditions: $ mes_{1}\left( I\right) >0$ and
$mes_{n-1}\left( G_{\beta }\right) >0$ for any $\beta \in I$. \end{proof}
Lemma \ref{L_2.1} shows that for the study of the measure of some subset $
\Omega
\subseteq R^{n}$ it is enough to study its stratifications by a class of corresponding hyperplanes.
\begin{lemma} \label{L_2.2}Let problem (1.1
${{}^1}$
) - (\ref{3}) has, at least, two different solutions $u,v$ that are contained in $\mathcal{V}\left( Q^{T}\right) $ and assume that $ Q_{1}^{T}\subseteq Q^{T}$ is one of a subdomain of $Q^{T}$ where $u$ and $v$ are different. Then there exists, at least, one class of parallel hyperplanes $L_{\alpha }$, $\alpha \in I\subseteq \left( \alpha _{1},\alpha _{2}\right) \subset R^{1}$ ($\alpha _{2}>\alpha _{1}$)\ with $co\dim _{R^{3}}L_{\alpha }=1$ such, that $u\neq v$ on $Q_{L_{\alpha }}^{T}\equiv \left[ \left( 0,T\right) \times \left( \Omega \cap L_{\alpha }\right) \right] \cap Q_{1}^{T}$, and vice versa, here $mes_{1}\left( I\right) >0$, $ mes_{2}\left( \Omega \cap L_{\alpha }\right) >0$ and $L_{\alpha }$ are hyperplanes which are defined as follows: there is vector $x_{0}\in S_{1}^{R^{3}}\left( 0\right) $ such that \begin{equation*} L_{\alpha }\equiv \left\{ x\in R^{3}\left\vert \ \left\langle x_{0},x\right\rangle =\alpha ,\right. \ \forall \alpha \in I\right\} . \end{equation*} \end{lemma}
\begin{proof} Let problem (1.1
${{}^1}$
) - (\ref{3}) have two different solutions $u,v\in \mathcal{V}\left( Q^{T}\right) $ then there exist a subdomain of $Q^{T}$ on which these solutions are different. Then there are $t_{1},t_{2}>0$ such that \begin{equation}
mes_{3}\left( \left\{ x\in \Omega \left\vert \ \left\vert u\left( t,x\right) -v\left( t,x\right) \right\vert >0\right. \right\} \right) >0
\label{2.6} \end{equation} holds for any $t\in J\subseteq \left[ t_{1},t_{2}\right] \subseteq \left[ 0,T\right) $, where $mes_{1}\left( J\right) >0$ by the virtue of the condition \begin{equation*} mes_{4}\left( \left\{ (t,x)\in Q^{T}\left\vert \ \left\vert u(t,x)-v(t,x)\right\vert \right. >0\right\} \right) >0 \end{equation*} and of Lemma \ref{L_2.1}.
Whence follows, that there exist, at least, one class of the parallel hyperplanes $L_{\alpha }$, $\alpha \in I\subseteq \left( \alpha _{1},\alpha _{2}\right) \subset R^{1}$ such that $co\dim _{R^{3}}L_{\alpha }=1$ and \begin{equation}
mes_{2}\left( \left\{ x\in \Omega \cap L_{\alpha }\left\vert \ \left\vert u\left( t,x\right) -v\left( t,x\right) \right\vert >0\right. \right\} \right) >0,\ \forall \alpha \in I
\label{2.7} \end{equation} hold for $\forall t\in J$, where subsets $I$ and $J$ are satisfy inequations: $mes_{1}\left( I\right) >0$, $mes_{1}\left( J\right) >0$, and also (\ref{2.7}) holds, by virtue of (\ref{2.6}). This proves the "if" part of Lemma.
Now consider the converse assertion. Let there exist a class of hyperplanes $ L_{\alpha }$, $\alpha \in I_{1}\subseteq \left( \alpha _{1},\alpha _{2}\right) \subset R^{1}$ with $co\dim _{R^{3}}L_{\alpha }=1$ that fulfills the condition of Lemma and the subset $I_{1}$\ satisfies of same condition as $I$. Then there exist, at least, one subset $J_{1}$ of $\left[ 0,T\right) $ such that $mes_{1}\left( J_{1}\right) >0$ and the inequation $u\left( t,x\right) \neq v\left( t,x\right) $ holds onto $Q_{2}^{T}$ with $ mes_{4}\left( Q_{2}^{T}\right) >0$, which is defined as $Q_{2}^{T}\equiv J_{1}\times U_{L}$, where \begin{equation}
U_{L}\equiv \underset{\alpha \in I_{1}}{\bigcup }\left\{ x\in \Omega \cap L_{\alpha }\left\vert \ u\left( t,x\right) \neq v\left( t,x\right) \right. \right\} \subset \Omega ,\ t\in J_{1}
\label{2.8} \end{equation} for which the inequation $mes_{R^{3}}\left( U_{L}\right) >0$ is fulfilled by virtue of the condition and of Lemma \ref{L_2.1}.
So we get \begin{equation*} u\left( t,x\right) \neq v\left( t,x\right) \text{ onto }Q_{2}^{T}\equiv J_{1}\times U_{L},\text{ with }mes_{4}\left( Q_{2}^{T}\right) >0. \end{equation*} Thus, we obtain the fact that $u\left( t,x\right) $ and $v\left( t,x\right) $ are different functions in $\mathcal{V}\left( Q^{T}\right) $. \end{proof}
It is not difficult to see that result of Lemma \ref{L_2.2} is independent of assumption: $Q_{1}^{T}\subset Q^{T}$ or $Q_{1}^{T}=Q^{T}$.
Likely one could be to prove more general results of such type using of the regularity properties of weak solutions of this problem (see, \cite{Sch1}, \cite{CafKohNir}, \cite{Lin1}, \cite{ChL-RiMay}, etc.).
\section{\label{Sec_I.3}Investigation of the auxiliary problem}
In this section we will transform problem (1.1
${{}^1}$
) - (\ref{3}) to the auxiliary problems in order to use of the result of Lemma \ref{L_2.2}. In other words, here our concept of the investigation of the posed question will presented. This concept is based to result of Lemma \ref{L_2.2}, which shows, that for study of posed problem it is enough to investigate this problem on the cross-sections of the domain $Q^{T}\equiv \left( 0,T\right) \times \Omega $.
So, we will begin with the definition of the domain \textit{\ }$
\Omega
\subset R
{{}^3}
$ on which will be study of the problem.
\begin{definition} \label{D_3.1}\textit{A bounded open domain }$
\Omega
\subset R
{{}^3}
$\textit{\ with the boundary }$\partial
\Omega
$\textit{\ is spoken from the class }$Lip_{loc}$\textit{\ iff }$\partial
\Omega
$\textit{\ is a locally Lipschitz hypersurface. (This means: any point }$ x\in \partial
\Omega
$\textit{\ possesses a neighbourhood in }$\partial
\Omega
$\textit{\ that admits a }representation as a hypersurface $y_{3}=\psi \left( y_{1},y_{2}\right) $, where $\psi $ is a Lipschitz function, and $ \left( y_{1},y_{2},y_{3}\right) $ are rectangular coordinates in $R
{{}^3}
$. In a coordinate basis that may be different from the canonical basis $ \left( e_{1},e_{1},e_{3}\right) $.) \end{definition}
According to $\Omega $\textit{\ }is a locally Lipschitz and bounded one can draw the conclusion: each point $x_{j}\in \partial \Omega $, has an open neighbourhood $U_{j}$ such that $U_{j}^{\prime }=\Omega \cap U_{j}$ , moreover, $\partial \Omega $ can be covered by a finite family of such sets $ U_{j}^{\prime }$, $j\in J$, that boundary $U_{j}^{\prime }$, $j\in J$ is Lipschitz, or $\partial \Omega \in Lip_{loc}$. Consequently\textit{\ }for every "cross-section" $\Omega _{L}\equiv \Omega \cap L\neq \varnothing $\ of $
\Omega
$\ with arbitrary hyperplain $L$\ exists, at least, one coordinate subspace ( $\left( x_{j},x_{k}\right) $)\ which possesses a domain $P_{x_{i}}\Omega _{L} $\ (or union of domains) whit the Lipschitz class boundary since $ \partial \Omega _{L}\equiv \partial \Omega \cap L\neq \varnothing $ and isomorphically defined of $\Omega _{L}$\ with the affine representation, \textit{\ }in addition $\partial \Omega _{L}\Longleftrightarrow \partial P_{x_{i}}\Omega _{L}$.
Thus, with use of the representation $P_{x_{i}}L$ of the hyperplane $L$ we get that $\Omega _{L}$ can be written in the form $P_{x_{i}}\Omega _{L}$, therefore an integral on $\Omega _{L}$ also will defined by the respective representation, i. e. as the integral on $P_{x_{i}}\Omega _{L}$.
It should be noted that $\Omega _{L}$ can consist of many parts then $ P_{x_{i}}\Omega _{L}$ will be such as $\Omega _{L}$. Consequently in this case $\Omega _{L}$ will be as the union of domains and the following relation will be holds \begin{equation*} \Omega _{L}=\underset{r=1}{\overset{m}{\cup }}\Omega _{L}^{r}\ \Longleftarrow \Longrightarrow \ P_{x_{i}}\Omega _{L}=\underset{r=1}{\overset {m}{\cup }}P_{x_{i}}\Omega _{L}^{r},\quad \infty >m\geq 1, \end{equation*} by virtue of the definition \ref{D_3.1}. Therefore, each of $P_{x_{i}}\Omega _{L}^{r}$ will be the domain and one can investigate these separately, as the following inclusions take place: $\Omega _{L}^{r}\subset \Omega $ and $ \partial \Omega _{L}^{r}\subset \partial \Omega $.
So, we will define subdomains of $Q^{T}\equiv \left( 0,T\right) \times \Omega $ as follows $Q_{L}^{T}\equiv \left( 0,T\right) \times \left( \Omega \cap L\right) $, where $L$ is arbitrary fixed hyperplane of the dimension two and $\Omega \cap L\neq \varnothing $. Therefore, we will study the problem onto the subdomain defined by use of the "cross-section" of $\Omega $ whit arbitrary fixed hyperplane of the dimension two $L$, i.e. the $co\dim _{R^{3}}L=1$ ($\Omega \cap L$, namely on $Q_{L}^{T}\equiv \left( 0,T\right) \times \left( \Omega \cap L\right) $).
Consequently, we will investigate uniqueness of the problem (1.1
${{}^1}$
) - (\ref{3}) on the "cross-section" $Q_{L}^{T}$ defined by the "cross-section" of $\Omega $, where $\Omega \subset R^{3}$. This "cross-section" is understood in the following sense: Let $L$ be a hyperplane in $R^{3}$ with $co\dim _{R^{3}}L=1$, clearly that $L$ is certain shift of $R^{2}$ or $R^{2}$. Denote by $\Omega _{L}$ of the "cross-section" $ \Omega _{L}\equiv \Omega \cap L\neq \varnothing $, $mes_{R^{2}}\left( \Omega _{L}\right) >0$, e.g. $L$ can be $L\equiv \left\{ \left( x_{1},x_{2},0\right) \left\vert \ x_{1},x_{2}\in R^{1}\right. \right\} $. In other words, if $L$ is the hyperplane in $R^{3}$ then we can determine it as \begin{equation*} L\equiv \left\{ x\in R^{3}\left\vert \ \left\langle a,x\right\rangle =a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}=b\right. \right\} , \end{equation*} where $a\in S_{1}^{R^{3}}\left( 0\right) $ is arbitrary fixed unit vector of $R^{3}$and $b\in R^{1}$ is arbitrary fixed constant, furthermore each $a\in S_{1}^{R^{3}}\left( 0\right) $ and $b\in R^{1}$ define of single $ L_{b}\left( a\right) $ and vice versa. Whence follows $ a_{3}x_{3}=b-a_{1}x_{1}-a_{2}x_{2}$, if we assume $a_{3}\neq 0$ then $x_{3}= \frac{1}{a_{3}}\left( b-a_{1}x_{1}-a_{2}x_{2}\right) $ , moreover, if we takes of substitutions: $\frac{b}{a_{3}}\Longrightarrow b,\frac{a_{1}}{a_{3}} \Longrightarrow a_{1}$ and $\frac{a_{2}}{a_{3}}\Longrightarrow a_{2}$ then we derive $x_{3}\equiv \psi _{3}\left( x_{1},x_{2}\right) =b-a_{1}x_{1}-a_{2}x_{2}$ in the new coefficients.
Since we will investigate the problem (1.1$^{1}$)-(\ref{3}) on $Q_{L}^{T}$, in the beginning we need define the problem that we will derive after using this projection to the problem (1.1$^{1}$)-(\ref{3}). In other words, if we denote by $F:D\left( F\right) \subseteq V\left( Q^{T}\right) \longrightarrow L^{2}\left( 0,T;V^{\ast }\left( \Omega \right) \right) \times L^{2}\left( \Omega \right) $ the operator generated by problem (1.1$^{1}$)-(\ref{3}), then we must determine of the derived problem after projection of the operator $F$ on $Q_{L}^{T}$. Clearly under this projection some of the expressions in the problem (1.1$^{1}$)-(\ref{3}) will change according of above relation, and we will derive the problem that we need to study. Consequently, now we will derive these expressions.
Thus, we get \begin{equation}
D_{3}\equiv \frac{\partial x_{1}}{\partial x_{3}}D_{1}+\frac{\partial x_{2}}{\partial x_{3}}D_{2}=-a_{1}^{-1}D_{1}-a_{2}^{-1}D_{2}\quad \&
\label{3.1} \end{equation} \begin{equation}
D_{3}^{2}=a_{1}^{-2}D_{1}^{2}+a_{2}^{-2}D_{2}^{2}+2a_{1}^{-1}a_{2}^{-1}D_{1}D_{2},\quad D_{i}=\frac{\partial }{\partial x_{i}},i=1,2,3,
\label{3.2} \end{equation} according to above mentioned reasoning.
We will assume that functions $u_{0}$ and $f$ satisfy of conditions of Theorem \ref{Th_1} , namely $u_{0}\in H^{1/2}\left( \Omega \right) $, $f\in L^{2}\left( 0,T;H^{1/2}\left( \Omega \right) \right) $ that are needed for the application of our approach. Consequently, functions $u_{0}$ and $f$\ are correctly defined on $\left( 0,T\right] \times \Omega _{L}$.
Let $L$ be arbitrary hyperplane intersecting with $\Omega $, i.e. $\Omega _{L}\neq \varnothing $ and $u\in \mathcal{V}\left( Q^{T}\right) $ is the solution of the problem (1.1
${{}^1}$
) - (\ref{3}). We will be investigate of the posed question according of Lemma \ref{L_2.2}. More precisely, we will study of the posed question for the problem generated by the "projection" (or "trace") of problem (1.1
${{}^1}$
) - (\ref{3}) onto $\left( 0,T\right] \times \Omega _{L}$.
So, we would like to apply of Lemma \ref{L_2.2} to solutions of the problem (1.1
${{}^1}$
) - (\ref{3}), for that it is necessary to study of properties of solutions of the problem (1.1
${{}^1}$
) - (\ref{3}) in "cross-section" $\left( 0,T\right] \times \Omega _{L}$. Consequently, one need to study the problem which is received from the problem (1.1
${{}^1}$
) - (\ref{3}) by "projection" (or "trace") it to $\left( 0,T\right] \times \Omega _{L}$ in order to investigate of the needed properties of solutions of the problem (1.1
${{}^1}$
) - (\ref{3}) on $\left( 0,T\right] \times \Omega _{L}$.
As function $u$ belong to $\mathcal{V}\left( Q^{T}\right) $, therefore the function $u$ on $\left( 0,T\right] \times \Omega _{L}$ is well defined. Thus, we obtain the following problem on $\left( 0,T\right] \times \Omega _{L}$ \begin{equation*} \frac{\partial u}{\partial t}-\nu \Delta u+\underset{j=1}{\overset{3}{\sum }} u_{j}D_{j}u=\frac{\partial u_{L}}{\partial t}-\nu \left( D_{1}^{2}+D_{2}^{2}+D_{3}^{2}\right) u_{L}+ \end{equation*} \begin{equation*} u_{L1}D_{1}u_{L}+u_{L2}D_{2}u_{L}+u_{L3}D_{3}u_{L}=\frac{\partial u_{L}}{ \partial t}-\nu \left[ D_{1}^{2}+D_{2}^{2}+a_{1}^{-2}D_{1}^{2}\right. + \end{equation*} \begin{equation*} \left. a_{2}^{-2}D_{2}^{2}+2a_{1}^{-1}a_{2}^{-1}D_{1}D_{2}\right] u_{L}+u_{L1}D_{1}u_{L}+u_{L2}D_{2}u_{L}-u_{L3}a_{1}^{-1}D_{1}u_{L}- \end{equation*} \begin{equation*} u_{L3}a_{2}^{-1}D_{2}u_{L}=\frac{\partial u_{L}}{\partial t}-\nu \left[ \left( 1+a_{1}^{-2}\right) D_{1}^{2}+\left( 1+a_{2}^{-2}\right) D_{2}^{2} \right] u_{L}- \end{equation*} \begin{equation} 2\nu a_{1}^{-1}a_{2}^{-1}D_{1}D_{2}u_{L}+\left( u_{L1}-a_{1}^{-1}u_{L3}\right) D_{1}u_{L}+\left( u_{L2}-a_{2}^{-1}u_{L3}\right) D_{2}u_{L}=f_{L} \label{3.3} \end{equation} on $\left( 0,T\right) \times \Omega _{L}$, by virtue of the above reasons, of the conditions of the main theorem, and also of the presentations (\ref {3.1}) and (\ref{3.2}). We get \begin{equation}
\func{div}u_{L}=D_{1}\left( u_{L}-a_{1}^{-1}u_{L3}\right) +D_{2}\left( u_{L}-a_{2}^{-1}u_{L3}\right) =0,\quad x\in \Omega _{L},\ t>0
\label{3.4} \end{equation} \begin{equation}
u_{L}\left( 0,x\right) =u_{L0}\left( x\right) ,\quad \left( t,x\right) \in \left[ 0,T\right] \times \Omega _{L};\quad u_{L}\left\vert \ _{\left( 0,T\right) \times \partial \Omega _{L}}\right. =0.
\label{3.5} \end{equation} by using of same way.
Thus, we derived the problem (\ref{3.3}) - (\ref{3.5}) the study of which will give we possibility to define properties of solutions $u$ of problem (1.1
${{}^1}$
) - (\ref{3}) on each "cross-section" $\left[ 0,T\right) \times \Omega _{L}\equiv Q_{L}^{T}$.
In the beginning it is necessary to investigate the existence of the solution of problem (\ref{3.3}) - (\ref{3.5}) and determine the space where the existing solutions are contained. Consequently, for study of the uniqueness of the solution of problem (1.1
${{}^1}$
) - (\ref{3}) at first it is necessary to investigate the existence and uniqueness of the solution for the derived problem (\ref{3.3}) - (\ref{3.5} ). Therefore we will to investigate of problem (\ref{3.3}) - (\ref{3.5}).
We must to note: For each hyperplane $L\subset R^{3}$ there exists, at least, one $2$-dimensional subspace of $R^{3}$ that in the given coordinat system one can determine as $\left( x_{i},x_{j}\right) $ and $ P_{x_{k}}L=R^{2}$ (e.g. $i,j,k=1,2,3$), i.e. \begin{equation*} L\equiv \left\{ x\in R^{3}\left\vert \ x=\left( x_{i},x_{j},\psi _{L}\left( x_{i},x_{j}\right) \right) ,\right. \left( x_{i},x_{j}\right) \in R^{2}\right\} \end{equation*} and \begin{equation*} \Omega \cap L\equiv \left\{ x\in \Omega \left\vert \ x=\left( x_{i},x_{j},\psi _{L}\left( x_{i},x_{j}\right) \right) ,\right. \left( x_{i},x_{j}\right) \in P_{x_{k}}\left( \Omega \cap L\right) \right\} \end{equation*} hold, where $\psi _{L}$ is the affine function that is the bijection.
Thereby, in this case functions $u(t,x),\ f(t,x)$ and$\ u_{0}(x)$ can be represented as \begin{equation*} u(t,x_{i},x_{j},\psi _{L}(x_{i},x_{j}))\equiv v(t,x_{i},x_{j})\text{, } f(t,x_{i},x_{j},\psi _{L}(x_{i},x_{j})\equiv \phi (x_{i},x_{j}) \end{equation*} and \begin{equation*} u_{0}(x_{i},x_{j},\psi _{L}(x_{i},x_{j}))\equiv v_{0}(x_{i},x_{j})\text{ \ \ on }(0,T)\times P_{x_{k}}\Omega _{L}, \end{equation*} respectively.
So, each of these functions can be represented as functions from the independent variables: $t$, $x_{i}$ and $x_{j}$.
\subsection{\label{SS_I.3.1}\textbf{On Dirichlet to Neumann map}}
As is known (\cite{Nac}, \cite{BehEl}, \cite{DePrZac}, \cite{H-DR}, \cite {BelCho} etc.) the Dirichlet to Neumann map is single-value maping if the homogeneous Dirichlet problem for elliptic equation has only trivial solution, i.e. zero not is eigenvalue of this problem. Consequently, it is enough to show that the homogeneous Dirichlet problem for elliptic equation assosiated to considered problem satisfies of the corresponding conditions of the results of the mentioned articles. So, we will prove the following
\begin{proposition} \label{Pr_4.1}The homogeneous Dirichlet problem for elliptic part of problem (3.3) - (3.5) has only trivial solution. \end{proposition}
\begin{proof} If consider the elliptic part of problem (3.3) - (3.5) then we get\ the problem \begin{equation*} -\Delta u_{L}+Bu_{L}\equiv -\nu \left[ \left( 1+a_{1}^{-2}\right) D_{1}^{2}+\left( 1+a_{2}^{-2}\right) D_{2}^{2}+2a_{1}^{-1}a_{2}^{-1}D_{1}D_{2}\right] u_{L}+ \end{equation*} \begin{equation*} \left( u_{L1}-a_{1}^{-1}u_{L3}\right) D_{1}u_{L}+\left( u_{L2}-a_{2}^{-1}u_{L3}\right) D_{2}u_{L}=0,\ x\in \Omega _{L},\quad u_{L}\left\vert _{\ \partial \Omega _{L}}\right. =0, \end{equation*} where $\Omega _{L}=\Omega \cap L$.
Let 's show that this problem cannot have nontrivial solutions. This will be to prove by method of contradiction. Let $u_{L}\in V\left( \Omega _{L}\right) $ be nontrivial solution of this problem then we get the following equation \begin{equation*} 0=\left\langle -\Delta u_{L}+Bu_{L},u_{L}\right\rangle _{P_{x_{3}}\Omega _{L}} \end{equation*} hence \begin{equation*} =-\underset{i=1}{\overset{3}{\nu \sum }}\left\langle \left[ \left( D_{1}^{2}+D_{2}^{2}\right) +\left( a_{1}^{-1}D_{1}+a_{2}^{-1}D_{2}\right) ^{2}\right] u_{Li},u_{Li}\right\rangle _{P_{x_{3}}\Omega _{L}}+ \end{equation*} \begin{equation*} \underset{i=1}{\overset{3}{\sum }}\underset{P_{x_{3}}\Omega _{L}}{\int } \left[ u_{L1}D_{1}u_{Li}u_{Li}+u_{L2}D_{2}u_{Li}u_{Li}+\right. \end{equation*} \begin{equation*} \left. u_{L3}\left( -a_{1}^{-1}D_{1}-a_{2}^{-1}D_{2}\right) u_{Li}u_{Li} \right] dx_{1}dx_{2}= \end{equation*} \begin{equation*} \underset{i=1}{\overset{3}{\nu \sum }}\underset{P_{x_{3}}\Omega _{L}}{\int } \left\{ \left( D_{1}u_{Li}\right) ^{2}+\left( D_{2}u_{Li}\right) ^{2}+\left[ \left( a_{1}^{-1}D_{1}+a_{2}^{-1}D_{2}\right) u_{Li}\right] ^{2}\right\} dx_{1}dx_{2}+ \end{equation*} \begin{equation*} \frac{1}{2}\underset{i=1}{\overset{3}{\sum }}\underset{P_{x_{3}}\Omega _{L}}{ \int }\left[ u_{L1}D_{1}\left( u_{Li}\right) ^{2}+u_{L2}D_{2}\left( u_{Li}\right) ^{2}+\right. \end{equation*} \begin{equation*} \left. u_{L3}\left( -a_{1}^{-1}D_{1}-a_{2}^{-1}D_{2}\right) \left( u_{Li}\right) ^{2}\right] dx_{1}dx_{2}\geq \end{equation*} \begin{equation*} \underset{i=1}{\overset{3}{\nu \sum }}\underset{P_{x_{3}}\Omega _{L}}{\int } \left[ \left\vert D_{1}u_{Li}\right\vert ^{2}+\left\vert D_{2}u_{Li}\right\vert ^{2}\right] dx_{1}dx_{2}+ \end{equation*} \begin{equation*} -\frac{1}{2}\underset{i=1}{\overset{3}{\sum }}\underset{P_{x_{3}}\Omega _{L}} {\int }\left[ D_{1}u_{L1}+D_{2}u_{L2}+\left( -a_{1}^{-1}D_{1}-a_{2}^{-1}D_{2}\right) u_{L3}\right] \left\vert u_{Li}\right\vert ^{2}dx_{1}dx_{2}= \end{equation*} by (\ref{3.4}) \begin{equation*} \underset{i=1}{\overset{3}{\nu \sum }}\underset{P_{x_{3}}\Omega _{L}}{\int } \left[ \left\vert D_{1}u_{Li}\right\vert ^{2}+\left\vert D_{2}u_{Li}\right\vert ^{2}\right] dx_{1}dx_{2}-\frac{1}{2}\underset{i=1}{ \overset{3}{\sum }}\underset{P_{x_{3}}\Omega _{L}}{\int }\left\vert u_{Li}\right\vert ^{2}\func{div}u_{L}dx_{1}dx_{2}= \end{equation*} \begin{equation*} \underset{i=1}{\overset{3}{\nu \sum }}\underset{P_{x_{3}}\Omega _{L}}{\int } \left[ \left\vert D_{1}u_{Li}\right\vert ^{2}+\left\vert D_{2}u_{Li}\right\vert ^{2}\right] dx_{1}dx_{2}>0. \end{equation*} Thus, the obtained contradiction shows that function $u_{L}$ need be zero, i.e. $u_{L}=0$ holds.
Consequently, the Dirichlet to Neumann map is single-value operator. \end{proof}
It is well-known that operator $-\Delta :H_{0}^{1}\left( \Omega _{L}\right) \longrightarrow $ $H^{-1}\left( \Omega _{L}\right) $ generates of the $C_{0}$ semigroup on $H\left( \Omega _{L}\right) $, and since inclusion $ H_{0}^{1}\left( \Omega _{L}\right) \subset H^{-1}\left( \Omega _{L}\right) $ is compact, therefore $\left( -\Delta \right) ^{-1}$ is the compact operator in $H^{-1}\left( \Omega _{L}\right) $. Moreover, $-\Delta :H^{1/2}\left( \partial \Omega _{L}\right) \longrightarrow H^{-1/2}\left( \partial \Omega _{L}\right) $ and the operator $B:$ $H^{1/2}\left( \partial \Omega _{L}\right) \longrightarrow H^{-1/2}\left( \partial \Omega _{L}\right) $ also possess appropriate properties of such types.
\section{\label{Sec_I.4}Existence of Solution of Problem (3.3) - (3.5)}
So, assume conditons of Theorem \ref{Th_1} fulfilled, i. e. \begin{equation*} u_{0}\in H^{1/2}\left( \Omega \right) ,\quad f\in L^{2}\left( 0,T;H^{1/2}\left( \Omega \right) \right) , \end{equation*} then these functions on $\Omega _{L}$, $Q_{L}^{T}$ are correctly defined and belong to $H\left( \Omega _{L}\right) $, $L^{2}\left( 0,T;H\left( \Omega _{L}\right) \right) $, respectively. Consequently, we can study problem (\ref {3.3}) - (\ref{3.5}) under conditions $u_{0L}\in H\left( \Omega _{L}\right) $ and $f_{L}\in L^{2}\left( 0,T;H\left( \Omega _{L}\right) \right) $, as independent problem.
By executing according the known argument started by Leray (\cite{Ler1}, see, also \cite{Lio1}, \cite{FoiManRosTem}, \cite{Gal}), the space $V\left( \Omega _{L}\right) $ of the vector functions $u$ one can determine by same way as in Definition \ref{D_2.1}: the space $V\left( \Omega _{L}\right) $ is the closure in $\left( H_{0}^{1}\left( \Omega _{L}\right) \right) ^{3}$\ of \begin{equation*} \left\{ \varphi \left\vert \ \varphi \in \left( C_{0}^{\infty }\left( \Omega _{L}\right) \right) ^{3},\right. \func{div}\varphi =0\right\} \end{equation*} $\left( W_{0}^{1,2}\left( \Omega _{L}\right) \right) ^{3}$, where $\func{div} $ is regarded in the sense (\ref{3.4}), in this case the dual space $V\left( \Omega _{L}\right) $ is determined as $V^{\ast }\left( \Omega _{L}\right) $, the space $H\left( \Omega _{L}\right) $ also is determined as the closure in $\left( L^{2}\left( \Omega _{L}\right) \right) ^{3}$ of \begin{equation*} \left\{ \varphi \left\vert \ \varphi \in \left( C_{0}^{\infty }\left( \Omega _{L}\right) \right) ^{3},\right. \func{div}\varphi =0\right\} \end{equation*} in the topology of $\left( L^{2}\left( \Omega _{L}\right) \right) ^{3}$.
Consequently, one can determine of space $V\left( Q_{L}^{T}\right) $ as \begin{equation*} V\left( Q_{L}^{T}\right) \equiv L^{2}\left( 0,T;V\left( \Omega _{L}\right) \right) \cap L^{\infty }\left( 0,T;H\left( \Omega _{L}\right) \right) . \end{equation*}
Here $\Omega \subset R^{3}$ is bounded domain of $Lip_{loc}$ and $\Omega _{L}\subset R^{2}$ is subdomain defined in the beginning of Section \ref {Sec_I.3} therefore, $\Omega _{L}$ is Lipschitz, $Q_{L}^{T}\equiv \left( 0,T\right) \times \Omega _{L}$.
Let $f_{L}\in L^{2}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $ and $u_{0L}\in H\left( \Omega _{L}\right) $. Consequently, a solution of problem (\ref{3.3}) - (\ref{3.5}) will be understood as follows.
So, we can call the solution of this problem: A function $u_{L}\in \mathcal{V }\left( Q_{L}^{T}\right) $ is called a solution of the problem (\ref{3.3}) - (\ref{3.5}) if $u_{L}(t,x^{\prime })$ satisfy the equality \begin{equation} \frac{d}{dt}\left\langle u_{L},v\right\rangle _{\Omega _{L}}-\left\langle \nu \Delta u_{L},v\right\rangle _{\Omega _{L}}+\left\langle \underset{j=1}{ \overset{3}{\sum }}u_{Lj}D_{j}u_{L},v\right\rangle _{\Omega _{L}}=\left\langle f_{L},v\right\rangle _{\Omega _{L}}, \label{3.6} \end{equation} for any $v\in V\left( \Omega _{L}\right) $ and almost everywhere in $\left( 0,T\right) $ and initial condition \begin{equation*} \left\langle u_{L}\left( t\right) ,v\right\rangle \left\vert _{t=0}\right. =\left\langle u_{0L},v\right\rangle , \end{equation*} in the sense of $H$, where $\left\langle \circ ,\circ \right\rangle _{\Omega _{L}}$ is the dual form for the pair of spaces $\left( V\left( \Omega _{L}\right) ,V^{\ast }\left( \Omega _{L}\right) \right) $ and $\Omega _{L}$ is Lipschitz. Where $x^{\prime }\in \Omega _{L}$ is $x^{\prime }\equiv \left( x_{1},x_{2}\right) $ (according to our selection of the $L$) and $ \mathcal{V}\left( Q_{L}^{T}\right) $ is \begin{equation*} \mathcal{V}\left( Q_{L}^{T}\right) \equiv \left\{ w\left\vert \ w\in V\left( Q_{L}^{T}\right) ,\ w^{\prime }\in L^{\frac{4}{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) \right. \right\} . \end{equation*}
We will lead of the proof of this problem in five-steps as indepandent problem.
\subsection{\label{SS_I.4.1}\textbf{A priori estamates}}
In order to derive of the a priori estimates for the possible solutions of the problem we will apply of the usual approach. By substituting in (\ref {3.6}) of the function $u_{L}$ instead of the function $v$, we get \begin{equation} \frac{d}{dt}\left\langle u_{L},u_{L}\right\rangle _{\Omega _{L}}-\left\langle \nu \Delta u_{L},u_{L}\right\rangle _{\Omega _{L}}+\left\langle \underset{j=1}{\overset{3}{\sum }}u_{Lj}D_{j}u_{L},u_{L} \right\rangle _{\Omega _{L}}=\left\langle f_{L},u_{L}\right\rangle _{\Omega _{L}}. \label{3.6'} \end{equation} Thence, by making the known calculations, taking into account of the condition on $\Omega _{L}$ and (\ref{3.4}), and also of calculations (\ref {3.1}) that carried out in the previous Section, we derive \begin{equation*} \frac{1}{2}\frac{d}{dt}\left\Vert u_{L}\right\Vert _{H\left( \Omega _{L}\right) }^{2}\left( t\right) +\nu \left( 1+a_{1}^{-2}\right) \left\Vert D_{1}u_{L}\right\Vert _{H\left( \Omega _{L}\right) }^{2}\left( t\right) + \end{equation*} \begin{equation} \nu \left( 1+a_{2}^{-2}\right) \left\Vert D_{2}u_{L}\right\Vert _{H\left( \Omega _{L}\right) }^{2}\left( t\right) +2\nu a_{1}^{-1}a_{2}^{-1}\left\langle D_{1}u_{L},D_{2}u_{L}\right\rangle _{\Omega _{L}}\left( t\right) =\left\langle f_{L},u_{L}\right\rangle _{\Omega _{L}}, \label{3.7} \end{equation} where $\left\langle g,h\right\rangle _{\Omega _{L}}=\underset{i=1}{\overset{3 }{\sum }}\underset{P_{x_{3}}\Omega _{L}}{\int }g_{i}h_{i}dx_{1}dx_{2}$ for any $g,h\in H\left( \Omega _{L}\right) $, or $g\in \left( W^{1,2}\left( \Omega _{L}\right) \right) ^{3}$ and $h\in \left( W^{-1,2}\left( \Omega _{L}\right) \right) ^{3}$, respectively. We will show the correctness of ( \ref{3.7}), and to this end we shall prove the correctness of each term of this sum, separately.
So, using of (\ref{3.6'}) we get \begin{equation*} -\nu \left\langle \Delta u_{L}\left( t\right) ,u_{L}\left( t\right) \right\rangle _{\Omega _{L}}= \end{equation*} \begin{equation*} -\underset{i=1}{\overset{3}{\nu \sum }}\left\langle \left[ \left( 1+a_{1}^{-2}\right) D_{1}^{2}+\left( 1+a_{2}^{-2}\right) D_{2}^{2}+2a_{1}^{-1}a_{2}^{-1}D_{1}D_{2}\right] u_{Li},u_{Li}\right\rangle _{P_{x_{3}}\Omega _{L}}= \end{equation*} \begin{equation*} \underset{i=1}{\overset{3}{\nu \sum }}\underset{P_{x_{3}}\Omega _{L}}{\int } \left[ \left( 1+a_{1}^{-2}\right) \left( D_{1}u_{Li}\right) ^{2}+\left( 1+a_{2}^{-2}\right) \left( D_{2}u_{Li}\right) ^{2}+\right. \end{equation*} \begin{equation*} \left. 2a_{1}^{-1}a_{2}^{-1}D_{1}u_{Li}D_{2}u_{Li}\right] dx_{1}dx_{2} \end{equation*} thus is obtained the sum reducible in (\ref{3.7}).
Whence isn`t difficult to seen, that if to estimate of the last adding in the above mentioned sum then one will received \begin{equation*} -\nu \left\langle \Delta u_{L}\left( t\right) ,u_{L}\left( t\right) \right\rangle _{\Omega _{L}}\geq \end{equation*} \begin{equation} \nu \left[ \left\Vert D_{1}u_{L}\right\Vert _{H\left( \Omega _{L}\right) }^{2}\left( t\right) +\left\Vert D_{2}u_{L}\right\Vert _{H\left( \Omega _{L}\right) }^{2}\left( t\right) \right] . \label{3.8} \end{equation}
Now consider the trilinear form from (\ref{3.6'}) \begin{equation*} \left\langle \underset{j=1}{\overset{3}{\sum }}u_{Lj}D_{j}u_{L},u_{L}\right \rangle _{\Omega _{L}}= \end{equation*} due to (\ref{3.3}) we get \begin{equation*} \underset{i=1}{\overset{3}{\sum }}\underset{P_{x_{3}}\Omega _{L}}{\int } \left[ u_{L1}D_{1}u_{Li}u_{Li}+u_{L2}D_{2}u_{Li}u_{Li}+\right. \end{equation*} \begin{equation*} \left. u_{L3}\left( -a_{1}^{-1}D_{1}-a_{2}^{-1}D_{2}\right) u_{Li}u_{Li} \right] dx_{1}dx_{2}= \end{equation*} \begin{equation*} \frac{1}{2}\underset{i=1}{\overset{3}{\sum }}\underset{P_{x_{3}}\Omega _{L}}{ \int }\left[ u_{L1}D_{1}\left( u_{Li}\right) ^{2}+u_{L2}D_{2}\left( u_{Li}\right) ^{2}+\right. \end{equation*} \begin{equation*} \left. u_{L3}\left( -a_{1}^{-1}D_{1}-a_{2}^{-1}D_{2}\right) \left( u_{Li}\right) ^{2}\right] dx_{1}dx_{2}= \end{equation*} \begin{equation*} -\frac{1}{2}\underset{i=1}{\overset{3}{\sum }}\underset{P_{x_{3}}\Omega _{L}} {\int }\left[ D_{1}u_{L1}+D_{2}u_{L2}+\left( -a_{1}^{-1}D_{1}-a_{2}^{-1}D_{2}\right) u_{L3}\right] \left( u_{Li}\right) ^{2}dx_{1}dx_{2}= \end{equation*} hence by (\ref{3.4}) \begin{equation} -\frac{1}{2}\underset{i=1}{\overset{3}{\sum }}\underset{P_{x_{3}}\Omega _{L}} {\int }\left( u_{Li}\right) ^{2}\func{div}u_{L}dx_{1}dx_{2}=0. \label{3.9} \end{equation}
Consequently, the correctness of equation (\ref{3.7}) is proved.
From (\ref{3.7}) in view of (\ref{3.8})-(\ref{3.9}) is derived the following inequality \begin{equation*} \frac{1}{2}\frac{d}{dt}\left\Vert u_{L}\right\Vert _{H\left( \Omega _{L}\right) }^{2}\left( t\right) + \end{equation*} \begin{equation} \nu \underset{i=1}{\overset{3}{\sum }}\underset{P_{x_{3}}\Omega _{L}}{\int } \left[ \left( D_{1}u_{Li}\right) ^{2}+\left( D_{2}u_{Li}\right) ^{2}\right] dx_{1}dx_{2}\leq \underset{P_{x_{3}}\Omega _{L}}{\int }\left\vert \left( f_{L}\cdot u_{L}\right) \right\vert dx_{1}dx_{2} \label{3.10} \end{equation} which give we the following a priori estimates \begin{equation} \left\Vert u_{L}\right\Vert _{H\left( \Omega _{L}\right) }\left( t\right) \leq C\left( f_{L},u_{L0},mes\Omega \right) , \label{3.11} \end{equation} \begin{equation} \left\Vert D_{1}u_{L}\right\Vert _{H\left( \Omega _{L}\right) }+\left\Vert D_{2}u_{L}\right\Vert _{H\left( \Omega _{L}\right) }\leq C\left( f_{L},u_{L0},mes\Omega \right) , \label{3.12} \end{equation} where $C\left( f_{L},u_{L0},mes\Omega \right) >0$ is the constant that is independent of $u_{L}$. Consequently, any possible solution of this problem belong to a bounded subset of the space $V\left( Q_{L}^{T}\right) $.
Thus, it is remain to receive of the necessary a priori estimate for $\frac{ \partial u_{L}}{\partial t}$ and to study of properties of the thrilinear term in order to prove of the existence theorem. \footnote{ It should be noted that if the represantation of $
\Omega
_{L}$ by coordinates $(x_{1},x_{2})$ not is best for the definition of the appropriate integral, then we will select other coordinates: either $ (x_{1},x_{3})$ or $(x_{2},x_{3})$ instead of $(x_{1},x_{2})$, which is best for our goal (that must exist by virtue of the definition of $
\Omega
$). \par {}}
\subsection{\label{SS_I.4.2}Boundedness of the trilinear form}
Boundedness of the trilinear form $b_{L}\left( u_{L},u_{L},v\right) $ from ( \ref{3.6}) follows from the next result.
\begin{proposition} \label{P_3.1}Let $u_{L}\in V\left( Q_{L}^{T}\right) \cap L^{\infty }\left( 0,T;H\right) $, $v\in V\left( \Omega _{L}\right) $ and $B$ is the operator defined by \begin{equation*} \left\langle B\left( u_{L}\right) ,v\right\rangle _{\Omega _{L}}=b_{L}\left( u_{L},u_{L},v\right) =\left\langle \underset{j=1}{\overset{3}{\sum }} u_{Lj}D_{j}u_{L},v\right\rangle _{\Omega _{L}} \end{equation*} then $B\left( u_{L}\right) $ belongs to bounded subset of $L^{\frac{3}{2} }\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $. \end{proposition}
\begin{proof} At first we will show boundedness of the operator $B$ acting from $V\left( \Omega _{L}\right) \times V\left( \Omega _{L}\right) $ to $V^{\ast }\left( \Omega _{L}\right) $ for a. e. $t\in \left( 0,T\right) $. We have \begin{equation*} \left\langle B\left( u_{L}\right) ,v\right\rangle _{\Omega _{L}}=\left\langle \underset{j=1}{\overset{3}{\sum }}u_{Lj}D_{j}u_{L},v \right\rangle _{\Omega _{L}}= \end{equation*} \begin{equation*} \underset{i=1}{\overset{3}{\sum }}\underset{P_{x_{3}}\Omega _{L}}{\int } \left[ u_{L1}D_{1}u_{Li}v_{i}+u_{L2}D_{2}u_{Li}v_{i}+u_{L3}\left( -a_{1}^{-1}D_{1}-a_{2}^{-1}D_{2}\right) u_{Li}v_{i}\right] dx_{1}dx_{2}= \end{equation*} \begin{equation*} \underset{i=1}{\overset{3}{\sum }}\underset{P_{x_{3}}\Omega _{L}}{\int } \left[ \left( u_{L1}-a_{1}^{-1}u_{L3}\right) D_{1}u_{Li}v_{i}+\left( u_{L2}-a_{2}^{-1}u_{L3}\right) D_{2}u_{Li}v_{i}\right] dx_{1}dx_{2}= \end{equation*} \begin{equation} \underset{i=1}{\overset{3}{\sum }}\underset{P_{x_{3}}\Omega _{L}}{\int } \left[ \left( u_{L1}-a_{1}^{-1}u_{L3}\right) D_{1}+\left( u_{L2}-a_{2}^{-1}u_{L3}\right) D_{2}\right] u_{Li}v_{i}dx_{1}dx_{2} \label{3.13} \end{equation} due of (\ref{3.4}) and of the definition \ref{D_3.1}.
Hence follows \begin{equation*} \left\vert \left\langle B\left( u_{L}\right) ,v\right\rangle \right\vert \leq \underset{i=1}{\overset{3}{\sum }}\underset{P_{x_{3}}\Omega _{L}}{\int } c\left( \left\vert u_{L1}\right\vert +\left\vert u_{L2}\right\vert +\left\vert u_{L3}\right\vert \right) \left( \left\vert D_{1}u_{Li}\right\vert +\left\vert D_{2}u_{Li}\right\vert \right) v_{i}dx_{1}dx_{2}\leq \end{equation*} \begin{equation} c\left\Vert u_{L}\right\Vert _{L^{4}\left( \Omega _{L}\right) }\left\Vert u_{L}\right\Vert _{V}\left\Vert v\right\Vert _{L^{4}\left( \Omega _{L}\right) }\Longrightarrow \left\Vert B\left( u_{L}\right) \right\Vert _{V^{\ast }}\leq c\left\Vert u_{L}\right\Vert _{L^{4}\left( \Omega _{L}\right) }\left\Vert u_{L}\right\Vert _{V} \label{3.14} \end{equation} due of $V\left( \Omega _{L}\right) \subset L^{4}\left( \Omega _{L}\right) $. This also shows that operator $B:V\left( \Omega _{L}\right) \longrightarrow V^{\ast }\left( \Omega _{L}\right) $ is bounded, and continuous for a. e. $ t>0$.
Finally, we obtain needed result using above inequality and the well-known inequality, which is valid in two-dimension space (see, \cite{Lad1}, \cite {Lio1}, \cite{Tem1}) \begin{equation*} \overset{T}{\underset{0}{\int }}\left\Vert B\left( u_{L}\left( t\right) \right) \right\Vert _{V^{\ast }}^{\frac{4}{3}}dt\leq c\overset{T}{\underset{0 }{\int }}\left( \left\Vert u_{L}\left( t\right) \right\Vert _{L^{4}}\left\Vert u_{L}\right\Vert _{V}\right) ^{\frac{4}{3}}dt\leq \end{equation*} according to Gagliardo--Nirenberg inequality we get \begin{equation*} c_{1}\overset{T}{\underset{0}{\int }}\left\Vert u_{L}\left( t\right) \right\Vert _{L^{2}}^{\frac{2}{3}}\left\Vert u_{L}\right\Vert _{V}^{2}dt\leq c_{1}\left\Vert u_{L}\right\Vert _{L^{\infty }\left( 0,T;H\right) }^{\frac{2 }{3}}\overset{T}{\underset{0}{\int }}\left\Vert u_{L}\right\Vert _{V}^{2}dt\Longrightarrow \end{equation*} \begin{equation} \left\Vert B\left( u_{L}\right) \right\Vert _{L^{\frac{4}{3}}\left( 0,T;V^{\ast }\right) }\leq c_{1}\left\Vert u_{L}\right\Vert _{L^{\infty }\left( 0,T;H\right) }^{\frac{1}{2}}\left\Vert u_{L}\right\Vert _{L^{2}\left( 0,T;V\right) }^{\frac{3}{2}}. \label{3.15} \end{equation} \end{proof}
Moreover, is proved that \begin{equation*} B:L^{2}\left( 0,T;V\left( \Omega _{L}\right) \right) \cap L^{\infty }\left( 0,T;H\right) =V\left( Q_{L}^{T}\right) \longrightarrow L^{\frac{4}{3}}\left( 0,T;V^{\ast }\right) \end{equation*} is bounded operator.
\subsection{\label{SS_I.4.3}Boundedness of $u^{\prime }$}
Sketch of the proof that $u^{\prime }$ belongs to bounded subset of the space $L^{\frac{4}{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $ . It is possible to draw the following conclusion based due received of a priori estimates, proposition \ref{P_3.1} and reflexivity of all used spaces: If we will use of the Faedo-Galerkin's method for investigation then for the approximate solutions we obtain estimates of such type as (\ref{3.11} ), (\ref{3.12}) and (\ref{3.15}). Indeed since $V\left( \Omega _{L}\right) $ is a separable there exists a countable subset of linearly independent elements $\left\{ w_{i}\right\} _{i=1}^{\infty }\subset V\left( \Omega _{L}\right) $, which is total in $V\left( \Omega _{L}\right) $. For each $m$ we can define an approximate solution of $u_{Lm}$ (\ref{3.6}) as follows
\begin{equation} u_{Lm}=\overset{m}{\underset{i=1}{\sum }}u_{Lm}^{i}\left( t\right) w_{i},\quad m=1,\ 2,...., \label{5.4} \end{equation} where $u_{Lm}^{i}\left( t\right) $, $i=\overline{1,m}$ be unknown functions that will be determined as solutions of the following system of the differential equations that is received according to equation (\ref{3.6}) \begin{equation*} \left\langle \frac{d}{dt}u_{Lm},w_{j}\right\rangle _{\Omega _{L}}=\left\langle \nu \Delta u_{Lm},w_{j}\right\rangle _{\Omega _{L}}+b_{L}\left( u_{Lm},u_{Lm},w_{j}\right) + \end{equation*} \begin{equation} +\left\langle f_{L},w_{j}\right\rangle _{\Omega _{L}},\quad t\in \left( 0,T \right] ,\quad j=\overline{1,m},\quad \label{5.5} \end{equation} \begin{equation*} u_{Lm}\left( 0\right) =u_{0Lm}. \end{equation*} Here we assume $\left\{ u_{0Lm}\right\} _{m=1}^{\infty }\subset H\left( \Omega _{L}\right) $ be such sequence that $u_{0Lm}\longrightarrow u_{0L}$ in $H\left( \Omega _{L}\right) $ as $m\longrightarrow \infty $. (Since $ V\left( \Omega _{L}\right) $ is everywhere dense in $H\left( \Omega _{L}\right) $ one can determine $u_{0Lm}$ by using the total system $\left\{ w_{i}\right\} _{i=1}^{\infty }$). \
So, with use (\ref{5.4}) in (\ref{5.5}) we obtain \begin{equation*} \overset{m}{\underset{j=1}{\sum }}\left\langle w_{j},w_{i}\right\rangle _{\Omega _{L}}\frac{d}{dt}u_{Lm}^{j}\left( t\right) -\nu \overset{m}{ \underset{j=1}{\sum }}\left\langle \Delta w_{j},w_{i}\right\rangle _{\Omega _{L}}u_{Lm}^{j}\left( t\right) + \end{equation*} \begin{equation*} \overset{m}{\underset{j,k=1}{\sum }}b_{L}\left( w_{j},w_{k},w_{i}\right) u_{Lm}^{j}\left( t\right) u_{Lm}^{k}\left( t\right) =\left\langle f_{L}\left( t\right) ,w_{i}\right\rangle _{\Omega _{L}},\ i=\overline{1,m}. \end{equation*} As the matrix generated by $\left\langle w_{i},w_{j}\right\rangle _{\Omega _{L}}$, $i,j=\overline{1,m}$ is nonsingular then its inverse exists. Thanks this from the previous equations we will derive the following Cauchy problem for the system of the nonlinear ordinary differential equations for unknown functions $u_{Lm}^{i}\left( t\right) $, $i=1,...,m$. \begin{equation*} \frac{du_{Lm}^{i}\left( t\right) }{dt}=\overset{m}{\underset{j=1}{\sum }} c_{i,j}\left\langle f_{L}\left( t\right) ,w_{j}\right\rangle _{\Omega _{L}}-\nu \overset{m}{\underset{j=1}{\sum }}d_{i,j}u_{Lm}^{j}\left( t\right) + \end{equation*} \begin{equation} \overset{m}{\underset{j,k=1}{\sum }}h_{ijk}u_{Lm}^{j}\left( t\right) u_{Lm}^{k}\left( t\right) , \label{5.6} \end{equation} \begin{equation*} u_{Lm}^{i}\left( 0\right) =u_{0Lm}^{i},\quad i=1,...,m,\quad m=1,\ 2,... \end{equation*} where $u_{0Lm}^{i}$ is $i^{th}$ component of $u_{0L}$ in representation $ u_{0L}=\overset{\infty }{\underset{k=1}{\sum }}u_{0Lm}^{k}w_{k}$.
The Cauchy problem for the system of the nonlinear ordinary differential equations (\ref{5.6}) has solution, which defined on whole of interval $ (0,T] $ due of uniformity of estimations received in subsections \ref {SS_I.4.1} and \ref{SS_I.4.2}. Consequently, the approximate solutions $ u_{Lm}$ exist and belong to a bounded subset of $W^{1,\frac{4}{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $ for every $m=1,\ 2,...$ since the right side of (\ref{5.6}) belong to a bounded subset of $ L^{2}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $ as were proved in subsections \ref{SS_I.4.1} and \ref{SS_I.4.2}, and also by virtue of the next lemma.
\begin{lemma} (\cite{Tem1}) Let $X$ be a given Banach space with dual $X^{\ast }$ and let $ u$ and $g$ be two functions belonging to $L^{1}\left( a,b;X\right) $. Then, the following three conditions are equivalent
\textit{(i)} $u$ is a. e. equal to a primitive function of $g$, \begin{equation*} u\left( t\right) =\xi +\underset{a}{\overset{t}{\int }}g\left( s\right) ds,\quad \xi \in X,\quad \text{a.e. }t\in \left[ a,b\right] \end{equation*} \textit{(ii)} For each test function $\varphi \in D\left( \left( a,b\right) \right) $, \begin{equation*} \underset{a}{\overset{b}{\int }}u\left( t\right) \varphi ^{\prime }\left( t\right) dt=-\underset{a}{\overset{b}{\int }}g\left( t\right) \varphi \left( t\right) dt,\quad \varphi ^{\prime }=\frac{d\varphi }{dt} \end{equation*} (iii) For each $\eta \in X^{\ast }$, \begin{equation*} \frac{d}{dt}\left\langle u,\eta \right\rangle =\left\langle g,\eta \right\rangle \end{equation*} in the scalar distribution sense, on $(a,b)$. If \textit{(i) - (iii)} are satisfied $u$, in particular, is a. e. equal to a continuous function from $ [a,b]$ into $X$. \end{lemma}
It isn
\'{}
t difficult to see that if take $\forall v\in V\left( \Omega _{L}\right) $ instead of $w_{k}$ and pass to limit according to $m\longrightarrow \infty $ in equation (\ref{5.5}) (may be by subsequence $\left\{ u_{Lm_{\mathit{l} }}\right\} _{\mathit{l}=1}^{\infty }$ of this sequence, is known that such subsequence exists) then we get \ \begin{equation} \left\langle \frac{d}{dt}u_{L},v\right\rangle _{\Omega _{L}}=\left\langle f_{L}+\nu \Delta u_{L}-\chi ,v\right\rangle _{\Omega _{L}}, \label{5.6.1} \end{equation} due of fullness of the class $\left\{ w_{i}\right\} _{i=1}^{\infty }$ in $ V\left( \Omega _{L}\right) .$ Where function $\chi $ belongs to $L^{\frac{4}{ 3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $ and is determined by equality \begin{equation*} \underset{\mathit{l}\longrightarrow \infty }{\lim }\left\langle B\left( u_{Lm_{\mathit{l}}}\right) ,v\right\rangle _{\Omega _{L}}=\left\langle \chi ,v\right\rangle _{\Omega _{L}} \end{equation*} that shown in the above section. So, we obtain that in (\ref{5.6.1}) the right side belong to $L^{\frac{4}{3}}\left( 0,T\right) $ then the left side also belongs to $L^{2}\left( 0,T\right) $ according to above a priori estimates and Proposition \ref{P_3.1}, i. e. \begin{equation*} \frac{du_{L}}{dt}\in L^{\frac{4}{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) . \end{equation*} Consequently, the following result is proven.
\begin{proposition} \label{P_4.1}Under above mentioned conditions $u_{L}^{\prime }$ belongs to a bounded subset of the space $L^{\frac{4}{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $. \end{proposition}
From above results of this section by virtue of the abstract form of Riesz-Fischer theorem follows
\begin{corollary} \label{C_4.1}Under above mentioned conditions function $u_{L}$ belongs to a bounded subset of the space $\mathcal{V}\left( Q_{L}^{T}\right) $, where \begin{equation} \mathcal{V}\left( Q_{L}^{T}\right) \equiv V\left( Q_{L}^{T}\right) \cap W^{1, \frac{4}{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) . \label{3.16} \end{equation} \end{corollary}
Thus for the proof that $u_{L}$ is the solution of poblem (\ref{3.3}) - (\ref {3.5}) or (\ref{3.6}) remains to show that $\chi =B\left( u_{L}\right) $ or $ \left\langle \chi ,v\right\rangle _{\Omega _{L}}=b_{L}\left( u_{L},u_{L},v\right) $ for $\forall v\in V\left( \Omega _{L}\right) $. \
\subsection{\label{SS_I.4.4}Weakly compactness of operator $B$}
\begin{proposition} \label{P_3.2}Operator $B:V\left( Q_{L}^{T}\right) \longrightarrow L^{\frac{4 }{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $ is weakly compact operator, i.e. any weakly convergent sequence $\left\{ u_{L}^{m}\right\} _{1}^{\infty }\subset V\left( Q_{L}^{T}\right) $ posses such subsequence $\left\{ u_{L}^{m_{k}}\right\} _{1}^{\infty }\subset \left\{ u_{L}^{m}\right\} _{1}^{\infty }$, that $\left\{ B\left( u_{L}^{m_{k}}\right) \right\} _{1}^{\infty }$ weakly converged in $L^{\frac{4 }{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $. \end{proposition}
\begin{proof} Let sequence $\left\{ u_{L}^{m}\right\} _{1}^{\infty }\subset V\left( Q_{L}^{T}\right) $ be weakly converge to $u_{L}^{0}$ in $V\left( Q_{L}^{T}\right) $. Then there exists such subsequence $\left\{ u_{L}^{m_{k}}\right\} _{1}^{\infty }\subset \left\{ u_{L}^{m}\right\} _{1}^{\infty }$ that $u_{L}^{m_{k}}\longrightarrow u_{L}^{0}$ in $ L^{2}\left( 0,T;H\right) $, due of the known theorems on the compactness of the embedding, particullary, as known the following embedding \begin{equation*} \mathcal{V}\left( Q_{L}^{T}\right) \equiv L^{2}\left( 0,T;V\left( \Omega _{L}\right) \right) \cap W^{1,\frac{4}{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) \subset L^{2}\left( 0,T;H\right) \end{equation*} is compact (see, e. g. \cite{Lio1}, \cite{Tem1}, \cite{Sol3}).
Actually it is enough to show that the operator defined by expression $ \underset{j=1}{\overset{3}{\sum }}u_{Lj}D_{j}u_{L}$ is weakly compact from $ \mathcal{V}\left( Q_{L}^{T}\right) $ to $L^{\frac{4}{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $. From a priori estimations and Proposition \ref{P_3.1} follow that operator $B:\mathcal{V}\left( Q_{L}^{T}\right) \longrightarrow L^{\frac{4}{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $ is bounded, i.e. the image of operator $B$ of each bounded subset of space $\mathcal{V}\left( Q_{L}^{T}\right) $ is the bounded subset of space $L^{\frac{4}{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $.\
From above compactness theorem follows the sequence $\left\{ u_{L}^{m}\right\} _{1}^{\infty }$ posses some subsequence $\left\{ u_{L}^{m_{k}}\right\} _{1}^{\infty }\subset \left\{ u_{L}^{m}\right\} _{1}^{\infty }$ strongly convergent to some element $u_{L}$ of $L^{2}\left( 0,T;H\right) $ in the space $L^{2}\left( 0,T;H\right) $. Consequently, $ B\left( \left\{ u_{L}^{m_{k}}\right\} _{1}^{\infty }\right) $ belongs of bounded subset of space $L^{\frac{4}{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $. Thence lead that there is such element $\chi \in L^{ \frac{4}{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $ that sequence $B\left( u_{L}^{m_{k}}\right) $ weakly converges to $\chi $ when $ m_{k}\nearrow \infty $, i.e. \begin{equation} B\left( u_{L}^{m_{k}}\right) \rightharpoonup \chi \quad \text{ in }L^{\frac{4 }{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) \label{3.17} \end{equation} due of the reflexivity of this space (there exists, at least, such subsequence that this occurs).
If we set the vector space \begin{equation*} \mathcal{C}^{1}\left( \overline{Q}_{L}\right) \equiv \left\{ v\left\vert \ v_{i}\in C^{1}\left( \left[ 0,T\right] ;C_{0}^{1}\left( \overline{\Omega _{L} }\right) \right) ,\right. i=1,2,3\right\} \end{equation*} and consider the trilinear form \begin{equation*} \underset{0}{\overset{T}{\int }}\left\langle B\left( u_{L}^{m}\right) ,v\right\rangle _{\Omega _{L}}dt=\underset{0}{\overset{T}{\int }}b\left( u_{L}^{m},u_{L}^{m},v\right) dt=\underset{0}{\overset{T}{\int }}\left\langle \underset{j=1}{\overset{3}{\sum }}u_{Lj}^{m}D_{j}u_{L}^{m},v\right\rangle _{\Omega _{L}}dt= \end{equation*} for $v\in \mathcal{C}^{1}\left( \overline{Q}_{L}\right) $, then we get \begin{equation} -\underset{i=1}{\overset{3}{\sum }}\underset{0}{\overset{T}{\int }}\underset{ P_{x_{3}}\Omega _{L}}{\int }\left[ \left( u_{Li}^{m}u_{L1}^{m}-a_{1}^{-1}u_{Li}^{m}u_{L3}^{m}\right) D_{1}v_{i}+\left( u_{Li}^{m}u_{L2}^{m}-a_{2}^{-1}u_{Li}^{m}u_{L3}^{m}\right) D_{2}v_{i}\right] dx_{1}dx_{2}dt. \label{3.17a} \end{equation}
according to (\ref{3.13}). Now if we take arbitrary term in this sum separately then it isn't difficult to see that the following convergences are true, because $u_{Li}^{m_{k}}\longrightarrow u_{Li}$ in $L^{2}\left( 0,T;H\right) $ and $u_{Li}^{m_{k}}\rightharpoonup u_{Li}$ in $L^{\infty }\left( 0,T;H\right) $ $\ast -$ weakly\ since $u_{L}^{m}$ belong to a bounded subset of $\mathcal{V}\left( Q_{L}^{T}\right) $ and (\ref{3.17a}) is fulfill for each term.
Thus passing to the limit when $m_{k}\nearrow \infty $ we obtain \begin{equation*} \chi =B\left( u_{L}\right) \Longrightarrow B\left( u_{L}^{m_{k}}\right) \rightharpoonup B\left( u_{L}\right) \quad \text{ in the distribution sense}. \end{equation*} Whence using the density of $\mathcal{C}^{1}\left( \overline{Q}_{L}\right) $ in $\mathcal{V}\left( Q_{L}^{T}\right) $, and as $B\left( u_{L}^{m_{k}}\right) \rightharpoonup \chi $ takes place in the space $L^{ \frac{4}{3}}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) $ we get that $\chi =B\left( u_{L}\right) $ also takes place in this space. \end{proof}
Consequently, we proved the existence of the function $u_{L}\in \mathcal{V} \left( Q_{L}^{T}\right) $ that satisfies equation (\ref{3.6}) by applying to this problem of the Faedo-Galerkin method and using the above mentioned results.
\subsection{\label{SS_I.4.5}Realisation of the initial condition}
We will lead the proof of the realisation of initial condition according to same way as in \cite{Tem1} (see, also \cite{Lad1}, \cite{Lio1}).
Let $\phi $ be a continuously differentiable function on $[0,T]$ with $\phi (T)=0$. With multiplying (\ref{5.5}) by $\phi (t)$, and then the first term integrating by parts we leads to equation \begin{equation*} -\underset{0}{\overset{T}{\int }}\left\langle u_{Lm},\frac{d}{dt}\phi (t)w_{j}\right\rangle _{\Omega _{L}}dt=\underset{0}{\overset{T}{\int }} \left\langle \nu \Delta u_{Lm},\phi (t)w_{j}\right\rangle _{\Omega _{L}}dt+ \end{equation*} \begin{equation*} \underset{0}{\overset{T}{\int }}b\left( u_{Lm},u_{Lm},\phi (t)w_{j}\right) dt+\underset{0}{\overset{T}{\int }}\left\langle f_{L},\phi (t)w_{j}\right\rangle _{\Omega _{L}}dt+\left\langle u_{0Lm},\phi (0)w_{j}\right\rangle _{\Omega _{L}}. \end{equation*}
One can pass to the limit with respect to subsequence $\left\{ u_{Lm_{l}}\right\} _{l=1}^{\infty }$ of the sequence $\left\{ u_{Lm}\right\} _{m=1}^{\infty }$ in the equality mentioned above owing to the results proved in the previous subsections. Then we find the equation \begin{equation*} -\underset{0}{\overset{T}{\int }}\left\langle u_{L},\frac{d}{dt}\phi (t)w_{j}\right\rangle _{\Omega _{L}}dt=\underset{0}{\overset{T}{\int }} \left\langle \nu \Delta u_{L},\phi (t)w_{j}\right\rangle _{\Omega _{L}}dt+ \end{equation*} \begin{equation} \underset{0}{\overset{T}{\int }}b\left( u_{L},u_{L},\phi (t)w_{j}\right) dt+ \underset{0}{\overset{T}{\int }}\left\langle f_{L},\phi (t)w_{j}\right\rangle _{\Omega _{L}}dt+\left\langle u_{0L},\phi (0)w_{j}\right\rangle _{\Omega _{L}}, \label{5.8} \end{equation} that holds for each $w_{j}$, $j=1,2,...$. Consequently, this equality holds for any finite linear combination of the $w_{j}$ and moreover due of continuity (\ref{5.8}) remains true and for any $v\in V\left( \Omega _{L}\right) $.
Whence, one can draw conclusion that function $u_{L}$ satisfies equation ( \ref{3.6}) in the distribution sense.
Now if multiply (\ref{3.6}) by $\phi (t)$, and integrate with respect to $t$ after integrating the first term by parts, then we get \begin{equation*} -\underset{0}{\overset{T}{\int }}\left\langle u_{L},v\frac{d}{dt}\phi (t)\right\rangle _{\Omega _{L}}dt-\underset{0}{\overset{T}{\int }} \left\langle \nu \Delta u_{L},\phi (t)v\right\rangle _{\Omega _{L}}dt+ \end{equation*} \begin{equation*} \underset{0}{\overset{T}{\int }}\left\langle \underset{j=1}{\overset{3}{\sum }}u_{Lj}D_{j}u_{L},\phi (t)v\right\rangle _{\Omega _{L}}dt=\underset{0}{ \overset{T}{\int }}\left\langle f_{L},\phi (t)v\right\rangle _{\Omega _{L}}dt+\left\langle u_{L}\left( 0\right) ,\phi (0)v\right\rangle _{\Omega _{L}}. \end{equation*}
If we compare this with (\ref{5.8}) after replacing $w_{j}$ with any $v\in V\left( \Omega _{L}\right) $ then we obtain \begin{equation*} \phi (0)\left\langle u_{L}\left( 0\right) -u_{0L},v\right\rangle _{\Omega _{L}}=0. \end{equation*} Whence, we get the realisation of the initial condition by virtue of arbitrariness of $v\in V\left( \Omega _{L}\right) $ and $\phi $, since function $\phi $ one can choose as $\phi (0)\neq 0$.
Consequently, the following result is proven.
\begin{theorem} \label{Th_2.1}Under above mentioned conditions for any \begin{equation*} u_{0L}\in \left( H\left( \Omega _{L}\right) \right) ^{3},\quad f_{L}\in L^{2}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) \end{equation*} problem (\ref{3.3}) - (\ref{3.5}) has weak solution $u_{L}\left( t,x\right) $ that belongs to $\mathcal{V}\left( Q_{L}^{T}\right) $. \end{theorem}
\begin{remark} From the obtained a priori estimates and Propositions \ref{P_3.1} and \ref {P_3.2} follows of the fulfilment of all conditions of the general theorem of the compactness method (see, e. g. \cite{Sol3}, \cite{SolAhm}, and for complementary informations see, \cite{Sol2}, \cite{Sol4}). Consequently, one could be to study the solvability of problem (\ref{3.3}) - (\ref{3.5}) with use of this general theorem. \end{remark}
\section{\label{Sec_I.5}Uniqueness of Solution of Problem (3.3) - (3.5)}
For the study of the uniqueness of the solution as usually: we will assume that posed problem have, at least, two different solutions $u=\left( u_{1},u_{2},u_{3}\right) $, $v=\left( v_{1},v_{2},v_{3}\right) $. Below will show that this isn't possible, and for which one need to investigate their difference, i.e. $w=u-v$.\ (Here for brevity we won't specify indexes for functions, which showing that here is investigated the system of equations ( \ref{3.3}) - (\ref{3.5}) on $Q_{L}^{T}$.)
So, we obtain the following problem for $w=u-v$ \begin{equation*} \frac{\partial w}{\partial t}-\nu \left[ \left( 1+a_{1}^{-2}\right) D_{1}^{2}+\left( 1+a_{2}^{-2}\right) D_{2}^{2}\right] w-2\nu a_{1}^{-1}a_{2}^{-1}D_{1}D_{2}w+ \end{equation*} \begin{equation*} \left( u_{1}-a_{1}^{-1}u_{3}\right) D_{1}u-\left( v_{1}-a_{1}^{-1}v_{3}\right) D_{1}v+\left( u_{2}-a_{2}^{-1}u_{3}\right) D_{2}u- \end{equation*} \begin{equation*} \left( v_{2}-a_{2}^{-1}v_{3}\right) D_{2}v=0, \end{equation*} \begin{equation*} \func{div}w=D_{1}\left[ \left( u-a_{1}^{-1}u_{3}\right) -\left( v-a_{1}^{-1}v_{3}\right) \right] +D_{2}\left[ \left( u-a_{2}^{-1}u_{3}\right) \right. - \end{equation*} \begin{equation}
\left. \left( v-a_{2}^{-1}v_{3}\right) \right] =D_{1}w+D_{2}w-\left( a_{1}^{-1}D_{1}+a_{2}^{-1}D_{2}\right) w_{3}=0,
\label{3.18} \end{equation} \begin{equation}
w\left( 0,x\right) =0,\quad x\in \Omega \cap L;\quad w\left\vert \ _{\left( 0,T\right) \times \partial \Omega _{L}}\right. =0.
\label{3.19} \end{equation}
Hence we derive \begin{equation*} \frac{1}{2}\frac{d}{dt}\left\Vert w\right\Vert _{2}^{2}+\nu \left[ \left( 1+a_{1}^{-2}\right) \left\Vert D_{1}w\right\Vert _{2}^{2}+\left( 1+a_{2}^{-2}\right) \left\Vert D_{2}w\right\Vert _{2}^{2}\right] + \end{equation*} \begin{equation*} 2\nu a_{1}^{-1}a_{2}^{-1}\left\langle D_{1}w,D_{2}w\right\rangle _{\Omega _{L}}+\left\langle \left( u_{1}-a_{1}^{-1}u_{3}\right) D_{1}u-\left( v_{1}-a_{1}^{-1}v_{3}\right) D_{1}v,w\right\rangle _{\Omega _{L}}+ \end{equation*} \begin{equation*} \left\langle \left( u_{2}-a_{2}^{-1}u_{3}\right) D_{2}u-\left( v_{2}-a_{2}^{-1}v_{3}\right) D_{2}v,w\right\rangle _{\Omega _{L}}=0 \end{equation*} or \begin{equation*} \frac{1}{2}\frac{d}{dt}\left\Vert w\right\Vert _{2}^{2}+\nu \left( \left\Vert D_{1}w\right\Vert _{2}^{2}+\left\Vert D_{2}w\right\Vert _{2}^{2}\right) +\nu \left[ a_{1}^{-2}\left\Vert D_{1}w\right\Vert _{2}^{2}+a_{2}^{-2}\left\Vert D_{2}w\right\Vert _{2}^{2}+\right. \end{equation*} \begin{equation*} \left. 2a_{1}^{-1}a_{2}^{-1}\left\langle D_{1}w,D_{2}w\right\rangle _{\Omega _{L}}\right] +\left\langle u_{1}D_{1}u-v_{1}D_{1}v,w\right\rangle _{\Omega _{L}}+\left\langle u_{2}D_{2}u-v_{2}D_{2}v,w\right\rangle _{\Omega _{L}}- \end{equation*} \begin{equation} a_{1}^{-1}\left\langle u_{3}D_{1}u-v_{3}D_{1}v,w\right\rangle _{\Omega _{L}}-a_{2}^{-1}\left\langle u_{3}D_{2}u-v_{3}D_{2}v,w\right\rangle _{\Omega _{L}}=0. \label{3.20} \end{equation}
If the last 4 terms in the sum of left part (\ref{3.20}) consider separately and if these simplify by calculations then we get \begin{equation*} \left\langle w_{1}D_{1}u,w\right\rangle _{\Omega _{L}}+\left\langle v_{1}D_{1}w,w\right\rangle _{\Omega _{L}}+\left\langle w_{2}D_{2}u,w\right\rangle _{\Omega _{L}}+\left\langle v_{2}D_{2}w,w\right\rangle _{\Omega _{L}}- \end{equation*} \begin{equation*} a_{1}^{-1}\left\langle w_{3}D_{1}u,w\right\rangle _{\Omega _{L}}-a_{1}^{-1}\left\langle v_{3}D_{1}w,w\right\rangle _{\Omega _{L}}-a_{2}^{-1}\left\langle w_{3}D_{2}u,w\right\rangle _{\Omega _{L}}-a_{2}^{-1}\left\langle v_{3}D_{2}w,w\right\rangle _{\Omega _{L}}= \end{equation*} \begin{equation*} \left\langle w_{1}D_{1}u,w\right\rangle _{\Omega _{L}}+\frac{1}{2} \left\langle v_{1},D_{1}w^{2}\right\rangle _{\Omega _{L}}+\left\langle w_{2}D_{2}u,w\right\rangle _{\Omega _{L}}+\frac{1}{2}\left\langle v_{2},D_{2}w^{2}\right\rangle _{\Omega _{L}}- \end{equation*} \begin{equation*} a_{1}^{-1}\left\langle w_{3}D_{1}u,w\right\rangle _{\Omega _{L}}-\frac{1}{2} a_{1}^{-1}\left\langle v_{3},D_{1}w^{2}\right\rangle _{\Omega _{L}}-a_{2}^{-1}\left\langle w_{3}D_{2}u,w\right\rangle _{\Omega _{L}}- \end{equation*} \begin{equation*} \frac{1}{2}a_{2}^{-1}\left\langle v_{3},D_{2}w^{2}\right\rangle =\frac{1}{2} \left\langle v_{1}-a_{1}^{-1}v_{3},D_{1}w^{2}\right\rangle _{\Omega _{L}}+ \frac{1}{2}\left\langle v_{2}-a_{2}^{-1}v_{3},D_{2}w^{2}\right\rangle _{\Omega _{L}}+ \end{equation*} \begin{equation*} \left\langle \left( w_{1}-a_{1}^{-1}w_{3}\right) w,D_{1}u\right\rangle _{\Omega _{L}}+\left\langle \left( w_{2}-a_{2}^{-1}w_{3}\right) w,D_{2}u\right\rangle _{\Omega _{L}}= \end{equation*} \begin{equation*} \left\langle \left( w_{1}-a_{1}^{-1}w_{3}\right) w,D_{1}u\right\rangle _{\Omega _{L}}+\left\langle \left( w_{2}-a_{2}^{-1}w_{3}\right) w,D_{2}u\right\rangle _{\Omega _{L}}. \end{equation*} In the last equality were used the equation $\func{div}v=0$ (see, (\ref{3.4} )) and the condition (\ref{3.19}).
If takes into account this equality in equation (\ref{3.20}) then we get \begin{equation*} \frac{1}{2}\frac{d}{dt}\left\Vert w\right\Vert _{2}^{2}+\nu \left( \left\Vert D_{1}w\right\Vert _{2}^{2}+\left\Vert D_{2}w\right\Vert _{2}^{2}\right) +\nu \left[ a_{1}^{-2}\left\Vert D_{1}w\right\Vert _{2}^{2}+\right. \end{equation*} \begin{equation*} \left. a_{2}^{-2}\left\Vert D_{2}w\right\Vert _{2}^{2}+2a_{1}^{-1}a_{2}^{-1}\left\langle D_{1}w,D_{2}w\right\rangle _{\Omega _{L}}\right] +\left\langle \left( w_{1}-a_{1}^{-1}w_{3}\right) w,D_{1}u\right\rangle _{\Omega _{L}}+ \end{equation*} \begin{equation} \left\langle \left( w_{2}-a_{2}^{-1}w_{3}\right) w,D_{2}u\right\rangle _{\Omega _{L}}=0,\quad \left( t,x\right) \in \left( 0,T\right) \times \Omega _{L}. \label{3.21} \end{equation}
Thus we obtain the Cauchy problem for equation (\ref{3.21}) with the initial condition \begin{equation} \left\Vert w\right\Vert _{2}\left( 0\right) =0. \label{3.22} \end{equation}
We get the following Cauchy problem for the differential inequation using the appropriate estimates \begin{equation*} \frac{1}{2}\frac{d}{dt}\left\Vert w\right\Vert _{2}^{2}+\nu \left( \left\Vert D_{1}w\right\Vert _{2}^{2}+\left\Vert D_{2}w\right\Vert _{2}^{2}\right) \leq \end{equation*} \begin{equation} \left\vert \left\langle \left( w_{1}-a_{1}^{-1}w_{3}\right) w,D_{1}u\right\rangle _{\Omega _{L}}\right\vert +\left\vert \left\langle \left( w_{2}-a_{2}^{-1}w_{3}\right) w,D_{2}u\right\rangle _{\Omega _{L}}\right\vert , \label{3.21'} \end{equation}
with the initial condition (\ref{3.22}).
Then for the right side of (\ref{3.21'}) we get the following estimate \begin{equation*} \left\vert \left\langle \left( w_{1}-a_{1}^{-1}w_{3}\right) w,D_{1}u\right\rangle _{\Omega _{L}}\right\vert +\left\vert \left\langle \left( w_{2}-a_{2}^{-1}w_{3}\right) w,D_{2}u\right\rangle _{\Omega _{L}}\right\vert \leq \end{equation*} \begin{equation*} \left( \left\Vert w_{1}-a_{1}^{-1}w_{3}\right\Vert _{4}+\left\Vert w_{2}-a_{2}^{-1}w_{3}\right\Vert _{4}\right) \left\Vert w\right\Vert _{4}\left\Vert \nabla u\right\Vert _{2}\leq \end{equation*} whence we derive \begin{equation*} \left( 1+\max \left\{ \left\vert a_{1}^{-1}\right\vert ,\left\vert a_{2}^{-1}\right\vert \right\} \right) \left\Vert w\right\Vert _{4}^{2}\left\Vert \nabla u\right\Vert _{2}\leq c\left\Vert w\right\Vert _{2}\left\Vert \nabla w\right\Vert _{2}\left\Vert \nabla u\right\Vert _{2} \end{equation*} thanks of Gagliardo-Nirenberg inequality (\cite{BesIlNik}).
It need to note that \begin{equation*} \left( w_{1}-a_{1}^{-1}w_{3}\right) w,\ \left( w_{2}-a_{2}^{-1}w_{3}\right) w\in L^{2}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) , \end{equation*} by virtue of (\ref{3.16}).
Now taking into account this in (\ref{3.21'}) one can arrive the following Cauchy problem for differential inequation \begin{equation*} \frac{1}{2}\frac{d}{dt}\left\Vert w\right\Vert _{2}^{2}\left( t\right) +\nu \left\Vert \nabla w\right\Vert _{2}^{2}\left( t\right) \leq c\left\Vert w\right\Vert _{2}\left( t\right) \left\Vert \nabla w\right\Vert _{2}\left( t\right) \left\Vert \nabla u\right\Vert _{2}\left( t\right) \leq \end{equation*} \begin{equation*} C\left( c,\nu \right) \left\Vert \nabla u\right\Vert _{2}^{2}\left( t\right) \left\Vert w\right\Vert _{2}^{2}\left( t\right) +\nu \left\Vert \nabla w\right\Vert _{2}^{2}\left( t\right) ,\quad \left\Vert w\right\Vert _{2}\left( 0\right) =0, \end{equation*} since $w\in L^{\infty }\left( 0,T;H\left( \Omega _{L}\right) \right) $. Consequently, $\left\Vert w\right\Vert _{2}\left\Vert \nabla w\right\Vert _{2}\in L^{2}\left( 0,T\right) $ by virtue of the proved above existence theorem $w\in \mathcal{V}\left( Q_{L}^{T}\right) $, where $C\left( c,\nu \right) >0$ is constant.
Thus we obtain the problem \begin{equation*} \frac{d}{dt}\left\Vert w\right\Vert _{2}^{2}\left( t\right) \leq 2C\left( c,\nu \right) \left\Vert \nabla u\right\Vert _{2}^{2}\left( t\right) \left\Vert w\right\Vert _{2}^{2}\left( t\right) ,\quad \left\Vert w\right\Vert _{2}\left( 0\right) =0. \end{equation*} If to denote $\left\Vert w\right\Vert _{2}^{2}\left( t\right) \equiv y\left( t\right) $ then \begin{equation*} \frac{d}{dt}y\left( t\right) \leq 2C\left( c,\nu \right) \left\Vert \nabla u\right\Vert _{2}^{2}\left( t\right) y\left( t\right) ,\quad y\left( 0\right) =0. \end{equation*}
Whence follows $\left\Vert w\right\Vert _{2}^{2}\left( t\right) \equiv y\left( t\right) =0$, and consequently the following result is proven: \
\begin{theorem} \label{Th_2.2}Under above mentioned conditions for any \begin{equation*} \left( f,u_{0}\right) \in L^{2}\left( 0,T;V^{\ast }\left( \Omega _{L}\right) \right) \times H\left( \Omega _{L}\right) \end{equation*} problem (\ref{3.3}) - (\ref{3.5}) has a unique weak solution $u\left( t,x\right) $ that is contained in $\mathcal{V}\left( Q_{L}^{T}\right) $. \end{theorem}
\section{\label{Sec_I.6}Proof of Theorem \protect\ref{Th_1}}
\begin{proof} (of Theorem \ref{Th_1}). As were noted in introduction, under the above mentioned conditions problem (1.1
${{}^1}$
) - (\ref{3}) is weakly solvable and any solution belongs to the space $ \mathcal{V}\left( Q^{T}\right) $. Consequently, under the conditions of Theorem \ref{Th_1} this problem also has weak solution that belongs, at least, to the space $\mathcal{V}\left( Q^{T}\right) $. But as shown in Sections \ref{Sec_I.4} under conditions of Theorem \ref{Th_1} the auxiliary problems of problem (1.1
${{}^1}$
) - (\ref{3}) are weakly solvable and any solution belongs to the space $ \mathcal{V}\left( Q_{L}^{T}\right) $. Moreover, as shown in Section \ref {Sec_I.5} weak solution of each of these problems is unique. Hence follows, that we can employ of Lemma \ref{L_2.2} to solutions of problem (1.1
${{}^1}$
) - (\ref{3}) on $Q_{L}^{T}$ due of the smoothness of solutions of this problem.
So, assume problem (1.1
${{}^1}$
) - (\ref{3}) has, at least, two different weak solutions under conditions of Theorem \ref{Th_1}. It is clear that if the problem have more than one solution then there is, at least, some subdomain of $Q^{T}\equiv \left( 0,T\right) \times \Omega $, on which this problem have, at least, two solutions that different.Consequently, starting from the above Lemma \ref {L_2.2} is sufficiently to investigate the existence and uniqueness of the posed problem on arbitrary fixed subdomain in order to shows that exist or unexist such subdomens, on which the studied problem can possess more than one solutions. More exactly it is sufficiently to study of this question in the case when subdomains are generated by arbitrary fixed hyperplanes by virtue of Lemma \ref{L_2.2}. For this aim it is enough to prove, that isn't exist such subdomains, on which the problem (1.1
${{}^1}$
) - (\ref{3}) could has of more than one solution by virtue of Lemma \ref {L_2.2}. Thus, in order to end of the proof is remains to use the above results (i.e. Theorems \ref{Th_2.1} and \ref{Th_2.2}).
Indeed, as follows from theorems that were proved in the previous sections there not are exist subdomains, on which the problem (1.1
${{}^1}$
) - (\ref{3}) could be possesses more than one weak solution.
Consequently, according of Lemma \ref{L_2.2} we obtain, that the problem (1.1
${{}^1}$
) - (\ref{3}) under conditions of Theorem \ref{Th_1} possesses only one weak solution. \end{proof}
Whence can make the following conclusion.
\subsection{Conclusion}
Let's \begin{equation*} f\in L^{2}\left( 0,T;V^{\ast }\left( \Omega \right) \right) ,\ u_{0}\in H\left( \Omega \right) . \end{equation*} It well-known that following inclusions are dense \begin{equation*} L^{2}\left( 0,T;H^{1/2}\left( \Omega \right) \right) \subset L^{2}\left( Q^{T}\right) ;\ H^{1/2}\left( \Omega \right) \subset H\left( \Omega \right) \ \ \& \end{equation*} \begin{equation*} L^{2}\left( 0,T;H^{1/2}\left( \Omega \right) \right) \subset L^{2}\left( 0,T;H^{-1}\left( \Omega \right) \right) . \end{equation*} Hence, there exist such sequences \begin{equation*} \left\{ u_{0m}\right\} _{m=1}^{\infty }\subset H^{1/2}\left( \Omega \right) ;\left\{ f_{m}\right\} _{m=1}^{\infty }\subset L^{2}\left( 0,T;H^{1/2}\left( \Omega \right) \right) \end{equation*} that $u_{0m}\longrightarrow u_{0}$ in $H\left( \Omega \right) $ , $ f_{m}\longrightarrow f$ in $L^{2}\left( 0,T;H^{-1}\left( \Omega \right) \right) $.
Thus, we establish following result.
\begin{theorem} \label{Th_8}Let $\Omega $ be a Lipschitz open bounded domain in $R^{3}$ and the given functions $f$ and $u_{0}$\ satisfy of conditions $f$ $\in L^{2}\left( 0,T;H^{1/2}\left( \Omega \right) \right) $ and $u_{0}\in H^{1/2}\left( \Omega \right) $, respectively. Then there exists unique function $u\in \mathcal{V}\left( Q^{T}\right) $ that is the weak solution of the considered problem, in the sense of Definition \ref{D_2.2}. \end{theorem}
Roughly speaking, since $L^{2}\left( 0,T;H^{1/2}\left( \Omega \right) \right) $ and $H^{1/2}\left( \Omega \right) $ are everywhere dense in spaces $L^{2}\left( 0,T;H^{1/2}\left( \Omega \right) \right) $ and $H^{1/2}\left( \Omega \right) $, respectively, then if functions $f$ and $u_{0}$ are any given functions from $L^{2}\left( 0,T;V^{\ast }\left( \Omega \right) \right) $ and $H\left( \Omega \right) $, respectively then in their any neighbohoods there are functions $\widetilde{f}$ and $\widetilde{u}_{0}$ from $ L^{2}\left( 0,T;H^{1/2}\left( \Omega \right) \right) $ and $H^{1/2}\left( \Omega \right) $, respectively that the problem (1.1
${{}^1}$
) - (\ref{3}) has unique weak solution $u$, that belongs to a bounded subset of $\mathcal{V}\left( Q^{T}\right) $, where a weak solution be understood in the sense of Definition \ref{D_2.2}.
So, under conditions of Theorem \ref{Th_1} the uniqueness of weak solution $ u(x,t)$ (of velocity vector) of the problem (1.1
${{}^1}$
) - (\ref{3}) obtained from the mixed problem for the incompressible Navier-Stokes $3D$-equation proved (explanations of the last proposition see Notation \ref{N_1} and next paragraph of this Notation \ref{N_1}).
\part{\label{Part II}Employment of modified approach to study of uniqueness}
\section{\label{Sec_II.7}One conditional uniqueness theorem for problem (1.1$ ^{1}$) - (1.3)}
We believed there have the sense to provide here yet one result connected with same question for problem (\ref{1}) - (\ref{3}), but with conditions onto the given functions under which the existence theorem of the weak solution of this problem is proven. Here the known approach for the investigation of the uniqueness of solution of problem (1.1
${{}^1}$
) - (\ref{3}) is applied, but with use also other properties of this problem.
Let posed problem have two different solutions: $u,v\in \mathcal{V}\left( Q^{T}\right) $, then within known approach we get the following problem for vector function $w(t,x)=u(t,x)-v(t,x)$ \begin{equation}
\frac{1}{2}\frac{\partial }{\partial t}\left\Vert w\right\Vert _{2}^{2}+\nu \left\Vert \nabla w\right\Vert _{2}^{2}+\underset{j,k=1}{\overset{3}{\sum }}\left\langle \frac{\partial v_{k}}{\partial x_{j}}w_{k},w_{j}\right\rangle =0,
\label{2.9} \end{equation} \begin{equation} w\left( 0,x\right) =w_{0}\left( x\right) =0,\quad x\in \Omega ;\quad w\left\vert \ _{\left[ 0,T\right] \times \partial \Omega }=0\right. , \label{2.10} \end{equation} where $\Omega \subset
\mathbb{R}
^{3}$ is above-mentioned domain.\
So, for the proof of triviality of solution of problem (\ref{2.9})-(\ref {2.10}), as usually will used method of contradiction. Consequently, one will start with assume that problem have nontrivial solution.
In addition, it is need to noted here will used of the peculiarity of having nonlinearity of this problem.
In the beginning we will study the following quadratic form (\cite{Gan}) for examination of problem (\ref{2.9})-(\ref{2.10}) \begin{equation*} B\left( w,w\right) =\underset{j,k=1}{\overset{3}{\sum }}\left( \frac{ \partial v_{k}}{\partial x_{j}}w_{k}w_{j}\right) \left( t,x\right) , \end{equation*} denote it as \begin{equation*} B\left( w,w\right) \equiv \underset{j,k=1}{\overset{3}{\sum }}\left( a_{jk}w_{k}w_{j}\right) \left( t,x\right) . \end{equation*} It is clear that behavior of the surface generated by function $B\left( w,w\right) $ respect to the variables $w_{k},\ k=1,2,3$ depende of the accelerations of the flow on the different directions.
Consider the question: it would possible to transform the quadratic form $ B\left( w,w\right) $ to the canonical form, namely to the following form \begin{eqnarray*} B\left( w,w\right) &\equiv &\underset{i=1}{\overset{3}{\sum }}\left( b_{i}w_{i}^{2}\right) \left( t,x\right) ,\quad b_{i}\left( t,x\right) \equiv b_{i}\left( \overline{D_{j}v_{k}}\right) , \\ \text{where \ }D_{i}v_{k} &\equiv &\frac{\partial v_{k}}{\partial x_{i}} ,\quad i,k=1,2,3,\quad b_{i}:
\mathbb{R}
^{9}\longrightarrow
\mathbb{R}
\text{ be functions?} \end{eqnarray*}
The matrix $\left\Vert a_{jk}\right\Vert $ of coefficients of the quadratic form $B\left( w,w\right) $\ can be represented in the following form \begin{equation*} \left\Vert a_{jk}\right\Vert _{j,k=1}^{3}=\left\Vert \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right\Vert ,\quad \text{where }a_{jk}=a_{kj}=\frac{1}{2}\left( D_{j}v_{k}+D_{k}v_{j}\right) , \end{equation*} hence it is symmetric matrix. As known in this case the above transformation exists according of symmetricness of matrix $\left\Vert a_{jk}\right\Vert _{j,k=1}^{3}$ (see, \cite{Gan}). Consequently, coefficients $b_{i}$ are defined and have the following presentations \begin{equation*} b_{1}=D_{1}v_{1};\ b_{2}=D_{2}v_{2}-\frac{\left( D_{1}v_{2}+D_{2}v_{1}\right) ^{2}}{4b_{1}};\ b_{3}=\frac{\det \left\Vert D_{i}v_{k}\right\Vert _{i,k=1}^{3}}{\det \left\Vert D_{i}v_{k}\right\Vert _{i,k=1}^{2}}, \end{equation*} where $\left\Vert D_{i}v_{k}\right\Vert _{i,k=1}^{3}$\ and $\left\Vert D_{i}v_{k}\right\Vert _{i,k=1}^{2}$ define by equalities
\begin{center} $\left\Vert D_{i}v_{k}\right\Vert _{i,k=1}^{3}\equiv \left\Vert \begin{array}{ccc} D_{1}v_{1} & \frac{1}{2}\left( D_{1}v_{2}+D_{2}v_{1}\right) & \frac{1}{2} \left( D_{1}v_{3}+D_{3}v_{1}\right) \\ \frac{1}{2}\left( D_{1}v_{2}+D_{2}v_{1}\right) & D_{2}v_{2} & \frac{1}{2} \left( D_{2}v_{3}+D_{3}v_{2}\right) \\ \frac{1}{2}\left( D_{1}v_{3}+D_{3}v_{1}\right) & \frac{1}{2}\left( D_{2}v_{3}+D_{3}v_{2}\right) & D_{3}v_{3} \end{array} \right\Vert $ \end{center}
and
\begin{center} $\left\Vert D_{i}v_{k}\right\Vert _{i,k=1}^{2}\equiv \left\Vert \begin{array}{cc} D_{1}v_{1} & \frac{1}{2}\left( D_{1}v_{2}+D_{2}v_{1}\right) \\ \frac{1}{2}\left( D_{1}v_{2}+D_{2}v_{1}\right) & D_{2}v_{2} \end{array} \right\Vert $ \ \end{center}
for any $\left( t,x\right) \in Q^{T}\equiv \left( 0,T\right) \times \Omega $.
Therefore we have \begin{equation} \left( B\left( w,w\right) \right) \left( t,x\right) \equiv \underset{j,k=1}{ \overset{3}{\sum }}\left( a_{jk}w_{k}w_{j}\right) \left( t,x\right) \equiv \underset{j=1}{\overset{3}{\sum }}b_{j}\left( t,x\right) \cdot w_{j}^{2}\left( t,x\right) \label{2.11} \end{equation}
that one can rewrite in the following open form \begin{equation*} B\left( w,w\right) \equiv \frac{1}{D_{1}v_{1}}\left[ 2D_{1}v_{1}w_{1}+\left( D_{1}v_{2}+D_{2}v_{1}\right) w_{2}+\left( D_{1}v_{3}+D_{3}v_{1}\right) w_{3} \right] ^{2}+ \end{equation*} \begin{equation*} \frac{1}{\left( 4D_{1}v_{1}\right) ^{2}}\left( 4D_{1}v_{1}D_{2}v_{2}-\left( D_{1}v_{2}+D_{2}v_{1}\right) ^{2}\right) \times \end{equation*} \begin{equation*} \left[ \left( 4D_{1}v_{1}D_{2}v_{2}-\left( D_{1}v_{2}+D_{2}v_{1}\right) ^{2}\right) w_{2}\right. + \end{equation*} \begin{equation*} \left. \left( 2D_{1}v_{1}\left( D_{2}v_{3}+D_{3}v_{2}\right) -\left( D_{1}v_{2}+D_{2}v_{1}\right) \left( D_{1}v_{3}+D_{3}v_{1}\right) \right) w_{3}\right] ^{2}+ \end{equation*} \begin{equation*} \frac{1}{4}\left[ 4D_{1}v_{1}D_{2}v_{2}D_{3}v_{3}+\left( D_{1}v_{2}+D_{2}v_{1}\right) \left( D_{1}v_{3}+D_{3}v_{1}\right) \left( D_{2}v_{3}+D_{3}v_{2}\right) \right. - \end{equation*} \begin{equation*} \left. D_{1}v_{1}\left( D_{2}v_{3}+D_{3}v_{2}\right) ^{2}-D_{2}v_{2}\left( D_{1}v_{3}+D_{3}v_{1}\right) ^{2}-D_{3}v_{3}\left( D_{1}v_{2}+D_{2}v_{1}\right) ^{2}\right] w_{3}^{2}. \end{equation*}
If take account (\ref{2.11}) in the equation (\ref{2.9}) then we get \begin{equation*} \frac{1}{2}\frac{\partial }{\partial t}\left\Vert w\right\Vert _{2}^{2}+\nu \left\Vert \nabla w\right\Vert _{2}^{2}+\underset{j=1}{\overset{3}{\sum }} \left\langle b_{j}w_{j},w_{j}\right\rangle =0,\quad \left\Vert w_{0}\right\Vert _{2}=0, \end{equation*} or \begin{equation} \frac{1}{2}\frac{\partial }{\partial t}\left\Vert w\right\Vert _{2}^{2}=-\nu \left\Vert \nabla w\right\Vert _{2}^{2}-\underset{j=1}{\overset{3}{\sum }} \left\langle b_{j}w_{j},w_{j}\right\rangle ,\quad \left\Vert w_{0}\right\Vert _{2}=0. \label{2.12} \end{equation}
This shows that if $b_{j}\left( t,x\right) \geq 0$ for a.e. $\left( t,x\right) \in Q^{T}$ then the posed problem have unique solution. It is need noted that images of functions $b_{j}\left( t,x\right) $ and $ D_{i}v_{k} $ belong to the bounded subset of the same space.
So, is remains to investigate the cases when the mentioned isn't fulfill.
Here the following variants are possible:
1. Integral of $B\left( w,w\right) $ is determined and non-negative \begin{equation*} \underset{\Omega }{\int }B\left( w,w\right) dx=\underset{j=1}{\overset{3}{ \sum }}\left\langle b_{j}w_{j},w_{j}\right\rangle \equiv \underset{j=1}{ \overset{3}{\sum }}{}\underset{\Omega }{\int }b_{j}w_{j}^{2}dx\geq 0; \end{equation*}
In this case one can conclude the main problem have unique solution (and this solution is stable).
2. Integral of is undetermined and $\underset{j=1}{\overset{3}{\sum }}{} \underset{\Omega }{\int }b\ w_{j}^{2}dx\neq 0$.
In this case for investigation of problem (\ref{2.12}) it is necessary to derive suitable estimates for $B\left( w,w\right) \equiv \underset{j,k=1}{ \overset{3}{\sum }}\left( D_{i}v_{k}w_{k}w_{j}\right) $.
So, let $\underset{\Omega }{\int }B\left( w,w\right) dx$ is undetermined. Therefore we need estimate the right part of the equation from (\ref{2.12})
\begin{equation*} \frac{1}{2}\frac{\partial }{\partial t}\left\Vert w\right\Vert _{2}^{2}=-\nu \left\Vert \nabla w\right\Vert _{2}^{2}+\left\vert \underset{j,k=1}{\overset{ 3}{\sum }}\left\langle D_{i}v_{k}w_{k},w_{j}\right\rangle \right\vert \leq \end{equation*} \begin{equation} -\underset{j=1}{\overset{3}{\sum }}{}\underset{\Omega }{\int }\nu \left\vert \nabla w_{j}\left( t,x\right) \right\vert ^{2}dx+\underset{j,k=1}{\overset{3} {\sum }}{}\underset{\Omega }{\int }\left\vert \left( D_{i}v_{k}w_{k}w_{j}\right) \left( t,x\right) \right\vert dx, \label{2.13} \end{equation} more precisely, we need estimate the second adding in the right part of (\ref {2.13}). So, for one of the trilinear terms we obtain \footnote{ It is known that (\cite{Lad1}, \cite{Lio1}) $\left\vert \left\langle u_{k}D_{i}v_{j},w_{l}\right\rangle \right\vert \leq \left\Vert u_{k}\right\Vert _{q}\left\Vert D_{i}v_{j}\right\Vert _{2}\left\Vert w_{l}\right\Vert _{n},\quad n\geq 3;$ \par $\left\Vert v_{j}\right\Vert _{4}\leq C\left( mes\ \Omega \right) \left\Vert Dv_{j}\right\Vert _{2}^{\frac{1}{2}}\left\Vert v_{j}\right\Vert _{2}^{\frac{1 }{2}},\quad n=2$} \begin{equation*} \left\vert \left\langle D_{i}v_{j}w_{i},w_{j}\right\rangle \right\vert \leq \left\Vert D_{i}v_{j}\right\Vert _{2}\left\Vert w_{i}\right\Vert _{p_{1}}\left\Vert w_{j}\right\Vert _{p_{2}}, \end{equation*} with use of the H\={o}lder inequality, where is sufficient to choose, $ p_{1}=p_{2}=4$. Consequently, one can estimate $\underset{\Omega }{\int } B\left( w,w\right) dx$ as follows \begin{equation*} \underset{\Omega }{\int }\left\vert B\left( w,w\right) \right\vert dx\leq \underset{i,j=1}{\overset{3}{\sum }}\left\Vert D_{j}v_{i}\right\Vert _{2}\left\Vert w_{i}\right\Vert _{4}\left\Vert w_{j}\right\Vert _{4}. \end{equation*}
{}Hence, use Gagliardo-Nirenberg inequality (see, e.g., \cite{BesIlNik}) we get \begin{equation*} \left\Vert w_{j}\right\Vert _{4}\leq c\left\Vert w_{j}\right\Vert _{2}^{1-\sigma }\left\Vert \nabla w_{j}\right\Vert _{2}^{\sigma },\quad \sigma =\frac{3}{4}, \end{equation*} where $c\equiv C\left( 4,2,2,0,1\right) $, and for this case \begin{equation*} \left\Vert w_{j}\right\Vert _{4}\leq c\left\Vert w_{j}\right\Vert _{2}^{ \frac{1}{4}}\left\Vert \nabla w_{j}\right\Vert _{2}^{\frac{3}{4} }\Longrightarrow \left\Vert w_{j}\right\Vert _{4}^{2}\leq c^{2}\left\Vert w_{j}\right\Vert _{2}^{\frac{1}{2}}\left\Vert \nabla w_{j}\right\Vert _{2}^{ \frac{3}{2}}. \end{equation*} Therefore \begin{equation*} \underset{\Omega }{\int }\left\vert B\left( w,w\right) \right\vert dx\leq c^{2}\underset{i,j=1}{\overset{3}{\sum }}\left\Vert D_{j}v_{i}\right\Vert _{2}\left\Vert w_{i}\right\Vert _{2}^{\frac{1}{4}}\left\Vert \nabla w_{i}\right\Vert _{2}^{\frac{3}{4}}\left\Vert w_{j}\right\Vert _{2}^{\frac{1 }{4}}\left\Vert \nabla w_{j}\right\Vert _{2}^{\frac{3}{4}} \end{equation*} holds. Now taking into account the above estimate in (\ref{2.14}) we derive \begin{equation*} \frac{1}{2}\frac{\partial }{\partial t}\left\Vert w\left( t\right) \right\Vert _{2}^{2}\leq -\underset{j=1}{\overset{3}{\sum }}{}\nu \left\Vert \nabla w_{j}\left( t\right) \right\Vert _{2}^{2}+c^{2}\underset{i,j=1}{ \overset{3}{\sum }}{}\left\Vert D_{j}v_{i}\left( t\right) \right\Vert _{2}\left\Vert w_{i}\left( t\right) \right\Vert _{2}^{\frac{1}{2}}\left\Vert \nabla w_{i}\left( t\right) \right\Vert _{2}^{\frac{3}{2}} \end{equation*} \begin{equation*} \leq -\underset{j=1}{\overset{3}{\sum }}\left\Vert \nabla w_{j}\left( t\right) \right\Vert _{2}^{\frac{3}{2}}\left[ \nu \left\Vert \nabla w_{j}\left( t\right) \right\Vert _{2}^{\frac{1}{2}}-c^{2}\underset{i=1}{ \overset{3}{\sum }}\left\Vert D_{i}v_{j}\left( t\right) \right\Vert _{2}\left\Vert w_{j}\left( t\right) \right\Vert _{2}^{\frac{1}{2}}\right] \end{equation*} \begin{equation*} \leq -\underset{j=1}{\overset{n}{\sum }}\left\Vert \nabla w_{j}\left( t\right) \right\Vert _{2}^{\frac{3}{2}}\left[ \nu \lambda _{1}^{\frac{1}{4} }-c^{2}\underset{i=1}{\overset{n}{\sum }}\left\Vert D_{i}v_{j}\left( t\right) \right\Vert _{2}\right] \left\Vert w_{j}\left( t\right) \right\Vert _{2}^{\frac{1}{2}}. \end{equation*} Whence follows, that if $\nu \lambda _{1}^{\frac{1}{4}}\geq c^{2}\underset{ i=1}{\overset{3}{\sum }}\left\Vert D_{i}v_{j}\left( t\right) \right\Vert _{2} $ then problem (1.1$^{1}$)-(\ref{3}) has only unique solution (and solution is stable), where $\lambda _{1}$ is minimum of the spectrum of the operator Laplace. Thus is proved
\begin{theorem} \label{Th_3.1}Let $\Omega \in R^{3}$ be a open bounded domain of Lipschitz class, $\left( u_{0},f\right) \in H\left( \Omega \right) \times L^{2}\left( 0,T;V^{\ast }\left( \Omega \right) \right) $ and weak solution $u\left( t,x\right) $ of problem (1.1$^{1}$)-(\ref{3}) exists and $u\in \mathcal{V} \left( Q^{T}\right) $. Then if either $\underset{\Omega }{\int }\left\vert B\left( w,w\right) \right\vert dx\geq 0$ or $\underset{\Omega }{\int } \left\vert B\left( w,w\right) \right\vert dx\neq 0$ (is undetermined) and $ \nu \lambda _{1}^{\frac{1}{4}}\geq c^{2}\underset{i=1}{\overset{3}{\sum }} \left\Vert D_{i}u_{j}\left( t\right) \right\Vert _{2}$ fulfilled then weak solution $u\left( t,x\right) $ is unique.
\end{theorem}
\end{document} | arXiv |
annex_A.tex in branches/2017/dev_merge_2017/DOC/tex_sub – NEMO
source: branches/2017/dev_merge_2017/DOC/tex_sub/annex_A.tex @ 9414
Last change on this file since 9414 was 9414, checked in by nicolasmartin, 5 years ago
Fix multiple defined references
\documentclass[../tex_main/NEMO_manual]{subfiles}
% ================================================================
% Chapter Ñ Appendix A : Curvilinear s-Coordinate Equations
\chapter{Curvilinear $s-$Coordinate Equations}
\label{apdx:A}
\minitoc
\newpage
$\ $\newline % force a new ligne
% Chain rule
\section{Chain rule for $s-$coordinates}
\label{sec:A_chain}
In order to establish the set of Primitive Equation in curvilinear $s$-coordinates
($i.e.$ an orthogonal curvilinear coordinate in the horizontal and an Arbitrary Lagrangian
Eulerian (ALE) coordinate in the vertical), we start from the set of equations established
in \autoref{subsec:PE_zco_Eq} for the special case $k = z$ and thus $e_3 = 1$, and we introduce
an arbitrary vertical coordinate $a = a(i,j,z,t)$. Let us define a new vertical scale factor by
$e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and the horizontal
slope of $s-$surfaces by :
\begin{equation} \label{apdx:A_s_slope}
\sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s
\quad \text{and} \quad
\sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s
The chain rule to establish the model equations in the curvilinear $s-$coordinate
system is:
\begin{equation} \label{apdx:A_s_chain_rule}
\begin{aligned}
&\left. {\frac{\partial \bullet }{\partial t}} \right|_z =
\left. {\frac{\partial \bullet }{\partial t}} \right|_s
-\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\
&\left. {\frac{\partial \bullet }{\partial i}} \right|_z =
\left. {\frac{\partial \bullet }{\partial i}} \right|_s
-\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}=
-\frac{e_1 }{e_3 }\sigma _1 \frac{\partial \bullet }{\partial s} \\
&\left. {\frac{\partial \bullet }{\partial j}} \right|_z =
\left. {\frac{\partial \bullet }{\partial j}} \right|_s
- \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}=
- \frac{e_2 }{e_3 }\sigma _2 \frac{\partial \bullet }{\partial s} \\
&\;\frac{\partial \bullet }{\partial z} \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} \\
\end{aligned}
In particular applying the time derivative chain rule to $z$ provides the expression
for $w_s$, the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate:
\begin{equation} \label{apdx:A_w_in_s}
w_s = \left. \frac{\partial z }{\partial t} \right|_s
= \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t}
= e_3 \, \frac{\partial s}{\partial t}
% continuity equation
\section{Continuity equation in $s-$coordinates}
\label{sec:A_continuity}
Using (\autoref{apdx:A_s_chain_rule}) and the fact that the horizontal scale factors
$e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of
the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows
in order to obtain its expression in the curvilinear $s-$coordinate system:
\begin{subequations}
\begin{align*} {\begin{array}{*{20}l}
\nabla \cdot {\rm {\bf U}}
&= \frac{1}{e_1 \,e_2 } \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z
+\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z \right]
+ \frac{\partial w}{\partial z} \\
& = \frac{1}{e_1 \,e_2 } \left[
\left. \frac{\partial (e_2 \,u)}{\partial i} \right|_s
- \frac{e_1 }{e_3 } \sigma _1 \frac{\partial (e_2 \,u)}{\partial s}
+ \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s
- \frac{e_2 }{e_3 } \sigma _2 \frac{\partial (e_1 \,v)}{\partial s} \right]
+ \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z} \\
+ \left. \frac{\partial (e_1 \,v)}{\partial j} \right|_s \right]
+ \frac{1}{e_3 }\left[ \frac{\partial w}{\partial s}
- \sigma _1 \frac{\partial u}{\partial s}
- \sigma _2 \frac{\partial v}{\partial s} \right] \\
& = \frac{1}{e_1 \,e_2 \,e_3 } \left[
\left. \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
-\left. e_2 \,u \frac{\partial e_3 }{\partial i} \right|_s
+ \left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s
- \left. e_1 v \frac{\partial e_3 }{\partial j} \right|_s \right] \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+ \frac{1}{e_3 } \left[ \frac{\partial w}{\partial s}
\intertext{Noting that $
\frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s
=\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s
=\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right)
=\frac{\partial \sigma _1}{\partial s}
$ and $
\frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s
$, it becomes:}
+\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right] \\
& \qquad \qquad \qquad \qquad \quad
+\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-u\frac{\partial \sigma _1 }{\partial s}-v\frac{\partial \sigma _2 }{\partial s}-\sigma _1 \frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}} \right] \\
+\left. \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s \right]
+ \frac{1}{e_3 } \; \frac{\partial}{\partial s} \left[ w - u\;\sigma _1 - v\;\sigma _2 \right]
\end{array} }
\end{subequations}
Here, $w$ is the vertical velocity relative to the $z-$coordinate system.
Introducing the dia-surface velocity component, $\omega $, defined as
the volume flux across the moving $s$-surfaces per unit horizontal area:
\begin{equation} \label{apdx:A_w_s}
\omega = w - w_s - \sigma _1 \,u - \sigma _2 \,v \\
with $w_s$ given by \autoref{apdx:A_w_in_s}, we obtain the expression for
the divergence of the velocity in the curvilinear $s-$coordinate system:
&= \frac{1}{e_1 \,e_2 \,e_3 } \left[
+ \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
+ \frac{1}{e_3 } \frac{\partial w_s }{\partial s} \\
+ \frac{1}{e_3 } \frac{\partial}{\partial s} \left( e_3 \; \frac{\partial s}{\partial t} \right) \\
+ \frac{\partial}{\partial s} \frac{\partial s}{\partial t}
+ \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s} \\
+ \frac{1}{e_3 } \frac{\partial e_3}{\partial t} \\
As a result, the continuity equation \autoref{eq:PE_continuity} in the
$s-$coordinates is:
\begin{equation} \label{apdx:A_sco_Continuity}
\frac{1}{e_3 } \frac{\partial e_3}{\partial t}
+ \frac{1}{e_1 \,e_2 \,e_3 }\left[
{\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s
+ \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right]
+\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0
A additional term has appeared that take into account the contribution of the time variation
of the vertical coordinate to the volume budget.
% momentum equation
\section{Momentum equation in $s-$coordinate}
\label{sec:A_momentum}
Here we only consider the first component of the momentum equation,
the generalization to the second one being straightforward.
$\bullet$ \textbf{Total derivative in vector invariant form}
Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum
equation in the vector invariant form. Its total $z-$coordinate time derivative,
$\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain
its expression in the curvilinear $s-$coordinate system:
\left. \frac{D u}{D t} \right|_z
&= \left. {\frac{\partial u }{\partial t}} \right|_z
- \left. \zeta \right|_z v
+ \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z
+ w \;\frac{\partial u}{\partial z} \\
+ \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z
-\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v
+ \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z
\intertext{introducing the chain rule (\autoref{apdx:A_s_chain_rule}) }
- \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s
-\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right.
\left. {-\frac{e_1}{e_3}\sigma _1 \frac{\partial (e_2 \,v)}{\partial s}
+\frac{e_2}{e_3}\sigma _2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v \\
& \qquad \qquad \qquad \qquad
{ + \frac{1}{2e_1} \left( \left. \frac{\partial (u^2+v^2)}{\partial i} \right|_s
- \frac{e_1}{e_3}\sigma _1 \frac{\partial (u^2+v^2)}{\partial s} \right)
+ \frac{w}{e_3 } \;\frac{\partial u}{\partial s} } \\
+ \left. \zeta \right|_s \;v
+ \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\
&\qquad \qquad \qquad \quad
+ \frac{w}{e_3 } \;\frac{\partial u}{\partial s}
- \left[ {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s}
- \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}} \right]\;v
- \frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\
+ \frac{1}{e_3} \left[ {w\frac{\partial u}{\partial s}
+\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s}
- \sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\
+ \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s
+ \frac{1}{e_3} \left[ w - \sigma _2 v - \sigma _1 u \right]
\; \frac{\partial u}{\partial s} \\
\intertext{Introducing $\omega$, the dia-a-surface velocity given by (\autoref{apdx:A_w_s}) }
+ \frac{1}{e_3 } \left( \omega - w_s \right) \frac{\partial u}{\partial s} \\
Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule}))
to $u$ and using (\autoref{apdx:A_w_in_s}) provides the expression of the last term
of the right hand side,
\begin{equation*} {\begin{array}{*{20}l}
w_s \;\frac{\partial u}{\partial s}
= \frac{\partial s}{\partial t} \; \frac{\partial u }{\partial s}
= \left. {\frac{\partial u }{\partial t}} \right|_s - \left. {\frac{\partial u }{\partial t}} \right|_z \quad ,
\end{equation*}
leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative,
$i.e.$ the total $s-$coordinate time derivative :
\begin{align} \label{apdx:A_sco_Dt_vect}
\left. \frac{D u}{D t} \right|_s
= \left. {\frac{\partial u }{\partial t}} \right|_s
+ \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s}
Therefore, the vector invariant form of the total time derivative has exactly the same
mathematical form in $z-$ and $s-$coordinates. This is not the case for the flux form
as shown in next paragraph.
$\bullet$ \textbf{Total derivative in flux form}
Let us start from the total time derivative in the curvilinear $s-$coordinate system
we have just establish. Following the procedure used to establish (\autoref{eq:PE_flux_form}),
it can be transformed into :
%\begin{subequations}
\left. \frac{D u}{D t} \right|_s &= \left. {\frac{\partial u }{\partial t}} \right|_s
& - \zeta \;v
+ \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i}
+ \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s} \\
&= \left. {\frac{\partial u }{\partial t}} \right|_s
&+\frac{1}{e_1\;e_2} \left( \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i}
+ \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j} \right)
+ \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\
&&- \,u \left[ \frac{1}{e_1 e_2 } \left( \frac{\partial(e_2 u)}{\partial i}
+ \frac{\partial(e_1 v)}{\partial j} \right)
+ \frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\
&&- \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i}
-u \;\frac{\partial e_1 }{\partial j} \right) \\
Introducing the vertical scale factor inside the horizontal derivative of the first two terms
($i.e.$ the horizontal divergence), it becomes :
%\begin{align*} {\begin{array}{*{20}l}
%{\begin{array}{*{20}l}
&+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u^2 )}{\partial i}
+ \frac{\partial( e_1 e_3 \,u v )}{\partial j}
- e_2 u u \frac{\partial e_3}{\partial i}
- e_1 u v \frac{\partial e_3 }{\partial j} \right)
+ \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\
&& - \,u \left[ \frac{1}{e_1 e_2 e_3} \left( \frac{\partial(e_2 e_3 \, u)}{\partial i}
+ \frac{\partial(e_1 e_3 \, v)}{\partial j}
- e_2 u \;\frac{\partial e_3 }{\partial i}
- e_1 v \;\frac{\partial e_3 }{\partial j} \right)
-\frac{1}{e_3} \frac{\partial \omega}{\partial s} \right] \\
&& - \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i}
&+ \frac{1}{e_1\,e_2\,e_3} \left( \frac{\partial( e_2 e_3 \,u\,u )}{\partial i}
+ \frac{\partial( e_1 e_3 \,u\,v )}{\partial j} \right)
+ \frac{\partial(e_1 e_3 \, v)}{\partial j} \right)
-\frac{1}{e_3} \frac{\partial \omega}{\partial s} \right]
- \frac{v}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i}
\intertext {Introducing a more compact form for the divergence of the momentum fluxes,
and using (\autoref{apdx:A_sco_Continuity}), the $s-$coordinate continuity equation,
it becomes : }
&+ \left. \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) \right|_s
+ \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t}
which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative,
$i.e.$ the total $s-$coordinate time derivative in flux form :
\begin{flalign}\label{apdx:A_sco_Dt_flux}
\left. \frac{D u}{D t} \right|_s = \frac{1}{e_3} \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s
+ \left. \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) \right|_s
-u \;\frac{\partial e_1 }{\partial j} \right)
\end{flalign}
which is the total time derivative expressed in the curvilinear $s-$coordinate system.
It has the same form as in the $z-$coordinate but for the vertical scale factor
that has appeared inside the time derivative which comes from the modification
of (\autoref{apdx:A_sco_Continuity}), the continuity equation.
$\bullet$ \textbf{horizontal pressure gradient}
The horizontal pressure gradient term can be transformed as follows:
\begin{equation*}
\begin{split}
-\frac{1}{\rho _o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z
& =-\frac{1}{\rho _o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial p}{\partial s}} \right] \\
& =-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma _1 }{\rho _o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\
&=-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho _o }\sigma _1
\end{split}
Applying similar manipulation to the second component and replacing
$\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes:
\begin{equation} \label{apdx:A_grad_p_1}
-\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z
&=-\frac{1}{\rho _o \,e_1 } \left( \left. {\frac{\partial p}{\partial i}} \right|_s
+ g\;\rho \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) \\
-\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
&=-\frac{1}{\rho _o \,e_2 } \left( \left. {\frac{\partial p}{\partial j}} \right|_s
+ g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) \\
An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for the
tilt of $s-$surfaces with respect to geopotential $z-$surfaces.
As in $z$-coordinate, the horizontal pressure gradient can be split in two parts
following \citet{Marsaleix_al_OM08}. Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$,
and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$.
The pressure is then given by:
p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left( \rho_o \, d + 1 \right) \; e_3 \; dk \\
&= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk
Therefore, $p$ and $p_h'$ are linked through:
\begin{equation} \label{apdx:A_pressure}
p = \rho_o \; p_h' + g \, ( z + \eta )
and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is:
\frac{\partial p_h'}{\partial k} = - d \, g \, e_3
Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and using the definition of
the density anomaly it comes the expression in two parts:
&=-\frac{1}{e_1 } \left( \left. {\frac{\partial p_h'}{\partial i}} \right|_s
+ g\; d \;\left. {\frac{\partial z}{\partial i}} \right|_s \right) - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\
&=-\frac{1}{e_2 } \left( \left. {\frac{\partial p_h'}{\partial j}} \right|_s
+ g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s \right) - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\
This formulation of the pressure gradient is characterised by the appearance of a term depending on the
the sea surface height only (last term on the right hand side of expression \autoref{apdx:A_grad_p_2}).
This term will be loosely termed \textit{surface pressure gradient}
whereas the first term will be termed the
\textit{hydrostatic pressure gradient} by analogy to the $z$-coordinate formulation.
In fact, the the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, and
$\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of
the vertical integration.
$\bullet$ \textbf{The other terms of the momentum equation}
The coriolis and forcing terms as well as the the vertical physics remain unchanged
as they involve neither time nor space derivatives. The form of the lateral physics is
discussed in \autoref{apdx:B}.
$\bullet$ \textbf{Full momentum equation}
To sum up, in a curvilinear $s$-coordinate system, the vector invariant momentum equation
solved by the model has the same mathematical expression as the one in a curvilinear
$z-$coordinate, except for the pressure gradient term :
\begin{subequations} \label{apdx:A_dyn_vect}
\begin{multline} \label{apdx:A_PE_dyn_vect_u}
\frac{\partial u}{\partial t}=
+ \left( {\zeta +f} \right)\,v
- \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right)
- \frac{1}{e_3} \omega \frac{\partial u}{\partial k} \\
- \frac{1}{e_1 } \left( \frac{\partial p_h'}{\partial i} + g\; d \; \frac{\partial z}{\partial i} \right)
- \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
+ D_u^{\vect{U}} + F_u^{\vect{U}}
\end{multline}
\begin{multline} \label{apdx:A_dyn_vect_v}
\frac{\partial v}{\partial t}=
- \left( {\zeta +f} \right)\,u
- \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left( u^2+v^2 \right)
- \frac{1}{e_3 } \omega \frac{\partial v}{\partial k} \\
- \frac{1}{e_2 } \left( \frac{\partial p_h'}{\partial j} + g\; d \; \frac{\partial z}{\partial j} \right)
- \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
+ D_v^{\vect{U}} + F_v^{\vect{U}}
whereas the flux form momentum equation differ from it by the formulation of both
the time derivative and the pressure gradient term :
\begin{subequations} \label{apdx:A_dyn_flux}
\begin{multline} \label{apdx:A_PE_dyn_flux_u}
\frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t} =
\nabla \cdot \left( {{\rm {\bf U}}\,u} \right)
+ \left\{ {f + \frac{1}{e_1 e_2 }\left( v \;\frac{\partial e_2 }{\partial i}
-u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,v \\
\begin{multline} \label{apdx:A_dyn_flux_v}
\frac{1}{e_3}\frac{\partial \left( e_3\,v \right) }{\partial t}=
- \nabla \cdot \left( {{\rm {\bf U}}\,v} \right)
-u \;\frac{\partial e_1 }{\partial j} \right)} \right\} \,u \\
Both formulation share the same hydrostatic pressure balance expressed in terms of
hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$:
\begin{equation} \label{apdx:A_dyn_zph}
It is important to realize that the change in coordinate system has only concerned
the position on the vertical. It has not affected (\textbf{i},\textbf{j},\textbf{k}), the
orthogonal curvilinear set of unit vectors. ($u$,$v$) are always horizontal velocities
so that their evolution is driven by \emph{horizontal} forces, in particular
the pressure gradient. By contrast, $\omega$ is not $w$, the third component of the velocity,
but the dia-surface velocity component, $i.e.$ the volume flux across the moving
$s$-surfaces per unit horizontal area.
% Tracer equation
\section{Tracer equation}
\label{sec:A_tracer}
The tracer equation is obtained using the same calculation as for the continuity
equation and then regrouping the time derivative terms in the left hand side :
\begin{multline} \label{apdx:A_tracer}
\frac{1}{e_3} \frac{\partial \left( e_3 T \right)}{\partial t}
= -\frac{1}{e_1 \,e_2 \,e_3}
\left[ \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right)
+ \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] \\
+ \frac{1}{e_3} \frac{\partial }{\partial k} \left( Tw \right)
+ D^{T} +F^{T}
The expression for the advection term is a straight consequence of (A.4), the
expression of the 3D divergence in the $s-$coordinates established above. | CommonCrawl |
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Preventing depression using a smartphone app: a randomized controlled trial
Mark Deady, Nicholas Glozier, Rafael Calvo, David Johnston, Andrew Mackinnon, David Milne, Isabella Choi, Aimee Gayed, Dorian Peters, Richard Bryant, Helen Christensen, Samuel B. Harvey
Published online by Cambridge University Press: 06 July 2020, pp. 1-10
There is evidence that depression can be prevented; however, traditional approaches face significant scalability issues. Digital technologies provide a potential solution, although this has not been adequately tested. The aim of this study was to evaluate the effectiveness of a new smartphone app designed to reduce depression symptoms and subsequent incident depression amongst a large group of Australian workers.
A randomized controlled trial was conducted with follow-up assessments at 5 weeks and 3 and 12 months post-baseline. Participants were employed Australians reporting no clinically significant depression. The intervention group (N = 1128) was allocated to use HeadGear, a smartphone app which included a 30-day behavioural activation and mindfulness intervention. The attention-control group (N = 1143) used an app which included a 30-day mood monitoring component. The primary outcome was the level of depressive symptomatology (PHQ-9) at 3-month follow-up. Analyses were conducted within an intention-to-treat framework using mixed modelling.
Those assigned to the HeadGear arm had fewer depressive symptoms over the course of the trial compared to those assigned to the control (F3,734.7 = 2.98, p = 0.031). Prevalence of depression over the 12-month period was 8.0% and 3.5% for controls and HeadGear recipients, respectively, with odds of depression caseness amongst the intervention group of 0.43 (p = 0.001, 95% CI 0.26–0.70).
This trial demonstrates that a smartphone app can reduce depression symptoms and potentially prevent incident depression caseness and such interventions may have a role in improving working population mental health. Some caution in interpretation is needed regarding the clinical significance due to small effect size and trial attrition.
Trial Registration Australian and New Zealand Clinical Trials Registry (www.anzctr.org.au/) ACTRN12617000548336
The GLEAM 4-Jy (G4Jy) Sample: I. Definition and the catalogue
Sarah V. White, Thomas M. O Franzen, Chris J. Riseley, O. Ivy Wong, Anna D. Kapińska, Natasha Hurley-Walker, Joseph R. Callingham, Kshitij Thorat, Chen Wu, Paul Hancock, Richard W. Hunstead, Nick Seymour, Jesse Swan, Randall Wayth, John Morgan, Rajan Chhetri, Carole Jackson, Stuart Weston, Martin Bell, Bi-Qing For, B. M. Gaensler, Melanie Johnston-Hollitt, André Offringa, Lister Staveley-Smith
Published online by Cambridge University Press: 01 June 2020, e018
The Murchison Widefield Array (MWA) has observed the entire southern sky (Declination, $\delta< 30^{\circ}$ ) at low radio frequencies, over the range 72–231MHz. These observations constitute the GaLactic and Extragalactic All-sky MWA (GLEAM) Survey, and we use the extragalactic catalogue (EGC) (Galactic latitude, $|b| >10^{\circ}$ ) to define the GLEAM 4-Jy (G4Jy) Sample. This is a complete sample of the 'brightest' radio sources ( $S_{\textrm{151\,MHz}}>4\,\text{Jy}$ ), the majority of which are active galactic nuclei with powerful radio jets. Crucially, low-frequency observations allow the selection of such sources in an orientation-independent way (i.e. minimising the bias caused by Doppler boosting, inherent in high-frequency surveys). We then use higher-resolution radio images, and information at other wavelengths, to morphologically classify the brightest components in GLEAM. We also conduct cross-checks against the literature and perform internal matching, in order to improve sample completeness (which is estimated to be $>95.5$ %). This results in a catalogue of 1863 sources, making the G4Jy Sample over 10 times larger than that of the revised Third Cambridge Catalogue of Radio Sources (3CRR; $S_{\textrm{178\,MHz}}>10.9\,\text{Jy}$ ). Of these G4Jy sources, 78 are resolved by the MWA (Phase-I) synthesised beam ( $\sim2$ arcmin at 200MHz), and we label 67% of the sample as 'single', 26% as 'double', 4% as 'triple', and 3% as having 'complex' morphology at $\sim1\,\text{GHz}$ (45 arcsec resolution). We characterise the spectral behaviour of these objects in the radio and find that the median spectral index is $\alpha=-0.740 \pm 0.012$ between 151 and 843MHz, and $\alpha=-0.786 \pm 0.006$ between 151MHz and 1400MHz (assuming a power-law description, $S_{\nu} \propto \nu^{\alpha}$ ), compared to $\alpha=-0.829 \pm 0.006$ within the GLEAM band. Alongside this, our value-added catalogue provides mid-infrared source associations (subject to 6" resolution at 3.4 $\mu$ m) for the radio emission, as identified through visual inspection and thorough checks against the literature. As such, the G4Jy Sample can be used as a reliable training set for cross-identification via machine-learning algorithms. We also estimate the angular size of the sources, based on their associated components at $\sim1\,\text{GHz}$ , and perform a flux density comparison for 67 G4Jy sources that overlap with 3CRR. Analysis of multi-wavelength data, and spectral curvature between 72MHz and 20GHz, will be presented in subsequent papers, and details for accessing all G4Jy overlays are provided at https://github.com/svw26/G4Jy.
The GLEAM 4-Jy (G4Jy) Sample: II. Host galaxy identification for individual sources
Sarah V. White, Thomas M. O. Franzen, Chris J. Riseley, O. Ivy Wong, Anna D. Kapińska, Natasha Hurley-Walker, Joseph R. Callingham, Kshitij Thorat, Chen Wu, Paul Hancock, Richard W. Hunstead, Nick Seymour, Jesse Swan, Randall Wayth, John Morgan, Rajan Chhetri, Carole Jackson, Stuart Weston, Martin Bell, B. M. Gaensler, Melanie Johnston–Hollitt, André Offringa, Lister Staveley–Smith
The entire southern sky (Declination, $\delta< 30^{\circ}$ ) has been observed using the Murchison Widefield Array (MWA), which provides radio imaging of $\sim$ 2 arcmin resolution at low frequencies (72–231 MHz). This is the GaLactic and Extragalactic All-sky MWA (GLEAM) Survey, and we have previously used a combination of visual inspection, cross-checks against the literature, and internal matching to identify the 'brightest' radio-sources ( $S_{\mathrm{151\,MHz}}>4$ Jy) in the extragalactic catalogue (Galactic latitude, $|b| >10^{\circ}$ ). We refer to these 1 863 sources as the GLEAM 4-Jy (G4Jy) Sample, and use radio images (of ${\leq}45$ arcsec resolution), and multi-wavelength information, to assess their morphology and identify the galaxy that is hosting the radio emission (where appropriate). Details of how to access all of the overlays used for this work are available at https://github.com/svw26/G4Jy. Alongside this we conduct further checks against the literature, which we document here for individual sources. Whilst the vast majority of the G4Jy Sample are active galactic nuclei with powerful radio-jets, we highlight that it also contains a nebula, two nearby, star-forming galaxies, a cluster relic, and a cluster halo. There are also three extended sources for which we are unable to infer the mechanism that gives rise to the low-frequency emission. In the G4Jy catalogue we provide mid-infrared identifications for 86% of the sources, and flag the remainder as: having an uncertain identification (129 sources), having a faint/uncharacterised mid-infrared host (126 sources), or it being inappropriate to specify a host (2 sources). For the subset of 129 sources, there is ambiguity concerning candidate host-galaxies, and this includes four sources (B0424–728, B0703–451, 3C 198, and 3C 403.1) where we question the existing identification.
David Alan Alexander
Susan Johnston, Richard Williams
Journal: BJPsych Bulletin / Volume 44 / Issue 4 / August 2020
An ultra-wide bandwidth (704 to 4 032 MHz) receiver for the Parkes radio telescope
George Hobbs, Richard N. Manchester, Alex Dunning, Andrew Jameson, Paul Roberts, Daniel George, J. A. Green, John Tuthill, Lawrence Toomey, Jane F. Kaczmarek, Stacy Mader, Malte Marquarding, Azeem Ahmed, Shaun W. Amy, Matthew Bailes, Ron Beresford, N. D. R. Bhat, Douglas C.-J. Bock, Michael Bourne, Mark Bowen, Michael Brothers, Andrew D. Cameron, Ettore Carretti, Nick Carter, Santy Castillo, Raji Chekkala, Wan Cheng, Yoon Chung, Daniel A. Craig, Shi Dai, Joanne Dawson, James Dempsey, Paul Doherty, Bin Dong, Philip Edwards, Tuohutinuer Ergesh, Xuyang Gao, JinLin Han, Douglas Hayman, Balthasar Indermuehle, Kanapathippillai Jeganathan, Simon Johnston, Henry Kanoniuk, Michael Kesteven, Michael Kramer, Mark Leach, Vince Mcintyre, Vanessa Moss, Stefan Osłowski, Chris Phillips, Nathan Pope, Brett Preisig, Daniel Price, Ken Reeves, Les Reilly, John Reynolds, Tim Robishaw, Peter Roush, Tim Ruckley, Elaine Sadler, John Sarkissian, Sean Severs, Ryan Shannon, Ken Smart, Malcolm Smith, Stephanie Smith, Charlotte Sobey, Lister Staveley-Smith, Anastasios Tzioumis, Willem van Straten, Nina Wang, Linqing Wen, Matthew Whiting
We describe an ultra-wide-bandwidth, low-frequency receiver recently installed on the Parkes radio telescope. The receiver system provides continuous frequency coverage from 704 to 4032 MHz. For much of the band ( ${\sim}60\%$ ), the system temperature is approximately 22 K and the receiver system remains in a linear regime even in the presence of strong mobile phone transmissions. We discuss the scientific and technical aspects of the new receiver, including its astronomical objectives, as well as the feed, receiver, digitiser, and signal processor design. We describe the pipeline routines that form the archive-ready data products and how those data files can be accessed from the archives. The system performance is quantified, including the system noise and linearity, beam shape, antenna efficiency, polarisation calibration, and timing stability.
Lead pollution and the Roman economy
Damian Pavlyshyn, Iain Johnstone, Richard Saller
Journal: Journal of Roman Archaeology / Volume 33 / 2020
More than a decade ago, the Oxford Roman Economy Project (OXREP)1 and the Cambridge economic history of the Greco-Roman world put the question of the performance of the Roman economy at the center of historical debate, prompting a flood of books and articles attempting to assess the degree of growth in the economy.2 The issue is of sufficient importance that it has figured in the narratives of economists analyzing the impact of institutional frameworks on the potential for growth.3 As the debate has continued, there has been some convergence: most historians would agree that there was some Smithian growth as evidenced by urbanization and trade, while acknowledging that production remained predominantly agricultural and based primarily on somatic energy (i.e., human and animal).4 This is, of course, a very broad framework that does not differentiate the Roman empire from other complex pre-industrial societies. The challenge is to refine the analysis in order to put content into the broad description of "modest though significant growth"5 and to offer a deeper understanding of the dynamics of the economy.
Bridging Industry to Beamline through an Advanced Laboratory-Based Characterisation Facility
Richard E Johnston, Cameron Pleydell-Pearce, Alan Clarke, Kyriakos Mouzakitis, Leon Wechie, Ling Xu, Ric Allott
Journal: Microscopy and Microanalysis / Volume 25 / Issue S2 / August 2019
What Lies Beneath: 3D vs 2D Correlative Imaging Challenges and How to Overcome Them
Ria L. Mitchell, Stefanie Freitag, Tobias Volkenandt, James Russell, Peter Davies, Cameron Pleydell-Pearce, Richard Johnston
Scan Strategies for Electron Energy Loss Spectroscopy at Optical and Vibrational Energies in Perylene Diimide Nanobelts
Sean M. Collins, Demie M. Kepaptsoglou, Duncan N. Johnstone, Tom Willhammar, Raj Pandya, Jeffrey Gorman, Richard Friend, Akshay Rao, Paul A. Midgley, Quentin Ramasse
Published online by Cambridge University Press: 05 August 2019, pp. 1738-1739
Correlating Microstructure to in situ Micromechanical Behaviour and Toughening Strategies in Biological Materials
Richard E Johnston, Ria L Mitchell, Cameron Pleydell-Pearce, Mark Coleman, Laura North, David LaBonte, Michelle Oyen, Rachel Board, Edward C Pope, Hari Arora, David Howells
Liberal Leaders and Liberal Success: The Impact of Alternation
Richard Johnston
Journal: Canadian Journal of Political Science/Revue canadienne de science politique / Volume 52 / Issue 3 / September 2019
A leader from Quebec boosts the fortunes of the Liberal party in that province. This, in turn, has helped make Quebec the veto player in twentieth-century Canadian elections and the Liberals the "natural" governing party. Although Quebec is no longer as critical as before, a leader from the province still makes a big difference. Full impact from the pattern requires more than one election to unfold. Patterns outside Quebec are similar, if fainter: the Liberal party is not punished for choosing a Quebecker and may even be helped. The early success of the pattern moved the Liberals to alternate between Quebec and non-Quebec leaders, such that the party is now led by a Quebecker more often than not. Maintaining alternation has never been easy and is only getting harder.
Health Care Provision during a Sporting Mass Gathering: A Structure and Process Description of On-Site Care Delivery
Amy Johnston, Jasmine Wadham, Josea Polong-Brown, Michael Aitken, Jamie Ranse, Alison Hutton, Brent Richards, Julia Crilly
Journal: Prehospital and Disaster Medicine / Volume 34 / Issue s1 / May 2019
Published online by Cambridge University Press: 06 May 2019, p. s134
Print publication: May 2019
During mass gatherings, such as marathons, the provision of timely access to health care services is required for the mass gathering population as well as the local community. However, effective provision of health care during sporting mass gatherings is not well understood.
To describe the structures and processes developed for an emergency team to operate an in-event acute health care facility during one of the largest mass sporting participation events in the southern hemisphere, the Gold Coast marathon.
A pragmatic qualitative methodology was used to describe the structures and processes required to operate an in-event acute health care facility providing services for marathon runners and spectators. Content analysis from 12 semi-structured interviews with Emergency Department (ED) clinical staff working during the two-day event was undertaken in 2016.
Structural elements that underpinned the in-event health care facility included: physical spaces such as the clinical zones in the marathon health tent, tent access, and egress points; and resources such as bilingual staff, senior medical staff, and equipment such as electrocardiograms. Critical processes included: clear communication pathways, interprofessional care coordination, and engagement involving shared knowledge of and access to resources. Distinct but overlapping clinical scope between nurses and doctors was also noted as important for timely care provision and appropriate case management. Staff outlined many perceived benefits and opportunities of in-event health care delivery including ED avoidance and disaster training.
This in-event model of emergency care delivery enabled acute out-of-hospital health care to be delivered in a portable and transportable facility. Clinical staff reported satisfaction with their ability to provide a meaningful contribution to hospital avoidance and to the local community. With the number of sporting mass gatherings increasing, this temporary, in-event model of health care provision is one option for event and health care planners to consider.
Amy N. B. Johnston, Jasmine Wadham, Josea Polong-Brown, Michael Aitken, Jamie Ranse, Alison Hutton, Brent Richards, Julia Crilly
Journal: Prehospital and Disaster Medicine / Volume 34 / Issue 1 / February 2019
Mass gatherings such as marathons are increasingly frequent. During mass gatherings, the provision of timely access to health care services is required for the mass-gathering population, as well as for the local community. However, the nature and impact of health care provision during sporting mass gatherings is not well-understood.
The aim of this study was to describe the structures and processes developed for an emergency health team to operate an in-event, acute health care facility during one of the largest mass-sporting participation events in the southern hemisphere, the Gold Coast Marathon (Queensland, Australia).
A pragmatic, qualitative methodology was used to describe the structures and processes required to operate an in-event, acute health care facility providing services for marathon runners and spectators. Content analysis from 12 semi-structured interviews with emergency department (ED) clinical staff working during the two-day event was undertaken in 2016.
Important structural elements of the in-event health care facility included: physical spaces, such as the clinical zones in the marathon health tent and surrounding area, and access and egress points; and resources such as bilingual staff, senior medical staff, and equipment such as electrocardiograms (ECGs) and intravenous fluids. Process elements of the in-event health care facility included clear communication pathways, as well as inter-professional care coordination and engagement involving shared knowledge of and access to resources, and distinct but overlapping clinical scope between nurses and doctors. This was seen to be critical for timely care provision and appropriate case management. Staff reported many perceived benefits and opportunities of in-event health care delivery, including ED avoidance and disaster training.
This in-event model of emergency care delivery, established in an out-of-hospital location, enabled the delivery of acute health care that could be clearly described and defined. Staff reported satisfaction with their ability to provide a meaningful contribution to hospital avoidance and to the local community. With the number of sporting mass gatherings increasing, this temporary, in-event model of health care provision is one option for event and health care planners to consider.
JohnstonANB, WadhamJ, Polong-BrownJ, AitkenM, RanseJ, HuttonA, RichardsB, CrillyJ.Health Care Provision During a Sporting Mass Gathering: A Structure and Process Description of On-Site Care Delivery. Prehosp Disaster Med. 2019;34(1):62–71.
Infection prevention in the operating room anesthesia work area
L. Silvia Munoz-Price, Andrew Bowdle, B. Lynn Johnston, Gonzalo Bearman, Bernard C. Camins, E. Patchen Dellinger, Marjorie A. Geisz-Everson, Galit Holzmann-Pazgal, Rekha Murthy, David Pegues, Richard C. Prielipp, Zachary A. Rubin, Joshua Schaffzin, Deborah Yokoe, David J. Birnbach
Journal: Infection Control & Hospital Epidemiology / Volume 40 / Issue 1 / January 2019
Published online by Cambridge University Press: 11 December 2018, pp. 1-17
Print publication: January 2019
Utility of the Addenbrooke's Cognitive Examination in Amyotrophic Lateral Sclerosis
Sneha Chenji, Dennell Mah, Wendy Johnston, Richard Camicioli, Nancy Fisher, Sanjay Kalra
Journal: Canadian Journal of Neurological Sciences / Volume 45 / Issue 5 / September 2018
Amyotrophic lateral sclerosis (ALS) is a neurodegenerative condition that primarily affects motor neurons. Cognitive changes are reported in 25%-50% of patients, secondary to frontotemporal involvement. The objective of this study was to evaluate the utility of a screening tool, the Addenbrooke's Cognitive Examination (ACE), in ALS patients.
In this retrospective cross-sectional study, performance on the ACE was compared between 55 ALS patients and 49 healthy controls. The validation of the ACE in ALS patients was explored using a neuropsychometric battery. Correlations between the ACE and clinical variables such as the ALS Functional Rating Scale-Revised (ALSFRS-R) and forced vital capacity were computed.
A higher percentage of patients were below cut-off scores, although this remained non-significant between the patient and control groups. The ACE did not reveal significant differences between ALS patients and controls. The scores on the ACE displayed moderate correlations with our neuropsychometric battery for some domains, whereas others showed poor or no associations. Poor ACE Total was associated with lower ALSFRS-R and finger-tapping scores.
Performance on the ACE was comparable between patients and controls. Associations with motor function pose a challenge to accurate interpretation of ACE performance. It is likely that patients with poor cognition have greater disability, or that poor ACE performance reflects reduced motor ability to perform the task. This raises concern for the utility of the ACE as a screening tool in ALS patients, especially since recent versions of the ACE continue to include motor-based tasks.
All Mixed Up: Using Machine Learning to Address Heterogeneity in (Natural) Materials
J. F. Einsle, Ben Martineau, Iris Buisman, Zoja Vukmanovic, Duncan Johnstone, Alex Eggeman, Paul A. Midgley, Richard J. Harrison
Correlative Imaging and Bio-inspiration: Multi-scale and Multi-modal Investigations of the Acorn Barnacle (Semibalanus balanoides)
Ria L. Mitchell, Cameron Pleydell-Pearce, Mark P. Coleman, Peter Davies, Laura North, Richard E. Johnston, Will Harris
An Assessment of Polarized Light Microscopy for the Quantification of Grain Size and Orientation in Titanium Alloys via Microanalytical Correlative Light to Electron Microscopy (CLEM)
Hamed Safaie, Ria L. Mitchell, Richard Johnston, James Russell, Cameron Pleydell-Pearce
Ethnoreligious Identity, Immigration, and Redistribution
Stuart Soroka, Matthew Wright, Richard Johnston, Jack Citrin, Keith Banting, Will Kymlicka
Journal: Journal of Experimental Political Science / Volume 4 / Issue 3 / Winter 2017
Print publication: Winter 2017
Do increasing, and increasingly diverse, immigration flows lead to declining support for redistributive policy? This concern is pervasive in the literatures on immigration, multiculturalism and redistribution, and in public debate as well. The literature is nevertheless unable to disentangle the degree to which welfare chauvinism is related to (a) immigrant status or (b) ethnic difference. This paper reports on results from a web-based experiment designed to shed light on this issue. Representative samples from the United States, Quebec, and the "Rest-of-Canada" responded to a vignette in which a hypothetical social assistance recipient was presented as some combination of immigrant or not, and Caucasian or not. Results from the randomized manipulation suggest that while ethnic difference matters to welfare attitudes, in these countries it is immigrant status that matters most. These findings are discussed in light of the politics of diversity and recognition, and the capacity of national policies to address inequalities.
Multidisciplinary Management of Pediatric Sports-Related Concussion
Michael J. Ellis, Lesley J. Ritchie, Patrick J. McDonald, Dean Cordingley, Karen Reimer, Satnam Nijjar, Mark Koltek, Shahid Hosain, Janine Johnston, Behzad Mansouri, Scott Sawyer, Norm Silver, Richard Girardin, Shannon Larkins, Sara Vis, Erin Selci, Michael Davidson, Scott Gregoire, Angela Sam, Brian Black, Martin Bunge, Marco Essig, Peter MacDonald, Jeff Leiter, Kelly Russell
Journal: Canadian Journal of Neurological Sciences / Volume 44 / Issue 1 / January 2017
Published online by Cambridge University Press: 24 October 2016, pp. 24-34
Objectives: To summarize the clinical characteristics and outcomes of pediatric sports-related concussion (SRC) patients who were evaluated and managed at a multidisciplinary pediatric concussion program and examine the healthcare resources and personnel required to meet the needs of this patient population. Methods: We conducted a retrospective review of all pediatric SRC patients referred to the Pan Am Concussion Program from September 1st, 2013 to May 25th, 2015. Initial assessments and diagnoses were carried out by a single neurosurgeon. Return-to-Play decision-making was carried out by the multidisciplinary team. Results: 604 patients, including 423 pediatric SRC patients were evaluated at the Pan Am Concussion Program during the study period. The mean age of study patients was 14.30 years (SD: 2.32, range 7-19 years); 252 (59.57%) were males. Hockey (182; 43.03%) and soccer (60; 14.18%) were the most commonly played sports at the time of injury. Overall, 294 (69.50%) of SRC patients met the clinical criteria for concussion recovery, while 75 (17.73%) were lost to follow-up, and 53 (12.53%) remained in active treatment at the end of the study period. The median duration of symptoms among the 261 acute SRC patients with complete follow-up was 23 days (IQR: 15, 36). Overall, 25.30% of pediatric SRC patients underwent at least one diagnostic imaging test and 32.62% received referral to another member of our multidisciplinary clinical team. Conclusion: Comprehensive care of pediatric SRC patients requires access to appropriate diagnostic resources and the multidisciplinary collaboration of experts with national and provincially-recognized training in TBI. | CommonCrawl |
The polynomial $f(x)$ is divided by the polynomial $d(x)$ to give a quotient of $q(x)$ and a remainder of $r(x)$. If $\deg f = 9$ and $\deg r = 3$, what is the maximum possible value of $\deg q$?
We have $f(x) = d(x)q(x) +r(x)$. Since $\deg f = 9$ and $\deg r = 3$, we must have $\deg q + \deg d = 9$. We know that in division $\deg r < \deg d$, which means that $\deg d \ge 4$. So
$$\deg q \le 9-4 = \boxed{5}.$$ | Math Dataset |
What is the product of the numerator and the denominator when $0.\overline{009}$ is expressed as a fraction in lowest terms?
Let $x=0.\overline{009}$. Then $1000x=9.\overline{009}$ and $1000x-x=999x=9$. Therefore, $0.\overline{009}=\frac{9}{999}$, which in lowest terms is $\frac{1}{111}$. The product of the numerator and the denominator is $1\cdot 111=\boxed{111}$. | Math Dataset |
Has the cesarean epidemic in Czechia been reversed despite fertility postponement?
Tomáš Fait1,2,
Anna Šťastná2,
Jiřina Kocourková ORCID: orcid.org/0000-0003-1339-85082,
Eva Waldaufová2,
Luděk Šídlo2 &
Michal Kníže1
Although the percentage of cesarean sections (CS) in Czechia is below the average of that of other developed countries (23.6%), it still exceeds WHO recommendations (15%). The first aim of the study is to examine the association between a CS birth and the main health factors and sociodemographic characteristics involved, while the second aim is to examine recent trends in the CS rate in Czechia.
Anonymized data on all mothers in Czechia for 2018 taken from the National Register of Expectant Mothers was employed. The risk of cesarean delivery for the observed factors was tested via the construction of a binary logistic regression model that allowed for adjustments for all the other covariates in the model.
Despite all the covariates being found to be statistically significant, it was determined that health factors represented a higher risk of a CS than sociodemographic characteristics. A previous CS was found to increase the risk of its recurrence by 33 times (OR = 32.96, 95% CI 30.95–35.11, p<0.001). The breech position increased the risk of CS by 31 times (OR = 31.03, 95% CI 28.14–34.29, p<0.001). A multiple pregnancy increased the odds of CS six-fold and the use of ART 1.8-fold. Mothers who suffered from diabetes before pregnancy were found to be twice as likely to give birth via CS (OR = 2.14, 95% CI 1.76–2.60, p<0.001), while mothers with gestational diabetes had just 23% higher odds of a CS birth (OR = 1.23, 95% CI 1.16–1.31, p<0.001). Mothers who suffered from hypertension gave birth via CS twice as often as did mothers without such complications (OR = 2.01, 95% CI 1.86–2.21, p<0.001).
The increasing age of mothers, a significant risk factor for a CS, was found to be independent of other health factors. Accordingly, delayed childbearing is thought to be associated with the increase in the CS rate in Czechia. However, since other factors come into play, further research is needed to assess whether the recent slight decline in the CS rate is not merely a temporal trend.
Cesarean section (CS), when used appropriately, should account for 10–15% of births [1, 2]. In recent years, however, the trend toward the use of CS in obstetric practice has been on the increase worldwide. Eastern Europe witnessed one of the highest increases (two-fold) in the use of CS in the period 2000–2015 [3, 4]. This trend was particularly marked in Czechia, where the CS rate increased from 10.3% in 1994 to a maximum value of 26.1% in 2015, followed by a slight decrease to 23.6% in 2018 (Fig. 1). Currently, the CS rate in Czechia is below the average of other developed countries [3,4,5] (Fig. 2).
Mean age of mothers at birth, first births and the CS rate, Czechia, 1994–2018. Source: [6,7,8,9,10,11]
Cesarean section rate in OECD countries in 2017. Source: [4]
One of the most likely reasons for this phenomenon concerns the dynamic increase in the age of mothers [12], which represents a significant recent demographic trend in Czechia [13,14,15]. Between 1994 and 2015 the mean age of mothers increased on a continuous basis, as did the share of CS, both of which stagnated only recently (Fig. 1). Fertility postponement is further connected with a decrease in the probability of having a second child [16, 17], the increased use of assisted reproduction methods [18, 19], and health risks for both mothers and their children [20, 21], i.e. factors which are also related to the increased use of CS [22, 23].
The reasons for the increase in the CS are multifactorial and include health care practices [2, 3]. The care of pregnant women in Czechia is fully entrusted to gynecologists and obstetricians. It is strongly recommended that the birth should take place in a medical facility and, even if it is conducted by a midwife, the doctor remains the legally responsible person. The decision on a planned CS cannot be based on a request from the mother. While some maternity facilities are run by private companies, all the health care facilities used by Czech citizens are covered by the public health insurance system under the same conditions.
The first aim of the article is to evaluate, taking Czechia as an example, the association between the use of CS and the main medical factors related to the increased use of CS (complications during pregnancy and childbirth, diabetes, gestational age, the birth weight, the breech position, repeat CS, singleton/multiple pregnancy, and conception method) and to subsequently compare these associations with those between the use of CS and sociodemographic characteristics (the age of the mother, the birth order, marital status and the mother's level of education). The second aim is to examine recent trends in the CS rate in Czechia.
Data and methodology
The study employed a unique data source that contains anonymized data on all mothers in Czechia for 2018 obtained from the National Registry of Mothers at Childbirth (NRMC), which is managed by the Institute of Health Information and Statistics of the Czech Republic (IHIS CR) [7]. The data contained in the National Register is based on the so-called report on the mother at childbirth, a mandatory statistical report that is completed on all mothers, including foreigners, who give birth in Czechia. Data on the CS rate in the private sector is not reported separately.
In 2018, a total of 111,749 mothers gave birth to 113,234 children; 6.2% of them had non-Czech citizenship [7]. Since one of the most important considerations concerning the study of cesarean births is whether ART was used to achieve pregnancy, information on the date of embryo transfer was added to the data set by linking the file from the NRMC with the respective file obtained from the National Register of Assisted Reproduction (NRAR) using the mothers' so-called birth numbers (a unique number that is assigned to all Czechs at birth). Based on the comparison of the date of birth and the date of embryo transfer, it was possible to estimate those pregnancies that resulted from the use of ART.
Firstly, a descriptive analysis of the relationships between the observed variables was conducted so as to evaluate the distribution of the increased incidence of CS births according to the various factors considered. Most of the monitored variables contained data on all the mothers, with the exception of marital status and level of education; the completion of these questions is optional. There was a lack of information on 673 mothers concerning marital status (0.6% of the total sample) and on 23,113 mothers in the case of the level of education (20.7% of the total sample). Cesarean deliveries were divided into planned and acute.
In order to assess the association between the various covariates and the risk of CS, a binary logistic regression model was constructed, which enabled the testing of the association of the various variables on the incidence of CS births (1 yes, 0 no), assuming all the other characteristics of the mothers were equal. The application of the binary logistic regression model allowed for the removal of the mutual influence of the covariates and the testing of whether they also acted individually, all else being equal.
Two logistic regression models were constructed. Model 1 included all the mothers except for those for whom no data was available on the marital status and level of education (N = 88,041, i.e. 79% of the total number of mothers), while Model 2 included only those mothers who had already given birth in the past (a total of 46,127 mothers, i.e. 80% of repeat mothers after excluding women with no data for marital status and/or education).
The binary logistic regression model was used to explain the effects of the explanatory variables on the dependent variable "having a childbirth via cesarean section" (Y = 1 for cesarean section, otherwise Y = 0). x = (x1, …. xk)' is the vector of the explanatory variables:
$$\mathrm{logit}\left({\text{Pr}}\langle Y=1|x\rangle \right)={\text{log}}\left\{\frac{{\text{Pr}}\langle Y=1|x\rangle }{1-{\text{Pr}}\langle Y=1|x\rangle }\right\}={\beta }_{0}+{x}^{\mathrm{^{\prime}}}\beta ,$$
where β0 is the intercept parameter and β is the vector of the slope parameters.
For the sake of clarity, the results were interpreted in terms of odds ratios (OR), which qualify the variables that indicate the odds of cesarean delivery for each category compared to the given reference category.
A number of demographic, health and socio-economic characteristics were included in the models as explanatory variables. With the exception of the age of the mother at childbirth (continuous), all the following covariates were categorical and were transformed into dummy variables:
Marital status was divided into four categories: single, married (ref.), divorced and widowed.
The highest attained level of education was divided into four groups: basic (including incomplete), secondary without the school leaving certificate (SLC), secondary with the SLC (ref.) and tertiary.
The WHO classification of premature babies [24] was used for the categorization of the gestational age, namely: extremely preterm (less than 28 weeks), very preterm (28 to < 32 weeks), and moderate to late preterm (32 to < 37 weeks). The post-term birth category was defined according to a report by Spong [25], which indicates a post-term delivery as occurring from the 42nd week of pregnancy. A gestational age of 37–41 weeks was used as the reference category.
The birth order was divided into 3 categories, namely women who had not yet given birth (ref.), women who had given birth for a second time, and women with third and higher order births.
Singleton (ref.) versus multiple pregnancy.
Previous CS birth (only in model 2, in which first-time mothers did not feature): no (ref.), yes.
The probable method of pregnancy of the women was estimated based on the embryo transfer date reported for 4,018 mothers. This method of assisted reproduction was used by 3.6% of mothers. This variable was then divided into two categories – without the use of ART (ref.) and following ART.
The incidence of diabetes in the mothers was divided into three categories: not detected (ref.), detected prior to pregnancy, detected during pregnancy.
Hypertension and threatened preterm labor, which are among the most common health complications, were identified as serious complications during pregnancy and childbirth. Other complications (bleeding in the first, second and third trimesters, placenta previa, placental abruption and other placental abnormalities, cardiovascular complications, preeclampsia, intra-uterine growth restriction and others) were combined in the "other complications" category.
The models also considered the incidence of a breech presentation. This variable was assigned the values: no (ref.) and yes. In the case of multiple pregnancies, the pregnancy was classified as "yes" if at least one of the children was in the breech position.
The birth weight was not included in the regression model since it is not considered to be a key indicator of CS. The preferred routine adopted by the field of obstetrics in Czechia comprises the evaluation of placental functioning applying the ultrasonographic measurement of flow through the umbilical artery and the middle cerebral artery. The birth weight is recorded only following the birth of the child; data on the estimated birth weight of the child prior to the birth does not form a part of the official data on which a decision on a CS is based.
In addition, the variable birth weight, which has a high degree of multicollinearity with the gestational age, was monitored in the descriptive analysis via the following 5 categories: extremely low birth weight (< 1,000 g), very low birth weight (1,000–1,499 g), low birth weight (1,500–2,499 g), normal birth weight (2,500–3,999 g) and high birth weight (≥ 4,000 g) [26]. The analysis considered the lowest birth weight in the case of multiple births.
The analysis was performed using SPSS Statistics 26 software. The findings and the discussion are reported according to STROBE Statement guidelines [27].
In 2018, a total of 111,749 mothers gave birth to 113,234 children in Czechia [7]. The highest proportion of mothers comprised the 30–34 age group (34.6%), followed by the 25–29 age group (30.5%).
In 2018, 6.9% of mothers gave birth prematurely in Czechia and 48.2% of all mothers gave birth to their first child. 35.3% of mothers had second-order births and the remaining 16.5% had third-order and higher births. 10.5% of mothers had experienced at least one previous CS birth. 1,464 sets of twins were born in Czechia in 2018, i.e. 1.3% of all births.
The share of CS births in 2018 was 23.6%. The highest proportion of CS births concerned elective CS planned during pregnancy (42.9%) and, together with elective CS, but performed during labour (7.8%), accounted for a total of 50.7%. Emergent cesarean sections performed during labor accounted for 33.5%, and during pregnancy 15.8%. Of all women who gave birth via CS (23,341) 62% were aged 30 and over. 31.8% of all CS births were repeat CS births, of which 18.9% were breech presentations, 6.4% followed ART and 4.4% were multiple pregnancies.
Differences in the frequency of CS deliveries by socio-demographic characteristics
The distribution of CS according to age categories indicated an increasing risk with the age of the mother (Table 1). The lowest share of CS births referred to the up to 19 years age category (15.8%), with higher proportions in each subsequent age category. Compared to the total proportion of births via CS of 23.6%, the up to 29 years age group had a lower share than the average, and the 30–34 years category corresponded to the average. A significantly higher proportion of CS births concerned mothers aged 35–39 and over 40, for whom 37.2% of pregnancies ended in CS births. Conversely, the share of women with a vaginal delivery decreased with age from 84.2% before the age of 20 to 62.8% for women aged 40 and over.
Table 1 Cesarean delivery by the age of the mother at delivery, Czechia, 2018
In addition, the ratio of planned and emergent CS also varied depending on the age of the mother (Fig. 3), i.e. the share of planned CS births increased with age. Compared to the youngest mother age group (up to 19 years of age), concerning whom 30.2% of all CS births were planned, the over 40 years age group featured more than twice the percentage of mothers with planned CS.
Elective and emergent CS by maternal age at delivery, Czechia, 2018. Source: [7], own calculations
The proportion of CS births varied slightly depending on the birth order (Fig. 4). The highest share concerned first-time mothers (25.0%) and the lowest share mothers of third and higher birth orders (21.6%). Slight differences were also observed with respect to the education attained and the marital status of the mothers (Fig. 4). Divorced women (27.9%) and widows (28.3%) gave birth via CS more frequently than did single (23.6%) and married (23.2%) women. A lower share of CS births was observed for women with basic and incomplete education levels (22.2%), and the highest share for secondary school (with SLC) graduates (24.5%).
Percentage of CS of all deliveries for the given category of mothers, Czechia, 2018. Source: [7], own calculations
Differences in the frequency of CS deliveries by health indication
Significant differences were determined with respect to the number of pregnancies and whether or not the mother had previously given birth via CS (Table 2). Multiple births were predominantly via CS (78.7%) as were births by women who had previously had a CS birth (71.2%).
Table 2 Percentage of CS according to a previous CS, singleton/multiple pregnancy and breech presentation, Czechia, 2018
The proportion of CS births increased in proportion to the occurrence of complications during pregnancy and childbirth (Fig. 5). The risk of CS was higher for those mothers at risk of a pre-term delivery (30.1%). Women with hypertension (37.6%) and other complications (39.0%) gave birth via CS almost twice as often as did women without health complications (20.9%).
Percentage of CS according to maternal health complications and ART usage, Czechia, 2018. Source: [7], own calculations
The incidence of various types of complications during pregnancy and childbirth also varied depending on the age of the mother. Complications such as first and third trimester bleeding, placenta previa, placental abruption and other placental abnormalities, cardiovascular complications and hypertension primarily affected mothers over 35 years of age, and even more significantly mothers over 40 years of age. Conversely, some complications were characteristic of younger mothers under 24 years of age, specifically the occurrence of a significantly higher proportion of the threat of pre-term birth and intra-uterine growth restriction. The incidence of other health complications (preeclampsia, bleeding in the second trimester) did not differ significantly according to the age of the mother.
However, the proportion of diabetes, especially diabetes that was detected during pregnancy, increased with the age of the mothers (Table 3); while 6.2% of mothers aged 25–29 were affected, 11.6% of mothers aged over 40 suffered from this condition. 40% of women with preexisting diabetes gave birth via CS, while 29.9% of women with gestational diabetes and 23% of women without diabetes had CS births.
Table 3 Diabetes by the age of the mother at delivery, Czechia, 2018
CS delivery was observed to be less frequent for women who became pregnant without ART (22.9%) than those who underwent assisted reproduction techniques (42.2%) (Fig. 5).
Of all children born via CS, 18.6% were in the breech presentation, 3% in the transverse and oblique lie and 78.4% were in the vertex presentation. Only 9.8% of children in the breech presentation were born spontaneously.
The proportion of CS births varied significantly according to the birth weight of the child (Fig. 6). CS was significantly more common in the case of newborns who weighed less than 2,500 g than for those with normal birth weights. The highest share of CS births concerned the very low weight category (68.5%). A higher proportion of CS births was also recorded for children with higher birth weights (25.9%) than for those with normal birth weights (21.5%).
Percentage of CS of all deliveries according to birth weight and gestational age, Czechia, 2018. Source: [7], own calculations
The final observed change concerned the gestational age. Significant differences were observed between mothers of gestational ages ranging from 22 to 45 weeks. The proportion of CS was significantly higher for pre-term births than for term births, while the share was slightly higher for post-term births (Fig. 6). Extremely pre-term and very pre-term births took place via CS in more than half of all such cases, while moderate to late pre-term births involved CS in 40.4% of such cases. The incidence of pre-term births was higher for women younger than 25 years and older than 40 years than for those aged 20–34.
Health factors versus the socio-demographics associated with the increased odds of a CS delivery
The sociodemographic characteristics and health status of all the women were analyzed together employing binary logistic regression in order to identify the covariates associated with the increased odds of a CS delivery. All the covariates were entered into Model 1 for 88,041 mothers (i.e. 79% of all mothers).
The results revealed that the odds of a cesarean birth increases with the maternal age (Table 4 – Model 1). Thus, the increasing age of mothers is an important covariate associated with the increasing incidence of CS births, even when it is adjusted for relevant confounders—other age-dependent risk characteristics (e.g. pregnancy complications, the use of ART and multiple births).
Table 4 Odds Ratios (Exp(B)) of undergoing a cesarean delivery, Czechia, 2018
The odds of a CS decreased with the birth order: for second-time mothers the odds were 11% lower than for first-time mothers (OR = 0.89, 95% CI 0.86–0.93, p<0.001) and 25% lower for mothers of higher order births (OR = 0.75, 95% CI 0.71–0.79, p<0.001). Women who gave birth to multiple children had 6-times higher odds of a CS (OR = 6.08, 95% CI 5.13–7.21, p<0.001) than women with singleton pregnancies. Slightly higher odds of a CS birth were detected for single women (OR = 1.06, 95% CI 1.02–1.10, p<0.01) than for married women. No significant difference was observed with respect to the other categories. In terms of the level of education attained, lower odds of giving birth via CS were detected for women with a tertiary education (OR = 0.89, 95% CI 0.85–0.93, p<0.001) compared to women with secondary education with SLC.
In terms of health characteristics, the breech position comprises a decisive indication for a CS birth; the odds of a CS birth were more than 30 times higher than for women whose child was in a different position (OR = 31.06, 95% CI 28.14–34.29, p<0,001). Women who gave birth pre-term also had higher odds of giving birth via CS, especially those who had very pre-term births (OR = 2.94, 95% CI 2.33–3.71, p<0.001). Further, women who most likely became pregnant following embryo transfer had significantly higher odds of a cesarean delivery, even after adjusting for the mother's age and the birth order and frequency; the odds of giving birth via CS were 1.8-times higher than for women who did not undergo ART (OR = 1.83, 95% CI 1.69–1.99, p<0.001). Mothers who suffered from diabetes prior to pregnancy were more than twice as likely to give birth via CS than women who did not have the condition (OR = 2.14, 95% CI 1.76–2.60, p<0.001), while those with gestational diabetes had only 1.2-times higher odds (OR = 1.23, 95% CI 1.16–1.31, p<0.001). Furthermore, mothers who suffered from hypertension had twice the odds of a CS birth than those without such complications (OR = 2.01, 95% CI 1.86–2.21, p<0.001).
Personal history of cesarean section
Model 2 included only those women who had already given birth (57,960 mothers, i.e. 51.9%), which allowed for the addition of the very significant variable of whether the woman had given birth via cesarean section in the past (Table 4 – Model 2). It was revealed that a previous CS birth comprises an absolutely crucial explanatory variable for a subsequent cesarean delivery. Second and higher-order mothers with previous experience of CS had 32-times higher odds of giving birth via CS than those who had previously given birth vaginally (OR = 32.96, 95% CI 30.95–35.11, p<0.001). Either no change was observed with respect to the association of the other monitored variables (diabetes, complications in pregnancy and childbirth, education) or the odds even increased (gestational age, multiple pregnancy and ART use). The odds of CS for women who gave birth very pre-term was 3.5-times higher (OR = 3.56, 95% CI 2.32–5.45, p<0,001) than for those who gave birth within term, and the odds of CS for women with a multiple pregnancy was almost 9-times higher (OR = 8.94, 95% CI 6.93–11.54, p<0,001) than for those who had a singleton pregnancy.
In accordance with the Robson classification of CS [28], which is accepted as the global standard for the monitoring of the CS indication spectrum [29], a previous CS birth and the breech presentation were confirmed as the highest risk factors for CS birth in Czechia (31-times higher odds of a CS birth for a breech position and 35-times higher odds of a CS birth for a breech position for multiparous women; 32-times higher odds of a CS birth following a previous CS birth) followed by multiple pregnancies (6-times higher odds and 9-times higher odds for multiparous women) and ART use (2-times higher odds). Our analysis also confirmed the importance of the other health and socio-demographic factors examined, i.e. they evinced statistical significance after adjustment for all the other covariates: gestational age, diabetes, complications in pregnancy and childbirth, the mother's age, marital status and education. The differences in the risk of a CS birth according to marital status and education were statistically significant only for certain categories. A slightly higher risk of CS (1.6-times higher odds) was observed for single compared to married women, and a lower risk of CS (0.89-times lower odds) was observed for tertiary-educated women than for those with a secondary education. Our results confirmed the age factor as an independent risk with concern to a CS birth. With respect to the explanation for the increase in the CS rate in Czechia since the 1990s, both clinical (higher maternal ages at birth, an increase in ART use, multiple pregnancies) and non-clinical factors (health provider practices and guidelines, legislation) played noticeable roles.
Despite the use of a comprehensive dataset, the study has a number of limitations. The design does not allow for the causal interpretation of the associations studied. The covariates in the models were restricted to those available in the register. Information on education and marital status is not provided for all the women in the dataset; hence, for this part of the analysis, it was necessary to reduce the dataset by 21%, although no differences were observed between the two groups in terms of the structure of the mothers by age and CS births. Moreover, information on the use of ART methods was estimated based on information on ART cycles performed in Czechia only; foreign women and women who underwent ART abroad were thus classified as non-ART. Given that Czechia is more likely to be a destination country for cross-border reproductive care, we did not anticipate any bias in the results from this point of view. As for the explanatory factors, since we had no information on the maternal pre-pregnancy weight and height, we were unable to adjust for the body mass index.
Our results are consistent with literature in terms of reporting significant associations between the studied risk factors and a CS birth. It is reasonable to conclude that these factors have, to various extents, been behind the growth in the CS rate in Czechia since the 1990s. It is important to prevent the further growth in the CS rate and to determine the optimal percentage of CS, especially concerning the elective cesarean delivery of planned primarily indicated CS. It is clear that the underuse of CS results in hypoxic neonatal injury, stillbirth, uterine rupture and obstetrics fistulas [30], while the overuse of CS is associated with the increased risk of anesthesiologic and cardiovascular complications, infection complications and hysterectomy [31], as well as with adverse perinatal outcomes [32].
The incidence of serious complications is so rare due to advances in health care that many obstetricians lack the relevant experience. Nevertheless, the data clearly indicates a higher risk of morbidity and mortality as a result of a CS than a spontaneous delivery, even with respect to VBAC (vaginal birth after cesarean) [33]. However, in Czechia many patients and some obstetricians appear to believe that the opposite is the case, as reflected by the fact that a previous CS was found to be a key risk factor for a subsequent CS. The fact that 71.2% of Czech women with a history of CS give birth again via CS serves to confirm the low chance of a VBAC in such cases. This is in line with another study that documented that a high percentage of births via CS are followed by a subsequent birth via the same method without the option of TOLAC (the trial of labor after cesarean) [34]. Increased maternal age [35] also contributes to the indication of ERCS (elective repeat CS). Enforcing this practice in Czechia may also have contributed to the increase in the CS rate. The increase in women giving birth via CS in their first pregnancy results in an ongoing increase in the repeat CS birth rate [36]. If the CS rate increases for first-time mothers, it can be expected that this will generate a higher proportion of repeat CS. Accordingly, it can be assumed that a change in practice has the potential to reduce the CS rate in Czechia [37].
A further reason for the increase in CS births concerns the move away from spontaneous delivery when the fetus is in the breech presentation. This trend, initiated by the Term Breech Trial Collaborative Group study [38], has gradually led to a decline in the experience of such births and, thus, to a further increase in the use of CS. This approach has begun to be applied consistently in Czechia and is frequently referred to in medical study materials. However, spontaneous delivery when the fetus is in the breech presentation remains inadvisable, especially in the case of pre-term births [39].
The literature shows that a number of maternal health risks are age-related and that the risk of a cesarean birth increases with the maternal age [23, 40, 41]. For example, older mothers are associated with higher risks of the incidence of diabetes mellitus [42], pre-term births [24, 43], lower child birth weights [20, 21, 44, 45] and pre-term births associated with diabetes mellitus [46, 47]. Mothers over 30 years of age also face the increased risk of child health complications, spend longer times in hospital following the birth and face a higher risk of more frequent and longer hospital stays in the first two years of the child's life [48]. The application of logistic regression confirmed that both pregnancy health complications (preterm-birth, diabetes, hypertension) and the mother´s age comprise independent risk factors for a CS birth. The Czech results confirmed that mothers who gave birth very pre-term (28–31 weeks) had 3-times higher odds of a CS than women who had an in-term birth [49]. Mothers who suffered from diabetes before pregnancy had more than two-times higher odds of giving birth via CS than women who did not suffer from this condition, while mothers with gestational diabetes had 1.23-times higher odds; these results correspond to those of other published studies [50]. As expected, mothers who suffer from hypertension gave birth via CS twice as often as did those with no such complications [51].
Furthermore, our results confirmed the age factor as an independent risk for CS birth. Similar results were reported in a British study [52], the sample population of which comprised 76,158 singleton pregnancies with a live fetus at 11 + 0 to 13 + 6 weeks. After adjusting for potential maternal and pregnancy confounding variables, advanced maternal age (defined as ≥ 40 years) was associated with an increased risk of cesarean section (OR, 1.95 (95% CI, 1.77–2.14); P < 0.001). A recent Danish study [12] showed that nulliparous women aged 35–39 years had twice the risk of a CS (adjusted OR, 2.18 (95% CI, 2.11–2.26); P < 0.001).
Thus, one of today's most important population trends – fertility postponement – also comprises one of the significant independent factors associated with the risk of a CS birth. According to Timofeev et al. [22], the ideal age of mothers at birth is 25–29 years, at which time the risk of complications in pregnancy and the neonatal period is lowest. The increased risk of an adverse pregnancy is evident as early as between 30 and 34 years and continues to increase with age [20]. The question thus concerns the age that marks the limit in terms of the increased health risks associated with the mother's age. The association becomes significant from the age of 40 onwards [52], sometimes even after the age of 35 [12]. In any case, the risks associated with age are of a progressive character [20, 41].
The highest fertility rate in Czechia in 2018 was attained by women aged 30, in contrast to the early 1990s when maximum fertility was attained at the age of 22 [14]. The shift in fertility to older women in Czechia is further illustrated via a comparison of the share of fertility achieved by the age of 30. In 1989, the proportion stood at 86.6%, whereas by 2018 the share had dropped to 48.6% [17]. Thus, the trend toward delayed childbearing is apparent in Czechia as a result of the second demographic transition [53, 54], which indicates that reverse changes in fertility trends are highly unlikely. Nevertheless, fertility postponement can be decelerated or halted by the introduction of effective measures that act to remove barriers to starting a family [14]. To sum up, the strength of the association between advanced maternal age and CS and the fact that the trend in the share of CS births in Czechia has copied the trend in the mean age of mothers at childbirth (Fig. 1) support the hypothesis of a causal relationship between the maternal age and CS. However, as other factors come into play, further research is required so as to assess whether the recent slight decline in the CS rate is not merely a temporal trend.
A further risk factor that is closely connected with fertility postponement concerns the use of ART. Our results confirmed that mothers who most likely became pregnant following embryo transfer also had 1.83 higher odds of a cesarean delivery, even when controlling for the age, order and frequency of birth. According to the meta-analysis of the Medline, EMBASE and CINAHL databases [55], IVF/ICSI pregnancies are associated with a 1.90-fold increase in the odds of a CS (95% CI 1.76–2.06) compared to spontaneous conceptions. Since the late 1990s, Czechia has registered a significant increase in the use of ART and it has become a country with a relatively high proportion of ART live births [18, 19]. Accordingly, the increased use of ART in Czechia may have contributed to the explanation of the increase in the CS rate.
It is noteworthy that, despite the decline in marriage, marital status continues to comprise a relevant variable. In Czechia a slightly higher risk of CS (OR 1.06) was observed for single compared to married women despite the control of variables such as the age of the mother and the birth order. The higher risk of giving birth via CS for single women may be due to the fact that marital status is related to the health status, i.e. married persons have a higher level of self-esteem than do single people [56].
With regard to the level of the woman's education, no significant differences were detected in terms of the risk of a CS between women with a basic but incomplete education, secondary without the SLC (school leaving certificate) and secondary with the SLC. The controlling of the age and other variables revealed lower odds of a CS birth (OR 0.89) for tertiary-educated women than those with the SLC. The higher odds of CS for women with lower levels of education could be explained by their working in riskier professions, a higher incidence of smoking or obesity or generally poorer living conditions [57]. Conversely, tertiary-educated women are, in general, more open to practicing a healthy life style and receptive to the promotion of the benefits of natural childbirth in contrast to the numerous risks of CS for the subsequent health of both mothers and their children [58]. Thus, the introduction of health education as a component of the antenatal care process as a form of non-clinical intervention should be considered aimed at reducing the unnecessary use of CS [59].
The trend toward an increase in CS in Czechia can also be understood from the legislation perspective, in particular with concern to the introduction of the new Civil Code in 2014, which replaced clearly-defined compensation levels for personal injury with the decision on the amount thereof being decided solely by the courts. The courts continue to maintain the misconception that CS is the best form of intervention in terms of assuring the health of the child and mother. A similar situation has been reported by Longo with respect to Italy [5, 60, 61].
The share of CS births in Czechia (23.6%) exceeds WHO recommendations of 2015 on the optimal proportion of CS births (10–15%). Based on our results, we doubt whether the WHO recommendations reflect the increasingly older ages of mothers, especially first-time mothers and the high degree of institutionalization of deliveries in developed countries. Trusting the delivery to physicians is usually accompanied by a significantly higher degree of monitoring, with the associated risks of false-positive indications of hypoxia, a higher rate of medication use, and the loss of faith in normal childbirth [62].
Some women prefer a CS since they consider it to be safer for both themselves and the baby, an opinion that runs contrary to current scientific knowledge. A history of CS is associated with a higher risk of uterus rupture, placenta accreta, ectopic pregnancy, stillbirth, pre-term birth, and bleeding and the need for a blood transfusion, injury during surgery and hysterectomy in subsequent pregnancies. A higher birth order CS also increases the risk of maternal mortality and morbidity compared to a vaginal delivery [63].
CS may also lead to enhanced health risks for the baby – altered immune development, the increased likelihood of allergies, atopy, asthma, a reduction in intestinal microbiome diversity [64] and late childhood obesity [65]. The risk is higher for planned CS. Few studies have been conducted to date on the influence of CS on the cognitive and educational outcomes of CS-born children [63].
Thus, it is important that all the indications concerning birth via CS are carefully considered and that this method is not overused. Czechia makes no effort to contribute to efforts to reduce the percentage of cesarean sections; on the contrary, the reimbursement of costs by health insurance companies is higher for a cesarean section than for a spontaneous birth. One of the measures that might significantly prevent the expansion of CS use concerns a recommendation from the relevant professional authorities to strictly refuse cesarean sections on request [66]. Although this recommendation has been mentioned frequently in various professional forums in Czechia [67], efforts persist internationally to enforce dubious indications for a CS birth such as the protection of the pelvic floor [68], which also enjoys some support in Czechia. Nevertheless, in Czechia, CS on request is not legally permitted. Furthermore, the implementation of clinical practice guidelines combined with a mandatory second opinion for a CS indication is also relevant to the reduced risk of CS in Czechia [66].
In conclusion, despite the international concern surrounding the increasing CS rate, the Czech CS rate decreased from 26.1% in 2015 to 23.6% in 2018. Interestingly, this has not been attributed to any particular Czech health strategy aimed at reducing the CS rate. Although it has been perceived as a significant success for the field of Czech obstetrics, further research is needed in order to assess whether this is not merely a temporal trend.
Meaning of the study: possible mechanisms and implications for clinicians and policymakers
Delayed childbearing appears to be associated with the increasing use of CS in parallel with the expansion of defensive obstetrics that imply a high risk of CS in cases of a breech presentation and following a previous CS. In addition, the increased use of CS also reflects social demand, an increasing trend toward the prosecution of obstetricians in the event of childbirth complications and the erroneous lay perception of CS as the safest and least painful childbirth method. On the other hand, clinical practice based on the official refusal of CS on request could well prevent the overuse of CS. As regards obstetric practice, measures to encourage TOLAC, albeit with a careful eligibility assessment, may also help to reduce CS. As regards non-clinical interventions targeted at women, the support of training programs and health education on the indications and contra-indications of CS may also serve to improve the CS rate.
The aim of the study was to contribute to the explanation of recent trends in the CS rate in Czechia based on the examination of the association between a CS birth and selected health factors and sociodemographic characteristics. Our analysis confirmed that the mother's age comprises an independent risk factor for a CS birth in addition to pregnancy health complications and other, sociodemographic, characteristics. Accordingly, delayed childbearing appears to be associated with the increase in the CS rate in Czechia. However, the recent slight decline in the CS rate may be related to the completion of the fertility postponement process in Czechia. Nevertheless, since other factors come into play, further research is required in order to assess whether the recent slight decline in the CS rate is not merely a temporal trend.
The data that supports the findings of this study is available from The Institute of Health Information and Statistics of the Czech Republic (IHIS CR); however, restrictions apply to the availability of the data, which was used under license for the current study; hence the data is not publicly available. The data is, however, available from the authors upon reasonable request and with the permission of the IHIS CR.
CS:
NRMC:
National Registry of Mothers at Childbirth
IHIS CR:
Institute of Health Information and Statistics of the Czech Republic
Assisted reproductive technologies
NRAR:
National Register of Assisted Reproduction
VBAC:
Vaginal birth after cesarean
TOLAC:
Trial of labor after cesarean
TFR:
IVF:
In vitro fertilization
ICSI:
Intracytoplasmic sperm injections
SLC:
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The authors thank the General Health Insurance Company of the Czech Republic for providing detailed data sources and the Department of Demography and Geodemography, Faculty of Science, Charles University for providing general support in the processing of the research.
This paper was supported by the Czech Science Foundation (No. 18-08013S) "Transition towards the late childbearing pattern: individual prospects versus societal costs" project and by the Charles University Research Centre program (UNCE/HUM/018).
Department of Gynecology and Obstetrics, Second Faculty of Medicine, Charles University and Motol University Hospital, Prague, Czechia
Tomáš Fait & Michal Kníže
Department of Demography and Geodemography, Faculty of Science, Charles University, Prague, Czechia
Tomáš Fait, Anna Šťastná, Jiřina Kocourková, Eva Waldaufová & Luděk Šídlo
Tomáš Fait
Anna Šťastná
Jiřina Kocourková
Eva Waldaufová
Luděk Šídlo
Michal Kníže
Conceptualization and design of the research: TF, AŠ and JK. Methodology: AŠ and TF. Formal analysis and interpretation of the data: AŠ and EW. Interpretation of the data: LŠ and MK. Writing – original draft preparation: TF and JK. Writing—reviewing and editing: AŠ. Data curation and visualization: LŠ. Manuscript revision for important intellectual content: TF, AŠ, JK, LŠ and MK. All the authors have read and approved the final manuscript.
Correspondence to Jiřina Kocourková.
Ethical approval was not required for this study. This study did not involve the use of human tissues or animal experimentation. Anonymized data was obtained directly from the Institute of Health Information and Statistics of the Czech Republic by signing a declaration of confidentiality. The Institute of Health Information and Statistics of the Czech Republic is mandated by Act No. 372/2011 Coll., on Health Services and the Conditions of their Provision (the Act on Health Services) and by Act No. 89/1995 Coll., on the National Statistical Service, as subsequently amended, to administrate the National Health Information System (NHIS) and to collect statistical data based on the mandatory statistical reporting of all mothers in Czechia. When processing personal data in the NHIS, the Institute of Health Information and Statistics of the Czech Republic follows Regulation (EU) 2016/679 of the European Parliament and of the Council of 27 April 2016 on the protection of natural persons concerning the processing of personal data and the free movement of such data, and repealing Directive 95/46/EC (the General Data Protection Regulation). The Institute does not provide personal data from the NHIS to any other subjects.
Fait, T., Šťastná, A., Kocourková, J. et al. Has the cesarean epidemic in Czechia been reversed despite fertility postponement?. BMC Pregnancy Childbirth 22, 469 (2022). https://doi.org/10.1186/s12884-022-04781-1
Cesarean section (CS)
Fertility postponement
Health status
Breech delivery | CommonCrawl |
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Yet More Geometric Langlands News
Posted on February 27, 2021 by woit
It has only been a couple weeks since my last posting on this topic, but there's quite a bit of new news on the geometric Langlands front.
One of the great goals of the subject has always been to bring together the arithmetic Langlands conjectures of number theory with the geometric Langlands conjectures, which involved curves over function fields or over the complex numbers. Fargues and Scholze for quite a few years now have been working on a project that realizes this vision, relating the arithmetic local Langlands conjecture to geometric Langlands on the Fargues-Fontaine curve. Their joint paper on the subject has just appeared [arXiv version here]. It weighs in at 348 pages and absorbing its ideas should keep many mathematicians busy for quite a while. There's an extensive introduction outlining the ideas used in the paper, including a long historical section (chapter I.11) explaining the story of how these ideas came about and how the authors overcame various difficulties in trying to realize them as rigorous mathematics.
In other geometric Langlands news, this weekend there's an ongoing conference in Korea, videos here and here. The main topic of the conference is ongoing work by Ben-Zvi, Sakellaridis and Venkatesh, which brings together automorphic forms, Hamiltonian spaces (i.e classical phase spaces with a G-action), relative Langlands duality, QFT versions of geometric Langlands, and much more. One can find many talks by the three of them about this over the last year or so, but no paper yet (will it be more or less than 348 pages?). There is a fairly detailed write up by Sakellaridis here, from a talk he gave recently at MIT.
In Austin, Ben-Zvi is giving a course which provides background for this work, bringing number theory and quantum theory together, conceptualizing automorphic forms as quantum mechanics on arithmetic locally symmetric spaces. Luckily for all of the rest of us, he and the students seem to have survived nearly freezing to death and are now back at work, with notes from the course via Arun Debray.
For something much easier to follow, there's a wonderful essay on non-fundamental physics at Nautilus, The Joy of Condensed Matter. No obvious relation to geometric Langlands, but who knows?
Update: Arun Debray reports that there is a second set of notes for the Ben-Zvi course being produced, by Jackson Van Dyke, see here.
Update: David Ben-Zvi in the comments points out that a better place for many to learn about his recent work with Sakellaridis and Venkatesh is his MSRI lectures from last year: see here and here, notes from Jackson Van Dyke here.
Update: Very nice talk by David Ben-Zvi today (3/22/21) about this, see slides here, video here.
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14 Responses to Yet More Geometric Langlands News
Fen Zuo says:
Geometric Langlands is reflected in integer/fractional quantum Hall effect and the so-called Hofstadter's butterfly, according to recent work of Kazuki Ikeda.
Jim Eadon says:
Can anyone provide a (hand-wavy) summary for the educated layman (Physics post-grad) of what a Fargues-Fontaine curve is, and what's special about, e.g. in the context of the Langlands programme?
Arun Debray says:
Re: DBZ's course, I am not the only student taking notes. Jackson Van Dyke is also posting his notes online: https://web.ma.utexas.edu/users/vandyke/notes/langlands_sp21/langlands.pdf (Github link: https://github.com/jacksontvd/langlands_sp21). Jackson's notes go into quite a bit more detail, and he's gone back and added more references and figures than I have. In the end we'll hopefully combine our notes into one document. If anyone has any questions, comments, or corrections about either set of notes they're welcome to get in touch with me or Jackson.
Arun,
Thanks a lot, both for producing the notes, and for letting us know about the other ones.
David Ben-Zvi says:
Peter – thanks for the references!
Maybe let me note the series of talks in Korea (by Sakellaridis, Venkatesh and me) are aimed at a more arithmetic audience, some previous talks (eg at MSRI last March) might be more accessible to the audience here.
Also I might add that the perspective on automorphic forms as quantum mechanics is very old and widely used. A newer perspective I'm trying to advertise in the course and talks is to think of automorphic forms (and the Langlands program) as being really about 4d QFT — an arithmetic elaboration of the Kapustin-Witten picture for geometric Langlands. This accounts for many of the special features of quantum mechanics on arithmetic locally symmetric spaces – e.g., dependence on a number field is the analog of considering states on different 3-manifolds, Hecke operators — a form of quantum integrability– come from 't Hooft line operators, the choice of level (congruence subgroup) corresponds to consideration of surface defects, and most importantly the relation with Galois representations (the Langlands program) can be viewed as electric-magnetic duality.
The new feature of the work with Sakellaridis and Venkatesh (the first paper should appear relatively soon..) is that the theories of periods of automorphic forms and L-functions of Galois representations can fruitfully be understood as considering boundary conditions in the two dual TQFT, and that the electric-magnetic duality of boundary conditions (as studied by Gaiotto-Witten) can be used to explain the relation between the two (the theory of integral representations of L-functions).
Peter Scholze says:
Jim Eadon,
let me try to answer. This paper is about the (local) Langlands correspondence over the $p$-adic numbers $\mathbb Q_p$. Recall that $p$-adic numbers can be thought of as power series $a_{-n}p^{-n} + \ldots + a_0 + a_1 p + a_2p^2 + \ldots$ in the "variable" $p$ — they arise by completing the rational numbers $\mathbb Q$ with respect to a distance where $p$ is small. They are often thought of as analogous to the ring of meromorphic functions on a punctured disc $\mathbb D^*$ over the complex numbers, which admit Laurent series expansions $a_{-n} t^{-n} + \ldots + a_0 + a_1 t + a_2t^2 + \ldots$. More precisely, there is this "Rosetta stone" going back to Weil between meromorphic functions over $\mathbb C$, their version $\mathbb F_p((t))$ over a finite field $\mathbb F_p$, and $\mathbb Q_p$.
However, there is an important difference: $t$ is an actual variable, while $p$ is just a completely fixed number — how should $p=2$ ever vary? In geometric Langlands over $\mathbb C$, it is critical to take several points in the punctured disc $\mathbb D^\ast$ and let them move, and collide, etc. What should the analogue be over $\mathbb Q_p$, where there seems to be no variable that can vary?
In one word, what the Fargues–Fontaine curve is about is to build an actual curve in which $p$ is the variable, so "turn $\mathbb Q_p$ into the functions on an actual curve". It then even becomes possible to take two independent points on the curve, and let them move, and collide. With this, it becomes possible to adapt all (well, at least a whole lot of) the techniques of geometric Langlands to this setup.
This idea of "turning $p$ into a variable and allowing several independent points" is something that number theorists have long been aiming for, and is basically the idea behind the hypothetical "field with one element". I would however argue that our paper is the first paper to really make profitable use of this idea.
Thanks! I've added some links to your MSRI talks, which do look like a better place for people to start.
I've always been fascinated by analogies between number theory and QM/QFT, the new angle on this you're pointing out is really remarkable, looks like a significant deep link between the subjects.
Could you (or Peter Scholze!) comment on any relation of this to the other topic of the posting (local arithmetic Langlands as geometric Langlands on the Fargues-Fontaine curve)?
Professor Scholze,
You give a sense of how exciting it is, to, literally? connect the dots between different fields. I'm glad I studied pure mathematics as a hobby enough to get the gist of your explanation. I will re-read your reply a few times, as it's deep, and connects several fascinating objects and techniques.
I really appreciate you taking the time to engage, it means a lot. And thanks too, to Professor Woit, for bringing such mathematics to my (and others) attention, I enjoy the blog.
Peter – the way I see it (somewhat metaphorically) is as follows. In extended 4d topological field theory we seek to attach vector spaces to 3-manifolds, categories to 2-manifolds etc. The Langlands program fits beautifully into this if you accept the "arithmetic topology" analogy: besides ordinary 3-manifolds we consider (Spec of ring of integers of) global fields (number fields and function fields over finite fields) as "3-manifolds". Besides surfaces we also admit local fields (such as p-adics or Laurent series over finite fields) and curves over the algebraic closure of finite fields as "2-manifolds" (this is the theme I'd like to get to in my course, though still a way to go).
If you accept this ansatz, there's no "geometric Langlands" and "arithmetic Langlands", we're just considering different kinds of "manifolds" as inputs. For example geometric Langlands on surfaces and local (arithmetic) Langlands both concern equivalences of categories (in the latter case, one seeks descriptions of categories of reps of reductive groups over local fields are described in terms of spaces of Galois representations).
The Fargues-Scholze work is (among many other things!) a spectacular realization of this kind of idea. They show that the local Langlands program can be (and arguably is best) considered as geometric Langlands on an actual curve attached to the local field (the Fargues-Fontaine curve). Moreover the most crucial structure here, the Hecke operators, are miraculously described in a geometric way (factorization — the colliding points in Peter's response) that descends directly (via Beilinson-Drinfeld) from the structure of operator products in 2d QFT.
The wonderful recent work of Arinkin, Gaitsgory, Kazhdan, Raskin, Rozenblyum and Varshavsky that Will Sawin mention in a recent comment also fits into this general paradigm, in that they show that [unramified] arithmetic and geometric Langlands in the function field setting are precisely related by "dimensional reduction" – you pass from "2-manifolds" (curves over alg closure of finite fields) to "3-manifolds" (curves over the finite fields — which you should think of as mapping tori of the Frobenius map, so 3-manifolds fibering over the circle) by taking trace of Frobenius, just as TFT would tell you.
Laurent Fargues says:
I'm late but I'm going to say a few words to complement Peter's comments. Since this is a Physics blog I'm going to give a few key words that may speak to physicists. A lot of things work by analogies in this work, trying to put together some ideas from arithmetic and geometry together, make some mental jumps and trying to fill the gaps.
When Peter is saying "turning into a variable and allowing several independent points" this is analog to the fusion rules in conformal fields theory. There is the possibility in the world of diamonds to take different copies of the prime number p and fuse them into one copy. Here there reference, if I dig in my mind the first time I heard about this, is the work of Beilinson Drinfeld on factorization sheaves in terms of D-modules. You will find this fusion process in the Verlinde formula too in a coherent sheaves setting for compact Riemann surfaces in the work of Beauville "Conformal blocks, fusion rules and the Verlinde formula" for example where you fuse different points on a Riemann surface. I typically remember a talk by Kapranov about 'The formalism of factorizability' and did not get why the Russian peoples, who are known to have a huge background in physics, were such obsessed with this. No doubt this is linked to vertex algebras, where there are fusion rules, too and plenty of things of interest for physicists. For arithmeticians the declic, I remember saying to myself "at least I understand why peoples are obsessed with those factorization stuff", came from Vincent Lafforgue who remarked that if you work in an étale setting instead of a D-module setting, the moduli spaces of Shtukas (a vast generalization of modular curves for functions fields over a finite field) admits a factorization structure and this factorization structure gives you the Langlands parameter for global Langlands over a function field over a finite field.
For the curve here is what I can say. If you take an hyperbolic Riemann surface it is uniformized by the half plane on which you have a complex coordinate z. You can imagine the same type of things for the curve where the variable is the prime number p.
By the way there is an object in the article that may speak to physicists : Bun_G, the moduli of principal G-bundles on the curve. The analog for physicists would be the moduli of principal G-bundles on a compact Riemann surface, where now G is a compact Lie group, that shows up in the work of Atiyah and Bott. Still by the way, one of the origin of my geometrization conjecture is trying to understand the reduction theory "à la Atiyah Bott" for principal G-bundles on the curve (the analog for any G of the work on the indian school (Narasimhan-Seshadri) + Harder for GL_n i.e. usual vector bundles).
There are other objects that may speak, by analogies, to physicists in this paper. Typically the so called local Shtuka moduli spaces "with one paw" (i.e. only one copy of the prime number p). The archimedian analogs are hermitian symmetric spaces. Realizing local Langlands in the cohomology of those local Shtuka moduli spaces has an archimedian analog : Schmid realization of Harisch-Chandra discrete series in the L^2 cohomology of symmetric spaces. Schmid uses the Atiyah-Singer index formula to obtain his result, a tool well know to Physicists. Hermitian symmetric spaces are moduli of Hodge structures and this has been a great thing to realize that p-adic Hodge structures à la Fontaine are the same as "geometric Hodge structures" linked to the curve.
I could speak about this during hours and do some name-dropping that speaks to physicists but one thing is sure: there is no link with the multiverse, this I'm sure. Anyway, I have no idea how the curve looks like in other universes of the multiverse.
Thanks Laurent!
The Atiyah-Bott story (involving gauge fields + the Yang-Mills equations) and the Atiyah-Schmid story (involving, for SL(2,R), the Dirac operator on a 2d space) are two of my favorite topics. They're essentially the two main components of the Standard model (the Dirac equation for matter fields, the Yang-Mills equation for gauge fields). Only difference is that they're in 2d rather than 4d….
I hope you're still planning to come to Columbia for fall 2022, look forward to seeing you then!
My Eillenberg lectures are reported to 2023 sadly, because of the virus. I'll try not to enter into the technical details and give some general picture of the objects showing up in this work.
By the way, the cancelled program "The Arithmetic of the Langlands program" jointly organized with Calegari, Caraiani and Scholze is officially reported to 2023, same period of the year as before.
A very nice popular sketch of the idea of "the curve" by Matthew Morrow:
https://webusers.imj-prg.fr/~matthew.morrow/Morrow,%20Raconte-moi%20la%20Courbe.pdf
& more:
https://webusers.imj-prg.fr/~matthew.morrow/Exp-1150-Morrow,%20La%20Courbe%20de%20FF.pdf
@Thomas thanks for the link. | CommonCrawl |
Logarithmically concave function
In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality
$f(\theta x+(1-\theta )y)\geq f(x)^{\theta }f(y)^{1-\theta }$
for all x,y ∈ dom f and 0 < θ < 1. If f is strictly positive, this is equivalent to saying that the logarithm of the function, log ∘ f, is concave; that is,
$\log f(\theta x+(1-\theta )y)\geq \theta \log f(x)+(1-\theta )\log f(y)$
for all x,y ∈ dom f and 0 < θ < 1.
Examples of log-concave functions are the 0-1 indicator functions of convex sets (which requires the more flexible definition), and the Gaussian function.
Similarly, a function is log-convex if it satisfies the reverse inequality
$f(\theta x+(1-\theta )y)\leq f(x)^{\theta }f(y)^{1-\theta }$
for all x,y ∈ dom f and 0 < θ < 1.
Properties
• A log-concave function is also quasi-concave. This follows from the fact that the logarithm is monotone implying that the superlevel sets of this function are convex.[1]
• Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function f(x) = exp(−x2/2) which is log-concave since log f(x) = −x2/2 is a concave function of x. But f is not concave since the second derivative is positive for |x| > 1:
$f''(x)=e^{-{\frac {x^{2}}{2}}}(x^{2}-1)\nleq 0$
• From above two points, concavity $\Rightarrow $ log-concavity $\Rightarrow $ quasiconcavity.
• A twice differentiable, nonnegative function with a convex domain is log-concave if and only if for all x satisfying f(x) > 0,
$f(x)\nabla ^{2}f(x)\preceq \nabla f(x)\nabla f(x)^{T}$,[1]
i.e.
$f(x)\nabla ^{2}f(x)-\nabla f(x)\nabla f(x)^{T}$ is
negative semi-definite. For functions of one variable, this condition simplifies to
$f(x)f''(x)\leq (f'(x))^{2}$
Operations preserving log-concavity
• Products: The product of log-concave functions is also log-concave. Indeed, if f and g are log-concave functions, then log f and log g are concave by definition. Therefore
$\log \,f(x)+\log \,g(x)=\log(f(x)g(x))$
is concave, and hence also f g is log-concave.
• Marginals: if f(x,y) : Rn+m → R is log-concave, then
$g(x)=\int f(x,y)dy$
is log-concave (see Prékopa–Leindler inequality).
• This implies that convolution preserves log-concavity, since h(x,y) = f(x-y) g(y) is log-concave if f and g are log-concave, and therefore
$(f*g)(x)=\int f(x-y)g(y)dy=\int h(x,y)dy$
is log-concave.
Log-concave distributions
Log-concave distributions are necessary for a number of algorithms, e.g. adaptive rejection sampling. Every distribution with log-concave density is a maximum entropy probability distribution with specified mean μ and Deviation risk measure D.[2] As it happens, many common probability distributions are log-concave. Some examples:[3]
• The normal distribution and multivariate normal distributions.
• The exponential distribution.
• The uniform distribution over any convex set.
• The logistic distribution.
• The extreme value distribution.
• The Laplace distribution.
• The chi distribution.
• The hyperbolic secant distribution.
• The Wishart distribution, where n >= p + 1.[4]
• The Dirichlet distribution, where all parameters are >= 1.[4]
• The gamma distribution if the shape parameter is >= 1.
• The chi-square distribution if the number of degrees of freedom is >= 2.
• The beta distribution if both shape parameters are >= 1.
• The Weibull distribution if the shape parameter is >= 1.
Note that all of the parameter restrictions have the same basic source: The exponent of non-negative quantity must be non-negative in order for the function to be log-concave.
The following distributions are non-log-concave for all parameters:
• The Student's t-distribution.
• The Cauchy distribution.
• The Pareto distribution.
• The log-normal distribution.
• The F-distribution.
Note that the cumulative distribution function (CDF) of all log-concave distributions is also log-concave. However, some non-log-concave distributions also have log-concave CDF's:
• The log-normal distribution.
• The Pareto distribution.
• The Weibull distribution when the shape parameter < 1.
• The gamma distribution when the shape parameter < 1.
The following are among the properties of log-concave distributions:
• If a density is log-concave, so is its cumulative distribution function (CDF).
• If a multivariate density is log-concave, so is the marginal density over any subset of variables.
• The sum of two independent log-concave random variables is log-concave. This follows from the fact that the convolution of two log-concave functions is log-concave.
• The product of two log-concave functions is log-concave. This means that joint densities formed by multiplying two probability densities (e.g. the normal-gamma distribution, which always has a shape parameter >= 1) will be log-concave. This property is heavily used in general-purpose Gibbs sampling programs such as BUGS and JAGS, which are thereby able to use adaptive rejection sampling over a wide variety of conditional distributions derived from the product of other distributions.
• If a density is log-concave, so is its survival function.[3]
• If a density is log-concave, it has a monotone hazard rate (MHR), and is a regular distribution since the derivative of the logarithm of the survival function is the negative hazard rate, and by concavity is monotone i.e.
${\frac {d}{dx}}\log \left(1-F(x)\right)=-{\frac {f(x)}{1-F(x)}}$ which is decreasing as it is the derivative of a concave function.
See also
• logarithmically concave sequence
• logarithmically concave measure
• logarithmically convex function
• convex function
Notes
1. Boyd, Stephen; Vandenberghe, Lieven (2004). "Log-concave and log-convex functions". Convex Optimization. Cambridge University Press. pp. 104–108. ISBN 0-521-83378-7.
2. Grechuk, B.; Molyboha, A.; Zabarankin, M. (2009). "Maximum Entropy Principle with General Deviation Measures". Mathematics of Operations Research. 34 (2): 445–467. doi:10.1287/moor.1090.0377.
3. See Bagnoli, Mark; Bergstrom, Ted (2005). "Log-Concave Probability and Its Applications" (PDF). Economic Theory. 26 (2): 445–469. doi:10.1007/s00199-004-0514-4. S2CID 1046688.
4. Prékopa, András (1971). "Logarithmic concave measures with application to stochastic programming". Acta Scientiarum Mathematicarum. 32: 301–316.
References
• Barndorff-Nielsen, Ole (1978). Information and exponential families in statistical theory. Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley \& Sons, Ltd. pp. ix+238 pp. ISBN 0-471-99545-2. MR 0489333.
• Dharmadhikari, Sudhakar; Joag-Dev, Kumar (1988). Unimodality, convexity, and applications. Probability and Mathematical Statistics. Boston, MA: Academic Press, Inc. pp. xiv+278. ISBN 0-12-214690-5. MR 0954608.
• Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter. ISBN 3-11-013863-8. MR 1291393.
• Pečarić, Josip E.; Proschan, Frank; Tong, Y. L. (1992). Convex functions, partial orderings, and statistical applications. Mathematics in Science and Engineering. Vol. 187. Boston, MA: Academic Press, Inc. pp. xiv+467 pp. ISBN 0-12-549250-2. MR 1162312.
| Wikipedia |
Concrete number
A concrete number or numerus numeratus is a number associated with the things being counted, in contrast to an abstract number or numerus numerans which is a number as a single entity. For example, "five apples" and "half of a pie" are concrete numbers, while "five" and "one half" are abstract numbers. In mathematics the term "number" is usually taken to mean an abstract number. A denominate number is a type of concrete number with a unit of measure attached with it. For example, "5 inches" is a denominate number because it has the unit inches after it.
History
Mathematicians in ancient Greece were primarily interested in abstract numbers, while writers of instructional books for practical use were not concerned with such distinctions, so the terminology distinguishing the two types of number was slow to appear. In the 16th century textbooks began to make the distinction. This has appeared with increasing frequency until modern times.[1]
Denominate numbers
Denominate numbers are further classified as either simple, meaning a single unit is given, or compound, meaning multiple units are given. For example, 6 kg is a simple denominate number, while 324 yards 1 foot 8 inches is a compound denominate number. The process of converting a denominate number to an equivalent form that uses a different unit is called reduction. More specifically, reduction to a lower or higher unit of measurement is called reduction to lower or higher denominations. Reduction to a lower denomination is accomplished by multiplying by the number of lower units contained in each higher unit. In the case of a compound denominant number, the products are then added together. For example, 1 hour 23 minutes 20 seconds is 1 h × 3600 s/h + 23 min × 60 s/min + 20 s = 5000 seconds. Similarly, a division is used to reduce to a higher denomination, and remainders can be applied to the next highest unit to form compound denominant numbers. Addition and subtraction of compound numbers can be performed by grouping the amounts associated with each unit and performing the necessary carry and borrow operations. Multiplication and division by a pure number are again similar.
See also
• Dimensional analysis
• Units of measurement
References
1. Smith, D.E. (1953). History of Mathematics. Vol. II. Dover. pp. 11–12. ISBN 0-486-20430-8. (for section)
Authority control: National
• Japan
| Wikipedia |
\begin{document}
\begin{center} {\LARGE
Numerical Conformal Mapping to One-Tooth Gear-Shaped Domains and Applications} \end{center} \begin{center} \today \end{center} \begin{center}
Philip R. Brown\footnote{Partially supported by CONACyT grant
166183} \\
R. Michael Porter\footnotemark[1] \end{center}
\noindent Abstract. We study conformal mappings from the unit disk (or a rectangle) to one-tooth gear-shaped planar domains from the point of view of the Schwarzian derivative, with emphasis on numerical considerations. Applications are given to evaluation of a singular integral, mapping to the complement of an annular rectangle, and symmetric multitooth domains.
\noindent Keywords: conformal mapping, accessory parameter, Schwarzian derivative, gearlike domain, Sturm-Liouville problem, spectral parameter power series, conformal modulus, topological quadrilateral, Weierstrass elliptic function.
\noindent AMS Subject Classification: Primary 30C30; Secondary 30C20, 33E05.
\section{Introduction \label{sec:intro}}
In \cite{BrP2} we initiated a study of conformal mappings to a \textit{gearlike} domain with a single tooth: a starlike open set in the complex plane bounded by arcs of two circles centered at the origin and segments of two lines passing through the origin. The approach was to examine the Schwarzian derivative of such a mapping (a rational function of a particular form). Relationships were found between the auxiliary parameters in the Schwarzian derivative and the geometry of the gearlike domain, and the set of parameters producing a univalent mapping (``region of gearlikeness'') was determined.
Using the theoretical results in \cite{BrP2}, we focus here on the computational aspects of the conformal mappings to one-tooth gear domains from both a disk and a rectangle. We begin in section~\ref{sec:gearlike} with the relationship between the parameters in the Schwarzian derivatives for these two cases. In section~\ref{sec:formulas} we work out the transformations or normalizations necessary for the solution of the Schwarzian equation to produce a gear (rather than just a ``pregear'' in the terminology of \cite{BrP2}). In section~\ref{sec:compu} we describe a variety of approaches for carrying out the computations, and report some numerical results in section~\ref{sec:numerical}. This includes a ``map'' of the internal structure of the region of gearlikeness.
Finally, in Section~\ref{sec:appl} we give three applications of our results on gear mappings: evaluation of a singular integral, mapping to the complement of an annular rectangle, and symmetric multitooth domains.
An Appendix contains some background material on one of the numerical methods employed.
\section{One-tooth gear domains\label{sec:gearlike}}
General facts stated below concerning conformal mapping, Schwarzian derivatives, etc., may be found in many standard texts such as \cite{DT,Hen,Hi,Neh}. A \emph{one-tooth gear domain} is a topological quadrilateral in the complex plane which is the union of an open disk and a sector of a concentric disk of larger radius. Thus up to an affine equivalence $az+b$, a one-tooth gear domain is the standard domain $G_{\beta,\gamma}$ with vertices at $w_1=\beta e^{i\gamma}$, $w_2=e^{i \gamma}$, $w_3=e^{-i \gamma}$, $w_4= \beta e^{-i \gamma}$, where the edges $[w_1,w_2]$ and $[w_3,w_4]$ are straight segments called the \emph{tooth edges} of the gear, and the edges from $w_2$ to
$w_3$ and from $w_4$ to $w_1$ are arcs of the circles $\{|w|=1\}$ and
$\{|w|=\beta\}$ subtending angles in the ranges $\gamma<\arg w<2\pi-\gamma$ and $-\gamma<\arg w<\gamma$ respectively. We say that $\beta$ is the \emph{gear ratio} and $\gamma$ is the \emph{gear
angle}.
From our perspective, the first thing one needs is to have available is the Schwarzian derivative $S_f=(f''/f')'-(1/2)(f''/f')^2$ of the conformal mapping. When $S_f$ is known, a unique solution $f$ is determined by the triple $J_f(0)$, where the \emph{2-jet} of $f$ at any point $z$ is defined to be \begin{equation} \label{eq:def2jet}
J_f(z) = (f(z),f'(z),f''(z)), \end{equation} as long as we assure that $f'(0)\not=0$.
As we note in Proposition 2.1 of \cite{BrP}, if $f(0)$ is the tooth center (the common center of the concentric circles containing the circular boundary arcs of the gear) and $f'(0)=1$, we have \begin{equation} \label{eq:standard2jet}
J_f(0) = (0,1,2(\cos t_2-\cos t_1)) \end{equation} where $0<t_1<t_2<\pi$ and $e^{\pm i\pi t_1}$ and $e^{\pm i\pi t_2}$ are the prevertices. However, when we solve for $f$ given $S_f$ and with the normalization (\ref{eq:standard2jet}), $f(0)$ does not generally turn out to be the tooth center. Unless we know precisely what 2-jet to use, the solution $f$ will be a M\"obius transformation of a one-tooth gear domain, which we term a \emph{pregear} (see Figure \ref{fig:pregearillustration}). Although pregears are less rigid objects than gears, they have some very restrictive properties. Note that the tooth edges of any pregear may be uniquely identified by having different interior angles at their endpoints. Applying Euclidean transformations, we restrict the discussion to pregears which are symmetric in $\mathbb{R}$ and have no vertices on $\mathbb{R}$. In \cite{BrP} the following was shown.
\begin{prop} \label{prop:pregearcondition} (a) Let $D$ be a circular
quadrilateral with the above symmetries, having two interior angles
equal to $\pi/2$ and two interior angles equal to $3\pi/2$. Assume
that one tooth edge of $D$ lies in the upper and the other in the
lower half-plane. Then $D$ is a pregear if and only if the full
circles $C^+$, $C^-$ containing the tooth edges intersect in two
points.
(b) Let $D$ be a pregear. Then $D$ is a gear if and only if its
tooth edges are straight, or equivalently, if the non-tooth edges
are arcs of concentric circles. \end{prop}
\subsection{Conformal mapping from disk to a one-tooth gear\label{subsec:diskschw}}
Let $\mathbb{D}=\{z\in\mathbb{C}\colon\ |z|<1\}$ denote the unit disk, and let $f\colon\mathbb{D}\to G_{\beta,\gamma}$ be a conformal mapping. Suppose the prevertices $z_i=f^{-1}(w_i)$ are located at points of the form $e^{\pm it_1}$ and $e^{\pm it_2}$, for $0<t_1<t_2<\pi$. The expression for the Schwarzian derivative ${\mathcal{S}}_f$ of $f$ as a rational function $S_f=R_{t_1,t_2,\lambda}$ in terms of $t_1$, $t_1$, and an auxiliary parameter $\lambda$ was worked out explicitly in \cite{BrP2}. There it was also noted that by means of precomposition of $f$ with a M\"obius transformation \begin{equation} \label{eq:Tq}
T_q(z)= \frac{z-q}{1-qz} \end{equation} ($-1<q<1$) which leaves $\mathbb{D}$ invariant, the prevertices can be symmetrized with respect to the imaginary axis. Then (writing now $f$ in place of $f\circ T_q$) the Schwarzian derivative takes the form \begin{equation}\label{eq:SfRtlambda0} S_f=R_{t,\lambda} \end{equation} where \begin{eqnarray} \label{eq:Rtlambda}
\frac{1}{2}R_{t,\lambda}(z) &=& \psi_{0,t}(z) - \lambda
\psi_{1,t}(z) \end{eqnarray} and
\begin{eqnarray} \label{eq:psi0tpsi1t}
\psi_{0,t}(z)
&=& \frac{ (\sin^2t)(z^4-(16\cos t)z^3 + (4+2\cos2t) z^2 - (16\cos t) z + 1) }
{2(z^4-(2 \cos2t)z^2+1)^2}, \nonumber \\
\psi_{1,t}(z) &=& \frac{-8\cos t}{z^4-(2\cos2t)z^2+1}. \label{eq:psit} \end{eqnarray} One advantage of the symmetrized form is that the conformal $M(t)$ of the gear domain is easily related to the single parameter $t$ via an elliptic integral, as we will see in a moment. In \cite[section 3]{BrP2} this was exploited to derive qualitative relationships between the pair of geometric parameters $\beta,\gamma$ that prescribe the gear and the mapping parameters $t,\lambda$. In this paper we are more concerned with quantitative information concerning this relationship.
\subsection{Conformal mapping from rectangle to gear}\label{subsec:rectangle}
In the study of circular quadrilaterals with two symmetries in \cite{BrP}, mappings from a rectangle to the quadrilateral were investigated as well as mappings from a disk. For numerical work such mappings have certain advantages over mapping from the disk, among them the fact that certain Schwarzian derivatives are real on the boundary (or on horizontal or vertical sections of the rectangle). The relationships between conformal mappings to a gear from a disk and from a rectangle given below are somewhat more complicated than in the situation discussed in \cite{BrP} but follow the same general line of reasoning.
As in \cite{BrP}, we let $E$ denote the elliptic integral \begin{equation} \label{eq:ellipticintegral}
E(z) = \int_0^z \frac{dz}{ \sqrt{
(z-e^{it})(z+e^{-it})(z+e^{it})(z-e^{-it})} },\quad |z|<1. \end{equation} This is a particular case of a Schwarz-Christoffel mapping \cite{DT}, and the image $R=E(\mathbb{D})=[-\omega_1,\omega_1]\times[-\,{\rm Im}\,\omega_2,\,{\rm Im}\,\omega_2]$ is a rectangle centered at the origin. Here $\omega_1$, $\omega_2$ are half-periods of the related Weierstass $\wp$-function, with
$\omega_1>0$, $\,{\rm Im}\, \omega_2=|\omega_2|$, and $\tau=\omega_2/\omega_1$. Thus the composition $f\circ E^{-1}$ is a vertex-preserving conformal mapping from a rectangle to the gear $G_{\beta,\gamma}$, whose conformal module \cite{LV} is by definition the ratio \begin{equation} \label{eq:M(t)}
M(t) = \frac{ \,{\rm Im}\, E(i) }{ E(1) }. \end{equation} of the sides of the rectangle.
As customary, write $e_i=\wp(\omega_i)$, $i=1,2,3$ where $\omega_3=\omega_1+\omega_2$. These values are real and satisfy $e_2 <e_3<e_1$. Also as in \cite{BrP} for definiteness we will have $\omega_1$, $\omega_2$ normalized so that $e_1-e_2=4$, so the periods are in fact uniquely determined by their ratio $\tau$. The change of variable \begin{equation} \label{eq:zetafromz}
\zeta= \frac{E(z)}{2} \end{equation} maps $\mathbb{D}$ onto the subrectangle \begin{equation}
R_0 = \{\zeta\colon\ -\frac{\omega_1}{2} < \,{\rm Re}\, \zeta < \frac{\omega_1}{2} ,\
-\,{\rm Im}\,\frac{\omega_2}{2} < \,{\rm Im}\, \zeta < \,{\rm Im}\,\frac{\omega_2}{2} \} \end{equation} of $R$, and $\pm e^{\pm it}$ are mapped to the vertices of $R_0$. We define \begin{equation} \label{eq:varphi}
\varphi_{\tau,\mu}(\zeta) =
- 4\left(\wp\bigg(\zeta+\frac{\omega_1+\omega_2}{2}\bigg)
+ \wp\bigg(\zeta+\frac{\omega_1-\omega_2}{2}\bigg)\right) + 4\mu \end{equation} where $\mu$ is real (Figure \ref{fig:rectangleschwarzian}). \begin{figure}
\caption{ $|\varphi_{\tau,\mu}|$ for Schwarzian derivative (\ref{eq:varphi})}
\label{fig:rectangleschwarzian}
\end{figure} It is straightforward to verify, using standard properties of Weierstrass elliptic functions \cite{WW} that $\varphi_{\tau,\mu}(\zeta)$ is real when $\zeta\in\mathbb{R}$ and also when $\zeta\in\partial R_0$ (although not for imaginary $\zeta$ in general). The composition \begin{equation} \label{eq:def:g}
g(\zeta) = f\circ E^{-1}(2\zeta) \end{equation} is our mapping $g\colon R_0\to G_{\beta,\gamma}$.
\begin{prop}\label{prop:recgear}
Let $G=G_{\beta,\gamma}$ be a gear domain and let $g\colon R_0\to G$ be a conformal
mapping which respects vertices, taking the horizontal edges of $R_0$
to the tooth edges of $G$ and the left vertical edge of $R_0$ to the
inner circumference of $G$. Then the Schwarzian derivative of $g$ is
equal to ${\mathcal{S}}_g=\varphi_{\tau,\mu}$ for some values of $\tau$ and
$\mu$. \end{prop}
\par
\noindent\textit{Proof. } The argument is quite similar to that of \cite[eq.\ (12)]{BrP}, so we will only mention the salient points. One extends $g$ by reflection across the edges of $R_0$ and notes that the composition of reflections along opposite edges in the $\zeta$-plane is a translation of magnitude $2\omega_1$ or $2\omega_2$. By the Chain Rule for Schwarzian derivatives, one sees that ${\mathcal{S}}_g(\zeta)$ is a function which has periods of both of these magnitudes. Due to the fact that $g$ sends the right angles at the vertices $\omega_3/2=(\omega_1+\omega_2)/2$ and $\overline{\omega_3}$ to the right interior angles of $G$, a calculation of the series expansion centered at these vertices shows that $g$ is holomorphic at these vertices. In contrast, due to the angles of $3\pi/2$ at the remaining vertices of $G$, the Schwarzian derivative $S_g$ has double poles at the left vertices of $R_0$, with singularities \[ \frac{-4}{(\zeta+\omega_3/2)^2},\
\frac{-4}{(\zeta+\overline{\omega_3}/2)^2}. \] One checks that the function $\varphi_{\tau,0}$ given explicitly by (\ref{eq:varphi}) is an elliptic function with the same singularities, so the difference is a constant, which is then seen to be real due to the symmetry. \hskip1em\raise3.5pt\hbox{\framebox[2mm]{\ }}
The parameter $\mu$ plays a role similar to that of the parameter of the same name in \cite{BrP}, just as does $\lambda$ for our mappings of the disk, although the Schwarzians are now different functions in the present work.
\subsection{Relation between disk and rectangle Schwarzians}
\begin{prop}
Let $0<t<\pi/2$, let $\tau/i>0$, and let $f\colon\mathbb{D}\to\mathbb{C}$ and
$g\colon R_0\to\mathbb{C}$ be conformal mappings with Schwarzian derivatives \[ S_f(z) = R_{t,\lambda}(z), \quad S_g(\zeta) = \varphi_{\tau,\mu}(\zeta) \] respectively. Suppose that $g(\zeta)=f(z)$ where (\ref{eq:zetafromz}) holds. Then $\tau/i=M(t)$ and
\begin{equation} \label{eq:mufromlambda}
\mu = 16 \lambda \cos t + \frac{3 + \cos2t}{6} . \end{equation} \end{prop}
\par
\noindent\textit{Proof. } The relation between $t$ and $\tau$ is evident by the definition of conformal module. Define \[ \varphi_1(z) = \wp (\zeta+\frac{\omega_1+\omega_2}{2}), \quad
\varphi_2(z) = \wp (\zeta+\frac{\omega_1-\omega_2}{2}), \] so $\varphi_{\tau,0}(\zeta) = -4(\varphi_1(z)+\varphi_2(z))$. From the symmetries and the well-known fact \cite{WW} that $\wp\colon (\mathbb{C}\bmod\{2\omega_1,2\omega_2\})\to\mathbb{C}\cup\{\infty\}$ is a 2-to-1 covering branched at the vertices of $R$, we see that $\varphi_1$ and $\varphi_2$ map $\mathbb{D}$ onto the upper and lower half planes, respectively. By the Reflection Principle, it can be deduced that $\varphi_1$ and $\varphi_2$ are M\"obius transformations. Further, they send the quadruple $(e^{it},-e^{-it},-e^{it},e^{-it})$ to \[ (e_3,\ e_2,\ \infty,\ e_1), \quad (e_2,\ e_3,\ e_1,\ \infty) \] respectively. Note that cross ratios of the above quartets are equal due to the relation $e_1+e_2+e_3=0$. From this it follows that $\varphi_2=(e_2\varphi_1+e^2+e_1e_3)/(\varphi_1-e_2)$, and then a straightforward calculation verifies that \begin{eqnarray*}
\varphi_1(z) &=& e_3 + i(e_1-e_3)(\cot t)\frac{z-e^{it}}{z+e^{it}},\\
\varphi_2(z) &=& e_3 - i(e_1-e_3)(\cot t)\frac{z-e^{-it}}{z+e^{-it}}. \end{eqnarray*} We now substitute these formulas in (\ref{eq:varphi}) to obtain a formula for $\varphi_{\tau,\mu}(\zeta)$ as a function of $z$, \begin{equation}
\varphi_{\tau,\mu}(\zeta) = \frac{ (\frac{8}{3}\cos2t)z^2+
(\frac{40}{3}\cos2t\cos t-8\cos t)z + \frac{8}{3}\cos2t}{z^2+ (2\cos2t)z +1 } + 4\mu. \end{equation}
The Chain Rule for Schwarzian derivatives says that \[ S_f(z) = S_g(\zeta)\left(\frac{E'(z)}{2}\right)^2 + S_E(z) \] and the terms involving $E$ are easily evaluated from the definition (\ref{eq:ellipticintegral}) as in \cite{BrP}. After simplification, we find that \[ S_f(z) =
\frac{ c_0 y +c_1 z + c_2 z^2+c_3 z^3 + c_4z^4 }{ 3 \Delta^2} +\frac{\mu}{\Delta} \] where $c_0=c_4=-2\cos2t$, $c_1=c_3=12(\cos3t-\cos t)$, $c_2=5-\cos4t$, $\Delta=z^4-(2\cos2t)z^2+1$. On comparison with (\ref{eq:Rtlambda})--(\ref{eq:psit}), we arrive at the desired relationship between $\lambda$ and $\mu$. \hskip1em\raise3.5pt\hbox{\framebox[2mm]{\ }}
\section{Formulas for calculating a gear mapping\label{sec:formulas}}
Our numerical work will rest heavily on the following fact.
\begin{figure}
\caption{Structure of pregears as in the proof of Proposition
\ref{prop:pregeartogear}. Note that precisely one of $b^{\pm}$ is
interior to the pregear.}
\label{fig:pregearillustration}
\end{figure}
\begin{prop}\label{prop:pregeartogear}
Let $D$ be a pregear symmetric in $\mathbb{R}$. Suppose that $D$ is not a
gear. Let $p_{-1},p_1\in\partial D$ ($p_{-1}<p_1$) be the extreme
points of the real interval $D\cap\mathbb{R}$ (i.e., the midpoints of
the non-tooth edges of $D$). Let $\rho$ be the common radius of the
tooth edges, and let $p\in\partial D$ be an interior point of
either tooth edge. Let $v$ be the inward pointing unit normal to the tooth
edge at $p$. Define \[ c = p+rv ,\quad d= \left( \rho^2 - (\,{\rm Im}\, c)^2\right)^{1/2},\quad b^\pm=\,{\rm Re}\, c\pm d, \] and \[ T_{b^\pm}(z) = \left\{\begin{array}{rl}
\displaystyle -\frac{z-b^-}{z-b^+} &
\mbox{ if } p_{-1}<b^-<p_1,\\[3ex]
\displaystyle \frac{z-b^+}{z-b^-} &
\mbox{ if } p_{-1}<b^+<p_1.
\end{array}\right.
\]
Then the image $G=T_{b^\pm}(D)$ is a gear with gear center at the
origin. Its gear parameters are given by \begin{eqnarray*}
\beta &=& \frac{T_{b^\pm}(p_1)}{T_{b^\pm}(p_{-1})} ,\\
\gamma &=& \arg T_{b^\pm}(p). \end{eqnarray*} \end{prop}
\par
\noindent\textit{Proof. } By construction, $c$ is the center of a tooth edge. By Proposition \ref{prop:pregearcondition}, the tooth edges must meet, which is impossible unless $|\,{\rm Im}\, c| < \rho$ (Figure \ref{fig:pregearillustration}). Thus $d$ is real (we assume positive), and the intersection points of the tooth edges are thus $b^{\pm}$, with $b^-<b^+$. The definition of $T_{b^\pm}(z)$ is arranged so that $T_{b^\pm}$ leaves $\mathbb{R}\cup\{\infty\}$ invariant, it has its pole exterior to $D$, and $T_{b^\pm}'(x)>0$ for $x\in\mathbb{R}$. The points $b^\pm$ are sent by $T_{b^\pm}$ to $0,\infty$, and hence the images of the edges are straight, so $G$ is a gear by Proposition \ref{prop:pregearcondition}, which further implies that the images of the remaining edges are circles centered at the origin of radii $-T_{b^\pm}(p_{-1})$ and $T_{b^\pm}(p_1)$. Since $T_{b^\pm}(p)$ lies on the upper tooth edge and this edge prolongs to pass through the origin, the formulas stated for $\beta$, $\gamma$ hold. \hskip1em\raise3.5pt\hbox{\framebox[2mm]{\ }}
An operation equivalent to changing the 2-jet at the origin may be effected with the aid of a self-mapping $T_q$ of $\mathbb{D}$ defined by (\ref{eq:Tq}). We use the following, which is verified by a direct application of the Chain Rule.
\begin{prop}\label{prop:2jetcomp}
Let $f$ and $g$ be two mappings such that $J_f(z_0)=(a_0,a_1,a_2)$ and
$J_g(a_0)=(b_0,b_1,b_2)$. Then \[ J_{g\circ f}(z_0) = (b_0, a_1b_1,\ a_1^2b_2+a_2b_1). \] In particular, \[ J_{f\circ T_q}(0)= \left(f(-q),\ (1\!-\!q^2)f'(-q),\ (1\!-\!q^2)\left((1\!-\!q^2)f''(-q)+2q\,f'(-q)\right)\right). \] \end{prop}
The mapping $f\circ T_q$ does not in general fix the origin. To achieve this condition we must use $h=T_{b^\pm}\circ f\circ T_q$ where $T_{b^\pm}$ was defined in Proposition \ref{prop:pregeartogear}. If $p_{-1}<b^-<p_1$, then the 2-jet of the M\"obius transformation $T_{b^\pm}$ at $b^-$ is \[ J_{T_{b^\pm}}(b^-)= \left(0,\ \frac{1}{b^+-b^-},\ \frac{-2}{(b^+-b^-)^2} \right). \] Otherwise, if $p_{-1}<b^+<p_1$, then the 2-jet of the M\"obius transformation $T_{b^\pm}$ at $b^+$ is \[ J_{T_{b^\pm}}(b^+)= \left(0,\ \frac{1}{b^+-b^-},\ \frac{2}{(b^+-b^-)^2} \right). \] We now apply Proposition \ref{prop:2jetcomp} to obtain \begin{eqnarray} \label{eq:2jetzerotozero}
J_h(0) &=& \left(0,\ \frac{1-q^2}{b^+-b^-}f'(-q),\
\frac{\pm 2(1-q^2)^2}{(b^+-b^-)^2}f'(-q)^2 + \right. \nonumber\\
&& \quad\quad\quad \left. \frac{1-q^2}{b^+-b^-}((1-q^2)f''(-q)+2qf'(-q) \right) \end{eqnarray} and as a result we have the following full description of the normalized gear mapping. \begin{prop} \label{prop:zerotozero} Given $t,\lambda$, let $f$ denote
the solution of ${\mathcal{S}}_f=R_{t,\lambda}$ normalized by $J_f(0)=(0,1,0)$.
Suppose that the image $f(\mathbb{D})$ is a pregear. Let $q=-f^{-1}(b^-)$
or $q=-f^{-1}(b^+)$
where the pregear center $b^-\in\mathbb{R}$ or $b^+\in\mathbb{R}$ is as described in Proposition
\ref{prop:pregeartogear}. Then the solution $h$ of the Schwarzian
differential equation ${\mathcal{S}}_h=(R_{t,\lambda}\circ T_q)(T_q')^2$
normalized by the 2-jet of (\ref{eq:2jetzerotozero}) is a gear (not
only a pregear) mapping satisfying $h(0)=0$, $h'(0)>0$. \end{prop}
In order to apply Proposition \ref{prop:zerotozero}, it is necessary to know the radius $\rho$ in Proposition \ref{prop:pregeartogear}. The following result will fill this need.
\begin{prop} \label{prop:curv}
Let $f$ be holomorphic near $z_0=e^{it_0}$, $f'(z_0)\not=0$, and suppose that for $|z|=1$ near $e^{it_0}$, $f(z)$ lies in an arc of a circle of some radius $\rho$. Then the curvature of this arc is
\[ \frac{1}{\rho} = \frac{\pm 1}{|f'(z_0)|}\,{\rm Re}\,\left(1+z_0\frac{f''(z_0)}{f'(z_0)} \right). \] \end{prop}
A simpler version of this result was used in \cite{BrP,KP2}. The choice of sign in this curvature formula corresponds to whether $f(rz_0)$ enters or leaves the circle containing the arc when $r<1$ increases to $r>1$.
\par
\noindent\textit{Proof. } It is sufficient to consider the special case that $z_0=1$ (and then apply the result to $f(e^{it_0}z)$).
Let $c$ denote the center of the circle $|w-c|=\rho$ containing the image arc. By the Reflection Principle, \[ (f(x)-c)(\overline{f(1/x)-c}) = \rho^2 \] for $1-\epsilon<x<1+\epsilon$. From the derivative
\[ f'(x)(\overline{f(1/x)-c}) - {1
\over x^2}(f(x)-c)\overline{f'(1/x)} = 0 \] we see that $f'(1)(\overline{f(1)-c})=(f(1)-c)\overline{f'(1)}$, or equivalently \begin{equation}\label{imarg}
\,{\rm Im}\, \overline{f'(1)}(f(1)-c)=0,\quad \arg(f(1)-c) \equiv \arg f'(1) \bmod \pi. \end{equation} Take a further derivative and evaluate at $x=1$: \[ f''(1)(\overline{f(1)-c})
- 2|f'(1)|^2 + 2\overline{f'(1)}(f(1)-c)+\overline{f''(1)}(f(1)-c) =0, \] from which it follows that
\[ 2\,{\rm Re}\, \overline{f''(1)}(f(1)-c) - 2|f'(1)|^2 + 2\,{\rm Re}\, \overline{f'(1)}(f(1)-c) = 0 \] so
\[ \,{\rm Re}\,( (f''(1)+f'(1)) \overline{(f(1)-c)} ) = |f'(1)|^2. \] As a consequence of (\ref{imarg}), \begin{eqnarray*}
\,{\rm Re}\, (f''(1)+f'(1))\overline{(f(1)-c)} &=& \,{\rm Re}\,\left(
(f''(1)+f'(1))|f(1)-c|\frac{\pm|f'(1)|}{f'(1)}\right) \\
&=& \pm |f(1)-c| \,{\rm Re}\,\left( (f''(1)+f'(1))\frac{|f'(1)|}{f'(1)} \right). \end{eqnarray*} We therefore have
\[ \frac{\pm1}{ |f(1)-c|} = \frac{\,{\rm Re}\,\left((f''(1)+f'(1))\frac{|f'(1)|}{f'(1)}\right) }
{|f'(1)|^2} = \frac{1}{|f'(1)|} \,{\rm Re}\,\left( 1+\frac{f''(1)}{f'(1)}\right). \hskip1em\raise3.5pt\hbox{\framebox[2mm]{\ }} \]
\section{Numerical computation of gear parameters\label{sec:compu}}
\subsection{Auxiliary second-order linear ODE}
The basic facts relating the second-order linear differential equation $2y''+R_{t,\lambda}y=0$ to the Schwarzian derivative of a conformal mapping are widely known \cite{Neh} and have been applied in many contexts. A conformal $f$ mapping with Schwarzian derivative $R_{t,\lambda}$ is obtained as a quotient \begin{equation} \label{eq:y2/y1}
f = \frac{y_2}{y_1} \end{equation}
of any two linearly independent solutions of the differential equation \begin{equation} \label{eq:SL}
2y'' + R_{t,\lambda}y =0 \end{equation} or equivalently, as an antiderivative of $y_1^{-2}$ (we assume $y_1$ nonvanishing).
Thus given parameters $t,\lambda$, it is a straightforward matter to compute $f$ numerically such that \begin{equation} \label{eq:SfRtlambda}
{\mathcal{S}}_f = R_{t,\lambda}. \end{equation} A particular solution will depend on the normalization chosen for $y_1$, $y_2$ at a given base point. In general the image $D=f(\mathbb{D})$ will not be a gear. Assuming it is at least a pregear, in order to apply Proposition \ref{prop:curv} and Proposition \ref{prop:pregeartogear}, we must be prepared to calculate the 2-jet $J_f$ at certain boundary points, which may be accomplished by the relations \begin{equation} \label{eq:2jetfromode}
f=\frac{y_2}{y_1},\quad f'= \frac{1}{y_1^2} \quad
f'' = \frac{-2y_1'}{y_1^3} \end{equation} where we assume that the constant $y_1y_2'-y_2y_1'$ is equal to 1.
\subsection{Computational procedures for the disk\label{subsec:compdisk}}
We describe here some aspects of the calculation for mappings defined in the disk $\mathbb{D}$. Similar considerations apply to mappings defined in the rectangle $R_0$, as well as further observations we will make in \ref{subsec:comprec}. First we discuss calculation along radii.
Consider a fixed value of $t_0$. When we write equation (\ref{eq:SL}) by parametrizing along the radius from $0$ to $z_0=e^{it_0}$, it takes the form \begin{equation} \label{eq:SLt}
\eta''(r) + e^{2it_0} R_{t,\lambda}(re^{it_0})\eta(r) = 0 \end{equation} with $\eta(r)=y(re^{it_0})$. We use $t_0=0,\pi/2,\pi$, i.e., we calculate the values of $f$ along the rays from 0 to $1$, $i$, $-1$. We consider two general approaches to the computation.
\subsubsection{Calculation via the 2-jet.\label{subsubsec:diskradial}} The first, direct approach is simply to solve the equations (\ref{eq:SLt}) numerically along radii. The values of the solutions at $r=1$ together with relation (\ref{eq:2jetfromode}) and Proposition \ref{prop:curv} and Proposition \ref{prop:pregeartogear} permit us to calculate the transformation $T(z)$ which sends the pregear to a gear. Then $F=T\circ f$ is a gear mapping with $S_f=R_{t,\lambda}$.
\subsubsection{Calculation via the spectral parameter power series.\label{subsubsec:diskspps}} In the second approach, using (\ref{eq:Rtlambda})--(\ref{eq:psit}) we rewrite (\ref{eq:SL}) as \begin{equation} \label{eq:SLpsi}
y'' + \psi_{0,t}y = \lambda \psi_{1,t}y \end{equation}
and again integrate along the three rays from 0 to $1$, $i$, $-1$. We are following very closely the notation used in \cite{BrP,KP2}, but now the functions $\psi_0,\psi_1$ are somewhat different since we are considering a different class of quadrilaterals. Then we make use of the SPPS representation of the solutions of (\ref{eq:SLpsi}), as was done in \cite{BrP,KP2}. A summary of the main result establishing this representation is given in the Appendix. The procedure for gear mappings is as follows.
\begin{itemize} \item Calculate the SPPS integrals routinely $\widetilde X^{(n)}$,
$X^{(n)}$ in terms of the data $\psi_{0,t}$, $\psi_{1,t}$. These are
functions on $0\le r\le 1$ which depend on $t$ but not on
$\lambda$. These indefinite integrals produce the coefficients of
power series (\ref{eq:SPPSseries}) in $\lambda$ defining linearly
independent solutions $\eta_1,\eta_2$ of (\ref{eq:SLpsi}) in $\mathbb{D}$,
normalized by 1-jets $(\eta_1(0),\eta_1'(0))$ and
$(\eta_2(0),\eta_2'(0))$ equal to $(1,0)$ and $(0,e^{it_0})$,
respectively. This normalization corresponds via (\ref{eq:y2/y1})
to the conformal mapping $f$ normalized by the 2-jet \[ J_f(0) = (0,1,0) \] at $z=0$. \item Evaluate the coefficients $\Xt{n}$, $\X{n}$ of the power series at $r=1$. This
defines $\eta_1(1)$, $\eta_2(1)$ as functions of $\lambda$ (separately for each
of the three rays). \item Express the 2-jet $J_f$ as a function of $\lambda$ by means of (\ref{eq:2jetfromode}). \end{itemize}
Some variants of these approaches will be mentioned later.
\subsection{Computational procedure for the rectangle \label{subsec:comprec}}
\subsubsection{Condition to make pregear into a gear}
Analogously to the mappings of the disk, the ${\mathcal{S}}_g$ can be expressed in terms of solutions of the ordinary differential equation \begin{equation} \label{eq:odeR0}
2y''(\zeta) + \varphi_{\tau,\mu}(\zeta) y(\zeta) = 0. \end{equation} Two particular solutions are normalized by $J_{y_1}(0)=(1,0)$, $J_{y_2}(0)=(0,1)$. The quotient $g_0=y_2/y_1$, normalized by $J_{g_0}(0)=(0,1,0)$, generally does not have a gear as image; the gear mapping must be sought within the general solution of $S_g=\varphi_{\tau,\mu}$, i.e. \begin{equation}
\label{eq:gdef}
g = \frac{ a\frac{y_2}{y_1}+b}{c\frac{y_2}{y_1}+d}
= \frac{by_1+ay_2}{dy_1+cy_2} \end{equation} with $ad-bc=1$. Thus \[ g(0)=\frac{b}{c},\quad g'(0)=\frac{ad-bc}{d^2},\quad
g''(0)=\frac{-2(ad-bc)c}{d^3} \] We may assume $a,b,c,d\in\mathbb{R}$ because we are only interested in the case that $g'(\zeta)\in\mathbb{R}$ for $\zeta\in\mathbb{R}$. We may further suppose that $g(0)=0$, $g'(0)$=1 without affecting whether the image is a gear or not, so with this simplification we have $b=0$ and $a=d$. Therefore we write \begin{equation} \label{eq:ggeneral}
g = \frac{y_2}{y_1+\alpha y_2} \end{equation} where $\alpha=c/a\in\mathbb{R}$, and the question can be restated as how to choose $\alpha$ to make the pregear a gear.
\begin{prop} \label{prop:gearconditiong}
Let the mapping $g$ be determined by (\ref{eq:ggeneral}) where
$y_1$, $y_2$ are normalized solutions of (\ref{eq:odeR0}) in $R_0$.
Suppose that the image $g_0(R_0)$ of $g_0=y_2/y_1$ is a pregear.
Then $g$ is a gear mapping if and only if\/ $\alpha$ is a root of
the quadratic equation
\[ \left( \,{\rm Im}\, y_2\overline{y_2'}\,|_{\omega_3/2}\right)\alpha^2 +
\left( \,{\rm Im}\,(y_1\overline{y_2'}+ y_2\overline{y_1'})\,|_{\omega_3/2}\right)\alpha +
\,{\rm Im}\, y_1\overline{y_1'}\,|_{\omega_3/2} = 0 \] and the image $g(\mathbb{D})$ is bounded. If the quadratic equation has no roots, then $g_0$ does not even produce a pregear. \end{prop}
\par
\noindent\textit{Proof. } By Proposition \ref{prop:pregearcondition} the condition for $g(R_0)$ to be a gear is that the image of the upper edge of $\partial R_0$ should be straight. Parametrize this edge by $\omega_3/2 - s$ for $s\in[0,\omega_1)$. Since generally $y_1,y_2$ do not take real values here, we introduce an auxiliary function $v(s)=e^{-i\gamma}y(\omega_3/2 - s)$ as the general solution to \[ 2v'' + \varphi(\frac{\omega_3}{2}-s)v =0. \] This differential equation has real coefficients, and the initial values are \[ v(0) = e^{-i\gamma}y(\frac{\omega_3}{2}),\quad
v'(0) = -e^{-i\gamma}y'(\frac{\omega_3}{2}) \] where $y$ is the general solution to (\ref{eq:odeR0}). If we can find $\gamma$ so that both $v(0)$ and $v'(0)$ are real, then we have $v(t)$ real for all $t$. Clearly the condition is that $y'(\omega_3/2)$ \textit{is a real multiple of} $y(\omega_3/2)$. Further, when this holds, we have $y(\omega_3/2-s)=e^{i\gamma}v(s)$, so $\arg y(\omega_3/2-s)$ is constant. Therefore $\arg 1/y^2$ is constant on the top edge of $\partial R_0$.
From (\ref{eq:ggeneral}) we have that $g'=(y_1+\alpha y_2)^{-2}$. Writing $y=y_1+\alpha y_2$, our problem is to choose $\alpha$ so that \[ y_1'+\alpha y_2' \/\mbox{ \it is a real multiple of }\/ y_1+\alpha y_2
\/ \mbox{ \it at } \zeta=\frac{\omega_3}{2}. \] The statement of the Proposition now follows from the fact that given any $z_1,z_2,w_1,w_2\in\mathbb{C}$, $\alpha\in\mathbb{R}$, then $w_1+\alpha w_2$ is a real multiple of $z_1+\alpha z_2$ if and only if \[ (\,{\rm Im}\, z_2 \overline{w_2})\alpha^2 + (\,{\rm Im}\,(z_1 \overline{w_2} + z_2 \overline{w_1}))\alpha + \,{\rm Im}\, z_1 \overline{w_1} =0, \] as can be seen by multiplying the numerator of $(w_1+\alpha w_2)/(z_1+\alpha z_2)$ by the conjugate of the denominator and then taking the imaginary part. \hskip1em\raise3.5pt\hbox{\framebox[2mm]{\ }}
\subsubsection{Computational procedure\label{subsubsec:comprect}}
Now we can specify the numerical implementation of Proposition \ref{prop:gearconditiong}. While (\ref{eq:odeR0}) holds for $\zeta\in R_0$, numerically we solve it first along the real interval $[0,\omega_1/2]$ and evaluate the 1-jets of $y_1$, $y_2$ the endpoint to obtain \begin{equation} \label{eq:yright}
J_{y_1}\left(\frac{\omega_1}{2}\right)=(b_1,b_1'),\quad
J_{y_2}\left(\frac{\omega_1}{2}\right)=(b_2,b_2'). \end{equation} Then we solve the initial value problem
$2u''(s) -\varphi(\omega_1/2 + is) u(s) \ = \ 0$,
$J_{u_1}(0) = (1,0)$, $J_{u_2}(0) = (0,1)$
for $t\in[0,|\omega_2|/2]$, obtaining at the endpoint \begin{equation} \label{eq:uright}
J_{u_1}\left(\frac{|\omega_2|}{2}\right)= (c_1,c_1'), \quad
J_{u_2}\left(\frac{|\omega_2|}{2}\right)= (c_2,c_2'). \end{equation} It follows from this that $y_1(\omega_1/2+is)=b_1 u_1(s) + ib_1'u_2(s)$, $y_2(\omega_1/2+is)=b_2 u_1(s) + ib_2'u_2(s)$, because the left and right sides satisfy identical initial conditions. Evaluate this at
$s=|\omega_2|/2$ using (\ref{eq:yright}), (\ref{eq:uright}) to obtain \begin{eqnarray}
J_{y_1}\left(\frac{|\omega_3|}{2}\right) &=& (b_1c_1+ib_1'c_2,\ b_1'c_2'-ib_1c_1') ,
\nonumber\\
J_{y_2}\left(\frac{|\omega_3|}{2}\right) &=& (b_2c_1+ib_2'c_2,\ b_2'c_2'-ib_2c_1') . \label{eq:ycorner} \end{eqnarray} The values $b_1,b_1',b_2,b_2',c_1,c_1',c_2,c_2'$ are all real. They give in (\ref{eq:ycorner}) the values required in Proposition \ref{prop:gearconditiong}, which can thus be found by two integrations on a real interval and two integrations on the right edge of $\partial R_0$.
\section{Numerical results\label{sec:numerical}}
We give here some specific results of the numerical application of the operations described in the previous section.
\subsection{Computing $\beta$ and $\gamma$ as functions of $t$ and $\lambda$}
\noindent\emph{Radial integration.} The first, most direct method discussed in \ref{subsubsec:diskradial} above is to substitute into (\ref{eq:2jetfromode}) the boundary values obtained by solving (\ref{eq:SLt}) along three specific radii in order to apply Proposition~\ref{prop:curv} and Proposition \ref{prop:pregeartogear}. Going back and solving (\ref{eq:SfRtlambda0}) numerically on any radius, we calculate $f(z)$ for any $z\in\mathbb{D}$. In particular, we have all boundary points of the image gear $f(\mathbb{D})$ as long as we stay away from the prevertices (where $R_{t,\lambda}$ has poles). Then it is easy to find the slope of the tooth edges, the gear center, and the extreme points \begin{figure}
\caption{Pairs $(\gamma,\beta)$ for fixed $t=\pi/n$ with $n=3,4,\dots,10$. Larger dashing indicates larger value of $t$.}
\label{fig:gammabetafixedt}
\end{figure} $f(-1),f(1)$ of the other two edges. From this we have a way of calculating $\beta,\gamma$ as a function of $t,\lambda$. This method of calculation was used to produce Figure \ref{fig:gammabetafixedt}, which suggested Conjecture~3.5 of \cite{BrP2} to the effect that for fixed $\beta$, there are precisely two values of $\gamma$ for each $t$ below a threshold value $t_\beta$.
\noindent\emph{SPPS integrals.} The approach described in \ref{subsubsec:diskspps} was carried out in many examples. To apply Proposition \ref{prop:SPPS}, it is necessary to have a nonvanishing solution of the differential equation. This amounts to having a good initial guess for $\lambda_\infty$, which is a simple matter in the light of formulas (\ref{eq:lambdalimits}) below.
Once this procedure is carried out, we again obtain the data of Proposition \ref{prop:curv} and Proposition \ref{prop:pregeartogear}, but now expressed as a function of $\lambda$. This permits to study the behavior of $\beta$ and $\gamma$ as functions of $\lambda$ (for fixed $t$), and also to solve for $\lambda$ with desired properties. In particular, Figure \ref{fig:kappagraphs} gives an illustration of the curvature $\kappa(\lambda)$ of the tooth edges for a fixed value of $t$. When $\kappa(\lambda)=0$, the mapping $f$ with ${\mathcal{S}}_f=R_{t,\lambda}$ sends the arcs $[z_1,z_2]$ and $[z_3,z_4]$ to subsets of straight lines. However, in general the solution $f$ is not univalent: one must choose here the largest value of $\lambda$. It is interesting that by this method, one can solve for any desired value (not necessarily zero) and thus obtain pregears with prescribed radii for the tooth edges.
\begin{figure}
\caption{Graphs of the curvature $\kappa(\lambda)$ of the tooth edges of the family of pregears corresponding to $R_{t\lambda}$ for $t/\pi=0.1$, $0.2$, $0,3$, $0,4$.}
\label{fig:kappagraphs}
\end{figure}
As described in \ref{subsubsec:diskspps}, we may substitute the SPPS formulas into those of Proposition \ref{prop:pregeartogear}, to obtain the coefficients of the M\"obius transformation $T(z)$ as functions of $\lambda$. Applying this ``symbolic'' $T$ to the SPPS series for $p_{-1}(\lambda)$, $p_1(\lambda)$, $p(\lambda)$, we obtain formulas $\beta(\lambda)$ and $\gamma(\lambda)$ as combinations of power series (one could apply the Cauchy rule for products of series, together with inversion of series, to obtain a single power series in each case, but this is is rather complicated and fortunately is not necessary). An example is shown in Figure \ref{fig:betagamaSPPS}. When combined, $(\gamma(\lambda),\beta(\lambda))$ parametrize a graph such as in Figure \ref{fig:gammabetafixedt}.
\begin{figure}
\caption{Graphs of $\beta(\lambda)$ and $\gamma(\lambda)$ produced by SPPS formulas, for $t=\pi/4$. }
\label{fig:betagamaSPPS}
\end{figure}
Inasmuch as this calculation depends on the values of $R_{t,\lambda}(z)$ for $z$ on the radius from $z=0$ to $z_0=i$ it necessarily involves complex arithmetic. As an alternative to this method, we can also use the other criterion of Proposition \ref{prop:pregearcondition}: that the non-tooth edges be concentric. By integrations along the real axis, which only involve real values of $R_{t,\lambda}$, we can calculate the centers of the circles containing these edges as $f(-1)+1/\kappa_{-1}$ and $f(1)+1/\kappa_{1}$, where $\kappa_{-1}$ and $\kappa_{1}$ are the corresponding curvatures. Subtracting the SPPS formulas for the values of the centers produces a function of $\lambda$ which vanishes when the centers coincide. Numerical experiments indicate that either approach seems to work equally well.
\noindent\emph{Repositioning of gear center.} To avoid possible confusion we reiterate that with the condition $J_f(0)=(0,1,0)$, the value $f(0)$ is not likely to be the gear center $w_0$ of $f(\mathbb{D})$. Therefore such mappings are quite different from the solutions $F$ of the classical integral representation of gear mappings as in \cite{BPea,Goo,Pea}. Consider a general configuration of prevertices $\pm e^{\pm it_1},\pm e^{\pm it_2}$ (recall the discussion at the beginning of \ref{subsec:diskschw}), and let $f$ be the solution of ${\mathcal{S}}_f=R_{t_1,t_2,\lambda}$ normalized by $J_f(0)=(0,1,0)$. Having calculated the data of Proposition \ref{prop:pregeartogear} we can find $w_0$, and presumably having already solved $f=y_2/y_1$ along $[-1,1]$, it is possible now to approximate $p=f^{-1}(w_0)$ by numerical inversion. Then the composition $F=f\circ T_{-p}$ (which has a different 2-jet at the origin) sends 0 to $w_0$. Further, $F(\mathbb{D})=f(\mathbb{D})$. Clearly $S_F=R_{t_1',t_2',\lambda}$ where $T_{-p}(e^{it_1'})=e^{it_1}$, $T_{-p}(e^{it_2'})=e^{it_2}$. Based on numerical examples of this procedure we are led to conjecture the following.
\begin{conj} \label{conj:uniquelambda} Given $t_1,t_2$,
$0<t_1<t_2<\pi/2$, there is a unique $\lambda\in\mathbb{R}$ such that the
solution $f$ of ${\mathcal{S}}_f=R_{t_1,t_2,\lambda}$ normalized by
$J_f(0)=(0,1,0)$ is a gear mapping. \end{conj} This is equivalent to the statement that if there is a conformal mapping of gears $G_{\beta,\gamma}\to G_{\beta',\gamma'}$ respecting the vertices and with 2-jet of the form $(0,r,0)$ at the origin ($r>0$), then $\beta=\beta'$ and $\gamma=\gamma'$.
\noindent\emph{Integration in a rectangle.} The method described in \ref{subsubsec:comprect} is easily applied, integrating from 0 to $\omega_1/2$ and from $\omega_1/2$ to $(\omega_1+\omega_2)/2$ to find the required parameter $\alpha$ for converting the gear to a pregear. In the first integration one may save the values of $y_1,y_2$ at points along $[0,\omega_1]$ and then use them as initial values for integrating upwards or downward on vertical segments passing through those points. An example is shown in Figure \ref{fig:rectanglemap}. The image at the right results from the value of $\alpha$ which produces an unbounded domain. The complement of this unbounded domain is again a gear domain; we know of no relation between this and the bounded image.
\begin{figure}
\caption{ Mapping of rectangle to gear ($\tau=1.5$). }
\label{fig:rectanglemap}
\end{figure}
\subsection{Inverse problem: prescribed gear parameters}\label{inverseproblem}
As occurs in many contexts in conformal mapping, the interesting problem is to find the auxiliary parameters which produce a given geometry. First we consider the following simpler question: Given $t$, determine $\lambda$ so that the image of the gear mapping $f_{t,\lambda}$ has gear ratio $\beta$ (or alternatively, gear angle $\gamma$). Since $t$ is fixed, we can calculate the formula approximating $\kappa(\lambda)$, solve $\kappa(\lambda) = 0$, and then having $t,\lambda$ we determine $\beta$. This can be repeated as necessary, by a process of successive approximations, to make $\beta$ have the desired value within specified accuracy. We have carried out this approach successfully. A more direct method is to use the SPPS formulas for $\beta(\lambda)$, $\gamma(\lambda)$ (recall Figure \ref{fig:betagamaSPPS}), and solve them directly for the desired values of $\beta,\gamma$. However, for extremely small values of $t$ this does not give good results unless a great number of powers are taken in the SPPS formulas.
\begin{figure}
\caption{Experimental region of success of Broyden's method for inverting the
parameter correspondence, applied for a rectangular grid of
$1.1\le\beta\le10.0$ and $0.1\le\gamma\le\pi-0.1$. }
\label{fig:failbroyden}
\end{figure}
The method of Broyden \cite{FKS} may also be used to find zeros of the mapping $(t,\lambda)\to(\beta-\beta_0,\gamma-\gamma_0)$, without recourse to the SPPS formulas. This turns out to be extremely fast (thousands of solutions in less than a second on an ordinary portable computer). However, the Broyden method works by jumping around unpredictably in $\mathbb{R}^2$ from the initial guess for $(t,\lambda)$, and may fail by leaving the $(t,\lambda)$ region where $(\beta,\gamma)$ is well defined. Figure \ref{fig:failbroyden} shows values where this method succeeds starting from the initial guess $(t,\lambda)=(\pi/4,0)$. We will not pursue further the question of improving the initial guess.
\section{Internal structure of the region of gearlikeness}\label{sec:internal}
In \cite{BrP2} it was shown that the region of gearlikeness in the $(t,\lambda)$-plane is \[ \mathcal{G} = \{ (t,\lambda)\colon\ \lambda_t^-<\lambda<\lambda_t^+ \} , \] where \begin{eqnarray} \label{eq:lambdalimits}
\lambda_t^- = -\frac{1}{4} - \frac{1}{16}\left(\cos t +
\frac{1}{\cos t}\right) ,\quad
\lambda_t^+ = \frac{1}{4} - \frac{1}{16}\left(\cos t +
\frac{1}{\cos t}\right). \end{eqnarray}
We use our numerical methods to obtain a very illuminating picture of the structure of $\mathcal{G}$ as related to $(\beta,\gamma)$. Observe that the vertical cross-sections of $\mathcal{G}$ (representing gears of a given conformal modulus $M(t)$) are of common height $\lambda_t^+ - \lambda_t^-=1/2$ for all $t$, and \[ \lambda_0^-=\lim_{t\to0} \lambda_t^- = -\frac{3}{8}; \quad
\lambda_0^+=\lim_{t\to0} \lambda_t^+ = \frac{1}{8}. \]
We use the methods of the previous section for $(\beta,\gamma)\mapsto(t,\lambda)$ to calculate the level curves of the geometric parameters. Figure \ref{fig:gearregionstructure} shows the subsets of $\mathcal{G}$ of constant $\log\beta=$ 0.2, 0.4, 0.6, 0.8, 1.0; 1.25, 1.5, 2.0\dots\ and of constant $\gamma=0.1\pi,\ 0.2\pi,\ \dots, 0.9\pi$.
Individually, the $\beta$-curves accumulate only at the two extreme points $(0,\lambda_0^-)$ and $(0,\lambda_0^+)$, never at interior points $(t,\lambda_t^\pm)$ of the lower and upper boundaries. At these extreme points $\beta\to1$ while $\gamma\to\pi,0$ respectively, and the gear $G_{\beta,\gamma}$ degenerates to a disk in either case. It may appear paradoxical that the limiting Schwarzian derivatives $R_{t,\lambda_t^\pm}(z)$ are not identically zero; however, the pullbacks of the Schwarzian derivatives according to Proposition \ref{prop:zerotozero} by appropriate $T_q$ (with $q$ depending on $\lambda$) do vanish in the limit. For fixed $\gamma$, as $t\to0$ or $t\to\pi/2$ we have $\beta\to\infty$ or $\beta\to1$ respectively. For fixed $\beta$, as $t\to0$ we have $\lambda\to\lambda_0^-$ or $\lambda\to\lambda_0^+$ and then $\gamma\to\pi$ or $\gamma\to0$. This is one of the qualitative results on the conformal module proved in \cite{BrP2}.
\begin{figure}
\caption{Region of gearlikeness $\mathcal{G}$ foliated by $\beta$ level curves (left) and $\gamma$ level curves (right). }
\label{fig:gearregionstructure}
\end{figure}
Each level curve for fixed $\gamma$ intersects the $\lambda$-axis in a value, which might be termed $\lim_{t\to0}\lambda^{[t,\gamma]}$, corresponding to a degenerate gear. As $\beta\to\infty$ for fixed $\gamma$, the gear $G_{\beta,\gamma}$ tends to the union of $\mathbb{D}$ with a full sector of angle $2\gamma$. Letting $t_1\to0$ in the gear mapping, we are led to apply the general formula (4) to this circular triangle with angles $3\pi/2$, $3\pi/2$, $\gamma$ and prevertices $e^{\pm i t_2}$, $1$, to obtain \begin{eqnarray*}
{\mathcal{S}}_f(z) &=& \frac{(2\gamma/\pi)^2(\cos t_2-1)}{(z-1)^2(z^2-(2\cos t_2)z+1)}\\
&&\ \ -\ \frac{ (\cos t_2-1)((5 z^2-14 z+5)\cos t_2+7z^2-10z+7)}
{2(z-1)^2(z^2-(2\cos t_2)z+1)^2} \end{eqnarray*} whereas \begin{eqnarray*}
R_{0,t_2,\lambda}(z) = \frac{8\lambda(\cos t_2-1)}{(z-1)^2(z^2-(2\cos t_2)z+1)} -
\frac{5\sin^2t_2} {2(z-1)^2(z^2-(2\cos t_2)z+1)^2}. \end{eqnarray*} Equating these two Schwarzian derivatives we find that \begin{equation} \label{eq:limitlambda}
\lim_{t\to0}\lambda^{[t,\gamma]} =
\frac{1}{8}\left( 1 - \left(\frac{2\gamma}{\pi}\right)^2\right) \end{equation} independently of $t_2$. This formula is confirmed by Figure \ref{fig:gearregionstructure} (right).
Goodman noted in \cite{Goo} that when one limits the discussion to a particular $\gamma$ (and fixes the normalization as $f(0)=0=f'(0)-1$), a relation is determined between the parameters $t_1$ and $t_2$, stating that he could not calculate it except for the particular case $\gamma=\pi/2$, where he found that \[ \cos t_1=1,\quad \cos t_2=\frac{1}{2}, \] i.e.\ $t_1=0$, $\beta=\infty$ as the outer vertices of the gear have coalesced at $\infty$, while $t_2=\pi/3$. Using Goodman's explicit mapping formula \[ f(z) = \frac{4}{27}\frac{2(1-z+z^2)-2+3z+3z^2-2z^3}{z(1-z)} \] and comparing ${\mathcal{S}}_f$ with $R_{0,\pi/3,\lambda}$, one finds readily that $\lambda=0$, which is thus the value of $\lim_{t\to0}\lambda^{[t,\pi/2]}$, confirming (\ref{eq:limitlambda}) for this case.
\section{Applications and conclusions}\label{sec:appl}
We close with some brief applications of our results on gear mappings.
\subsection{The first Maclaurin series coefficient}\label{subsec:maclaurin}
In \cite{Goo} Goodman left unsolved the problem of calculating the ratio $b_1/f(1)$, where $b_1=f'(0)$ is the first Maclaurin coefficient of the conformal mapping $h$ of $\mathbb{D}$ onto a one-tooth gear domain that maps the origin to the gear center. This ratio was expressed in terms of singular integrals in \cite{Pea} and later in \cite{Br3}, where it is shown that \[ f'(0)=f(1)\int_0^1 \left( \frac{1}{x} -
\frac{\sqrt{1-(\cos t_2)x}}{x\sqrt{1-(\cos t_1)x}\sqrt{1-x^2}} \right)\,dx . \] The integrand presents rather complicated singularities at the endoints of integration.
In the construction of Proposition \ref{prop:zerotozero} we obtained the 2-jet $J_f(0)$ via (\ref{eq:2jetzerotozero}), so we have $f'(0)$. It is also a simple matter to calculate $f(1)$ numerically. Thus one may evaluate integrals of the form given above via solutions of gear mapping problems.
\subsection{Module of the complement of annular rectangle}
Consider the bounded region $A=A_{\beta,\gamma}$ with boundary $ \partial A = \{e^{i\theta}\colon\ \gamma<\theta<2\pi-\gamma\}\cup \{re^{i\gamma}\colon\ 1\leq r \leq \beta^2\} \cup \{\beta^2e^{i\theta}\colon\ \gamma<\theta<2\pi-\gamma\} \cup\{re^{-i\gamma}\colon\ 1\leq r\leq \beta^2\}, $ i.e., an ``annular rectangle'' as in Figure \ref{fig:annularrectangle} subtending an angle $2(\pi-\gamma)$ within an annulus of radii of ratio $\beta^2$. While it is easy to map a disk or rectangle to $A$, it is not known (cf.\ \cite[p.\ 122]{Ku}) how to obtain an expression for the conformal module of the exterior $A^*$ of $A$ in closed form in terms of of the modulus of $A$. (This is the situation with quadrilaterals in general. There has been a surge of interest recently \cite{DP,
HRV1, HRV2, VZ} in the question of numerical calculation of exterior moduli of topological quadrilaterals.) Here we give a numerical solution to this problem.
\begin{figure}
\caption{Complement of annular rectangle is formed of a gear and its reflection along its B-arc. }
\label{fig:annularrectangle}
\end{figure}
\begin{theo}\label{thm:annularrectangle}
Let $A$ be the annular rectangle with inner radius $1$ and outer
radius $\beta^2$ and angle $2(\pi-\gamma)$. The conformal module of
the exterior $A^*=(\mathbb{C}\cup\{\infty\})\setminus \mbox{cl}\,A$ is half the
module of the gear domain with gear angle $\gamma$ and gear ratio
$\beta$. \end{theo} \par
\noindent\textit{Proof. } Consider the conformal mapping $g\colon R_0\to G_{\gamma,\beta}$ of Proposition~\ref{prop:recgear}. The Schwarz reflection applied across the right vertical edge of $R_0$ produces a mapping onto $A^*$ from a rectangle having double the width of $R_0$. \hskip1em\raise3.5pt\hbox{\framebox[2mm]{\ }}
Given an annular rectangle $A_{\beta,\gamma}$, the values $t$ and $\lambda$ corresponding to $\beta$ and $\gamma$ can be computed as described in Section~\ref{inverseproblem}. This means that $f_{t,\lambda}$ is a mapping onto a gear $G_{\beta,\gamma}$ with conformal module $M(t)$. According to Theorem~\ref{thm:annularrectangle}, the module of $A_{\beta,\gamma}$ is therefore calculated numerically to be $M(t)/2$ .
\subsection{Multitooth gears}
Let $f\colon\mathbb{D}\to G_{\beta,\gamma}$ be a conformal mapping to a normalized gear domain, $f(0)=0$. The function $f_n(z) = \sqrt[n]{f(z^n)}$ is a mapping to a regular $n$-toothed gear. Applying the Chain Rule to $P_n\circ f_n = f \circ P_n$ where $P_n(z)=z^n$, and using ${\mathcal{S}}_{P_n}= (1-n^2)z^{-2}/2$ we find
\[ \frac{1-n^2}{2}\frac{f_n'(z)^2}{f_n(z)^2} +{\mathcal{S}}_{f_n}(z) =
{\mathcal{S}}_f(z^n)\, n^2z^{2(n-1)} + \frac{1-n^2}{2}{z^2}. \] Since both ${\mathcal{S}}_f$ and ${\mathcal{S}}_{f_n}$ are rational functions, it follows that $(f_n'/f_n)^2$ is a rational function. With somewhat more work one recovers the formula of \cite{Goo} of which equation (2) of \cite{BrP2} is a particular case.
The image $f_n(G_{\beta,\gamma})$ has gear ratio $\beta^{1/n}$, and each tooth subtends an angle of $2\gamma/n$ with spacing of $2(\pi-\gamma)/n$ between consecutive teeth. By means of these facts it is simple to calculate any desired regular multitooth domain. We illustrate with the example of $n=10$ teeth, having intertooth space equal to the tooth width, and gear ratio arbitrarily chosen as 1.3. Thus we have $\beta=1.3^{10}\approx13.79$, $\gamma\approx\pi/2$. By the methods given in Section~\ref{inverseproblem}, we find that the one-tooth gear with these parameters is obtained by $t\approx0.6024$, $\lambda\approx-0.0029$. The result is in Figure \ref{fig:multitooth}. We stress that the renormalization worked out in Proposition~\ref{prop:zerotozero} is essential, since without the condition that $f(0)=0$ is the gear center, the application of $P_n$ will not work.
\begin{figure}
\caption{Single-tooth gear (left) calculated to generate a multitooth
version of prescribed geometry (center). Detail (right) includes level curves corresponding to
$|z|=0.9,\ 0.91,\dots,\ 0.99,\ 0.991,\dots,.999,\
0.9991,\dots,0.9999$.}
\label{fig:multitooth}
\end{figure} It may be noted that when the number $n$ of teeth is large, the circular arcs are approximated by straight lines, and the gear mapping may be approximated by a Schwarz=Christoffel integral.
\appendix \section{Appendix: SPPS method} The sequence $I_n$ of \textit{iterated integrals} generated by an arbitrary pair of functions $(q_0,q_1)$ is defined recursively by setting $I_0=1$ identically and for $n\ge1$, \begin{equation}
I_n(z) = \int_0^z I_{n-1}(\zeta)\,q_{n-1}(\zeta)\,d\zeta \end{equation} where $q_{n+2j}=q_n$ for $j=1,2,\dots$
\begin{prop}{\rm \cite{KP1}} \label{prop:SPPS}
Let\/ $\psi_0$ and $\psi_1$ be given, and suppose that\/ $y_\infty$ is a nonvanishing solution of \[ y_\infty''+\psi_0\,y_\infty=\lambda_\infty\psi_1\,y_\infty \] on the interval $[0,1]$, where $\lambda_\infty$ is any constant. Choose\/ $q_0=1/y_\infty^2$, $q_1=\psi_1\,y_\infty^2$ and define\/ $\X{n}$, $\Xt{n}$ to be the two sequences of iterated integrals generated by\/ $(q_0,q_1)$ and by\/ $(q_1,q_0)$, respectively. Then for each\/ $\lambda\in\mathbb{C}$ the functions \begin{eqnarray}
y_1 &=& y_\infty \sum_{k=0}^\infty(\lambda-\lambda_\infty)^k
\Xt{2k}, \nonumber\\
y_2 &=& y_\infty \sum_{k=0}^\infty(\lambda-\lambda_\infty)^k
\X{2k+1} \label{eq:SPPSseries} \end{eqnarray} are linearly independent solutions of the equation \begin{equation} \label{eq:y''}
y''+\psi_0y=\lambda\psi_1y \end{equation} on $[0,1]$. Further, the series for $y_1$ and $y_2$ converge uniformly on $[0,1]$ for every $\lambda$. \end{prop}
It is a straightforward matter to obtain the appropriate linear combination of solutions with desired 2-jets at $z=0$, for example $(0,1)$ and $(1,0)$.
\noindent Philip R. Brown\\ Department of General Academics\\ Texas A\&M University at Galveston\\ PO Box 1675, Galveston, Texas 77553 -1675 \\ \texttt{ [email protected] }
\noindent R. Michael Porter \\ Departamento de Matem\'aticas, CINVESTAV--I.P.N.\\ Apdo.\ Postal 1-798, Arteaga 5 \\ Santiago de Queretaro, Qro., 76000 MEXICO \\ \texttt{[email protected]}
\end{document} | arXiv |
A simple and effective algorithm for the maximum happy vertices problem
Marco Ghirardi ORCID: orcid.org/0000-0002-4222-83751 &
Fabio Salassa1
TOP volume 30, pages 181–193 (2022)Cite this article
In a recent paper, a solution approach to the Maximum Happy Vertices Problem has been proposed. The approach is based on a constructive heuristic improved by a matheuristic local search phase. We propose a new procedure able to outperform the previous solution algorithm both in terms of solution quality and computational time. Our approach is based on simple ingredients implying as starting solution generator an approximation algorithm and as an improving phase a new matheuristic local search. The procedure is then extended to a multi-start configuration, able to further improve the solution quality at the cost of an acceptable increase in computational time.
Vertex coloring problems are one of the most popular and extensively studied subjects in the field of graph theory. They have received wide attention in the literature, not only for their real-world applications but also for their theoretical aspects and for the computational hardness (Malaguti and Toth 2010). Traditional vertex coloring problems consist of coloring all vertices of a graph G with different colors in such a way that any pair of adjacent vertices are labeled with different colors. Recently, interest has been also devoted to vertex coloring problems where the coloring of adjacent vertices is desired to be the same. This is the case of the problem called Maximum Happy Vertices Problem (MHV) considered in this paper. Given a set of precolored vertices, the problem asks to extend the coloring to the remaining vertices with the objective to maximize the number of nodes colored with the same color of their adjacent vertices.
The MHV problem and the concept of "happiness" related to vertices have been proposed in Zhang and Li (2015). A vertex is considered happy if all its neighbors are of the same color. The problem objective is the maximization of the number of happy vertices.
More formally, the MHV problem considers an undirected graph \(G = (V, E)\) with n vertices and m edges (with \(\varGamma (i)\) defined as the set of neighbors of vertex i), a color set \(K=\{ 1,\ldots ,k\}\), a subset of vertices \(A \subseteq V\) where \(|A| \ge k\) and a partial coloring \(c : A \rightarrow \{1,\dots ,k\}\) such that \(\forall \ i \in \{1,\dots ,k\}, \exists \ v \in A : c(v) = i\). The problem asks to extend the coloring c to the remaining non-precolored vertices to a complete graph coloring \(\bar{c} : V \rightarrow \{1,\dots ,k\}\) such that the total number of happy vertices is maximized.
In a recent paper Lewis et al. (2019), the MHV problem has been addressed and a solution approach based on the Construct, Merge, Solve & Adapt (CMSA) framework of Blum et al. (2016) has been applied to deal with 380 computationally hard instances.
The problem has also been tackled from a theoretical point of view, see the proof of NP-hardness in Zhang and Li (2015), approximation algorithms in Zhang et al. (2018) and complexity results in Agrawal (2017) and Aravind et al. (2016), where polynomial algorithms for simple special cases have been proposed.
From a computational perspective, to the best of our knowledge, the work of Lewis et al. (2019) is the first attempt to propose solution procedures dealing with large-size instances. Moreover, the authors of Lewis et al. (2019) made freely available both the instance generator Lewis et al. (2018a, b) and the source code (except the part related to the mixed integer linear programming solver GUROBI) Lewis et al. (2019). The work of Lewis et al. (2019) proposes a hybrid heuristic approach, based on a constructive heuristic improved by a matheuristic local search phase.
Matheuristics are solution methods that have been successfully applied to several combinatorial optimization problems [see for instance Ball (2011), Della Croce et al. (2013)], giving rise to an impressive amount of research in recent years. Matheuristics have been applied to routing Macrina et al. (2019) Shahmanzari et al. (2020), packing Billaut et al. (2015), Martinez-Sykora et al. (2017), rostering Della Croce and Salassa (2014), Doi et al. (2018), lot sizing Ghirardi and Amerio (2019) and machine scheduling Della Croce et al. (2014), Croce et al. (2019), Fanjul-Peyro et al. (2017) just to cite a few of them. Matheuristics rely on the general idea of exploiting the strength of both metaheuristic algorithms and exact methods.
In the present work, we developed a simple but effective matheuristic algorithm, along the same line of CMSA in Lewis et al. (2019), to deal with the Maximum Happy Vertices Problem. The proposed matheuristic algorithm is based on an overarching neighborhood search approach with an intensification search phase realized by a MILP solver. The main advantages of our approach, with respect to the one of Lewis et al. (2019), are:
Better performances in terms of solution quality,
Much better performances in terms of computational times (few seconds against 1 h),
Simple design of the solution procedure,
Simple integration in a multi-start version able to further improve the solutions quality.
The paper is organized as follows. In Sect. 2 , the integer linear programming formulations of the problem are provided. Section 3 is devoted to the description of the proposed solution algorithms. In Sect. 4, computational results and benchmarks are presented. Sect. 5 concludes the paper with final remarks.
MIP models
Two mixed integer linear programming formulations are provided in Lewis et al. (2019).
In the first model (M1), integer variables \(x_i \in \{1,\ldots , k\}\) define the color assigned to each vertex, while variables \(y_i\) are forced to be one only if vertex i is unhappy. The set A represents the set of precolored vertices while c(i) is the color assigned to each vertex in A. Recall that \(\varGamma (i)\) is defined as the set of neighbors of vertex i. The overall model is:
$$\begin{aligned}&\displaystyle \max n - \sum _{i=1}^{n} y_i \end{aligned}$$
$$\begin{aligned}&\text{ subject } \text{ to: } \nonumber \\&x_i = c(i)&\qquad \forall v_i \in A \end{aligned}$$
$$\begin{aligned}&y_i \ge \frac{|x_i-x_j|}{n}&\qquad \forall i \in V, \forall j \in \varGamma (i) \end{aligned}$$
$$\begin{aligned}&x_i \in \{1,\ldots , k\}&\qquad \forall i \in V \end{aligned}$$
$$\begin{aligned}&y_i \in \{0,1\}&\qquad \forall i \in V \end{aligned}$$
where (1) maximizes the number of happy vertices, (2) assigns colors to all the precolored vertices, (3) sets \(y_i=1\) for unappy vertices. (4) and (5) define the optimization variables.
Note that constraints (3) are not linear, and hence they require a linearization [not explicitated in Lewis et al. (2019)], which results in their substitution with:
$$\begin{aligned}&y_i \ge \frac{x_i-x_j}{n}&\qquad \forall i \in V, \forall j \in \varGamma (i) \end{aligned}$$
$$\begin{aligned}&y_i \ge \frac{x_j-x_i}{n}&\qquad \forall i \in V, \forall j \in \varGamma (i) \end{aligned}$$
The second model (M2) uses binary variables \(x_{ij}\) where \(x_{ij}=1\) if and only if color j is assigned to vertex i. Variables \(y_i\) have the same meaning as in the first model.
$$\begin{aligned}&\text{ subject } \text{ to: } \nonumber \\&x_{ij} = 1&\qquad \forall i \in A : c(i) = j \end{aligned}$$
$$\begin{aligned}&\sum _{j=1}^{k} x_{ij} = 1&\qquad \forall i \in V \end{aligned}$$
$$\begin{aligned}&y_i \ge |x_{ij}-x_{lj}|&\qquad \forall i \in V, \forall l \in \varGamma (i), \forall j \in K \end{aligned}$$
$$\begin{aligned}&x_{ij} \in \{0,1\}&\qquad \forall i \in V , \forall j \in K \end{aligned}$$
Here, (8) maximizes the number of happy vertices, constraints (9) specifies the precolorings, constraints (10) ensures that one color is assigned to any vertex, and constraints (11) forces \(y_i = 1\) if vertex i is unhappy. (12) and (13) define the optimization variables.
As before, constraints (11) are not linear, and we propose the following linearization:
$$\begin{aligned}&y_i \ge x_{ij}-x_{lj}&\qquad \forall i \in V, \forall l \in \varGamma (i), \forall j \in K \end{aligned}$$
$$\begin{aligned}&y_i \ge x_{lj}-x_{ij}&\qquad \forall i \in V, \forall l \in \varGamma (i), \forall j \in K \end{aligned}$$
We also point out that \(y_i\) variables, for this second model, not necessarily need to be defined as binary. It is, in fact, sufficient to define them as \(0 \le y_i \le 1\) given that they are constrained by (14) and (15) which enforce \(y_i\) to be 0 or 1.
Despite the fact that in Lewis et al. (2019), it is reported that model M2 is far less reliable w.r.t. solution quality, we tested it against all instances using CPLEX 12.7 as MIP solver and found out that 80 instances out of the whole dataset made of 380 instances were solved to optimality. All instances with 250, 500, 750 and 1000 nodes and \(k = 10\) had been solved to optimality within the time limit of 3600 s, the same time limit used in Lewis et al. (2019). Thus, in our experiments, model M2 outperforms model M1. From now on, in our algorithms, we use model M2 for benchmarks since, overall, it gives better solutions within the same time limit with respect to the model M1.
A simple solution approach
We propose here a simple but effective matheuristic improvement approach. Starting from a given solution, the algorithm iteratively improves it with a scheme based on the neighborhood search approach. Each iteration explores the neighborhood by constructing a problem where the variables to be optimized refer to a subset of the variables of the original problem, while other ones are fixed to the value they have in the current solution. The detailed procedure is described in Algorithm 1. The algorithm starts with a given feasible solution \(\bar{c}\) (step 1). A counter \(no\_improvement\) of iterations passed without finding an improving solution is set to 0 (step 2). At each iteration of the main loop (cycle 3–16), a subset of candidate colors for each node is selected and an exact method is employed to build a possibly improved solution. In the current solution (steps \(4--8\)), candidate colors for each node i are the current color and all the colors assigned to nodes adjacent to i with a path of length L, i.e. nodes colors that can be recognized following an \(L-length\) edge path. The resulting problem is then optimally solved through model (8)–(13), obtaining solution \(\bar{c}'\) (step 9). If the new solution is better than the previous one, the counter of non-improving iterations is reset to 0 (step 11). Otherwise, it is increased by one (step 13). Note that solution \(\bar{c}'\) cannot be worse than the current solution \(\bar{c}\) because the latter is a feasible solution of the ILP model. Hence, it is always accepted as the new current solution (step 15). The improvement phase is repeated if less than S iterations have been performed without improving the current solution.
CMSA solution approach proposed in Lewis et al. (2019) is based on a similar improvement scheme as the one we propose, with a different neighborhood definition. We highlight here the main differences:
In CMSA, candidate colors for each node i to be chosen for reoptimization are only the color of i plus, with a given probability, a subset of the colors of the neighbor nodes of i, while our matheuristic procedure considers, as possible candidates for each node i, all colors of the nodes that could be reached from node i with a path of a given length. Hence, the neighborhood dimension of the proposed algorithm is larger than the one of CMSA.
CMSA uses model M1, while in our case, model M2 has been selected. This choice does not affect the algorithm results in terms of solution quality (all ILPs are solved to optimality) but influences the running time.
In Lewis et al. (2019), two constructive methods are proposed for the initial solution generation, namely \(Greedy-MHV\) and \(Growth-MHV\). We point out that \(Greedy-MHV\) is the same procedure as the approximation algorithm \(\mathcal {G}\) proposed in Zhang et al. (2018). The approximation algorithm \(\mathcal {G}\) has been used in our approach as starting solution. During preliminary testings, we tried the best among algorithm \(\mathcal {G}\) (a.k.a. \(Greedy-MHV\)) and \(Growth-MHV\), but we experienced no improvements in the solutions quality.
The rationale of algorithm \(\mathcal {G}\) of Zhang et al. (2018) is to label all the uncolored vertices with the same color and testing all possible k colors obtaining, in such way, k different vertices colorings. The starting solution is then chosen among all k colorings, i.e. the one exhibiting the largest number of happy vertices (Algorithm 2).
To further test the matheuristic improvement algorithm in order to assess the quality of the proposed approach, we tested it in a multi-start setting. In this new configuration, the improvement procedure depicted in Algorithm 1 is applied not only on the solution obtained by the best coloring of algorithm \(\mathcal {G}\), but on all possible k different vertices colorings. The best final result is then returned. Algorithm 3 resumes the main steps of the multi-start procedure.
Computational experiments
We decided to test different configurations of algorithms over a dataset generated thanks to the instance generator in Lewis et al. (2018a, b). According to Lewis et al. (2019), instances were generated as random graphs using values of \(p = 5 / (n - 1)\), where n is the total number of nodes and p the probability of two vertices being adjacent. This value of p induces an average vertex degree of 5. In all instances, \(10\%\) of the vertices were precolored. Authors of Lewis et al. (2019) state that these configurations lead to the creation of the most difficult-to-solve instances. As in Lewis et al. (2019), we considered classes of instances with a number of colors k equal to 10 or 50 on graphs having a number of nodes n of 500, 750, 1000, 2000, 3000, 4000, 5000, 7500 and 10000. Since the solver is able to solve to optimality four classes (namely all the ones with \(k=10\) and with n equal to 250, 500, 750 and 1000), these were not considered in our dataset. For each of the remaining 15 classes, 20 instances were generated. For tuning the algorithm parameters, we generated an additional smaller dataset, composed of 10 instances for each of the classes with k equal to 10 or 50 and n equal to 3000, 5000 and 10000.
The algorithms have been implemented in C++ and the source code is available upon request to the authors. All tests have been performed on an i5-8500 3 GHz CPU system with 16 GB of RAM and CPLEX 12.7 as MIP solver. CPLEX solver has been applied with no parameters tuning and in multi-threaded mode.
The following two subsections present the results of the experiments aiming to tune the algorithm parameters, and the comparison between the results of the proposed algorithms and CMSA, proposed in Lewis et al. (2019).
Parameter tuning
In order to tune the values of parameters L and S of Algorithm 1, a set of computational experiments has been performed.
Table 1 summarizes the results. For each class of instances (k colors and n nodes), we present the average percentage of happy nodes \(H\%\) and computational time T over the 10 tuning instances, with different parameters values L and S. The best entries for each line are highlighted in bold.
Parameter L defines the neighborhood size, and has been considered equal to 1 or 2. Setting a value of 3 or more will result in the creation of ILP models with too many free variables, sometimes exceeding a time limit of 3600 s without finding the optimal solution. It is clear from the table that the best choice is \(L=2\), having better results at cost of an acceptable increase of computational time.
Paramenter S configures the algorithm stopping criterion and ranges from 1 to 3. While an improvement is clear in results obtained increasing S from 1 to 2, the results are, for most instances, the same when \(S=3\). Hence, we decided to set \(S=2\).
Table 1 Improvement procedure algorithm parameter tuning
Algorithms results comparison
As previously pointed out, authors of Lewis et al. (2019) made freely available the source code of their algorithms except the part related to the mixed integer linear programming model and solver. Then, in order to benchmark our procedure with the reference algorithm CMSA, we re-implemented it, integrating their source code with a mixed integer linear programming model. In the description of CMSA, it is not clear how single-color labels could be efficiently excluded from the list of possible colors since the variables used are of integer type (model M1 is used) and no constraints sets (i.e. disjunctive constraints) have been explicitated to deal with values exclusion. Hence, we contacted the authors of Lewis et al. (2019) asking for details on CMSA implementation which is slightly different with respect to the published paper. Thanks to their help we reconstructed CMSA as originally implemented. Each time the LP model M1 is run, the following rules are used:
If a node \(\bar{i}\) has only one candidate color \(\bar{c}\), the corresponding variable is set to that color (\(x_{\bar{i}} = \bar{c}\)).
If a node \(\bar{i}\) has more candidate colors, the corresponding variable is left free to get any value (\(x_{\bar{i}} \in \{1,...,k\}\)).
For other details about CMSA refer to Lewis et al. (2019). On the other side, excluding values implying model M2 as in our matheuristic is rather simple: it is, in fact, sufficient to add constraints like \(x_{\bar{i}\bar{j}} = 0\) if we want to prevent node \(\bar{i}\) to be labeled with color \(\bar{j}\).
We tested the following approaches:
CPLEX: Lower Bound and Upper Bound after 3600 s calculated by CPLEX solver with model M1.
CMSA: original CMSA using as starting solution the best among \(Greedy-MHV\) and \(Growth-MHV\) with a time limit of 3600 s [as in Lewis et al. (2019)]. Considering that CMSA is not a deterministic algorithm, we present here the best result obtained with 10 different executions.
MH-G: matheuristic algorithm 1, configured with \(L=2\) and \(S=2\), using as starting solution the approximation Algorithm \(\mathcal {G}\).
MS: multi-start version of the procedure, depicted in algorithm 3.
Table 2 summarizes the results. The meaning of the columns of Table 2 is the following:
Column 1: number of different colors k.
Column 2: number of nodes n of the specific class of instances.
Column 3: percentage of "happy" vertices w.r.t the total number of nodes of the upper bound provided by CPLEX after 3600 s of run.
Column 4: percentage of "happy" vertices w.r.t the total number of nodes of the lower bound provided by CPLEX after 3600 s of run.
Column 5: average values of the percentage of "happy" vertices given by the CMSA approach after 3600 s—best of 10 executions.
Column 6: average values of the percentage of "happy" vertices given by the proposed \(MH-G\) algorithm (bold characters if \(MH-G\) is better than CMSA).
Column 7: average maximum CPU time needed to compute the result of \(MH-G\) algorithm, in seconds.
Column 8: average values of the percentage of "happy" vertices given by the MS configuration of the proposed algorithm (bold characters if MS is better than \(MH-G\)).
Column 9: average maximum CPU time needed to compute the result of MS procedure, in seconds.
Table 2 Computational results: algorithms results comparison
Table 3 Number of improved instances with respect to CMSA algorithm
As can be seen, the simple proposed approach \(MH-G\) outperforms CMSA both in terms of solution quality and CPU effort. We recall that the stopping criterion used in Lewis et al. (2019) is the time limit of 3600 s. Our approach gives better results in about two order of magnitude less CPU time. Moreover with algorithm MS, we gain even more solution quality, largely within the 3600 s limit. These results illustrate the effectiveness of our approach which shows up to improve with respect to the current literature.
To further assess the effectiveness of our approaches, Table 3 is reported. Even if the averages improvements of objectives function values may seem limited, the number of improvements is definitely clear. Here, Columns 1 and 2 are the same as in Table 3, while column 3 explicits the number of instances per class. Columns 4 and 5 are dedicated to enlight the number of instances improved with respect to the CMSA procedure of \(MH-G\) and MS algorithms, respectively. As can be seen, apart from one case where the number of improved instances is very limited, \(MH-G\) (and consequently MS) approaches consistently improve over CMSA. It is important to note that there are no instances where CMSA is better than any of our approaches. Globally, we could improve 180 out of 300 instances and we point out that our approach is consistently better on the larger-size instances. This again confirms the effectiveness of the proposed approach.
A simple procedure has been developed to deal with the Maximum Happy Vertices Problem. A starting solution generation obtained thanks to an approximation algorithm is improved via a large-scale neighborhood exploration made with an MILP formulation of the problem. The procedure is then extended in a multi-start configuration. Both approaches have been tested over 300 instances from the literature and compared with a reference algorithm, namely CMSA from Lewis et al. (2019). Solution quality and very limited running times confirm the effectiveness of our approach which is based on simple elements and shows up to improve with respect to the current literature.
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Open access funding provided by Politecnico di Torino within the CRUI-CARE Agreement.
DIGEP, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy
Marco Ghirardi & Fabio Salassa
Marco Ghirardi
Fabio Salassa
Correspondence to Marco Ghirardi.
Ghirardi, M., Salassa, F. A simple and effective algorithm for the maximum happy vertices problem. TOP 30, 181–193 (2022). https://doi.org/10.1007/s11750-021-00610-4
Issue Date: April 2022
Happy coloring
Matheuristics
Mathematics Subject Classification
90C27 Combinatorial Optimization
90C11 Mixed Integer Programming
90C59 Approximation methods and heuristics in mathematical programming | CommonCrawl |
Hanan grid
In geometry, the Hanan grid H(S) of a finite set S of points in the plane is obtained by constructing vertical and horizontal lines through each point in S.
The main motivation for studying the Hanan grid stems from the fact that it is known to contain a minimum length rectilinear Steiner tree for S.[1] It is named after Maurice Hanan, who was first[2] to investigate the rectilinear Steiner minimum tree and introduced this graph.[3]
References
1. Martin Zachariasen, A Catalog of Hanan Grid Problems Networks, vol. 38, 2000, pp. 200-221
2. Christine R. Leverenz, Miroslaw Truszczynski, The Rectilinear Steiner Tree Problem: Algorithms and Examples using Permutations of the Terminal Set, 1999 ACM Southeast Regional Conference, 1999, doi:10.1145/306363.306402
3. M. Hanan, On Steiner's problem with rectilinear distance, J. SIAM Appl. Math. 14 (1966), 255 - 265.
| Wikipedia |
We establish the optimal quantization problem for probabilities under constrained Rényi-$\alpha$-entropy of the quantizers. We determine the optimal quantizers and the optimal quantization error of one-dimensional uniform distributions including the known special cases $\alpha = 0$ (restricted codebook size) and $\alpha = 1$ (restricted Shannon entropy).
A. Gersho and R. M. Gray: Vector Quantization and Signal Compression. Kluwer, Boston 1992.
G. H. Hardy, J. E. Littlewood and G. Polya: Inequalities. Second edition. Cambridge University Press, Cambridge 1959. | CommonCrawl |
Less-is-more effect
The less-is-more effect refers to the finding that heuristic decision strategies can yield more accurate judgments than alternative strategies that use more pieces of information. Understanding these effects is part of the study of ecological rationality.
Examples
One popular less-is-more effect was found in comparing the take-the-best heuristic with a linear decision strategy in making judgments about which of two objects has a higher value on some criterion. Whereas the linear decision strategy uses all available cues and weighs them, the take-the-best heuristic uses only the first cue that differs between the objects. Despite this frugality, the heuristic yielded more accurate judgments than the linear decision strategy.[1]
Beyond this first finding, less-is-more effects were found for other heuristics, including the recognition heuristic[2] and the hiatus heuristic.[3]
Explanations
Some less-is-more effects can be explained within the framework of bias and variance. According to the bias-variance tradeoff, errors in prediction are due to two sources. Consider a decision strategy that uses a random sample of objects to make a judgment about an object outside of this sample. Due to sampling variance, there is a large number of hypothetical predictions, each based on a different random sample. Bias refers to the difference between the average of these hypothetical predictions and the true value of the object to be judged. In contrast, variance refers to the average variation of the hypothetical judgments around their average.[4]
Determinants of variance
The variance component of judgment error depends on the degree to which the decision strategy adapts to each possible sample. One major determinant of this degree is a strategy's number of free parameters. Therefore, (heuristic) strategies that use with fewer pieces of information and have fewer parameters tend to have lower error from variance than strategies with more parameters.[5]
Determinants of bias
At the same time, fewer parameters tend to increase the error from bias, implying that heuristic strategies are more likely to be biased than strategies that use more pieces of information. The exact amount of bias, however, depends on the specific problem to which a decision strategy is applied. If the decision problem has a statistical structure that matches the structure of the heuristic strategy, the bias can be surprisingly small. For example, analyses of the take-the-best heuristic and other lexicographic heuristics have shown that the bias of these strategies is equal to the bias of the linear strategy when the weights of the linear strategy show specific regularities[6][7] that were found to be prevalent in many real-life situations.[8]
References
1. Czerlinski, Jean; Goldstein, Daniel G.; Gigerenzer, Gerd (1999). "How good are simple heuristics?". Simple Heuristics that make us smart. New York: Oxford University Press. pp. 97–118.
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\title[Shifted moments of L-functions and moments of theta functions] {\bf Shifted moments of L-functions and moments of theta functions} \author{Marc Munsch} \address{CRM, Universit\'e de Montr\'eal, 5357 Montr\'eal, Qu\'ebec } \email{[email protected]}
\date{\today}
\subjclass[2000]{11M06, 11M26, 11L20} \keywords{$L$- functions, moments, Theta function, Generalized Riemann Hypothesis}
\begin{abstract}
Assuming the Riemann Hypothesis, Soundararajan showed in \cite{SoundRiemann} that $\displaystyle{\int_{0}^{T} \vert \zeta(1/2 + it)\vert^{2k} \ll T(\log T)^{k^2 + \epsilon}}$ . His method was used by Chandee \cite{ChandeeShifts} to obtain upper bounds for shifted moments of the Riemann Zeta function. Building on ideas of \cite{ChandeeShifts} and \cite{SoundRiemann}, we obtain, conditionally, upper bounds for shifted moments of Dirichlet $L$- functions which allow us to derive upper bounds for moments of theta functions.
\end{abstract}
\maketitle
\section{Introduction}
For any integer $q$, we denote by $X_q$ the group of multiplicative characters modulo $q$. Denote by $X_q^+$ the subset of $X_q$ consisting of primitive even characters $\chi$ (those satisfying $\chi(-1) = 1$) and $X_q^-$ the subset of $X_q$ consisting of primitive odd characters $\chi$ (those satisfying $\chi(-1) = -1$). Furthermore, we use $X_q^*$ to denote the set of primitive characters modulo $q$.
This paper is divided into two parts. Firstly, we study shifted moments of Dirichlet $L$-functions and secondly, we apply this study to obtain upper bounds on moments of theta functions.
A standard problem in analytic number theory is the study moments of the Riemann zeta function or more generally $L$-functions on the critical line. For instance, it is conjectured (see~\cite[Chapter~5]{MM}) that the moments at the central point satisfy the following asymptotic formulas: \begin{equation} \label{eq:Conj L} M_{2k}(q) =\sum_{\chi\in X_q^*} \vert L(1/2,\chi )\vert^{2k} \sim C_k q\log^{k^2}q,\hspace{1cm} C_k>0. \end{equation} Even though the asymptotic formulas are not known for $k\geq 3$, lower bounds of the expected order of magnitude $$\sum_{\chi\in X_q^*} \vert L(1/2,\chi )\vert^{2k} \gg q\log^{k^2}q,$$ have been given by Rudnick and Soundararajan~\cite{RS1} for $q$ prime. Assuming the Generalized Riemann Hypothesis and building on Soundararajan work \cite{SoundRiemann}, we can show that $\displaystyle{M_{2k}(q) \ll q\log^{k^2+\epsilon}q.}$ We can generalize in some way these moments using shifts and consider
\begin{equation} \label{shifts}\sum_{\chi \in X_q^*} \left\vert L\left(\frac{1}{2}+ it_1,\chi\right)\cdots L\left(\frac{1}{2}+ it_{2k},\chi\right)\right\vert \end{equation} where $(t_1,\cdots,t_{2k})$ is a sequence of real numbers. It is expected that if the $t_i$ are reasonably small, we should be able to obtain an asymptotic formula for (\ref{shifts}) (see for instance \cite{Conrey} for a survey about shifted moments in families of $L$- functions). Although, we cannot prove such a general result even assuming GRH, we are able to give a conditional upper bound of nearly the conjectured order of magnitude.
\begin{theorem}\label{shiftedmoments}Assume that the Dirichlet $L$-functions modulo $q$ satisfy the Generalized Riemann Hypothesis. Suppose $q$ is large and the $2k$-tuple $t=(t_1,\cdots, t_{2k})$ is such that $t_i \ll \log q$. Then, for all $\epsilon>0$, we have the uniform bound:
$$ \sum_{\chi \in X_q^*} \left\vert L\left(\frac{1}{2}+it_1,\chi\right)\cdots L\left(\frac{1}{2}+it_{2k},\chi\right)\right\vert \ll \phi(q)(\log q)^{k/2+\epsilon} \prod_{i<j} E_{i,j}$$where
$$E_{i,j}=\begin{cases}\left(\min\left\{\frac{1}{\vert t_i-t_j\vert}, \log q\right\}\right)^{1/2} &\mbox{if } \vert t_i-t_j\vert \leq \frac{1}{100}, \\ \sqrt{\log \log q}&\mbox{if }\vert t_i-t_j\vert \geq \frac{1}{100}.\\ \end{cases}$$
\end{theorem} This can be related to the main result of \cite{ChandeeShifts} and enlightens the fact that $L\left(\frac{1}{2}+it_i,\chi\right)$ and $L\left(\frac{1}{2}+it_j,\chi\right)$ are essentially correlated when $\vert t_i -t_j\vert \approx \frac{1}{\log q}$ and ``independent" as long as $\vert t_i - t_j \vert$ is significantly larger than $1/\log q$.
For real $x> 0$ and $\eta\in\{ 0,1\}$ we set $$\theta (\eta, x,\chi) =\sum_{n=1}^\infty \chi (n)n^{\eta}e^{-\pi n^2x/q}, \qquad \chi\in X_q. $$ We note that, if we set $\eta_{\chi}=1$ if $\chi$ is odd and $\eta_{\chi}=0$ otherwise, then $$\theta_{q} (\eta_{\chi}, x,\chi)=\theta_{q} (x,\chi) $$ is the classical theta-function of the character $\chi$ (see~\cite{Dav} for a background and basic properties). We can express these values using Mellin transforms of $L$- functions which make the use of our result about moments of shifted $L$- functions very appropriate.
When computing the root number of $\chi$ appearing in the functional equation of the associated Dirichlet $L$- function, the question of whether $\theta_{q} (1,\chi) \neq 0$ appears naturally (see~\cite{Lou99} for details). Numerical computations led to the conjecture that it never happens if $\chi$ is primitive (see~\cite{CZ} for a counterexample with $\chi$ unprimitive). In order to investigate the non-vanishing of theta functions at their central point, the study of moments has been initiated in \cite{LoMu}, \cite{LoMu1} and \cite{MS}. Let us define $$S_{2k}^+(q) =\sum_{\chi\in X_q^+\backslash \chi{_0}}\vert\theta (1,\chi)\vert^{2k} \qquad\mbox{and}\qquad S_{2k}^-(q) =\sum_{\chi\in X_q^-}\vert\theta (1,\chi)\vert^{2k}.$$ It is conjectured in \cite{MS}, based on numerical computation and some theoretical support, that \begin{equation} \label{eq:Conj TS} \begin{split} &S_{2k}^+(q)\sim a_k\phi(q)q^{k/2}\(\log q\)^{(k-1)^2},\\ & S_{2k}^-(q)\sim b_k\phi(q)q^{3k/2}\(\log q\)^{(k-1)^2} \end{split} \end{equation} for some positive constants $a_k$ and $b_k$, depending only on $k$. Recently, a lower bound of expected order for $S_{2k}^+(q)$ and $S_{2k}^-(q)$ has been proven unconditionally in \cite{MS}. In the second part of the paper, we will derive upper bounds giving good support towards Conjecture (\ref{eq:Conj TS}).
\noindent Precisely, we prove
\begin{theorem}\label{upperGRH} Assume the Generalized Riemann Hypothesis for all Dirichlet $L$- functions modulo $q$. Then, for all $\epsilon>0$, we have $$S_{2k}^+(q) \ll \phi(q)q^{k/2}(\log q)^{(k-1)^2+\epsilon} \qquad\mbox{and}\qquad S_{2k}^-(q) \ll \phi(q)q^{3k/2}(\log q)^{(k-1)^2+\epsilon}.$$
\end{theorem}
This can be related to recent results of~\cite{HarperMaks} (see also~\cite{Heap}), where the authors obtain the asymptotic behaviour of a Steinhaus random multiplicative function (basically a multiplicative random variable whose values at prime integers are uniformly distributed on the unit circle). This can be viewed as a random model for $\theta_{q} (x,\chi)$. In fact, the rapidly decaying factor $e^{-\pi n^2/q}$ is mostly equivalent to restrict the sum over integers $n \le n_0(q)$ for some $n_0(q) \approx \sqrt{q}$ and the averaging behavior of $\chi(n)$ with $n\ll q^{1/2}$ is essentially similar to that of a Steinhaus random multiplicative function. Hence, these results are a good support for Conjecture~\eqref{eq:Conj TS}. Upper bounds of Theorem \ref{upperGRH} together with lower bounds obtained in \cite[Theorem $1.1$]{MS} confirm this heuristic.
The method of the proof of Theorem \ref{upperGRH} relies on the bound obtained for moments of shifted $L$- functions.
\section{Moments of shifted L-functions}
In that section, we mostly adapt results and ideas of \cite{SoundRiemann} to our situation. These techniques build on ideas of Selberg about the distribution of $\vert \log \zeta(1/2+it)\vert$ (see \cite{Selberg}).The starting point is the following equality
$$ \int_{T}^{2T} \vert \zeta(1/2 + it)\vert^{2k} dt = -\int_{-\infty}^{+\infty} e^{2kV} d \text{ meas}(S(T,V))=2k\int_{-\infty}^{+\infty} e^{2kV} \text{meas}(S(T,V)) dV$$ where $S(T,V)=\left\{t\in \left[T,2T\right]: \log \vert \zeta(1/2+it)\vert \geq V\right\}$. From this, an upper bound for the moment can be directly deduced from the upper bound of $\text{meas } (S(T,V))$. In our case, we have to study the frequency (in terms of characters) of large values of $L$-functions. Thus, we will proceed in the same way by estimating the measure of $$S_t(q,V)=\left\{\chi (\bmod \,q), \chi^2 \neq \chi_0,: \log\left\vert L\left(\frac{1}{2}+it_1,\chi\right)\right\vert+\cdots+\log\left\vert L\left(\frac{1}{2}+it_{2k},\chi\right)\right\vert \geq V\right\}$$ for $V>0$ and a $2k$-tuple $t=(t_1,\cdots, t_{2k})$. Most of the work consists in keeping the dependence both in terms of the modulus $q$ and the height of the shifts. If the shifts are not too large, we are able to obtain a quasi-optimal upper bound under the Generalized Riemann Hypothesis. This result will be sufficient for our application to moments of theta functions. It should be noticed that the recent method of Harper (see \cite{Harper}) may be used to remove the $\epsilon$ factor in Theorem \ref{shiftedmoments}.
Let set $N_t(q,V)= \# S_t(q,V)$. We can express the shifted moments of $L$- functions as the following
\begin{align}\label{momentshifted} \sum_{\chi \in X_q^*}\left\vert L\left(\frac{1}{2}+it_1,\chi\right)\cdots L\left(\frac{1}{2}+it_{2k},\chi\right)\right\vert = \sum_{\chi \in X_q^*} e^{\log\left\vert L(\frac{1}{2}+it_1,\chi)\right\vert+\cdots+\log\left\vert L(\frac{1}{2}+it_{2k},\chi)\right\vert} \notag \\ =\sum_{\chi \in X_q^*}\int_{-\infty}^{\log\left\vert L(\frac{1}{2}+it_1,\chi)\right\vert+\cdots+\log\left\vert L(\frac{1}{2}+it_{2k},\chi)\right\vert} e^{V}dV=\int_{-\infty}^{+\infty} e^{V} N_t(q,V)dV + q^{o(1)}. \end{align} The error term comes from the contribution of quadratic characters which can easily be bounded, using Corollary \ref{upperboundline} by $$O\left(\int_{-\infty}^{4ck\frac{\log q}{\log \log q}}e^V dV \right) \ll q^{o(1)}.$$ Hence, the problem of estimating the moments boils down to getting precise bounds for $N_t(q,V)$. In order to do that, let us define the following quantity
$$W=2k \log\log q + 2\sum_{i,j \atop i<j}F_{i,j}$$ where
$$F_{i,j}=\begin{cases}\log\left(\min\left\{\frac{1}{\vert t_i-t_j\vert}, \log q\right\}\right) &\mbox{if } \vert t_i-t_j\vert \leq \frac{1}{100}, \\ \log \log \log q&\mbox{if }\vert t_i-t_j\vert \geq \frac{1}{100}.\\ \end{cases}$$
We will prove the following theorem which estimates the measure of $S_t(q,V)$ for large $q$ and all $V$.
\begin{theorem}\label{largevalues}Assume that the Dirichlet $L$-functions modulo $q$ satisfy the Generalized Riemann Hypothesis. Suppose that $\vert t\vert\leq T \leq \log^A q$ where $A>0$ and $V$ is a large real number. If $4\sqrt{\log \log q}\leq V\leq W$ then
$$N_t(q,V) \ll \phi(q)\frac{V}{\sqrt{W}}\exp\left(-\frac{V^2}{W}\left(1-\frac{18k}{5\log W}\right)^2\right);$$ if $W<V<\frac{1}{4k}W\log W$ we have $$N_t(q,V)\ll \phi(q)\frac{V}{\sqrt{W}}\exp\left(-\frac{V^2}{W}\left(1-\frac{18kV}{5W\log W}\right)^2\right); $$ and if $\frac{1}{4k}W\log W<V$ we have $$N_t(q,V)\ll \phi(q)\exp\left(-\frac{V}{801k}\log V\right) .$$ \end{theorem}
\textit{\bf{Proof of Theorem \ref{shiftedmoments}}} Inserting the bounds of Theorem \ref{largevalues} in Equation (\ref{momentshifted}) gives the upper bound in Theorem \ref{shiftedmoments}. Precisely, it is appropriate for this computation to use Theorem \ref{largevalues} in the weakest form
$$N_t(q,V) \ll \phi(q)(\log q)^{o(1)} exp(-V^2/W) \text{ for } 3\leq V\leq 200W, $$ $$N_t(q,V) \ll \phi(q)(\log q)^{o(1)} exp(-2V) \text{ for } V> 200W .$$ This allows us to bound the moments by $\phi(q)(\log q)^{o(1)}e^{W/4}$ which concludes the proof.
\subsection{Preliminary results}
We regroup in that subsection all the technical results that we will use in the proof of Theorem \ref{largevalues}. These are mainly suitable adaptations to our case of Lemmas of \cite{SoundRiemann}. In the sequel, we will always write $s=\sigma + it$ for a complex number $s$. We write $\log^{+}(x):=\max (\log x,0)$.
\begin{lemma} \label{upsigma} Unconditionally, for any $s$ not coinciding with $1$, $0$ or a zero of $L(s,\chi)$, and for any $x \geq 2,$ we have \begin{eqnarray*} -\frac{L'}{L}(s,\chi) &=& \sum_{n \leq x} \frac{\chi(n)\Lambda(n)}{n^{s}}\frac{\log\frac{x}{n}}{\log x} + \frac{1}{\log x}\left( \frac{L'}{L}(s,\chi) \right)' + \frac{1}{\log x} \sum_{\rho \neq 0,1} \frac{x^{\rho - s}}{(\rho - s)^2} \\ && + \frac{1}{\log x} \sum_{n=0}^{\infty} \frac{x^{-2n-a-s}}{(2n + a +s)^2}. \end{eqnarray*} \end{lemma}
\begin{proof}This is Lemma $2.4$ of \cite{ChandeeCritical} with $a(n)=\chi(n)\Lambda(n)$, $d=1$ and $k(j)=a$ (here $a = 0$ or $1$ is the number given by $\chi(-1) = (-1)^a$). \end{proof}
\begin{proposition}\label{large values} Assume GRH for all Dirichlet L-functions of modulus $p$. Let $T$ be a parameter and let $x\ge 2$. Let $\lambda_0=0.56\ldots$ denote the unique positive real number satisfying $e^{-\lambda_0} = \lambda_0$. For all $\lambda \ge \lambda_0$, the following estimate $$
\log |L(\sigma + it,\chi)| \le \text{\rm Re } \sum_{n\le x} \frac{\chi(n)\Lambda(n)}{n^{\frac{1}{2}+ \frac{\lambda}{\log x} +it} \log n} \frac{\log (x/n)}{\log x} + \frac{(1+\lambda)}{2} \frac{\log (q)+\log^{+}(T)}{\log x} + O\Big( \frac{1}{\log x}\Big) $$ holds uniformly for $\vert t\vert \leq T$ and $1/2\leq \sigma\leq \sigma_0=1/2 + \frac{\lambda}{\log x}$.
\end{proposition}
\begin{proof} Let $a = 0$ or $1$ be again the number given by $\chi(-1) = (-1)^a$. Letting $\rho=1/2 +i\gamma$ run over the non-trivial
zeros of $L(s,\chi)$, we define
$$
F_{\chi}(s) = \text{Re }\sum_{\rho} \frac{1}{s-\rho} = \sum_{\rho} \frac{\sigma-1/2}{(\sigma-1/2)^2+
(t-\gamma)^2}.
$$Obviously $F_{\chi}(s)$ is non-negative in the half-plane $\sigma \ge 1/2$. By Hadamard's factorization (see \cite[Chapter 12, Eq. (17)]{Dav}), we have \begin{equation}\label{Hadamard} \frac{L'(s,\chi)}{L(s,\chi)}=-\frac{1}{2}\log \frac{q}{\pi}-\frac{1}{2}\frac{\Gamma^{'}}{\Gamma}\left(\frac{s+a}{2}\right)+B(\chi)+\sum_{\rho} \left(\frac{1}{s-\rho} + \frac{1}{\rho}\right).\end{equation} Here, $B(\chi)$ is a constant depending only on $\chi$, whose real part is given by
$$\text{Re }(B(\chi))=-\sum_{\rho}\frac{1}{\rho}.$$ By taking the real parts of both sides of (\ref{Hadamard}), an application of Stirling's formula yields
\begin{equation}\label{ineqReal}
-\text{Re }\frac{L'(s,\chi)}{L(s,\chi)}=\frac{\log(q)+\log^{+}(t)}{2} -F_{\chi}(s) + O(1) \leq \frac{\log(q)+\log^{+}(T)}{2}+O(1)
\end{equation} where we used the positivity of $F_{\chi}(s)$ in that region. Integrating (\ref{ineqReal}) as $\sigma=\text{Re }(s)$ varies from $\sigma$ to $\sigma_0(>1/2)$, we obtain, setting $s_0=\sigma_0 +it$,
\begin{equation}\label{deuxabscisses} \log\vert L(s,\chi)\vert -\log\vert L(s_0,\chi)\vert \leq \left(\frac{\log(q)+\log^{+}(T)}{2}+O(1)\right)(\sigma_0 - \sigma).\end{equation} On the other hand, using Lemma \ref{upsigma}, we get
\begin{align}\label{logderivative} -\frac{L'}{L}(s,\chi) &= &\sum_{n \leq x} \frac{\chi(n)\Lambda(n)}{n^{s}}\frac{\log\frac{x}{n}}{\log x} + \frac{1}{\log x}\left( \frac{L'}{L}(s,\chi) \right)' + \frac{1}{\log x} \sum_{\rho \neq 0,1} \frac{x^{\rho - s}}{(\rho - s)^2} \\ &&+\frac{1}{\log x} \sum_{n=0}^{\infty} \frac{x^{-2n-a-s}}{(2n + a +s)^2} \end{align} for any $s$ not coinciding with a zero of $L(s,\chi)$ and for any $x \geq 2$. Taking $s=\sigma+it$, integrating (\ref{logderivative}) over $\sigma$ from $\sigma_0$ to $\infty$ and extracting the real parts, we have, for $x \geq 2$,
\begin{align}\label{integrate}
\log |L(s_0,\chi)| = \text{Re }\Big( \sum_{2\le n\le x} \frac{\chi(n)\Lambda(n)}{n^{s_0} \log n} \frac{\log (x/n)}{\log x}
&- \frac{1}{\log x} \frac{L^{\prime}}{L}(s_0,\chi) \notag \\
& \hspace{-8mm} + \frac{1}{\log x} \sum_{\rho}
\int_{\sigma_0}^{\infty} \frac{x^{\rho-s}}{(\rho-s)^2} d\sigma +O\Big(\frac{1}{\log x}\Big)\Big).
\\ \notag
\end{align} The integral in (\ref{integrate}) is bounded as follows:
$$
\sum_{\rho}\Big|\int_{\sigma_0}^{\infty} \frac{x^{\rho -s}}{(\rho -s)^2} d\sigma\Big|
\le \sum_{\rho}\int_{\sigma_0}^{\infty}\frac{ x^{\frac 12-\sigma}}{|s_0-\rho|^2} d\sigma
= \sum_{\rho}\frac{x^{\frac 12-\sigma_0}}{|s_0-\rho|^2 \log x}= \frac{x^{\frac 12-\sigma_0}F_{\chi}(s_0)}{(\sigma_0-\frac 12)\log x}.$$ Thus, using (\ref{ineqReal}), we deduce that for $x\geq 2$
\begin{align}\label{majorations0}\log |L(s_0,\chi)| &\leq \text{Re } \sum_{2\le n\le x} \frac{\chi(n)\Lambda(n)}{n^{s_0} \log n} \frac{\log (x/n)}{\log x} \\
&+ \frac{\log(q)+\log^+(t)}{2\log x}
- \frac{F_{\chi}(s_0)}{\log x}
+ \frac{x^{\frac 12-\sigma_0}}{\log^2 x}\frac{F_{\chi}(s_0)}{(\sigma_0-\frac 12)} \notag
+ O\left(\frac{1}{\log x}\right). \notag \\ \notag \end{align} Hence, combining (\ref{deuxabscisses}) together with (\ref{majorations0}), the following inequality
\begin{align}\label{majlog}
\log |L(\sigma+ it,\chi)| &\le \frac{\log(q)+\log^{+}(T)}{2}\left(\sigma_0 - \sigma + \frac{1}{\log x}\right) + \text{Re }\sum_{2 \le n\le x} \frac{\chi(n)\Lambda(n)}{n^{s_0} \log n} \frac{\log (x/n)}{\log x} \notag \\
&\hskip .5 in +F_{\chi}(s_0) \Big( \frac{x^{\frac 12-\sigma_0}}{(\sigma_0-\frac 12) \log^2 x} -\frac{1}{\log x} \Big) + O\Big(\frac{1}{\log x}\Big) \end{align} holds for $x\geq 2$ and uniformly for $1/2\leq \sigma \leq \sigma_0 \leq 3/2$, $\vert t\vert \leq T$. We choose $\sigma_0 =\frac12 + \frac{\lambda}{\log x}$, where $\lambda \ge \lambda_0$. This restriction on $\lambda$ ensures that the term involving $F_{\chi}(s_0)$ in (\ref{majlog}) makes a negative contribution
and may therefore be omitted. The proposition follows easily.
\end{proof}
\begin{corollary}\label{upperboundline} Let $\chi$ be a primitive character modulo $q$ and assume GRH for $L(s,\chi)$. Then if $q$ is large enough, there exists an absolute constant $c>0$ such that
$$ \left\vert L\left(\frac{1}{2}+it,\chi\right)\right\vert \ll \exp\left(c \frac{\log q + \log^{+} t}{\log \log q}\right).$$ \end{corollary}
\begin{proof} This follows directly from the above proposition by setting $x=\log^{2-\epsilon} q$.
\end{proof}
\begin{remark} This inequality is less precise than \cite[Corollary 1.2]{ChandeeCritical} when $qt$ is large. Nevertheless, this covers the case when $t$ is relatively small compared to $q$ which is suitable for our applications.\end{remark}
Our proof of Theorem \ref{shiftedmoments} rests upon our main Proposition \ref{large values}. We begin by showing that the sum over prime powers appearing in that proposition may be in fact restricted over primes.
\begin{lemma}\label{primepowers} Assume that the Dirichlet $L$-functions modulo $q$ satisfy the Generalized Riemann Hypothesis. Let $t \le \log^A q$ with $A>0$, $x\ge 2$ and $\sigma\ge \frac 12$. Then, if $\chi$ is a Dirichlet character modulo $q$ such that $\chi \neq \chi_0^{2}$, we have $$
\left| \sum_{n\le x \atop n \neq p} \frac{\chi(n)\Lambda(n)}{n^{\sigma+it}\log n}
\frac{\log x/n}{\log x} \right| \ll \log \log\log q +O(1). $$
\end{lemma}
\begin{proof} Clearly, the contribution coming from the prime powers $p^k$ with $k\geq 3$ is $\ll 1$. It remains to handle the terms $n=p^2$. Hence, we have to bound
\begin{equation}\label{partial}\sum_{p\leq \sqrt{x}} \frac{\chi^2(p)}{p^{2\sigma + 2it}}\frac{\log(\sqrt{x}/p)}{\log \sqrt{x}}.\end{equation} We split this sum into ranges $2\leq p \leq \log^{8+4A+\epsilon} q$ and $\log^{8+4A+\epsilon} q \le p \le \sqrt{x}$. Then the first sum is easily bounded by $\displaystyle{\sum_{p\leq \leq \log^{8+4A+\epsilon} q} 1/p \ll \log\log\log q}$.
To treat the second sum, let us recall (see for instance \cite[p. 125]{Dav}) that under GRH, the estimate
$$ \sum_{n\leq x} \chi(n)\Lambda(n) \ll x^{1/2}\log^2 (qx)$$ holds for $x\ge 2$ and $\chi$ a non trivial character. By partial summation, we can deduce that
$$ \sum_{p\le x}\frac{\chi(p)\log p}{p^{2it}} \ll \vert t\vert x^{1/2}\log^2 (qx).$$ Thus, again by partial summation, we derive (using our restriction on $t$ and the fact that $\chi^2$ is non trivial) that the sum over primes $\geq \log^{8+4A+\epsilon} q $ is $O(1)$, which concludes the proof.
\end{proof}
Proposition \ref{large values} together with Lemma \ref{primepowers} give directly \begin{corollary}\label{boundprimes} For a Dirichlet character $\chi$ modulo $q$ such that $\chi^2 \neq \chi_0$, the inequality
$$\log |L(\sigma +it,\chi)| \le \text{\rm Re } \sum_{p\le x} \frac{\chi(p)}{p^{\sigma+ \frac{\lambda}{\log x} +it}} \frac{\log (x/p)}{\log x} + \frac{(1+\lambda)}{2} \frac{\log q + \log^{+} T}{\log x} + O\Big(\log \log \log q\Big)$$ holds uniformly for $\vert t\vert \leq T<\log^A q$ and $1/2\leq \sigma \leq 1/2+ \frac{\lambda}{\log x}$. \end{corollary}
The next lemma is a $q$-analogue of \cite[Lemma $3$]{SoundRiemann}.
\begin{lemma}\label{momentsprimes} Suppose $x\geq 2$ and $k$ is an integer such that $x^k<q$. Then for any $t\in\mathbb{R}$ and any complex numbers $a(p)$ we have
$$ \sum_{\chi \in X_q}\left|\sum_{p\leq x}\frac{\chi(p)a(p)}{p^{1/2+it}}\right|^{2k}\leq \phi(q)k!\left(\sum_{p\leq x}\frac{|a(p)|^2}{p}\right)^{k}.$$ Hence, there exist positive constants $c_{\chi}$ such that $\sum_{\chi \bmod q}c_{\chi}=\phi(q)$ and the following inequality holds:
$$ \left|\sum_{p\leq x}\frac{\chi(p)a(p)}{p^{1/2+it}}\right|^{2k}\leq c_{\chi}k!\left(\sum_{p\leq x}\frac{|a(p)|^2}{p}\right)^{k}.$$ \end{lemma}
\begin{proof}The proof follows the same lines as in the proof of Lemma $3$ of \cite{SoundRiemann}. After expanding the $2k$-th power, we use the orthogonality of characters modulo $q$ (here the inequality $x^k<q$ ensures that $m=n \bmod\, q$ implies $m=n$) instead of the orthogonality in $t$-aspect.
\end{proof}
We will need the following adaptation of \cite[Lemma $3.5$]{ChandeeShifts}.
\begin{lemma}\label{cos}
$$\sum_{p\leq z} \frac{\cos(a\log p)}{p} \leq \begin{cases} \log\left(\min\left\{\frac{1}{\vert a\vert},\log z\right\}\right)+ O(1) &\mbox{if } \vert a\vert \leq \frac{1}{100}, \\ \log\log(2+\vert a\vert)+O(1)& \mbox{if } \vert a\vert\geq \frac{1}{100}. \\\end{cases}$$
\end{lemma}
\begin{proof} If $\vert a\vert \leq \frac{1}{\log z}$, it follows from Mertens' Theorem. Otherwise, we use inequality $(2.1.6)$, p.$57$ of \cite{GS}. \end{proof}
\subsection{Proof of Theorem \ref{largevalues}}
First, remark that if $-\infty\leq V\leq 4\sqrt{\log \log q}$, then trivially we have
$$\int_{-\infty}^{+\infty} e^{V} N_t(q,V)dV \leq \phi(q) e^{4\sqrt{\log \log q}} = o(\phi(q)\log q).$$ In view of Corollary \ref{upperboundline}, we can assume $4\sqrt{\log \log q} \leq V \leq 4ck \frac{\log q}{\log \log q}$ using the fact that $t_0=\max(t_i, i=1\cdots, 2k) \leq \log^A q.$ It remains to estimate $N_t(q,V)$ for large $q$ with an explicit dependence on $t_0$. Choosing $\lambda=0.6$ in Corollary \ref{boundprimes}, we obtain if $\chi^2 \neq \chi_0$ that
\begin{flalign}\label{somme} &\log\left\vert L\left(\frac{1}{2}+it_1,\chi\right)\right\vert+\cdots+\log\left\vert L\left(\frac{1}{2}+it_{2k},\chi\right)\right\vert & \notag \\ \notag &\leq \text{\rm Re } \left(\sum_{p\le x} \frac{\chi(p)p^{-it_1}}{p^{\frac{1}{2}+ \frac{0.6}{\log x}}} \frac{\log (x/p)}{\log x}+\cdots+ \frac{\chi(p)p^{-it_{2k}}}{p^{\frac{1}{2}+ \frac{0.6}{\log x}}} \frac{\log (x/p)}{\log x}\right)+ \frac{8k}{5} \frac{\log q + \log^+ T}{\log x}& \\ \notag &+ O\Big(\log \log \log q\Big). &\\ \notag \end{flalign} Following \cite{ChandeeShifts} and \cite{SoundRiemann}, we define the quantity $A$ as
$$ A = \begin{cases}\frac{\log W}{2} &\mbox{if } 4\sqrt{\log \log q}\leq V\leq W, \\ \frac{W\log W}{2V}& \mbox{if } W\leq V\leq \frac{1}{4k}W\log W, \\ 2k & \mbox{if } V>\frac{1}{4k}W\log W.\end{cases} $$Let $x=(q\max(T,1))^{A/V}$ and $z=x^{1/\log \log q}$. From the previous bounds, we have
\begin{flalign*}&\log\left\vert L\left(\frac{1}{2}+it_1,\chi\right)\right\vert+\cdots+\log\left\vert L\left(\frac{1}{2}+it_{2k},\chi\right)\right\vert & \\ &\leq S_1(\chi)+S_2(\chi)+\frac{8k}{5} \frac{\log q + \log^{+} T}{\log x}+O\Big(\log \log \log q\Big),&\\ \end{flalign*} where
$$ S_1(\chi)=\left\vert\sum_{p\le z} \frac{\chi(p)(p^{-it_1}+\cdots+p^{-it_{2k}})}{p^{\frac{1}{2}+ \frac{0.6}{\log x}}}\frac{\log (x/p)}{\log x}\right\vert$$ and
$$ S_2(\chi)=\left\vert\sum_{z< p\le x} \frac{\chi(p)(p^{-it_1}+\cdots+p^{-it_{2k}})}{p^{\frac{1}{2}+ \frac{0.6}{\log x}}}\frac{\log (x/p)}{\log x}\right\vert .$$ It remains to study how often with respect to characters these quantities could be large. Firstly, if $\chi \in S_t(q,V)$, we must have \footnote{Compare $\frac{V}{A}$ to $\log\log\log q$.}
$$ S_1(\chi) \geq V_1:=V\left(1-\frac{9k}{5A}\right)\text{ or } S_2(\chi)\geq V_2:=\frac{kV}{5A}.$$ Let $N_i(q)=\# S_i(q):= \left\{ \chi (\bmod \,q), \chi^2 \neq \chi_0 : S_i(\chi)\geq V_i\right\}$ for $i=1,2$. We want to find upper bounds for $N_i(q)$ with a certain uniformity in $t$ \footnote{We have to keep in mind that for our applications $t$ will be at most of size $\log q$.}. By Lemma \ref{momentsprimes}, we see that, for any natural number $l\leq \frac{3}{4}\frac{V}{A}$, \footnote{Here we use that $T\leq \log^A q$.} we have
$$ \left\vert S_2(\chi)\right\vert^{2l} \leq c_{\chi}l!\left(\sum_{z< p\le x} \frac{4k^2}{p} \right)^{l} \ll c_{\chi}(4lk^2(\log\log\log q+ O(1))^{l}.$$ Choosing $l=\lfloor 3V/4A \rfloor$ and observing that
\begin{equation}\label{N2} \sum_{\chi \in S_2(q)}\left\vert S_2(\chi)\right\vert^{2l} \geq N_2(q)V_2^{2l}\end{equation} we derive
\begin{equation*} N_2(q) \ll \sum_{\chi \in S_2(q)} c_{\chi} \left(\frac{5A}{kV}\right)^{2l}(4lk^2(\log\log\log q+ O(1))^{l} \ll \phi(q) \exp\left(-\frac{V}{2A}\log V\right). \end{equation*} It remains to find an upper bound for $N_1(q)$. By Lemma \ref{momentsprimes}, for any $l < \frac{\log q}{\log z}$,
\begin{align*} \left\vert S_1(\chi)\right\vert^{2l}& \leq c_{\chi}l!\left(\sum_{ p\le z} \frac{\vert p^{-it_1}+\cdots+p^{-it_{2k}}\vert^2}{p} \right)^{l} \\ &\ll c_{\chi}l!\left(\sum_{ p\le z} \frac{2k+2\sum_{i<j}\cos((t_i-t_j)\log p)}{p} \right)^{l} \\ & \ll c_{\chi}l!\left(2k\log\log z+2\sum_{i,j\atop i<j}F_{i,j}\right)^l \\ &\ll c_{\chi}l! \,W^l \ll c_{\chi}\sqrt{l}\left(\frac{lW}{e}\right)^l \\ \end{align*} where we applied Lemma \ref{cos} and used together Stirling's formula and the fact that $z<q$.
\begin{remark} If $\vert t_i-t_j\vert \geq \frac{1}{100}$, by hypothesis this quantity is at most $2\log^A q$. Hence, the second case of Lemma \ref{cos} implies that the sum over primes is $\ll \log\log\log q$. \end{remark}Proceeding as in (\ref{N2}) for $N_2$, we deduce that
$$ N_1(q) \ll V_1^{-2l} \sum_{\chi \in S_1(q)}\left\vert S_1(\chi)\right\vert^{2l} \ll \phi(q)\sqrt{l}\left(l\frac{W}{eV_1^2}\right)^l. $$ When $V\leq \frac{W^2}{4k^3}$, we choose $l=\lfloor\frac{V_1^2}{W}\rfloor$, and when $V>\frac{W^2}{4k^3}$, we choose $l=\lfloor 8V\rfloor$. We easily verify, using the definition of $A$, that the condition $l<\frac{\log q}{\log z}$\footnote{Use the fact that $W/4k^2 \leq \log\log q.$} holds in both cases. Finally we get
$$N_1(q) \ll \phi(q)\frac{V}{\sqrt{W}}\exp\left(-\frac{V_1^2}{W}\right)+\phi(q)\exp(-3V\log V).$$
Using our bounds on $N_1(q)$ and $N_2(q)$, elementary computations lead to the proof of Theorem \ref{largevalues}.
\section{Application to upper bounds for moments of theta functions}
In that section, we will prove Theorem \ref{upperGRH} in the case of even characters. The proof for odd characters goes exactly along the same lines. The method is the following, we express theta values as Mellin transform of $L$- functions and then we use our previous result about moments of shifted $L$- functions.
For every even primitive character $\chi$ modulo $q$, recall the following relation for $c>1/2$
$$\theta(1,\chi)=\int_{c-i\infty}^{c+\infty} L(2s,\chi)\left(\frac{q}{2\pi}\right)^{s}\Gamma(2s)ds.$$
Shifting the line of integration to $\Re(s)=1/4$ and using the decay of $\Gamma(s)$ in vertical strips, we end up with
$$\theta(1,\chi)=\left(\frac{q}{\pi}\right)^{\frac{1}{4}}\int_{-\infty}^{\infty}L\left(\frac{1}{2}+2it,\chi\right)\left(\frac{q}{\pi}\right)^{2it}\Gamma\left(\frac{1}{2}+2it\right)dt.$$
We express the moments as
\begin{equation}\label{moments2k}\sum_{\chi \in X_q^+ \backslash\chi_0}\vert \theta(1,\chi)\vert^{2k}=\left(\frac{q}{\pi}\right)^{\frac{k}{2}} \sum_{\chi \in X_q^+\backslash\chi_0}\left\vert \int_{-\infty}^{\infty}L\left(\frac{1}{2}+2it,\chi\right)\left(\frac{q}{\pi}\right)^{2it}\Gamma\left(\frac{1}{2}+2it\right)dt\right\vert^{2k}.\end{equation} Hence, the problem boils down to getting a bound of size $\log^{(k-1)^2+\epsilon} q$ for the $2k$-fold integral. In the following, we can sum over $X_q^*$ without substantial loss.
\subsection{Cutting part}
The strategy is the following: we will cut up to a certain reasonable height, for instance $\log^{\epsilon} q$. Precisely, using the decay of $\Gamma\left(\frac{1}{2}+2it\right)$, we bound the tail:
\begin{lemma}\label{resteintegral} Fix $\epsilon>0$. There exists an absolute constant $c$ such that $$\sum_{\chi \in X_q^*}\left\vert \int_{-\infty}^{\infty}L\left(\frac{1}{2}+2it,\chi\right)\left(\frac{q}{\pi}\right)^{2it}\Gamma\left(\frac{1}{2}+2it\right)\mathds{1}_{\vert t\vert \geq\log^{\epsilon}(q)}(t)dt\right\vert^{2k} \ll \phi(q) e^{-c\log^{\epsilon}q}.$$ \end{lemma}
\begin{proof}
Using H\"older inequality with parameters $\frac{1}{2k}+\frac{2k-1}{2k}=1$, the problem reduces to bound
$$\sum_{\chi \in X_q^*}\left(\int_{\vert t\vert \geq\log^{\epsilon} q}\left\vert L\left(\frac{1}{2}+2it,\chi\right)\right\vert^{2k}\left\vert\Gamma\left(\frac{1}{2}+2it\right)\right\vert dt\right)\left(\int_{\vert t\vert \geq\log^{\epsilon} q}\left\vert \Gamma\left(\frac{1}{2}+2it\right)\right\vert dt\right)^{2k-1}.$$ We decompose dyadically the range of integration in the left hand side and use the convergence of the right hand side to end up with
$$\sum_{n\geq \log^{\epsilon} q} \sum_{\chi \in X_q^*}\int_{n}^{2n}\left\vert L\left(\frac{1}{2}+2it,\chi\right)\right\vert^{2k}\left\vert\Gamma\left(\frac{1}{2}+2it\right)\right\vert dt.$$Using Stirling's formula and Proposition $2.9$ of \cite{LiChandee}\footnote{The method is the same as our proof of Theorem \ref{shiftedmoments}.}, we get for $c_1>0$ an absolute constant
$$\begin{array}{l} \displaystyle{\sum_{n\geq \log^{\epsilon} q} \sum_{\chi \in X_q^*}\int_{n}^{2n}\left\vert L\left(\frac{1}{2}+2it,\chi\right)\right\vert^{2k}\left\vert\Gamma\left(\frac{1}{2}+2it\right)\right\vert dt} \\ \displaystyle{\ll \sum_{n\geq \log^{\epsilon} q} e^{-c_1n} \sum_{\chi \in X_q^*}\int_{n}^{2n}\left\vert L\left(\frac{1}{2}+2it,\chi\right)\right\vert^{2k} dt} \\ \displaystyle{\ll \phi(q)(\log q)^{k^2+\epsilon} \sum_{n\geq \log^{\epsilon} q} e^{-c_1n} n(\log n)^{k^2+\epsilon} \ll \phi(q) e^{-c\log^{\epsilon}q}.} \end{array}$$
\end{proof}
\subsection{Bound for the hypercube integral}
It remains to bound optimally the integral on the $2k$-hypercube $\mathcal{H}$ of size $\log^{\epsilon} q$. First, observing that $\Gamma\left(\frac{1}{2}+2it\right)$ is bounded on $\mathcal{H}$ and expanding the integral in (\ref{moments2k}), we get
\begin{eqnarray}\label{hypercube} \sum_{\chi \in X_q^*}\left\vert \int_{-\infty}^{\infty}L\left(\frac{1}{2}+2it,\chi\right)\left(\frac{q}{\pi}\right)^{2it}\Gamma\left(\frac{1}{2}+2it\right)\mathds{1}_{\vert t\vert \leq\log^{\epsilon}(q)}(t)dt\right\vert^{2k} \\
\ll \sum_{\chi \in X_q^*}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\left\vert L\left(\frac{1}{2}+2it_1,\chi\right) \cdots L\left(\frac{1}{2}-2it_{2k},\chi\right)\right\vert \mathds{1}_{||t|| \leq\log^{\epsilon}(q)}(t)dt_1\cdots dt_{2k} \nonumber \\ \nonumber \end{eqnarray} where $||t||= \max_{i=1,\cdots,2k} \vert t_i\vert$. We will use Theorem \ref{momentshifted} to handle that integral. In order to do this, we have to control how the shifts $t_i$ are close to each other.
By a permutation change of the variables, we can assume that $t_1\leq t_2\leq \cdots\leq t_{2k}$. Indeed, the integral on $\mathcal{H}$ is equal to $(2k)!$ times the integral with this additional restriction. \\
For every $2k$- tuple $\overline{t}=(t_1,\cdots,t_{2k})$, define a $(2k-1)$- tuple $\overline{j}=(j_1,\cdots, j_{2k-1})$ where $j_{i}= \min\{ i+1\leq j\leq 2k, \vert t_i-t_j\vert >\frac{1}{\log q}\}$. If for some $i$, no such $j$ exists, we set $j_i=2k+1$. In the following, we will say that $\overline{t}=(t_1,\cdots,t_{2k})$ is of type $\overline{j}$. Let us give few remarks about that definition. First of all, we have to think about $j_i$ as the first occurrence of a shift lying far from $t_i$. Furthermore, notice that $2\leq j_1\leq j_2 \leq \cdots \leq j_{2k-1} \leq 2k+1$ and that we can split the domain of integration $\mathcal{H}$ in a disjoint union $\mathcal{H}=\cup \mathcal{H}_{\overline{j}}$ of $2k$-tuples $\overline{t}=(t_1,\cdots,t_{2k})$ where $\overline{t}$ is of type $\overline{j}$. Hence, proving Theorem \ref{upperGRH} reduces to bound the contribution of the integral over $\overline{t}$ of type $\overline{j}$ for all possible choices of $\overline{j}$. \\
The strategy is to apply Theorem \ref{shiftedmoments} in a appropriate way to obtain the expected bound. Using Theorem \ref{shiftedmoments}, we get that the contribution in (\ref{hypercube}) of $\overline{t}$ of type $\overline{j}$ is bounded by
\begin{equation}\label{integral shift} \phi(q) (\log q)^{k/2+\epsilon} \int\cdots\int_{\mathcal{H}_{\overline{j}}} \,\, \left(\prod_{i<j} E_{i,j}\right) \,dt_1 \cdots dt_{2k} \end{equation} where $E_{i,j}$ is defined in Theorem \ref{shiftedmoments}. For every $i=1,2,\cdots,2k-1$, we will essentially bound $\displaystyle{\prod_{j=i+1}^{2k}E_{i,j}}$ in two different ways, depending on whether the variable $t_i$ possesses a close shift or not.
$\bullet$ \underline{Case 1}: Close shifts.
If $t_i$ admits a close shift then $j_i>i+1$. Using the first case of Theorem \ref{shiftedmoments}, we have the following trivial bound
\begin{equation}\label{closeshifts}\left\vert \prod_{j=i+1}^{2k}E_{i,j} \right\vert \leq (\log q)^{\frac{2k-i}{2}}.\end{equation}
$\bullet$ \underline{Case 2}: Isolated shifts.
For those indices $i$, $t_i$ does not admit a close shift, which means that $j_i=i+1$. We remark that $\frac{1}{\vert t_i - t_j\vert} \leq \frac{1}{\vert t_i - t_{j_i}\vert}$ for $j\geq j_i$, since we have $t_1\leq t_2\leq \cdots\leq t_{2k}$. Hence, using again both cases of Theorem \ref{shiftedmoments}, we derive the following bound:
\begin{equation}\label{isolated} \left\vert \prod_{j=i+1}^{2k}E_{i,j} \right\vert \leq \frac{1}{\vert t_i-t_{i+1}\vert^{\frac{(2k-i)}{2}}}(\log\log q)^{\frac{2k-i}{2}}. \end{equation} To deal with the integral in (\ref{integral shift}), we can make the following linear change of variables:
\begin{equation}\label{changevar}u_i=\begin{cases}t_i-t_{i+1} \,\,\,\text{ if } i\leq 2k-1, \\ t_{2k} \hspace{1cm}\text{ if } i=2k. \\
\end{cases}\end{equation} Thus, the determinant of the Jacobian being equal to $1$, the integral in (\ref{integral shift}) becomes
\begin{equation}\label{afterchange}(\log\log q)^{k(2k-1)}\prod_{i=1\atop i\vert j_i\neq i+1}^{2k-1}(\log q)^{\frac{2k-i}{2}} \int\cdots\int_{\mathcal{D}_{\overline{j}}}\prod_{i=1\atop i \vert j_i=i+1}^{2k-1}\frac{1}{\vert u_i\vert^{\frac{(2k-i)}{2}}}du_1\cdots du_{2k} \end{equation} where the domain $\mathcal{D}_{\overline{j}}$ is included in
$$\prod_{i\vert j_i\neq i+1} ] -1/ \log q, 0] \prod_{i\vert j_i=i+1} ] -\log^{\epsilon} q, -1/ \log q].$$ For those $i$ such that $j_i \neq i+1$, we bound the integral over $u_i$ by the length of the interval of integration $1/\log q$. For the other indices, we integrate explicitly on $] -\log^{\epsilon} q, -1/ \log q]$. In order to obtain the expected bound, we need to ``save" a logarithm for each integration $du_i$ for $i=1\cdots 2k-1$. An additional problem arises when the variable does not admit a close shift and we integrate $u_{2k-1}^{-1/2}$. Let us first treat the easiest case.
$\spadesuit \hspace{3mm}$ \underline{Subcase 1}: $j_{2k-1}\neq 2k$.
In that case, all the exponents in the denominator of the integral in (\ref{afterchange}) are greater than $1$. Therefore, we obtain after explicit integration that $(\ref{afterchange})$ is bounded by
\begin{equation}\label{winlog} (\log\log q)^{k(2k-1)}\prod_{i\vert j_i\neq i+1} \frac{(\log q)^{\frac{2k-i}{2}} }{\log q}\prod_{i=1\atop i \vert j_i=i+1}^{2k-1}\frac{(\log q)^{\frac{2k-i}{2}} }{\log q} \log\log q \end{equation} where the factor $\log \log q$ comes from the possible integration of $1/u$ when $i=2k-2$. Hence, (\ref{integral shift}) is bounded by
$$ \phi(q) (\log q)^{f(k)+2k^2\epsilon}$$ where
\begin{align*} f(k) &=\frac{k}{2}+ \frac{1}{2} \sum_{i=1}^{2k-1} (2k-i-2) \\
& = \frac{k}{2}+ \frac{1}{2} \sum_{i=-1}^{2k-3} i \\
& = \frac{k}{2}+ \frac{(2k-3)(2k-2)}{4} -\frac{1}{2} \\
& = (k-1)^2
\end{align*} which proves Theorem \ref{upperGRH} in that case.
$\spadesuit \hspace{3mm}$ \underline{Subcase 2}: $j_{2k-1}= 2k$.
The only remaining problem arises when $t_{2k-1}$ does not have a close shift. In that case, an explicit integration in (\ref{afterchange}) is not sufficient to save $\log q$ after integration, but only saves $\log^{1/2} q$. We are going to split the proof in two subcases depending on whether $t_1$ admits a close shift or not.
$\clubsuit \hspace{3mm}$ \underline{Subsubcase 1}: $\fbox{$j_1 \neq 2$}$
We will use exactly the same bounds as before except for $i=1$. The trivial inequality $\vert t_1-t_{2k}\vert^{-1/2} \leq \vert t_{2k-1}-t_{2k}\vert^{-1/2}$ together with the simple observation that $\vert t_{2k-1}-t_{2k}\vert$ is large (by hypothesis $j_{2k-1}=2k$) implies the following bound
\begin{equation}\label{modifE1}\prod_{j> 1} E_{1,j} \leq (\log q)^{\frac{2k-2}{2}}\frac{\log \log q}{\vert t_{2k-1}-t_{2k}\vert^{1/2}}.\end{equation} Doing the same change of variables as before in (\ref{changevar}) and using (\ref{modifE1}), we end up with the bound
\begin{equation*}\label{afterchangebis}(\log \log q)^{k(2k-1)}(\log q)^{\frac{2k-2}{2}}\prod_{i=2\atop i\vert j_i\neq i+1}^{2k-2}(\log q)^{\frac{2k-i}{2}} \int\cdots\int_{\mathcal{D}_{\overline{j}}}\prod_{2=1\atop i \vert j_i=i+1}^{2k-2}\frac{1}{\vert u_i\vert^{\frac{(2k-i)}{2}}}\frac{1}{\vert u_{2k-1}\vert}du_1\cdots du_{2k}.\end{equation*} A slightly modification of the computation following (\ref{winlog}) enables us to obtain the expected bound $(\log q)^{(k-1)^2+\epsilon}$.
$\clubsuit \hspace{3mm}$ \underline{Subsubcase 2}: $\fbox{$j_1 = 2$}$
We proceed as in the previous subsubcase with the following bound (the $\log\log$ factor coming from the possible case where the shifts are far away from each other)
$$ \prod_{j> 1} E_{1,j}\ll \frac{(\log\log q)^{\frac{2k-1}{2}}}{\displaystyle{\prod_{j=2}^{2k}\vert t_1-t_j \vert^{1/2}}} \leq \frac{(\log\log q)^{\frac{2k-1}{2}}}{\vert t_1-t_2\vert^{\frac{2k-2}{2}}} \frac{1}{\vert t_1-t_{2k}\vert^{1/2}} \leq \frac{(\log\log q)^{\frac{2k-1}{2}}}{\vert t_1-t_2\vert^{\frac{2k-2}{2}}} \frac{1}{\vert t_{2k-1}-t_{2k}\vert^{1/2}} $$
Doing the same change of variables as before in (\ref{changevar}), we end up with the same integral as in the previous subsubcase. Hence, the same computation works and this concludes the proof of Theorem \ref{upperGRH}.
\end{document} | arXiv |
Resources development and tourism environmental carrying capacity of ecotourism industry in Pingdingshan City, China
Yufeng Zhao1 &
Lei Jiao1
Pingdingshan City has unique and rich ecotourism landscape. To realize the sustainable development of ecotourism industry in Pingdingshan and simultaneously achieve economic development and environmental protection in the development of ecotourism resources, literature analysis and field investigation methods were used to deeply explore the advantages and problems of Pingdingshan in the process of resources development in this study. Then, the indicator system of environmental carrying capacity of Pingdingshan was established and environmental carrying capacity was calculated to understand the local tourism environmental carrying capacity.
The development of ecotourism resources in Pingdingshan has advantages of natural conditions, convenient transportation, and sound infrastructure, but there were also problems, such as insufficient resources protection and unscientific management system. After calculation, it was found that the environmental carrying capacity of ecotourism in Pingdingshan was overloaded.
This study reveals the development of the tourism industry in Pingdingshan and provides a basis for the future development of Pingdingshan tourism resources, which is beneficial to the sustainable development of the local ecotourism industry.
With the change of people's travel idea, ecotourism has developed rapidly. It has received widespread attention since its appearance. The ecotourism industry in Indonesia has gained a rapid development by virtue of its good ecotourism resources, indicating that the protection of ecotourism resources can effectively promote the sustainable development of the tourism industry (Hengky 2017). Siswanto (2015) concluded that the ecotourism industry had created more income for local residents and business owners and brought certain economic benefits to the local community. The development of ecotourism resources could raise people's awareness of environmental protection while increasing economic income. Pingdingshan City is located in the south-central part of Henan Province, China. The geographical position is very unique, and the ecotourism resources are quite abundant. With the development of resources, the ecotourism industry in Pingdingshan City has developed rapidly, and many problems have arisen. Evaluating the carrying capacity of the ecological environment can effectively promote the development of ecotourism (Daneshvar and Sheybani 2011). Shi et al. (2015) calculated the carrying capacity of ecotourism in Shang-La County and found that the reasonable development of natural resources in Shang-La effectively promoted the long-term development of its ecotourism. Many resource-based cities have developed rapidly by virtue of abundant natural resources, but they also face problems of over-exploitation and resource exhaustion. As a resource-based city, ecotourism industry of Pingdingshan has developed rapidly, and its state of resource development needs to be paid attention to. However, the current research on tourism resources development and environmental carrying capacity of Pingdingshan has not been quite sufficient.
Based on the research on resource development of Pingdingshan City, this study established the evaluation system of ecotourism environmental carrying capacity of Pingdingshan and carried out corresponding evaluation to promote the sound and rapid development of ecotourism resources in Pingdingshan.
Development status of tourism resources in Pingdingshan City
Pingdingshan has been listed as a resource-based city by virtue of its rich and diversified natural resources. In addition to coal, steel, and other natural resources, Pingdingshan also has abundant tourism resources. Pingdingshan has superior geographical position and unique natural conditions. Its tourism resources are characterized by diverse contents, unique scenery, and concentrated resources. In recent years, the ecotourism industry of Pingdingshan has developed rapidly. The construction of scenic spots has almost covered all the important tourism resources of the Pingdingshan. The tourism resources are protected to drive the economic development in the process of development (Nuzula et al. 2017). However, the development of ecotourism resources in Pingdingshan also exists many problems. First of all, the destruction of ecotourism resources occurs frequently due to insufficient resources protection. Then, many management systems in the ecotourism industry are unscientific, resulting in the waste and inefficient use of many resources in the tourism industry, which cannot guarantee the sustainable development of tourism resources. In addition, the development and utilization of ecotourism resources in Pingdingshan is low. The lack of infrastructure is an important factor affecting the protection of tourism resources and low level of development and utilization, which is not conducive to the long-term development of tourist attractions.
Calculation method of tourism environmental carrying capacity
The ecotourism environmental carrying capacity refers to the acceptable number of tourists that can be accepted in a certain area, which can not only meet the needs of tourists and benefit the tourism industry, but also protect the environment and reduce the impact. In this study, the indicator system of environmental carrying capacity was constructed, and the calculation was made on that basis.
The system of ecotourism environmental carrying capacity consists of four parts: resource environmental carrying capacity, ecology environmental carrying capacity, psychology environmental carrying capacity, and tourism environmental carrying capacity. The value of ecotourism environmental carrying capacity is not the sum of the parts, but the minimum value of each part.
Resource environmental carrying capacity (RECC) refers to the number of tourists that can be reached in a region on the premise of realizing the time requirements of tourists, which is a major manifestation of environmental protection (Ye et al. 2016). Its expression is:
RECC = (daily turnover rate × total resource space)/per capita basic space standard
Further calculations are based on the per capita area of the tourist area:
$$ \mathrm{RECC}=\left(S\times T\right)/\left(s\times t\right), $$
where S stands for tourist area (m2), T stands for opening hours (h), s stands for per capita area occupied by tourists (m2), and t stands for time required for a visitor to visit (h).
Ecology environmental carrying capacity (EECC) is the amount of tourism that a tourist area can bear without affecting the environment, which is expressed by the number of tourists. Its expression is:
$$ \mathrm{EECC}=\min \left(\mathrm{WECC},\mathrm{AECC},\mathrm{SWCC}\right), $$
where WECC stands for water environmental carrying capacity, WECC = daily sewage treatment level/daily sewage production per person, AECC stands for air environmental carrying capacity, \( \mathrm{AECC}=\frac{S\times f}{s} \), where S stands for tourist area (m2), f stands for forest coverage ratio (%), s stands for per capita green areas (m2/person), and SWCC stands for solid waste carrying capacity, SWCC = total amount of daily disposed solid waste/per capita daily.
Selection criteria of indicators are different for tourist areas of different kinds or with different properties (Shen et al. 2017). According to the actual situation, water environmental capacity was regarded as the main aspect (Xu and Yan 2016).
Psychology environmental carrying capacity (PECC) includes the psychological capacity of residents and the psychological capacity of tourists in tourist areas (Zhang 2014). The psychological capacity of residents refers to the maximum number of tourists that residents of tourist areas can bear psychologically. The psychological carrying capacity of tourists refers to the maximum crowding condition that tourists can accept in the process of traveling. The result can be obtained through a questionnaire survey.
Tourism environmental carrying capacity (TECC) refers to the maximum number of tourists that the basic service facilities in the tourism area can afford (Zelenka and Kacetl 2014). Its expression is:
$$ \mathrm{TECC}=\min \left({\mathrm{TECC}}_1,{\mathrm{TECC}}_2,\dots \mathrm{TECC}{}_i\right), $$
where TECCi stands for economic burden part of supply amount i, \( {\mathrm{TECC}}_i=\frac{S_i}{D_i} \), Si stands for daily supply amount of i (amount/day), and Di stands for the per capita demand of i (amount/person/day).
In conclusion, the calculation formula of ecotourism environmental carrying capacity is:
$$ \mathrm{ETEC}=\min\;\left(\mathrm{RECC},\mathrm{EECC},\mathrm{PECC},\mathrm{TECC}\right). $$
Calculation of tourism environmental carrying capacity of Pingdingshan City
Firstly, RECC of Pingdingshan City was calculated. After checking the relevant literature, it was found that the effective tourist area of Pingdingshan City is about 972 km2, the open time is 12 h, the per capita occupied area was 5 m2, and the time needed for visit is 12 h. According to Eq. (1), we have:
$$ \mathrm{RECC}=\left(S\times T\right)/\left(s\times t\right)=194,400. $$
Then, EECC of Pingdingshan City was calculated. Taking water environmental carrying capacity (WECC) as an example, WECC = daily treatment level of sewage/daily output of sewage = 3,360,000/60 = 56,000.
Through checking the relevant literature and investigation, it was found that the psychology environmental carrying capacity of citizens was 90,000 person/day, the psychology environmental carrying capacity of tourists was 70,000 person/day, and the average value was 80,000 person/day.
The calculation of economy environmental bearing capacity was represented by traffic and accommodation:
$$ \mathrm{TECC}=\min \left({\mathrm{TECC}}_{\mathrm{traffic}},{\mathrm{TECC}}_{\mathrm{accommodation}}\right)=\min \left(76,000,78,647\right)=76,000. $$
The results of ecotourism environmental carrying capacity of Pingdingshan City were obtained, as shown in Table 1.
Table 1 The calculation result of the ecotourism environmental carrying capacity in Pingdingshan City
According to the minimum principle, the ecotourism environmental carrying capacity of Pingdingshan City was 20.44 million people each year, which was obtained by ecology environmental carrying capacity. It was found that ecology environmental carrying capacity could represent the ecotourism environmental carrying capacity, which was the most important and basic part in the indicator system. According to relevant reports, the number of tourists in Pingdingshan City has reached 20.96 million in 2016; hence, the ecotourism environment of Pingdingshan City was slightly overloaded.
In addition, the evaluation indicators of each carrying capacity module were further processed and analyzed, and 40 evaluation units were classified.
And then, nine evaluation factors were selected based on the data to establish the model, and evaluation indicators of two types were analyzed by using Statistical Product and Service Solutions (SPSS). The results are as follows.
The clustering and standardization results of resource environmental carrying capacity indicators in each scenic spot are shown in Fig. 1. In classification 1, the annual precipitation, river area ratio, and > 0 °C accumulated temperature were all negative values, indicating that the natural conditions of this type of scenic spot had natural advantages, and the corresponding resource environmental carrying capacity was high. In classification 2, the resource environmental carrying capacity was restricted by the river area, which was significantly different from the other four types of scenic spots, but it was also high. In classification 3, the resource environmental carrying capacity was restricted by the accumulated temperature. In classification 4 and 5, only the proportion of vegetation area was negative and the others were positive, indicating that the resource environmental carrying capacity of these two types of regions was low (Table 2).
Standard value of resource environmental carrying capacity clustering results
Table 2 Classification and grade of each indicator's carrying capacity
The clustering and standardization results of psychology environmental carrying capacity indicators in each scenic spot are shown in Fig. 2. In classification 1 and 2, the number of service industry units, domestic highway mileage, and settled ratio were all negative, indicating that the psychology environmental carrying capacity was very high. Pingdingshan City has a dense expressway network and 400-km-long railway, and there are three airports around. Convenient transportation is one of the great advantages of developing tourism in Pingdingshan City. The standardized values of classification 3 and 4 were both relatively small, and the psychology environmental carrying capacity of them was smaller than that of classification 5. Moreover, the two kinds of areas could be planned together because of the similar values of the relevant indicators. The indicators in classification 5 were relatively large, indicating that the psychology environmental carrying capacity was low, and it was likely to be restricted by ecological resettlement.
Standard value of psychology environmental carrying capacity clustering results
Discussion and conclusion
The development of resource-based cities is based on resources. The excessive exploitation of resources will lead the development of cities to decline gradually. In order to guide the scientific development of tourism industry in Pingdingshan City, the local government should attach great importance to the development of local tourism industry and provide relatively sufficient funds for the construction of scenic spots. Convenient transportation and relatively sufficient supply of water and electric are both the development advantages of ecotourism industry in Pingdingshan City, but the resident level of tourists are far from being up to the required standards. In addition, due to the late and rapid development of some scenic spots in Pingdingshan City and under the effects of local conditions, the environmental pollution of tourist attractions is prominent. The relevant infrastructure of the scenic spots is generally backward, which directly increases the pollution of scenic spots. With the rapid growth of industrial and domestic pollutants and the backwardness of treatment facilities in Pingdingshan City, environmental pollution will gradually be serious and the remaining environmental capacity will be smaller and smaller (Solís et al. 2014). The solid waste of scenic areas in Pingdingshan City will be uniformly transported to the outside of the scenic spot for unified treatment, so the solid waste carrying capacity was relatively large. Therefore, it is necessary to focus on solving these two problems, rationally allocate tourism resources, and promote the long-term development of the ecotourism industry. The ecotourism industry in Pingdingshan City is developing rapidly. In order to achieve better and long-term tourism development, it is necessary to strengthen the protection of resources and improve the scientific management of the system to solve problems existing in the development. Due to the large scenic area of Pingdingshan City, the rich types of ecotourism resources, and different opening degree of each scenic spot, there exist some uncertainties in the ecological carrying capacity of scenic areas in Pingdingshan City. According to the calculation results, the resource environmental carrying capacity, tourism environmental carrying capacity, and psychology environmental carrying capacity have not reached the limit, and more tourists can be accepted, but the ecology environmental carrying capacity has been seriously overloaded, resulting in overall overload. Therefore, the key to the development of tourism industry in Pingdingshan City is to improve the ecology environmental carrying capacity (Cheng et al. 2016).
Under the premise of adhering to the necessary principles, the development of tourism industry in Pingdingshan needs to take corresponding measures for the existing problems. By establishing an indicator system of ecotourism environmental carrying capacity with four levels and nine evaluation factors, this study analyzed from four aspects and found that the ecology environmental carrying capacity of Pingdingshan City has seriously affected the development of the tourism industry and positive measures need to be taken to improve it. The study results provide a basis for the development of tourism industry resources, which is conducive to the sustainable development of resource-based cities and has reference values for the further development of the tourism industry in Pingdingshan City.
Cheng J, Zhou K, Chen D, Fan J (2016) Evaluation and analysis of provincial differences in resources and environment carrying capacity in China. Chin Geogr Sci 26(4):539–549.
Daneshvar MRM, Sheybani S (2011) GIS based evaluation of ecotourism zonation using ecological carrying capacity model (Study area: Kalat county, north of Khorasan–e–Razavi province) (In Persian). International Conference on Tourism Management and Sustainable Development, Marvdasht, pp 1–17.
Hengky SH (2017) Probing coastal eco-tourism in Pasir Putih Beach, Indonesia. Bus Manag Horiz 5(1):1–11.
Nuzula NI, Armono HD, Rosyid DM (2017) Management of Baluran National Park resources for coastal ecotourism based on suitability and carrying capacity. Appl Mech Mater 862:161–167.
Shen SY, Niu EX, Meng B (2017) Evaluation of ecological environment carrying capacity in coastal waters of Liaoning based on Grey relation. J Dalian Maritime Univ 43(3):112–118.
Shi LY, Zhao HB, Li YL, Ma H, Yang SC, Wang HW (2015) Evaluation of Shangri-La County's tourism resources and ecotourism carrying capacity. Int J Sust Dev World 2(2):103–109.
Siswanto A (2015) Eco-tourism development strategy Balurannational Park in the regency of Situbondo, East Java, Indonesia. Int J Eval Res Educ 4(4):185.
Solís D, Corral JD, Perruso L, Agar JJ (2014) Evaluating the impact of individual fishing quotas (IFQs) on the technical efficiency and composition of the US Gulf of Mexico red snapper commercial fishing fleet. Food Policy 46(6):74–83.
Xu ZL, Yan W (2016) Carrying capacity of water environment in public tourism resources based on matter-element model. Ecol Econ 3:296–300.
Ye W, Xu X, Wang H, Wang H, Yang H, Yang Z (2016) Quantitative assessment of resources and environmental carrying capacity in the northwest temperate continental climate ecotope of China. Environ Earth Sci 75(10):868.
Zelenka J, Kacetl J (2014) The concept of carrying capacity in tourism. Amfiteatru Econ 16(36):641–654.
Zhang B (2014) A research on residents' social-psychological carrying capacity of tourism destinations: a case study on Asakusa area, Tokyo, Japan. Tour Tribune 29(12):55–65.
This study was supported by Henan Provincial Government Decision Making Research (project number: 2012B116).
All the data have been included in the article.
Henan University of Urban Construction, Longxiang Avenue, Pingdingshan City, 467000, Henan Province, China
Yufeng Zhao
& Lei Jiao
Search for Yufeng Zhao in:
Search for Lei Jiao in:
YFZ analyzed the low-carbon tourism mode, looked up the advantages of developing low-carbon tourism in natural conditions and tourism resources in Guizhou, estimated the carbon emissions of Guizhou tourism industry in 2011–2015, discussed and analyzed the results together with LJ, and drafted the article. LJ assisted YFZ to collect the data needed for estimating the carbon emissions of Guizhou tourism industry from 2011 to 2015 and calculate and analyze them. He put forward the main problems existing in the development of low-carbon tourism; put forward corresponding suggestions from the perspectives of government, tourism enterprises, and tourists; and perfected the article. Both authors read and approved the final manuscript.
Correspondence to Yufeng Zhao.
Zhao, Y., Jiao, L. Resources development and tourism environmental carrying capacity of ecotourism industry in Pingdingshan City, China. Ecol Process 8, 7 (2019) doi:10.1186/s13717-019-0161-0
Received: 13 November 2018
Accepted: 21 February 2019
Ecotourism resources
Environmental carrying capacity
Resource-based city | CommonCrawl |
Find the number of real solutions $(x,y,z,w)$ of the simultaneous equations
\begin{align*}
2y &= x + \frac{17}{x}, \\
2z &= y + \frac{17}{y}, \\
2w &= z + \frac{17}{z}, \\
2x &= w + \frac{17}{w}.
\end{align*}
By inspection, $(\sqrt{17},\sqrt{17},\sqrt{17},\sqrt{17})$ and $(-\sqrt{17},-\sqrt{17},-\sqrt{17},-\sqrt{17})$ are solutions. We claim that these are the only solutions.
Let
\[f(x) = \frac{1}{2} \left( x + \frac{17}{x} \right) = \frac{x^2 + 17}{2x}.\]Then the given equations become $f(x) = y,$ $f(y) = z,$ $f(z) = w,$ and $f(w) = x.$ Note that none of these variables can be 0.
Suppose $t > 0.$ Then
\[f(t) - \sqrt{17} = \frac{t^2 + 17}{2t} - \sqrt{17} = \frac{t^2 - 2t \sqrt{17} + 17}{2t} = \frac{(t - \sqrt{17})^2}{2t} \ge 0,\]so $f(t) \ge \sqrt{17}.$ Hence, if any of $x,$ $y,$ $z,$ $w$ are positive, then they are all positive, and greater than or equal to $\sqrt{17}.$
Furthermore, if $t > \sqrt{17},$ then
\[f(t) - \sqrt{17} = \frac{(t - \sqrt{17})^2}{2t} = \frac{1}{2} \cdot \frac{t - \sqrt{17}}{t} (t - \sqrt{17}) < \frac{1}{2} (t - \sqrt{17}).\]Hence, if $x > \sqrt{17},$ then
\begin{align*}
y - \sqrt{17} &< \frac{1}{2} (x - \sqrt{17}), \\
z - \sqrt{17} &< \frac{1}{2} (y - \sqrt{17}), \\
w - \sqrt{17} &< \frac{1}{2} (z - \sqrt{17}), \\
x - \sqrt{17} &< \frac{1}{2} (w - \sqrt{17}).
\end{align*}This means
\[x - \sqrt{17} < \frac{1}{2} (w - \sqrt{17}) < \frac{1}{4} (z - \sqrt{17}) < \frac{1}{8} (y - \sqrt{17}) < \frac{1}{16} (x - \sqrt{17}),\]contradiction.
Therefore, $(\sqrt{17},\sqrt{17},\sqrt{17},\sqrt{17})$ is the only solution where any of the variables are positive.
If any of the variables are negative, then they are all negative. Let $x' = -x,$ $y' = -y,$ $z' = -z,$ and $w' = -w.$ Then
\begin{align*}
2y' &= x' + \frac{17}{x'}, \\
2z' &= y' + \frac{17}{y'}, \\
2w' &= z' + \frac{17}{z'}, \\
2x' &= w' + \frac{17}{w'},
\end{align*}and $x',$ $y',$ $z',$ $w'$ are all positive, which means $(x',y',z',w') = (\sqrt{17},\sqrt{17},\sqrt{17},\sqrt{17}),$ so $(x,y,z,w) = (-\sqrt{17},-\sqrt{17},-\sqrt{17},-\sqrt{17}).$
Thus, there are $\boxed{2}$ solutions. | Math Dataset |
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On privacy, research, and privacy research.
Demystifying the US Census Bureau's reconstruction attack
2021-05-26 — updated 2021-07-26
This post is part of a series on differential privacy. Check out the table of contents to see the other articles!
Every 10 years, the US Census Bureau asks every American household a series of questions. How many people are living here? How old are they? What is their race and ethnicity? What is their relationship to each other?
The collected answers have very high quality, for two reasons. First, all households have to respond truthfully to these questions, by law. Second, the US Census Bureau has a legal duty to keep the answers secret for 72 years after each survey. Both aspects are key to convince everyone to answer truthfully. Appealing to people's sense of civic duty also helps!
What is the data used for, then? The Census Bureau aggregates it and publishes statistics about the US population. A lot of statistics: in 2010, it released over 150 billion statistics about the US population. These statistics then have many applications: scientific research, informing public policy, determining election districts, etc.
Confidentiality is central to the Census Bureau's mission. By law, they have to make sure that nobody can use their data to find out information about individuals. Disclosing such information even comes with criminal penalties! This has led to the creation of an entire field of study: statistical disclosure control. It predates even the oldest definition of privacy mentioned on this blog (k-anonymity).
How did statistical disclosure control work in practice? Before 1990, the method was pretty blunt: they removed the tables with fewer than five individuals or households in them. Then, from 1990 to 2010, the Census Bureau used a more complex technique called swapping. Swapping works in two steps. First, it selects households at random in small geographic areas: those are the ones most at risk of re-identification. Then, it exchanges records between these households and others before generating the statistics. The details of this swapping process were secret, to make it harder for people to design attacks.
Nowadays, the Census Bureau is moving towards formal notions: the statistics published for the 2020 Census will be differentially private. If you've read the previous articles of this blog, this might not surprise you. Differential privacy (DP) is designed to solve this exact problem: publishing statistics without revealing information about individuals. But this choice was far from obvious! Adding noise to statistics is quite scary for data users1. Using DP for such a complex release is also unprecedented.
So, what convinced the Census Bureau to take this decision? Their scientists ran an attack on some of the 2010 data, to better understand the privacy risks. And they realized that the attack was much more successful than they expected. The attack is simple, not very expensive, and pretty scary. The scientists then measured how much more swapping it would take for the attack to fail… and observed that the utility loss would be catastrophic. Older techniques like suppression were also ineffective. The only way to mitigate the risks and limit accuracy loss was differential privacy.
This choice, of course, has other benefits. Differential privacy provides quantifiable guarantees. It composes well, and protects even against very powerful attackers. It's also good for transparency: for the first time, the Census Bureau will be able to publish the details of their process. But these nice theoretical properties weren't the main factor in their decision. Instead, the choice of DP was pragmatic: it worked best to prevent realistic attacks without sacrificing too much utility.
In this blog post, we'll take a look at the details of this compelling attack, at the core of this decision. The attack has two stages: reconstruction and reidentification.
The first step in the attack is to reconstruct records, using statistical data. The statistical data is as follows. In each geographic area (like Census blocks), we can get the answers to questions like: how many people of age 47 live there? How many people between the ages of 25 and 29 self-identify as white? How many households with 2 adults and 2 children are there?
How can we use this kind of information and reconstruct the original records? Let's take a simplified example. Say that a hypothetical Census block has four people in it.
17 Asian
Now, suppose that we have the following statistical data about this Census block.
There are four people in total.
Two of these people have age 17.
Two of these people self-identify as White.
Two of these people self-identify as Asian.
The average age of people who self-identify as White is 30.
The average age of people who self-identify as Asian is 32.
This data is statistical in nature: these numbers are all aggregated over several people (here, two). Yet, it's not difficult to guess what the original table looks like based on the numbers.
Not obvious yet? Take the two people aged 17. Points 1, 3 and 4 tell us that:
either they both self-identify as White,
either they both self-identify as Asian,
either one of them self-identifies as White and the other as Asian.
The first option is impossible: if they both self-identified as White, then their average age should be 17, not 30 (point 5). The second option is also impossible, for the same reason (with point 6). So the third option is correct. We now know the first two records, and we can find the age of both others using the average age of each race group. It's like a fun puzzle!
That's the idea behind reconstruction attacks: taking statistical information and using simple reasoning to reverse-engineer the original records. Of course, when we have billions of statistics to work with, we don't do this by hand. Instead, we convert the data into a massive system of equations. Each piece of individual information is an unknown variable. The data gives us the relationships between them. Then, we can use a big computer to find a possible solution to this large system. This solution, in turn, gives us reconstructed records.
So, the team at the Census Bureau did exactly that, with statistical data from the 2010 Census. They transformed it into many equations, and used Gurobi to reconstruct the raw data. The records they obtained matched 46% of the original records exactly. That's pretty impressive! Especially since only a small fraction of the statistics were used in the attack (6.2 billion out of 150 billion). Swapping was not enough to prevent accurate reconstruction.
At first glance, that result looks pretty scary. But let's take a step back: how bad is it really? What does this 46% number actually tell us?
How bad is reconstruction?
Imagine that a given Census block has a particularly homogeneous population: out of 100 habitants, 95 all self-identify as White, and are evenly distributed between the ages of 20 and 393. The other 5% of people in this block do not belong to this demographic, and we don't have any information about them.
Can we "reconstruct" this dataset with high accuracy? Yes, and it will be easier than we expect: simply use the data on the majority group, and ignore the minority population. In practice, for each age between 20 and 39, we output 5 records with this age, and White as a race.
Given the statistics on this block, it's likely that our naive guess is pretty close to the truth. After all, only 5% of people don't belong to the majority demographic, and we know the distribution of the other 95%. Recall the success metric from the previous section: the percentage of matching records between real data and reconstructed data. According to this metric, our naive strategy performs very well! Accuracy is about 95%, if the age distribution is very uniform.
Has this process revealed sensitive information? Well… not really, right? All reconstructed records are identical across many people. So they don't seem to reveal very sensitive information… And the people in demographic minorities are safe from the attack.
It's also not clear yet how problematic these reconstructed records are. All we get is a list of records that are compatible with the published statistics. But how do we know which ones are actually correct? And how can we use them to learn something new and sensitive about individuals?
These questions show that on its own, the 46% number from the Census Bureau doesn't tell us much. But the Census Bureau didn't stop at reconstruction. The attack has a second step, re-identification, and this one gets much scarier results. Let's take a look at how it works.
Re-identification
In the Census Bureau attack, the reconstruction step outputs records with five characteristics: Census block, age, sex, race, and ethnicity. The idea of the re-identification attack is simple. First, they buy the kind of commercial data that an attacker could have access to. Second, they link this information with reconstructed records.
This "commercial data" is exactly what it sounds like: information about people, compiled by random companies, and made available to anyone who pays for it (or steals it). These companies, called "data brokers", are largely unregulated in the US. Their privacy practices are about as awful as you can imagine4. For their attack, Census obtained data from five different companies. Their goal was to simulate what an attacker would likely have access to in 2010.
These commercial datasets typically contain people's names, associated with demographic information: location of residence, age (or date of birth), and sex5. We will use these three characteristics to re-identify reconstructed records.
The technique is simple. We look at each record of the commercial dataset, one after the other. And we try to match this record's characteristics with the reconstructed record. Is there a single reconstructed record with the same location, age, and sex? If so, we link both records together. Here is a visualization of that process (with made-up data). The commercial dataset is in blue, the reconstructed Census records are in green.
Linking datasets in this way achieves two goals.
It confirms that the reconstructed record corresponds to a specific individual. In fact, it allows us to re-identify the reconstructed record. Here, the commercial data tells us that this person is James Link.
It gives us more information about this person, which the commercial data didn't have. Here, we learn James Link's race and ethnicity.
How do we quantify the success of this attack? We can look at two numbers. First, how many records can be linked between datasets in this way? Second, out of these linked records, how many are accurate? After all, some of these re-identifications might be wrong: both datasets might be incomplete or inaccurate. The percentage of correct answers is the precision of the attack:
$$ \text{precision} = \frac{\text{true re-identifications}}{\text{number of linked records}}. $$
So how does the attack perform? Census Bureau scientists linked 138 million records between both datasets. And their average precision was 38%. This means that the attack successfully re-identified 52 million records. Scary!
One could argue that the attacker can't know which records are correctly re-identified. Some of them will be false positives! After all, 38% aren't great odds. But the analysis from Census Bureau scientists doesn't stop there. Two further considerations make the attack even more compelling.
The precision of re-identification goes up for people in small Census blocks: it's 72% on the smallest Census blocks (in which 8 million people live). This makes sense: statistics across a few people are more revealing than aggregates over large groups. It's not surprising, but it's still bad news: it means that folks in minority populations are more at risk of being re-identified. But disclosure avoidance is precisely trying to protect these people!
The precision also goes up if one has better-quality data. Census Bureau scientists use a neat trick to find worst-case guarantees: they use the raw Census data itself as an approximation of the best possible data an attacker could find. Using this, they show that the global precision increases to 75%, and even goes up to 97% on small Census blocks.
Is this assumption of high-quality data unrealistic? No, for two reasons.
The commercial data used in the attack is what someone could have had access to in 2010 — more than 10 years ago. Data brokers have much better data available for sale by now.
The re-identification step only requires high-quality data about our targets. The attack works just fine even if we're trying to re-identify a single person, or a handful of people. With high-quality information about them, the precision goes up to these worst-case numbers.
There's nothing preventing businesses from running this attack on their employees or customers. In this kind of context, access to high-quality data isn't an optimistic assumption: it's a given.
This attack is bad news.
It proves two points beyond doubt. First, anyone can reconstruct Census records. Many of these reconstructed records are accurate, especially in small Census blocks. This does not require massive computing power: you can run smaller-scale attacks in minutes on your average laptop. Second, high-quality data about some people is enough to re-identify their Census records. This re-identification has high precision, and reveals previously-unknown information about these people.
In the attack, the attacker could learn race and ethnicity information. Data brokers might want to do this to augment their datasets, for example. But the risks can get even more tangible. Remember: the attack only used a small fraction of all published statistics. The targeted tables had only demographic information. What if someone were to attack household tables instead? This could likely reveal intimate details about the people you live with. Do you have a have a partner of the same sex? Children of a different race? Anyone with basic information about you — employer, acquaintance, data broker… — might find out.
Some of that information is particularly sensitive, and otherwise unavailable via data brokers. For example, reconstruction also works on children's data, which is illegal to sell in the US. Someone could combine household reconstruction with re-identification, and locate kids of a specific race, sex and age range in a given area. No need to spell out the possible harms this could enable.
Risks to individuals can translate to risks to data quality in the long run. Suppose Census data can no longer be considered as confidential. How will it impact people's willingness to answer future Census surveys? Minority groups are already more likely to have privacy concerns about their answers. This will get worse if the Census Bureau can no longer guarantee the confidentiality of the data.
This explains the Census Bureau's decision to move to differential privacy. Everything we described is just one possible attack, and it used only a fraction of the released data. The Census must protect all other attacks that people might come up with in the future! And that's exactly what differential privacy provides: provable privacy guarantees, even against attacks that haven't invented yet.
If you'd like to learn more about differential privacy, I have good news for you: this blog also contains an entire article series about this notion, introducing it in friendly, non-technical terms. Head over to the table of contents to see the other posts!
Nice, you made it all the way to the end of this article! Here are some more links and disclaimers.
This article is almost entirely sourced using documents from a lawsuit. The main one is this declaration from John Abowd, the Census Bureau's chief scientist. It explains the history and context behind the use of DP for the 2020 Census. It starts on page 85 of this PDF, and Appendix B (starting on page 147) describes the attack in more detail. Some of the numbers also come from Abowd's supplemental declaration.
This last declaration has been filed as a response to a filing by a couple of demographers. They make an argument similar to the one outlined in the second part of this article: you can reconstruct records by picking them randomly, so reconstruction doesn't mean anything. Hopefully, this post managed to convinced you that this argument has two flaws: it ignores the higher risks for minority groups, and it fails to address re-identification, the second part of the attack.
Still hungry for more legal filings? Many leading data privacy experts filed an amicus brief supporting the use of DP for the 2020 Census. It's clear, concise, and makes a lot of great points. Worth a read!
In case this wasn't obvious, this article made a lot of simplifications. I rounded all numbers so they would look nicer. I optimized the reconstruction example for clarity and fun, not for accuracy: in particular, the Census Bureau doesn't actually release statistics like "average age". The real attack doesn't use only a system of equations for reconstruction, but inequalities as well; I also suppose that there is some optimization done, but the details aren't public. The linking attack has some additional logic to do fuzzy matching. And I'm sure I made other shortcuts along the way. If you're looking for more accurate information, you will probably find it in the documents linked above.
There's a lot more to say about swapping, too. For example, it has surprising negative effects on data quality! Since folks in demographic minorities in each location are more likely to have their record swapped, the process biases the data: it makes all areas seem more homogeneous than they actually are. Recall that the details of swapping are secret: data users couldn't quantify such effects to take them into account in their analysis! Interestingly, the attack run by Census Bureau scientists ignored swapping entirely. A more clever attack might take it into account, and attempt to reverse it. This could make the attack even more accurate, especially for folks in demographic minorities.
Finally: I have not been involved in any way with Census work. I'm thankful to Aloni Cohen, Cynthia Dwork, Thomas Steinke, Kunal Talwar, and Yuan Yuan Zheng for helpful comments and suggestions on drafts of this post. Of course, if there are inaccuracies or errors left, that's entirely on me — please let me know if you find any!
The people using the data: scientists, people drawing electoral district boundaries, public agencies, businesses, etc. ↩
We're using the classification from the Census here. Census data also has a separate Hispanic origin field, called "ethnicity". Don't ask me to explain this, I have no idea. ↩
This hypothetical Census block is basically a tech company. ↩
To learn more about this industry, this EFF paper is a pretty solid resource. ↩
Or gender, depending on the dataset. These are not the same thing, but the Census asks about sex, so we'll assume the commercial datasets use the same notion. This inevitably introduces errors and feels a bit icky. ↩
All opinions here are my own, not my employer's. | Feedback on these posts are very welcome! Please reach out via e-mail (se.niatnofsed@neimad) or Twitter (@TedOnPrivacy) for comments and suggestions. | Interested in deploying formal anonymization methods? My colleagues and I at Tumult Labs can help. Contact me at oi.tlmt@neimad, and let's chat!
by Damien Desfontaines — — propulsed by Pelican | CommonCrawl |
\begin{document}
\title{The Number of Point-Splitting Circles} \author{Federico Ardila M.} \date{August 3, 2001} \maketitle
\begin{abstract} Let $S$ be a set of $2n+1$ points in the plane such that no three are collinear and no four are concyclic. A circle will be called {\it point-splitting} if it has 3 points of $S$ on its circumference, $n-1$ points in its interior and $n-1$ in its exterior. We show the surprising property that $S$ always has exactly $n^2$ point-splitting circles, and prove a more general result. \end{abstract}
\section{Introduction}
Our starting point is the following problem, which first appeared in the 1962 Chinese Mathematical Olympiad \cite{4}.
{\bf Problem 1.1.} Let $S$ be a set of $2n+1$ points in the plane such that no three are collinear and no four are concyclic. Prove that there exists a circle which has 3 points of $S$ on its circumference, $n-1$ points in its interior, and $n-1$ in its exterior.
Following \cite[p.48]{3}, we call such a circle {\it point-splitting} for the given set of points. For the rest of sections 1 and 2, $S$ denotes an arbitrary set of $2n+1$ points in general position in the plane, where $n$ is a fixed integer.
There are several solutions to problem 1.1. Perhaps the easiest one is the following. Let $A$ and $B$ be two consecutive vertices of the convex hull of $S$. We claim that some circle going through $A$ and $B$ is point-splitting. All circles through $A$ and $B$ have their centers on the perpendicular bisector $\ell$ of the segment $AB$. Pick a point $O$ on $\ell$ which lies on the same side of $AB$ as $S$, and is so far away from $AB$ that the circle $\Gamma$ with center $O$ and going through $A$ and $B$ completely contains $S$. This can clearly be done. Now slowly ``push" $O$ along $\ell$, moving it towards $AB$. The circle $\Gamma$ will change continuously with $O$. As we do this, $\Gamma$ will stop containing some points of $S$. In fact, it will lose the points of $S$ one at a time: if it lost $P$ and $Q$ simultaneously, then points $P,Q,A$ and $B$ would be concyclic. We can move $O$ so far away past $AB$ that, in the end, the circle will not contain any points of $S$.
Originally, $\Gamma$ contained all the points of $S$. Now, as it loses one point of $S$ at a time, we can decide how many points we want it to contain. In particular, if we stop moving $O$ when the circle is about to lose the $n$-th point $P$ of $S$, then the resulting $\Gamma$ will be point-splitting: it will have $A$, $B$ and $P$ on its circumference, $n-1$ points inside it, and $n-1$ outside it, as shown in Figure 1.
\begin{figure}
\caption{A point-splitting circle through $A, B$ and $P$}
\end{figure}
The above proof hints that any set $S$ has several different point-splitting circles. We can certainly construct one for each pair of consecutive vertices of the convex hull of $S$. In fact, the argument above can be modified to show that, for {\bf any} two points of $S$ we can find a point-splitting circle going through them. The reader might find it instructive to work out a proof.
This suggests that we ask the following question. What can we say about the number $N_S$ of point-splitting circles of $S$? At first sight, it seems that we really cannot say very much about this number. Point-splitting circles seem hard to ``control", and harder to count.
We should be able to find upper and lower bounds for $N_S$ in terms of $n$. Right away we know that $N_S \geq n(2n+1)/3$, since we can find a point-splitting circle for each pair of points of $S$, and each such circle is counted by three different pairs. Computing an upper bound seems more difficult. If we fix points $A$ and $B$ of $S$, it is indeed possible that all $2n-1$ circles through $A$, $B$ and another point of $S$ are point-splitting. The reader is invited to check this. This is not likely to happen very often in a set $S$, and we can get {\it some} upper bound out of this. However, it is hard to make this precise and get a non-trivial upper bound.
When $S$ consists of 5 points, the situation is simple enough that we can actually show that $N_S=4$ always. This was done in \cite{2}. It was also proposed, but not chosen, as a problem for the 1999 International Mathematical Olympiad. Notice that our lower bound above gives $N_S \geq 4$.
In a different direction, problem 5 of the 1998 Asian-Pacific Mathematical Olympiad, proposed by the author, stated the following.
{\bf Proposition 1.2.} $N_S$ has the same parity as $n$.
This result follows easily from the nontrivial observation that, for any $A$ and $B$ in $S$, the number of point-splitting circles that go through $A$ and $B$ is odd.
The following result brings together the above considerations.
{\bf Theorem 1.3.} Any set $S$ of $2n+1$ points in the plane in general position has exactly $n^2$ point-splitting circles.
Theorem 1.3 is the main result of this paper. In section 2 we prove that every set of $2n+1$ points in the plane in general position has the same number of point-splitting circles. In section 3 we prove that this number is exactly $n^2$. In section 4 we present some questions that arise from our work.
\section{$N_S$ is Constant}
At this point, we could go ahead and prove the very counterintuitive Theorem 1.3, suppressing the motivation behind its discovery. With the risk of making the argument seem longer, we believe that it is worthwhile to present a natural way of realizing and proving that the number of point-splitting circles of $S$ depends only on $n$. Therefore, we ask the reader to forget momentarily the punchline of this article.
Suppose that we are trying to find out whatever we can about the number $N_S$. As mentioned in Section 1, this number does not seem very tractable and it is not clear how much we can say about it. Being optimistic, we can hope to be able to answer the following two questions.
{\bf Question 2.1}. What are the sharp lower and upper bounds $m = m_{2n+1}$ and $M = M_{2n+1}$ for $N_S$?
{\bf Question 2.2}. What are all the values that $N_S$ takes in the interval $[m, M]$?
Question 2.1 seems considerably difficult. To answer it completely, we would first need to prove an inequality $m \leq N_S \leq M$, and then construct suitable sets $S_{min}$ and $S_{max}$ which achieve these bounds. To see how difficult this is, the reader is invited to try to construct {\it any} set $S$ of $2n+1$ points for which the number $N_S$ can be easily computed.
At this point question 2.1 seems very hard, so let us focus on Question 2.2 instead. Here is a first approach.
Intuitively, since the set $S$ can be transformed continuously, we should expect the value of $N_S$ to change ``continuously" with it. Suppose we start with the set $S_{min}$ (with $N_S=m$) and move its points continuously so that we end up with $S_{max}$ (with $N_S=M$). The value of $N_S$ should change ``continuously" as $S$ changes continuously. By ``continuity" we would guess that $N_S$ sweeps all the integers between $m$ and $M$ as $S$ changes from $S_{min}$ to $S_{max}$.
Right away, we know that this is not entirely true. By Proposition 1.2 we know that the parity of $N_S$ is determined by $n$, so $N_S$ will not sweep {\bf all} the integers between $m$ and $M$. This is not too surprising, since we haven't made precise the meaning of the statement that the value of $N_S$ should change ``continuously" as $S$ changes continuously. The above guess assumed that the value of $N_S$ can only jump by 1 as $S$ is transformed continuously. (That is, if we have a set with $k-1$ point-splitting circles and we deform it continuously into a set with $k+1$ point-splitting circles, then somewhere in the middle we must have had a set with $k$ point-splitting circles.) We have no reason to assume that.
We can still hope that, as $S$ changes, $N_S$ sweeps all the integers {\bf of the right parity} between $m$ and $M$. To show this, we would have to show that the value of $N_S$ can only jump by 2 as $S$ is transformed continuously. This is a reasonable statement which we can try to prove.
In any case, the natural question to ask is what kind of ``continuity" the value of $N_S$ satisfies as $S$ changes continuously. We certainly expect that if two sets $S$ and $T$ look very very much alike, then the difference $N_S-N_T$ should be small. We have to find a way to make this statement precise.
Suppose we have sets $S_{min}=\{P_1, \ldots, P_{2n+1}\}$ and $S_{max}=\{Q_1, \ldots, Q_{2n+1}\}$ that achieve the upper and lower bounds for $N_S$, respectively. Now slowly transform $S_{min}$ into $S_{max}$: first send $P_1$ to $Q_1$ continously along some path, then send $P_2$ to $Q_2$ continuously along some other path, and so on. We can think of our set $S$ as changing with time. At the initial time $t=0$, our set is $S(0)=S_{min}$. At the final time $t=T$, our set is $S(T)=S_{max}$. In between, $S(t)$ varies continously with respect to $t$. How does $N_{S(t)}$ vary ``continuously" with time? How small can we make $N_{S(t + \Delta t)} - N_{S(t)}$ for small enough $\Delta t$? This is the question we need to ask.
{\bf Technical Remark.} As we move from $S(0)$ to $S(T)$ continuously, it is likely that several intermediate sets $S(t)$, with $0 < t < T$, are not in general position. Strictly speaking, we should only consider those times $t$ when $S(t)$ is in general position; when $S(t)$ is not in general position, we should decree that $S(t)$ is undefined, and have a discontinuity at $t$.
We shall see that we can go from $S(0)$ to $S(T)$ with only finitely many such discontinuous points. At such a discontinuity $t$, we still need to know how small we can make $N_{S(t + \Delta t)} - N_{S(t - \Delta t)}$ for small $\Delta t$.
For small enough $\Delta t$, the set $S(t + \Delta t)$ is a very slight deformation of $S(t)$. What is missing is an understanding of what can make $N_S$ change as the set $S$ changes very slightly from $S(t)$ to $S(t+\Delta t)$, and how small this change is. Let us answer this question.
Notice that, in the way we defined the deformation from $S_{min}$ to $S_{max}$, the points of $S$ moved only one at a time. Let us focus for now on the interval of time where $P_1$ moves towards $Q_1$.
Suppose that the number $N_S$ changes between time $t$ and time $t + \Delta t$. Then it must be the case that for some $i,j,k$ and $l$ the circle $P_iP_jP_k$ contained (or did not contain) point $P_l$ at time $t$, but at time $t + \Delta t$ it does not (or does) contain it. For this to be true, it must have happened that somewhere between times $t$ and $t + \Delta t$, these four points must have been concyclic, or three of them must have been collinear. Since $P_1$ is the only point that has moved, we can conclude that $P_1$ must have crossed a circle or a line determined by the other points; this is what caused $N_S$ to change. We will call the circles and lines determined by the points $P_2, P_3, \ldots, P_{2n+1}$ the {\it boundaries}.
We can choose the path along which $P_1$ is going to move towards $Q_1$. To make things easier, we may assume that $P_1$ never crosses two of the boundaries at the same time. This can clearly be guaranteed: we know that these boundaries intersect pairwise in finitely many points, and all we have to do is avoid these intersection points in the path from $P_1$ to $Q_1$. We can also assume that $\Delta t$ is small enough that $P_1$ crosses exactly one boundary between times $t$ and $t+\Delta t$. Let us see how $N_S$ changes in this time interval.
It will be convenient to call a circle $P_iP_jP_k$ {\it $(a,b)$-splitting} (where $a+b = 2n-2$) if it has $a$ points of $S$ inside it and the remaining $b$ outside it. For example, an $(n-1,n-1)$-splitting circle is just a point-splitting circle.
\begin{figure}
\caption{$P_1$ crosses line $P_iP_j$.}
\end{figure}
First assume that $P_1$ crosses line $P_iP_j$, going from position $P_1(t) = A$ to position $P_1(t+\Delta t) = B$. From the remarks made above, we know that only circle $P_1P_iP_j$ can change the value of $N_S$ by becoming or ceasing to be point- splitting. Assume that circle $AP_iP_j$ was $(a,b)$-splitting. Since $P_1$ only crossed the boundary $P_iP_j$ when going from $A$ to $B$, the region common to circles $AP_iP_j$ and $BP_iP_j$ cannot contain any points of $S$, as indicated in Figure 2. The region outside of both circles cannot contain points of $S$ either. For circle $AP_iP_j$ to be $(a,b)$-splitting, the other two regions must then contain $a$ and $b$ points respectively, as shown. Therefore, circle $BP_iP_j$ is $(b,a)$-splitting. It follows that $AP_iP_j$ was point-splitting if and only if $BP_iP_j$ is point-splitting (if and only if $a=b=n-1$). We conclude that the value of $N_S$ doesn't change when $P_1$ crosses a line determined by the other points; it can only change when $P_1$ crosses a circle.
Now assume that $P_1$ crosses circle $P_iP_jP_k$, going from position $P_1(t) = A$ inside the circle to position $P_1(t+\Delta t) = B$ outside it. (The other case, when $P_1$ moves inside the circle, is analogous.) We can assume that $P_1$ crossed the arc $P_iP_j$ of the circle that doesn't contain point $P_k$. Notice that $A$ must be outside triangle $P_iP_jP_k$ if we want $P_1$ to cross only one boundary in the time interval considered. Assume that circle $AP_iP_j$ was $(a,b)$-splitting. As before, we know that the only regions of Figure 3 containing points of $S$ are the one common to circles $AP_iP_j$ and $BP_iP_j$, and the one outside both of them. They must contain $a-1$ and $b$ points respectively, for circle $AP_iP_j$ to be $(a,b)$-splitting. In this case, the value of $N_S$ can change only by circles $P_iP_jP_k$, $P_1P_jP_k$, $P_1P_kP_i$ and $P_1P_iP_j$ becoming or ceasing to be point-splitting. It is clear that circle $P_iP_jP_k$ went from being $(a,b)$-splitting to being $(a-1,b+1)$-splitting. The same is true of circle $P_1P_iP_j$.
\begin{figure}
\caption{$P_1$ crosses circle $P_iP_jP_k$.}
\end{figure}
It is also not hard to see, by a similar argument, that circles $P_1P_jP_k$ and $P_1P_kP_i$ both went from being $(a-1,b+1)$-splitting to being $(a,b)$-splitting. Again, the key assumption is that $P_1$ only crossed the boundary $P_iP_jP_k$ in this time interval.
So, by having $P_1$ cross circle $P_iP_jP_k$, we have traded two $(a,b)$-splitting and two $(a-1,b+1)$-splitting circles for two $(a-1,b+1)$-splitting and two $(a,b)$-splitting circles, respectively. It follows that the number $N_S$ of point-splitting circles remains constant when $P_1$ crosses a circle $P_iP_jP_k$ also.
We had shown that, as we moved $P_1$ to $Q_1$, $N_S$ could only possibly change in a time interval when $P_1$ crossed a boundary determined by the other points. But now we see that, even in such a time interval, $N_S$ does not change! Therefore moving $P_1$ to $Q_1$ doesn't change the value of $N_S$. Similarly, moving $P_i$ to $Q_i$ doesn't change $N_S$ either, for any $1 \leq i \leq 2n+1$. It follows that $N_S$ is the same for $S_{min}$ and $S_{max}$. In fact, $N_S$ is the same for any set $S$ of $2n+1$ points in general position!
\section {$N_S = n^2$}
Now that we know that the number $N_S$ depends only on the number of points in $S$, define $N_{2n+1}$ to be the number of point-splitting circles for a set of $2n+1$ points in general position. We compute $N_{2n+1}$ recursively.
Construct a set $S$ of $2n+1$ points as follows. First consider the vertices of a regular $2n-1$-gon with center $O$. Now move them very slightly to positions $P_1, \ldots, P_{2n-1}$ so that they are in general position. The difference will be so slight that all the lines $OP_i$ still split the remaining points into two sets of equal size, and all the circles $P_iP_jP_k$ still contain $O$. Also consider a point $Q$ which is so far away from the others that it lies outside of all the circles formed by the points considered so far. Of course, we need $Q$ to be in general position with respect to the remaining points. Let us count the number of point-splitting circles of $S=\{O, P_1, \ldots, P_{2n-1}, Q\}$.
First consider the circles of the form $P_iP_jP_k$. These circles contain $O$ and don't contain $Q$; so they are point-splitting for $S$ if and only if they are point-splitting for $\{P_1, \ldots, P_{2n-1}\}$. Thus there are $N_{2n-1}$ such circles.
Next consider the circles $OP_iP_j$. It is clear that these circles contain at most $n-2$ other $P_k$'s. They do not contain $Q$, so they contain at most $n-2$ points, and they are not point-splitting.
Finally consider the circles that go through $Q$ and two other points $X$ and $Y$ of $S$. Circle $QXY$ splits the remaining points in the same way that line $XY$ does. More specifically, circle $QXY$ contains a point $P$ of $S$ if and only if $P$ is on the same side of line $XY$ that $Q$ is. This follows easily from the fact that $Q$ lies outside circle $PXY$. Therefore we have to determine how many lines determined by two of the points of $S-\{Q\}$ split the remaining points of this set into two sets of $n-1$ points each. This question is much easier; it is clear from our construction that the lines $OP_i$ do this and the lines $P_iP_j$ do not. It follows that the $2n-1$ circles $OP_iQ$ are point-splitting, and the circles $P_iP_jQ$ are not.
Summarizing, the point-splitting circles of $S$ are the $N_{2n-1}$ point-splitting circles of $S-\{O,Q\}$ and the $2n-1$ circles $OP_iQ$. Therefore $N_{2n+1}=N_{2n-1}+2n-1$. Since $N_3=1$, it follows inductively that $N_{2n+1}=n^2.$ This completes the proof of Theorem 1.3.
{\bf Theorem 3.1.} Consider a set of $2n+1$ points in general position in the plane, and two non-negative integers $a<b$ such that $a+b=2n-2$. There are exactly $2(a+1)(b+1)$ circles which are either $(a,b)$-splitting or $(b,a)$-splitting for the set of points.
{\it Sketch of Proof.} The argument of Section 2 carries directly to this situation, to show that the number of circles in consideration, which we denote $N(a,b)$, only depends on $a$ and $b$. Therefore it suffices to compute it recursively, using the set $S$ above. It is essential in the proof that $a<n-1$.
Just as above, there are $N(a-1,b-1)$ such circles among the circles $P_iP_jP_k$. Among the $OP_iP_j$ there are exactly $2n-1$ such circles, namely the circles $OP_iP_{i+a+1}$ (taking subscripts modulo $2n-1$). There are also $2n-1$ such circles among the $QP_iP_j$, namely the circles $QP_iP_{i+a+1}$. Finally, there are no such circles among the $OP_iQ$. Therefore $N(a,b) = N(a-1,b-1) + 4n - 2 = N(a-1,b-1) + 2a + 2b + 2$.
Repeating the above argument for $a=0$, we get that $N(0,b)=2b+2$. If we combine this and the recursive relation obtained, Theorem 3.1 follows by induction.
It is worth mentioning at this point that Theorems 1.3 and 3.1 are closely related to a beautiful result of D.T. Lee, which gives a sharp bound for the number of vertices of an order $j$ Voronoi diagram. The connection is obtained if we embed our set $S$ of points on the surface of a sphere. Then the point-splitting circles of $S$ are put in correspondence with the ``point-splitting planes" of a three-dimensional convex polytope with $2n+1$ vertices. These are known to be related to Voronoi diagrams. See \cite[p. 397]{1} for more details on this, and a proof of a result essentially equivalent to Theorems 1.3 and 3.1.
\section{Questions}
Our work completely determines the number of point-splitting circles, as well as the total number of $(a,b)$-splitting and $(b,a)$-splitting circles for a set of points in general position in the plane. However, we know very little about these numbers for sets of points that are not in general position.
The situation here is much more subtle. For example, the number of point-splitting circles of a set $S$ is not uniquely determined by the subsets of $S$ which are concyclic. Consider the following example. Let $S_1$ and $S_2$ be the two sets of seven points shown in Figure 4. Both of them are almost in general position; the only exception is that, for each of the two sets, there is a circle going through four points of the set. In $S_1$, this circle $\Gamma_1$ contains exactly one point of $S_1$ inside it. In $S_2$, this circle $\Gamma_2$ contains no points of $S_2$ inside it. In analogy with Theorem 1.3, where it did not matter which points were inside which circles, we might hope that $S_1$ and $S_2$ have the same number of point-splitting circles.
\begin{figure}
\caption{$N_{S_1} = 8$ and $N_{S_2} = 9$.}
\end{figure}
Unfortunately this is not the case. If we move $A,B,C$ or $D$ very slightly to put $S_1$ in general position, the resulting set will have $9$ point-splitting circles by Theorem 1.3. It is easy to see that exactly two of $ABC, BCD, CDA$ and $DAB$ are among these circles. When we deform the set back to $S_1$, these $9$ circles will still be point-splitting, but two of them will deform into $\Gamma_1$. So $S_1$ has $8$ point-splitting circles.
Similarly, if we move $a,b,c$ or $d$ very slightly to put $S_2$ in general position, the resulting set will have $9$ point-splitting circles. But now we can see that when we deform the set back to $S_2$, none of these circles will deform into $\Gamma_2$, because $\Gamma_2$ contains no points of $S_2$. Therefore $S_2$ has $9$ point-splitting circles.
Even if the number of point-splitting circles is not constant, we might be able to say something about it. As a small example, consider all sets of seven points which are almost in general position, except that four of them are concyclic. It is possible to show, by an argument similar to the above, that such a set can only have $8$ or $9$ point-splitting circles. It seems reasonable that, in general, one might be able to define some measure of how far a set $S$ is from being in general position, and to obtain bounds for $N_S$ in terms of that measure.
\end{document}
\end{document} | arXiv |
\begin{document}
\title{A weak reduction of the Erd{\"o} \begin{abstract} We introduce the theory of \textit{div point sets}, which aims to provide a framework to study the combinatoric nature of any set of points in general position on an Euclidean plane. We then show that proving the unsatisfiability of some first-order logic formulae concerning some sets of multisets of uniform cardinality over boolean variables would prove the Erd{\"o}s-Szekeres conjecture, which states that for any set of $2^{n-2}+1$ points in general position, there exists $n$ points forming a convex polygon, where $n \geq 3$. \end{abstract}
\section{Introduction} In early 20th century Erd{\"o}s and Szekeres \cite{erdos1935} showed that for all $n \geq 3$, there exists some integer $N \geq n$ such that, among any $N$ points in general position on an Euclidean plane, there are $n$ points forming a convex polygon, and conjectured that the smallest value for such $N$ is determined by the function $g$ where $g(n)=2^{n-2}+1$. This is now known as the Erd{\"o}s-Szekeres conjecture (and the problem of determining the smallest $N$ is often referred to as the \textit{Happy Ending Problem} since it led to the marriage of Szekeres and Klein, who first proposed the question). In their second paper, Erd{\"o}s and Szekeres \cite{erdos1960} showed that $g(n)$ is certainly greater than $2^{n-2}$. Currently, the best known bounds are $$2^{n-2}+1 \leq g(n) \leq {2n-5 \choose n-2} +1$$ Throughout the decades many improvements for the upper bound have been made. The current upper bound was obtained by T{\'o}th and Valtr \cite{toth1998} in 1998 as an improvement to the previous one by Kleitman and Pachter \cite{kleitman1998} in the same year.
In 2002, using an exhaustive search algorithm, Szekeres and Peters \cite{szekeres2006} were able to demonstrate that the conjecture holds for $n=6$. To this day it remains the greatest $n$ for which we know for certainty that the smallest $N$ is indeed $2^{n-2}+1$.
The aim of this article is to show that for every $n \geq 5$, there exists an instance of a constraint unsatisfiability problem, which, if solved (i.e. proving that some FOL propositions about certain multisets are unsatisfiable), would prove that the conjecture holds for $n$, through the theory of \textit{div point sets}.
\subsection{preliminary} Throughout the article we would assume Zermelo-Fraenkel set theory (ZF). The word \textit{class} would be used to denote a collection of sets satisfying some predicate $\phi$. Everything would be formulated under first order logic (FOL). $\mathbb{N}_{\geq c}$ would be used to refer to the set of natural numbers greater or equal to some $c \in \mathbb{N}$. For any 2 natural numbers $a$, $b$, $\binom{a}{b}$ denotes the binomial coefficient $a \; choose \; b$. $\land$,$\lor$,$\neg$,$\Rightarrow$ and $\Leftrightarrow$ would be used to mean \textit{and}, \textit{or}, \textit{not}, \textit{imply} and \textit{iff} respectively. We write $A := B$ to mean $A$ is defined to be equivalent to $B$. $\forall x_1\in A \; \forall x_2 \in A \; \forall x_3 \in A \; ... \forall x_n \in A$ would be abbreviated to \[ \forall x_1,x_2,x_3 ... x_n \in A \] and $\exists x_1 \in A \; \exists x_2 \in A \; \exists x_3 \in A \; ... \exists x_n \in A $ to \[ \exists x_1,x_2,x_3 ... x_n \in A \]
For any set $V$, $|V|$ would be used to denote its cardinality. When a set $V$ has a cardinality of $k$, we may describe it as a \textit{$k$-cardinality set}. $\mathcal{P}(V)$ would be used to denote its power set. The subscript of a set union or intersection may be omitted to indicate that the union or intersection is applied to each element in the set i.e. \begin{align*} &\text{For any set } A \\ & \bigcup A = \bigcup_{a \in A} a = \bigcup_{k=1}^n a_k = a_1 \cup a_2 \cup .. \cup a_n \\ & \bigcap A = \bigcap_{a \in A} a = \bigcap_{k=1}^n a_k = a_1 \cap a_2 \cap .. \cap a_n \\
&\text{where } |A| = n \text{ and } a_1, a_2 ... a_n \text{ are unique elements of } A \end{align*} This use of notation applies to $\bigvee$ and $\bigwedge$ as well: \begin{align*} &\text{For any set of formulae } A_L \\ & \bigvee A_L = \bigvee_{a \in A_L} a = \bigvee_{k=1}^n a_k = a_1 \vee a_2 \vee .. \vee a_n \\ & \bigwedge A_L = \bigwedge_{a \in A_L} a = \bigwedge_{k=1}^n a_k = a_1 \wedge a_2 \wedge .. \wedge a_n \\
&\text{where } |A_L| = n \text{ and } a_1, a_2 ... a_n \text{ are unique formulae in } A_L \end{align*} For any $k$-tuple $T$, $\pi_i(T)$ would be used to denote the $i$-th element of $T$ where $i \leq k$ e.g. \begin{align*} A = (\pi_1(A),\pi_2(A)) \quad \end{align*} where $A$ is a 2-tuple (often referred to as an ordered pair).
To avoid ambiguity, for any function $f : X \longrightarrow Y$, we would use $f^{members}$ to denote a new function, from $\mathcal{P}(X)$ to $\mathcal{P}(Y)$, such that \begin{align*} &\forall A \in \mathcal{P}(X) \\ &\qquad f^{members}(A) = \bigcup_{a \in A} \{ f(a)\} \end{align*} Here is a generalization of it, $f^{members^n}$, defined recursively: \begin{gather*} \begin{split} &f^{members^1} \coloneqq f^{members} \\ &f^{members^n}(x) = \bigcup_{a \in x} \{ f^{members^{n-1}}(a) \} \text{ where } n \in \mathbb{N}_{\geq 2} \end{split} \end{gather*}
Intuitively, multiset can be viewed as a generalization of set, where the same element can occur multiple times. Two multisets are the same iff both multisets contain the same distinct elements and every distinct element occurs the same number of times in both multisets. More formally, a multiset is defined as an ordered pair $(A,m_{\mathfrak{m}})$ where $m_{\mathfrak{m}} : A \longrightarrow \mathbb{N}_{\geq 1}$ describes the number of occurrences of each element in the multiset, and $A$ is the set of all distinct elements in the multiset. The cardinality of a multiset $(A,m_{\mathfrak{m}})$ is the sum of all $m_{\mathfrak{m}}(x)$ for $x \in A$. Multisets are expressed using square brackets. Here is an example: let $f$ be a function that always outputs 1, \begin{align*} &[f(x) : x \in \mathbb{N}_{\geq 1} : x \leq 3] = [1,1,1] = (\{1\},\{(1,3)\}) \\ &\text{as compared to} \\ &\{f(x) : x \in \mathbb{N}_{\geq 1} : x \leq 3\} = \{1\} \end{align*}
Intuitively, hypergraph can be viewed as a generalization of graph, where an edge can contain any number of vertices. A hypergraph is defined as an ordered pair $(V,E)$ where $E$ is a subset of $\mathcal{P}(V) \setminus \varnothing$. Elements in $V$ are referred to as vertices while elements in $E$ are referred to as edges or hyperedges. A hypergraph is $k$-uniformed when all of its hyperedges have the same cardinality. A graph in the conventional sense can thus be defined as a 2-uniformed hypergraph.
A full vertex coloring on some hypergraph $(V,E)$ is defined as a function, $C : V \longrightarrow cDom$, where $cDom$ is a non-empty subset of $\mathbb{N}$, often referred to as the set of colors. When $|Dom|=2$, we say the coloring is monochromatic. We would use $FullCol(G,cDom)$ to denote the set of all possible full vertex colorings on a hypergraph $G$ of the set of colors $cDom$. That is to say, for any hypergraph $G$ of $n$ vertices and any $cDom$, \begin{align*}
|FullCol(G,cDom)| = n^{|cDom|} \end{align*}
The boolean satisfiability problem (SAT) is the problem of determining if there exists some value-assignment for the variables in a propositional logic formula such that it yields $True$ i.e. it is satisfiable.
A formula is referred to as a tautology when there exists no value-assignment for the variables such that it yields $False$ e.g. $a \lor \neg a$.
We say that a formula is in \textit{Disjunctive Normal Form} (DNF) when it is a disjunction of conjunctions. Let $S$ be a set of formuale, a disjunction is a formula that can be expressed as $\bigwedge S$, while a conjunction is a formula that can be expressed as $\bigvee S$.
\section{\textit{Div point set} as a representation for any set of points in general position on an Euclidean plane}
We start off by introducing an object which we would be referring to as \textit{div point set}.
\begin{defn} A \textit{div point set} is any ordered pair $(P,\Theta_P)$ satisfying \begin{alignat}{2}
\label{def1_1}
&\mathrlap{\lvert\Theta_P\rvert = \binom{\lvert P\rvert}{2} \land P \not= \varnothing} \\[1.5ex]
\label{def1_2}
& \forall D_n \in \Theta_P & \quad & \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{|@{\hskip0.6em}l} \color{black}
D_n \text{ is an ordered pair. } \\
d_n \coloneqq \pi_1(D_n) \\
\delta_n \coloneqq \pi_2(D_n) \\
\lvert d_n\rvert = \lvert\delta_n\rvert =2\\
d_n \in \mathcal{P}(P)\\
\bigcup \delta_n = P \setminus d_n \\
\bigcap \delta_n = \varnothing
\end{array}
\end{aligned}\\[1.5ex]
\label{def1_3}
& \forall D_n, D_m \in \Theta_P &\quad & D_n=D_m \Leftrightarrow \pi_1(D_n)= \pi_1(D_m) \end{alignat} We would be using $\mathscr{DPS}^*$ to denote the class of all ordered pairs satisfying \reff{def1_1}, \reff{def1_2} and \reff{def1_3} i.e. $\mathscr{X}$ is a \textit{div point set} iff $\mathscr{X} \in \mathscr{DPS}^*$. \end{defn}
\begin{remark}
Since $\lvert\Theta_P\rvert = \binom{\lvert P\rvert}{2}$ and, for every $D \in \Theta_P$, $|\pi_1(D)| = 2$ and $\pi_1(D) \in \mathcal{P}(P)$, by \reff{def1_3}, we can conclude that \begin{align}\label{power_set_car_2_as_divider}
\bigcup_{D \in \Theta_P} \pi_1(D) = \{ d \in \mathcal{P}(P): |d| = 2\} \end{align} \end{remark} \begin{prelude}[Axiom 1] For any $n$ points in general position in $\mathbb{E}^2$, where $n \geq 2$, we can always select 2 arbitrary points and draw a line across them, dividing the remaining $n-2$ points into 2 disjoint sets. We shall refer to these 2 disjoint sets as \textit{divs} produced by a \textit{divider} made up of the 2 points, and the points in the \textit{divs} as \textit{TBD points} (short for \textit{to-be-distributed-among-divs}).
Any set of points $P$ in general position on an Euclidean plane where $\lvert P\rvert \geq 2$ can be represented by some \textit{div point set} $(P,\Theta_P)$: we shall refer to each $D_n \in \Theta_P$ as a \textit{dividon}, to be interpreted as follows: \begin{equation}
\begin{aligned}
&d_n \coloneqq \pi_1(D_n) & \quad & \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{|@{\hskip0.6em}l}
\text{Let the 2 elements in } d_n \text{ be } a, b \\
a \text{ and } b \text{ represent the 2 points making up the \textit{divider}} \end{array} \end{aligned} \\
&\delta_n \coloneqq \pi_2(D_n) & \quad & \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{|@{\hskip0.6em}l}
\text{Let the 2 elements in } \delta_n \text{ be } div_1, div_2\\
div_1 \text{ and } div_2 \text{ represent the 2 \textit{divs} produced by the \textit{divider}} \end{array} \end{aligned}
\end{aligned}
\end{equation} Basically, each \textit{dividon} describes the relative positions of the corresponding \textit{TBD points} in terms of how they are distributed between the 2 \textit{divs} produced by each \textit{divider}.
The sets of points in \textit{Figures I, II} and \textit{III} can be represented by any \textit{div point set} $(A, \Theta_A)$ as long as $A$ is a set of 4 elements $a$, $b$, $c$, $d$ and \begin{align*}
\Theta_{A} = & \{(\{a,b\},\{(\{c\},\{d\}\}), \\
&\;\; (\{a,c\},\{(\{b\},\{d\}\}), \\
&\;\; (\{a,d\},\{(\{b\},\{c\}\}), \\
&\;\; (\{b,c\}, \{(\{a,d\},\varnothing\}), \\
&\;\; (\{b,d\},\{(\{a,c\},\varnothing\}), \\
&\;\; (\{c,d\}, \{(\{a,b\},\varnothing\})\}) \end{align*} To make sense of the above \textit{div point set} representation, we label the third point from the bottom in \textit{Figure I} and the second point from the bottom in \textit{Figures II} and \textit{III} as $a$ (note that each of these points is surrounded by the remaining 3 points in the figure). For the rest of the points in each figure we shall label them arbitrarily as $b$, $c$, and $d$. Notice how in all figures for the 3 \textit{dividers} made up of $a$ and an arbitrary point, we have 2 \textit{divs} of 1 cardinality, and how for the remaining 3 \textit{dividers}, we have rest of the points in a single \textit{div} - precisely that of what $D \in \Theta_A$ describes.
Only a handful of \textit{div point sets} can be used to represent points in general position in $\mathbb{E}^2$. For majority of $\mathscr{X} \in \mathscr{DPS}^*$, there exists no meaningful interpretation for $\pi_1(\mathscr{X})$ as some set of points in $\mathbb{E}^2$ such that $\pi_2(\mathscr{X})$ describe their relative positions. A classical example would be $(Q, \Theta_Q)$ where $Q$ is any set of 4 elements $a$, $b$, $c$, $d$ and \begin{align*}
\Theta_{Q} = & \{(\{a,b\},\{(\{c,d\},\varnothing\}), \\
&\;\; (\{a,c\},\{(\{b,d\},\varnothing\}), \\
&\;\; (\{a,d\},\{(\{b,c\},\varnothing\}), \\
&\;\; (\{b,c\}, \{(\{a,d\},\varnothing\}), \\
&\;\; (\{b,d\},\{(\{a,c\},\varnothing\}), \\
&\;\; (\{c,d\}, \{(\{a,b\},\varnothing\})\}) \end{align*}
For any $\mathscr{X} \in \mathscr{DPS}^*$ to have a meaningful interpretation for $\pi_1(\mathscr{X})$ as some set of points in $\mathbb{E}^2$, it has to satisfy certain conditions. After some experimentation with points in $\mathbb{E}^2$, one would make the observation that the following formulae always hold for any distinct points $a$, $b$, $c$, $d$:
\begin{alignat}{2}
\label{general_position_law1}
&a\not=b\not=c\not=d \quad \Rightarrow & \quad & \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{|@{\hskip0.6em}l} \color{black}
a \in \langle b,c \rangle^{\scalebox{0.75}[1.0]{-}d} \Leftrightarrow d \in \langle b,c \rangle^{\scalebox{0.75}[1.0]{-}a} \\
a \in \langle b,c \rangle^d \Leftrightarrow d \in \langle b,c \rangle^a
\end{array}
\end{aligned}\\[2.5ex]
\label{general_position_law2}
&a\not=b\not=c\not=d \quad \Rightarrow & \quad & \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{|@{\hskip0.6em}l} \color{black}
c \in \langle a,b \rangle^{\scalebox{0.75}[1.0]{-}d} \\
\quad \Leftrightarrow ((a \in \langle b,c \rangle^d \land a \in \langle b,d \rangle^c) \\
\quad \qquad \lor (a \in \langle b,c \rangle^{\scalebox{0.75}[1.0]{-}d} \land a \in \langle b,d \rangle^{\scalebox{0.75}[1.0]{-}c}) )
\end{array}
\end{aligned}\\[2.5ex]
\label{general_position_law3}
&a\not=b\not=c\not=d \quad \Rightarrow & \quad & \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{|@{\hskip0.6em}l} \color{black}
c \in \langle a,b \rangle^{d} \\
\quad \Leftrightarrow ((a \in \langle b,c \rangle^d \land a \in \langle b,d \rangle^{\scalebox{0.75}[1.0]{-}c}) \\
\quad\qquad \lor (a \in \langle b,c \rangle^{\scalebox{0.75}[1.0]{-}d} \land a \in \langle b,d \rangle^{c}) )
\end{array}
\end{aligned}\\[2.5ex]
\label{general_position_law4}
&a\not=b\not=c\not=d \quad \Rightarrow & \quad & \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{|@{\hskip0.6em}l} \color{black}
a \in \langle b,c \rangle^{-d} \land a \in \langle b,d \rangle^{-c} \Rightarrow a \in \langle c,d \rangle^{b}
\end{array}
\end{aligned}\end{alignat} wherein $\langle x,y \rangle^z$ denote the \textit{div} containing $z$ produced by the \textit{divider} made up of the point $x$ and $y$, and $\langle x,y \rangle^{\scalebox{0.75}[1.0]{-}z}$ denote the \textit{div} not containing $z$ produced by the \textit{divider} (here $x$, $y$, and $z$ are metavariable). \reff{general_position_law1} is trivially true. \reff{general_position_law2}, \reff{general_position_law3} and \reff{general_position_law4} are demonstrated in \textit{Figures IV}, \textit{V} and\textit{VI} respectively.
In the context of \textit{div point sets}, \reff{general_position_law1} is always true by \reff{def1_2} (recall $ \bigcap \delta = \varnothing$), while \reff{general_position_law2}, \reff{general_position_law3} and \reff{general_position_law4} can each be rewritten as constraints on the \textit{dividons} as shown in \reff{dividon_law1}, \reff{dividon_law2}, and \reff{dividon_law3}.
\begin{align} \begin{split} \label{dividon_law1}
&\forall R \in \{ S \in \mathcal{P}(P) : |S| =4 \} \\ &\begin{aligned}\qquad &\forall D_1, D_2, D_3 \in \Theta_{P} \\
&\qquad \bigcup_{n=1}^{3} \pi_1(D_n) = R \land |\bigcap_{n=1}^{3} \pi_1(D_n)|= 1 \\ &\begin{aligned}\qquad \Rightarrow ( \; &\phi(\pi_2(D_1),R \setminus \pi_1(D_1)) = 0 \\ &\Leftrightarrow \phi(\pi_2(D_2),R \setminus \pi_1(D_2)) = \phi(\pi_2(D_3),R \setminus \pi_1(D_3)) \; ) \\ \end{aligned} \end{aligned} \end{split} \\[2.5ex] \begin{split}\label{dividon_law2}
&\forall R \in \{ S \in \mathcal{P}(P) : |S| =4 \} \\ &\begin{aligned}\qquad &\forall D_1, D_2, D_3 \in \Theta_{P} \\
&\qquad \bigcup_{n=1}^{3} \pi_1(D_n) = R \land |\bigcap_{n=1}^{3} \pi_1(D_n)|= 1 \\ &\begin{aligned}\qquad \Rightarrow ( \; &\phi(\pi_2(D_1),R \setminus \pi_1(D_1)) = 1 \\ &\Leftrightarrow \phi(\pi_2(D_2),R \setminus \pi_1(D_2)) \not= \phi(\pi_2(D_3),R \setminus \pi_1(D_3)) \; ) \\ \end{aligned} \end{aligned} \end{split} \\[2.5ex] \begin{split}\label{dividon_law3}
&\forall R \in \{ S \in \mathcal{P}(P) : |S| =4 \} \\ &\begin{aligned}\qquad &\forall D_1, D_2, D_3 \in \Theta_{P} \\
&\qquad \bigcup_{n=1}^{3} \pi_1(D_n) \subset R \land |\bigcup_{n=1}^{3} \pi_1(D_n)| = 3 \land D_1\not=D_2\not=D_3\\ &\begin{aligned}\qquad \Rightarrow ( \; &\phi(\pi_2(D_1),R \setminus \pi_1(D_1)) = \phi(\pi_2(D_2),R \setminus \pi_1(D_2)) = 0 \\ &\Rightarrow \phi(\pi_2(D_3),R \setminus \pi_1(D_3)) = 1 \; ) \\ \end{aligned} \end{aligned} \end{split} \end{align} Here $\phi$ determines if two distinct \textit{TBD} points belong to the same \textit{div} i.e. \begin{equation}\label{phi}
\phi(\delta,TBD_2)= \begin{cases}
1 & \text{if } \; \exists div \in \delta \; \; |TBD_2 \cap div| = 2 \\
0 & \text{if }\; \forall div \in \delta \; \; |TBD_2 \cap div| = 1 \end{cases} \end{equation} \end{prelude} \begin{note}
In \reff{dividon_law1} and \reff{dividon_law2}, it is not necessary to write down $D_1\not=D_2\not=D_3$ explicitly as a part of the conjunction in the antecedent since $\bigcup_{n=1}^{3} \pi_1(D_n) = R \land |\bigcap_{n=1}^{3} \pi_1(D_n)|= 1$ ensures that $D_1$, $D_2$, and $D_3$ are distinct. \end{note}
\begin{ax} Some $\mathscr{X} \in \mathscr{DPS}^*$ has an interpretation for $\pi_1(\mathscr{X})$ as some set of points in $\mathbb{E}^2$ such that $\pi_2(\mathscr{X})$ describes the relative positions of these points iff $\mathscr{X}$ is in $\mathscr{DPS}^+$, the class of \textit{div point sets} $(P,\Theta_P)$ satisfying \reff{dividon_law1}, \reff{dividon_law2}, and \reff{dividon_law3}. \end{ax} \begin{remark} For \textit{div point sets} of 3 or less points, it is vacuously true that they satisfy \reff{dividon_law1}, \reff{dividon_law2}, and \reff{dividon_law3} and they are all thus in the class $\mathscr{DPS}^+$. This is consistent with Euclidean geometry: any set of 3 points in general position can be represented by any \textit{div point set} of 3 points, and the same goes to any set of 2 or less points. \end{remark}
\begin{defn} We say that two \textit{div point sets} $(A,\Theta_A)$ and $(B, \Theta_B)$ are isomorphic iff there exists a bijection $f:A \stackrel{\rm{1:1}}{\longrightarrow} B$ preserving the structure of the \textit{dividons}, notationally, \begin{alignat}{2}\label{isomorphism}
&(A,\Theta_A) \cong (B,\Theta_B) && \Leftrightarrow \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{@{\hskip0em}l}
\exists f:A \stackrel{\rm{1:1}}{\longrightarrow} B\\
\quad\forall D_A \in \Theta_A \\
\quad\quad\exists D_B \in \Theta_B \\
\quad\quad\quad f^{members}(\pi_1(D_A)) = \pi_1(D_B)\\
\quad\quad\quad \Leftrightarrow f^{members^{2}}(\pi_2(D_A)) = \pi_2(D_B) \\
\end{array}
\end{aligned} \end{alignat} \end{defn}
\begin{remark}It is trivially true that for any two distinct \textit{div point sets} of 3 or less points, they are isomorphic to each other if they are of the same number of points. \end{remark}
\begin{theo} $\neg ( \mathscr{X} \cong Conc_4^1 ) \Leftrightarrow ( \mathscr{X} \cong Conv_4 )$ for all $\mathscr{X} \in \mathscr{DPS}^+_4$ where $\mathscr{DPS}^+_4$ denotes the class of \textit{div point sets} of 4 points in $\mathscr{DPS}^+$ and \begin{gather} \begin{split} \label{c4}
Conc_4^1 = &(Cc_4^1, \Theta_{Cc_4^1}) \\
Cc_4^1 = &\{1,2,3,4\} \\
\Theta_{Cc_4^1} = & \{(\{1,2\},\{\{3\},\{4\}\}), \\
&\;\; (\{1,3\},\{\{2\},\{4\}\}), \\
&\;\; (\{1,4\},\{\{2\},\{3\}\}), \\
&\;\; (\{2,3\}, \{\{1,4\},\varnothing\}), \\
&\;\; (\{2,4\},\{\{1,3\},\varnothing\}), \\
&\;\; (\{3,4\}, \{\{1,2\},\varnothing\})\}
\end{split} \begin{split}
Conv_4 = &(Cv_4, \Theta_{Cv_4}) \\
Cv_4 = &\{1,2,3,4\} \\
\Theta_{Cv_4} = & \{(\{1,2\},\{\{3,4\},\varnothing\}), \\
&\;\; (\{1,3\},\{\{2\},\{4\}\}), \\
&\;\; (\{1,4\},\{\{2,3\},\varnothing\}), \\
&\;\; (\{2,3\}, \{\{1,4\},\varnothing\}), \\
&\;\; (\{2,4\},\{\{1\},\{3\}\}), \\
&\;\; (\{3,4\}, \{\{1,2\},\varnothing\})\}
\end{split} \end{gather} \end{theo} \begin{proof}[Proof for \textit{Theorem 1}] \begin{summary}In Part 1 of the proof we would define a function $\psi$ that returns 0 or 1 based on the \textit{divs} of a \textit{dividon} of some \textit{div point set} in $\mathscr{DPS}^+_4$. In Part 2 we would define a class $\mathscr{DPS}^\mathbb{N}_4$, a set of vertices for a hypergraph $H$, and a function $Col$ that uses $\psi$, and show that for every $\mathscr{X} \in \mathscr{DPS}^\mathbb{N}_4$, there exists a unique full vertex monochromatic coloring $Col(\pi_2(\mathscr{X}))$ on $H$. In Part 3 we would define a set of edges for $H$ in such a manner that the coloring $Col(\pi_2(\mathscr{X}))$ on $H$ satisfies some conditions iff $\mathscr{X}$ satisfies \reff{dividon_law1} and \reff{dividon_law2}. In Part 4 we would demonstrate that for the coloring to satisfy the conditions, there exists only 3 $\color{black}\mathcal{Scenarios} $. The colorings described in $\color{black}\mathcal{Scenarios}$ II and III are isomorphic$\color{black}^*$ to $Col(\pi_2(Conc_4^1))$ and $Col(\pi_2(Conv_4))$ respectively, and $Conc_4^1$ and $Conv_4$ both satisfy \reff{dividon_law3}, but the \textit{div point set} the coloring described in $\color{black}\mathcal{Scenario}\; I$ is based on does not satisfy \reff{dividon_law3}. Therefore we conclude that \textit{div point sets} of 4 points satisfying \reff{dividon_law1}, \reff{dividon_law2} and \reff{dividon_law3} are either isomorphic to $Conc_4^1$ or $Conv_4$, thus proving \textit{Theorem 1}. \end{summary} \begin{proofpart} Since every \textit{dividon} of any \textit{div point set} in $ \mathscr{DPS}^+_4$ has $4-2=2$ \textit{TBD} points, we can be certain that, let $D$ be a \textit{dividon} and $a$ and $b$ be the \textit{TBD} points, $\pi_2(D) \in \{ type_0, type_1 \}$ where \begin{gather} \begin{split}\label{implies_phi_to_delta} & type_0 \coloneqq \{\{a\},\{b\}\}\\ &type_1 \coloneqq \{\{a,b\},\varnothing\} \\
\end{split} \end{gather} By exploiting the fact every $\pi_2(D)$ is either $type_0$ or $type_1$, for $\mathscr{X} \in \mathscr{DPS}^+_4$, we can define a new function $\psi$, a simpler version of $\phi$ (recall \reff{phi}) that does basically the same thing: \begin{gather}\label{coloring}\begin{split}
\psi(\delta)= \begin{cases}
0 & \text{if } \; \forall \textit{div} \in \delta \quad |div| = 1 \\
1 & \text{if } \; \exists \textit{div} \in \delta \quad |div| = 2 \end{cases} \end{split}\end{gather} For every $D \in \pi_2(\mathscr{X})$ where $\mathscr{X} \in \mathscr{DPS}^+_4$, we thus have \begin{gather}\label{phi_to_delta}\begin{split} \phi(\pi_2(D), P \setminus \pi_1(D) ) = \psi(\pi_2(D)) \end{split}\end{gather} \end{proofpart} \begin{proofpart} Let $\mathscr{DPS}^\mathbb{N}_4$ be the class of all \textit{div point sets} $(P,\Theta_P)$ for which $P = \{1,2,3,4\}$. All $\mathscr{X} \in \mathscr{DPS}^\mathbb{N}_4$ would have the same set of \textit{dividers} (recall \reff{power_set_car_2_as_divider}). Now let $H=(V,E)$ be a hypergraph whose vertices are the \textit{dividers} of $\mathscr{X} \in \mathscr{DPS}^\mathbb{N}_4$. Using $\psi$, we can define a bijective function $Col$ that transforms the set of \textit{dividons} of any $\mathscr{X} \in \mathscr{DPS}^\mathbb{N}_4$ into some full vertex monochromatic coloring for H. \begin{equation} \begin{gathered}\label{coloring1} Col : \{\pi_2(\mathscr{X}) : \mathscr{X} \in \mathscr{DPS}^\mathbb{N}_4\} \stackrel{\rm{1:1}}{\longrightarrow} FullCol(H,\{0,1\}) \\ Col(\Theta_P) = \{ (\pi_1(D),\psi(\pi_2(D))) : D \in \Theta_P \} \\ \end{gathered} \end{equation}
It is bijective since $|FullCol(H,\{0,1\})| = |\mathscr{DPS}^\mathbb{N}_4|$ and \begin{gather}\begin{split} &\forall \mathscr{X_1}, \mathscr{X_2} \in \mathscr{DPS}^{\mathbb{N}}_4 \\ &\quad Col(\mathscr{\pi_2(X_1)}) = Col(\mathscr{\pi_2(X_2)}) \Leftrightarrow \mathscr{X_1} = \mathscr{X_2} \\ \end{split}\end{gather} due to the fact that for any 2 \textit{dividons}, $D_1$ and $D_2$, made up of the same divider, belonging to 2 \textit{div point set} in $\mathscr{DPS}^{\mathbb{N}}_4$ respectively, $\psi(\pi_2(D_1))=\psi(\pi_2(D_2))$ iff $D_1 = D_2$. \end{proofpart} \begin{proofpart} Let any set of 3 \textit{dividers} having 1 point in common to be an edge of $H$ i.e. \begin{gather}\begin{split}\label{edges}
E \coloneqq \{ e \in \mathcal{P}(V) : |e| = 3 \land |\bigcap e| = 1 \} \end{split}\end{gather} for some $\mathscr{X} \in \mathscr{DPS}^\mathbb{N}_4$ to satisfy \reff{dividon_law1} and \reff{dividon_law2} is equivalent to having $Col(\pi_2(\mathscr{X})) \in FullCol(H,\{0,1\})$ to satisfy I and II: \begin{enumerate}[I.] \item For any vertex $v$ colored 0, the other 2 vertices belonging to the same edge as $v$ must be colored the same. \item For any vertex $v$ colored 1, the other 2 vertices belonging to the same edge as $v$ must be colored differently. \end{enumerate} This is in virtue of fact that for any $\mathscr{X} \in \mathscr{DPS}^\mathbb{N}_4$, \reff{dividon_law1} and \reff{dividon_law2} can be rewritten as having the coloring $C \coloneqq Col(\pi_2(\mathscr{X}))$ to satisfy some formulae, namely \reff{graph_law1} and \reff{graph_law2}. \begin{align} \begin{split} \label{graph_law1} &\forall e \in E \\ &\qquad \forall d_1, d_2, d_3 \in e \\ &\qquad \qquad d_1 \not = d_2 \not = d_3 \Rightarrow (C(d_1) = 0 \Leftrightarrow C(d_2) = C(d_3)) \end{split} \end{align} \begin{align} \begin{split} \label{graph_law2} &\forall e \in E \\ &\qquad \forall d_1, d_2, d_3 \in e \\ &\qquad \qquad d_1 \not = d_2 \not = d_3 \Rightarrow (C(d_1) = 1 \Leftrightarrow C(d_2) \not= C(d_3)) \end{split} \end{align}
The above rewriting works because \begin{align}\begin{split}\label{phi_to_psi}
&\forall R \in \{ S \in \mathcal{P}(P) : |S| =4 \} \\ &\qquad \forall D \in \Theta_{P} \\ &\qquad \qquad \phi(\pi_2(D),R \setminus \pi_1(D)) = \phi(\pi_2(D),P \setminus \pi_1(D)) = \psi(\pi_2(D)) \end{split}\end{align}
holds for any \textit{div point set} $(P,\Theta_P)$ for which $|P|=4$, and any \textit{dividons} $D_1, D_2$ and $D_3$ satisfying $|\bigcap_{n=1}^{3} \pi_1(D_n)| = 1 \land |\bigcup_{n=1}^{3} \pi_1(D_n)| = 4$ would respectively have three \textit{dividers} $d_1$, $d_2$ and $d_3 $ where \begin{gather}
|\bigcap_{n=1}^{3} d_n| = 1 \land d_1 \not= d_2 \not= d_3 \end{gather} which are precisely what make up an edge of $H$. Therefore some $\mathscr{X} \in \mathscr{DPS}^\mathbb{N}_4$ satisfies \reff{dividon_law1} and \reff{dividon_law2} iff $Col(\mathscr{\pi_2(X)})$ satisfies \textit{I} and \textit{II}. \end{proofpart} \begin{proofpart} To satisfy I and II, for every edge of $H$, the 3 vertices it contains must be colored either $[0,0,0]$ or $[0,1,1]$.
Suppose we start off by giving three arbitrary vertices belonging to the same edge the coloring of $[0,0,0]$, by I, the rest of the vertices have to be colored the same (recall that each vertex belongs to 2 different edges). We either end up with $H$ having all vertices colored 0 (let's call it $\color{black}\mathcal{Scenario}\; I$), or 3 vertices colored 0 and 3 vertices colored 1 (let's call it $\color{black}\mathcal{Scenario}\; II$).
Now suppose we start off by giving three arbitrary vertices belonging to the same edge the coloring of $[0,1,1]$. By I, the remaining 2 vertices of another edge the vertex colored 0 belongs to needs to be colored the same. If we color them both 0, the last uncolored vertex of $H$ must then be colored 1 as it belongs to edges wherein both the other 2 vertices are colored differently. We would end up in $\color{black}\mathcal{Scenario}\; II$ again. On the other hand, if we colored them both 1, the last uncolored vertex must then be colored 0 as it belongs to edges wherein both the other 2 vertices are colored the same. Let's call this $\color{black}\mathcal{Scenario}\; III$, where 2 vertices are colored 0 and 4 vertices are colored 1.
A pictorial description of the colorings is shown in Figure VII.
$\color{black} \mathcal{Scenario}$ I describes a coloring isomorphic$\color{black}^*$ to $Col(\pi_2(Conc_4^1))$ to $Col(\pi_2(\mathscr{X}_{\varnothing}))$ where $\mathscr{X}_{\varnothing} \in \mathscr{DPS}^{\mathbb{N}}_4$ and \begin{align*}
\pi_2(\mathscr{X}_{\varnothing}) = & \{(\{1,2\},\{(\{3,4\},\varnothing\}), \\
&\;\; (\{1,3\},\{(\{2,4\},\varnothing\}), \\
&\;\; (\{1,4\},\{(\{2,3\},\varnothing\}), \\
&\;\; (\{2,3\}, \{(\{1,4\},\varnothing\}), \\
&\;\; (\{2,4\},\{(\{1,3\},\varnothing\}), \\
&\;\; (\{3,4\}, \{(\{1,2\},\varnothing\})\}) \end{align*} while $\color{black}\mathcal{Scenario}$ II describes a coloring isomorphic$\color{black}^*$ to $Col(\pi_2(Conc_4^1))$ and $\color{black} \mathcal{Scenario}$ III descrbies a coloring isomorphic$\color{black}^*$ to $Col(\pi_2(Conv_4))$. $Conc_4^1$ and $Conv_4$ both satisfy \reff{dividon_law3}, and $\mathscr{X}_{\varnothing}$ does not. Since every $\mathscr{X} \in \mathscr{DPS^\mathbb{*}_4}$ is isomorphic to some $\mathscr{X} \in \mathscr{DPS^\mathbb{N}_4}$, and in $\mathscr{DPS^\mathbb{N}_4}$ only $Conc_4^1$ and $Conv_4$ satisfy all \reff{dividon_law1}, \reff{dividon_law2}, and \reff{dividon_law3}, we conclude that \begin{gather}\begin{split} \forall X \in \mathscr{DPS}^{+}_4 \quad \exists a \in \{Conc_4^1,Conv_4 \} \quad X \cong a \end{split} \end{gather} \end{proofpart}
\end{proof} \begin{note} \textit{isomorphic$\color{black}^*$}: the isomorphism we are talking about here is that of colorings, which can be defined as follows: \begin{alignat}{2}
&C_1 \cong C_2 && \Leftrightarrow \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{@{\hskip0em}l}
\exists f_{C}: \{ \pi_1(c) : c \in C_1\} \stackrel{\rm{1:1}}{\longrightarrow} \{ \pi_1(c) : c \in C_2\}\\
\qquad\forall c_1 \in C_1 \\
\qquad\qquad\exists c_2 \in C_2 \\
\qquad\qquad\qquad f_{C}(\pi_1(c_1)) = \pi_1(c_2) \Rightarrow \pi_2(c_1) = \pi_2(c_2) \\
\end{array}
\end{aligned} \end{alignat} \end{note}
\begin{remark} In Euclidean geometry, \textit{Theorem 1} is equivalent as stating that for any set of 4 distinct points in general position, it is either the case that a point can be found inside a triangle formed by connecting the remaining 3 points, or the case that a convex quadrilateral can be created by connecting all 4 points, which can be verified rather easily by a human child with a pen, a piece of paper and a love for geometry. \end{remark}
\subsection{ \textit{unit div point set} and \textit{sub div point set}} For \textit{div point sets} of 5 or more points, the function $\psi$ would not be really useful since there would be 3 or more \textit{TBD points} in each \textit{dividon}. That means we cannot use the same approach as above to derive \textit{div point sets} of 5 or more points satisfying \reff{dividon_law1}, \reff{dividon_law2} and \reff{dividon_law3}. With that in mind, we introduce the object \textit{unit div point set}.
\begin{defn} A \textit{unit div point set} is any ordered pair $(P,\Omega_P)$ satisfying \reff{def3_1}, \reff{def3_2} and \reff{def3_3}.
\begin{alignat}{2}\label{def3_1}
&\mathrlap{\lvert\Omega_P\rvert = \binom{\lvert P\rvert}{2} \binom{|P|-2}{2} \land P \not= \varnothing}\\[1.5ex] \label{def3_2}
& \forall D_n \in \Omega_P & \quad & \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{|@{\hskip0.6em}l}
D_n \text{ is an ordered pair. } \\
d_n \coloneqq \pi_1(D_n) \\
\delta_n \coloneqq \pi_2(D_n) \\
\lvert d_n\rvert = \lvert\delta_n\rvert = \lvert\bigcup \delta_n\rvert =2\\
d_n \in \mathcal{P}(P)\\
\bigcup \delta_n \in \mathcal{P}(P \setminus d_n) \\
\bigcap \delta_n = \varnothing
\end{array}
\end{aligned}\\[1.5ex] \label{def3_3}
& \forall D_n, D_m \in \Omega_P & \quad & \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{|@{\hskip0.6em}l}
\xi(D_n) = \xi(D_m) \Leftrightarrow D_n=D_m
\end{array}
\end{aligned} \end{alignat} where \begin{align} \xi(D) = \pi_1(D) \cup \bigcup \pi_2(D) \end{align} We would be using $\mathscr{UDPS}^*$ to denote the class of all \textit{unit div point set}. Each $D \in \Omega_P$ would be referred to as a \textit{unit dividon}. \end{defn} \begin{remark} Similar to how every \textit{dividon} of $\mathscr{X} \in \mathscr{DPS^*}$ is either $type_0$ or $type_1$ as illustrated in \reff{implies_phi_to_delta}, every \textit{unit dividon} $D$ of any \textit{unit div point set} of points $P$ always satisfies \begin{gather} \begin{split} \label{implies_phi_to_delta2} & \exists a,b \in P \setminus \pi_1(D) \\ & \qquad type_0 \coloneqq \{\{a\},\{b\}\}\\ & \qquad type_1 \coloneqq \{\{a,b\},\varnothing\} \\ & \qquad \pi_2(D) \in \{ type_0, type_1 \} \\
\end{split} \end{gather} \end{remark} \begin{prelude}[Defintion 4] One may immediately notice that any \textit{div point sets} of 4 points also satisfy \reff{def3_1}, \reff{def3_2} and \reff{def3_3}, similar to how any \textit{unit div point set} of 4 points also satisfy \reff{def1_1}, \reff{def1_2} and \reff{def1_3}, which is to say, \begin{align}
\{\mathscr{X_{udps}} \in \mathscr{UDPS}^* : |\pi_1(\mathscr{X_{udps}})| = 4 \} = \{\mathscr{X_{dps}} \in \mathscr{DPS}^* : |\pi_1(\mathscr{X_{dps}})| = 4\} \end{align} by virtue of the fact that $\binom{4}{2} \binom{4-2}{2} = \binom{4}{2}$ and \begin{alignat}{2}
& \forall \mathscr{X} \in \mathscr{UDPS}^* \notag \\
&\qquad |\pi_1(\mathscr{X})| = 4 \quad \Rightarrow & \quad & \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{|@{\hskip0.6em}l}
\forall D_n \in \pi_2(\mathscr{X}) \\
\qquad \bigcup \pi_2(D_n) = P \setminus \pi_1(D_n) \\
\forall D_n,D_m \in \pi_2(\mathscr{X}) \\
\qquad \pi_1(D_n)= \pi_1(D_m) \Leftrightarrow D_n=D_m
\end{array}
\end{aligned} \end{alignat}
As we can see, the difference between a \textit{div point set} and a \textit{unit div point set} lies in that the former relies on a single \textit{dividon} to describe the distribution of the $|P|-2$ \textit{TBD points} between 2 \textit{divs} for each \textit{divider}, while the later relies on $\binom{|P|-2}{2}$ \textit{unit dividons} for that (since each \textit{unit dividon} only describes the distribution of 2 \textit{TBD points}). For every $\mathscr{X_{dps}} \in \mathscr{DPS}^*$ there exists a unique $\mathscr{X_{udps}} \in \mathscr{UDPS}^*$ which $\mathscr{X_{dps}}$ can be transformed into, by breaking down each \textit{dividon} into $\binom{|P|-2}{2}$ \textit{unit dividons} containing the same \textit{divider}, achievable using the function $\mathscr{bd}$ defined as follows
\begin{align}
\begin{split}\label{bd_d_u}
&\mathscr{bd}(D,P) = \{(\pi_1(D),\mathscr{d_u}(\pi_2(D),P_{TBD}) : P_{TBD} \in \mathcal{P}(P \setminus \pi_1(D)) : |P_{TBD}| = 2\}
\\ &\mathscr{d_u}(\delta,TBD_2) = \begin{cases}
\{P,\varnothing\}& \text{if }\; \phi(\delta,TBD_2) = 1 \\
\{ x \in \mathcal{P}(P) : |x| = 1 \}& \text{if }\; \phi(\delta,TBD_2) = 0
\end{cases} \end{split}
\end{align}
$\mathscr{bd}$ takes in a \textit{dividon} and a set of points, and returns a set of \textit{unit dividons}. It makes use of $\mathscr{d_u}$ that takes in a set of \textit{divs} from a \textit{dividon} and a set of 2 points, and returns a set of \textit{divs} for a \textit{unit dividon}.
\end{prelude} \begin{defn} The function $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}$ transforms a \textit{div point set} into a \textit{unit div point set}. \begin{align}\begin{split}\label{def4}
\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X_{dps}}) = (\pi_1(\mathscr{X_{dps}}), \bigcup \{\mathscr{bd}(D,\pi_1(\mathscr{X_{dps}})) : D \in \pi_2(\mathscr{X_{dps}}) \}) \end{split}\end{align} \\ $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}$ can be implemented in Haskell as follows: \begin{lstlisting} import Control.Monad import Data.List ((\\)) powerList = filterM (const [True, False])
f:: ([Int],[([Int],[[Int]])]) -> ([Int],[([Int],[[Int]])]) f (points,dividons) = (points,unit_dividons)
where
unit_dividons = foldl (++) [] $ map get_unit_dividons dividons
get_unit_dividons (d,(delta1:_)) = [(d,(\(a:b:_)->
if a `in_same_div_as_b` b
then [[a,b],[]] else [[a],[b]])
x ) |
x <- powerList (points \\ d), length x == 2,
let (in_same_div_as_b) a b = (a `elem` delta1) == (b `elem` delta1)] \end{lstlisting}
\end{defn} \begin{remark} It is no surprise that \begin{align} \forall \mathscr{X} \in \mathscr{DPS}^* \qquad \undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X}) = \mathscr{X} \Leftrightarrow \lvert \pi_1(\mathscr{X}) \rvert = 4 \end{align} since for each $D\in \mathscr{X}_{4}$ where $\mathscr{X}_{4} \in \mathscr{DPS}_4^*$, $\mathscr{bd}(D,\pi_1(\mathscr{X_{4}}))$ is a singleton and the one element it contains is $D$.\end{remark} \begin{remark} On the other hand, \begin{align} \forall \mathscr{X} \in \mathscr{DPS}^* \qquad \undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X}) = (\pi_1(\mathscr{X}),\varnothing) \Leftrightarrow 1 \leq \lvert \pi_1(\mathscr{X}) \rvert \leq 3 \end{align} and that is not going to be useful. So it is more sensible to define $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}$ over \textit{div point sets} of 4 or more points i.e. $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}} : \mathscr{DPS}^*_{\geq4} \longrightarrow \mathscr{UDPS}^*$. \end{remark} \begin{lem} $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}} : \mathscr{DPS}^*_{\geq4} \longrightarrow \mathscr{UDPS}^*$ is injective but not surjective. \end{lem}
\begin{proof}[Proof for Lemma 1] It is injective because for every \textit{dividon} $D$ of any \textit{div point set} of 4 or more points, $\mathscr{bd}(D,\pi_1(\mathscr{X_{dps}}))$ in \reff{def4} differs depending on $D$. By $I$ and $II$ below, we can see that it is not surjective onto the co-domain $\mathscr{UDPS}^*$. \begin{enumerate}[I.]
\item There exists $\mathscr{X_{udps}} \in \mathscr{UDPS}^*$, where $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{W})=\mathscr{X_{udps}}$ iff $\mathscr{W}$ is an ordered pair satisfying all the conditions to be a \textit{div point set} except that, some \textit{dividon} has more than 2 \textit{divs}, and, as a result, such $\mathscr{W} \not\in \mathscr{DPS}^*$ (Recall $|\delta_n| = 2$ in \reff{def1_2}). E.g. \textit{unit div point sets} with \textit{unit dividons} such as \begin{align*}\begin{split} \{(a,b),(\{c\},\{d\})\},\{(a,b),(\{c\},\{e\})\},\{(a,b),(\{e\},\{d\})\} \end{split}\end{align*}
can only be transformed from a \textit{div-point-set}-like object where $|\pi_2(D)| =3 $ for some \textit{dividion} $D$, in this case: $\{(a,b),(\{c\},\{d\},\{e\})\}$. That is to say, for any $\mathscr{X_{udps}}\prime \in \mathscr{UDPS}^*$, where $\mathscr{X_{udps}}\prime = \undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X_{dps}})$ for some $\mathscr{X_{dps}} \in \mathscr{DPS}^*$, $\mathscr{X_{udps}}\prime$ satisfies \begin{align}\begin{split}\label{2divs} &\forall D_1,D_2,D_3 \in \pi_2(\mathscr{X_{udps}}\prime) \\
&\qquad D_1 \not = D_2 \not = D_3 \land \pi_1(D_1) = \pi_1(D_2) = \pi_1(D_3) \land |\bigcup_{n=1}^3 \bigcup \pi_2(D_n)| = 3\\ & \qquad \Rightarrow \neg (( \psi(\pi_2(D_1)) = \psi(\pi_2(D_2)) = \psi(\pi_2(D_3)) = 0 ) \\
\end{split} \end{align} \item As a consequence of $\bigcap \delta_n = \varnothing$ in \reff{def1_2}, for any distinct \textit{TBD} points $c$, $d$, and $e$, of some \textit{divider} of a \textit{div point set}, if $c$ and $d$ are in the same \textit{div}, and $d$ and $e$ are in the same \textit{div}, it is certainly the case for $c$ and $e$ to be found in the same \textit{div}. So \textit{unit div point sets} with \textit{unit dividons} such as \begin{gather*}\begin{split} \{(a,b),(\{c,d\},\varnothing)\},\{(a,b),(\{c,e\},\varnothing)\},\{(a,b),(\{e\},\{d\})\} \end{split}\end{gather*} can not be transformed from any \textit{div point set}. That is to say, for any $\mathscr{X_{udps}}\prime \in \mathscr{UDPS}^*$, where $\mathscr{X_{udps}}\prime = \undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X_{dps}})$ for some $\mathscr{X_{dps}} \in \mathscr{DPS}^*$, $\mathscr{X_{udps}}\prime$ satisfies \begin{align}\begin{split}\label{associativity} &\forall D_1,D_2,D_3 \in \pi_2(\mathscr{X_{udps}}\prime) \\
&\qquad D_1 \not = D_2 \not = D_3 \land \pi_1(D_1) = \pi_1(D_2) = \pi_1(D_3) \land |\bigcup_{n=1}^3 \bigcup \pi_2(D_n)| = 3\\ & \qquad \Rightarrow \neg ( \psi(\pi_2(D_1)) = \psi(\pi_2(D_2)) = 1 \land \psi(\pi_2(D_3)) = 0)
\end{split} \end{align} \end{enumerate}
\end{proof} \begin{remark} Combining \reff{associativity} and \reff{2divs} above gives \reff{unit_dividon_law0}. \begin{align}\begin{split}\label{unit_dividon_law0} &\forall D_1,D_2,D_3 \in \Omega_P \\
&\qquad (D_1 \not = D_2 \not = D_3 \land \pi_1(D_1) = \pi_1(D_2) = \pi_1(D_3) \land |\bigcup_{n=1}^3 \bigcup \pi_2(D_n)| = 3)\\ &\begin{aligned}\qquad \Rightarrow (&\psi(\pi_2(D_1)) = 1 \Leftrightarrow \psi(\pi_2(D_2)) = \psi(\pi_2(D_3)) ) \\ &\land ( \psi(\pi_2(D_1)) = 0 \Leftrightarrow \psi(\pi_2(D_2)) \not= \psi(\pi_2(D_3)) ) \end{aligned} \end{split}\end{align} Let's define $\mathscr{UDPS}^\Theta$ to be a subclass of $\mathscr{UDPS}^*$ for which $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}} : \mathscr{DPS}^*_{\geq4} \longrightarrow \mathscr{UDPS}^{\Theta}$ is bijective. We can be certain that $\mathscr{UDPS}^\Theta \subseteq \mathscr{UDPS}^{\Theta\prime}$, where $\mathscr{UDPS}^{\Theta\prime}$ is the class of \textit{unit div point sets} of 4 or more points satisfying \reff{unit_dividon_law0}. It is likely the case that \reff{unit_dividon_law0} is all that a \textit{unit div point set} must satisfy to be in the class $\mathscr{UDPS}^\Theta$ (i.e. $\mathscr{UDPS}^{\Theta\prime} = \mathscr{UDPS}^{\Theta}$), but that is not important in the current discussion and we would not be going into that. \end{remark} \begin{lem} A \textit{unit div point set} $(P,\Omega_P)$ has an interpretation for $P$ as some set of 4 or more points in $\mathbb{E}^2$ such that $\Omega_P$ describes the relative positions of the points iff it is in $\mathscr{UDPS}^+$ wherein each \textit{unit div point set} satisfies \reff{unit_dividon_law1}, \reff{unit_dividon_law2}, and \reff{unit_dividon_law3}. \begin{align} \begin{split} \label{unit_dividon_law1}
&\forall R \in \{ S \in \mathcal{P}(P) : |S| =4 \} \\ &\begin{aligned}\qquad &\forall D_1, D_2, D_3 \in \Omega_P \\ &\qquad (\; \xi(D_1) =\xi(D_2) =\xi(D_3) = R \land D_1 \not= D_2 \not= D_3 \\
&\qquad \land |\bigcap_{n=1}^{3} \pi_1(D_n)|= 1 \; )\\ &\begin{aligned}\qquad \Rightarrow ( \; &\psi(\pi_2(D_1)) = 0 \\ &\Leftrightarrow \psi(\pi_2(D_2)) = \psi(\pi_2(D_3)) \; ) \\ \end{aligned} \end{aligned} \end{split} \end{align} \begin{align}\begin{split} \label{unit_dividon_law2}
&\forall R \in \{ S \in \mathcal{P}(P) : |S| =4 \} \\ &\begin{aligned}\qquad &\forall D_1, D_2, D_3 \in \Omega_P \\ &\qquad (\; \xi(D_1) =\xi(D_2) =\xi(D_3) = R \land D_1 \not= D_2 \not= D_3 \\
&\qquad \land |\bigcap_{n=1}^{3} \pi_1(D_n)|= 1 \;) \\ &\begin{aligned}\qquad \Rightarrow ( \; &\psi(\pi_2(D_1)) = 1 \\ &\Leftrightarrow \psi(\pi_2(D_2)) \not= \psi(\pi_2(D_3)) \; ) \\ \end{aligned} \end{aligned} \end{split} \end{align} \begin{align} \begin{split}\label{unit_dividon_law3}
&\forall R \in \{ S \in \mathcal{P}(P) : |S| =4 \} \\ &\begin{aligned}\qquad &\forall D_1, D_2, D_3 \in \Omega_P \\ &\qquad (\; \xi(D_1) =\xi(D_2) =\xi(D_3) = R \land D_1 \not= D_2 \not= D_3 \\
&\qquad \land |\bigcup_{n=1}^{3} \pi_1(D_n)|= 3 \; ) \\ &\begin{aligned}\qquad \Rightarrow ( \; &\psi(\pi_2(D_1)) = \psi(\pi_2(D_2)) = 0 \\ &\Rightarrow \psi(\pi_2(D_3)) = 1 \; ) \end{aligned} \end{aligned} \end{split} \end{align} \end{lem} \begin{proof}[Proof for Lemma 2] A \textit{div point set} $\mathscr{X_{dps}}$ satisfies \reff{dividon_law1}, \reff{dividon_law2}, and \reff{dividon_law3} iff the \textit{unit div point set} $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X_{dps}})$ satisfies \reff{unit_dividon_law1}, \reff{unit_dividon_law2}, and \reff{unit_dividon_law3}. Firstly we make the following observation similar to that of \reff{phi_to_psi}: for any \textit{unit divdion} $D_{\mathscr{u}}$ of some \textit{unit div point set} $\mathscr{A_{udps}}$ and its corresponding \textit{divdion} $D$ of the \textit{div point set} $\mathscr{A_{dps}}$ where $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{A_{dps}}) = \mathscr{A_{udps}}$ - corresponding in the sense that $D_{u} \in \mathscr{bd}(D,\pi_2(\mathscr{A_{dps}}))$ and so $\pi_1(D_{u}) = \pi_1(D)$ - let $R := \xi(D_\mathscr{u}) $, we would have \begin{align}\begin{split}\label{D} &\phi(\pi_2(D_\mathscr{u}),R \setminus \pi_1(D)) = \phi(\pi_2(D_\mathscr{u}),\bigcup \pi_2(D_\mathscr{u})) = \psi(\pi_2(D_\mathscr{u})) \end{split}\end{align} By restricting some \textit{unit dividons} $D_1$, $D_2$ and $D_3$ into satisfying $\xi(D_1) = \xi(D_2) = \xi(D_3) = R$ for some set of 4 points $R$, we can replace every occurrence of $\phi(\pi_2(D_n),R \setminus \pi_1(D_n))$ with $\psi(\pi_2(D_n))$ (for $n \in \{1,2,3\}$) in \reff{dividon_law1}, \reff{dividon_law2}, and \reff{dividon_law3}, and ensure the satisfiability of $ \bigcup_{n=1}^{3} \pi_1(D_n) = R$ (in \reff{dividon_law1} and \reff{dividon_law2}) by further restricting these \textit{unit dividons} to be distinct (i.e $D_1 \not= D_2 \not= D_3$). This would give \reff{unit_dividon_law1}, \reff{unit_dividon_law2}, and \reff{unit_dividon_law3}: they are basically a different way of expressing \reff{dividon_law1}, \reff{dividon_law2}, and \reff{dividon_law3} in the case of \textit{unit div point sets}.
Therefore any $\mathscr{X_{udps}} \in \mathscr{UDPS}^+$ has an interpretation for $\pi_1(\mathscr{X_{udps}})$ as some set of 4 or more points in $\mathbb{E}^2$ similar to how any $\mathscr{X_{dps}} \in \mathscr{DPS}^+$ has an interpretation for $\pi_1(\mathscr{X_{dps}})$.
\end{proof} \begin{note}
In \reff{unit_dividon_law3}, it is not necessary to write down $\bigcup_{n=1}^{3} \pi_1(D_n) \subset R$ explicitly as a part of the conjunction in the antecedent like how it is in \reff{dividon_law3}, since $\xi(D_1) =\xi(D_2) =\xi(D_3) = R \land |\bigcup_{n=1}^{3} \pi_1(D_n)|= 3$ ensures that the union of $\pi_1(D_1)$, $\pi_1(D_2)$, and $\pi_1(D_3)$ is a proper subset of $R$. \end{note}
\begin{lem}\label{hypergraph-for-all-udps} If $(P,\Omega_P)$ is in $\mathscr{UDPS}^+$, $Col_{\mathscr{udps}}(\Omega_P)$, a full vertex monochromatic coloring on $H_{\mathscr{udps}}$, satisfies \reff{hypergraph_law1} and \reff{hypergraph_law2}. Here $Col_{\mathscr{udps}}$ is a function similar to $Col$ in \reff{coloring1}: \begin{equation} \begin{gathered} Col_{\mathscr{udps}} : \{\pi_2(\mathscr{X}) : \mathscr{X} \in \mathscr{UDPS}^+\} \stackrel{\rm{1:1}}{\longrightarrow} FullCol(H_{\mathscr{udps}},\{0,1\}) \\ Col_{\mathscr{udps}}(\Omega_P)= \{ ((\pi_1(D),\bigcup \pi_2(D)),\psi(\pi_2(D)) : D \in \Omega_P\} \end{gathered} \end{equation} and $H_{\mathscr{udps}}$ is a 3-and-6-uniform hypergraph with 2 sets of hyperedges, $E_1$ and $E_2$, defined as a 3-tuple $H_{\mathscr{udps}} = (V_{\mathscr{udps}},E_1,E_2)$, constructed based on $P$: \begin{align} \begin{split}\label{3-6-hypergraph}
V_{\mathscr{udps}} & \coloneqq \bigcup \{ V_{of}(d,P) : d \in \mathcal{P}(P): |d| = 2\} \\
E_1 & \coloneqq \{ e \in \mathcal{P}(V_{\mathscr{udps}}) : |e| = 6 \land \forall v_1,v_2 \in e \; \; \xi(v_1) = \xi(v_2) \} \\
E_2 & \coloneqq \{ e \in \mathcal{P}(V_{\mathscr{udps}}) : |e| = 3 \land |\bigcup_{v \in e} \pi_2(v)| = 3 \land \forall v_1,v_2 \in e \; \; \pi_1(v_1) = \pi_1(v_2) \} \\ \end{split}\end{align} with $\xi$ as defined in \textit{Defintion 3} and $V_{of}$ being a function that returns a set of ordered pairs consisting of \textit{divider} and \textit{TBD points} of \textit{unit dividons} of that \textit{divider}, notationally, \begin{align} \begin{split}
V_{of}(d,P)= \{ (d, P_{TBD}): P_{TBD} \in \mathcal{P}(P \setminus d) : |P_{TBD}| = 2 \} \\ \end{split}\end{align} and, finally, we have \begin{alignat}{2} &\begin{aligned} \label{hypergraph_law1} &\forall e \in E_1\\
&\qquad \exists v_1, v_2 \in e & \quad & \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{|@{\hskip0.6em}l} v_1 \not = v_2 \\ \pi_1(v_1) = \pi_2(v_2) \\ \pi_1(v_2) = \pi_2(v_1) \\
C(v_1) = C(v_2) = 0 \\
C^{members}(e \setminus \{v_1,v_2\}) = \{1\} \\ \end{array} \end{aligned} \\[1.5ex]
& \qquad \Leftrightarrow \neg \exists v_1, v_2, v_3 \in e & \quad & \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{|@{\hskip0.6em}l} v_1 \not = v_2 \not= v_3 \\
|\pi_1(v_1) \cap \pi_1(v_2) \cap \pi_1(v_3)| = 1 \\
C(v_1) = C(v_2) = C(v_3) = 0 \\
C^{members}(e \setminus \{v_1,v_2,v_3\}) = \{1\} \end{array} \end{aligned} \end{aligned} \\ &\begin{aligned} \label{hypergraph_law2} &\forall e \in E_2\\
&\qquad \forall v_1,v_2,v_3 \in e & \quad & \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{|@{\hskip0.6em}l} v_1 \not= v_2 \not= v_3 \\ \Rightarrow ( C(v_1) = 1 \Leftrightarrow C(v_2) = C(v_3) )\\ \qquad \land ( C(v_1) = 0 \Leftrightarrow C(v_2) \not= C(v_3) )\\ \end{array} \end{aligned} \\[1.5ex] \end{aligned} \end{alignat} wherein $C = Col_{\mathscr{udps}}(\Omega_P)$. \end{lem}
\begin{remark} If $\mathscr{UDPS}^{\Theta\prime} = \mathscr{UDPS}^{\Theta}$, a stronger version of \textit{Lemma \ref{hypergraph-for-all-udps}} is then true: $(P,\Omega_P)$ is in $\mathscr{UDPS}^+$ iff $Col_{\mathscr{udps}}(\Omega_P)$ satisfies \reff{hypergraph_law1} and \reff{hypergraph_law2}. \end{remark}
\begin{remark} One may notice that the construction of $H_{\mathscr{udps}}$ depends solely on $\pi_1(\mathscr{X_{udps}})$ (i.e. the points of a \textit{unit div point set}), as different from the full vertex coloring, which depends solely on $\pi_2(\mathscr{X_{udps}})$ (i.e. the set of \textit{unit dividons}), similar to how the hypergraph $H$ and its coloring are defined back in the proof for \textit{Theorem 1}. However, the vertices of $H_{\mathscr{udps}}$ are ordered pairs, structurally different from vertices of $H$ which are 2-cardinality sets. Such definition for the vertices of $H_{\mathscr{udps}}$ in terms of not only the \textit{divider} of a \textit{unit dividon} but also its \textit{TBD points} is necessary. This is because for any \textit{unit div point set} $(P,\Omega_P)$, there exists $\binom{|P|-2}{2}$ distinct \textit{unit dividons} sharing a common \textit{divider}. In order to distinguish \textit{unit dividons} from one another in a \textit{unit div point set} of 5 or more points, we would need to take into account both the \textit{divider} and the \textit{TBD points}. \end{remark} \begin{remark} For any \textit{unit div point set} of 4 points, $\mathscr{X_4}$, the second set of edges, $E_2$, of $H_{\mathscr{upds}}$ constructed based on $\pi_1(\mathscr{X_4})$ is an empty set, and thus \reff{hypergraph_law2} is vacuously true for any coloring on such $H_{\mathscr{upds}}$. $E_1$ of such $H_{\mathscr{upds}}$ on the other hand is a singleton. For such $H_{\mathscr{upds}}$, in \reff{hypergraph_law1}, the existential predicate before the logical operator $\Leftrightarrow$ is true iff $\mathscr{X_4}$ is isomorphic to $Conv^4$, while the existential predicate after the logical operator $\neg$ at the right hand side of $\Leftrightarrow$ is true iff $\mathscr{X_4}$ is isomorphic to $Conc^4_1$. Thus for $Col_{\mathscr{udps}}(\pi_2(\mathscr{X_4}))$ to satisfiies \reff{hypergraph_law1} is equivalent to having $\mathscr{X_4}$ isomorphic to either $ Conv^4$ or $Conc^4_1$, which is consistent with \textit{Theorem 1}. \end{remark}
\begin{proof}[Proof for Lemma 3] \begin{summary} In Part 1 we show that $Col(\pi_2(\mathscr{X}))$ satisfies \reff{hypergraph_law2} iff $\mathscr{X}$ satisfies \reff{unit_dividon_law0}, and in Part 2 we show that $Col(\pi_2(\mathscr{X}))$ satisfies \reff{hypergraph_law1} iff $\mathscr{X}$ satisfies \reff{unit_dividon_law1}, \reff{unit_dividon_law2}, and \reff{unit_dividon_law3}, for any $\mathscr{X} \in \mathscr{UDPS}^*$. \end{summary} \setcounter{proofpart}{0} \begin{proofpart}
\reff{hypergraph_law2} is simply a different way of expressing \reff{unit_dividon_law0} in the context of coloring: the ordered pairs $(d_n,\bigcup \delta_n)$ of some \textit{unit dividons} $D_n=(d_n,\delta_n)$ that satisfy $(D_1 \not = D_2 \not = D_3 \land \pi_1(D_1) = \pi_1(D_2) = \pi_1(D_3) \land |\bigcup_{n=1}^3 \bigcup \pi_2(D_n)| = 3)$ are defined in \reff{3-6-hypergraph} to be the vertices of an edge in $E_2$. \end{proofpart} \begin{proofpart} The constraints described in \reff{unit_dividon_law1}, \reff{unit_dividon_law2}, and \reff{unit_dividon_law3} revolve around $R$ where $R$ is some 4-cardinality subset of a set of points $P$. For every such $R \subseteq P$, there are a total of $\binom{4}{2}=6$ \textit{unit dividons} $D \in \Theta_P$ where $\xi(D)=R$, for any \textit{unit div point set} $(P,\Theta_P)$. By \textit{Theorem 1}, a \textit{unit div point set} of 4 points (recall that \textit{div point sets} of 4 points are their own \textit{unit div point sets}) satisfies \reff{unit_dividon_law1}, \reff{unit_dividon_law2}, and \reff{unit_dividon_law3} iff it is isomorphic to either $Conc_4^1$ or $Conc_4$. More fundamentally, this means that any \textit{unit div point set} $(P,\Omega_P)$ satisfies \reff{unit_dividon_law1}, \reff{unit_dividon_law2}, and \reff{unit_dividon_law3} iff for every 4-cardinality subset $R$ of $P$, the 6-cardinality subset $\Omega_{of 6}$ of $\Omega_P$ (where $\xi(D) = R$ for every $D \in \Omega_{of 6}$) is isomorphic$\color{black}^*$ to either $\pi_2(Conc_4^1)$ or $\pi_2(Conv_4)$.
Therefore a \textit{unit div point set} $(P, \Omega_P)$ satisfies \reff{unit_dividon_law1}, \reff{unit_dividon_law2}, and \reff{unit_dividon_law3} iff for every such $R \subseteq P$, let $C'$ be a subset of $Col_{\mathscr{udps}}(\Omega_P)$ where $\xi(\pi_1(c)) = R$ for all $c \in C'$, $C'$ is isomorphic to either $Col(\pi_2(Conc_4^1))$ or $Col(\pi_2(Conv_4))$. Notationally, \begin{alignat}{2} \begin{aligned}
&\forall R \in \{ P' \in \mathcal{P}(P) : |P'| = 4 \} \\ &\qquad C' \coloneqq \{ c \in Col_{\mathscr{udps}}(\Omega_P) : \xi(\pi_1(c))= R\} \\ &\qquad C' \cong Col(\pi_2(Conc_4^1)) \Leftrightarrow \neg ( C' \cong Col(\pi_2(Conv_4)) ) \end{aligned} \end{alignat} which is exactly what is expressed in \reff{hypergraph_law1}, considering that \begin{align}\begin{split}
E_1 = \{ \mathscr{UDs}(R) : R \in \mathcal{P}(P) : |R| = 4 \} \end{split} \end{align} where $\mathscr{UDs}$ returns a set of ordered pairs each consisting of the \textit{divider} and the \textit{TBD} points of every such \textit{unit dividon} for each $R$: \begin{align}\begin{split}\label{uds}
&\mathscr{UDs}(R) = \{ \mathscr{ud}(d) : d \in \mathcal{P}(R) : |d| = 2 \}\\ &\mathscr{ud}(d) = (d,\ R \setminus d) \end{split} \end{align} \end{proofpart}
\end{proof} \begin{note} \textit{isomorphic$\color{black}^*$}: the isomorphism we are talking about here is that of sets of \textit{unit dividons}, which can be defined as follows: \begin{alignat}{2}
&\Omega_1 \cong\Omega_2 && \Leftrightarrow \begin{aligned}[t] \renewcommand\arraystretch{1.25}\begin{array}[t]{@{\hskip0em}l}
|\Omega_1| = |\Omega_2| \\
\land \exists f_{\Omega}: \bigcup_{D \in \Omega_1} \pi_1(D) \stackrel{\rm{1:1}}{\longrightarrow} \bigcup_{D \in \Omega_2} \pi_1(D)\\
\qquad\forall D_1 \in \Omega_1 \; \; \exists D_2 \in \Omega_2 \\
\qquad\qquad f_{\Omega}^{members}(\pi_1(D_1)) = \pi_1(D_2) \Leftrightarrow f_{\Omega}^{members^{2}}(\pi_2(D_1)) = \pi_2(D_2) \\
\end{array}
\end{aligned} \end{alignat} \end{note} \begin{defn} We say that $\mathscr{X_1 \in \mathscr{DPS}^*}$ is a \textit{sub div point set} of $\mathscr{X_2} \in \mathscr{DPS}^*$ (denoted by $\leq$) iff the set of \textit{unit divdions} of the corresponding \textit{unit div point set} of $\mathscr{X_1}$ is a subset of that of $\mathscr{X_2}$. Notationally, \begin{gather}\begin{split}\label{sub-div-point-set} &\forall \mathscr{X_1}, \mathscr{X_2} \in \mathscr{DPS}^*\\ &\quad \mathscr{X_1} \leq \mathscr{X_2} \Leftrightarrow \pi_2(\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X_1})) \subseteq \pi_2(\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X_2})) \end{split}\end{gather} \end{defn}
\begin{defn} $\mathscr{Sdps_{of}}$ is a function that returns the set of all \textit{sub div point sets} of $m$ points for some \textit{div point set}, or an empty set depending on $m$. \begin{align} \begin{split} \mathscr{Sdps_{of}}(\mathscr{X_{dps}},m) = \begin{cases}
\{ \mathscr{Sdps}(\mathscr{X_{dps}},P_s) : P_s \in \mathcal{P}(\mathcal{\pi_1(\mathscr{X_{dps}})}) : |P_s| = m \} & \text{ if } m \geq 4 \\
\varnothing & \text{ otherwise} \end{cases} \end{split} \end{align} where $\mathscr{Sdps}$ returns the \textit{sub div point set} of some set of points $P_s$ of a \textit{div point set}: \begin{align} \begin{split}
&\mathscr{Sdps}(\mathscr{X_{dps}},P_s) = \undervec{\mathscr{F}}_{\mathscr{dps}}^{\mathscr{UDPS}}((P_s,\{ D : D \in \pi_2(\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X_{dps}})) : \xi(D) \subseteq P_s \}) ) \\
&\text{where } \undervec{\mathscr{F}}_{\mathscr{dps}}^{\mathscr{UDPS}} \text{ is the inverse of } \undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}} : \mathscr{DPS}^*_{\geq4} \longrightarrow \mathscr{UDPS}^{\Theta}. \end{split} \end{align}
Since a \textit{div point set} of $n$ points always has $\binom{n}{m}$ distinct \textit{sub div point sets} of $m$ points and $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}$ is defined over \textit{div point sets} of 4 or more points, $\mathscr{Sdps_{of}}(\mathscr{X_{dps}},m)$ has the cardinality of $\binom{|\pi_1(\mathscr{X_{dps}})|}{m}$ for all $\mathscr{X_{dps}} \in \mathscr{DPS}^*$ and $m \geq 4$. \end{defn} \begin{lem} For any \textit{div point set} $\mathscr{X}$, let $\mathscr{A}$ and $\mathscr{B}$ be any 2 \textit{sub div point sets} of $\mathscr{X}$, and $k$ be the number of points $\mathscr{A}$ and $\mathscr{B}$ have in common, $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{A})$ and $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{B})$ always have $6 \binom{k}{4}$ \textit{unit dividons} in common. Notationally, \begin{align} \begin{split} & \forall \mathscr{X} \in \mathscr{DPS^*} \\ & \qquad \forall m, n \in \mathbb{N}_{\geq 1} \\ & \qquad \qquad \forall \mathscr{A} \in \mathscr{Sdps_{of}}(\mathscr{X},m) \;\; \forall \mathscr{B} \in \mathscr{Sdps_{of}}(\mathscr{X},n) \\
& \qquad \qquad \qquad |\pi_2(\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{A})) \cap \pi_2(\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{B}))| = 6 \binom{|\pi_1(\mathscr{A}) \cap \pi_1(\mathscr{B})|}{4} \end{split} \end{align} \end{lem} \begin{remark}
In the case that both the \textit{sub div point sets} $\mathscr{A}$ and $\mathscr{B}$ are the \textit{div point set} $\mathscr{X}$ itself i.e. $\mathscr{A}=\mathscr{B}=\mathscr{X}$, \textit{Lemma 4} is equivalent to stating that for any \textit{div point set} $\mathscr{X}$, $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X})$ has $6 \binom{|\pi_1(\mathscr{X})|}{4}$ \textit{unit dividons}, which is true by \reff{def3_1} since $\binom{k}{2} \binom{k-2}{2}=6 \binom{k}{4}$. \end{remark} \begin{proof}[Proof for Lemma 4]
For any $m \geq |\pi_1(\mathscr{X})|$ or $m < 4$, the proposition is vacuously true, since $\mathscr{Sdps_{of}}(\mathscr{X},m)$ would be an empty set. For any $m$ less than $|\pi_1(\mathscr{X})|$ but greater than or equal to $4$, the proposition can be proven by first observing that $\mathscr{UDs}(R) \cap \mathscr{UDs}(R') = \varnothing \Leftrightarrow R\not=R'$ (recall \reff{uds}) for any sets $R$ and $R'$ with a cardinality of 4, indicating that no 2 \textit{unit div point sets} of 4 points have in common \textit{unit dividons} of the same \textit{divider} and \textit{TBD points}. Notationally,
\begin{align}\begin{split} \label{non-cap}
&\forall \mathscr{A},\mathscr{B} \in \{ \mathscr{X} : \mathscr{X} \in \mathscr{UDPS}^* : |\pi_1({\mathscr{X}})| = 4 \} \\ &\qquad \qquad \xi^{memebers}(\pi_2(\mathscr{A})) \cap \xi^{memebers}(\pi_2(\mathscr{B})) = \varnothing \Leftrightarrow \pi_1(\mathscr{A}) \not= \pi_1(\mathscr{B}) \end{split}\end{align}
For any 2 \textit{unit div point sets} of 5 or more points, $\mathscr{A_{udps}}$ and $\mathscr{B_{udps}}$, if they have 4 points in common, let the set of such 4 points be $R$ i.e. $ R=\pi_1(\mathscr{A_{udps}})\cap\pi_1(\mathscr{B_{udps}})$, for each $D_\xi \in \mathscr{UDs}(R)$, there exists $D_a \in \mathscr{A_{udps}}$ and $D_b \in \mathscr{B_{udps}}$ where $\xi(D_a) = \xi(D_b) = D_\xi$. In the case when $\mathscr{A_{udps}}=\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{A_{dps}})$ and $\mathscr{B_{udps}}=\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{B_{dps}})$ for some $\mathscr{A_{dps}}$ and $\mathscr{B_{dps}}$ that are both \textit{sub div point sets} of a certain $\mathscr{X} \in \mathscr{DPS}^*$, $D_a = D_b$ for every pair of such \textit{unit dividons} of $\mathscr{A_{udps}}$ and $\mathscr{B_{udps}}$, and thus for every set of 4 points 2 \textit{sub div point set} have in common, they have 6 \textit{unit dividons} in common. Let $k$ be the number of points these 2 \textit{sub div point set} have in common, the number of such distinct sets of 4 points is $\binom{k}{4}$ and thus, by \reff{non-cap}, the number of \textit{unit dividon} they have in common is precisely $6 \binom{|k|}{4}$.
\end{proof} \begin{theo} Let $\mathscr{DPS}^+_5$ denote the class of all \textit{div point sets} of 5 points in $\mathscr{DPS}^+$, all $\mathscr{X} \in \mathscr{DPS}^+_5$ either have 4, 2 or 0 distinct \textit{sub div point set} of 4 points isomorphic to $Conc_4^1$ (with the remaining \textit{sub div point sets} of 4 points isomorphic to $Conv_4$). \end{theo} \begin{proof}[Proof for Theorem 2] \begin{summary} In \textit{Part 1} we show that there exists no $\mathscr{X} \in \mathscr{DPS}^+_5$ where $\mathscr{Sdps_{of}}(\mathscr{X},4)$ has precisely 1, 3 or 5 elements isomorphic to $Conc_4^1$. In \textit{Part 2} we show that there exists $\mathscr{X} \in \mathscr{DPS}^+_5$ where $\mathscr{Sdps_{of}}(\mathscr{X},4)$ has precisely 0, 2 or 4 elements isomorphic to $Conc_4^1$. \end{summary} \begin{proofpart} By \textit{Lemma 2} it is clear that a \textit{div point set} $\mathscr{X_{dps}}$ is in $\mathscr{DPS}^{+}$ iff $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X_{dps}})$ is in $\mathscr{UDPS}^{+}$, which, by \textit{Lemma 3}, implies that $Col_{\mathscr{udps}}(\pi_2(\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X_{dps}}))$ satisfies \reff{hypergraph_law2}. For a full vertex monochromatic coloring on some hypergraph $H_{\mathscr{udps}}$ to satisfy \reff{hypergraph_law2}, every $e \in E_2$ of $H_{\mathscr{udps}}$ must has its vertices colored $[1,0,0]$ or $[1,1,1]$, which mean it would have an even number of vertices colored $0$: 2 multiplying by any number gives an even number and all edges in $ E_2$ of such $H_{\mathscr{udps}}$ are disjoint (for any \textit{unit div point set} of 5 points, there exists exactly $\binom{5-2}{2}=3$ distinct \textit{unit dividon} with the same \textit{divider}).
$Conc_4^1$ has an odd number of \textit{unit dividons} $D$ where $\psi(\pi_2(D)) = 0$, while $Conv_4$ has an even number for such \textit{unit dividons}. By \textit{Lemma 4}, we can see that 2 distinct \textit{sub div point sets} of 4 points always have no \textit{unit dividons} in common. Therefore, for any \textit{div point set} of 5 points $\mathscr{X_{dps}}$, if $Col_{\mathscr{udps}}(\pi_2(\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X_{dps}}))$ satisfies \reff{hypergraph_law2}, $\mathscr{X_{dps}}$ would not have an odd number of \textit{sub div point sets} of 4 points isomorphic to $Conc_4^1$. We thereby conclude that there exists no $\mathscr{X} \in \mathscr{DPS}^+_5$ where $\mathscr{Sdps_{of}}(\mathscr{X},4)$ has precisely 1, 3 or 5 elements isomorphic to $Conc_4^1$. \end{proofpart} \begin{proofpart} We shall now demonstrate that it is possible to construct \textit{unit div point sets} of 5 points $\mathscr{X_{udps}}$, where $Col_{\mathscr{udps}}(\pi_2(\mathscr{X_{udps}}))$ satisfies \reff{hypergraph_law1} and \reff{hypergraph_law2} and there are precisely 4, 2, or 0 distinct $\Omega_{of 6} \in All_{\Omega_{of 6}}(\mathscr{X_{udps}})$ isomorphic to $\pi_2(Conc_4^1)$, (with the remaining $\Omega_{of 6}$ isomorphic to $Conv_4$), and that such $\mathscr{X_{udps}}$ is in $\mathscr{UDPS}^{\Theta}$. Here $All_{\Omega_{of 6}}$ is a function that returns a set of 6-cardinality sets of \textit{unit dividons} where for any 2 \textit{unit dividons} $D_1$ and $D_2$ in each set, $\xi(D_1)=\xi(D_2)$, defined as follows \begin{align} \begin{split} \label{allOmega}
&All_{\Omega_{of 6}}(\mathscr{X_{udps}}) = \{ \Omega_{of_{based\_on}}(R,\pi_2(\mathscr{X_{udps}})) : R \in \mathcal{P}(\pi_1(\mathscr{X_{udps}})) : |R| = 4 \} \\ &\Omega_{of_{based\_on}}(R,\Omega) = \{ D \in \Omega : \xi(D) = R \} \end{split} \end{align} \begin{enumerate}[I.] \item To construct a \textit{unit div point set} $\mathscr{X_{udps}}$ where no $\Omega_{of 6}$ in $All_{\Omega_{of 6}}(\mathscr{X_{udps}})$ is isomorphic to $\pi_2(Conc_4^1)$, we would need to make sure there are only 2 \textit{unit dividons} $D$ in $\Omega_{of 6}$ where $\psi(\pi_2(D)) = 0$ for all 5 $\Omega_{of 6}$ in $All_{\Omega_{of 6}}(\mathscr{X_{udps}})$, notationally \begin{align} \begin{split}\label{all_2} &\forall \Omega_{of 6} \in All_{\Omega_{of 6}}(\mathscr{X_{udps}}) \\
&\qquad |\{D \in \Omega_{of 6} : \psi(\pi_2(D)) = 0 \}| = 2 \end{split} \end{align} which can also be expressed as \begin{align} \begin{split} \label{all_2_alt}
|\{ \Omega_{of 6} \in All_{\Omega_{of 6}}(\mathscr{X_{udps}}) : |\{D \in \Omega_{of 6} : \psi(\pi_2(D)) = 0 \}| = 2 \}| = 5 \end{split} \end{align} (Formulating it in terms of the cardinality (instead of using the universal quantifier as in \reff{all_2}) would make things a lot simpler as we go on to II and III.) \\ Let $D^*$ be the set of all \textit{such unit dividons} i.e. \begin{align} \begin{split} &\forall D \in \pi_2(\mathscr{X_{udps}}) \\ &\qquad D \in D^* \Leftrightarrow \psi(\pi_2(D)) = 0 \end{split} \end{align} and $\Omega_{of 6_1}, \Omega_{of 6_2}, \Omega_{of 6_3}, \Omega_{of 6_4}, \Omega_{of 6_5}$ be the 5 elements in $All_{\Omega_{of 6}}(\mathscr{X_{udps}})$, and $D_n^1$ and $D_n^2$ be every 2 such unit divdions in $\Omega_{of 6_n}$, for $n \in \{1,2,3,4,5\}$, i.e. \begin{align} \{D_n^1,D_n^2\} = \Omega_{of 6_n} \cap D^* \end{align} In order for the coloring to satisfy \reff{hypergraph_law1}, we need to ensure that \begin{align}\label{same-divider-diving-points} \begin{split} \pi_1(D_n^1) = \bigcup \pi_2(D_n^2) \\ \pi_1(D_n^2) = \bigcup \pi_2(D_n^1) \end{split} \end{align} holds for $n \in \{1,2,3,4,5\}$. And to satisfy \reff{hypergraph_law2}, we need to ensure that if some \textit{unit dividon} is in $D^*$, we would be able to find another \textit{unit dividon} in $D^*$ that has the same divider, and there exists exactly 1 such \textit{unit dividon}, notationally, \begin{align}\label{exists-2-dividon-of-same-divider} \begin{split} &\forall D' \in D^* \\
&\quad | \{ D \in D^* : \pi_1(D) = \pi_1(D') \} | = 2 \end{split} \end{align} One way to go about achieving that is to let $D_n^1 \not= D_n^2$ for every $n \in \{1,2,3,4,5\}$ while avoiding $D_n^1 = D_m^1 \land D_n^2 = D_m^2$ for any distinct $m$ and $n$. Starting from $D_1$ and going all the way to $D_5$ and we would have \begin{equation} \begin{gathered} \pi_1(D_1^1) = \bigcup \pi_2(D_1^2) = \pi_1(D_2^1) =\bigcup \pi_2(D_2^2) = A\\ \pi_1(D_1^2) = \bigcup \pi_2(D_1^1) = \pi_1(D_3^1) =\bigcup \pi_2(D_3^2) = B\\ \pi_1(D_2^2) = \bigcup \pi_2(D_2^1) = \pi_1(D_4^1) =\bigcup \pi_2(D_4^2) = C\\ \pi_1(D_3^2) = \bigcup \pi_2(D_3^1) = \pi_1(D_6^1) =\bigcup \pi_2(D_6^2) = D \\ \pi_1(D_4^2) = \bigcup \pi_2(D_4^1) = \pi_1(D_5^1) =\bigcup \pi_2(D_5^2) = E\\ \pi_1(D_5^2) = \bigcup \pi_2(D_5^1) = \pi_1(D_6^2) =\bigcup \pi_2(D_6^1) = F \end{gathered} \end{equation} for some 2-cardinality subsets $A,B,C,D,E,F$ of $\pi_1(\mathscr{X_{udps}})$, where \begin{gather*} A \not=B \not=C \not=D \not=E \not=F \\ (A \cap B) = (A \cap C) = (B \cap F) = (C \cap D) = (D \cap E) = (E \cap F) = \varnothing \end{gather*} $\mathscr{X_{udps}}$ described above is in $\mathscr{UDPS}^{\Theta}$ because there exists $\mathscr{X_{dps}} \in \mathscr{DPS}^+$ where $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X_{dps}})=\mathscr{X_{udps}}$: such $\mathscr{X_{dps}}$ would be isomorphic to $Conv_5$ defined in \reff{stronger-theorem-2}. \item To construct a \textit{unit div point set} $\mathscr{X_{udps}}$ where precisely 2 $\Omega_{of 6}$ in $All_{\Omega_{of 6}}(\mathscr{X_{udps}})$ are isomorphic to $\pi_2(Conc_4^1)$ (with the remaining 3 $\Omega_{of 6}$ in $All_{\Omega_{of 6}}(\mathscr{X_{udps}})$ isomorphic to $\pi_2(Conv_4^1)$), we would need to make sure that, for exactly 2 $\Omega_{of 6}$ in $All_{\Omega_{of 6}}(\mathscr{X_{udps}})$, there are precisely 3 \textit{unit dividons} $D$ in $\Omega_{of 6}$ where $\psi(\pi_2(D)) = 1$, and, for the remaining 3 $\Omega_{of 6}$ in $All_{\Omega_{of 6}}(\mathscr{X_{udps}})$, there are precisely 2 \textit{unit dividons} $D$ in $\Omega_{of 6}$ where $\psi(\pi_2(D)) = 0$, notationally \begin{align} \begin{split}
& |\{ \Omega_{of 6} \in All_{\Omega_{of 6}}(\mathscr{X_{udps}}) : |\{D \in \Omega_{of 6} : \psi(\pi_2(D)) = 0 \}| = 3 \}| = 2 \\
& |\{ \Omega_{of 6} \in All_{\Omega_{of 6}}(\mathscr{X_{udps}}) : |\{D \in \Omega_{of 6} : \psi(\pi_2(D)) = 0 \}| = 2 \}| = 3 \\ \end{split} \end{align} Using the same notation above, this time we would have $\{D_n^1,D_n^2\} = \Omega_{of 6_n} \cap D^*$ for $n \in \{1,2,3\}$ and $ \{D_n^1,D_n^2,D_n^3\} = \Omega_{of 6_n} \cap D^*$ for $n \in \{4,5\}$. In order for the coloring to satisfy \reff{hypergraph_law1} we need to ensure that \reff{same-divider-diving-points} holds for $n \in \{1,2,3\}$ and \begin{equation} \begin{gathered}\label{divider_one_in_common}
|\pi_1(D_n^1) \cap \pi_1(D_n^2) \cap \pi_1(D_n^3)| =1 \end{gathered} \end{equation} holds for $n \in \{4,5\}$. And to satisfy \reff{hypergraph_law2}, we also need to ensure that \reff{exists-2-dividon-of-same-divider} holds as well. One way to go about achieving that is to let $D_4^x$ to have a common \textit{divider} as $D_5^x$ for all $x \in \{1,2\}$, while letting the remaining \textit{unit dividons} in $D_4$ and $D_5$, namely $D_4^3$ and $D_5^3$, to have a common \textit{divider} as $D_1^1$ and $D_2^1$ respectively, and the remaining \textit{unit dividons} in $D_1$ and $D_2$, namely $D_1^2$ and $D_2^2$, to have a common \textit{dividers} as the two \textit{dividons} in $D_3$ respectively. That is to say, for some subsets of 2 cardinality, $A,B,C,D,E,F$ of $\pi_1(X_{udps})$, we have \begin{equation} \begin{gathered} \pi_1(D_4^1) = \pi_1(D_5^1) = A\\ \pi_1(D_4^2) = \pi_1(D_5^2) = B\\ \pi_1(D_4^3) = \pi_1(D_1^1) = \bigcup \pi_2(D_1^2) = C\\ \pi_1(D_5^3) = \pi_1(D_2^1) = \bigcup \pi_2(D_2^2) = D\\ \pi_1(D_1^2) = \bigcup \pi_2(D_1^1) = \pi_1(D_3^1) =\bigcup \pi_2(D_3^2) = E \\ \pi_1(D_2^2) = \bigcup \pi_2(D_2^1) = \pi_1(D_3^2) =\bigcup \pi_2(D_3^1) = F \\ \end{gathered} \end{equation} where \begin{gather*} A \not=B \not=C \not=D \not=E \not=F \\
|A \cap B \cap C| = 1 \\
|A \cap B \cap D| = 1 \\ (C \cap E) = (D \cap F) = (E \cap F) = \varnothing \end{gather*} $\mathscr{X_{udps}}$ described above is in $\mathscr{UDPS}^{\Theta}$ because there exists $\mathscr{X_{dps}} \in \mathscr{DPS}^+$ where $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X_{dps}})=\mathscr{X_{udps}}$: such $\mathscr{X_{dps}}$ would be isomorphic to $Conc_5^1$ defined in \reff{stronger-theorem-2}. \item To construct a \textit{unit div point set} $\mathscr{X_{udps}}$ where precisely 4 $\Omega_{of 6}$ in $All_{\Omega_{of 6}}(\mathscr{X_{udps}})$ are isomorphic to $\pi_2(Conc_4^1)$ (with the remaining 1 $\Omega_{of 6}$ in $All_{\Omega_{of 6}}(\mathscr{X_{udps}})$ isomorphic to $\pi_2(Conv_4^1)$), this time we would need to make sure that \begin{align} \begin{split}
& |\{ \Omega_{of 6} \in All_{\Omega_{of 6}}(\mathscr{X_{udps}}) : |\{D \in \Omega_{of 6} : \psi(\pi_2(D)) = 0 \}| = 3 \}| = 4 \\
& |\{ \Omega_{of 6} \in All_{\Omega_{of 6}}(\mathscr{X_{udps}}) : |\{D \in \Omega_{of 6} : \psi(\pi_2(D)) = 0 \}| = 2 \}| = 1 \\ \end{split} \end{align} Using the same notation above, we would have $\{D_n^1,D_n^2\} = \Omega_{of 6_n} \cap D^*$ for $n \in \{1\}$ and $ \{D_n^1,D_n^2,D_n^3\} = \Omega_{of 6_n} \cap D^*$ for $n \in \{2,3,4,5\}$. In order for the coloring to satisfy \reff{hypergraph_law1}, we need to ensure that \reff{same-divider-diving-points} holds for $n \in \{1\}$ and \reff{divider_one_in_common} holds for $n \in \{2,3,4,5\}$. And to satisfy \reff{hypergraph_law2}, we also need to ensure that \reff{exists-2-dividon-of-same-divider} holds as well. One way to go about satisfying these conditions is to let $D_4^x$ and $D_2^x$ to have a common \textit{divider} as $D_5^x$ and $D_3^x$ respecitvely, for $x \in \{1,2\}$, while letting the remaining \textit{unit dividons} in $D_2$, $D_4$ and $D_5$, namely $D_2^3$, $D_4^3$ and $D_5^3$, to have a common \textit{divider} as $D_1^2$, $D_3^3$ and $D_1^1$ respectively. That is to say, for some subsets of 2 cardinality, $A,B,C,D,E,F$ of $\pi_1(X_{udps})$, we have \begin{equation} \begin{gathered} \pi_1(D_4^1) = \pi_1(D_5^1) = A \\ \pi_1(D_4^2) = \pi_1(D_5^2) = B \\ \pi_1(D_4^3) = \pi_1(D_3^3) = C\\ \pi_1(D_3^1) = \pi_1(D_2^1) = D \\ \pi_1(D_3^2) = \pi_1(D_2^2) = E \\ \pi_1(D_5^3) = \pi_1(D_1^1) = \bigcup \pi_2(D_1^2) = F\\ \pi_1(D_2^3) = \pi_1(D_1^2) = \bigcup \pi_2(D_1^1) = G \end{gathered} \end{equation} where \begin{gather*} A \not=B \not=C \not=D \not=E \not=F \not= G \\
|A \cap B \cap C| = 1 \\
|A \cap B \cap F| = 1 \\
|C \cap D \cap E | = 1 \\
|D \cap E \cap G | = 1 \\ F \cap G = \varnothing \end{gather*} $\mathscr{X_{udps}}$ described above is in $\mathscr{UDPS}^{\Theta}$ because there exists $\mathscr{X_{dps}} \in \mathscr{DPS}^+$ where $\undervec{\mathscr{F}}_{\mathscr{udps}}^{\mathscr{DPS}}(\mathscr{X_{dps}})=\mathscr{X_{udps}}$: such $\mathscr{X_{dps}}$ would be isomorphic to $Conc_5^2$ defined in \reff{stronger-theorem-2}. \end{enumerate} \end{proofpart} \end{proof} \begin{remark} A stronger version of \textit{Theorem 2} would state that for all $\mathscr{X_{dps}} \in \mathscr{DPS}^+_5$, $\mathscr{X_{dps}}$ is either isomorphic to $Conv_5$, $Conc_5^1$ or $Conc_5^2$, where \begin{gather}\label{stronger-theorem-2} \Scale[0.9]{ \begin{split}
Conv_5 = &(Cv_5, \Theta_{Cv_5}) \\
Cv_5 = &\{1,2,3,4,5\} \\
\Theta_{Cv_5} = & \{(\{1,2\},\{(\{3,4,5\},\varnothing\}), \\
&\;\; (\{1,3\},\{\{2\},\{4,5\}\}), \\
&\;\; (\{1,4\},\{\{2,3\},\{5\}\}), \\
&\;\; (\{1,5\},\{\{2,3,4\},\varnothing\}), \\
&\;\; (\{2,3\},\{\{1,4,5\},\varnothing\}), \\
&\;\; (\{2,4\},\{\{1,5\},\{3\}\}), \\
&\;\; (\{2,5\},\{\{1\},\{3,4\}\}), \\
&\;\; (\{3,4\}, \{\{1,2,5\},\varnothing\}) \\
&\;\; (\{3,5\}, \{\{1,2\},\{4\}\}) \\
&\;\; (\{4,5\}, \{\{1,2,3\},\varnothing\})
\end{split} \begin{split}
Conc_5^1 = &(Cc_5^1, \Theta_{Cv_5}) \\
Cc_5^1 = &\{1,2,3,4,5\} \\
\Theta_{Cv_5^1} = & \{(\{1,2\},\{(\{3,4,5\},\varnothing\}), \\
&\;\; (\{1,3\},\{\{2\},\{4,5\}\}), \\
&\;\; (\{1,4\},\{\{2,3,5\},\varnothing\}), \\
&\;\; (\{1,5\},\{\{2,3\},\{4\}), \\
&\;\; (\{2,3\},\{\{1,4,5\},\varnothing\}), \\
&\;\; (\{2,4\},\{\{1,5\},\{3\}\}), \\
&\;\; (\{2,5\},\{\{1\},\{3,4\}\}), \\
&\;\; (\{3,4\}, \{\{1,2,5\},\varnothing\}) \\
&\;\; (\{3,5\}, \{\{1,2\},\{4\}\}) \\
&\;\; (\{4,5\}, \{\{1\},\{2,3\}\})
\end{split}
\begin{split}
Conc_5^2 = &(Cc_5^2, \Theta_{Cc_5^2}) \\
Cc_5^2 = &\{1,2,3,4,5\} \\
\Theta_{Cv_5^2} = & \{(\{1,2\},\{(\{3,4,5\},\varnothing\}), \\
&\;\; (\{1,3\},\{\{2\},\{4,5\}\}), \\
&\;\; (\{1,4\},\{\{2,3,5\},\varnothing\}), \\
&\;\; (\{1,5\},\{\{2,3\},\{4\}), \\
&\;\; (\{2,3\},\{\{1,4\},\{5\}\}), \\
&\;\; (\{2,4\},\{\{1,3,5\},\varnothing\}), \\
&\;\; (\{2,5\},\{\{1\},\{3,4\}\}), \\
&\;\; (\{3,4\}, \{\{1,5\},\{2\}\}) \\
&\;\; (\{3,5\}, \{\{1,2\},\{4\}\}) \\
&\;\; (\{4,5\}, \{\{1\},\{2,3\}\}) \end{split}
} \end{gather} To prove this version of \textit{Theorem 2} we would need to prove that there exists no \textit{div point sets} in $\mathscr{DPS}^+_5$ not isomorphic to $Conv_5$, $Conc_5^1$ or $Conc_5^2$. \end{remark} \begin{remark} Let $All_{of\Omega}$ be a generalization of $All_{of \Omega_{6}}$ where $All_{of\Omega}(\mathscr{X},4) = All_{of\Omega_{6}}(\mathscr{X})$ i.e. \begin{align}
All_{of\Omega}(\mathscr{X},n) = \{ \Omega_{of_{based\_on}}(R, \pi_2(\mathscr{X})) : R \in \mathcal{P}(\pi_1(\mathscr{X})) : |R| = n \} \end{align} by \textit{Theorem 2}, it is clear that following proposition is false: \proposition{
A \textit{unit div point set} of 5 or more points $\mathscr{X_{udps}}$ is in $\mathscr{UDPS}^+$ iff all members of $All_{of\Omega_{of 6}}(\mathscr{X_{udps}},n)$ are in $\mathscr{UDPS}^+$, for any $n \in \mathbb{N}_{\geq 4}$ less than $|\pi_1(\mathscr{X_{udps}})|$. } However, this weaker version of it still holds true: \proposition{
If $\mathscr{X_{udps}}$ is in $\mathscr{UDPS}^+$, all members of $All_{of\Omega_{of 6}}(\mathscr{X_{udps}},n)$ are also in $\mathscr{UDPS}^+$ for any $n \in \mathbb{N}_{\geq 4}$ less than $|\pi_1(\mathscr{X_{udps}})|$. } There is undoubtedly some similarity between the false proposition above, and the following proposition which is too false: \proposition{
A \textit{div point set} of 4 or more points, $\mathscr{X_{dps}}$, is in $\mathscr{DPS}^+$ iff all elements in $\mathscr{Sdps_{of}}(\mathscr{X_{dps}},n)$ are in $\mathscr{DPS}^+$, for any $n \in \mathbb{N}_{\geq 3}$ less than $|\pi_1(\mathscr{X_{dps}})|$. }
Since it is vacuously true that any \textit{div point sets} of 3 points satisfy \reff{dividon_law1}, \reff{dividon_law2}, and \reff{dividon_law3}, we cannot conclude that a certain \textit{div point set} satisfies \reff{dividon_law1}, \reff{dividon_law2}, and \reff{dividon_law3} just because all its \textit{sub div point sets} of 3 points satisfy them. Now recall \textit{Lemma 3} where $E_2$ of the hypergraph based on $P$ is an empty set in the case when $|P|=4$ and, as a result, for such $E_2$, it is vacuously true that \reff{hypergraph_law2} always holds for any coloring, and thus we cannot conclude that a certain \textit{unit div point set} $\mathscr{X_{udps}}$ where $Col_{\mathscr{udps}}(\pi_2(\mathscr{X_{udps}}))$ satisfies \reff{hypergraph_law2}, just because all memebers of $All_{of\Omega_{of 6}}(\mathscr{X_{udps}},4)$ are isomorphic to some \textit{unit div point set} $\mathscr{A_{udps}}$ where $Col_{\mathscr{udps}}(\pi_2(\mathscr{X_{udps}}))$ satisfies \reff{hypergraph_law2}.
It can be proven that the proposition regarding \textit{unit div point sets} above is true in the case when $n \in \mathbb{N}_{\geq 5}$, similar to how the proposition regarding \textit{div point sets} is true in the case when $n \in \mathbb{N}_{\geq 4}$. \end{remark} \subsection{\textit{convexity}} The notion that there exists $n$ points forming a convex polygon among some set of points in $\mathbb{E}^2$ can be expressed through \textit{convexity} in the context of \textit{div point sets}. \begin{defn} A \textit{div point set} $\mathscr{X}$ has a \textit{convexity} of $n$ iff there exists a \textit{div point set} $\mathscr{X_{sub}}$ such that $\mathscr{X_{sub}} \leq \mathscr{X} $ and $\mathscr{X_{sub}}$ is isomorphic to $Conv_n$ defined as follow \begin{align} \begin{split}\label{conv}
& Conv_n = (P, \{ (d,\delta_{conv}(d,P)) : d \in \mathcal{P}(P) : |d| = 2\}) \\
&\text{where }
\begin{cases}
&P = \{ x \in \mathbb{N}_{\geq 1} : x \leq n \} \\
&\delta_{conv}(d,P) = \{ \{ p : p \in P : inside(p,d) \}, \{ p : p \in P : outside(p,d) \}\} \\
&\text{where }
\begin{cases}
& inside(p,d) = ( p > min(d) \land p < max(d) ) \\
& outside(p,d) = ( p < min(d) \lor p > max(d) ) \\
&\text{where }
\begin{cases}
&\text{$min(d)$ returns the smallest number in $d$} \\
&\text{$max(d)$ returns the biggest number in $d$}.
\end{cases}
\end{cases}
\end{cases}
\end{split}
\end{align} for any $n \in \mathbb{N}_{\geq 3}$. Here is an implementation of $Conv_n$ as a function in Haskell: \begin{lstlisting} import Data.List
combine :: Int -> [a] -> [[a]] combine 0 _ = [[]]
combine n xs = [ y:ys | y:xs' <- tails xs, ys <- combine (n-1) xs']
convex:: Int -> ([Int],[([Int],[[Int]])]) convex n = (points, dividons)
where
points = [1..n]
dividers = combine 2 points
dividons = [(divider,[div1,div2])
| divider@(a:b:_) <- dividers,
let divs = points \\ divider,
let div1 = [ x | x <- divs, x > a, x < b ],
let div2 = divs \\ div1 ] \end{lstlisting} \end{defn} \begin{ax}
For any $\mathscr{X} \in \mathscr{DPS^+}$, $\mathscr{X}$ has an interpretation for $\pi_1(\mathscr{X})$ as some set of points in $E^2$ among which there exists $n$ points forming a convex polygon, iff $\mathscr{X}$ has a convexity of $n$. More precisely, there exists an interpretation for $P' \subseteq \pi_1(\mathscr{X})$ as some set of $|P'|$ points in $E^2$ forming a convex polygon iff $\mathscr{Sdps}(\mathscr{X},P')$ is isomorphic to $Conv_n$, for any $n \geq 3$. \end{ax} \begin{remark} One may notice that for $n \geq 4$, all \textit{sub div point sets} of $n-1$ points of $Conv_{n}$ are isomorphic to $Conv_{n-1}$, and as a consequence, a \textit{div point set} with a \textit{convexity} of $k$ would also have a convexity of $m$, for all $k$, $m$ in $\mathbb{N}_{\geq 3}$ where $m < k$. In Euclidean geometry, by \textit{Axiom 2}, that is equivalent to the following proposition: for any $n \geq 4$, after removing any one point from a set of $n$ points that are the vertices of a convex polygon, the remaining points too forms a convex polygon, and as a consequence, any set of points in general position containing $k$ points forming a convex polygon would also contain $m$ points forming a convex polygon, for all $k$, $m$ in $\mathbb{N}_{\geq 3}$ where $m < k$. \end{remark} \begin{remark} We can conclude from \textit{Theorem 2} that a \textit{div point set} of 5 or more points always has a convexity of 4. By \textit{Axiom 2}, this means that we can always find 4 points forming a convex polygon in any set of 5 or more points in general position on an Euclidean plane, as stated in the Erdos-Szekeres conjecture (for the case when $n=4$). \end{remark} \section{ A reduction to a \textit{multiset unsatisfiability problem}} The Erd{\"o}s-Szekeres conjecture can be expressed as a conjunction of \reff{lowerbound} and \reff{upperbound} in the theory of \textit{div point sets}. \begin{align} \begin{split} \label{lowerbound} &\forall n \in \mathbb{N}_{\geq 3} \\
&\qquad \exists \mathscr{A} \in \mathscr{DPS}^+ \quad |\pi_1(A)| = 2^{n-2} \ \land \forall \mathscr{A_s} \leq \mathscr{A} \quad \mathscr{A_s} \not\cong Conv_n \end{split} \\ \begin{split} \label{upperbound} &\forall n \in \mathbb{N}_{\geq 3} \\
&\qquad \forall \mathscr{A} \in \mathscr{DPS}^+ \quad |\pi_1(A)| > 2^{n-2} \Leftrightarrow \exists \mathscr{A_s} \leq \mathscr{A} \quad \mathscr{A_s} \cong Conv_n \end{split} \end{align} Since the lower bound has been proven to be $2^{n-2}+1$, all is left is to prove \reff{upperbound} and the conjecture would be proven.
\subsection{a combinatorial characteristics of \textit{sub div point sets}}
As we examine \textit{div point sets} of $v$ points for $v > 5$, we would notice this pretty interesting fact about \textit{sub div point sets}: for any natural number $a \geq 1$, let $\mathscr{SDPSS}$ be the set of all \textit{sub div point set} of $v-a$ points of any \textit{div point set} of $v$ points, for any $\mathscr{X_{SDPS}}$ in $\mathscr{SDPSS}$, we can always select $v-a$ distinct $(a+1)$-cardinality subsets of $\mathscr{SDPSS}$, each of which contains $\mathscr{X_{SDPS}}$ and other \textit{div point sets}, and all these \textit{div point sets} that it contains all have $\binom{v-a-1}{t}$ \textit{sub div point sets} of $t$ points in common, for any natural number $t \geq 1$. What is cool about this is that it can be generalized from $a+1$ to $a+b$ for any $b \geq 1$ as long as $a+b$ is smaller than $v$ (and in which case the \textit{div point sets} would have $\binom{v-a-b}{t}$ \textit{sub div point sets} of $t$ points in common). Notationally, \begin{align} \begin{split}\label{combinatorics} &\forall \mathscr{X} \in \mathscr{DPS}^+ \\
&\qquad v \coloneqq |\pi_1(\mathscr{X})| \\ & \qquad \forall a \in \mathbb{N}_{\geq 1} \\ &\qquad \qquad \mathscr{SDPSS} \coloneqq \mathscr{Sdps_{of}}(\mathscr{X}, v-a)\\ &\qquad \qquad \forall \mathscr{X_{SDPS}} \in \mathscr{SDPSS} \\ &\qquad \qquad \qquad \forall b \in \{ x: x \in \mathbb{N}_{\geq 1} : x < v-a \} \\ &\qquad \qquad \qquad \qquad \exists \mathscr{S} \in \mathcal{P_n}( \mathcal{P_n}(\mathscr{SDPSS}, a+b),v-a)\\ &\qquad \qquad \qquad \qquad \qquad \forall \mathscr{s} \in \mathscr{S} \\ &\qquad \qquad \qquad \qquad \qquad \qquad \mathscr{X_{SDPS}} \in \mathscr{s}\\ & \qquad \qquad \qquad \qquad \qquad \qquad \forall t \in \mathbb{N}_{\geq 1} \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad |\bigcap_{\mathcal{l} \in \mathscr{s}} \mathscr{Sdps_{of}}(\mathcal{l},t)| = \binom{v-a-b}{t}\\ \end{split} \end{align} where \begin{align}
\mathcal{P_n}(S,n) = \{ x : x \in \mathcal{P}(S) : |x| = n \} \end{align} To understand why such combinatorial characteristic exists, consider this: any 2 \textit{sub div point sets}, $\mathscr{S}_1$ and $\mathscr{S}_2$ of a certain \textit{div point set} is distinct iff they are of distinct points i.e. $\mathscr{S}_1\not=\mathscr{S}_2 \Leftrightarrow \pi_1(\mathscr{S}_1)\not= \pi_1(\mathscr{S}_2)$, and thus \reff{combinatorics} is equivalent as stating that for any set $\mathcal{N}$ with the same cardinality as $\mathbb{N}$, \begin{align} \begin{split}\label{natural-combinatorics} &\forall X \in \mathcal{P}(\mathcal{N}) \\
&\qquad v \coloneqq |X| \\ & \qquad \forall a \in \mathbb{N}_{\geq 1} \\ &\qquad \qquad X_{subset\_set} \coloneqq \mathcal{P_n}(X, v-a)\\ &\qquad \qquad \forall X_{subset} \in X_{subset\_set} \\ &\qquad \qquad \qquad \forall b \in \{ x: x \in \mathbb{N}_{\geq 1} : x < v-a \} \\ &\qquad \qquad \qquad \qquad \exists S \in \mathcal{P_n}( \mathcal{P_n}(X_{subset\_set}, a+b),v-a)\\ &\qquad \qquad \qquad \qquad \qquad \forall s \in S \\ &\qquad \qquad \qquad \qquad \qquad \qquad X_{subset} \in s\\ & \qquad \qquad \qquad \qquad \qquad \qquad \forall n \in \mathbb{N}_{\geq 1} \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad |\bigcap_{l \in s} \mathcal{P_n}(l,n)| = \binom{v-a-b}{n}\\ \end{split} \end{align}
For the purpose of illustration, suppose we have some \textit{div point set} of 9 points $\mathscr{X_9}$, let $isom$ be a bijective function from $\mathscr{Sdps_{of}}(\mathscr{X_9},4)$ to a set of natural numbers $N$ where $N = \{ n: n \in \mathbb{N}_{\geq 1} : n \leq |\mathscr{Sdps_{of}}(\mathscr{X_9},4)|\}$, the set \begin{align} \{ \{ iso(\mathscr{X_4}) : \mathscr{X_4} \in \mathscr{Sdps_{of}}(\mathscr{X_5},4) \} :\mathscr{X_5} \in \mathscr{Sdps_{of}}(\mathscr{X_9},5) \} \end{align} shows how \textit{sub div point sets} of 4 points of $\mathscr{X_9}$ (each represented by a distinct natural number) would be disturbed among \textit{sub div point sets} of 5 points of $\mathscr{X_9}$ and is isomorphic$\color{black}^*$ to: {\tiny \begin{align*}\begin{split} &\{\{1,2,7,22,57\},\{1,3,8,23,58\},\{1,4,9,24,59\},\{1,5,10,25,60\},\{1,6,11,26,61\},\{2,3,12,27,62\},\\ &\{2,4,13,28,63\},\{2,5,14,29,64\},\{2,6,15,30,65\},\{3,4,16,31,66\},\{3,5,17,32,67\},\{3,6,18,33,68\},\\ &\{4,5,19,34,69\},\{4,6,20,35,70\},\{5,6,21,36,71\},\{7,8,12,37,72\},\{7,9,13,38,73\},\{7,10,14,39,74\},\\ &\{7,11,15,40,75\},\{8,9,16,41,76\},\{8,10,17,42,77\},\{8,11,18,43,78\},\{9,10,19,44,79\},\{9,11,20,45,80\},\\ &\{10,11,21,46,81\},\{12,13,16,47,82\},\{12,14,17,48,83\},\{12,15,18,49,84\},\{13,14,19,50,85\},\{13,15,20,51,86\},\\ &\{14,15,21,52,87\},\{16,17,19,53,88\},\{16,18,20,54,89\},\{17,18,21,55,90\},\{19,20,21,56,91\},\{22,23,27,37,92\},\\ &\{22,24,28,38,93\},\{22,25,29,39,94\},\{22,26,30,40,95\},\{23,24,31,41,96\},\{23,25,32,42,97\},\{23,26,33,43,98\},\\ &\{24,25,34,44,99\},\{24,26,35,45,100\},\{25,26,36,46,101\},\{27,28,31,47,102\},\{27,29,32,48,103\},\{27,30,33,49,104\},\\ &\{28,29,34,50,105\},\{28,30,35,51,106\},\{29,30,36,52,107\},\{31,32,34,53,108\},\{31,33,35,54,109\},\{32,33,36,55,110\},\\ &\{34,35,36,56,111\},\{37,38,41,47,112\},\{37,39,42,48,113\},\{37,40,43,49,114\},\{38,39,44,50,115\},\{38,40,45,51,116\},\\ &\{39,40,46,52,117\},\{41,42,44,53,118\},\{41,43,45,54,119\},\{42,43,46,55,120\},\{44,45,46,56,121\},\{47,48,50,53,122\},\\ &\{47,49,51,54,123\},\{48,49,52,55,124\},\{50,51,52,56,125\},\{53,54,55,56,126\},\{57,58,62,72,92\},\{57,59,63,73,93\},\\ &\{57,60,64,74,94\},\{57,61,65,75,95\},\{58,59,66,76,96\},\{58,60,67,77,97\},\{58,61,68,78,98\},\{59,60,69,79,99\},\\ &\{59,61,70,80,100\},\{60,61,71,81,101\},\{62,63,66,82,102\},\{62,64,67,83,103\},\{62,65,68,84,104\},\{63,64,69,85,105\},\\ &\{63,65,70,86,106\},\{64,65,71,87,107\},\{66,67,69,88,108\},\{66,68,70,89,109\},\{67,68,71,90,110\},\{69,70,71,91,111\},\\ &\{72,73,76,82,112\},\{72,74,77,83,113\},\{72,75,78,84,114\},\{73,74,79,85,115\},\{73,75,80,86,116\},\{74,75,81,87,117\},\\ &\{76,77,79,88,118\},\{76,78,80,89,119\},\{77,78,81,90,120\},\{79,80,81,91,121\},\{82,83,85,88,122\},\{82,84,86,89,123\},\\ &\{83,84,87,90,124\},\{85,86,87,91,125\},\{88,89,90,91,126\},\{92,93,96,102,112\},\{92,94,97,103,113\},\{92,95,98,104,114\},\\ &\{93,94,99,105,115\},\{93,95,100,106,116\},\{94,95,101,107,117\},\{96,97,99,108,118\},\{96,98,100,109,119\}, \\ &\{97,98,101,110,120\}, \{99,100,101,111,121\},\{102,103,105,108,122\},\{102,104,106,109,123\},\{103,104,107,110,124\},\\ &\{105,106,107,111,125\}, \{108,109,110,111,126\}, \{112,113,115,118,122\},\{112,114,116,119,123\},\{113,114,117,120,124\},\\ &\{115,116,117,121,125\}, \{118,119,120,121,126\},\{122,123,124,125,126\}\}\end{split} \end{align*} }Notice how for every $\mathscr{X_5} \in \mathscr{Sdps_{of}}(\mathscr{X_9},5)$, there exists a set in $\mathcal{P_n}(\mathcal{P_n}(\mathscr{Sdps_{of}}(\mathscr{X_9},5),5),5)$ whose elements are distinct subsets of $\mathscr{Sdps_{of}}(\mathscr{X_9},5)$, each containing $\mathscr{X_5}$ and other \textit{div point sets} all having a common \textit{sub div point set} of 4 points. \\ \\ We believe that \reff{upperbound} is simply an elegant result of having a structure, whose sub-structures possess the combinatorial characteristic described above, that satisfies a certain constraint, which, in this case, is that described in \reff{assign5} below.
\begin{note} \textit{isomorphic$\color{black}^*$}: the isomorphism here is defined as a bijective function $f$ from $\mathbb{N}$ to $\mathbb{N}$ such that $f^{members^{2}}(A) = B$. \end{note}
\subsection{the problem $\color{black}UNSAT_{multiset}^{\mathscr{DPS}^+}$} \begin{defn}$UNSAT_{multiset}$ is the decision problem of determining if there exists no value-assignment for all variables in $V$, distributed in a certain manner among the multisets in $M$, such that it satisfies the FOL formulae in $C$, where the value-assignment is defined to be a function $Z$: for all $v$ in $V$, $Z(v)= x$ for some $x \in D$. Here $D$, often referred to as the domain, is the set of values a variable can be assigned to. An instance of $UNSAT_{multiset}$ can thus be represented as a 4-tuple $(V,D,M,C)$. \end{defn} \begin{prelude}[Definition 9] We shall now present the problem $UNSAT_{multiset}^{\mathscr{DPS}^+}$, a special case of $UNSAT_{multiset}$, of which, if an instance is solved (\textit{solved} in the sense that it is proven that the formulae in $F$ are unsatisfiable), it would prove that, for a particular $n \in \mathbb{N}_{\geq 5}$ (depending on which instance of $UNSAT_{multiset}^{\mathscr{DPS}^+}$ is solved), there exists no \textit{div point set} of $2^{n-2}+1$ points $\mathscr{X}$ that satisfies \begin{align} \begin{split}\label{assign5} &\forall \mathscr{X_5} \in \mathscr{Sdps_{of}}(\mathscr{X},5) \\ &\qquad [Assign(\mathscr{X_4}) : \mathscr{X_4} \in \mathscr{Sdps_{of}}(\mathscr{X_5},4) ] \in \{ [1,1,1,1,0], [1,1,0,0,0], [0,0,0,0,0] \} \end{split} \end{align} but does not satisfy \begin{align} \begin{split}\label{n-conv} & \exists \mathscr{A_s} \in \mathscr{Sdps_{of}}(\mathscr{X},n) \\ & \qquad \forall \mathscr{A_{ss}} \in \mathscr{Sdps_{of}}( \mathscr{A_s} ,4) \quad Assign(\mathscr{A_{ss}}) = 0 \end{split} \end{align} where \begin{align} \begin{split} Assign(\mathscr{A}) = \begin{cases}
1 & \text{if } \; \quad \mathscr{A} \cong Conc_4^1 \\
0 & \text{if } \; \quad \mathscr{A} \cong Conv_4 \\ \end{cases} \end{split} \end{align} consequently proving that the proposition after the universal quantifier in \reff{upperbound} holds for that particular $n$, and thus the $n$-instance of Erd{\"o}s-Szekeres conjecture. This is because if a \textit{div point set} of 5 or more points $\mathscr{X}$ is in $\mathscr{DPS}^+$, by \textit{Theorem 2}, $\mathscr{X}$ satisfy \reff{assign5}. If there exists no \textit{div point set} of $2^{n-2}+1$ points that satisfies \reff{assign5} but not \reff{n-conv}, it would indicate that every \textit{div point set} of $2^{n-2}+1$ or more points in $\mathscr{DPS}^+$ satisfy \reff{n-conv} and therefore has a convexity of $n$. (Note that \reff{upperbound} can be rewritten as follows \begin{align}\label{upperbound2} \begin{split} &\forall n \in \mathbb{N}_{\geq 3} \\ & \qquad\forall \mathscr{A} \in \mathscr{DPS}^+ \\
& \qquad\qquad |\pi_1(A)| > 2^{n-2} \\ & \qquad\qquad \Leftrightarrow \exists \mathscr{A_s} \in \mathscr{Sdps_{of}}(\mathscr{A},n) \\ & \qquad \qquad \qquad \forall \mathscr{A_{ss}} \in \mathscr{Sdps_{of}}(\mathscr{A_s},4) \quad Assign(\mathscr{A_{ss}}) = 0 \end{split} \end{align} since any \textit{sub div point set} of $k$ points of any $Conv_n$ is isomorphic to $Conv_k$ for all $k,n \in \mathbb{N}_{\geq 3}$ where $n \geq k$). \end{prelude} \begin{defn} $UNSAT_{multiset}^{\mathscr{DPS}^+}$ is a special case, or subproblem, of $UNSAT_{multiset}$ (\textit{subproblem} in the sense that all instances of $UNSAT_{multiset}^{\mathscr{DPS}^+}$ are instances of $UNSAT_{multiset}$). An instance of $UNSAT_{multiset}$, $(V,D,M,C)$, is an instance of $UNSAT_{multiset}^{\mathscr{DPS}^+}$ iff for some $n \geq 5$, \begin{equation} \begin{gathered}
|V| = \binom{2^{n-2}+1}{4} \\ D = \{0,1\} \\ M = A \cup B \\ \end{gathered} \end{equation} and $A$ is a set of 5-cardinality multisets while $B$ is a set of $n$-cardinality multisets, and the variables in $V$ are distributed in $m \in A$ the same way as how elements in $\mathscr{Sdps_{of}}(\mathscr{X},4)$ are distributed in $\mathscr{X_{SPDPS}} \in\mathscr{Sdps_{of}}(\mathscr{X},5)$, while the variables are distributed in $m \in B$ the same way as how elements in $\mathscr{Sdps_{of}}(\mathscr{X},4)$ are distributed in $\mathscr{X_{SPDPS}} \in\mathscr{Sdps_{of}}(\mathscr{X},n)$, where $\mathscr{X}$ is a \textit{div point set} of $2^n+1$ points, and $C$ consists of formulae \ref{constraint1} and \ref{constraint2}. \begin{align} \label{constraint1} &\forall a \in A \qquad a \in \{ [1,1,1,1,0], [1,1,0,0,0] ,[0,0,0,0,0] \} \\ \label{constraint2} &\forall b \in B \qquad b \not = \underbrace{[0,0,0,...,0,0]}_{\binom{n}{4} \; 0's} \end{align} The distribution of variables in $A$ and $B$ can be implement in Haskell as follows: \begin{lstlisting} import Data.List import Data.Maybe type Multiset = [Integer]
merge (a:x) (b:y) = (a,b) : merge x y merge [] _ = []
choose :: Integer -> Integer -> Integer n `choose` k
| k < 0 = 0
| k > n = 0
| otherwise = factorial n `div` (factorial k * factorial (n-k))
factorial :: Integer -> Integer factorial n = foldl (*) 1 [1..n]
combine :: Integer -> [Integer] -> [[Integer]] combine 0 _ = [[]]
combine n xs = [ y:ys | y:xs' <- tails xs, ys <- combine (n-1) xs']
number_of_points = (\n->(2^(n-2)+1))
n_setOf_m_Multisets:: Integer -> Integer -> [Multiset] n_setOf_m_Multisets m n = [ map fromJust $ map ((flip lookup) encoding)
(combine 4 m_points) | m_points <- combine n [1..m] ]
where
encoding = merge (combine 4 [1..m]) [1..(m `choose` 4)]
setA :: Integer -> [Multiset] setA n = n_setOf_m_Multisets (number_of_points n) 5
setB :: Integer -> [Multiset]
setB n = [ x | x <- n_setOf_m_Multisets (number_of_points n) n, 2 `elem` x ] \end{lstlisting} \begin{remark} A different implementation may result in a different $M$ for the same $n$. Nonetheless, the different $M$ obtained from a different implementation would be isomorphic to the $M$ obtained from this implementation, in which case we would consider that distribution to be the same. Thus as far as unsatisfiability is concerned, for every $n \in \mathbb{N}_{\geq 5}$, there exists exactly one instance of $UNSAT_{multiset}^{\mathscr{DPS}^+}$. \end{remark} \begin{remark} Each variable in $V$ represents $Assign(\mathscr{X_4})$ for a particular element $\mathscr{X_4} \in \mathscr{Sdps_{of}}(\mathscr{X},4)$ where $\mathscr{X}$ is a \textit{div point set} of $2^{n-2}+1$ points for some $n \in \mathbb{N}_{\geq 5}$. If there exists no value-assignment $Z$ satisfying formulae in $C$, we can be certain that there exists no \textit{div point set} of $2^{n-2}+1$ points $\mathscr{X}$ satisfying \reff{assign5} but not \reff{n-conv} as mentioned above and consequently proving the $n$-instance of conjecture. \end{remark} \begin{remark}
Here is the simplest instance of $UNSAT_{multiset}^{\mathscr{DPS}^+}$ (when $n=5$): since $A=B$, we have $|M|=|A|=|B|=\binom{2^{5-2}+1}{5}=126$ multisets, and $|V| = \binom{2^{5-2}+1}{4} = 126$ variables as well (with each denoted by $v_n$ below), distributed among the multisets in $M$ as follows: {\tiny \begin{align*}\begin{split} &M = \{[v_{1},v_{2},v_{7},v_{22},v_{57}],[v_{1},v_{3},v_{8},v_{23},v_{58}],[v_{1},v_{4},v_{9},v_{24},v_{59}],[v_{1},v_{5},v_{10},v_{25},v_{60}],[v_{1},v_{6},v_{11},v_{26},v_{61}],[v_{2},v_{3},v_{12},v_{27},v_{62}],\\ &[v_{2},v_{4},v_{13},v_{28},v_{63}],[v_{2},v_{5},v_{14},v_{29},v_{64}],[v_{2},v_{6},v_{15},v_{30},v_{65}],[v_{3},v_{4},v_{16},v_{31},v_{66}],[v_{3},v_{5},v_{17},v_{32},v_{67}],[v_{3},v_{6},v_{18},v_{33},v_{68}],\\ &[v_{4},v_{5},v_{19},v_{34},v_{69}],[v_{4},v_{6},v_{20},v_{35},v_{70}],[v_{5},v_{6},v_{21},v_{36},v_{71}],[v_{7},v_{8},v_{12},v_{37},v_{72}],[v_{7},v_{9},v_{13},v_{38},v_{73}],[v_{7},v_{10},v_{14},v_{39},v_{74}],\\ &[v_{7},v_{11},v_{15},v_{40},v_{75}],[v_{8},v_{9},v_{16},v_{41},v_{76}],[v_{8},v_{10},v_{17},v_{42},v_{77}],[v_{8},v_{11},v_{18},v_{43},v_{78}],[v_{9},v_{10},v_{19},v_{44},v_{79}],[v_{9},v_{11},v_{20},v_{45},v_{80}],\\ &[v_{10},v_{11},v_{21},v_{46},v_{81}],[v_{12},v_{13},v_{16},v_{47},v_{82}],[v_{12},v_{14},v_{17},v_{48},v_{83}],[v_{12},v_{15},v_{18},v_{49},v_{84}],[v_{13},v_{14},v_{19},v_{50},v_{85}],[v_{13},v_{15},v_{20},v_{51},v_{86}],\\ &[v_{14},v_{15},v_{21},v_{52},v_{87}],[v_{16},v_{17},v_{19},v_{53},v_{88}],[v_{16},v_{18},v_{20},v_{54},v_{89}],[v_{17},v_{18},v_{21},v_{55},v_{90}],[v_{19},v_{20},v_{21},v_{56},v_{91}],[v_{22},v_{23},v_{27},v_{37},v_{92}],\\ &[v_{22},v_{24},v_{28},v_{38},v_{93}],[v_{22},v_{25},v_{29},v_{39},v_{94}],[v_{22},v_{26},v_{30},v_{40},v_{95}],[v_{23},v_{24},v_{31},v_{41},v_{96}],[v_{23},v_{25},v_{32},v_{42},v_{97}],[v_{23},v_{26},v_{33},v_{43},v_{98}],\\ &[v_{24},v_{25},v_{34},v_{44},v_{99}],[v_{24},v_{26},v_{35},v_{45},v_{100}],[v_{25},v_{26},v_{36},v_{46},v_{101}],[v_{27},v_{28},v_{31},v_{47},v_{102}],[v_{27},v_{29},v_{32},v_{48},v_{103}],[v_{27},v_{30},v_{33},v_{49},v_{104}],\\ &[v_{28},v_{29},v_{34},v_{50},v_{105}],[v_{28},v_{30},v_{35},v_{51},v_{106}],[v_{29},v_{30},v_{36},v_{52},v_{107}],[v_{31},v_{32},v_{34},v_{53},v_{108}],[v_{31},v_{33},v_{35},v_{54},v_{109}],[v_{32},v_{33},v_{36},v_{55},v_{110}],\\ &[v_{34},v_{35},v_{36},v_{56},v_{111}],[v_{37},v_{38},v_{41},v_{47},v_{112}],[v_{37},v_{39},v_{42},v_{48},v_{113}],[v_{37},v_{40},v_{43},v_{49},v_{114}],[v_{38},v_{39},v_{44},v_{50},v_{115}],[v_{38},v_{40},v_{45},v_{51},v_{116}],\\ &[v_{39},v_{40},v_{46},v_{52},v_{117}],[v_{41},v_{42},v_{44},v_{53},v_{118}],[v_{41},v_{43},v_{45},v_{54},v_{119}],[v_{42},v_{43},v_{46},v_{55},v_{120}],[v_{44},v_{45},v_{46},v_{56},v_{121}],[v_{47},v_{48},v_{50},v_{53},v_{122}],\\ &[v_{47},v_{49},v_{51},v_{54},v_{123}],[v_{48},v_{49},v_{52},v_{55},v_{124}],[v_{50},v_{51},v_{52},v_{56},v_{125}],[v_{53},v_{54},v_{55},v_{56},v_{126}],[v_{57},v_{58},v_{62},v_{72},v_{92}],[v_{57},v_{59},v_{63},v_{73},v_{93}],\\ &[v_{57},v_{60},v_{64},v_{74},v_{94}],[v_{57},v_{61},v_{65},v_{75},v_{95}],[v_{58},v_{59},v_{66},v_{76},v_{96}],[v_{58},v_{60},v_{67},v_{77},v_{97}],[v_{58},v_{61},v_{68},v_{78},v_{98}],[v_{59},v_{60},v_{69},v_{79},v_{99}],\\ &[v_{59},v_{61},v_{70},v_{80},v_{100}],[v_{60},v_{61},v_{71},v_{81},v_{101}],[v_{62},v_{63},v_{66},v_{82},v_{102}],[v_{62},v_{64},v_{67},v_{83},v_{103}],[v_{62},v_{65},v_{68},v_{84},v_{104}],[v_{63},v_{64},v_{69},v_{85},v_{105}],\\ &[v_{63},v_{65},v_{70},v_{86},v_{106}],[v_{64},v_{65},v_{71},v_{87},v_{107}],[v_{66},v_{67},v_{69},v_{88},v_{108}],[v_{66},v_{68},v_{70},v_{89},v_{109}],[v_{67},v_{68},v_{71},v_{90},v_{110}],[v_{69},v_{70},v_{71},v_{91},v_{111}],\\ &[v_{72},v_{73},v_{76},v_{82},v_{112}],[v_{72},v_{74},v_{77},v_{83},v_{113}],[v_{72},v_{75},v_{78},v_{84},v_{114}],[v_{73},v_{74},v_{79},v_{85},v_{115}],[v_{73},v_{75},v_{80},v_{86},v_{116}],[v_{74},v_{75},v_{81},v_{87},v_{117}],\\ &[v_{76},v_{77},v_{79},v_{88},v_{118}],[v_{76},v_{78},v_{80},v_{89},v_{119}],[v_{77},v_{78},v_{81},v_{90},v_{120}],[v_{79},v_{80},v_{81},v_{91},v_{121}],[v_{82},v_{83},v_{85},v_{88},v_{122}],[v_{82},v_{84},v_{86},v_{89},v_{123}],\\ &[v_{83},v_{84},v_{87},v_{90},v_{124}],[v_{85},v_{86},v_{87},v_{91},v_{125}],[v_{88},v_{89},v_{90},v_{91},v_{126}],[v_{92},v_{94},v_{97},v_{103},v_{113}],[v_{92},v_{95},v_{98},v_{104},v_{114}],\\ &[v_{92},v_{93},v_{96},v_{102},v_{112}],[v_{93},v_{94},v_{99},v_{105},v_{115}],[v_{93},v_{95},v_{100},v_{106},v_{116}],[v_{94},v_{95},v_{101},v_{107},v_{117}],[v_{96},v_{97},v_{99},v_{108},v_{118}], \\ &[v_{96},v_{98},v_{100},v_{109},v_{119}],[v_{97},v_{98},v_{101},v_{110},v_{120}], [v_{99},v_{100},v_{101},v_{111},v_{121}],[v_{102},v_{103},v_{105},v_{108},v_{122}],[v_{102},v_{104},v_{106},v_{109},v_{123}],\\ &[v_{103},v_{104},v_{107},v_{110},v_{124}],[v_{105},v_{106},v_{107},v_{111},v_{125}], [v_{108},v_{109},v_{110},v_{111},v_{126}], [v_{112},v_{113},v_{115},v_{118},v_{122}],[v_{112},v_{114},v_{116},v_{119},v_{123}],\\ &[v_{113},v_{114},v_{117},v_{120},v_{124}],[v_{115},v_{116},v_{117},v_{121},v_{125}], [v_{118},v_{119},v_{120},v_{121},v_{126}],[v_{122},v_{123},v_{124},v_{125},v_{126}]\} \end{split} \end{align*} } It is no surprise that the distribution of variables in $m \in M$ above is exactly that of \textit{sub div point sets} of 4 points in $\mathscr{X}_5 \in \mathscr{Sdps_{of}}(\mathscr{X_9},5)$ as shown above. \end{remark} \end{defn}
\begin{remark} $UNSAT_{multiset}^{\mathscr{DPS}^+}$ can be reduced into the boolean unsatisfiability problem, the complement of $SAT$, in a rather straightforward manner by first converting each multiset in $A$ into the DNF formula below: \begin{align}
\bigvee_{v_0 \in \mathcal{A}} (\neg v_0 \land \bigwedge_{v_1 \in \mathcal{A} \setminus \{v_0\}} v_1 ) \lor \bigvee_{\mathcal{A}_{|3|} \in \mathcal{A}_{|3|}^*}( \bigwedge_{v_0 \in \mathcal{A}_3} \neg v_0 \land \bigwedge_{v_1 \in V \setminus \mathcal{A}_{|3|}} v_1 ) \lor ( \bigwedge_{v_0 \in \mathcal{A}} \neg v_0 ) \end{align}
where $\mathcal{A}_{|3|}^* = \{ \mathcal{A}_{|3|} \in \mathbb{P}(\mathcal{A}): |\mathcal{A}_{|3|}|=3 \}$ and $\mathcal{A}$ denotes the set of variables in each multiset, and each multiset in $B$ into the DNF formula
below:\begin{align} \bigvee_{v \in \mathcal{B}} v \end{align} where $\mathcal{B}$ denotes the set of variables in each multiset, then joining all these DNF formulae conjunctively. One may realize that, in the case when $\mathcal{B}=\mathcal{A}$ ,the conjunction of $\bigvee_{v \in \mathcal{B}} v$ and $\bigwedge_{v \in \mathcal{A}} \neg v$ gives a tautology, and thus for the instance of $UNSAT_{multiset}^{\mathscr{DPS}^+}$ where $n=5$, we would have a simpler propositional formula. The same observation can be made in the FOL formulae of such $UNSAT_{multiset}^{\mathscr{DPS}^+}$ instance wherein satisfying both \reff{constraint1} and \reff{constraint2} is equivalent to satisfying $\forall a \in A \; \; a \in \{ [1,1,1,1,0], [1,1,0,0,0] \}$. \end{remark} \begin{remark} We thereby conclude that a plausible approach to proving the upper-bound of the Erd{\"o}s-Szekeres conjecture through $UNSAT_{multiset}^{\mathscr{DPS}^+}$ is by induction i.e. we start of by solving the instance of $UNSAT_{multiset}^{\mathscr{DPS}^+}$ where $n=5$ - apparently accomplishable with a modern SAT solver - and then we prove the inductive hypothesis that $\forall m \in \mathbb{N}_{\geq 5} \; \; \mathcal{UNSAT}(m) \Rightarrow \mathcal{UNSAT}(m+1)$ where $\mathcal{UNSAT}(n)$ denotes the unsatisfiability of the $n$-instance of $UNSAT_{multiset}^{\mathscr{DPS}^+}$. \end{remark} \begin{remark} The Erd{\"o}s-Szekeres conjecture would not be disproven even if a certain instance of $UNSAT_{multiset}^{\mathscr{DPS}^+}$ turns out to yield $False$ (i.e. it is satisfiable), since satisfying the constraints only implies that there exists a \textit{div point set} of $2^{n-2}+1$ points for a particular $n \in \mathbb{N}_{\geq 5}$ where \begin{enumerate}[I.] \item none of its \textit{sub div point sets} of $n$ points is isomorphic to $Conv_n$ \item each of its \textit{sub div point sets} of 5 points has 4, 2 or 0 distinct \textit{sub div point sets} of 4 points isomorphic to $Conc_4^1$ \end{enumerate} from which we cannot conclude that such \textit{div point set} is in $\mathscr{DPS}^+$, unless it too satisfies the stronger version of \textit{Theorem 2} i.e. unless proven so, we should not rule out the possibility for some of its \textit{sub div point sets} of 5 points to not be in $\mathscr{DPS}^+$ despite themselves having 4, 2 or 0 distinct \textit{sub div point sets} of 4 points isomorphic to $Conc_4^1$ (with the remaining isomorphic to $Conv_4$).
To disprove the Erd{\"o}s-Szekeres conjecture, not only do we need to show that \reff{upperbound} is false, we need to demonstrate there exists no other constraints besides \reff{dividon_law1}, \reff{dividon_law2}, and \reff{dividon_law3} $\mathscr{X} \in \mathscr{DPS}^*$ has to satisfy such that there exists an interpretation for $\pi_1(\mathscr{X})$ as some set of points in $\mathbb{E}^2$ i.e. \textit{Axiom 1}'s consistency with Euclidean geometry. \end{remark}
\end{document} | arXiv |
\begin{document}
\title[Recurrence and ergodicity] {On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces}
\author{Jean-Pierre Conze and Eugene Gutkin}
\address{IRMAR, CNRS UMR 6625, Universit\'e de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France} \email{[email protected]}
\address{Copernicus University, Chopina 12/18, Torun 87-100; IMPAN, Sniadeckich 8, Warszawa 10, Poland} \email{[email protected],[email protected]}
\keywords{noncompact, periodic polygonal surfaces, billiard flow, billiard map, skew products, centered displacement functions, recurrence, transience, wind-tree model, small obstacles condition, ergodic cocycles, quasi-periods, periods}
\subjclass{37A25, 37A40, 37C40, 37E35}
\date{\today}
\begin{abstract} We study the recurrence and ergodicity for the billiard on noncompact polygonal surfaces with a free, cocompact action of $\Z$ or $\Z^2$. In the $\Z$-periodic case, we establish criteria for recurrence. In the more difficult $\Z^2$-periodic case, we establish some general results. For a particular family of $\Z^2$-periodic polygonal surfaces, known in the physics literature as the wind-tree model, assuming certain restrictions of geometric nature, we obtain the ergodic decomposition of directional billiard dynamics for a dense, countable set of directions. This is a consequence of our results on the ergodicity of $\ZZ$-valued cocycles over irrational rotations. \end{abstract}
\maketitle
\tableofcontents
\section*{Introduction} \label{intro}
Beginning with Boltzmann's {\em ergodic hypothesis}, mathematicians have been investigating the ergodicity of dynamical systems of physical origin. Among them are the geodesic flows on riemannian configuration spaces describing mathematically the physical models at hand. If the configuration space has a boundary, we arrive at a billiard.
It is notoriously difficult to study the ergodicity of these dynamical systems, in particular the famous Boltzmann-Sinai model. On the contrary, the conservativeness of a dynamical system of this kind is guaranteed by the Poincar\'e recurrence theorem, provided its phase space has finite volume. This holds, for instance, if the configuration space is compact.
The situation changes drastically if the configuration space has infinite volume. This happens, in particular, if the space is invariant under a free action of an infinite group, say $\Z^d$. Not only the ergodicity, but even the conservativeness of these dynamical systems is a challenging question; it is open in many relevant examples.
Some physical models correspond to the geodesic flows on polygonal surfaces invariant under free actions of infinite groups \cite{Gut09}. We will speak of {\em periodic polygonal surfaces} or {\em $G$-periodic polygonal surfaces}, where $G$ is the group in question. For instance, the space of the classical wind-tree model in statistical physics \cite{HW80} is a $\ZZ$-periodic polygonal surface.
In this work we study the recurrence and ergodicity for geodesic flows on noncompact polygons and noncompact polygonal surfaces. For reader's convenience, we will briefly survey the relevant material in the compact case. We refer to \cite{Gut84,Gut96} for details. Let $P$ be a compact polygonal surface, e. g., a polygon. If $P$ is rational, the study of the geodesic flow on $P$ is equivalent to the study of the geodesic flow on a compact translation surface, say $S$. That flow decomposes as a one-parameter family of directional translation flows, say $T_{\theta}^t:S\to S,\,0\le\theta\le 2\pi$. The celebrated result in this subject says that for Lebesgue almost all directions $\theta$ the flows $T_{\theta}^t$ are (uniquely) ergodic \cite{KMS}. This theorem has far reaching applications to the billiard in irrational polygons \cite{KMS,Vo97}. See \cite{Gut96,Gut03} for details.
Let now $S$ be a noncompact translation surface. Let $T_{\theta}^t:S\to S,\,0\le\theta\le 2\pi,$ be the one-parameter family of directional translation flows \cite{Gut09}. For the purposes of this discussion we assume that $S$ is a periodic translation surface. The following question naturally arises: Is there an analog of the unique ergodicity theorem in \cite{KMS} for noncompact, periodic translation surfaces? This question is mainly open. In fact, it is not known whether the flows $T_{\theta}^t:S\to S$ are conservative for typical directions $\theta$. The examples from our sections~~\ref{one_periodic}, ~~\ref{two_periodic} and~~\ref{rect_lorenz} might be useful to formulate conjectures about the flows $T_{\theta}^t:S\to S$ for noncompact translation surfaces.
We will now informally describe some of our results. Let $\tilde{P}$ be a $\Z$-periodic polygonal surface with a boundary. Let $P=\tilde{P}/\Z$ be the compact quotient. Suppose that the billiard flow on $P$ is ergodic. Then the billiard flow on $\tilde{P}$ is conservative. See Theorem~~\ref{one_period_thm}.
With any polygon $O$ inside the unit square we associate the $\Z$-periodic strip $\tilde{P}_{O}$ with polygonal obstacles. Then for a dense $G_{\de}$-set of obstacles, the billiard flow on $\tilde{P}_{O}$ is conservative. See Theorem~~\ref{dens_G_del_thm}. Let $O$ be an {\em irrational polygon}. Suppose that its angles admit a superexponentially fast approximation by numbers in $\pi\QQ$. Then the geodesic flow on $\tilde{P}_O$ is conservative. See Theorem~~\ref{vorob_cond_thm}.
Let $\tilde{P}=\tilde{P}(a,b)$ be the {\em rectangular Lorenz gas} obtained by deleting from $\RR$ the $\ZZ$-periodic family of $a\times b$ rectangles. This corresponds to the {\em wind-tree model} \cite{HW80}. Let $p,q\in\N$ be relatively prime; denote by $\tilde{T}(a,b;p,q)$ the billiard flow in the direction $\arctan{(q/p)}$. We say that the {\em obstacles are small} if $qa+pb \le 1$. Assuming the small obstacles condition, we analyze the flow $\tilde{T}(a,b;p,q)$. Let $\tilde{T}_{\mbox{cons}}(a,b;p,q)$ and $\tilde{T}_{\mbox{diss}}(a,b;p,q)$ be the conservative and the dissipative parts of $\tilde{T}(a,b;p,q)$ respectively. We show that $\tilde{T}_{\mbox{diss}}(a,b;p,q)$ is trivial iff $qa+pb=1$. Assume now that $a/b$ is irrational. Then
we obtain an ergodic decomposition of $\tilde{T}_{\mbox{cons}}(a,b;p,q)$. The $2pq$ ergodic components are isomorphic; they have a simple geometric meaning. Thus, $\tilde{T}_{\mbox{cons}}(a,b;p,q)$ is a finite multiple of an ergodic flow. See Theorem~~\ref{erg_decom_thm}, Theorem~~\ref{bil_erg_decom_thm}, and Proposition~~\ref{bil_erg_decom_cor}.
For instance, the conservative part of the wind-tree billiard flow in direction $\pi/4$ is the flow $\tilde{T}_{\mbox{cons}}(a,b;1,1)$; it has two ergodic components. Figure~~\ref{fig7} shows a typical orbit of this flow. It encounters only a half of the set of rectangular obstacles. Loosely speaking, the orbit skips every other obstacle. The skipped obstacles are visited by a typical orbit from the other ergodic component of the flow.
This is a special case of the general situation, as we explain in Theorem~~\ref{erg_decom_thm} and Theorem~~\ref{bil_erg_decom_thm}.
\begin{figure}\label{fig7}
\end{figure}
We will now comment on our methods. The billiard flows are suspensions of billiard maps. Exploiting the periodicity under a group $G$, we identify a billiard map of this kind with a skew product over an interval exchange with the fiber $G$. In our setting, we obtain skew products with the fibers $\Z$ and $\ZZ$. Thus, we reduce the questions concerning the ergodicity of the billiard on noncompact polygonal surfaces to the ergodicity of particular $\Z^d$-valued cocycles over interval exchanges.
This work is not concerned with the subject of ergodicity for cocycles over general interval exchanges. We establish the ergodicity of a class of $\Z^2$-valued cocycles over irrational rotations. See Theorems~~\ref{wdd-erg-thm},~~\ref{erg-beta}, and ~~\ref{ergo-Psi} in section~~\ref{ergodic}. Theorem~~\ref{wdd-erg-thm} states that a cocycle of this kind is ergodic if the continued fraction decomposition of the rotation number satisfies certain genericity assumptions which hold for almost all numbers. Theorem~~\ref{erg-beta} and Theorem~~\ref{ergo-Psi} strengthen Theorem~~\ref{wdd-erg-thm} by removing these assumptions.
We apply these results in section~~\ref{rect_lorenz} to obtain the ergodic decompositions of directional flows in the wind-tree model. Let $T^t$ be one of the $2pq$ geometric components of $\tilde{T}_{\mbox{cons}}(a,b;p,q)$. The Poincar\'e map for $T^t$ is a skew product with fibre $\ZZ$ over a circle rotation; this rotation is irrational iff $a/b$ is irrational. The corresponding $\ZZ$-valued cocycle belongs to the class of cocycles studied in section~~\ref{ergodic}. Theorem~~\ref{ergo-Psi} implies the ergodicity of $T^t$.
We will now outline the structure of our exposition. Section~~\ref{cadre} contains the information to be used in the body of the paper. In section~~\ref{ergodic_sub} we review the material on skew products, cocycles, recurrence and transience. In section~~\ref{trans_surf_sub} we establish the framework of polygonal surfaces. In section~~\ref{bill_sub} we recall the basic facts about the billiard flow and the billiard map.
Section~~\ref{one_periodic} is about the $\Z$-periodic case. This framework is naturally divided into two extreme situations: the generic case and the rational case. In the former situation we show the conservativeness and some ergodic properties for the generic $\Z$-periodic polygonal surface; in the latter we establish these properties for the directional flows in almost all directions.
Section~~\ref{two_periodic} and section~~\ref{rect_lorenz} are devoted to the billiard on $\Z^2$-periodic polygonal surfaces. In section~~\ref{two_periodic} we establish some ergodic properties for arbitrary $\Z^2$-periodic polygonal surfaces. In section~~\ref{rect_lorenz} we consider a particular family of such surfaces: The rectangular Lorenz gas or the wind-tree model of the physics literature. Applying the results of section~~\ref{ergodic}, we obtain the ergodic decomposition of directional billiard flows for a dense, countable set of directions, under restrictions of geometric nature, namely the smallness of obstacles condition. Section~~\ref{ergodic} is a study of ergodicity for a class of cocycles over irrational rotations. The results, besides being of interest on their own, are instrumental for the material in section~~\ref{rect_lorenz}.
\section{The setting and preliminaries} \label{cadre}
For convenience of the reader, we recall the basic material about recurrence \cite{Aa97, CorFomSin, Sc77}. We will consider two kinds of {\em dynamical systems}: transformations and flows. In the former case, we have the standard Borel space $(X, {\mathcal A})$ endowed with a possibly infinite measure $\nu$, and a transformation $\tau: (X, {\mathcal A})\to(X, {\mathcal A})$ preserving $\nu$. For simplicity, we will assume that $\tau$ is invertible. The setting for flows is analogous \cite{CorFomSin}. We will use the notation $(X,\tau,\nu)$ (resp. $(Y,T^t,\mu)$) for transformations (resp. flows).
The dynamical system $(X,\tau,\nu)$ is {\em recurrent} or {\em conservative} if for every measurable set $B\subset X$ and for $\nu$-a.e. point $x\in B$ there is $n=n(x)>0$ such that $\tau^n x \in B$. Recurrence for flows $(Y,T^t,\mu)$ is defined analogously. A dynamical system uniquely decomposes as a disjoint union of the {\em conservative} part and the {\em dissipative part}. For simplicity, we describe this decomposition only for a transformation, $(X,\tau,\nu)$. The conservative, dissipative subsets $C,D\subset X$ are measurable and $\tau$-invariant. If $\nu(C)>0$ then $(C,\tau,\nu)\subset(X,\tau,\nu)$ is recurrent. Suppose that $\nu(D)>0$. Then there is a measurable set $A\subset D$ such that $D = \cup_{n \in \Z} \tau^n A$; moreover, $\tau^p A \cap \tau^q A = \emptyset$ for $p \ne q$.
If $\nu(X)=\infty$, this decomposition is, in general, nontrivial. We will use the following observation. Let $(Y,T^t,\mu)$ be a flow, let $X\subset Y$ be a {\em cross-section}, and let $(X,\tau,\nu)$ be the induced transformation. Then the transformation $(X,\tau,\nu)$ is recurrent iff the flow $(Y,T^t,\mu)$ is recurrent.
The geometric spaces that we work with in the body of the paper are differentiable manifolds, possibly with boundary and corners. The transformations and flows are piecewise differentiable.
\subsection{Ergodic theory for skew products} \label{ergodic_sub}
\break We will represent our dynamical systems as skew products over dynamical systems with finite invariant measures. Their fibers will be infinite abelian groups.
In this section we recall the relevant material on $G$-valued cocycles, where $G$ is an infinite abelian group; we will write the group operation additively. We restrict the discussion mostly to the groups $G=\R^m\times\Z^n$. We denote by $\leb_G$ a Haar measure on $G$, suppressing the subscript if the group is clear from the context.
\begin{defin} \label{cocycle_def} Let $(X,\tau,\nu)$ be a dynamical system, and let $\varphi:X\to G$ be a measurable function. It determines a {\em cocycle} $\varphi(n, x)$, also denoted by $\varphi_n(x)$ or simply $(\varphi_n)$, as follows. We set $\varphi(0, x) = 0$. For $n\ne 0$ we set
\begin{equation} \label{cocycle_eq} \varphi_n(x) = \sum_{j=0}^{n-1} \varphi(\tau^j x), \mbox{if}\ n > 0; \ \varphi_n(x) = - \sum_{j=n}^{-1} \varphi(\tau^j x), \mbox{if}\ n < 0. \end{equation}
\end{defin}
Thus, $\varphi(n, x)$ are the ergodic (or Birkhoff) sums of $\varphi$ with respect to the transformation $(X,\tau,\nu)$. The cocycle $(\varphi_n)$ can be viewed as the random walk on the group $G$ driven by the dynamical system $(X,\tau,\nu)$. Set
\begin{equation} \label{produit_gauche_eq} \tilde{\tau}(x,g) = (\tau x, g+\varphi(x)). \end{equation}
Then $\tilde{\tau}$, or $\tau_\varphi$ to emphasize the dependence on $\varphi$, is a transformation of $\tilde{X}=X\times G$ preserving the product measure $\tilde{\nu} =\nu\times\leb$. The dynamical system $(\tilde{X},\tilde{\tau},\tilde{\nu})$ (or $(\tilde{X},\tau_\varphi,\tilde{\nu})$) is the {\it skew product} over $(X,\tau,\nu)$ with the {\it displacement function} $\varphi$. Equation~~\eqref{cocycle_eq} corresponds to the iterates of $\tilde{\tau}$. Namely, for $n\in\Z$ we have
\begin{equation} \label{prod_gauche_iter_eq} \tilde{\tau}^n(x,g)=(\tau^n(x),g+\varphi_n(x)). \end{equation}
\begin{defin} \label{recurr1_def} Let $(X,\tau,\nu)$ be a dynamical system with $\nu(X)<\infty$. Let $\varphi:X\to G$ be a measurable function. The cocycle $(\varphi_n)$ is {\it transient at $x\in X$} if $\varphi_n(x)\to \infty$; otherwise, the cocycle is {\em recurrent at $x$}.\footnote{We will also say that $x$ is a transient (resp. recurrent) point for the cocycle.} The cocycle is recurrent (resp. transient) if it is recurrent (resp. transient) at a.e. $x \in X$. \end{defin}
We point out a subtlety in Definition~~\ref{recurr1_def}. Let $\alpha_n, \beta_n$ be $\Z$-valued cocycles over $(X,\tau,\nu)$. Their direct sum $\varphi_n=(\alpha_n,\beta_n)$ is a $\ZZ$-valued cocycle. The recurrence of $\alpha_n,\beta_n$ does not necessarily imply that $\varphi_n$ is recurrent. See, e. g., \cite{ChCo09} for an example.
The sets of transient and recurrent points for a cocycle are measurable and invariant. Hence, any cocycle $\varphi_n$ over an ergodic $(X,\tau,\nu)$ is either recurrent or transient. Let $(X,\tau,\nu)$ be arbitrary, let $(\varphi_n)$ be a cocycle, and let $R\subset X$ be the set of recurrent points for $\varphi_n$. Suppose that $\nu(R)>0$, and let $\nu_{R}$ be the restriction of $\nu$ to $R$; set $\tilde{\nu}_{R}=\nu_{R}\times\leb$, $\tilde{R}=R\times G$. Then the skew product $(\tilde{R},\tilde{\tau},\tilde{\nu}_{R})$ is a conservative dynamical system \cite{Sc77}. Assume, moreover, that $X$ is a separable metric space and that $\tau:X\to X$ is compatible with the topological structure.\footnote{These assumptions will be satisfied in our applications.} Then for a.e. $x\in R$ there is an infinite sequence $n_k=n_k(x)$ such that $\tau^{n_k} x \to x$ and $\varphi(n_k, x) \to 0$ \cite{Sc77}. Therefore, for almost every point in $R$ the sequence $(\varphi_k(x))_{k \ge 0}$ visits arbitrarily close to $0\in G$. If $G=\Z^d$, then we have $\varphi_k(x) = 0$ infinitely many times. Suppose now that $(\tilde{X},\tilde{\tau},\tilde{\nu})$ comes from a billiard model. Then $R$ is the conservative part of the billiard phase space. The billiard ball emanating from a point in $R$ almost surely returns infinitely often to the obstacle from which it started, and arbitrarily close to the point of departure.
\vskip 3mm We refer the reader to \cite{Sc77} for criteria of recurrence for cocycles. See also \cite{Ke75}, \cite{At76}. The following lemma from \cite{ChCo09} gives a simple sufficient condition for recurrence of $\R^d$-valued cocycles.
\begin{lem} \label{recurrence_lem} Let $(X,\tau,\nu)$ be a dynamical system with a finite measure, let $\varphi:X\to\R^d$ be a measurable function, and let $\varphi(n,x)$ be the corresponding cocycle.
Let $| \ \ |$ be a norm on $\R^d$. Suppose that there exists a strictly increasing sequence of integers $k_n$ and a sequence of nonnegative functions $\delta_n(x)$ converging to $0$ for almost every $x$ such that \begin{equation} \label{lyapunov_eq}
\lim_n \nu(\{x: |\varphi(k_n, x)| \ge \delta_n(x) n^{\frac{1}{d}}\})=0. \end{equation}
Then the cocycle is recurrent. \end{lem}
Let $(X,\tau,\nu)$ be a dynamical system, and let $\varphi:X\to G$ be a measurable function. Recall that $\varphi$ is a {\em coboundary} if there exists a measurable function $\psi:X\to G$ such that $\varphi=\psi-\tau\psi$. Let $(\tilde{X},\tilde{\tau},\tilde{\nu})$ be the dynamical system defined by equation~~\eqref{produit_gauche_eq}. Let $H\subset G$ be a closed subgroup; set $\tilde{X}_H=X\times G/H,\tilde{\nu}_H=\nu\times\leb_{G/H}$. Define $\tilde{\tau}_ H: \tilde{X}_H \to \tilde{X}_H$ by $\tilde{\tau}_ H(x,g +H) = (\tau x, (\varphi(x)+ g) + H)$. Then $(\tilde{X}_H,\tilde{\tau}_ H,\tilde{\nu}_H)$ is the skew product over $(X,\tau,\nu)$ with the fibre $G/H$ and the displacement function $\varphi_H(x)=\varphi(x)+H$.
\begin{lem} \label{cobord_lem} If there is a closed, proper subgroup $H\subset G$ such that the dynamical system $(\tilde{X}_H,\tilde{\tau}_H,\tilde{\nu}_H)$ is ergodic, then $\varphi$ is not a coboundary. \begin{proof} Assume the opposite, and let $\varphi=\psi-\tau\psi$ where $\psi:X\to G$ is a measurable function. Set $\Psi(x,g)=(x,\psi(x)+g)$. Then $\Psi:X\times G\to X\times G$ is an automorphism of the measure space $(X\times G,\tilde{\nu})$; it conjugates $\tilde{\tau}$ and the product transformation $\tau\times\text{Id}$. Dividing by $H$, we obtain the automorphism $\Psi_H:X\times G/H\to X\times G/H$ conjugating $\tilde{\tau}_H$ and $\tau\times\text{Id}_{G/H}$. This contradicts the ergodicity of $\tilde{\tau}_H$. \end{proof} \end{lem}
We introduce a terminology to express the property that a walk in $G$ visits any compact set with zero asymptotic frequency.
\begin{defin} \label{NRUO_def} Let $(X,\tau,\nu)$ be a dynamical system with finite invariant measure, let $G\subset\R^d$ or $\Z^d$ be a closed subgroup, and let $\varphi:X\to G$ be a measurable function. The associated cocycle $(\varphi_n)$ is {\em zero-recurrent} if it is recurrent, and for a.e. point $x \in X$ and any compact set $K\subset G$ we have
\begin{equation} \label{freq0} \lim _{n\to\infty}\left\{{1\over n} \sum_{k = 0}^{n-1} 1_K(\varphi(k, x))\right\} = 0. \end{equation}
\end{defin}
\begin{prop} \label{recurrence_prop} Let $(X,\tau,\nu)$ be a dynamical system with finite invariant measure, let $\varphi:X\to\R$ be an integrable function, let $(\varphi_n)$ be the associated cocycle, and let $(\tilde{X},\tilde{\tau},\tilde{\nu})$ be the corresponding skew product. Denote by ${\mathcal J}$ the $\sigma$-algebra of measurable $\tau$-invariant subsets. Let
${\mathbb E}(\varphi |{\mathcal J})$ be the conditional expectation of $\varphi$ with respect to ${\mathcal J}$. Let $R\subset X$ be the set of recurrent points for the cocycle $(\varphi_n)$.
Then the following properties hold.
\begin{itemize}
\item[1.] The set $R$
and the set $\{x: {\mathbb E}(\varphi |{\mathcal J})(x) = 0 \}$ coincide up to a set of $\nu$-measure zero.
\item[2.] If the dynamical system $(X,\tau,\nu)$ is ergodic and $\int_X \varphi \ d\nu = 0$, then the cocycle $(\varphi_n)$ is recurrent.
\item[3.] If, moreover, $\varphi$ is not a coboundary, then the cocycle $(\varphi_n)$ is zero-recurrent. \end{itemize} \begin{proof} Let $A,B\subset X$ be measurable sets. By $A=B$ we will mean that $A$ and $B$ are equal in the measure-theoretic sense.\footnote{I. e., their symmetric difference is of $\nu$-measure $0$.} We will also say, simply, that $A$ and $B$ coincide.
The first two claims are classical. For $\nu$-a.e. $x$ the conditional expectation ${\mathbb E}(\varphi |{\mathcal J})(x)$ is defined, and, by the Birkhoff ergodic theorem,
$\frac{1}{n}\varphi(n,x) \to {\mathbb E}(\varphi |{\mathcal J})(x)$. Hence, the $\tau$-invariant sets $\{x:{1\over n} \varphi(n,x)
\rightarrow 0\}$ and $\{x: {\mathbb E}(\varphi |{\mathcal J})(x) = 0\}$ coincide. We denote this $\tau$-invariant set by $X_0$. By Lemma~~\ref{recurrence_lem}, the cocycle $(\varphi_n)$ is recurrent on
$X_0$. By the Birkhoff ergodic theorem, almost every $x\in X\setminus X_0$ is transient for $(\varphi_n)$; moreover, on $X\setminus X_0$, the cocycle has linear dissipation. Thus, $R=X_0=\{x: {\mathbb E}(\varphi |{\mathcal J})(x) = 0\}$, proving claim 1. Claim 2 directly follows from claim 1. We will now prove claim 3.
Set $\tilde{X}=X\times\R$ and let $(\tilde{X},\tilde{\tau},\tilde{\nu})$ be the skew product equation~~\eqref{produit_gauche_eq}. Let $K\subset \R$ be any compact. Set $$u_K(x,g)=\lim_{n\to\infty} {1\over n} \sum_{k = 0}^{n-1} 1_K(\varphi(k, x) + g).$$
The ergodic theorem applied to $(\tilde{X},\tilde{\tau},\tilde{\nu})$ ensures the existence of the limit for a.e. $(x,g)$ and that $u_K$ is an integrable, nonnegative, $\tilde{\tau}$-invariant function on $\tilde{X}$. Suppose that $u_K\ne 0$ on a set of positive measure. Then $u_K\tilde{\nu}$ is a finite $\tilde{\tau}$-invariant measure on $\tilde{X}$, absolutely continuous with respect to $\tilde{\nu}$. Since $(X,\tau,\nu)$ is ergodic, this implies that $\varphi$ is a coboundary \cite{Co79}, contrary to the assumption. Thus, for any compact $K\subset \R$ we have $\lim \frac1n \sum_{k = 0}^{n-1}1_K(\varphi(k, x) + g) = 0$ for $\tilde{\nu}$-a.e. $(x,g)\in\tilde{X}$. Since $\R$ is a countable union of compacta, there exists $\tilde{Y}\subset\tilde{X}$, $\tilde{\nu}(\tilde{X}\setminus\tilde{Y})=0$, such that for $(x,g)\in\tilde{Y}$ and any compact $K\subset\R$ we have $u_K(x,g)=0$.
By Fubini's theorem, there exists $g_0\in\R$ such that, for $\nu$ a.e. $x\in X$ we have $\lim \frac1n \sum_{k = 0}^{n-1} 1_K(\varphi(k, x) + g_0) = 0$ for any compact $K\subset\R$. But $g\mapsto g+g_0$ is a self-homeomorphism of $\R$. \end{proof} \end{prop}
\begin{rem} \label{zero_rec_rem} {\em The same argument proves claim 3 for $\R^d$-valued cocycles. See \cite{Co79} for a generalization to cocycles with values in locally compact groups.
} \end{rem}
Proposition~~\ref{recurrence_prop} and Lemma~~\ref{cobord_lem} imply the following.
\begin{corol} Let $(X,\tau,\nu)$ be a dynamical system with finite measure. Let $\varphi:X\to\R$ be a measurable function such that $\int_X \varphi\, d\nu = 0$, and let $(\varphi_n)$ be the associated cocycle. Let $(\tilde{X},\tilde{\tau},\tilde{\nu})$ be the skew product corresponding to $\varphi$. For $p\in\R$ let $(\tilde{X}_p,\tilde{\tau}_p,\tilde{\nu}_p)$ be the reduction of $(\tilde{X},\tilde{\tau},\tilde{\nu})$ with respect to the subgroup $H=p\Z$, as in Lemma~~\ref{cobord_lem}.
Suppose that for some $p\ne0$ the dynamical system $(\tilde{X}_p,\tilde{\tau}_p,\tilde{\nu}_p)$ is ergodic. Then the cocycle $(\varphi_n)$ is zero-recurrent. \end{corol}
We will use the following proposition.
\begin{prop} \label{lem-k_n} Let $(X,\tau,\nu)$ be an ergodic dynamical system with a finite measure, let $\varphi:X\to\R^d$ be a measurable function, and let $\varphi(n,x)$ be the corresponding cocycle. Then the following dichotomy holds: i) The cocycle $(\varphi_n)$ is recurrent; ii) Let $k_n\in\N$ be any strictly increasing sequence. Then there exists $c>0$\footnote{In general, it depends on the sequence.} such that for a.e. $x\in X$ we have
\begin{eqnarray} \label{lim-k_n}
\limsup_n (n^{-1/d} |\varphi(k_n, x)|) = c. \end{eqnarray} \begin{proof}
The quantity $\limsup_n (n^{-{1\over d}} |\varphi(k_n, x)|)$ is an invariant, measurable function. By ergodicity, it is equal to a constant $c \ge 0$. The claim is now immediate from Lemma~~\ref{recurrence_lem}. \end{proof} \end{prop}
\subsection{Noncompact, periodic polygonal surfaces} \label{trans_surf_sub}
\break We will now establish the geometric framework for our study. Let $G$ be an infinitely countable group acting freely and cocompactly by isometries on a noncompact riemannian manifold $\tilde{P}$.\footnote{In general, with boundary and corners.} Then $P=\tilde{P}/G$ is compact; the projection $p:\tilde{P}\to P$ is a riemannian covering. Let $U\tilde{P},UP$ be the unit tangent bundles for $\tilde{P},P$; let $\tilde{T}^t,T^t$ be the respective geodesic flows; let $\tilde{\mu},\mu$ be the Liouville measures for $U\tilde{P},UP$ respectively. The action of $G$ on $\tilde{P}$ uniquely extends to a free, cocompact action on $U\tilde{P}$. We have $UP=U\tilde{P}/G$; let $q:U\tilde{P}\to UP$ be the projection. Then $q:(U\tilde{P},\tilde{T}^t,\tilde{\mu})\to(UP,T^t,\mu)$ is a covering of flows.
Let $X\subset UP$ be a compact submanifold which is a cross-section for $(UP,T^t,\mu)$. Then the manifold $\tilde{X}=p^{-1}(X)\subset U\tilde{P}$ is a cross-section for the flow $(U\tilde{P},\tilde{T}^t,\tilde{\mu})$. Let $\nu,\tilde{\nu}$ be the induced measures on $X,\tilde{X}$ respectively; let $\tau:X\to X,\,\tilde{\tau}:\tilde{X}\to \tilde{X}$ be the respective Poincar\'e maps. Then $\tilde{X} = X\times G$ measure theoretically, and $\tilde{\nu}=\nu\times\leb$. There is a unique mapping $\varphi:X\to G$ such that $(\tilde{X},\tilde{\tau},\tilde{\nu})$ is the skew product over $(X,\tau,\nu)$ with the displacement function $\varphi$. See equation~~\eqref{produit_gauche_eq}.
Let the manifold $\tilde{P}$ be a {\em noncompact polygonal surface}. See \cite{Gut84,Gut09} for the background. We will say that $\tilde{P}$ is a {\em $G$-periodic} polygonal surface. When the group $G$ is implicit, we will say that $\tilde{P}$ is a periodic polygonal surface. When $G=\Z$ or $G=\ZZ$, we will say that $\tilde{P}$ is $\Z$-periodic or $\ZZ$-periodic respectively. The projection $p:\tilde{P}\to P$ is a {\em covering of polygonal surfaces} \cite{Gut09}.
If $\tilde{P}$ (or, equivalently, $P$) is a {\em rational polygonal surface}, we associate with it a finite subgroup $\Ga=\Ga(P)\subset O(2)$. For $N\ge 1$ let $R_N\subset O(2)$ be the dihedral group of order $2N$, i. e., the group generated by two orthogonal reflections, with the angle $\pi/N$ between their axes. If $\partial P\neq\emptyset$, then $\Ga(P)=R_N$, where $N$ is determined by $P$. Note that $R_1$ consists of a reflection and the identity. The associated {\em translation surface} $S=S(P)$ \cite{GJ} is a compact riemann surface endowed with a riemannian metric, flat everywhere except for a finite number of {\em cone points}. The group $\Ga(P)$ acts on $S(P)$ by isometries, and we have $P=S(P)/\Ga(P)$ \cite{Gut84}. Thus, a polygonal surface is a translation surface iff $\Ga(P)=\text{Id}$. Let $\Ga=\Ga(P)$ and let $S=S(P)$. Then there is a unique noncompact, $G$-periodic translation surface $\tilde{S}$ such that the following conditions hold. The groups $\Ga$ and $G$ act on $\tilde{S}$ by isometries; the two actions commute. We have $\tilde{S}/G=S$, $\tilde{S}/\Ga=\tilde{P}$, $S/\Ga=P$, $\tilde{P}/G=P$; the projections $\tilde{S}\to S$, $\tilde{P}\to P$, $\tilde{S}\to\tilde{P}$, $S\to P$ are compatible.
The group $\Ga$ acts on the unit circle $U\subset\RR$; let $U/\Ga$ be the quotient. The flows $(U\tilde{P},\tilde{T}^t,\tilde{\mu})$ and $(UP,T^t,\mu)$ decompose as one-parameter families of {\em directional geodesic flows} $(U\tilde{P}_{\theta},\tilde{T}^t_{\theta},\tilde{\mu}_{\theta})$ and $(UP_{\theta},T^t_{\theta},\mu_{\theta})$ where $\theta\in U/\Ga$. The projection $q$ is compatible with the decompositions, inducing the directional projections $q_{\theta}:(U\tilde{P}_{\theta},\tilde{T}^t_{\theta}, \tilde{\mu}_{\theta})\to(UP_{\theta},T^t_{\theta},\mu_{\theta})$. The flow $(UP_{\theta},T^t_{\theta},\mu_{\theta})$ is naturally isomorphic to the linear flow in direction $\theta$ on the translation surface $S$; the measure $\mu_{\theta}$ corresponds to the Lebesgue measure on $S$. Analogously, $(U\tilde{P}_{\theta},\tilde{T}^t_{\theta},\tilde{\mu}_{\theta})$ is the {\em linear flow in direction $\theta$ on the periodic translation surface} $\tilde{S}$ \cite{Gut09}.
\begin{exa} \label{strip_exa} {\em Let $B\subset\RR$ be the horizontal strip bounded by the lines $\{y=0\}$ and $\{y=1\}$. For $0\le a,b <1,a+b>0$ let $R_0=R_0(a,b)$ be the $(a\times b)$-rectangle centered at $(1/2,1/2)$. Set $\tilde{P}(a,b)=B\setminus\cup_{k\in\Z}(R_0+(k,0))$. The $\Z$-periodic polygonal surface $\tilde{P}(a,b)$ is a strip with a periodic sequence of rectangular obstacles. See figure~~\ref{rect_loren_band}. Let $Q$ be the unit square $0\le x,y \le 1$; let $C$ be the cylinder obtained by identifying the vertical sides of $Q$. Then $P(a,b)=C\setminus R_0(a,b)$ is the unit cylinder with a rectangular obstacle. In the limit cases $a=0$ or $b=0$ the obstacles degenerate into {\em barriers}. We have $\Ga=R_2$ if $b\ne 0$ and $\Ga=R_1$ if $b=0$.
\begin{figure}
\caption{\it An infinite band with a periodic configuration of rectangular obstacles.}
\label{rect_loren_band}
\end{figure}
Set $P=P(a,b)$ and $S=S(a,b)$. Let $a,b\ne 0$. The translation surface $S$ is constructed from 4 copies of $Q\setminus R_0(a,b)$ via identifications of their sides shown in figure~~\ref{quotient_S(a,b)}. It has $4$ cone points with cone angles $6\pi$. This yields $g(S)=5$ \cite{GJ}. The genus of $S$ can also be computed directly from the angles in $P$ \cite{Gut84}. The same analysis applies when $a=0$ or $b=0$. If $a=0$, the surface $S$ is made from $4$ rectangles with vertical barriers. It has $4$ cone points with cone angles $4\pi$, yielding $g(S)=3$. If $b=0$, then $S$ is made from $2$ rectangles with horizontal barriers. There are $2$ cone points with cone angles $4\pi$, yielding $g(S)=2$.
Set $\tilde{P}=\tilde{P}(a,b)$ and $\tilde{S}=\tilde{S}(a,b)$. The noncompact translation surface $\tilde{S}$ is obtained by analogous identifications of pairs of sides in the disjoint union of 4 copies of $\tilde{P}$. Since $\tilde{P}$ is $\Z$-periodic, and since these identifications are compatible with the action of $\Z$, the translation surface $\tilde{S}$ is $\Z$-periodic. It has infinite genus. } \end{exa}
\begin{figure}
\caption{\it The translation surface made from {\em 4} copies of $Q\setminus R_0(a,b)$ by identifying the sides bearing the same labels.}
\label{quotient_S(a,b)}
\end{figure}
\subsection{The billiard flow and the billiard map} \label{bill_sub}
\break Let $P$ be a compact polygonal surface, and let $\partial P$ be its boundary. Orbits of the geodesic flow $(UP,T^t,\mu)$, viewed as curves in $P$, are the geodesics. We will use the term {\em billiard curves} for those geodesics that intersect $\partial P$ at regular points.
Virtually every polygonal surface has singular points. Points $z\in\text{interior}(P)$ (resp. $z\in\partial(P)$) are singular if they are cone points (resp. corner points). A geodesic in $P$ may start or end at a singular point, but it cannot pass through a singular point. A phase point $v\in UP$ is singular if the geodesic it defines arrives at a singular point, and hence is defined on a proper subinterval of $(-\infty,\infty)$. Therefore the geodesic flow $T^t,-\infty<t<\infty,$ is defined only on the set of regular (i. e., non-singular) points in $UP$. We will use the terms {\em regular and singular sets} for the sets of regular and singular points, respectively.
The singular set has codimension one; thus, the Liouville measure of the singular set is zero, and the regular set has full measure. As is usual in the billiard literature \cite{Gut84,GK}, we will use the notation $(UP,T^t,\mu)$ for the geodesic flow on the regular set. Analogous notational conventions are used for the billiard map, which we will now define. The reader should mentally substitute ``regular phase points'' whenever we speak of phase points in what follows. The billiard map will be defined on the regular set which has full measure in the canonical cross-section that we will now describe.
Let $A\subset P$. We denote by $U_AP\subset UP$ the set of vectors
$v\in UP$ whose base points belong to $A$. Let $BUP\subset UP$ be the smallest $T^t$-invariant set containing $U_{\partial P}P$. If $\partial P\ne\emptyset$, we set $\mu_B=\mu|_{BUP}$. By definition, $U_{\partial P}P$ is a cross-section for the flow $(BUP,T^t,\mu_B)$. Let $\nu$ be the induced measure on $U_{\partial P}P$. We will use the following terminology: The flow $(BUP,T^t,\mu_B)$ is the {\em billiard flow} of $P$; the set $U_{\partial P}P\subset BUP$ is the {\em standard cross-section} for the billiard flow; the induced transformation $(U_{\partial P}P,\tau,\nu)$ is the {\em billiard map} for $P$.
Let now $P$ be a rational polygonal surface with a boundary, and let $\Ga(P)=R_N$. We identify $U/\Ga$ and $[0,\pi/N]$. For $\theta\in[0,\pi/N]$ set $U_{\partial P}P_{\theta}=UP_{\theta}\cap U_{\partial P}P$, $BUP_{\theta}=BUP\cap UP_{\theta}$. Suppose that $\mu_{\theta}(BUP_{\theta})>0$. Let $b\mu_{\theta}$ be the restriction of $\mu_{\theta}$ to $BUP_{\theta}$; then $(BUP_{\theta},T^t_{\theta},b\mu_{\theta})$ is the {\em billiard flow in direction $\theta$}. The set $U_{\partial P}P_{\theta}\subset BUP_{\theta}$ is the {\em standard directional cross-section}. Let $\nu_{\theta}$ be the induced measure on $U_{\partial P}P_{\theta}$. The induced dynamical system $(U_{\partial P}P_{\theta},\tau_{\theta},\nu_{\theta})$ is the {\em directional billiard map} \cite{Gut09}. When $P\subset\RR$ is a compact polygon, this is the standard terminology \cite{Gut03}.
Let $S$ be a compact translation surface. Let $O\subset S$ be a polygon, not necessarily connected, such that $S\setminus O$ is connected. The polygonal surface $P=S\setminus\text{interior}(O)$ is a {\em translation surface with polygonal obstacles}. Some of the components of $O$ may be linear segments, hence we will also speak of {\em translation surface with polygonal obstacles and/or barriers}.
\begin{lem} \label{bil_type_lem} 1. Let $P$ be a compact translation surface with polygonal obstacles and/or barriers. Then $\mu(UP\setminus BUP)=0$. 2. Let $P$ be a compact, rational polygonal surface with a boundary; let $R_N$ be the corresponding reflection group. Let $\E_{\text{erg}}(P)\subset[0,\pi/N]$ be the set of uniquely ergodic directions. i) If $\partial P$ contains intervals with distinct directions, then for every $\theta\in\E_{\text{erg}}(P)$ we have $BUP_{\theta}=UP_{\theta}$. ii) Suppose that $\partial P$ consists of intervals with the same direction, say $\theta_0$. Then for $\theta\in\E_{\text{erg}}(P)\setminus\{\theta_0\}$ we have $BUP_{\theta}=UP_{\theta}$. \begin{proof} 1. Let $P=S\setminus O$. Let $\ga$ be an infinite geodesic in $P$ that does not intersect $O$. Then $\ga$ is an infinite geodesic in $S$, and $\ga\cap O=\emptyset$. If $\text{interior}(O)\ne\emptyset$, then $\ga$ is not dense, and hence its direction is not minimal. Suppose $\text{interior}(O)=\emptyset$, i. e., $O$ consists of barriers. Assume that $O$ contains segments of distinct directions. A geometric argument which we leave to the reader implies that $\ga$ is not dense in $S$, and hence its direction is not minimal. Let $O$ consist of segments with direction $\theta_0$. Then the direction of $\ga$ is either nonminimal or it is $\theta_0$. But the set of nonminimal directions is countable, implying the claim.
2. There is a compact translation surface with obstacles and/or barriers, say $S\setminus O$, and a finite covering $p:(S\setminus O)\to P$. Let $\beta\subset P$ be an infinite geodesic in direction $\theta$, and let $\alpha\subset S$ be its pull back by $p$. The preceding argument shows that in the case i) (resp. case ii)) the direction of $\alpha$ belongs to $U\setminus\E_{\text{erg}}$ (resp. $U\setminus(\E_{\text{erg}}\cup\{\theta_0\})$). \end{proof} \end{lem}
Let $\tilde{P}$ be a noncompact, $G$-periodic polygonal surface, and let $P=\tilde{P}/G$ be the compact quotient. Suppose that $\partial P\ne\emptyset$. By Lemma~~\ref{bil_type_lem}, $U_{\partial P}P$ and $U_{\partial\tilde{P}}\tilde{P}$ are the cross-sections for the billiard flows of $P$ and $\tilde{P}$ respectively. Let $(U_{\partial P}P,\tau,\nu)$ and $(U_{\partial\tilde{P}}\tilde{P},\tilde{\tau},\tilde{\nu})$ be the respective billiard maps. Then $(U_{\partial\tilde{P}}\tilde{P},\tilde{\tau},\tilde{\nu})$ is a skew product over $(U_{\partial P}P,\tau,\nu)$ with the fibre $G$ and a displacement function $\varphi: U_{\partial P}P \to G$. See equation~~\eqref{produit_gauche_eq}.
\begin{lem} \label{centr_lem}
The displacement function is centered:
\begin{equation} \label{integr_eq} \int_{U_{\partial P}P}\varphi d\nu=0. \end{equation}
\begin{proof} For $v\in U_{\partial P}P$ let $\ga_v=\{\ga_v(t):0\le t\le t(v)\}$ be the segment of the geodesic ray $\{\ga_v(t)\}$ determined by $v$ ending when $\{\ga_v\}$ first returns to $\partial P$. Note that $ t(v)$ is the {\em first return time function} for the billiard map. The tangent vector $\ga'_v(t)$ is defined for $0<t<t(v)$ and the left limit $\lim_{t\to t(v)-}\ga'_v(t)$ exists. Set $\si(v)=-\lim_{t\to t(v)-}\ga'_v(t)$. The transformation $\si:U_{\partial P}P\to U_{\partial P}P$ is the {\em canonical involution} for the billiard map $(U_{\partial P}P,\tau,\nu)$ \cite{Gut03}.
The canonical involution $\tilde{\si}$ for the billiard map $(U_{\partial\tilde{P}}\tilde{P},\tilde{\tau},\tilde{\nu})$ is defined the same way. Let $\tilde{v}=(v,g)\in U_{\partial \tilde{P}}\tilde{P}$. Then $$\tilde{\si}(\tilde{v})=\tilde{\si}(v,g)=(\si(v),g+\varphi(v)).$$ The identity $$(v,g)=\tilde{\si}^2(v,g)=\tilde{\si}(\si(v),g+\varphi(v))=(\si^2(v),g+\varphi(v)+\varphi(\si(v)))$$ yields \begin{equation} \label{symm_eq} \varphi(\si(v))=-\varphi(v). \end{equation} Since $\si$ preserves the Liouville measure, the claim follows. \end{proof} \end{lem}
Let now $\tilde{P}$ be a rational polygonal surface. Let $\Ga(P)=R_N$; we identify $U/\Ga(P)$ and $[0,\pi/N]$. For $\theta\in[0,\pi/N]$ let $(U_{\partial P}P_{\theta},\tau_{\theta},\mu_{\theta})$ and $(U_{\partial\tilde{P}}\tilde{P}_{\theta},\tilde{\tau}_{\theta},\tilde{\mu}_{\theta})$ be the directional billiard maps for $P$ and $\tilde{P}$ respectively. Then
$(U_{\partial\tilde{P}}\tilde{P}_{\theta},\tilde{\tau}_{\theta},\tilde{\mu}_{\theta})$ is the skew product over $(U_{\partial P}P_{\theta},\tau_{\theta},\mu_{\theta})$ with the displacement function $ \varphi_{\theta}=\varphi|_{U_{\partial P}P_{\theta}}$. By equation~~\eqref{produit_gauche_eq}
\begin{equation} \label{skew_prod_dir_eq} \tilde{\tau}_{\theta}(v,g)=(\tau_{\theta}(v),g+\varphi_{\theta}(v)). \end{equation}
\begin{lem} \label{integr_prop} Let $N$ be even. Then for every $\theta\in[0,\pi/N]$ the {\em directional displacement function} $\varphi_{\theta}$ is centered: \begin{equation} \label{integr_eq1} \int_{U_{\partial P}P_{\theta}}\varphi_{\theta}d\nu_{\theta}=0. \end{equation} If $N$ is odd, then the function $\varphi_{\pi/(2N)}$ is centered.
\begin{proof} The canonical involution $\si:U_{\partial P}P\to U_{\partial P}P$ induces {\em directional involutions} $\si_{\theta}:U_{\partial P}P_{\theta}\to U_{\partial P}P_{\eta(\theta)}$. The central symmetry of $U$ and the identification $U/R_N=[0,\pi/N]$ induce the transformation $\theta\mapsto\eta(\theta)$ of $[0,\pi/N]$. The proof of Lemma~~\ref{centr_lem} yields
\begin{equation} \label{integr_eq2} \int_{U_{\partial P}P_{\eta(\theta)}}\varphi_{\eta(\theta)}d\nu_{\eta(\theta)} = -\int_{U_{\partial P}P_{\theta}}\varphi_{\theta}d\nu_{\theta}. \end{equation} If $N$ is even, then $R_N$ contains the central symmetry, hence $\eta(\theta)=\theta$. If $N$ is odd, then $\eta(\theta)=\pi/N-\theta$. Both claims now follow from equation~~\eqref{integr_eq2}. \end{proof} \end{lem}
\section{$\Z$-periodic polygonal surfaces} \label{one_periodic}
We will use the setting and the notation of section~~\ref{cadre}, with $G=\Z$. Let $\tilde{P}$ be a $\Z$-periodic polygonal surface, and let $P=\tilde{P}/\Z$. If $P$ is a rational polygonal surface, we will denote by $\tilde{S}$ and $S$ the translation surfaces of $\tilde{P}$ and $P$ respectively. Then $\tilde{S}$ is $\Z$-periodic, and $S=\tilde{S}/\Z$.
\subsection{Main result} \label{result_sub}
\break A compact translation surface $S$ is {\em arithmetic} \cite{GJ} if it admits a translation covering $\pi:S\to\TT$ onto a flat torus whose branch locus is a single point. Via an affine renormalization, we can assume that $S$ covers the standard torus $\TT_0=\RR/\ZZ$ and that the branch locus is $\{0\}+\ZZ$. These translation surfaces are also known as {\em square-tiled} and as {\em origamis}.
The surface $\tilde{P}$ is arithmetic iff $P=\tilde{P}/\Z$ is arithmetic \cite{Gut09}. Let $P$ be a compact, arithmetic polygonal surface; let $\pi:S\to\TT_0$ be as above. Let $\Ga=\Ga(P)$. A direction $\theta\in U$ is rational if $\tan\theta\in\QQ$. Using the covering $\pi:S\to\TT$ and the natural action of ${\text{GL}(2,\R)}$ on translation surfaces \cite{GJ}, we extend the notion of {\em rational directions} to all arithmetic translation surfaces, and hence to arithmetic polygonal surfaces.\footnote{We will say {\em $P$-rational} to emphasize that the set of rational directions depends on the surface in question.}
Let $(U/\Ga)_{\text{rat}}\subset U/\Ga$ be the set of $P$-rational directions. Then $\theta\in(U/\Ga)_{\text{rat}}$ iff every geodesic in $P$ in direction $\theta$ is periodic or a {\em saddle connection} \cite{Gut84}. The set $(U/\Ga)_{\text{rat}}$ is countable. We set $(U/\Ga)_{\text{irr}}=U/\Ga\setminus(U/\Ga)_{\text{rat}}$; we say that $\theta\in (U/\Ga)_{\text{irr}}$ are the {\em irrational directions}.
\begin{thm} \label{one_period_thm} Let $\tilde{P}$ be a $\Z$-periodic polygonal surface with a boundary, and let $P=\tilde{P}/\Z$.
\noindent 1. If the flow $(UP,T^t,\mu)$ is ergodic, then the geodesic flow for $\tilde{P}$ is recurrent.
\noindent 2. Let $P$ be a rational polygonal surface, and let
$\Ga=\Ga(P)$. Suppose that $|\Ga|$ is divisible by $4$. Then for a full measure set of directions $\theta\in U/\Ga$ the directional geodesic flow $(U\tilde{P}_{\theta},\tilde{T}^t_{\theta}, {\tilde{\lambda}}_{\theta})$ is zero-recurrent. (See definition~~\ref{NRUO_def}.)
\noindent 3. Let $P$ be an arithmetic polygonal surface. i) For an irrational direction $\theta$ the flow $(U\tilde{P}_{\theta},\tilde{T}^t_{\theta}, {\tilde{\lambda}}_{\theta})$ is zero-recurrent. ii) Let $\theta$ be a rational direction. Then the set of orbits of $\tilde{T}_{\theta}^t$ is a disjoint union of periodic bands and bands of orbits that are dissipative with a positive rate. The boundaries of these bands are concatenations of saddle connections. \begin{proof} 1. The claim follows from Lemma~~\ref{centr_lem} and claim 2 in Proposition~~\ref{recurrence_prop}.
\noindent 2. Let $\theta\in\E_{\text{erg}}(P)$, the set of uniquely ergodic directions. By Lemma~~\ref{bil_type_lem}, Lemma~~\ref{integr_prop}, and Proposition \ref{recurrence_prop}, the flow $(U\tilde{P}_{\theta},\tilde{T}^t_{\theta}, {\tilde{\lambda}}_{\theta})$ is conservative. Since the set $\E_{\text{erg}}(P)\subset S^1$ has full lebesgue measure \cite{KMS}, we obtain that $(U\tilde{P}_{\theta},\tilde{T}^t_{\theta}, {\tilde{\lambda}}_{\theta})$ is conservative for a.e. $\theta$.
For $k\in\N$ set $P_k=\tilde{P}/k\Z$. Let $l,k\in\N$ and let $\ell$ divide $k$. Then there is a covering $p_{k,\ell}:P_k\to P_\ell$, implying $\E_{\text{erg}}(P_k)\subseteq \E_{\text{erg}}(P_\ell)$. In particular, $\E_{\text{erg}}(P_k)\subseteq \E_{\text{erg}}(P)$ for any $k>1$. By Lemma~~\ref{cobord_lem} and Proposition~~\ref{recurrence_prop}, if $\theta\in\E_{\text{erg}}(P_k)$, and $k>1$, then the flow $(U\tilde{P}_{\theta},\tilde{T}^t_{\theta}, {\tilde{\lambda}}_{\theta})$ is zero-recurrent. Thus, $(U\tilde{P}_{\theta},\tilde{T}^t_{\theta}, {\tilde{\lambda}}_{\theta})$ is zero-recurrent for $\theta\in\cup_{k>1}\E_{\text{erg}}(P_k)$, a full measure subset of $\E_{\text{erg}}(P)$.
\noindent 3. In this case all surfaces $P_k$ are arithmetic and $\E_{\text{erg}}(P_k)=\E_{\text{erg}}(P)=(U/\Ga)_{\text{irr}}$ \cite{Gut84}. We can assume without loss of generality that $\E_{\text{erg}}(P)=[0,\pi/N]\setminus\QQ$. The preceding argument yields claim i).
Let now $\theta\in[0,\pi/N]\cap\QQ$. By \cite{Gut84}, the flow $(UP,T_{\theta}^t,\mu)$ decomposes into periodic bands whose boundaries are made from saddle connections. Depending on whether ergodic sums of the displacement function along a periodic orbit vanish or not, the preimage of a periodic band in $U\tilde{P}$ is a union of periodic and transient bands. Claim ii) follows. \end{proof} \end{thm}
\begin{corol} \label{one_period_cor}
Let $\tilde{P}$ be a $\Z$-periodic, rational polygonal surface with a boundary. If $|\Ga(P)|$ is divisible by $4$, then the flow $(U\tilde{P},\tilde{T}^t,\tilde{\mu})$ is zero-recurrent. \begin{proof} Follows from the decomposition of $(U\tilde{P},\tilde{T}^t,\tilde{\mu})$ into the directional flows $(U\tilde{P}_{\theta},\tilde{T}^t_{\theta}, {\tilde{\lambda}}_{\theta})$, a Fubini-type argument, and claim 2 in Theorem~~\ref{one_period_thm}. \end{proof} \end{corol}
\subsection{Examples and applications} \label{exa_sub}
\break We will now illustrate the preceding material with a few examples.
\begin{exa} \label{more_strip_exa} {\em Let $0<h<1$ and $0\le a,b <1$ be such that $0<h\pm\frac{b}2<1$ and $a+b>0$. Let $R(a,b;h)\subset\RR$ be the closed $a\times b$ rectangle centered at $(\frac12,h)$, whose sides are parallel to the coordinate axes. Then $R(a,b;h)$ belongs to the interior of the unit square $Q=\{(x,y):0\le x,y \le 1\}$. Let $P(a,b;h)$ be the polygonal surface obtained by deleting from $Q$ the interior of $R(a,b;h)$, and identifying the sides $\{x=0\},\{x=1\}$. If $0<a,b$, then $P(a,b;h)$ is the flat unit cylinder with a rectangular obstacle. The obstacle is the $a\times b$ rectangle centered in the cylinder at the height $h$. See figure~~\ref{rect_obst}. If $b=0$ (resp. $a=0$) then the rectangular obstacle degenerates into a horizontal (resp. vertical) barrier.
For $k\in\Z$ let $R_k(a,b;h)=R(a,b;h)+(k,0)$; let $B=\{(x,y):-\infty<x<\infty, 0 \le y \le 1\}$. Set $\tilde{P}(a,b;h)=B\setminus\cup_{k\in\Z}R_k(a,b;h)$. Then $\tilde{P}(a,b;h)$ is a $\Z$-periodic polygonal surface, and $P(a,b;h)=\tilde{P}(a,b;h)/\Z$. When $h=\frac{1}2$, we recover Example~~\ref{strip_exa}.
Let $\Ga=\Ga(P(a,b;h))$. If $b\ne0$, then $|\Ga|=4$; when $b=0$, then $|\Ga|=2$. Thus, for $b\ne 0$ the surface $\tilde{P}(a,b;h)$ satisfies the assumptions of claim 2 in Theorem~~\ref{one_period_thm}. The surface $P(a,b;h)$ is arithmetic iff $a,b\in\QQ$ \cite{GJ}. Theorem~~\ref{one_period_thm} and Corollary~~\ref{one_period_cor} imply the following statement. } \end{exa}
\begin{figure}
\caption{\it Flat cylinder with a rectangular obstacle.}
\label{rect_obst}
\end{figure}
\begin{corol} \label{rect_arithm_thm} Let $(U\tilde{P}(a,b;h),\tilde{T}^t,\tilde{\mu})$ be the geodesic flow for $\tilde{P}(a,b;h)$; let $(U\tilde{P}(a,b;h)_{\theta},\tilde{T}^t_{\theta},\tilde{\mu}_{\theta})$ be the directional flows. We will refer to them as $\tilde{T}^t$ and $\tilde{T}^t_{\theta}$. Let $b\ne 0$. Then the following claims hold.
\noindent 1. The flow $\tilde{T}^t$ is zero-recurrent.
\noindent 2. For a.e. $\theta\in[0,\pi/2]$ the flow $\tilde{T}^t_{\theta}$ is zero-recurrent.
\noindent 3. Let $a,b,h\in\QQ$. Then, for every $\theta\in[0,\pi/2]$ such that $\tan\theta\notin\QQ$, the flow $\tilde{T}^t_{\theta}$ is zero-recurrent.
\end{corol}
\begin{rem} \label{horiz_bar_rem} {\em If $b=0$, then $\tilde{P}(a,0;h)$ is the horizontal band with a periodic configuration of horizontal barriers of length $a$. Thus $N=1$ and $U/\Ga=[0,\pi]$. See Example~~15 in \cite{Gut09}. For $\theta\ne\pi/2$ the flows $\tilde{T}^t_{\theta}$ are transient: Every orbit of $\tilde{T}^t_{\theta}$ drifts horizontally with the rate $\sin\theta$. The flow $\tilde{T}^t_{\pi/2}$ is periodic. This example fits into the framework of Lemma~~\ref{integr_prop}. } \end{rem}
Let $Q\subset\RR$ be a polygon satisfying for an integer $t \geq 1$ the following conditions. i) There is a nonzero vector $\vec{v}\in\RR$, and for $1\le i \le t$ there are sides $s_i,s_i'$ of $Q$ such that $s_i'=s_i+\vec{v}$. ii) We have $Q\cap(Q+\vec{v})=\cup_{1\le i \le t}s_i'$. We denote by $\tilde{P}=\tilde{P}(Q)$ the $\Z$-periodic polygon obtained by deleting from $\cup_{k\in\Z}(Q+k\vec{v})$ the sides of the form $s_i+k\vec{v}:1\le i \le t,k\in\Z$. We say that $\tilde{P}$ is the {\em stairway based on $Q$} or, simply, a {\em stairway}. The compact, polygonal surface $P=\tilde{P}/\Z$ is obtained by identifying the sides
$s_i$ and $s_i'$ of $Q$ for $1\le i \le t$. Since $\cup_{1\le i \le t}(s_i\cup s_i')\subset\partial Q$ is a proper subset, $\partial P\ne\emptyset$. Let $\Ga\subset O(2)$ be the group generated by reflections about the sides of $Q$ other than $s_i,s_i':1\le i \le t$. If $|\Ga|<\infty$, then $\tilde{P}$ is a {\em rational stairway}. The following is immediate from claims 2 and 3 in Theorem~~\ref{one_period_thm}.
\begin{thm} \label{stair_ration_thm} Let $\tilde{P}\subset\RR$ be a rational stairway, and let $\Ga=R_N$. If $N$ is even, then the following claims hold.
\noindent 1. For a full measure set of directions the flow $(U\tilde{P}_{\theta},\tilde{T}^t_{\theta},\tilde{\mu}_{\theta})$ is zero-recurrent.
\noindent 2. Suppose, in addition, that the surface $\tilde{P}$ is arithmetic. Then for every irrational direction the flow $(U\tilde{P}_{\theta},\tilde{T}^t_{\theta},\tilde{\mu}_{\theta})$ is zero-recurrent. \end{thm}
\begin{exa} \label{stair_exa} {\em Let $a,b>0$. Let $Q=Q(a,b)$ be the $2a\times b$ rectangle. We view $\partial Q$ as a union of $6$ sides: 2 vertical sides of length $b$ and 4 horizontal sides of length $a$. Let $s,s'$ be the lower left and the upper right horizontal sides respectively. Then $\tilde{P}(a,b)$ based on $Q$ is the infinite stairway, with the stairs of length $a$ and height $b$. Its quotient $P(a,b)$ is the rectangle $Q$ with two sides of length $a$ identified. See figure~~\ref{stairway}. The corresponding group is $R_2$. By \cite{Gut84,GJ}, $\tilde{P}(a,b)$ is arithmetic iff $a,b\in\QQ$.
} \end{exa}
\begin{figure}
\caption{\it A stairway polygonal surface and its quotient.}
\label{stairway}
\end{figure}
\begin{corol} \label{stair_cor} Let $\tilde{P}(a,b)$ be the stairway in Example~~\ref{stair_exa}. Then for a.e. $\theta\in[0,\pi/2]$ the flow $\tilde{T}^t_{\theta}$ is zero-recurrent. If $a,b\in\QQ$, then $\tilde{T}^t_{\theta}$ is zero-recurrent if $\tan\theta\notin\QQ$. \end{corol}
The claims of Corollary~~\ref{stair_cor} are immediate, by Theorem~~\ref{stair_ration_thm}. See \cite{HoWe09} for another proof of recurrence of $\tilde{T}^t_{\theta}$ and \cite{HuWe09} for a study of ergodic invariant measures.
\subsection{Generalizations and further applications} \label{exten_sub}
\break Let $R_0$ be the unit square; let $O\subset R_0$ be a polygon such that $R_0\setminus O$ is connected. Then $\tilde{P}_O=\cup_{k\in\Z}\{(R_0\setminus\text{interior}(O))+(k,0)\}$ is a periodic band with obstacles and/or barriers. We will study the recurrence for the billiard in $\tilde{P}_O$.
Let $R(a,b;h;\alpha)$ be the rectangle in Example~~\ref{more_strip_exa} rotated by $\alpha$ about its center point. We assume that $a,b,h$ are such that $R(a,b;h;\alpha)$ belongs to the interior of the unit square for all $\alpha$. Set $\tilde{P}(\alpha)=\tilde{P}_{R(a,b;h;\alpha)}$.
Recall that a subset in a topological space is {\em residual} if it contains a dense $G_{\de}$ set.
\begin{prop} \label{rot_rect_thm} The set of $\alpha\in S^1$ such that the flow $(U\tilde{P}(\alpha),\tilde{T}^t,\tilde{\mu})$ is recurrent is residual. \begin{proof} Set $P(\alpha)=\tilde{P}(\alpha)/\Z$. By \cite{KMS}, the billiard flow for $P(\alpha)$ is ergodic for a dense $G_{\de}$ set of angles $\alpha$. The statement now follows from claim 1 in Theorem~~\ref{one_period_thm}. \end{proof} \end{prop}
The set of planar polygons has a natural topology \cite{Gut03}. In this topology, polygons with a fixed number of sides form closed subsets in euclidean spaces. Imposing upper bounds on the sizes of polygons, we obtain (relatively) compact subsets in euclidean spaces. In what follows, whenever we invoke topological notions for spaces of polygons, we mean the natural topology.
\begin{prop} \label{tri_obst_thm} Let $\ttt$ be the space of triangles in the interior of the unit square. For $O\in\ttt$ let $\tilde{P}_O$ be the corresponding periodic band with triangular obstacles. Then the set of triangles such that the flow $(U\tilde{P}_{O},\tilde{T}^t,\tilde{\mu})$ is recurrent is residual.
\begin{proof} The space $\ttt$ is a relatively compact subset in $\R^6$. For $O\in\ttt$ the quotient $P_O=\tilde{P}_O/\Z$ is the standard cylinder with a triangular obstacle. By \cite{KMS}, $\ttt$ contains a dense $G_{\de}$ set of triangles such that the geodesic flow on $P_O$ is ergodic. Now we apply claim 1 in Theorem~~\ref{one_period_thm}. \end{proof} \end{prop}
Let ${\mathcal C}$ be a closed set of polygons inside the unit square. For $O\in{\mathcal C}$ let $\tilde{P}_O$ be the corresponding periodic band with obstacles. Propositions~~\ref{rot_rect_thm} and~~\ref{tri_obst_thm} are special cases of the following.
\begin{thm} \label{dens_G_del_thm} The set of $O\in{\mathcal C}$ such that the geodesic flow on $\tilde{P}_O$ is recurrent is residual. \end{thm}
The proof of Theorem~~\ref{dens_G_del_thm} is analogous to the proofs of Propositions~~\ref{rot_rect_thm},~~\ref{tri_obst_thm}. It invokes the result in \cite{KMS} that the ergodicity is topologically typical. It is not known if ergodicity is typical measure theoretically \cite{Gut03}.
Y. Vorobets found a sufficient condition for the ergodicity of a (compact) polygon \cite{Vo97}. The condition invokes the speed of approximation of $\pi$-irrational angles of $P$ by rationals. Referring the reader to \cite{Vo97} for a precise formulation, we will say that the angles {\em admit the Vorobets approximation}.
\begin{thm} \label{vorob_cond_thm} Let $O$ be a polygon inside the unit square. Suppose that all irrational angles of $O$ and those between $O$ and the horizontal axis admit the Vorobets approximation. Suppose, moreover, that not all of these angles are $\pi$-rational. Let $\tilde{P}_O$ be the corresponding periodic band. Then the geodesic flow on $\tilde{P}_O$ is zero-recurrent. \begin{proof} For $k\in\N$ set $P_k=\tilde{P}_O/k\Z$. Since all irrational angles of $P_k$ admit the Vorobets approximation, the billiard flow on $P_k$ is ergodic. If $l$ divides $k$, we have the covering $p_{k,l}:P_k\to P_l$. It remains to invoke the proof of claim 2 in Theorem~~\ref{one_period_thm}. \end{proof} \end{thm}
We will now define a property of cocycles that has a simple geometric meaning. Let $(X,\tau,\nu)$ be a dynamical system with a finite invariant measure. Let $\varphi:X\to\R$ be a measurable function, and let $(\varphi_n)$ be the corresponding cocycle.
\begin{defin} \label{unb_osc_def} The cocycle $(\varphi_n)$ has (the property of) {\it unbounded oscillations} if for a. e. $x\in X$ we have
\begin{equation} \label{unb_osc_eq} \sup_n \varphi(n,x) = +\infty, \ \inf_n \varphi(n,x) = -\infty. \end{equation} \end{defin}
Let $(\tilde{X},\tau_{\varphi},\tilde{\tau})$ be the skew product over $(X,\tau,\nu)$ with the displacement function $\varphi$. If the cocycle $(\varphi_n)$ has unbounded oscillations, then it is recurrent. Suppose that $(\tilde{X},\tau_{\varphi},\tilde{\tau})$ is the Poincar\'e map of a skew product flow. Then a. e. orbit of the flow has unbounded oscillations in the obvious sense. See \cite{BaKhMaPl} for examples of physical systems corresponding to the billiard with unbounded oscillations in periodic polygons.
\begin{prop} \label{unb_osc_thm}
Let $(X,\tau,\nu)$ be an ergodic dynamical system with finite invariant measure. Let $\varphi:X\to\R$ be a measurable function satisfying $\int_X \varphi \ d\nu = 0$; let $(\varphi_n)$ be the corresponding cocycle. If $\varphi$ is not a coboundary, then $(\varphi_n)$ has unbounded oscillations. \begin{proof}
Suppose that the property $\sup_n\varphi(n,x)=+\infty$ for a. e. $x\in X$ is not satisfied. Then, by ergodicity, $\sup_n\varphi(n,x)<\infty$ for a. e. $x\in X$. Set
\begin{equation} \label{relat_eq} h(x)= \sup_{k \ge 1} \varphi(k, x), \ g(x) = \sup_{k \ge 2} \varphi(k, x) - h(x). \end{equation}
Since $\tau \varphi(k, x) = \varphi(k+1, x) - \varphi(x)$, we have
\begin{equation} \label{phi_h_g_eq} \varphi(x)=\sup_{k \ge 2} \varphi(k, x) - \tau \sup_{k \ge 1} \varphi(k, x) = h(x) - h(\tau x) + g(x). \end{equation}
Iterating equation~~\eqref{phi_h_g_eq}, we obtain $\varphi(n, x) = h(x) - h(\tau^n x) + \sum_0^{n-1} g(\tau^jx)$.
\vskip 3mm By claim 2 in Proposition~~\ref{recurrence_prop}, the cocycle $(\varphi_n)$ is recurrent. Therefore, for a.e. $x$ there is an infinite sequence $n_k=n_k(x)$ such that $\varphi (n_k, x)$ and $h(\tau^{n_k} x)$ are bounded. Since, by equation~~\eqref{relat_eq}, $g(x)\le 0$, the above formula implies that the series $\sum_{j=0}^\infty g(\tau^j x)$ converges for a.e. $x$. By the recurrence of the cocycle, it implies $g=0$ a.e. Hence, by equation~~\eqref{phi_h_g_eq}, $\varphi$ is a coboundary, contrary to our assumption. Assuming that the condition $\inf_n\varphi_n(x)=-\infty$ for a. e. $x\in X$ is not satisfied, we derive that $\varphi$ is a coboundary in a similar fashion. \end{proof} \end{prop}
Combining Proposition~~\ref{recurrence_prop} and Proposition~~\ref{unb_osc_thm} with the statements on ergodicity in \cite{KMS} and \cite{Vo97}, we strengthen the preceding results. Below we formulate the strengthened versions of Theorem~~\ref{dens_G_del_thm} and Theorem~~\ref{vorob_cond_thm}. We leave the analogous strengthenings of Proposition~~\ref{rot_rect_thm} and Proposition~~\ref{tri_obst_thm} to the reader.
\begin{corol} \label{dens_G_del_cor} Let ${\mathcal C}$ be a closed set of polygons inside the unit square. For $O\in{\mathcal C}$ let $\tilde{P}_O$ be the corresponding periodic band with obstacles.
The set of $O\in{\mathcal C}$ such that the geodesic flow on $\tilde{P}_O$ is zero-recurrent and has unbounded oscillations, is residual. \end{corol}
\begin{corol} \label{vorob_cond_cor} Let $O$ be a polygon inside the unit square; let $\tilde{P}_O$ be the corresponding periodic band.
Suppose that each $\pi$-irrational angle of $O$ and each $\pi$-irrational angle between $O$ and the horizontal axis admits the Vorobets approximation. Suppose, moreover, that not all of these angles are $\pi$-rational. Then the geodesic flow on $\tilde{P}_O$ is zero-recurrent, with unbounded oscillations. \end{corol}
We point out that there is a considerable interest in the physics literature in the conservativeness and related properties for $\Z$-periodic billiards. See, for instance, \cite{BaKhMaPl}, \cite{CLS}, and the references there.
\section{$\ZZ$-periodic polygonal surfaces: Dichotomies} \label{two_periodic}
Let $R_0$ be the unit square; let $O\subset\text{interior}(R_0)$ be a polygon. Set $\tilde{P}_O=\RR\setminus\cup_{(p,q)\in\ZZ}(O+(p,q))$. Thus, the $\ZZ$-periodic polygonal surface $\tilde{P}_O$ is the euclidean plane with a doubly periodic configuration of obstacles. It is the $\ZZ$-version of the band with obstacles studied in section~~\ref{exa_sub} and section~~\ref{exten_sub}. The space $\tilde{P}_O$ may be called the {\em polygonal Lorenz gas}. When $O$ is a rectangle, $\tilde{P}_O$ is the wind-tree model. See \cite{Eh} and \cite{HW80}.
The study of geodesic flows for $\ZZ$-periodic polygonal surfaces is less complete that the corresponding study for $\Z$-periodic polygonal surfaces. In section~~\ref{rect_lorenz} we will study in detail the directional flows for special directions in the wind-tree model.
In this section we expose a few general results on the conservativeness of arbitrary $\ZZ$-periodic polygonal surfaces $\tilde{P}_O$.
For concreteness of exposition, we will consider the surfaces $\tilde{P}_O$ when $O$ is a triangle. We may then call $\tilde{P}_O$ a {\em triangular Lorenz gas}. The reader will easily extend the results that follow to arbitrary polygons in the unit square. We denote by $\ttt$ the topological space of triangles inside the unit square.
\begin{prop} \label{cas_irrat_thm} There is a dense $G_{\de}$ set $\ddd\subset\ttt$ of triangles such that for $O\in\ddd$ the following dichotomy holds: i) the billiard in $\tilde{P}_{O}$ is zero-recurrent or ii) the billiard in $\tilde{P}_{O}$ is transient and satisfies equation~~\eqref{lim-k_n}, with $d=2, c > 0$. \begin{proof} The billiard in $\tilde{P}_O$ fits into the framework of sections~~\ref{trans_surf_sub},~~\ref{bill_sub}. The compact polygonal surface $P_O=\tilde{P}_O/\ZZ$ is the standard torus with a triangular obstacle. The flow $(U\tilde{P}_O,\tilde{T}^t,\tilde{\mu})$ is a skew product over the geodesic flow $(UP_O,T^t,\mu)$ with the fibre $\ZZ$. Set $\tilde{O}=\cup_{(p,q)\in\ZZ}(O+(p,q))$. The boundaries $\partial O,\partial\tilde{O}$ yield canonical cross-sections for the respective flows. Let $(U_{\partial O}P,\tau,\nu)$ and $(U_{\partial\tilde{O}}\tilde{P},\tilde{\tau},\tilde{\nu})$ be the respective billiard maps. Then $(U_{\partial\tilde{O}}\tilde{P},\tilde{\tau},\tilde{\nu})$ is a skew product over $(U_{\partial O}P,\tau,\nu)$; let $\varphi:U_{\partial O}P\to\ZZ$ be the corresponding displacement function. The flow $(U\tilde{P}_O,\tilde{T}^t,\tilde{\mu})$ is zero-recurrent (resp. transient) iff the map $(U_{\partial\tilde{O}}\tilde{P},\tilde{\tau},\tilde{\nu})$ is zero-recurrent (resp. transient).
For $(p,q)\in\NN$ set $P_{(p,q)}=\tilde{P}_{O}/(p\Z\times q\Z)$. Then $P_{(p,q)}$ are compact polygonal surfaces and $P_{(1,1)}=P_{O}$. If $(p,q),(p',q')\in\NN$ are such that $p$ divides $p'$ and $q$ divides $q'$, then there is a finite covering $\pi_{(p,q)}^{(p',q')}:P_{(p',q')}\to P_{(p,q)}$. The set of $O\in\ttt$ such that the billiard map $(U_{\partial O}P_{(p,q)},\tau_{(p,q)},\nu_{(p,q)})$ is ergodic contains a dense $G_{\de}$ set $\ddd_{(p,q)}$ \cite{KMS}.
For $O\in \cup_{(p>1,q)}\cup_{(p,q>1)}\ddd_{(p,q)}$, which is a dense $G_{\de}$, the map $(U_{\partial\tilde{O}}\tilde{P},\tilde{\tau},\tilde{\nu})$ satisfies the conditions of Proposition~~\ref{lem-k_n}. \end{proof} \end{prop}
Let now $O\subset R_0$ be a rational triangle; thus, the polygonal surface $P_O$ is rational; let $R_N$ be the corresponding dihedral group. For $\theta\in[0,\pi/N]$ let $(UP_{\theta},T^t_{\theta},\mu_{\theta})$ and $(U\tilde{P}_{\theta},\tilde{T}^t_{\theta},\tilde{\mu}_{\theta})$ be the directional geodesic flows on $P_O$ and $\tilde{P}_O$ respectively. Let $\tau_{\theta}$ and $\tilde{\tau}_{\theta}$ be the respective directional billiard maps. See section~~\ref{bill_sub}.
\begin{prop} \label{rat_case_thm} Let $O\subset R_0$ be a rational triangle such that $N=N(O)$ is even. Then for a.e. $\theta\in[0,\pi/N]$ the following dichotomy holds: i) The map $\tilde{\tau}_{\theta}$ is zero-recurrent or ii) the map $\tilde{\tau}_{\theta}$ is transient and satisfies equation~~\eqref{lim-k_n}, with $d=2, c > 0$.
\begin{proof} Let $S,\tilde{S}$ be the translation surface corresponding to $P_O,\tilde{P}_O$ respectively \cite{Gut84}. Then $\tilde{S}$ is a $\ZZ$-periodic translation surface, and $S=\tilde{S}/\ZZ$. We view $T_{\theta}^t,\tilde{T}_{\theta}^t$ as flows on $S_O,\tilde{S}_O$ respectively.
For $(p,q)\in\N^2$ set $S_{(p,q)}=\tilde{S}/(p\Z\times q\Z)$. If $(p,q),(p',q')\in\NN$ are such that $p$ divides $p'$ and $q$ divides $q'$, then there is a finite covering $g_{(p,q)}^{(p',q')}:S_{(p',q')}\to S_{(p,q)}$. We denote by $(S_{(p,q)},T^t_{(p,q)},\mu_{(p,q)})$ the flow in direction $\theta$ for the surface $S_{(p,q)}$. These flows are compatible with the coverings $g_{(p,q)}^{(p',q')}$; they form a projective family.
Let $E\subset[0,\pi/N]$ be the set of directions $\theta$ such that all directional flows $T^t_{(p,q)}$ are ergodic. For $\theta\in E$, by Lemma~~\ref{bil_type_lem}, $\partial P_O$ yields cross-sections for the flows $(S_{(p,q)},T^t_{(p,q)},\mu_{(p,q)})$. Let $\tau_{(p,q)}$ be the Poincar\'e maps; let $\tilde{\tau}$ be the Poincar\'e map with respect to the corresponding cross-section for the flow $(\tilde{S},\tilde{T}^t_{\theta},\tilde{\mu})$. Then $\tilde{\tau}$ is a skew product over $\tau_{(p,q)}$. By Lemma~~\ref{integr_prop}, the corresponding displacement functions are centered. Proposition~~\ref{lem-k_n} implies the claim for $\tilde{\tau}$.
It remains to show that the set $E$ has full measure. For each surface $S_{(p,q)}$ the set $E_{(p,q)}$ of uniquely ergodic directions has full measure \cite{KMS}. Since $E=\cap_{(p,q)\in\N^2}E_{(p,q)}$, the claim follows. \end{proof} \end{prop}
\section{$\ZZ$-periodic polygonal surfaces: Rectangular Lorenz gas} \label{rect_lorenz}
We will study the polygonal Lorenz gas $\tilde{P}_O$ of section~~\ref{two_periodic} when $O$ is a rectangle: The rectangular Lorenz gas. In the physics literature this is known as the {\em wind-tree model} \cite{HW80}; it is of some interest for foundations of statistical physics \cite{Eh}. We begin by introducing notation. Let $0< a, b< 1$. For $(m,n)\in\ZZ$ let $R_{(m,n)}(a,b)\subset\RR$ be the $a\times b$ rectangle centered at $(m,n)$ whose sides are parallel to the coordinate axes. The Lorenz gas with rectangular obstacles of size $a\times b$ corresponds to the polygonal surface
\begin{equation} \label{lorenz_eq} \tilde{P}(a,b) = \RR \setminus \bigcup_{(m,n)\in\ZZ}R_{(m,n)}(a,b). \end{equation}
The quotient surface $P(a,b) = \tilde P(a,b)/\ZZ$ is the unit torus with a rectangular hole.\footnote{The polygonal surface $\tilde{P}(0,b)$ or $\tilde{P}(a,0)$ is the euclidean plane with a $\ZZ$-periodic family of barriers. The billiard flow is then transient. Hence, we assume that $a,b> 0$.}
We will modify the notation of section~~\ref{cadre} as follows. The surface $P(a,b)$ is rational and its dihedral group is $R_2$. We identify $U/R_2$ with $[0,\pi/2]$. We will suppress $(a,b)$ from our notation whenever this does not cause confusion. For $\theta\in[0,\pi/2]$ we denote by $(\tilde{Z}_{\theta},\tilde{T}^t_{\theta},\tilde{\mu}_{\theta})$ and $(Z_{\theta},T^t_{\theta},\mu_{\theta})$ (resp. $(\tilde{X}_{\theta},\tilde{\tau}_{\theta},\tilde{\nu}_{\theta})$ and $(X_{\theta},\tau_{\theta},\nu_{\theta})$) the billiard flow (resp. billiard map) for $\tilde{P}(a,b)$ and $P(a,b)$ respectively.
\subsection{Rational directions and small obstacles} \label{small_obst_sub}
\break A direction $\theta\in[0,\pi/2]$ is rational if $\tan\theta\in\QQ$. Rational directions $\theta(p,q) = arctan (q/p)$ correspond to pairs $(p,q)\in\NN$ with relatively prime $p,q$. When there is no danger of confusion, we will use the notation $(p,q)$ instead of $\theta(p,q)$.
Let $R(a,b)=ABCD$ be the rectangle in the unit torus. Let $\theta\in(0,\pi/2)$. The space $X_{\theta}$ consists of unit vectors pointing outward, whose base points belong to $ABCD$ and whose directions belong to the set $\{\pm\theta,\pi\pm\theta\}$. See figure~~{\ref{bill-fig1}.
We say that the Lorenz gas $\tilde P(a,b)$ has {\it small obstacles with respect to $(p,q)$} if the geodesics in $P(a,b)$ emanating from $A$ or $C$ in the direction $\theta(p,q)$ return to either point without encountering $R(a,b)$ on the way.
\begin{figure}
\caption{\it The cross-section for the conservative part of the billiard flow in direction $\pi/4$.}
\label{bill-fig1}
\end{figure}
\begin{lem} \label{small_obst_lem} The small obstacles condition is satisfied iff \begin{equation} qa+pb \leq 1. \label{small_obst_eq} \end{equation}
The inequality in equation~~\eqref{small_obst_eq} is strict iff the directional geodesic flow $(\tilde{Z}_{(p,q)},\tilde{T}^t_{(p,q)},\tilde{\mu}_{(p,q)})$ has a set of positive measure of orbits that do not encounter obstacles. \begin{proof} The condition is satisfied iff $R(a,b)$ fits between two parallel lines with slopes $q/p$ and vertical displacement $1/p$. By an elementary calculation, this is possible iff $$ a\frac{q}{p}+b\le\frac{1}{p}. $$ Moreover, the equality in equation~~\eqref{small_obst_eq} holds iff $R(a,b)$ takes all of the space between the boundary components of the strip. See figure~~\ref{small_obst}. \end{proof} \end{lem}
\begin{figure}
\caption{\it The smallness of obstacles condition: Fitting the rectangle in a strip.}
\label{small_obst}
\end{figure}
In what follows we fix $(p,q)$ and assume that the inequality equation~~\eqref{small_obst_eq} is satisfied. We identify $X_{(p,q)}$ with 2 copies of the rectangle $ABCD$; the copy denoted by $X_+=(ABCD)_+$ (resp. $X_-=(ABCD)_-$) carries the outward pointing vectors in the directions $\theta,\pi+\theta$ (resp. $\pi-\theta,2\pi-\theta$). Figure~~\ref{rectangles} illustrates this. We will now investigate the Poincar\'e map $\tau_{(p,q)}:X_{(p,q)}\to X_{(p,q)}$; we suppress the subscripts when it causes no confusion.
\begin{figure}
\caption{\it The cross-section $X_{(p,q)}$ and the two corresponding rectangles.}
\label{rectangles}
\end{figure}
\begin{lem} \label{poinc_map_lem} {\em 1.} There are natural identifications of $X_+$ and $X_-$ with the circle $\R/\Z$ endowed with distinguished points $A, B, C, D$; their relative positions are given by \begin{equation} \label{arcs_eq1}
|AB| = |CD| = \frac{qa}{2(qa+pb)},\ |BC| = |DA| = \frac{pb}{2(qa+pb)}. \end{equation}
{\em 2.} Set $\tau_{\pm}=\tau|_{X_{\pm}}$. Then $\tau_+:X_+\to X_-, \ \tau_-:X_-\to X_+$. Set $S=\R/\Z$. With the identifications $X_{\pm}=S$, the maps $\tau_+:S\to S$ and $\tau_-:S\to S$ are the orthogonal reflections about the axes $AC$ and $BD$ respectively. The maps $\tau_-\tau_+:X_+\to X_+$ and $\tau_+\tau_-:X_-\to X_-$ are the rotations of $S$ by $\displaystyle {qa\over (qa+pb)}$ and $\displaystyle {pb\over (qa+pb)}$ respectively. \begin{proof} Vectors emanating from the rectangular obstacle in direction $\eta$ at the first return assume the direction $r(\eta)$, where $r$ is a reflection in $R_2$. Thus, $\tau:X_+\to X_-,\ X_-\to X_+$.
Let $s$ and $\gamma$ be the arclength and the angle coordinates on the billiard cross-section. Up to a constant factor, the invariant measure for the billiard map has density $d\nu = \sin \gamma\, ds\, d\gamma$.\footnote{See, e.g., \cite{GK} for this material.} Integrating, we have $\nu(AB)=\nu(CD)=qa,\nu(BC)=\nu(DA)=pb$, up to a constant factor. Normalizing $\nu(S)=1$, we obtain equation~~\eqref{arcs_eq1}.
By construction, the maps $\tau_{\pm}:S\to S$ are orientation reversing diffeomorphisms. Since they preserve the arclength, they are isometries. Thus, $\tau_{\pm}:S\to S$ are orthogonal reflections. By construction, $\tau_+$ (resp. $\tau_-$) fixes the points $A,C$ (resp. $B,D$). These pairs of points correspond to the axes of reflections when we identify $S$ with the unit circle in $\RR$. \end{proof} \end{lem}
In what follows we will sometimes view $\tau_{\pm}$ as isometries of the unit circle, $\tau_{\pm}:S\to S$, and sometimes as mappings between the two copies of the circle, $\tau_+:S_+\to S_-, \ \tau_-:S_-\to S_+$.
Set $Z_{\pm}=S_{\pm}\times\ZZ$ and $Z=Z_+\cup Z_-$. We set $\tilde{\tau}=\tilde{\tau}_{(p,q)}$. Then $\tilde{\tau}:Z\to Z$ is the Poincar\'e map; it interchanges the sets $Z_+,Z_-$. We use the notation $\tilde{\tau}_{\pm}(x,g)=(\tau_{\pm}(x),g+\varphi_{\pm}(x))$. Thus, $\varphi_{\pm}:S\to\ZZ$ are the displacement functions. The following is immediate from Lemma~~\ref{poinc_map_lem} and figure~~\ref{circle}.
\begin{figure}
\caption{\it The circle and the two orthogonal reflections.}
\label{circle}
\end{figure}
\begin{lem} \label{Poinc_map_lem} The displacement functions $\varphi_{\pm}:S\to\ZZ$ are constant on the circular arcs $ABC$, $CDA$, $DAB$, $BCD$. We have
$$\varphi_+|_{ABC}=(p,q),\ \varphi_+|_{CDA}=(-p,-q),
\ \varphi_-|_{DAB}=(-p,q),\ \varphi_-|_{BCD}=(p,-q).$$ \end{lem}
Set $$ A_1=\tau_-^{-1}(A),B_1=\tau_+^{-1}(B),C_1=\tau_-^{-1}(C),D_1=\tau_+^{-1}(D). $$
Our next result describes the transformation $\tilde{\tau}^2:Z\to Z$. We set $\tilde{\tau}^2_{\pm}=\tilde{\tau}^2|_{Z_{\pm}}$. Recall that we have identified $S$ and $\R/\Z$. We will usually denote by $x+y$ the operation in $\R/\Z$. If the danger of confusion arises, we will write $x+y\mod 1$.
\begin{prop} \label{poinc_map_prop} We have \begin{eqnarray*} (\tilde{\tau}^2)_+(x,g)&=&(x+\frac{qa}{qa+pb},g+\psi_+(x)),\\ (\tilde{\tau}^2)_-(x,g)&=&(x+\frac{pb}{qa+pb},g+\psi_-(x)). \end{eqnarray*}
The displacement functions $\psi_{\pm}$ take values $(\pm 2p,0),(0,\pm 2q)$. Each $\psi_{\pm}$ determines a partition of $S$ into four intervals such that $\psi_{\pm}=\mbox{const}$ on each interval. The endpoints of these intervals belong to the set $A,B,C,D,A_1,B_1,C_1,D_1$. \begin{proof} We have $(\tau^2)_+=\tau_-\tau_+,(\tau^2)_-=\tau_+\tau_-$. The product of two orthogonal reflections is the rotation by twice the angle between their axes. The values of angles follow from Lemma~~\ref{poinc_map_lem}.
We have $\psi_{\pm}(x)=\varphi_{\pm}(x)+\varphi_{\mp}(\tau_{\pm}(x))$. By Lemma~~\ref{Poinc_map_lem}, $\psi_{\pm}$ are constant on the circular arcs which are the intersections of half-circles $ABC,CDA,DAB,BCD$ with half-circles $A_1BC_1,C_1DA_1,D_1AB_1,B_1CD_1$. We have $$
\psi_+|_{ABC\cap D_1AB_1}=\varphi_+|_{ABC}+\varphi_-|_{DAB}=(p,q)+(-p,q)=(0,2q); $$ $$
\psi_+|_{ABC\cap B_1CD_1}=\varphi_+|_{ABC}+\varphi_-|_{BCD}=(p,q)+(p,-q)=(2p,0); $$ $$
\psi_+|_{CDA\cap D_1AB_1}=\varphi_+|_{CDA}+\varphi_-|_{DAB}=(-p,-q)+(-p,q)=(-2p,0); $$ $$
\psi_+|_{CDA\cap B_1CD_1}=\varphi_+|_{CDA}+\varphi_-|_{BCD}=(-p,-q)+(p,-q)=(0,-2q). $$ Analogously $$
\psi_-|_{DAB\cap A_1BC_1}=\varphi_-|_{DAB}+\varphi_+|_{ABC}=(-p,q)+(p,q)=(0,2q); $$ $$
\psi_-|_{DAB\cap C_1DA_1}=\varphi_-|_{DAB}+\varphi_+|_{CDA}=(-p,q)+(-p,-q)=(-2p,0); $$ $$
\psi_-|_{BCD\cap A_1BC_1}=\varphi_-|_{BCD}+\varphi_+|_{ABC}=(p,-q)+(p,q)=(2p,0); $$ $$
\psi_-|_{BCD\cap C_1DA_1}=\varphi_-|_{BCD}+\varphi_+|_{CDA}=(p,-q)+(-p,-q)=(0,-2q). $$ \end{proof} \end{prop}
We set
\begin{equation} \label{rot_angle_eq} \alpha=\frac{qa}{qa+pb},\ \beta=\frac{pb}{qa+pb}. \end{equation}
Then $0<\alpha,\beta<1$ and $\alpha + \beta=1$. In what follows we assume that $\alpha<\beta$ or, equivalently, $qa<pb$. This assumption allows us to avoid extra computations. The case $\beta<\alpha$ reduces to this by switching the coordinate axes. We identify $S_+$ (resp. $S_-$) with $[0,1]$ so that the points $A,B,C,D$ (resp. $D,A,B,C$) go to $0,{\alpha\over 2},{1 \over 2},{1 \over 2}+{\alpha\over 2}$ (resp. $0,{1 \over 2}-{\alpha\over 2},{1 \over 2},1-{\alpha\over 2}$) respectively. With these identifications, $\psi_{\pm}:S\to\ZZ$ are piecewise constant functions on $[0,1]$. We will now explicitly describe them. The formulas below follow from Proposition~~\ref{poinc_map_prop} by straightforward calculations; we leave them to the reader.
\begin{prop} \label{Poinc_map_prop} The function $\psi_+:[0,1]\to\ZZ$ is given by
\begin{equation} \label{cocy_plus_eq} \psi_+(x) = \left \{ \begin{array}{clcr} (0, 2q) &{\rm on \ }]0,{1\over 2}-{\alpha\over 2}[,\\ (2p, 0) & {\rm on \ } ]{1\over 2}-{\alpha\over 2},{1\over 2}[,\\ (0, -2q) & {\rm on \ } ]{1\over 2},1-{\alpha\over 2}[, \\ (-2p, 0) & {\rm on \ } ]1-{\alpha\over 2},1[. \end{array} \right . \end{equation} The function $\psi_-:[0,1]\to\ZZ$ is given by
\begin{equation} \label{cocy_minus_eq} \psi_-(x) = \left \{ \begin{array}{clcr} (-2p, 0) &{\rm on \ }]0,{\alpha\over 2}[,\\ (0, 2q) & {\rm on \ } ]{\alpha\over 2},{1\over 2}[,\\ (2p, 0) & {\rm on \ } ]{1\over 2},{1\over 2}+{\alpha\over 2}[, \\ (0, -2q) & {\rm on \ } ]{1\over 2}+{\alpha\over 2},1[. \end{array} \right . \end{equation} \end{prop}
The following properties of $\psi_{\pm}:[0,1]\to\ZZ$ are immediate from equations~~\eqref{cocy_plus_eq}, ~~\eqref{cocy_minus_eq}. The lengths of intervals of continuity are ${\alpha \over 2},{\beta \over 2}$, and they alternate. Each function takes four values which generate the subgroup $H_{(p,q)}=2p\Z\oplus 2q\Z\subset\ZZ$. Using the isomorphism $(a, b)\mapsto(2pa, 2qb)$ of $\Z^2$ and $H_{(p,q)}$, we replace the displacement functions $\psi_+$ and $\psi_-$ by piecewise constant functions on $[0,1]$ that do not depend on $p,q$. Let $\Psi$ be the function corresponding to $\psi_+$. Then
\begin{equation} \label{cocy_Psi} \Psi(x) = \left \{ \begin{array}{clcr} (0, 1) &{\rm on \ }]0,{1 \over 2}-{\alpha\over 2}[,\\ (1, 0) & {\rm on \ } ]{1 \over 2}-{\alpha\over 2},{1 \over 2}[,\\ (0, -1) & {\rm on \ } ]{1 \over 2},1-{\alpha\over 2}[, \\ (-1, 0) & {\rm on \ } ]1-{\alpha\over 2},1[. \end{array} \right . \end{equation}
\subsection{Ergodic decompositions for the billiard dynamics} \label{decomp_sub}
\break Let $\tilde{\tau}:X\times\ZZ\to X\times\ZZ$ be the billiard map in direction $(p,q)$ for the Lorenz gas with rectangular obstacles of size $a\times b$. Recall that we have identified $X$ with 2 copies of the unit circle: $X=S_+\cup S_-$. Let $G_{(p,q)}\subset\ZZ$ be the group generated by $(p,q)$ and
$(p,-q)$. Then $|\Z^2/G_{(p,q)}|=2pq$. If $G$ is any countable group, we will denote by $\tilde{\nu}$ the measure on $X\times G$ which is the product of the Lebesgue measure on $X$ and the counting measure on $G$.
\begin{thm} \label{erg_decom_thm} Let $(p,q)\in\NN$ with $p,q$ relatively prime. Let $a,b>0$ satisfy $qa+pb \leq 1$; suppose that $a/b$ is irrational. For $\bg\in\Z^2/G_{(p,q)}$ denote by $\bg+G_{(p,q)}\subset\ZZ$ the corresponding cosets. Then the following holds.
\noindent{\em 1.} For $\bg\in\Z^2/G_{(p,q)}$ the sets $X\times(\bg+G_{(p,q)})\subset X\times\ZZ$ are $\tilde{\tau}$-invariant. The dynamical systems $(X\times(\bg+G_{(p,q)}),\tilde{\tau},\tilde{\nu})$ are ergodic; they are isomorphic for all $\bg\in\Z^2/G_{(p,q)}$.
\noindent{\em 2.} The partition
\begin{equation} \label{erg_dec_eq} X\times\ZZ=\cup_{\bg\in\ZZ/G_{(p,q)}}X\times(\bg+G_{(p,q)}). \end{equation}
yields the decomposition of the dynamical system $(X\times\ZZ,\tilde{\tau},\tilde{\nu})$ into $2pq$ isomorphic ergodic components. \begin{proof} Set $\bG=\ZZ/G_{(p,q)}$. Lemma~~\ref{poinc_map_lem} and Lemma~~\ref{Poinc_map_lem} identify $\tilde{\tau}$ with the collection of transformations
$\tilde{\tau}|_{\bg}:S_{\pm}\times(\bg+G_{(p,q)})\to S_{\mp}\times(\bg+G_{(p,q)})$, where $\bg\in\bG$. This implies the first part of claim 1.
Propositions~~\ref{poinc_map_prop} and~~\ref{Poinc_map_prop} represent the restrictions
$\tilde{\tau}^2_{\pm}|_{\bg}:S_{\pm}\times(\bg+G_{(p,q)})\to S_{\pm}\times(\bg+G_{(p,q)})$ as skew product transformations $\rho_{\si,\psi}$ over certain rotations $s\mapsto s+\si$ on $S=\R/\Z$ with particular displacement functions $\psi$. They do not depend on $\bg\in\bG$. This proves the third part of claim 1.
With the notation of equations~~\eqref{rot_angle_eq},~~\eqref{cocy_plus_eq},~~\eqref{cocy_minus_eq}, we have
\begin{equation} \label{gbar_eq}
\tilde{\tau}^2_+|_{\bg}=\rho_{\alpha,\psi_+},\ \tilde{\tau}^2_-|_{\bg}=\rho_{\beta,\psi_-}. \end{equation}
Since $\tilde{\tau}$ interchanges $S_+\times(\bg+G_{(p,q)})$ and
$S_-\times(\bg+G_{(p,q)})$, the ergodicity of $\tilde{\tau}|_{\bg}$ would follow from the ergodicity of skew products $\rho_{\alpha,\psi_+},\rho_{\beta,\psi_-}$. By symmetry, it suffices to prove the ergodicity of $\rho_{\alpha,\psi_+}$. Let $\Psi:S\to\ZZ$ be given by equation~~\eqref{cocy_Psi}, and let $\rho_{\alpha,\Psi}:S\times\ZZ\to S\times\ZZ$ be the corresponding skew product. The isomorphism $G_{(p,q)}=\ZZ$ and equation~~\eqref{cocy_Psi} yield $\rho_{\alpha,\psi_+}=\rho_{\alpha,\Psi}$. By Theorem~~\ref{ergo-Psi} in section~~\ref{spec_cocy_sub}, $\rho_{\alpha,\Psi}$ is ergodic for any irrational $\alpha$. We have established claim 1. Claim 2 is immediate from it. \end{proof} \end{thm}
Theorem~~\ref{erg_decom_thm} describes the ergodic decomposition of the billiard map in direction $(p,q)$ on the polygonal surface $\tilde{P}(a,b)$. We will now describe the decomposition of the geodesic flow in direction $(p,q)$.\footnote{To simplify notation, we will suppress the dependence on $(p,q)$ whenever this does not cause confusion.} The configuration space for the directional flow $(\tilde{Z},\tilde{T}^t_{(p,q)},\tilde{\mu})$ consists of unit vectors in directions $(\pm p,\pm q)$ with base points in $\tilde{P}(a,b)$. For $\tilde{z}\in\tilde{Z}$ we denote by $\ga(\tilde{z})\subset\tilde{P}(a,b)$ the geodesic it generates. Let $\tilde{C}\subset\tilde{Z}$ (resp. $\tilde{D}\subset\tilde{Z}$) be the set of $\tilde{z}\in\tilde{Z}$ such that $\ga(\tilde{z})$ encounters (resp. does not encounter) rectangular obstacles. Then $\tilde{Z}=\tilde{C}\cup\tilde{D}$, a disjoint union. For $\bg\in\bG$ set $\tilde{O}(\bg)=\cup_{(m,n)\in(\bg+G_{(p,q)})}R_{(m,n)}(a,b)$. Thus, $\tilde{O}(\bg)$ is the union of obstacles $R_{(m,n)}(a,b)$, as $(m,n)$ varies in the coset $\bg+G_{(p,q)}$. Let $\tilde{C}(\bg)\subset\tilde{C}$ be the set of phase points $\tilde{z}\in\tilde{Z}$ such that $\ga(\tilde{z})$ encounters obstacles in $\tilde{O}(\bg)$. Let $\tilde{\mu}_{\bg}$ be the restriction of $\tilde{\mu}$ to $\tilde{C}(\bg)$.
\begin{thm} \label{bil_erg_decom_thm} Let $(p,q)\in\NN$ with $p,q$ relatively prime. Let $a,b>0$ satisfying equation~~\eqref{small_obst_eq} be such that $a/b$ is irrational. Let $(\tilde{Z},\tilde{T}^t_{(p,q)},\tilde{\mu})$ be the directional flow. Then the following holds.
\noindent{\em 1}. The sets $\tilde{C}(\bg),\bg\in\bG,$ are $\tilde{T}^t$-invariant; the dynamical systems $(\tilde{C}(\bg),\tilde{T}^t,\tilde{\mu}_{\bg})$ are ergodic and pairwise isomorphic. The partition $\tilde{C}=\cup_{\bg\in\bG}\tilde{C}(\bg)$ yields the ergodic decomposition
\begin{equation} \label{bil_erg_dec_eq} (\tilde{C},\tilde{T}^t_{(p,q)},\tilde{\mu})=\cup_{\bg\in\bG}(\tilde{C}(\bg),\tilde{T}^t_{(p,q)},\tilde{\mu}_{\bg}) \end{equation}
of the conservative part of the flow $(\tilde{Z},\tilde{T}^t_{(p,q)},\tilde{\mu})$.
\noindent{\em 2}. The dissipative part $\tilde{D}$ is trivial iff we have equality in equation~~\eqref{small_obst_eq}. Suppose that the inequality in equation~~\eqref{small_obst_eq} holds. Then $\tilde{D}=L\times\R$, where $L$ is a countable union of disjoint intervals of the same length. The restriction of $\mu$ to $\tilde{D}$ is the product of lebesgue measures on $L$ and $\R$; the flow $(L\times\R,\tilde{T}^t_{(p,q)},\tilde{\mu})$ is the translation flow along $\R$. \begin{proof} By definition, the restriction of the flow $\tilde{T}^t_{(p,q)}$ to $\tilde{D}$ is dissipative. Claim 2 is immediate from Lemma~~\ref{small_obst_lem}.
The flow $(\tilde{C},\tilde{T}^t_{(p,q)},\tilde{\mu})$ is a suspension flow over the transformation $(X\times\ZZ,\tilde{\tau},\tilde{\nu})$. Claim 1 now follows directly from Theorem~~\ref{erg_decom_thm}. In particular, equation~~\eqref{bil_erg_dec_eq} follows from the ergodic decomposition of $(X\times\ZZ,\tilde{\tau},\tilde{\nu})$ given by equation~~\eqref{erg_dec_eq}. \end{proof} \end{thm}
The proposition below relates the ergodic decomposition equation~~\eqref{bil_erg_dec_eq} to an equidistribution of billiard orbits. It holds under the assumptions of Theorem~~\ref{bil_erg_decom_thm}.
The geodesic $\ga(\tilde{z})$ generated by $\tilde{z}\in\tilde{Z}$ is a curve in the polygonal surface $\tilde{P}(a,b)$. We will use the notation $\ga_{\tilde{z}}(t),0\le t,$ for this curve, parameterized by the arclength. For $(m,n)\in\ZZ,\tilde{z}\in\tilde{C}$ and $T>0$ let $N(\tilde{z},T;(m,n))$ be the number of times $0\le t \le T$ such that the billiard orbit $\ga_{\tilde{z}}(t)$ encounters the obstacle $R_{(m,n)}(a,b)$.
\begin{prop} \label{bil_erg_decom_cor}
Let $(m,n),(m',n')\in\ZZ$. Then the following dichotomy holds.
\noindent 1. Suppose that the numbers $\frac{m-m'}{p},\frac{n-n'}{q}$ are integers of the same parity. Then there is a $\tilde{T}^t$ invariant subset $\tilde{E}\subset\tilde{C}$ of infinite measure determined by the coset $(m,n)+G_{(p,q)}$, and such that for $\tilde{\mu}$-almost every $\tilde{z}\in\tilde{E}$ both functions $N(\tilde{z},T;(m,n)),N(\tilde{z},T;(m',n'))$ go to infinity as $T\to\infty$. Moreover, for $\tilde{\mu}$-almost every $\tilde{z}\in\tilde{E}$ we have
\begin{equation} \label{bil_erg_lim_eq} \lim_{T \to \infty} \frac{N(\tilde{z},T;(m,n))}{N(\tilde{z},T;(m',n'))} = 1 \end{equation}
The set $\tilde{C}\setminus\tilde{E}$ also has infinite measure. For $\tilde{\mu}$-almost every $\tilde{z}\in\tilde{C}\setminus\tilde{E}$ we have $$ N(\tilde{z},T;(m,n))=N(\tilde{z},T;(m',n'))=0. $$
\noindent 2. Suppose that the above assumption on $(m,n),(m',n')$ is not satisfied. Then for $\tilde{\mu}$-almost every $\tilde{z}\in\tilde{C}$ one of the following possibilities holds:
\noindent a) $N(\tilde{z},T;(m,n))=N(\tilde{z},T;(m',n'))=0$;
\noindent b) $N(\tilde{z},T;(m,n))=0$, $N(\tilde{z},T;(m',n'))\to\infty$;
\noindent c) $N(\tilde{z},T;(m',n'))=0$, $N(\tilde{z},T;(m,n))\to\infty$.
\begin{proof} 1. Recall that $(\tilde{C},\tilde{T}^t_{(p,q)},\tilde{\mu})$ is a suspension flow over the billiard map $(X\times\ZZ,\tilde{\tau},\tilde{\nu})$. The set $X$ consists of unit vectors with directions $(\pm p,\pm q)$ based on the boundary of the rectangle $R(a,b)$. Let $\tilde{X}=X\times\ZZ$ and for $(m,n)\in\ZZ$ set $\tilde{X}_{(m,n)}=X\times\{(m,n)\}$. Then $$ \tilde{X}=\cup_{(m,n)\in\ZZ}\tilde{X}_{(m,n)}, $$ a disjoint union. We will use the ergodic decomposition of $(X\times\ZZ,\tilde{\tau},\tilde{\nu})$ established in Thorem~~\ref{erg_decom_thm}. For $\bg\in\bG$ let $\tilde{X}(\bg)\subset\tilde{X}$ be the corresponding ergodic component of $(X\times\ZZ,\tilde{\tau},\tilde{\nu})$. We denote by $(m,n)\mapsto\overline{(m,n)}$ the projection of $\ZZ$ onto $\bG=\ZZ/G_{(p,q)}$. Then, by equation~~\eqref{erg_dec_eq}, $$ \tilde{X}(\bg)=\cup_{\overline{(m,n)}=\bg}\tilde{X}_{(m,n)}. $$ Denote by $1_{(m,n)}(\tilde{x})$ the function $1_{\tilde{X}_{(m,n)}}:\tilde{X}\to\Z$. For $k\in\N$ set $$ f(\tilde{x},k;(m,n))=\sum_{i=0}^k1_{(m,n)}(\tilde{\tau}^i(\tilde{x})). $$
Our assumption on $(m,n),(m',n')$ is equivalent to $\overline{(m,n)}=\overline{(m',n')}$. Set $\overline{(m,n)}=\bg\in\bG$. Then $\tilde{X}_{(m,n)},\tilde{X}_{(m',n')}\subset\tilde{X}(\bg)$. We use the ergodic theorem for dynamical systems with infinite invariant measure \cite{Aa97}. It states that for a.e. $\tilde{x}\in\tilde{X}(\bg)$
$$ \lim_{k \to \infty} \frac{f(\tilde{x},k;(m,n))}{f(\tilde{x},k;(m',n'))} = \lim_{k \to \infty} \frac{\sum_{i=0}^k1_{(m,n)}(\tilde{\tau}^i(\tilde{x}))}{\sum_{i=0}^k1_{(m',n')}(\tilde{\tau}^i(\tilde{x}))}= \frac{\tilde{\nu}(\tilde{X}_{(m,n)})}{\tilde{\nu}(\tilde{X}_{(m',n')})}. $$
The volume $\tilde{\nu}(\tilde{X}_{(k,l)})$ does not depend on $(k,l)\in\ZZ$. We have $0<\tilde{\nu}(\tilde{X}_{(k,l)})<\infty$, and $\tilde{\nu}(\tilde{X}_{(k,l)})$ is determined by $a\times b$ and $(p,q)$. See Lemma~~\ref{poinc_map_lem}. Hence, the preceding equation implies the formula
\begin{equation} \label{erg_lim_eq} \lim_{k \to \infty} \frac{f(\tilde{x},k;(m,n))}{f(\tilde{x},k;(m',n'))} = 1, \end{equation}
which holds, as usual, for a.e. $\tilde{x}\in\tilde{X}(\bg)$.
We will now outline an asymptotic relationship between the functions $N(\tilde{z},T;(m,n))$ and $f(\tilde{x},k;(m,n))$. Let $P=\tilde{P}/\ZZ$ be the compact polygonal surface; to simplify our notation, we suppress the dependence on $a\times b$ and on $(p,q)$. Recall that $P$ is the standard torus with a rectangular obstacle. Let $(Z,T^t,\mu)$ and $(X,\tau,\nu)$ be the billiard flow and the billiard map in the direction $(p,q)$ for $P$. Then $(Z,T^t,\mu)$ is a suspension flow over $(X,\tau,\nu)$; the roof function $r(x):X\to\R_+$ is the time it takes for the forward billiard orbit $\ga_x(t), 0<t,$ to return to the cross-section. The mean time of return is given by
$$ \bar{r}=\frac{\int_Xr(x)d\nu}{\nu(X)}. $$ Let $A\subset X$ be a measurable set. We associate with $A$ two functions. The function $N_A(z,T):Z\times\R_+\to\N$ is the number of times $0<t<T$ the billiard flow orbit $\ga_z(t)$ encounters $A$; the function $f_A(x,k):X\times\N\to\N$ is the number of times $0\le i <k$ the billiard map orbit $\tau^i(x)$ returns to $A$. Suppose that $(X,\tau,\nu)$ is ergodic. Then for $\nu$-a. e. $x\in X$ we have
\begin{equation} \label{flow_map_eq} N_A(x,T) = f_A(x,\lfloor\frac{T}{\bar{r}}\rfloor)+o(T). \end{equation}
We point out that equation~~\eqref{flow_map_eq} holds for ergodic suspension flows, in general. In our setting the roof function has a simple geometric meaning. Let $|\cdot|$ denote the euclidean norm on $\RR$. Recall that $\varphi:X\to\ZZ\subset\RR$ is the displacement function. The elements $\tilde{x}\in\tilde{X}$ are vectors in $\RR$ based at boundary points of the rectangles $R_{(m,n)}(a,b)$ as $(m,n)\in\ZZ$. Let $\tilde{x}\mapsto x$ be the projection of $\tilde{X}$ onto $X$. For $x\in X$ let $b(x)\in\RR$ be the base point. Then $\tilde{x}\mapsto b(\tau(x))-b(x)+\varphi(x)$ is a well defined mapping of $\tilde{X}$ to $\RR$. We then have
\begin{equation} \label{bil_ret_eq}
r(\tilde{x})=|b(\tilde{\tau}(x))-b(x)+\varphi(x)|. \end{equation}
Note that the function $r:\tilde{X}\to\R_+$ is $\ZZ$-invariant. The mean return time to the cross-section $\tilde{X}$ is equal to the mean return time to the quotient cross-section $X$. We have
\begin{equation} \label{mean_ret_eq}
\bar{r}=\frac{\int_X|b(\tau(x))-b(x)+\varphi(x)|d\nu(x)}{\nu(X)}. \end{equation}
Thus, $0<\bar{r}<\infty$. Set $\tilde{E}=\tilde{C}(\bg)$. Combining equations~~\eqref{erg_lim_eq},~~\eqref{flow_map_eq},~~\eqref{mean_ret_eq}, we obtain the former part of our claim. We have $\tilde{C}\setminus\tilde{E}=\cup_{\bar{h}\in\bG\setminus\{\bg\}}\tilde{C}(\bar{h})$. The remaining part of our claim follows from the preceding discussion and equation~~\eqref{flow_map_eq}.
\noindent 2. Set $\bg=\overline{(m,n)},\bg'=\overline{(m',n')}$. The assumption on $(m,n),(m',n')$ does not hold iff $\bg\ne\bg'$. Thus, $\tilde{X}(\bg)$ and $\tilde{X}(\bg')$ (resp. $\tilde{C}(\bg)$ and $\tilde{C}(\bg')$) are distinct ergodic components of $\tilde{X}$ (resp. $\tilde{C}$). Equations a), b), c) follow from the preceding discussion. Equation a) holds when $\tilde{z}\in\tilde{C}\setminus(\tilde{C}(\bg)\cup\tilde{C}(\bg')$; equation b) (resp. c)) holds when $\tilde{z}\in\tilde{C}(\bg')$ (resp. $\tilde{z}\in\tilde{C}(\bg)$). \end{proof} \end{prop}
The following is immediate from Proposition~~\ref{bil_erg_decom_cor}.
\begin{corol} \label{equidistr_cor} Let $(m,n),(m',n')\in\ZZ$. Then for almost every $\tilde{z}\in\tilde{C}$ the ratio $N(\tilde{z},T;(m,n))/N(\tilde{z},T;(m',n'))$ converges to either $1$, or $0$, or infinity, as $T\to\infty$. \end{corol}
\section{Ergodicity of cocycles over irrational rotations} \label{ergodic}
The subject of this section is the ergodic theory for a certain class of skew product transformations. The results are instrumental in obtaining ergodic decompositions for directional flows in the rectangular Lorenz gas model. See Theorem~~\ref{erg_decom_thm}, Theorem~~\ref{bil_erg_decom_thm}, and Corollary~\ref{bil_erg_decom_cor}.
Throughout this section, we will use the following setting. Let $G$ be a locally compact abelian group.\footnote{In our applications $G\simeq\R^{d_1}\times\Z^{d_2}$.} Set $X = \R/\Z$. For $0<\alpha<1$ let $\rho_\alpha: X\to X$ be the rotation $x \mapsto x+ \alpha {\rm \ mod \ 1 \ }$. Let $\Phi:X\to G$ be a piecewise constant function. Define the transformation $\rho_{\alpha,\Phi}:X\times G\to X\times G$ by $(x,g) \mapsto (\rho_\alpha(x), g+\Phi(x))$. Let $\leb$ denote the Lebesgue measure on $X$; let $\mu$ be the measure on $X\times G$ which is the cartesian product of $\leb$ and a Haar measure on $G$. The dynamical system $(X\times G,\rho_{\alpha,\Phi},\mu)$ is the {\em skew product} over $\rho_\alpha$, with the fibre $G$ and the displacement function $\Phi$. Let $(\Phi_n)$ be the cocycle corresponding to $\Phi$ and $\rho_{\alpha}$. See Definition~~\ref{cocycle_def}. If the dynamical system $(X\times G,\rho_{\alpha,\Phi},\mu)$ is ergodic, we will say that {\em the cocycle $(\Phi_n)$ is ergodic}.
In the studies of ergodicity for $(X\times G, \rho_{\alpha, \Phi}, \mu)$ it is common to assume that the points of discontinuity of $\Phi$ are not arithmetically related to $\alpha$. In view of equation~~\eqref{cocy_Psi}, this assumption does not hold for our applications in section~~\ref{rect_lorenz}. Thus, we will expose two approaches to proving the ergodicity of cocycles over irrational rotations. One of them is based on the ``well distributed discontinuities'' property for a cocycle.\footnote{We will use the abbreviation (wdd).} See Definition~~\ref{propert_def}. The cocycles $(\Psi_n)$, needed for our applications, satisfy (wdd) for generic $\alpha$. This approach allows us to establish the ergodicity of $(\Psi_n)$, and similar cocycles, for generic rotation angles. We develop this approach in section~~\ref{erg_cocy_sub}, see especially Corollary~~\ref{D-wdd}.
In our applications in section~~\ref{rect_lorenz}, $\alpha$ is determined by the parameters $a$ and $b$, i. e., the sizes of billiard obstacles. See equation~~\eqref{rot_angle_eq}. Hence, the results of section~~\ref{erg_cocy_sub} prove the claims of section~~\ref{decomp_sub} for generic obstacles. However, (wdd) may fail for some parameters $a,b$. Our other approach is geared specifically to the cocycles $(\Psi_n)$. We establish their ergodicity for all irrational $\alpha$ in section~~\ref{spec_cocy_sub}.
\subsection{The inequality of Denjoy-Koksma} \label{denj_koks_sub}
\break We recall basic facts about continued fractions. See, for instance, \cite{Kh37} for this material. Let $\alpha \in ]0,1[$ be an irrational number, let $[0;a_1,..., a_n,...]$ be its continued fraction representation, and let $(p_n/q_n)_{n \ge 0}$ be the sequence of its convergents. The integers $p_n$ (resp. $q_n$) are the {\em numerators} (resp. {\em denominators}) of $\alpha$. Thus $p_{-1}=1$, $p_0=0$, $q_{-1}=0$, $q_0=1$. For $n \ge 1$ we have
\begin{equation} \label{converg_eq} p_n = a_n p_{n-1}+p_{n-2}, q_n = a_n q_{n-1}+q_{n-2}, (-1)^n = p_{n-1} q_n - p_n q_{n-1}. \end{equation}
For $u\in\R$ set $\|u\|= \inf_{n \in \Z} |u - n|$. Then for $n \ge 0$ we have $\|q_n \alpha\| = (-1)^n (q_n \alpha - p_n)$. We also have \begin{eqnarray}
1 &=& q_n\|q_{n+1} \alpha \| + q_{n+1} \|q_n \alpha\|, \label{f_1} \\
{1\over q_{n+1}+q_n} &\le& \|q_n \alpha\| \le {1\over q_{n+1}} = {1 \over a_{n+1} q_n+q_{n-1}}, \label{f_3} \\
\|q_n \alpha \| &\leq& \|k \alpha \|\ \mbox{for}\ 1\le k < q_{n+1} \label{f_4}. \end{eqnarray}
We denote by $V(\varphi)$ the variation of functions; we will use the shorthand BV for functions of bounded variation. A function is
{\em centered} if $\int_X\varphi(x)\,dx = 0$. Let $\varphi$ be a centered BV function on $X$. Let $p/q$ be a rational number in lowest terms such that $\|\alpha - p/q\| < {1 / q^2}$. The {\em Denjoy-Koksma inequality} says that for any $x\in X$ we have
\begin{eqnarray}
|\sum_{\ell = 0}^{q-1}\varphi(x+\ell \alpha)| \le V(\varphi). \label{f_8} \end{eqnarray}
The following is immediate from equations~~\eqref{f_3} and \eqref{f_8}.
\begin{cor} \label{rec1} Let $\Phi:X\to G$ be a centered BV function. Then the cocycle $(\Phi_n)$ over any irrational rotation is recurrent. \end{cor}
We will use the following properties of the sequence $p_k/q_k$. At least one of any two consecutive numerators (resp. denominators) is odd. If both $p_n, q_n$ are odd, then one of $p_{n+1}, q_{n+1}$ is even.
\begin{lem} \label{4odd}
Let $\alpha$ and $p_k,q_k$ for $k\ge 0$ be as above. Then the following holds. {\em 1}. In any pair of consecutive denominators at least one satisfies $q_n\|q_n\alpha\| < 1/2$. {\em 2}. Out of any four consecutive denominators at least one is odd and satisfies $q_n
\|q_n \alpha\| < 1/2$.
\begin{proof} 1. For any $n\in\N$ define $\delta_1$ and $\delta_2$ by
$q_n\|q_n\alpha\| = {1\over 2} - \delta_1$,
$q_{n+1}\|q_{n+1}\alpha\| = {1\over 2} - \delta_2$. By equation~~\eqref{f_1}, we have $$(q_{n+1} - q_n)^2 = 2\delta_1 q_{n+1}^2 + 2\delta_2q_{n}^2.$$ Hence, $\delta_1$ and $\delta_2$ cannot be both negative.
\noindent 2. The a priori possible parities for any four consecutive denominators $q_{n-1}, q_n, q_{n+1}, q_{n+2}$ are as follows:
\begin{eqnarray*} &&(0, \ 1, \ 0, \ 1) \ \ (0, \ 1, \ 1, \ 0) \ \ (1, \ 0, \ 1, \ 0) \ \ (0, \ 1, \ 1, \ 1) \\ &&(1, \ 0, \ 1, \ 1) \ \ (1, \ 1, \ 0, \ 1) \ \ (1, \ 1, \ 1, \ 0) \ \ (1, \ 1, \ 1, \ 1). \end{eqnarray*}
If there are two consecutive odd denominators, then the statement follows from claim 1. It remains to consider the possibilities $(0, \ 1, \ 0, \ 1)$ and $(1, \ 0, \ 1, \ 0)$. Then we have, respectively, $q_n$ is odd, $a_{n+1} \not =1$, and $q_{n+1}$ is odd, $a_{n+2} \not =1$. Set $q=q_{n}$ (resp. $q = q_{n+1}$) in the former
(resp. latter) case. Then $q\|q\alpha\| < 1/2$. \end{proof} \end{lem}
\subsection{Ergodicity of generic cocycles} \label{erg_cocy_sub}
\break Let $d(\cdot,\cdot)$ be an invariant distance on $G$.
\begin{defin} \label{def-period} {\em Let $a\in G$. 1. Suppose that for $n \ge 1$ there exist $\ell_n\in\N$, $\varepsilon_n > 0,$ and $\delta>0$ such that
\begin{itemize}
\item[i)] We have $\lim_n \varepsilon_n = 0$, $\lim_n \ell_n \alpha {\rm \ mod \ 1 \ } = 0$,
\item[ii)] We have $\leb(\{ x: d(\Phi_{\ell_n}(x) , a) < \varepsilon_n \}) \ge \delta$.
\end{itemize}
Then we say that $a$ is a {\it quasi-period} for the cocycle $(\Phi_n)$.
\noindent 2. We say that $a$ is a {\it period} if for every $\rho_{\alpha, \Phi}$-invariant measurable function $f$ on $X\times G$ and for a.e. $(x,g) \in X \times G$ we have \begin{equation} \label{period_eq} f(x,g+a) = f(x,g). \end{equation}
} \end{defin}
We will use the following fact \cite{Co09}.
\begin{lem} \label{lem-period} Every quasi-period is a period. \end{lem}
The set of periods is a closed subgroup of $G$ which coincides with the group of finite essential values of the cocycle \cite{Sc77}. A cocycle is ergodic iff its group of periods is $G$.
We introduce more notation. Let $\Phi:X\to G$ be a non constant piecewise constant function. Denote by $R(\Phi)\subset G$ the range of $\Phi$, i. e., $a\in R(\Phi)$ iff $\Phi(x) =a$ on a nontrivial interval. Denote by ${\mathcal D} = \{t_i: i=1,...,d\}$ the set of discontinuities of $\Phi$. We assume without loss of generality that $0\in{\mathcal D}$. For $N \in \N$ let ${\mathcal D}_N=\{t_i - j\alpha {\rm \mod 1}:\,1\le i\le d,0\le j < N\}$ be the set of discontinuities for $\Phi_N(t)= \sum_{k=0}^{N-1} \Phi(t + k\alpha)$. We set ${\mathcal D}_N=\{0= \gamma_{N,1} < ... <\gamma_{N,dN}<1\}$; thus, for $1\le\ell\le dN$ the elements $\ga_{N,\ell}$ run through ${\mathcal D}_N$ in the natural order. We set $\gamma_{N,dN+1} = \gamma_{N,1}$. The following notions will be important in what follows.
\begin{defin} \label{propert_def} {\em 1. Let $0< \alpha <1$ be irrational. Let $\Phi:X\to G$ be a piecewise constant function; let $(\Phi_n)$ be the corresponding cocycle over $\rho_\alpha$. Suppose that there is $c >0$ and an infinite set $W$ of denominators of $\alpha$ such that for all $q \in W,\ell \in \{1,\dots,dq\}$ we have
\begin{eqnarray} \label{wddPrty} \gamma_{q,\ell+1} - \gamma_{q,\ell} \geq {c \over q}. \end{eqnarray}
Then the cocycle has {\em well distributed discontinuities}. We will use the shorthand {\em (wdd)}.
\noindent 2. Let $\alpha \in ]0,1[$ be irrational, and let $[0;a_1,..., a_n,...]$ be its continued fraction. Then $\alpha$ has {\em property (D)} if there is $M\in\N$ such that for \ infinitely many $n$ either $a_{n} \in [2, M]$ or $a_{n}= a_{n+1} =1$. } \end{defin}
\begin{lem} \label{min-ri2} Let $t\in\frac12(\Z\alpha+\Z) \setminus (\Z\alpha+\Z)$. Then there exist $c>0$ and $L\in\N$ such that, if $n \geq L$ and either i) $a_{n+1} \in [2, M]$ or ii) $a_{n+1} = a_{n+2} = 1$, then for $1\le k \le q_n-1$ we have
\begin{equation} \label{twell-dist}
\|k \alpha - t\| \geq {c\over q_n}. \end{equation} \begin{proof} Let $t= {1\over 2} \ell \alpha + {1\over 2}r$, with $\ell, r \in
\Z$, and $\ell$ or $r$ odd. Let $L$ be such that $|\ell| < q_{n-1}$ for $n \geq L$. Let $k \in [1, q_n-1]$. For $n \geq L$, we have if $a_{n+1} \geq 2$,
$$ |2k - \ell| \leq 2k + |\ell| < 2 q_n+q_{n-1} \leq a_{n+1}q_n+q_{n-1} = q_{n+1}.$$
If $a_{n+1} \in [2, M]$, then, by equations~~\eqref{f_3} and~~\eqref{f_4}, for all $j \in [1, q_{n+1}[$, we have
\begin{equation} \label{minorJ}
\|j \alpha \| \geq \|q_n \alpha \| \geq {1\over q_n+q_{n+1}} = {1\over (a_{n+1}+1)q_n+q_{n-1}} \geq {1\over (2+M)q_n}. \end{equation}
If $2k - \ell \not = 0$, equation~~\eqref{minorJ} implies
$$\|k \alpha - t\| \geq {1\over 2} \|(2k - \ell)\alpha\| \geq \frac{1}{2q_n(2 + M)}.$$
If $a_{n+1} = a_{n+2} = 1$, we have $q_{n+1} = q_{n}+ q_{n-1}$,
$q_{n+2} = q_{n+1}+ q_n = 2 q_n +q_{n-1}$ and $|2k - \ell| < 2 q_n+q_{n-1} = q_{n+2}$. Thus, if $2k - \ell \not = 0$
$$\|(2k - \ell) \alpha\| \geq \|q_{n+1} \alpha\| \geq {1\over q_{n+2}+q_{n+1}} = {1\over 3q_{n}+2q_{n-1}} \geq {1 \over 5 q_{n}}.$$
\vskip 3mm If $\ell$ is even then $r$ is odd, so that for $2k =
\ell$ we have $\|k \alpha - t\| = \|(k-{1\over 2}\ell) \alpha -
{1\over 2}\| = {1\over 2}$.
\vskip 3mm In each case equation~~\eqref{twell-dist} holds with $c= \inf(\frac{1}{2(2 + M)}, \frac1{10})$. \end{proof} \end{lem}
\vskip 3mm The following is immediate from Lemma~~\ref{min-ri2}.
\begin{prop} \label{wdd-ri2} Let $\alpha$ be irrational, let $\Phi:X\to G$ be a piecewise constant function, and let ${\mathcal D}\subset X$ be its set of discontinuities. If $\alpha$ has property (D) and if ${\mathcal D}\subset\{\frac12(\Z\alpha+\Z) \setminus (\Z\alpha+\Z)\}\mod 1$, then the cocycle $(\Phi_n)$ has (wdd). \end{prop}
The following theorem is the main result of this subsection.
\begin{thm} \label{wdd-erg-thm} Let $0<\alpha<1$ be irrational. Let $\Phi:X\to\Z^r$ be a piecewise constant, centered function, and let $R(\Phi)\subset\Z^r$ be the range of $\Phi$.
Let $(\Phi_n)$ be the corresponding cocycle. If $(\Phi_n)$ has property (wdd), then the group of its periods contains $R(\Phi)$.
\begin{proof} Let $W\subset\N$ be the infinite set of denominators of $\alpha$ introduced in Definition~~\ref{propert_def}. We will study the family of functions $\{\Phi_q(x)=\sum_{0\le k \le q-1}\Phi(x+k\alpha\mod 1)\,:q\in W\}$. For $1\le \ell \le dq$ let $I_{q,\ell}= ]\gamma_{q,\ell}, \gamma_{q,\ell+1}[$ be the intervals of continuity. Set ${\mathcal I}_q=\{I_{q,\ell}:1\le \ell \le dq\}$.
For $t_i$ in the set ${\mathcal D}$ of discontinuities of $\Phi$, let $\sigma_i=\lim_{\varepsilon \rightarrow 0^+} [\Phi(t_i +\varepsilon) - \Phi(t_i - \varepsilon)]$ be the jump at $t_i$; let $\Si(\Phi)=\{\si_i:1\le i \le d\}$ be the set of jumps. Set $R=\cup_{q\in W}R(\Phi_q) \subset \Z^r$. By equation~~\eqref{f_8}, $R$ is a finite set.
Each interval $[{k \over q}, {k+1 \over q}[$, $0 \leq k < q$, contains an element $j\alpha \mod 1, 0\le j \le q-1$. Thus, for any $t\in X$, the elements $\{t+j\alpha\mod 1:j=0,\dots,q-1\}$ partition $X$ into intervals of lengths less than $2/q$. Hence, any interval $J\subset X$ of length $\ge 2/q$ contains at least one point of the set $\{t+j\alpha\mod 1:j=0,\dots,q-1\}$.
Let $c$ be the constant from equation~~\eqref{wddPrty}; set $L = \lfloor\frac{2}{c}\rfloor+1$. Let $I_{q,\ell} \in {\mathcal I}_q$ be arbitrary. Set $J_{q, \ell} \subset X$ be the union of $L$ consecutive intervals in ${\mathcal I}_q$ starting with $I_{q,\ell}$. By equation~~\eqref{wddPrty}, the length of $J_{q,\ell}$ is greater than or equal to $2/q$. Thus, for any $t_i\in\ddd$, the interval $J_{q, \ell}$ contains a point in the set $\{t_i+j\alpha\mod 1:j=0,\dots,q-1\}$. Therefore, for any $\sigma \in \Sigma(\Phi)$, there is $v\in R$ and two consecutive intervals $I,I'\in{\mathcal I}_q$ such that $I\cup I'\subset J_{q, \ell}$ and such that $\Phi_q$ takes values $v$ and $v+\si$ on $I$ and $I'$ respectively.
Let $\sigma \in \Sigma(\Phi)$, $v\in R$. Let ${\mathcal F}_q(\sigma)\subset{\mathcal I}_q$ be the family of intervals $I\in{\mathcal I}_q$ such that the jump at the right endpoint of $I$ is $\sigma$. Let ${\mathcal A}_q(\sigma,v)\subset{\mathcal F}_q(\sigma)$ be the set of intervals $I\in{\mathcal F}_q(\sigma)$ such that the value of $\Phi_q$ on $I$ is $v$; let ${\mathcal A}'_q(\sigma,v)\subset{\mathcal F}_q(\sigma)$ be the set of intervals $I'\in{\mathcal F}_q(\sigma)$ adjacent on the right to the intervals $I\in{\mathcal A}_q(\sigma,v)$. Let $A_q(\sigma,v)\subset X$ (resp. $A'_q(\sigma,v)\subset X$) be the union of intervals $I\in{\mathcal A}_q(\sigma,v)$ (resp. $I'\in{\mathcal A}'_q(\sigma,v)$). Thus, $\Phi_q$ takes value $v$ (resp. $v+\si$) on $A_q(\sigma,v)$ (resp. $A'_q(\sigma,v)$).
Denote by $|\cdot|$ the cardinality of a set. There is $v_0 \in R$ and an infinite subset of $W$ (which we denote by $W$ again) such that for $q\in W$ we have
\begin{equation} \label{proport_eq}
|{\mathcal A}_q(\sigma,v_0)|,|{\mathcal A}'_q(\sigma,v_0)|\ge\frac{1}{|R|}|{\mathcal F}_q(\sigma)|. \end{equation}
We have $|{\mathcal F}_q(\sigma)|\ge qd/L$. By equation~~\eqref{wddPrty} and equation~~\eqref{proport_eq} $$ \leb\left(A_q(\sigma,v_0)\right),\,\leb\left(A'_q(\sigma,v_0)\right)\ge
\frac{1}{|R|}\frac{qd}{L}\frac{c}{q}\ge \frac{dc^2}{(2+c)|R|}. $$ Thus, both $v_0$ and $v_0+\sigma$ are quasi-periods for the cocycle $(\Phi_n)$. By Lemma \ref{lem-period}, they are periods. Hence $\sigma$ is a period for the cocycle $(\Phi_n)$. Since $\si\in\Sigma(\Phi)$ was arbitrary, the group of periods for $\rho_{\alpha,\Phi}$ contains $\Si(\Phi)$.
\vskip 3mm If $H\subset G$ is a subgroup, we will denote by ``bar'' the reduction modulo $H$. Thus, $\Phi:X\to G/H$ is a piecewise constant function. The dynamical system $(X\times G/H, \rho_{\alpha,\bar{\Phi}},\bar{\mu})$ is the skew product over $\rho_{\alpha}$ with the fiber $G/H$ and the displacement function $\bar{\Phi}$. To simplify notation, we will denote it by $\overline{\rho_{\alpha,\Phi}}$.
Let $H\subset G$ (resp. $H'\subset G$) be the group generated by $\Si(\Phi)$ (resp. $R(\Phi)$). We have shown that $H$ is contained in the group of periods for $\rho_{\alpha,\Phi}$. The function $\bar{\Phi}:X\to G/H$ is constant. Let $a\in H'$ be such that $\bar{\Phi}=\bar{a}$. Then $$ \overline{\rho_{\alpha,\Phi}}(x,\bg)=(\rho_{\alpha}(x),\bg+\bar{a}). $$
Observe that $H'/H\subset G/H$ is the cyclic group generated by
$\bar{a}$. If $|H'/H|=\infty$, then $\rho_{\alpha,\bar{\Phi}}$ is dissipative, contrary to Corollary~~\ref{rec1}. Thus, $H'/H$ is a finite cyclic group. Let $|H'/H|=n$.
Let $f$ be a $\rho_{\alpha,\Phi}$-invariant measurable function. Then $f$ defines a $\overline{\rho_{\alpha,\Phi}}$-invariant function; we denote it by $\bar{f}$. By the above equation
$$ \bar{f}(\rho^n_{\alpha}(x),\bg)=\bar{f}(x,\bg). $$
Hence $\bar{f}(x,\bg)$ depends only on $\bg$. In the self-explanatory notation, $\bar{f}(x,\bg)=\bar{f}(\bg)$. Therefore, $\bar{a}$ is a period for $\bar{f}$, and hence $a$ is a period for $f$. \end{proof} \end{thm}
The following is immediate from Proposition~~\ref{wdd-ri2} and Theorem~~\ref{wdd-erg-thm}.
\begin{cor} \label{D-wdd} Let $\Phi:X\to G$ be a piecewise constant, centered function such that $R(\Phi)$ generates $G$. Let ${\mathcal D}\subset X$ be the set discontinuities of $\Phi$. Let $0<\alpha<1$ be irrational. Suppose that ${\mathcal D}\subset\{\frac12(\Z\alpha+\Z) \setminus (\Z\alpha+\Z)\}\mod1$. If $\alpha$ has property (D), then the skew product $\rho_{\alpha,\Phi}$ is ergodic. \end{cor}
\begin{rem} \label{generic_rem} {\em Let $\Psi:X\to\ZZ$ be the piecewise constant function that arose in our analysis of the rectangular Lorenz gas. See equation~~\eqref{cocy_Psi}. It satisfies the assumptions of Corollary~~\ref{D-wdd}. Almost every irrational $\alpha$ has property (D). Thus, Corollary~~\ref{D-wdd} implies the claim of Theorem~~\ref{erg_decom_thm} for generic small obstacles. The results in the next section will allow us to prove Theorem~~\ref{erg_decom_thm} for arbitrary small obstacles.
} \end{rem}
\subsection{Removing the genericity assumptions} \label{spec_cocy_sub}
\break Let $0<\alpha<1$. Set
\begin{equation} \label{gamma_zeta_eq} \gamma = 1_{[0,{1\over 2}[} - 1_{[{1\over 2}, 1[},\ \zeta = 1_{[0, {1\over 2} - {\alpha \over 2}[} - 1_{[{1\over 2}, 1- {\alpha \over 2}[}. \end{equation}
Thus, $\ga,\zeta:X\to\Z$ are piecewise constant functions. See Figure~~\ref{bill-fig2}.
\vspace {-4.cm} \begin{figure}\label{bill-fig2}
\end{figure}
\vspace {1.cm}
We will establish the ergodicity of cocycles $(\ga_n),(\zeta_n)$ for any irrational $\alpha$. First, we explain the heuristics. Suppose that $\alpha$ does not satisfy property (D). Then, passing to a subsequence, if need be, we have $a_n\to\infty$. Suppose that for all sufficiently large $n$ the numbers $q_{2n}$ and $p_{2n}$ are odd, while $q_{2n+1}$ is even and $p_{2n+1}$ is odd. Then for $n>n_0$ the inequality $\zeta(q_n,\cdot)\ne 0$ holds only on sets of very small measure. Thus, we cannot use the method of Theorem~~\ref{wdd-erg-thm}. Instead, we will consider
$\zeta(tq,\cdot)$ for such $t$ that $||tq \alpha ||$ is close to zero but big enough to ensure that $\zeta(tq,x) = \sum_{k=0}^{t-1} \zeta(q,x+ kq\alpha)$ does not vanish on a set of measure bounded away from zero.
We will informally refer to this idea as the ``filling method''. Figure~~\ref{bill-fig6} illustrates it.
\vspace {-4 cm} \begin{figure}
\caption{\it Illustration of the filling method: Graph of $\zeta(t_n q_n,\cdot)$.}
\label{bill-fig6}
\end{figure}
From now on, $0<\alpha<1$ is an arbitrary irrational number. We leave the proof of the following lemma to the reader.
\begin{lem} \label{undemi-lem}
If $q$ is odd and $q\|q\alpha\| < 1/2$, then for all $x$ we have $\sum_{j=0}^{q-1}\ga(x +j\alpha)=\pm1$. \end{lem}
By Lemma~~\ref{undemi-lem}, $1$ is a quasi-period for the cocycle $(\gamma_n)$ over $\rho_{\alpha}$. By Lemma~~\ref{lem-period}, $(\gamma_n)$ is ergodic.
\begin{thm} \label{erg-beta} Let $\zeta:X\to\Z$ be given by equation~~\eqref{gamma_zeta_eq}. Then the cocycle $(\zeta_n)$ over $\rho_{\alpha}$ is ergodic. \begin{proof} Let $p_n/q_n$ be the convergents of $\alpha$. Let $p_n',q_n'\in\N$ be such that $q_n = 2q_n'$ or $q_n =2q_n'+1$ and $p_n = 2p_n'$ or $p_n=2p_n'+1$, depending on the parity. Set $\alpha = {p_n/q_n} + \theta_n$.
The set of discontinuities of $\zeta$ is $\{0,\beta=\frac12 - \frac{\alpha}{2},\frac12,\beta'= 1 - \frac{\alpha}{2}\}$; the respective jumps are $1,\,-1,\,-1,\,1$. If $t \in \{0, \beta, \frac12, \beta' \}$, the corresponding discontinuities of $\zeta_{q}$ are $\{ t - j \alpha: j= 0,..., q-1\}$.
Depending on the parities of $p_n,q_n$, we define partitions $\{0, \beta, \frac12, \beta'\}=P_1\cup P_2$ as follows:
\noindent 1) For $q_n$ odd and $p_n$ even, we set $P_1 = \{0, \beta' \}$, $P_2 = \{{1 \over 2}, \beta \}$;
\noindent 2) For $q_n$ even and $p_n$ odd, we set $P_1 = \{0, {1\over 2} \}$, $P_2 = \{\beta, \beta'\}$;
\noindent 3) For $q_n$ odd and $p_n$ odd, we set $P_1 = \{0, \beta \}$, $P_2 = \{{1\over 2}, \beta' \}$.
Discontinuities of $\zeta_{q_n}$ which come from points in the same atom of the partition are very close to each other; discontinuities which come from points in distinct atoms of the partition are well separated from each other.
We will consider in detail only case 2). The analysis of other cases is similar, and we leave it to the reader. In what follows, all numbers and equalities are understood ${\rm { \ mod \ 1}}$. To simplify notation, we will suppress the subscript $n$. Thus, we write $\alpha = p/q +\theta$, etc.
\noindent a) The set of discontinuities, $D_0\subset X$, corresponding to $t=0$ is $D_0=\{- j \alpha: j= 0, ..., q-1\}$. For each integer $r$ there is $j_1(r) \in \{0, ..., q-1\}$ such that $-j_1(r)p=r {\rm { \ mod \ }} q$. Hence, $D_0=\{ {r \over q} - j_1(r) \theta: \ r= 0, ..., q-1 \}$.
\noindent b) Let $t=\frac12$. The corresponding set of discontinuities is $D_{\frac12}=\{\frac{(q' -jp)}{q} - j\theta: j= 0, ..., q-1\}$. For each integer $r$ there is $j_2(r) \in \{0, ..., q-1\}$ such that $q'-j_2(r)p=r {\rm { \ mod \ }} q$. Thus, $D_{\frac12}=\{ {r \over q} - j_2(r) \theta, \ r= 0, ..., q-1 \}$.
\noindent c) Let $t=1 - \frac{\alpha}{2}$. Then the set of discontinuities is $D_{1 - \frac{\alpha}{2}}=\{\frac{1}{2q} - \frac{(p'+1 +jp)}{q} - (j+ \frac12) \theta: j= 0, ..., q-1\}$. For each integer $r$ there is $j_3(r) \in \{0, ..., q-1\}$ such that $-(p'+1+j_3(r)p)=r {\rm \ mod \ } q$. Hence $D_{1 - \frac{\alpha}{2}}=\{ \frac{r}{q} + \frac{1}{2q} - (j_3(r) + \frac12) \theta: \ r= 0, ..., q-1 \}$.
\noindent d) Let $t=\frac12 - \frac{\alpha}{2}$. The set of discontinuities is $D_{\frac12 - \frac{\alpha}{2}}=\{\frac{1}{2q} - \frac{(-q'+p'+1 + jp)}{q} - (j + \frac12) \theta: j= 0,\dots,q-1\}$. For each integer $r$ there is $j_4(r) \in \{0, ..., q-1\}$ such that $q'-(p'+1+j_4(r)p)=r {\rm \ mod \ } q$. Hence $D_{\frac12 - \frac{\alpha}{2}}=\{\frac{r}{q} + \frac{1}{2q} - (j_4(r) + \frac12) \theta: \ r= 0, ..., q-1 \}$.
The set of discontinuities of $\zeta_{q}$ is $D_0\cup D_{\frac12}\cup D_{1 - \frac{\alpha}{2}}\cup D_{\frac12 -
\frac{\alpha}{2}}$. Observe that in all cases $|j_i(r) \theta| \leq
|q\theta|$; since $(j_2 - j_1) p = \frac12 q {\rm \ mod \ } q$ and $(j_4 - j_3) p = \frac12 q {\rm \ mod \ } q$, we have
\begin{eqnarray} j_2(r) = j_1(r) \pm {1\over 2} q, \ j_4(r) = j_3(r) \pm {1\over 2} q. \label{j1j2} \end{eqnarray}
We are going to determine the values taken by the cocycle $\zeta_q(x)$ for $x$ in a neighborhood of the typical interval $[\frac{r}{q}, \frac{r+1}{q}]$, where $r$ is an integer in $\{1, ..., q-1\}$. Assume, for concreteness, that $\theta < 0$, $j_1 = j_2+ \frac12 q$, $j_4 = j_3 + \frac12 q$.\footnote{The analysis for $\theta >0$ and/or $j_1 = j_2 - \frac12 q$, and/or $j_4 = j_3 - \frac12 q$ is analogous.} Let $x$ start at $\frac{r}{q}$ and let it move to the right; set $\zeta_q(x)=a$. The value of the cocycle $\zeta_q(x)$ is constant until $x$ crosses the discontinuity (corresponding to $t=0$) at $\frac{r}{q} - j_1(r)\theta$, where $\zeta_q(x)$ increases by $1$. After that the cocycle does not change until $x$ crosses the discontinuity at $\frac{r}{q} - j_2(r) \theta$ (corresponding to $t=\frac12$) where the cocycle decreases by $1$, returning to the value $a$.
The first two discontinuities occur before $x$ crosses the two other discontinuities under the condition that $|j_i(r)
\theta|$ is less than ${1 \over 2q}$. This takes place if
$q^2|\theta| < \frac12$, a condition which holds below because we consider the case when $q^2|\theta|$ is small. As $x$ continues to move to the right, the cocycle remains at the value $a$ until, near $\frac{r}{q} + \frac{1}{2q}$, it increases by $1$ at the point $\frac{r}{q} + \frac{1}{2q} - (j_3(r) + \frac12) \theta$, a discontinuity corresponding to $t=1 - \frac{\alpha}{2}$, and then decreases by $1$ at $\frac{r}{q} + \frac{1}{2q} - (j_4(r) + \frac12) \theta$, a discontinuity corresponding to $t=\frac12 - \frac{\alpha}{2}$.
Therefore, we have
\begin{eqnarray*} \zeta_{q}&=&a \pm 1 {\rm \ on \ } ]\frac{r}{q} - j_1(r) \theta,\frac{r}{q} - j_2(r) \theta[, \\ \zeta_{q}&=& a \pm 1 {\rm \ on \ } ]\frac{r}{q} + \frac{1}{2q} - (j_3(r) + \frac12) \theta, \frac{r}{q} + \frac{1}{2q} - (j_4(r) + \frac12) \theta[. \end{eqnarray*}
Elsewhere, $\zeta_{q}= a$.
This analysis is valid for every interval $[\frac{r}{q}, \frac{r+1}{q}]$. The order of the discontinuity points may change, but not the order between the groups of discontinuity. In particular, this implies that $\zeta_{q}=a$ on a subset of large measure in $[0,1]$. Since the mean value of $\zeta_{q}$ is zero, we have $a=0$.
We will now finish the proof. If $\alpha$ satisfies condition (D), then the claim holds, by Corollary~~\ref{D-wdd}. Thus, we assume that $\alpha$ does not satisfy condition (D). Then one of the cases 1), 2), 3) materializes for an infinite sequence $(n_k)$ such that $a_{n_k+1} \to \infty$. We will then say, for brevity, that a case {\em occurs infinitely often}. If case 1) occurs infinitely often, then $1$ is a quasi-period for $(\zeta_n)$. See figure~~\ref{bill-fig3}.
\begin{figure}
\caption{\it Graph of the cocycle $\zeta_q$ for $q$ odd, $p$ even. Its value is $1$ on a set of measure $\geq \delta > 0$.}
\label{bill-fig3}
\end{figure}
Suppose now that case 2) occurs infinitely often. We will use the preceding analysis. For $1\le r \le q_{n_k}-1$ set
\begin{eqnarray*} I_{k,r} &=& ]\frac{r}{q_{n_k}} - j_1(r) \theta_{n_k}, \frac{r}{q_{n_k}} - j_2(r) \theta_{n_k}[,\\ J_{k,r} &=& ]\frac{r}{q_{n_k}} + \frac{1}{2q_{n_k}} - (j_3(r) + \frac12) \theta_{n_k}, \frac{r}{q_{n_k}} + \frac{1}{2q_{n_k}} - (j_4(r) + \frac12) \theta_{n_k}[. \end{eqnarray*}
By equation~~\eqref{j1j2}, these intervals have length $\frac12 q_{n_k}
|\theta_{n_k}|$. At the scale $\frac{1}{q_{n_k}}$, they are close to $\frac{r}{q_{n_k}}$ and to $\frac{r}{q_{n_k}} + \frac{1}{2q_{n_k}}$ respectively. Outside of these intervals, $\zeta(q_{n_k}, \cdot)=0$.
\vskip 3mm Let $\delta \in ]0, {1\over 4}[$. Set $t_{k} = \lfloor \delta a_{{n_k}+1}\rfloor$. For $J\subset X$ and $u\in\R$, we set $(J + u)=J+u \mod 1$. Set $$A_k = \cup_{j=0}^{q_{n_k}-1} \, \cup_{s=0}^{t_k-1} \, (I_{k,j} - sq_{n_k} \alpha), \ B_k = \cup_{j=0}^{q_{n_k}-1} \, \cup_{s=0}^{t_k-1} \, (J_{{k},j} - sq_{n_k}\alpha).$$
The distance between the intervals $I_{{k},r}$ and $J_{{k},r}$ is at least $\frac{1}{2q_{n_k}} - q_{n_k}|\theta_{n_k}|$. Since, by the choice of $t_k$, we have $q_{n_k} |\theta_{n_k}| t_k \leq
\frac{1}{2q_{n_k}} - q_{n_k}|\theta_{n_k}|$, the translated intervals in the definition of $A_k$ and $B_k$ do not overlap.
\vskip 3mm Let us consider the cocycle at time $t_{k}q_{n_k}$. We have $\zeta(t_{k}q_{n_k}, x) = \sum_{s=0}^{t_k-1} \zeta(q_{n_k}, x + s q_{n_k}\alpha)$. By the preceding analysis of the values of $\zeta_q$, we have $\zeta(t_k q_{n_k}, \cdot)= \pm 1$ on $A_k$ and $B_k$. Also
$$
\leb(A_k) = {1\over 2} t_k q_{n_k} q_{n_k}|\theta_{n_k}| \geq {1\over 2} \delta a_{{n_k}+1} \frac{q_{n_k}}{q_{{n_k}+1}} \geq {1\over 2} \delta > 0. $$
Since $t_k q_{n_k} \alpha {\rm \ mod \ } 1 \to 0$, and since on $A_k$ the cocycle takes at most the two values $\pm 1$, we have shown that $1$ or $-1$ is a quasi-period for the cocycle $(\zeta_n)$.
\vskip 3mm The possibility that case 3) occurs infinitely often is analyzed the same way, with the conclusion that $1$ is a quasi-period.
Thus, no matter which of the three cases occurs infinitely often, $1$ is a quasi-period for the cocycle $(\zeta_n)$. The claim now follows, by Lemma~~\ref{lem-period}. \end{proof} \end{thm}
\vskip 3mm We will now establish the main result of this subsection.
\begin{thm} \label{ergo-Psi} Let $\Psi:X\to\ZZ$ be the function defined by equation~~\eqref{cocy_Psi}. Then, over any irrational rotation $\rho_{\alpha}: x \to x+\alpha {\rm \ mod \ } 1$ on the circle, the corresponding cocycle is ergodic. \begin{proof} Set $\Psi=(\psi_1,\psi_2)$ and $\beta= \frac12 - \frac{\alpha}{2}$. The functions $\psi_1, \psi_2$ satisfy $$\psi_1(x)+\psi_2(x) = \gamma(x),\ \psi_1(x) -\psi_2(x) = \gamma(x+\beta).$$
By Lemma~~\ref{4odd} and Lemma~~\ref{undemi-lem}, for an infinite sequence $(n_k)$ we have $\gamma(q_{n_k},x), \gamma(q_{n_k}, x+\beta)\in\{\pm 1\}$. This implies the existence of measurable sets $A_{k} \subset X$ satisfying $\leb(A_{k})> \frac14$, and such that for $x \in A_{k}$ the vector function $(\gamma(q_{n_k},x), \gamma(q_{n_k},x+\beta))$ is constant. Its values are $(+1,+1)$, ($+1, -1)$, $(-1,+1)$, or $(-1,-1)$.
Thus, for $x\in A_{k}$, the vector function $(\psi_1(q_{n_k},x), \psi_2(q_{n_k},x))$ is identically $(1,0)$, $(0,1)$, $(-1,0)$, or $(0,-1)$. Hence, one of the elements $(\pm 1,0),(0,\pm1)\in\ZZ$ is a quasi-period for the cocycle $(\Psi_n)$. Suppose, for instance, that $(1,0)$ is a quasi-period. Hence, by Lemma~~\ref{lem-period}, $(1,0)$ is a period. Let $f$ be a $\rho_{\alpha, \Psi}$-invariant function. It defines a $\rho_{\alpha, \zeta}$-invariant function on $X \times\Z$. Here $\zeta:X\to\Z$ is given by equation~~\eqref{gamma_zeta_eq}. By Theorem~~\ref{erg-beta}, $f=\mbox{const}$, i. e., the cocycle $(\Psi_n)$ is ergodic. The other cases are disposed of the same way. \end{proof} \end{thm}
\noindent{\bf Acknowledgements} In the course of preparation of this work the coauthors have made several visits to each other's home institutions. It is a pleasure to thank IRMAR and UMK for making these visits possible. The work of E.G. was partially supported by MNiSzW grant N N201 384834.
\end{document} | arXiv |
\begin{document}
\author{Zafer \c{S}iar \and Bing\"{o}l University, Department of Mathematics, Bing\"{o}l/TURKEY \and [email protected]} \title{An exponential Diophantine equation related to the difference of powers of two Fibonacci numbers} \maketitle
\begin{abstract} In this paper, we prove that there is no $x\geq 4$ such that the difference of $x$-th powers of two consecutive Fibonacci numbers greater than $0$ is a Lucas number. Also we show that the Diophantine equation \[ F_{n+l}^{x}-F_{n}^{x}=L_{r} \] with $l\in \left \{ 2,3,4\right \} ,$ $n>0,$ and $r\geq 0$ has no solutions for $x\geq 4.$ Finally, we conjecture that the Diophantine equation \[ F_{n}^{x}-F_{m}^{x}=L_{r} \] with $(n,m)\neq (1,0),(2,0),$ and $r\geq 0$ has no solutions for $x\geq 4.$ \end{abstract}
Keywords: Fibonacci and Lucas numbers, Exponential Diophantine equations, Linear forms in logarithms; Baker's method
AMS Subject Classification(2010): 11B39, 11D61, 11J86,
\section{\protect
Introduction}
Let $(F_{n})$ and $(L_{n})$ be the sequences of Fibonacci numbers and of Lucas numbers defined by $F_{0}=0,~F_{1}=1,$ $F_{n}=F_{n-1}+F_{n-2}$ and $ L_{0}=2,L_{1}=1,~L_{n}=L_{n-1}+L_{n-2}$ for $n\geq 2,$ respectively. Binet formulas for these numbers are \[ F_{n}=\frac{\alpha ^{n}-\beta ^{n}}{\sqrt{5}}\text{ and }L_{n}\text{ } =\alpha ^{n}+\beta ^{n}, \] where $\alpha =\dfrac{1+\sqrt{5}}{2}$ and $\beta =\dfrac{1-\sqrt{5}}{2},$ which are the roots of the characteristic equation $x^{2}-x-1=0.$ It can be seen that $1<\alpha <2,$ $-1<\beta <0$ and $\alpha \beta =-1.$ The most known identity related to these numbers is \begin{equation} L_{n}=F_{n-1}+F_{n+1}. \label{2.8} \end{equation} If $n\geq 1$, then the relation between $F_{n}$ and $\alpha $ is given by \begin{equation} \alpha ^{n-2}\leq F_{n}\leq \alpha ^{n-1} \label{1.1} \end{equation} and\ similarly, the relation between $n$-th Lucas number $L_{n}$ and $\alpha $ is \ \begin{equation} \alpha ^{n-1}\leq L_{n}<2\alpha ^{n}. \label{1.2} \end{equation} The inequalities (\ref{1.1}) and (\ref{1.2}) can be proved by induction. For more information about the Fibonacci and Lucas sequences with their applications, one can see \cite{Ko}.
The problem of finding the perfect powers in the Fibonacci sequence was a classical problem that attracted much attention over the last decades. One can consult \cite{cohn} for Fibonacci numbers that are a square or twice a square, and \cite{Bgud,Bgud2,togbe} for the similar studies. In all these works, the authors have used elementary methods, congruences, modular approach, and linear forms in logarithms. But, in recent years, many mathematicians started to use particularly linear forms in logarithms of algebraic numbers in order to solve some Diophantine equations including Fibonacci, Lucas, Pell, and balancing numbers. For example, in \cite{togbe}, Marques and Togbe showed that if $s\geq 1$ is an integer such that $ F_{m}^{s}+F_{m+1}^{s}$ is a Fibonacci number for all sufficiently large $m,$ then $s\in \left\{ 1,2\right\} .$ Then, Luca and Oyono, in \cite{oyono}, solved completely this problem. That is, they proved that the equation $ F_{m}^{s}+F_{m+1}^{s}=F_{n}$ has no solutions $(m,n,s)$ with $m\geq 2$ and $ s\geq 3.$ After that, in \cite{Ruiz}, the authors extended this problem to the $k-$generalized Fibonacci numbers. In \cite{Rihane1}, Rihane et al. tackled the Diophantine equation \[ P_{n}^{x}+P_{n+1}^{x}=P_{m} \] and gave all the solutions of this equation in nonnegative integers $m,n,x.$ Same authors, in \cite{Rihane2}, proved that the Diophantine equation \[ B_{n+1}^{x}-B_{n}^{x}=B_{m} \] has the solutions $(m,n,x)=(2n+2,n,2),(1,0,x),(0,n,0)$ in nonnegative integers $m,n,$ and $x.$
On the other hand, the relation \begin{equation} F_{n+2}-F_{n-2}=L_{n} \label{1.3} \end{equation} is well known. Motivated by this equality and the above mentioned studies, we present a new problem. We will try to answer the question such that when does the difference of $x$-th powers of\ any two Fibonacci numbers become a Lucas number? Clearly, a trivial solution of this question for $x=1$ is seen immediately from (\ref{1.3}). In this study, we show that the Diophantine equation \[ F_{n+l}^{x}-F_{n}^{x}=L_{r} \] with $l\in \left\{ 1,2,3,4\right\} ,$ $n>0,$ and $r\geq 0$ has no solutions for $x\geq 4.$ Finally, we conjecture that the Diophantine equation \begin{equation} F_{n}^{x}-F_{m}^{x}=L_{r} \label{1.4} \end{equation} with $(n,m)\neq (1,0),(2,0),$ and $r\geq 0$ has no solutions for $x\geq 4.$ Here, we will prove this conjecture for $n\leq 2m+4.$ But, the proof of this conjecture for $n>2m+4$ is really difficult.
Now let us give some inequalities, which will be useful for the proof of our main theorem. We observe that the inequality \[ \dfrac{F_{n-1}}{F_{n}}\leq \frac{2}{3} \] holds for all $n\geq 3.$ This implies that \begin{equation} \dfrac{F_{n}}{F_{m}}\geq \frac{3}{2} \label{1.5} \end{equation} for $m<n$ and $n\geq 3.$ Also, it follows that \[ \left( \dfrac{F_{m}}{F_{n}}\right) ^{x}\leq \left( \dfrac{F_{m}}{F_{n}} \right) ^{2}=\frac{1}{\left( F_{n}/F_{m}\right) ^{2}}\leq \frac{4}{9} \] for $x\geq 2$ and $n\geq 3.$ And thus, \begin{equation} 1-\left( \dfrac{F_{m}}{F_{n}}\right) ^{x}\geq \frac{1}{2}. \label{1.6} \end{equation}
Our main theorem is\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{theorem} \label{T4} Let $x$ be a positive integer and let $n,m,r$ be non-negative integers such that $n\leq 2m+4$ if $m\neq 0.$ Then all solutions $\left( n,m,r,x\right) $ of the Diophantine equation $F_{n}^{x}-F_{m}^{x}=L_{r}$ are the elements of the sets \[ \left \{ \begin{array}{c} (1,0,1,x),(2,0,1,x),\left( 3,0,3,2\right) ,\left( 3,0,0,1\right) , \\ \left( 4,0,2,1\right) ,\left( 3,1,1,1\right) ,\left( 3,2,1,1\right) ,\left( 3,1,2,2\right) ,\left( 3,2,2,2\right) , \\ \left( 3,1,4,3\right) ,\left( 3,2,4,3\right) ,\left( 4,1,0,1\right) ,\left( 4,2,0,1\right) ,\left( 5,4,0,1\right) , \\ \left( 4,3,1,1\right) ,\left( 5,2,3,1\right) ,\left( 6,5,2,1\right) ,\left( 6,1,4,1\right) ,\left( 5,3,2,1\right) \end{array} \right \} \] or $\left \{ \left( k+1,k-3,k-1,1\right) \right \} $ with $k\geq 4.$ \end{theorem}
\section{Auxiliary results}
In order to solve some Diophantine equations as in (\ref{1.4}), many mathematicians have used Baker's theory of lower bounds for a nonzero linear form in logarithms of algebraic numbers. Since such bounds are of crucial importance in effectively solving of the Diophantine equation (\ref{1.4}), we start with recalling some basic notions from algebraic number theory.
Let $\eta $ be an algebraic number of degree $d$ with minimal polynomial \[ a_{0}x^{d}+a_{1}x^{d-1}+\cdots +a_{d}=a_{0}\dprod\limits_{i=1}^{d}\left( x-\eta ^{(i)}\right) \in \mathbb{Z}[x], \] where the $a_{i}$'s are relatively prime integers with $a_{0}>0$ and $\eta ^{(i)}$'s are conjugates of $\eta .$ Then \begin{equation} h(\eta )=\frac{1}{d}\left( \log a_{0}+\dsum\limits_{i=1}^{d}\log \left( \max
\left\{ |\eta ^{(i)}|,1\right\} \right) \right) \label{2.1} \end{equation} is called the logarithmic height of $\eta .$ In particularly, if $\eta =a/b$ is a rational number with $\gcd (a,b)=1$ and $b>1,$ then $h(\eta )=\log
\left( \max \left\{ |a|,b\right\} \right) .$
The following properties of logarithmic height are found in many works stated in the references:
\begin{equation} h(\eta \pm \gamma )\leq h(\eta )+h(\gamma )+\log 2, \label{2.2} \end{equation} \begin{equation} h(\eta \gamma ^{\pm 1})\leq h(\eta )+h(\gamma ), \label{2.3} \end{equation} \begin{equation}
h(\eta ^{m})=|m|h(\eta ). \label{2.4} \end{equation} The following theorem is deduced from Corollary 2.3 of Matveev \cite{Mtv} and provides a large upper bound for the subscripts in the equation (\ref {1.4}) (also see Theorem 9.4 in \cite{Bgud}).
\begin{theorem} \label{T1} Assume that $\gamma _{1},\gamma _{2},...,\gamma _{t}$ are positive real algebraic numbers in a real algebraic number field $\mathbb{K}$ of degree $D$, $b_{1},b_{2},...,b_{t}$ are rational integers, and \[ \Lambda :=\gamma _{1}^{b_{1}}\cdots \gamma _{t}^{b_{t}}-1 \] is not zero. Then \[
|\Lambda |>\exp \left( -1.4\cdot 30^{t+3}\cdot t^{4.5}\cdot D^{2}(1+\log D)(1+\log B)A_{1}A_{2}\cdots A_{t}\right) , \] where \[
B\geq \max \left \{ |b_{1}|,...,|b_{t}|\right \} , \]
and $A_{i}\geq \max \left \{ Dh(\gamma _{i}),|\log \gamma _{i}|,0.16\right \} $ for all $i=1,...,t.$ \end{theorem}
The following lemma, which will be used in the main theorem, gives a sufficient condition for a rational number to be a convergent of a given real number.
\begin{lemma} \label{L7}(\cite{yann})Let $\gamma $ be a real number. Any non-zero rational number $\frac{a}{b}$ with \[ \left \vert \gamma -\frac{a}{b}\right \vert <\frac{1}{2b^{2}} \] is a convergent of $\gamma .$ \end{lemma}
\begin{lemma} \label{L3}(\cite{weger}, Weger's Lemma 2.2) Let $a,u\in
\mathbb{R}
$ and $0<$ $a<1$. If $\left \vert u\right \vert <a,$ then \[ \left \vert \log (1+u)\right \vert <\frac{-\log (1-a)}{a}\cdot \left \vert u\right \vert \] and \[ \left \vert u\right \vert <\frac{a}{1-e^{-a}}\cdot \left \vert e^{u}-1\right \vert . \] \end{lemma}
The following lemma can be found in \cite{peter}.
\begin{lemma}
\label{L4}If $F_{n}|L_{m},$ then $n\leq 4.$ \end{lemma}
The following two theorems are given in \cite{cohn} and \cite{zfr}, respectively.
\begin{theorem} \label{T2} If $L_{n}=x^{2},$ then $n=1,\,3$ and if $L_{n}=2x^{2},$ then $ n=0,6.$ \end{theorem}
\begin{theorem} \label{T2.1}If $L_{n}=L_{m}x^{2}$ with $m\geq 2,$ then $n=m.$ \end{theorem}
The following two theorems are proved in \cite{zfr2}.
\begin{theorem} \label{T7}Let $1\leq m\leq n.$ Then all solutions of the equation $ F_{n}+F_{m}=F_{r}$ are \[ (n,m,r)=(n,n-1,n+1),(1,1,3),(2,1,3),(2,2,3),(3,1,4). \] \end{theorem}
\begin{theorem} \label{T3}All solutions $(n,m,r,k)$ of the Diophantine equation $ F_{k}=F_{n}+F_{m}+F_{r}$ with $1\leq r\leq m\leq n$ are the elements of the sets \[ \left \{ \left( n,n-2,n-3,n+1\right) ,\left( n,n,n-1,n+2\right) \right \} \] and \[ \left \{ \begin{array}{c} \left( 1,1,1,4\right) ,\left( 4,1,1,5\right) ,\left( 4,2,2,5\right) ,\left( 5,3,1,6\right) , \\ \left( 2,1,1,4\right) ,\left( 2,2,2,4\right) ,\left( 3,3,1,5\right) \end{array} \right \} . \] \end{theorem}
The following lemma can be deduced from Theorem $2$ given in \cite{Bravo}.
\begin{lemma} \label{L5}The Diophantine equation $L_{n}=2^{x}-1$ for some nonnegative integers $n,x$ has only the solutions $(n,x)=(1,1),(2,2),(4,3).$ \end{lemma}
\section{The proof of Theorem\ \protect\ref{T4}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }
\proof
Let $x>0$ be an integer and let $n,m,r$ be non-negative integers such that $ n\leq 2m+4$ if $m\neq 0.$ Assume that $F_{n}^{x}-F_{m}^{x}=L_{r}.$ It is clear that $n\neq m$ since $L_{r}\neq 0$ for all integer $r.$ Then, $n>m.$ If $m=0,$ then we have $F_{n}^{x}=L_{r}.$ Since $F_{n}\mid L_{r},$ it follows that $n\leq 4$ by Lemma \ref{L4}. It is obvious that $n\neq 0.$ If $ n=1$ or $n=2,$ then $(n,m,r,x)=(1,0,1,x),(2,0,1,x).$ If $n=3$ or $n=4,$ then we have $L_{r}=\square $ or $L_{r}=2\square ,$ or $L_{r}=L_{2}\square .$ In these cases, we get the solutions $(n,m,r,x)=\left( 3,0,3,2\right) ,\left( 3,0,0,1\right) ,\left( 4,0,2,1\right) $ by Theorems \ref{T2} and \ref{T2.1}. Now let $m\geq 1.$ If $m=1$ or $m=2,$ then we see that $n\geq 3.$ Assume that $n=3.$ Hence, we have the equation $L_{r}=2^{x}-1.$ Then by Lemma \ref {L5}, we get \[ (n,m,r,x)\in \left \{ \left( 3,1,1,1\right) ,\left( 3,2,1,1\right) ,\left( 3,1,2,2\right) ,\left( 3,2,2,2\right) ,\left( 3,1,4,3\right) ,\left( 3,2,4,3\right) \right \} . \] From now on, assume that $m\geq 1,$ $n\geq 4,$ and $x\geq 1.$ Now let $x=1.$ Then we have the equation $F_{n}-F_{m}=L_{r},$ i.e., $ F_{n}=F_{m}+F_{r-1}+F_{r+1}$ by (\ref{2.8}). By Theorems \ref{T7} and \ref {T3}, we obtain \[ (n,m,r,x)\in \left \{ \begin{array}{c} \left( 3,1,1,1\right) ,\left( 3,2,1,1\right) ,\left( 4,3,1,1\right) ,\left( 4,1,0,1\right) ,\left( 4,2,0,1\right) , \\ \left( 5,4,0,1\right) ,\left( 5,2,3,1\right) ,\left( 6,5,2,1\right) ,\left( 6,1,4,1\right) ,\left( 5,3,2,1\right) \end{array} \right \} , \] and $(n,m,r,x)\in \left \{ \left( k+1,k-3,k-1,1\right) \right \} $ with $ k\geq 4.$ Thus, we can suppose that $x\geq 2.$ Since $n\geq 4$ and $n>m,$ it follows that $5\leq 3^{x}-2^{x}\leq F_{n}^{x}-F_{m}^{x}=L_{r},$ which implies that $r\geq 4.$ On the other hand, using (\ref{1.1}), (\ref{1.2}) and (\ref{1.6}), we get \begin{equation} \alpha ^{r-1}\leq L_{r}=F_{n}^{x}-F_{m}^{x}\leq F_{n}^{x}\leq \alpha ^{(n-1)x} \label{a} \end{equation} and also, \begin{equation} 2\alpha ^{r}>L_{r}=F_{n}^{x}-F_{m}^{x}=F_{n}^{x}\left( 1-\left( \frac{F_{m}}{ F_{n}}\right) ^{x}\right) \geq \frac{F_{n}^{x}}{2}\geq \frac{\alpha ^{(n-2)x} }{2}. \label{b} \end{equation} If we make necessary calculations by using the inequalities (\ref{a}) and ( \ref{b}), we get \begin{equation} (m-3)x\leq (n-3)x\leq r<nx\leq \left( 2m+4\right) x, \label{3.1} \end{equation} where we used the facts that $n\geq 4$ and $m<n\leq 2m+4.$ Rearranging the equation $F_{n}^{x}-F_{m}^{x}=L_{r}$ as $F_{n}^{x}-\alpha ^{r}=F_{m}^{x}+\beta ^{r}$ and taking absolute values of both sides of last equality, we get \begin{equation} \left \vert F_{n}^{x}-\alpha ^{r}\right \vert \leq F_{m}^{x}+\left \vert \beta \right \vert ^{r}. \label{3.2} \end{equation} Dividing both sides of (\ref{3.2}) by $F_{n}^{x}$ yields to \begin{equation} \left \vert 1-F_{n}^{-x}\alpha ^{r}\right \vert \leq \frac{1}{\left( F_{n}/F_{m}\right) ^{x}}+\frac{\left \vert \beta \right \vert ^{r}}{F_{n}^{x} }\leq \frac{1}{\left( 1.5\right) ^{x}}+\frac{1}{\alpha ^{(n-2)x}}<\frac{2}{ (1.5)^{x}}, \label{3.3} \end{equation} where we used the inequality (\ref{1.5}) and the fact that $\alpha ^{(n-2)x}\geq \left( \frac{3}{2}\right) ^{x}$ for $m\geq 1$ and $n\geq 4.$ Put \[ \Lambda _{1}:=1-F_{n}^{-x}\alpha ^{r}. \] If $\Lambda _{1}=0,$ then we get $F_{n}^{x}=\alpha ^{r},$ which is impossible since $\alpha ^{r}$ is irrational for all positive integers $r.$ So $\Lambda _{1}\neq 0.$ Now,\ let us apply Theorem \ref{T1} with $\gamma _{1}:=\alpha ,~\gamma _{2}:=F_{n}$ and $b_{1}:=-x,~b_{2}:=r.$ Note that the numbers $\gamma _{1}$ and $\gamma _{2}$ are positive real numbers and elements of the field $\mathbb{K}=\mathbb{Q}(\sqrt{5}).$ It is obvious that the degree of the field $\mathbb{K}$ is $2.$ So $D=2.$\ Moreover, since \[ h(\gamma _{1})=h(\alpha )=\dfrac{\log \alpha }{2}=\dfrac{0.4812...}{2} \] and \[ h(\gamma _{2})=h(F_{n})=n\log \alpha \] by (\ref{2.3}), we can take $A_{1}:=0.5$ and$~A_{2}:=2n\log \alpha .$ Also, it is obvious that $r\geq x$ by (\ref{3.1}) since $n\geq 4.$ Therefore, we can take $B=r.$ Thus, taking into account the inequality (\ref{3.3}) and using Theorem \ref{T1}, we obtain \[ 2(1.5)^{x}>\left \vert \Lambda _{1}\right \vert >\exp \left( -1.4\cdot 30^{5}\cdot 2^{4.5}\cdot 2^{2}(1+\log 2)(1+\log r)\left( 0.5\right) 2n\log \alpha \right) . \] Taking logarithms in the above inequality, we get \[ x\log (1.5)-\log 2<2.51\cdot 10^{9}\cdot (1+\log r)\cdot n. \] Thus, it follows that \begin{equation} x<6.2\cdot 10^{9}\cdot (1+\log nx)\cdot n \label{z} \end{equation} by (\ref{3.1}). Now we assume that $n\leq 270.$ Then, the inequality (\ref{z} ) gives us that $x<6.43\cdot 10^{13}.$
Let \[ z_{1}:=r\log \alpha -x\log F_{n} \] and $u=e^{z_{1}}-1.$ By considering the inequality (\ref{3.3}), we have \[ \left \vert u\right \vert =\left \vert e^{z_{1}}-1\right \vert <\frac{2}{ (1.5)^{x}}<0.9 \] for $x\geq 2.$ Choosing $a=0.9$ in Lemma \ref{L3}, we get \[
|z_{1}|=\left \vert \log (u+1)\right \vert <\frac{-\log (1-0.9)}{(0.9)}\cdot \frac{2}{(1.5)^{x}}<\frac{5.12}{(1.5)^{x}}. \] Hence, it follows that \[ 0<\left \vert r\log \alpha -x\log F_{n}\right \vert <\frac{5.12}{(1.5)^{x}}. \] Dividing both sides of this inequality by $x\log \alpha ,$ we get \begin{equation} 0<\left \vert \frac{\log F_{n}}{\log \alpha }-\frac{r}{x}\right \vert <\frac{ 11}{x\cdot (1.5)^{x}}. \label{3.5} \end{equation} Now assume that $x\geq 103.$ Then it can be seen that \[ \dfrac{(1.5)^{x}}{22}>6.23\cdot 10^{16}>6.43\cdot 10^{13}>x, \] and so we have \[ \left \vert \frac{\log F_{n}}{\log \alpha }-\frac{r}{x}\right \vert <\frac{11 }{x\cdot (1.5)^{x}}<\dfrac{1}{2x^{2}}. \] Lemma \ref{L7} tells us that the rational number $\frac{r}{x}$ is a convergent to $\gamma =\frac{\log F_{n}}{\log \alpha }.$ Then, let $ [a_{0},a_{1},a_{2},...]$ be the continued fraction of $\gamma $ and let $ p_{k}/q_{k}$ be its $k$-th convergent. Assume that $\frac{r}{x}=\dfrac{p_{t} }{q_{t}}$ for some $t.$ Then we have $1.7\cdot 10^{23}>q_{34}>6.43\cdot 10^{13}>x$ for every $n\in \left[ 4,270\right] .$ Thus $t\in
\{0,1,2,...,33\}.$ Furthermore, $a_{M}=\max \{a_{i}|i=0,1,2,...,33\}=1598.$ From the known properties of continued fraction, we get \begin{eqnarray*} \left \vert \frac{\log F_{n}}{\log \alpha }-\frac{r}{x}\right \vert &=&\left \vert \gamma -\dfrac{p_{t}}{q_{t}}\right \vert =\dfrac{1}{(\gamma _{t+1}q_{t}+q_{t-1})q_{t}} \\ &=&\dfrac{1}{(\gamma _{t+1}+\frac{q_{t-1}}{q_{t}})q_{t}^{2}}>\dfrac{1}{ (a_{t+1}+2)q_{t}^{2}}>\dfrac{1}{(a_{M}+2)q_{t}^{2}}\geq \dfrac{1}{1600x^{2}}, \end{eqnarray*} where we have used the facts that $a_{t}=\left \lfloor \gamma _{t}\right \rfloor $ and $q_{t-1}<q_{t}.$ Thus, from (\ref{3.5}), we obtain \[ \frac{11}{x\cdot (1.5)^{x}}>\dfrac{1}{1600x^{2}}, \] that is, \[ \frac{1}{12.46\cdot 10^{16}}>\frac{11}{(1.5)^{x}}>\dfrac{1}{1600x}\geq \frac{ 1}{10.288\cdot 10^{16}}, \] a contradiction. Therefore $x\leq 102.$ Taking into account the inequality ( \ref{3.1}), a quick computation with Mathematica gives us that the equation $ F_{n}^{x}-F_{m}^{x}=L_{r}$ has no solutions for $n\in \left[ 4,270\right] $ and $x\in \left[ 2,102\right] .$ Since this completes the analysis in the case $n\in \left[ 4,270\right] ,$ from now on, we can assume that $n>270.$ This implies that $m\geq 134.$ Since $n\leq 2m+4$, from (\ref{z}), we can write \begin{equation} x<6.2\cdot 10^{9}\cdot (1+\log \left( 2m+4\right) x)\cdot \left( 2m+4\right) . \label{3.4} \end{equation} Here, since $\left( m+2\right) x\geq 6$ implies that $(1+\log \left( 2m+4\right) x)\leq 2\log \left( m+2\right) x,$ we can rewrite the inequality (\ref{3.4}) as \begin{equation} x<24.8\cdot 10^{9}\cdot \left( m+2\right) \log \left( \left( m+2\right) x\right) \label{3.6} \end{equation} If $x\leq m+2,$ then we have an inequality, which is better than inequality ( \ref{3.6}). So, we are through. Contrast to this, if $x>m+2,$ then (\ref{3.6} ) yields to us that $x<49.6\cdot 10^{9}\cdot \left( m+2\right) \log x,$ which can be rearranged as \begin{equation} \frac{x}{\log x}<49.6\cdot 10^{9}\cdot \left( m+2\right) . \label{3.7} \end{equation} Using the fact that \[ \text{if }A\geq 3\text{ and }\frac{x}{\log x}<A,\text{ then }x<2A\log A, \] we obtain \[ x<99.2\cdot 10^{9}\cdot \left( m+2\right) \log \left( 49.6\cdot 10^{9}\cdot \left( m+2\right) \right) , \] or \begin{equation} x<595.2\cdot 10^{9}\left( m+2\right) \log \left( m+2\right) . \label{3.8} \end{equation} Now, put $y:=\dfrac{x}{\alpha ^{2m}}.$ Then, since $m\geq 134,$ from the inequality (\ref{3.8}), we get \begin{equation} y<\dfrac{595.2\cdot 10^{9}\left( m+2\right) \log \left( m+2\right) }{\alpha ^{2m}}<\dfrac{1}{\alpha ^{m}}. \label{3.9} \end{equation} Particularly, note that $y<\dfrac{1}{\alpha ^{m}}\leq \alpha ^{-134}<10^{-28}.$ On the other hand, it can be seen that \[ F_{n}^{x}=\dfrac{\alpha ^{nx}}{5^{x/2}}\left( 1-\dfrac{(-1)^{n}}{\alpha ^{2n} }\right) ^{x} \] and \[ F_{m}^{x}=\dfrac{\alpha ^{mx}}{5^{x/2}}\left( 1-\dfrac{(-1)^{m}}{\alpha ^{2m} }\right) ^{x}. \] Furthermore, we have \begin{eqnarray*} 0 &<&\left( 1-\dfrac{1}{\alpha ^{2n}}\right) ^{x}<1<\left( 1+\dfrac{1}{ \alpha ^{2n}}\right) ^{x}=1+\frac{x}{\alpha ^{2n}}+\frac{x(x-1)}{\alpha ^{4n} }+... \\ &<&e^{y}<1+2y \end{eqnarray*} because $y<10^{-28}.$ If we write $m$ instead of $n$ in the above inequality, it holds. Thus, we see that \begin{equation} \max \left \{ \left \vert F_{n}^{x}-\dfrac{\alpha ^{nx}}{5^{x/2}}\right \vert ,\left \vert F_{m}^{x}-\dfrac{\alpha ^{mx}}{5^{x/2}}\right \vert \right \} <\frac{2y\alpha ^{nx}}{5^{x/2}}. \label{3.10} \end{equation} Let us rearrange the equation $F_{n}^{x}-F_{m}^{x}=L_{r}$ as \[ \alpha ^{r}-\dfrac{\alpha ^{nx}}{5^{x/2}}+\dfrac{\alpha ^{mx}}{5^{x/2}} =F_{n}^{x}-\dfrac{\alpha ^{nx}}{5^{x/2}}-F_{m}^{x}+\dfrac{\alpha ^{mx}}{ 5^{x/2}}-\beta ^{r}. \] Taking absolute values of both sides of the above equality and using (\ref {3.10}), we get \[ \left \vert \alpha ^{r}-\dfrac{\alpha ^{nx}}{5^{x/2}}\left( 1-\alpha ^{(m-n)x}\right) \right \vert \leq \left \vert F_{n}^{x}-\dfrac{\alpha ^{nx} }{5^{x/2}}\right \vert +\left \vert F_{m}^{x}-\dfrac{\alpha ^{mx}}{5^{x/2}} \right \vert +\left \vert \beta \right \vert ^{r}<\frac{4y\alpha ^{nx}}{ 5^{x/2}}+\left \vert \beta \right \vert ^{r}. \] Dividing both sides of this inequality by $\dfrac{\alpha ^{nx}}{5^{x/2}},$ we obtain \begin{equation} \left \vert \alpha ^{r-nx}\cdot 5^{x/2}+\alpha ^{(m-n)x}-1\right \vert < \dfrac{4}{\alpha ^{m}}+\frac{1}{\alpha ^{r}}\cdot \left( \frac{\sqrt{5}}{ \alpha ^{n}}\right) ^{x}, \label{3.11} \end{equation} where we used the fact that $\alpha \beta =-1$ and $y<\dfrac{1}{\alpha ^{m}} . $ Since \[ \left \vert \alpha ^{r-nx}\cdot 5^{x/2}-1\right \vert \leq \left \vert \alpha ^{r-nx}\cdot 5^{x/2}+\alpha ^{(m-n)x}-1\right \vert +\alpha ^{(m-n)x}, \] from (\ref{3.11}), we get \begin{equation} \left \vert \alpha ^{r-nx}\cdot 5^{x/2}-1\right \vert <\dfrac{4}{\alpha ^{m}} +\frac{1}{\alpha ^{r}}\cdot \left( \frac{\sqrt{5}}{\alpha ^{n}}\right) ^{x}+ \frac{1}{\alpha ^{(n-m)x}}<\dfrac{4}{\alpha ^{m}}+\frac{1.05}{\alpha ^{x}} \text{,} \label{3.12} \end{equation} where we used the fact that $\left( \frac{\sqrt{5}}{\alpha ^{n}}\right) ^{x}\leq 0.05$ for $n>270$ and $r\geq x.$ Put \[ \Lambda _{2}:=1-\alpha ^{r-nx}\cdot 5^{x/2}. \] If $\Lambda _{2}=0,$ then we see that $\alpha ^{2(nx-r)}=5^{x},$ which is possible only when $nx=r$ since $5^{x}\in
\mathbb{Z}
.$ This is impossible since $r<nx$ by (\ref{3.1}). Therefore $\Lambda _{2}\neq 0.$ Also, \begin{equation} \left \vert \Lambda _{2}\right \vert <\dfrac{4}{\alpha ^{m}}+\frac{1.05}{ \alpha ^{x}}<\frac{1}{2} \label{c} \end{equation} since $m\geq 134$ and $x\geq 2.$ Thus $\alpha ^{r-nx}\cdot 5^{x/2}\in \left[ \frac{1}{2},\frac{3}{2}\right] .$ Particularly, making necessary calculations, it is seen that \begin{equation} 1.659x-0.86<nx-r<1.7x+1.46. \label{3.13} \end{equation} Let $k_{1}=\min \left \{ m,x\right \} .$ Then we see from (\ref{3.12}) that \begin{equation} \left \vert \alpha ^{r-nx}\cdot 5^{x/2}-1\right \vert <\frac{5.05}{\alpha ^{k_{1}}}. \label{3.14} \end{equation} Now, let us apply Theorem \ref{T1} to the inequality (\ref{3.14}). Take $ \gamma _{1}:=\sqrt{5},~\gamma _{2}:=\alpha ,$ and $b_{1}:=x,~b_{2}:=r-nx.$ Observe that the numbers $\gamma _{1}$ and $~\gamma _{2}$ are positive real numbers and belong to the field $K=Q(\sqrt{5}).$ Therefore $D=2.$ Also, since $h(\gamma _{1})=\log \sqrt{5}=0.804...,$ and $h(\gamma _{2})=\dfrac{ \log \alpha }{2}=\dfrac{0.4812...}{2}$ by (\ref{2.1}), we can take $ A_{1}:=1.61$ and $~A_{2}:=0.5.$ Besides, from (\ref{3.13}), it is clear that $nx-r<\left( 2.5\right) x$ for $x\geq 2.$ So, we can take $B=\left(
2.5\right) x\geq \max \left \{ |x|,|r-nx|\right \} .$ Thus, Theorem \ref{T1} tells us that \[ \frac{5.05}{\alpha ^{k_{1}}}>\left \vert \Lambda _{2}\right \vert >\exp \left( -C(1+\log 2)(1+\log \left( 2.5\right) x)\left( 1.61\right) \left( 0.5\right) \right) \] or \begin{equation} k_{1}\log \alpha -\log 5.05<C(1+\log 2)(1+\log \left( 2.5\right) x)\left( 1.61\right) \left( 0.5\right) , \label{3.15} \end{equation} where $C=1.4\cdot 30^{5}\cdot 2^{4.5}\cdot 2^{2}.$ If $k_{1}=x,$ then a computer search with Mathematica gives us that $x<2.46\cdot 10^{11}.$ If $ k_{1}=m,$ then, by\ using(\ref{3.8}), we get{\footnotesize \begin{equation} m\log \alpha -\log 5.05<C(1+\log 2)(1+\log \left( 2.5\right) +\log (595.2\cdot 10^{9}\left( m+2\right) \log \left( m+2\right) ))\left( 1.61\right) \left( 0.5\right) . \label{3.16} \end{equation} } With the help of a program in Mathematica\textit{, }the inequality (\ref {3.16}) gives us that $m<5.18\cdot 10^{11}.$ Substituting this value of $m$ into (\ref{3.8}), we obtain $x<8.32\cdot 10^{24}.$
Now, let \[ z_{2}:=x\log \sqrt{5}-(nx-r)\log \alpha \] and $u:=e^{z_{2}}-1.$ Then $\left \vert u\right \vert =\left \vert e^{z_{2}}-1\right \vert =\left \vert \Lambda _{2}\right \vert <\frac{1}{2}$ by (\ref{c}). Thus, taking $a=1/2$ in Lemma \ref{L7} and making necessary calculations, we get \[
|z_{2}|=\left \vert \log (1+u)\right \vert <\frac{-\log (1-\frac{1}{2})}{1/2} \left \vert u\right \vert <1.4\left( \dfrac{4}{\alpha ^{m}}+\frac{1.05}{ \alpha ^{x}}\right) . \] That is, \[ 0<\left \vert x\log \sqrt{5}-(nx-r)\log \alpha \right \vert <1.4\left( \dfrac{4}{\alpha ^{m}}+\frac{1.05}{\alpha ^{x}}\right) . \] Dividing both sides of the above inequality by $x\log \alpha ,$ we obtain \begin{equation} \left \vert \frac{\log \sqrt{5}}{\log \alpha }-\frac{(nx-r)}{x}\right \vert < \frac{2.91}{x}\left( \dfrac{4}{\alpha ^{m}}+\frac{1.05}{\alpha ^{x}}\right) . \label{3.17} \end{equation} Since $m\geq 134,$ it follows that $\alpha ^{m}\geq \alpha ^{134}>10^{28}>1000x.$ Now we suppose that $x>100.$ Then it can be seen that $\alpha ^{x}>1000x.$ Hence, we can rewrite (\ref{3.17}) as \[ \left \vert \frac{\log \sqrt{5}}{\log \alpha }-\frac{(nx-r)}{x}\right \vert < \frac{2.91}{x}\left( \dfrac{4}{1000x}+\frac{1.05}{1000x}\right) <\frac{1}{ 66x^{2}}. \] This implies by Lemma \ref{L7} that the rational number $\frac{(nx-r)}{x}$ is a convergent to $\gamma =\frac{\log \sqrt{5}}{\log \alpha }.$ Now let $ [a_{0},a_{1},a_{2},...]$ be the continued fraction of $\gamma $ and let $ p_{k}/q_{k}$ be its $k$-th convergent. Assume that $\frac{(nx-r)}{x} =p_{t}/q_{t}$ for some $t.$ Then we have $2\cdot 10^{26}>q_{48}>8.32\cdot 10^{24}>x.$ Thus $t\in \{0,1,2,...,47\}.$ Furthermore, $a_{M}=\max
\{a_{i}|i=0,1,2,...,47\}=29.$ From the known properties of continued fraction, we get \[ \frac{1}{66x^{2}}>\left \vert \frac{\log \sqrt{5}}{\log \alpha }-\frac{(nx-r) }{x}\right \vert =\left \vert \gamma -\dfrac{p_{t}}{q_{t}}\right \vert > \dfrac{1}{(a_{M}+2)q_{t}^{2}}\geq \dfrac{1}{31x^{2}}, \] a contradiction. So, $x\leq 100.$ Then $x<m.$ Hence, from (\ref{3.12}), we get \begin{equation} \left \vert \alpha ^{r-nx}\cdot 5^{x/2}-1\right \vert <\frac{5.05}{\alpha ^{x}}. \label{3.18} \end{equation} From (\ref{3.13}), we know that \[ 1.659x-0.86<nx-r<1.7x+1.46. \] Put $t=nx-r.$ We found that the inequality (\ref{3.18}) is not satisfied\ for all $x\in \lbrack 2,100]$ and $t\in \left[ \left \lfloor 1.659x-0.86\right \rfloor ,\left \lceil 1.7x+1.46\right \rceil \right] .$ Thus the proof is completed.
Thus, we can give the following result.
\begin{corollary} Let $x>0$ be an integer and let $n,r$ be nonnegative integers. Then all the solutions $(n,r,x)$ of the Diophantine equation $F_{n+l}^{x}-F_{n}^{x}=L_{r}$ with $l\in \left \{ 1,2,3,4\right \} $ are given by \begin{eqnarray*} (n,r,x) &=&(0,1,x),\left( 2,1,1\right) ,\left( 2,2,2\right) ,\left( 2,4,3\right) ,\left( 4,0,1\right) ,\left( 3,1,1\right) ,\left( 5,2,1\right) \text{ if }l=1, \\ (n,r,x) &=&(0,1,x),\left( 1,1,1\right) ,\left( 1,2,2\right) ,\left( 1,4,3\right) ,\left( 2,0,1\right) ,\left( 3,2,1\right) \text{ if }l=2, \\ (n,r,x) &=&\left( 0,3,2\right) ,\left( 0,0,1\right) ,\left( 1,0,1\right) ,\left( 2,3,1\right) \text{ if }l=3, \\ (n,r,x) &=&\left( 0,2,1\right) \text{ if }l=4. \end{eqnarray*} \end{corollary}
As one can see from the above result, the Diophantine equation \[ F_{n+l}^{x}-F_{n}^{x}=L_{r} \] with $l\in \left \{ 1,2,3,4\right \} ,$ $n>0,$ and $r\geq 0$ has no solutions for $x\geq 4.$ If we pay attention, this equation has solutions only for $n\leq 5.$ From the equations obtained by substituting these values of $n$ (except for $n=0$) into the last equation, the equations, which have a solution are given as follows:
\begin{corollary} The Diophantine equation $3^{x}-2^{x}=L_{r}$ in nonnegative integers $r,x$ has only the solution $(r,x)=\left( 1,1\right) .$ \end{corollary}
\begin{corollary} The Diophantine equation $5^{x}-3^{x}=L_{r}$ in nonnegative integers $r,x$ has only the solution $(r,x)=\left( 0,1\right) .$ \end{corollary}
\begin{corollary} The Diophantine equation $8^{x}-5^{x}=L_{r}$ in nonnegative integers $r,x$ has only the solution $(r,x)=\left( 2,1\right) .$ \end{corollary}
\begin{corollary} The Diophantine equation $3^{x}-1=L_{r}$ in nonnegative integers $r,x$ has only the solution $(r,x)=\left( 0,1\right) .$ \end{corollary}
\begin{corollary} The Diophantine equation $5^{x}-2^{x}=L_{r}$ in nonnegative integers $r,x$ has only the solution $(r,x)=\left( 2,1\right) .$ \end{corollary}
\begin{corollary} The Diophantine equation $5^{x}-1=L_{r}$ in nonnegative integers $r,x$ has only the solution $(r,x)=\left( 3,1\right) .$ \end{corollary}
\section{Concluding Remarks}
We were not able to solve Diophantine equation (\ref{1.4}) for $n>2m+4.$ But we conjecture that the Diophantine equation (\ref{1.4}) has no solutions in nonnegative integers $n,m,r,$ and $x$ when $n>2m+4.$ We think the following conjecture is true and a computer search with Mathematica enables us to give it.
\begin{conjecture} The Diophantine equation (\ref{1.4}) with $(n,m)\neq (1,0),(2,0)$ has no solutions for $x\geq 4.$ \end{conjecture}
\end{document} | arXiv |
Proof of why BODMAS (or BIDMAS) works?
In my first full-time teaching post, it is very likely that I'll need to be teaching a small amount of GCSE Mathematics to students retaking it. One thing that has been bugging me is that I can't seem to find any sort of "proof" or explanation of why the BODMAS (PEMDAS, for Americans) rule for doing calculations works. I learned this rule in school, and applying it is second nature. It's obvious to me that multiplication precedes addition, and brackets proceeds powers, via elementary properties of real numbers.
What would be a good way of explaining this rule to students who hadn't seen it before, and does anyone have any understanding-focused ways of explaining why calculations are ordered this way?
notation arithmetic-operations calculations
JesseTG
omegaSQU4REDomegaSQU4RED
$\begingroup$ It doesn't really have a reason for working; it is just a convention to resolve ambiguities. (It does make writing polynomials a bit easier.) There are less ambiguous notations available, like Reverse Polish Notation, but they aren't in common use. $\endgroup$ – Adam May 8 '16 at 19:24
$\begingroup$ As @Adam says, it's a convention so that we could write operations without so many brackets and still have other people understand what we mean. So in that sense it can't be "proved". $\endgroup$ – DavidButlerUofA May 8 '16 at 20:05
$\begingroup$ I think the distinction between conventions and mathematical laws is often blurred in school. How can a student who is taught mathematics instrumentally know that BODMAS is just a convention, while FOIL is a mathematical result that can be proved? $\endgroup$ – Dag Oskar Madsen May 8 '16 at 23:25
$\begingroup$ @ToddWilcox "Orders" as in "$x^4$ is a fourth-order polynomial". $\endgroup$ – R.M. May 9 '16 at 15:14
$\begingroup$ Its always easy and dandy until you have to explain operator precedence.Take for instance this maths question that went viral in Japan iflscience.com/editors-blog/… $\endgroup$ – Gandalf May 10 '16 at 2:30
It's purely a matter of how we choose to define the notation. The main reason for it is that it lets us write polynomial expressions (which are extremely common) without parentheses, e.g., $x^3 + 3x^2 y - 41x + 2z$ rather than $(x^3) + (3(x^2)y) - (41x) + (2z)$.
However, what really matters is that the notation is clear and unambiguous, so expressions like $a/bc$ should be avoided (and replaced with something like $\frac{a}{b} c$ or $\frac{a}{bc}$ depending on what's intended) rather than trying to rigidly adhere to one convention or the other.
Trying to "prove" a notational convention — or treating it as anything more than an agreed-upon way to efficiently communicate meaning — would represent a fundamental misunderstanding of the way mathematical notation works.
Daniel HastDaniel Hast
$\begingroup$ A student who doesn't understand basic arithmetic operations has more pressing issues than not knowing order of operations. Not much point in knowing order of operations for operations you don't understand, is there? $\endgroup$ – Daniel Hast May 9 '16 at 16:47
$\begingroup$ Wait, do you mean $(3(x^2))y$ or $3((x^2)y)$ in your second term? :-) $\endgroup$ – Michael Joyce May 9 '16 at 18:36
$\begingroup$ @MichaelJoyce Also, a couple of extra parentheses to indicate the order of addition and subtraction would be in order $\endgroup$ – Hagen von Eitzen May 9 '16 at 20:22
$\begingroup$ @omegaSQU4RED I've had to face that challenge a few times. The answer is really that most people who are working with equations are working with them at a level where the order of operations proves beneficial, and that at the lower levels you just have to "trust" that it will be a useful detail later. I've found that algebra based physics is the best point to start showing the value of the order of operations, because you see SO many polynomials like 1/2at^2+vt+x that you start to appreciate the value of said groupings. Up until that point, I haven't found many good examples. $\endgroup$ – Cort Ammon - Reinstate Monica May 9 '16 at 20:27
$\begingroup$ @omegaSQU4RED On the other hand, it may be a good introduction to students about the difference between things that are simply conventions and things that are objectively provable (given a set of axioms). This is a concept you want them to understand at some point in their education. Why not sooner rather than later? $\endgroup$ – jpmc26 May 10 '16 at 1:56
Perhaps it is worth pointing out that every programming language defines an operator precedence structure to avoid ambiguities. An example table for C and C++ can be found here. Ambiguities must be avoided in order for the language parser to create the correct compiled (or interpreted) machine code to implement the expression.
For example, the expression $4+3*2$ could be interpreted as either $(4+3)*2=14$ or $4+(3*2)=10$. Operator precedence in most languages follows the mathematics convention of ranking multiplication higher than addition, leading to the second interpretation.
The operator precedence structure goes well beyond arithmetic, dealing with complex expressions such as (in C):
a = b < c ? * p + b * c : 1 << d ()
Here the first * is a unary operator, whereas the second * is a binary operator. (Example from here.)
To build on other answers, you might show how other conventions exist. Use an H.P. calculator for example (postfix), the LISP family of languages (prefix), and the APL language (all right-associative), all of which do not have differing precedence of operators at all, and write expressions in different ways.
Given 4 parallel translations of the same expression, the students may better appreciate that the notation is a communications convention, different from the underlying meaning.
If you think "BODMAS" is a chore to remember, take a look at this chart!
The best way to learn it is not to use a silly nmenonic, but to grasp that the order is a convention adoped because people found it handy in their work. Other answers have pointed out that this comes from polynomials. Just knowing of polynomials you know the convention from that.
Also look how Einstein came up with his own convention which again entails leaving out explicit symbols and groupings and just writing stuff next to each other: in that kind of problem domain, that's a common thing, so the notation can be simplified to that end. Consider this the same idea as the polynomial: in $2x^3$ where are the symbols and grouping notes? And then Direc invented a notation that covers the kind of work he was doing. People will, and continue to, streamline the notation to match the kind of work being done.
I understand this is not a realistic suggestion, but can you avoid "teaching" "PEMDAS" or "BOMDAS" altogether, and teach your students just the math instead? As pretty much everybody already said, this is not actually a rule -- this is a mnemonic device that's supposed to help students remember the actual rules of the order of operations (in the traditional math sense; programming languages and software packages are a whole different story). And then the order of operations is again not as much rules as conventions, intended to simplify notation. The problem I have with "PEMDAS" is that in my experience it does more harm than good. Way too many students follow it literally and actually evaluate 8-2+1 to 5 because "A" in the "PEMDAS" "rule" is before "S".
Of course, this is merely a single example, and not even the worst one, of a much deeper problem when students learn math superficially by rote memorization of rules instead of understanding and internalizing the content.
Disclaimer: as a college math instructor, I'm on the receiving side of what comes out of school education, but I'm not involved in school education.
zipirovichzipirovich
$\begingroup$ I've seen elementary school teachers who actually thought 8-2+1=5 because the A is before the S. Then they would come and tell me that the book had 7 as the answer and was clearly wrong. This is a big problem. $\endgroup$ – Amy B May 14 '16 at 18:40
$\begingroup$ +1 Agree so very much. MadMath, PEMDAS: Exterminate With Extreme Prejudice $\endgroup$ – Daniel R. Collins May 15 '16 at 5:01
$\begingroup$ But the order of operations really is just a convention to be memorized. There's nothing (or little) deeper to understand here - it's just the way we have decided to write mathematics. $\endgroup$ – Austin Mohr May 15 '16 at 17:15
$\begingroup$ The problem with $8-2+1$ is not understanding that $8-2=8+(-2)$. $\endgroup$ – JP McCarthy Mar 6 '19 at 10:32
It works by avoiding the ambiguity that
2 + 3 x 6
would otherwise have. If we simply said we calculate left to right, we'd have a result of 30. With the priority, multiplication higher, we have agreement the above resolves to 20. There's no more complicated origin than this.
JTP - Apologise to MonicaJTP - Apologise to Monica
$\begingroup$ Note that "if we simply said we calculate left to right", there would also be no ambiguity. We would have agreement the above resolves to 30, always. In the Smalltalk programming language, there are no operators, and thus no operator precedence, everything is simply evaluated left to right (+, *, /, etc. are just legal names for functions exactly like foo or multiply), and there is no problem with ambiguity. The point is that there has to be a rule, what the rule is, is largely irrelevant. $\endgroup$ – Jörg W Mittag May 9 '16 at 8:46
It is kind of arbitrary which operation goes first, but my guess is that our current system is just a little more concise in most problems. As an example imagine two systems: a system A that is like our current system, multiplication goes before addition and the multiplication sign can be left out and a system B where addition goes first and the addition sign can be left out.
First imagine you want to multiply a bunch of numbers and then add $x$ to it.
In system A it would look like this:
$$a\times b\times c\times d+x=abcd+x$$ and in system B it would look like this: $$(a\times b\times c\times d)+x=(a\times b\times c\times d)x$$ The parentheses need to be added, because otherwise $x$ would be added to $d$ before being multiplied. Clearly system A wins in this example, but look what happens when the numbers are added first and then multiplied by $x$.
System A: $$(a+b+c+d)x$$ System B: $$abcd\times x$$ This time system B wins, so it appears it just depends on the situation. Also note that system B looks quite ugly, but only because we are so used to system A. System A still has a couple of tricks up its sleeve though, due to distributivity the second example can be expanded whereas in system B there is no such thing possible. $$(a+b+c+d)x=ax+bx+cx+dx$$ And lastly polynomials (as mentioned in other answers) just don't work as well in system B.
System A:$$f(x)=ax^2+bx+c$$ System B:$$f(x)=(a\times x^2)(b\times x)c$$
BramBram
$\begingroup$ Distributivity still works fine in both systems. You've only changed the notation, not the operators themselves, so it has to work the same — we're just not used to the alternate notation. $\endgroup$ – Daniel Hast May 10 '16 at 17:13
You can prove certain properties of the operations, such as commutativity and associativity. From that we can identify that there are ambiguities in standard notation that need to be resolved. However, the resolution of those ambiguities could be entirely arbitrary, there is no proof for any particular resolution as it is merely a selection, a convention. It may be worth discussing the reasoning behind the convention, though.
Brackets take highest priority by virtue of enabling overriding of the convention. If they did not take first precedence they would be ineffective.
Exponentiation (ordinals) is next due to being the strongest operator.
Multiplication and division are of the same relative strength, however, multiplication is commutative whereas division is not, so we give multiplication priority.
Addition and subtraction are last in strength, but again we prioritize addition over subtraction due to its commutativity.
With these heuristics in place, we can guess at where other operators might fall in order of precedence, for example roots are the same strength as exponents, whereas a factorial we would prioritize above exponents.
DanikovDanikov
$\begingroup$ Addition is not prioritized over subtraction. For example, $1 - 2 + 3$ is interpreted as $(1 - 2) + 3$, not as $1 - (2 + 3)$. Think of this instead as $1 + (-2) + 3$, so only addition is involved and the order of evaluation doesn't matter (by associativity). Likewise for multiplication and division, though there's the subtle ambiguity (mentioned in my answer) that multiplication by juxtaposition is often (but not always) taken to have higher priority than division with a horizontally written operator (such as / or ÷). $\endgroup$ – Daniel Hast May 9 '16 at 17:09
$\begingroup$ How do you propose to prove associativity or commutativity of operations? $\endgroup$ – Jessica B May 10 '16 at 6:06
$\begingroup$ @JessicaB at GCSE level, by example? 2 + 4 = 4 + 2 but 2 - 4 != 4 - 2. (12 * 6) * 3 = 12 * (6 * 3) but (12 / 6) / 3 != 12 / (6 / 3). And so on. Maybe there's a nice formal proof, but they're probably keen to get on with the good stuff :) $\endgroup$ – Andy Mortimer May 10 '16 at 10:59
$\begingroup$ For nonnegative reals, one can define addition as length of a concatenation of line segments and multiplication as area of a rectangle — this perhaps isn't fully rigorous but is enough to use as a basis for reasoning for most questions at this level — and then commutativity and associativity are easy to see. This can be extended without too much trouble to negative reals as well. $\endgroup$ – Daniel Hast May 10 '16 at 17:10
It is convention, but is it arbitrary?
Let us consider just multiplication and addition.
Suppose we are in a shop and buy 4 tins of beans, 3 loaves of bread, and 2kg of cheese. The beans are £0.50/tin, the bread is £1/loaf, and the cheese £1.50/kg.
So that is $4\times0.50+3\times1+2\times1.5$
Or if addition has precedence, or we evaluate left to right, then we need brackets. $(4\times0.50)+(3\times1)+(2\times1.5)$
Thus on a simple calculator that evaluates left to right, and has no brackets, or memory feature (as attached to the trolleys at my local supermarket), it is impossible to calculate the sum of you shop (unless you do multiplication as repeated addition, this is ok if you only by a few of any item, and not good for imprecisely cut cheese.)
Additionally I do not believe that there is a use case, where left to right evaluation does better.
ctrl-alt-delorctrl-alt-delor
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ST/H001042/1 01/10/2009 31/03/2011 £1,088,256
ST/H001042/2 Transfer ST/H001042/1 01/10/2010 30/09/2012 £2,686,975
Impact Summary
Description We have discovered the Higgs Boson the fundamental scalar boson that is predicted to give mass to all other particles.
Exploitation Route Further research is required to establish if this is the Higgs Boson or if it is one of many (possibly Supersymmetric) Higgs Bosons.
Sectors Education
URL https://twiki.cern.ch/twiki/bin/view/AtlasPublic
Description The discovery of the Higgs Boson captured the imagination of millions of people. It will lead to an increased interest in science among the general public and lead to more students studying science at University.
First Year Of Impact 2012
Sector Education
Impact Types Societal | CommonCrawl |
Codeforces Round #519 by Botan Investments
string suffix structures
G. Speckled Band
Ildar took a band (a thin strip of cloth) and colored it. Formally, the band has $$$n$$$ cells, each of them is colored into one of $$$26$$$ colors, so we can denote each color with one of the lowercase letters of English alphabet.
Ildar decided to take some segment of the band $$$[l, r]$$$ ($$$1 \le l \le r \le n$$$) he likes and cut it from the band. So he will create a new band that can be represented as a string $$$t = s_l s_{l+1} \ldots s_r$$$.
After that Ildar will play the following game: he cuts the band $$$t$$$ into some new bands and counts the number of different bands among them. Formally, Ildar chooses $$$1 \le k \le |t|$$$ indexes $$$1 \le i_1 < i_2 < \ldots < i_k = |t|$$$ and cuts $$$t$$$ to $$$k$$$ bands-strings $$$t_1 t_2 \ldots t_{i_1}, t_{i_1 + 1} \ldots t_{i_2}, \ldots, {t_{i_{k-1} + 1}} \ldots t_{i_k}$$$ and counts the number of different bands among them. He wants to know the minimal possible number of different bands he can get under the constraint that at least one band repeats at least two times. The result of the game is this number. If it is impossible to cut $$$t$$$ in such a way, the result of the game is -1.
Unfortunately Ildar hasn't yet decided which segment he likes, but he has $$$q$$$ segments-candidates $$$[l_1, r_1]$$$, $$$[l_2, r_2]$$$, ..., $$$[l_q, r_q]$$$. Your task is to calculate the result of the game for each of them.
The first line contains one integer $$$n$$$ ($$$1 \le n \le 200\,000$$$) — the length of the band Ildar has.
The second line contains a string $$$s$$$ consisting of $$$n$$$ lowercase English letters — the band Ildar has.
The third line contains a single integer $$$q$$$ ($$$1 \le q \le 200\,000$$$) — the number of segments Ildar has chosen as candidates.
Each of the next $$$q$$$ lines contains two integer integers $$$l_i$$$ and $$$r_i$$$ ($$$1 \le l_i \le r_i \le n$$$) denoting the ends of the $$$i$$$-th segment.
Output $$$q$$$ lines, where the $$$i$$$-th of them should contain the result of the game on the segment $$$[l_i, r_i]$$$.
abcabcdce
Consider the first example.
If Ildar chooses the segment $$$[1, 6]$$$, he cuts a string $$$t = abcabc$$$. If he cuts $$$t$$$ into two bands $$$abc$$$ and $$$abc$$$, the band $$$abc$$$ repeats two times and the number of different tapes is $$$1$$$. So, the result of this game is $$$1$$$.
If Ildar chooses the segment $$$[4, 7]$$$, he cuts a string $$$t = abcd$$$. It is impossible to cut this band in such a way that there is at least one band repeating at least two times. So, the result of this game is $$$-1$$$.
If Ildar chooses the segment $$$[3, 6]$$$, he cuts a string $$$t = cabc$$$. If he cuts $$$t$$$ into three bands $$$c$$$, $$$ab$$$ and $$$c$$$, the band $$$c$$$ repeats two times and the number of different bands is $$$2$$$. So, the result of this game is $$$2$$$. | CommonCrawl |
\begin{document}
\author{Valentin Blomer} \author{Andrew Corbett}
\address{Mathematisches Institut, Endenicher Allee 60, 53115 Bonn} \email{[email protected]}
\address{The Innovation Centre, University Of Exeter, Exeter, UK, EX4 4RN}
\email{[email protected]}
\title{A symplectic restriction problem}
\thanks{The first author was supported in part by the DFG-SNF lead agency program grant BL 915/2-2}
\begin{abstract}
We investigate the norm of a degree 2 Siegel modular form of asymptotically large weight whose argument is restricted to the 3-dimensional subspace of its imaginary part. On average over Saito--Kurokawa lifts an asymptotic formula is established that is consistent with the mass equidistribution conjecture on the Siegel upper half space as well as the Lindel\"of hypothesis for the corresponding Koecher--Maa{\ss} series. The ingredients include a new relative trace formula for pairs of Heegner periods.
\end{abstract}
\subjclass[2010]{Primary: 11F37, 11F55, 11F67, 11F72} \keywords{restriction norm, Koecher--Maa{\ss} series, Saito--Kurokawa lift, Heegner points, relative trace formula}
\setcounter{tocdepth}{2} \maketitle
\maketitle
\section{Introduction}
\subsection{Restriction norm of eigenfunctions} The question, `to what extent can the mass of a Laplace eigenfunction $\phi$ on a Riemannian manifold $X$ localize?', is a classical problem in analysis and is often quantified by upper (or lower) bounds for $L^p$-bounds for the restriction of $\phi$ to suitable submanifolds $Y \subseteq X$. The prototypical example is the case where $X$ is a surface and $Y$ is a curve, often a geodesic; see e.g.\ \cite{B, BGT, CS, GRS, Ma2, R1, R2} and references therein.
If $X$ is the quotient of a symmetric space by an arithmetic lattice (often called an arithmetic manifold), an additional layer of number theoretic structure enters. Not only can this be used to obtain stronger bounds \cite{Ma1}, but sometimes the period integrals can be expressed in terms of special values of $L$-functions. A typical such case is the $L^2$-restriction of a Maa{\ss} form for the group ${\rm SL}_{n+1}(\mathbb{Z})$ to its upper left $n$-by-$n$ block, which can be expressed as an average of central values of ${\rm GL}(n) \times {\rm GL}(n+1)$ Rankin--Selberg $L$-functions \cite{LY, LLY}. Other potential cases arise from the Gross-Prasad conjecture \cite{II}.
Often in this context, optimal restriction norm bounds are equivalent to the Lindel\"of conjecture on average over the spectral family of $L$-functions in question.
In this paper we consider certain Siegel modular forms $F$ for the symplectic group ${\rm Sp}_4(\mathbb{Z})$: we investigate the $L^2$-restriction of a Saito--Kurokawa lift $F(Z)$ on the 6-dimensional Siegel upper half space $\mathbb{H}^{(2)}$ to the 3-dimensional subspace where the argument $Z = X+iY$ is restricted to its imaginary part. This is a very natural set-up, it is a direct higher-dimensional analogue of the classical problem of bounding a cusp form $f$ for ${\rm SL}_2(\mathbb{Z})$ on the vertical geodesic, mentioned at the beginning; cf.\ \cite[Section 7]{BKY}. While the latter leads, via Hecke's integral representation, directly to the corresponding $L$-function $L( f, s)$, things become much more involved for Siegel modular forms.
We start by stating the corresponding period formula. For an even positive integer $k$ let $S_k^{(2)}$ denote the space of Siegel modular forms of degree 2 of weight $k$ for the group ${\rm Sp}_4(\mathbb{Z})$, equipped with the standard Petersson inner product; see Section \ref{sec2}. We think of $k$ as tending to infinity and are interested in asymptotic results with respect to $k$. We restrict the argument of a cusp form $F\in S_k^{(2)}$ to its imaginary part $iY$ with $Y \in \mathcal{P}(\mathbb{R})$ where $\mathcal{P}(\mathbb{R})$, equipped with the measure $dY/(\det Y)^{3/2}$, is the set of positive definite symmetric $2$-by-$2$ matrices. Consider the \emph{restriction norm} \begin{equation}\label{NF}
\mathcal{N}(F) := \frac{\pi^2}{90} \cdot \frac{1}{\| F \|^2_2} \int_{{\rm SL}_2(\mathbb{Z}) \backslash \mathcal{P}(\mathbb{R})} |F(i Y)|^2 (\det Y)^{k} \frac{dY}{(\det Y)^{3/2}}, \end{equation} where ${\rm SL}_2(\mathbb{Z})$ acts on $\mathcal{P}(\mathbb{R})$ by $\gamma \mapsto \gamma^{\top} Y \gamma$. Letting $\mathbb{H}$ denote the usual upper half plane, we observe that $${\rm SL}_2(\mathbb{Z}) \backslash \mathcal{P}(\mathbb{R}) \cong {\rm SL}_2(\mathbb{Z}) \backslash\mathbb{H} \times \mathbb{R}_{>0}$$ has infinite measure; see \eqref{product} below. The factor $$\frac{\pi^2}{90} = \frac{3}{\pi} \cdot \frac{\pi^3}{270} = \frac{\text{vol}({\rm Sp}_4(\mathbb{Z}) \backslash \mathbb{H}^{(2)}) }{ \text{vol} ({\rm SL}_2(\mathbb{Z}) \backslash \mathbb{H})}$$ accounts for the fact that, in accordance with the literature, we choose the standard measures on ${\rm Sp}_4(\mathbb{Z}) \backslash \mathbb{H}^{(2)}$ and ${\rm SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ which are not probability measures.
Let $\Lambda$ denote a set of spectral components of $L^2({\rm SL}_2(\mathbb{Z})\backslash \mathbb{H})$ consisting of the constant function $\sqrt{3/\pi}$, an orthonormal basis of Hecke--Maa{\ss} cusp forms and the Eisenstein series $E(., 1/2 + it)$ for $t \in \mathbb{R}$. The set $\Lambda$ is equipped with the counting measure on its discrete part and with the measure $dt/4\pi$ on its continuous part. We denote by $\int_{\Lambda}$ the corresponding combined sum/integral. For $F \in S_k^{(2)}$ and ${\tt u} \in \Lambda$ let $L(F \times {\tt u}, s)$ denote the Koecher--Maa{\ss} series defined in \eqref{KM}. This series has a functional equation featuring the gamma factors $G(F \times {\tt u}, s)$ as defined in \eqref{G-KM}, but has \emph{no} Euler product. The following proposition is proved in Section \ref{sec6}.
\begin{prop}\label{prop1} For $F \in S_k^{(2)}$ with even $k$ we have
\begin{displaymath}
\begin{split}
& \mathcal{N}(F) = \frac{\pi^2}{90 }\cdot \frac{1}{32} \cdot \frac{1}{ \| F \|^2_2} \int_{-\infty}^{\infty} \int_{\Lambda_{\text{\rm ev}}} |G(F \times \overline{{\tt u}}, 1/2 + it) L(F \times \overline{{\tt u}}, 1/2 + it) |^2 d{\tt u} \, dt ,
\end{split}
\end{displaymath}
where $\Lambda_{\text{\rm ev}}$ denotes the set of all even ${\tt u} \in \Lambda$.
\end{prop}
An interesting subfamily of Siegel modular forms are the Saito--Kurokawa lifts $F_h$ (sometimes called the \emph{Maa{\ss} Spezialschar}) of half-integral weight modular forms $h \in S^+_{k-1/2}(4)$ in Kohnen's plus-space or equivalently their Shimura lifts $f_h \in S_{2k-2}$ (see Section \ref{sec2} for details). In this case, the Koecher--Maa{\ss} series $L(F_h \times{\tt u}, s)$ roughly becomes a Rankin--Selberg $L$-function of two half-integral weight cusp forms, namely of $h$ and the weight $1/2$ automorphic form whose Shimura lift equals ${\tt u}$; see Proposition \ref{prop2} below and cf.\ \cite{DI}. Of course, this series also has no Euler product. The convexity bound for $L$-functions along with trivial bounds implies $$\mathcal{N}(F_h) \ll k^{2+\varepsilon},$$ whilst the statement \begin{equation}\label{1} \mathcal{N}(F_h) \ll k^{\varepsilon} \end{equation} would follow from the Lindel\"of hypothesis for these $L$-functions. It should be noted, however, that in absence of an Euler product it is not expected that these $L$-functions satisfy the Riemann hypothesis, but one may still hope that the Lindel\"of hypothesis is true; see \cite{Ki} for some support of this conjecture. However, even if it is, then proving \eqref{1} appears to be far out of reach by current technology -- it corresponds to an average of size $k^{3/2}$ of a family of $L$-functions of conductor $k^8$. This is analogous to the genus 1 situation in which the $L^2$-restriction norm of a holomorphic cusp form of weight $k$ leads to an average of size $k^{1/2}$ of a family of $L$-functions of conductor $k^4$; see \cite[(1.12)]{BKY}. These problems belong to the hard cases where sharp bounds for the $L^2$-restriction norm imply very strong subconvexity bounds.
A different symplectic restriction problem was treated in \cite{LiuY} and \cite{BKY}, where the argument $Z \in \mathbb{H}^{(2)}$
of Saito-Kurokawa lifts was restricted to the diagonal, a four-dimensional subspace of $\mathbb{H}^{(2)}$. The corresponding analogue of Proposition \ref{prop1}, due to Ichino \cite{Ic}, leads to an average of size $k$ of $L$-functions
of conductor $k^4$.
\subsection{The main result and mass equidistribution in higher rank} Fix a smooth, non-negative test function $W$ with non-empty support in $[1, 2]$. Let $\omega := \int_1^2 W(x) x\, dx$ and consider \begin{equation}\label{Nav} \mathcal{N}_{\text{av}}(K) := \frac{1}{\omega} \cdot \frac{12}{ K^2} \cdot \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \sum_{h \in B_{k-1/2}^+(4)} \mathcal{N}(F_h) \end{equation} for a large parameter $K$ and a Hecke eigenbasis $B_{k-1/2}^+(4)$ of $S_{k-1/2}^+(4)$. Note that $\dim S_{k-1/2}^+(4) \sim k/6$. The first main result of this paper is the following asymptotic formula. \begin{theorem}\label{thm1} We have $\mathcal{N}_{\text{{\rm av}}}(K) = 4\log K + O(1)$ as $K \rightarrow \infty$. \end{theorem}
This may be interpreted as an average version of the Lindel\"of hypothesis for twisted Koecher--Maa{\ss} series. This restriction problem, however, is structurally quite different from all previously considered restriction problems with connections to $L$-functions, as the period formula features $L$-functions that are not in the Selberg class and the restriction norm does not remain bounded.
More importantly, there is a strong connection between Theorem \ref{thm1} and the \textit{mass equidistribution conjecture} that we now explain. Let $g$ be a test function on ${\rm Sp}_4(\mathbb{Z}) \backslash \mathbb{H}^{(2)}$. Then the (arithmetic) mass equidistribution conjecture for the Siegel upper half space states that
$$ \frac{1}{\| F \|^2} \int_{{\rm Sp}_4(\mathbb{Z}) \backslash \mathbb{H}^{(2)}} g(Z) |F(Z)|^2 (\det Y)^{k} \frac{dX\, dY}{(\det Y)^{3}} \longrightarrow \int_{{\rm Sp}_4(\mathbb{Z}) \backslash \mathbb{H}^{(2)}} g(Z) \frac{dX\, dY}{(\det Y)^{3}} $$ as $F$ traverses a sequence of Hecke--Siegel cusp forms of growing weight. While the corresponding statement for classical cusp forms of degree 1 was proved by Holowinsky and Soundararajan \cite{HS}, no such statement has been obtained for Siegel modular forms of higher degree (but see \cite{SV} for certain cases of the quantum unique ergodicity conjecture in higher rank). Nevertheless, one may even go one step further and conjecture that the above limit holds when one restricts the full space ${\rm Sp}_4(\mathbb{Z}) \backslash \mathbb{H}^{(2)}$ to a submanifold. In particular, one might conjecture that
$$ \frac{\text{vol}({\rm Sp}_4(\mathbb{Z}) \backslash \mathbb{H}^{(2)}) }{\| F \|^2} \int_{{\rm SL}_2(\mathbb{Z}) \backslash \mathcal{P}(\mathbb{R})} g(Y) |F(iY)|^2 (\det Y)^{k} \frac{ dY}{(\det Y)^{3/2}} \longrightarrow\int_{{\rm SL}_2(\mathbb{Z}) \backslash \mathcal{P}(\mathbb{R})} g(Y) \frac{ dY}{(\det Y)^{3/2}} $$
holds. As the right hand side has infinite measure, we cannot simply replace $g$ with the constant function. This is precisely the reason why $\mathcal{N}_{\text{av}}(K)$ is unbounded as $K \rightarrow \infty$. However, since $F$ is a cusp form, the $L^2$-normalized and ${\rm Sp}_4(\mathbb{Z})$-invariant function $|F(iY)|^2 (\det Y)^{k}/\| F \|^2$ decays exponentially quickly if $Y$ is (in a precise sense) very large or very small. So effectively $g$ may be restricted to the characteristic function of a compact set depending on $k$. We quantify this in Appendix \ref{appc} and show that, for such $g$, the right hand side equals $$\text{vol}({\rm SL}_2(\mathbb{Z})\backslash \mathbb{H})\cdot 4 \log k + O(1).$$ In this case the previous asymptotic reads
\begin{equation}\label{vol}
\mathcal{N}(F) \sim 4\log k
\end{equation}
as $k \rightarrow \infty$. The asymptotic \eqref{vol} is, of course, highly conjectural, and as mentioned above even the ordinary mass equidistribution conjecture (without restricting to a thin subset) is currently out of reach. Theorem \ref{thm1} provides an unconditional proof of \eqref{vol} on average over Saito--Kurokawa lifts in agreement with the mass equidistribution conjecture. In particular, the constant 4 in Theorem \ref{thm1} is very relevant, and this constant has a story of its own. It is the outcome of several archimedean integrals, numerical values in period formulae and a gigantic Euler product whose special value can be expressed in terms of zeta values (cf.\ \eqref{euler}). In deducing its value we have corrected several numerical constants in the literature. We shall come back to this point in due course; see for instance the remark after Lemma \ref{lem3}. The authors would like to thank Gergely Harcos for useful and clarifying discussions in this respect.
Theorem \ref{thm1} opens the door for several other related problems. The reader may wonder what happens for generic, i.e.\ non-CAP Siegel modular forms. Any reasonable spectral average would include at least the space of Siegel modular forms $S_k^{(2)}$ of weight $k$ which is of dimension $\sim ck^{3}$ for some constant $c$ (in fact $c = 1/8640$). This leads to a bigger average than the one presently considered over about $k^2$ Saito-Kurokawa lifts. The starting point for the $L^2$-restriction norm of generic Siegel modular forms is again the period formula in Proposition \ref{prop1}. Coupled with an approximate functional equation (as in Lemma \ref{approx1}), this is amenable to the Kitaoka-Petersson formula \cite{Kit} and an analysis along the lines of \cite{Bl2}. We hope to return to this interesting problem soon.
Whilst the proof of Theorem \ref{thm1} rests on many ingredients, to which we address in detail in the coming sections, there are a few highlights which may be of stand alone interest. We describe these in the remainder of the introduction.
\subsection{A relative trace formula for pairs of Heegner periods}
Here we focus on a novel trace formula of independent interest beyond its application in proving Theorem \ref{thm1}. Let $D$ be a discriminant, i.e.\ a non-square integer $\equiv 0, 1$ (mod 4). For a discriminant $D < 0$ let
$H_D \subseteq {\rm SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ denote the set of all Heegner points; that is, the set of all $z = (\sqrt{|D|}i - B)/(2A)$ where $AX^2 + BXY + CY^2$ is a $\Gamma$-equivalence class of integral quadratic forms of discriminant $D = B^2 - 4AC$. For a function $f : {\rm SL}_2(\mathbb{Z}) \backslash \mathbb{H} \rightarrow \mathbb{C}$ define the period \begin{equation}\label{defP} P(D; f) = \sum_{z \in H_D} \frac{f(z)}{\epsilon(z)}
\end{equation} where $\epsilon(z) \in \{1, 2, 3\}$ is the order of the stabilizer of $z$ in ${\rm PSL}_2(\mathbb{Z})$. Its counterparts for positive discriminants $D$ are periods over geodesic cycles. These periods are classical objects with myriad interwoven connections to half-integral weight modular forms, base change $L$-functions, quadratic fields and quadratic forms. An interesting special case is the constant function $f = 1$ in which case $P(D; 1) = H(D)$ is, by definition, the Hurwitz class number.
With applications to the above mentioned symplectic restriction problem in mind, we are interested in pairs of Heegner periods in the spectral average $$ \int_{\Lambda_{\text{ev}}} P(D_1; {\tt u}) \overline{P(D_2; {\tt u})} h(t_{\tt u}) d{\tt u} $$ for a suitable test function $h$ and two discriminants $D_1, D_2 < 0$. While pairs of geodesics have been studied in a few situations \cite{Pi1, Pi2, MMW}, to the best of our knowledge, nothing seems to be known about spectral averages of pairs of Heegner periods. Opening the sums in the definition of $P(D_1; {\tt u})$ and $P(D_2; {\tt u})$, this can be expressed as a double sum of an automorphic kernel $$\sum_{z_1 \in H_{D_1}}\sum_{z_2 \in H_{D_2}} \frac{1}{\epsilon(z_1)\epsilon(z_2)}\sum_{\gamma \in \Gamma} k(z_1, \gamma z_2)$$ in the usual notation which resembles the set-up of a relative trace formula. However, the standard methods in this situation (e.g.\ \cite{Go}) do not easily apply here as the stabilizers of $z_1$ and $z_2$ are essentially trivial. We thus take a different approach to establish the following relative trace formula for which we need some notation. For $n >0$ and $t \in \mathbb{R}$ let \begin{equation*} W_t( n) := \frac{1}{2\pi i} \int_{(2)} \frac{\Gamma(\frac{1}{2}(\frac{1}{2} + s + 2it))\Gamma(\frac{1}{2}(\frac{1}{2} + s -2 it))}{\Gamma( \frac{1}{4} + it)\Gamma( \frac{1}{4} -it)\pi^s} e^{s^2} n^{-s} \frac{ds}{s}. \end{equation*}
For $t\in \mathbb{R}$, $x > 0$ and $\kappa\in \mathbb{R}$ let \begin{equation}\label{defF} F(x, t, \kappa) = J_{it}(x) \cos(\pi\kappa/2 - \pi i t/2) - J_{-it}(x) \cos(\pi\kappa/2 + \pi i t/2) \end{equation} where $J_{it}(x)$ is the Bessel function. Finally, for $\kappa \in \mathbb{Z} + 1/2$, $n, m \in \mathbb{Z}$ and $c \in \mathbb{N}$ define the modified Kloosterman sums \begin{equation}\label{defKlo} K^+_{\kappa}(m, n, c) = \sum_{\substack{d\, (\text{mod }c)\\ (d, c) = 1}} \epsilon^{2\kappa}_d \left(\frac{c}{d}\right) e\left(\frac{md + n\bar{d}}{c}\right) \cdot \begin{cases} 0, & 4 \nmid c,\\ 2, & 4 \mid c, \, 8 \nmid c,\\ 1, & 8 \mid c, \end{cases} \end{equation} where \begin{equation}\label{epsd}
\epsilon_d = \begin{cases} 1, & d \equiv 1\, ({\rm mod }\,4), \\ i, & d \equiv 3\, ({\rm mod }\, 4).\end{cases}
\end{equation} Note that $K^+_{\kappa}(n, m, c)$ is symmetric in $m$ and $n$ and $2$-periodic in $\kappa$. They satisfy the Weil-type bound \begin{equation}\label{weil} K_{\kappa}^+(m, n, c) \ll c^{1/2 + \varepsilon}(m, n, c)^{1/2}, \end{equation} see e.g.\ \cite[Lemma 4]{Wa} in the case $n=m$, the general case being analogous. In order to simplify the notation we assume that $D_1, D_2$ are fundamental discriminants. In Section \ref{secproof2} we state the general version for arbitrary negative discriminants.
\begin{theorem}\label{thm2} Let $\Delta_1, \Delta_2$ be negative fundamental discriminants and let $h$ be an even function, holomorphic in $|\Im t | < 2/3$ with $h(t) \ll (1+|t|)^{-10}$. Then \begin{displaymath} \begin{split}
&\frac{1}{|\Delta_1\Delta_2|^{1/4}}\int_{\Lambda_{\text{{\rm ev}}}} P(\Delta_1; {\tt u}) \overline{P(\Delta_2; {\tt u})} h(t_{\tt u}) d{\tt u} = \frac{3}{\pi} \frac{H(\Delta_1)H(\Delta_2)}{|\Delta_1\Delta_2|^{1/4}} h(i/2)\\
& \quad + \int_{-\infty}^{\infty} \Big|\frac{\Delta_1\Delta_2}{4}\Big|^{ it/2} \frac{ \Gamma(-\frac{1}{4} + \frac{it}{2}) e^{(1/2- it)^2}}{ \sqrt{8\pi} \Gamma(\frac{1}{4} + \frac{i t}{2})} \frac{L( \chi_{\Delta_1}, 1/2 + it)L(\chi_{\Delta_2}, 1/2 + it)}{ \zeta(1+ 2it)} h(t) \frac{dt}{4\pi}\\ & \quad + \delta_{\Delta_1= \Delta_2} \sum_m\frac{\chi_{\Delta_1}(m)}{m} \int_{-\infty}^{\infty} W_{t}(m) h(t) t \tanh(\pi t) \frac{dt}{4\pi^2} \\
&\quad + e(3/8) \sum_{n, c, m} \frac{ K_{3/2}^+(|\Delta_1| n^2, |\Delta_2|, c)\chi_{\Delta_1}(m)}{ n^{1/2}cm} \int_{-\infty}^{\infty} \frac{F(4\pi n\sqrt{|\Delta_1\Delta_2|}/c, t, 1/2)}{\cosh(\pi t)} h(t) W_{t}(nm) t \frac{dt}{\pi}. \end{split} \end{displaymath}
\end{theorem} The experienced reader will spot the strategy of the proof from the shape of the formula: A Katok-Sarnak-type formula translates $P(\Delta; {\tt u})$ into a product of a first and a $\Delta$-th half-integral weight Fourier coefficient. In this way, a pair of two Heegner periods becomes a product of \emph{four} half-integral weight Fourier coefficients. A quadrilinear form of half-integral weight Fourier coefficients is not directly amenable to any known spectral summation formula, but we can use a Waldspurger-type formula a second time, now in the other direction, to translate the two first coefficients into a central $L$-value. This $L$-value can be written explicitly as a sum of Hecke eigenvalues by an approximate functional equation. We can now use the correspondence between half-integral and integral weight forms a \emph{third} time, namely by combining the Hecke eigenvalues into the half-integral weight coefficients by means of metaplectic Hecke relations. Finally, the Kuznetsov formula for the Kohnen plus space provides the desired geometric evaluation of the relative trace. This particular version of the Kuznetsov formula is also new and will be stated and proved in Section \ref{summation}.
\subsection{Mean values of \texorpdfstring{$L$}{L}-functions} We highlight another ingredient of independent interest. This is a hybrid Lindel\"of-on-average bound for central values of twisted $L$-functions, its proof is deferred to Section \ref{proof}. \begin{prop}\label{Lfunc} Let $\mathcal{D}, \mathcal{T} \geqslant 1$ and $\varepsilon > 0$. For a fundamental discriminant $\Delta$ let $\chi_{\Delta} = (\frac{\Delta}{.})$ be the Jacobi--Kronecker symbol.
{\rm (a)} We have
$$\sum_{t_{ u} \leqslant \mathcal{T}}\alpha(u) \sum_{\substack{|\Delta| \leqslant \mathcal{D}\\ \Delta \text{ {\rm fund. discr.}}}} L({u} \times \chi_{\Delta}, 1/2) \ll \Big(\sum_{t_{ u} \leqslant \mathcal{T}}|\alpha(u)|^2\Big)^{1/2} (\mathcal{T}\mathcal{D})^{1+\varepsilon}$$ where the sum is over an orthonormal basis of Hecke--Maa{\ss} cusp forms ${u}$ with spectral parameter $t_{u}$, and $\alpha(u)$ is any sequence of complex numbers, indexed by Maa{\ss} forms.
{\rm (b)} We have
$$\int_{-T}^T \alpha(t) \sum_{\substack{|\Delta| \leqslant \mathcal{D}\\ \Delta \text{ {\rm fund. discr.}}}} |L( \chi_{\Delta}, 1/2 + it)|^2 dt \ll \Big(\int_{-T}^T|\alpha(t)|^2 dt\Big)^{1/2} (\mathcal{T}^{1/2}\mathcal{D})^{1+\varepsilon}$$ for an arbitrary function $\alpha : [-T, T] \rightarrow \mathbb{C}$. \end{prop}
The proof uses, among other things, the spectral large sieve of Deshouillers-Iwaniec \cite{DesIw} and Heath-Brown's large sieve for quadratic characters \cite{HB}. Note that $L({ u} \times \chi_{\Delta}, 1/2) \geqslant 0$ is non-negative \cite[Corollary 1]{KS}. The key point here is that there is complete uniformity in $\mathcal{T}$ and $\mathcal{D}$. We give an immediate application. Let us choose $\alpha(u) = L(u, 1/2)$ and note that
$L(u, 1/2) L(u \times \chi_{\Delta}, 1/2) = L({\rm BC}_{K}(u), 1/2)$ where the right hand side is the base change $L$-function to $K = \mathbb{Q}(\sqrt{\Delta})$. We can now use a standard mean value bound for $L(u, 1/2)$, e.g.\ \cite[Theorem 3]{Iw} to conclude \begin{cor}\label{Lfunc-cor} For $\mathcal{T}, \mathcal{D} \geqslant 1$ and $\varepsilon > 0$ we have
$$\sum_{t_{ u} \leqslant \mathcal{T}} \sum_{\substack{\deg K/\mathbb{Q} = 2\\ |\text{{\rm disc}}(K)| \leqslant \mathcal{D}}} L({\rm BC}_{K}(u), 1/2) \ll (\mathcal{T}^2\mathcal{D})^{1+\varepsilon}$$ where the first sum runs over a basis of Hecke--Maa{\ss} cusp forms $u$ with spectral parameter $t_u \leqslant \mathcal{T}$. \end{cor}
Again this bound is completely uniform and best-possible in the $\mathcal{D}$ and $\mathcal{T}$ aspect. We give another interpretation of Proposition \ref{Lfunc}(a). For odd $u$, the root number of $L(u \times \chi_{\Delta}, s)$ is $-1$ (see \cite[Lemma 2.1]{BFKMMS}), so the central value vanishes. For even $u$ the central $L$-values $L(u \times \chi_{\Delta}, 1/2)$, as in \eqref{BarMao} below, are proportional to squares of Fourier coefficients $b_v(\Delta)$ of weight 1/2 Maa{\ss} forms $v$ in Kohnen's subspace for $\Gamma_0(4)$, normalized as in \eqref{four-half}. We refer to Section \ref{sec4} for the relevant definitions. In particular, for the usual choice of the Whittaker function the normalized Fourier coefficient $\tilde{b}_v(\Delta) = e^{-\pi |t_v|/2} |t_v|^{ \text{sgn}(\Delta)/4} |\Delta|^{1/2} b_v(\Delta) $ is of size one on average. In this way we conclude bounds for linear forms in half-integral weight \emph{Rankin--Selberg coefficients}: \begin{equation}\label{interpret}
\sum_{1/4 \leqslant t_{ v} \leqslant \mathcal{T}}\alpha(v) \sum_{\substack{|\Delta| \leqslant \mathcal{D}\\ \Delta \text{ {\rm fund. discr.}}}} |\tilde{b}_v(\Delta)|^2 \ll \Big(\sum_{t_{ v} \leqslant \mathcal{T}}|\alpha(v)|^2\Big)^{1/2} (\mathcal{T}\mathcal{D})^{1+\varepsilon}, \end{equation} where the $v$-sum runs over an $L^2$-normalized Hecke eigenbasis of non-exceptional weight 1/2 Maa{\ss} forms in Kohnen's subspace for $\Gamma_0(4)$ with spectral parameter $t_v$. We refer to the remark after the proof of Lemma \ref{lem3} for more details.
\subsection{Organization of the paper}\label{15}
Section \ref{sec2} -- \ref{summation} prepare the stage and compile all necessary automorphic information. New results include versions of the half-integral Kuznetsov formula and a Voronoi formula for Hurwitz class numbers. Proposition \ref{prop1}, Theorem \ref{thm2}, and Proposition \ref{Lfunc} are proved in Sections \ref{sec6}, \ref{secproof2}, \ref{proof} respectively. This is followed by an interlude on the analysis of oscillatory integrals. In the remainder we complete the proof of Theorem \ref{thm1}. In Section \ref{weakversion} we first prove an upper bound $\mathcal{N}_{\text{av}}(K) \ll K^{\varepsilon}$ by a preliminary argument. This will be useful to control certain auxiliary variables and error terms later. Due to several applications of certain spectral summation formulae, we have various diagonal and off-diagonal terms. Section \ref{diag-diag} treats the total diagonal term that extracts the leading term $4\log K$ in Theorem \ref{thm1}. Sections \ref{diag-off} and \ref{off-off} deal with the diagonal off-diagonal and the off-off-diagonal term.
\subsection{Common notation}\label{16}
For $ c\not= 0$ we extend the Jacobi-Symbol $\chi_c(d)= (\frac{c}{d})$ for positive odd integers $d $ to all integers $d\not= 0$ as the completely multiplicative function defined by $\chi_c(-1) = \text{sign}( c)$ and $\chi_c(2) = 1$ if $c \equiv 1 \, (\text{mod }8)$, $\chi_c(2) = -1$ if $c \equiv 5 \, (\text{mod }8)$, $\chi_c(2) = 0$ if $c$ is even. The value of $\chi_c(2)$ remains undefined only if $c \equiv 3 \, (\text{mod }4)$.
We call an integer $D \in \mathbb{Z} \setminus \{0\}$ a discriminant if $D \equiv 0$ or $1$ (mod 4). Every discriminant $D$ can uniquely be written as $D = \Delta f^2$ for some $f\in \mathbb{N}$ and some fundamental discriminant $\Delta$ (possibly $\Delta = 1$). For each discriminant $D$, the map $\chi_D$ is a quadratic character of modulus $|D|$ that is induced by the character $\chi_{\Delta}$ corresponding to the field $\mathbb{Q}(\sqrt{\Delta})$. (If $\Delta = 1$, then $\chi_{\Delta}$ is the trivial character.) Throughout, the letters $D$ and $\Delta$ are always reserved for discriminants resp.\ fundamental discriminants, usually negative.
The letter $\Gamma$ is used for the gamma function and also for the group $\Gamma = {\rm SL}_2(\mathbb{Z})$; confusion will not arise. We write $\overline{\Gamma} = {\rm PSL}_2(\mathbb{Z})$.
For $a, b \in \mathbb{N}$ we write $a \mid b^{\infty}$ to mean that all prime divisors of $a$ divide $b$. We also write $(a, b^{\infty}) = a/a_1$ where $a_1$ is the largest divisor of $a$ that is coprime to $b$.
We use the usual exponential notation $e(z):=e^{2\pi i z}$ for $z\in\mathbb{C}$. The letter $\varepsilon$ denotes an arbitrarily small positive constant, not necessarily the same at every occurrence. The Kronecker symbol $\delta_{S}$ takes the value 1 if the statement $S$ is true and $0$ otherwise. The notation $\int_{(\sigma)}$ denotes a complex contour integral over the vertical line with real part $\sigma$. We use the usual Vinogradov symbols $\ll$ and $\gg$, and we use $\asymp$ to mean both $\ll$ and $\gg$. We always assume that the number $K$ in Theorem \ref{thm1} is sufficiently large. \\
\section{Holomorphic forms of degree one and two}\label{sec2}
For a positive integer $k$ let $S^+_{k-1/2}(4)$ denote Kohnen's plus \cite{Ko1} space of holomorphic cusp forms
of weight $k-1/2$ and level 4. These have a Fourier expansion of the form \begin{equation}\label{four} h(z) = \sum_{(-1)^k n \equiv 0, 3\, (\text{mod 4})} c_h(n) e(nz) \end{equation} and form a finite-dimensional Hilbert space with the inner product $$\langle h_1, h_2 \rangle = \int_{\Gamma_0(4) \backslash \mathbb{H}} h_1(z) \overline{h_2(z)} y^{k-1/2} \frac{dx\, dy}{y^2}.$$ This space is isomorphic (as a module of the Hecke algebra) to the space $S_{2k-2}$ of holomorphic cusp forms of weight $2k-2$ and level 1 \cite[Theorem 1]{Ko1}. We denote by $f_h \in S_{2k-2}$ the (unique up to scaling) image of a newform $h \in S^+_{k-1/2}(4)$. The Hecke algebra on $S^+_{k - 1/2}$ is generated by the operators $T(p^2)$, $p$ prime, and for $p=2$ we follow Kohnen's definition \cite[p.\ 250]{Ko1} of $T(4)$ that allows a uniform treatment of all primes including $p=2$.
If $\lambda(p)$ are the Hecke eigenvalues of $f_h$ (normalized so that the Deligne's bound reads $|\lambda(p)| \leqslant 2$), then \begin{equation}\label{fourier-relation} \lambda(p) c_h(n) = p^{3/2 - k} c_h(p^2 n) + p^{-1/2} \chi_{ (-1)^{k+1} n}(p) c_h(n) + p^{k - 3/2} c_h(n/p^2) \end{equation} for all primes $p$ with the convention $c_h(x) = 0$ for $x \not \in\{n \in \mathbb{N} \mid (-1)^k n \equiv 0, 3\, (\text{mod 4}) \}$. Iterating this formula gives \begin{equation}\label{fourier-relation1} \lambda(m) c_h(n) = \sum_{\substack{d_1 \mid d_2 \mid m\\ (d_1d_2)^2 \mid m^2n}} \left(\frac{d_1}{d_2} \right)^{1/2} \chi_{ (-1)^{k+1} n}(d_1d_2) c_h\left( \frac{m^2}{(d_1d_2)^2}n\right) \left(\frac{m}{d_1d_2}\right)^{3/2 - k} \end{equation} for squarefree $m \in \mathbb{N}$.
The space $S^{+}_{k-1/2}(4)$ can be characterized as an eigenspace of a certain operator acting on the space $S_{k-1/2}(4)$ of all holomorphic cusp forms of weight $k-1/2$ and level 4 \cite[Proposition 2]{Ko1}. It possesses Poincar\'e series $P^+_n \in S^+_{k-1/2}(4)$ satisfying the usual relation \cite[(4)]{Ko2} $$\langle h, P^+_n\rangle = \frac{\Gamma(k-3/2)}{(4\pi n)^{k-3/2}} c_h(n)$$ for all $(-1)^k n \equiv 0, 3\, (\text{mod 4})$ and all $h \in S^+_{k-1/2}(4)$ with Fourier expansion \eqref{four}. These Poincar\'e series are the orthogonal projections of the Poincar\'e series $P_n \in S_{k-1/2}(4)$ onto $S^{+}_{k-1/2}(4)$ and their Fourier coefficients are computed explicitly in \cite[Proposition 4]{Ko2}. This gives us the following Petersson formula for Kohnen's plus space.
\begin{lemma}\label{lem1} Let $k \geqslant 3$ be an integer, $\kappa = k-1/2$. Let $\{h_j\}$ be an orthogonal basis of $S^+_{\kappa}(4)$ with Fourier coefficients $c_j(n)$ as in \eqref{four}. Let $n, m$ be positive integers with $(-1)^kn, (-1)^km \equiv 0, 3\, ({\rm mod } \,\,4)$. Then \begin{displaymath} \begin{split}
&\frac{\Gamma(\kappa - 1)}{(4\pi)^{\kappa - 1}} \sum_j \frac{c_j(n) \overline{c_j(m)}}{\| h_j \|^2 ( \sqrt{nm})^{\kappa-1}} = \frac{2}{3}\left(\delta_{m=n} + 2 \pi e(-\kappa/4) \sum_{c} \frac{K^+_{\kappa}(n, m, c)}{c} J_{\kappa - 1}\left(\frac{4\pi \sqrt{mn}}{c}\right)\right). \end{split} \end{displaymath} \end{lemma}
For a positive integer $k$ we denote by $S^{(2)}_k$ the space of Siegel cusp forms of degree 2 of weight $k$ for the symplectic group ${\rm Sp}_4(\mathbb{Z})$ with Fourier expansion \begin{equation}\label{four1} F(Z) = \sum_{T \in \mathcal{P}(\mathbb{Z})} a(T) e(\text{tr}(TZ)) \end{equation} for $Z = X + iY \in \mathbb{H}^{(2)}$ on the Siegel upper half plane, where $\mathcal{P}(\mathbb{Z})$ is the set of symmetric, positive definite 2-by-2 matrices with integral diagonal elements and half-integral off-diagonal elements. This is a finite-dimensional Hilbert space with respect to the inner product
\begin{equation*}
\langle F, G \rangle = \int_{{\rm Sp}_4(\mathbb{Z})\backslash \mathbb{H}^{(2)}} F(Z) \overline{G(Z)} (\det Y)^k \frac{dX \, dY}{(\det Y)^3}.
\end{equation*}
The norm $\| F \|^2_2$ can be expressed in terms of the adjoint $L$-function at $s=1$, a connection that has only very recently been proved by Chen and Ichino \cite{ChIc}. There is a special family of Siegel cusp forms that are derived from elliptic cusp forms (Saito--Kurokawa lifts or Maa{\ss} Spezialschar). Let $k$ be an even positive integer. Let $h \in S^+_{k-1/2}(4)$ be a Hecke eigenform of weight $k-1/2$ with Fourier expansion as in \eqref{four}, and let
$f_h \in S_{2k-2}$ denote the corresponding Shimura lift. The Saito--Kurokawa lift associates to $h$ (or $f_h$) a Siegel cusp form $F_h$ of weight $k$ for ${\rm Sp}_4(\mathbb{Z})$ with Fourier expansion \eqref{four1}, where
\begin{equation}\label{four2}
a(T) = \sum_{d \mid (n, r, m)}d^{k-1} c \left(\frac{4 \det T}{d^2}\right), \quad T = \left(\begin{matrix} n & r/2\\ r/2 & m\end{matrix}\right) \in \mathcal{P}(\mathbb{Z}),
\end{equation} see e.g.\ \cite[\S 6]{EZ}. If $L (f_h, s)$ is the standard $L$-function of $f_h$ (normalized so that the functional equation sends $s$ to $1-s$), then the norms of $F_h$ and $h$ are related by
(\cite[p.\ 551]{KoSk}, \cite[Lemma 4.2 \& 5.2 with $M=1$]{Br})
\begin{equation}\label{normF}
\| F_h \|^2 = \| h \|^2 \frac{\Gamma(k) L(f_h, 3/2)}{4\pi^k}.
\end{equation} \emph{Remarks:} 1) Note that the formula three lines after (4) in \cite{KoSk} is off by a factor of 2. \\ 2) This inner product relation was generalized to Ikeda lifts in \cite{KK}. \\ 3) For future reference we note that for $\Re s > 1$ we have \begin{equation}\label{1overL} \frac{1}{L(f_h, s)} = \prod_p \left(1 - \frac{\lambda(p)}{p^s} + \frac{1}{p^{2s}}\right) = \sum_{(n, m) = 1} \frac{\lambda(n) \mu(n)\mu^2(m)}{n^sm^{2s}} \end{equation} if $\lambda(n)$ denotes the $n$-th Hecke eigenvalue of $f_h$.
\section{Non-holomorphic automorphic forms}\label{sec4}
We recall the spectral decomposition of $L^2(\Gamma\backslash \mathbb{H})$ with $\Gamma = {\rm SL}_2(\mathbb{Z})$, consisting of the constant function $u_0 = \sqrt{3/\pi}$, a countable orthonormal basis $\{u_j, j = 1, 2, \ldots \}$ of Hecke--Maa{\ss} cusp forms
and Eisenstein series $E(., 1/2 + it)$, $t \in \mathbb{R}$. As in the introduction we call the collection of these functions $\Lambda$ and the subset of even members $\Lambda_{\text{ev}}$, and we write $\int_{\Lambda}$ resp.\ $\int_{\Lambda_{\text{ev}}}$ for the corresponding spectral averages. We also introduce the notation $\int^{\ast}_{\Lambda_{\text{ev}}} d{\tt u}$ for a spectral average without the residual spectrum which in this case consists only of the constant function. We use the general notational convention that an element in $\Lambda$ or $\Lambda_{\text{ev}}$ is denoted by ${\tt u}$ while a cusp form is usually denoted by $u$.
For $t \in \mathbb{R}$ let $U_t^{\text{ev}}$ denote the space of even weight zero Maa{\ss} cusp forms for $\Gamma$ with Laplacian eigenvalue $1/4 + t^2$. It is equipped with the inner product \begin{equation}\label{inner-maass} \langle u_1, u_2 \rangle = \int_{\Gamma \backslash \mathbb{H}} u_1(z) \overline{u_2(z)} \frac{dx \, dy}{y^2}. \end{equation} We write the Fourier expansion as
$$u(z) = \sum_{n \not= 0} a(n) W_{0, it}(4 \pi |n| y) e(nx)$$ with $a(-n) = a(n)$,
where $W_{0, it}(4\pi y) = 2 y^{1/2} K_{it}(2\pi y)$ is the Whittaker function. The Hecke operators $T(n)$, normalized as in \cite[(1.1)]{KS}, act on $U_t^{\text{ev}}$ as a commutative family of normal operators. We call $t = t_u$ the spectral parameter of $u$. The Eisenstein series $$E(z, s) = \sum_{\gamma \in \overline{\Gamma}_{\infty} \backslash \overline{\Gamma}} (\Im \gamma z)^s = \sum_{\substack{ (c, d)\in \mathbb{Z}^2/\{\pm 1\}\\ {\rm gcd}(c, d) = 1}} \frac{y^s}{|cz+d|^{2s}}$$ for $\overline{\Gamma} = {\rm PSL}_2(\mathbb{Z})$ are eigenfunctions of all $T(n)$ with eigenvalue \begin{equation}\label{defrho} \rho_s(n) := \sum_{ab = n} (a/b)^{s - 1/2} = n^{s - 1/2} \sigma_{1 - 2s}(n), \quad \sigma_s(n) = \sum_{d \mid n} d^s, \end{equation} and an eigenfunction of the Laplacian with eigenvalue $s(1-s)$. We call $(s - 1/2)/i$ the spectral parameter of $E(., s)$. If ${\tt u}$ is an Eisenstein series or a Hecke--Maa{\ss} cusp form with Hecke eigenvalues $\lambda_{\tt u}(n)$ we define the corresponding $L$-function $L({\tt u}, s) = \sum_n \lambda_{\tt u}(n) n^{-s}$. In particular \begin{equation}\label{eisen-L}
L(E(., 1/2 + it), s) = \zeta(s + it) \zeta(s - it).
\end{equation}
If $u \in U_t^{\text{ev}}$ is a cuspidal Hecke eigenform with eigenvalues $\lambda(n)$, then $n^{1/2}a(n) = a(1) \lambda(|n|)$, and by Rankin--Selberg theory (and \cite[6.576.4]{GR} with $a = b = 4\pi$, $\nu = \mu = it$) \begin{equation}\label{norm} \begin{split}
\| u \|^2 & = \frac{\pi}{3} \underset{s=1}{\text{res} }\langle |u|^2, E(., s) \rangle = \frac{2\pi}{3} \underset{s=1}{\text{res} } \sum_{n > 0} \frac{|a(n)|^2}{|n|^{s-1}} \int_0^{\infty} W_{0, it}(4 \pi y)^2 y^{s-2} dy\\
&= \frac{2\pi |a(1)|^2}{3} \underset{s=1}{\text{res} } \sum_{n > 0} \frac{|\lambda(n)|^2}{|n|^{s}} \frac{\pi^{1/2} \Gamma(s/2) \Gamma(s/2 - it)\Gamma(s/2 + it)}{(2\pi)^s \Gamma((1 + s)/2)}
= \frac{2 |a(1)|^2 L(\text{sym}^2 u, 1)}{\cosh(\pi t)}. \end{split} \end{equation} We recall the Kuznetsov formula \cite{Ku} and combine directly the ``same sign'' and the ``opposite sign'' formula to obtain a version for the even part of the spectrum. The conversion between Hecke eigenvalues and Fourier coefficients in the cuspidal case follows from \eqref{norm}. \begin{lemma}\label{kuz-even}
Let $n, m \in \mathbb{N}$. Let $h$ be an even holomorphic function in $|\Im t| \leqslant 2/3$ with $h(t) \ll (1 + |t|)^{-3}$. For non-constant ${\tt u} \in \Lambda$ let $\mathcal{L}({\tt u}) = L(\text{{\rm sym}}^2 {\tt u}, 1)$ if ${\tt u}$ is cuspidal and\footnote{Note that the measure $d{\tt u}$ is $dt/4\pi$ in the Eisenstein case which explains the factor $1/2$ in the definition of $\mathcal{L}(E(., 1/2 + it))$.} $\mathcal{L}({\tt u}) = \frac{1}{2}|\zeta(1 + 2it)|^2$ if ${\tt u} = E(., 1/2 + it)$ is Eisenstein\footnote{with the obvious interpretation in \eqref{kuz-even-form} for $t=0$}. Then \begin{equation}\label{kuz-even-form} \begin{split} \int_{\Lambda_{\text{{\rm ev}}}}^{\ast} \frac{\lambda_{\tt u}(n) \lambda_{\tt u}(m) }{\mathcal{L}({\tt u})} h(t_{\tt u})d{\tt u}= &\delta_{n= m} \int_{-\infty}^{\infty} h(t) t \tanh(\pi t) \frac{dt}{4\pi^2} \\ &+ \sum_c \frac{S(m, n, c)}{c}h^*\Big( \frac{\sqrt{nm}}{c}\Big)+ \sum_c \frac{S(m, -n, c)}{c} h^{\ast\ast}\Big( \frac{\sqrt{nm}}{c}\Big) \end{split} \end{equation} where \begin{equation}\label{hast}
h^{\ast}(x) = 2i \int_{-\infty}^{\infty} \frac{J_{2it}(4 \pi x)}{\sinh(\pi t)} h(t) t \tanh(\pi t) \frac{dt}{4\pi}, \quad h^{\ast\ast}(x) = \frac{4}{\pi}\int_{-\infty}^{\infty} K_{2it}(4 \pi x)\sinh(\pi t) h(t) t \frac{dt}{4\pi}. \end{equation} \end{lemma}
We turn to half-integral weight forms. Let $V_t^+(4)$ denote the (``Kohnen'') space of weight $1/2$ Maa{\ss} cusp forms for $\Gamma_0(4)$ with eigenvalue $1/4 + t^2$ with respect to the weight $1/2$ Laplacian and Fourier expansion \begin{equation}\label{four-half}
v(z) = \sum_{ \substack{n\not= 0 \\ n \equiv 0, 1 \, (\text{mod } 4)}} b(n) W_{\frac{1}{4} \text{sgn}(n), it}(4 \pi |n|y) e(nx). \end{equation} The congruence condition on the indices can be encoded in an eigenvalue equation: the functions $v \in V_t^+(4)$ are invariant under the operator $L$ defined in \cite[(0.7), (0.8)]{KS}, cf.\ also \cite[(A.1)]{Bi1}. The space $V_t^{+}(4)$ is a finite-dimensional Hilbert space with respect to the inner product \begin{equation}\label{inner} \langle v_1, v_2 \rangle = \int_{\Gamma_0(4)\backslash \mathbb{H}} v_1(z) \overline{v_2(z)} \frac{dx\, dy}{y^2}. \end{equation}
The Hecke operators $T(p^2)$, $p$ prime, act on $V^+_t(4)$ as a commutative family of normal operators that commute with the weight $1/2$ Laplacian (again we use Kohnen's modification for $T(4)$ in order to treat all primes uniformly). Explicitly, if $T(p^2) v = \lambda(p) v$, then (see \cite[(1.3)]{KS}) the Fourier coefficients of $v$ satisfy \begin{equation}\label{hecke-half} \lambda(p) b(n) = p b(np^2) + p^{-1/2} \chi_n(p) b(n) + p^{-1} b(n/p^2) \end{equation} for all primes $p$ and all $n \in \mathbb{Z} \setminus \{0\}$ with $n \equiv 0, 1$ (mod 4) with the convention $b(x) = 0$ for $x \not\in \mathbb{Z}$. If $v \in V_t^+(4)$ is an eigenfunction of all Hecke operators $T(p^2)$ with eigenvalues $\lambda(p)$, the relation \eqref{hecke-half} can be captured in the identity $$ \sum_{f=1}^{\infty} \frac{b(\Delta f^2 )}{f^{s-1}} = b(\Delta) \prod_p \frac{1 - \chi_{\Delta}(p)p^{-s-1/2}}{1 - \lambda(p) p^{-s} + p^{-2s}}$$ for a fundamental discriminant $\Delta$. Extending $\lambda(p)$ to all $n$ by the usual Hecke relations, we see that the denominator is just $\sum_{\nu} \lambda(p^{\nu})p^{-\nu s}$, so that \begin{equation}\label{non-fund}
f b(\Delta f^2) = b(\Delta) \sum_{d \mid f} \mu(d) \chi_{\Delta}(d) \lambda(f/d) d^{-1/2} \end{equation} for a fundamental discriminant $\Delta$ and $f \in \mathbb{N}$. \\
\section{Period formulae}
Let $v \in V_t^+(4)$.
Katok and Sarnak proved in \cite[Proposition 4.1]{KS} that there is a linear map (a theta lift) $\mathscr{S}$ sending $v$ to a non-zero element in $U_{2t}^{\text{ev}}$ if $b(1)\not= 0$ and to $0$ otherwise. A calculation \cite[p.\ 221-223]{KS} shows that if $v$ is an eigenform of $T(p^2)$, then $\mathscr{S} v$ is an eigenform of $T(p)$ with the same eigenvalue, and this computation works verbatim for $p=2$, too. Conversely, given an eigenform $u\in U_{2t}^{\text{ev}}$ with Hecke eigenvalues $\lambda(p)$, by \cite[Theorem 1.2]{BM}\footnote{Hecke operators are normalized differently in \cite{BM}, but this plays no role.} there is a unique (up to scaling) $v \in V_t^+(4)$ having eigenvalues $\lambda(p)$ for $T(p^2)$, $p > 2$, (and then automatically also for $T(4)$, since $T(4)$ commutes with the other operators) which may or may not satisfy $b(1) \not= 0$. In particular, for a given eigenform $u\in U_{2t}^{\text{ev}}$ there is at most one eigenform $v \in V_t^+$ (up to scaling) with $\mathscr{S}v = u \in U_{2t}^{\text{ev}}$. If it exists, we normalize it to be $L^2$-normalized and denote its Fourier coefficients $b(n)$ as in\footnote{This determines $b(n)$ only up to a constant of absolute value 1, but we will only encounter products of the type $b(n_1)\overline{b(n_2)}$, so that this constant is irrelevant.} \eqref{four-half}. If no such $v$ exists, we define $b(n)$ to be 0.
If ${\tt u}$ is a Hecke--Maa{\ss} cusp form or an Eisenstein series, the absolute square of the periods $P(D; {\tt u})$ can be expressed in terms of central $L$-functions. We introduce the relevant notation. For a discriminant $D = \Delta f^2$ with a fundamental discriminant $\Delta$ let \begin{equation}\label{basicL} L(D, s) := L(\chi_{\Delta}, s) \sum_{d \mid f} \mu(d) \chi_{\Delta}(d) \sigma_{1-2s}(f/d) d^{-s}=: \sum_{n=1}^{\infty} \frac{\varepsilon_D(n)}{n^s}. \end{equation}
With $\rho_s$ as in \eqref{defrho} we can re-write this as \begin{equation}\label{basicL1} L(D, s) = L(\chi_{\Delta}, s) f^{1/2 - s} \sum_{d \mid f} \mu(d) \chi_{\Delta}(d) \rho_{s}(f/d) d^{-1/2}. \end{equation} Since $\rho_s = \rho_{1-s}$, we see that $L(D, s)$ satisfies the same type of functional equation as $L(\Delta, s)$ namely \begin{equation}\label{Lfuncteq}
\Lambda(D, s) := L(D, s) |D|^{s/2} \Gamma\Big(\frac{s+\mathfrak{a}}{2}\Big) \pi^{-s/2} = \Lambda(D, 1-s) \end{equation} with $\mathfrak{a} = 1$ if $D < 0$ and $\mathfrak{a} = 0$ if $D > 0$. For a Hecke--Maa{\ss} cusp form or an Eisenstein series ${\tt u} $ define \begin{equation}\label{alt} L({\tt u}, D, s) = \sum_{n=1}^{\infty} \frac{\varepsilon_D(n) \lambda_{\tt u}(n)}{n^s}. \end{equation} The key point is that \begin{equation}\label{key} L({\tt u}, \Delta f^2, 1/2) = L({\tt u}, \Delta, 1/2) \Bigl(\sum_{d \mid f} \mu(d) \chi_{\Delta}(d) \lambda_{\tt u}(f/d) d^{-1/2}\Bigr)^2 \end{equation} as one can check by a formal computation with Euler products using the Hecke relation for the eigenvalues $\lambda_{\tt u}$ (which are identical for Maa{\ss} forms and Eisenstein series), see \cite[p.\ 188-189]{KZ}. For later purposes we record the simple bound \begin{equation}\label{key-simple} \Bigl(\sum_{d \mid f} \mu(d) \chi_{\Delta}(d) \lambda_{\tt u}(f/d) d^{-1/2}\Bigr)^2 \ll f^{1/3} \end{equation} uniformly in $\Delta$ and ${\tt u}$, which follows from the Kim-Sarnak bound with $ 2 \cdot 7/64 < 1/3$.
From \eqref{basicL1}, the fact that $|L(\chi_{\Delta}, 1/2 + it)|^2 = L(E(., 1/2 + it), \Delta, 1/2)$ and \eqref{key} we have \begin{equation}\label{eisen-L2} \begin{split}
|L(\Delta f^2, 1/2+it)|^2 &= |L(\chi_{\Delta}, 1/2+it)|^2 \Bigl|\sum_{d \mid f} \mu(d) \chi_{\Delta}(d) \lambda_{E(., 1/2 + it)}(f/d) d^{-1/2}\Bigr|^2\\ & = L(E(., 1/2 + it), \Delta f^2, 1/2). \end{split}
\end{equation}
The next key lemma expresses the periods $P(D; u)$ defined in \eqref{defP} for cusp forms $u$ as half-integral weight Fourier coefficients, and then their squares as $L$-functions. The first formula \eqref{mixed} is essentially a formula of Katok-Sarnak \cite[(0.16) \& (0.19)]{KS}, the passage from squares of metaplectic Fourier coefficients to $L$-functions in \eqref{BarMao} is a Kohnen-Zagier type formula of Baruch-Mao \cite[Theorem 1.4]{BM}. The combination \eqref{katok-Sarnak} of these two is a special case of a formula of Zhang \cite[Theorem 1.3.2]{Zh1} or \cite[Theorem 7.1]{Zh2}, derived independently by a different method.
\begin{lemma}\label{lem3} If ${ u} \in U_{2t}^{\text{{\rm ev}}}$ is an even Hecke--Maa{\ss} cusp form and $D_1, D_2 <0$ are two discriminants, then \begin{equation}\label{mixed}
\frac{P(D_1; { u})\overline{P(D_2; { u})} \|{ u}\|^{-2} }{|D_1D_2|^{1/4}} = \frac{3}{\pi} L({ u}, 1/2) \Gamma(1/4 + it)\Gamma(1/4 - it) |D_1D_2|^{1/2}b(D_1) \overline{b(D_2)}. \end{equation}
For a discriminant $D$ of either sign we have \begin{equation}\label{BarMao}
|b(D)|^2 = \frac{1}{24\pi} \frac{L({ u}, D, 1/2)}{L(\text{{\rm sym}}^2 { u}, 1)} \frac{\cosh(2 \pi t)\Gamma( \frac{1}{2} - \frac{1}{4} \text{{\rm sgn}}(D)+ it)\Gamma( \frac{1}{2} - \frac{1}{4} \text{{\rm sgn}}(D)- it)}{ |D|}. \end{equation}
For $D < 0$ we have \begin{equation}\label{katok-Sarnak}
\frac{|P(D; { u})|^2 \|{ u}\|^{-2} }{|D|^{1/2}} = \frac{L({ u}, 1/2) L({ u}, D, 1/2)}{4L(\text{{\rm sym}}^2 { u}, 1)}. \end{equation} \end{lemma}
\emph{Remark:} The exact shape of these formulas is an unexpectedly subtle matter, and the attentive reader might well be confused by the various and slightly contradictory versions in the literature. There are at least four sources of possible conflict: \begin{itemize} \item the Whittaker functions can be normalized in different ways; \item the inner products can be normalized in different ways; \item the translation from adelic language to classical can cause problems; \item there can be ambiguities related to the groups ${\rm GL}(2)$ vs.\ ${\rm SL}(2)$ vs.\ ${\rm PSL}(2)$. \end{itemize} The Katok-Sarnak formula exists in the literature with proofs given in at least in three different versions: \cite[(0.16)]{KS}, \cite[Theorem A1]{Bi2} and \cite[Theorem 4]{DIT}. The original version of Katok-Sarnak was carefully revised by Bir\'o, but the latter seems to be still off by a factor 2 compared to the version in Duke-Imamo\u{g}lu-Toth, which was checked numerically.
The Baruch-Mao formula \cite[Theorem 1.4]{BM} is quoted in \cite[(5.17)]{DIT} with an additional factor 2. Zhang's result \cite[Theorem 7.1]{Zh2} (and also the remark after \cite[Theorem 1.3.2]{Zh1}, the theorem itself being correct) is missing the stabilizer $\epsilon(z)$ in the period $P(D; u)$. This formula is slightly incorrectly reproduced in \cite{LMY} and several follow-up papers, based on a different normalization of the Whittaker function. Finally, neither combination of one of the three Katok-Sarnak formulae with the Baruch-Mao formula in \cite[Theorem 1.4]{BM} coincides with Zhang's formula.
We therefore feel that these beautiful and important results should be stated with correct constants and normalizations. For the proof of Theorem \ref{thm1} and its connection to the mass equidistribution conjecture this is absolutely crucial. As \cite[Theorem 4]{DIT} was checked numerically by the authors, we follow their version of the Katok-Sarnak formula. This gives \eqref{mixed}. We verified and confirmed the constant in Zhang's formula independently by proving an averaged version in Appendix \ref{appb}. This gives \eqref{katok-Sarnak}. By backwards engineering, we established the numerical constant in the Baruch-Mao formula, which gives \eqref{BarMao} and coincides with \cite[(5.17)]{DIT}.
Note that \eqref{katok-Sarnak} is essentially universal: the right hand side of \eqref{katok-Sarnak} is independent of any normalization, the left hand side depends only on the normalization of the inner product \eqref{inner-maass} which is standard.
\begin{proof} We start with the formula \cite[(0.16)]{KS} for a general discriminant $D < 0$, but use the numerical constants as in
\cite[Theorem 4]{DIT} (proved only for fundamental discriminants there). This formula expresses $P(D; { u})$ for an arbitrary discriminant $D < 0$ as a sum over Fourier coefficients of all $v$ with $\mathscr{S}v = { u}$. By the above remarks, there is at most one such $v$. If there is none, then both sides of \eqref{mixed} and \eqref{katok-Sarnak} vanish by \cite[(0.16), (0.19)]{KS} and our convention that $b(n) = 0$ in this case, and there is nothing to prove. Also note that the left hand side of \eqref{mixed} and both sides of \eqref{katok-Sarnak} are independent of the normalization of ${ u}$, so without loss of generality we may assume that ${ u}$ is Hecke-normalized as in \cite{KS}. We obtain
$$\frac{P(D_1; { u})\overline{P(D_2; { u})} \|{ u}\|^{-2} }{|D_1D_2|^{1/4}}
= 6 |D_1D_2|^{1/2} b(D_1)\overline{b (D_2)} |b(1)|^2 \| { u}\|^2.$$ Next we insert \cite[(0.19)]{KS} (again keeping in mind the different normalization of \eqref{inner} and observing that this is coincides with the numerically checked version of \cite[Theorem 4]{DIT}) getting \eqref{mixed}.
If $D$ is a fundamental discriminant, then \eqref{BarMao} follows from \cite[(5.17)]{DIT} together with \eqref{norm} with $a(1) = 1$ and $2t$ in place of $t$. By \eqref{non-fund} and \eqref{key} this remains true for arbitrary discriminants. The formula \eqref{katok-Sarnak} is a direct consequence of \eqref{mixed}, \eqref{BarMao} and well-known properties of the gamma function, and was proved independently by Zhang \cite[Theorem 1.3.2]{Zh1}.
\end{proof}
\emph{Remark:} The argument at the beginning of this proof shows that the $u$-sum in Proposition \ref{Lfunc}(a), up to terms of size 0, can be replaced with the $v$-sum in \eqref{interpret}, using \eqref{BarMao}. This completes the proof of \eqref{interpret}.
\\
A similar result holds for Eisenstein series. If $aX^2 + bXY + cY^2$ is an integral quadratic form of discriminant $D = b^2 -4ac < 0$ with Heegner point $z = (\sqrt{|D|}i - b)/(2a)$, then
$$E(z, s) = \frac{1}{ \zeta(2s)} \sum_{(u, v) \in (\mathbb{Z}^2 \setminus (0, 0))/\{\pm 1\}} \frac{(\sqrt{|D|}/2)^s}{(au^2 - buv + cv^2)^s}.$$
Hence $$P(D; E(., s)) = \frac{1}{ \zeta(2s)} \Big(\frac{\sqrt{|D|}}{2}\Big)^s \zeta(D, s)$$ where $\zeta(D, s)$ is defined\footnote{Note that Zagier defines $\Gamma = {\rm PSL}_2(\mathbb{Z})$, so his definition of equivalence coincides with ours. The quotient by $\{\pm 1\}$ in the $u, v$-sum is not spelled out explicitly in \cite[(6)]{Z}, but implicitly used in the proof of \cite[Proposition 3]{Z} on p.\ 131. } in \cite[(6)]{Z}. By \cite[Proposition 3 iii]{Z} (or \cite[Theorem 3]{DIT}) we obtain the following lemma in analogy to Lemma \ref{lem3}.
\begin{lemma}\label{lem-eisen} If $D < 0$ is a discriminant, then
\begin{equation}\label{eisen1}
P(D; E(., s)) = \frac{1}{ \zeta(2s)} \Big(\frac{\sqrt{|D|}}{2}\Big)^s \zeta(s) L(D, s), \end{equation} and hence \begin{equation}\label{eisen2}
\frac{|P(D; E(., 1/2 + it))|^2}{|D|^{1/2}} = \frac{L(E(., 1/2 + it), 1/2) L(E(., 1/2 + it), D, 1/2)}{ 2|\zeta(1 + 2 it)|^2} . \end{equation}
\end{lemma}
\emph{Remarks:} 1) The second formula follows from the first by \eqref{eisen-L} and \eqref{eisen-L2}.
2) Here the verification of the numerical constants is much easier than in the cuspidal case. Taking residues at $s=1$ in \eqref{eisen1} for a fundamental discriminant $\Delta < 0$ returns the class number formula for $\mathbb{Q}(\sqrt{\Delta})$, which confirms the numerical constants.
3) For future reference we recall the standard bounds \begin{equation}\label{lower}
\zeta(1 + it) \gg |t|^{-\varepsilon}, \quad |t_{ u}|^{\varepsilon} \gg L(\text{sym}^2 { u}, 1) \gg |t_{ u}|^{-\varepsilon} \end{equation} for $\varepsilon > 0$. This is in particular relevant to obtain upper bounds in \eqref{eisen2} and \eqref{katok-Sarnak}. \\
We close this section by stating a standard approximate functional equation \cite[Theorem 5.3]{IK} for the $L$-functions occurring in the previous period formulae. For $u \in U_t^{\text{ev}}$ and a fundamental discriminant $\Delta$ (possibly $\Delta = 1$) we have \begin{equation}\label{approx-basic}
L(u, \Delta, 1/2) = L(u \times \chi_{\Delta}, 1/2) = 2\sum_n \frac{\lambda(n)\chi_{\Delta}(n)}{n^{1/2}} W_{t}\Big(\frac{n}{|\Delta|}\Big) \end{equation} where \begin{equation}\label{v-t} W_{t}(n) = \frac{1}{2\pi i} \int_{(2)} \frac{\Gamma(\frac{1}{2}(\frac{1}{2}+ \mathfrak{a} + s + it))\Gamma(\frac{1}{2}(\frac{1}{2} + \mathfrak{a}+ s - it))}{\Gamma(\frac{1}{2}(\frac{1}{2} + \mathfrak{a} + it))\Gamma(\frac{1}{2}(\frac{1}{2} + \mathfrak{a} - it)) \pi^{s} }e^{s^2} n^{-s}
\frac{ds}{s}
\end{equation} with $\mathfrak{a} = 1$ if $\Delta < 0$ and $\mathfrak{a} = 0$ if $\Delta > 0$. Note that $W_t$ depends on $\Delta$ only in terms of its sign. If we want to emphasize this we write $W_t^+$ and $W_t^-$ with $\pm = \text{sgn}(\Delta)$.
A similar expression holds for $E(., 1/2 + it)$ in place of $u$ except that in the case $\Delta = 1$ we have $L(E(., 1/2 + it), 1, s) = \zeta(s+it)\zeta(s- it)$ and there is an additional polar term\footnote{Note that there is a sign error in \cite[Theorem 5.3]{IK}: the residue $R$ should be subtracted.}. We have \begin{equation}\label{approx-basic1} \begin{split}
L(E(., 1/2 + it), \Delta, 1/2)& = 2\sum_n \frac{\rho_{1/2 + it}(n) \chi_{\Delta}(n)}{n^{1/2}} W_{t}\Big(\frac{n}{|\Delta|}\Big)\\ & -\delta_{\Delta = 1} \sum_{\pm} \frac{\zeta(1 \pm 2it)\Gamma(\frac{1}{2} \pm it)\pi^{\mp it} e^{(1/2 \pm it)^2}}{(\frac{1}{2} \pm i t)\Gamma(\frac{1}{4} + \frac{it}{2})\Gamma(\frac{1}{4} - \frac{it}{2})} \end{split} \end{equation} with $\rho_{1/2 + it}$ as in \eqref{defrho}.
\section{Half-integral weight summation formulae}\label{summation}
In this section we compile the Voronoi summation and the Kuznetsov formula for half-integral weight forms.
\subsection{Voronoi summation} As before let \begin{equation}\label{51}
v(z) = \sum_{ n\not= 0 } b(n) W_{\frac{k}{2} \text{{\rm sgn}}(n), it}(4 \pi |n|y) e(nx) \end{equation}
be a Maa{\ss} form of weight $k \in \{1/2, 3/2\}$ and spectral parameter $t$ for $\Gamma_0(4)$ with respect to the usual theta multiplier. We start with the Voronoi summation formula \cite[Theorem 3]{By}. \begin{lemma}\label{Vor} Let $c\in \mathbb{N}$, $4 \mid c$, $(a, c) = 1$. Let $\phi$ be a smooth function with compact support in $(-\infty, 0) \cup (0, \infty)$ and for $y > 0$ define
\begin{equation}\label{defPhi}
\Phi(\pm y) = \int_0^{\infty} \mathcal{J}^{\pm, +}(ty) \phi(t) + \mathcal{J}^{\pm, -}(ty) \phi(-t) dt,
\end{equation}
where
\begin{displaymath}
\begin{split} & \mathcal{J}^{\pm, \pm}(x) = \frac{\cos(\pi k/2 \mp i \pi r)}{\sin(2\pi i r)} J_{-2 ir}(2\sqrt{x}) - \frac{\cos(\pi k/2\pm i \pi r)}{\sin(2\pi i r)} J_{2 ir}(2\sqrt{x}) = - \frac{F(2 \sqrt{x}, 2r, \mp k)}{\sin(2\pi i r)},\\
& \mathcal{J}^{\pm, \mp}(x) = \frac{2 K_{2ir}(2\sqrt{x})}{\Gamma(1/2 \pm k/2 + ir)\Gamma(1/2 \pm k/2 - ir)} \end{split}
\end{displaymath} with $F$ as in \eqref{defF}. Then
$$\sum_{n \not = 0} b(n)\sqrt{|n|} e\Big(\frac{an}{c}\Big) \phi(n) = \Big(\frac{-c}{a}\Big) \epsilon^{2k}_a e\Big(\frac{k}{4}\Big) \sum_{n \not= 0} b(n) \sqrt{|n|} \frac{2\pi}{c} e\Big(-\frac{\bar{a}n}{c}\Big) \Phi\Big(\frac{(2\pi)^2 n}{c^2}\Big)$$
with $\epsilon_a$ as in \eqref{epsd}. \end{lemma}
The proof of the Voronoi formula (Lemma \ref{Vor}) follows from a certain vector-valued functional equation satisfied by the $L$-functions with coefficients $b(\pm n) e(\pm an/c)$. The same functional equation holds if $v$ is not cuspidal (this is clear from general principles and worked out explicitly in \cite{DG} along the same lines), but in this case the $L$-functions are not entire; they have various poles. We use this observation for two non-cuspidal modular forms. The first is a half-integral weight Eisenstein series \begin{equation} \begin{split} E^{\ast}\Big(z, \frac{1}{2} + it\Big) =&\frac{ 2^{1 + 2it} \pi^{-it} \Gamma(1/2 + 2it) \zeta(1+4it) }{\Gamma(3/4 + it)\Gamma(1/4 + it)} y^{1/2 + it} + \frac{2^{1 - 2it} \pi^{3it} \Gamma(1/2 - 2it) \zeta(1-4it)}{\Gamma(3/4 + it)\Gamma(1/4 + it)} y^{1/2 - it} \\
&+ \sum_{D } \frac{L(D,1/2 + 2it) |D|^{it} }{|D|^{1/2} }\frac{W_{\frac{1}{4} \text{sgn}(D), it}(4 \pi |D|y)}{\Gamma(\frac{1}{2} + \frac{1}{4} \text{sgn}(D) + it)}e(Dx)\\
\end{split} \end{equation}
which transforms under $\Gamma_0(4)$ as a weight $1/2$ automorphic form with theta multiplier; see \cite[p.\ 964]{DIT}. As usual, $D$ runs over all discriminants. The other is Zagier's weight 3/2 Eisenstein series\footnote{We have multiplied Zagier's definition by a factor $(4\pi y)^{3/4}$ in order to make $|\mathcal{H}(z)|$ invariant under $\Gamma_0(4)$.} \cite[Theorem 2]{HZ} \begin{equation}\label{zag} \begin{split}
\mathcal{H}(z) = &\sum_{D < 0} \frac{H(D)}{|D|^{3/4}} e(|D|x) W_{3/4, 1/4}(4\pi |D|y) - \frac{(4\pi y)^{3/4}}{12} \\ & + \frac{1}{4\sqrt{\pi}} \sum_{n = \square} \frac{e(-nx)}{n^{1/4}} W_{-3/4, 1/4}(4\pi n y)+ \frac{y^{1/4}}{\sqrt{8} \pi^{1/4}} \end{split} \end{equation} which transforms under $\Gamma_0(4)$ as a weight $3/2$ automorphic form with theta multiplier. As before $H(D)$ is the Hurwitz class number. For $\epsilon \in \{\pm 1\}$, $(a, c) = 1$, $4\mid c$, the Dirichlet series
$$\sum_{\epsilon D > 0} \frac{L(D, 1/2 + 2it)e(aD/c) |D|^{it}}{|D|^{1/2 + w}}$$ has poles at $w = 1/2 \pm it$ with certain residues $R_{\epsilon}(\pm t, a/c)$, say, and consequently we obtain the following analogue of Lemma \ref{Vor}.
\begin{lemma}\label{Vor-eis} Let $c\in \mathbb{N}$, $4 \mid c$, $(a, c) = 1$, $t \in \mathbb{R} \setminus \{0\}$. Let $\phi$ be a smooth function with compact support in $(-\infty, 0) \cup (0, \infty)$ and for $y > 0$ define $\Phi$ as in \eqref{defPhi}. Then
\begin{displaymath}
\begin{split}
\sum_{D \not = 0}\frac{L(D, 1/2 + 2it)|D|^{it} }{\Gamma(\frac{1}{2} + it + \frac{1}{4} \text{{\rm sgn}}(D))} e\Big(\frac{aD}{c}\Big) \phi(D) = &\Big(\frac{-c}{a}\Big) \epsilon_a e\Big(\frac{1}{8}\Big)\Big[\sum_{\epsilon \in \{\pm 1\}} \sum_{\pm} \frac{R_{\epsilon}(\pm t, a/c)}{\Gamma(\frac{1}{2} + it + \frac{1}{4}\epsilon)}\int_0^{\infty} \phi(\epsilon x) x^{ \pm it} dx \\
& + \sum_{D \not= 0}\frac{L(D, 1/2 + 2it)|D|^{it} }{\Gamma(\frac{1}{2} + it + \frac{1}{4} \text{{\rm sgn}}(D))} \frac{2\pi}{c} e\Big(-\frac{\bar{a}D}{c}\Big) \Phi\Big(\frac{(2\pi)^2 D}{c^2}\Big) \Big]
\end{split}
\end{displaymath} \end{lemma}
For $t=0$ one combines the $\pm$-terms and takes the limit as $t \rightarrow 0$. In our application, the values $R_{\epsilon}(t, a/c)$ are irrelevant (as long as they are polynomial in $c$ and $t$) since we apply the formula with a function $\phi$ that oscillates much more strongly that $x^{\pm it}$ so that the integral is negligible.
We obtain a similar summation formula for Hurwitz class numbers. Although we do not need it for the present result, we compute in Appendix \ref{appa} the residues explicitly and get the following handsome formula. \begin{lemma}\label{class-num1} Let $c\in \mathbb{N}$, $4 \mid c$, $(a, c) = 1$. Let $\phi$ be a smooth function with compact support in $(0, \infty) $. Then \begin{displaymath} \begin{split}
\sum_{D < 0}\frac{H(D)}{|D|^{1/4}} e\Big(\frac{a|D|}{c}\Big) \phi(|D|) = \Big(\frac{-c}{a}\Big) \bar{\epsilon}_a e\Big(\frac{3}{8}\Big)&\Bigg[ \sum_{D <0} \frac{H(D)}{|D|^{1/4}} \frac{2\pi}{c} e\Big(-\frac{\bar{a}|D|}{c}\Big) \int_0^{\infty} \mathcal{J}^+ \Big(t \frac{(2\pi)^2 |D|}{c^2}\Big) \phi(t) dt\\ & + \frac{1}{4\sqrt{\pi}} \sum_{ n = \square} n^{1/4} \frac{2\pi}{c} e\Big(\frac{\bar{a}n}{c}\Big) \int_0^{\infty} \mathcal{J}^{-}\Big(t \frac{(2\pi)^2 n}{c^2}\Big) \phi(t) dt \\ & + \int_0^{\infty} \phi(x) \Big( \frac{1}{\sqrt{8}c^{1/2}} x^{-1/4}- \frac{\sqrt{2}\pi}{3 c^{3/2}} x^{1/4} \Big) dx \Bigg] \end{split} \end{displaymath} where \begin{equation}\label{Jclean} \mathcal{J}^+(x) = \frac{\sin(2\sqrt{x}) }{\sqrt{\pi} x^{1/4}}, \quad \mathcal{J}^-(x) = \frac{2e^{-2\sqrt{x}}}{x^{1/4}}. \end{equation} \end{lemma}
\emph{Remark:} Observing that for negative $D \equiv 0, 1$ (mod 4) we have
$$e(|D|/4) + e(3|D|/4) = 2 \delta_{D \equiv 0 \, (\text{mod } 4)}, \quad -e(|D|/4) + e(3|D|/4) = 2i \delta_{D \equiv 1 \, (\text{mod } 4)},$$ it is a straightforward exercise to conclude $$\sum_{\substack{-X < D < 0\\ D\equiv \delta \, (\text{mod } 4)}} H(D) = \frac{\pi}{36} X^{3/2} - \frac{1}{8} X + O(X^{3/4})$$ for $\delta = 0, 1$. Further congruence conditions on $D$ can be imposed, and the error term can be improved by a more careful treatment of the dual term in the Voronoi formula. See \cite{Vi} for the corresponding result for the ordinary class number $h(d)$. The following table provides some numerical results (here we combined the cases $\delta = 0$ and $\delta = 1$). \\%The error term matches the basic Landau-type error term for sums of three squares, cf.\ \cite[(4.46)]{IK} (and can be improved if the dual term in the Voronoi formula is estimated non-trivially as in \cite{CIw, HB1}), but the interesting fact is the secondary main term from the pole of $L^+(s, a/c)$ at $s = 3/4$.
{\footnotesize
\begin{tabular}{l|l|l|l|l|l|l} $X$ & 1000 & 2000 & 4000 & 6000 & 8000 & 10000\\ \hline $\displaystyle\sum^{\substack{\quad\\ \quad}}_{D \leqslant X} H(D)$ & 5280.5 &15131.3& 43189.5& 79685.7 & 122967 & 172106 \\ \hline $\displaystyle \frac{\pi}{18} \overset{\substack{\quad\\ \quad}}{X^{3/2}} - \frac{1}{4}X\quad$ & 5269.22 & 15110.7 & 43153.7 & 79615.6 & 122885.6 & $\underset{\substack{\quad\\\quad\\\quad}}{172032.9}$ \\ \hline \end{tabular}}
\subsection{The Kuznetsov formula} The Kuznetsov formula was generalized by Proskurin \cite{Pr} to arbitrary weights and by Andersen-Duke \cite{AD} to Kohnen's subspace. Interestingly, only the direction from Kloosterman sums to spectral sums appears to be in the literature, but no complete version in the other direction. Bir\'o \cite[p.\ 151]{Bi1} has a version only valid for test functions on the spectral side whose spectral mean value is 0, Ahlgren-Andersen \cite[Section 3]{AA} use an approximate version.
We take this opportunity to state and prove the relevant Kuznetsov formula both for the full space of half-integral weight forms and for Kohnen's subspace. For $\kappa \in \{1/2, 3/2\}$ the relevant Kloosterman sums are \begin{equation}\label{klooster2} K_{\kappa}(n, m, c) = \sum_{\substack{d\, (\text{mod }c)\\ (d, c) = 1}} \epsilon^{2\kappa}_d \left(\frac{c}{d}\right) e\left(\frac{nd + m\bar{d}}{c}\right) \end{equation} for $4 \mid c$. The Eisenstein series belong to the two essential cusps $\mathfrak{a} = \infty, 0$. We normalize and denote their Fourier coefficients by $\phi_{\mathfrak{a}m}(1/2 + it) = \phi^{(\kappa)}_{\mathfrak{a}m}(1/2 + it)$ as in \cite[(12) - (14)]{Pr}. We denote by $\left.\sum\right.^{(\kappa)}$ a sum over an orthonormal basis of the space of cusp forms of weight $\kappa$ and label the members by $v_j$, $j = 1, 2, \ldots$ with Fourier coefficients $b_j(n)$ as in \eqref{51} and spectral parameters by $t_j$. \begin{prop}\label{kuz-full}
Let $\kappa \in \{1/2, 3/2\}$, $m, n > 0$. Let $h$ be an even function, holomorphic in $|\Im t| < 2/3$ with $h(t) \ll (1 + |t|)^{-4}$. Then \begin{displaymath}
\begin{split} & \left.\sum_j \right.^{(\kappa)}\frac{\sqrt{mn}\overline{b_j(m)} b_j(n)}{\cosh(\pi t_j)} h(t_j) + \sum_{\mathfrak{a}} \int_{-\infty}^{\infty} \Big(\frac{n}{m}\Big)^{it} \frac{ \overline{\phi^{(\kappa)}_{\mathfrak{a}m}(1/2 + it)} \phi^{(\kappa)}_{\mathfrak{a}n}(1/2 + it)}{4\cosh(\pi t) \Gamma(\frac{1+\kappa}{2} + it)\Gamma(\frac{1+\kappa}{2}- it)} h(t) dt \\
&= \delta_{n=m}\int_{-\infty}^{\infty} h(t) t \sinh(\pi t)\Gamma \Big( \frac{1-\kappa}{2} + it\Big)\Gamma \Big( \frac{1-\kappa}{2} - it\Big) \frac{dt}{4\pi^3} \\
&+ e\Big(\frac{1-\kappa}{4}\Big)\sum_c \frac{K_{\kappa}(m, n, c)}{c} \int_0^{\infty} \frac{F( 4\pi\sqrt{nm}/c, 2t, -\kappa)}{\cosh(\pi t)} \Gamma \Big( \frac{1-\kappa}{2} + it\Big)\Gamma \Big( \frac{1-\kappa}{2} - it\Big) h(t) t\, \frac{dt}{2\pi^2}
\end{split} \end{displaymath} if in addition $h(\pm i/4) = 0$. Moreover, regardless of the value of $h(\pm i/4)$ we have \begin{displaymath}
\begin{split} & \left.\sum_j\right.^{(\kappa)} \frac{\sqrt{mn}\overline{b_j(-m)} b_j(-n)}{\cosh(\pi t_j)} h(t_j) + \sum_{\mathfrak{a}} \int_{-\infty}^{\infty} \Big(\frac{n}{m}\Big)^{it} \frac{ \overline{\phi^{(\kappa)}_{\mathfrak{a}, -m}(1/2 + it)} \phi^{(\kappa)}_{\mathfrak{a}, -n}(1/2 + it)}{4\cosh(\pi t)\Gamma(\frac{1-\kappa}{2} + it)\Gamma(\frac{1-\kappa}{2}- it)} h(t) dt \\
&= \delta_{n=m}\int_{-\infty}^{\infty} h(t) t \sinh(\pi t)\Gamma\Big(\frac{1+\kappa}{2} + it\Big)\Gamma\Big(\frac{1+\kappa}{2}- it\Big) \frac{dt}{4\pi^3} \\&+ e\Big(\frac{1+\kappa}{4}\Big)\sum_c \frac{K_{-\kappa}(m, n, c)}{c} \int_0^{\infty} \frac{F( 4\pi\sqrt{nm}/c, 2t, \kappa)}{\cosh(\pi t)}\Gamma\Big(\frac{1+\kappa}{2} + it\Big)\Gamma\Big(\frac{1+\kappa}{2}- it\Big) h(t) t\, \frac{dt}{2\pi^2}.
\end{split} \end{displaymath}
\end{prop}
\emph{Remark:} Note that the space of weight 1/2 Maa{\ss} forms $v$ with spectral parameter $i/4$ are in the kernel of the Maa{\ss} lowering operator. Hence $y^{-1/4}v$ is holomorphic, so that $v$ has no non-vanishing negative Fourier coefficients. This is consistent with the fact that the Eisenstein contribution vanishes in this case because of the gamma factors in the denominator.
\begin{proof} By \cite[Lemma 3]{Pr} with $\sigma = 1$, $t = 2\tau \in \mathbb{R}$ and the first formula in \cite[Lemma 6]{Pr} we have the ``pre-Kuznetsov'' formula \begin{displaymath} \begin{split} &-\sum_c \frac{K_{\kappa}(m, n, c)}{c^2} x^{-\kappa}\frac{\pi}{\sinh(2\pi \tau)} \int_0^{x} F(y, 2\tau, 1-\kappa)y^{\kappa - 1} dy+ \frac{\delta_{n=m}e((1+\kappa)/4)}{4\pi(n+m)} \\ &= \frac{\pi^2e((1+\kappa)/4)}{2\Gamma(1 - \frac{\kappa}{2} + i\tau)\Gamma(1 - \frac{\kappa}{2} - i\tau)}\Bigg(\sum_j \frac{\overline{b_j}(m)b_j(n) }{\cosh(\pi(t_j - \tau))\cosh(\pi(t_j + \tau))}\\ & + \frac{1}{4\sqrt{nm}} \sum_{\mathfrak{a}} \int_{-\infty}^{\infty} \Big(\frac{m}{n}\Big)^{-it} \frac{\overline{\phi_{\mathfrak{a}m}(1/2 + it)}\phi_{\mathfrak{a}n}(1/2 + it) }{\Gamma(\frac{1+\kappa}{2} + it) \Gamma(\frac{1+\kappa}{2} - it) \cosh(\pi(t- \tau))\cosh(\pi(t + \tau))} dt\Bigg). \end{split} \end{displaymath} where $$x = 4\pi \sqrt{mn}/c.$$
Note that taking $\sigma = 1$ is admissible in the present situation because we have the same Weil-type bounds for the Kloosterman sums $K_{\kappa}(n, m, c)$ as for $K^+_{\kappa}(n, m, c)$ in \eqref{weil}. Also note that there is a typo in \cite[Lemma 6]{Pr} in the upper limit of the integral.
For $h$ as in the lemma and $t \in \mathbb{R}$ we have the following inversion formula $$\int_{-\infty}^{\infty} \big(h(\tau + i/2)+ h(\tau - i/2)\big) \frac{\cosh(\pi \tau)}{\cosh(\pi(\tau - t)) \cosh(\pi(\tau + t))} d\tau = \frac{2 h(t)}{\cosh(\pi t)}.$$ This is the lemma on p.327 of \cite{Ku}\footnote{We have corrected a sign error. This sign error is cancelled by another sign error in \cite[(6.6)]{Ku}. The inversion formula was re-produced in \cite[Lemma 16.4]{IK} with the same sign error. There the sign error is cancelled by sign errors in the first and fifth display on p.410.} which is readily proved by residue calculus. We now integrate the pre-Kuznetsov formula against $$ \big(h(\tau + i/2)+ h(\tau - i/2)\big) \cosh(\pi \tau) \Gamma (1 - \kappa/2 + i\tau) \Gamma (1 - \kappa/2 - i\tau).$$ Our assumptions on $h$ ensure absolute convergence and the possibility to shift the contour up and down to $\Im t = \pm 1/2$ without crossing poles. In this way the
$\delta$-term becomes $$\delta_{m=n} \frac{e((1+\kappa)/4)}{4\pi (m+n)}\int_{-\infty}^{\infty} h(\tau) \Gamma(1/2 - \kappa/2 + i\tau)\Gamma(1/2 - \kappa/2 - i\tau) 2\tau\sinh(\pi \tau) d\tau.$$ For the Kloosterman term we insert the definition in \eqref{defF} getting \begin{displaymath} \begin{split} - \sum_c \frac{K_{\kappa}(m, n, c)}{c^2}&\frac{\pi i}{2 x^{\kappa}} \int_{-\infty}^{\infty}\int_0^x h(\tau) \frac{\Gamma((1-\kappa)/2 + i\tau)\Gamma((1-\kappa)/2 - i\tau)}{\cosh(\pi \tau)} \\ &\sum_{\epsilon_1, \epsilon_2 \in \{\pm 1\}} J_{\epsilon_1 + \epsilon_2 2 i \tau}(y) \cos(\pi(\kappa/2 + \epsilon_2 i \tau)) ((1 - \kappa )/2+ \epsilon_1\epsilon_2 i \tau) y^{\kappa - 1} dy \, d\tau \end{split} \end{displaymath} We note that \begin{displaymath} \begin{split}
y^{1-\kappa}\frac{d}{dy} \Big(\frac{J_{2i\tau}(y)}{y^{1-\kappa}}\Big) &= J'_{2i\tau}(y) - \frac{(\kappa - 1)J_{2i\tau}(y)}{y} \\ &= \frac{J_{2i\tau+1}(y)(i\tau + (1-\kappa)/2) + J_{2i\tau-1}(y)((1 - \kappa)/2 - i\tau)}{-2i\tau} \end{split} \end{displaymath} where the last equality follows from the recurrence relations \cite[8.471.1\&2]{GR}. Substituting this, we can evaluate the $y$-integral by the fundamental theorem of calculus, arriving at $$-\sum_c \frac{K_{\kappa}(m, n, c)}{c^2} \frac{\pi }{ x } \int_{-\infty}^{\infty} \tau h(\tau) \frac{\Gamma((1-\kappa)/2 + i\tau)\Gamma((1-\kappa)/2 - i\tau)}{\cosh(\pi r)} F(x, 2\tau, -\kappa)
d\tau$$ after some elementary manipulations. We multiply the resulting expression by $\sqrt{mn}$
to obtain the first formula. Note that the $c$-sum is absolutely convergent by the power series expansion of the Bessel function contained in $F(x, 2t, -\kappa)$ and the fact that $h(\pm i/4) = 0$, as we can shift the $t$-contour up and down to $|\Im t| = 1/2 - \varepsilon$.
There are two ways to derive the second formula from the first. We can either observe that in Proskurin's notation we can compute $\langle \overline{\mathcal{U}_m(., s_1)}, \overline{\mathcal{U}_n(., \bar{s}_2)}\rangle$ instead of $\langle \mathcal{U}_m(., s_1), \mathcal{U}_n(., \bar{s}_2)\rangle$. This changes the signs of the coefficients in the spectral expansion and has the effect of changing $\kappa$ into $-\kappa$. Note that in this case we do not need the extra condition $h(\pm i/4) = 0$ to shift contours. Alternatively, we use the following fact (cf.\ \cite[(4.17), (4.18), (4.27), (4.28), (4.64), p.507, p.509]{DFI}): the map $$T_{\kappa} = ((\textstyle \frac{\kappa - 1}{2})^2 + t^2) X\Lambda_{\kappa}$$ with $X : f(z) \mapsto f(-\bar{z})$ and $\Lambda_{\kappa} = \kappa/2 + y(i\partial_x - \partial_y)$ the weight lowering operator is a bijective isometry between weight $\kappa$ and weight $2-\kappa$ that exchanges positive and negative Fourier coefficients: \begin{equation*} \begin{split}
T_{\kappa} & \Bigg(\sum_{n \not=0} b(n) e(nx)W_{\text{sgn}(n)\frac{\kappa}{2}, it}(4 \pi|n|y) \Bigg) \\
& = \sum_{n \not= 0} b(-n) \text{sgn}(n) \Big( \Big(\frac{\kappa - 1}{2}\Big)^2+t^2\Big)^{-\frac{1}{2}\text{sgn}(n)} e(nx)W_{\text{sgn}(n)\frac{2-\kappa}{2}, it}(4 \pi|n|y).
\end{split} \end{equation*} This yields again the second formula for the first. \end{proof}
In order to get a corresponding formula for the Kohnen space, we apply the $L$ operator $$\frac{1}{2(1 + i^{2\kappa})} \sum_{w \, \text{mod } 4} \left( \begin{matrix} 1+w & 1/4\\ 4w & 1\end{matrix}\right)$$ to the formula. As in the case of the Petersson formula (Lemma \ref{lem1}), this has the effect that \begin{itemize} \item the cuspidal term is restricted to forms in the Kohnen space; \item the $\delta$-term is multiplied by $2/3$; the reason for the number 2/3 is that the dimension of the Kohnen space is 1/3 of the full space, but only half of the coefficients appear; \item the Kloosterman sums \eqref{klooster2} are replaced with the Kloosterman sums \eqref{defKlo} and the Kloosterman term is also multiplied by 2/3. \end{itemize} The hardest part is to compute the Eisenstein coefficients. All of this has been worked out in detail in \cite[Section 5]{AD}. The corresponding formula \cite[Theorem 5.3]{AD} can be inverted in the same way. In the following lemma let $\sum^+$ denote a sum over an orthonormal basis of weight 1/2 Maa{\ss} cusp forms in Kohnen's space. We recall the definition \eqref{basicL} and \eqref{basicL1} of $L(D, s)$ for a discriminant $D$.
\begin{lemma}\label{lem10} Let $\kappa = 1/2$ and let $n, m \in \mathbb{Z}$ that are congruent $0$ or $1$ modulo $4$. Let $h$ be an even function, holomorphic in $|\Im t| < 2/3$ with $h(t) \ll (1 + |t|)^{-4}$. Then \begin{displaymath}
\begin{split}
& \left.\sum_j \right.^{+}\frac{\sqrt{mn}\overline{b_j(m)} b_j(n)}{\cosh(\pi t_j)} h(t_j) + \frac{1}{12}\int_{-\infty}^{\infty} \Big(\frac{n}{m}\Big)^{it} \frac{L(m, 1/2 - 2it)L(n, 1/2 + it) }{|\zeta(1 + 4it)|^2\cosh(\pi t) \Gamma(\frac{1+\kappa}{2} + it)\Gamma(\frac{1+\kappa}{2}- it)} h(t) dt \\
&= \frac{2}{3}\delta_{n=m}\int_{-\infty}^{\infty} h(t) t \sinh(\pi t)\Gamma \Big( \frac{1-\kappa}{2} + it\Big)\Gamma \Big( \frac{1-\kappa}{2} - it\Big) \frac{dt}{4\pi^3} \\
&+ \frac{2}{3}e\Big(\frac{1-\kappa}{4}\Big)\sum_c \frac{K^+_{\kappa}(m, n, c)}{c} \int_0^{\infty} \frac{F( 4\pi\sqrt{nm}/c, 2t, -\kappa)}{\cosh(\pi t)} \Gamma \Big( \frac{1-\kappa}{2} + it\Big)\Gamma \Big( \frac{1-\kappa}{2} - it\Big) h(t) t\, \frac{dt}{2\pi^2}
\end{split} \end{displaymath} for $n, m > 0$ if in addition $h(\pm i/4) = 0$. Moreover, regardless of the value of $h(\pm i/4)$ we have \begin{displaymath}
\begin{split}
& \left.\sum_j\right.^{(+)} \frac{\sqrt{|mn|} \, \overline{b_j(m)} b_j(n)}{\cosh(\pi t_j)} h(t_j) + \frac{1}{12}\int_{-\infty}^{\infty} \Big(\frac{|n|}{|m|}\Big)^{it} \frac{L(m, 1/2 - 2it)L(n, 1/2 + it) }{|\zeta(1 + 4it)|^2\cosh(\pi t)\Gamma(\frac{1-\kappa}{2} + it)\Gamma(\frac{1-\kappa}{2}- it)} h(t) dt \\
&= \frac{2}{3} \delta_{n=m}\int_{-\infty}^{\infty} h(t) t \sinh(\pi t)\Gamma\Big(\frac{1+\kappa}{2} + it\Big)\Gamma\Big(\frac{1+\kappa}{2}+ it\Big) \frac{dt}{4\pi^3} \\&+ \frac{2}{3}e\Big(\frac{1+\kappa}{4}\Big)\sum_c \frac{K^+_{-\kappa}(|m|, |n|, c)}{c} \int_0^{\infty} \frac{F( 4\pi\sqrt{|nm|}/c, 2t, \kappa)}{\cosh(\pi t)}\Gamma\Big(\frac{1+\kappa}{2} + it\Big)\Gamma\Big(\frac{1+\kappa}{2}- it\Big) h(t) t\, \frac{dt}{2\pi^2}
\end{split} \end{displaymath} if $n, m < 0$. \end{lemma}
\section{Harmonic analysis on positive definite matrices}\label{sec6}
Here we prove Proposition \ref{prop1}. We identify the Hilbert spaces \begin{equation}\label{product}
(\mathbb{H}, y^{-2} dx\, dy) \times (\mathbb{R}_{>0}, r^{-1} dr) \cong (\mathcal{P}(\mathbb{R}), (\det Y)^{-3/2} dY)
\end{equation} via $$\iota : (x + iy, r) \mapsto \sqrt{r}\left(\begin{matrix} y^{-1} & -xy^{-1}\\-x y^{-1} & y^{-1} (x^2 + y^2)\end{matrix}\right).$$ Note that for $ Y = \iota(x+iy, r)$ with $ r = \det Y$ we have
$$\Big|\det \frac{dY}{d(r, x, y)} \Big| = \frac{\sqrt{r}}{y^2},$$ so that the measures coincide. The group $\overline{\Gamma} = {\rm PSL}_2(\mathbb{Z})$ acts faithfully on $\mathcal{P}(\mathbb{R})$ and $\mathcal{P}(\mathbb{Z})$ by $T \mapsto U^{\top} T U$ for $U \in \overline{\Gamma}$. This is compatible with the action of $\overline{\Gamma}$ on $\mathbb{H}$ by M\"obius transforms. Every smooth function $f \in L^2(\Gamma\backslash \mathbb{H})$ has a spectral decomposition $$f(z) = \sum_{j \geqslant 0} \langle f, u_j \rangle u_j(z) + \int_{-\infty}^{\infty} \langle f, E(., 1/2 + it)\rangle E(z, 1/2 + it) \frac{dt}{4\pi} = \int_{\Lambda} \langle f, {\tt u} \rangle {\tt u}(z) d{\tt u}. $$
We recall the notion of the spectral parameter $t_{\tt u}$ for ${\tt u} \in \Lambda$; the constant function has spectral parameter $i/2$. Combining the spectral decomposition on $\Gamma \backslash \mathbb{H}$ with Mellin inversion, we conclude that, for a smooth function $\Phi \in L^2(\Gamma \backslash \mathcal{P}(\mathbb{R}))$, the spectral decomposition $$\Phi(Y) = \Phi(\iota(x + iy, r)) = \int_{(0)} \int_{\Lambda} \langle \widehat{\Phi}(s), {\tt u} \rangle {\tt u}(x + iy) d {\tt u} \,\, r^{-s} \frac{ds}{2\pi i}$$ holds provided $\Phi$ is sufficiently rapidly decaying as $r\rightarrow 0$ and $r \rightarrow \infty$. Here, $$\widehat{\Phi}(s)(x+iy) = \int_0^{\infty} \Phi(\iota(x + iy, r)) r^{s} \frac{dr}{r}$$ is the Mellin transform with respect to the $r$-variable. This gives the Parseval formula \begin{equation}\label{pars}
\| \Phi(Y) \|^2 = \int_{\Gamma \backslash \mathcal{P}(\mathbb{R})} |\Phi(Y)|^2 \frac{dY}{(\det Y)^{3/2}} = \int_{-\infty}^{\infty} \int_{\Lambda} |\langle \widehat{\Phi}(it), {\tt u} \rangle|^2 d{\tt u}\, \frac{dt }{2\pi}. \end{equation}
For an automorphic form ${\tt u} \in \Lambda$ and a Siegel cusp form $F \in S^{(2)}_k$ with Fourier expansion \eqref{four1} we define the twisted Koecher--Maa{\ss} series by \begin{equation}\label{KM} L(F \times {\tt u}, s) := \sum_{T\in \mathcal{P}(\mathbb{Z})/\overline{\Gamma}} \frac{a(T)}{\epsilon(T)( \det T)^{s + (k-1)/2} }{\tt u}\left( (\det T)^{-1/2}T\right) \end{equation} where $\epsilon(T) = \{U \in \overline{\Gamma} \mid U^{\top} T U = T\}$ is the stabilizer and $\Re s$ is (initially) sufficiently large. This series is often defined in terms of ${\rm GL}_2(\mathbb{Z})$-equivalence instead of $\overline{\Gamma}$-equivalence, but for us the present version is more convenient. This function has no Euler product, but it does have a functional equation. Let \begin{equation}\label{G-KM} G(F \times {\tt u}, s) := 4(2\pi)^{-k+1-2s}\prod_{\pm} \Gamma\left(\frac{k-1}{2} +s - \frac{1}{4} \pm \frac{it_{\tt u}}{2}\right). \end{equation} Let us assume that $k$ is even. Then for all even ${\tt u}$ (including Eisenstein series and the constant function), the function $L(F \times {\tt u}, s)$ has an analytic continuation to $\mathbb{C}$ that is bounded in vertical strips and satisfies the functional equation \cite[Theorem 3.5]{Im} \begin{equation*} L(F \times {\tt u}, s) G(F \times {\tt u}, s) = L(F \times {\tt u}, 1-s) G(F \times {\tt u}, 1-s). \end{equation*} The functional equation is a consequence of the following period formula. For $\Re s$ sufficiently large \cite[p.\ 927-928]{Im} we have \begin{displaymath} \begin{split} \int_0^{\infty} & \langle F(i \cdot (., r)), {\tt u}\rangle r^{\frac{k-1}{2} + s} \frac{dr}{r} = \int_{\overline{\Gamma} \backslash \mathbb{H}} \int_0^{\infty} F(i \cdot (z, r)) r^{\frac{k-1}{2} + s} \frac{dr}{r} \overline{{\tt u}}(z) \frac{dx \,dy}{y^2}\\ & = \int_{\overline{\Gamma} \backslash \mathcal{P}(\mathbb{R})} \sum_{T \in \mathcal{P}(\mathbb{Z})} a(T) e^{-2\pi \text{tr}(YT)} (\det Y)^{\frac{k-1}{2} + s}\overline{{\tt u}}\left( (\det Y)^{-1/2}Y\right) \frac{dY}{(\det Y)^{3/2}}. \end{split} \end{displaymath} Now splitting the $T$-sum into equivalence classes modulo $\overline{\Gamma}$ and unfolding the integral, this equals $$ \sum_{T\in \mathcal{P}(\mathbb{Z})/\overline{\Gamma}} \frac{a(T)}{\epsilon(T) } \int_{ \mathcal{P}(\mathbb{R})} e^{-2\pi \text{tr}(YT)} (\det Y)^{\frac{k-1}{2} + s} \overline{{\tt u}}\left( (\det Y)^{-1/2}Y \right) \frac{dY}{(\det Y)^{3/2}}.$$ Note that it is important that the action of $\overline{\Gamma}$ is faithful. The last integral over $\mathcal{P}(\mathbb{R})$ was evaluated by Maa{\ss} \cite[p.\ 85 and p.\ 94]{Ma}: $$\int_{ \mathcal{P}(\mathbb{R})} e^{- \text{tr}(YT)} (\det Y)^{ s} \overline{{\tt u}}\left( (\det Y)^{-1/2}Y \right) \frac{dY}{(\det Y)^{3/2}} = \frac{\pi^{1/2}}{(\det T)^{s}} \overline{\tt u}\left((\det T)^{-1/2} T\right)\prod_{\pm} \Gamma\Big(s - \frac{1}{4} \pm \frac{it_{\tt u}}{2}\Big) $$ for any ${\tt u} \in \Lambda$. Thus we obtain $$\int_0^{\infty} \langle F(i \cdot (., r)), {\tt u}\rangle r^{\frac{k-1}{2} + s} \frac{dr}{r} = \frac{\sqrt{\pi}}{4} L(F \times \overline{{\tt u}}, s) G(F \times \overline{{\tt u}}, s), $$ initially for $\Re s$ sufficiently large, but then by analytic continuation for all $s \in \mathbb{C}$. For odd ${\tt u}$, the left hand side vanishes, since $F(i \cdot (., r)), {\tt u}\rangle = 0$ for every $r$. The Parseval formula \eqref{pars} now implies Proposition \ref{prop1} in the introduction. \\%the following period formula.
In the special case where $F$ is a Saito--Kurokawa lift, the Koecher--Maa{\ss} series simplifies.
\begin{lemma}\label{lem7} If $k$ is even and $F = F_h \in S_k^{(2)}$ is a Saito--Kurokawa lift with Fourier expansion as in \eqref{four1} and \eqref{four2} and ${\tt u} \in \Lambda$, then
$$L(F_h \times {\tt u}, s) = 4^{s + \frac{k-1}{2}} \zeta(2s) \sum_{\substack{D< 0\\ D \equiv 0, 1 \, (\text{{\rm mod }}4)}} \frac{c_h(|D|)P(D; {\tt u}) }{|D|^{s + \frac{k-1}{2}}}$$ for $\Re s$ sufficiently large and $P(D; {\tt u})$ as in \eqref{defP}. \end{lemma}
\emph{Remark:} One would expect that $c_h(|D|)$ is roughly of size $|D|^{\frac{k}{2} - \frac{3}{4}}$ and that $P(D; {\tt u})$ is roughly of size $|D|^{\frac{1}{4}}$ (for non-constant ${\tt u}$) so that typically $c_h(|D|)P(D; {\tt u})|D|^{-\frac{k-1}{2}}$ is roughly of constant size (with respect to $D$). Using trivial bounds for $P(D; {\tt u})$ and $c_h(|D|)$, the quantity $c_h(|D|)P(D; {\tt u})|D|^{-\frac{k-1}{2}}$ is certainly $\ll |D|^{3/4+\varepsilon}$.
\begin{proof} We copy the argument from \cite[p.\ 22]{Bo}. Let $\Delta$ be a negative fundamental discriminant, $D = \Delta f^2$ a negative discriminant and $T \in \mathcal{P}(\mathbb{Z})/\overline{\Gamma}$. For such $T = \left(\begin{smallmatrix} n & r/2\\ r/2 & m\end{smallmatrix}\right)$ we write $e(T) = (n, r, m)$. It follows from \eqref{four2} that \begin{displaymath} \begin{split}
L(F_h \times {\tt u}, s) &= 4^{s + \frac{k-1}{2}} \sum_{D = \Delta f^2} \sum_{t \mid f} \frac{1}{|D|^{s + \frac{k-1}{2}}} \sum_{\substack{ \text{det}(T) = - D/4\\ e(T) = t}} \frac{a(T)}{\epsilon(T)} {\tt u}((\det T)^{-1/2} T)\\
&= 4^{s + \frac{k-1}{2}} \sum_{D = \Delta f^2} \sum_{d \mid t \mid f} \frac{d^{k-1} c_h(|D|/d^2)}{|D|^{s + \frac{k-1}{2}}} \sum_{\substack{ \text{det}(T) = - D/4\\ e(T) = t}} \frac{ {\tt u}((\det T)^{-1/2} T)}{\epsilon(T)} . \end{split} \end{displaymath} Writing $t = t'd$ with $t' \mid f/d$, we can evaluate the $t'$-sum getting \begin{displaymath} \begin{split}
L(F_h \times {\tt u}, s) & = 4^{s + \frac{k-1}{2}} \sum_{D = \Delta f^2} \sum_{d \mid f} \frac{d^{k-1} c_h(|D|/d^2)}{|D|^{s + \frac{k-1}{2}}} P(D/d^2, {\tt u}) = 4^{s + \frac{k-1}{2}} \zeta(2s) \sum_{D} \frac{c_h(|D|)P(D; {\tt u}) }{|D|^{s + \frac{k-1}{2}}} \end{split} \end{displaymath} as desired. \end{proof}
We combine Proposition \ref{prop1} and Lemma \ref{lem7} with \eqref{normF} and use the notation of these formulas to derive the following basic spectral formula for the restricted norm $\mathcal{N}(F_h)$ of a Saito--Kurokawa lift $F_h \in S_k^{(2)}$.
\begin{prop}\label{prop2} Let $k$ be even and let $F = F_h \in S_k^{(2)}$ be a Saito--Kurokawa lift with Fourier expansion as in \eqref{four1} and \eqref{four2}. Let $f_h \in S_{2k-2}$ denote the corresponding Shimura lift of $h$.
Then \begin{displaymath}
\mathcal{N}(F_h) = \frac{\pi^2}{90 } \cdot \frac{18\sqrt{\pi}}{2 L(f_h, 3/2)} \int_{-\infty}^{\infty} \int_{\Lambda_{\text{\rm ev}}} |\mathcal{G}(k, t_{ {\tt u}}, 1/2 + it)\mathcal{L}(h, \overline{{\tt u}}, 1/2 + it)|^2 d{\tt u} \, dt, \end{displaymath} where \begin{equation}\label{defG} \begin{split} \mathcal{G}(k, t_{\tt u}, s) &= \pi^{ - 2s} \frac{2^k\Gamma(\frac{k-1}{2} + s - \frac{1}{4} + \frac{i t_{\tt u}}{2})\Gamma(\frac{k-1}{2} + s - \frac{1}{4} - \frac{i t_{\tt u}}{2})}{(\Gamma(k) \Gamma(k-3/2))^{1/2}} \end{split} \end{equation} and $\mathcal{L}(h, {\tt u}, s) $ is the analytic continuation of \begin{equation}\label{curlyL}
\mathcal{L}(h, {\tt u}, s) = \left(\frac{\Gamma(k - 3/2)}{(4\pi)^{k-3/2}}\right)^{1/2} \zeta(2s) \sum_{\substack{D< 0\\ D \equiv 0, 1 \, (\text{{\rm mod }}4)}} \frac{c_h(|D|)P(D; {\tt u}) }{ \| h \| \cdot |D|^{s + \frac{k-1}{2}}} \end{equation} for $\Re s$ sufficiently large. \end{prop}
The renormalized functions still satisfy the functional equation $$\mathcal{L}(h,\overline{ {\tt u}} , s) \mathcal{G}(k, t_{{\tt u}}, s) = \mathcal{L}(h ,\overline{{\tt u}}, 1-s) \mathcal{G}(k, t_{ {\tt u}}, 1-s).$$ The inclusion of the gamma factor in \eqref{curlyL} is motivated by the formula in Lemma \ref{lem1}. \\
From the Dirichlet series expansion and the functional equation we obtain an approximate functional equation, cf.\ \cite[Theorem 5.3]{IK}.
\begin{lemma}\label{approx1} Let $F = F_h\in S_k^{(2)}$ be a Saito--Kurokawa lift and ${\tt u}\in \Lambda$ with spectral parameter $t_{\tt u}$. For $t, x \in \mathbb{R}$ let \begin{equation}\label{defV3}
V_t(x; k, t_{\tt u}) := \frac{1}{2\pi i}\int_{(3)} e^{v^2} \mathcal{G}(k, t_{\tt u}, v + 1/2 + it) \mathcal{G}(k, t_{\tt u}, v + 1/2 - it) x^{-v} \frac{dv}{v}.
\end{equation} Then \begin{displaymath} \begin{split}
& |\mathcal{G}(k, t_{\tt u}, 1/2 + it)\mathcal{L}(h, {\tt u}, 1/2 + it)|^2 \\
&=2 \frac{\Gamma(k - 3/2)}{(4\pi)^{k-3/2}} \sum_{f_1, f_2} \sum_{D_1, D_2 < 0} \frac{c_h(|D_1|)c_h(|D_2|)P(D_1; {\tt u}) \overline{P(D_2; {\tt u}) }}{ \| h \| \cdot f_1^{1 +2it}f_2^{1-2it}|D_1|^{ k/2 + it}|D_2|^{ k/2 + it}}V_{ t}(|D_1D_2| (f_1f_2)^2; k, t_{\tt u}).
\end{split} \end{displaymath} \end{lemma} Combining Proposition \ref{prop2}, Lemma \ref{approx1} and \eqref{1overL} we obtain the following basic formula: \begin{equation}\label{Fh} \begin{split}
\mathcal{N}(F_h) &= \frac{\pi^2}{90} \cdot \frac{18\sqrt{\pi}}{2 }\cdot 2\sum_{(n, m) = 1} \frac{\lambda(n) \mu(n)\mu^2(m)}{n^{3/2}m^{3}} \int_{-\infty}^{\infty} \int_{\Lambda_{\text{\rm ev}}} \frac{\Gamma(k - 3/2)}{(4\pi)^{k-3/2} \| h \|^2} \\
&\sum_{f_1, f_2, D_1, D_2} \frac{c(|D_1|) c(|D_2|) P(D_1; {\tt u})\overline{P(D_2; {\tt u})}}{f_1f_2 |D_1D_2|^{k/2}} \Big(\frac{|D_2|f_2^2}{|D_1|f_1^2}\Big)^{it} V_{t}(|D_1D_2|(f_1f_2)^2; k, t_{\tt u}) d{\tt u} \, dt. \end{split} \end{equation}
\section{A relative trace formula}\label{secproof2}
In this section we prove a slightly more general formula than that stated in Theorem \ref{thm2}. Let $D_1 = \Delta_1f_1^2, D_2 = \Delta_2f_2^2 < 0$ be two arbitrary negative discriminants and $h$ as specified in Theorem \ref{thm2}. Combining \eqref{mixed} (with $t/2$ in place of $t$) and \eqref{eisen1} we have
\begin{displaymath}
\begin{split}
\frac{1}{|D_1D_2|^{1/4}} &\int_{\Lambda_{\text{ev}}} P(D_1; {\tt u}) \overline{P(D_2; {\tt u})} h(t_{\tt u}) d{\tt u} = \frac{3}{\pi} \frac{H(D_1) H(D_2)}{|D_1D_2|^{1/4}} h(i/2) \\
&+ \sum_{u \text{ cuspidal, even}} \frac{3}{\pi} L(u, 1/2) \Gamma(1/4 + it_u/2)\Gamma(1/4 - it_u/2) |D_1D_2|^{1/2} b(D_1)\overline{b(D_2)} h(t_u)\\
&+ \int_{-\infty}^{\infty}\left( \frac{|D_1|}{|D_2|}\right)^{it/2} |\zeta(1/2 + it)|^2 \frac{L(D_1, 1/2 + it)L(D_2, 1/2 - it)}{2|\zeta(1 + 2it)|^2} h(t) \frac{dt}{4\pi}.
\end{split}
\end{displaymath} By the argument of the proof of Lemma \ref{lem3} we can re-write the sum over even cusp forms $u$ as a sum over weight 1/2 cusp forms $v$ in Kohnen's space. Thus the last two terms of the preceding display become \begin{displaymath} \begin{split}
& \left.\sum_{j}\right.^{(+)} \frac{3}{\pi} L(u_j, 1/2) \Gamma(1/4 + it_j)\Gamma(1/4 - it_j) |D_1D_2|^{1/2} b_j(D_1)\overline{b_j(D_2)} h(2t_j)\\
&+ \int_{-\infty}^{\infty}\left( \frac{|D_1|}{|D_2|}\right)^{it} |\zeta(1/2 + 2it)|^2 \frac{L(D_1, 1/2 + 2it)L(D_2, 1/2 - 2it)}{2|\zeta(1 + 4it)|^2} h(2t) \frac{dt}{2\pi}. \end{split} \end{displaymath} Here, as in Lemma \ref{lem10}, $\sum^{(+)}$ indicates a sum over an orthonormal basis of weight 1/2 cusp forms $v_j$ in Kohnen's space with spectral parameter $t_j$ and Fourier coefficients $b_j(D)$, and $u_j$ is the corresponding Shimura lift with spectral parameter $2t_j = t_u$. If $\lambda_j(n)$ are the Hecke eigenvalues of $u_j$, then by the approximate functional equations \eqref{approx-basic} and \eqref{approx-basic1} with $\Delta = 1$ we can write $$L(u_j, 1/2) = 2\sum_n \frac{\lambda_j(n)}{n^{1/2}} W_{2t_j}(n)$$ and $$
|\zeta(1/2 + 2it)|^2 = 2\sum_n \frac{\rho_{1/2 + 2it}(n)}{n^{1/2}} W_{2t}(n) -\sum_{\pm} \frac{\zeta(1 \pm 4it)\Gamma(\frac{1}{2} \pm 2it)\pi^{\mp 2it} e^{(1/2 \pm 2 it)^2}}{(\frac{1}{2} \pm 2i t)\Gamma(\frac{1}{4} + it)\Gamma(\frac{1}{4} - it)} $$
with $W_t$ as in \eqref{v-t} with $\mathfrak{a} = 0$. Note that $W_{2t}(x)$ is even and holomorphic in $|\Im t | <2/3$, satisfying the uniform bound $W_{2t}(x)\ll (1+|t|^2)/x^2 $ in this region (by trivial bounds). Moreover $W_{2t}(x)$ vanishes at $t = \pm i/4$.
We first deal with residue term in the formula for $|\zeta(1/2 + 2it)|^2$ and substitute this back into the Eisenstein term. This gives
$$- \sum_{\pm} \int_{-\infty}^{\infty} \left( \frac{|D_1|}{|D_2|}\right)^{it} \frac{ \Gamma(\frac{1}{2} \pm 2it)\pi^{\mp 2it} e^{(1/2 \pm 2 it)^2}}{(\frac{1}{2} \pm 2i t)\Gamma(\frac{1}{4} + it)\Gamma(\frac{1}{4} - it)} \frac{L(D_1, 1/2 + 2it)L(D_2, 1/2 - 2it)}{2\zeta(1 \mp 4it)} h(2t) \frac{dt}{2\pi}.$$ We can slightly simplify this by applying in the plus-term the functional equation for $L(D_1, 1/2 + 2it)$ and changing $t$ to $-t$, and in the minus-term the functional equation \eqref{Lfuncteq} for $L(D_2, 1/2 - 2 i t)$. In this way we see that the two terms are equal and after some simplification we obtain
$$2 \int_{-\infty}^{\infty} |D_1D_2|^{ it} \frac{ 2^{-\frac{3}{2} - 2 i t}\Gamma(-\frac{1}{4} + it) e^{(1/2- 2 it)^2}}{ \sqrt{\pi} \Gamma(\frac{1}{4} + i t)} \frac{L(D_1, 1/2 + 2it)L(D_2, 1/2 + 2it)}{2\zeta(1+ 4it)} h(2t) \frac{dt}{2\pi}.$$
Our next goal is to use the half-integral weight Hecke relations to combine $\lambda_j(n) b_j(D_1)$. For notational simplicity let us define $\tilde{b}_j(D_1) = \sqrt{|D_1|}b_j(D_1)$. From \eqref{non-fund} we obtain \begin{displaymath} \begin{split} \lambda(n) \tilde{b}_j(D_1)& = \tilde{b}_j(D_1) = \tilde{b}(\Delta_1) \sum_{d \mid f_1} \frac{\mu(d) \chi_{\Delta_1}(d)}{d^{1/2}} \lambda\Big(\frac{f_1}{d}\Big) \lambda(n)
= \tilde{b}_j(\Delta_1) \sum_{\substack{d_1rs = f_1\\ r \mid n}} \frac{\mu(d_1) \chi_{\Delta_1}(d_1)}{d_1^{1/2}} \lambda\Big(\frac{sn}{r}\Big) \\
& = \sum_{\substack{d_1rs = f_1\\ r \mid n}} \frac{\mu(d_1) \chi_{\Delta_1}(d_1)}{d_1^{1/2}} \sum_{m \mid sn/r} \frac{\chi_{\Delta_1}(m)}{\sqrt{m}} \tilde{b}_j\Big(\Delta_1 \Big(\frac{sn}{rm}\Big)^2\Big).
\end{split} \end{displaymath} In the last step we used \eqref{non-fund} again together with M\"obius inversion. This yields
\begin{displaymath} \begin{split}
&\sum_n \frac{\lambda_j(n)}{n^{1/2}} W_{2t}(n)\tilde{b}_j(D_1) = \sum_{ d_1rs = f_1 }\frac{\mu(d_1) \chi_{\Delta_1}(d_1)}{d_1^{1/2}} \sum_n \frac{W_{2t}(rn)}{\sqrt{rn}} \sum_{m \mid sn} \frac{\chi_{\Delta_1}(m)}{\sqrt{m}} \tilde{b}_j\Big(\Delta_1 \Big(\frac{sn}{m}\Big)^2\Big) \\ &= \sum_{ d_1rs = f_1 }\frac{\mu(d_1) \chi_{\Delta_1}(d_1)}{d_1^{1/2}} \sum_{n, m} \frac{W_{2t}(rn m/(m, s))}{\sqrt{rn m/(m, s)}} \frac{\chi_{\Delta_1}(m)}{\sqrt{m}} \tilde{b}_j\Big(\Delta_1 \Big(\frac{sn}{(m, s)}\Big)^2\Big) \\ &= \sum_{ d_1rs = f_1 }\frac{\mu(d_1) \chi_{\Delta_1}(d_1)}{d_1^{1/2}} \sum_{\tau u = s} \sum_{n}\sum_{(m, u) = 1} \frac{W_{2t}(rn m)}{\sqrt{rn m }} \frac{\chi_{\Delta_1}(m\tau)}{\sqrt{m\tau}} \tilde{b}_j(\Delta_1 (un)^2)\\ &= \sum_{ d_1rs = f_1 }\frac{\mu(d_1) \chi_{\Delta_1}(d_1)}{d_1^{1/2}} \sum_{\tau u = s} \sum_{n, m} \sum_{v\mid u} \mu(v) \frac{W_{2t}(rn vm)}{\sqrt{rn vm }} \frac{\chi_{\Delta_1}(vm\tau)}{\sqrt{vm\tau}} \tilde{b}_j(\Delta_1 (un)^2)\\ &= \sum_{ d_1r\tau vw = f_1 }\sum_{n, m} \frac{\mu(d_1)\mu(v) \chi_{\Delta_1}(d_1vm\tau)}{\sqrt{d_1rn\tau} vm} W_{2t}(rn vm) \tilde{b}_j(\Delta_1 (vwn)^2).
\end{split} \end{displaymath} Comparing \eqref{non-fund} and \eqref{basicL1}, we see that the same Hecke relations hold for Eisenstein series, and we therefore have \begin{displaymath} \begin{split}
&\sum_n \frac{\rho_{1/2 + 2it}(n)}{n^{1/2}} W_{2t}(n) |D_1|^{it} L(D_1, 1/2 + 2 it) \\ &= \sum_{ d_1r\tau vw = f_1 }\sum_{n, m} \frac{\mu(d_1)\mu(v) \chi_{\Delta_1}(d_1vm\tau)}{\sqrt{d_1rn\tau} vm} W_{2t}(rn vm) (\Delta_1(vwn)^2)^{it} L(\Delta_1 (vwn)^2, 1/2 + 2it). \end{split} \end{displaymath} We are now in a position to apply the second Kohnen-Kuznetsov formula in Lemma \ref{lem10} with $$\frac{6}{\pi} \Gamma(1/4 + it)\Gamma(1/4 - it) \cosh(\pi t) W_{2t}(rnvm) h(2t)$$ in place of $h(t)$. This function satisfies the hypotheses of that formula (and decays rapidly enough in $n$ and $m$), recall that $W_{2t}(n)$ vanishes at $t = \pm i/4$. The diagonal exists only if $\Delta_1 = \Delta_2$ and $vwn = f_2$.
For a function $H$ we introduce the integral transform \begin{equation}\label{h0} H^{\dagger}(x) = \int_{-\infty}^{\infty} \frac{F(x, t, 1/2)}{\cosh(\pi t)} H(t) t \frac{dt}{\pi} \end{equation} with $F$ as in \eqref{defF}.
\begin{theorem}\label{thm5} Let Let $D_1 = \Delta_1 f_1^2, D_2 = \Delta_2f^2_2$ be negative discriminants and let $h$ be an even function, holomorphic in $|\Im t | < 2/3$ with $h(t) \ll (1+|t|)^{-10}$. Define $W_t$ as in \eqref{v-t}, $L(D, s)$ as in \eqref{basicL} and $K^+_{3/2}(a, b, c)$ as in \eqref{defKlo}. Then \begin{displaymath} \begin{split}
&\frac{1}{|D_1D_2|^{1/4}}\int_{\Lambda_{\text{ev}}} P(D_1; {\tt u}) \overline{P(D_2; {\tt u})} h(t_{\tt u}) d{\tt u} = \frac{3}{\pi} \frac{H(D_1)H(D_2)}{|D_1D_2|^{1/4}} h(i/2)\\
& \quad + \int_{-\infty}^{\infty} \Big|\frac{D_1D_2}{4}\Big|^{ it/2} \frac{ \Gamma(-\frac{1}{4} + \frac{it}{2}) e^{(1/2- it)^2}}{ \sqrt{8\pi} \Gamma(\frac{1}{4} + \frac{i t}{2})} \frac{L(D_1, 1/2 + it)L(D_2, 1/2 + it)}{ \zeta(1+ 2it)} h(t) \frac{dt}{4\pi}\\ & \quad + \delta_{\Delta_1= \Delta_2} \sum_{\substack{d_1r \tau v w = f_1\\ vwn = f_2}} \sum_m \frac{\mu(d_1)\mu(v) \chi_{\Delta_1}(d_1vm\tau)}{\sqrt{d_1rn\tau} vm}
\int_{-\infty}^{\infty} W_{t}( rnvm) h(t) t \tanh(\pi t) \frac{dt}{4\pi^2} \\
&\quad + e(3/8)\sum_{ d_1r\tau vw = f_1 }\sum_{n, c, m} \frac{\mu(d_1)\mu(v) \chi_{\Delta_1}(d_1vm\tau)}{\sqrt{d_1rn\tau} vm} \frac{ K_{3/2}^+(|\Delta_1|(vwn)^2, |D_2|, c)}{ c} H_{ rnvm}^{\dagger}\Big(\frac{4\pi vwn\sqrt{|\Delta_1D_2|}}{c}\Big)\\
\end{split} \end{displaymath} where $H_{b}(t) = h(t)W_{t}(b)$ and $H^{\dagger}$ is given by \eqref{h0}. \end{theorem}
\emph{Remarks:}\\
1) Specializing $f_1 = f_2 = 1$ we obtain Theorem \ref{thm2}. \\ 2) Recall again that $W_t(x) = 0$ for $t = \pm i/2$, so that contour shifts in the $t$-integral ensure that the $c, n$-sum is absolutely convergent.\\
3) The first term on the right hand side corresponds to the constant function and is obviously indispensable. In all practical applications, the $t$-integral in the second term is of bounded length due to the factor $\exp((1/2 - it)^2)$, so that by subconvexity bounds for $L(D, 1/2 + it)$ this term is dominated by the class number term. The last term can be analyzed as the off-diagonal term in the Kuznetsov formula except that it contains an extra $n$-sum of length $\approx |t|$ from the approximate functional equation, cf.\ \eqref{bound-wt} below. Contrary to its appearance, the diagonal term is symmetric in $f_1, f_2$ (as it should be) and can be written as $$\delta_{\Delta_1 = \Delta_2} \int_{-\infty}^{\infty} \int_{(2)} \frac{\Gamma(\frac{1}{2}(\frac{1}{2} + s +2 it))\Gamma(\frac{1}{2}(\frac{1}{2} + s - 2it))}{\Gamma(\frac{1}{4}+it)\Gamma(\frac{1}{4}-it) \pi^{s} }\frac{e^{s^2} }{s} L(\chi_{\Delta_1}, s)P(f_1, f_2, s) h(t) t \tanh(\pi t) \frac{ds}{2\pi i} \frac{dt}{4\pi^2} $$ where $$P(f_1, f_2, s) = \prod_p \frac{p^{-\alpha_p(s + 1/2)} - p^{(\beta_p - 2)(s + 1/2)} - \chi_{\Delta_1}(p)p^{2s-1/2}(p^{-\alpha_p(s+1/2)} - p^{-\beta_p(s+1/2)} ) }{1 - p^{2s + 1}}$$ with $\alpha_p = \max(v_p(f_1), v_p(f_2))$ and $\beta_p = \min(v_p(f_1), v_p(f_2))$ for the usual $p$-adic valuation $v_p$.
\section{Mean values of \texorpdfstring{$L$}{L}-functions}\label{proof}
This section is devoted to the proof of Proposition \ref{Lfunc}. To begin with, we recall Heath-Brown's large sieve in two variations \cite[Corollaries 3 \& 4]{HB}\footnote{In the original version of \cite[Corollary 4]{HB}, the $n$-sum is restricted to odd numbers $n$, but in the case of fundamental discriminants $\Delta$, the symbol $(\frac{\Delta}{n})$ is also defined for even $n$, and the proof works in the same way.}
\begin{lemma}\label{HBlemma} {\rm a)} Let $N, Q \geqslant 1$, let $S(Q)$ denote the set of real primitive characters of conductor up to $Q$ and let $(a_n)$ be a sequence of complex numbers with $|a_n| \leqslant 1$. Then
$$\sum_{\chi \in S(Q)} \Big|\sum_{n \leqslant N} a_n\chi(n)\Big|^2 \ll N(Q+N) (QN)^{\varepsilon}$$ for every $\varepsilon > 0$.
{\rm b)} Let $D, N \geqslant 1$, $(a_n), (b_{\Delta})$ be two sequences of complex numbers with $|a_m|, |b_{\Delta}| \leqslant 1$, where $b_{\Delta}$ is supported on the set of fundamental discriminants $\Delta$. Then
$$\sum_{|\Delta| \leqslant D} \sum_{n \leqslant N}a_n b_{\Delta} \chi_{\Delta}(n) \ll (DN)^{1+\varepsilon} (D^{-1/2} + N^{-1/2}).$$
\end{lemma}
We start with part (a) of the proposition. As mentioned in the introduction, $L(u \times \chi, 1/2) = 0$ if $u$ is odd for root number reasons. For even $u$ we use the approximate functional equation \eqref{approx-basic} and write \begin{equation}\label{approx}
L({ u} \times \chi_{\Delta}, 1/2 ) = \frac{1}{2\pi i}\int_{(2)} \sum_{n} \frac{\lambda_{ u}(n) \chi_{\Delta}(n)}{n^{1/2 + s}} |\Delta|^s G(s, t_{ u} ) ds \end{equation} with $$G(s, t_{ u} ) = \frac{2 e^{s^2}\Gamma((1/2 + \mathfrak{a} + s + it_{ u})/2)\Gamma((1/2 + \mathfrak{a} + s - it_{ u})/2)}{\Gamma((1/2 + \mathfrak{a}+ it_{ u})/2)\Gamma((1/2 + \mathfrak{a}- it_{ u})/2)\pi^s s} $$ where $\mathfrak{a} = 1$ if $\Delta < 0$ and $\mathfrak{a} = 0$ if $\Delta > 0$. We can treat positive and negative discriminants separately, so that we may assume that $G(s, t_{ u} ) $ is independent of $\Delta$. Eventually we would like to apply the Cauchy-Schwarz inequality, the spectral large sieve inequality and Heath-Brown's large sieve for quadratic characters. The latter requires that the $n$-sum is restricted to odd squarefree integers. Therefore we uniquely factorise $n = 2^{\alpha} n_1n_2^2$ with $n_1, n_2$ odd, $n_1$ squarefree and use the Hecke relations to write \begin{displaymath} \begin{split} &\sum_{n }\frac{\lambda_{ u}(n) \chi_{\Delta}(n)}{n^{1/2 + s}} = \sum_{\alpha}\frac{\lambda_{ u}(2^{\alpha}) \chi_{\Delta}(2^{\alpha})}{2^{\alpha(1/2 + s)}} \sum_{2 \nmid n_1 } \frac{\mu^2(n_1)\lambda_{ u}(n_1) \chi_{\Delta}(n_1)}{n_1^{1/2 + s}} \sum_{(n_2, 2\Delta) = 1 } \frac{ \lambda_{ u}(n_2^2) }{n_2^{1 + 2s}} \sum_{2 \nmid d} \frac{\mu(d) \chi_{\Delta}(d)}{d^{3/2 + 3s}}.\\
\end{split} \end{displaymath} We use M\"obius inversion to detect the condition $(n_2, \Delta) = 1$ and we observe that the $\alpha$-sum depends only on $\Delta$ modulo 8. Therefore \begin{displaymath} \begin{split}
& \sum_{ |\Delta| \leqslant \mathcal{D}} \sum_{n } \frac{\lambda_{ u}(n) \chi_{\Delta}(n)}{n^{1/2 + s}} |\Delta|^s \\
&= \sum_{2 \nmid d f} \frac{\mu(d)\mu(f) \chi_f(d)}{d^{3/2 +3s}f^{1+s}} \sum_{\delta \in \{0, 1, 4, 5\}} P(s; { u}, \delta, f) \sum_{ \substack{ |\Delta'| \leqslant\mathcal{D}/f\\ \Delta' \equiv \bar{f}\delta\, (\text{mod } 8)}} \sum_{ 2 \nmid n_1 } \frac{\mu^2(n_1) \chi_f(n_1)\lambda_{ u}(n_1) (\frac{\Delta'}{n_1d})}{n_1^{1/2 + s}} |\Delta'|^s \end{split} \end{displaymath} where $\Delta' f$ is restricted to negative fundamental discriminants and $$P(s; { u}, \delta, f) := \sum_{\alpha}\frac{\lambda_{ u}(2^{\alpha}) \chi(\delta, 2^{\alpha})}{2^{\alpha(1/2 + s)}} \sum_{2 \nmid n_2 } \frac{ \lambda_{ u}(f^2n_2^2) }{n_2^{1 + 2s}} \ll f^{2\theta+\varepsilon}$$ uniformly in $\Re s \geqslant \varepsilon$ for $\theta = 7/64$ by the Kim-Sarnak bound. In the above formula we define $\chi(\delta, 2^{\alpha}) := \chi_{\Delta}(2^{\alpha})$ for any $\Delta \equiv \delta$ (mod 8).)
We substitute this back into \eqref{approx}. Shifting the contour to the far right, we can truncate the $n_1$-sum at $n_1 \leqslant ( \mathcal{D}\mathcal{T})^{1+\varepsilon} $ for $|t_u| \leqslant \mathcal{T}$ at the cost of a negligible error. Having done this, we shift the contour back to $\Re s = \varepsilon $, truncate the integral at $|\Im s| \leqslant ( \mathcal{D}\mathcal{T})^{\varepsilon}$, again with a negligible error, so that \begin{displaymath} \begin{split}
\sum_{t_{ u} \leqslant \mathcal{T}}& \sum_{\substack{|\Delta| \leqslant \mathcal{D}\\ \Delta \text{ {\rm fund. discr.}}}} \alpha(u) L({u} \times \chi_{\Delta}, 1/2) \ll ( \mathcal{D}\mathcal{T})^{O(\varepsilon)} \sup_{ \substack{N \leqslant ( \mathcal{D}\mathcal{T})^{1+\varepsilon}\\ \Re s = \varepsilon}} \sum_{2 \nmid df} \frac{\mu^2(d)}{d^{3/2+\varepsilon }f^{1-2\theta}} \\
& \times \sum_{t_{ u} \leqslant \mathcal{T}} \Bigg| \alpha(u) \sum_{\substack{ N \leqslant n_1 \leqslant 2N\\ 2 \nmid n_1} } \frac{\mu^2(n_1) \chi_f(n_1)\lambda_{ u}(n_1)}{n_1^{1/2 + s}} \sum_{ \substack{ |\Delta'| \leqslant \mathcal{D}/f\\ \Delta' \equiv \bar{f}\delta\, (\text{mod } 8)}}\Big(\frac{\Delta'}{n_1d}\Big) |\Delta'|^s \Bigg| .
\end{split}
\end{displaymath}
A priori, the right hand side is restricted to even $u$, but by positivity we can extend it to all $u$.
Next we apply the Cauchy-Schwarz inequality.
In the second factor we artificially insert $1/L(\text{sym}^2{ u}, 1)$ at the cost of a factor of $\mathcal{T}^{\varepsilon}$ (by \eqref{lower}) to convert the Hecke eigenvalues into Fourier coefficients and apply
the spectral large sieve inequality \cite[Theorem 2]{DesIw}. This leaves us with bounding
\begin{equation}\label{largesieve}
\begin{split}
& ( \mathcal{D}\mathcal{T})^{O(\varepsilon)} \Big(\sum_{t_u \leqslant \mathcal{T}} |\alpha(u)|^2\Big)^{1/2} \sum_{2 \nmid df} \frac{\mu^2(d)}{d^{3/2+\varepsilon }f^{1-2\theta}} \Bigg( \frac{\mathcal{T}^2 + N}{N} \sum_{\substack{ N \leqslant n_1 \leqslant 2N \\ 2 \nmid n_1}} \mu^2(n_1) \Big| \sum_{ \substack{ |\Delta'| \leqslant \mathcal{D}/f\\ \Delta' \equiv \bar{f}\delta\, (\text{mod } 8)}} \Big(\frac{\Delta'}{n_1d}\Big)|\Delta'|^s \Big|^2\Bigg)^{1/2}
\end{split}
\end{equation}
for $N \leqslant ( \mathcal{D}\mathcal{T})^{1+\varepsilon}$.
For a given odd, squarefree $d \in\mathbb{N}$, the $n_1$-sum equals
\begin{displaymath}
\begin{split}
& \sum_{ r_1r_2 = d } \sum_{\substack{N/r_1 \leqslant n_1 \leqslant 2N/r_1\\ ( n_1, 2 r_2) = 1}} \mu^2(r_1n_1) \Big| \sum_{ \substack{ |\Delta'| \leqslant \mathcal{D}/f\\ \Delta' \equiv \bar{f}\delta\, (\text{mod } 8)\\ (\Delta', r_1) = 1}} \Big(\frac{\Delta'}{n_1r_2}\Big) |\Delta'|^s \Big|^2\\
& \leqslant \sum_{ r_1r_2 = d } \sum_{\substack{ n \leqslant 2Nr_2\\ 2 \nmid n}} \tau(n) \mu^2(n) \Big| \sum_{ \substack{ |\Delta'| \leqslant \mathcal{D}/f\\ \Delta' \equiv \bar{f}\delta\, (\text{mod } 8)\\ (\Delta', r_1) = 1}} \Big(\frac{\Delta'}{n}\Big) |\Delta'|^s \Big|^2.
\end{split}
\end{displaymath} For odd squarefree $n \not= 1$, the map $\Delta' \mapsto (\frac{\Delta'}{n})$ is a primitive quadratic character of conductor $n$, so that by Heath-Brown's large sieve for quadratic characters (Lemma \ref {HBlemma}a) this expression is bounded by $$\ll (\mathcal{D}\mathcal{T})^{\varepsilon}\Big(Nd + \frac{\mathcal{D}}{f}\Big) \frac{\mathcal{D}}{f}.$$ Putting everything together, we complete the proof of Proposition \ref{Lfunc}(a).\\
The proof of part (b) is almost identical except that the spectral large sieve is replaced with the standard bound \cite[Theorem 9.1]{IK} for Dirichlet polynomials. Here we use the approximate functional equation
$$L( \chi_{\Delta}, 1/2 + i t )^2 = \frac{1}{2\pi i}\int_{(2)} \Big( \sum_{n} \frac{\tau(n) \chi_{\Delta}(n)}{n^{1/2 + it + s}} |\Delta|^s \tilde{G}(s, t ) + \epsilon(t)\sum_{n} \frac{\tau(n) \chi_{\Delta}(n)}{n^{1/2 - it + s}} |\Delta|^s \tilde{G}(s, -t ) \Big) ds$$
where $|\epsilon(t)| = 1$ and \begin{equation*} \tilde{G}(s, t) = \frac{ e^{s^2}\Gamma(1/2 +\mathfrak{a} + s + it/2)^2}{\Gamma(1/2+\mathfrak{a} + it/2)^2 \pi^s s} \frac{1}{(\frac{1}{4} + t^2)^2} \prod_{\epsilon_1, \epsilon_2 \in \{\pm 1\}} \hspace{-0.3cm}\Big(\epsilon_1 s - \Big(\frac{1}{2} +\epsilon_2 it\Big)\Big). \end{equation*}
We included the polynomial in order to counteract the pole at $s=1/2 - it$ of $L(\chi_{\Delta}, s + 1/2 + it)^2$ for $\Delta = 1$, so that no residual term arises in the approximate functional equation. The function $\tilde{G}$ has similar analytic properties as $G$ above, and the divisor function $\tau$ satisfies the same Hecke relations as $\lambda_u$. The proof is now almost literally the same, except that the factor $(\mathcal{T}^2 + \mathcal{N})/\mathcal{N}$ in \eqref{largesieve} is $(\mathcal{T} + \mathcal{N})/\mathcal{N}$. Thus the proof of Proposition \ref{Lfunc} is concluded.
\section{Interlude: special functions and oscillatory integrals} In this rather technical section we compile various sums and integrals over Bessel functions and other oscillatory integrals that we need as a preparation for the proof of Theorem \ref{thm1}. To start with, the following lemma is a half-integral weight version of \cite[Lemma 5.8]{Iw1}, but with a somewhat different proof. \begin{lemma}\label{lem2}
Let $x > 0$ $ A \geqslant 0$, $K > 1$. Let $w$ be a smooth function with support in $[1, 2]$ satisfying $w^{(j)}(x) \ll_{\varepsilon} K^{j\varepsilon}$ for $j \in \mathbb{N}_0$. Then there exist smooth functions $w_0, w_+, w_-$ such that for every $j \in \mathbb{N}_0$ we have $$w_0(x) \ll_A K^{-A},$$ \begin{equation}\label{boundsw} \frac{d^j}{dx^j}w_{\pm}(x) \ll_{j, A } \Big(1 + \frac{K^2}{x}\Big)^{-A} \frac{1}{x^j} \end{equation} and \begin{equation}\label{bessel-sum} \sum_{k\text{ {\rm even}}} i^k w\left(\frac{k}{K}\right) J_{k-3/2}(x) = \sum_{\pm} e^{\pm ix} w_{\pm}(x) + w_0(x). \end{equation} The implied constants in \eqref{boundsw} depend on the $B$-th Sobolev norm of $w$ for a suitable $B= B(A, j)$. The functions $w_{\pm}$ are explicitly given in \eqref{W2pm} and \eqref{W2m}. \end{lemma}
\begin{proof} We denote the left hand side of \eqref{bessel-sum} by $W(x)$. For $k \in \mathbb{N}$, by \cite[8.411.13]{GR} we have \begin{equation*} J_{k - 3/2}(x) = \int_{-1/2}^{1/2} e(k \theta) e(-3\theta/2) e^{-ix \sin(2\pi \theta)} d\theta - \frac{(-1)^k}{\pi} \int_0^{\infty} e^{-(k - 3/2)\theta - x \sinh\theta} d\theta. \end{equation*} We put $$W_1(x) =- \sum_{k\text{ {\rm even}}} i^k w\left(\frac{k}{K}\right)\frac{1}{\pi} \int_0^{\infty} e^{-(k - 3/2)\theta - x \sinh\theta} d\theta,$$ $$W_1^{\pm}(x) = \frac{1}{2}e^{\mp i x} \sum_{k \in \mathbb{Z}} (\pm i)^k w\left(\frac{k}{K}\right) \int_{-1/2}^{1/2} e(k \theta) e(-3\theta/2) e^{-ix \sin(2\pi \theta)} d\theta. $$
Applying Poisson summation modulo 4, we have $$ \sum_{k\text{ even}} i^k w\left(\frac{k}{K}\right) e^{-k\theta} = \frac{1}{4} \sum_{\substack{\kappa \, (\text{mod }4)\\ \kappa \equiv 0, 2 \, (\text{mod } 4)}} i^{\kappa} \sum_{h \in \mathbb{Z}} e\left(\frac{h\kappa}{4}\right) \int_{-\infty}^{\infty} w\left(\frac{y}{K}\right) e^{-y\theta} e\left(\frac{y h}{4}\right) dy.$$
The $h = 0$ term vanishes (as do all even $h$), and by partial integration the other terms are bounded by $O_n( K |h|^{-n}( K^{-(1-\varepsilon)n} + \theta^n) e^{-K\theta})$ for any $n \in \mathbb{N}$, so that $$W_1(x) \ll_n K \int_0^{\infty} (K^{-(1-\varepsilon)n} + \theta^n) e^{-K\theta} e^{-x \sinh \theta} d\theta \ll_n K^{-(1-\varepsilon)n}.$$ Again by Poisson summation we have \begin{displaymath} \begin{split} W^+_1(x) & =\frac{1}{2}e^{- i x} \int_{-1/2}^{1/2} e(-3\theta/2) e^{-ix \sin(2\pi \theta)} \sum_{h \in \mathbb{Z}} \int_{-\infty}^{\infty} e\left(\frac{y}{4}\right) w\left(\frac{y}{K}\right) e(y\theta) e(-hy) dy \, d\theta\\ & = \frac{e(3/8)}{2}e^{- i x} \int_{-1/4}^{3/4} e(-3\theta/2) e^{ix \cos(2\pi \theta)} \sum_{h \in \mathbb{Z}} \int_{-\infty}^{\infty} w\left(\frac{y}{K}\right) e(y\theta) e(-hy) dy \, d\theta. \end{split} \end{displaymath} Since $-1/4 < \theta \leqslant 3/4$, we see by partial integration in the $y$-integral that the contribution of $h \not= 0$ is $O_A(K^{-A})$. Let $v$ be a smooth function with compact support in $[-2, 2]$, identically equal to $1$ on $[-1, 1]$. Then for $h = 0$ we can smoothly truncate the $\theta$-integral by inserting the function $v(\theta K^{9/10})$, the error being again $O_A(K^{-A})$ by partial integration. We obtain $W_1^+(x) = W_2^+(x) + W_2(x)$, where $W_2(x) \ll_A K^{-A}$ and after changing variables \begin{equation}\label{W2pm} \begin{split} W_2^+(x)& = \frac{e(3/8)}{2}e^{- i x} \int_{-\infty}^{\infty} v(\theta K^{-1/10}) e\Big(\frac{-3\theta}{2K}\Big) e^{ix \cos(2\pi \theta/K)} \int_{-\infty}^{\infty} w(y) e(y\theta) dy \, d\theta. \end{split} \end{equation} Then for $j \in \mathbb{N}_0$ we have \begin{equation}\label{dW} \begin{split} \frac{d^j}{dx^j}W_2^+(x) = \frac{e(3/8)}{2} \int_{-\infty}^{\infty} w(y) \int_{-\infty}^{\infty} v(\theta K^{-1/10}) e^{i \phi( \theta; x, y)} (i(\cos(2\pi \theta/K) - 1))^j d\theta \, dy \end{split} \end{equation} with $\phi(\theta; x, y) = -3\pi \theta/K+ 2\pi \theta y + x ( \cos(2\pi \theta/K)-1)$ satisfying $$\frac{d}{d\theta} \phi(\theta; x, y) = -\frac{3\pi}{K} + 2\pi y + 2\pi \frac{x}{K} \sin\Big(2\pi \frac{\theta}{K}\Big) \quad \text{and} \quad \frac{d^j}{d\theta^j} \phi(\theta; x, y) \ll \frac{x}{K^j}, \,\,j \geqslant 2.$$ In the following we frequently use the Taylor expansions $\sin(t) = t + O(t^3)$ and $\cos(t) = 1 + t^2/2 + O(t^4)$.
We first extract smoothly the range $|\theta| \leqslant \frac{1}{100} K^2/x$. Here we observe that the derivative $\frac{d}{d\theta} \phi(\theta; x, y)$ cannot be too small (it is important that $w$ is supported on $[1, 2]$, not on $[0, 1]$), and we
apply \cite[Lemma 8.1]{BKY} with $$\beta - \alpha \ll \frac{K^2}{x}, \quad X = \Big(\frac{K}{x}\Big)^{2j}, \quad U = \min(K^{1/10}, K^2/x), \quad R = 1 , \quad Y = x, \quad Q = K.$$ In this way we obtain a contribution of $$\ll_n (\beta - \alpha)X[(QR/\sqrt{Y})^{-n} + (RU)^{-n}] \ll \frac{1}{x^j} \Big(\frac{K^2}{x}\Big)^{1 + j} \Big( \Big(\frac{K}{\sqrt{x}}\Big)^{-n} + \Big(\frac{K^2}{x}\Big)^{-n} + K^{- n/10}\Big)$$ to \eqref{dW} for every $n \geqslant 0$. This is easily seen to be
$$\ll_{j, A} \frac{1}{x^j} \Big(1 + \frac{K^2}{x}\Big)^{-A}$$
for every $A > 0$. For the portion $|\theta| \gg K^2/x$ we integrate by parts in the $y$ integral and apply trivial estimates to obtain a bound
$$\ll_{j, A} \int_{|\theta| \gg K^2/x} (1 + \theta)^{-A} \Big(\frac{\theta}{K}\Big)^{2j} d\theta $$ which is easily seen to be $$\ll_{j, A} \min\Big(\Big(\frac{K^2}{x}\Big)^{-A} \frac{1}{x^j}, \frac{1}{K^{2j}}\Big) \ll \frac{1}{x^j} \Big(1 + \frac{K^2}{x}\Big)^{-A}.$$ The same analysis works for $W_1^-(x) = W_2^-(x) + \tilde{W}_2(x)$ where \begin{equation}\label{W2m} \begin{split} W_2^-(x)& = \frac{e(-3/8)}{2}e^{+ i x} \int_{-\infty}^{\infty} v(\theta K^{-1/10}) e\Big(\frac{-3\theta}{2K}\Big) e^{-ix \cos(2\pi \theta/K)} \int_{-\infty}^{\infty} w(y) e(y\theta) dy \, d\theta. \end{split} \end{equation}
We put $w_{\pm} = W_2^{\pm}$ and $w_0 = W_1 + W_2 + \tilde{W}_2$, and the lemma follows on noting that $\frac{1}{2}(i^k + (-i)^k) = i^k \delta_{2\mid k}$.
\end{proof}
\emph{Remarks:} 1) It is clear from the proof that if $w$ depends on other parameters in a real- or complex-analytic way with control on derivatives, then $w_{\pm} = W_2^{\pm}$, defined in \eqref{W2pm}, depends on these parameters in the same way. We will use this observation in Section \ref{thm1} and \eqref{off-off}.
2) The bound \eqref{boundsw} remains true for $A \geqslant -1/2$. In the case, the claim follows for $x \geqslant K^2$ from the asymptotic formula \cite[8.451.1 \& 7 \& 8]{GR}. We state this for completeness, but we do not need it here. \\
We need a similar formula for the transforms occurring in \eqref{hast} and \eqref{h0}. \begin{lemma}\label{bessel-kuz} Let $A, T \geqslant 2$ and let ${\tt h}$ be a smooth function with support in $[T, 2T]$ satisfying ${\tt h}^{(j)}(t) \ll T^{-j}$ for $j \in \mathbb{N}_0$.
{\rm a)} We have \begin{displaymath} \begin{split} & {\tt h}^*(x) = \frac{T^2}{\sqrt{x}} \Big(1 + \frac{T^2}{x}\Big)^{-A} \sum_{\pm} e(\pm x) H^{\pm}_A(x) + K^{\pm}_A(x)\\
\end{split}
\end{displaymath} where $K^{\pm}_A(x) \ll_A (T+x)^{-A}$ and $x^j \partial_x^j H^{\pm}_A(x) \ll_{A, j} 1$. An analogous asymptotic formula holds for ${\tt h}^{\dagger}(x)$.
{\rm b)} We have $$h^{**}(x) = T \Big(\frac{x}{T} + \frac{T}{x}\Big)^{-A} H_A(x) + \tilde{K}_A(x)$$ where $\tilde{K}_A(x) \ll_A (T+ x)^{-A}$ and $x^j \partial_x^j H_A(x) \ll_{A, j} 1$. \end{lemma}
\begin{proof} a) For the first part we recall the uniform asymptotic formula \cite[7.13.2(17)]{EMOT}.
$$\frac{\pi i}{\cosh(\pi t)} J_{2 it}(x) = \sum_{\pm} \frac{ e^{\pm ix \mp i \omega(x, t)}}{x^{1/2} } f^{\pm}_M(x, t) + O_M((|t| + x)^{-M} )$$ where
$\omega(x, t) = |t| \cdot \text{arcsinh} (|t|/x) - \sqrt{t^2 + x^2} + x$ and \begin{equation}\label{flat}
x^i |t|^j\frac{\partial^i}{\partial x^i} \frac{\partial^j}{\partial t^j} f^{\pm}_{M}(x, t) \ll_{i, j, M} 1\end{equation}
for every $M \geqslant 0$. The error term in \cite{EMOT} is $O(x^{M })$, but for small $x$ the error term $O(|t|^{-M} )$ follows from the power series expansion \cite[8.440]{GR}.
Partial integration in the form of \cite[Lemma 8.1]{BKY} with $U = T$, $Y = Q = T+x$, $R= \text{arcsinh}(T/x) \gg T/x$ shows that $$x^j \frac{\partial^j}{\partial x^j} \int_{\mathbb{R}} e^{ \mp i \omega(x, t)} f^{\pm}_{ M}(x, t) h(t) t \frac{dt}{4\pi^2} \ll_{j, A, M} T^2 \Big( 1 + \frac{T^2}{x}\Big)^{-A}$$ and the claim follows.
b) For the proof of the second part we distinguish 3 ranges. For $x > 10 T$ the claim follows easily from the rapid decay of the Bessel $K$-function and its derivatives. For $x < T/10$ we use the uniform asymptotic expansion \cite[7.13.2(19)]{EMOT} (along with the power series expansion \cite[8.485, 8.445]{GR} for very small $x$) $$ \cosh(\pi t) {K}_{2it}(x) = \sum_{\pm} e^{\pm i \tilde{\omega}(x, t)} \tilde{f}^{\pm}_M(x, t)
+ O(|t|^{-M}), \quad \tilde{\omega}(x, t) = |t| \cdot \text{arccosh} \frac{|t|}{x} - \sqrt{t^2 - x^2}$$
where $\tilde{f}^{\pm}_M$ satisfies the analogous bounds in \eqref{flat}. Again integration by parts (\cite[Lemma 8.1]{BKY} with $U = Y = Q = T$, $R= \text{arccosh}(T/x) \geqslant 1$) confirms the claim in this range. Finally, for $x \asymp T$ we use the integral representation \cite[8.432.4]{GR}
$$\cosh(\pi t) K_{2it}(x) = \frac{\pi}{2} \int_{-\infty}^{\infty} \cos(x \sinh \pi u) e(tu) du.$$
This integral is not absolutely convergent, but partial integration shows that the tail is very small, and we can in fact truncate the integral at $|u| \leqslant \varepsilon \log T$ at the cost of an admissible error $O(T^{-A})$. Thus we are left with bounding \begin{displaymath} \begin{split} & \frac{d^j}{dx^j} \int_{-\infty}^{\infty} \int_{-\varepsilon \log T}^{\varepsilon \log T} \cos(x \sinh \pi u) e(tu) h(t) t \, du \, dt \\
& \ll \int_{T \leqslant |t| \leqslant 2T} \int_{-\varepsilon \log T}^{\varepsilon \log T}
|\sinh(\pi u)|^j (1+| u|T)^{-B} t \, du\, dt \ll T.\\ \end{split} \end{displaymath}
if $B$ is chosen sufficiently large with respect to $j$. \end{proof}
For large arguments, the Bessel function $J_{ir}(y)$ behaves like an exponential. More precisely, by \cite[8.451.1 \& 7 \& 8]{GR} we have an asymptotic expansion which we will need later: \begin{equation}\label{bessel-approx} \begin{split}
\sum_{\pm} (\mp) &\frac{J_{ir}(2\pi x) \cos(\pi/4 \pm \pi i r/2)}{\sin(\pi i r)} = \sum_{\pm} \frac{e(\pm x)}{2\pi\sqrt{x}}\sum_{k=0}^{n-1} \frac{i^k(\pm 1)^k}{(4\pi x)^{k}} \frac{\Gamma(ir + k + 1/2)}{k! \Gamma(ir - k + 1/2)} + O\Big(\Big(\frac{|r|^2}{x}\Big)^{-n}\Big)
\end{split} \end{equation}
for $r \in \mathbb{R}$, $x \geqslant 1$ and fixed $n \in \mathbb{N}$. This is useful as soon as $x \geqslant r^2$.\\
The following lemma is essentially an application of Stirling's formula.
\begin{lemma} Let $k \geqslant 1$, $s = \sigma + it \in \mathbb{C}$ with $k + \sigma \geqslant 1/2$, $M \in \mathbb{N}$. Then
$$\frac{\Gamma(k+s)}{\Gamma(k) }= k^s G_{M, \sigma}(k, t) + O_{\sigma, M}((k + |t|)^{-M})$$ where \begin{equation}\label{flat1} k^{\frac{i}{2} + j} \frac{d^{i}}{dt^{i}} \frac{d^{j} }{dk^{j}} G_{M, \sigma}(k, t) \ll_{M, \sigma, i, j} \Big( 1 + \frac{t^2}{k}\Big)^{-M} \end{equation} for $i, j \in \mathbb{N}_0$. Moreover, \begin{equation}\label{asymp}
\frac{\Gamma(k+\sigma + it)}{\Gamma(k) }= k^s \exp\Big(-\frac{t^2}{2k}\Big)\Big(1 + O_{\sigma}\Big(\frac{|t|}{k} + \frac{t^4}{k^3}\Big)\Big). \end{equation} \end{lemma}
\begin{proof} This is a standard application of Stirling's formula. First of all, since $$\frac{\Gamma(k+s)}{\Gamma(k) } k^{-s}= \frac{\Gamma(k+\sigma)}{\Gamma(k) }k^{-\sigma} \frac{\Gamma(k+\sigma + it)}{\Gamma(k + \sigma) } (k+\sigma)^{-it} (1 + \sigma/k)^{it}$$ with \begin{equation}\label{sigma}
\frac{d^{j_1}}{dk^{j_2}} \frac{d^{j_1} }{dt^{j_2}}(1 + \sigma/k)^{it} \ll_{\sigma, j_1, j_2} \frac{1}{k^{j_1+j_2}} \Big(1 + \frac{|t|}{k}\Big)^{j_1}, \quad (1 + \sigma/k)^{it} = 1 + O_{\sigma}(|t|/k),
\end{equation}
it suffices for both statements to treat the two cases $s= \sigma \in \mathbb{R}$ fixed and $s = it \in i\mathbb{R}$. The first case is very simple, so we display the details for the second case. We have $$\frac{\Gamma(k+it)}{\Gamma(k) } k^{-it} = \exp\big(\alpha (k, t) + i\beta (k, t)\big)
\Big(\tilde{G}_{M }(k, t) + O_{M }((k+|t|)^{-M})\Big) $$ where $\tilde{G}_{M }$ satisfies \begin{equation}\label{G}
(k+|t|)^{j_1+j_2} \frac{d^{j_1}}{dk^{j_1}} \frac{d^{j_2} }{dt^{j_2}} \tilde{G}_{M }(k, t) \ll_{M, j_1, j_2} 1, \quad \tilde{G}_{M }(k, t) = 1 + O((k+|t|)^{-1})
\end{equation}
and \begin{displaymath} \begin{split} \alpha (k, t) & = - t \arctan\frac{t}{k } + \frac{k - 1/2}{2} \log\Big(1 + \frac{ t^2}{k^2}\Big),\\
\beta (k, t) &= t \Big(\log\sqrt{1 + \frac{ t^2}{k^2}} - 1\Big) + \Big(k - \frac{1}{2}\Big) \arctan\frac{t}{k }. \end{split} \end{displaymath}
It is not hard to see that
$$ \alpha(k, t) \leqslant - c \min\Big(\frac{t^2}{k}, |t|\Big)$$
for some absolute constant $c$ (in fact, $c = (\pi - \log 4)/4 = 0.438\ldots$ is the optimal constant). In particular, $\Gamma(k+it)/\Gamma(k)$ is exponentially decreasing as soon as $|t| \geqslant k^{1/2}$. Moreover, by a Taylor argument we have
$$\alpha(k, t) = -\frac{t^2}{2k} + O\Big(\frac{t^2}{k^2} + \frac{t^4}{k^3}\Big), \quad \beta(k, t) \ll \frac{|t|}{k} + \frac{|t|^3}{k^2}.$$
This proves \eqref{asymp}. To prove \eqref{flat1}, we need to bound the derivatives of $\alpha$ and $\beta$ which is most quickly done by using Cauchy's integral formula. Note that both $\alpha$ and $\beta$ have a branch cut at the two rays $\pm t/k \in [ i, i \infty)$. We assume that $k$ is sufficiently large (otherwise there is noting to prove) and we choose a circle $C_1$ about $k$ of radius $k/100$ and a circle $C_2$ about $t$ of radius $\sqrt{k}/10$. Then $w/z$ is away from the branch cuts for $z \in C_1$, $w \in C_2$, and we have $$\alpha(z, w) \ll \frac{|w|^2}{|z|} + |w| \ll \frac{|t|^2 +k}{k}, \quad \beta(z, w) \ll \frac{|w|}{|z|} + \frac{|w|^3}{|z|^2} \ll \frac{|t| }{k} + \frac{|t|^3}{k^2} + \frac{1}{k^{1/2}}.$$ for $z \in C_1$, $w\in C_2$. From Cauchy's integral formula we conclude \begin{equation}\label{alpha} \frac{d^{i}}{dt^{i}} \frac{d^{j} }{dk^{j}}\alpha (k, t)\ll_{i, j} \Big(1 + \frac{t^2}{k}\Big)k^{-\frac{i}{2} - j} , \quad
\frac{d^{i}}{dt^{i}} \frac{d^{j} }{dk^{j}}\beta (k, t)\ll_{i, j} \Big(1 + \frac{t^2}{k}\Big)^2k^{-\frac{i}{2} - j} \end{equation} for $i, j \in \mathbb{N}_0$. Combining \eqref{sigma}, \eqref{G}, \eqref{alpha} completes the proof of \eqref{flat1}. \end{proof}
We apply this to the function $\mathcal{G}(k, t_{\tt u}, s)$ defined in \eqref{defG}.
\begin{cor}\label{cor19} Let $A \geqslant 0$, $ \sigma \geqslant -1/4$ and let $t \in \mathbb{R}$, $t_{\tt u} \in \mathbb{R} \cup [-i/2, i/2]$, $k \in 2\mathbb{N}$. Then \begin{equation}\label{G1}
\mathcal{G}(k, t_{\tt u}, \sigma + 1/2 + it) \ll_{A, \sigma} k^{-1/4 + 2\sigma} \left(1 + \frac{|t|^2 + |t_{\tt u}|^2}{k} \right)^{-A} . \end{equation} Moreover, for $v \in \mathbb{C}$
we have $$ \mathcal{G}(k, t_{\tt u}, v + 1/2 + it)\mathcal{G}(k, t_{\tt u}, v + 1/2 - it) = {\tt G}_M(k, t_{\tt u}, t, v) + O_{\Re v, \Re w, M}(k^{-M})$$ with \begin{equation}\label{G2} \begin{split} k^{j_1 + \frac{j_2}{2} + \frac{j_3}{2}}& \frac{d^{j_1}}{dk^{j_1}} \frac{d^{j_2} }{dt^{j_2}}\frac{d^{j_3} }{d\tau^{j_3}}
{\tt G}_M(k,\tau, t, v, w) \ll_{ \textbf{j}, \Re v, \Re w, M} k^{-1/2 +2 \Re v + 2\Re w} (1 + |\Im v|)^{j_1}
\end{split} \end{equation}
for $\textbf{j} \in \mathbb{N}_0^3$. Finally, for $t, \tau \ll k^{2/3}$ we have
\begin{equation}\label{taylor}
\mathcal{G}(k, \tau, 1/2 + it) \mathcal{G}(k, \tau, 1/2 - it) = \frac{16}{\pi k^{1/2}} \exp\Big(-\frac{2(t + \tau/2)^2+ 2(t - \tau/2)^2}{k}\Big) \Big(1 + O(k^{-1/3})\Big).
\end{equation} \end{cor} Recalling the definition \eqref{defV3} of $V_t(x; k, t_{\tt u})$ we conclude from \eqref{G1} and appropriate contour shifts the uniform bounds \begin{equation}\label{size-restr} \begin{split}
k^{j_1 + \frac{j_2}{2} + \frac{j_3}{2}}x^{j_4} \frac{d^{j_1}}{dk^{j_1}} \frac{d^{j_2} }{dt^{j_2}}\frac{d^{j_3} }{d\tau^{j_3}}\frac{d^{j_4}}{dx^{j_4}} V_t(x; k, \tau) \ll_{A, \textbf{j}} k^{-1/2} \Big(1 + \frac{x}{k^4}\Big)^{-A} \Big(1 + \frac{|t|^2 + |\tau|^2}{k} \Big)^{-A} \end{split}
\end{equation}
for $A > 0$, $\textbf{j} \in \mathbb{N}_0^4$.
In a similar, but simpler fashion we also apply this to the weight function $W_t$ defined in \eqref{v-t} and state the bound \begin{equation}\label{bound-wt}
(1+|t|)^{j_1} x^{j_2} \frac{d^{j_1}}{dt^{j_1}}\frac{d^{j_2}}{dx^{j_2}} W_t(x) \ll_{A, j_1, j_2} \Big(1 + \frac{x}{1 + |t|}\Big)^{-A} \end{equation} for $A \geqslant 0$, $j_1, j_2 \in \mathbb{N}_0$.
\section{A weak version of Theorem \ref{thm1}}\label{weakversion}
In this section we present a relatively soft argument that provides the upper bound $\mathcal{N}_{\text{av}}(K) \ll K^{\varepsilon}$. This will useful later in order to estimate certain error terms later. By \eqref{Nav} and \eqref{Fh} we have \begin{displaymath} \begin{split}
\mathcal{N}_{\text{av}}(K) \ll &\frac{1}{K^2}\sum_{k \in 2\mathbb{N}} W\Big(\frac{k}{K}\Big) \sum_{h \in B_{k-1/2}^+(4)} \int_{-\infty}^{\infty} \int_{\Lambda_{\text{\rm ev}}} \frac{\Gamma(k-3/2)}{(4\pi)^k \| h \|^2} \sum_{f_1, f_2} \frac{1}{f_1f_2}\\
& \sum_{D_1, D_2 < 0} \frac{c_h(|D_1|)c_h(|D_2|)P(D_1; {\tt u}) \overline{P(D_2; {\tt u})} }{ |D_1D_2|^{ k/2 }} \Big(\frac{|D_2|f_2^2}{|D_1|f_1^2}\Big)^{it}V_{ t}(|D_1D_2|(f_1f_2)^2 ; k, t_{\tt u}) d{\tt u} \, dt. \end{split} \end{displaymath}
By \eqref{size-restr} we have $t, t_{\tt u} \ll K^{1/2 + \varepsilon}$ (up to a negligible error). We insert a smooth partition of unity into the $t_u$-integral and attach a factor $w(|t_{\tt u}|/\mathcal{T}_{\text{spec}})$
where $w$ has support in $[1, 2]$ unless $\mathcal{T}_{\text{spec}} = 1$,
in which case $w$ has support in $[0, 2]$. Let \begin{displaymath} \begin{split}
\mathcal{N}_{\text{av}}(K, ; &\mathcal{T}_{\text{spec}}):=\frac{1}{K^2}\sum_{k \in 2\mathbb{N}} W\Big(\frac{k}{K}\Big) \sum_{h \in B_{k-1/2}^+(4)} \int_{-\infty}^{\infty} \int_{\Lambda_{\text{\rm ev}}} \frac{\Gamma(k-3/2)}{(4\pi)^k \| h \|^2} w\Big( \frac{|t_{\tt u}|}{\mathcal{T}_{\text{spec}}}\Big)
\sum_{f_1, f_2} \frac{1}{f_1f_2}\\
& \sum_{D_1, D_2 < 0} \frac{c_h(|D_1|)c_h(|D_2|)P(D_1; {\tt u}) \overline{P(D_2; {\tt u})} }{ |D_1D_2|^{ k/2 }} \Big(\frac{|D_2|f_2^2}{|D_1|f_1^2}\Big)^{it}V_{ t}(|D_1D_2|(f_1f_2)^2 ; k, t_{\tt u}) d{\tt u} \, dt. \end{split} \end{displaymath} for $\nu = 0, 1, 2, \ldots$ and $ \mathcal{T}_{\text{spec}} = 2^{\nu}\ll K^{1/2 + \varepsilon}$.
This section is then devoted to the proof of the bound \begin{equation}\label{weak}
\mathcal{N}_{\text{av}}(K; \mathcal{T}_{\text{spec}}) \ll_{\varepsilon} 1 + \frac{\mathcal{T}_{\text{spec}}^2}{K^{1-\varepsilon}}
\end{equation} for $\varepsilon>0$. We will now sum over $h$ using Lemma \ref{lem1}. This yields a diagonal term and an off-diagonal term that we treat separately in the following two subsection. Throughout, the letter $D$, with or without subscripts, shall always denote a \emph{negative} discriminant unless stated otherwise. Let letter $A$ shall denote an arbitrarily large fixed constant, not necessarily the same on every occurrence.
\subsection{The diagonal term}\label{10.1}
The contribution to $\mathcal{N}_{\text{av}}(K; \mathcal{T}_{\text{spec}})$ of the diagonal term from Lemma \ref{lem1} is bounded by
\begin{displaymath}
\begin{split}
& \ll\frac{1}{K }\int_{-\infty}^{\infty} \int_{ \Lambda_{\text{\rm ev}} }w\Big( \frac{|t_{\tt u}|}{\mathcal{T}_{\text{spec}}}\Big)
\sum_{f_1,f_2} \frac{1}{f_1f_2} \sum_{D} \frac{ |P(D ; {\tt u})|^2 }{ |D|^{3/2}} \sup_{k \ll K} |V_t((|D|f_1f_2)^2; k, t_{\tt u}) |d{\tt u} \, dt .
\end{split}
\end{displaymath}
For the constant function ${\tt u} = \sqrt{3/\pi}$ we have $P(D; \sqrt{3/\pi}) \ll H(D)$. Recalling \eqref{size-restr} and Lemma \ref{hur2}, this gives a total contribution of
$$\ll \frac{1}{K } \int_{-\infty}^{\infty} \sum_{f_1 ,f_2} \frac{1}{f_1f_2} \sum_D \frac{H(D)^2}{|D|^{3/2}} \Big( 1 + \frac{|D|f_1 f_2}{K^2}\Big)^{-10} \Big(1 + \frac{|t|^2}{K}\Big)^{-10}dt \ll \sum_{f_1 ,f_2} \frac{1}{(f_1f_2)^{3/2}} \ll 1$$
if $\mathcal{T}_{\text{spec}} = 1$ and otherwise the contribution vanishes.
Similarly, by \eqref{eisen2}, \eqref{eisen-L}, \eqref{eisen-L2} and \eqref{lower}, the Eisenstein spectrum contributes
\begin{displaymath}
\begin{split}
& \ll \max_{1 \leqslant R \leqslant K^{2+\varepsilon}} \frac{1}{RK^{1-\varepsilon}} \int_{|\tau| \ll \mathcal{T}_{\text{spec}}} |\zeta(1/2 + i\tau)|^2 \sum_{R \leqslant |\Delta| \leqslant 2R} |L(\chi_{\Delta}, 1/2 + i\tau)|^2 d\tau.
\end{split}
\end{displaymath}
We recall our convention that $\Delta$ denotes a negative fundamental discriminant, $D = \Delta f^2$ an arbitrary negative discriminant (where $f$ here has nothing to do with $f_1, f_2$ above). In the above bound we have already executed the sum over $f$. By Proposition \ref{Lfunc}(b) and
a standard bound for the fourth moment of the Riemann zeta-function this is $O(K^{-1/2 + \varepsilon})$.
By \eqref{katok-Sarnak}, \eqref{key}, \eqref{key-simple} and \eqref{lower}, the contribution of the cuspidal spectrum is at most
\begin{displaymath}
\begin{split}
& \ll \max_{1 \leqslant R \leqslant K^{2+\varepsilon}} \frac{1}{R K^{1-\varepsilon}} \sum_{t_{ u} \ll \mathcal{T}_{\text{spec}}} L({ u}, 1/2) \sum_{ R \leqslant |\Delta| \leqslant 2R} L({ u} \times \chi_{\Delta}, 1/2 ) \ll \mathcal{T}^2_{\text{spec}}K^{\varepsilon-1} . \end{split}
\end{displaymath} by Corollary \ref{Lfunc-cor}.
All of these bounds are consistent with \eqref{weak}.
\subsection{The off-diagonal term: generalities}\label{112} We now consider the off-diagonal term in Lemma \ref{lem1} from the sum over $h$. Here we need to bound \begin{displaymath} \begin{split}
\frac{1}{K^2}\sum_{k \in 2\mathbb{N}} &W\left(\frac{k}{K}\right) i^k \int_{-\infty}^{\infty} \int_{\Lambda_{\text{\rm ev}}} w\Big( \frac{|t_{\tt u}|}{\mathcal{T}_{\text{spec}}}\Big)
\sum_{f_1, f_2} \sum_{D_1, D_2 } \frac{P(D_1; {\tt u})\overline{P(D_2; {\tt u})} }{ f_1f_2 |D_1 D_2|^{ 3/4 }} \Big(\frac{|D_2|f_2^2}{|D_1|f_1^2}\Big)^{it}\\
& V_t(|D_1D_2|(f_1f_2)^2, k, t_{\tt u}) \sum_{4 \mid c} \frac{K^+_{3/2}(|D_1|, |D_2|, c)}{c} J_{k-3/2}\Big(\frac{4\pi \sqrt{|D_1D_2|}}{c}\Big) d{\tt u} \, dt . \end{split} \end{displaymath}
We first sum over $k$ using Lemma \ref{lem2}. Up to a negligible error, we obtain
\begin{displaymath} \begin{split}
\frac{1}{K^2} \int_{-\infty}^{\infty} & \int_{\Lambda_{\text{\rm ev}}} w\Big( \frac{|t_{\tt u}|}{\mathcal{T}_{\text{spec}}}\Big)
\sum_{f_1, f_2} \sum_{D_1, D_2 } \frac{P(D_1; {\tt u})\overline{P(D_2; {\tt u})} }{ f_1f_2 |D_1 D_2|^{ 3/4 }} \Big(\frac{|D_2|f_2^2}{|D_1|f_1^2}\Big)^{it}\\
& \sum_{4 \mid c} \frac{K^+_{3/2}(|D_1|, |D_2|, c)}{c} e\Big(\pm \frac{2\sqrt{|D_1D_2|}}{c}\Big) \tilde{V}\Big(|D_1D_2|(f_1f_2)^2, \frac{\sqrt{|D_1D_2|}}{c}, t, t_{\tt u}\Big) d{\tt u} \, dt \end{split} \end{displaymath} where
\begin{equation}\label{tildeV} \begin{split} y^{j_1} K^{\frac{1}{2}(j_2+j_3)} & x^{j_4} \frac{d^{j_1}}{dy^{j_1}} \frac{d^{j_2} }{dt^{j_2}}\frac{d^{j_3} }{d\tau^{j_3}}\frac{d^{j_4}}{dx^{j_4}} \tilde{V}(x, y, t, \tau) \\
&\ll_{A, \textbf{j}} K^{-1/2} \Big(1 + \frac{x}{K^4}\Big)^{-A} \Big(1 + \frac{|t|^2 + |\tau|^2}{K} \Big)^{-A} \Big(1 + \frac{K^2}{y}\Big)^{-A} \end{split} \end{equation} for any $A \geqslant 0$, $\textbf{j} \in \mathbb{N}_0^4$, cf.\ \eqref{size-restr} and the remark after Lemma \ref{lem2}. In order to apply Voronoi summation, we open the Kloosterman sum and are left with bounding
\begin{equation}\label{leftwith} \begin{split}
\frac{1}{K^2}\sum_{4 \mid c}& \underset{\substack{d\,(\text{mod }c)\\ (d, c) = 1}}{\max}\Big| \int_{\Lambda_{\text{\rm ev}}} w\Big( \frac{|t_{\tt u}|}{\mathcal{T}_{\text{spec}}}\Big)
\sum_{f_1, f_2} \sum_{D_1, D_2 } \frac{P(D_1; {\tt u})\overline{P(D_2; {\tt u})} }{ f_1f_2 |D_1 D_2|^{ 3/4 }} \\
& e\Big(\pm \frac{2\sqrt{|D_1D_2|}}{c}\Big) e\Big( \frac{|D_1|d + |D_2|\bar{d}}{c}\Big)V^{\ast}\Big(|D_1|f_1^2, |D_2|f_2^2, \frac{\sqrt{|D_1D_2|}}{c}, t, t_{\tt u}\Big) d{\tt u} \Big| \end{split} \end{equation} where \begin{equation}\label{Vastdef}
V^{\ast}(x_1, x_2, y, \tau) = \int_{-\infty}^{\infty}
\Big(\frac{x_2}{x_1}\Big)^{it} \tilde{V}(x_1x_2, y, t, \tau) dt.
\end{equation} Integration by parts shows that
\begin{equation}\label{astV} \begin{split} &y^{j_1} K^{\frac{1}{2}j_2} x_1^{j_3}x_2^{j_4} \frac{d^{j_1}}{dy^{j_1}} \frac{d^{j_2} }{d\tau^{j_2}}\frac{d^{j_3} }{dx_1^{j_3}}\frac{d^{j_4}}{dx_2^{j_4}} V^*(x_1, x_2, y, \tau) \\
&\ll_{A, \textbf{j}} \Big(1 + \frac{x_1x_2}{K^4}\Big)^{-A} \Big(1 + \frac{ |\tau|^2}{K} \Big)^{-A} \Big(1 + \frac{K^2}{y}\Big)^{-A}\Big(1 + K^{1/2}|\log x_2/x_1|\Big)^{-A} \end{split} \end{equation}
for any $A \geqslant 0$, $\textbf{j} \in \mathbb{N}_0^4$. The first and third factor on the right hand side of the last display imply $|D_1 D_2| = K^{4+o(1)}$ and $c, f_1, f_2 = K^{o(1)}$ up to a negligible error. On the other hand, the last factor implies $D_1f_1^2 =D_2f_2^2(1 + O(1/K^{1/2}))$, so that in effect $D_1, D_2 = K^{2+o(1)}$.
In the following we treat the cuspidal part, the Eisenstein part and the constant function separately. In principle we could treat them on equal footing and we will do this in Section \ref{off-off}, but for now we keep the prerequisites as simple as possible.
\subsection{The Eisenstein contribution} We start with the contribution of the Eisenstein spectrum. As this is much smaller in size than the cuspidal spectrum, very simple bounds suffice. By \eqref{eisen1}, \eqref{lower} and \eqref{astV} we obtain a contribution of
$$\frac{K^{\varepsilon} }{K^{4} } \int_{-2\mathcal{T}_{\text{spec}}}^{2\mathcal{T}_{\text{spec}}} \sum_{\substack{D_1f_1^2 = D_2 f_2^2 (1 + O(K^{\varepsilon-1/2} ))\\ |D_1|, |D_2| = K^{2 + o(1)}\\ f_1, f_2 \ll K^{\varepsilon}}} |\zeta(1/2 + it)|^2 |L(D_1, 1/2 + it)L(D_2, 1/2 + it)| dt. $$
We use the basic inequality $|L(D_1, 1/2 + it)L(D_2, 1/2 + it)| \leqslant\frac{1}{2}( |L(D_1, 1/2 + it)|^2 + |L(D_2, 1/2 + it)|^2) $. For fixed $f_1, f_2, D_1$ there are $O( K^{3/2+\varepsilon} )$ values of $D_2$ satisfying the summation condition. Thus we obtain the bound
$$\frac{K^{\varepsilon} }{K^{5/2 } } \int_{-2\mathcal{T}_{\text{spec}}}^{2\mathcal{T}_{\text{spec}}} \sum_{ |D| = K^{2 +o(1)}} |\zeta(1/2 + it)|^2 |L(D, 1/2 + it)|^2 dt. $$ By \eqref{basicL} and Proposition \ref{Lfunc}(b) along with a bound for the fourth moment of the Riemann zeta function we obtain the desired bound
$$\frac{K^{\varepsilon} }{K^{5/2 } } \int_{-2\mathcal{T}_{\text{spec}}}^{2\mathcal{T}_{\text{spec}}} \sum_{ |\Delta| \leqslant K^{2+\varepsilon}} |\zeta(1/2 + it)|^2 |L(\Delta, 1/2 + it)|^2 dt \ll \frac{\mathcal{T}_{\text{spec}}}{K^{1/2 - \varepsilon}}.$$
\subsection{The cuspidal contribution}\label{104} Next we consider the cuspidal contribution and consider the following portion \begin{equation}\label{portion}
\frac{\overline{P(D_2; u)}}{|D_2|^{ 3/4 }}\sum_{D_1 } \frac{P(D_1; u)}{|D_1|^{3/4 }} e\Big( \frac{|D_1|d}{c}\Big) e\Big( \pm \frac{2\sqrt{|D_1D_2|}}{c}\Big) V^{\ast}\Big(|D_1|f_1^2, |D_2| f_2^2, \frac{\sqrt{|D_1D_2|}}{c}, t_{\tt u}\Big) \end{equation}
of \eqref{leftwith}, i.e.\ we freeze $c, f_1, f_2$ and $u$ for the moment. For notational simplicity we consider only the plus case, the minus case may be treated similarly. We insert \eqref{mixed} with $t = t_u/2$ and re-write \eqref{portion} as
\begin{displaymath}
\begin{split}
& \frac{3}{\pi}\overline{b(D_2)}\Big|\Gamma\Big(\frac{1}{4} + \frac{it_u}{2}\Big)\Big|^2 L(u, 1/2) \sum_{D_1 } b(D_1)e\Big(- \frac{D_1d}{c}\Big) e\Big( \frac{2\sqrt{|D_1D_2|}}{c}\Big)V^{\ast}\Big(|D_1|f_1^2, |D_2|f_2^2, \frac{\sqrt{|D_1D_2|}}{c}, t_u\Big) .
\end{split}
\end{displaymath} To the $D_1$-sum we apply the Voronoi formula (Lemma \ref{Vor}) with weight function \begin{equation}\label{phi-func}
\phi(x) = \phi_{c, f_1, f_2}(x; t, t_u, D_2)= \frac{1}{ |x|^{1/2 }} e\Big( \frac{2\sqrt{|xD_2|}}{c}\Big)V^{\ast}\Big(|x|f_1^2, |D_2|f_2^2, \frac{\sqrt{|xD_2|}}{c}, t_u\Big) \end{equation} for $x < 0$ and $\phi(x) = 0$ for $x > 0$.
We recall that $c, f_1, f_2 \ll K^{\varepsilon}$ are essentially fixed from the decay conditions of $V^{\ast}$, but we need to be uniform in $|D_2| = K^{2+o(1)}$. We define $$\Phi(\pm y) = \Phi_{c, f_1, f_2}(\pm y; t, t_u, D_2) = \int_0^{\infty} \mathcal{J}^{\pm, -}(x y) \phi(-x) dx$$ as in \eqref{defPhi} with $r = t_u/2$ and obtain that \eqref{portion} is equal to \begin{equation}\label{D1sum} \begin{split}
&\frac{3}{\pi} \overline{b(D_2)}\Big|\Gamma\Big(\frac{1}{4} + \frac{it_u}{2}\Big)\Big|^2 L(u, 1/2) \frac{2\pi}{c} \left( \frac{-c}{-d}\right)\epsilon_{-d} e\left( \frac{1}{8}\right) \sum_{D} b(D) \sqrt{|D|}e\left(\frac{\bar{d}D}{c}\right) \Phi\left( \frac{(2\pi)^2 D}{c^2}\right), \end{split} \end{equation} where only in the above sum do we allow $D$ to be either positive or negative. If $D > 0$, then the integral transform $\Phi((2\pi)^2 D/c^2 )$ contains a factor \begin{equation}\label{BesselK}
\frac{K_{ i t_u} ( 4\pi \sqrt{|xD|}/c) }{\Gamma(3/4 + it_u/2)\Gamma(3/4 - it_u/2)}, \end{equation}
and we recall that $|x| = K^{2+o(1)}$, $c \ll K^{\varepsilon}$ up to a negligible error. Thus the argument of the Bessel function is $\gg K^{2-\varepsilon}$, while the index is $\ll K^{1/2+\varepsilon}$. By the rapid decay of the Bessel $K$-function this contribution is easily seen to be negligible (we use \eqref{BarMao} and bound $b(D)$ trivially), and we may restrict from now on to $D< 0$. In this case \begin{equation*} \begin{split}
\Phi\left( \frac{(2\pi)^2 D}{c^2}\right) = \int_0^{\infty} & \sum_{\pm} (\mp) \frac{\cos(\pi/4 \pm \pi i t_u/2)}{\sin(\pi i t_u)} J_{\pm i t_u}\Big( \frac{4\pi \sqrt{|Dx|}}{c}\Big) e\Big( \frac{2\sqrt{|xD_2|}}{c}\Big)\\
&V^{\ast}\Big(xf_1^2, |D_2|f_2^2, \frac{\sqrt{|xD_2|}}{c}, t_u\Big) \frac{dx}{ x^{1/2 }}.
\end{split}
\end{equation*}
Using \eqref{bessel-approx}, up to a negligible error we can write \begin{equation*} \begin{split}
\Phi\left( \frac{(2\pi)^2 D}{c^2}\right) = \frac{c^{1/2}}{|D|^{1/4}} \sum_{\pm} \int_0^{\infty} & e\Big( \frac{2 \sqrt{x} (\sqrt{|D_2|} \pm \sqrt{|D|})}{c}\Big)\\
&f^{\pm}\Big(\frac{2 \sqrt{|Dx|}}{c}, t_u\Big) V^{\ast}\Big(xf_1^2, |D_2|f_2^2, \frac{\sqrt{|xD_2|}}{c}, t_u\Big) \frac{dx}{ x^{3/4 }}
\end{split}
\end{equation*}
with \begin{equation}\label{besseldecay} x^j \frac{\partial^j}{\partial x^j} f^{\pm}(x, r) \ll_{j} 1 \end{equation} for any $ j \in \mathbb{N}_0$.
We substitute this back into \eqref{D1sum} which equals \eqref{portion}. We substitute this back into \eqref{leftwith}. In this way we see that the cuspidal contribution to \eqref{leftwith} is at most
\begin{equation}\label{isnow} \begin{split}
\frac{1}{K^2}&\sum_{4 \mid c} \frac{1}{c^{1/2}} \sum_{u \text{ even}} w\Big( \frac{|t_{\tt u}|}{\mathcal{T}_{\text{spec}}}\Big) \sum_{f_1, f_2} \frac{1}{f_1f_2} \\
& \sum_{D, D_2} \overline{|b(D_2)|}\Big|\Gamma\Big(\frac{1}{4} + \frac{it_u}{2}\Big)\Big|^2 L(u, 1/2) |b(D)| |D|^{1/4} | \Psi_{c, f_1, f_2}( D, D_2, t_u)| \end{split} \end{equation} with
\begin{displaymath}
\begin{split}
\Psi_{c, f_1, f_2}( D, D_2, t_u) & =
\int_0^{\infty} e\Big( \frac{2\sqrt{x} (\sqrt{|D_2|} \pm \sqrt{|D|})}{c}\Big) f^{\pm}\Big(\frac{2 \sqrt{|Dx|}}{c}, t_u\Big)V^{\ast}\Big(xf_1^2, |D_2|f_2^2, \frac{\sqrt{|xD_2|}}{c}, t_u\Big) \frac{dx}{ x^{3/4 }} \\
&= 2 \int_0^{\infty} e\Big( \frac{2x (\sqrt{|D_2|} \pm \sqrt{|D|})}{c}\Big) f^{\pm}\Big(\frac{2 \sqrt{|Dx^2|}}{c}, t_u\Big) V^{\ast}\Big(x^2f_1^2, |D_2|f_2^2, \frac{\sqrt{|x^2D_2|}}{c}, t_u\Big) \frac{dx}{ x^{1/2 }}.
\end{split}
\end{displaymath}
By \eqref{astV} and \eqref{besseldecay}, each integration by parts with respect to $x$ introduces an additional factor \begin{equation}\label{add1}
\frac{c K^{1/2}}{x(\sqrt{|D_2|} \pm \sqrt{|D|})}, \end{equation} and we conclude that \begin{equation}\label{add2} \begin{split}
\Psi_{c, f_1, f_2}( D, D_2,& t_u) \ll_A \Big(1 + \frac{ |t_{\tt u}|^2}{K} \Big)^{-A} \int_0^{\infty} \Big( 1 + \frac{x(\sqrt{|D_2|} \pm \sqrt{|D|})}{c K^{1/2}}\Big)^{-A} \\
& \Big(1+K^{1/2}\log \frac{f_2^2 |D_2|}{f_1^2 x^2} \Big)^{-A}
\Big(1 + \frac{|D_2|(x f_1f_2)^2}{K^4}\Big)^{-A} \Big(1 + \frac{K^2c}{x |D_2|^{1/2}}\Big)^{-A} \frac{dx}{ x^{1/2 }} \end{split} \end{equation}
for every $A \geqslant 0$. Here only the negative part in the $\pm$ sign is relevant, since otherwise the expression is trivially negligible. The limiting factor for the size of the $x$-integral is the first factor in the second line of the previous display, so that we obtain
\begin{equation*} \begin{split}
\Psi_{c, f_1, f_2}( D, D_2, t_u) \ll_A & \frac{f_2 ^{1/2}|D_2|^{1/4} }{f_1^{1/2}K^{1/2}} \Big(1 + \frac{ |t_{\tt u}|^2}{K} \Big)^{-A} \Big( 1 + \frac{f_2|D_2|^{1/2}(\sqrt{|D_2|}- \sqrt{|D|})}{f_1cK^{1/2}}\Big)^{-A}\\
& \Big(1 + \frac{|D_2|f_2^2}{K^2}\Big)^{-A} \Big(1 + \frac{K^2f_1c}{ |D_2|f_2}\Big)^{-A} \end{split} \end{equation*} for every $A \geqslant 0$. It is not hard to see that this can be simplified as
\begin{equation*} \begin{split}
\Psi_{c, f_1, f_2}( D, D_2, t_u) \ll_A & (1 + f_1f_2c)^{-A} \Big(1 + \frac{ |t_{\tt u}|^2}{K} \Big)^{-A} \Big(1 + \frac{|D_2|}{K^2}\Big)^{-A} \Big( 1 + \frac{ |D_2| - |D|}{K^{1/2}}\Big)^{-A}. \end{split} \end{equation*}
With this bound we return to \eqref{isnow}, apply the simple bound $|b(D)b(D_2)| \leqslant |b(D)|^2 + |b(D_2)|^2$ together with \eqref{BarMao}, getting an upper bound of the shape
\begin{equation*} \begin{split}
\frac{1}{K^2}\sum_{4 \mid c} \frac{1}{c^{1/2}}& \sum_{u \text{ even}} w\Big( \frac{|t_{\tt u}|}{\mathcal{T}_{\text{spec}}}\Big) \sum_{f_1, f_2} \sum_{D, D_2} \frac{ |D|^{1/4} }{ f_1f_2|D_2|} \frac{L(u, 1/2)L(u, D_2, 1/2)}{L(\text{sym}^2 u, 1)} | \Psi_{c, f_1, f_2}( D, D_2, t_u)| \end{split} \end{equation*}
plus a similar expression that with $L(u, D, 1/2)/|D|$ in place of $L(u, D_2, 1/2)/|D_2|$ which can be treated in the same way. We sum over $D, f_1, f_2, c$ and end up with (after changing the value of $A$)
\begin{equation*} \begin{split}
\frac{1}{K^{3}} & \sum_{u \text{ even}}w\Big( \frac{|t_{\tt u}|}{\mathcal{T}_{\text{spec}}}\Big) \Big(1 + \frac{ |t_{\tt u}|^2}{K} \Big)^{-A} \sum_ { D_2} \Big(1 + \frac{|D_2| }{K^2}\Big)^{-A} \frac{L(u, 1/2)L(u, D_2, 1/2)}{L(\text{sym}^2 u, 1)} . \end{split} \end{equation*} We write $D_2 = \Delta f^2$ with a fundamental discriminant $\Delta$ and use \eqref{key}, \eqref{key-simple}. Summing over $f$, we obtain \begin{equation*} \begin{split}
\frac{1}{K^{3}} & \sum_{u \text{ even}}w\Big( \frac{|t_{\tt u}|}{\mathcal{T}_{\text{spec}}}\Big) \Big(1 + \frac{ |t_{\tt u}|^2}{K} \Big)^{-A} \sum_ { \Delta } \frac{K^{4/3}}{|\Delta|^{2/3} } \Big(1 + \frac{|\Delta| }{K^2}\Big)^{-A} \frac{L(u, 1/2)L(u, \Delta, 1/2)}{L(\text{sym}^2 u, 1)} . \end{split} \end{equation*} We estimate the denominator $L(\text{sym}^2 u, 1)$ by \eqref{lower} and apply Corollary \ref{Lfunc-cor} to finally obtain the upper bound $\mathcal{T}_{\text{spec}}^2 K^{\varepsilon-1}$ in agreement with \eqref{weak}.
\subsection{The constant function} This is very similar to the preceding subsection, so we can be brief. In short, we win a factor $\mathcal{T}_{\text{spec}} \ll K^{1+\varepsilon}$ from the fact that the spectrum is reduced to one element, and we lose a factor $|D|^{1/2} \ll K^{1+\varepsilon}$ since each class number is a factor $|D|^{1/4}$ bigger than the generic period $P(D;u)$. The key point is that we end up with a pure bound without a $K^{\varepsilon}$-power. This $K^{\varepsilon}$-power is unavoidable when we apply Corollary \ref{Lfunc-cor}, but for a sum over class numbers $H(D)$ alone we can apply Lemma \ref{hur2} below which avoids a $K^{\varepsilon}$-power.
The analogue of \eqref{portion} is
$$\frac{3}{\pi} \frac{\overline{H(D_2)}}{|D_2|^{ 3/4 }}\sum_{D_1 } \frac{H(D_1)}{|D_1|^{3/4 }} e\Big( \frac{|D_1|d}{c}\Big) e\Big( \pm \frac{2\sqrt{|D_1D_2|}}{c}\Big) V^{\ast}\Big(|D_1|f_1^2, |D_2|f_2^2, \frac{\sqrt{|D_1D_2|}}{c}, t_{\tt u}\Big). $$
We apply the Voronoi formula (Lemma \ref{class-num1}) to the $D_1$-sum as before. Due to the oscillatory behaviour of $\phi$ in \eqref{phi-func} the main terms are easily seen to be negligible, and as in the previous argument also one of the osciallatory terms is negligible due to the exponential decay of $\mathcal{J}^-$ in \eqref{Jclean}. The behaviour of $\mathcal{J}^+$ similar, but much simpler, as no asymptotic formula of a Bessel function is necessary. The analogue of \eqref{isnow} then becomes \begin{equation*} \begin{split}
\frac{1}{K^2}\sum_{4 \mid c} \frac{1}{c^{1/2}}& \sum_{f_1, f_2} \sum_{D, D_2} \frac{H(D_2) H(D) |D|^{1/4} }{ |DD_2|^{3/4} f_1f_2} | \Psi_{c, f_1, f_2}( D, D_2, t_u)|. \end{split} \end{equation*} In the same way as above this leads to \begin{equation}\label{leadsto} \begin{split}
\frac{1}{K^{3}} & \sum_ { D} \Big(1 + \frac{D }{K^2}\Big)^{-A} \frac{H(D)^2}{|D|^{1/2}} \ll 1. \end{split} \end{equation} The last step is justified by the following simple lemma. \begin{lemma}\label{hur2} For $x \geqslant 1$ we have
$$\sum_{D \leqslant x} H(D)^2 \ll x^2.$$ \end{lemma}
\begin{proof} Let $h(D)$ denote the usual class number. Since
$$H(D) \leqslant \sum_{n^2 \mid D} h(D/n^2)$$
we have
$$\sum_{|D| \leqslant x} H(D)^2 \leqslant \sum_{n_1, n_2} \sum_{[n_1^2, n_2^2] \mid |D| < x} h\left(\frac{D}{n_1^2}\right)h\left(\frac{D}{n_2^2}\right) \leqslant \sum_{n_1, n_2, m} \sum_{|D| \leqslant \frac{x}{n_1^2n_2^2m^2} } h(n_2^2D) h(n_1^2D).$$ Since $h(\Delta n^2) = h(\Delta) n \prod_{p \mid n} (1 - \chi_{\Delta}(p)/p) \leqslant h(\delta) n\tau(n)$ for a fundamental discriminant $\Delta$, we get from bounds for moments of the ordinary class number \cite{Ba} that
\begin{displaymath}
\begin{split}
\sum_{|D| \leqslant x} H(D)^2 &\leqslant\sum_{n_1, n_2, m, f} \sum_{|\Delta| \leqslant x/(n_1 n_2 m f )^2 } h(\Delta)^2 n_1n_2f^2 \tau(n_1)\tau(n_2)\tau(f)^2\\
& \ll x^2\sum_{n_1, n_2, m, f} \frac{n_1n_2f^2 \tau(n_1)\tau(n_2)\tau(f)^2}{(n_1n_2mf)^4} \ll x^2.
\end{split}
\end{displaymath}
\end{proof}
The bound \eqref{leadsto} is in agreement with, and completes the proof of, \eqref{weak}.\\
We conclude this section with a brief discussion. The bound \eqref{weak} along with $\mathcal{T}_{\text{spec}} \ll K^{1/2 + \varepsilon}$ (from the decay of $V_t$) implies immediately the upper bound $N_{\text{av}}(K) \ll K^{\varepsilon}$. The $\varepsilon$-power is unavoidable at this point because of the use of Heath-Brown's large sieve in the proof of Proposition \ref{Lfunc}. Except for the spectral large sieve implicit in proof of Proposition \ref{Lfunc}, we have not touched the spectral ${\tt u}$-sum, so any further improvement must involve a treatment of this sum. This is precisely the purpose of the relative trace formula for Heegner periods given in Theorem \ref{thm2}. The weaker result \eqref{weak} is nevertheless useful: it allows us to discard small eigenvalues $t_{\tt u} \ll K^{1/2 - \varepsilon}$ and it allows us to estimate efficiently some error terms later. The following sections are devoted to the proof of Theorem \ref{thm1}.
\section{Proof of Theorem \ref{thm1}: the preliminary argument}\label{sec12}
By \eqref{Nav} and \eqref{Fh} we have \begin{displaymath} \begin{split}
\mathcal{N}_{\text{av}}(K) = & \frac{12}{\omega K^2} \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \sum_{h \in B_{k-1/2}^+(4)} \frac{\pi^2}{90} \cdot \frac{18\sqrt{\pi}}{2 }\cdot 2\sum_{(n, m) = 1} \frac{\lambda(n) \mu(n)\mu^2(m)}{n^{3/2}m^{3}} \int_{-\infty}^{\infty} \int_{\Lambda_{\text{\rm ev}}} \frac{\Gamma(k - 3/2)}{(4\pi)^{k-3/2} \| h \|^2} \\
& \sum_{f_1, f_2, D_1, D_2} \frac{c(|D_1|) c(|D_2|) P(D_1; {\tt u})\overline{P(D_2; {\tt u})}}{f_1f_2 |D_1D_2|^{k/2}}\Big(\frac{|D_2|f_2^2}{|D_1|f_1^2}\Big)^{it} V_t(|D_1D_2|(f_1f_2)^2; k, t_{\tt u})
d{\tt u} \, dt.
\end{split}
\end{displaymath}
Throughout we agree on the convention that $D$ (with or without indices) denotes a negative discriminant and $\Delta$ denotes a negative fundamental discriminant. We make two immediate manipulations. Fix some $0 < \eta < 1/100$. By the concluding remark of the preceding section we can insert the function \begin{equation}\label{def-omega} \omega(t_{\tt u}) = 1 - e^{-(t_{\tt u}/K^{1/2- \eta})^{10^6/\eta}} \end{equation} (not to be confused with the constant $\omega$ in the previous display) into the ${\tt u}$-integral, and we can truncate the $n, m$-sum at $$n,m \ll K^{\eta}. $$ Both transformations induce an admissible error, the former due to that $\omega(t_{\tt u}) - 1 \ll_A K^{-A}$ for every $A > 0$ and $t_{\tt u} \gg K^{\frac{1}{2} - \frac{\eta}{2}}$. Note that $\omega$ is even, holomorphic, within $[0, 1]$ for $t_{\tt u} \in \mathbb{R} \cup \{i/2, -i/2\}$ and satisfies \begin{equation}\label{prop-omega} \begin{split}
\omega(t_{\tt u}) & \ll K^{-10^6}, \quad |t_{\tt u}| \leqslant K^{\frac{1}{2} - 2\eta},\\
|t_{\tt u}|^j \frac{d^j}{dt^j_{\tt u}} \omega(t_{\tt u}) & \ll_j 1 \end{split} \end{equation} for $j \in \mathbb{N}_0$. In particular, up to a negligible error we may ignore the constant function ${\tt u} = \sqrt{3/\pi}$ with $t_{\tt u} = i/2$. As before we denote by $\int_{\Lambda_{\text{ev}}}^{\ast}$ a spectral sum/integral over the non-residual spectrum, i.e.\ everything except the constant function.
Note that $\lambda(n)$ depends on $h$, as it is a Hecke eigenvalue of $f_h$. Before we can sum over $h$ using Lemma \ref{lem1}, we must first combine $\lambda(n)$ with $c(|D_2|)$. To this end we recall \eqref{fourier-relation1} and recast $\mathcal{N}_{\text{av}}(K)$, up to a small error, coming from the truncation of the $n, m$-sum and the ${\tt u}$-integral, as
\begin{equation}\label{slightly-simp} \begin{split}
& \frac{12}{ \omega K^2} \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \sum_{h \in B_{k-1/2}^+(4)} \frac{\pi^2}{90} \cdot \frac{18\sqrt{\pi}}{2 }\cdot 2\sum_{\substack{(n, m) = 1\\ n, m \leqslant K^{\eta}}} \frac{ \mu(n)\mu^2(m)}{n^{3/2}m^{3}} \\
& \int_{-\infty}^{\infty} \int^{\ast}_{\Lambda_{\text{\rm ev}}} \omega(t_{\tt u}) \frac{\Gamma(k - 3/2)}{(4\pi)^{k-3/2} \| h \|^2}
\sum_{f_1, f_2, D_1, D_2} \sum_{\substack{d_1 \mid d_2 \mid n\\ (d_1d_2)^2 \mid n^2D_2}} \left(\frac{d_1}{d_2} \right)^{1/2} \chi_{ D_2}\Big(\frac{d_2}{d_1}\Big)\left(\frac{n}{d_1d_2}\right)^{3/2 - k}\\
& \frac{c(|D_1|) c(|D_2|n^2/(d_1d_2)^2) P(D_1; {\tt u})\overline{P(D_2; {\tt u})}}{f_1f_2 |D_1D_2|^{k/2}} \Big(\frac{|D_2|f_2^2}{|D_1|f_1^2}\Big)^{it} V_t(|D_1D_2|(f_1f_2)^2; k, t_{\tt u})
d{\tt u} \, dt .
\end{split}
\end{equation}
We can now sum over $h$ using Lemma \ref{lem1}, and we start with an analysis of the diagonal term which is given by
\begin{displaymath} \begin{split}
\mathcal{N}^{\text{diag}}&(K) = \frac{12}{\omega K^2} \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \frac{\pi^2}{90} \cdot \frac{18\sqrt{\pi}}{2 } \cdot 2\cdot \frac{2}{3}\sum_{\substack{(n, m) = 1\\ n, m \leqslant K^{\eta}}} \frac{ \mu(n)\mu^2(m)}{n^{3/2}m^{3}} \int_{-\infty}^{\infty} \int^{\ast}_{\Lambda_{\text{\rm ev}}} \omega(t_{\tt u}) \sum_{f_1, f_2, D} \\
& \sum_{\substack{d_1 \mid d_2 \mid n\\ (d_1d_2)^2 \mid n^2D}} \left(\frac{d_1}{d_2} \right)^{1/2} \chi_{ D}\Big(\frac{d_2}{d_1}\Big) \frac{ P(Dn^2/(d_1d_2)^2; {\tt u})\overline{P(D; {\tt u})}}{f_1f_2 (|D|n/(d_1d_2))^{3/2}}\Big(\frac{d_1d_2f_2}{nf_1}\Big)^{2it} V_t\Big(\Big(\frac{|D|nf_1f_2}{d_1d_2}\Big)^2; k, t_{\tt u}\Big) d{\tt u} \, dt .
\end{split}
\end{displaymath}
The analysis of this term occupies this and the following two sections, and we will eventually show that $ \mathcal{N}^{\text{diag}}(K) = 4\log K + O(1)$. The discussion of the off-diagonal term is postponed to Section \ref{off-off}.
We write $d_2 = d_1\delta$, $n = d_1\delta \nu$, so that $d_1^2 \mid \nu^2 D$. Since $n$ is squarefree, this implies $d_1^2 \mid D$. We write $d_1 = d$ and $D d^2$ in place of $D$. With this notation we recast $ \mathcal{N}^{\text{diag}}(K)$ as
\begin{displaymath} \begin{split}
\frac{12}{ \omega K^2} & \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \frac{\pi^2}{90} \cdot \frac{18\sqrt{\pi}}{2 } \cdot 2 \cdot \frac{2}{3}\sum_{\substack{(d\delta \nu, m) = 1\\ d\delta \nu, m \leqslant K^{\eta}}} \frac{ \mu(d\delta \nu)\mu^2(m)}{(d\delta\nu)^{3/2}m^{3}} \int_{-\infty}^{\infty} \int^{\ast}_{\Lambda_{\text{\rm ev}}} \omega(t_{\tt u}) \sum_{f_1, f_2, D} \frac{\chi_D(\delta)}{\delta^{1/2}} \\ &\frac{ P(D\nu^2; {\tt u})\overline{P(Dd^2; {\tt u})}}{f_1f_2 (d|D|\nu)^{3/2}} \Big(\frac{df_2}{\nu f_1}\Big)^{2it} V_t ( ( |D|\nu df_1f_2 )^2; k, t_{\tt u} ) d{\tt u} \, dt.
\end{split}
\end{displaymath}
From the Katok-Sarnak formula in combination with \eqref{non-fund} in the cuspidal case and from \eqref{eisen1} in combination with \eqref{basicL1} in the Eisenstein case we conclude\footnote{This remains true in the excluded case ${\tt u} = \text{const}$ if we define $\lambda_{\tt u} = \rho_1$.}
$$ P(\Delta f^2, {\tt u}) = P(\Delta , {\tt u})\alpha_{\tt u}(f), \quad \alpha_{\tt u}(f) = f^{1/2} \sum_{d \mid f} \mu(d) \chi_{\Delta}(d) \lambda_{\tt u}(f/d) d^{-1/2}$$ for a fundamental discriminant $\Delta$ where $\alpha_{\tt u}(f)$ depends also on $\Delta$, which we suppress from the notation. Using this notation along with \eqref{katok-Sarnak} in the cuspidal case and \eqref{eisen2} in the Eisenstein case we obtain
\begin{equation*} \begin{split}
\mathcal{N}^{\text{diag}}(K) &=
\frac{12}{ \omega K^2} \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \frac{\pi^2}{90} \cdot \frac{18\sqrt{\pi}}{2 }\cdot 2 \cdot \frac{2}{3} \cdot \frac{1}{4}\sum_{\substack{(d\delta \nu, m) = 1\\ d\delta \nu, m \leqslant K^{\eta}}} \frac{ \mu(d\delta \nu)\mu^2(m)}{(d\delta\nu)^{3/2}m^{3}} \int_{-\infty}^{\infty} \int^{\ast}_{\Lambda_{\text{\rm ev}}} \omega(t_{\tt u})\\
& \sum_{f_1, f_2}\sum_{\substack{D = \Delta f^2\\ (\delta, f) = 1}} \frac{\chi_{\Delta}(\delta) \alpha_{\tt u} (f\nu)\overline{ \alpha_{\tt u}(f d) } }{\delta^{1/2}f_1f_2 (df^2\nu)^{3/2}|\Delta|} \frac{L({\tt u}, 1/2) L({\tt u}, \Delta, 1/2)}{\mathcal{L}({\tt u})} \Big(\frac{df_2}{\nu f_1}\Big)^{2it} V_t ( ( |D|\nu df_1f_2 )^2; k, t_{\tt u} ) d{\tt u} \, dt
\end{split}
\end{equation*}
where $\mathcal{L}({\tt u}) = L(\text{sym}^2 {\tt u}, 1)$ if ${\tt u}$ is cuspidal and $\mathcal{L}({\tt u}) = \frac{1}{2}|\zeta(1 + 2 i t)|^2$ if ${\tt u} = E(., 1/2 + it)$ is Eisenstein (with the obvious interpretation in the case $t = 0$). With later transformations in mind, we also restrict the $f$-sum to $f \leqslant K^{\eta}$. By trivial estimates along with Corollary \ref{Lfunc-cor} and \eqref{size-restr}, this induces an error of $O(K^{\varepsilon - \eta})$. By the usual Hecke relations we have $$ \alpha_{\tt u} (f\nu)\overline{ \alpha_{\tt u}(f d)} = f (d\nu)^{1/2} \sum_{\substack{d_1 \mid fd\\ d_2 \mid f\nu}} \sum_{d_3 \mid (\frac{fd}{d_1}, \frac{f\nu}{d_2}) }\frac{\mu(d_1)\chi_{\Delta}(d_1) \mu(d_2)\chi_{\Delta}(d_2) }{\sqrt{d_1d_2}} \lambda_{\tt u}(f^2 d\nu/(d_1d_2d_3^2)) .$$ We summarize the previous discussion as \begin{equation}\label{weobt} \begin{split} \mathcal{N}^{\text{diag}}(K) & = \frac{12}{ \omega K^2} \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \frac{\pi^2}{90} \cdot \frac{18\sqrt{\pi}}{2 }\cdot 2 \cdot \frac{2}{3} \cdot \frac{1}{4}\sum_{\substack{(d\delta \nu, m) = 1\\ d\delta \nu, m \leqslant K^{\eta}}} \frac{ \mu(d\delta \nu)\mu^2(m)}{(d\delta\nu)^{3/2}m^{3}} \\ &\int_{-\infty}^{\infty}
\sum_{f_1, f_2, \Delta}\sum_{\substack{ f\leqslant K^{\eta}\\ (\delta, f) = 1}} \frac{\chi_{\Delta}(\delta) }{\delta^{1/2}f_1f_2 df^2\nu|\Delta| }\Big(\frac{df_2}{\nu f_1}\Big)^{2it} \\
&\sum_{\substack{d_1 \mid fd\\ d_2 \mid f\nu}} \sum_{d_3 \mid (\frac{fd}{d_1}, \frac{f\nu}{d_2}) }\frac{\mu(d_1)\chi_{\Delta}(d_1) \mu(d_2)\chi_{\Delta}(d_2) }{\sqrt{d_1d_2}} \mathcal{I}\Big(\Delta, t, k, \frac{f^2 d \nu}{d_1d_2d_3^2}\Big) dt
+ O(K^{\varepsilon - \eta}) \end{split} \end{equation} where $$ \mathcal{I}(\Delta, t,k, r) = \int^{\ast}_{\Lambda_{\text{\rm ev}}} \frac{L({\tt u}, 1/2) L({\tt u}, \Delta, 1/2)}{\mathcal{L}({\tt u})} \lambda_{\tt u}(r) h(t_{\tt u}) d{\tt u}$$ with \begin{equation}\label{htau}
h(\tau) = \omega(\tau) V_t ( ( |\Delta|f^2\nu df_1f_2 )^2; k,\tau ) . \end{equation} The expression $\mathcal{I}$ depends also on $f^2 \nu d f_1f_2$, but we suppress this from the notation.
We insert the approximate functional equations \eqref{approx-basic} and \eqref{approx-basic1} getting \begin{displaymath} \begin{split}
\mathcal{I}(\Delta, t,k, r) = & \ 4\int^{\ast}_{\Lambda_{\text{\rm ev}}}\sum_{{\tt n},{\tt m}} \frac{\lambda_{\tt u}({\tt n}) \lambda_{\tt u}({\tt m}) \chi_{\Delta}({\tt m})\lambda_{\tt u}(r)}{({\tt n}{\tt m})^{1/2}}W^+_{t_{\tt u}}({\tt n}) W^-_{t_{\tt u}}\Big(\frac{{\tt m}}{|\Delta|}\Big) \frac{1}{\mathcal{L}({\tt u})} h(t_{\tt u}) d{\tt u}\\
& - \int_{-\infty}^{\infty} \sum_{\pm} \frac{\zeta(1 \pm 2i\tau)\Gamma(\frac{1}{2} \pm i\tau)\pi^{\mp i\tau} e^{(1/2 \pm i\tau)^2}}{(\frac{1}{2} \pm i \tau)\Gamma(\frac{1}{4} + \frac{i\tau}{2})\Gamma(\frac{1}{4} - \frac{i\tau}{2})}\frac{|L(\chi_{\Delta}, 1/2 + i\tau)|^2}{|\zeta(1 + 2i\tau)|^2} \rho_{1/2 + i\tau}(r) h(\tau) \frac{d\tau}{2\pi}.
\end{split}
\end{displaymath} Since the $\tau$-integral is rapidly converging, it is easy to see that the polar term contributes at most $O(K^{\varepsilon - 1})$ to \eqref{weobt}, so from now on we focus on the first term of the preceding display. By the Hecke relations we can recast it as
\begin{equation}\label{diagdiag}
4 \sum_{d \mid r} \int^{\ast}_{\Lambda_{\text{\rm ev}}}\sum_{{\tt n},{\tt m}} \frac{\lambda_{\tt u}({\tt n}r/d) \lambda_{\tt u}({\tt m}) \chi_{\Delta}({\tt m}) }{(d{\tt n}{\tt m})^{1/2}} W^+_{t_{\tt u}}(d{\tt n}) W^-_{t_{\tt u}}\Big(\frac{{\tt m}}{|\Delta|}\Big) \frac{1}{\mathcal{L}({\tt u})} h(t_{\tt u}) d{\tt u} .
\end{equation}
This is now in shape to apply the Kuznetsov formula for the even spectrum, Lemma \ref{kuz-even}. We treat the three terms on the right hand side of \eqref{kuz-even-form} separately and start with the diagonal term to which the next section is devoted. The two off-diagonal terms are treated in Section \ref{diag-off}.
\section{The diagonal diagonal term}\label{diag-diag} The diagonal contribution equals
$$ \mathcal{I}^{\text{diag}}(\Delta, t, k, r) = 4 \sum_{d \mid r} \sum_{{\tt n}} \frac{ \chi_{\Delta}({\tt n}r/d) }{{\tt n}r^{1/2}} \int_{-\infty}^{\infty} W^+_{\tau}(d{\tt n}) W^-_{\tau}\Big(\frac{{\tt n}r/d}{|\Delta|}\Big) \tau \tanh(\pi \tau) h(\tau) \frac{d\tau}{4\pi^2} .$$ Opening up the Mellin transform in the definition \eqref{v-t}, this equals \begin{displaymath} \begin{split} \frac{1}{\sqrt{r}} \int_{-\infty}^{\infty} & h(\tau)
\int_{(2)} \int_{(2)} \prod_{\pm}\Big( \frac{\Gamma(\frac{1}{2}(\frac{1}{2} + s_1 \pm i\tau))\Gamma(\frac{1}{2}(\frac{3}{2} + s_2 \pm i\tau))}{\Gamma(\frac{1}{2}(\frac{1}{2} \pm i\tau))\Gamma(\frac{1}{2}(\frac{3}{2} \pm i\tau)) } \Big) |\Delta|^{s_2} L( \chi_{\Delta}, 1 + s_1 + s_2)
\\
& \sum_{r_1r_2 = r} \frac{\chi_{\Delta}(r_2) }{r_1^{s_1} r_2^{s_2}} \frac{e^{s_2^2 +s_1^2}}{\pi^{s_1+s_2}s_1s_2} \frac{ds_1 \, ds_2}{(2\pi i)^2}
\tau \tanh(\pi \tau) \frac{d\tau}{\pi^2}. \end{split} \end{displaymath} We shift the $s_1, s_2$-contours to $\Re s_1 = \Re s_2 = -1/4$ getting \begin{equation}\label{i-diag} \begin{split} \mathcal{I}^{\text{diag}}&(\Delta, t, k, r) = \frac{1}{\sqrt{r}} L( \chi_{\Delta}, 1) \sum_{r_1r_2 = r} \chi_{\Delta}(r_2) \int_{-\infty}^{\infty} h(\tau) \tau \tanh(\pi \tau) \frac{d\tau}{\pi^2} \\
&+ O\Big(\int_{-\infty}^{\infty} |h(\tau)| |\tau|^{3/4} d\tau \int_{-\infty}^{\infty} e^{\xi^2}\Big(\frac{| L(\chi_{\Delta}, 1/2 + i\xi)|}{|\Delta|^{1/4}|} +| L(\chi_{\Delta}, 3/4 + i\xi)|\Big) d\xi \Big) .
\end{split} \end{equation}
We first deal with the error term and substitute it back into \eqref{weobt}. Roughly speaking the factor $|\tau|^{3/4}$ saves a factor $K^{1/8}$ from the trivial bound, while on average over $\Delta$ the $L$-values on the lines $1/2$ and $3/4$ are still bounded. More precisely, recalling \eqref{size-restr} the error term gives a total contribution of at most
$$\frac{1}{K^{3/2}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \sum_{\Delta} \Big(\frac{| L(\chi_{\Delta}, 1/2 + i\xi)|}{|\Delta|^{5/4}} +\frac{| L(\chi_{\Delta}, 3/4 + i\xi)|}{|\Delta|} \Big)\Big( 1 + \frac{|t|^2 + |\tau|^2}{K}\Big)^{-10} e^{\xi^2} d\xi \, |\tau|^{3/4}d\tau \, dt.$$ By a standard mean value bound for the $\Delta$-sum (e.g. \cite[Theorem 2]{HB}), the previous display can be bounded by $\ll K^{-1/8 + \varepsilon}$. We substitute the main term in \eqref{i-diag} into \eqref{weobt} getting \begin{equation*} \begin{split} &\mathcal{N}^{\text{diag}, \text{diag}}(K) = \frac{12}{ \omega K^2} \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \frac{\pi^2}{90} \cdot \frac{18\sqrt{\pi}}{2 }\cdot 2 \cdot \frac{2}{3} \cdot \frac{1}{4}\sum_{\substack{(d\delta \nu, m) = 1\\ d\delta \nu, m \leqslant K^{\eta}}} \frac{ \mu(d\delta \nu)\mu^2(m)}{(d\delta\nu)^{3/2}m^{3}} \\ &\sum_{f_1, f_2, \Delta}\sum_{\substack{ f \leqslant K^{\eta}\\ (\delta, f) = 1}} L( \chi_{\Delta}, 1) \int_{-\infty}^{\infty}
\sum_{\substack{d_1 \mid fd\\ d_2 \mid f\nu}} \sum_{d_3 \mid (\frac{fd}{d_1}, \frac{f\nu}{d_2}) } \sum_{r_1r_2 = \frac{f^2 d \nu}{d_1d_2d_3^2}} \frac{\chi_{\Delta}(\delta) \mu(d_1)\chi_{\Delta}(d_1) \mu(d_2)\chi_{\Delta}(d_2) d_3 \chi_{\Delta}(r_2) }{\delta^{1/2}f_1f_2 (df^2\nu)^{3/2} |\Delta|(f_1\nu/(f_2d))^{\pm 2it} }\\
& \int_{-\infty}^{\infty} h(\tau) \tau \tanh(\pi \tau) \frac{d\tau}{\pi^2} dt \end{split} \end{equation*} where we recall that $h(\tau)$ is given by \eqref{htau} and depends in particular on $t$, $k$ and $\Delta$ (as well as on $f, f_1, f_2, d, \nu$). A trivial estimate at this point using \eqref{size-restr} shows \begin{equation}\label{trivial} \begin{split} \mathcal{N}^{\text{diag}, \text{diag}}(K)\ll &
\frac{1}{K^2}\sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \sum_{f_1, f_2,\Delta} \frac{L(\chi_{\Delta}, 1)}{|\Delta| f_1f_2}\\
&\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{1}{k^{1/2}} \Big(1 + \frac{|\Delta| (f_1f_2)^2}{k^4}\Big)^{-A} \Big(1 + \frac{|t|^2 + |\tau|^2}{k} \Big)^{-A} |\tau| d\tau \, dt \ll \log K, \end{split} \end{equation} (by standard mean value results for $L(\chi_{\Delta}, 1)$), but eventually we want an asymptotic formula, not an upper bound.
Our next goal is to show that the $t$-integral forces $f_1\nu = f_2d$, up to a negligible error. To make this precise, we first observe that by the same computation as in \eqref{trivial} the portion $|t| \leqslant K^{2/5}$ contributes at most $O(K^{-1/10 + \varepsilon})$ to $ \mathcal{N}^{\text{diag}, \text{diag}}(K) $. We can therefore insert a smooth weight function that vanishes on $|t| \leqslant \frac{1}{2} K^{2/5}$ and is one on $|t| \geqslant K^{2/5}$. Integrating by parts sufficiently often using \eqref{size-restr}, we can then restrict to $$f_1\nu = f_2d (1 + O(K^{\varepsilon - 2/5}))$$
up to a negligible error. It is then easy to see that the terms $f_1 d \not = f_2 \nu$ contribute $O(K^{\varepsilon - 2/5})$ to $ \mathcal{N}^{\text{diag}, \text{diag}}(K) $. Having excluded these, we re-insert the portion $|t| \leqslant K^{2/5}$ to the $t$-integral, again at the cost of an error $O(K^{\varepsilon-1/10})$. Finally we complete the $d, \delta, \nu, m, f$-sum at the cost of an error $O(K^{\varepsilon-\eta})$. Since $(\nu, d) = 1$, the equation $f_1d = f_2 \nu$ implies $f_1 = dg$, $f_2 = \nu g$ for some $g\in \mathbb{N}$. Substituting all this, we recast $\mathcal{N}^{\text{diag}, \text{diag}}(K) $ as \begin{equation*} \begin{split}
& \frac{12}{ \omega K^2} \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \frac{\pi^2}{90} \cdot \frac{18\sqrt{\pi}}{2 }\cdot 2 \cdot \frac{2}{3} \cdot \frac{1}{4}\sum_{ (d\delta \nu, m) = 1 } \frac{ \mu(d\delta \nu)\mu^2(m)}{(d\delta\nu)^{3/2}m^{3}} \sum_{\substack{g, f, \Delta \\ (\delta, f) = 1}} L( \chi_{\Delta}, 1) \\ &\int_{-\infty}^{\infty}
\sum_{\substack{d_1 \mid fd\\ d_2 \mid f\nu}} \sum_{d_3 \mid (\frac{fd}{d_1}, \frac{f\nu}{d_2}) } \sum_{r_1r_2 = \frac{f^2 d \nu}{d_1d_2d_3^2}} \frac{\chi_{\Delta}(\delta) \mu(d_1)\chi_{\Delta}(d_1) \mu(d_2)\chi_{\Delta}(d_2) d_3 \chi_{\Delta}(r_2) }{\delta^{1/2}g^2d\nu (df^2\nu)^{3/2} |\Delta|} \int_{-\infty}^{\infty} h(\tau) \tau \tanh(\pi \tau) \frac{d\tau}{\pi^2} dt
\end{split} \end{equation*} up to an error of $O(K^{\varepsilon-\eta})$, where in the definition \eqref{htau} of $h$ we replace $f_1 = dg$, $f_2 = \nu g$. We substitute the definition in \eqref{htau} and open the Mellin transform \eqref{defV3}. So that the previous display becomes \begin{equation}\label{previousdisplay} \begin{split} & \frac{12}{ \omega K^2} \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \frac{\pi^2}{90} \cdot \frac{18\sqrt{\pi}}{2 }\cdot 2 \cdot \frac{2}{3} \cdot \frac{1}{4} \int_{(3)} \frac{e^{v^2 }}{v} \mathscr{L}(2v) \\ &\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \mathcal{G}(k, \tau, v + 1/2 + it) \mathcal{G}(k, \tau, v + 1/2 - it) \omega(\tau) \tau \tanh(\pi \tau) \frac{d\tau}{\pi^2} dt \frac{dv }{2\pi i}
\end{split} \end{equation} where \begin{displaymath} \begin{split}
\mathscr{L}(v) = \sum_{ (d\delta \nu, m) = 1 } \sum_{\substack{g, f, \Delta\\ (f, \delta) = 1}} \sum_{\substack{d_1 \mid fd\\ d_2 \mid f\nu}} \sum_{d_3 \mid (\frac{fd}{d_1}, \frac{f\nu}{d_2}) } \sum_{r_1r_2 = \frac{f^2 d \nu}{d_1d_2d_3^2}} \frac{ \mu(d\delta \nu)\mu^2(m) \mu(d_1)\mu(d_2) \chi_{\Delta}(\delta d_1 d_2r_2) d_3 }{(d\nu)^{4 + 2v}( \delta g) ^{2+2v}f^{3+2v} m^3 |\Delta|^{1+v}} L( \chi_{\Delta}, 1) . \end{split} \end{displaymath} Recall our general assumption that $\Delta$ runs over negative fundamental discriminants. The main term will come from the residue at $v = 0$ and we need to analytically continue $\mathscr{L}$ to $\Re v < 0$ and compute the residue at $v = 0$. The critical portion is the $\Delta$-sum which we analyze in the following lemma. \begin{lemma} \label{lemsip} Let $\alpha \in \mathbb{N}$ and uniquely write $\alpha = \alpha_1^2 \alpha_2$ with $\mu^2(\alpha_2) = 1$. Then $$\sum_{-X \leqslant \Delta < 0} \chi_{\Delta}(\alpha) L( \chi_{\Delta}, 1) = \frac{\zeta(2) X}{2\alpha_2} \prod_{p \mid \alpha} \Big(1 + \frac{1}{p} - \frac{1}{p^{3}}\Big)^{-1} \prod_{p }\Big( 1 - \frac{1}{p^{2}} - \frac{1}{p^{3}} + \frac{1}{p^{4}}\Big) + O\big(X^{13/18}\alpha^{1/4} (X\alpha)^{\varepsilon}\big).$$ In particular, the Dirichlet series
$$\mathscr{K}(v; \alpha) = \sum_{\Delta < 0} \frac{\chi_{\Delta}(\alpha) L( \chi_{\Delta}, 1)}{|\Delta|^{1+v}}$$ has analytic continuation to $\Re v > -5/18$ except for a simple pole at $v=0$ with residue $$ \frac{\zeta(2)}{2\alpha_2} \prod_{p \mid \alpha} \Big(1 + \frac{1}{p} - \frac{1}{p^{3}}\Big)^{-1} \prod_{p }\Big( 1 - \frac{1}{p^{2}} - \frac{1}{p^{3}} + \frac{1}{p^{4}}\Big) $$
and is bounded by $O_{\varepsilon}(|v| \alpha^{1/4+\varepsilon})$ in the region $\Re v \geqslant -5/18 + \varepsilon$, $|v| \geqslant \varepsilon$. \end{lemma}
\emph{Remark:} The computation of the leading constant in Lemma \ref{lemsip} seems to be a new result even in the case $\alpha = 1$ and features an interesting Euler product. See \cite{Ju} for similar Euler products for averages at the point 1/2. \\
\begin{proof} Let $w$ be a fixed smooth function that is equal to $1$ on $[0, 1]$ and vanishes on $[2, \infty)$. Let $Y \geqslant 1$. We have
$$\sum_n\frac{\chi_{\Delta}(n)}{n} w\left( \frac{n}{Y}\right) = \int_{(2)} L(\chi_{\Delta}, 1+s) \widehat{w}(s)Y^s \frac{ds}{2\pi i} = L( \chi_{\Delta}, 1) + O(Y^{-1/2} |\Delta|^{1/6 + \varepsilon})$$
where $\widehat{w}$ in the present case denotes the Mellin transform and the left hand side comes from a contour shift to $\Re s = -1/2$ and the Conrey-Iwaniec \cite{CoIw} subconvexity bound for real characters. We obtain
\begin{equation}\label{char-sum}
\sum_{-X \leqslant \Delta < 0} \chi_{\Delta}(\alpha) L( \chi_{\Delta}, 1)= \sum_n \frac{1}{n}w\left( \frac{n}{Y}\right) \sum_{-X \leqslant \Delta < 0} \chi_{\Delta}(\alpha n) + O(X^{7/6+\varepsilon} Y^{-1/2}).
\end{equation} We decompose the main term as $S_{\square} + S_{\not= \square}$ depending on whether $\alpha n$ is a square or not. We first consider the portion $S_{\square}$ where $n$ is restricted to $n = \alpha_2k^2$ with $k \in \mathbb{N}$. This gives
\begin{equation}\label{er1}
\begin{split} S_{\square} & = \sum_k \frac{1}{\alpha_2 k^2} w\left( \frac{\alpha_2k^2}{Y}\right) \sum_{\substack{-X \leqslant \Delta < 0\\ (\Delta, \alpha k) = 1}} 1 = \sum_k \frac{1}{\alpha_2 k^2} \sum_{\substack{-X \leqslant \Delta < 0\\ (\Delta, \alpha k) = 1}} 1 + O\Big(\frac{X}{Y}\Big). \end{split}
\end{equation} We decompose the main term as $ S_{\square}^{\text{odd}} + S_{\square}^{\text{even, 4}} + S_{\square}^{\text{even, 8}}$ depending on whether $\Delta$ is odd, exactly divisible 4 or exactly divisible by 8. We have $$S_{\square}^{\text{odd}} = \frac{1}{2\alpha_2} \sum_k \frac{1}{k^2} \sum_{\substack{m \leqslant X\\ (m, \alpha k) = 1}} \mu^2(m)(\chi_0(m) - \chi_{-4}(m))$$ where $\chi_0$ is the trivial character modulo 4 and $\chi_{-4}$ the non-trivial character modulo 4. For $\chi \in \{\chi_0, \chi_{-4}\}$ we consider the Dirichlet series \begin{displaymath}
\begin{split} &\frac{1}{2\alpha_2}\sum_k \frac{1}{k^2} \sum_{ (m, \alpha k) = 1} \frac{\mu^2(m)\chi(m)}{m^s} = \frac{\zeta(2)}{2\alpha_2} \prod_{p \nmid \alpha} \Big(1 + \frac{\chi(p)}{p^s} - \frac{\chi(p)}{p^{s+2}}\Big) \\ &= \frac{\zeta(2)}{2\alpha_2} \prod_{p \mid \alpha} \Big(1 + \frac{\chi(p)}{p^s} - \frac{\chi(p)}{p^{s+2}}\Big)^{-1} L(\chi, s) \prod_p\Big( 1 - \frac{\chi(p)^2}{p^{2s}} - \frac{\chi(p)}{p^{s+2}} + \frac{\chi(p)^2}{p^{2s+2}}\Big).
\end{split}
\end{displaymath} A standard application of Perron's formula (e.g.\ \cite[Corollary II.2.4]{Te}) shows now that \begin{equation}\label{er2} S_{\square}^{\text{odd}} = \frac{\zeta(2) X}{4\alpha_2} \prod_{\substack{p \mid \alpha\\ p \nmid 2}} \Big(1 + \frac{1}{p} - \frac{1}{p^{3}}\Big)^{-1} \prod_{p \nmid 2}\Big( 1 - \frac{1}{p^{2}} - \frac{1}{p^{3}} + \frac{1}{p^{4}}\Big) + O\Big( \frac{X^{1/2}}{\alpha_2}(\alpha_2 X)^{\varepsilon}\Big). \end{equation} We have $S_{\square}^{\text{even, 4}} \not= 0$ and $S_{\square}^{\text{even, 8}} \not= 0$ only if $\alpha$ is odd, which we assume from now on. Then $$S_{\square}^{\text{even, 4}} = \frac{1}{\alpha_2} \sum_{(k, 2) = 1} \frac{1}{k^2} \sum_{\substack{m \leqslant X/4\\ (m, \alpha k) = 1\\ m \equiv 1 (\text{mod } 4)}} \mu^2(m) = \frac{1}{2\alpha_2} \sum_{(k, 2) = 1} \frac{1}{k^2} \sum_{\substack{m \leqslant X/4\\ (m, \alpha k) = 1 }} \mu^2(m)(\chi_0(m) + \chi_{-4}(m)) $$ and by the same computation we obtain $$S_{\square}^{\text{even, 4}} = \frac{\zeta(2) X}{16\alpha_2} \Big(1 - \frac{1}{4}\Big) \prod_{p \mid \alpha} \Big(1 + \frac{1}{p} - \frac{1}{p^{3}}\Big)^{-1} \prod_{p \nmid 2}\Big( 1 - \frac{1}{p^{2}} - \frac{1}{p^{3}} + \frac{1}{p^{4}}\Big) + O\Big( \frac{X^{1/2}}{\alpha_2}(\alpha_2 X)^{\varepsilon}\Big).$$ Finally, $$S_{\square}^{\text{even, 8}} = \frac{1}{\alpha_2} \sum_{(k, 2) = 1} \frac{1}{k^2} \sum_{\substack{m \leqslant X/8\\ (m, \alpha k) = 1\\ m \equiv 1 (\text{mod } 2)}} \mu^2(m) = \frac{1}{\alpha_2} \sum_{(k, 2) = 1} \frac{1}{k^2} \sum_{\substack{m \leqslant X/8\\ (m, \alpha k) = 1 }} \mu^2(m)\chi_0(m)$$ where now $\chi_0$ is the unique character modulo 2, and we get the same main term for $S_{\square}^{\text{even, 8}} $. Putting everything together, we obtain for $\alpha_2$ odd that \begin{displaymath} \begin{split} S_{\square}& = \frac{\zeta(2) X}{\alpha_2} \prod_{p \mid \alpha} \Big(1 + \frac{1}{p} - \frac{1}{p^{3}}\Big)^{-1} \prod_{p }\Big( 1 - \frac{1}{p^{2}} - \frac{1}{p^{3}} + \frac{1}{p^{4}}\Big) \cdot \frac{16}{11} \Big( \frac{1}{4} + \frac{1}{16} \cdot\frac{3}{4} \cdot 2\Big) + O\Big(\frac{X}{Y}+ \frac{X^{1/2}}{\alpha_2}(\alpha_2 X)^{\varepsilon}\Big)\\ & = \frac{\zeta(2) X}{2\alpha_2} \prod_{p \mid \alpha} \Big(1 + \frac{1}{p} - \frac{1}{p^{3}}\Big)^{-1} \prod_{p }\Big( 1 - \frac{1}{p^{2}} - \frac{1}{p^{3}} + \frac{1}{p^{4}}\Big) + O\Big(\frac{X}{Y}+ \frac{X^{1/2}}{\alpha_2}(\alpha_2 X)^{\varepsilon}\Big) \end{split} \end{displaymath} and for $\alpha_2$ even that \begin{displaymath} \begin{split} S_{\square}& = \frac{\zeta(2) X}{\alpha_2} \prod_{p \mid \alpha} \Big(1 + \frac{1}{p} - \frac{1}{p^{3}}\Big)^{-1} \prod_{p }\Big( 1 - \frac{1}{p^{2}} - \frac{1}{p^{3}} + \frac{1}{p^{4}}\Big) \cdot \frac{16}{11}\cdot \frac{1}{4}\Big(1 + \frac{1}{2} -\frac{1}{8} \Big) + O\Big(\frac{X}{Y}+ \frac{X^{1/2}}{\alpha_2}(\alpha_2 X)^{\varepsilon}\Big)\\ & = \frac{\zeta(2) X}{2\alpha_2} \prod_{p \mid \alpha} \Big(1 + \frac{1}{p} - \frac{1}{p^{3}}\Big)^{-1} \prod_{p }\Big( 1 - \frac{1}{p^{2}} - \frac{1}{p^{3}} + \frac{1}{p^{4}}\Big) + O\Big(\frac{X}{Y}+ \frac{X^{1/2}}{\alpha_2}(\alpha_2 X)^{\varepsilon}\Big). \end{split} \end{displaymath} Note how beautifully the constants fit together. We return to \eqref{char-sum} and study $S_{\not=\square}$ where the $n$-sum is restricted to $\alpha n \not= \square$. We write $n = 2^{\nu} n'$ and $\alpha = 2^a \alpha'$ with $n', \alpha'$ odd. We need to bound
$$ \sum_{\substack{2^{\nu} n' \leqslant 2Y\\ 2^{\nu+a}n'\alpha' \not= \square}} \frac{1}{2^{\nu} n' } \Big| \sum_{- X\leqslant \Delta < 0} \chi_{\Delta}(2^{a + \nu}) \chi_{\Delta}(\alpha'n')\Big|.$$ The number $\alpha'n'$ is not a square, unless it is 1, in which case $a+\nu$ is necessarily odd, so that the $\Delta$-sum and hence the entire expression is $O(1)$. The number $\chi_{\Delta}(2^{a + \nu})$ depends only on $\Delta$ modulo 8, and we can split the sum into residue classes modulo 8 to make it independent of that factor. If $\alpha' n'$ is not a square and odd, then $\Delta \mapsto (\frac{\Delta}{\alpha' n'})$ is defined for every negative $\Delta$ and in fact a possibly imprimitive, but certainly non-trivial character modulo $\alpha'n'$. Splitting even into residue classes modulo 16, we can detect the condition that $\Delta$ is a fundamental discriminant by requiring $\Delta$ or $\Delta/4$ be squarefree. Thus it suffices to bound $$\sum_{m \leqslant X'} \mu^2(m) \psi(m)$$
for a character $\psi$ of conductor $\ll \alpha Y$ and $X' \leqslant X$. Writing $\mu^2 (m) = \sum_{d^2 \mid m} \mu(d)$ and using the P\'olya-Vinogradov inequality, we bound the previous display by
\begin{equation}\label{er3}
\ll \sum_{d \leqslant X} \min\Big(\frac{X}{d^2}, (\alpha Y)^{1/2+\varepsilon}\Big) \ll (X^{1/2}( \alpha Y)^{1/4})^{1+\varepsilon}.
\end{equation}
Collecting error terms in \eqref{char-sum}, \eqref{er1}, \eqref{er2}, \eqref{er3}, the total error becomes
$$\ll \Big(\frac{X^{7/6}}{Y^{1/2}} + \frac{X}{Y} + \frac{X^{1/2}}{\alpha_2} + X^{1/2}(\alpha Y)^{1/4}\Big)(Xy\alpha)^{\varepsilon}.$$
We choose $Y = X^{8/9}\alpha^{-1/3} + 1$ to recover a total error of $(X^{13/18} \alpha^{1/6} + X^{1/9} \alpha^{1/3} + X^{1/2} \alpha^{1/4})(X\alpha)^{1+\varepsilon}$. This completes the proof of the asymptotic formula. The claim on the Dirichlet series $\mathscr{K}(v; \alpha)$ follows by partial summation.
\end{proof}
With the notation of the previous lemma we can write $$\mathscr{L}(v) = \sum_{ (d\delta \nu, m) = 1 } \sum_{\substack{g, f\\ (f, \delta) = 1}} \sum_{\substack{d_1 \mid fd\\ d_2 \mid f\nu}} \sum_{d_3 \mid (\frac{fd}{d_1}, \frac{f\nu}{d_2}) } \sum_{r_1r_2 = \frac{f^2 d \nu}{d_1d_2d_3^2}} \frac{ \mu(d\delta \nu)\mu^2(m) \mu(d_1)\mu(d_2) d_3 }{(d\nu)^{4 + 2v}( \delta g) ^{2+2v}f^{3+4v} m^3 } \mathscr{K}(v, \delta d_1 d_2r_2 ), $$ and we conclude that $\mathscr{L}$ has analytic continuation to $\Re v > -13/18$ except for a simple pole at $v = 0$ and polynomial (in fact linear) bounds on vertical lines. If $\text{rad}(n)$ denotes the squarefree kernel of $n$, then \begin{equation}\label{euler} \begin{split} &\underset{v=0}{\text{res}} \mathscr{L}(v) = \frac{\zeta(2)^2}{2}\prod_{p }\Big( 1 - \frac{1}{p^{2}} - \frac{1}{p^{3}} + \frac{1}{p^{4}}\Big) \\ &\sum_{ (d\delta \nu, m) = 1 } \sum_{ (f, \delta) = 1} \sum_{\substack{d_1 \mid fd\\ d_2 \mid f\nu}} \sum_{d_3 \mid (\frac{fd}{d_1}, \frac{f\nu}{d_2}) } \sum_{r_1r_2 = \frac{f^2 d \nu}{d_1d_2d_3^2}} \frac{ \mu(d\delta \nu)\mu^2(m) \mu(d_1)\mu(d_2) d_3 }{(d\nu)^{4 }\delta^2 f^{3 } m^3 \text{rad}(\delta d_1 d_2r_2) } \prod_{p \mid \delta d_1 d_2r_2} \Big(1 + \frac{1}{p} - \frac{1}{p^3}\Big)^{-1} \end{split} \end{equation}
where we have implicitly computed the $g$-sum as $\zeta(2)$. Now a massive computation with Euler products, best performed with a computer algebra system, shows gigantic cancellation, and we obtain the beautiful formula
$$\underset{v=0}{\text{res}} \mathscr{L}(v) = \frac{\zeta(2)^2}{2\zeta(4)}.$$
With this we return to \eqref{previousdisplay} and shift the $v$-contour to $\Re v = -1/10$. By \eqref{G1} and trivial bounds the remaining integral expression is bounded by
$$\ll \frac{1}{K^2} \sum_{K \leqslant k \leqslant 2K} K^{-\frac{1}{2} - \frac{4}{10}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \Big(1 + \frac{|t|^2 + |\tau|^2}{K}\Big) |\tau| d\tau \, dt \ll K^{-4/10}.$$ It remains to deal with the double pole at $v = 0$ whose residue is given by
\begin{equation*} \begin{split} \mathcal{R} = & \frac{12}{ \omega K^2} \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \frac{\pi^2}{90} \cdot \frac{18\sqrt{\pi}}{2 }\cdot 2 \cdot \frac{2}{3} \cdot \frac{1}{4} \\ &\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \underset{v=0}{\text{res}}\Big( \frac{e^{v^2 }}{v} \mathscr{L}(2v) \mathcal{G}(k, \tau, v+ 1/2 + it) \mathcal{G}(k, \tau, v + 1/2 - it)\Big) \omega(\tau) \tau \tanh(\pi \tau) \frac{d\tau}{\pi^2} dt .
\end{split} \end{equation*} We can remove the factor $\omega(\tau)\tanh(\pi \tau)$ at the cost of an error $O(K^{-\eta/2})$, cf.\ the definition \eqref{def-omega}. Using the definition \eqref{defG}, at $v=0$ we have the following Taylor expansion \begin{displaymath} \begin{split}
& \mathcal{G}(k, \tau, v+ 1/2 + it) \mathcal{G}(k, \tau, v + 1/2 - it)\\
&= \mathcal{G}(k, \tau, 1/2 + it) \mathcal{G}(k, \tau, 1/2 - it) \Big\{1 + v\Big(\sum_{\pm, \pm} \frac{\Gamma'}{\Gamma}\Big( \frac{k-1/2}{2} \pm it \pm \frac{i\tau}{2}\Big) - 4\log \pi\Big) + O(v^2)\Big\}. \end{split} \end{displaymath}
We have $\frac{\Gamma'}{\Gamma}(z) = \log z +O(|z|^{-1})$ (for $\Re z \geqslant 1$, say) and so $$\sum_{\pm} \frac{\Gamma'}{\Gamma}\Big( \frac{k-1/2}{2} - it \pm \frac{i\tau}{2}\Big) - 4\log \pi = 4 \log k + O(1)$$ for $t, \tau \ll k^{2/3}$, say. Otherwise, the $t$ and $\tau$ integrals are negligible outside this region. In this range we can insert \eqref{taylor}
to conclude that \begin{equation*} \begin{split}
\mathcal{R} = & \frac{12}{\omega K^2} \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \frac{\pi^2}{90} \cdot \frac{18\sqrt{\pi}}{2 } \cdot 2 \cdot \frac{2}{3} \cdot \frac{1}{4} \cdot \frac{1}{2} \cdot 4\log k \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{\zeta(2)^2}{2\zeta(4)}\frac{16}{\pi k^{1/2}} e^{-(4t^2 + \tau^2)/k} | \tau| \frac{d\tau}{\pi^2} dt + O(1).
\end{split} \end{equation*} We evaluate the two integrals, sum over $k$ (by Poisson, for instance) and recall the definition of $\omega$ just before \eqref{Nav} getting \begin{equation*} \begin{split} \mathcal{R} &= \frac{12}{\omega K^2} \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \frac{\pi^2}{90} \cdot \frac{18\sqrt{\pi}}{2 } \cdot 2\cdot \frac{2}{3} \cdot \frac{1}{4} \cdot \frac{1}{2} \cdot 4\log k \frac{\zeta(2)^2}{2\zeta(4)}\frac{16}{\pi k^{1/2}} k \cdot \frac{\sqrt{\pi k}}{2} \frac{1}{\pi^2} + O(1)\\ & = \frac{12}{2} \cdot \frac{\pi^2}{90} \cdot \frac{18\sqrt{\pi}}{2 } \cdot 2 \cdot \frac{2}{3} \cdot \frac{1}{4} \cdot \frac{1}{2} \cdot 4 \log K \cdot \frac{\zeta(2)^2}{2\zeta(4)}\frac{16}{\pi} \cdot \frac{\sqrt{\pi }}{2} \frac{1}{\pi^2} + O(1) = 4 \log K + O(1).
\end{split} \end{equation*} We have now detected the main term and we conclude this section by stating that $$\mathcal{N}^{\text{diag}, \text{diag}}(K) = \mathcal{R} + O(1) = 4\log K + O(1).$$
\section{The diagonal off-diagonal term}\label{diag-off}
\subsection{Preparing the stage} We return to \eqref{diagdiag} and consider the off-diagonal terms on the right hand side of \eqref{kuz-even-form}. We treat the first off-diagonal term in detail in the following three subsections. In Section \ref{144} show the minor modifications to treat the second off-diagonal term. The first off-diagonal term is given by \begin{displaymath} \begin{split} \mathcal{I}^{\text{off-diag,1}}(\Delta, t,k, r):= 8i \sum_{d \mid r} &\sum_{{\tt n},{\tt m}} \frac{ \chi_{\Delta}({\tt m}) }{(d{\tt n}{\tt m})^{1/2}} \sum_c \frac{S({\tt m}, {\tt n}r/d, c)}{c} \\
&\int_{-\infty}^{\infty} \frac{J_{2i\tau}(4\pi \sqrt{({\tt m}{\tt n}r/d)}/c)}{\sinh(\pi \tau)} W^+_{\tau}(d{\tt n}) W^-_{\tau}\Big(\frac{{\tt m}}{|\Delta|}\Big)h(\tau) \tau \tanh(\pi \tau) \frac{d\tau}{4\pi} \end{split} \end{displaymath} where $h$ was defined in \eqref{htau} and depends in particular on $t$, $k$ and $\Delta$. This needs to be inserted into \eqref{weobt} with $r = f^2d\nu/(d_1 d_2 d_3^2) \leqslant K^{4\eta}$. In \eqref{weobt} we apply a smooth partition of unity to the $\Delta$-sum and consider a typical portion
$$\mathcal{J}^{(1)}(X, t,k, r) := \sum_{\Delta}w\Big( \frac{|\Delta|}{X}\Big) \mathcal{I}^{\text{off-diag,1}}(\Delta, t,k, r)$$ for a smooth function $w$ with compact support in $[1, 2]$. Our aim in this section is to prove the bound \begin{equation}\label{J-bound}
\mathcal{J}^{(1)}(X, t,k, r) \ll X K^{1/2 - \eta} \end{equation} uniformly in $k \asymp K$, $t \leqslant K^{1/2 + \varepsilon}$, \begin{equation}\label{X}
X \leqslant K^{2+\varepsilon}, \quad r \leqslant K^{4\eta}
\end{equation}
and $f, f_1, f_2, d, \nu \in \mathbb{N}$ which are implicit in the definition \eqref{htau}. Taking \eqref{J-bound} for granted, we estimate \eqref{weobt} trivially to obtain a contribution of $O(K^{\varepsilon - \eta})$, which is admissible. So it remains to show \eqref{J-bound}, and to this end we start with some initial discussion.
The $c$-sum in $\mathcal{I}^{\text{off-diag,1}}(\Delta, t,k, r)$ is absolutely convergent, as can be seen by using the Weil bound for the Kloosterman sum and shifting the $t$-contour to $\Re i\tau = 1/3$, say, without crossing any poles. By the power series expansion for the Bessel function \cite[8.440]{GR} we have $$J_{2i\tau}(x) \ll_{\Im \tau} x^{-2\Im \tau}e^{\pi|\tau|} (1+|\tau|)^{-1/2}, \quad x \leqslant 1$$ and we can therefore truncate the $c$-sum at $c \leqslant K^{10^6}$, say, at the cost of a very small error. Having done this, we can sacrifice holomorphicity of the integrand in the $\tau$-integral, and we insert a smooth partition of unity into the $\tau$-integral restricting to $\tau \asymp T$, say, with \begin{equation}\label{T} K^{1/2 - 2\eta} \leqslant T \leqslant K^{1/2+\varepsilon}, \end{equation} otherwise $h(\tau)$ is negligible by \eqref{htau}, \eqref{size-restr} and \eqref{prop-omega}. We insert smooth partitions of unity into the ${\tt n}, {\tt m}, c$-sums and thereby restrict to ${\tt n} \asymp N$, ${\tt m} \asymp M$ $c \asymp C$, say, where \begin{equation}\label{NM} N \leqslant T^{1+\varepsilon} , M \leqslant X T^{1 + \varepsilon} \end{equation} by \eqref{bound-wt} and initially $C \leqslant K^{10^6}$. Next we want to evaluate asymptotically the $\tau$-integral. To this end we use Lemma \ref{bessel-kuz} with the weight function ${\tt h} = {\tt h}_{{\tt n}, {\tt m}, \Delta}$ given by
$$W^+_{\tau}(d{\tt n}) W^-_{\tau}\Big(\frac{{\tt m}}{|\Delta|}\Big)h(\tau) w\Big( \frac{|\tau|}{T}\Big) = W^+_{\tau}(d{\tt n}) W^-_{\tau}\Big(\frac{{\tt m}}{|\Delta|}\Big)\omega(\tau) V_{\pm t}((|\Delta|f^2f_1f_2d\nu)^2; k, \tau) $$ where $w$ is the weight function occurring in the smooth partition of unity of the $\tau$-integral. By \eqref{bound-wt}, \eqref{prop-omega} and \eqref{size-restr}, the function ${\tt h}$ is ``flat'' in all variables, i.e.\
$$\frac{d^{j_1}}{d{\tt n}^{j_1}}\frac{d^{j_2}}{d{\tt m}^{j_2}}\frac{d^{j_3}}{d\Delta^{j_3}} \frac{d^j}{dt^j}{\tt h}_{{\tt n}, {\tt m}, \Delta}(t) \ll_{\textbf{j}} K^{-1/2} T^{-j}N^{-j_1} M^{-j_2} X^{-j_3}$$ for $k \asymp K$, $|\Delta| \asymp X$, ${\tt n} \asymp N$, ${\tt m} \asymp M$ and $\textbf{j} \in \mathbb{N}_0^4$, uniformly in all other variables. Lemma \ref{bessel-kuz}a implies that we can restrict, up to a negligible error, to \begin{equation}\label{C} C \leqslant \frac{\sqrt{NMr_2}}{T^2} K^{\varepsilon}, \end{equation} where $r_2 = r/d$, and up to a negligible error we are left with bounding \begin{equation}\label{E1} \begin{split} \mathscr{J}^{(1)}_r(X, N, M, C, T) = &\frac{K^{\varepsilon}T^2}{(NM)^{3/4}(KC)^{1/2}}\sum_{r_1r_2=r} \frac{1}{(rr_1)^{1/4}} \\
&\sum_{\Delta, {\tt n} , {\tt m}, c} \chi_{\Delta}({\tt m}) S({\tt m}, {\tt n} r_2, c) e\Big( \pm 2 \frac{\sqrt{{\tt n} {\tt m} r_2}}{ c}\Big) W\Big( \frac{|\Delta|}{X}, \frac{{\tt n}}{N}, \frac{{\tt m}}{M}, \frac{c}{C}\Big) \end{split} \end{equation}
for a smooth function $W$ with compact support in $[1, 2]^4$ and bounded Sobolev norms with variables $X, N, M, C, T$ satisfying \eqref{X}, \eqref{NM}, \eqref{C}, \eqref{T}, respectively.
The basic strategy is now to apply Poisson summation first in the ${\tt n}$-sum and then in the ${\tt m}$-sum in Section \ref{132}. This shortens the variables in generic ranges, so that a trivial bound turns out to be of size $XK^{1/2 + 2\eta+\varepsilon}$. This is very close to our target \eqref{J-bound}. In Section \ref{133} we will extract a character sum from this expression where the P\'olya-Vinogradov inequality produces the final saving, at least in generic ranges of the variables. In order to also treat non-generic ranges where some of the variables are relatively short, at each step we also apply trivial bounds along with Heath-Brown's large sieve. In particular, by Lemma \ref{HBlemma}b) we can bound
\eqref{E1} by
\begin{equation}\label{E2}
\begin{split} \mathscr{J}^{(1)}_r(X, N, M, C, T) &\ll \frac{K^{\varepsilon}T^2}{(NM)^{3/4}(KC)^{1/2}} X NMC^{3/2} \Big( \frac{1}{X^{1/2}} + \frac{1}{M^{1/2}}\Big) \\ &= \frac{K^{\varepsilon}T^2}{ K^{1/2}} CN^{1/4} \Big(M^{1/4} X^{1/2} + \frac{X}{M^{1/4}} \Big). \end{split}
\end{equation}
\subsection{Poisson summation}\label{132}
Now we open the Kloosterman sum in \eqref{E1} and apply Poisson summation in ${\tt n}$ in residue classes modulo $c$. In this way we re-write $\mathscr{J}^{(1)}_r(X, N, M, C, T) $ as
\begin{equation}\label{E3}
\begin{split}
&\frac{K^{\varepsilon}T^2}{(NM)^{3/4}(KC)^{1/2}}\sum_{r_1r_2 = r} \frac{1}{(rr_1)^{1/4}} \sum_{\Delta, {\tt m}, c} \chi_{\Delta}({\tt m}) \underset{\gamma\, (\text{mod } c)}{\left.\sum\right.^{\ast}}e\Big(\frac{{\tt m}\gamma}{c}\Big) \sum_{\nu \, (\text{mod } c)}e\Big( \frac{\bar{\gamma}\nu r_2}{ c}\Big)\\
& \quad\quad\quad\quad\frac{1}{c} \sum_{n \in \mathbb{Z}}e\Big(\frac{n \nu}{c}\Big) \int_{0}^{\infty} e\Big( \pm 2 \frac{\sqrt{x {\tt m} r_2}}{ c}\Big) W\Big( \frac{|\Delta|}{X}, \frac{x}{N}, \frac{{\tt m}}{M}, \frac{c}{C}\Big) e\Big( - \frac{x n}{c}\Big) dx.
\end{split} \end{equation} We consider the character sum \begin{equation}\label{char1} \frac{1}{c}\underset{\gamma\, (\text{mod } c)}{\left.\sum\right.^{\ast}}e\Big(\frac{{\tt m}\gamma}{c}\Big) \sum_{\nu \, (\text{mod } c)}e\Big( \frac{\bar{\gamma}\nu r_2}{ c}\Big)e\Big(\frac{n \nu}{c}\Big) = \underset{\substack{\gamma\, (\text{mod } c)\\ \bar{\gamma}r_2 \equiv -n\, (\text{mod } c)}}{\left.\sum\right.^{\ast}}e\Big(\frac{{\tt m}\gamma}{c}\Big). \end{equation} This is non-zero only if $(n, c) = (r_2, c)$. We write $(r_2, c) = \delta$, $r_2 = \delta r_2'$, $c = \delta c'$, $n = \delta n'$ with $(n'r_2', c') = 1$ and recast \eqref{char1} as $$ \underset{\substack{\gamma\, (\text{mod } c)\\ \gamma \equiv -r_2'\overline{n'}\, (\text{mod } c')}}{\left.\sum\right.^{\ast}}e\Big(\frac{{\tt m}\gamma}{c}\Big).$$ We decompose $\delta = \delta_1 \delta_2$ with $\delta_2$ maximal so that $(\delta_2, c') = 1$. Note that $r_2' \bar{n'}$ is coprime to $c'$, so the condition $(c, \gamma) = 1$ is equivalent to $(\delta_2, \gamma) = 1$. We factor $c = c'\delta_1 \cdot \delta_2$ with $(c'\delta_1, \delta_2) = 1$ and apply the Chinese Remainder Theorem to see that the previous display vanishes unless $\delta_1 \mid {\tt m}$, say ${\tt m} = \delta_1{\tt m}'$, in which case it equals $$\delta_1 R_{\delta_2}({\tt m}') e\Big( -\frac{{\tt m}'r_2' \overline{n' \delta_2}}{c'}\Big)$$ where $R$ denotes the Ramanujan sum.
Next we consider the $x$-integral in \eqref{E3}. The phase has a unique stationary point at $x = r_2{\tt m}/n^2$ if $\text{sgn}(n) = \pm$ and no stationary point if $\text{sgn}(n) = \mp$. If $N \leqslant r_2{\tt m}/n^2 \leqslant 2N$ and $\text{sgn}(n) = \pm$, we can apply the stationary phase lemma \cite[Proposition 8.2]{BKY} with $X=1$, $V = V_1 = Q = N$, $Y = \sqrt{NMr_2}/C \geqslant T^2K^{-\varepsilon} \geqslant K^{1/2 - \varepsilon}$ to see that the integral is given by
$$\Big(\frac{cr_2{\tt m}}{|n|^3}\Big)^{1/2} e\Big( \pm \frac{r_2{\tt m}}{c|n|}\Big)W_1\Big( \frac{|\Delta|}{X}, \frac{r_2{\tt m}}{n^2N}, \frac{{\tt m}}{M}, \frac{c}{C}\Big) $$ for a smooth function $W_1$ with compact support in $[1, 2]^4$ and bounded Sobolev norms, up to a negligible error from truncating the series in \cite[(8.9)]{BKY}. Otherwise we apply integration by parts in the form of \cite[Lemma 8.1]{BKY} with $X = 1$, $U = Q = N$, $Y = R = \sqrt{NMr_2}/C$ to conclude that the integral is negligible.
Noting that with our previous notation
$$e\Big( \pm \frac{r_2{\tt m}}{c|n|}\Big) = e\Big( \frac{d'_2{\tt m}' }{c'n' \delta_2}\Big)$$ for $\text{sgn}(n) = \pm$, we can now apply the additive reciprocity formula $e(1/ab) = e(\bar{a}/b)e(\bar{b}/a)$ for $(a, b) = 1$ to conclude that $\mathscr{J}^{(1)}_r(X, N, M, C, T) $ equals, up to a negligible error term, \begin{equation}\label{E2a}
\begin{split}
\frac{K^{\varepsilon}T^2}{(NM)^{3/4}(KC)^{1/2}}\sum_{r_1\delta_1\delta_2 d'_2 = r} &\frac{1}{(rr_1)^{1/4}} \underset{\substack{(c', n'r_2'\delta_2) = 1\\ \delta_1 \mid (c')^{\infty}}}{\sum_{\Delta, {\tt m}', c'} \sum_{\pm n' \in \mathbb{N}}} \chi_{\Delta}({\tt m}'\delta_1) \delta_1 R_{\delta_2}({\tt m}') e\Big( \frac{{\tt m}'r_2' \overline{c'}}{n' \delta_2}\Big) \\
&\Big(\frac{c'd'_2{\tt m}'}{\delta_2|n'|^3}\Big)^{1/2}W_1\Big( \frac{|\Delta|}{X}, \frac{d'_2{\tt m}'}{\delta_2 (n')^2N}, \frac{{\tt m}'\delta_1}{M}, \frac{c'\delta_1\delta_2}{C}\Big).
\end{split} \end{equation} Here we recall the notation conventions from Section \ref{15} regarding expressions $\delta \mid c^{\infty}$ etc.
For easier readability we remove all the dashes at the variables, and we define $$W_2(x, y, z, w) = w^{1/2} y^{-3/2} z^{1/2} W_1(x, z/y^2, z, w).$$ We also open the Ramanujan sum $R_{\delta_2}({\tt m}) = \sum_{\delta_3 \mid (\delta_2, {\tt m})} \delta_3\mu(\delta_2/\delta_3)$, write $\delta_2 = \delta_3\delta_4$ and replace $m$ with $m\delta_3$. Finally we drop the $\pm$-sign in the summation condition on $n$ (both cases are identical). With this notation we can re-write \eqref{E2a} as \begin{equation}\label{E4}
\begin{split}
\frac{K^{\varepsilon}T^2}{M K^{1/2}}&\sum_{r_1\delta_1\delta_3\delta_4 r_2 = r} \frac{\delta_1^{3/4} \delta_3 \mu(\delta_4)}{(rr_1r_2 \delta_4)^{1/4} } \\
&\underset{\substack{(c, nr_2\delta_3\delta_4) = 1\\ \delta_1 \mid c^{\infty}}}{\sum_{\Delta, {\tt m}, c, n} } \chi_{\Delta}({\tt m}\delta_1\delta_3) e\Big( \frac{{\tt m} r_2 \overline{c}}{n \delta_4}\Big) W_2\Big( \frac{|\Delta|}{X}, \frac{n\delta_3 \sqrt{N \delta_1\delta_4}}{\sqrt{Mr_2}}, \frac{{\tt m}\delta_1\delta_3}{M}, \frac{c\delta_1\delta_3\delta_4}{C}\Big).
\end{split} \end{equation} Before we continue to transform this expression, we estimate trivially with the large sieve (Lemma \ref{HBlemma}b) \begin{equation}\label{E4a}
\begin{split} &\mathscr{J}^{(1)}_r(X, N, M, C, T) \\ & \ll \frac{K^{\varepsilon}T^2}{M K^{1/2}}\sum_{r_1\delta_1\delta_3\delta_4 r_2 = r} \frac{\delta_1^{3/4}\delta_3}{(rr_1r_2\delta_4)^{1/4} } X \frac{\sqrt{Mr_2} }{\delta_3\sqrt{N\delta_1\delta_4}} \frac{M}{\delta_1\delta_3} \frac{C}{\delta_1\delta_3\delta_4} \Big( \Big(\frac{\delta_1}{M}\Big)^{1/2} + X^{-1/2}\Big) \\ &\ll \frac{K^{\varepsilon} T^2 C}{(KN)^{1/2}} (X + (XM)^{1/2}).
\end{split} \end{equation}
Our next goal is to apply Poisson summation in ${\tt m}$ restricted to residue classes $\delta_4 n |\Delta|$. For $m \in \mathbb{Z}$, this leads to the character sum
$$\sum_{\mu \, (\text{mod } \delta_4 n |\Delta|)} \chi_{\Delta}(\mu) e\Big( \frac{\mu r_2 \overline{c}}{n \delta_4}\Big) e\Big(\frac{\mu m}{\delta_4 n|\Delta|}\Big) = \sum_{\mu \, (\text{mod } \delta_4 n |\Delta|)} \chi_{\Delta}(\mu) e\Big( \frac{\mu ( r_2 \overline{c}|\Delta| + m)}{ \delta_4n|\Delta|}\Big) . $$ We decompose both $n$ and $\delta_4$ according to their common divisor with $\Delta$. We write $n = n_1 n_2$ and $\delta_4 = \delta_5 \delta_6$ where $n_1\delta_5 \mid \Delta^{\infty}$, $(n_2\delta_6, \Delta) = 1$. We obtain by the Chinese Remainder Theorem that the above character sum equals \begin{displaymath} \begin{split}
& \sum_{\mu \, (\text{mod } \delta_6 n_2)} e\Big( \frac{\mu\overline{\delta_5 n_1|\Delta|} ( r_2 \overline{c}|\Delta| + m)}{n_2 \delta_6 }\Big)\sum_{\mu \, (\text{mod } \delta_5 n_1 |\Delta|)} \chi_{\Delta}(\mu) e\Big( \frac{\mu \overline{\delta_6n_2}( r_2 \overline{c}|\Delta| + m)}{ \delta_5n_1|\Delta|}\Big).
\end{split} \end{displaymath}
The first sum vanishes unless $n_2\delta_6 \mid r_2|\Delta| + m c$ in which case it equals $n_2\delta_6$. Since $\Delta$ is a negative fundamental discriminant, the second sum vanishes unless $n_1\delta_5 \mid r_2|\Delta| + m c$ in which case it equals
$$i\sqrt{|\Delta|} n_1 \delta_5\chi_{\Delta}\Big( n_2\delta_6 \frac{ r_2 \bar{c}|\Delta| + m}{n_1\delta_5}\Big). $$ Having this evaluation available, the Poisson summation formula transforms \eqref{E4} into \begin{equation}\label{E5}
\begin{split}
\frac{K^{\varepsilon}T^2}{M K^{1/2}}&\sum_{r_1\delta_1\delta_3\delta_5\delta_6 r_2 = r} \frac{\delta_1^{3/4} \delta_3 \mu(\delta_5\delta_6)}{(rr_1r_2 \delta_5\delta_6)^{1/4} } \underset{\substack{(c, n_1n_2r_2\delta_3\delta_5\delta_6) = 1\\ \delta_1 \mid c^{\infty}, n_1\delta_5 \mid \Delta^{\infty}, (n_2 \delta_6, \Delta) = 1\\ n_1n_2\delta_5\delta_6 \mid r_2|\Delta| + mc }}{\sum_{\Delta, c, n_1, n_2} \sum_{ m\in \mathbb{Z}}} \chi_{\Delta}( \delta_1\delta_3) \frac{\chi_{\Delta}\Big( n_2\delta_6 \frac{ r_2 \bar{c}|\Delta| + m}{n_1\delta_5}\Big)}{ |\Delta|^{1/2}} \\
&\int_0^{\infty} W_2\Big( \frac{|\Delta|}{X}, \frac{n_1n_2 \delta_3\sqrt{N \delta_1 \delta_5\delta_6}}{\sqrt{Mr_2}}, \frac{x\delta_1\delta_3}{M}, \frac{c\delta_1\delta_3\delta_5\delta_6}{C}\Big) e\Big(- \frac{mx}{n_1n_2\delta_5\delta_6 |\Delta|}\Big)dx,
\end{split} \end{equation} up to a factor $i$.
The above integral is just the Fourier transform of $W_2$ with respect to the third variable which we write as
$$ \frac{M}{\delta_1\delta_3}W_3\Big( \frac{|\Delta|}{X}, \frac{n_1n_2 \delta_3\sqrt{N \delta_1\delta_5\delta_6}}{\sqrt{Mr_2}}, \frac{mM }{n_1n_2\delta_5\delta_6 |\Delta|\delta_1\delta_3}, \frac{c\delta_1\delta_3\delta_5\delta_6}{C}\Big) . $$ Defining $W_4(x, y, z, w) = W_3(x, y, z/(xy), w)$ we recast the previous display as
$$ \frac{M}{\delta_1\delta_3}W_4\Big( \frac{|\Delta|}{X}, \frac{n_1n_2\delta_3 \sqrt{N \delta_1\delta_5\delta_6}}{\sqrt{Mr_2}}, \frac{m\sqrt{NM } }{ (r_2\delta_1 \delta_5\delta_6 )^{1/2}X}, \frac{c\delta_1\delta_3\delta_5\delta_6}{C}\Big) . $$ The function $W_4$ is compactly supported in the first, second and fourth variable and rapidly decaying in the third.
We first estimate the $m=0$ contribution to be \begin{displaymath} \begin{split}
\ll \frac{K^{\varepsilon}T^2}{M K^{1/2}}\sum_{r_1\delta_1\delta_3\delta_5\delta_6 r_2 = r} \frac{\delta_1^{3/4} \delta_3 }{(rr_1r_2 \delta_5\delta_6)^{1/4} } \frac{M}{\sqrt{X}\delta_1\delta_3} X \frac{C}{\delta_1\delta_3\delta_5\delta_6} \ll K^{\varepsilon} \frac{X^{1/2}CT^2}{K^{1/2} } \ll K^{\varepsilon} \frac{(XNM)^{1/2}}{K^{1/2} } \ll XK^{\varepsilon} \end{split} \end{displaymath} by \eqref{C}, \eqref{NM}, \eqref{X}, \eqref{T} and a divisor estimate for $n_1n_2$. This is clearly admissible for \eqref{J-bound}, so from now on we assume $m\not= 0$.
By the same argument, we obtain for the $m \not= 0$ contribution the bound \begin{equation}\label{E6} \begin{split}
\frac{K^{\varepsilon}T^2}{M K^{1/2}}\sum_{r_1\delta_1\delta_3\delta_5\delta_6 r_2 = r} \frac{\delta_1^{3/4} \delta_3 }{(rr_1r_2 \delta_5\delta_6)^{1/4} } \frac{M}{\sqrt{X}\delta_1\delta_3} X \frac{C}{\delta_1\delta_3\delta_5\delta_6} \frac{(r_2\delta_1 \delta_5\delta_6 )^{1/2}X}{\sqrt{NM } } \ll K^{\varepsilon} \frac{X^{3/2}CT^2}{(MNK)^{1/2} }. \end{split} \end{equation} By \eqref{C} and \eqref{X}, this is only a factor $K^{3\eta + \varepsilon}$ away from our target \eqref{J-bound}, so a very small additional saving suffices to win. For easier readability we consider only the case $m > 0$, the other case being entirely analogous.
\subsection{The endgame}\label{133} Up until now we have not touched the long $\Delta$-sum, which we will now use to obtain some additional saving. Before we do this, we must exclude the case that $C$ is very small. To this end we combine \eqref{E6} and \eqref{E4a} to obtain $$\mathscr{J}^{(1)}_r(X, N, M, C, T)\ll K^{\varepsilon} \frac{T^2C}{(NK)^{1/2}}\min\Big(\frac{X^{3/2}}{M^{1/2}} + X + (XM)^{1/2}\Big) \ll K^{\varepsilon} \frac{T^2C}{(NK)^{1/2}}X.$$ Similarly we can combine \eqref{E6} and \eqref{E2} to obtain \begin{displaymath} \begin{split} \mathscr{J}^{(1)}_r(X, N, M, C, T)& \ll K^{\varepsilon} \frac{T^2C}{K^{1/2}}\min\Big( \frac{X^{3/2}}{(MN)^{1/2}}, (MN)^{1/4} X^{1/2} + \Big(\frac{N}{M}\Big)^{1/4} X\Big) \\ &\leqslant K^{\varepsilon} \frac{T^2CX}{K^{1/2}} \Big( \frac{1}{(MN)^{1/8}}+ \Big(\frac{N}{M}\Big)^{1/4} \Big) \ll K^{\varepsilon} \frac{T^2CX}{K^{1/2}} \Big( \frac{r^{1/8}}{C^{1/4}T^{1/2}}+ \frac{r^{1/4}N^{1/2}}{C^{1/2}T}\Big) \end{split} \end{displaymath} using \eqref{C}. Combining the previous two bounds we finally obtain $$\mathscr{J}^{(1)}_r(X, N, M, C, T) \ll K^{\varepsilon} \frac{T^2CX}{K^{1/2}} \cdot \frac{r^{1/8}}{C^{1/4}T^{1/2}} \ll XK^{\frac{1}{4} + \frac{1}{2}\eta + \varepsilon} C^{\frac{3}{4}}$$ by \eqref{T} and \eqref{X} which meets our target \eqref{J-bound} unless $$C \geqslant K^{1/3 - 3\eta}$$ which we assume from now on. We recall that $c$ is automatically coprime to $n_1n_2 r_2\delta_3\delta_5\delta_6$. For fixed $k \in \mathbb{N}$ let $\mathscr{S}(k) = \{k_1 x^2 : k_1 \mid k, x \in \mathbb{N}\}$ be the set of square classes of all divisors of $k$. From \eqref{E5} we remove all $c\in \mathscr{S}(2r_1\delta_1m)$. Since $C$ is large, the $O(K^{\varepsilon})$ square classes are only a thin subset of all $c$ and by the same computation as in \eqref{E6}, they contribute no more than \begin{equation}\label{square-c} \begin{split} & \ll K^{\varepsilon} \frac{X^{3/2}C^{1/2}T^2}{(MNK)^{1/2} } \ll r^{1/2}XK^{1/2 + 2\eta + \varepsilon} C^{-1/2} \ll XK^{1/3 + 6\eta} \end{split} \end{equation} to$ \mathscr{J}^{(1)}_r(X, N, M, C, T) $ which for $\eta < 1/40$ is admissible.
With this we return to \eqref{E5} and explain the idea how to obtain additional savings. Ignoring (for the purpose of these heuristic remarks) the secondary variables $\delta_1, \delta_3, \delta_5, \delta_6, r_2, n_1$, we have to sum
$$\sum_{n_2 \mid |\Delta| + mc} \chi_{\Delta}(n_2m)$$
Writing $|\Delta| + m c = n_2 s$ for $s \in \mathbb{N}$, and assuming also for simplicity that $n_2, m$ are odd, we obtain a sum over $$\Big( \frac{-sn_2 + mc}{n_2m}\Big) = \Big( \frac{-sn_2 + mc}{n_2 }\Big) \Big( \frac{-sn_2 + mc}{m }\Big) = \Big( \frac{ mc}{n_2 }\Big) \Big( \frac{-sn_2 }{m }\Big) = \Big( \frac{ m}{n_2 }\Big)\Big( \frac{n_2 }{m }\Big) \cdot \Big( \frac{ c}{n_2 }\Big) \Big( \frac{-s }{m }\Big) $$ where $-n_2s +mc$ is essentially restricted to squarefree numbers. By quadratic reciprocity, the first two factors are essentially constant. Since $c$ is not a square, the map $n_2 \mapsto (\frac{c}{n_2})$ is a non-trivial character, and since typically the length of $n_2$ is much longer than the length of $c$, the $n_2$-sum has some saving from the P\'olya-Vinogradov inequality (here we also need to deal with the squarefree condition). We now make this precise. For clarity, we repeat \eqref{E5} with the small amendments we have made so far: \begin{equation}\label{E7}
\begin{split}
\frac{K^{\varepsilon}T^2}{ (XK)^{1/2}}&\sum_{r_1\delta_1\delta_3\delta_5\delta_6 r_2 = r} \frac{ \ \mu(\delta_5\delta_6)}{(rr_1r_2 \delta_1\delta_5\delta_6)^{1/4} } \underset{\substack{(c, n_1n_2r_2\delta_3\delta_5\delta_6) = 1\\ \delta_1 \mid c^{\infty}, n_1\delta_5 \mid \Delta^{\infty}, (n_2 \delta_6, \Delta) = 1\\ n_1n_2\delta_5\delta_6 \mid r_2|\Delta| + mc }}{\sum_{\Delta, n_1, n_2, m} \sum_{ c\not\in \mathscr{S}(2r_1\delta_1m)} } \chi_{\Delta}\Big( \delta_1\delta_3 n_2\delta_6 \frac{ r_2 \bar{c}|\Delta| + m}{n_1\delta_5}\Big) \\
& W_5\Big( \frac{|\Delta|}{X}, \frac{n_1n_2 \delta_3\sqrt{N \delta_1\delta_5\delta_6}}{\sqrt{Mr_2}}, \frac{m\sqrt{NM } }{ (r_2\delta_1 \delta_5\delta_6 )^{1/2}X}, \frac{c\delta_1\delta_3\delta_5\delta_6}{C}\Big)
\end{split} \end{equation} where $W_5(x, y, z, w) = x^{-1/2} W_4(x, y, z, w)$.
We define $s$ through the equation
\begin{equation}\label{s}
n_1n_2 \delta_5\delta_6 s = r_2|\Delta| + mc.
\end{equation}
Note that, up to a negligible error, \begin{equation}\label{unbal} m c \ll K^{\varepsilon}\frac{CX r_2 ^{1/2}}{\sqrt{NM\delta_1\delta_5\delta_6}\delta_3 } \ll K^{\varepsilon} \frac{r_2 X}{T^2} \end{equation}
by \eqref{C}, so that by \eqref{T} we conclude that $mc $ is substantially smaller than $| r_2\Delta|$. In particular, $n_1n_2 \delta_5\delta_6 s \asymp r_2 X$, so that \begin{equation}\label{Ss} s \asymp \frac{ X\delta_3 \sqrt{N \delta_1r_2}}{\sqrt{M\delta_5\delta_6} }. \end{equation} We first argue that we can truncate the $n_1$-sum in \eqref{E5} at $n_1 \leqslant K^{4\eta}$. Indeed, since $(n_1, c) = 1$, but $n_1 \mid \Delta^{\infty}$, the squarefree kernel $\text{rad}(n_1)$ must divide $m$. Summing trivially over $n_1, n_2, s, c, m$ in \eqref{E5} as in \eqref{E6}, the portion $n_1 \geqslant Y$ contributes at most \begin{equation}\label{similar} \begin{split} &K^{\varepsilon} \frac{T^2}{(XK)^{1/2}} \sum_{r_1\delta_1\delta_3\delta_5\delta_6 r_2 = r} \frac{ 1 }{(rr_1r_2 \delta_1\delta_5\delta_6)^{1/4} } \\ & \times \sum_{n_1 \geqslant Y} \frac{\sqrt{Mr_2}}{\delta_3\sqrt{N\delta_1\delta_5\delta_6}n_1} \frac{ X\delta_3 \sqrt{N \delta_1r_2}}{\sqrt{M\delta_5\delta_6} } \frac{C}{\delta_1\delta_3\delta_5\delta_6} \frac{X(r_2\delta_1\delta_5\delta_6)^{1/2}}{ \sqrt{NM } \text{rad}(n_1)} \ll K^{\varepsilon} \frac{X^{3/2} CT^2}{(MNK)^{1/2} Y^{1-\varepsilon}} \end{split} \end{equation}
by applying Rankin's trick and using that $\sum_n \text{rad}(n)^{-1} n^{-\sigma}$ is absolutely convergent for $\sigma > 0$. If $Y \geqslant K^{4\eta}$, this is $\ll X K^{1/2-2\eta + \varepsilon}$ by \eqref{C} and \eqref{X}, hence admissible for \eqref{J-bound}. Having truncated the $n_1$-sum, we decompose $\Delta = \Delta_1\Delta_2$ into two fundamental discriminants of suitable signs where $(\Delta_2, 2n_1\delta_5) = 1$ and $\Delta_1 \mid 8n_1 \delta_5 $, in particular $|\Delta_1| \leqslant 8K^{8\eta}$ and $(\Delta_1, cn_2\delta_6) = 1$. With this notation and recalling \eqref{s} we can write
$$\chi_{\Delta}\Big( \delta_1\delta_3 n_2\delta_6 \frac{ r_2 \bar{c}|\Delta| + m}{n_1\delta_5}\Big) = \chi_{\Delta_1}(\delta_1\delta_3cs) \chi_{\Delta_2}(\delta_1\delta_3\delta_5\delta_6 n_1n_2m).$$
Next we make $n_2m$ coprime to $2r_2\Delta_1$ by factoring $n_2 = n_2'n_2''$ and $m=m'm''$ with $(n_2'm', 2r_2\Delta_1) = 1$, $n_2''m'' \mid (2r_2\Delta_1)^{\infty} \mid (2r_2n_1\delta_5)^{\infty}$ so that the previous display equals
\begin{equation}\label{char}
\begin{split}
& \chi_{\Delta_1}(\delta_1\delta_3cs) \chi_{\Delta_2}(\delta_1\delta_3\delta_5\delta_6 n_1 n_2''m'') \chi_{ (n_1n_2\delta_5\delta_6 s - mc)( r_2|\Delta_1|)} (n_2'm') \\
= & \chi_{\Delta_1}(\delta_1\delta_3cs) \chi_{\Delta_2}(\delta_1\delta_3\delta_5\delta_6 n_1 n_2''m'') \Big(\frac{- r_2|\Delta_1|mc}{n_2'}\Big) \Big(\frac{ r_2|\Delta_1|n_1n_2\delta_5\delta_6 s}{m'}\Big)\\
= & \chi_{\Delta_1}(\delta_1\delta_3cs) \chi_{\Delta_2}(\delta_1\delta_3\delta_5\delta_6 n_1 n_2''m'') \Big(\frac{ r_2|\Delta_1|n_1n''_2\delta_5\delta_6 s}{m'}\Big) \Big(\frac{- r_2|\Delta_1|m'' }{n_2'}\Big)\Big(\frac{m'}{n_2'}\Big) \Big(\frac{n_2'}{m'}\Big) \Big(\frac{c}{n_2'}\Big).
\end{split}
\end{equation} By a computation similar to \eqref{similar}, this time using that $$\sum_{\substack{a \mid b^{\infty}\\ a \leqslant X}} 1 \ll (bX)^{\varepsilon}$$ for $X \geqslant 1$, $b \in \mathbb{N}$ (which follows in the same way by Rankin's trick), we can assume $n_2'', m'' \leqslant K^{4\eta}$, the remaining portion to \eqref{E5} being $\ll XK^{1/2 - 2\eta + \varepsilon}$. We are left with short (i.e.\ $\ll K^{4\eta}$) variables \begin{equation}\label{short} r_1, \delta_1, \delta_3, \delta_5, \delta_6, r_2, n_1, n_2'', m'', \Delta_1 \end{equation} and potentially long variables $$\Delta_2, c, n_2', m', s$$ subject to $c \not\in \mathscr{S}(2r_1\delta_1m'm'')$ as well as \begin{displaymath} \begin{split}
& (c, n_1n_2'n_2'' r_2 \delta_3\delta_5\delta_6) = 1, \quad \delta_1 \mid c^{\infty} , \quad n_1\delta_5 \mid (\Delta_1\Delta_2)^{\infty}, \quad (n_2'n_2'' \delta_6, \Delta_1\Delta_2) = 1,\quad \Delta_1 \mid 8n_1\delta_5,\\&(\Delta_2, 2n_1\delta_5) = 1, \quad (n_2'm', 2r_2\Delta_1) = 1, \quad n_2''m'' \mid(2r_2\Delta_1)^{\infty}, \quad n_1n_2'n_2'' \delta_5\delta_6s = r_2 \Delta_1\Delta_2 + mc. \end{split} \end{displaymath}
We can eliminate $\Delta_2$ from the last equation, so that a congruence
$$ n_1n_2'n_2'' \delta_5\delta_6s = mc \, (\text{mod } r_2\Delta_1)$$
remains. Then the conditions $(\Delta_2, 2n_1\delta_5) = (n_2'n_2''\delta_6, \Delta_2) = 1$ are re-phrased as
$$(n_1n_2'n_2'' \delta_5\delta_6s - mc , 2r_2n_1\delta_5n_2' n_2'' \delta_6) = r_2$$
which is equivalent to
$$(n_1n_2'n_2'' \delta_5\delta_6, mc) = r_2', \quad (2r_2, n_1n_2'n_2'' \delta_5\delta_6s - mc) = r_2'', \quad r_2'r_2'' = r_2.$$
The condition $n_1\delta_5 \mid (\Delta_1\Delta_2)^{\infty}$ reads
$$\text{rad}(n_1\delta_5) \mid \frac{n_1n_2'n_2'' \delta_5\delta_6s - mc}{r_2}.$$
All of these conditions on $n_2'$ can be detected by congruences modulo ``short'' variables in \eqref{short} (and some powers of 2) as well as $(n_2', m) = 1$. Finally we need to remember that $\Delta_1\Delta_2$ is a fundamental discriminant. To this end we split into residue classes $\Delta_1\Delta_2 \equiv 1, 5, 8, 9, 12, 13$ (mod 16) and insert a factor $\mu^2((n_1n_2'n_2'' \delta_5\delta_6s - mc)/(\alpha r_2))$ with $\alpha \in \{1, 4\}$. We use the convolution formula
$$\mu^2\Big(\frac{n_1n_2'n_2'' \delta_5\delta_6s - mc}{\alpha r_2}\Big)= \sum_{y^2 \mid \frac{n_1n_2'n_2'' \delta_5\delta_6s - mc}{\alpha r_2}} \mu(y),$$ and insert all of this back into \eqref{E7}. We claim that we can restrict $y \leqslant K^{A\eta}$ for some constant $A$. Indeed, summing over all short variables, as well as $ c, m'$, we get a congruence for $n_2's$ modulo $y^2$, so that the portion $y > Y$ contributes at most $$K^{O(\eta)} \sum_{y \geqslant Y}\frac{T^2}{(XK)^{1/2}} C \frac{X}{\sqrt{NM}} \frac{X}{y^2} \ll \frac{XK^{1/2+ O(\eta)}}{Y}$$ which is acceptable if $Y = K^{A\eta}$ for $A$ sufficiently large. Note that \eqref{unbal} implies that $n_1n_2'n_2'' \delta_5\delta_6s - mc $ never vanishes, so there is no ``$1+$'' in the congruence count. In addition, for fixed $y$ we also remove all $c \in \mathscr{S}(y^2)$ at the cost of an error $XK^{1/3+ O(\eta)}$ as in \eqref{square-c}.
We are finally ready to return to \eqref{char} and split the sum over $n_2'$ into residue classes modulo $\nu$ modulo $H := 32 r_1\delta_1\delta_3\delta_5\delta_6r_2n_1n_2''m''\Delta_1 y^2 = K^{O(\eta)}$. By assumption, $c$ is not in a square class of any divisor of $H$. Thus we consider $$\sum_{\substack{n'_2 \equiv \nu \, (\text{mod } H)\\ (n'_2, m') = 1}}\Big(\frac{c}{n_2'}\Big)W\Big(\frac{n'_2}{R}\Big)$$ for $\nu$ odd, $c \ll C$ and $R \ll \sqrt{M/N} K^{O(\eta)}$. We can detect the congruence condition by characters, none of which conspires with $n_2' \mapsto (\frac{c}{n_2'})$ to become the trivial character. By the P\'olya-Vinogradov inequality, we can bound the previous display by $C^{1/2} K^{O(\eta)}$, and by trivial estimates over $c, m', s$ and the present estimate for the sum over $n_2'$ we obtain the final bound $$K^{O(\eta)} \frac{T^2}{(XK)^{1/2}} C \frac{X}{\sqrt{NM}}\frac{X\sqrt{N}}{\sqrt{M}} C^{1/2} \ll K^{O(\eta)} \frac{N^{3/4} X^{3/2}}{K^{1/2}TM^{1/4}} \ll K^{O(\eta)}\frac{X^{3/2}}{K^{1/2}T^{1/4}} \ll XK^{3/8+ O(\eta)}$$ by \eqref{C}, \eqref{NM}, \eqref{X} and \eqref{T}. For sufficiently small $\eta$ this is in agreement with \eqref{J-bound} and completes the analysis of the first off-diagonal term in \eqref{diagdiag}.
\subsection{The second off-diagonal term}\label{144} The analysis of the second off-diagonal term in \eqref{kuz-even-form} is very similar, so we can be brief. Here we need to consider \begin{displaymath} \begin{split} \mathcal{I}^{\text{off-diag,2}}(\Delta, t,k, r):= \frac{16}{\pi} \sum_{d \mid r} &\sum_{{\tt n},{\tt m}} \frac{ \chi_{\Delta}({\tt m}) }{(d{\tt n}{\tt m})^{1/2}} \sum_c \frac{S({\tt m}, {\tt n}r/d, c)}{c} \\
&\int_{-\infty}^{\infty} K_{2i\tau}(4\pi \sqrt{({\tt m}{\tt n}r/d)}/c) \sinh(\pi \tau) W^+_{\tau}(d{\tt n}) W^-_{\tau}\Big(\frac{{\tt m}}{|\Delta|}\Big)h(\tau) \tau \frac{d\tau}{4\pi} \end{split} \end{displaymath} with $h$ as in \eqref{htau}. This needs to be inserted into \eqref{weobt} with $r = f^2d\nu/(d_1 d_2 d_3^2) \leqslant K^{4\eta}$. Under the same size restrictions \eqref{X} as before we want to show that
$$\mathcal{J}^{(2)}(X, t,k, r) := \sum_{\Delta}w\Big( \frac{|\Delta|}{X}\Big) \mathcal{I}^{\text{off-diag,2}}(\Delta, t,k, r) \ll XK^{1/2 - \eta}.$$ As before we first use holomorphicity ot ensure absolute convergence of the $c$-sum and obtain a very coarse truncation. Then we apply smooth partitions of unity restricting to $\tau \asymp T$ satisfying \eqref{T}, $n \asymp N$, $m \asymp M$ satisfying \eqref{NM} and $c\asymp C$. This time we use Lemma \ref{bessel-kuz}b to conclude that \begin{equation}\label{C1} C \leqslant K^{\varepsilon} \frac{\sqrt{NM}}{T}, \end{equation} and by an analogue of \eqref{E1} we need to bound the quantity \begin{equation*} \begin{split}
\mathscr{J}^{(2)}_r(X, N, M, C, T) = &\frac{K^{\varepsilon}T}{(NMK)^{1/2}C}\sum_{r_1r_2=r} \frac{1}{(rr_1)^{1/4}} \sum_{\Delta, {\tt n} , {\tt m}, c} \chi_{\Delta}({\tt m}) S({\tt m}, {\tt n} r_2, c) W\Big( \frac{|\Delta|}{X}, \frac{{\tt n}}{N}, \frac{{\tt m}}{M}, \frac{c}{C}\Big) \end{split} \end{equation*}
where $W$ satisfies the same properties. The trivial bound using the large sieve (Lemma \ref{HBlemma}) is now \begin{equation}\label{F1}
\mathscr{J}^{(2)}_r(X, N, M, C, T) \ll \frac{K^{\varepsilon}T}{(NMK)^{1/2}C} XNMC^{3/2}\Big( \frac{1}{X^{1/2}} + \frac{1}{M^{1/2}}\Big)= \frac{K^{\varepsilon}T}{ K^{1/2}} (XNC)^{1/2} ( X^{1/2} + M^{1/2} ).
\end{equation} Next we apply Poisson summation in ${\tt n}$ in residue classes modulo $c$ which is simpler than before because there is no exponential $e(\pm 2\sqrt{{\tt n}{\tt m}r_2}/c)$. This transforms $ \mathscr{J}^{(2)}_r(X, N, M, C, T)$ into \begin{displaymath}
\begin{split} \frac{K^{\varepsilon}T}{(NMK)^{1/2}C}\sum_{r_1r_2=r} \frac{1}{(rr_1)^{1/4}} \sum_{\Delta, {\tt m}, c} \chi_{\Delta}({\tt m}) N \sum_{n \in \mathbb{Z}} \underset{\substack{\gamma \, (\text{mod }c)\\ r_2\bar{\gamma} \equiv -n \, (\text{mod } c)}}{\left.\sum\right.^{\ast}} e\Big( \frac{{\tt m} \gamma }{c}\Big) W_1\Big( \frac{|\Delta|}{X}, \frac{nN}{C}, \frac{{\tt m}}{M}, \frac{c}{C}\Big) \end{split} \end{displaymath} for a weight function $W_1$ that is compactly supported in the first, third and fourth variable and rapidly decaying in the second. This term contains the same character sum as in \eqref{char1}. By the same manipulation we obtain \begin{equation}\label{F2} \begin{split} \mathscr{J}^{(2)}_r(X, N, M, C, T) = & \frac{K^{\varepsilon}TN^{1/2}}{(MK)^{1/2}C}\sum_{r_1\delta_1\delta_3\delta_4 r_2 = r} \frac{(\delta_1 \delta_3)^{3/4}\mu(\delta_4)}{(rr_1r_2 \delta_4)^{1/4} } \\
&\underset{\substack{(c, nr_2\delta_3\delta_4) = 1\\ \delta_1 \mid c^{\infty}}}{\sum_{\Delta, {\tt m}, c} \sum_{n\in \mathbb{Z}}} \chi_{\Delta}({\tt m}\delta_1\delta_3) e\Big( -\frac{{\tt m}r_2 \overline{n\delta_2}}{c}\Big)W_1\Big( \frac{|\Delta|}{X}, \frac{n\delta_1\delta_2 N}{C}, \frac{{\tt m}\delta_1}{M}, \frac{c\delta_1\delta_2}{C}\Big). \end{split} \end{equation} This is arithmetically analogous to \eqref{E4} except that the roles of $\delta_2n$ and $c$ are reversed in the exponential. This makes good sense since in generic ranges we have $c \asymp K^{1/2}$, $ n \asymp K$ in \eqref{E4}, but $c \asymp K$, $n \asymp K^{1/2}$ in \eqref{F2}. The large sieve now gives the bound \begin{equation}\label{F3}
\mathscr{J}^{(2)}_r(X, N, M, C, T) \ll \frac{K^{\varepsilon}TN^{1/2}}{(MK)^{1/2}C} X\frac{C}{N}MC\Big(\frac{1}{\sqrt{X}} + \frac{1}{\sqrt{M}}\Big) =\frac{K^{\varepsilon}TX^{1/2} C }{(NK)^{1/2}} ( \sqrt{X} +\sqrt{M} ). \end{equation}
Next we apply Poisson summation in ${\tt m}$ in residue classes modulo $|\Delta| c$. In the character sum
$$\sum_{\mu\, (\text{mod }c|\Delta|)} \chi_{\Delta}(\mu) e\Big(\frac{\mu(m- r_2\overline{n\delta_2}|\Delta|)}{c|\Delta|}\Big)$$
we decompose $c = c_1 c_2$ where $c_1 \mid \Delta^{\infty}$, $(c_2, \Delta) = 1$. The character sum vanishes unless $c_1c_2 \mid mn\delta_2 - r_2 |\Delta|$ in which case it equals
$$i \sqrt{|\Delta|} c_1 \chi_{\Delta}\Big(c_2 \frac{m- r_2 \overline{n\delta_2}|\Delta|}{c_1}\Big).$$ Estimating trivially at this point yields \begin{equation}\label{F4}
\mathscr{J}^{(2)}_r(X, N, M, C, T) \ll K^{\varepsilon}\frac{T(MN)^{1/2}C^2 X^2 }{C(XK)^{1/2}MN} = \frac{X^{3/2}CT}{(KNM)^{1/2}} \ll X K^{1/2 +2 \eta + \varepsilon} \end{equation}
by \eqref{C1} and \eqref{X}, matching the bound in \eqref{E6}. With the roles of $c$ and $n$ reversed, we now want to make sure that $n \ll C(N\delta_1\delta_2)^{-1}$ is large enough. By the trivial bound \eqref{F4} we can assume that $n \geqslant C(N\delta_1\delta_2)^{-1} K^{-4\eta}$. Now combining \eqref{F4} and \eqref{F1} we obtain
$$ \mathscr{J}^{(2)}_r(X, N, M, C, T) \ll K^{\varepsilon} \frac{(CX)^{1/2} T }{K^{1/2} }\min\Big(\frac{XC^{1/2}}{(NM)^{1/2}} , (NX)^{1/2} + (NM)^{1/2}\Big) \leqslant K^{\varepsilon} \frac{TXC^{1/2}}{K^{1/2}}(N^{1/2} + C^{1/4}).$$ Combining \eqref{F4} and \eqref{F3} we obtain $$ \mathscr{J}^{(2)}_r(X, N, M, C, T) \ll K^{\varepsilon} \frac{CTX^{1/2}}{(MNK)^{1/2}}\min(\sqrt{MX} + M, X)\ll K^{\varepsilon} \frac{CTX }{(NK)^{1/2}} . $$ Combining the previous two bounds, we obtain $$ \mathscr{J}^{(2)}_r(X, N, M, C, T) \ll K^{\varepsilon}\frac{TXC^{3/4}}{K^{1/2}} = K^{\varepsilon}\frac{TXN^{3/4}}{K^{1/2}} \Big( \frac{C}{N}\Big)^{3/4} \ll XK^{1/4+\varepsilon} \Big(\frac{C}{N}\Big)^{3/4} $$ by \eqref{NM} and \eqref{T}. This is acceptable unless $$C/N \geqslant K^{1/3 - 2\eta} \quad \text{and} \quad C \geqslant K^{2/3 - 2\eta}$$ which we assume from now on, so that in particular $n \gg C(N\delta_1\delta_2)K^{-4\eta} \gg N^{1/3 -10\eta}$. By the same argument as in the previous subsection we can now extract certain square classes in the $n$-sum and then save from the P\'olya-Vinogradov inequality. In effect, we replace the factor $C$ from a trivial bound of the $c_2$-sum by a factor $ K^{O(\eta)}\sqrt{N/C}$ of the square root of the conductor of $c_2 \mapsto (\frac{n}{c_2})$. This leads to the final bound $$ \mathscr{J}^{(2)}_r(X, N, M, C, T) \ll K^{O(\eta)} \frac{X^{3/2} T \sqrt{N/C}}{(KNM)^{1/2}} \leqslant K^{O(\eta)} \frac{X^{3/2} T }{K^{1/2}C^{1/2}} \ll XK^{1/6 +O(\eta)}$$ by \eqref{X}, \eqref{T} and our assumption $C \geqslant K^{2/3 - 2\eta}$. This is in agreement with \eqref{J-bound} and completes the analysis of the the second diagonal off-diagonal term, hence the analysis of the complete diagonal term.
\section{The off-diagonal term}\label{off-off}
\subsection{Initial steps} We return to \eqref{slightly-simp} and analyze the off-diagonal term in Lemma \ref{lem1} applied to the $h$-sum. Here we are only interested in upper bounds, so dropping all numerical constants it suffices to estimate \begin{displaymath} \begin{split}
\frac{1}{ K^2} & \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \sum_{\substack{(n, m) = 1\\ n, m \leqslant K^{\eta}}} \frac{ \mu(n)\mu^2(m)}{n^{3/2}m^{3}} \int_{-\infty}^{\infty} \int^{\ast}_{\Lambda_{\text{\rm ev}}} \omega(t_{\tt u}) \sum_{f_1, f_2, D_1, D_2} \sum_{\substack{d_1 \mid d_2 \mid n\\ (d_1d_2)^2 \mid n^2D_2}} \left(\frac{d_1}{d_2} \right)^{1/2} \chi_{ D_2}\Big(\frac{d_2}{d_1}\Big) \\
&\frac{ P(D_1; {\tt u})\overline{P(D_2; {\tt u})}}{f_1f_2 |D_1D_2|^{3/4}} \Big(\frac{|D_2|f_2^2}{|D_1|f_1^2}\Big)^{it} i^k\sum_c \frac{K^+(|D_1|, |D_2|n^2/(d_1d_2)^2, c)}{c} J_{k-3/2}\Big( \frac{ 4\pi \sqrt{|D_1D_2|}n}{cd_1d_2}\Big) \\
& V_{t}(|D_1D_2|(f_1f_2)^2; k, t_{\tt u}) d{\tt u} \, dt,
\end{split} \end{displaymath}
and our target bound is $K^{-\eta}$. We recall that $V_t(x, k, \tau)$ was defined in \eqref{defV3} and besides the decay properties it is important to note that $V_t(x, k, \tau)$ is holomorphic in, say, $|\Im \tau| \leqslant 1$. Since we want to apply the trace formula (Theorem \ref{thm5}) to the spectral sum later, we must not destroy holomorphicity in the third variable.
As in Section \ref{sec12} we write $d_2 = d_1\delta$, $n = d_1\delta \nu$. Then $d_1^2 \mid \nu^2 D_2$ and $d_1^2 \mid D_2$ since $n$ is squarefree. Again we write $d_1 = d$ and $D_2 d^2$ in place of $D_2$ and bound the preceding display as
\begin{displaymath} \begin{split}
\frac{1}{ K^2} & \sum_{ d\delta \nu, m \leqslant K^{\eta}} \frac{ \mu^2(d\delta \nu m) }{d^3 \delta^2 \nu^{3/2} m^3 }\Big| \sum_{k \in 2\mathbb{N}} W\left(\frac{k}{K}\right) \int_{-\infty}^{\infty} \int^{\ast}_{\Lambda_{\text{\rm ev}}} \omega(t_{\tt u}) \sum_{f_1, f_2, D_1, D_2} \frac{\chi_{D_2}(\delta) P(D_1; {\tt u})\overline{P(D_2d^2; {\tt u})}}{f_1f_2 |D_1D_2|^{3/4}} \\
& \Big(\frac{|D_2|(df_2)^2}{|D_1| f_1^2}\Big)^{it} i^k\sum_c \frac{K_{3/2}^+(|D_1|, |D_2|\nu^2 , c)}{c} J_{k-3/2}\Big( \frac{ 4\pi \sqrt{|D_1D_2|}\nu}{c }\Big) V_{t}(|D_1D_2|( d f_1f_2)^2; k, t_{\tt u}) d{\tt u} \, dt\Big|.
\end{split} \end{displaymath}
We sum over $k$ using Lemma \ref{lem2} and open the Kloosterman sum. As in Section \ref{112}, up to a negligible error we obtain the upper bound
\begin{displaymath} \begin{split}
& \sum_{ d\delta \nu, m \leqslant K^{\eta}} \sum_{4 \mid c} \sum_{f_1, f_2} \frac{ \mu^2(d\delta \nu m) }{d^3 \delta^2 \nu^{3/2} m^3 cf_1f_2 }\underset{\substack{\gamma\, (\text{mod } c)\\ (\gamma, c) = 1}}{\max} | \mathcal{I}^{\text{off}}(K)|
\end{split} \end{displaymath}
where $\mathcal{I}^{\text{off}}(K) = \mathcal{I}_{d, \delta, \nu, m, c, f_1, f_2, \gamma}^{\text{off}}(K)$ is given by
\begin{displaymath} \begin{split} \mathcal{I}^{\text{off}}(K)
= & \frac{1}{K^2} \int_{-\infty}^{\infty} \int^{\ast}_{\Lambda_{\text{\rm ev}}} \omega(t_{\tt u}) \sum_{ D_1, D_2} \frac{ \chi_{D_2}(\delta)P(D_1; {\tt u})\overline{P(D_2d^2; {\tt u})}}{ |D_1D_2|^{3/4}} \Big(\frac{|D_2|(df_2)^2}{|D_1| f_1^2}\Big)^{it} \\
& e\Big( \frac{|D_1|\gamma + |D_2|\nu^2\bar{\gamma}}{c}\Big) e\Big(\pm \frac{ 2 \sqrt{|D_1D_2|}\nu}{c }\Big) \tilde{V}\Big(|D_1D_2|(df_1f_2)^2, \frac{\sqrt{|D_1D_2|}\nu}{c}, t, t_{\tt u}\Big) d{\tt u} \, dt;
\end{split} \end{displaymath}
here $\tilde{V}$ satisfies \eqref{tildeV} and is holomorphic in $|\Im t_{\tt u}| \leqslant 1$. The bounds contained in \eqref{tildeV} imply in particular $c, f_1, f_2 \leqslant K^{\eta}$ up to a negligible error. For notation simplicity we consider only the plus-case, the minus case being entirely similar.
As in Section \ref{sec12} we start with a Voronoi step, but in the present situation (since we have already excluded the constant function that requires a very careful treatment) we can afford to lose small powers of $K$ on the way. We introduce the notation
$$A \preccurlyeq B \quad :\Longleftrightarrow \quad A \ll K^{c\eta} B$$
for some constant $c$, not necessarily the same on every occasion. We always assume that $\eta$ is sufficiently small.
To begin with, we integrate over $t$ which by the properties of \eqref{tildeV} induces the condition $|D_1| f_1^2 - |D_2|(df_2)^2 \preccurlyeq K^{1/2}$ up to a negligible error. This now implies $K^2 \preccurlyeq D_1, D_2 \preccurlyeq K^{2 }$, up to a negligible error. In $\tilde{V}$ we can separate the variables $D_1, D_2$ from $t_{\tt u}$ by Mellin inversion (keeping holomorphicity in $t_{\tt u}$). Since $\tilde{V}$ is of size $K^{-1/2}$ and we also get a factor $K^{1/2}$ from the $t$-integration, we are left with bounding
\begin{displaymath} \begin{split} \tilde{\mathcal{I}}^{\text{off}}(K)
= & \frac{1}{K^2} \int^{\ast}_{\Lambda_{\text{\rm ev}}} \Omega(t_{\tt u}) \sum_{ D_1, D_2} \frac{ \chi_{D_2}(\delta)P(D_1; {\tt u})\overline{P(D_2d^2; {\tt u})}}{ |D_1D_2|^{3/4}} V_1\Big(\frac{|D_1|}{K^2}\Big) V_2\Big(\frac{|D_2|}{K^2}\Big) \\
& V_3\Big(K^{1/2} \log\frac{|D_2|(df_2)^2}{|D_1| f_1^2}\Big) e\Big( \frac{|D_1|\gamma + |D_2|\nu^2\bar{\gamma}}{c}\Big) e\Big( \frac{ 2 \sqrt{|D_1D_2|}\nu}{c }\Big) d{\tt u}
\end{split} \end{displaymath}
where $V_1, V_2$ have support in $[K^{-O(\eta)}, K^{O(\eta)}]$ with Sobolev norms bounded by $ \preccurlyeq 1$, $V_3$ is rapidly decaying, and $\Omega(\tau)$ is holomorphic in $|\Im \tau| \leqslant 1$, satisfies the conditions \eqref{prop-omega} and is non-negligible only in the range $K^{1/2} \preccurlyeq |\tau| \preccurlyeq K^{1/2}$.
We now consider the $D_1$-sum \begin{equation}\label{D1-sum}
\sum_{ D_1} \frac{ P(D_1; {\tt u})\overline{P(D_2d^2; {\tt u})}}{ |D_1D_2|^{3/4}} V_1\Big(\frac{|D_1|}{K^2}\Big) V_3\Big(K^{1/2} \log\frac{|D_2|(df_2)^2}{|D_1| f_1^2}\Big) e\Big( \frac{|D_1|\gamma }{c}\Big) e\Big( \frac{ 2 \sqrt{|D_1D_2|}\nu}{c }\Big)
\end{equation} and insert \eqref{mixed} with $t = t_{\tt u}/2$ if ${\tt u}$ is cuspidal and \eqref{eisen1} if ${\tt u} = E(., 1/2 + it_{\tt u})$ is Eisenstein. For clarity we recall that \begin{displaymath} \begin{split}
\frac{ P(D_1; {\tt u}) \overline{P(D_2d^2; {\tt u})}}{|D_1D_2 |^{3/4}} = \frac{d^{1/2}}{|D_1|^{1/2}|D_2|^{3/4}} \begin{cases} b(D_1) \sqrt{|D_1|} A(D_2d^2, {\tt u}), & {\tt u} \text{ cuspidal},\\
L(D_1, 1/2 + it_{\tt u}) |D_1|^{it_{\tt u}/2} A(D_2, {\tt u}), & {\tt u} = E(., 1/2 + it_{\tt u}),\end{cases} \end{split} \end{displaymath} where
$$A(D, {\tt u}) = \begin{cases} \frac{3}{\pi} L({\tt u}, 1/2) \Gamma(\frac{1}{4} + \frac{it_{\tt u}}{2})\Gamma(\frac{1}{4} - \frac{it_{\tt u}}{2})b(D) \sqrt{|D|}, & {\tt u} \text{ cuspidal},\\
\frac{\zeta(1/2 + it_{\tt u})\zeta(1/2 - it_{\tt u}) L(D, 1/2 - it_{\tt u}) |D|^{-it_{\tt u}/2}}{2|\zeta(1 + 2 it_{\tt u})|^2}, & {\tt u} = E(., 1/2 + it_{\tt u}).\end{cases}$$ Even though the normalization in the cuspidal and the Eisenstein case is quite different, the Voronoi formulae in Lemma \ref{Vor} (with $r = t_{\tt u}/2)$ and \ref{Vor-eis} (with $t = t_{\tt u}/2$) can deal in the same fashion with the sums \begin{equation}\label{d1}
\sum_{D_1 < 0} \left\{ \begin{array}{l} b(D_1) \sqrt{|D_1|} \\ L(D_1, 1/2 + it_{\tt u}) |D_1|^{it_{\tt u}/2}\end{array} \right\} e\Big( \frac{|D_1|\gamma}{c}\Big) \phi(D_1) \end{equation} with
$$\phi(x) = \frac{1}{|x|^{1/2}}V_1\Big(\frac{|x|}{K^2}\Big) V_3\Big(K^{1/2} \log\frac{|D_2|(df_2)^2}{|x| f_1^2}\Big) e\Big( \frac{ 2 \sqrt{|xD_2|}\nu}{c }\Big) $$
for $x < 0$ and $\phi(x) = 0$ for $x > 0$. It is easy to see that with this choice of $\phi$ and for $t \ll K^{1/2 + \varepsilon}$ the two polar terms in Lemma \ref{Vor-eis} are negligible, due to the strong oscillatory behaviour of the exponential $e(\pm 2\sqrt{|xD_2|}\nu/c)$ for $K^2 \preccurlyeq x, D_2 \preccurlyeq K^{2 }$, $c \preccurlyeq 1$. By the same argument as in Section \ref{104}, see in particular \eqref{BesselK}, the terms with $D > 0$ on the right hand side of the Voronoi summation formula are negligible. Hence, up to a negligible error, \eqref{d1} becomes \begin{equation*} \begin{split}
\Big( \frac{-c}{\gamma}\Big)&\epsilon_{\gamma} e(1/8) \frac{2\pi}{c}\sum_{D_1 < 0} \left\{ \begin{array}{l} b(D_1) \sqrt{|D_1|} \\ L(D_1, 1/2 + it_{\tt u}) |D_1|^{it_{\tt u}/2}\end{array} \right\} e\Big(- \frac{|D_1|\bar{\gamma}}{c}\Big) \int_0^{\infty}V_1\Big(\frac{|x|}{K^2}\Big) \\
& \sum_{\pm} (\mp) \frac{\cos(\pi/4 \pm \pi i t_u/2)}{\sin(\pi i t_u)} J_{\pm i t_u}\Big( \frac{4\pi \sqrt{|D_1x|}}{c}\Big) V_3\Big(K^{1/2} \log\frac{|D_2|(df_2)^2}{|x| f_1^2}\Big) e\Big( \frac{2\sqrt{|xD_2|}\nu}{c}\Big) \frac{dx}{ x^{1/2 }}.
\end{split} \end{equation*} Using \eqref{bessel-approx} we approximate the Bessel $J$-function for $t_{\tt u} \ll K^{1/2 + \varepsilon}$ by an exponential and replace up to a negligible error the preceding display by \begin{equation*} \begin{split}
\Big( \frac{-c}{\gamma}\Big)&\epsilon_{\gamma} e(1/8) \frac{2\pi}{c} \sum_{D_1 < 0} \left\{ \begin{array}{l} b(D_1) \sqrt{|D_1|} \\ L(D_1, 1/2 + it_{\tt u}) |D_1|^{it_{\tt u}/2}\end{array} \right\} e\Big( -\frac{|D_1|\bar{\gamma}}{c}\Big) \sum_{\pm} \int_0^{\infty} \frac{c^{1/2}}{|D_1|^{1/4}} V_1\Big(\frac{|x|}{K^2}\Big) \\
& f^{\pm}\Big( \frac{2\sqrt{|D_1x|}}{c}, t_{\tt u}\Big) V_3\Big(K^{1/2} \log\frac{|D_2|(df_2)^2}{|x| f_1^2}\Big) e\Big( \frac{2\sqrt{|xD_2|}\nu \pm 2 \sqrt{|D_1x|}}{c}\Big) \frac{dx}{ x^{3/4 }}
\end{split} \end{equation*} with $$f^{\pm}(x, \tau) = \frac{ 1}{2\pi }\sum_{k=0}^{n-1} \frac{i^k(\pm 1)^k}{(4\pi x)^{k}} \frac{\Gamma(i\tau+ k + 1/2)}{k! \Gamma(i\tau - k + 1/2)}
$$
for $n$ sufficiently large, but fixed. In particular $f^{\pm}$ satisfies \eqref{besseldecay}, and we see from the asymptotic expansion \eqref{bessel-approx} that $f^{\pm}$ is holomorphic in the second variable in, say, $|\Im \tau| < 1$. For notational simplicity we consider only the leading term $k=0$, the lower order terms being completely analogous (but easier).
We restore the periods $P(D; {\tt u})$, so that the leading term of \eqref{D1-sum} has the shape \begin{equation*} \begin{split}
\sum_{\pm} \Big( \frac{-c}{\gamma}\Big)&\epsilon_{\gamma} e(1/8) \frac{2\pi }{c^{1/2}}\sum_{D_1 < 0} \frac{P(D_1; {\tt u}) \overline{P(D_2d^2; {\tt u})}}{|D_2|^{3/4}|D_1|^{1/2}} e\Big(- \frac{|D_1|\bar{\gamma}}{c}\Big) \frac{1}{2\pi}\int_0^{\infty} V_1\Big(\frac{|x|}{K^2}\Big) \\
& V_3\Big(K^{1/2} \log\frac{|D_2|(df_2)^2}{|x| f_1^2}\Big) e\Big( \frac{2\sqrt{|xD_2|}\nu \pm 2 \sqrt{|D_1x|}}{c}\Big) \frac{dx}{ x^{3/4 }} \end{split}
\end{equation*}
with lower order terms of similar form. Recall that $K^2 \preccurlyeq D_2 \preccurlyeq K^{2 }$.
Integrating by parts, we see as in \eqref{add1} -- \eqref{add2} that only the minus-term in the exponential is relevant (the plus-term is negligible) and the $x$-integral restricts to $\sqrt{|D_2|}\nu - \sqrt{|D_1|} \preccurlyeq K^{-1/2}$. We therefore introduce a new variable $h \in \mathbb{Z}$ by
$$|D_1| = |D_2|\nu^2 + h$$ with $h \preccurlyeq K^{1/2 }$. Changing variables in the $x$-integral, we obtain \begin{equation*} \begin{split}
& \Big( \frac{-c}{\gamma}\Big)\epsilon_{\gamma} e(1/8) \frac{1 }{c^{1/2}}\sum_{h \preccurlyeq K^{1/2}} \frac{P(-|D_2|\nu^2 - h, {\tt u}) \overline{P(D_2d^2; {\tt u})} }{|D_2|^{1/2}(|D_2|\nu^2 + h)^{1/2}} e\Big( -\frac{(|D_2|\nu^2 + h)\bar{\gamma}}{c}\Big) \\
&\int_0^{\infty} V_1\Big(\frac{|xD_2|}{K^2}\Big) V_3\Big(K^{1/2} \log\frac{ (df_2)^2}{|x| f_1^2}\Big) e\Big( \frac{2 \sqrt{x D_2}(\sqrt{|D_2|}\nu- \sqrt{(|D_2|\nu^2 + h)})}{c}\Big) \frac{dx}{ x^{3/4 }} \end{split}
\end{equation*}
where $h$ is restricted to numbers such that $-(|D_2|\nu^2 + h)$ is a negative discriminant. The weight function $V_3$ forces $x \preccurlyeq 1$ and more precisely \begin{equation}\label{x} x - (df_2/f_1)^2 \preccurlyeq K^{-1/2}. \end{equation} By a Taylor expansion we can write
$$ e\Big( \frac{2 \sqrt{x|D_2|}(\sqrt{|D_2|}\nu- \sqrt{(|D_2|\nu^2 + h)})}{c}\Big) = e\Big( -\frac{\sqrt{x}h}{c\nu}\Big) F(D_2)$$
where $$F(D) = F_{x, h, \nu, c}(D) = 1 + \frac{i\pi \sqrt{x} h^2}{2 |D| \nu^3 c} - \frac{i\pi \sqrt{x}h^3}{4|D|^2 \nu^5 c} +\ldots $$ Again we only keep the leading term (the lower order terms being similar, but easier). We substitute all of this back into $\tilde{\mathcal{I}}^{\text{off}}(K)$ and pull the $x$-integration outside which is subject to \eqref{x}. In this way we see that it suffices to bound \begin{equation}\label{I1} \begin{split}
\mathcal{I}_1^{\text{off}}(K)= \frac{1}{K^{5/2}} \int^{\ast}_{\Lambda_{\text{\rm ev}}} \Omega(t_{\tt u}) & \sum_{ D_2} \sum_{h \preccurlyeq K^{1/2}} \frac{P(-|D_2|\nu^2 - h, {\tt u}) \overline{P(D_2d^2; {\tt u})}}{|D_2|^{1/2} (|D_2|\nu^2 + h)^{1/2}}\chi_{D_2}(\delta) \\
& V_x\Big(\frac{|D_2|}{K^2}\Big) e\Big( -\frac{ h\bar{\gamma}}{c} -\frac{\sqrt{x}h}{c\nu}\Big)d{\tt u}
\end{split}
\end{equation} where $V_x(z) = V_2(z) V_1(xz)$, uniformly in $$x, \nu, c, \delta, d \preccurlyeq 1, \quad (\gamma, c) = 1.$$ A trivial bound using Cauchy-Schwarz and Proposition \ref{Lfunc} gives $ \mathcal{I}_1^{\text{off}}(K) \preccurlyeq 1$, as in Section \ref{weakversion}. In order to make progress and get additional savings, we must now treat the ${\tt u}$-integral non-trivially. This is where the trace formula has its appearance.
\subsection{Application of the trace formula} We now apply Theorem \ref{thm5} to the spectral expression \begin{displaymath} \begin{split}
& \int^{\ast}_{\Lambda_{\text{\rm ev}}} \Omega(t_{\tt u}) \frac{P(-|D_2|\nu^2 - h, {\tt u}) \overline{P(D_2d^2; {\tt u})}}{|D_2|^{1/4} (|D_2|\nu^2 + h)^{1/4}} d{\tt u} \\
&= \int_{\Lambda_{\text{\rm ev}}} \Omega(t_{\tt u}) \frac{P(-|D_2|\nu^2 - h, {\tt u}) \overline{P(D_2d^2; {\tt u})}}{|D_2|^{1/4} (|D_2|\nu^2 + h)^{1/4}} d{\tt u} - \frac{3}{\pi} \frac{H(D_2d^2)H(-|D_2|\nu^2 - h)}{|D_2|^{1/4} (|D_2|\nu^2 + h)^{1/4}} \Omega(i/2).
\end{split}
\end{displaymath}
We discuss the four terms on the right hand side of the trace formula.
1) The class number term gets immediately cancelled.
2) The $t$-integral in the polar term is rapidly decaying and by a Burgess-type subconvexity bound $L(D, 1/2 + it) \ll |D|^{3/16+\varepsilon}(|1+|t|)^{10}$, say, its contribution to \eqref{I1} is at most\footnote{Of course, instead of subconvexity, we could also used mean value bounds on average over $D_2$ to get an even stronger saving.}
$$
\preccurlyeq \frac{1}{K^{5/2}} \sum_{ D_2 \preccurlyeq K^2} \sum_{h \preccurlyeq K^{1/2}} \frac{1}{|D_2|^{1/16} (|D_2|\nu^2 + h)^{1/16 }} \preccurlyeq K^{-1/4}.$$
3) For the diagonal term we observe that $\sum_m m^{-1} \int |\Omega(t)W_{t}(rnvm)t| dt \preccurlyeq K$ uniformly in $n, r, v$ by \eqref{bound-wt}.
If the fundamental discriminants underlying $D_2d^2$ and $-|D_2|\nu^2 - h$ coincide, we have at most $\preccurlyeq K^{1/4}$ choices for $h$ (in most cases much fewer), so the diagonal term contributes at most
$$\preccurlyeq \frac{1}{K^{5/2}} K \sum_{ D_2\preccurlyeq K^2} \frac{K^{1/4}}{|D_2|^{1/2} } \preccurlyeq K^{-1/4}$$
to \eqref{I1}.
4) It remains to deal with the fourth term, and to this end we write $D_2 = \Delta_2 f_2$ with a fundamental discriminant $\Delta_2$. Then the Kloosterman term can be bounded by
\begin{equation*} \begin{split}
\mathcal{I}_1^{\text{off}, \text{off}}(K)= &\frac{1}{K^{5/2}} \sum_{h \preccurlyeq K^{1/2}} \Big| \sum_{ D_2 = \Delta_2 f_2^2} \frac{ \chi_{D_2}(\delta) }{|D_2|^{1/4} (|D_2|\nu^2 + h)^{1/4}} V_x\Big(\frac{|D_2|}{K^2}\Big) e\Big( -\frac{h\bar{\gamma}}{c}\Big) \\
& \sum_{ d_1r\tau vw = f_2 }\sum_{n, c, m} \frac{\mu(d_1)\mu(v) \chi_{\Delta_2}(d_1vm\tau)}{\sqrt{d_1rn\tau} vm} \frac{ K_{3/2}^+(|\Delta_2|(vwn)^2, |D_2|\nu^2 + h, c)}{ c} \\
&\quad\quad\quad \times \int_{-\infty}^{\infty} \frac{F(4\pi vwn\sqrt{|\Delta_2|(|D_2|\nu^2 + h)}/c, t, 1/2)}{\cosh(\pi t)} \Omega(t) W_{t}( rnvm) t \frac{dt}{\pi}\Big|.
\end{split}
\end{equation*} (Here we exchanged the roles of $D_1$ and $D_2$ in Theorem \ref{thm5}.) The general strategy is now as follows: We evaluate the $t$-integral by Lemma \ref{bessel-kuz} and simplify the expression by using suitable Taylor expansions. It is a lucky coincidence that this step yields rational phases in the exponentials. We are then ready to apply Poisson summation in the long $\Delta_2$-sum which will eventually give enough savings. We now make these ideas precise.
We recall that $\Omega$ satisfies the conditions stated in \eqref{prop-omega} and is negligible for $|t| \gg K^{1/2+\varepsilon}$. The $n, m$-sum is absolutely convergent by \eqref{bound-wt}, and we can truncate it at $rnvm \preccurlyeq K^{1/2}$ at the cost of a negligible error. We split the $n, m$-sum into dyadic ranges $N \leqslant n \leqslant 2N$, $M \leqslant m \leqslant 2M$ where \begin{equation}\label{N-M}
NM \preccurlyeq (rv)^{-1}K^{1/2}.
\end{equation}
By the remark after Theorem \ref{thm5} and the properties of $\Omega$, the integrand of the $t$-integral is holomorphic in, say, $|\Im t| < 2/3$, so by contour shifts, Weil's bound \eqref{weil} and the power series expansion of the Bessel $J$-function we see that the $c$-sum is absolutely convergent and can be truncated at $c \ll K^{10^6}$, say, at the cost of a negligible error. Having truncated the $c$-sum in this very coarse way, we can sacrifice holomorphicity and include a smooth partition of unity into the $t$-integral, where a typical portion is weighted by $w(|t|/T)$ with a smooth compactly function $w$ localizing $|t| \asymp T$ with $K^{1/2-2\eta} \ll T \ll K^{1/2+\varepsilon}$. We apply Lemma \ref{bessel-kuz}a) to evaluate asymptotically the $t$-integral which in particular restricts the size of $c$. Splitting also the $c$-sum into dyadic ranges $c \asymp C$, we can assume, up to a negligible error, \begin{equation}\label{CC}
C \preccurlyeq \frac{vwn \sqrt{|\Delta_2|(|D_2|\nu^2 + h)}}{T^2} \preccurlyeq \frac{KN}{d_1r\tau}. \end{equation}
Having recorded these conditions, we can write the $t$-integral (as usual, up to a negligible error) as $$\sum_{\pm} \frac{T^2c^{1/2}}{\sqrt{vwn} |\Delta_2|^{1/4}(|D_2|\nu^2 + h)^{1/4}} e\Big(\pm \frac{2 vwn\sqrt{|\Delta_2|(|D_2|\nu^2 + h)}}{c}\Big) H^{\pm}\Big( \frac{2 vwn\sqrt{|\Delta_2|(|D_2|\nu^2 + h)}}{c}\Big) $$
for a flat function $H$, i.e.\ $x^j\frac{d^j}{dx^j} H^{\pm}(x) \ll_j 1$ for $j \in \mathbb{N}_0$. Substituting back, it remains to bound
\begin{equation}\label{145} \begin{split}
& \frac{1}{K^{3/2}} \sum_{h \preccurlyeq K^{1/2}} \sum_{ d_1,r,\tau ,v,w } \sum_{\substack{n \asymp N\\ m \asymp M}} \sum_{c \asymp C} \Big| \sum_{ \Delta_2 } \frac{ \chi_{D_2}(\delta) }{|D_2|^{1/2} (|D_2|\nu^2 + h)^{1/2}} V_x\Big(\frac{|D_2|}{K^2}\Big) \frac{ \chi_{\Delta_2}(d_1vm\tau)}{nvm} \\
& \frac{ K_{3/2}^+(|\Delta_2|(vwn)^2, |D_2|\nu^2 + h, c)}{ c^{1/2}} e\Big(\pm \frac{2 vwn\sqrt{|\Delta_2|(|D_2|\nu^2 + h)}}{c}\Big) H^{\pm}\Big( \frac{2 vwn\sqrt{|\Delta_2|(|D_2|\nu^2 + h)}}{c}\Big)\Big|.
\end{split}
\end{equation} where for given $r, v, d_1, \tau$ the parameters $N, M, C$ are subject to \eqref{N-M} and \eqref{CC} and $D_2 = \Delta_2 (d_1r\tau vw )^2$. Estimating trivially at this point using the Weil bound \eqref{weil}, we obtain the bound \begin{equation}\label{tri} \preccurlyeq \frac{1}{K^{3/2}} K^{1/2} C \preccurlyeq K^{1/2}. \end{equation} We see that applying the trace formula was a gambit in the sense that the trivial bound is now roughly a factor $K^{1/2}$ off our target. On the other hand, all automorphic information is now gone, and we may hope to get enough saving from the long character sums. In particular, we can assume that $C \geqslant K^{1 - a\eta}$ for some sufficiently large constant $a$, otherwise the trivial bound \eqref{tri} suffices. For such $C$, we can use a Taylor expansion
$$e\Big(\pm \frac{2 vwn\sqrt{|\Delta_2|(|D_2|\nu^2 + h)}}{c}\Big) = e\Big(\pm \frac{2 vwn|\Delta_2d_1r\tau vw\nu}{c} \pm \frac{vwnh}{d_1r\tau vw\nu c}\Big)\Phi(\Delta_2)$$
with
$$\Phi(\Delta) - 1 \ll \frac{nh^2}{c|D_2|} \preccurlyeq K^{-3/2}$$ in the current range of variables. The error term contributes $\preccurlyeq K^{-1}$ to \eqref{145}. Similarly, we also have
$$\frac{H^{\pm}( 2 vwn\sqrt{|\Delta_2|(|D_2|\nu^2 + h)}/c) }{|D_2|^{1/2} (|D_2|\nu^2 + h)^{1/2}} = \frac{H^{\pm}( 2 vwn |\Delta_2|d_1r\tau vw\nu/c)}{|D_2|\nu} + O\Big(\frac{h}{|D_2|^2}\Big)$$ and again the error term contributes $\preccurlyeq K^{-1}$ to \eqref{145}. Defining $\tilde{V}_x(z) = z^{-1} V_x(z)$, we are left with bounding
\begin{equation*} \begin{split}
\frac{1}{K^{7/2}} & \sum_{h \preccurlyeq K^{1/2}} \sum_{ (d_1r\tau vw, \delta) = 1 } \sum_{\substack{n \asymp N\\ m \asymp M}} \sum_{c \asymp C} \Big| \sum_{ \Delta_2 } \tilde{V}_x\Big(\frac{|\Delta_2|(d_1r\tau vw)^2}{K^2}\Big) \frac{ \chi_{\Delta_2}(\delta d_1vm\tau)}{nvmc^{1/2}} \\
& K_{3/2}^+(|\Delta_2|(vwn)^2, |\Delta_2|(d_1r\tau vw\nu)^2 + h, c) e\Big(\pm \frac{2 (vw)^2n |\Delta_2|d_1r\tau \nu }{c}\Big) H^{\pm}\Big( \frac{2 (vw)^2n |\Delta_2|d_1r\tau \nu}{c}\Big)\Big|.
\end{split}
\end{equation*}
Note the very fortunate fact that the algebraic phase $ e(\pm 2 vwn\sqrt{|\Delta_2|(|D_2|\nu^2 + h)}/c) $ in \eqref{145} has become a rational phase. As usual, the $\Delta_2$-sum runs over negative fundamental discriminants, and we split the sum into residue classes $\Delta_2 \equiv 1, 5, 8, 9, 12, 13$ (mod 16) and insert a factor $\mu^2(\Delta_2/\alpha)$ with $\alpha \in \{1, 4\}$ to detect squarefreeness. For notational simplicity let us treat the case of odd $\Delta$, the case of even $\Delta$ being similar. Using the well-known convolution formula for $\mu^2$, this leaves us with bounding
\begin{equation}\label{almostdone} \begin{split} & \frac{1}{K^{7/2}} \sum_{h \preccurlyeq K^{1/2}} \sum_{ (d_1r\tau vw, \delta) = 1 } \sum_{(d_2, \delta d_1vm\tau) = 1} \sum_{\substack{n \asymp N\\ m \asymp M}} \sum_{c \asymp C} \frac{1}{vnmc^{1/2}}\\
& \Big| \sum_{ \Delta_2 } \psi(\Delta_2) \tilde{V}_x\Big(\frac{|\Delta_2|(d_2d_1r\tau vw)^2}{K^2}\Big) \Big( \frac{\Delta_2}{\delta d_1vm\tau}\Big) K_{3/2}^+(|\Delta_2|(d_2vwn)^2, |\Delta_2|(d_2d_1r\tau vw\nu)^2 + h, c) \\
& e\Big(\pm \frac{2 (vw)^2nd_2^2 |\Delta_2|d_1r\tau \nu }{c}\Big) H^{\pm}\Big( \frac{2 (vwd_2)^2n |\Delta_2|d_1r\tau \nu}{c}\Big)\Big|.
\end{split}
\end{equation}
for a character $\psi$ modulo 4. Recall that the Kloosterman sum is nonzero only if $4 \mid c$. Estimating trivially at this point (using \eqref{weil}), we can assume that
$$d_2d_1r\tau v w \preccurlyeq C/K \preccurlyeq K^{1/2}$$
by \eqref{CC} and \eqref{N-M}, the remaining portion being $O(K^{-\eta})$ if the $K^{O(\eta)}$ factor in the previous $ \preccurlyeq$ sign is sufficiently large. We now open the Kloosterman sum and apply Poisson summation in $\Delta_2$ in residue classes modulo $ c \delta d_1vm\tau$. If ${\tt D}$ denotes the dual variable, this yields the character sum \begin{equation}\label{charsum} \begin{split}
&\sum_{\Delta_2 \, (\text{mod } c \delta d_1vm\tau)} \psi(\Delta_2) \Big( \frac{\Delta_2}{\delta d_1vm\tau}\Big) e\Big( \pm \frac{2 (vw)^2nd_2^2 |\Delta_2|d_1r\tau \nu }{c}\Big) \\
& \sum_{\substack{\gamma\, (\text{mod }c)\\ (\gamma, c) = 1}} \epsilon^{2\kappa}_{\gamma} \left(\frac{c}{\gamma}\right) e\left(\frac{|\Delta_2|(d_2vwn)^2\gamma + (|\Delta_2|(d_2d_1r\tau vw\nu)^2 + h)\bar{\gamma}}{c}\right)e\Big( \frac{\Delta_2 {\tt D}}{c \delta d_1vm\tau}\Big). \end{split} \end{equation} We write $c = c_1c_2$ where $(c_1, 2\delta d_1vm\tau) = 1$ and $c_2 \mid (2\delta d_1vm\tau)^{\infty}$. Then both sums split off a sum modulo $c_1$ given by \begin{displaymath} \begin{split}
&\sum_{\Delta_2 \, (\text{mod } c_1)} e\Big( \pm \frac{2 (vw)^2nd_2^2 |\Delta_2|d_1r\tau \nu \bar{c}_2}{c_1}\Big) \\
& \sum_{\substack{\gamma\, (\text{mod }c_1)\\ (\gamma, c_1) = 1}} \left(\frac{\gamma}{c_1}\right) e\left(\frac{(|\Delta_2|(d_2vwn)^2\gamma + (|\Delta_2|(d_2d_1r\tau vw\nu)^2 + h)\bar{\gamma})\bar{c}_2}{c_1}\right)e\Big( \frac{\Delta_2 {\tt D} \overline{c_2 \delta d_1vm\tau}}{c_1 }\Big), \end{split} \end{displaymath} cf.\ \cite[Lemma 2]{Iw0} for the treatment of the theta-multiplier. Summing over $\Delta_2$ bounds this double sum modulo $c_1$ by \begin{displaymath} \begin{split} & \leqslant c_1\#\{ \gamma \in (\mathbb{Z}/c_1\mathbb{Z}_1)^{\ast} \mid (d_2vwn)^2\gamma + (d_2d_1r\tau vw\nu)^2 \bar{\gamma} \pm 2 (vw)^2nd_2^2 d_1r\tau \nu + {\tt D} \overline{ \delta d_1vm\tau} = 0 \}\\ & \ll_{\varepsilon} c_1^{1+\varepsilon} (c_1, (d_2vwn, d_2d_1r\tau vw\nu)^2). \end{split} \end{displaymath} We estimate the remaining part of the character sum \eqref{charsum} trivially by $c_2^2\delta d_1vm\tau$. By the properties of the (essentially non-oscillating) weight functions $\tilde{V}_x$ and $H^{\pm}$, the dual variables ${\tt D}$ can be truncated at $${\tt D} \preccurlyeq \frac{ c \delta d_1vm\tau}{K^2/(d_2d_1r\tau vw)^2},$$ and so the $\Delta_2$-sum in \eqref{almostdone} can be bounded by \begin{displaymath} \begin{split} & \preccurlyeq \Big(\frac{K^2/(d_2d_1r\tau vw)^2}{ c \delta d_1vm\tau} + 1\Big) c^{1+\varepsilon} (c_1, (d_2vwn, d_2d_1r\tau vw\nu)^2) c_2\delta d_1vm\tau\\ & = \Big( \frac{K^2c^{\varepsilon}}{ (d_2d_1r\tau vw)^2 } + c^{1+\varepsilon} \delta d_1vm\tau\Big) \Big(c, (d_2w)^2( n, r \nu)^2 (2\delta d_1vm\tau)^{\infty}\Big) \end{split} \end{displaymath} using the notation explained in Section \ref{15}. The first term in the first parenthesis accounts for the zero frequency in the Poisson summation formula. We substitute this back into \eqref{almostdone} getting the (generous) upper bound
\begin{equation}\label{generous}
\begin{split} \preccurlyeq & \frac{1}{K^{3}} \sum_{d_2 d_1r\tau v w\preccurlyeq K^{1/2} } \sum_{\substack{n \asymp N\\ m \asymp M}} \sum_{c \asymp C} \frac{ (c, (2d_2 d_1r\tau v wnm)^{\infty})}{vnmc^{1/2}}\Big( \frac{K^2c^{\varepsilon}}{ (d_2d_1r\tau vw)^2 } + c^{1+\varepsilon} d_1vm\tau\Big).
\end{split}
\end{equation} Here we dropped the variable $\delta \preccurlyeq 1$. The rest is book-keeping. By Rankin's trick we have $$\sum_{c \asymp C} (c, x^{\infty}) \ll C\sum_{\substack{ d \ll C\\ d \mid x^{\infty}}} 1 \ll C\sum_{ d \mid x^{\infty}} \Big( \frac{C}{d} \Big)^{\varepsilon} \ll C(Cx)^{\varepsilon}$$
for every $\varepsilon > 0$ and $x \in \mathbb{N}$. Using \eqref{CC} and \eqref{N-M}, the bound \eqref{generous} becomes
\begin{equation*}
\begin{split} &\preccurlyeq \frac{1}{K^{3}} \sum_{d_2 d_1r\tau v w\preccurlyeq K^{1/2} } \frac{C^{1/2}}{v }\Big( \frac{K^2 }{ (d_2d_1r\tau vw)^2 } + C M d_1v\tau\Big) \\ &\preccurlyeq \frac{1}{K^{3}} \sum_{d_2 d_1r\tau v w\preccurlyeq K^{1/2} } \frac{(KN)^{1/2}}{(d_1r\tau)^{1/2}v }\Big( \frac{K^2 }{ (d_2d_1r\tau vw)^2 } +\frac{ KN M v }{r}\Big)\\ & \preccurlyeq \frac{1}{K^{3}} \sum_{d_2 d_1r\tau v w\preccurlyeq K^{1/2} } \frac{K^{3/4}}{(d_1r^2v\tau)^{1/2} }\Big( \frac{K^2 }{ v (d_2d_1r\tau vw)^2 } +\frac{ K^{3/2} }{r^2v}\Big) \preccurlyeq K^{-1/4}.
\end{split}
\end{equation*} This is the desired power saving and completes the proof of Theorem \ref{thm1}.
\appendix \section{ A period formula on average}\label{appb}
In order to verify the constant $1/4$ in the period formula \eqref{katok-Sarnak}, we take a large parameter $T$, a very small $\varepsilon > 0$ and consider the two averages $$A_1(T) = \sum_{u} \frac{L(u, 1/2) L(u \times \chi_{\Delta}, 1/2)}{L(\text{sym}^2 u, 1)} h_{T, \varepsilon}(t_u)$$ and
$$A_2(T) =\sum_{u} \frac{|P(\Delta, u)|^2}{\sqrt{|\Delta|}} h_{T, \varepsilon}(t_u)$$ for a (fixed) negative fundamental discriminant $\Delta$ with class number 1 (for simplicity) and $$ h_{T, \varepsilon}(t) = \frac{t^2 + 1/4}{T^2} \exp\Big(-\Big(\frac{t - T}{T^{1-\varepsilon}}\Big)^2 - \Big(\frac{t + T}{T^{1-\varepsilon}}\Big)^2\Big).$$ The computation is relatively standard, so we can be brief. Using the same approximate functional equation as in \eqref{approx-basic} we have
$$L(u, 1/2) = 2 \sum_n \frac{\lambda_u(n)}{n^{1/2}} W^+_{t_u}(n), \quad L( u \times \chi_{\Delta}, 1/2) = 2 \sum_m \frac{\lambda_u(m) \chi_{\Delta}(m)}{m^{1/2}} W^-_{t_u}\Big(\frac{m}{|\Delta|}\Big)$$ for even $u$ with $W_t$ as in \eqref{v-t}.
For odd $u$, each summand in $A_1(T)$ vanishes. This gives $$A_1(T ) = 4 \sum_{nm} \frac{\chi_{\Delta}(m)}{\sqrt{nm}} \sum_{u \text{ even}} \frac{\lambda_u(n) \lambda_u(m)}{L(\text{sym}^2 u, 1)} V_{t_u}(n) W_{t_u}(m) h_{T, \varepsilon}(t_u).$$ To make this spectrally complete, we artificially add the corresponding Eisenstein contribution
$$ 4 \sum_{nm} \frac{\chi_{\Delta}(m)}{\sqrt{nm}} \int_{-\infty}^{\infty} \frac{\rho_{it}(n) \rho_{-it}(m)}{|\zeta(1 + 2it)|^2} W^+_{t}(n) W^-_t\Big(\frac{m}{|\Delta|}\Big) h_{T, \varepsilon}(t) \frac{dt}{2\pi} .$$ Using \eqref{approx-basic1}, this can be written in terms of moments of the Riemann zeta function and Dirichlet $L$-functions. By standard mean value bounds the contribution is $O(T^{1+\varepsilon})$ (recall that $\Delta$ is fixed). We apply the Kuznetsov formula for the even spectrum given in Lemma \ref{kuz-even}. In this way we get a main term \begin{displaymath} \begin{split} &4 \sum_{nm} \frac{\chi_{\Delta}(n)}{n} \int_{-\infty}^{\infty} W^+_{t}(n) W^-_{t}(n) h_{T, \varepsilon}(t) t \tanh(\pi t) \frac{dt}{4\pi^2} \\ &= \int_{-\infty}^{\infty} h_{T, \varepsilon}(t) \int_{(2)} \int_{(2)} \prod_{\pm} \frac{\Gamma(1/2 + s_1 \pm it_u)}{\Gamma(1/2 \pm it_u) \pi^{s_1} s_1} \frac{\Gamma(3/2 + s_2 \pm it_u)}{\Gamma(3/2 \pm it_u) \pi^{s_2} s_2}\\
&\quad\quad\quad |\Delta|^{s_2} L( \chi_{\Delta}, 1 + s_1 + s_2)
e^{s_2^2 +s_1^2} \frac{ds_1 \, ds_2}{(2\pi i)^2}
t \tanh(\pi t) \frac{dt}{\pi^2}.
\end{split}
\end{displaymath}
We shift the $s_1, s_2$-contour to real part $-1/4$, obtaining
\begin{equation}\label{A1}
L( \chi_{\Delta}, 1) \int_{-\infty}^{\infty} h_{T, \varepsilon}(t)t \tanh(\pi t) \frac{dt}{\pi^2} + O(T^{7/4 + \varepsilon})
\end{equation} with main term $\gg T^{2-\varepsilon}$. It remains to deal with the off-diagonal terms in \eqref{kuz-even-form} and we briefly sketch why both of them are negligible. By \eqref{bound-wt} we can restrict $n, m \ll T^{1+\varepsilon}$ at the cost of a negligible error. The first off-diagonal term contributes a term of the shape $$ \sum_{nm} \frac{\chi_{\Delta}(m)}{\sqrt{nm}} \sum_c \frac{S(n, m, c)}{c}\int_{-\infty}^{\infty} J_{2it}\Big(4\pi \frac{\sqrt{nm}}{c}\Big) W^+_{t}(n) W^-_{t}(m) h_{T, \varepsilon}(t_u) t \, \frac{dt}{\cosh(\pi t)}.$$ By Lemma \ref{bessel-kuz}, the $t$-integral is negligible unless $c \leqslant \sqrt{nm} T^{-2+\varepsilon}$ which does not happen for $n, m \ll T^{1+\varepsilon}$. The second off-diagonal term contributes a term of the shape $$ \sum_{nm} \frac{\chi_{\Delta}(m)}{\sqrt{nm}} \sum_c \frac{S(n, m, c)}{c}\int_{-\infty}^{\infty} K_{2it}\Big(4\pi \frac{\sqrt{nm}}{c}\Big) V_{t}(n) W_{t}(m) h_{T, \varepsilon}(t_u) \sinh(\pi t) t \, dt.$$ Again by Lemma \ref{bessel-kuz}, up to a small error the $t$-integral is negligible unless $c \leqslant \sqrt{nm} T^{-1+\varepsilon}$ in which case it is essentially non-oscillating in $n, m$, so we can restrict to $n, m = T^{1+o(1)}$, $c \ll T^{\varepsilon}$. Poisson summation in the $m$-variable now shows that the entire expression is negligible, since $\chi_{\Delta}$ is a non-trivial character.
We continue with the analysis of $A_2(T)$. Since we assume class number 1, we have
$$A_2(T) = \frac{1}{\sqrt{\Delta} \epsilon_{\Delta}^2} \sum_u |u(z_{\Delta})|^2 h_{T, \varepsilon}(t_u)$$ where $z_{\Delta}$ is the unique Heegner point (modulo $\Gamma$) and $\epsilon_{\Delta} \in \{1, 2, 3\}$ is half the number of roots of unity in $\mathbb{Q}(\sqrt{\Delta})$. We artificially add the constant function and the Eisenstein series at a cost of $O(T)$ and apply the pre-trace formula for the entire spectrum--for odd $u$ we have $u(z_{\Delta}) = 0$ automatically. This gives $$A_2(T) = \frac{1}{\sqrt{\Delta} \epsilon_{\Delta}^2} \sum_{\gamma \in \overline{\Gamma}} k(\gamma z_{\Delta}, z_{\Delta}) + O(T)$$ where
$$k(z, w) =\frac{1}{4\pi} \int_{-\infty}^{\infty} F(1/2 + it, 1/2 - it, 1, -v) h_{T, \varepsilon}(t) \tanh(\pi t) t\, dt, \quad v = v(z, w) = \frac{|z-w|^2}{4\Im z \Im w}. $$ The stabilizer of $z_{\Delta}$ contributes \begin{equation}\label{A2} \frac{1}{\sqrt{\Delta} \epsilon_{\Delta}}\cdot \frac{1}{4\pi} \int_{-\infty}^{\infty} h_{T, \varepsilon}(t) \tanh(\pi t) t\, dt. \end{equation}
It is easy to see that $u(\gamma z, z) \leqslant \delta$ implies $\| \gamma \| \ll \sqrt{\delta + 1}$, cf.\ e.g. \cite[(A.7) with $n=1$]{IS}. From \cite[(1.64)]{Iw3} we see that $k(z, w)$ is negligible as soon as $u \gg T^{\varepsilon - 2}$, so that the contribution of all matrices not in the stabilizer is negligible. Combining \eqref{A1}, \eqref{A2} and the class number formula in the case $h_{\Delta} = 1$, we obtain $$A_2(T) \sim \frac{1}{4} A_1(T), \quad T \rightarrow \infty$$ in accordance with \eqref{katok-Sarnak}.
\section{A Dirichlet series with Hurwitz class numbers}\label{appa}
The aim of this section is an analysis of the $L$-function
$$L_+(s, a/c) = \sum_{D < 0} \frac{H(D)e(a|D|/c)}{|D|^{1/4 + s}}$$ for $4 \mid c$, $(a, c) = 1$. As before, $H(D)$ denotes the Hurwitz class number, and the series converges absolutely in $\Re s > 5/4$. The results are probably known to specialists, but do not seem to be in the literature and may be of independent interest. We recall the notation \eqref{epsd}.
\begin{lemma}\label{class-num} Let $c > 0$, $4 \mid c$, $(a, c) = 1$. The Dirichlet series $L_+(s, a/c)$ has meromorphic continuation to all $\mathbb{C}$. It has two simple poles at $s = 5/4$, $s = 3/4$ (and no other poles) with residues
\begin{displaymath}
\begin{split} & \underset{s = 5/4}{\text{{\rm res}}} L_+(s, a/c) =- \frac{\sqrt{2} \pi}{3c^{3/2}} \left( \frac{-c}{a}\right) \bar{\epsilon}_a e(3/8), \quad \underset{s = 3/4}{\text{{\rm res}}} L_+(s, a/c) = \frac{1}{\sqrt{8}c^{1/2}} \left( \frac{-c}{a}\right) \bar{\epsilon}_a e(3/8).
\end{split}
\end{displaymath}
\end{lemma}
\begin{proof} We recall the definition of $\mathcal{H}(z)$ in \eqref{zag} and compute
\begin{equation}\label{defI}
\mathcal{I}(s, a/c) := \int_0^{\infty} \Big(\mathcal{H}(a/c + iy) - (c_1 y^{3/4} +c_2y^{1/4})\Big)y^{s-1/2} \frac{dy}{y}
\end{equation}
with $c_1 = - (4\pi)^{3/4}/12$, $c_2 = 1/(\sqrt{8} \pi^{1/4})$ in two ways. Let
$$L_-(s, a/c) = \frac{1}{4 \sqrt{\pi}} \sum_{n=1}^{\infty} \frac{e(-an^2/c)}{ n^{2s-1/2}}.$$ This $L$-function is obviously holomorphic in $\Re s > 3/4$. It has a simple pole at $s=3/4$ with residue \begin{equation}\label{resl-} \underset{s=3/4}{\text{res}}L_-(s, a/c) =\frac{1}{4 \sqrt{\pi}} \frac{1}{2c} \sum_{n \,(\text{mod } c)} e\Big(-\frac{an^2}{c}\Big) = \frac{1}{4 \sqrt{\pi}} (1+i) \bar{\epsilon}_{-a} \left( \frac{c}{-a}\right) \frac{1}{2\sqrt{c}}, \end{equation} and it has a functional equation \begin{equation}\label{functheta}
L_-(s, a/c) = \Big(\frac{c}{2\pi}\Big)^{1-2s} \left( \frac{-c}{-a}\right) \epsilon_{-a} e(1/8) \frac{\Gamma( 3/4-s)}{\Gamma(s - 1/4)} L_-(1-s, -\bar{a}/c). \end{equation} This follows from the corresponding properties of Hurwitz zeta function and standard computations with Gau{\ss} sums (or the transformation behaviour of one-dimensional theta series). In particular we obtain the analytic continuation of $L_-(s, a/c)$ to all of $\mathbb{C}$ with only a simple pole at $s=3/4$.
Moreover, $L_-(s, a/c) = 0$ for $s = 1/4 - n$, $n = 1, 2, 3 \ldots$.
Returning to \eqref{defI}, we have \begin{equation}\label{I} \mathcal{I}(s, a/c) = L_+(s, a/c) G_{3/4}(s) + L_-(s, a/c) G_{-3/4}(s) \end{equation} where $$G_{c}(s) = \int_0^{\infty} W_{c, 1/4}(4\pi y) y^{s-1/2} \frac{dy}{y}.$$ Since $W_{3/4, 1/4}(x) = e^{-x/2} x^{3/4}$, we can explicitly compute \begin{equation}\label{g+} G_{3/4}(s) = 2^{s+1/4}(4\pi)^{1/2 - s} \Gamma(s + 1/4). \end{equation} For the analysis of $G_{-3/4}$, we combine \cite[7.621.3, 9.131.1, 9.111]{GR} to obtain \begin{displaymath} \begin{split} G_{-3/4}(s) &= (4\pi)^{1/2 - s} \frac{\Gamma(s -1/4)\Gamma(s + 1/4) }{\Gamma(s + 5/4)}F(s + 1/4, s-1/4, s + 5/4; 1/2) \\ &= (4\pi)^{1/2 - s} \Gamma(s -1/4)\int_0^1 t^{s-3/4}(1 - t/2)^{1/4 - s} dt. \end{split} \end{displaymath} In particular, by repeated partial integration in the $t$-integral we see that $G_{-3/4}(s)$ is meromorphic with simple poles at most at $s= (2n+1)/4$, for integers $n \leqslant 0$. From \cite[9.121.24]{GR} we get \begin{equation}\label{g-} G_{-3/4}(3/4) = 2 (\sqrt{2} - 1)\pi^{1/4}. \end{equation}
On the other hand, we may complete the pair $(a, c)$ to a matrix $(\begin{smallmatrix} a & b\\ c & d\end{smallmatrix})\in\Gamma_0(4)$. Then
$$\mathcal{H}(z) = \mathcal{H}\left( \frac{dz - b}{-cz + a}\right) \left( \frac{-c}{a}\right) \bar{\epsilon}_a \left( \frac{-cz + a}{|-cz + a|}\right)^{-3/2},$$ in particular $$\mathcal{H}\left( \frac{a}{c} + iy\right) = \mathcal{H}\left( - \frac{d}{c} + \frac{i}{c^2 y}\right) \left( \frac{-c}{a}\right) \bar{\epsilon}_a e(3/8).$$ Splitting the integral in \eqref{defI} into $\int_0^{1/c}$ and $\int_{1/c}^{\infty}$ and applying the functional equation in the former, we obtain \begin{equation}\label{funcI} \begin{split} \mathcal{I}(s, a/c)= & c^{1-2s} \left( \frac{-c}{a}\right) \bar{\epsilon}_a e(3/8) \int_{1/c}^{\infty} \Big(\mathcal{H}\left( - \frac{d}{c} + iy \right) - (c_1 y^{3/4} + c_2 y^{1/4})\Big)y^{-s+1/2} \frac{dy}{y} \\ &- \frac{c_1 c^{-s-1/4}}{s +1/4} -\frac{c_2 c^{-s+1/4}}{s - 1/4} + c^{1-2s} \left( \frac{-c}{a}\right) \bar{\epsilon}_a e(3/8) \Big( \frac{c_1c^{s-5/4}}{s-5/4} + \frac{c_2c^{s-3/4}}{s-3/4}\Big)\\
& + \int_{1/c} ^{\infty} \Big(\mathcal{H}(a/c + iy) - (c_1 y^{3/4} + c_2 y^{1/4})\Big)y^{s-1/2} \frac{dy}{y} . \end{split} \end{equation} This establishes the analytic continuation of $\mathcal{I}(s, a/c)$ to all of $\mathbb{C}$ except for poles at $s = 5/4, 3/4, 1/4, -1/4$. From the preceding analysis we conclude the meromorphic continuation of $L^+(s, a/c)$ as a function of finite order with possible poles at most at $5/4, 3/4, 1/4$. Since $L_{-}(s, a/c) G_{-3/4}(s)$ is holomorphic at $s = 5/4$, we obtain the formula for the residue at $s = 5/4$ from \eqref{I}, \eqref{g+} and \eqref{funcI}.
Using in addition \eqref{resl-}, \eqref{g-}, we obtain \begin{displaymath} \begin{split} \underset{s = 3/4}{\text{res}} L_+(s, a/c)& = \frac{\pi^{1/4}}{\sqrt{2}c^{1/2}} \left( \left( \frac{-c}{a}\right) \bar{\epsilon}_a e(3/8) \frac{1}{\sqrt{8}\pi^{1/4}} - (\sqrt{2} - 1)\pi^{1/4} \frac{1}{4 \sqrt{\pi}} (1+i) \bar{\epsilon}_{-a} \left( \frac{c}{-a}\right)\right) \\
\end{split} \end{displaymath} which confirms the residue formula at $s = 3/4$, since $\bar{\epsilon}_{-a}(\frac{c}{-a}) = (-i) (\frac{-c}{a}) \bar{\epsilon}_a$. Similarly, using also \eqref{functheta} and the simple formula $F(1/2, 0, 3/2; 1/2) = 1$, we get \begin{displaymath} \begin{split} \underset{s = 1/4}{\text{res}} L_+(s, a/c)& = \frac{1}{2\pi^{3/4}} \Big( - \frac{1}{\sqrt{8}\pi^{1/4}} - \sqrt{8} \pi^{1/4} L_-(1/4, a/c)\Big) \\ &= \frac{1}{2\pi^{3/4}} \Big( - \frac{1}{\sqrt{8}\pi^{1/4}} + \sqrt{8} \pi^{1/4}\Big( \frac{c}{2\pi}\Big)^{1/2} \left( \frac{-c}{-a}\right) \epsilon_{-a} e(1/8)\sqrt{\pi} \frac{1}{4 \sqrt{\pi}} (1+i) \bar{\epsilon}_{d} \left( \frac{c}{d}\right) \frac{1}{2\sqrt{c}} \Big)
\end{split} \end{displaymath} with $d \equiv -\bar{a}$ (mod $c$), which vanishes. \end{proof}
\emph{Remark:} One can show that away from the two poles, the function $L^+(s, a/c)$ satisfies the growth condition $L^+(s, a/c) \ll_{\Re s} ((1 + |s|) c)^{\max(0, 5/4 - \Re s, 1 - 2\Re s) + \varepsilon}.$\\
\section{A volume computation}\label{appc}
In this appendix we justify \eqref{vol} for a Saito--Kurokawa lift $F \in S_k^{(2)}$ associated with a Hecke eigenform $f \in S_{2k-2}$. Following \cite[Section 2]{Bl}, we write its Fourier expansion at $Z = iY$ as $$F(iY) = \sum_{T\in \mathcal{P}(\mathbb{Z}) } \alpha(T) (\det 2T)^{\frac{k}{2} - \frac{3}{4}} e^{-2\pi \text{tr}(TY)}, $$
normalized such that $\alpha(T)^2 = L(f \times \chi_{-\det 2T}, 1/2)$ if $-\det 2T$ is a fundamental discriminant. In this case, $\| F \| = (2\pi)^{-k}\Gamma(k) k^{-1/4 + o(1)}$ by \cite[(2.8)]{Bl}. We conclude that
$$\mathcal{F}(Y) := \frac{(\det Y)^{k/2}F(iY) }{\| F \|_2}=k^{1/4 + o(1)} \sum_{T\in \mathcal{P}(\mathbb{Z}) } \frac{\alpha(T)}{\det(2T)^{3/4}} \frac{(4\pi)^{k} (\det TY)^{\frac{k}{2}} e^{-2\pi \text{tr}(TY)}}{\Gamma(k)}. $$
The function $$X \mapsto \frac{(4\pi)^k (\det X)^{k/2} e^{-2\pi \text{tr}X}}{\Gamma(k)}$$ is invariant under conjugation, and for a diagonal matrix $X = \text{diag}(x_1, x_2)$ it is negligible unless $x_1, x_2 = k/4\pi + O(\sqrt{k}\log k)$, cf. \cite[Section 4]{Bl}. For large $k$, we conclude that $\mathcal{F}(Y)$ is negligible unless there exists $T \in \mathcal{P}(\mathbb{Z})$ such that the two eigenvalues of $TY$ are $k/4\pi + O(\sqrt{k}\log k)$. In particular, the essential support of $\mathcal{F}$ can be restricted to matrices $Y$ whose maximal eigenvalue $\lambda_{\max}(Y)$ satisfies $\lambda_{\max}(Y) \ll k$. Since $|\mathcal{F}|$ is invariant under $Y \mapsto Y^{-1}$, its minimal eigenvalue $\lambda_{\min}(Y)$ must satisfy $\lambda_{\min}(Y) \gg 1/k$. The implied constants will not be relevant. As in \eqref{product}, we write $$Y = Y(r, x, y) = \sqrt{r}\left(\begin{matrix} y^{-1} & -xy^{-1}\\-x y^{-1} & y^{-1} (x^2 + y^2)\end{matrix}\right)$$ with $x + iy \in \Gamma \backslash \mathbb{H}$. Let $\mathcal{Y}$ be the set of such matrices $Y$ with $1/k \ll \lambda_{\min}(Y) \leqslant \lambda_{\max}(Y) \ll k$.
Without loss of generality we may assume that $Y$ is Minkowski-reduced, equivalently $x+iy$ is in the standard fundamental domain $|x|\leqslant 1/2$, $x^2 + y^2 \geqslant 1$. The two eigenvalues of $Y$ are given by $$\sqrt{r} \frac{1 + x^2 + y^2 \pm \sqrt{(1+x)^2 + 2(x^2 - 1)y^2 + y^4}}{2y}.$$ Thus we have \begin{displaymath} \begin{split} \text{vol}(\mathcal{Y})& = \underset{Y(r, x, y) \in \mathcal{Y}}{\int_{-1/2}^{1/2} \int_{\sqrt{1 - x^2}}^{\infty} \int_{\mathbb{R}} } \frac{dr}{r} \frac{dy \, dx}{y^2}= \int_{-1/2}^{1/2} \int_{\sqrt{1 - x^2}}^{\infty} \big(4\log (k/y) + O(1) \big) \frac{dy \, dx}{y^2} \\ &= \text{vol}({\rm SL}_2(\mathbb{Z}) \backslash \mathbb{H}) \cdot 4 \log k + O(1), \end{split} \end{displaymath} as desired.
\end{document} | arXiv |
\begin{document} \thispagestyle{empty}
{\Large{\bf
\begin{center} A sequential update algorithm for computing\\ the stationary distribution vector in upper block-Hessenberg Markov chains\footnote[1]{ Published online in {\it Queueing Systems} on February 21, 2019 (doi: 10.1007/s11134-019-09599-x) }
\if0 \footnote[1]{ This research was supported in part by JSPS KAKENHI Grant Numbers JP18K11181. } \fi
\end{center} } }
\begin{center} { Hiroyuki Masuyama \footnote[2]{E-mail: [email protected]} }
{\small Department of Systems Science, Graduate School of Informatics, Kyoto University\\ Kyoto 606-8501, Japan }
{\small \textbf{Abstract}
\begin{tabular}{p{0.85\textwidth}} This paper proposes a new algorithm for computing the stationary distribution vector in continuous-time upper block-Hessenberg Markov chains. To this end, we consider the last-block-column-linearly-augmented (LBCL-augmented) truncation of the (infinitesimal) generator of the upper block-Hessenberg Markov chain. The LBCL-augmented truncation is a linearly-augmented truncation such that the augmentation distribution has its probability mass only on the last block column. We first derive an upper bound for the total variation distance between the respective stationary distribution vectors of the original generator and its LBCL-augmented truncation. Based on the upper bound, we then establish a series of linear fractional programming (LFP) problems to obtain augmentation distribution vectors such that the bound converges to zero. Using the optimal solutions of the LFP problems, we construct a matrix-infinite-product (MIP) form of the original (i.e., not approximate) stationary distribution vector and develop a sequential update algorithm for computing the MIP form. Finally, we demonstrate the applicability of our algorithm to BMAP/M/$\infty$ queues and M/M/$s$ retrial queues. \end{tabular} }
\end{center}
\begin{center} \begin{tabular}{p{0.90\textwidth}} {\small
{\bf Keywords:} Upper block-Hessenberg Markov chain; level-dependent M/G/1-type Markov chain; Matrix-infinite-product (MIP) form; Last-block-column-linearly-augmented truncation (LBCL-augmented truncation); BMAP/M/$\infty$ queue; M/M/$s$ retrial queue
{\bf Mathematics Subject Classification:} 60J22; 60K25
} \end{tabular}
\end{center}
\section{Introduction}\label{sec-intro}
This paper considers an upper block-Hessenberg Markov chain in continuous time. To describe such a Markov chain, we first introduce some symbols. Let $\mathbb{R}_+$ denote the set of all nonnegative real numbers, i.e., $\mathbb{R}_+=[0,\infty)$. Let $\mathbb{N} = \{1,2,3,\dots\}$, $\mathbb{Z}_+ = \{0,1,2,\dots\}$, and $\mathbb{Z}_n = \{0,1,\dots,n\}$ for $n \in \mathbb{Z}_+$. We then introduce some sets of pairs of integers:
\begin{eqnarray*} \mathbb{S} &=& \bigcup_{k=0}^{\infty} \mathbb{L}_k, \quad \mathbb{S}_n = \bigcup_{k=0}^n \mathbb{L}_k, \quad \overline{\mathbb{S}}_n = \mathbb{S} \setminus \mathbb{S}_n, \quad n \in \mathbb{Z}_+, \\ \mathbb{L}_k &=& \{k\} \times \mathbb{M}_k, \quad k \in \mathbb{Z}_+, \end{eqnarray*}
where $\mathbb{M}_k = \{1,2,\dots,M_k\} \subset \mathbb{N}$. We also define $(k,i;\ell,j)$ as an ordered pair $((k,i),(\ell,j))$ in $\mathbb{S}^2$. Furthermore, we define $\bm{e}=(1,1,\dots)^{\top}$, which has an appropriate (finite or infinite) number of ones.
Let $\{(X(t),J(t));t \in \mathbb{R}_+\}$ denote a regular-jump bivariate Markov chain with state space $\mathbb{S}$ (see \cite[Chapter 8, Definition 2.5]{Brem99} for the definition of regular-jump Markov chains). Let $\bm{Q}:=(q(k,i;\ell,j))_{(k,i;\ell,j) \in\mathbb{S}^2}$ denote the (infinitesimal) generator of the Markov chain $\{(X(t),J(t))\}$, which is in an upper block-Hessenberg form:
\begin{equation} \bm{Q} = \bordermatrix{
& \mathbb{L}_0 & \mathbb{L}_1 & \mathbb{L}_2 & \mathbb{L}_3 & \cdots \cr \mathbb{L}_0 & \bm{Q}_{0,0} & \bm{Q}_{0,1} & \bm{Q}_{0,2} & \bm{Q}_{0,3} & \cdots \cr \mathbb{L}_1 & \bm{Q}_{1,0} & \bm{Q}_{1,1} & \bm{Q}_{1,2} & \bm{Q}_{1,3} & \cdots \cr \mathbb{L}_2 & \bm{O} & \bm{Q}_{2,1} & \bm{Q}_{2,2} & \bm{Q}_{2,3} & \cdots \cr \mathbb{L}_3 & \bm{O} & \bm{O} & \bm{Q}_{3,2} & \bm{Q}_{3,3} & \cdots \cr ~\vdots & \vdots & \vdots & \vdots & \vdots & \ddots }. \label{defn-Q-MG1-type} \end{equation}
We refer to $\{(X(t),J(t))\}$ as the {\it upper block-Hessenberg Markov chain} (which may be called the {\it level-dependent M/G/1-type Markov chain}) and refer to $X(t)$ and $J(t)$ as the {\it level variable} and the {\it phase variable}, respectively. Note that if $(X(t),J(t)) \in \mathbb{L}_k$ then $X(t) = k$ and thus $\mathbb{L}_k$ is called {\it level} $k$.
Throughout the paper, unless otherwise stated, we assume that $\{(X(t),J(t))\}$ is ergodic (i.e., irreducible, aperiodic and positive recurrent). We then define $\bm{\pi}:=(\pi(k,i))_{(k,i)\in\mathbb{S}} > \bm{0}$ as the unique stationary distribution vector of the ergodic generator $\bm{Q}$ (see, e.g., \cite[Chapter 5, Theorems 4.4 and 4.5]{Ande91}). By definition, $\bm{\pi}\bm{Q} = \bm{0}$ and $\bm{\pi}\bm{e}=1$. For later use, we also define $\bm{\pi}_k =(\pi(k,i))_{i\in\mathbb{M}_k}$ for $k \in \mathbb{Z}_+$ and partition $\bm{\pi}$ as
\begin{eqnarray*} \bm{\pi} &=& \bordermatrix{
& \mathbb{L}_0 & \mathbb{L}_1 & \cdots \cr
& \bm{\pi}_0 & \bm{\pi}_1 & \cdots }. \end{eqnarray*}
It is, in general, difficult to obtain an explicit expression of $\bm{\pi}=(\bm{\pi}_0,\bm{\pi}_1,\dots)$. Thus, we study the computation of the stationary distribution vector $\bm{\pi}$ through a {\it linearly augmented truncation} of the ergodic generator $\bm{Q}$. The linearly augmented truncation is described below.
Let $\presub{(n)}\bm{Q}:=(\presub{(n)}q(k,i;\ell,j))_{(k,i;\ell,j)\in (\mathbb{S}_n)^2}$, $n \in \mathbb{Z}_+$, denote the northwest corner truncation of the ergodic generator $\bm{Q}$, which is given by
\begin{eqnarray*} \presub{(n)}\bm{Q} = \left( \begin{array}{ccccccc} \bm{Q}_{0,0} & \bm{Q}_{0,1} & \bm{Q}_{0,2} & \cdots & \bm{Q}_{0,n-2} & \bm{Q}_{0,n-1} & \bm{Q}_{0,n} \\ \bm{Q}_{1,0} & \bm{Q}_{1,1} & \bm{Q}_{1,2} & \cdots & \bm{Q}_{1,n-2} & \bm{Q}_{1,n-1} & \bm{Q}_{1,n} \\ \bm{O} & \bm{Q}_{2,1} & \bm{Q}_{2,2} & \cdots & \bm{Q}_{2,n-2} & \bm{Q}_{2,n-1} & \bm{Q}_{2,n} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ \bm{O} & \bm{O} & \bm{O} & \cdots & \bm{Q}_{n-1,n-2} & \bm{Q}_{n-1,n-1} & \bm{Q}_{n-1,n} \\ \bm{O} & \bm{O} & \bm{O} & \cdots & \bm{O} & \bm{Q}_{n,n-1} & \bm{Q}_{n,n} \\ \end{array} \right).
\end{eqnarray*}
We then define $\presub{(n)}\overline{\bm{Q}}:=(\presub{(n)}\overline{q}(k,i;\ell,j))_{(k,i;\ell,j)\in(\mathbb{S}_n)^2}$, $n\in\mathbb{Z}_+$, as a $Q$-matrix (diagonally dominant matrix with nonpositive diagonal elements and nonnegative off-diagonal ones; see, e.g., \cite[Section~2.1]{Ande91}) such that
\begin{equation} \presub{(n)}\overline{\bm{Q}} = \presub{(n)}\bm{Q} - \presub{(n)}\bm{Q}\bm{e} \presub{(n)}\bm{\alpha}, \qquad n\in\mathbb{Z}_+, \label{defn-(n)ol{Q}} \end{equation}
where $\presub{(n)}\bm{\alpha}:=(\presub{(n)}\alpha(k,i))_{(k,i) \in \mathbb{S}_n}$ is a probability vector. We refer to $\presub{(n)}\overline{\bm{Q}}$ as the {\it linearly augmented truncation} of $\bm{Q}$. We also refer to $\presub{(n)}\bm{\alpha}$ as the {\it augmentation distribution vector}.
Let $\presub{(n)}\overline{\bm{\pi}}=(\presub{(n)}\overline{\pi}(k,i))_{(k,i) \in \mathbb{S}_n}$, $n\in\mathbb{Z}_+$, denote
\begin{equation} \presub{(n)}\overline{\bm{\pi}} = {\presub{(n)}\bm{\alpha} (- \presub{(n)}\bm{Q})^{-1} \over \presub{(n)}\bm{\alpha} (- \presub{(n)}\bm{Q})^{-1}\bm{e}}, \qquad n \in \mathbb{Z}_+, \label{defn-(n)ol{pi}} \end{equation}
where $(- \presub{(n)}\bm{Q})^{-1}$ exists due to the ergodicity of $\bm{Q}$. From (\ref{defn-(n)ol{Q}}) and (\ref{defn-(n)ol{pi}}), we have
\[ \presub{(n)}\overline{\bm{\pi}}\presub{(n)}\overline{\bm{Q}}=\bm{0}, \quad \presub{(n)}\overline{\bm{\pi}} \ge \bm{0},\quad \presub{(n)}\overline{\bm{\pi}}\bm{e}=1; \]
that is, $\presub{(n)}\overline{\bm{\pi}}$ is a stationary distribution vector of the linearly augmented truncation $\presub{(n)}\overline{\bm{Q}}$. Furthermore, as $n\to\infty$, each element of $\presub{(n)}\overline{\bm{Q}}$ converges to the corresponding one of $\bm{Q}$. Thus, we can expect $\presub{(n)}\overline{\bm{\pi}}$ to be an approximation to $\bm{\pi}$. This is why we refer to $\presub{(n)}\overline{\bm{\pi}}$ as the {\it linearly augmented truncation approximation} to $\bm{\pi}$.
We note that if the augmentation distribution vector $\presub{(n)}\bm{\alpha}$ has its probability mass only on the last block (i.e., $\mathbb{L}_n$) then the linearly augmented truncation $\presub{(n)}\overline{\bm{Q}}$ inherits upper block-Hessenberg structure from the original generator $\bm{Q}$. To utilize this tractable structure, we focus on a special linearly augmented truncation $\presub{(n)}\overline{\bm{Q}}$ with $\presub{(n)}\bm{\alpha} = \presub{(n)}\widehat{\bm{\alpha}}$, where $\presub{(n)}\widehat{\bm{\alpha}}$ is a probability vector such that
\begin{equation} \presub{(n)}\widehat{\bm{\alpha}} = \bordermatrix{
& \mathbb{S}_{n-1} & \mathbb{L}_n \cr
& \bm{0} & \bm{\alpha}_n }. \label{cond-(n_s)alpha} \end{equation}
For convenience, we refer to such a linearly augmented truncation as a {\it last-block-column-linearly-augmented truncation (LBCL-augmented truncation)}.
We now define $\presub{(n)}\widehat{\bm{Q}}:=(\presub{(n)}\widehat{q}(k,i;\ell,j)_{(k,i;\ell,j) \in (\mathbb{S}_n)^2}$, $n\in\mathbb{Z}_+$, as the LBCL-augmented truncation of $\bm{Q}$, that is, a $Q$-matrix such that
\begin{equation} \presub{(n)}\widehat{\bm{Q}} = \presub{(n)}\bm{Q} - \presub{(n)}\bm{Q}\bm{e} \presub{(n)}\widehat{\bm{\alpha}}, \qquad n\in\mathbb{Z}_+. \label{defn-(n)wh{Q}} \end{equation}
We also define $\presub{(n)}\widehat{\bm{\pi}}:=(\presub{(n)}\widehat{\pi}(k,i))_{(k,i) \in \mathbb{S}_n}$, $n\in\mathbb{Z}_+$, as
\begin{equation} \presub{(n)}\widehat{\bm{\pi}} = {\presub{(n)}\widehat{\bm{\alpha}} (- \presub{(n)}\bm{Q})^{-1} \over \presub{(n)}\widehat{\bm{\alpha}} (- \presub{(n)}\bm{Q})^{-1}\bm{e}}, \qquad n \in \mathbb{Z}_+. \label{defn-(n)ol{pi}_n} \end{equation}
Note that $\presub{(n)}\widehat{\bm{\pi}}$ is equal to $\presub{(n)}\overline{\bm{\pi}}$ in (\ref{defn-(n)ol{pi}}) with $\presub{(n)}\bm{\alpha} = \presub{(n)}\widehat{\bm{\alpha}}$; that is, $\presub{(n)}\widehat{\bm{\pi}}$ is a stationary distribution vector of the LBCL-augmented truncation $\presub{(n)}\widehat{\bm{Q}}$. Hence, we call $\presub{(n)}\widehat{\bm{\pi}}$ the {\it last-block-column-linearly-augmented truncation approximation (LBCL-augmented truncation approximation)} to $\bm{\pi}$.
In this paper, we propose a new algorithm for computing the original stationary distribution vector $\bm{\pi}$ by using the LBCL-augmented truncation approximation $\presub{(n)}\widehat{\bm{\pi}}$. In fact, $\presub{(n)}\widehat{\bm{\pi}}$ does not necessarily converge to $\bm{\pi}$ as $n \to \infty$ (see Section~\ref{subsec-example}). We solve such a problem by choosing $\presub{(n)}\widehat{\bm{\alpha}}$ adaptively for each $n \in \mathbb{Z}_+$. To achieve this, we first derive an upper bound for the total variation distance between $\bm{\pi}$ and $\presub{(n)}\widehat{\bm{\pi}}$. With this upper bound, we establish a series of linear fractional programming (LFP) problems for finding $\{\presub{(n)}\widehat{\bm{\alpha}};n\in\mathbb{Z}_+\}$ such that $\{\presub{(n)}\widehat{\bm{\pi}};n\in\mathbb{Z}_+\}$ converges to $\bm{\pi}$. Fortunately, the optimal solutions of the LFP problems are explicitly obtained. Thus, we can readily construct a convergent sequence of LBCL-augmented truncation approximations, which yields a matrix-infinite-product (MIP) form of $\bm{\pi}$. We note that the LFP problems are not given in advance but are formulated successively while constructing the MIP form. As a result, we can develop a sequential update algorithm for computing $\bm{\pi}$.
We now review related work. Some researchers \cite{Baum12-Procedia,Brig95,Phun10-QTNA} have studied the computation of level-dependent quasi-birth-and-death processes (LD-QBDs), which belong to a special case of upper block-Hessenberg Markov chains. These previous studies propose algorithms for computing the {\it conditional stationary distribution vector} $\bm{\pi}^{(N)}$:
\[ \bm{\pi}^{(N)} = {(\bm{\pi}_0,\bm{\pi}_1,\dots,\bm{\pi}_N) \over \sum_{\ell=0}^N \bm{\pi}_{\ell}}, \]
where $N \in \mathbb{N}$ is the truncation parameter that should be determined so that $\bm{\pi}^{(N)}$ is sufficiently close to $\bm{\pi}$. Takine~\cite{Taki16} develops an algorithm for computing $\bm{\pi}^{(N)}$ of a special upper block-Hessenberg Markov chain, which assumes that, for all sufficiently large $n\in\mathbb{Z}_+$, the $\bm{Q}_{n,n-1}$ are nonsingular and the $\bm{Q}_{n,n}$ are of the same order (see Assumption 1 therein). These additional assumptions in \cite{Taki16} are removed by Kimura and Takine \cite{M.Kimu18}. Besides, Shin and Pearce~\cite{Shin98}, Li et al.~\cite{LiQuan05}, and Klimenok and Dudin~\cite{Klim06} modify transition rates (or transition probabilities) such that they are eventually level independent, and then these researchers establish algorithms for computing approximately the stationary distribution vectors of upper block-Hessenberg Markov chains.
The algorithms proposed in \cite{Brig95,Shin98} have update procedures to improve their outputs, like our algorithm. However, their update procedures need to recompute, from scratch, most components of their new outputs every time. On the other hand, our algorithm utilizes the components of the current result, together with some additional computation, to generate an updated result. This is a remarkable feature of our algorithm.
The rest of this paper is divided into four sections. Section~\ref{sec-MIP-form-solution} describes preliminary results on the LBCL-augmented truncation approximation for upper block-Hessenberg Markov chains. Section~\ref{sec-algorithm} proposes a sequential update algorithm that generates a sequence of LBCL-augmented truncation approximations converging to the original stationary distribution vector. Section~\ref{sec-discussion} demonstrates the applicability of the proposed algorithm. Finally, Section~\ref{sec-remarks} provides concluding remarks.
\section{The LBCL-augmented truncation approximation}\label{sec-MIP-form-solution}
This section consists of three subsections. In Section~\ref{subsec-matrix-product-form}, we show a matrix-product form of the LBCL-augmented truncation approximation $\presub{(n)}\widehat{\bm{\pi}}$. In Section~\ref{subsec-error-bound}, we derive an error bound for $\presub{(n)}\widehat{\bm{\pi}}$, more specifically, an upper bound for the total variation distance between $\presub{(n)}\widehat{\bm{\pi}}$ and $\bm{\pi}$. In Section~\ref{subsec-example}, we provide an example such that $\presub{(n)}\widehat{\bm{\pi}}$ does not converge to $\bm{\pi}$ as $n \to \infty$.
Before entering the body of this section, we describe our notation. For any matrix $\bm{M}$ (resp.~vector $\bm{m}$), let $|\bm{M}|$ (resp.~$|\bm{m}|$) denote the matrix (resp.~vector) obtained by taking the absolute value of each element of $\bm{M}$ (resp.~$\bm{m}$). A finite matrix is treated, if necessary, as an infinite matrix that keeps the existing elements in their original positions and has an infinite number of zeros in the other positions. Such treatment is also applied to finite vectors. Thus, for example, it follows from (\ref{defn-(n)wh{Q}}) that
\begin{eqnarray} \lefteqn{
\presub{(n)}\widehat{\bm{\pi}} \,| \presub{(n)}\widehat{\bm{Q}} - \bm{Q} | } \quad && \nonumber \\ &=& \bordermatrix{
& \mathbb{S}_n & \overline{\mathbb{S}}_n \cr
& \presub{(n)}\widehat{\bm{\pi}} & \bm{0} } \nonumber \\ && {} \times
\left| \bordermatrix{
& \mathbb{S}_n & \overline{\mathbb{S}}_n \cr \mathbb{S}_n & \presub{(n)}\bm{Q} -\presub{(n)}\bm{Q} \bm{e} \presub{(n)}\widehat{\bm{\alpha}} & \bm{O} \cr \overline{\mathbb{S}}_n & \bm{O} & \bm{O} } - \bordermatrix{
& \mathbb{S}_n & \overline{\mathbb{S}}_n \cr \mathbb{S}_n & \presub{(n)}\bm{Q} & \presub{(n)}\bm{Q}_{>n} \cr \overline{\mathbb{S}}_n & \ast & \ast }
\right| \nonumber \\ &=& \left(
\begin{array}{c@{~}|@{~}c} \presub{(n)}\widehat{\bm{\pi}} & \bm{0} \end{array} \right) \left(
\begin{array}{c@{~}|@{~}c} -\presub{(n)}\bm{Q} \bm{e} \presub{(n)}\widehat{\bm{\alpha}} & \presub{(n)}\bm{Q}_{>n} \rule[-2.5mm]{0mm}{2mm}{} \\ \hline * & * \end{array} \right), \label{eqn-(n)wh{pi}((n)wh{Q}-Q)} \end{eqnarray}
where $\presub{(n)}\bm{Q}_{>n} = (q(k,i;\ell,j))_{(k,i;\ell,j) \times \mathbb{S}_n \times \overline{\mathbb{S}}_n}$. It also follows from (\ref{cond-(n_s)alpha}) that, for any column vector $\bm{v}:= (v(k,i))_{(k,i)\in\mathbb{S}}$,
\begin{equation} \presub{(n)}\widehat{\bm{\alpha}}\bm{v} = \bordermatrix{
& \mathbb{S}_n & \overline{\mathbb{S}}_n \cr
& \presub{(n)}\widehat{\bm{\alpha}} & \bm{0} } \bm{v} = \bordermatrix{
& \mathbb{S}_{n-1} & \mathbb{L}_n & \overline{\mathbb{S}}_n \cr
& \bm{0} & \bm{\alpha}_n & \bm{0} } \bm{v} = \bm{\alpha}_n\bm{v}_n, \label{eqn-(n)wh{alpha}v} \end{equation}
where $\bm{v}_n = (v(k,i))_{(k,i) \in\mathbb{L}_n} = (v(n,i))_{i\in\mathbb{M}_n}$ for $n \in \mathbb{Z}_+$. Furthermore, we use the following notation: If a sequence $\{\bm{Z}_n; n \in \mathbb{Z}_+\}$ of finite matrices (or vectors) converges element-wise to an infinite matrix (or vector) $\bm{Z}$, then we denote this convergence by $\lim_{n\to\infty}\bm{Z}_n = \bm{Z}$. We also define the empty sum as zero (e.g., $\sum_{k=1}^0 \cdot = 0$).
\subsection{A matrix-product form}\label{subsec-matrix-product-form}
We partition $\presub{(n)}\widehat{\bm{\pi}}$ and $(-\presub{(n)}\bm{Q})^{-1}$ level-wise as follows:
\begin{align} \presub{(n)}\widehat{\bm{\pi}} &= \bordermatrix{
& \mathbb{L}_0 & \mathbb{L}_1 & \cdots & \mathbb{L}_n \cr
& \presub{(n)}\widehat{\bm{\pi}}_{0} & \presub{(n)}\widehat{\bm{\pi}}_{1} & \cdots & \presub{(n)}\widehat{\bm{\pi}}_{n} }, & n &\in \mathbb{Z}_+, \nonumber \\ (-\presub{(n)}\bm{Q})^{-1} &= \bordermatrix{
& \mathbb{L}_0 & \mathbb{L}_1 & \cdots & \mathbb{L}_n \cr \mathbb{L}_0 & \presub{(n)}\bm{X}_{0,0} & \presub{(n)}\bm{X}_{0,1} & \cdots & \presub{(n)}\bm{X}_{0,n} \cr \mathbb{L}_1 & \presub{(n)}\bm{X}_{1,0} & \presub{(n)}\bm{X}_{1,1} & \cdots & \presub{(n)}\bm{X}_{1,n} \cr ~\vdots & \vdots & \vdots & \ddots & \vdots \cr \mathbb{L}_n & \presub{(n)}\bm{X}_{n,0} & \presub{(n)}\bm{X}_{n,1} & \cdots & \presub{(n)}\bm{X}_{n,n} }, & n &\in \mathbb{Z}_+. \label{defn-(-Q)^{-1}-MG1-type} \end{align}
From (\ref{defn-(-Q)^{-1}-MG1-type}) and (\ref{cond-(n_s)alpha}), we have
\begin{equation} \presub{(n)}\widehat{\bm{\alpha}} (-\presub{(n)}\bm{Q})^{-1} =\bm{\alpha}_n (\presub{(n)}\bm{X}_{n,0},\presub{(n)}\bm{X}_{n,1},\dots,\presub{(n)}\bm{X}_{n,n}),\qquad n \in \mathbb{Z}_+. \label{eqn-(n)wh{alpha}(-(n)Q)^{-1}} \end{equation}
Substituting (\ref{eqn-(n)wh{alpha}(-(n)Q)^{-1}}) into (\ref{defn-(n)ol{pi}_n}) yields
\[ \presub{(n)}\widehat{\bm{\pi}} = {\bm{\alpha}_n (\presub{(n)}\bm{X}_{n,0},\presub{(n)}\bm{X}_{n,1},\dots,\presub{(n)}\bm{X}_{n,n}) \over \bm{\alpha}_n \sum_{\ell=0}^n \presub{(n)}\bm{X}_{n,\ell}\bm{e} },\qquad n \in \mathbb{Z}_+, \]
which leads to
\begin{eqnarray} \presub{(n)}\widehat{\bm{\pi}}_{k} = {\bm{\alpha}_n \presub{(n)}\bm{X}_{n,k} \over \bm{\alpha}_n \sum_{\ell=0}^n \presub{(n)}\bm{X}_{n,\ell}\bm{e} }, \qquad n \in \mathbb{Z}_+,\ k \in \mathbb{Z}_n. \label{eqn-(n)pi_{n,k}} \end{eqnarray}
Note that, because $(-\presub{(n)}\bm{Q})^{-1}\presub{(n)}\bm{Q}=-\bm{I}$, the inverse matrix $(-\presub{(n)}\bm{Q})^{-1} \ge \bm{O}$ has no zero rows. Therefore,
\begin{eqnarray} \sum_{\ell=0}^n \presub{(n)}\bm{X}_{n,\ell}\bm{e} &>& \bm{0}, \qquad n \in \mathbb{Z}_+,\ \ell \in \mathbb{Z}_n. \label{ineqn-(n)X_{n,l}e>0} \end{eqnarray}
We derive a matrix-product form of $\presub{(n)}\widehat{\bm{\pi}}_{k}$, $k\in\mathbb{Z}_n$, from (\ref{eqn-(n)pi_{n,k}}). To do this, we need some preparation. We first partition $\presub{(n)}\bm{Q}$ as
\[ \presub{(n)}\bm{Q} = \left(
\begin{array}{cccc|c}
& & & & \,\bm{Q}_{0,n} \\
& & \presub{(n-1)}\bm{Q} & & \,\vdots \\ \rule[-2mm]{0mm}{4mm} & & & & \,\bm{Q}_{n-1,n} \\ \hline \rule{0mm}{4mm}\bm{O} & \cdots & \bm{O} & \bm{Q}_{n,n-1} & \, \bm{Q}_{n,n} \end{array} \right),\qquad n \in \mathbb{N}. \]
From this equation and (\ref{defn-(-Q)^{-1}-MG1-type}), we have the following (see the last two equations in \cite[Section~0.7.3]{Horn13}): For $n \in \mathbb{N}$,
\begin{eqnarray} \presub{(n)}\bm{X}_{n,n} &=& \left[ - \bm{Q}_{n,n} - (\bm{O},\dots,\bm{O},\bm{Q}_{n,n-1}) (-\presub{(n-1)}\bm{Q})^{-1} \left( \begin{array}{c} \bm{Q}_{0,n} \\ \bm{Q}_{1,n} \\ \vdots \\ \bm{Q}_{n-1,n} \end{array} \right) \right]^{-1} \nonumber \\ &=& \left( - \bm{Q}_{n,n} - \bm{Q}_{n,n-1} \sum_{\ell=0}^{n-1} \presub{(n-1)}\bm{X}_{n-1,\ell}\bm{Q}_{\ell,n} \right)^{-1}, \label{defn-(n)X_{n,n}} \end{eqnarray}
and
\begin{eqnarray} \presub{(n)}\bm{X}_{n,k} &=& \presub{(n)}\bm{X}_{n,n} \cdot (\bm{O},\dots,\bm{O},\bm{Q}_{n,n-1}) (-\presub{(n-1)}\bm{Q})^{-1} \nonumber \\ &=& \presub{(n)}\bm{X}_{n,n} \cdot \bm{Q}_{n,n-1}\presub{(n-1)}\bm{X}_{n-1,k}, \qquad k \in \mathbb{Z}_{n-1}. \label{defn-(n)X_{n,k}} \end{eqnarray}
We also define
\begin{align} \bm{U}_n^* &= \presub{(n)}\bm{X}_{n,n}, \label{eqn-U_k^*=(n)X_{n,n}} \\ \bm{U}_{n,k} &= \left\{ \begin{array}{ll} \bm{Q}_{n,n-1}\presub{(n-1)}\bm{X}_{n-1,k}, & \quad k \in \mathbb{Z}_{n-1}, \\ \bm{I}, & \quad k=n, \end{array} \right. \label{defn-U_{n,k}} \end{align}
for $n \in \mathbb{Z}_+$. It then follows from (\ref{defn-(n)X_{n,n}}), (\ref{eqn-U_k^*=(n)X_{n,n}}), and (\ref{defn-U_{n,k}}) that
\begin{equation} \bm{U}_n^* = \left\{ \begin{array}{l@{~~~}l} (-\bm{Q}_{0,0})^{-1}, & n=0, \\ \left( -\bm{Q}_{n,n} - \displaystyle\sum_{\ell=0}^{n-1} \bm{U}_{n,\ell} \bm{Q}_{\ell,n} \right)^{-1}, & n \in \mathbb{N}. \end{array} \right. \label{defn-U_k^*} \end{equation}
Using $\bm{U}_n^*$ and $\bm{U}_{n,k}$, we can express $\presub{(n)}\bm{X}_{n,k}$, $k\in\mathbb{Z}_n$, as follows.
\begin{lem}\label{lem-(n_s)X_{s,l}} For $n \in \mathbb{Z}_+$,
\begin{equation} \presub{(n)}\bm{X}_{n,k} = \bm{U}_n^* \bm{U}_{n,k}, \qquad k \in \mathbb{Z}_n, \label{eqn-(n)X_{n,k}} \end{equation}
and
\begin{equation} \bm{U}_{n,k} = \left\{ \begin{array}{l@{~~}l} (\bm{Q}_{n,n-1} \bm{U}_{n-1}^*) (\bm{Q}_{n-1,n-2} \bm{U}_{n-2}^*) \cdots (\bm{Q}_{k+1,k}\bm{U}_{k}^*), & k \in \mathbb{Z}_{n-1}, \\ \bm{I}, & k = n. \end{array} \right. \label{defn-U_{k,l}} \end{equation}
\end{lem}
\begin{proof} Combining (\ref{defn-(n)X_{n,k}}) with (\ref{eqn-U_k^*=(n)X_{n,n}}) and (\ref{defn-U_{n,k}}), we have (\ref{eqn-(n)X_{n,k}}). Furthermore, applying (\ref{eqn-(n)X_{n,k}}) to (\ref{defn-U_{n,k}}) yields
\[ \bm{U}_{n,k} = \bm{Q}_{n,n-1} \bm{U}_{n-1}^* \bm{U}_{n-1,k},\qquad k \in \mathbb{Z}_{n-1}, \]
which leads to (\ref{defn-U_{k,l}}). \hspace*{\fill}$\Box$ \end{proof}
\begin{rem} A result similar to Lemma~\ref{lem-(n_s)X_{s,l}} is presented in Shin~\cite{Shin09} under the condition that $\presub{(n)}\bm{Q}$ is block tridiagonal (see Theorem~2.1 therein). \end{rem}
\begin{rem}\label{rem-(n)X_{k,l}} The matrices $\presub{(n)}\bm{X}_{k,\ell}$, $\ell \in \mathbb{Z}_k$, and $\bm{U}_{n,\ell}$, $\ell \in \mathbb{Z}_{n-1}$, have probabilistic interpretations. The $(i,j)$-th element of $\presub{(n)}\bm{X}_{k,\ell}$ represents the expected total sojourn time in state $(\ell,j)$ before the first visit to $\overline{\mathbb{S}}_n$ (i.e., to any state above level $n$) starting from state $(k,i)$ (see, e.g., \cite[Theorem~2.4.3]{Lato99}). Furthermore, the $(i,j)$-th element of $\bm{U}_{n,\ell}$ represents the expected total sojourn time in state $(\ell,j)$ before the first visit to $\overline{\mathbb{S}}_n$ starting from state $(n,i)$, measured per unit of time spent in state $(n,i)$. Thus, we have (see \cite[Equation~(5.33)]{Lato99})
\begin{equation} \bm{\pi}_{\ell} = \bm{\pi}_n \bm{U}_{n,\ell}, \qquad n \in \mathbb{N},\ \ell \in \mathbb{Z}_n. \label{eqn-pi_l-pi_k*U_{k,l}} \end{equation} \end{rem}
We now obtain a matrix-product form of $\presub{(n)}\widehat{\bm{\pi}}_{k}$, $k\in\mathbb{Z}_n$, by substituting (\ref{eqn-(n)X_{n,k}}) into (\ref{eqn-(n)pi_{n,k}}).
\begin{lem}\label{lem-product-form-(s)ol{pi}_s}
\begin{eqnarray} \presub{(n)}\widehat{\bm{\pi}}_{k} &=& { \bm{\alpha}_n \bm{U}_n^* \bm{U}_{n,k} \over \bm{\alpha}_n \sum_{\ell=0}^n \bm{U}_n^* \bm{U}_{n,\ell}\bm{e} }, \qquad n \in \mathbb{Z}_+,\ k \in \mathbb{Z}_n. \label{eqn-(n_s)pi_{n_s,l}} \end{eqnarray} \end{lem}
\begin{rem} Equations (\ref{ineqn-(n)X_{n,l}e>0}) and (\ref{eqn-(n)X_{n,k}}) lead to
\begin{equation} \sum_{\ell=0}^n \bm{U}_n^* \bm{U}_{n,\ell}\bm{e}>\bm{0}, \qquad n \in \mathbb{Z}_+,\ \ell \in \mathbb{Z}_n. \label{ineqn-U_n^*-U_{n,l}e>0} \end{equation} \end{rem}
\subsection{An error bound}\label{subsec-error-bound}
In this subsection, we present an error bound for the LBCL-augmented truncation approximation $\presub{(n)}\widehat{\bm{\pi}}$ to $\bm{\pi}$. The error bound is used to develop an algorithm for computing $\bm{\pi}$ in the next section.
To derive the error bound, we assume a Foster-Lyapunov drift condition.
\begin{cond}\label{cond-01} The generator $\bm{Q}$ is irreducible, and there exist a constant $b \in (0,\infty)$, a finite set $\mathbb{C} \subset \mathbb{S}$, and a positive column vector $\bm{v}:=(v(k,i))_{(k,i)\in\mathbb{S}}$ such that $\inf_{(k,i)\in\mathbb{S}}v(k,i) > 0$ and
\begin{equation} \bm{Q}\bm{v} \le - \bm{e} + b \bm{1}_{\mathbb{C}}, \label{ineqn-QV<=-f+b1_C} \end{equation}
where $\bm{1}_{\mathbb{B}}:=(1_{\mathbb{B}}(k,i))_{(k,i)\in\mathbb{S}}$, $\mathbb{B} \subseteq \mathbb{S}$, denotes a column vector defined by
\[ 1_{\mathbb{B}}(k,i) =\left\{ \begin{array}{l@{~~~}l} 1, & (k,i) \in \mathbb{S}, \\ 0, & (k,i) \in \mathbb{S}\setminus\mathbb{B}. \end{array} \right. \]
\end{cond}
\begin{rem}
Recall that $\bm{Q}$ is the generator of the regular-jump Markov chain $\{(X(t),J(t))\}$ (see Section~\ref{sec-intro}) and thus $\bm{Q}$ is stable, i.e., $|q(\ell,j;\ell,j)| < \infty$ for all $(\ell,j) \in \mathbb{S}$ (see, e.g., \cite[Chapter~8, Definition~2.4 and Theorem~3.4]{Brem99}). The irreducibility of $\bm{Q}$ and the finiteness of $\mathbb{C}$ imply that
\[ \inf_{(k,i)\in\mathbb{C}} p^t(k,i;\ell,j) > 0\quad\mbox{for all $t > 0$ and $(\ell,j)\in \mathbb{S}$}, \]
which shows that $\mathbb{C}$ is a {\it small set} (see, e.g., \cite{Kont16}). Therefore, if Condition~\ref{cond-01} holds, then the irreducible generator $\bm{Q}$ is ergodic (see, e.g., \cite[Theorem~1.1]{Kont16}). \end{rem}
Let $\bm{\Phi}^{(\beta)}:=(\phi^{(\beta)}(k,i;\ell,j)_{(k,i;\ell,j)\in\mathbb{S}}$ denote a stochastic matrix such that
\begin{equation*} \bm{\Phi}^{(\beta)} = \int_0^{\infty} \beta {\rm e}^{- \beta t} \bm{P}^{(t)} {\rm d} t, \qquad \beta > 0,
\end{equation*}
where $\bm{P}^{(t)}:=(p^{(t)}(k,i;\ell,j))_{(k,i;\ell,j)\in\mathbb{S}}$, $t \in \mathbb{R}_+$, is the transition matrix function of the Markov chain $\{(X(t),J(t))\}$ with generator $\bm{Q}$, i.e.,
\[ \mathsf{P}((X(t),J(t)) = (\ell,j) \mid (X(0),J(0)) = (k,i)),\qquad (k,i;\ell,j)\in\mathbb{S}^2. \]
Because $\bm{Q}$ is ergodic, we have $\bm{\Phi}^{(\beta)} > \bm{O}$. We also define $\overline{\phi}_{\mathbb{C}}^{(\beta)}$, $\beta>0$, as
\begin{eqnarray*} \overline{\phi}_{\mathbb{C}}^{(\beta)} &=& \sup_{(\ell,j) \in \mathbb{S}}\min_{(k,i)\in\mathbb{C}} \phi^{(\beta)}(k,i;\ell,j) > 0,\qquad \beta > 0.
\end{eqnarray*}
We then have the following result from \cite[Theorem~2.1]{LLM2018} with $\bm{f}=\bm{g}=\bm{e}$.
\begin{prop}\label{prop-Lem-2.1-LLM2018} Under Condition~\ref{cond-01}, the following holds:
\begin{equation}
\| \presub{(n)}\widehat{\bm{\pi}} - \bm{\pi} \| \le 2 \cdot \presub{(n)}\widehat{\bm{\pi}}\hspace{0.1em}
| \presub{(n)}\widehat{\bm{Q}} - \bm{Q} | \left( \bm{v} + { b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} } \bm{e} \right),\quad n\in\mathbb{Z}_+,~\beta > 0,
\label{bound-|D|g} \end{equation}
where, for any vector $\bm{m}:=(m(i))$, $\|\bm{m}\|$ denotes the total variation norm of $\bm{m}$, i.e., $\|\bm{m}\| = \sum_i |m(i)|$. \end{prop}
From Proposition~\ref{prop-Lem-2.1-LLM2018}, we derive a more informative bound for $\| \presub{(n)}\widehat{\bm{\pi}} - \bm{\pi} \|$. For this purpose, we define
$\bm{U}^*_{n,k}$, $n \in \mathbb{Z}_+$, $k\in\mathbb{Z}_n$, as
\begin{equation} \bm{U}^*_{n,k} = \bm{U}_n^* \bm{U}_{n,k}, \qquad n \in \mathbb{Z}_+,\ k\in\mathbb{Z}_n. \label{defn-U_{n,k}^*} \end{equation}
We also define $\bm{u}^*_n:=(u_n^*(i))_{i\in\mathbb{M}_n}$, $n \in \mathbb{Z}_+$, as
\begin{equation} \bm{u}^*_n = \sum_{\ell=0}^n \bm{U}^*_{n,\ell}\bm{e} = \sum_{\ell=0}^n \bm{U}_n^* \bm{U}_{n,\ell}\bm{e} > \bm{0}, \qquad n \in \mathbb{Z}_+, \label{defn-u_n^*} \end{equation}
where $\bm{u}^*_n > \bm{0}$ due to (\ref{ineqn-U_n^*-U_{n,l}e>0}). Using (\ref{defn-U_{n,k}^*}) and (\ref{defn-u_n^*}), we rewrite (\ref{eqn-(n_s)pi_{n_s,l}}) as
\begin{eqnarray} \presub{(n)}\widehat{\bm{\pi}}_k = { \bm{\alpha}_n \bm{U}^*_{n,k} \over \bm{\alpha}_n \bm{u}^*_n }, \qquad n \in \mathbb{Z}_+,\ k\in\mathbb{Z}_n. \label{eqn-(n)wh{pi}_k} \end{eqnarray}
\begin{thm}\label{thm-bound} If Condition~\ref{cond-01} holds, then
\begin{eqnarray}
\| \presub{(n)}\widehat{\bm{\pi}} - \bm{\pi} \| &\le& E(n), \qquad n \in \mathbb{Z}_+, \label{bound-pi-g} \end{eqnarray}
where $E(\,\cdot\,):=E^{(\beta)}(\,\cdot\,)$, called the {\it error bound function}, is given by
\begin{eqnarray} E(n) &=& {2 \over \bm{\alpha}_n \bm{u}_n^* } \! \left\{ \bm{\alpha}_n \! \left( \bm{v}_n + \sum_{k=0}^n \bm{U}_{n,k}^* \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \bm{v}_{\ell} \right) \! + { 2b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} } \right\},~ n \in \mathbb{Z}_+, \quad~~~~ \label{defn-E(n)} \end{eqnarray}
with $\beta > 0$. \end{thm}
\begin{rem} The error bound function $E$ has a free parameter $\beta>0$ involved in the intractable factor $\overline{\phi}_{\mathbb{C}}^{(\beta)}$. Thus, it is, in general, difficult to discuss theoretically how $\beta$ impacts on the decay speed of $E$. Through numerical experiments, Masuyama~\cite{Masu17-JORSJ} investigates such a problem for the {\it last-column block-augmented truncation}, though the function $E$ is referred to therein as the {\it error decay function}, instead of the error bound function. Note that the last-column block-augmented truncation belongs to the class of {\it block-augmented truncations} (see \cite{LiHai00} for details). Therefore, the last-column block-augmented truncation is indeed different from our LBCL-augmented truncation, though they are fairly similar. \end{rem}
\noindent {\it Proof of Theorem~\ref{thm-bound}~} Suppose that Condition~\ref{cond-01} holds. It then follows from (\ref{eqn-(n)wh{pi}((n)wh{Q}-Q)}) that
\begin{eqnarray} \lefteqn{ \presub{(n)}\widehat{\bm{\pi}}
|\presub{(n)}\widehat{\bm{Q}} - \bm{Q}| \left( \bm{v} + { b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} } \bm{e} \right) } \quad && \nonumber \\ &\le& \left(
\begin{array}{c@{~}|@{~}c} \presub{(n)}\widehat{\bm{\pi}} & \bm{0} \end{array} \right) \left(
\begin{array}{c@{~}|@{~}c} -\presub{(n)}\bm{Q} \bm{e} \presub{(n)}\widehat{\bm{\alpha}} & \presub{(n)}\bm{Q}_{>n} \rule[-2.5mm]{0mm}{2mm}{} \\ \hline * & * \end{array} \right) \left( \bm{v} + { b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} } \bm{e} \right) \nonumber \\ &=& \left(
\begin{array}{c@{~}|@{~}c} \presub{(n)}\widehat{\bm{\pi}} (-\presub{(n)}\bm{Q}\bm{e}) \cdot \presub{(n)}\widehat{\bm{\alpha}} & \presub{(n)}\widehat{\bm{\pi}} \presub{(n)}\bm{Q}_{>n} \end{array} \right) \left( \bm{v} + { b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} } \bm{e} \right), \quad n \in \mathbb{Z}_+. \qquad~ \label{eqn-diff-pi} \end{eqnarray}
Substituting (\ref{eqn-diff-pi}) into (\ref{bound-|D|g}), we have, for $n \in \mathbb{Z}_+$,
\begin{eqnarray}
\| \presub{(n)}\widehat{\bm{\pi}} - \bm{\pi} \| &\le& 2 \left(
\begin{array}{c@{~}|@{~}c} \presub{(n)}\widehat{\bm{\pi}} (-\presub{(n)}\bm{Q}\bm{e}) \cdot \presub{(n)}\widehat{\bm{\alpha}} & \bm{0} \end{array} \right) \left( \bm{v} + { b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} } \bm{e} \right) \nonumber \\ && {} + 2 \left(
\begin{array}{c@{~}|@{~}c} \bm{0} & \presub{(n)}\widehat{\bm{\pi}} \presub{(n)}\bm{Q}_{>n} \end{array} \right) \left( \bm{v} + { b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} } \bm{e} \right) \nonumber \\ &=& 2 \presub{(n)}\widehat{\bm{\pi}} (-\presub{(n)}\bm{Q} \bm{e}) \cdot \left( \bm{\alpha}_n \bm{v}_n + { b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} } \right) \nonumber \\ && {} + 2 \left[ \sum_{k=0}^n \presub{(n)}\widehat{\bm{\pi}}_k \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \left( \bm{v}_{\ell} + { b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} }\bm{e} \right) \right], \qquad~~~ \label{eqn-diff-pi-02} \end{eqnarray}
where the last equality holds due to (\ref{eqn-(n)wh{alpha}v}) and $\presub{(n)}\widehat{\bm{\alpha}}\bm{e} =1$ for $n \in \mathbb{Z}_+$. Because $\bm{Q}\bm{e} = \bm{0}$,
\[ \sum_{k=0}^n \presub{(n)}\widehat{\bm{\pi}}_k \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell}\bm{e} = \sum_{k=0}^n \presub{(n)}\widehat{\bm{\pi}}_k \left(-\sum_{\ell=0}^n \bm{Q}_{k,\ell}\bm{e} \right) = \presub{(n)}\widehat{\bm{\pi}} (-\presub{(n)}\bm{Q}\bm{e}). \]
Incorporating this into (\ref{eqn-diff-pi-02}), we have
\begin{eqnarray}
\| \presub{(n)}\widehat{\bm{\pi}} - \bm{\pi} \| &\le& 2 \presub{(n)}\widehat{\bm{\pi}} (-\presub{(n)}\bm{Q} \bm{e}) \cdot \left( \bm{\alpha}_n \bm{v}_n + { 2b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} } \right) \nonumber \\ && {} + 2 \left[ \sum_{k=0}^n \presub{(n)}\widehat{\bm{\pi}}_k \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \bm{v}_{\ell} \right]. \label{eqn-diff-pi-03} \end{eqnarray}
Note here that (\ref{defn-(n)ol{pi}_n}) and $\presub{(n)}\widehat{\bm{\alpha}}\bm{e}=1$ yield
\[
\presub{(n)}\widehat{\bm{\pi}} (-\presub{(n)}\bm{Q}\bm{e}) = { \presub{(n)}\widehat{\bm{\alpha}}\bm{e} \over \presub{(n)}\widehat{\bm{\alpha}} (-\presub{(n)}\bm{Q})^{-1}\bm{e} } = { 1 \over \presub{(n)}\widehat{\bm{\alpha}} (-\presub{(n)}\bm{Q})^{-1}\bm{e} }. \]
Thus, we can rewrite (\ref{eqn-diff-pi-03}) as
\begin{eqnarray}
\| \presub{(n)}\widehat{\bm{\pi}} - \bm{\pi} \| &\le& { 2 \over \presub{(n)}\widehat{\bm{\alpha}} (-\presub{(n)}\bm{Q})^{-1}\bm{e} } \left( \bm{\alpha}_n \bm{v}_n + { 2b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} } \right) \nonumber \\ && {} + 2 \sum_{k=0}^n \presub{(n)}\widehat{\bm{\pi}}_k \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \bm{v}_{\ell}. \label{eqn-diff-pi-04} \end{eqnarray}
Furthermore, from (\ref{eqn-(n)wh{alpha}(-(n)Q)^{-1}}), (\ref{eqn-(n)X_{n,k}}), and (\ref{defn-u_n^*}), we have
\begin{equation*} \presub{(n)}\widehat{\bm{\alpha}}(-\presub{(n)}\bm{Q})^{-1}\bm{e} = \bm{\alpha}_n \sum_{\ell=0}^n\bm{U}_n^*\bm{U}_{n,\ell}\bm{e} = \bm{\alpha}_n \bm{u}_n^*, \qquad n \in \mathbb{Z}_+. \end{equation*}
Applying this equation and (\ref{eqn-(n)wh{pi}_k}) to (\ref{eqn-diff-pi-04}), we obtain
\begin{eqnarray*}
\| \presub{(n)}\widehat{\bm{\pi}} - \bm{\pi} \| &\le& { 2 \over \bm{\alpha}_n \bm{u}_n^* } \left( \bm{\alpha}_n \bm{v}_n + { 2b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} } \right) \nonumber \\ && {} + 2 \sum_{k=0}^n {\bm{\alpha}_n \bm{U}_{n,k}^* \over \bm{\alpha}_n \bm{u}_n^*} \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \bm{v}_{\ell} \nonumber \\ &=& { 2 \over \bm{\alpha}_n \bm{u}_n^* } \left\{
\bm{\alpha}_n \left( \bm{v}_n + \sum_{k=0}^n \bm{U}_{n,k}^* \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \bm{v}_{\ell} \right) + { 2b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} } \right\}, \end{eqnarray*}
which results in (\ref{bound-pi-g}) together with (\ref{defn-E(n)}). \hspace*{\fill}$\Box$
\subsection{A counterexample to convergence}\label{subsec-example}
In the previous subsection, we have established the error bound for the LBCL-augmented truncation approximation $\presub{(n)}\widehat{\bm{\pi}}$. We note that, even if the truncation parameter $n$ goes to infinity, $\presub{(n)}\widehat{\bm{\pi}}$ does not necessarily converge to $\bm{\pi}$, in general. However, it always holds that $\lim_{n\to\infty}\presub{(n)}\widehat{\bm{\pi}}=\bm{\pi}$ for {\it special} upper block-Hessenberg Markov chains such that the block matrices $\bm{Q}_{k,\ell}$ are scalars. For such a special case, Gibson and Seneta \cite{Gibs87-JAP} prove that any augmented truncation approximation converges to the original stationary distribution as the truncation parameter goes to infinity (see Theorem 2.2 therein). Of course, this is not the case for {\it general} upper block-Hessenberg Markov chains. Indeed, we introduce a counterexample \cite{Kimura-Takine16}.
Fix $\mathbb{M}_n = \{1,2\}$ for all $n \in \mathbb{Z}_+$, and assume that the block matrices $\bm{Q}_{k,\ell}$ satisfy the following:
\begin{align} \bm{Q}_{n,n} &= \left( \begin{array}{cc} \star & \star \\ 0 & \star \end{array} \right), & \bm{Q}_{n,n+1} &= \left( \begin{array}{cc} \star & 0 \\ \star & \star \end{array} \right), & n &\in\mathbb{Z}_+, \label{special-Q_{n,n}} \\ \bm{Q}_{2k-1,2k-2} &= \left( \begin{array}{cc} \star & 0 \\ 0 & \star \end{array} \right), & \bm{Q}_{2k,2k-1} &= \left( \begin{array}{cc} \star & 0 \\ 0 & 0 \end{array} \right), & k &\in \mathbb{N}. \label{special-Q_{2k-1,2k-2}} \end{align}
where the symbol ``\, $\star$\, " denotes some nonzero element. In this case, $\bm{Q}$ is irreducible (see Figure~\ref{fig-transition}), but $\mathbb{S}_{2k-1}$ is not reachable from state $(2k,2)$ avoiding $\overline{\mathbb{S}}_{2k}$.
\begin{figure*}
\caption{Transition diagram}
\label{fig-transition}
\end{figure*}
Thus, the probabilistic interpretations (see Remark~\ref{rem-(n)X_{k,l}}) of the matrices $\bm{U}_{2k}^*=\presub{(2k)}\bm{X}_{2k,2k}$ and $\bm{U}_{2k,\ell}$, $\ell\in\mathbb{Z}_{2k-1}$, implies that
\begin{align} &&&& \bm{U}_{2k}^* &= \left( \begin{array}{cc} \star & \star \\ 0 & \star \end{array} \right),& k &\in \mathbb{N},&&&& \label{eqn-U_{2k}^*} \\ &&&& \bm{U}_{2k,\ell} &= \left( \begin{array}{cc} \star & \star \\ 0 & 0 \end{array} \right), & k &\in \mathbb{N},~\ell\in\mathbb{Z}_{2k-1}.&&&& \label{eqn-U_{2k,l}} \end{align}
We now assume that $\bm{Q}$ is ergodic. We then set
\begin{equation} \bm{\alpha}_n = (0,1),\qquad n \in \mathbb{Z}_+, \label{special-alpha_n} \end{equation}
which implies that $\presub{(n)}\widehat{\bm{\pi}}$ is the last-column-augmented truncation approximation to the stationary distribution vector $\bm{\pi} > \bm{0}$ of $\bm{Q}$. Applying (\ref{eqn-U_{2k}^*}), (\ref{eqn-U_{2k,l}}), and (\ref{special-alpha_n}) to (\ref{eqn-(n_s)pi_{n_s,l}}) yields
\[ \presub{(2k)}\widehat{\bm{\pi}} = (0,\dots,0,1),\qquad k \in \mathbb{N}, \]
which shows that $\{\presub{(n)}\widehat{\bm{\pi}};n\in\mathbb{Z}_+\}$ does not converge to $\bm{\pi}$ in the present setting.
The example presented here implies that, in some cases, the convergence of $\{\presub{(n)}\widehat{\bm{\pi}}\}$ to $\bm{\pi}$ can require an adaptive choice of
the augmentation distribution vector $\presub{(n)}\bm{\alpha}$, depending on $n$. We discuss this problem in the next section.
\section{Main results}\label{sec-algorithm}
This section is divided into three subsections. In Section~\ref{subsec-LFP}, we formulate linear fractional programming (LFP) problems for finding augmentation distribution vectors such that the error bound function $E$ converges to zero, i.e., $\lim_{n\to\infty}E(n) = 0$. In Section~\ref{subsec-MIP}, using the optimal solutions of these LFP problems, we construct an MIP form of $\bm{\pi}$. In Section~\ref{subsec-computation}, we present a sequential update algorithm for computing the MIP form.
In this section, we assume that Condition~\ref{cond-01} holds, as in Section~\ref{subsec-error-bound}. We also assume that $n$ takes an arbitrary value in $\mathbb{Z}_+$, unless otherwise stated.
\subsection{LFP problems for an MIP form of the stationary distribution vector}\label{subsec-LFP}
Consider the following LFP problem for each $n \in \mathbb{Z}_+$:
\begin{subequations}\label{prob-01} \begin{align} &&&& &\mbox{Minimize} & & r_n(\bm{\alpha}_n):= {\bm{\alpha}_n\bm{y}_n \over \bm{\alpha}_n\bm{u}_n^*}; \label{defn-r_n} &&&& \\ &&&& &\mbox{Subject to} & & \bm{\alpha}_n \ge \bm{0}, \label{const-x>=0} &&&& \\ &&&& & & & \bm{\alpha}_n\bm{e} = 1, &&&& \label{const-xe=1} \end{align}
where $\bm{y}_n:=(y_n(i))_{i\in\mathbb{M}_n}$ denotes
\begin{equation} \bm{y}_n = \bm{v}_n + \sum_{k=0}^n \bm{U}_{n,k}^* \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \bm{v}_{\ell} > \bm{0}. \label{defn-y_n} \end{equation}
\end{subequations}
It follows from (\ref{defn-E(n)}), (\ref{defn-r_n}), and (\ref{defn-y_n}) that
\begin{equation} E(n) = 2\left( r_n(\bm{\alpha}_n) + { 1 \over \bm{\alpha}_n \bm{u}_n^{\ast} } { 2b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} } \right), \qquad n \in \mathbb{Z}_+. \label{bound-(n)pi-pi} \end{equation}
Furthermore, let $\bm{\alpha}_n^*:=(\alpha_n^*(j))_{j\in\mathbb{M}_n}$ denote a probability vector such that
\begin{equation} \alpha_n^*(j) = \left\{ \begin{array}{l@{~~~}l} 1, & j=j_n^*, \\ 0, & j \neq j_n^*, \end{array} \right. \label{defn-psi_n^*} \end{equation}
where
\begin{equation} j_n^* \in \argmin_{j\in\mathbb{M}_n} {y_n(j) \over u_n^*(j)}. \label{defn-j_n^*} \end{equation}
We then have the following theorem.
\begin{thm}\label{thm-optima-solution-LFP} For each $n \in \mathbb{Z}_+$, the probability vector $\bm{\alpha}_n^*$ is an optimal solution of the LFP problem~(\ref{prob-01}). \end{thm}
\begin{proof} From (\ref{defn-psi_n^*}) and (\ref{defn-j_n^*}), we have
\[ \xi_n := {\bm{\alpha}_n^* \bm{y}_n \over \bm{\alpha}_n^* \bm{u}_n^*} = {y_n(j_n^*) \over u_n^*(j_n^*)} = \min_{j\in\mathbb{M}_n}{y_n(j) \over u_n^*(j)} > 0, \]
which leads to $\bm{y}_n \ge \xi_n \bm{u}_n^* > \bm{0}$. Thus, for any $1 \times M_n$ probability vector $\bm{p}_n$, we obtain
\[ {\bm{p}_n\bm{y}_n \over \bm{p}_n\bm{u}_n^*} \ge \xi_n = {\bm{\alpha}_n^* \bm{y}_n \over \bm{\alpha}_n^* \bm{u}_n^*}. \]
Therefore, $\bm{\alpha}_n^*$ is an optimal solution of the LFP problem~(\ref{prob-01}). \hspace*{\fill}$\Box$ \end{proof}
\subsection{An MIP form of the stationary distribution vector}\label{subsec-MIP}
Let $\presub{(n)}\widehat{\bm{\pi}}^* := ( \presub{(n)}\widehat{\bm{\pi}}_{0}^*, \presub{(n)}\widehat{\bm{\pi}}_{1}^*, \dots, \presub{(n)}\widehat{\bm{\pi}}_{n}^*)$ denote a probability vector such that
\begin{equation} \presub{(n)}\widehat{\bm{\pi}}_{k}^* = { \bm{\alpha}_n^*\bm{U}_{n,k}^* \over \bm{\alpha}_n^* \bm{u}_n^* } = { {\rm row}\{\bm{U}_{n,k}^* \}_{j_n^*} \over u_n^*(j_n^*) },\qquad k \in \mathbb{Z}_n, \label{defn-(n)wh{pi}_k^*} \end{equation}
where ${\rm row}\{\,\cdot\,\}_j$ denotes the $j$-th row of the matrix in the brackets. Note here that $\presub{(n)}\widehat{\bm{\pi}}_{k}^*$ is equal to $\presub{(n)}\widehat{\bm{\pi}}_{k}$ in (\ref{eqn-(n)wh{pi}_k}) with $\bm{\alpha}_n=\bm{\alpha}_n^*$. Therefore, it follows from Theorem~\ref{thm-bound} that
\begin{equation}
\| \presub{(n)}\widehat{\bm{\pi}}^* - \bm{\pi} \| \le E^*(n),\qquad n \in \mathbb{Z}_+, \label{bound-(n)wh{pi}^*-02} \end{equation}
where function $E^*$ is equal to $E$ given in (\ref{bound-(n)pi-pi}) with $\bm{\alpha}_n=\bm{\alpha}_n^*$; that is,
\begin{eqnarray} E^*(n) &=& 2\left( r_n( \bm{\alpha}_n^*) + { 1 \over \bm{\alpha}_n^* \bm{u}_n^{\ast} } { 2b \over \beta\overline{\phi}_{\mathbb{C}}^{(\beta)} } \right), \qquad n \in \mathbb{Z}_+. \label{defn-E^*(n)} \end{eqnarray}
To proceed further, we assume the following.
\begin{cond}\label{cond-02}
\begin{eqnarray} \sum_{n=0}^{\infty} \bm{\pi}_n \bm{\Delta}_n \bm{v}_n < \infty, \label{lim-pi_n-Delta_n-v_n} \end{eqnarray}
where $\bm{\Delta}_n:=(\Delta_n(i,j))_{i,j\in\mathbb{M}_n}$ denotes an $M_n \times M_n$ diagonal matrix such that
\begin{equation}
\Delta_n(i,i) = |q(n,i;n,i)|,\qquad i \in \mathbb{M}_n. \label{defn-Delta_n} \end{equation}
\end{cond}
\begin{lem}\label{lem-r^{(1)}} Suppose that Conditions~\ref{cond-01} and \ref{cond-02} hold. We then have
\begin{equation} \lim_{n\to\infty} r_n(\bm{\alpha}_n^*) = \lim_{n\to\infty} {\bm{\alpha}_n^*\bm{y}_n \over \bm{\alpha}_n^*\bm{u}_n^*} = 0. \label{lim-r_n(psi_n^*)} \end{equation} \end{lem}
\begin{rem}\label{rem-Q-exp-bounded} If $\bm{Q}$ is bounded, i.e., $\sup_{(n,i)\in\mathbb{S}}\Delta_n(i,i) < \infty$, then Condition~\ref{cond-02} is reduced to
\begin{equation*} \bm{\pi} \bm{v} = \sum_{n=0}^{\infty} \bm{\pi}_n \bm{v}_n < \infty.
\end{equation*}
\end{rem}
\noindent {\it Proof of Lemma~\ref{lem-r^{(1)}}~} To prove this lemma, we require the following proposition (which is proved in Appendix~\ref{appen-ineqn-pi-(U_n^*)^{-1}}).
\begin{prop}\label{lem-pi_n-Lambda_n} Under Condition~\ref{cond-01},
\begin{equation} \bm{\pi}_n \bm{\Delta}_n \ge \bm{\pi}_n (\bm{U}_n^*)^{-1} \ge \bm{0}, \neq \bm{0} \quad\mbox{for all $n \in \mathbb{Z}_+$}. \label{ineqn-pi-(U_n^*)^{-1}} \end{equation} \end{prop}
Let $\widetilde{\bm{\alpha}}_n$ denote
\begin{equation} \widetilde{\bm{\alpha}}_n = { \bm{\pi}_n (\bm{U}_n^*)^{-1} \over \bm{\pi}_n (\bm{U}_n^*)^{-1} \bm{e}}\ge \bm{0}, \neq \bm{0}, \label{fix-x-alpha_n} \end{equation}
which is well-defined due to Proposition~\ref{lem-pi_n-Lambda_n}. Note that $\widetilde{\bm{\alpha}}_n$ is a feasible solution of the LFP problem~(\ref{prob-01}). Thus, by the optimality of $\bm{\alpha}_n^*$, we have
\begin{equation*} r_n(\bm{\alpha}_n^*) \le r_n(\widetilde{\bm{\alpha}}_n)\quad \mbox{for all $n \in \mathbb{Z}_+$}.
\end{equation*}
It follows from (\ref{defn-r_n}) and (\ref{fix-x-alpha_n}) that
\begin{eqnarray} r_n(\widetilde{\bm{\alpha}}_n) &=& { \bm{\pi}_n (\bm{U}_n^*)^{-1} \bm{y}_n \over \bm{\pi}_n (\bm{U}_n^*)^{-1} \bm{u}_n^*} \nonumber\\ &=& {1 \over \bm{\pi}_n (\bm{U}_n^*)^{-1} \bm{u}_n^*} \nonumber \\ && \times \left( \bm{\pi}_n (\bm{U}_n^*)^{-1}\bm{v}_n + \bm{\pi}_n (\bm{U}_n^*)^{-1} \sum_{k=0}^n \bm{U}_{n,k}^* \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \bm{v}_{\ell} \right),\qquad \label{eqn-180326-01} \end{eqnarray}
where the second equality holds due to (\ref{defn-y_n}). It also follows from (\ref{eqn-pi_l-pi_k*U_{k,l}}), (\ref{defn-U_{n,k}^*}), and (\ref{defn-u_n^*}) that
\begin{eqnarray*} \bm{\pi}_n (\bm{U}_n^*)^{-1} \bm{U}_{n,k}^* &=& \bm{\pi}_n \bm{U}_{n,k} = \bm{\pi}_k,\qquad k \in \mathbb{Z}_n, \\ \bm{\pi}_n (\bm{U}_n^*)^{-1} \bm{u}_n^* &=& \sum_{\ell=0}^n \left\{ \bm{\pi}_n (\bm{U}_n^*)^{-1} \bm{U}_{n,\ell}^* \right\}\bm{e} = \sum_{\ell=0}^n \bm{\pi}_{\ell}\bm{e}. \end{eqnarray*}
Substituting these equations into (\ref{eqn-180326-01}), and using (\ref{ineqn-pi-(U_n^*)^{-1}}), we obtain
\begin{eqnarray} r_n(\widetilde{\bm{\alpha}}_n) &=& { 1 \over \sum_{\ell=0}^n \bm{\pi}_{\ell} \bm{e} } \left( \bm{\pi}_n (\bm{U}_n^*)^{-1}\bm{v}_n + \sum_{k=0}^n \bm{\pi}_k \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \bm{v}_{\ell} \right) \nonumber \\ &\le& { 1 \over \sum_{\ell=0}^n \bm{\pi}_{\ell} \bm{e} } \left( \bm{\pi}_n \bm{\Delta}_n \bm{v}_n + \sum_{k=0}^n \bm{\pi}_k \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \bm{v}_{\ell} \right). \label{ineqn-r_n(psi_n^*)} \end{eqnarray}
Consequently, the proof of (\ref{lim-r_n(psi_n^*)}) is completed by showing that the right-hand side of (\ref{ineqn-r_n(psi_n^*)}) converges to zero as $n\to\infty$.
It follows from (\ref{ineqn-QV<=-f+b1_C}) that, for all $n \in \mathbb{Z}_+$ and $k \in \mathbb{Z}_n$,
\begin{eqnarray*} \bm{0} \le \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \bm{v}_{\ell} &\le& - \sum_{\ell=0}^n \bm{Q}_{k,\ell} \bm{v}_{\ell} - \bm{e} + b\bm{e} \nonumber \\ &\le& - \bm{Q}_{k,k} \bm{v}_k + b\bm{e} \le \bm{\Delta}_k \bm{v}_k + b\bm{e}, \end{eqnarray*}
and thus
\begin{equation*} \sum_{k=0}^n \bm{\pi}_k \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \bm{v}_{\ell} \le \sum_{k=0}^{\infty} \bm{\pi}_k\bm{\Delta}_k \bm{v}_k + b< \infty \quad \mbox{for all $n \in \mathbb{Z}_+$}, \end{equation*}
where the last inequality holds due to (\ref{lim-pi_n-Delta_n-v_n}). Therefore, by the dominated convergence theorem,
\begin{eqnarray} \lim_{n\to\infty} \sum_{k=0}^n \bm{\pi}_k \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \bm{v}_{\ell} = \sum_{k=0}^{\infty} \bm{\pi}_k \lim_{n\to\infty} \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \bm{v}_{\ell} = \bm{0}. \label{eqn-180318-02} \end{eqnarray}
It also follows from (\ref{lim-pi_n-Delta_n-v_n}) that
\begin{equation} \lim_{n\to\infty} \bm{\pi}_n \bm{\Delta}_n \bm{v}_n = 0. \label{eqn-180318-03} \end{equation}
Combining (\ref{eqn-180318-02}), (\ref{eqn-180318-03}), and $\sum_{\ell=0}^{\infty} \bm{\pi}_{\ell} \bm{e}=1$, we obtain
\[ \lim_{n\to\infty} {1 \over \sum_{\ell=0}^n \bm{\pi}_{\ell} \bm{e} } \left( \bm{\pi}_n \bm{\Delta}_n \bm{v}_n + \sum_{k=0}^n \bm{\pi}_k \sum_{\ell=n+1}^{\infty} \bm{Q}_{k,\ell} \bm{v}_{\ell} \right) =0, \]
which completes the proof. \hspace*{\fill}$\Box$
The following theorem is a consequence of Lemma~\ref{lem-r^{(1)}} together with (\ref{bound-(n)wh{pi}^*-02}) and (\ref{defn-E^*(n)}).
\begin{thm}\label{thm-E^*-01} Suppose that Conditions~\ref{cond-01} and \ref{cond-02} hold. We then have
\begin{equation} \lim_{n\to\infty} E^*(n) = 0, \label{convergence-(n)ol{pi}_n^*} \end{equation}
and thus (\ref{bound-(n)wh{pi}^*-02}) yields
\begin{eqnarray}
\lim_{n\to\infty} \| \presub{(n)}\widehat{\bm{\pi}}^* - \bm{\pi} \| &=& 0. \label{convergence-(n)wh{pi}^*} \end{eqnarray} \end{thm}
\begin{proof} We prove only (\ref{convergence-(n)ol{pi}_n^*}). It follows from (\ref{defn-y_n}), (\ref{defn-psi_n^*}), and $\inf_{(k,i)\in\mathbb{S}}v(k,i) > 0$ (see Condition~\ref{cond-01}) that
\[ \bm{\alpha}_n^* \bm{y}_n \ge \bm{\alpha}_n^* \bm{v}_n = v(n,j_n^*) > 0, \qquad n \in \mathbb{Z}_+. \]
Therefore, (\ref{lim-r_n(psi_n^*)}) implies that
\[ \lim_{n\to\infty} \bm{\alpha}_n^* \bm{u}_n^* = \infty,
\]
which yields
\begin{equation} \lim_{n\to\infty} {1 \over \bm{\alpha}_n^* \bm{u}_n^* } {2b \over \beta \overline{\phi}_{\mathbb{C}}^{(\beta)} } = 0. \label{eqn-180318-04} \end{equation}
Applying (\ref{eqn-180318-04}) and Lemma~\ref{lem-r^{(1)}} to (\ref{defn-E^*(n)}) results in (\ref{convergence-(n)ol{pi}_n^*}). \hspace*{\fill}$\Box$ \end{proof}
Theorem~\ref{thm-E^*-01} yields a {\it matrix-infinite-product (MIP) form} of $\bm{\pi}=(\bm{\pi}_0,\bm{\pi}_1,\dots)$ under Conditions~\ref{cond-01} and \ref{cond-02}. This is summarized in the following corollary.
\begin{coro}\label{coro-MIP-form-solution} If Conditions~\ref{cond-01} and \ref{cond-02} hold, then
\begin{equation} \bm{\pi}_k = \lim_{n\to\infty} { \bm{\alpha}_n^* \bm{U}_{n,k}^* \over \bm{\alpha}_n^* \bm{u}_n^* }, \qquad k \in \mathbb{Z}_+, \label{limit-form-pi_k} \end{equation}
or equivalently,
\begin{equation} \bm{\pi}_k = \lim_{n\to\infty} { \bm{\alpha}_n^* \bm{U}_n^* \bm{U}_{n-1}\bm{U}_{n-2} \cdots \bm{U}_k \over \bm{\alpha}_n^* \sum_{\ell=0}^n \bm{U}_n^* \bm{U}_{n-1}\bm{U}_{n-2} \cdots \bm{U}_{\ell}\bm{e} }, \qquad k \in \mathbb{Z}_+, \label{MIP-form-solution} \end{equation}
where
\begin{equation} \bm{U}_k = \bm{Q}_{k+1,k}\bm{U}_k^*,\qquad k \in \mathbb{Z}_+. \label{defn-U_k} \end{equation}
\end{coro}
\begin{proof} Suppose that Conditions~\ref{cond-01} and \ref{cond-02} hold. It then follows from (\ref{defn-(n)wh{pi}_k^*}) and (\ref{convergence-(n)wh{pi}^*}) that
\[ \bm{\pi}_k = \lim_{n\to\infty} \presub{(n)}\widehat{\bm{\pi}}_k^* = \lim_{n\to\infty} {\bm{\alpha}_n^*\bm{U}_{n,k}^* \over \bm{\alpha}_n^*\bm{u}_n^*},\qquad k\in\mathbb{Z}_+, \]
which shows that (\ref{limit-form-pi_k}) holds. Furthermore, combining (\ref{defn-U_{n,k}^*}) with (\ref{defn-U_{k,l}}) and (\ref{defn-U_k}) yields, for $n \in \mathbb{Z}_+$,
\[ \bm{U}_{n,k}^* = \left\{ \begin{array}{ll} \bm{U}_n^*\bm{U}_{n-1}\bm{U}_{n-2} \cdots \bm{U}_k, & k\in\mathbb{Z}_{n-1}, \\ \bm{I},& k=n. \end{array} \right. \]
Using this and (\ref{defn-u_n^*}), we can rewrite (\ref{limit-form-pi_k}) as (\ref{MIP-form-solution}).\hspace*{\fill}$\Box$ \end{proof}
\begin{rem} Theorem~\ref{thm-E^*-01} ensures that the convergence in (\ref{limit-form-pi_k}) and (\ref{MIP-form-solution}) is uniform for $k \in \mathbb{Z}_+$. \end{rem}
\begin{rem} Another MIP form of $\bm{\pi}_k$ is presented in the preprint \cite{Masu16-arXiv:1603}, under some technical conditions different from Conditions~\ref{cond-01} and \ref{cond-02}. \end{rem}
\subsection{A sequential update algorithm for the MIP form}\label{subsec-computation}
In this subsection, we propose an algorithm for computing $\bm{\pi}$, based on Theorem~\ref{thm-E^*-01} and Corollary~\ref{coro-MIP-form-solution}. Our algorithm sequentially updates the LBCL-augmented truncation approximation so that it converges to the MIP form (\ref{limit-form-pi_k}) of $\bm{\pi}$.
To efficiently achieve this update procedure, we derive recursive formulas. Combining (\ref{defn-U_{n,k}^*}) with (\ref{defn-U_k^*}) and (\ref{defn-U_{k,l}}), we have
\begin{subequations}\label{recursion-U_{n,k}^*} \begin{eqnarray} \bm{U}_{0,0}^* &=& \bm{U}_0^* = (-\bm{Q}_{0,0})^{-1}, \label{eqn-U_{0,0}^*} \\ \bm{U}_{n,k}^* &=& \left\{ \begin{array}{l@{~~~}ll} \bm{U}_n^* \bm{Q}_{n,n-1} \cdot \bm{U}^*_{n-1,k}, & n \in \mathbb{N},\ & k \in \mathbb{Z}_{n-1}, \\ \bm{U}_n^*, & n \in \mathbb{N},\ & k = n. \end{array} \right. \label{eqn-U_{n,k}^*} \end{eqnarray} \end{subequations}
Using (\ref{defn-u_n^*}), (\ref{eqn-U_{0,0}^*}), and (\ref{eqn-U_{n,k}^*}), we also obtain
\begin{subequations}\label{recursion-u_n^*} \begin{eqnarray} \bm{u}^*_0 &=& \bm{U}_0^* \bm{e} = (-\bm{Q}_{0,0})^{-1}\bm{e}, \label{eqn-u_0^*} \\ \bm{u}^*_n &=& \bm{U}_n^* \left( \bm{e} + \bm{Q}_{n,n-1} \bm{u}^*_{n-1} \right), \qquad n \in \mathbb{N}. \label{eqn-u_n^*} \end{eqnarray} \end{subequations}
Furthermore, (\ref{defn-U_{k,l}}) and (\ref{defn-U_{n,k}^*}) yield
\[ \bm{U}_{n,\ell} = \bm{Q}_{n,n-1}\bm{U}_{n-1}^*\bm{U}_{n-1,\ell} = \bm{Q}_{n,n-1}\bm{U}_{n-1,\ell}^*,\qquad \ell \in \mathbb{Z}_{n-1}. \]
Substituting this into (\ref{defn-U_k^*}) leads to
\begin{eqnarray} \bm{U}_n^* &=& \left( - \bm{Q}_{n,n} - \bm{Q}_{n,n-1} \sum_{\ell=0}^{n-1} \bm{U}_{n-1,\ell}^*\bm{Q}_{\ell,n} \right)^{-1}, \qquad n \in \mathbb{N}. \label{eqn-U_n^*} \end{eqnarray}
Our algorithm is composed of the equations (\ref{recursion-U_{n,k}^*})--(\ref{eqn-U_n^*}), Theorem~\ref{thm-E^*-01}, and Corollary~\ref{coro-MIP-form-solution}.
\begin{algorithm}[H] \renewcommand{\thealgorithm}{}
\renewcommand{\arabic{enumi}.}{\arabic{enumi}.} \renewcommand{(\alph{enumii})}{(\alph{enumii})} \renewcommand{\roman{enumiii}.}{\roman{enumiii}.}
\caption{Computing the MIP form of $\bm{\pi}$}
{\bf Input}: $\bm{Q}$, $\varepsilon \in (0,1)$, and increasing sequence $\{n_{\ell};\ell\in \mathbb{Z}_+\}$ of positive integers. \\ {\bf Output}: $\presub{(n)}\widehat{\bm{\pi}}^* = (\presub{(n)}\widehat{\bm{\pi}}_{0}^*,\presub{(n)}\widehat{\bm{\pi}}_{1}^*,\dots,\presub{(n)}\widehat{\bm{\pi}}_{n}^*)$, where $n \in \mathbb{Z}_+$ is fixed when the iteration stops.
\begin{enumerate} \setlength{\parskip}{0cm} \setlength{\itemsep}{0cm}
\item Find $\bm{v}>\bm{0}$, $b>0$, and $\mathbb{C} \in \mathbb{S}$ such that Conditions~\ref{cond-01} and \ref{cond-02} hold.
\item Set $n = 0$ and $\ell=1$.
\item Compute $\bm{U}_0^*$ by (\ref{eqn-U_{0,0}^*}) and $\bm{u}_0^*$ by (\ref{eqn-u_0^*}).
\item Iterate (a)--(d) below:
\begin{enumerate} \setlength{\parskip}{0cm} \setlength{\itemsep}{0cm}
\item Increment $n$ by one.
\item Compute $\bm{U}_n^*=\bm{U}_{n,n}^*$ by (\ref{eqn-U_n^*}).
\item Compute $\bm{U}_{n,k}^*$, $k=0,1,\dots,n-1$, by
(\ref{eqn-U_{n,k}^*}) and $\bm{u}^*_n$ by
(\ref{eqn-u_n^*}).
\item If $n=n_{\ell}$, then perform the following:
\begin{enumerate}
\item Compute $\bm{y}_n$ by (\ref{defn-y_n}), and find $j_n^*$ satisfying (\ref{defn-j_n^*}).
\item Compute $\presub{(n)}\widehat{\bm{\pi}}_{k}^*$, $k=0,1,\dots,n$, by (\ref{defn-(n)wh{pi}_k^*}).
\item If $\| \presub{(n_{\ell})}\widehat{\bm{\pi}}^* -
\presub{(n_{\ell-1})}\widehat{\bm{\pi}}^*\| < \varepsilon$, then stop the iteration; otherwise increment $\ell$ by one and return to step~(a). \end{enumerate} \end{enumerate}
\end{enumerate}
\end{algorithm}
\begin{rem} Equation (\ref{convergence-(n)wh{pi}^*}) leads to
\begin{eqnarray*} \lim_{n\to\infty}
\| \presub{(n)}\widehat{\bm{\pi}}^*
- \presub{(n+m)}\widehat{\bm{\pi}}^*\| &=& 0 \quad \mbox{for any fixed $m \in \mathbb{N}$}.
\end{eqnarray*}
Therefore, our algorithm iterates Step~4 only a finite number of times. \end{rem}
\begin{rem}\label{rem-Taki16-00} Step (4.b) computes $\bm{U}_n^*$ by (\ref{eqn-U_n^*}). The $(i,j)$-th element of $\bm{U}_n^*$ is the expected total sojourn time in state $(n,j)$ before the first visit to $\overline{\mathbb{S}}_n$ starting from state $(n,i)$. Thus, $\bm{T}_n^* = (-\bm{U}_n^*)^{-1}$, defined in (\ref{defn-T_n^*}), is a non-conservative $Q$-matrix that governs the transient transitions of an absorbing Markov chain obtained by observing $\{(X(t),J(t))\}$ when it is in $\mathbb{L}_n$ during the first passage time to $\overline{\mathbb{S}}_n$ starting from $\mathbb{L}_n$. This consideration indicates $\bm{U}_n^* = (-\bm{T}_n^*)^{-1}$ can be efficiently computed (see \cite[Proposition 1]{Le-Boud91}), provided that $\bm{T}_n^*$ is given. \end{rem}
\begin{rem} Generally, our algorithm computes the infinite sum $\sum_{\ell=n+1}^{\infty}\bm{Q}_{k,\ell}\bm{v}_{\ell}$ to obtain $\bm{y}_n$ in (\ref{defn-y_n}). However,
this infinite sum can be calculated in many practical cases associated with queueing models (as implied by the examples in the next section). Moreover, if $\bm{Q}$ is an LD-QBD generator, or equivalently, $\bm{Q}_{k,\ell} = \bm{O}$ for $k \in \mathbb{Z}_+$ and $|\ell - k| > 1$, then $\bm{y}_n$ is expressed without any infinite sum:
\[ \bm{y}_n = \bm{v}_n + \bm{U}_{n,n}^*\bm{Q}_{n,n+1}\bm{v}_{n+1}, \qquad n \in \mathbb{Z}_+. \]
Furthermore, a noteworthy fact is that computing the infinite sum $\sum_{\ell=n+1}^{\infty}\bm{Q}_{k,\ell}\bm{v}_{\ell}$ is not always necessary even if $\bm{Q}$ is not an LD-QBD generator. To demonstrate this, suppose that we have an explicit expression for $\bm{w}_{k,n}$, $k,n\in\mathbb{Z}_+$, such that
\[ \lim_{n\to\infty} \sum_{k=0}^n \bm{\pi}_k\bm{w}_{k,n} = 0, \quad \sum_{\ell=n+1}^{\infty}\bm{Q}_{k,\ell}\bm{v}_{\ell} \le \bm{w}_{k,n}, \quad k,n\in\mathbb{Z}_+. \]
It then follows from (\ref{ineqn-r_n(psi_n^*)}) and (\ref{eqn-180318-03}) that
\[ r_n(\widetilde{\bm{\alpha}}_n) \le { 1 \over \sum_{\ell=0}^n \bm{\pi}_{\ell}\bm{e} } \left( \bm{\pi}_n\bm{\Delta}_n\bm{v}_n + \sum_{k=0}^n\bm{\pi}_k \bm{w}_{k,n} \right) \to 0\quad \mbox{as $n \to \infty$}. \]
Thus, we modify Step (4.d.i) as follows: Compute
\[ \breve{\bm{y}}_n := (\breve{y}_n(j))_{j\in\mathbb{M}_n} = \bm{v}_n + \sum_{k=0}^n \bm{U}_{n,k}^*\bm{w}_{k,n}, \]
and find
\[ j_n^* \in \argmin_{j\in\mathbb{M}_n} {\breve{y}_n(j) \over u_n^*(j) }. \]
Despite this modification, our update algorithm works well. \end{rem}
\section{Applicability of the proposed algorithm}\label{sec-discussion}
This section demonstrates the applicability of our algorithm. To this end, we consider a BMAP/M/$\infty$ queue and M/M/$s$ retrial queue, respectively, in Sections~\ref{subsec-BMAP-M-Infty} and \ref{subsec-MMss-retrial}. For each model, we present a sufficient condition for Conditions~\ref{cond-01} and \ref{cond-02}, under which our update algorithm works well.
\subsection{BMAP/M/$\infty$ queue}\label{subsec-BMAP-M-Infty}
This subsection considers a BMAP/M/$\infty$ queue. The system has an infinite number of servers. Customers arrive at the system according to a batch Markovian arrival process (BMAP) (see, e.g., \cite{Luca91}). Arriving customers are immediately served, and their service times are independent and identically distributed (i.i.d.) with an exponential distribution having mean $\mu^{-1}$.
Let $\{N(t);t \in \mathbb{R}_+\}$ denote the counting process of arrivals from the BMAP; that is, $N(t)$ is equal to the total number of arrivals during the time interval $[0,t]$, where $N(0) = 0$. Let $\{J(t);t \in \mathbb{R}_+\}$ denote the background Markov chain of the BMAP, which is defined on state space $\mathbb{M}=\{1,2,\dots,M\} \subset \mathbb{N}$. We assume that the bivariate stochastic process $\{(N(t),J(t));t\in\mathbb{R}_+\}$ is a continuous-time Markov chain which follows the transition law given by
\begin{eqnarray*} \lefteqn{ \mathsf{P}(N(t+\Delta t) = k,J(t+\Delta t) = j \mid J(t)=i) } \quad && \nonumber \\ &=& \left\{ \begin{array}{l@{~~~}ll} 1 + D_{0,i,i}\Delta t + o(\Delta t), & k=0, & i = j \in \mathbb{M}, \\ D_{0,i,j}\Delta t + o(\Delta t), & k=0, & i,j \in \mathbb{M},~i \neq j, \\ D_{k,i,j}\Delta t + o(\Delta t), & k \in \mathbb{N}, & i,j \in \mathbb{M}, \\ 0, & \mbox{otherwise}, \end{array} \right. \end{eqnarray*}
where $a(t) = o(b(t))$ represents $\lim_{t\to0}a(t)/b(t) = 0$. Thus, the BMAP is characterized by $\{\bm{D}_n;n\in\mathbb{Z}_+\}$, where $\bm{D}_n=(D_{n,i,j})_{i,j\in\mathbb{M}}$ for $n \in \mathbb{Z}_+$. Moreover, $\bm{D}:=\sum_{n\in\mathbb{Z}_+}\bm{D}_n$
is the generator of the background Markov chain $\{J(t);t\in\mathbb{R}_+\}$. As usual, we assume that $\bm{D}$ is irreducible and
\[ \bm{D}\bm{e} \ge \bm{0},\neq \bm{0}. \]
Let $X(t)$, $t \in \mathbb{R}_+$, denote the number of customers in the system at time $t$. It then follows that $\{(X(t),J(t));t\in\mathbb{R}_+\}$ is a continuous-time Markov chain on state space $\mathbb{S}:=\mathbb{Z}_+ \times \mathbb{M}$ with generator $\bm{Q}$ given by
\begin{equation} \bm{Q} = \bordermatrix{
& \mathbb{L}_0 & \mathbb{L}_1 & \mathbb{L}_2 & \mathbb{L}_3 & \cdots \cr \mathbb{L}_0 & \bm{D}_0 & \bm{D}_1 & \bm{D}_2 & \bm{D}_3 & \cdots \cr \mathbb{L}_1 & \mu\bm{I} & \bm{D}_0-\mu\bm{I} & \bm{D}_1 & \bm{D}_2 & \cdots \cr \mathbb{L}_2 & \bm{O} & 2\mu\bm{I} & \bm{D}_0-2\mu\bm{I} & \bm{D}_1 & \cdots \cr \mathbb{L}_3 & \bm{O} & \bm{O} & 3\mu\bm{I} & \bm{D}_0-3\mu\bm{I} & \cdots \cr ~\vdots & \vdots & \vdots & \vdots & \vdots & \ddots }, \label{defn-Q-BMAP-G-infty} \end{equation}
where $\mathbb{L}_k = \{k\} \times \mathbb{M}$ (i.e., $M_k = M$) for all $k\in\mathbb{Z}_+$ and
\begin{eqnarray} \bm{Q}_{k,\ell} = \left\{ \begin{array}{l@{~~~}ll} \bm{D}_{\ell-k}, & k \in \mathbb{Z}_+, & \ell = k+1,k+2,\dots, \\ \bm{D}_0 - k\mu\bm{I}, & k \in \mathbb{N}, & \ell=k, \\ k\mu\bm{I}, & k \in \mathbb{N}, & \ell=k-1. \end{array} \right. \label{eqn-Q-BMAP-M-infty} \end{eqnarray}
We now suppose that, for some $C>0$,
\begin{equation} \sum_{k=1}^{\infty}(k+{\rm e})\log(k + {\rm e}) \bm{D}_k \bm{e} < C\bm{e}, \label{cond-BMAP-M-infty} \end{equation}
and let
\begin{equation*} \bm{v}_k = \log(k + {\rm e}) \bm{e},\qquad k \in \mathbb{Z}_+, \end{equation*}
where ``${\rm e}$" denotes Napier's constant. Clearly, $\sum_{k=1}^{\infty}\log(k + {\rm e}) \bm{D}_k \bm{e} < C\bm{e}$ due to (\ref{cond-BMAP-M-infty}). Thus, Condition \ref{cond-01} holds for generator $\bm{Q}$ in (\ref{defn-Q-BMAP-G-infty}) (see \cite[Lemma~1]{Yaji16}).
It remains to verify that Condition \ref{cond-02} holds. From (\ref{defn-Delta_n}) and (\ref{defn-Q-BMAP-G-infty}), we have $\Delta_k(i,i) = |q(k,i;k,i)| = k\mu + |D_{0,i,i}|$ and thus Condition \ref{cond-02} is reduced to
\begin{equation} \sum_{(k,i)\in\mathbb{Z}_+\times\mathbb{M}} \pi(k,i) k \log(k + {\rm e}) < \infty. \label{ineqn-pi*v-03} \end{equation}
Therefore, we show that (\ref{ineqn-pi*v-03}) holds.
We begin with the following lemma.
\begin{lem}\label{lem-BMAP-M-infty-subexp} Let $V$ denote a function on $\mathbb{R}_+$ such that
\begin{equation} V(x) = (x + {\rm e}) \log(x + {\rm e}),\qquad x \in \mathbb{R}_+. \label{defn-V(x)} \end{equation}
If (\ref{cond-BMAP-M-infty}) holds, then there exist some $K \in \mathbb{Z}_+$ and $\theta > 0$ such that
\begin{eqnarray} \sum_{\ell=0}^{\infty}\bm{Q}_{k,\ell} V(\ell)\bm{e} &\le& -\theta V(k) \bm{e} \le -\bm{e}\quad \mbox{for all $k \ge K+1$}. \label{ineqn-sum-Q*v-subexp-k>=K} \end{eqnarray}
\end{lem}
\begin{proof} Because $\lim_{x\to\infty}V(x) = \infty$, it suffices to prove that
\begin{eqnarray} \limsup_{k\to\infty} {1 \over V(k)}\sum_{\ell=0}^{\infty}\bm{Q}_{k,\ell} V(\ell)\bm{e} &\le& -\mu \bm{e}. \label{limsup-sum-Q*v/V(k)} \end{eqnarray}
It follows from (\ref{eqn-Q-BMAP-M-infty}) that, for $k\in\mathbb{N}$,
\begin{eqnarray} {1 \over V(k)} \sum_{\ell=0}^{\infty}\bm{Q}_{k,\ell} V(\ell)\bm{e} &=& \left\{ -\mu k \left( 1 - {V(k-1) \over V(k)} \right) \bm{e} + \sum_{\ell=0}^{\infty} { V(k+\ell) \over V(k) }\bm{D}_{\ell} \bm{e} \right\}.\quad~~~~ \label{eqn-Qv-subexp} \end{eqnarray}
Furthermore, $V$ is differentiable and convex. Thus, we have
\[ V(k) \ge V(k-1) + V'(k-1),\qquad k \ge 1. \]
Using this inequality and (\ref{defn-V(x)}), we obtain
\begin{eqnarray} \liminf_{k\to\infty} k\left( 1 - {V(k-1) \over V(k)} \right) &\ge& \liminf_{k\to\infty} k {V'(k-1) \over V(k)} \nonumber \\ &=& \lim_{k\to\infty} {k \over k + {\rm e}}{\log (k - 1 + {\rm e}) + 1 \over \log (k + {\rm e})}
= 1. \label{liminf-k*V(k-1)/V(k)} \end{eqnarray}
Applying (\ref{liminf-k*V(k-1)/V(k)}) to (\ref{eqn-Qv-subexp}) yields
\begin{eqnarray*} \limsup_{k\to\infty} {1 \over V(k)} \sum_{\ell=0}^{\infty}\bm{Q}_{k,\ell} V(\ell)\bm{e} \le -\mu\bm{e} + \limsup_{k\to\infty} \sum_{\ell=0}^{\infty} { V(k+\ell) \over V(k) }\bm{D}_{\ell} \bm{e}, \end{eqnarray*}
and therefore (\ref{limsup-sum-Q*v/V(k)}) holds if
\begin{equation} \lim_{k\to\infty} \sum_{\ell=0}^{\infty} { V(k+\ell) \over V(k) }\bm{D}_{\ell}\bm{e} = \bm{0}. \label{lim-sum-V(k+l)/V(k)*D(l)} \end{equation}
Consequently, our goal is to prove (\ref{lim-sum-V(k+l)/V(k)*D(l)}).
We note that $V \ge 1$ is log-concave, which implies the following: For any $x,y \in \mathbb{R}_+$ such that $x+y>0$,
\begin{align*} \log V(x) &\ge {y \over x+y} \log V(0) + {x \over x+y} \log V(x+y), \\ \log V(y) &\ge {x \over x+y} \log V(0) + {y \over x+y} \log V(x+y). \end{align*}
These inequalities yield
\[ \log V(x) + \log V(y) \ge \log V(0) + \log V(x+y) \ge \log V(x+y),
\]
which leads to
\begin{equation} V(x+y) \le V(x)V(y),\qquad x,y \in \mathbb{R}_+. \label{indeqn-V(x+y)} \end{equation}
Using (\ref{indeqn-V(x+y)}) and (\ref{defn-V(x)}), we obtain, for all $k \in \mathbb{Z}_+$,
\begin{eqnarray*} \sum_{\ell=1}^{\infty} { V(k+\ell) \over V(k) }\bm{D}_{\ell}\bm{e} \le \sum_{\ell=1}^{\infty} V(\ell)\bm{D}_{\ell}\bm{e} = \sum_{\ell=1}^{\infty} (\ell + {\rm e}) \log (\ell + {\rm e})\bm{D}_{\ell}\bm{e} < C\bm{e}, \end{eqnarray*}
where the last inequality is due to (\ref{cond-BMAP-M-infty}). Thus, by the dominated convergence theorem and (\ref{defn-V(x)}), we obtain
\begin{equation*} \lim_{k\to\infty} \sum_{\ell=0}^{\infty} { V(k+\ell) \over V(k) }\bm{D}_{\ell}\bm{e} = \bm{D}_0\bm{e} + \sum_{\ell=1}^{\infty} \lim_{k\to\infty}{ V(k+\ell) \over V(k) }\bm{D}_{\ell}\bm{e} = \sum_{\ell=0}^{\infty}\bm{D}_{\ell}\bm{e} = \bm{0}, \end{equation*}
which shows that (\ref{lim-sum-V(k+l)/V(k)*D(l)}) holds. \hspace*{\fill}$\Box$ \end{proof}
Let $\widetilde{\bm{v}}:=(\widetilde{v}(k,i))_{(k,i) \in \mathbb{Z}_+ \times \mathbb{M}}$ and $\widetilde{\bm{f}}:=(\widetilde{f}(k,i))_{(k,i) \in \mathbb{Z}_+ \times \mathbb{M}}$ denote column vectors such that
\begin{eqnarray} \widetilde{v}(k,i) &=& V(k) = (k + {\rm e})\log (k + {\rm e}), ~~\qquad k \in \mathbb{Z}_+, \quad~~ i \in \mathbb{M}, \nonumber \\ \widetilde{f}(k,i) &=& \left\{ \begin{array}{l@{~~~}ll} 1, & 0 \le k \le K, & i \in \mathbb{M}, \\ \theta V(k) = \theta (k + {\rm e})\log (k + {\rm e}), & k \ge K+1, & i \in \mathbb{M}, \end{array} \right. \label{defn-f-Qv-BMAP-M-infty-heavy} \end{eqnarray}
where $K \in \mathbb{Z}_+$ and $\theta > 0$ satisfying (\ref{ineqn-sum-Q*v-subexp-k>=K}). It then follows from Lemma~\ref{lem-BMAP-M-infty-subexp} that, for some $\widetilde{b} > 0$,
\begin{equation*} \bm{Q}\widetilde{\bm{v}} \le -\widetilde{\bm{f}} + \widetilde{b}\bm{1}_{\mathbb{Z}_K \times \mathbb{M}},
\end{equation*}
which yields $\bm{\pi}\widetilde{\bm{f}} < \widetilde{b}$. Combining this inequality and (\ref{defn-f-Qv-BMAP-M-infty-heavy}) results in (\ref{ineqn-pi*v-03}). We have confirmed that Condition \ref{cond-02} is satisfied. As a result, our algorithm is always applicable to BMAP/M/$\infty$ queues satisfying (\ref{cond-BMAP-M-infty}).
\subsection{M/M/$s$ retrial queue}\label{subsec-MMss-retrial}
In this subsection, we consider an M/M/$s$ retrial queue (which is sometimes called an M/M/$s$/$s$ retrial queue). The system has $s$ ($s \in \mathbb{N}$) servers but no {\it real} waiting room. {\it Primary customers} (which originate from the exterior) arrive to the system according to a Poisson process with rate $\lambda \in (0,\infty)$. If an arriving primary customer finds an idle server, then the customer occupies the server, otherwise it joins the {\it orbit} (i.e., the {\it virtual} waiting room). Customers in the orbit are referred to as {\it retrial customers}. Each retrial customer stays in the orbit for an exponentially distributed time with mean $\eta^{-1} \in (0,\infty)$, independently of all the other events. After the sojourn in the orbit, a retrial customer tries to occupy one of idle servers. If such a retrial customer finds no idle servers, then it goes back to the orbit; that is, becomes a retrial customer again. We assume that the service times of primary and retrial customers are i.i.d.\ with an exponential distribution having mean $\mu^{-1} \in (0,\infty)$.
Let $X(t)$, $t\in\mathbb{R}_+$, denote the number of customers in the orbit at time $t$. Let $J(t)$, $t\in\mathbb{R}_+$, denote the number of busy servers at time $t$. The stochastic process $\{(X(t),J(t));t\in\mathbb{R}_+\}$ is a level-dependent quasi-birth-and-death process (LD-QBD) on state space $\mathbb{S}:= \mathbb{Z}_+ \times \mathbb{Z}_s$ with generator $\bm{Q}$ given by
\begin{equation} \bm{Q} = \bordermatrix{
& \mathbb{L}_0 & \mathbb{L}_1 & \mathbb{L}_2 & \mathbb{L}_3 & \cdots \cr \mathbb{L}_0 & \bm{Q}_{0,0} & \bm{Q}_{0,1} & \bm{O} & \bm{O} & \cdots \cr \mathbb{L}_1 & \bm{Q}_{1,0} & \bm{Q}_{1,1} & \bm{Q}_{1,2} & \bm{O} & \cdots \cr \mathbb{L}_2 & \bm{O}& \bm{Q}_{2,1} & \bm{Q}_{2,2} & \bm{Q}_{2,3} & \cdots \cr \mathbb{L}_3 & \bm{O} & \bm{O} & \bm{Q}_{3,2}& \bm{Q}_{3,3} & \ddots \cr ~~\vdots & \vdots & \vdots & \vdots & \ddots & \ddots }, \label{eqn-Q-LD-QBD} \end{equation}
where $\mathbb{L}_k = \{k\} \times \mathbb{Z}_s$ for $k \in \mathbb{Z}_+$, and where
\begin{align} &&&& \bm{Q}_{k,k-1} &= \left ( \begin{array}{llllll} 0 \ & k \eta \ & 0 & \cdots & 0 \\ 0 & 0 & k \eta \ & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ \vdots & & & 0 \ & k \eta \\ 0 & \cdots & \cdots & 0 & 0 \\ \end{array} \right ), & k &\in \mathbb{N},&&&& \label{eqn-Q_{k,k-1}} \\ &&&& \bm{Q}_{k,k+1} &= \left ( \begin{array}{cccccc} 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \cdots & 0 & \lambda \\ \end{array} \right ), & k &\in \mathbb{Z}_+,&&&& \label{eqn-Q_{k,k+1}} \end{align}
and
\begin{eqnarray} \bm{Q}_{k,k} &=& \left ( \begin{array}{cccccc} -\psi_{k,0} \ & \lambda\ & 0 & \cdots & \cdots & 0 \\ \mu & -\psi_{k,1} & \lambda & \ddots & & \vdots \\ 0 & 2 \mu \ & -\psi_{k,2} \ & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots \quad & -\psi_{k,s-1} \quad & \lambda \\ 0 & \cdots & \cdots & 0 & s \mu \ & -\psi_{k,s} \\ \end{array} \right ),\quad k \in \mathbb{Z}_+,\qquad \label{eqn-Q_{k,k}} \end{eqnarray}
with
\begin{align*} &&&& \psi_{k,i} &= \lambda + i \mu + k\eta, & k &\in \mathbb{Z}_+,\ i \in \mathbb{Z}_{s-1}, &&&& \\ &&&& \psi_{k,s} &= \lambda + s \mu, & k &\in \mathbb{Z}_+. &&&& \end{align*}
We now assume that the stability condition $\rho := \lambda / (s\mu) < 1$ holds. It then follows that the LD-QBD $\{(X(t),J(t))\}$ is ergodic (see, e.g., \cite[Section~2.2]{Fali97}) and thus has the unique stationary distribution vector $\bm{\pi}=(\pi(k,i))_{(k,i)\in\mathbb{S}}$. Under this stability condition, we show that Conditions~\ref{cond-01} and \ref{cond-02} are satisfied, which requires the following proposition.
\begin{prop}[{}{\cite[Lemma~4.1]{Masu17-JORSJ}}]\label{lem-MMC} Suppose that $\rho = \lambda / (s\mu) < 1$. For $k \in \mathbb{Z}_+$, let $\bm{v}_k=(v(k,j))_{j\in\mathbb{Z}_s}$ be given by
\begin{eqnarray} v(k,i) &=& \left\{ \begin{array}{l@{~~~}ll} \alpha^k/c, & k \in \mathbb{Z}_+,& i \in \mathbb{Z}_{s-1}, \\ \alpha^k/(c\gamma), & k \in \mathbb{Z}_+,& i = s, \end{array} \right. \label{eqn-v(k)-retrial} \end{eqnarray}
where $\alpha$, $\gamma$, and $c$ are positive constants such that
\begin{eqnarray} 1 &<& \alpha < \rho^{-1}, \label{defn-theta} \\ \alpha^{-1} &<& \gamma < 1 - \rho(\alpha-1), \nonumber \\
c &=& s\mu \left[ 1 - \rho(\alpha-1) - \gamma \right]. \nonumber
\end{eqnarray}
Furthermore, let
\begin{eqnarray*} b &=& \max_{k \in \mathbb{Z}_{K}}\alpha^k \left[1 - c^{-1}\{ k\eta ( 1 - \gamma^{-1}\alpha^{-1}) + \lambda ( 1 - \gamma^{-1} ) \} \right] \vee 0, \nonumber
\\ K &=& \left\lceil { c + \lambda ( \gamma^{-1} - 1) \over \eta ( 1 - \gamma^{-1}\alpha^{-1}) } \right\rceil \vee 1 - 1, \nonumber
\end{eqnarray*}
where $x \vee y = \max(x,y)$. Under these conditions, the generator $\bm{Q}$ of the LD-QBD, characterized by (\ref{eqn-Q-LD-QBD})--(\ref{eqn-Q_{k,k}}), satisfies
\begin{equation*} \bm{Q}\bm{v} \le - c\bm{v} + b \bm{1}_{\mathbb{S}_{K}}.
\end{equation*}
\end{prop}
We note that $c\bm{v} \ge \bm{e}.$ Proposition~\ref{lem-MMC} thus shows that Condition~\ref{cond-01} is satisfied. Moreover, Theorem~1 in \cite{KimJeri12} states that, for a certain constant $c_0 > 0$,
\begin{equation} \pi(k,i) \simhm{k} {c_0 \over i!} \left(\eta \over \mu \right)^i k^{-s+i+\lambda/(s\eta)} \rho^k,\qquad i \in \mathbb{Z}_s, \label{asymp-pi(k,i)} \end{equation}
where $a_1(x) \simhm{x} a_2(x)$ represents $\lim_{x\to\infty}a_1(x)/a_2(x) = 1$. Combining (\ref{eqn-v(k)-retrial})--(\ref{asymp-pi(k,i)}) yields
\[ \sum_{(k,i) \in \mathbb{S}} \pi(k,i) k v(k,i) < \infty, \]
which implies that Condition~\ref{cond-02} is satisfied. Consequently, our algorithm is always applicable to stable M/M/$s$ retrial queues.
\section{Concluding Remarks}\label{sec-remarks}
This paper has presented a sequential update algorithm for computing the stationary distribution vector in continuous-time upper block-Hessenberg Markov chains. The algorithm stops after finitely many iterations if Conditions~\ref{cond-01} and \ref{cond-02} are satisfied. These conditions hold in any stable M/M/$s$ retrial queue and the BMAP/M/$\infty$ queues satisfying the mild condition (\ref{cond-BMAP-M-infty}). Furthermore, the algorithm would be applicable (under some mild conditions) to MAP/PH/$s$ retrial queues, BMAP/PH/$\infty$ queues, and their variants.
\appendix
\section{Proof of Proposition~\ref{lem-pi_n-Lambda_n}}\label{appen-ineqn-pi-(U_n^*)^{-1}}
Let $\bm{T}_n^*$, $n \in \mathbb{Z}_+$, denote
\begin{equation} \bm{T}_n^* = \left\{ \begin{array}{l@{~~~}l} \bm{Q}_{0,0}, & n=0, \\
\bm{Q}_{n,n} + \displaystyle\sum_{\ell=0}^{n-1} \bm{U}_{n,\ell} \bm{Q}_{\ell,n}, & n \in \mathbb{N}. \end{array} \right. \label{defn-T_n^*} \end{equation}
It then follows from (\ref{defn-U_k^*}), (\ref{defn-Delta_n}), and (\ref{defn-T_n^*}) that
\begin{equation} \bm{\pi}_n\bm{\Delta}_n \ge \bm{\pi}_n(-\bm{Q}_{n,n}) \ge \bm{\pi}_n(-\bm{T}_n^*) = \bm{\pi}_n(\bm{U}_n^*)^{-1}, \qquad n \in \mathbb{Z}_+. \label{eqn-180909-01} \end{equation}
It also follows from (\ref{eqn-pi_l-pi_k*U_{k,l}}), (\ref{defn-T_n^*}), and $\sum_{\ell=0}^{\infty} \bm{\pi}_{\ell} \bm{Q}_{\ell,n} = \bm{0}$ ($n\in\mathbb{Z}_+$) that
\begin{eqnarray} \bm{\pi}_n(-\bm{T}_n^*) &=& -\bm{\pi}_n \bm{Q}_{n,n} -\bm{\pi}_n \displaystyle\sum_{\ell=0}^{n-1} \bm{U}_{n,\ell} \bm{Q}_{\ell,n} \nonumber \\ &=& - \sum_{\ell=0}^n \bm{\pi}_{\ell} \bm{Q}_{\ell,n} = \sum_{\ell=n+1}^{\infty} \bm{\pi}_{\ell} \bm{Q}_{\ell,n}
\ge \bm{0},\neq \bm{0},\qquad n \in \mathbb{Z}_+. \label{eqn-180909-02} \end{eqnarray}
Combining (\ref{eqn-180909-01}) and (\ref{eqn-180909-02}) yields (\ref{ineqn-pi-(U_n^*)^{-1}}). The proof has been completed.
\section*{Acknowledgments} The author thanks Mr. Masatoshi Kimura and Dr. Tetsuya Takine for providing the counterexample presented in Section~\ref{subsec-example}. The author also thanks an anonymous referee for his/her valuable comments that helped to improve the paper.
\end{document} | arXiv |
One hundred concentric circles with radii $1,2,3,\ldots,100$ are drawn in a plane. The interior of the circle of radius $1$ is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius 100 can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
The sum of the areas of the green regions is \begin{align*}
&\phantom{=}\
\left[(2^2-1^2)+(4^2-3^2)+(6^2-5^2)+\cdots+(100^2-99^2)\right]\pi\\
&=\left[(2+1)+(4+3)+(6+5)+\cdots+(100+99)\right]\pi\\
&={1\over2}\cdot100\cdot101\pi.
\end{align*}Thus the desired ratio is $${1\over2}\cdot{{100\cdot101\pi}\over{100^2\pi}}={101\over200},$$and $m+n=\boxed{301}$. | Math Dataset |
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How to calculate the ECEF coordinates of a satellite given its elevation and azimuth angle, plus the coordinates of a reference object on Earth
So I am looking for a way to 'reverse engineer' satellite ECEF coordinates given its angles of elevation and azimuth and given the coordinates of an object on Earth serving as a reference ground. The satellite is from the Galileo constellation (MEO). Ideally, the solution should be represented as
[x, y, z]=elaz2coords(el, az, xe, ye, ze),
where elaz2coords encapsulates the set of equations which transform coordinates. If someone could point me towards such a set of equations or towards a procedure towards deriving such a set, I would appreciate it greatly.
satellite-constellation
AkhaimAkhaim
$\begingroup$ Welcome to space! Have you searched this site for similar questions? $\endgroup$
– Organic Marble
$\begingroup$ @OrganicMarble, the closes to my question I found was this. Other questions I saw usually go in different direction, how to calculate azimuth and elevation from satellite coordinates $\endgroup$
– Akhaim
$\begingroup$ You need range (distance) in addition to elevation and azimuth. $\endgroup$
– David Hammen
$\begingroup$ @DavidHammen, you are absolutely right. I found some resources where elevation and azimuth are given as a function of satellite and ground reference point coordinates (opposite from what I am searching for), and from those equations it was rather clear that another equation is missing, which is the one for range. $\endgroup$
There isn't enough information to obtain the position. Those five numbers define a line, not a point. Azimuth and elevation tell you in what direction the satellite is, but not how far away it is. You can either get more measurements, or cheat.
If you get another line of sight from a far away observer at the same time, you can calculate where the lines cross, and that is where your target is. Of course, given experimental error, there won't actually be a point where they exactly cross, but you can compute the point nearest to both (or as many more as you can get) lines in a least-squares sense. If you make some other kind of observation, such as radar rather than optical, even if it's from the same place, you can get a second curve or surface with which to intersect your line of sight.
Cheating means assuming the result of some other measurement based on some constraint you think is obeyed, such as a circular orbit with a known semimajor axis, which gives you a sphere on which the satellite must lie, and you can find the intersection of the line of sight with that sphere. This also has errors, because real satellites feel many perturbing forces which make their orbits differ from ideal Keplerian ellipses in many different and rapidly changing ways, so whatever idealized orbit you assume they're in will also be inaccurate.
Ryan CRyan C
$\begingroup$ With the (often good) assumption that a satellite isn't actively maneuvering, you could also make many az/el measurements from the same location to derive the orbit $\endgroup$
– Erin Anne
$\begingroup$ @ErinAnne true, but that involves a lot more effort, and is not a project that beginners should try to start with. $\endgroup$
– Ryan C
As Ryan C and David Hammen pointed out to me, the original proposal is under-defined, as it lacks at least one more piece of information. Thus, one should be able to obtain the triplet of ECEF (geocentric) coordinates $(x^{(s)}, y^{(s)}, z^{(s)})$ given the elevation, $\varepsilon_r^{(s)}$, azimuth, $\alpha_r^{(s)}$, line-of-sight distance (a.k.a. range or slanted range) between the reference object on Earth and the observed satellite, $\rho_r^{(s)}$, together with the coordinates of the reference object on Earth, $\vec{r}_{r}$. The latter can be either in ECEF [$(x_{r}, y_{r}, z_{r})$] or geodetic (latitude, $\lambda_r$, longitude, $\varphi_r$, and altitude $H_r$) coordinates. Given this pair of triplets, $(\varepsilon_r^{(s)}, \alpha_r^{(s)}, \rho_r^{(s)})$ & $\vec{r}_{r}$, and the relationships given here and here, the following set of equations can be constructed: $$dz = cos(\lambda)cos(\varepsilon)cos(\alpha)+sin(\lambda)sin(\varepsilon)$$ $$dx = \frac{dz\times cos(\varphi)cos(\lambda)-cos(\varepsilon)\left[ sin(\varphi)sin(\lambda)sin(\alpha)+cos(\varphi)cos(\alpha) \right]}{sin(\lambda)}$$ $$dy = \frac{cos(\varepsilon)sin(\alpha)+dx\times sin(\varphi)}{cos(\varphi)}$$
Since $\vec{r}_{r}$ can be inter-converted between geodetic and ECEF (geocentric) coordinates, as shown in this Wikipedia post, with minor complication of calculating the latitude, which requires either an approximation or an iterative solution, then given $(\lambda_r, \varphi_r, H_r)$ one can obtain $(x_r, y_r, z_r)$ and vice-versa. Given $(x_r, y_r, z_r)$ (regardless of its origin [i.e. direct input or derived from geodetic coordinates]) and the solutions from the system of the equations above, $(dx, dy, dz)$, one can find the ECEF coordinates of the satellite as: $$x^{(s)}=x_r+\rho_r^{(s)}dx$$ $$y^{(s)}=y_r+\rho_r^{(s)}dy$$ $$z^{(s)}=z_r+\rho_r^{(s)}dz$$
In the meantime, I've also realized that MATLAB has a bunch of tools that can do this and related conversions (e.g. aer2ecef, aer2geodetic, ecef2geodetic, etc), but it is also good to know for myself how these transformations operate internally.
$\begingroup$ yes, adding the slant range will do nicely. one more thing you may want to include is Earth's oblateness, which complicates the relationship between the geodetic inputs and the geocentric outputs. you mentioned it, but haven't put it in your formulas yet, so I recommend you try it out. thanks for visiting space exploration stack exchange, and stay around if you like! $\endgroup$
$\begingroup$ @RyanC, thanks for the warm welcome. Just to make sure we use the same nomenclature, what do you mean by oblateness? In conversion between geocentric and geodetic, one can calculate latitude iteratively, and depending on the iterative scheme a parameter $N$ may be involved, which takes into account the eccentricity of the Earth, $e$ (its square, in fact), which is also a function of flattening $f=1-\frac{A_{semi-minor}}{A_{semi-major}}$, $e^2=f(1+\frac{A_{semi-minor}}{A_{semi-major}})$ Is then $f$ the oblateness you refer to? $\endgroup$
Thanks for contributing an answer to Space Exploration Stack Exchange!
Should we accept questions about information provided by ChatGPT?
Computing GEO satellite's longitude from elevation/azimuth from a given latitude/longitude?
Satellite position using 2 station ranging
Is it possible to plot the ground track of a satellite with just azimuth & elevation angles without range?
Why can't I just use dot product angles to get satellite vision of a point on Earth?
Ephemeris time and clock corrections in RINEX navigation files
How does one calculate the look angle for non-geo satellites (i.e. LEO, HEO, etc.) | CommonCrawl |
How do you prove that $p(n \xi)$ for $\xi$ irrational and $p$ a polynomial is uniformly distributed modulo 1?
The Weyl equidistribution theorem states that the sequence of fractional parts ${n \xi}$, $n = 0, 1, 2, \dots$ is uniformly distributed for $\xi$ irrational.
This can be proved using a bit of ergodic theory, specifically the fact that an irrational rotation is uniquely ergodic with respect to Lebesgue measure. It can also be proved by simply playing with trigonometric polynomials (i.e., polynomials in $e^{2\pi i k x}$ for $k$ an integer) and using the fact they are dense in the space of all continuous functions with period 1. In particular, one shows that if $f(x)$ is a continuous function with period 1, then for any $t$, $\int_0^1 f(x) dx = \lim \frac{1}{N} \sum_{i=0}^{N-1} f(t+i \xi)$. One shows this by checking this (directly) for trigonometric polynomials via the geometric series. This is a very elementary and nice proof.
The general form of Weyl's theorem states that if $p$ is a monic integer-valued polynomial, then the sequence ${p(n \xi)}$ for $\xi$ irrational is uniformly distributed modulo 1. I believe this can be proved using extensions of these ergodic theory techniques -- it's an exercise in Katok and Hasselblatt. I'd like to see an elementary proof.
Can the general form of Weyl's theorem be proved using the same elementary techniques as in the basic version?
Akhil MathewAkhil Mathew
$\begingroup$ Correct me if I'm wrong, but it seems like this would be better suited for MathOverflow.net? $\endgroup$ – Nick Jul 20 '10 at 21:47
$\begingroup$ Nick, I don't think it's research-level. Since many people committing to this site expressed a desire for an MO-like site that would admit questions in pure math at, say, an undergraduate level, I think it should stay here. $\endgroup$ – Jamie Banks Jul 20 '10 at 22:17
$\begingroup$ It's definitely not research-level, but I actually doubt it'd get closed on MO, since it's a fact that most books mention without proof; I don't really see why it would be terribly out of place on either site. $\endgroup$ – Akhil Mathew Jul 20 '10 at 23:16
$\begingroup$ fair, Akhil; I'm glad to have it here, though, so that we indicate the acceptability of higher-level math questions (non-grade school) $\endgroup$ – Jamie Banks Jul 21 '10 at 0:08
$\begingroup$ (I also think these questions will get slightly easier once we have jsMath or similar LaTeX support on math.SE) $\endgroup$ – Jamie Banks Jul 22 '10 at 6:14
There is a fairly good exposition in Terry Tao's post, see Corollaries 4-6. Here is a sketch:
We prove the more general statement: Let $p(n)= \chi n^d + a_{d-1} n^{d-1} + \cdots + a_1 n + a_0$ be any polynomial, with $\chi$ irrational. Then $p(n) \mod 1$ is equidistributed. Our proof is by induction on $d$; the base case $d=1$ is standard.
Set $e(x) = e^{2 \pi i x}$. By the standard trickery with exponential polynomials, it is enough to show $$\sum_{n=0}^{N-1} e(p(n)) = o(N).$$
Choose a positive integer $h$. With a small error, we can replace the sum by $$\sum_{n=0}^{N-1} (1/h) \left( e(p(n)) + e(p(n+1)) + \cdots + e(p(n+h-1)) \right).$$ By Cauchy-Schwarz, this is bounded by $$\frac{\sqrt{N}}{h} \left[ \sum_{n=0}^{N-1} \left( e(p(n)) + \cdots + e(p(n+h-1)) \right) \overline{ \left( e(p(n)) + \cdots + e(p(n+h-1)) \right)} \right]^{1/2}.$$
Expanding the inner sum, we get $h^2$ terms of the form $e(p(n) - p(n+k))$. There are $h$ terms where $k=0$; these each sum up to $N$. For the other $h^2-h$ terms, the sum is of the form $\sum_{n=0}^{N-1} e(q(n))$, where $q$ has leading term $\chi d n^{d-1}$. By induction, each of these sums is $o(N)$.
So the quantity in the square root is $$hN+o(N)$$ where the constant in the $o$ depends on $h$ and $\chi$. Putting it all together, we get a bound of $$N/\sqrt{h} + o(N).$$
Since $h$ was arbitrary, this proves the result.
David E SpeyerDavid E Speyer
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Testing the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \cos{n}}}$
can $\sin(n)$ be arbitrary small?
Weyl Equidistribution Theorem and a Limit
Relationship between Kronecker's Approximation Thm and Weyl's Equidistribution Thm?
Prove Sequences are Uniformly Distributed Modulo 1
Are polynomials modulo $1$ equidistributed?
Proof of a special case of Fermat's Last Theorem.
Non uniform continuity of a function and almost periodicity
Elementary proof that $\sum_{k=1}^{\infty}{\frac{1}{m^{k^2}}}$ is irrational (for any integer $m > 1$)
show that $ \limsup n\; | \;\{ (n+1)^2 \sqrt{2}\} - \{ n^2 \sqrt{2}\}\; | = \infty $
Hands-on applications of Weyl's criterion
Will these geometric means always converge to $1/e$? | CommonCrawl |
Is there a limit to frames per second to be captured?
Today we see ~100,000 fps cameras for hyper slo-mo.
Is there a physical limit for frames per second for a camera? Speed of light maybe? How?
speed-of-light speed camera
NeutronStar
stepssteps
I used a 8 million fps camera 25years ago - and the technology was old even then. I think purely electronic cameras can beat this by a factor of 10x today.
Those cameras used a rotating hexagonal mirror and an arc of film, each frame behind an individual lens. As the mirror rotated it reflected the incoming rays onto each lens, and so each frame of film in turn. The only moving part was the mirror and that ran on helium bearings at something like 10,000 rps. IIRC 200 frames filled 90deg of the arc.
So in 0.25 * 1/10,000 of a second the camera generated 200 individual frames, (might have another factor of 2x because the reflection angle is twice the rotation angle)
The main difficulty with these cameras was that one revolution (0.1ms) later the images would be overwritten by the mirror again. Designing a shutter that closed in that time was interesting.
This is the same principle as the ligthhouse paradox. So if you were to make the arc of the camera large enough (say on the moon) and had a fast enough rotating mirror (and enough light!) you can get essentially unlimited frames per second.
Martin BeckettMartin Beckett
$\begingroup$ Here is some footage at $10^{12}$ fps, fast enough to show the movement of a pulse of light. It's actually a bit of a cheat, since they're actually using lots of puslses of light (one per frame) rather than just one, but still it's incredibly impressive, and amazing to watch. $\endgroup$ – Nathaniel Oct 9 '13 at 5:57
$\begingroup$ Doing very short exposures is (relatively) easy - using short pulses of right down to the point where the uncertainty principle wins. $\endgroup$ – Martin Beckett Oct 9 '13 at 15:49
Martin Beckett is correct that frame rates can be much higher than 100,000 fps. In this supplement to his perfectly good answer, I want to point out that principle there is a fundamental limit to the number of frames per second that can be captured. It's just that the limit is incredibly high.
The limit is due to the fact that a higher frame rate means that the shutter has to be open for a shorter period of time for each frame, which means that less light hits the photosensor array. You can counter this by making the sensors more sensitive, but eventually you'll get to the point where only a few photons make it through the aperture before the shutter closes on each frame.
In this case, you wouldn't see a scene, but just a few bright dots where the photons hit the detector. Of course you can combine many such images to produce a full picture, but then that defeats the point of having such a high frame rate in the first place. The uncertainty principle also comes into play at this point - if you want to know exactly when the photon arrived, it becomes impossible to know its energy (i.e. its frequency), so there's a point at which colour photography would be become impossible.
The only way to solve this fundamental problem would be to light the scene more brightly, so that there are more photons around to build up an image with. But sooner or later you'll get to the point where the light has to be so bright that it destroys whatever you're trying to film. You can also use a larger aperture, but that makes it hard to keep things in focus, and there's a limit to how much it will help you, since any given scene only gives off so many photons to be captured.
A perfectly efficient $100\:\mathrm{W}$ light bulb would give off $100\:\mathrm{J}$ per second of light, made of photons in the visible range. Each photon has an energy given by $hf \approx 6\times 10^{-34} \times 5 \times 10^{14} Hz \sim 10^{-21}\:\mathrm{J}$. ($h$ is Planck's constant and $f$ is the frequency of the light.) This means that the bulb gives off about $100/10^{-21} = 10^{23}$ photons per second. So if the camera was pointing directly at the bulb you would still see a grainy low-res image with a frame rate of up to, say $10^{17}$ fps, which gives you a million photons to build an image out of on every frame. This is an incredibly fast frame rate, much faster than has been achieved with present technology. Even light moves sluggishly at this frame rate. If you point the camera directly at the sun, with a $\sim 1\:\mathrm{m^2}$ aperture, you could go 1000 times faster than this. But to get much faster than that there's no choice other than using brighter and brighter lights.
NathanielNathaniel
$\begingroup$ Usually ultra high speed is of high energy events or you can use laser pulses to provide quite a lot of light in a billionth of second $\endgroup$ – Martin Beckett Oct 9 '13 at 15:52
$\begingroup$ @MartinBeckett sure - I just wanted to make the point that there is a fundamental limit to how short an exposure can be, even if it's probably shorter than anyone might have a practical reason to use. $\endgroup$ – Nathaniel Oct 10 '13 at 1:14
Not the answer you're looking for? Browse other questions tagged speed-of-light speed camera or ask your own question.
Is there a maximum frames per second (FPS)?
Filming light in slo mo
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Would a camera that only takes light at 90 degrees work?
Would a speed camera register more speed if it moves in the opposite direction than the object it's measuring? | CommonCrawl |
Exploring governance for sustainability in contexts of violence: the case of the hydropower industry in Colombia
Jorge-Andrés Polanco ORCID: orcid.org/0000-0002-3469-16851
Energy, Sustainability and Society volume 8, Article number: 39 (2018) Cite this article
The hydropower industry in Colombia is developing in contexts of violence because of armed conflict. The companies that drive hydropower development are usually large and benefit today from lessons that have been learned around the world. However, there is little understanding of how these good management practices are addressed in contexts of violence. This paper contributes to the filling of a knowledge gap between the energy business practices and the local implications of the armed conflict. Large companies would have to incorporate a holistic view of the power generation business that connects financial performance with both environmental protection and social equity. The governance of business sustainability is analyzed within violence, drawing upon a case study from the hydropower industry to explore emergent issues, dominant players, and tools that may provide solutions.
The case study method is based on the hermeneutical analysis of 16 in-depth interviews with employees from the energy sector, the public municipalities, and local leaders. The interviews were coded and occurrence rates were used as ranking criteria. Two co-occurrence matrices were constructed in order to estimate the ranking of the interests of the players and the tools of action they prefer.
The results exhibited conventional problems such as climate change, dwindling biodiversity, and the deteriorating condition of natural resources, in addition to the characteristic difficulties of armed conflicts, such as illegality, distrustfulness, and lack of opportunity for local populations. In view of both the weakness of the state and the scarcity of social capital, energy companies emerge as a central player in association with nongovernmental organizations. The tools used are more geared toward planning than they are toward joint action and evaluation.
It was concluded that the management of hydropower stations in the contexts of violence requires companies to orient their actions toward results and evaluate the impact of its management. Such management must be based on transformational relationships aimed at reducing the existing asymmetry between players and distributing the costs and benefits of the hydropower station more equitably.
Most of the electric power available in Colombia is generated by large hydropower stations [1]. More than a third of this energy supply has been developed during the years since the 1980s, when the violence as a result of armed conflict in the country grew considerably; between 1985 and 2017, approximately 7 million Colombians were forcibly displaced to other areas of the country [2, 3]. The year 2002 is known as one of the bloodiest periods, which was a result of multiple conflicts between guerillas, paramilitary groups, and the army [4]. During that year, attacks on electric power transmission networks, oil pipelines, and road infrastructure were frequent; massacres were intensified; and electoral procedures were hindered [4, 5].
Globally, most hydropower development is taking place in emerging economies in Asia, Africa, and South America [6]. However, Colombia is not the only country in which this development has occurred amid violence because of armed conflicts as may be observed in cases involving countries such as Myanmar [7], Angola, and the Democratic Republic of Congo [8]. Little territorial control by the state is frequently observed in these countries, as there are insurgent groups that benefit from extortion, and the local population is affected by the conflict [9, 10].
The companies that drive hydropower development are usually large, and this development usually involves the construction and operation of stations that produce more than 1 MW of power [6]. Those who manage these organizations benefit today from lessons that have been learned around the world, such as that of participation in evaluating the shared benefit and that of the implementation of environmental and social management plans with the aim of complying with development policies in the energy sector and the regulations of each respective country [11].
However, these good management practices are put to the test in contexts of violence, requiring these companies to incorporate a holistic view of the power generation business that connects financial performance with both environmental protection and social equity. How do energy companies manage hydropower facilities in contexts of violence? This issue is particularly relevant for Colombia taking into consideration the fact that its electric power supply is heavily dependent on the operation of large hydropower plants that are vulnerable to armed conflict.
There is little understanding of the local implications of the armed conflict in the energy business practices in Colombia. The lack of trust and opportunities, illegality, forced displacement of people, and weakness on the part of state organizations might be the main issues of the armed conflict [4, 12]. The energy companies have voluntary initiatives to protect their reputations and better adapt to this violent context by addressing environmental externalities, human rights, employment, and community development [13, 14].
However, the context of violence forces the energy firms into a key position that could maintain the asymmetry between the players if it is exploited to socially legitimize the firms rather than to create mechanisms that more equally distribute the costs and benefits of the infrastructure [15]. This paper contributes to fill the gap of knowledge about the interdependence of the hydropower industry with the territory, to understand how to share benefits more equitably in contexts of violence.
The concepts of corporate sustainability and collaborative governance form the conceptual framework of this study. The former is considered because it gives meaning to a holistic view of the business [16,17,18] and the latter because it explains the relationship between the business and the surrounding area under circumstances of interdependence that are characteristic of contexts of violence [19, 9]. Through collaborative governance, firms would be able to surmount institutional constraints of corporate sustainability [20], such as social pressure, reputational effects of environmental impacts, or any other interdependency with stakeholders [21, 22].
The purpose of this study was to analyze governance for sustainability in the hydropower companies that are located in contexts of violence, using a case study from the energy industry to explore the emergent issues, dominant players, and tools used in their solution. The exploratory case method employed by Yin [23] is used, inspired by other, similar works [7, 15]. In all, 16 in-depth interviews were conducted, and the data were triangulated using secondary information. The interviews involved employees from one of the country's main power companies, public employees from state organizations, and local leaders from areas that had experienced armed conflict. The secondary information that was examined dealt with corporate sustainability, collaborative governance, and energy development in contexts of violence.
The study is divided into five sections. The first section lays in order the conceptual foundations of corporate sustainability and collaborative governance to subsequently outline the problem of managing hydropower stations in contexts of violence. The second section presents the methodological process used for characterizing the data and its qualitative analysis. The third section describes the main results of the interviews in terms of issues, players, and tools. The fourth section discusses these results in view of good management practice while highlighting specific details associated with contexts of violence. Finally, the "Conclusions" describes theoretical and practical repercussions of the main findings, the limitations of the study, and further topics for research.
The term corporate sustainability (CS) is more widely used in academic literature than in practitioner literature. Practitioner literature tends to be more prescriptive trying to provide guidelines to managers on how to pursue CS while academic literature tends to be more holistic, complex, and philosophical [24]. A standardized definition does not exist. However, for the renewable energy research, some authors have recently mentioned that CS definitions all emphasize the importance of meeting stakeholders' needs and balancing the economic, environmental, and social dimensions of corporate performance [25].
Based on the work of Montiel and Delgado-Ceballos [24], Baumgartner [26], and Chang et al. [25], we broadly define corporate sustainability as the ability of a firm to perform growth over time by effectively reducing economic, social, and environmental impacts and meeting the expectations of diverse stakeholders. Corporate sustainability tends to be considered in terms of company performance in conjunction with the well-being of society and environmental quality [24, 27, 28]. Therefore, future performance is seen as being dependent on the company's potential for improving its present performance [18].
Lozano [16] argues that this potential depends on factors that are internal, external, and associated with the company and its surroundings. Among the main internal factors, profitability, responsibility, and corporate culture are the most visible, whereas among the external factors, consumer interest, the growing consciousness of society, and regulations stand out. Lozano [16] emphasizes that reputation, sustainability reports, access to resources, and socio-environmental crises are connecting factors because of the fact that they provide a link between the company and its surroundings.
Ensuring the permanence of the business on the market while at the same time improving its relationship with the surroundings is a governance issue [29]. What is in play is the political and ethical position of the company, as well as its strategy, structure, and organizational culture [26, 30].
The concept of business ethics brings together the internal factors proposed by Lozano [16] in that it involves belief, responsibility, and virtue as expressed by the business in seeking a long-term profit [31]. On the one hand, Windolph et al. [32] speak of the belief of the company in sales, access to capital, and the assembly of a workforce. On the other hand, these authors associate the responsibility and virtue of the organization with its response to state regulation (operating licenses) and pressure brought to bear by communications media and society when they demand that corporate values be based on human rights and the protection of natural resources.
According to these terms, ethics in the context of corporate sustainability provides reliability and legitimacy because it evidences the effect that the perception of stakeholders has on company performance [33, 34]. The attainment of reliability and legitimacy through relationships with stakeholders is an issue that is related to governance of corporate sustainability, which has the potential to hinder the sustainability of companies [35,36,37].
Therefore, being perceived as being reliable and legitimate by the community is a sensitive issue (among others) for businesses and impinges on how benefits are distributed, in accordance with the following levels of commitment [38, 39]: (1) transactional: the business tends to compensate the community, confidence is limited, and the benefits are differentiated; (2) transitional: the community participates, resulting in greater trust, but the benefits are still differentiated; and (3) transformational: the business and the community become integrated to generate change, joint learning occurs, and the benefits are shared.
Thus, corporate sustainability refers to a firm that minimizes its impact and has an inclusive relationship with all of his stakeholders, sharing its created value for a common well-being [40]. In addition, the future performance of corporate sustainability also depends on the sustainability of the territory in which the companies are not necessarily central players as defined by business ethics [41]. Hence, territorial sustainability is more closely related to politics as the business is increasingly forced to negotiate its operation. Therefore, politics in business is perceived as being related mostly to external and connecting factors of corporate sustainability as proposed by Lozano [16].
This negotiation seeks institutional arrangements that reduce transaction costs and increase the persuasion of players to achieve mutual expediency of territorial sustainability [42, 43]. A territorial understanding of sustainability is thus necessary, as it is here that the resources (raw material, infrastructure, social capital, etc.) that make the business viable and provide for the population's livelihood are located. Therefore, these spaces for social construction must be equipped with political technologies that allow resources to be measured and their exploitation to be controlled [44]. This way, institutional arrangements will become political technologies if measures are put in place by which they may be evaluated.
Thus, collaborative governance emerges as a form of network organization in which the agents first seek to build trust and a shared understanding of the issues involved by means of deliberation and negotiation [45]. They subsequently seek to obtain resources and build skills and leadership, which translates into commitment to finding a joint solution to problems for which a unilateral solution is difficult to achieve [45, 46].
Collaborative governance is defined by Emerson et al. ([19], p. 2) as "the processes and structures of public policy decision making and management that engage people constructively across the boundaries of public agencies, levels of government, and/or the public, private and civic spheres in order to carry out a public purpose that could not otherwise be accomplished". According to these authors, the products of collaborative governance emerge from three types of inter-organizational dynamics:
Principled commitment – this describes how participants interact regarding shared interests and individual objectives, using face-to-face communication and achieving their own processes and arriving at meaningful decisions;
Shared motivation – this considers the dynamics and collaborative processes involving understanding, trust, legitimacy, and commitment;
Capacity for joint action – this is characterized by the resources necessary for sustaining the collaborative process and includes the leadership, process structure, rules of the game, quality information, budget, and time available to the collaborators.
There is a normative appeal of collaborative governance to solve complex public problems or deliver public goods and services, such as access to natural resources and energy supply, because these problems can be neither understood nor addressed by a single organization [22, 47]. Through collaborative governance, firms would be able to surmount institutional constraints of corporate sustainability [20], such as social pressure, reputational effects of environmental impacts, or any other interdependency with stakeholders [21, 22].
With this being the case, long-term business performance requires a company organization that links economics to the environment and society not only in a corporate system [17] but also with regard to ethics and politics [32, 46]. This is a challenge that would connect the businesses to their surroundings in a manner that is sensitive to contexts of violence, such as in the case with the hydropower industry in Colombia.
The case of the hydropower industry in Colombia
Hydropower project development is currently occurring primarily in emerging economies in Asia, South America, and Africa, although the Balkans and the Caucasus region also figure as centers of development [6, 48, 49]. These regions possess abundant natural resources, access to which has been the cause of armed conflict and violence in cases such as those of Myanmar in Asia [7], Angola and the Democratic Republic of Congo in Africa [8], and Colombia in South America [5, 15].
According to the studies by Le Billon [9] and Springer and Le Billon [10], it is common to encounter greater state control of resources that are concentrated geographically, such as petroleum and minerals, when compared with that of scattered resources, such as agricultural land holdings. In addition, insurgent leaders may appear who maximize the return on controlling these resources consequent to their practice of extortion.
Colombia possesses a hydropower potential of approximately 56.2 GW, one of the greatest in South America, and this potential is being exploited mostly by stations with a capacity of more than 1 MW [6, 1].
The current total capacity of all installed plants is 16.9 GW, of which 69.8% came from large hydropower stations [50]. Approximately, a third of this capacity was developed after 1980, the year in which the violence in the country greatly intensified in terms of the population that was forcibly displaced (Fig. 1).
The number of people who have been displaced by the violence in Colombia – This displacement may have been caused by threats against one's life, family, or property or against the freedom of association and political participation
Source: National Government Victims Unit [2] and the ONG Consultancy for Human Rights and Displacement [3].
The recent violence in Colombia is a result of armed conflict waged for the purpose of territorial control (access to natural resources, strategic infrastructure, illegal economies, etc.) among guerillas, paramilitary forces, and legitimate state authorities (the army and the police). The forced displacement of populations is an aspect of this violence that is frequently trivialized and reduced to mere discussion of statistics and data [51]. However, it is also seen as evidence of the dynamics of the armed conflict [52, 12].
Forced displacement uproots people from their homes, breaks relationships of trust and neighborliness, and limits opportunities for work and subsistence [53, 52]. Territorial occupation, the limitation of collective action, and the destruction of the fabric of society are strategies of warfare that dictate forced displacement [12].
The numbers involved in this forced displacement reveal a dynamic of the conflict that is characterized by four key stages in connection with difficulties in the peace processes [4, 5, 54]:
Guerilla expansion after the collapse of peace talks in 1984 during the government of Belisario Betancur (1982–1986) – this resulted in an increase in the mobilization of military forces from the periphery to areas of the country that were more integrated economically, as well as to large population centers.
The emergence of paramilitary forces in the 1990s in reaction to guerilla expansion – the confrontation between paramilitary forces and guerillas and the collapse of peace talks during the government of Andrés Pastrana (1998–2002) led to the greatest geographical expansion of the conflict, affecting more than half of the country.
The retreat of guerillas – as a result of joint efforts by the paramilitary forces and public forces during the government of Álvaro Uribe (2002–2010) and the consolidation of USA support through the Colombia Plan, the intensity of the violence decreased and part of the paramilitary forces demobilized in 2006, which was followed by the emergence of the phenomenon of "parapolitics," in which members of the Congress allegedly had ties with paramilitary groups.
The reorganization of guerilla forces and paramilitary groups - This came about as a result of the polarization unleashed by the Peace Accord signed in 2016 between the government of Juan Manuel Santos (2010–2018) and the leading guerilla group. The Peace Accord was endorsed by Congress after the failure of the referendum, difficulties appeared in its implementation, and violence returned to the country's periphery.
This dynamic of armed conflict makes the territory insecure, reduces the population to a condition of vulnerability, and hinders, even more, the presence of the state and energy companies. Attacks on electric power transmission networks, oil pipelines, and road infrastructure are frequent; massacres are committed; and electoral processes are hindered [4, 5]. This has resulted in increased pressure on the population, with the number of people forcibly displaced between 1985 and 2017 being approximately 7 million [2, 3].
A large part of Colombia's hydropower infrastructure is located in areas that are inhabited by vulnerable populations, that have little state presence, and that are extremely rich in natural resources. Before the year 2000, large hydropower projects were commonly developed despite rejection by the local populations because of ensuing evictions, little community consultation and participation, and few economic and social benefits [55]. This resulted in social movements that were exploited by guerillas in their territorial advance by their appropriating popular demands and capturing resources that have been acquired by exploiting natural resources [5].
After 2000, energy companies strengthened their position in the territory before, after, and during the construction of large hydropower facilities not only as a result of adaptation to the contexts of violence but also as a consequence of stronger legislation and requirements by multilateral development banks [55].
Initially, the mindset of energy companies is pragmatic, reacting to the law and to conditions for accessing the capital market [56]. They then become more proactive, investigating voluntary initiatives that allow them to protect their reputations and better adapt to violent contexts [13, 14], such as the management of environmental externalities, human rights, employment, and community development.
As is seen from multiple experiences from around the world [11], the management of hydropower facilities tends to consider, on the one hand, participation as a key factor in evaluating social impact and as a structural base for the shared benefit that is sought using this power generation technology. On the other hand, environmental management plans have increasingly consolidated as tools that are integrated with the social aspect in such a way that they contribute to compliance with the development policies of the sector and regulations of each country.
This holistic view of the business then requires it to consider governance for sustainability in the business realm, where collaboration with the surrounding area is the determining factor. As mentioned above, the products of this collaboration are the result of principled commitment, shared motivation, and capacity for joint action [19, 46]. Therefore, the identification of these products allows one to investigate the sustainability of the hydropower-generation company in contexts of violence as shown below.
The goal of this study is to analyze governance for the sustainability of power generation businesses in contexts of recurring violence using a case study from the power industry to explore the emergent issues, dominant players, and problem-solving tools used.
The exploratory case study method employed by Yin [23] is used, considering its application in similar studies [43, 57]. In view of the scarcity of information available and the interplay of interests involved, it was necessary to interview different players and triangulate the resulting data using secondary information. The paragraphs below present the data characteristics as their analysis strategy.
This methodology is based on the hermeneutical analysis of 16 in-depth interviews with employees from the energy sector (7 interviews), public municipal employees (4 interviews), and local leaders (5 interviews). The employees from the sector are from one of the country's main private power generation companies, the hydropower stations of which are located in areas with a history of violence. The public municipal employees and local leaders belong to the sphere of influence of these power stations in areas that have experienced armed conflict involving varying levels of intensity with regard to the forced displacement of the population.
Open interviews were preferred, with the aim of "investigating a problem and understanding it as it is conceived and interpreted by the subjects being studied, without imposing preconceived categories" ([58], p. 93). Therefore, the understanding of governance for corporate sustainability is inductive.
The interviews were structured according to three analytical categories (issues, players, and tools), and the questions were asked to gather information on the subcategories therein (Table 1). Then, the interviews were recorded, transcribed, and encoded with the aid of Atlas.ti software. Subsequently, 200 codes were selected and grouped into the three analytical categories:
Issues (88 codes): These emphasize the type, problem, and purpose of the actions carried out by the players.
Players (60 codes): These refer to the agents who either live or operate in the area of influence of the hydropower stations and represent the production sector, state organizations, and organizations from civil society.
Tools (52 codes): These bring together the means used by the agents to carry out their roles and evaluate their operation in the area of influence of the hydropower stations.
Table 1 Descriptions of issues, players, and tools for territorial intervention
These analytical categories were selected taking into consideration the holistic focus on sustainability [13, 59], the continuous improvement approach of corporate management [60, 61], and the productive performance focus on collaborative governance [19, 47]. First, the holistic perspective of sustainability concerns the triple bottom line (economy, society, and environment) and the interlinkages between dimensions. Second, the continuous improvement approach looks for the efficiency of planning, execution, and evaluation tools. Third, the performance of collaborative governance takes into account actions and outcomes of the principled commitment, the shared motivation, and the capacity for joint action of energy firms' stakeholders.
The encoding allowed citation frequencies to be estimated, which, in turn, were used to prioritize each issue in each of the analysis categories. The citation frequencies were used as ranking criteria—the more cited an issue the more important it is considered. Next, the citation frequencies were normalized using the minimum-maximum procedure according to [62, 63]:
$$ {C}_N=\frac{C-{C}_{\mathrm{min}}}{C_{\mathrm{max}}-{C}_{\mathrm{min}}} $$
In which:
CN: normalized code within the range (0,1)
C: citation frequency of non-normalized code
Cmin: minimum citation frequency of non-normalized code
Cmax: maximum citation frequency of non-normalized code
Once the codes were normalized, they were compared according to the perceptions of the company and the other territorial players. In this last instance, the perceptions of the local leaders and public employees at the municipal level were used. In addition, co-occurrence matrices were constructed by crossing both the "issues" and "players" categories and the "players" and "tools" categories. This type of data analysis provides the holistic and systemic vision required by the sustainability study [64, 16], estimating the ranking of the interests of the players and the tools of action they prefer.
The results are presented in three parts. First, the respective rankings for the issues, players, and instruments are compared according to the perceptions of the company and the other territorial players. Second, the ranking of interests and tools associated with each player is estimated. Third, the main statements made by the hydropower-generating company, public municipal employees, and local leaders concerning hydropower plant management and its relationship with violence are highlighted.
The environment is the most relevant issue, followed by economy-society and society. The importance of the issue of the environment is associated with the negative impacts of hydropower-generation on water flow downstream from the dam. The issue of economy-society appears to emerge because of transportation problems affecting economic and productive activity in the territory (livestock, fishing, and agriculture).
The issue of society emerges as a result of the relationship between the local community, state organizations, and the energy company, in which context training, trust, and the construction of social capital appear as important themes. Although of less importance, the issue of economy-environment also features because of soil use that may cause deforestation, erosion, and sediment buildup. In addition, the theme of society-environment seems to demonstrate the importance of training and environmental consciousness in the territory. All these items are understood as being nonmonetary benefits. The issue of the economy has little importance for the interviewees and relates to financial resources for monetary and nonmonetary compensation.
Taking this into consideration, the state organizations (particularly the regional environmental authority), the energy company, and nongovernmental organizations (NGOs) appear as dominant players. All these players primarily make use of planning but also use execution and evaluation tools. The most prominent among the planning tools may be plans and projects that seek to coordinate players and integrate operations involving social, economic, and environmental issues. In contrast, execution tools are oriented toward the formation of network-building alliances and partnerships and not so much execution committees. Finally, evaluation tools appear to be limited to conducting technical studies; establishing mechanisms for communication; and to a lesser extent, audits and the establishment of measurement parameters.
It is clear that this evidence involving issues, players, and tools is dependent on the perception of the energy company and the territorial agents, including municipalities and local leaders (Fig. 2). From a territorial standpoint, the most relevant issue should be that of economy-society as a result of difficulties in carrying out productive activities. However, the company appears to consider the issue of the environment to be more significant, possibly due to the impact of the dam on the water flow or its effects on the sediments in the reservoir.
Issues, players, and instruments, according to the perceptions of the territorial players and the energy company. Ranking based on the occurrence rates
The issue of society seems to have the same relative importance for the company and the other territorial players, meaning that the relationship between them would be a determining factor because of the need for trust. The state appears to be the most important player for both. However, the company seems to give more importance to NGOs most likely due to the leadership of the Catholic Church in the Peace and Development Programs (PDP). Municipalities and local leaders give more importance to local communities because of their need for training and social and productive organization.
Finally, no differences were observed between the perception of the business and other territorial players regarding the ranking of the tools. However, it is likely that preferences do exist among the players concerning issues and tools used for their management as shown below.
Issues and tools of the players
According to the results of the co-occurrence matrices, the state entities appear to be concerned mainly about the issues of society, the environment, and economy-environment. This relationship has come about probably because of the importance that regional environmental authorities attach to protecting natural resources and to their relationship with other territorial players (Fig. 3). The state entities appear to make use of all three tools–planning, execution, and evaluation (Fig. 4).
The emergent issues according to the territorial players. Ranking based on the co-occurrence matrices
The intervention tools according to the territorial players. Ranking based on the co-occurrence matrices
As for the energy company, it appears to prioritize the issues of the economy, economy-society, and society-environment perhaps due to the importance attached to the business of the monetary or nonmonetary compensation fees during the operation of hydropower stations (Fig. 3). This player appears to use the tools of evaluation and planning but not so much execution in the continuous improvement process of its management (Fig. 4).
As for NGOs, the emerging issues appear to be the environment and economy-society perhaps because of their commitment to protecting the environment and the development of productive community activities. The issue of society may also be prominent due to the trust-building efforts of the PDP (Fig. 3). Thus, these organizations would be more involved in plans and alliances than they would in intervention committees (Fig. 4).
The organizations for development and the local communities would be marginalized in the interplay between players, participating tangentially as executors of projects in the areas of economy-society and society (Fig. 3). Perhaps this is why they are only visibly using tools for execution (Fig. 4). Universities would be interested mainly in the issue of economy-environment (Fig. 3), contributing modestly in the form of technical studies (Fig. 4).
In summary, state organizations, the energy company, and NGOs would dominate the interplay between players, mainly in the issues of the environment, economy-society, and society. The players would be making the most use of planning tools, thus revealing limitations involving execution and evaluation. However, this situation begins to become differentiated when the perceptions of the company, municipality, and local leadership are analyzed. This differentiation illustrates each player's own concerns although all players would appear to show the importance of their relationships in view of the need for trust. The following section presents a more in-depth examination of the interviews to illustrate the relationship of these findings with contexts of violence.
Statements regarding governance for sustainability in contexts of violence
The hydropower-generation business encounters the most difficulty because of two principal consequences of armed conflict and violence—the lack of trust and the lack of opportunities for the population. First, the hydropower facility is built on a territory that was controlled by guerillas in the immediately preceding period and has historical and social debts as a result of state absence.
This situation creates an expectation for compensation on the part of the local community, which is what the energy company in question was apparently unwilling to satisfy, imputing responsibility to the state in some cases or with humanitarian efforts in others: "The construction of the hydropower plant generated expectations in the community that the company did not accept because it considered them to be the responsibility of the state or humanitarian programs." (energy company's statements).
The same situation may occur in the construction of auxiliary infrastructure (transfers) as the local community expresses displeasure with the environmental impacts and their management on the part of the company: "Tension exists between the communities and the energy company due to the reduction of water flow in some gorges, which is perceived to be the result of the construction of transfers that feed into the reservoir. However, the energy company is working with the communities to resolve the problem." (municipalities' statements).
Presumably, included in this opposition were community organizations that were sympathetic to leftist political ideologies, which they had probably inherited from the guerillas: "The local communities have lived together with two players that have transformed the territory. First, insurgent groups occupied the spaces left by the State. Second, the energy company builds the hydropower facility and the transfers. Mistrust and opposition exist in local communities toward the energy company, in some cases led by leftist organizations." (local leaders' statements).
The distrustfulness is probably one of the main issues of hydropower governance in contexts of violence. On the one hand, it might be the consequence of a project's life cycle assessment without a community participation in the construction stage: "The mistrust is also based on the lack of communication, particularly in the transfer construction stage." (local leaders' statements). On the other, however, it could also be seen as a result of both the armed conflict and the historical and social debts: "The building of trust must confront historical and social debts, but the company progresses with information, participation, and community development programs" (energy company's statements). Moreover, "After armed conflict and the construction of the hydropower facility, development and peace programs appeared, which were led by the Church, bringing together local communities, the State, and the private sector." (local leaders' statements).
Second, the lack of opportunities is presumably related to land access, economic activities, and productive organization in a climate of illegality. According to the statements of the energy company and the municipalities, economic activities are being developed with the support of the central government: "Community economic activities are encouraged in association with the municipalities, from production to commercialization. Attempts are being made to use the reservoir for the development of tourism by the communities." (energy company's statements). Additionally, "Mechanized farming development, agricultural inputs, and technical assistance are being supported. This work is carried out mainly by an alliance with the Ministry of Agriculture and also with energy companies in some cases." (municipalities' statements). However, absentee landowners of large territories (mostly livestock breeders) are observed along with small farmers who sometimes do not have deeds to their land and encounter difficulties in production and commercialization. Artisanal fishermen also exist, who occupy state or private territory, particularly riverbanks: "Both large and small landowners exist. The large landowners usually work with livestock, and neither live nor are interested in the area. The small landowners are farmers and fishers who have a subsistence economy. Many of the fishers are not landowners and occupy de facto state and private property. A large part of the work of those who dwell in the country is informal; they are usually paid on a daily basis." (local leaders' statements).
The illegality appears to be related not only to land ownership but also to economic activity as several statements call attention to the presence of illicit crops and illegal mining tied to criminal gangs: "Community-based ecotourism exists although its organization is incipient. These communities have difficulties with the energy company due to the river flow management. The guerillas had ties with the livestock business." (municipalities' statements). Furthermore, "The state and energy companies are finding it difficult to support these country economies due to the low level of local capabilities. In addition, illegal mining is carried out by criminal gangs in the context of insecurity." (local leaders' statements).
Thus, the main difficulty of this situation appears to be the lack of action and evaluation. Evidence presumably exists for integral and inter-institutional planning exercises geared mostly toward building trust and creating new opportunities for the population: "The institutional relationship has improved as a result of new integrated action plans, in which the municipality, regional environmental authority, energy company, and the Church participate. As a result of this relationship, economic, social, and environmental projects are formulated in which community organizations participate." (municipalities' statements).
Even so, the institutional relationship has not improved as much as it has in others hydropower plants where the energy company had had more experience in the territory: "Collaborative networks are being built through inter-institutional projects although conflicts of interest exist with regional environmental authorities. The results are not as visible as they are in other areas where the business has been established for more time because mistrust is still instilled in some cases by social leaders with leftist political ideology." (energy company's statements).
The power company appears to be preoccupied with the increase in compensation costs, and presumably, there are still no parameters in place for measuring and evaluating the impact of the investments in the territory: "The cost of the compensations has increased and the company does not yet have indicators that allow it to measure the impact of its management in the area of influence of the hydropower facility." (energy company's statements).
On the one hand, state organizations appear to depend on these resources: "Communities are informed concerning how to use monetary compensations (transfers from the electric sector by law) of which the municipalities are beneficiaries, which is not welcomed by the municipalities because it reduces their control of the budget." (energy company's statements). On the other, local leaders appear to have the sense that the resources are being used inefficiently: "The energy company generated many jobs directly during the construction of the hydropower facility and its transfers. During operation, the job offer reduced considerably, being limited to the management of environmental impact." (local leaders' statements).
With this being the case, the hydropower-generation company's governance for sustainability appears to be occurring in the context of mistrust and lack of opportunity, seeking as it does to coordinate interests between players with sharp differences in resources and capacity.
Some of the elements observed in this study are also found in the context of hydropower development in other countries, particularly regarding the economic, social, and environmental impact, as well as some problematic areas for their management by the power companies. However, the context of violence makes Colombia a territory in which these companies are experiencing greater interdependence with the dynamics of the surroundings. It is perhaps for this reason that they presumably need to build new organization models that are more collaborative despite the mistrust and differences emanating from other players with regard to resources and capacity.
The fragmentation of river flow; the limits put to access to natural resources, such as fish; and the disproportionate distribution of the costs and economic benefits between the energy companies, the state, and the local communities are some of the developmental impacts resulting from hydropower development in Colombia that also exists in other countries [6, 11].
Although in this case, managing these impacts involves compliance with regulations and the use of international standards such as the sustainability protocol of the International Hydropower Association [13], difficulties are observed in other aspects, such as the fragmentation of the life cycle of hydropower projects (construction problems inherited by the operation stage); nonexistent or uncertain data for making decisions; and dependence on technical solutions to the detriment of more holistic solutions [65, 57].
In addition, the management of hydropower projects in contexts of violence from armed conflict also faces the lack of trust, the lack of opportunities, illegality, and weakness on the part of state organizations. As in other cases [9, 10], the control of the state is visible only in the presence of the army at the hydropower facility because of the likely presence of illicit crops and illegal mining in the areas surrounding the towns. It is probably for this reason that the presence of the army does not necessarily contribute to the security of the local community as explained by Ibáñez and Vélez [12].
Other state entities, such as municipalities, are growing stronger as a result of NGO support led by the Catholic Church and financed by the national private sector, particularly by the energy company. This kind of alliance between NGOs and the energy company is common in Latin America [14], whereas in Asian countries such as China [57] and India [66], it is seen as an aspect that needs to be improved because of NGOs' opposition to hydropower development.
By occupying a marginal role between players in the territory, local communities appear not to be affecting state entities as appears to be the case in other countries with armed conflict, such as Myanmar [7]. In actuality, the lack of opportunities and mistrust appear to limit the development of community organization systems based on neighborliness and solidarity [67].
As the end of the armed conflict is being discussed and many displaced families are returning to their territories [4], the reorganization of guerilla and paramilitary forces is being observed as a result of difficulties in the legitimization and implementation of the La Habana Peace Accord with the main guerilla group in the country [54]. Daniel Pécaut, in an interview by Valencia Gutiérrez [54], argues that in this context, the state must respond to the challenge to go through with reforms that not only fulfill the commitments of the La Habana treaty but also address the great inequality that has characterized Colombia. According to Pécaut, the presence of state institutions throughout the territory must be a priority so as to accomplish this goal.
Although the energy company may be neither able nor willing to take on responsibilities of the state, the history of state absence and the marginalization of the local communities appear to force the energy industry into a key position in the construction of these institutions. Nevertheless, Duarte-Abadía et al. [15] indicate that this centrality could potentially maintain the asymmetry between the players if it is exploited to socially legitimize the hydropower company rather than to create mechanisms that more equally distribute the costs and benefits of the infrastructure.
In one way or another, the context of violence and the centrality of the energy company appear to be defining the interdependence of the power generation company with the territory in which the hydropower facility is located. This interdependence appears to be reflected also in the emergent products of collaborative governance as defined by the inter-organizational dynamic [19, 45, 46]. Also being developed are plans and projects, which are associated with principled commitment and shared motivation, more than alliances, partnerships, technical studies, and mechanisms for communication, which are more associated with the capacity for shared action.
In other words, on the one hand, territorial planning is presumably being carried out in an integrated and inter-institutional manner to overcome the differences between the players. On the other hand, however, illegality, mistrust, the transactional relationship, inefficient use of resources, and the lack of measures are limiting commitment, execution, and evaluation. These limitations are presumably reducing opportunities for transformational learning as is also the case in India [66, 68], given the continuing difference between costs and benefits and opposition from local communities. However, other cases stand out, mainly from Southeast Asia [69, 68], where the difficulties involving inclusion that are characteristic of multilateral partnerships are being overcome by the construction of regional and polycentric mechanisms for coordination and participation.
Hydropower-generation companies in contexts of violence due to armed conflict such as that occurring in Colombia appear to be intensifying their interdependence with the territories in which the hydropower facilities operate. This seems to be emerging as a result of collaborative governance in which the products contribute more to planning than they do to joint execution and evaluation.
Theoretically, this approximation toward the issue of governance for the sustainability of business reveals the factors for corporate sustainability proposed by Lozano [16] and its relationship with collaborative governance as put forth by Emerson et al. [19]. On the one hand, factors of sustainability appear to explain the link between the performance of energy companies and the sustainability of the territories in which hydropower facilities operate. However, this link has been dealt with in this work more from the standpoint of ethics [33, 34] and politics [41] than from that of the corporate system [17]. It has been dealt with more from an ethical and political perspective because of the evidence of problems involving legitimacy (trust) and power (centrality) on the part of the business considered in this study in the surroundings. The corporate system perspective was not emphasized because the configuration of the strategy and its relationship to the structure and corporate culture lie outside the scope of this study.
On the other hand, responsibility, access to resources, regulations, and the socio-environmental impact of the company under consideration were the main corporate sustainability factors that allowed a conceptual association with collaborative governance. Consequently, it was possible to propose a theoretical relationship between business sustainability and the inter-organizational dynamic characteristic of the surroundings with regard to principled commitment, shared motivation, and capacity for joint action. This relationship proved to be very useful in conducting the search for governance products. However, as established by the works of Ulibarri [45] and Ulibarri and Scott [46], it would be necessary to analyze the organizational networks to explain how collaborative governance could translate into joint action.
Thus, in practice, the features of collaborative governance observed here appear to suggest voluntary actions on the part of the energy company as a product of deliberation, negotiation, and even lobbying with state organizations [70]. However, hydropower development and the evolution of the armed conflict appear to be placing the company in a key position amid illegality, a state whose territorial possession is still weak, apparently inefficient use of resources, and little social capital for collaborative work.
To adapt to this context, collaborative governance, on the one hand, should be geared more toward results [47], requiring political technologies that contribute to the measurement and evaluation of the impact of the management [71, 44]. On the other hand, it should be directed toward reducing the asymmetry that exists between the players by building competencies and leadership [46] as well as toward distributing the costs and benefits of hydropower-generation in a more equitable manner [15].
A more equitable benefit distribution by collaborative actions is one of the most important hydropower sustainability issues in conflict areas, having also implications in the energy market because of its effects in the local security of energy supply [72]. Colombia does not have any energy deficit according to the public information [50], but most of the energy supply depends on power plants located in conflict areas. Thus, local security of energy supply should be addressed with the commitment of local community, state entities, and energy firms if more large dams are to be built or continue to operate in conflict areas.
Due to the exploratory nature of this case study, the presentation has been limited to hypothetical assertions concerning governance for sustainability in the hydropower generation business. These assertions are capable of defining the problem and providing clues for its solution.
However, it is suggested that future investigations analyze the incidence of inter-organizational dynamics on the corporate system; expand the discussion to include other case studies; and the opinion of academics, NGOs, and state organizations both on the regional and national levels. In addition, the use of quantitative studies would contribute toward explaining the extent to which the sustainability of the power generation business depends on the territory, as well as the contribution of collaborative governance products to joint action.
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Thanks are due to the students from the doctorate program of Management of the University of Medellín, who supported the data collection process and on-site field interviews, especially to Juliana Ceballos. Thank you very much to my colleague Fabián Ramírez. I am also grateful to Dr. Stephen Sparkes from Statkraft, Norway, who gave me a very important feedback of the manuscript.
This work was supported by the Colombian state agency of science - Colciencias, the University of Medellín, and the energy company (FP44842-034-2016).
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
Associate Professor of the Doctorate Program in Management, University of Medellín, Carrera 87 no. 30–65, Medellín, 050026, Colombia
Jorge-Andrés Polanco
The author read and approved the final manuscript.
Correspondence to Jorge-Andrés Polanco.
All respondents in the study have been informed about the usage of the information they provide through interviews and have given their consent to participate in the study.
The respondents of the study have given their consent for the data to be used and published in this scientific article.
The author declares that he has no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Polanco, JA. Exploring governance for sustainability in contexts of violence: the case of the hydropower industry in Colombia. Energ Sustain Soc 8, 39 (2018). https://doi.org/10.1186/s13705-018-0181-0
Hydropower industry | CommonCrawl |
A truncated real interpolation method and characterizations of screened Sobolev spaces
Huiying Fan 1,, and Tao Ma 2,
School of Mathematical Science, Zhejiang University, Hangzhou 310027, China
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Fund Project: The second author was supported by National Natural Science Foundation of China (Grant Nos. 11671308, 11971431)
In this paper, we study evolution equation $ \partial_t u = -L_\alpha u+f $ and the corresponding Cauchy problem, where $ L_\alpha $ represents the Laguerre operator $ L_\alpha = \frac 12(-\frac{d^2}{dx^2}+x^2+\frac 1{x^2}(\alpha^2-\frac 14)) $, for every $ \alpha\geq-\frac 12 $. We get explicit pointwise formulas for the classical solution and its derivatives by virtue of the parabolic heat-diffusion semigroup $ \{ e^{-\tau(\partial_t+L_\alpha)}\}_{\tau>0} $. In addition, we define the Poisson operator related to the fractional power $ (\partial_t+L_\alpha)^s $ and reveal weighted mixed-norm estimates for revelent maximal operators.
Keywords: Parabolic evolution equation, parabolic heat semigroups, Laguerre operator, mixed-norm estimates.
Mathematics Subject Classification: Primary: 35C15, 47D03; Secondary: 42B25.
Citation: Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249
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Huiying Fan Tao Ma | CommonCrawl |
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von)[lower-alpha 1] Leibniz[lower-alpha 2] (1 July 1646 [O.S. 21 June] – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is a prominent figure in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history and philology. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. In addition, he contributed to the field of library science by devising a cataloguing system whilst working at Wolfenbüttel library in Germany that would have served as a guide for many of Europe's largest libraries.[17] Leibniz's contributions to a wide range of subjects were scattered in various learned journals, in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primarily in Latin, French and German.[18][lower-alpha 3]
Gottfried Wilhelm Leibniz
Portrait by Christoph Bernhard Francke, 1695
Born1 July 1646
Leipzig, Saxony, Holy Roman Empire
Died14 November 1716(1716-11-14) (aged 70)
Hanover, Hanover, Holy Roman Empire
Education
• Alte Nikolaischule
(1655–1661)
• Leipzig University (1661–1666:
• B.A. in philosophy, December 1662
• M.A. in philosophy, February 1664
• LL.B., September 1665
• Dr. phil. hab., March 1666)
• University of Jena
(summer school, 1663)[1]
• University of Altdorf
(Dr. jur., November 1666)
Era17th-/18th-century philosophy
RegionWestern philosophy
School
• Rationalism
• Pluralistic idealism[2]
• Foundationalism[3]
• Conceptualism[4]
• Optimism
• Indirect realism[5]
• Correspondence theory of truth[6]
• Relationism
Theses
• De Arte Combinatoria (On the Combinatorial Art) (March 1666)
• Disputatio Inauguralis de Casibus Perplexis in Jure (Inaugural Disputation on Ambiguous Legal Cases) (November 1666)
Doctoral advisorBartholomäus Leonhard von Schwendendörffer (Dr. jur. thesis advisor)[7][8]
Other academic advisors
• Erhard Weigel (Jena)[1]
• Jakob Thomasius (B.A. advisor)[9]
• Christiaan Huygens
Notable students
• Jacob Bernoulli (epistolary correspondent)
• Christian Wolff (epistolary correspondent)
Main interests
Mathematics, physics, geology, medicine, biology, embryology, epidemiology, veterinary medicine, paleontology, psychology, engineering, linguistics, philology, sociology, metaphysics, ethics, economics, diplomacy, history, politics, music theory, poetry, logic, theodicy, universal language, universal science
Notable ideas
• Algebraic logic
• Binary code
• Calculus
• Differential equations
• Mathesis universalis
• Monads
• Best of all possible worlds
• Pre-established harmony
• Identity of indiscernibles
• Mathematical matrix
• Mathematical function
• Newton–Leibniz axiom
• Leibniz's notation
• Leibniz integral rule
• Integral symbol
• Leibniz harmonic triangle
• Leibniz's test
• Leibniz formula for π
• Leibniz formula for determinants
• Fractional derivative
• Chain rule
• Quotient rule
• Product rule
• Leibniz wheel
• Leibniz's gap
• Algebra of concepts
• Vis viva (principle of conservation of energy)
• Principle of least action
• Salva veritate
• Stepped reckoner
• Symbolic logic/Boolean algebra
• Semiotics
• Analysis situs
• Principle of sufficient reason
• Law of continuity
• Transcendental law of homogeneity
• Ars combinatoria (alphabet of human thought)
• Characteristica universalis
• Calculus ratiocinator
• Compossibility
• Partial fraction decomposition
• Protogaea
• Problem of why there is anything at all
• Pluralistic idealism
• Metaphysical dynamism
• Relationism
• Apperception
• A priori/a posteriori distinction
• Deontic logic
• Well-founded phenomenon
Influences
• Confucius
• Plato
• Aristotle
• Euclid
• Archimedes
• Apollonius
• Llull
• Suárez
• Grotius
• Hobbes
• Descartes
• Pallavicino[10]
• Pascal
• Bossuet
• Conway
• Spinoza
• Malebranche
Influenced
• Berkeley, Platner, Voltaire, Hume, Wolff, Kant, Émilie du Châtelet, Wiener, Riemann, Gibbon, Gauss, Lagrange, Euler, Boole, Newman, Peirce, Frege, Russell, Rousseau, Gödel, Tarski, Mandelbrot, Blondel, Heidegger, Ankersmit, Deleuze, Wundt, Rescher, Rauschenbusch[11]
Signature
Part of a series on
Theodicy
Key concepts
• Augustinian theodicy
• Irenaean theodicy
• Problem of evil
• Free will
• Absence of good
• Epicurean paradox
• Natural evil
• Moral evil
• Divine retribution
• Best of all possible worlds
• Inconsistent triad
• Dystheism
• Misotheism
Notable figures
• Saint Augustine
• Saint Irenaeus
• Gottfried Wilhelm Leibniz
• John Hick
• Alvin Plantinga
• William Rowe
• Friedrich Nietzsche
• David Hume
• Epicurus
• Bart D. Ehrman
• Elie Wiesel
• Richard Swinburne
As a philosopher, he was a leading representative of 17th-century rationalism and idealism. As a mathematician, his major achievement was the development of the main ideas of differential and integral calculus, independently of Isaac Newton's contemporaneous developments.[20] Mathematicians have consistently favored Leibniz's notation as the conventional and more exact expression of calculus.[21][22][23]
In the 20th century, Leibniz's notions of the law of continuity and transcendental law of homogeneity found a consistent mathematical formulation by means of non-standard analysis. He was also a pioneer in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685[24] and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is the foundation of nearly all digital (electronic, solid-state, discrete logic) computers. This includes the Von Neumann architecture, which represents the standard "computer architecture" through from the second half of the 20th century to the present. Leibniz has been called the "founder of computer science".[25]
In philosophy and theology, Leibniz is most noted for his optimism, i.e. his conclusion that our world is, in a qualified sense, the best possible world that God could have created, a view sometimes lampooned by other thinkers, such as Voltaire in his satirical novella Candide. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three influential early modern rationalists. His philosophy also assimilates elements of the scholastic tradition, notably the assumption that some substantive knowledge of reality can be achieved by reasoning from first principles or prior definitions. The work of Leibniz anticipated modern logic and still influences contemporary analytic philosophy, such as its adopted use of the term "possible world" to define modal notions.
Biography
Early life
Gottfried Leibniz was born on July 1, 1646, toward the end of the Thirty Years' War, in Leipzig, Saxony, to Friedrich Leibniz and Catharina Schmuck.
Friedrich noted in his family journal:
21. Juny am Sontag 1646 Ist mein Sohn Gottfried Wilhelm, post sextam vespertinam 1/4 uff 7 uhr abents zur welt gebohren, im Wassermann.
In English:
On Sunday 21 June [NS: 1 July] 1646, my son Gottfried Wilhelm was born into the world a quarter before seven in the evening, in Aquarius.[26][27]
Leibniz was baptized on 3 July of that year at St. Nicholas Church, Leipzig; his godfather was the Lutheran theologian Martin Geier.[28] His father died when he was six years old, and from that point on, Leibniz was raised by his mother.[29]
Leibniz's father had been a Professor of Moral Philosophy at the University of Leipzig, and the boy later inherited his father's personal library. He was given free access to it from the age of seven. While Leibniz's schoolwork was largely confined to the study of a small canon of authorities, his father's library enabled him to study a wide variety of advanced philosophical and theological works—ones that he would not have otherwise been able to read until his college years.[30] Access to his father's library, largely written in Latin, also led to his proficiency in the Latin language, which he achieved by the age of 12. At the age of 13 he composed 300 hexameters of Latin verse in a single morning for a special event at school.[31]
In April 1661 he enrolled in his father's former university at age 14,[32][1][33] and completed his bachelor's degree in Philosophy in December 1662. He defended his Disputatio Metaphysica de Principio Individui (Metaphysical Disputation on the Principle of Individuation),[34] which addressed the principle of individuation, on 9 June 1663. Leibniz earned his master's degree in Philosophy on 7 February 1664. In December 1664 he published and defended a dissertation Specimen Quaestionum Philosophicarum ex Jure collectarum (An Essay of Collected Philosophical Problems of Right),[34] arguing for both a theoretical and a pedagogical relationship between philosophy and law. After one year of legal studies, he was awarded his bachelor's degree in Law on 28 September 1665.[35] His dissertation was titled De conditionibus (On Conditions).[34]
In early 1666, at age 19, Leibniz wrote his first book, De Arte Combinatoria (On the Combinatorial Art), the first part of which was also his habilitation thesis in Philosophy, which he defended in March 1666.[34][36] De Arte Combinatoria was inspired by Ramon Llull's Ars Magna and contained a proof of the existence of God, cast in geometrical form, and based on the argument from motion.
His next goal was to earn his license and Doctorate in Law, which normally required three years of study. In 1666, the University of Leipzig turned down Leibniz's doctoral application and refused to grant him a Doctorate in Law, most likely due to his relative youth.[37][38] Leibniz subsequently left Leipzig.[39]
Leibniz then enrolled in the University of Altdorf and quickly submitted a thesis, which he had probably been working on earlier in Leipzig.[40] The title of his thesis was Disputatio Inauguralis de Casibus Perplexis in Jure (Inaugural Disputation on Ambiguous Legal Cases).[34] Leibniz earned his license to practice law and his Doctorate in Law in November 1666. He next declined the offer of an academic appointment at Altdorf, saying that "my thoughts were turned in an entirely different direction".[41]
As an adult, Leibniz often introduced himself as "Gottfried von Leibniz". Many posthumously published editions of his writings presented his name on the title page as "Freiherr G. W. von Leibniz." However, no document has ever been found from any contemporary government that stated his appointment to any form of nobility.[42]
1666–1676
Leibniz's first position was as a salaried secretary to an alchemical society in Nuremberg.[43] He knew fairly little about the subject at that time but presented himself as deeply learned. He soon met Johann Christian von Boyneburg (1622–1672), the dismissed chief minister of the Elector of Mainz, Johann Philipp von Schönborn.[44] Von Boyneburg hired Leibniz as an assistant, and shortly thereafter reconciled with the Elector and introduced Leibniz to him. Leibniz then dedicated an essay on law to the Elector in the hope of obtaining employment. The stratagem worked; the Elector asked Leibniz to assist with the redrafting of the legal code for the Electorate.[45] In 1669, Leibniz was appointed assessor in the Court of Appeal. Although von Boyneburg died late in 1672, Leibniz remained under the employment of his widow until she dismissed him in 1674.[46]
Von Boyneburg did much to promote Leibniz's reputation, and the latter's memoranda and letters began to attract favorable notice. After Leibniz's service to the Elector there soon followed a diplomatic role. He published an essay, under the pseudonym of a fictitious Polish nobleman, arguing (unsuccessfully) for the German candidate for the Polish crown. The main force in European geopolitics during Leibniz's adult life was the ambition of Louis XIV of France, backed by French military and economic might. Meanwhile, the Thirty Years' War had left German-speaking Europe exhausted, fragmented, and economically backward. Leibniz proposed to protect German-speaking Europe by distracting Louis as follows: France would be invited to take Egypt as a stepping stone towards an eventual conquest of the Dutch East Indies. In return, France would agree to leave Germany and the Netherlands undisturbed. This plan obtained the Elector's cautious support. In 1672, the French government invited Leibniz to Paris for discussion,[47] but the plan was soon overtaken by the outbreak of the Franco-Dutch War and became irrelevant. Napoleon's failed invasion of Egypt in 1798 can be seen as an unwitting, late implementation of Leibniz's plan, after the Eastern hemisphere colonial supremacy in Europe had already passed from the Dutch to the British.
Thus Leibniz went to Paris in 1672. Soon after arriving, he met Dutch physicist and mathematician Christiaan Huygens and realised that his own knowledge of mathematics and physics was patchy. With Huygens as his mentor, he began a program of self-study that soon pushed him to making major contributions to both subjects, including discovering his version of the differential and integral calculus. He met Nicolas Malebranche and Antoine Arnauld, the leading French philosophers of the day, and studied the writings of Descartes and Pascal, unpublished as well as published.[48] He befriended a German mathematician, Ehrenfried Walther von Tschirnhaus; they corresponded for the rest of their lives.
When it became clear that France would not implement its part of Leibniz's Egyptian plan, the Elector sent his nephew, escorted by Leibniz, on a related mission to the English government in London, early in 1673.[49] There Leibniz came into acquaintance of Henry Oldenburg and John Collins. He met with the Royal Society where he demonstrated a calculating machine that he had designed and had been building since 1670. The machine was able to execute all four basic operations (adding, subtracting, multiplying, and dividing), and the society quickly made him an external member.
The mission ended abruptly when news of the Elector's death (12 February 1673) reached them. Leibniz promptly returned to Paris and not, as had been planned, to Mainz.[50] The sudden deaths of his two patrons in the same winter meant that Leibniz had to find a new basis for his career.
In this regard, a 1669 invitation from Duke John Frederick of Brunswick to visit Hanover proved to have been fateful. Leibniz had declined the invitation, but had begun corresponding with the duke in 1671. In 1673, the duke offered Leibniz the post of counsellor. Leibniz very reluctantly accepted the position two years later, only after it became clear that no employment was forthcoming in Paris, whose intellectual stimulation he relished, or with the Habsburg imperial court.[51]
In 1675 he tried to get admitted to the French Academy of Sciences as a foreign honorary member, but it was considered that there were already enough foreigners there and so no invitation came. He left Paris in October 1676.
House of Hanover, 1676–1716
Leibniz managed to delay his arrival in Hanover until the end of 1676 after making one more short journey to London, where Newton accused him of having seen his unpublished work on calculus in advance.[52] This was alleged to be evidence supporting the accusation, made decades later, that he had stolen calculus from Newton. On the journey from London to Hanover, Leibniz stopped in The Hague where he met van Leeuwenhoek, the discoverer of microorganisms. He also spent several days in intense discussion with Spinoza, who had just completed his masterwork, the Ethics.[53]
In 1677, he was promoted, at his request, to Privy Counselor of Justice, a post he held for the rest of his life. Leibniz served three consecutive rulers of the House of Brunswick as historian, political adviser, and most consequentially, as librarian of the ducal library. He thenceforth employed his pen on all the various political, historical, and theological matters involving the House of Brunswick; the resulting documents form a valuable part of the historical record for the period.
Leibniz began promoting a project to use windmills to improve the mining operations in the Harz Mountains. This project did little to improve mining operations and was shut down by Duke Ernst August in 1685.[51]
Among the few people in north Germany to accept Leibniz were the Electress Sophia of Hanover (1630–1714), her daughter Sophia Charlotte of Hanover (1668–1705), the Queen of Prussia and his avowed disciple, and Caroline of Ansbach, the consort of her grandson, the future George II. To each of these women he was correspondent, adviser, and friend. In turn, they all approved of Leibniz more than did their spouses and the future king George I of Great Britain.[54]
The population of Hanover was only about 10,000, and its provinciality eventually grated on Leibniz. Nevertheless, to be a major courtier to the House of Brunswick was quite an honor, especially in light of the meteoric rise in the prestige of that House during Leibniz's association with it. In 1692, the Duke of Brunswick became a hereditary Elector of the Holy Roman Empire. The British Act of Settlement 1701 designated the Electress Sophia and her descent as the royal family of England, once both King William III and his sister-in-law and successor, Queen Anne, were dead. Leibniz played a role in the initiatives and negotiations leading up to that Act, but not always an effective one. For example, something he published anonymously in England, thinking to promote the Brunswick cause, was formally censured by the British Parliament.
The Brunswicks tolerated the enormous effort Leibniz devoted to intellectual pursuits unrelated to his duties as a courtier, pursuits such as perfecting calculus, writing about other mathematics, logic, physics, and philosophy, and keeping up a vast correspondence. He began working on calculus in 1674; the earliest evidence of its use in his surviving notebooks is 1675. By 1677 he had a coherent system in hand, but did not publish it until 1684. Leibniz's most important mathematical papers were published between 1682 and 1692, usually in a journal which he and Otto Mencke founded in 1682, the Acta Eruditorum. That journal played a key role in advancing his mathematical and scientific reputation, which in turn enhanced his eminence in diplomacy, history, theology, and philosophy.
The Elector Ernest Augustus commissioned Leibniz to write a history of the House of Brunswick, going back to the time of Charlemagne or earlier, hoping that the resulting book would advance his dynastic ambitions. From 1687 to 1690, Leibniz traveled extensively in Germany, Austria, and Italy, seeking and finding archival materials bearing on this project. Decades went by but no history appeared; the next Elector became quite annoyed at Leibniz's apparent dilatoriness. Leibniz never finished the project, in part because of his huge output on many other fronts, but also because he insisted on writing a meticulously researched and erudite book based on archival sources, when his patrons would have been quite happy with a short popular book, one perhaps little more than a genealogy with commentary, to be completed in three years or less. They never knew that he had in fact carried out a fair part of his assigned task: when the material Leibniz had written and collected for his history of the House of Brunswick was finally published in the 19th century, it filled three volumes.
Leibniz was appointed Librarian of the Herzog August Library in Wolfenbüttel, Lower Saxony, in 1691.
In 1708, John Keill, writing in the journal of the Royal Society and with Newton's presumed blessing, accused Leibniz of having plagiarised Newton's calculus.[55] Thus began the calculus priority dispute which darkened the remainder of Leibniz's life. A formal investigation by the Royal Society (in which Newton was an unacknowledged participant), undertaken in response to Leibniz's demand for a retraction, upheld Keill's charge. Historians of mathematics writing since 1900 or so have tended to acquit Leibniz, pointing to important differences between Leibniz's and Newton's versions of calculus.
In 1711, while traveling in northern Europe, the Russian Tsar Peter the Great stopped in Hanover and met Leibniz, who then took some interest in Russian matters for the rest of his life. In 1712, Leibniz began a two-year residence in Vienna, where he was appointed Imperial Court Councillor to the Habsburgs. On the death of Queen Anne in 1714, Elector George Louis became King George I of Great Britain, under the terms of the 1701 Act of Settlement. Even though Leibniz had done much to bring about this happy event, it was not to be his hour of glory. Despite the intercession of the Princess of Wales, Caroline of Ansbach, George I forbade Leibniz to join him in London until he completed at least one volume of the history of the Brunswick family his father had commissioned nearly 30 years earlier. Moreover, for George I to include Leibniz in his London court would have been deemed insulting to Newton, who was seen as having won the calculus priority dispute and whose standing in British official circles could not have been higher. Finally, his dear friend and defender, the Dowager Electress Sophia, died in 1714.
Death
Leibniz died in Hanover in 1716. At the time, he was so out of favor that neither George I (who happened to be near Hanover at that time) nor any fellow courtier other than his personal secretary attended the funeral. Even though Leibniz was a life member of the Royal Society and the Berlin Academy of Sciences, neither organization saw fit to honor his death. His grave went unmarked for more than 50 years. He was, however, eulogized by Fontenelle, before the French Academy of Sciences in Paris, which had admitted him as a foreign member in 1700. The eulogy was composed at the behest of the Duchess of Orleans, a niece of the Electress Sophia.
Personal life
Leibniz never married. He complained on occasion about money, but the fair sum he left to his sole heir, his sister's stepson, proved that the Brunswicks had, by and large, paid him well. In his diplomatic endeavors, he at times verged on the unscrupulous, as was all too often the case with professional diplomats of his day. On several occasions, Leibniz backdated and altered personal manuscripts, actions which put him in a bad light during the calculus controversy.[56]
He was charming, well-mannered, and not without humor and imagination.[57] He had many friends and admirers all over Europe. He was identified as a Protestant and a philosophical theist.[58][59][60][61] Leibniz remained committed to Trinitarian Christianity throughout his life.[62]
Philosopher
Leibniz's philosophical thinking appears fragmented, because his philosophical writings consist mainly of a multitude of short pieces: journal articles, manuscripts published long after his death, and many letters to many correspondents. He wrote only two book-length philosophical treatises, of which only the Théodicée of 1710 was published in his lifetime.
Leibniz dated his beginning as a philosopher to his Discourse on Metaphysics, which he composed in 1686 as a commentary on a running dispute between Nicolas Malebranche and Antoine Arnauld. This led to an extensive and valuable correspondence with Arnauld;[63] it and the Discourse were not published until the 19th century. In 1695, Leibniz made his public entrée into European philosophy with a journal article titled "New System of the Nature and Communication of Substances".[64] Between 1695 and 1705, he composed his New Essays on Human Understanding, a lengthy commentary on John Locke's 1690 An Essay Concerning Human Understanding, but upon learning of Locke's 1704 death, lost the desire to publish it, so that the New Essays were not published until 1765. The Monadologie, composed in 1714 and published posthumously, consists of 90 aphorisms.
Leibniz also wrote a short paper, "Primae veritates" ("First Truths"), first published by Louis Couturat in 1903 (pp. 518–523)[65] summarizing his views on metaphysics. The paper is undated; that he wrote it while in Vienna in 1689 was determined only in 1999, when the ongoing critical edition finally published Leibniz's philosophical writings for the period 1677–90.[66] Couturat's reading of this paper was the launching point for much 20th-century thinking about Leibniz, especially among analytic philosophers. But after a meticulous study of all of Leibniz's philosophical writings up to 1688—a study the 1999 additions to the critical edition made possible—Mercer (2001) begged to differ with Couturat's reading; the jury is still out.
Leibniz met Spinoza in 1676, read some of his unpublished writings, and has since been suspected of appropriating some of Spinoza's ideas. While Leibniz admired Spinoza's powerful intellect, he was also forthrightly dismayed by Spinoza's conclusions,[67] especially when these were inconsistent with Christian orthodoxy.
Unlike Descartes and Spinoza, Leibniz had a thorough university education in philosophy. He was influenced by his Leipzig professor Jakob Thomasius, who also supervised his BA thesis in philosophy.[9] Leibniz also eagerly read Francisco Suárez, a Spanish Jesuit respected even in Lutheran universities. Leibniz was deeply interested in the new methods and conclusions of Descartes, Huygens, Newton, and Boyle, but viewed their work through a lens heavily tinted by scholastic notions. Yet it remains the case that Leibniz's methods and concerns often anticipate the logic, and analytic and linguistic philosophy of the 20th century.
Principles
Leibniz variously invoked one or another of seven fundamental philosophical Principles:[68]
• Identity/contradiction. If a proposition is true, then its negation is false and vice versa.
• Identity of indiscernibles. Two distinct things cannot have all their properties in common. If every predicate possessed by x is also possessed by y and vice versa, then entities x and y are identical; to suppose two things indiscernible is to suppose the same thing under two names. Frequently invoked in modern logic and philosophy, the "identity of indiscernibles" is often referred to as Leibniz's Law. It has attracted the most controversy and criticism, especially from corpuscular philosophy and quantum mechanics.
• Sufficient reason. "There must be a sufficient reason for anything to exist, for any event to occur, for any truth to obtain."[69]
• Pre-established harmony.[70] "[T]he appropriate nature of each substance brings it about that what happens to one corresponds to what happens to all the others, without, however, their acting upon one another directly." (Discourse on Metaphysics, XIV) A dropped glass shatters because it "knows" it has hit the ground, and not because the impact with the ground "compels" the glass to split.
• Law of Continuity. Natura non facit saltus[71] (literally, "Nature does not make jumps").
• Optimism. "God assuredly always chooses the best."[72]
• Plenitude. Leibniz believed that the best of all possible worlds would actualize every genuine possibility, and argued in Théodicée that this best of all possible worlds will contain all possibilities, with our finite experience of eternity giving no reason to dispute nature's perfection.[73]
Leibniz would on occasion give a rational defense of a specific principle, but more often took them for granted.[74]
Monads
Leibniz's best known contribution to metaphysics is his theory of monads, as exposited in Monadologie. He proposes his theory that the universe is made of an infinite number of simple substances known as monads.[75] Monads can also be compared to the corpuscles of the mechanical philosophy of René Descartes and others. These simple substances or monads are the "ultimate units of existence in nature". Monads have no parts but still exist by the qualities that they have. These qualities are continuously changing over time, and each monad is unique. They are also not affected by time and are subject to only creation and annihilation.[76] Monads are centers of force; substance is force, while space, matter, and motion are merely phenomenal. It is said that he anticipated Albert Einstein by arguing, against Newton, that space, time, and motion are completely relative as he quipped,[77] "As for my own opinion, I have said more than once, that I hold space to be something merely relative, as time is, that I hold it to be an order of coexistences, as time is an order of successions."[78] Einstein, who called himself a "Leibnizian" even wrote in the introduction to Max Jammer's book Concepts of Space that Leibnizianism was superior to Newtonianism, and his ideas would have dominated over Newton's had it not been for the poor technological tools of the time; it has been argued that Leibniz paved the way for Einstein's theory of relativity.[79]
Leibniz's proof of God can be summarized in the Théodicée.[80] Reason is governed by the principle of contradiction and the principle of sufficient reason. Using the principle of reasoning, Leibniz concluded that the first reason of all things is God.[80] All that we see and experience is subject to change, and the fact that this world is contingent can be explained by the possibility of the world being arranged differently in space and time. The contingent world must have some necessary reason for its existence. Leibniz uses a geometry book as an example to explain his reasoning. If this book was copied from an infinite chain of copies, there must be some reason for the content of the book.[81] Leibniz concluded that there must be the "monas monadum" or God.
The ontological essence of a monad is its irreducible simplicity. Unlike atoms, monads possess no material or spatial character. They also differ from atoms by their complete mutual independence, so that interactions among monads are only apparent. Instead, by virtue of the principle of pre-established harmony, each monad follows a pre-programmed set of "instructions" peculiar to itself, so that a monad "knows" what to do at each moment. By virtue of these intrinsic instructions, each monad is like a little mirror of the universe. Monads need not be "small"; e.g., each human being constitutes a monad, in which case free will is problematic.
Monads are purported to have gotten rid of the problematic:
• interaction between mind and matter arising in the system of Descartes;
• lack of individuation inherent to the system of Spinoza, which represents individual creatures as merely accidental.
Theodicy and optimism
The Theodicy[82] tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds. It must be the best possible and most balanced world, because it was created by an all powerful and all knowing God, who would not choose to create an imperfect world if a better world could be known to him or possible to exist. In effect, apparent flaws that can be identified in this world must exist in every possible world, because otherwise God would have chosen to create the world that excluded those flaws.[83]
Leibniz asserted that the truths of theology (religion) and philosophy cannot contradict each other, since reason and faith are both "gifts of God" so that their conflict would imply God contending against himself. The Theodicy is Leibniz's attempt to reconcile his personal philosophical system with his interpretation of the tenets of Christianity.[84] This project was motivated in part by Leibniz's belief, shared by many philosophers and theologians during the Enlightenment, in the rational and enlightened nature of the Christian religion. It was also shaped by Leibniz's belief in the perfectibility of human nature (if humanity relied on correct philosophy and religion as a guide), and by his belief that metaphysical necessity must have a rational or logical foundation, even if this metaphysical causality seemed inexplicable in terms of physical necessity (the natural laws identified by science).
Because reason and faith must be entirely reconciled, any tenet of faith which could not be defended by reason must be rejected. Leibniz then approached one of the central criticisms of Christian theism:[85] if God is all good, all wise, and all powerful, then how did evil come into the world? The answer (according to Leibniz) is that, while God is indeed unlimited in wisdom and power, his human creations, as creations, are limited both in their wisdom and in their will (power to act). This predisposes humans to false beliefs, wrong decisions, and ineffective actions in the exercise of their free will. God does not arbitrarily inflict pain and suffering on humans; rather he permits both moral evil (sin) and physical evil (pain and suffering) as the necessary consequences of metaphysical evil (imperfection), as a means by which humans can identify and correct their erroneous decisions, and as a contrast to true good.[86]
Further, although human actions flow from prior causes that ultimately arise in God and therefore are known to God as metaphysical certainties, an individual's free will is exercised within natural laws, where choices are merely contingently necessary and to be decided in the event by a "wonderful spontaneity" that provides individuals with an escape from rigorous predestination.
Discourse on Metaphysics
For Leibniz, "God is an absolutely perfect being". He describes this perfection later in section VI as the simplest form of something with the most substantial outcome (VI). Along these lines, he declares that every type of perfection "pertains to him (God) in the highest degree" (I). Even though his types of perfections are not specifically drawn out, Leibniz highlights the one thing that, to him, does certify imperfections and proves that God is perfect: "that one acts imperfectly if he acts with less perfection than he is capable of", and since God is a perfect being, he cannot act imperfectly (III). Because God cannot act imperfectly, the decisions he makes pertaining to the world must be perfect. Leibniz also comforts readers, stating that because he has done everything to the most perfect degree; those who love him cannot be injured. However, to love God is a subject of difficulty as Leibniz believes that we are "not disposed to wish for that which God desires" because we have the ability to alter our disposition (IV). In accordance with this, many act as rebels, but Leibniz says that the only way we can truly love God is by being content "with all that comes to us according to his will" (IV).
Because God is "an absolutely perfect being" (I), Leibniz argues that God would be acting imperfectly if he acted with any less perfection than what he is able of (III). His syllogism then ends with the statement that God has made the world perfectly in all ways. This also affects how we should view God and his will. Leibniz states that, in lieu of God's will, we have to understand that God "is the best of all masters" and he will know when his good succeeds, so we, therefore, must act in conformity to his good will—or as much of it as we understand (IV). In our view of God, Leibniz declares that we cannot admire the work solely because of the maker, lest we mar the glory and love God in doing so. Instead, we must admire the maker for the work he has done (II). Effectively, Leibniz states that if we say the earth is good because of the will of God, and not good according to some standards of goodness, then how can we praise God for what he has done if contrary actions are also praiseworthy by this definition (II). Leibniz then asserts that different principles and geometry cannot simply be from the will of God, but must follow from his understanding.[87]
Fundamental question of metaphysics
Leibniz wrote: "Why is there something rather than nothing? The sufficient reason ... is found in a substance which ... is a necessary being bearing the reason for its existence within itself."[88] Martin Heidegger called this question "the fundamental question of metaphysics".[89][90]
Symbolic thought and rational resolution of disputes
Leibniz believed that much of human reasoning could be reduced to calculations of a sort, and that such calculations could resolve many differences of opinion:
The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate, without further ado, to see who is right.[91][92][93]
Leibniz's calculus ratiocinator, which resembles symbolic logic, can be viewed as a way of making such calculations feasible. Leibniz wrote memoranda[94] that can now be read as groping attempts to get symbolic logic—and thus his calculus—off the ground. These writings remained unpublished until the appearance of a selection edited by Carl Immanuel Gerhardt (1859). Louis Couturat published a selection in 1901; by this time the main developments of modern logic had been created by Charles Sanders Peirce and by Gottlob Frege.
Leibniz thought symbols were important for human understanding. He attached so much importance to the development of good notations that he attributed all his discoveries in mathematics to this. His notation for calculus is an example of his skill in this regard. Leibniz's passion for symbols and notation, as well as his belief that these are essential to a well-running logic and mathematics, made him a precursor of semiotics.[95]
But Leibniz took his speculations much further. Defining a character as any written sign, he then defined a "real" character as one that represents an idea directly and not simply as the word embodying the idea. Some real characters, such as the notation of logic, serve only to facilitate reasoning. Many characters well known in his day, including Egyptian hieroglyphics, Chinese characters, and the symbols of astronomy and chemistry, he deemed not real.[96] Instead, he proposed the creation of a characteristica universalis or "universal characteristic", built on an alphabet of human thought in which each fundamental concept would be represented by a unique "real" character:
It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and as exactly as arithmetic expresses numbers or geometry expresses lines, we could do in all matters insofar as they are subject to reasoning all that we can do in arithmetic and geometry. For all investigations which depend on reasoning would be carried out by transposing these characters and by a species of calculus.[97]
Complex thoughts would be represented by combining characters for simpler thoughts. Leibniz saw that the uniqueness of prime factorization suggests a central role for prime numbers in the universal characteristic, a striking anticipation of Gödel numbering. Granted, there is no intuitive or mnemonic way to number any set of elementary concepts using the prime numbers.
Because Leibniz was a mathematical novice when he first wrote about the characteristic, at first he did not conceive it as an algebra but rather as a universal language or script. Only in 1676 did he conceive of a kind of "algebra of thought", modeled on and including conventional algebra and its notation. The resulting characteristic included a logical calculus, some combinatorics, algebra, his analysis situs (geometry of situation), a universal concept language, and more. What Leibniz actually intended by his characteristica universalis and calculus ratiocinator, and the extent to which modern formal logic does justice to calculus, may never be established.[98] Leibniz's idea of reasoning through a universal language of symbols and calculations remarkably foreshadows great 20th-century developments in formal systems, such as Turing completeness, where computation was used to define equivalent universal languages (see Turing degree).
Formal logic
Main article: Algebraic logic
Leibniz has been noted as one of the most important logicians between the times of Aristotle and Gottlob Frege.[99] Leibniz enunciated the principal properties of what we now call conjunction, disjunction, negation, identity, set inclusion, and the empty set. The principles of Leibniz's logic and, arguably, of his whole philosophy, reduce to two:
1. All our ideas are compounded from a very small number of simple ideas, which form the alphabet of human thought.
2. Complex ideas proceed from these simple ideas by a uniform and symmetrical combination, analogous to arithmetical multiplication.
The formal logic that emerged early in the 20th century also requires, at minimum, unary negation and quantified variables ranging over some universe of discourse.
Leibniz published nothing on formal logic in his lifetime; most of what he wrote on the subject consists of working drafts. In his History of Western Philosophy, Bertrand Russell went so far as to claim that Leibniz had developed logic in his unpublished writings to a level which was reached only 200 years later.
Russell's principal work on Leibniz found that many of Leibniz's most startling philosophical ideas and claims (e.g., that each of the fundamental monads mirrors the whole universe) follow logically from Leibniz's conscious choice to reject relations between things as unreal. He regarded such relations as (real) qualities of things (Leibniz admitted unary predicates only): For him, "Mary is the mother of John" describes separate qualities of Mary and of John. This view contrasts with the relational logic of De Morgan, Peirce, Schröder and Russell himself, now standard in predicate logic. Notably, Leibniz also declared space and time to be inherently relational.[100]
Leibniz's 1690 discovery of his algebra of concepts[101][102] (deductively equivalent to the Boolean algebra)[103] and the associated metaphysics, are of interest in present-day computational metaphysics.[104]
Mathematician
Although the mathematical notion of function was implicit in trigonometric and logarithmic tables, which existed in his day, Leibniz was the first, in 1692 and 1694, to employ it explicitly, to denote any of several geometric concepts derived from a curve, such as abscissa, ordinate, tangent, chord, and the perpendicular (see History of the function concept).[105] In the 18th century, "function" lost these geometrical associations. Leibniz also believed that the sum of an infinite number of zeros would be equal to one half using the analogy of the creation of the world from nothing.[106] Leibniz was also one of the pioneers in actuarial science, calculating the purchase price of life annuities and the liquidation of a state's debt.[107]
Leibniz's research into formal logic, also relevant to mathematics, is discussed in the preceding section. The best overview of Leibniz's writings on calculus may be found in Bos (1974).[108]
Leibniz, who invented one of the earliest mechanical calculators, said of calculation: "For it is unworthy of excellent men to lose hours like slaves in the labor of calculation which could safely be relegated to anyone else if machines were used."[109]
Linear systems
Leibniz arranged the coefficients of a system of linear equations into an array, now called a matrix, in order to find a solution to the system if it existed.[110] This method was later called Gaussian elimination. Leibniz laid down the foundations and theory of determinants, although the Japanese mathematician Seki Takakazu also discovered determinants independently of Leibniz.[111][112] His works show calculating the determinants using cofactors.[113] Calculating the determinant using cofactors is named the Leibniz formula. Finding the determinant of a matrix using this method proves impractical with large n, requiring to calculate n! products and the number of n-permutations.[114] He also solved systems of linear equations using determinants, which is now called Cramer's rule. This method for solving systems of linear equations based on determinants was found in 1684 by Leibniz (Cramer published his findings in 1750).[112] Although Gaussian elimination requires $O(n^{3})$ arithmetic operations, linear algebra textbooks still teach cofactor expansion before LU factorization.[115][116]
Geometry
The Leibniz formula for π states that
$1\,-\,{\frac {1}{3}}\,+\,{\frac {1}{5}}\,-\,{\frac {1}{7}}\,+\,\cdots \,=\,{\frac {\pi }{4}}.$
Leibniz wrote that circles "can most simply be expressed by this series, that is, the aggregate of fractions alternately added and subtracted".[117] However this formula is only accurate with a large number of terms, using 10,000,000 terms to obtain the correct value of π/4 to 8 decimal places.[118] Leibniz attempted to create a definition for a straight line while attempting to prove the parallel postulate.[119] While most mathematicians defined a straight line as the shortest line between two points, Leibniz believed that this was merely a property of a straight line rather than the definition.[120]
Calculus
Leibniz is credited, along with Sir Isaac Newton, with the discovery of calculus (differential and integral calculus). According to Leibniz's notebooks, a critical breakthrough occurred on 11 November 1675, when he employed integral calculus for the first time to find the area under the graph of a function y = f(x).[121] He introduced several notations used to this day, for instance the integral sign ∫, representing an elongated S, from the Latin word summa, and the d used for differentials, from the Latin word differentia. Leibniz did not publish anything about his calculus until 1684.[122] Leibniz expressed the inverse relation of integration and differentiation, later called the fundamental theorem of calculus, by means of a figure[123] in his 1693 paper Supplementum geometriae dimensoriae....[124] However, James Gregory is credited for the theorem's discovery in geometric form, Isaac Barrow proved a more generalized geometric version, and Newton developed supporting theory. The concept became more transparent as developed through Leibniz's formalism and new notation.[125] The product rule of differential calculus is still called "Leibniz's law". In addition, the theorem that tells how and when to differentiate under the integral sign is called the Leibniz integral rule.
Leibniz exploited infinitesimals in developing calculus, manipulating them in ways suggesting that they had paradoxical algebraic properties. George Berkeley, in a tract called The Analyst and also in De Motu, criticized these. A recent study argues that Leibnizian calculus was free of contradictions, and was better grounded than Berkeley's empiricist criticisms.[126]
From 1711 until his death, Leibniz was engaged in a dispute with John Keill, Newton and others, over whether Leibniz had invented calculus independently of Newton.
The use of infinitesimals in mathematics was frowned upon by followers of Karl Weierstrass,[127][128] but survived in science and engineering, and even in rigorous mathematics, via the fundamental computational device known as the differential. Beginning in 1960, Abraham Robinson worked out a rigorous foundation for Leibniz's infinitesimals, using model theory, in the context of a field of hyperreal numbers. The resulting non-standard analysis can be seen as a belated vindication of Leibniz's mathematical reasoning. Robinson's transfer principle is a mathematical implementation of Leibniz's heuristic law of continuity, while the standard part function implements the Leibnizian transcendental law of homogeneity.
Topology
Leibniz was the first to use the term analysis situs,[129] later used in the 19th century to refer to what is now known as topology. There are two takes on this situation. On the one hand, Mates, citing a 1954 paper in German by Jacob Freudenthal, argues:
Although for Leibniz the situs of a sequence of points is completely determined by the distance between them and is altered if those distances are altered, his admirer Euler, in the famous 1736 paper solving the Königsberg Bridge Problem and its generalizations, used the term geometria situs in such a sense that the situs remains unchanged under topological deformations. He mistakenly credits Leibniz with originating this concept. ... [It] is sometimes not realized that Leibniz used the term in an entirely different sense and hence can hardly be considered the founder of that part of mathematics.[130]
But Hideaki Hirano argues differently, quoting Mandelbrot:[131]
To sample Leibniz' scientific works is a sobering experience. Next to calculus, and to other thoughts that have been carried out to completion, the number and variety of premonitory thrusts is overwhelming. We saw examples in "packing", ... My Leibniz mania is further reinforced by finding that for one moment its hero attached importance to geometric scaling. In Euclidis Prota ..., which is an attempt to tighten Euclid's axioms, he states ...: "I have diverse definitions for the straight line. The straight line is a curve, any part of which is similar to the whole, and it alone has this property, not only among curves but among sets." This claim can be proved today.[132]
Thus the fractal geometry promoted by Mandelbrot drew on Leibniz's notions of self-similarity and the principle of continuity: Natura non facit saltus.[71] We also see that when Leibniz wrote, in a metaphysical vein, that "the straight line is a curve, any part of which is similar to the whole", he was anticipating topology by more than two centuries. As for "packing", Leibniz told his friend and correspondent Des Bosses to imagine a circle, then to inscribe within it three congruent circles with maximum radius; the latter smaller circles could be filled with three even smaller circles by the same procedure. This process can be continued infinitely, from which arises a good idea of self-similarity. Leibniz's improvement of Euclid's axiom contains the same concept.
Scientist and engineer
Leibniz's writings are currently discussed, not only for their anticipations and possible discoveries not yet recognized, but as ways of advancing present knowledge. Much of his writing on physics is included in Gerhardt's Mathematical Writings.
Physics
Leibniz contributed a fair amount to the statics and dynamics emerging around him, often disagreeing with Descartes and Newton. He devised a new theory of motion (dynamics) based on kinetic energy and potential energy, which posited space as relative, whereas Newton was thoroughly convinced that space was absolute. An important example of Leibniz's mature physical thinking is his Specimen Dynamicum of 1695.[133]
Until the discovery of subatomic particles and the quantum mechanics governing them, many of Leibniz's speculative ideas about aspects of nature not reducible to statics and dynamics made little sense. For instance, he anticipated Albert Einstein by arguing, against Newton, that space, time and motion are relative, not absolute: "As for my own opinion, I have said more than once, that I hold space to be something merely relative, as time is, that I hold it to be an order of coexistences, as time is an order of successions."[78]
Leibniz held a relationist notion of space and time, against Newton's substantivalist views.[134][135][136] According to Newton's substantivalism, space and time are entities in their own right, existing independently of things. Leibniz's relationism, in contrast, describes space and time as systems of relations that exist between objects. The rise of general relativity and subsequent work in the history of physics has put Leibniz's stance in a more favorable light.
One of Leibniz's projects was to recast Newton's theory as a vortex theory.[137] However, his project went beyond vortex theory, since at its heart there was an attempt to explain one of the most difficult problems in physics, that of the origin of the cohesion of matter.[137]
The principle of sufficient reason has been invoked in recent cosmology, and his identity of indiscernibles in quantum mechanics, a field some even credit him with having anticipated in some sense. In addition to his theories about the nature of reality, Leibniz's contributions to the development of calculus have also had a major impact on physics.
The vis viva
Leibniz's vis viva (Latin for "living force") is mv2, twice the modern kinetic energy. He realized that the total energy would be conserved in certain mechanical systems, so he considered it an innate motive characteristic of matter.[138] Here too his thinking gave rise to another regrettable nationalistic dispute. His vis viva was seen as rivaling the conservation of momentum championed by Newton in England and by Descartes and Voltaire in France; hence academics in those countries tended to neglect Leibniz's idea. Leibniz knew of the validity of conservation of momentum. In reality, both energy and momentum are conserved (in closed systems), so both approaches are valid.
Other natural science
By proposing that the earth has a molten core, he anticipated modern geology. In embryology, he was a preformationist, but also proposed that organisms are the outcome of a combination of an infinite number of possible microstructures and of their powers. In the life sciences and paleontology, he revealed an amazing transformist intuition, fueled by his study of comparative anatomy and fossils. One of his principal works on this subject, Protogaea, unpublished in his lifetime, has recently been published in English for the first time. He worked out a primal organismic theory.[139] In medicine, he exhorted the physicians of his time—with some results—to ground their theories in detailed comparative observations and verified experiments, and to distinguish firmly scientific and metaphysical points of view.
Psychology
Psychology had been a central interest of Leibniz.[140][141] He appears to be an "underappreciated pioneer of psychology"[142] He wrote on topics which are now regarded as fields of psychology: attention and consciousness, memory, learning (association), motivation (the act of "striving"), emergent individuality, the general dynamics of development (evolutionary psychology). His discussions in the New Essays and Monadology often rely on everyday observations such as the behaviour of a dog or the noise of the sea, and he develops intuitive analogies (the synchronous running of clocks or the balance spring of a clock). He also devised postulates and principles that apply to psychology: the continuum of the unnoticed petites perceptions to the distinct, self-aware apperception, and psychophysical parallelism from the point of view of causality and of purpose: "Souls act according to the laws of final causes, through aspirations, ends and means. Bodies act according to the laws of efficient causes, i.e. the laws of motion. And these two realms, that of efficient causes and that of final causes, harmonize with one another."[143] This idea refers to the mind-body problem, stating that the mind and brain do not act upon each other, but act alongside each other separately but in harmony.[144] Leibniz, however, did not use the term psychologia.[145] Leibniz's epistemological position—against John Locke and English empiricism (sensualism)—was made clear: "Nihil est in intellectu quod non fuerit in sensu, nisi intellectu ipse." – "Nothing is in the intellect that was not first in the senses, except the intellect itself."[146] Principles that are not present in sensory impressions can be recognised in human perception and consciousness: logical inferences, categories of thought, the principle of causality and the principle of purpose (teleology).
Leibniz found his most important interpreter in Wilhelm Wundt, founder of psychology as a discipline. Wundt used the "… nisi intellectu ipse" quotation 1862 on the title page of his Beiträge zur Theorie der Sinneswahrnehmung (Contributions on the Theory of Sensory Perception) and published a detailed and aspiring monograph on Leibniz.[147] Wundt shaped the term apperception, introduced by Leibniz, into an experimental psychologically based apperception psychology that included neuropsychological modelling – an excellent example of how a concept created by a great philosopher could stimulate a psychological research program. One principle in the thinking of Leibniz played a fundamental role: "the principle of equality of separate but corresponding viewpoints." Wundt characterized this style of thought (perspectivism) in a way that also applied for him—viewpoints that "supplement one another, while also being able to appear as opposites that only resolve themselves when considered more deeply."[148][149] Much of Leibniz's work went on to have a great impact on the field of psychology.[150] Leibniz thought that there are many petites perceptions, or small perceptions of which we perceive but of which we are unaware. He believed that by the principle that phenomena found in nature were continuous by default, it was likely that the transition between conscious and unconscious states had intermediary steps.[151] For this to be true, there must also be a portion of the mind of which we are unaware at any given time. His theory regarding consciousness in relation to the principle of continuity can be seen as an early theory regarding the stages of sleep. In this way, Leibniz's theory of perception can be viewed as one of many theories leading up to the idea of the unconscious. Leibniz was a direct influence on Ernst Platner, who is credited with originally coining the term Unbewußtseyn (unconscious).[152] Additionally, the idea of subliminal stimuli can be traced back to his theory of small perceptions.[153] Leibniz's ideas regarding music and tonal perception went on to influence the laboratory studies of Wilhelm Wundt.[154]
Social science
In public health, he advocated establishing a medical administrative authority, with powers over epidemiology and veterinary medicine. He worked to set up a coherent medical training program, oriented towards public health and preventive measures. In economic policy, he proposed tax reforms and a national insurance program, and discussed the balance of trade. He even proposed something akin to what much later emerged as game theory. In sociology he laid the ground for communication theory.
Technology
In 1906, Garland published a volume of Leibniz's writings bearing on his many practical inventions and engineering work. To date, few of these writings have been translated into English. Nevertheless, it is well understood that Leibniz was a serious inventor, engineer, and applied scientist, with great respect for practical life. Following the motto theoria cum praxi, he urged that theory be combined with practical application, and thus has been claimed as the father of applied science. He designed wind-driven propellers and water pumps, mining machines to extract ore, hydraulic presses, lamps, submarines, clocks, etc. With Denis Papin, he created a steam engine. He even proposed a method for desalinating water. From 1680 to 1685, he struggled to overcome the chronic flooding that afflicted the ducal silver mines in the Harz Mountains, but did not succeed.[155]
Computation
Leibniz may have been the first computer scientist and information theorist.[156] Early in life, he documented the binary numeral system (base 2), then revisited that system throughout his career.[157] While Leibniz was examining other cultures to compare his metaphysical views, he encountered an ancient Chinese book I Ching. Leibniz interpreted a diagram which showed yin and yang and corresponded it to a zero and one.[158] More information can be found in the Sinophile section. Leibniz may have plagiarized Juan Caramuel y Lobkowitz and Thomas Harriot, who independently developed the binary system, as he was familiar with their works on the binary system.[159] Juan Caramuel y Lobkowitz worked extensively on logarithms including logarithms with base 2.[160] Thomas Harriot's manuscripts contained a table of binary numbers and their notation, which demonstrated that any number could be written on a base 2 system.[161] Regardless, Leibniz simplified the binary system and articulated logical properties such as conjunction, disjunction, negation, identity, inclusion, and the empty set.[162] He anticipated Lagrangian interpolation and algorithmic information theory. His calculus ratiocinator anticipated aspects of the universal Turing machine. In 1961, Norbert Wiener suggested that Leibniz should be considered the patron saint of cybernetics.[163] Wiener is quoted with "Indeed, the general idea of a computing machine is nothing but a mechanization of Leibniz's Calculus Ratiocinator."[164]
In 1671, Leibniz began to invent a machine that could execute all four arithmetic operations, gradually improving it over a number of years. This "stepped reckoner" attracted fair attention and was the basis of his election to the Royal Society in 1673. A number of such machines were made during his years in Hanover by a craftsman working under his supervision. They were not an unambiguous success because they did not fully mechanize the carry operation. Couturat reported finding an unpublished note by Leibniz, dated 1674, describing a machine capable of performing some algebraic operations.[165] Leibniz also devised a (now reproduced) cipher machine, recovered by Nicholas Rescher in 2010.[166] In 1693, Leibniz described a design of a machine which could, in theory, integrate differential equations, which he called "integraph".[167]
Leibniz was groping towards hardware and software concepts worked out much later by Charles Babbage and Ada Lovelace. In 1679, while mulling over his binary arithmetic, Leibniz imagined a machine in which binary numbers were represented by marbles, governed by a rudimentary sort of punched cards.[168][169] Modern electronic digital computers replace Leibniz's marbles moving by gravity with shift registers, voltage gradients, and pulses of electrons, but otherwise they run roughly as Leibniz envisioned in 1679.
Librarian
Later in Leibniz's career (after the death of von Boyneburg), Leibniz moved to Paris and accepted a position as a librarian in the Hanoverian court of Johann Friedrich, Duke of Brunswick-Luneburg.[170] Leibniz's predecessor, Tobias Fleischer, had already created a cataloging system for the Duke's library but it was a clumsy attempt. At this library, Leibniz focused more on advancing the library than on the cataloging. For instance, within a month of taking the new position, he developed a comprehensive plan to expand the library. He was one of the first to consider developing a core collection for a library and felt "that a library for display and ostentation is a luxury and indeed superfluous, but a well-stocked and organized library is important and useful for all areas of human endeavor and is to be regarded on the same level as schools and churches".[171] Leibniz lacked the funds to develop the library in this manner. After working at this library, by the end of 1690 Leibniz was appointed as privy-councilor and librarian of the Bibliotheca Augusta at Wolfenbüttel. It was an extensive library with at least 25,946 printed volumes.[171] At this library, Leibniz sought to improve the catalog. He was not allowed to make complete changes to the existing closed catalog, but was allowed to improve upon it so he started on that task immediately. He created an alphabetical author catalog and had also created other cataloging methods that were not implemented. While serving as librarian of the ducal libraries in Hanover and Wolfenbüttel, Leibniz effectively became one of the founders of library science. Seemingly, Leibniz paid a good deal of attention to the classification of subject matter, favoring a well-balance library covering a host of numerous subjects and interests.[172] Leibniz, for example, proposed the following classification system in the Otivm Hanoveranvm Sive Miscellanea (1737).[172][173]
Leibniz's Idea of Arranging a Narrower Library
• Theology
• Jurisprudence
• Medicine
• Intellectual Philosophy
• Philosophy of the Imagination or Mathematics
• Philosophy of Sensible Things or Physics
• Philology or Language
• Civil History
• Literary History and Libraries
• General and Miscellaneous
He also designed a book indexing system in ignorance of the only other such system then extant, that of the Bodleian Library at Oxford University. He also called on publishers to distribute abstracts of all new titles they produced each year, in a standard form that would facilitate indexing. He hoped that this abstracting project would eventually include everything printed from his day back to Gutenberg. Neither proposal met with success at the time, but something like them became standard practice among English language publishers during the 20th century, under the aegis of the Library of Congress and the British Library.
He called for the creation of an empirical database as a way to further all sciences. His characteristica universalis, calculus ratiocinator, and a "community of minds"—intended, among other things, to bring political and religious unity to Europe—can be seen as distant unwitting anticipations of artificial languages (e.g., Esperanto and its rivals), symbolic logic, even the World Wide Web.
Advocate of scientific societies
Leibniz emphasized that research was a collaborative endeavor. Hence he warmly advocated the formation of national scientific societies along the lines of the British Royal Society and the French Académie Royale des Sciences. More specifically, in his correspondence and travels he urged the creation of such societies in Dresden, Saint Petersburg, Vienna, and Berlin. Only one such project came to fruition; in 1700, the Berlin Academy of Sciences was created. Leibniz drew up its first statutes, and served as its first President for the remainder of his life. That Academy evolved into the German Academy of Sciences, the publisher of the ongoing critical edition of his works.[174]
Lawyer and moralist
Leibniz's writings on law, ethics, and politics[175] were long overlooked by English-speaking scholars, but this has changed of late.[176]
While Leibniz was no apologist for absolute monarchy like Hobbes, or for tyranny in any form, neither did he echo the political and constitutional views of his contemporary John Locke, views invoked in support of liberalism, in 18th-century America and later elsewhere. The following excerpt from a 1695 letter to Baron J. C. Boyneburg's son Philipp is very revealing of Leibniz's political sentiments:
As for ... the great question of the power of sovereigns and the obedience their peoples owe them, I usually say that it would be good for princes to be persuaded that their people have the right to resist them, and for the people, on the other hand, to be persuaded to obey them passively. I am, however, quite of the opinion of Grotius, that one ought to obey as a rule, the evil of revolution being greater beyond comparison than the evils causing it. Yet I recognize that a prince can go to such excess, and place the well-being of the state in such danger, that the obligation to endure ceases. This is most rare, however, and the theologian who authorizes violence under this pretext should take care against excess; excess being infinitely more dangerous than deficiency.[177]
In 1677, Leibniz called for a European confederation, governed by a council or senate, whose members would represent entire nations and would be free to vote their consciences;[178] this is sometimes considered an anticipation of the European Union. He believed that Europe would adopt a uniform religion. He reiterated these proposals in 1715.
But at the same time, he arrived to propose an interreligious and multicultural project to create a universal system of justice, which required from him a broad interdisciplinary perspective. In order to propose it, he combined linguistics (especially sinology), moral and legal philosophy, management, economics, and politics.[179]
Law
Leibniz trained as a legal academic, but under the tutelage of Cartesian-sympathiser Erhard Weigel we already see an attempt to solve legal problems by rationalist mathematical methods (Weigel's influence being most explicit in the Specimen Quaestionum Philosophicarum ex Jure collectarum (An Essay of Collected Philosophical Problems of Right)). For example, the Inaugural Disputation on Perplexing Cases[180] uses early combinatorics to solve some legal disputes, while the 1666 Dissertation on the Combinatorial Art[181] includes simple legal problems by way of illustration.
The use of combinatorial methods to solve legal and moral problems seems, via Athanasius Kircher and Daniel Schwenter to be of Llullist inspiration: Ramón Llull attempted to solve ecumenical disputes through recourse to a combinatorial mode of reasoning he regarded as universal (a mathesis universalis).[182]
In the late 1660s the enlightened Prince-Bishop of Mainz Johann Philipp von Schönborn announced a review of the legal system and made available a position to support his current law commissioner. Leibniz left Franconia and made for Mainz before even winning the role. On reaching Frankfurt am Main Leibniz penned The New Method of Teaching and Learning the Law, by way of application.[183] The text proposed a reform of legal education and is characteristically syncretic, integrating aspects of Thomism, Hobbesianism, Cartesianism and traditional jurisprudence. Leibniz's argument that the function of legal teaching was not to impress rules as one might train a dog, but to aid the student in discovering their own public reason, evidently impressed von Schönborn as he secured the job.
Leibniz's next major attempt to find a universal rational core to law and so found a legal "science of right",[184] came when Leibniz worked in Mainz from 1667-72. Starting initially from Hobbes' mechanistic doctrine of power, Leibniz reverted to logico-combinatorial methods in an attempt to define justice.[185] As Leibniz's so-called Elementa Juris Naturalis advanced, he built in modal notions of right (possibility) and obligation (necessity) in which we see perhaps the earliest elaboration of his possible worlds doctrine within a deontic frame.[186] While ultimately the Elementa remained unpublished, Leibniz continued to work on his drafts and promote their ideas to correspondents up until his death.
Ecumenism
Leibniz devoted considerable intellectual and diplomatic effort to what would now be called an ecumenical endeavor, seeking to reconcile the Roman Catholic and Lutheran churches. In this respect, he followed the example of his early patrons, Baron von Boyneburg and the Duke John Frederick—both cradle Lutherans who converted to Catholicism as adults—who did what they could to encourage the reunion of the two faiths, and who warmly welcomed such endeavors by others. (The House of Brunswick remained Lutheran, because the Duke's children did not follow their father.) These efforts included corresponding with French bishop Jacques-Bénigne Bossuet, and involved Leibniz in some theological controversy. He evidently thought that the thoroughgoing application of reason would suffice to heal the breach caused by the Reformation.
Philologist
Leibniz the philologist was an avid student of languages, eagerly latching on to any information about vocabulary and grammar that came his way. He refuted the belief, widely held by Christian scholars of the time, that Hebrew was the primeval language of the human race. He also refuted the argument, advanced by Swedish scholars in his day, that a form of proto-Swedish was the ancestor of the Germanic languages. He puzzled over the origins of the Slavic languages and was fascinated by classical Chinese. Leibniz was also an expert in the Sanskrit language.[106]
He published the princeps editio (first modern edition) of the late medieval Chronicon Holtzatiae, a Latin chronicle of the County of Holstein.
Sinophile
Leibniz was perhaps the first major European intellectual to take a close interest in Chinese civilization, which he knew by corresponding with, and reading other works by, European Christian missionaries posted in China. He apparently read Confucius Sinarum Philosophus in the first year of its publication.[188] He came to the conclusion that Europeans could learn much from the Confucian ethical tradition. He mulled over the possibility that the Chinese characters were an unwitting form of his universal characteristic. He noted how the I Ching hexagrams correspond to the binary numbers from 000000 to 111111, and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired.[189] Leibniz communicated his ideas of the binary system representing Christianity to the Emperor of China, hoping it would convert him.[106] Leibniz was the only major Western philosopher of the time who attempted to accommodate Confucian ideas to prevailing European beliefs.[190]
Leibniz's attraction to Chinese philosophy originates from his perception that Chinese philosophy was similar to his own.[188] The historian E.R. Hughes suggests that Leibniz's ideas of "simple substance" and "pre-established harmony" were directly influenced by Confucianism, pointing to the fact that they were conceived during the period when he was reading Confucius Sinarum Philosophus.[188]
Polymath
While making his grand tour of European archives to research the Brunswick family history that he never completed, Leibniz stopped in Vienna between May 1688 and February 1689, where he did much legal and diplomatic work for the Brunswicks. He visited mines, talked with mine engineers, and tried to negotiate export contracts for lead from the ducal mines in the Harz mountains. His proposal that the streets of Vienna be lit with lamps burning rapeseed oil was implemented. During a formal audience with the Austrian Emperor and in subsequent memoranda, he advocated reorganizing the Austrian economy, reforming the coinage of much of central Europe, negotiating a Concordat between the Habsburgs and the Vatican, and creating an imperial research library, official archive, and public insurance fund. He wrote and published an important paper on mechanics.
Posthumous reputation
When Leibniz died, his reputation was in decline. He was remembered for only one book, the Théodicée,[191] whose supposed central argument Voltaire lampooned in his popular book Candide, which concludes with the character Candide saying, "Non liquet" (it is not clear), a term that was applied during the Roman Republic to a legal verdict of "not proven". Voltaire's depiction of Leibniz's ideas was so influential that many believed it to be an accurate description. Thus Voltaire and his Candide bear some of the blame for the lingering failure to appreciate and understand Leibniz's ideas. Leibniz had an ardent disciple, Christian Wolff, whose dogmatic and facile outlook did Leibniz's reputation much harm. He also influenced David Hume, who read his Théodicée and used some of his ideas.[192] In any event, philosophical fashion was moving away from the rationalism and system building of the 17th century, of which Leibniz had been such an ardent proponent. His work on law, diplomacy, and history was seen as of ephemeral interest. The vastness and richness of his correspondence went unrecognized.
Much of Europe came to doubt that Leibniz had discovered calculus independently of Newton, and hence his whole work in mathematics and physics was neglected. Voltaire, an admirer of Newton, also wrote Candide at least in part to discredit Leibniz's claim to having discovered calculus and Leibniz's charge that Newton's theory of universal gravitation was incorrect.
Leibniz's long march to his present glory began with the 1765 publication of the Nouveaux Essais, which Kant read closely. In 1768, Louis Dutens edited the first multi-volume edition of Leibniz's writings, followed in the 19th century by a number of editions, including those edited by Erdmann, Foucher de Careil, Gerhardt, Gerland, Klopp, and Mollat. Publication of Leibniz's correspondence with notables such as Antoine Arnauld, Samuel Clarke, Sophia of Hanover, and her daughter Sophia Charlotte of Hanover, began.
In 1900, Bertrand Russell published a critical study of Leibniz's metaphysics.[193] Shortly thereafter, Louis Couturat published an important study of Leibniz, and edited a volume of Leibniz's heretofore unpublished writings, mainly on logic. They made Leibniz somewhat respectable among 20th-century analytical and linguistic philosophers in the English-speaking world (Leibniz had already been of great influence to many Germans such as Bernhard Riemann). For example, Leibniz's phrase salva veritate, meaning interchangeability without loss of or compromising the truth, recurs in Willard Quine's writings. Nevertheless, the secondary literature on Leibniz did not really blossom until after World War II. This is especially true of English speaking countries; in Gregory Brown's bibliography fewer than 30 of the English language entries were published before 1946. American Leibniz studies owe much to Leroy Loemker (1904–1985) through his translations and his interpretive essays in LeClerc (1973).
Nicholas Jolley has surmised that Leibniz's reputation as a philosopher is now perhaps higher than at any time since he was alive.[194] Analytic and contemporary philosophy continue to invoke his notions of identity, individuation, and possible worlds. Work in the history of 17th- and 18th-century ideas has revealed more clearly the 17th-century "Intellectual Revolution" that preceded the better-known Industrial and commercial revolutions of the 18th and 19th centuries.
In 1985, the German government created the Leibniz Prize, offering an annual award of 1.55 million euros for experimental results and 770,000 euros for theoretical ones. It was the world's largest prize for scientific achievement prior to the Fundamental Physics Prize.
The collection of manuscript papers of Leibniz at the Gottfried Wilhelm Leibniz Bibliothek – Niedersächische Landesbibliothek was inscribed on UNESCO's Memory of the World Register in 2007.[195]
Cultural references
Leibniz still receives popular attention. The Google Doodle for 1 July 2018 celebrated Leibniz's 372nd birthday.[196][197][198] Using a quill, his hand is shown writing "Google" in binary ASCII code.
One of the earliest popular but indirect expositions of Leibniz was Voltaire's satire Candide, published in 1759. Leibniz was lampooned as Professor Pangloss, described as "the greatest philosopher of the Holy Roman Empire".
Leibniz also appears as one of the main historical figures in Neal Stephenson's series of novels The Baroque Cycle. Stephenson credits readings and discussions concerning Leibniz for inspiring him to write the series.[199]
Leibniz also stars in Adam Ehrlich Sachs's novel The Organs of Sense.
Writings and publication
Leibniz mainly wrote in three languages: scholastic Latin, French and German. During his lifetime, he published many pamphlets and scholarly articles, but only two "philosophical" books, the Combinatorial Art and the Théodicée. (He published numerous pamphlets, often anonymous, on behalf of the House of Brunswick-Lüneburg, most notably the "De jure suprematum" a major consideration of the nature of sovereignty.) One substantial book appeared posthumously, his Nouveaux essais sur l'entendement humain, which Leibniz had withheld from publication after the death of John Locke. Only in 1895, when Bodemann completed his catalogue of Leibniz's manuscripts and correspondence, did the enormous extent of Leibniz's Nachlass become clear: about 15,000 letters to more than 1000 recipients plus more than 40,000 other items. Moreover, quite a few of these letters are of essay length. Much of his vast correspondence, especially the letters dated after 1700, remains unpublished, and much of what is published has appeared only in recent decades. The more than 67,000 records of the Leibniz Edition's Catalogue cover almost all of his known writings and the letters from him and to him. The amount, variety, and disorder of Leibniz's writings are a predictable result of a situation he described in a letter as follows:
I cannot tell you how extraordinarily distracted and spread out I am. I am trying to find various things in the archives; I look at old papers and hunt up unpublished documents. From these I hope to shed some light on the history of the [House of] Brunswick. I receive and answer a huge number of letters. At the same time, I have so many mathematical results, philosophical thoughts, and other literary innovations that should not be allowed to vanish that I often do not know where to begin.[200]
The extant parts of the critical edition[201] of Leibniz's writings are organized as follows:
• Series 1. Political, Historical, and General Correspondence. 25 vols., 1666–1706.
• Series 2. Philosophical Correspondence. 3 vols., 1663–1700.
• Series 3. Mathematical, Scientific, and Technical Correspondence. 8 vols., 1672–1698.
• Series 4. Political Writings. 9 vols., 1667–1702.
• Series 5. Historical and Linguistic Writings. In preparation.
• Series 6. Philosophical Writings. 7 vols., 1663–90, and Nouveaux essais sur l'entendement humain.
• Series 7. Mathematical Writings. 6 vols., 1672–76.
• Series 8. Scientific, Medical, and Technical Writings. 1 vol., 1668–76.
The systematic cataloguing of all of Leibniz's Nachlass began in 1901. It was hampered by two world wars and then by decades of German division into two states with the Cold War's "iron curtain" in between, separating scholars, and also scattering portions of his literary estates. The ambitious project has had to deal with writings in seven languages, contained in some 200,000 written and printed pages. In 1985 it was reorganized and included in a joint program of German federal and state (Länder) academies. Since then the branches in Potsdam, Münster, Hanover and Berlin have jointly published 57 volumes of the critical edition, with an average of 870 pages, and prepared index and concordance works.
Selected works
The year given is usually that in which the work was completed, not of its eventual publication.
• 1666 (publ. 1690). De Arte Combinatoria (On the Art of Combination); partially translated in Loemker §1 and Parkinson (1966)
• 1667. Nova Methodus Discendae Docendaeque Iurisprudentiae (A New Method for Learning and Teaching Jurisprudence)
• 1667. "Dialogus de connexione inter res et verba"
• 1671. Hypothesis Physica Nova (New Physical Hypothesis); Loemker §8.I (part)
• 1673 Confessio philosophi (A Philosopher's Creed); an English translation is available online.
• Oct. 1684. "Meditationes de cognitione, veritate et ideis" ("Meditations on Knowledge, Truth, and Ideas")
• Nov. 1684. "Nova methodus pro maximis et minimis" ("New method for maximums and minimums"); translated in Struik, D. J., 1969. A Source Book in Mathematics, 1200–1800. Harvard University Press: 271–81.
• 1686. Discours de métaphysique; Martin and Brown (1988), Ariew and Garber 35, Loemker §35, Wiener III.3, Woolhouse and Francks 1
• 1686. Generales inquisitiones de analysi notionum et veritatum (General Inquiries About the Analysis of Concepts and of Truths)
• 1694. "De primae philosophiae Emendatione, et de Notione Substantiae" ("On the Correction of First Philosophy and the Notion of Substance")
• 1695. Système nouveau de la nature et de la communication des substances (New System of Nature)
• 1700. Accessiones historicae[202]
• 1703. "Explication de l'Arithmétique Binaire" ("Explanation of Binary Arithmetic"); Carl Immanuel Gerhardt, Mathematical Writings VII.223. An English translation by Lloyd Strickland is available online.
• 1704 (publ. 1765). Nouveaux essais sur l'entendement humain. Translated in: Remnant, Peter, and Bennett, Jonathan, trans., 1996. New Essays on Human Understanding Langley translation 1896. Cambridge University Press. Wiener III.6 (part)
• 1707–1710. Scriptores rerum Brunsvicensium[202] (3 Vols.)
• 1710. Théodicée; Farrer, A. M., and Huggard, E. M., trans., 1985 (1952). Wiener III.11 (part). An English translation is available online at Project Gutenberg.
• 1714. "Principes de la nature et de la Grâce fondés en raison"
• 1714. Monadologie; translated by Nicholas Rescher, 1991. The Monadology: An Edition for Students. University of Pittsburgh Press. Ariew and Garber 213, Loemker §67, Wiener III.13, Woolhouse and Francks 19. An English translation by Robert Latta is available online.
Posthumous works
• 1717. Collectanea Etymologica, edited by the secretary of Leibniz Johann Georg von Eckhart
• 1749. Protogaea
• 1750. Origines Guelficae[202]
Collections
Six important collections of English translations are Wiener (1951), Parkinson (1966), Loemker (1969), Ariew and Garber (1989), Woolhouse and Francks (1998), and Strickland (2006). The ongoing critical edition of all of Leibniz's writings is Sämtliche Schriften und Briefe.[201]
See also
• General Leibniz rule
• Leibniz Association
• Leibniz operator
• List of German inventors and discoverers
• List of pioneers in computer science
• List of things named after Gottfried Leibniz
• Mathesis universalis
• Scientific revolution
• Leibniz University Hannover
• Bartholomew Des Bosses
• Joachim Bouvet
• Outline of Gottfried Wilhelm Leibniz
• Gottfried Wilhelm Leibniz bibliography
Notes
1. Leibniz himself never attached von to his name and was never actually ennobled.
2. Sometimes spelled Leibnitz. Pronunciation: /ˈlaɪbnɪts/ LYBE-nits,[12] German: [ˈɡɔtfʁiːt ˈvɪlhɛlm ˈlaɪbnɪts] (listen)[13][14] or German: [ˈlaɪpnɪts] (listen);[15] French: Godefroi Guillaume Leibnitz[16] [ɡɔdfʁwa ɡijom lɛbnits].
3. There is no complete gathering of the writings of Leibniz translated into English.[19]
References
Citations
1. Arthur 2014, p. 16.
2. Michael Blamauer (ed.), The Mental as Fundamental: New Perspectives on Panpsychism, Walter de Gruyter, 2013, p. 111.
3. Fumerton, Richard (21 February 2000). "Foundationalist Theories of Epistemic Justification". Stanford Encyclopedia of Philosophy. Retrieved 19 August 2018.
4. Stefano Di Bella, Tad M. Schmaltz (eds.), The Problem of Universals in Early Modern Philosophy, Oxford University Press, 2017, p. 207 n. 25: "Leibniz's conceptualism [is related to] the Ockhamist tradition..."
5. A. B. Dickerson, Kant on Representation and Objectivity, Cambridge University Press, 2003, p. 85.
6. David, Marian (10 July 2022). Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University – via Stanford Encyclopedia of Philosophy.
7. Kurt Huber, Leibniz: Der Philosoph der universalen Harmonie, Severus Verlag, 2014, p. 29.
8. Gottfried Wilhelm Leibniz at the Mathematics Genealogy Project
9. Arthur 2014, p. 13.
10. Knebel, Sven K. (2022). "Pallavicino the Optimist". Sforza Pallavicino: A Jesuit Life in Baroque Rome. Brill Publishers: 191–224. doi:10.1163/9789004517240_010. ISBN 978-90-04-51724-0.
11. McNab, John (1972). Towards a Theology of Social Concern: A Comparative Study of the Elements for Social Concern in the Writings of Frederick D. Maurice and Walter Rauschenbusch (PhD thesis). Montreal: McGill University. p. 201. Retrieved 6 February 2019.
12. "Leibniz" entry in Collins English Dictionary.
13. Mangold, Max, ed. (2005). Duden-Aussprachewörterbuch (Duden Pronunciation Dictionary) (in German) (7th ed.). Mannheim: Bibliographisches Institut GmbH. ISBN 978-3-411-04066-7.
14. Wells, John C. (2008), Longman Pronunciation Dictionary (3rd ed.), Longman, ISBN 9781405881180
15. Eva-Maria Krech; et al., eds. (2010). Deutsches Aussprachewörterbuch (German Pronunciation Dictionary) (in German) (1st ed.). Berlin: Walter de Gruyter GmbH & Co. KG. ISBN 978-3-11-018203-3.
16. See inscription of the engraving depicted in the "1666–1676" section.
17. Murray, Stuart A.P. (2009). The library : an illustrated history. New York, NY: Skyhorse Pub. ISBN 978-1-60239-706-4.
18. Roughly 40%, 35% and 25%, respectively.www.gwlb.de Archived 7 July 2011 at the Wayback Machine. Leibniz-Nachlass (i.e. Legacy of Leibniz), Gottfried Wilhelm Leibniz Bibliothek (one of the three Official Libraries of the German state Lower Saxony).
19. Baird, Forrest E.; Kaufmann, Walter (2008). From Plato to Derrida. Upper Saddle River, New Jersey: Pearson Prentice Hall. ISBN 978-0-13-158591-1.
20. Russell, Bertrand (15 April 2013). History of Western Philosophy: Collectors Edition (revised ed.). Routledge. p. 469. ISBN 978-1-135-69284-1. Extract of page 469.
21. Handley, Lindsey D.; Foster, Stephen R. (2020). Don't Teach Coding: Until You Read This Book. John Wiley & Sons. p. 29. ISBN 9781119602620. Extract of page 29
22. Apostol, Tom M. (1991). Calculus, Volume 1 (illustrated ed.). John Wiley & Sons. p. 172. ISBN 9780471000051. Extract of page 172
23. Maor, Eli (2003). The Facts on File Calculus Handbook. The Facts on File Calculus Handbook. p. 58. ISBN 9781438109541. Extract of page 58
24. David Smith, pp. 173–181 (1929)
25. "2021: 375th birthday of Leibniz, father of computer science". people.idsia.ch.
26. It is possible that the words "in Aquarius" refer to the Moon (the Sun in Cancer; Sagittarius rising (Ascendant)); see Astro-Databank chart of Gottfried Leibniz.
27. The original has "1/4 uff 7 uhr" and there is good reason to assume that also in the 17th century this meant a quarter to seven, since the "uff", in its modern form of "auf", is still, as of 2018 exactly in this vernacular, in use in several Low German speaking regions. The quote is given by Hartmut Hecht in Gottfried Wilhelm Leibniz (Teubner-Archiv zur Mathematik, Volume 2, 1992), in the first lines of chapter 2, Der junge Leibniz, p. 15; see H. Hecht, Der junge Leibniz; see also G. E. Guhrauer, G. W. Frhr. v. Leibnitz. Vol. 1. Breslau 1846, Anm. p. 4.
28. Kurt Müller, Gisela Krönert, Leben und Werk von Gottfried Wilhelm Leibniz: Eine Chronik. Frankfurt a.M., Klostermann 1969, p. 3.
29. Mates, Benson (1989). The Philosophy of Leibniz: Metaphysics and Language. Oxford University Press. ISBN 978-0-19-505946-5.
30. Mackie (1845), 21
31. Mackie (1845), 22
32. "Leibniz biography". www-history.mcs.st-andrews.ac.uk. Retrieved 8 May 2018.
33. Mackie (1845), 26
34. Arthur 2014, p. x.
35. Hubertus Busche, Leibniz' Weg ins perspektivische Universum: Eine Harmonie im Zeitalter der Berechnung, Meiner Verlag, 1997, p. 120.
36. A few copies of De Arte Combinatoria were produced as requested for the habilitation procedure; it was reprinted without his consent in 1690.
37. Jolley, Nicholas (1995). The Cambridge Companion to Leibniz. Cambridge University Press.:20
38. Simmons, George (2007). Calculus Gems: Brief Lives and Memorable Mathematics. MAA.:143
39. Mackie (1845), 38
40. Mackie (1845), 39
41. Mackie (1845), 40
42. Aiton 1985: 312
43. Ariew R., G.W. Leibniz, life and works, p. 21 in The Cambridge Companion to Leibniz, ed. by N. Jolley, Cambridge University Press, 1994, ISBN 0-521-36588-0. Extract of page 21
44. Mackie (1845), 43
45. Mackie (1845), 44–45
46. Benaroya, Haym; Han, Seon Mi; Nagurka, Mark (2 May 2013). Probabilistic Models for Dynamical Systems. CRC Press. ISBN 978-1-4398-5015-2.
47. Mackie (1845), 58–61
48. Gottfried Wilhelm Leibniz. 2017. {{cite book}}: |website= ignored (help)
49. Mackie (1845), 69–70
50. Mackie (1845), 73–74
51. Davis, Martin (2018). The Universal Computer : The Road from Leibniz to Turing. CRC Press. p. 9. ISBN 978-1-138-50208-6.
52. On the encounter between Newton and Leibniz and a review of the evidence, see Alfred Rupert Hall, Philosophers at War: The Quarrel Between Newton and Leibniz, (Cambridge, 2002), pp. 44–69.
53. Mackie (1845), 117–118
54. For a study of Leibniz's correspondence with Sophia Charlotte, see MacDonald Ross, George, 1990, "Leibniz's Exposition of His System to Queen Sophie Charlotte and Other Ladies." In Leibniz in Berlin, ed. H. Poser and A. Heinekamp, Stuttgart: Franz Steiner, 1990, 61–69.
55. Mackie (1845), 109
56. Leibniz, Gottfried Wilhelm Freiherr von (1920). The Early Mathematical Manuscripts of Leibniz: Translated from the Latin Texts Published by Carl Immanuel Gerhardt with Critical and Historical Notes. Open court publishing Company. ISBN 9780598818461.
57. See Wiener IV.6 and Loemker §40. Also see a curious passage titled "Leibniz's Philosophical Dream", first published by Bodemann in 1895 and translated on p. 253 of Morris, Mary, ed. and trans., 1934. Philosophical Writings. Dent & Sons Ltd.
58. "Christian Mathematicians – Leibniz – God & Math – Thinking Christianly About Math Education". 31 January 2012.
59. Gottfried Wilhelm Leibniz (2012). Loptson, Peter (ed.). Discourse on Metaphysics and Other Writings. Broadview Press. pp. 23–24. ISBN 978-1-55481-011-6. The answer is unknowable, but it may not be unreasonable to see him, at least in theological terms, as essentially a deist. He is a determinist: there are no miracles (the events so called being merely instances of infrequently occurring natural laws); Christ has no real role in the system; we live forever, and hence we carry on after our deaths, but then everything—every individual substance—carries on forever. Nonetheless, Leibniz is a theist. His system is generated from, and needs, the postulate of a creative god. In fact, though, despite Leibniz's protestations, his God is more the architect and engineer of the vast complex world-system than the embodiment of love of Christian orthodoxy.
60. Christopher Ernest Cosans (2009). Owen's Ape & Darwin's Bulldog: Beyond Darwinism and Creationism. Indiana University Press. pp. 102–103. ISBN 978-0-253-22051-6. In advancing his system of mechanics, Newton claimed that collisions of celestial objects would cause a loss of energy that would require God to intervene from time to time to maintain order in the solar system (Vailati 1997, 37–42). In criticizing this implication, Leibniz remarks: "Sir Isaac Newton and his followers have also a very odd opinion concerning the work of God. According to their doctrine, God Almighty wants to wind up his watch from time to time; otherwise it would cease to move." (Leibniz 1715, 675) Leibniz argues that any scientific theory that relies on God to perform miracles after He had first made the universe indicates that God lacked sufficient foresight or power to establish adequate natural laws in the first place. In defense of Newton's theism, Clarke is unapologetic: "'tis not a diminution but the true glory of his workmanship that nothing is done without his continual government and inspection"' (Leibniz 1715, 676–677). Clarke is believed to have consulted closely with Newton on how to respond to Leibniz. He asserts that Leibniz's deism leads to "the notion of materialism and fate" (1715, 677), because it excludes God from the daily workings of nature.
61. Hunt, Shelby D. (2003). Controversy in Marketing Theory: For Reason, Realism, Truth, and Objectivity. M. E. Sharpe. p. 33. ISBN 978-0-7656-0931-1. Consistent with the liberal views of the Enlightenment, Leibniz was an optimist with respect to human reasoning and scientific progress (Popper 1963, p. 69). Although he was a great reader and admirer of Spinoza, Leibniz, being a confirmed deist, rejected emphatically Spinoza's pantheism: God and nature, for Leibniz, were not simply two different "labels" for the same "thing".
62. Leibniz on the Trinity and the Incarnation: Reason and Revelation in the Seventeenth Century (New Haven: Yale University Press, 2007, pp. xix–xx).
63. Ariew & Garber, 69; Loemker, §§36, 38
64. Ariew & Garber, 138; Loemker, §47; Wiener, II.4
65. Later translated as Loemker 267 and Woolhouse and Francks 30
66. A VI, 4, n. 324, pp. 1643–1649 with the title: Principia Logico-Metaphysica
67. Ariew & Garber, 272–284; Loemker, §§14, 20, 21; Wiener, III.8
68. Mates (1986), chpts. 7.3, 9
69. Loemker 717
70. See Jolley (1995: 129–131), Woolhouse and Francks (1998), and Mercer (2001).
71. Gottfried Leibniz, New Essays, IV, 16: "la nature ne fait jamais des sauts". Natura non-facit saltus is the Latin translation of the phrase (originally put forward by Linnaeus' Philosophia Botanica, 1st ed., 1751, Chapter III, § 77, p. 27; see also Stanford Encyclopedia of Philosophy: "Continuity and Infinitesimals" and Alexander Baumgarten, Metaphysics: A Critical Translation with Kant's Elucidations, Translated and Edited by Courtney D. Fugate and John Hymers, Bloomsbury, 2013, "Preface of the Third Edition (1750)", p. 79 n.d.: "[Baumgarten] must also have in mind Leibniz's "natura non-facit saltus [nature does not make leaps]" (NE IV, 16)."). A variant translation is "natura non-saltum facit" (literally, "Nature does not make a jump") (Britton, Andrew; Sedgwick, Peter H.; Bock, Burghard (2008). Ökonomische Theorie und christlicher Glaube. LIT Verlag Münster. p. 289. ISBN 978-3-8258-0162-5. Extract of page 289.)
72. Loemker 311
73. Arthur Lovejoy, The Great Chain of Being. Harvard University Press, 1936, Chapter V "Plenitude and Sufficient Reason in Leibniz and Spinoza", pp. 144–182.
74. For a precis of what Leibniz meant by these and other Principles, see Mercer (2001: 473–484). For a classic discussion of Sufficient Reason and Plenitude, see Lovejoy (1957).
75. O'Leary-Hawthorne, John; Cover, J. A. (4 September 2008). Substance and Individuation in Leibniz. Cambridge University Press. p. 65. ISBN 978-0-521-07303-5.
76. Rescher, Nicholas (1991). G. W. Leibniz's Monadology: an edition for students. Pittsburgh: University of Pittsburgh Press. p. 40. ISBN 978-0-8229-5449-1.
77. Ferraro, Rafael (2007). Einstein's Space-Time: An Introduction to Special and General Relativity. Springer. p. 1. ISBN 978-0-387-69946-2.
78. See H. G. Alexander, ed., The Leibniz-Clarke Correspondence, Manchester: Manchester University Press, pp. 25–26.
79. Agassi, Joseph (September 1969). "Leibniz's Place in the History of Physics". Journal of the History of Ideas. 30 (3): 331–344. doi:10.2307/2708561. JSTOR 2708561.
80. Perkins, Franklin (10 July 2007). Leibniz: A Guide for the Perplexed. Bloomsbury Academic. p. 22. ISBN 978-0-8264-8921-0.
81. Perkins, Franklin (10 July 2007). Leibniz: A Guide for the Perplexed. Bloomsbury Academic. p. 23. ISBN 978-0-8264-8921-0.
82. Rutherford (1998) is a detailed scholarly study of Leibniz's theodicy.
83. Franklin, James (2022). "The global/local distinction vindicates Leibniz's theodicy". Theology and Science. 20 (4): 445–462. doi:10.1080/14746700.2022.2124481. S2CID 252979403.
84. Magill, Frank (ed.). Masterpieces of World Philosophy. New York: Harper Collins (1990).
85. Magill, Frank (ed.) (1990)
86. Anderson Csiszar, Sean (26 July 2015). The Golden Book About Leibniz. CreateSpace Independent Publishing Platform. p. 20. ISBN 978-1515243915.
87. Leibniz, Gottfried Wilhelm. Discourse on Metaphysics. The Rationalists: Rene Descartes – Discourse on Method, Meditations. N.Y.: Dolphin., n.d., n.p.,
88. Monadologie (1714). Nicholas Rescher, trans., 1991. The Monadology: An Edition for Students. Uni. of Pittsburgh Press, p. 135.
89. "The Fundamental Question". hedweb.com. Retrieved 26 April 2017.
90. Geier, Manfred (17 February 2017). Wittgenstein und Heidegger: Die letzten Philosophen (in German). Rowohlt Verlag. ISBN 978-3-644-04511-8. Retrieved 26 April 2017.
91. Kulstad, Mark; Carlin, Laurence (2020), "Leibniz's Philosophy of Mind", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Winter 2020 ed.), Metaphysics Research Lab, Stanford University, retrieved 22 June 2023
92. Gray, Jonathan. ""Let us Calculate!": Leibniz, Llull, and the Computational Imagination". The Public Domain Review. Retrieved 22 June 2023.
93. The Art of Discovery 1685, Wiener 51
94. Many of his memoranda are translated in Parkinson 1966.
95. Marcelo Dascal, Leibniz. Language, Signs and Thought: A Collection of Essays (Foundations of Semiotics series), John Benjamins Publishing Company, 1987, p. 42.
96. Loemker, however, who translated some of Leibniz's works into English, said that the symbols of chemistry were real characters, so there is disagreement among Leibniz scholars on this point.
97. Preface to the General Science, 1677. Revision of Rutherford's translation in Jolley 1995: 234. Also Wiener I.4
98. A good introductory discussion of the "characteristic" is Jolley (1995: 226–240). An early, yet still classic, discussion of the "characteristic" and "calculus" is Couturat (1901: chpts. 3, 4).
99. Lenzen, W., 2004, "Leibniz's Logic," in Handbook of the History of Logic by D. M. Gabbay/J. Woods (eds.), volume 3: The Rise of Modern Logic: From Leibniz to Frege, Amsterdam et al.: Elsevier-North-Holland, pp. 1–83.
100. Russell, Bertrand (1900). A Critical Exposition of the Philosophy of Leibniz. The University Press, Cambridge.
101. Leibniz: Die philosophischen Schriften VII, 1890, pp. 236–247; translated as "A Study in the Calculus of Real Addition" (1690) Archived 19 July 2021 at the Wayback Machine by G. H. R. Parkinson, Leibniz: Logical Papers – A Selection, Oxford 1966, pp. 131–144.
102. Edward N. Zalta, "A (Leibnizian) Theory of Concepts", Philosophiegeschichte und logische Analyse / Logical Analysis and History of Philosophy, 3 (2000): 137–183.
103. Lenzen, Wolfgang. "Leibniz: Logic". Internet Encyclopedia of Philosophy.
104. Jesse Alama, Paul E. Oppenheimer, Edward N. Zalta, "Automating Leibniz's Theory of Concepts", in A. Felty and A. Middeldorp (eds.), Automated Deduction – CADE 25: Proceedings of the 25th International Conference on Automated Deduction (Lecture Notes in Artificial Intelligence: Volume 9195), Berlin: Springer, 2015, pp. 73–97.
105. Struik (1969), 367
106. Agarwal, Ravi P; Sen, Syamal K (2014). Creators of Mathematical and Computational Sciences. Springer, Cham. p. 186. ISBN 978-3-319-10870-4.
107. Gowers, Timothy; Barrow-Green, June; Leader, Imre (2008). The Princeton Companion to Mathematics. Princeton: Princeton University Press. p. 745. ISBN 978-0-691-11880-2.
108. Jesseph, Douglas M. (1998). "Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes". Perspectives on Science. 6.1&2 (1–2): 6–40. doi:10.1162/posc_a_00543. S2CID 118227996. Retrieved 31 December 2011.
109. Goldstine, Herman H. (1972). The Computer from Pascal to von Neumann. Princeton: Princeton University Press. p. 8. ISBN 0-691-08104-2.
110. Jones, Matthew L. (1 October 2006). The Good Life in the Scientific Revolution: Descartes, Pascal, Leibniz, and the Cultivation of Virtue. University of Chicago Press. pp. 237–239. ISBN 978-0-226-40955-9.
111. Agarwal, Ravi P; Sen, Syamal K (2014). Creators of Mathematical and Computational Sciences. Springer, Cham. p. 180. ISBN 978-3-319-10870-4.
112. Gowers, Timothy; Barrow-Green, June; Leader, Imre, eds. (2008). The Princeton Companion to Mathematics. Princeton: Princeton University Press. p. 744. ISBN 978-0-691-11880-2.
113. Knobloch, Eberhard (13 March 2013). Leibniz's Theory of Elimination and Determinants. Springer. pp. 230–237. ISBN 978-4-431-54272-8.
114. Concise Dictionary of Mathematics. V&S Publishers. April 2012. pp. 113–114. ISBN 978-93-81588-83-3.
115. Lay, David C. (2012). Linear algebra and its applications (4th ed.). Boston: Addison-Wesley. ISBN 978-0-321-38517-8.
116. Tokuyama, Takeshi; et al. (2007). Algorithms and Computation: 18th International Symposium, ISAAC 2007, Sendai, Japan, December 17–19, 2007 : proceedings. Berlin [etc.]: Springer. p. 599. ISBN 978-3-540-77120-3.
117. Jones, Matthew L. (2006). The Good Life in the Scientific Revolution : Descartes, Pascal, Leibniz, and the Cultivation of Virtue ([Online-Ausg.] ed.). Chicago [u.a.]: Univ. of Chicago Press. p. 169. ISBN 978-0-226-40954-2.
118. Davis, Martin (28 February 2018). The Universal Computer : The Road from Leibniz to Turing, Third Edition. CRC Press. p. 7. ISBN 978-1-138-50208-6.
119. De Risi, Vincenzo (2016). Leibniz on the Parallel Postulate and the Foundations of Geometry. Birkhäuser. p. 4. ISBN 978-3-319-19863-7.
120. De Risi, Vincenzo (10 February 2016). Leibniz on the Parallel Postulate and the Foundations of Geometry. Birkhäuser, Cham. p. 58. ISBN 978-3-319-19862-0.
121. Leibniz, Gottfried Wilhelm Freiherr von; Gerhardt, Carl Immanuel (trans.) (1920). The Early Mathematical Manuscripts of Leibniz. Open Court Publishing. p. 93. Retrieved 10 November 2013.
122. For an English translation of this paper, see Struik (1969: 271–284), who also translates parts of two other key papers by Leibniz on calculus.
123. Dirk Jan Struik, A Source Book in Mathematics (1969) pp. 282–284
124. Supplementum geometriae dimensoriae, seu generalissima omnium tetragonismorum effectio per motum: similiterque multiplex constructio lineae ex data tangentium conditione, Acta Euriditorum (Sep. 1693) pp. 385–392
125. John Stillwell, Mathematics and its History (1989, 2002) p.159
126. Katz, Mikhail; Sherry, David (2012), "Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond", Erkenntnis, 78 (3): 571–625, arXiv:1205.0174, doi:10.1007/s10670-012-9370-y, S2CID 119329569
127. Dauben, Joseph W (December 2003). "Mathematics, ideology, and the politics of infinitesimals: mathematical logic and nonstandard analysis in modern China". History and Philosophy of Logic. 24 (4): 327–363. doi:10.1080/01445340310001599560. ISSN 0144-5340. S2CID 120089173.
128. Hockney, Mike (29 March 2016). How to Create the Universe. Lulu Press, Inc. ISBN 978-1-326-61200-9.
129. Loemker §27
130. Mates (1986), 240
131. Hirano, Hideaki. "Leibniz's Cultural Pluralism And Natural Law". Archived from the original on 22 May 2009. Retrieved 10 March 2010.
132. Mandelbrot (1977), 419. Quoted in Hirano (1997).
133. Ariew and Garber 117, Loemker §46, W II.5. On Leibniz and physics, see the chapter by Garber in Jolley (1995) and Wilson (1989).
134. Futch, Michael. Leibniz's Metaphysics of Time and Space. New York: Springer, 2008.
135. Ray, Christopher. Time, Space and Philosophy. London: Routledge, 1991.
136. Rickles, Dean. Symmetry, Structure and Spacetime. Oxford: Elsevier, 2008.
137. Arthur 2014, p. 56.
138. See Ariew and Garber 155–86, Loemker §§53–55, W II.6–7a
139. On Leibniz and biology, see Loemker (1969a: VIII).
140. L. E. Loemker: Introduction to Philosophical papers and letters: A selection. Gottfried W. Leibniz (transl. and ed., by Leroy E. Loemker). Dordrecht: Riedel (2nd ed. 1969).
141. T. Verhave: Contributions to the history of psychology: III. G. W. Leibniz (1646–1716). On the Association of Ideas and Learning. Psychological Report, 1967, Vol. 20, 11–116.
142. R. E. Fancher & H. Schmidt: Gottfried Wilhelm Leibniz: Underappreciated pioneer of psychology. In: G. A. Kimble & M. Wertheimer (Eds.). Portraits of pioneers in psychology, Vol. V. American Psychological Association, Washington, DC, 2003, pp. 1–17.
143. Leibniz, G. W. (2007) [1714/1720]. The Principles of Philosophy known as Monadology. Translated by Jonathan Bennett. p. 11.
144. Larry M. Jorgensen, The Principle of Continuity and Leibniz's Theory of Consciousness.
145. The German scholar Johann Thomas Freigius was the first to use this Latin term 1574 in print: Quaestiones logicae et ethicae, Basel, Henricpetri.
146. Leibniz, Nouveaux essais, 1765, Livre II, Des Idées, Chapitre 1, § 6. New Essays on Human Understanding Book 2. p. 36; transl. by Jonathan Bennett, 2009.
147. Wundt: Leibniz zu seinem zweihundertjährigen Todestag, 14. November 1916. Alfred Kröner Verlag, Leipzig 1917.
148. Wundt (1917), p. 117.
149. Fahrenberg, Jochen (2017). "The influence of Gottfried Wilhelm Leibniz on the Psychology, philosophy, and Ethics of Wilhelm Wundt" (PDF). Retrieved 28 June 2022.
150. D. Brett King, Wayne Viney and William Woody. A History of Psychology: Ideas and Context (2009), 150–153.
151. Nicholls and Leibscher Thinking the Unconscious: Nineteenth-Century German Thought (2010), 6.
152. Nicholls and Leibscher (2010).
153. King et al. (2009), 150–153.
154. Klempe, SH (2011). "The role of tone sensation and musical stimuli in early experimental psychology". Journal of the History of the Behavioral Sciences. 47 (2): 187–199. doi:10.1002/jhbs.20495. PMID 21462196.
155. Aiton (1985), 107–114, 136
156. Davis (2000) discusses Leibniz's prophetic role in the emergence of calculating machines and of formal languages.
157. See Couturat (1901): 473–478.
158. Ryan, James A. (1996). "Leibniz' Binary System and Shao Yong's "Yijing"". Philosophy East and West. 46 (1): 59–90. doi:10.2307/1399337. JSTOR 1399337.
159. Ares, J.; Lara, J.; Lizcano, D.; Martínez, M. (2017). "Who Discovered the Binary System and Arithmetic? Did Leibniz Plagiarize Caramuel?". Science and Engineering Ethics. 24 (1): 173–188. doi:10.1007/s11948-017-9890-6. hdl:20.500.12226/69. PMID 28281152. S2CID 35486997.
160. Navarro-Loidi, Juan (May 2008). "The Introductions of Logarithms into Spain". Historia Mathematica. 35 (2): 83–101. doi:10.1016/j.hm.2007.09.002.
161. Booth, Michael (2003). "Thomas Harriot's Translations". The Yale Journal of Criticism. 16 (2): 345–361. doi:10.1353/yale.2003.0013. ISSN 0893-5378. S2CID 161603159.
162. Lande, Daniel. "Development of the Binary Number System and the Foundations of Computer Science". The Mathematics Enthusiast: 513–540.
163. Wiener, N., Cybernetics (2nd edition with revisions and two additional chapters), The MIT Press and Wiley, New York, 1961, p. 12.
164. Wiener, Norbert (1948). "Time, Communication, and the Nervous System". Annals of the New York Academy of Sciences. 50 (4 Teleological): 197–220. Bibcode:1948NYASA..50..197W. doi:10.1111/j.1749-6632.1948.tb39853.x. PMID 18886381. S2CID 28452205. Archived from the original on 23 July 2021. Retrieved 23 July 2021.
165. Couturat (1901), 115
166. See N. Rescher, Leibniz and Cryptography (Pittsburgh, University Library Systems, University of Pittsburgh, 2012).
167. "The discoveries of principle of the calculus in Acta Eruditorum" (commentary, pp. 60–61), translated by Pierre Beaudry, amatterofmind.org, Leesburg, Va., September 2000. (pdf)
168. "The Reality Club: Wake Up Call for Europe Tech". www.edge.org. Archived from the original on 28 December 2005. Retrieved 11 January 2006.
169. Agarwal, Ravi P; Sen, Syamal K (2014). Creators of Mathematical and Computational Sciences. Springer, Cham. p. 28. ISBN 978-3-319-10870-4.
170. "Gottfried Wilhelm Leibniz | Biography & Facts". Encyclopedia Britannica. Retrieved 18 February 2019.
171. Schulte-Albert, H. (April 1971). "Gottfried Wilhelm Leibniz and Library Classification". The Journal of Library History. 6 (2): 133–152. JSTOR 25540286.
172. Schulte-Albert, H. G. (1971). "Gottfried Wilhelm Leibniz and Library Classification". The Journal of Library History. 6 (2): 133–152. JSTOR 25540286.
173. Otivm Hanoveranvm Sive Miscellanea Ex ore & schedis Illustris Viri, piæ memoriæ, Godofr. Gvilielmi Leibnitii ... / Quondam notata & descripta, Cum ipsi in collendis & excerpendis rebus ad Historiam Brunsvicensem pertinentibus operam navaret, Joachimvs Fridericvs Fellervs, Secretarius Ducalis Saxo-Vinariensis. Additæ sunt coronidis loco Epistolæ Gallicæ amœbeæ Leibnitii & Pellissonii de Tolerantia Religionum & de controversiis quibusdam Theologicis ... 1737.
174. On Leibniz's projects for scientific societies, see Couturat (1901), App. IV.
175. See, for example, Ariew and Garber 19, 94, 111, 193; Riley 1988; Loemker §§2, 7, 20, 29, 44, 59, 62, 65; W I.1, IV.1–3
176. See (in order of difficulty) Jolley (2005: ch. 7), Gregory Brown's chapter in Jolley (1995), Hostler (1975), Connelly (2021), and Riley (1996).
177. Loemker: 59, fn 16. Translation revised.
178. Loemker: 58, fn 9
179. Andrés-Gallego, José (2015). "Are Humanism and Mixed Methods Related? Leibniz's Universal (Chinese) Dream". Journal of Mixed Methods Research. 29 (2): 118–132. doi:10.1177/1558689813515332. S2CID 147266697. Archived from the original on 27 August 2016. Retrieved 24 June 2015.
180. Artosi ed.(2013)
181. Loemker, 1
182. Connelly, 2018, ch.5; Artosi et al. 2013, pref.
183. Connelly, 2021, ch.6
184. Christopher Johns, 2018
185. (Akademie Ed VI ii 35-93)
186. Connelly, 2021, chs.6-8
187. Perkins (2004), 117
188. Mungello, David E. (1971). "Leibniz's Interpretation of Neo-Confucianism". Philosophy East and West. 21 (1): 3–22. doi:10.2307/1397760. JSTOR 1397760.
189. On Leibniz, the I Ching, and binary numbers, see Aiton (1985: 245–248). Leibniz's writings on Chinese civilization are collected and translated in Cook and Rosemont (1994), and discussed in Perkins (2004).
190. Cook, Daniel (2015). "Leibniz, China, and the Problem of Pagan Wisdom". Philosophy East and West. 65 (3): 936–947. doi:10.1353/pew.2015.0074. S2CID 170208696.
191. "Vasilyev, 1993" (PDF). Archived from the original (PDF) on 23 February 2011. Retrieved 12 June 2010.
192. Russell, 1900
193. Jolley, 217–219
194. "Letters from and to Gottfried Wilhelm Leibniz within the collection of manuscript papers of Gottfried Wilhelm Leibniz". UNESCO Memory of the World Programme. 16 May 2008. Archived from the original on 19 July 2010. Retrieved 15 December 2009.
195. "Gottfried Wilhelm Leibniz's 372nd Birthday". Google Doodle Archive. 1 July 2018. Retrieved 23 July 2021.{{cite web}}: CS1 maint: url-status (link)
196. Musil, Steven (1 July 2018). "Google Doodle celebrates mathematician Gottfried Wilhelm Leibni". CNET.
197. Smith, Kiona N. (30 June 2018). "Sunday's Google Doodle Celebrates Mathematician Gottfried Wilhelm Leibniz". Forbes.
198. Stephenson, Neal. "How the Baroque Cycle Began" in P.S. of Quicksilver Perennial ed. 2004.
199. Letter to Vincent Placcius, 15 September 1695, in Louis Dutens (ed.), Gothofridi Guillemi Leibnitii Opera Omnia, vol. 6.1, 1768, pp. 59–60.
200. "Leibniz-Edition" (in German). Archived from the original on 7 January 2008.
201. Holland, Arthur William (1911). "Germany/History" . In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 11 (11th ed.). Cambridge University Press. pp. 828–901, see page 899, para two. The two chief collections which were issued by the philosopher are the Accessiones historicae (1698–1700) and the Scriptores rerum Brunsvicensium.....
Bibliographies
• Bodemann, Eduard, Die Leibniz-Handschriften der Königlichen öffentlichen Bibliothek zu Hannover, 1895, (anastatic reprint: Hildesheim, Georg Olms, 1966).
• Bodemann, Eduard, Der Briefwechsel des Gottfried Wilhelm Leibniz in der Königlichen öffentliche Bibliothek zu Hannover, 1895, (anastatic reprint: Hildesheim, Georg Olms, 1966).
• Cerqueiro, Daniel (2014). Leibnitz y la ciencia del infinito. Buenos Aires: Pequeña Venecia. ISBN 9789879239247.
• Ravier, Émile, Bibliographie des œuvres de Leibniz, Paris: Alcan, 1937 (anastatic reprint Hildesheim: Georg Olms, 1966).
• Heinekamp, Albert and Mertens, Marlen. Leibniz-Bibliographie. Die Literatur über Leibniz bis 1980, Frankfurt: Vittorio Klostermann, 1984.
• Heinekamp, Albert and Mertens, Marlen. Leibniz-Bibliographie. Die Literatur über Leibniz. Band II: 1981–1990, Frankfurt: Vittorio Klostermann, 1996.
An updated bibliography of more than 25.000 titles is available at Leibniz Bibliographie.
Primary literature (chronologically)
• Wiener, Philip, (ed.), 1951. Leibniz: Selections. Scribner.
• Schrecker, Paul & Schrecker, Anne Martin, (eds.), 1965. Monadology and other Philosophical Essays. Prentice-Hall.
• Parkinson, G. H. R. (ed.), 1966. Logical Papers. Clarendon Press.
• Mason, H. T. & Parkinson, G. H. R. (eds.), 1967. The Leibniz-Arnauld Correspondence. Manchester University Press.
• Loemker, Leroy, (ed.), 1969 [1956]. Leibniz: Philosophical Papers and Letters. Reidel.
• Morris, Mary & Parkinson, G. H. R. (eds.), 1973. Philosophical Writings. Everyman's University Library.
• Riley, Patrick, (ed.), 1988. Leibniz: Political Writings. Cambridge University Press.
• Niall, R. Martin, D. & Brown, Stuart (eds.), 1988. Discourse on Metaphysics and Related Writings. Manchester University Press.
• Ariew, Roger and Garber, Daniel. (eds.), 1989. Leibniz: Philosophical Essays. Hackett.
• Rescher, Nicholas (ed.), 1991. G. W. Leibniz's Monadology. An Edition for Students, University of Pittsburgh Press.
• Rescher, Nicholas, On Leibniz, (Pittsburgh: University of Pittsburgh Press, 2013).
• Parkinson, G. H. R. (ed.) 1992. De Summa Rerum. Metaphysical Papers, 1675–1676. Yale University Press.
• Cook, Daniel, & Rosemont, Henry Jr., (eds.), 1994. Leibniz: Writings on China. Open Court.
• Farrer, Austin (ed.), 1995. Theodicy, Open Court.
• Remnant, Peter, & Bennett, Jonathan, (eds.), 1996 (1981). Leibniz: New Essays on Human Understanding. Cambridge University Press.
• Woolhouse, R. S., and Francks, R., (eds.), 1997. Leibniz's 'New System' and Associated Contemporary Texts. Oxford University Press.
• Woolhouse, R. S., and Francks, R., (eds.), 1998. Leibniz: Philosophical Texts. Oxford University Press.
• Ariew, Roger, (ed.), 2000. G. W. Leibniz and Samuel Clarke: Correspondence. Hackett.
• Richard T. W. Arthur, (ed.), 2001. The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672–1686. Yale University Press.
• Richard T. W. Arthur, 2014. Leibniz. John Wiley & Sons.
• Robert C. Sleigh Jr., (ed.), 2005. Confessio Philosophi: Papers Concerning the Problem of Evil, 1671–1678. Yale University Press.
• Dascal, Marcelo (ed.), 2006. G. W. Leibniz. The Art of Controversies, Springer.
• Strickland, Lloyd, 2006 (ed.). The Shorter Leibniz Texts: A Collection of New Translations. Continuum.
• Look, Brandon and Rutherford, Donald (eds.), 2007. The Leibniz-Des Bosses Correspondence, Yale University Press.
• Cohen, Claudine and Wakefield, Andre, (eds.), 2008. Protogaea. University of Chicago Press.
• Murray, Michael, (ed.) 2011. Dissertation on Predestination and Grace, Yale University Press.
• Strickand, Lloyd (ed.), 2011. Leibniz and the two Sophies. The Philosophical Correspondence, Toronto.
• Lodge, Paul (ed.), 2013. The Leibniz-De Volder Correspondence: With Selections from the Correspondence Between Leibniz and Johann Bernoulli, Yale University Press.
• Artosi, Alberto, Pieri, Bernardo, Sartor, Giovanni (eds.), 2014. Leibniz: Logico-Philosophical Puzzles in the Law, Springer.
• De Iuliis, Carmelo Massimo, (ed.), 2017. Leibniz: The New Method of Learning and Teaching Jurisprudence, Talbot, Clark NJ.
Secondary literature up to 1950
• Du Bois-Reymond, Emil, 1912. Leibnizsche Gedanken in der neueren Naturwissenschaft, Berlin: Dummler, 1871 (reprinted in Reden, Leipzig: Veit, vol. 1).
• Couturat, Louis, 1901. La Logique de Leibniz. Paris: Felix Alcan.
• Heidegger, Martin, 1983. The Metaphysical Foundations of Logic. Indiana University Press (lecture course, 1928).
• Lovejoy, Arthur O., 1957 (1936). "Plenitude and Sufficient Reason in Leibniz and Spinoza" in his The Great Chain of Being. Harvard University Press: 144–182. Reprinted in Frankfurt, H. G., (ed.), 1972. Leibniz: A Collection of Critical Essays. Anchor Books 1972.
• Mackie, John Milton; Guhrauer, Gottschalk Eduard, 1845. Life of Godfrey William von Leibnitz. Gould, Kendall and Lincoln.
• Russell, Bertrand, 1900, A Critical Exposition of the Philosophy of Leibniz, Cambridge: The University Press.
• Smith, David Eugene (1929). A Source Book in Mathematics. New York and London: McGraw-Hill Book Company, Inc.
• Trendelenburg, F. A., 1857, "Über Leibnizens Entwurf einer allgemeinen Charakteristik," Philosophische Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin. Aus dem Jahr 1856, Berlin: Commission Dümmler, pp. 36–69.
• Adolphus William Ward (1911), Leibniz as a Politician: The Adamson Lecture, 1910 (1st ed.), Manchester, Wikidata Q19095295{{citation}}: CS1 maint: location missing publisher (link) (lecture)
Secondary literature post-1950
• Adams, Robert Merrihew. 1994. Leibniz: Determinist, Theist, Idealist. New York: Oxford, Oxford University Press.
• Aiton, Eric J., 1985. Leibniz: A Biography. Hilger (UK).
• Antognazza, Maria Rosa, 2008. Leibniz: An Intellectual Biography. Cambridge Univ. Press.
• Barrow, John D.; Tipler, Frank J. (1986). The Anthropic Cosmological Principle (1st ed.). Oxford University Press. ISBN 978-0-19-282147-8. LCCN 87028148.
• Bos, H. J. M. (1974). "Differentials, higher-order differentials and the derivative in the Leibnizian calculus". Archive for History of Exact Sciences. 14: 1–90. doi:10.1007/bf00327456. S2CID 120779114.
• Brown, Stuart (ed.), 1999. The Young Leibniz and His Philosophy (1646–76), Dordrecht, Kluwer.
• Cerqueiro, Daniel. Leibnitz y la ciencia del infinito(2014).Pequeña Venecia. Buenos Aires. isbn=9789879239247.
• Connelly, Stephen, 2021. ‘’Leibniz: A Contribution to the Archaeology of Power’’, Edinburgh University Press ISBN 978-1-4744-1808-9.
• Davis, Martin, 2000. The Universal Computer: The Road from Leibniz to Turing. WW Norton.
• Deleuze, Gilles, 1993. The Fold: Leibniz and the Baroque. University of Minnesota Press.
• Fahrenberg, Jochen, 2017. PsyDok ZPID The influence of Gottfried Wilhelm Leibniz on the Psychology, Philosophy, and Ethics of Wilhelm Wundt.
• Fahrenberg, Jochen, 2020. Wilhelm Wundt (1832 – 1920). Introduction, Quotations, Reception, Commentaries, Attempts at Reconstruction. Pabst Science Publishers, Lengerich 2020, ISBN 978-3-95853-574-9.
• Finster, Reinhard & van den Heuvel, Gerd 2000. Gottfried Wilhelm Leibniz. Mit Selbstzeugnissen und Bilddokumenten. 4. Auflage. Rowohlt, Reinbek bei Hamburg (Rowohlts Monographien, 50481), ISBN 3-499-50481-2.
• Grattan-Guinness, Ivor, 1997. The Norton History of the Mathematical Sciences. W W Norton.
• Hall, A. R., 1980. Philosophers at War: The Quarrel between Newton and Leibniz. Cambridge University Press.
• Hamza, Gabor, 2005. "Le développement du droit privé européen". ELTE Eotvos Kiado Budapest.
• Hoeflich, M. H. (1986). "Law & Geometry: Legal Science from Leibniz to Langdell". American Journal of Legal History. 30 (2): 95–121. doi:10.2307/845705. JSTOR 845705.
• Hostler, John, 1975. Leibniz's Moral Philosophy. UK: Duckworth.
• Ishiguro, Hidé 1990. Leibniz's Philosophy of Logic and Language. Cambridge University Press.
• Jolley, Nicholas, (ed.), 1995. The Cambridge Companion to Leibniz. Cambridge University Press.
• Kaldis, Byron, 2011. Leibniz' Argument for Innate Ideas in Just the Arguments: 100 of the Most Important Arguments in Western Philosophy edited by M Bruce & S Barbone. Blackwell.
• Karabell, Zachary (2003). Parting the desert: the creation of the Suez Canal. Alfred A. Knopf. ISBN 978-0-375-40883-0.
• LeClerc, Ivor (ed.), 1973. The Philosophy of Leibniz and the Modern World. Vanderbilt University Press.
• Luchte, James (2006). "Mathesis and Analysis: Finitude and the Infinite in the Monadology of Leibniz". Heythrop Journal. 47 (4): 519–543. doi:10.1111/j.1468-2265.2006.00296.x.
• Mates, Benson, 1986. The Philosophy of Leibniz: Metaphysics and Language. Oxford University Press.
• Mercer, Christia, 2001. Leibniz's Metaphysics: Its Origins and Development. Cambridge University Press.
• Perkins, Franklin, 2004. Leibniz and China: A Commerce of Light. Cambridge University Press.
• Riley, Patrick, 1996. Leibniz's Universal Jurisprudence: Justice as the Charity of the Wise. Harvard University Press.
• Rutherford, Donald, 1998. Leibniz and the Rational Order of Nature. Cambridge University Press.
• Schulte-Albert, H. G. (1971). Gottfried Wilhelm Leibniz and Library Classification. The Journal of Library History (1966–1972), (2). 133–152.
• Smith, Justin E. H., 2011. Divine Machines. Leibniz and the Sciences of Life, Princeton University Press.
• Wilson, Catherine, 1989. Leibniz's Metaphysics: A Historical and Comparative Study. Princeton University Press.
• Zalta, E. N. (2000). "A (Leibnizian) Theory of Concepts" (PDF). Philosophiegeschichte und Logische Analyse / Logical Analysis and History of Philosophy. 3: 137–183. doi:10.30965/26664275-00301008.
External links
Wikisource has the text of the 1911 Encyclopædia Britannica article "Leibnitz, Gottfried Wilhelm".
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• Works by Gottfried Wilhelm Leibniz at Project Gutenberg
• Works by or about Gottfried Wilhelm Leibniz at Internet Archive
• Works by Gottfried Wilhelm Leibniz at LibriVox (public domain audiobooks)
• Look, Brandon C. "Gottfried Wilhelm Leibniz". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
• Peckhaus, Volker. "Leibniz's Influence on 19th Century Logic". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
• Burnham, Douglas. "Gottfried Leibniz: Metaphysics". Internet Encyclopedia of Philosophy.
• Carlin, Laurence. "Gottfried Leibniz: Causation". Internet Encyclopedia of Philosophy.
• Lenzen, Wolfgang. "Leibniz: Logic". Internet Encyclopedia of Philosophy.
• O'Connor, John J.; Robertson, Edmund F., "Gottfried Wilhelm Leibniz", MacTutor History of Mathematics Archive, University of St Andrews
• Gottfried Wilhelm Leibniz at the Mathematics Genealogy Project
• Translations by Jonathan Bennett, of the New Essays, the exchanges with Bayle, Arnauld and Clarke, and about 15 shorter works.
• Gottfried Wilhelm Leibniz: Texts and Translations, compiled by Donald Rutherford, UCSD
• Leibnitiana, links and resources edited by Gregory Brown, University of Houston
• Philosophical Works of Leibniz translated by G.M. Duncan (1890)
• The Best of All Possible Worlds: Nicholas Rescher Talks About Gottfried Wilhelm von Leibniz's "Versatility and Creativity"
• "Protogæa" (1693, Latin, in Acta eruditorum) – Linda Hall Library
• Protogaea (1749, German) – full digital facsimile from Linda Hall Library
• Leibniz's (1768, 6-volume) Opera omnia – digital facsimile
• Leibniz's arithmetical machine, 1710, online and analyzed on BibNum [click 'à télécharger' for English analysis]
• Leibniz's binary numeral system, 'De progressione dyadica', 1679, online and analyzed on BibNum [click 'à télécharger' for English analysis]
Gottfried Wilhelm Leibniz
Mathematics and
philosophy
• Alternating series test
• Best of all possible worlds
• Calculus controversy
• Calculus ratiocinator
• Characteristica universalis
• Compossibility
• Difference
• Dynamism
• Identity of indiscernibles
• Individuation
• Law of continuity
• Leibniz wheel
• Leibniz's gap
• Leibniz's notation
• Lingua generalis
• Mathesis universalis
• Pre-established harmony
• Plenitude
• Sufficient reason
• Salva veritate
• Theodicy
• Transcendental law of homogeneity
• Rationalism
• Universal science
• Vis viva
• Well-founded phenomenon
Works
• De Arte Combinatoria (1666)
• Discourse on Metaphysics (1686)
• New Essays on Human Understanding (1704)
• Théodicée (1710)
• Monadology (1714)
• Leibniz–Clarke correspondence (1715–1716)
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Ancient and
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1920
postwar
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1970
1990
2010
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| Wikipedia |
Zero-sum Ramsey theory
In mathematics, zero-sum Ramsey theory or zero-sum theory is a branch of combinatorics. It deals with problems of the following kind: given a combinatorial structure whose elements are assigned different weights (usually elements from an Abelian group $A$), one seeks for conditions that guarantee the existence of certain substructure whose weights of its elements sum up to zero (in $A$). It combines tools from number theory, algebra, linear algebra, graph theory, discrete analysis, and other branches of mathematics.
The classic result in this area is the 1961 theorem of Paul Erdős, Abraham Ginzburg, and Abraham Ziv:[1] for any $2m-1$ elements of $\mathbb {Z} _{m}$, there is a subset of size $m$ that sums to zero.[2] (This bound is tight, as a sequence of $m-1$ zeroes and $m-1$ ones cannot have any subset of size $m$ summing to zero.[2]) There are known proofs of this result using the Cauchy-Davenport theorem, Fermat's little theorem, or the Chevalley–Warning theorem.[2]
Generalizing this result, one can define for any abelian group G the minimum quantity $EGZ(G)$ of elements of G such that there must be a subsequence of $o(G)$ elements (where $o(G)$ is the order of the group) which adds to zero. It is known that $EGZ(G)\leq 2o(G)-1$, and that this bound is strict if and only if $G=\mathbb {Z} _{m}$.[2]
See also
• Zero-sum problem
References
1. Erdős, Paul; Ginzburg, A.; Ziv, A. (1961). "Theorem in the additive number theory". Bull. Res. Council Israel. 10F: 41–43. Zbl 0063.00009.
2. "Erdös-Ginzburg-Ziv theorem - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2023-05-22.
Further reading
• Zero-sum problems - A survey (open-access journal article)
• Zero-Sum Ramsey Theory: Graphs, Sequences and More (workshop homepage)
• A. Bialostocki, "Zero-sum trees: a survey of results and open problems" N.W. Sauer (ed.) R.E. Woodrow (ed.) B. Sands (ed.), Finite and Infinite Combinatorics in Sets and Logic, Nato ASI Ser., Kluwer Acad. Publ. (1993) pp. 19–29
• Y. Caro, "Zero-sum problems: a survey" Discrete Math., 152 (1996) pp. 93–113
| Wikipedia |
\begin{document}
\title[Global stability]{Global stability\\for a coupled physics inverse problem} \author{Giovanni Alessandrini}
\address{Dipartimento di Matematica e Geoscienze, Universit\`a degli Studi di Trieste, Italy}
\email{[email protected]}
\thanks{This work is supported by FRA2012 `Problemi Inversi', Universit\`a degli Studi di Trieste}
\subjclass[2000]{} \keywords{}
\begin{abstract} We prove a global H\"older stability estimate for a hybrid inverse problem combining microwave imaging and ultrasound. The principal features of this result are that we assume to have access to measurements associated to a single, arbitrary and possibly sign changing solution of a Schr\"odinger equation, and that zero is allowed to be an eigenvalue of the equation. \end{abstract}
\maketitle
\section{Introduction} In this note we consider an inverse problem with internal measurements. Given the Schr\"odinger equation \begin{equation}\label{sch} \Delta u + q u = 0 \, \text{ in } \Omega \ , \end{equation}
in a bounded Lipschitz domain $\Omega$, the inverse problem consists of finding the coefficient $q=q(x)\ge const.>0$ given the interior measurement $q u^2$ and the boundary data $u|_{\partial \Omega}$ for one nontrivial solution $u$. This is a simplified version of an hybrid inverse problem introduced by
Ammari, Capdeboscq, De Gournay, Rozanova-Pierrat and Triki \cite{ammarieco} in which the internal electromagnetic parameters of a body are to be detected by illuminating it by microwaves and, simultaneously, by focusing ultrasonic waves on a small portion of it. In mathematical terms, they deal with the equation \begin{equation}\label{sch2} div ({a}\nabla u)+ {k^2}q u = 0 \, \text{ in } \Omega \ , \end{equation}
where $\Omega$ represents the body, $a^{-1}>0$ models the magnetic permeability, $q>0$ the electric permittivity and $u$ the electric field. The goal in \cite{ammarieco} is to find ${a}, q$ given the local energies $q u^2$ , $a|\nabla u|^2$ when such measurements are available for \emph{several} solutions $u$ and wavenumbers $k$.
For the simplified equation \eqref{sch}, Triki \cite {triki} obtained a uniqueness result for the determination of $q$ given the local energy $q u^2$ for one solution $u$ corresponding to prescribed Dirichlet data, in \cite {triki} also one result of local stability is found.
In this note we shall investigate global stability in the same setting.
In contrast to the general phenomenology of inverse boundary value problems and of inverse scattering problems, which typically show severe ill-posedness ( Mandache \cite{mandacheIP}, Di Cristo and Rondi \cite{dicrondi}), the emerging methodologies of imaging based on the coupling of different physical modalities (see Bal \cite{balinside} for an overview), enable to acquire interior measurements which lead to the expectation of a much better behavior in terms of stability. Such an expectation has indeed been confirmed in many cases, let us cite Bal and Uhlmann \cite{baluhl}, Kuchment and Steinhauer \cite{kush}, Montalto and Stefanov \cite{montstef}, just to mention, with no ambition of completeness, a few samples of an impressively growing literature. However, a recurrent feature of the available results of stability is that suitable nondegeneracy conditions on the solutions of the underlying equations are needed. Depending on the problem treated, and on the number of coefficients that are to be determined, it is required that the solution $u$\, or its gradient, does not vanish. In other cases it is required that the Jacobian matrix associated to an array of solutions is nonsingular, and even bigger matrices formed by an array of solutions and their derivatives may come into play. Such nondegeneracy conditions may be difficult to achieve in general. Typically, it is possible to prove the existence of nondegenerate sets of solutions, but, in presence of unknown coefficients, it may be unclear how one can drive the system from the exterior so that the solutions satisfy the desired nondegeneracy conditions. It should be mentioned that, if one is free to tune up one parameter in the governing equation, like the wavenumber $k$ in \eqref{sch2}, then nondegeneracy can be guaranteed for at least some choice of $k$, this is a remarkable recent result by Alberti \cite{alberti}.
Here, instead, we intend to examine the prototypical case \eqref{sch} when measurements can be taken at one fixed wavenumber only, conventionally set as $k=1$. For such an equation the associated, Dirichlet or Neumann, direct boundary value problems might be not well posed if $0$ is an eigenvalue, or well posed with very large costants, if $0$ is close to an eigenvalue. Hence it may be troublesome to control the interior behavior of a solution by appropriate choices of boundary data. And, generically, solutions to \eqref{sch} may vanish inside, and may have critical points.
Let us show, by a simple example in dimension $n=1$, what kind of pathologies one might encounter.
Let us fix $0<r<R$ and, for every $m=1,2,\ldots$, let us set \begin{eqnarray*} q_m(x) = \left\{ \begin{array}{rl}
A_m & \ \ \ \ \mbox{if}\ |x|<r \ ,\\
1 &\ \ \ \ \mbox{if}\ r\le |x| \le R\ , \end{array} \right. \end{eqnarray*} where \begin{equation*} A_m= \left(\frac{\pi}{2}+2m\pi\right)^2r^{-2} \ . \end{equation*} A solution to $u_{xx}+q_m u = 0$ in $(-R,R)$ is \begin{eqnarray*} u_m(x) = \left\{ \begin{array}{rl}
\frac{1}{\sqrt{A_m}}\cos (\sqrt{A_m}x) & \ \ \ \ \mbox{if}\ |x|<r \ ,\\
-\sin (|x|-r) &\ \ \ \ \mbox{if}\ r\le |x| \le R\ . \end{array} \right. \end{eqnarray*} Note that in the interval $(-r,r)$ $u_m$ becomes very small and highly oscillating as $m$ increases. We have \begin{equation*}
\|q_{2m}u_{2m}^2- q_{m}u_{m}^2\|_{\infty} \le 2 \text{ for every } m=1,2,\ldots \ , \end{equation*}
whereas, for any $p \ , 1 \le p \le \infty$ \begin{equation*}
\|q_{2m}- q_{m}\|_{p} \rightarrow \infty \text{ as } m\rightarrow \infty \ . \end{equation*} Thus, in other words, the error on the measurement $qu^2$ does not dominate the error on $q$.
Here we shall present a global stability result of conditional type, which makes use of measurements for only one arbitrary, nontrivial, but possibly sign changing, solution, and when no spectral condition of the underlying equation is assumed, that is, we admit that $0$ might be an eigenvalue for equation \eqref{sch}.
Let us remind that for a different hybrid problem, coupling elastography and magnetic resonance, Honda, McLaughlin and Nakamura \cite{nakajoice} also obtained a global stability of H\"older type, when the underlying solution may vanish somewhere. There are, however, substantial differences in the governing equations which impose much different a priori assumptions and methods.
Let us now describe the a priori assumptions that we shall use. On the unknown coefficient $q\in L^{\infty}(\Omega)$ we require that for a given $K\ge 1$ we have \begin{equation}\label{K} K^{-1} \le q \le K \ , \text{ a.e. in } \Omega \ . \end{equation} Also, we consider one weak solution $u\in W^{1,2}(\Omega)\cap C(\overline{\Omega})$ to \eqref{sch} and we prescribe for a given $E>0$ the following global energy bound \begin{equation}\label{E}
\int_\Omega\left(u^2+ |\nabla u|^2\right) \le E^2\ . \end{equation} Regarding the interior measurement $qu^2$ we require the following nondegeneracy in average, that is we are given $H>0$ such that \begin{equation}\label{H} \int_\Omega qu^2 \ge H^2\ . \end{equation} Further we shall also need to specify in a quantitative form the Lipschitz regularity of the domain $\Omega$. For this purpose some notation and definitions are needed.
Given $x\in \mathbb R^n$, we shall denote $x=(x',x_n)$, where $x'=(x_1,\ldots,x_{n-1})\in\mathbb R^{n-1}$, $x_n\in\mathbb R$. Given $x\in \mathbb R^n$, $r>0$, we shall use the following notation for balls and cylinders \begin{equation*}
B_r(x)=\{y\in \mathbb R^n\ |\ |y-x|<r\}, \quad B_r=B_r(0)\ , \end{equation*} \begin{equation*}
B'_r(x')=\{y'\in \mathbb R^{n-1}\ |\ |y'-x'|<r\}, \quad B'_r=B'_r(0)\ , \end{equation*} \begin{equation*}
\Gamma_{a,b}(x)=\{y=(y',y_n)\in \mathbb R^n\ |\ |y'-x'|<a, |y_n-x_n|<b\}, \quad \Gamma_{a,b}=\Gamma_{a,b}(0)\ . \end{equation*}
We shall say that $\Omega$ is of \emph{Lipschitz class} with constants $\rho$, $M>0$, if, for any $P \in \partial\Omega$, there exists a rigid transformation of coordinates under which $P=0$ and \begin{equation*}
\Omega \cap \Gamma_{\frac{\rho}{M},\rho}(P)=\{x=(x',x_n) \in \Gamma_{\frac{\rho}{M},\rho}\quad | \quad x_{n}>Z(x')
\}\ , \end{equation*} where $Z:B'_{\frac{\rho}{M}}\to\mathbb R$ is a Lipschitz function satisfying \begin{equation*}
Z(0)=0, \end{equation*} \begin{equation*}
\|Z\|_{{L}^{\infty}(B'_{\frac{\rho}{M}})}+
\rho\|\nabla Z\|_{{L}^{\infty}(B'_{\frac{\rho}{M}})} \leq M\rho \ . \end{equation*}
We shall also use the following notation.
For every $d>0$ we denote \begin{equation*}
\Omega_d =\left\{x\in \Omega\ | \ dist(x, \partial \Omega) > d \right\} \ . \end{equation*}
By $|\Omega|$ we shall denote the measure of $\Omega$.
We can now state our main result.
\begin{theorem}\label{main} Let $\Omega$ be of {Lipschitz class} with constants $\rho$, $M>0$. Let $q_1, q_2\in L^{\infty}(\Omega)$ satisfy \eqref{K}, let $u_1,u_2\in W^{1,2}(\Omega)\cap C(\overline{\Omega})$ be solutions to \eqref{sch} when $q=q_1, q_2$ respectively. Assume that $u_1,u_2$ satisfy the a priori assumptions \eqref{E}, \eqref{H} and suppose that, for a given $\varepsilon >0$ , \begin{equation}\label{interror}
\|q_1u_1^2-q_2u_2^2\|_{L^{\infty}(\Omega)} \le \varepsilon \ , \end{equation}
and also \begin{equation} \label{bdryerror}
\||u_1|-|u_2|\|_{L^{\infty}(\partial\Omega)} \le \sqrt{K\varepsilon} \ . \end{equation}
Then, for every $d>0$, there exists $\eta\in (0,1)$ and $C>0$, only depending on $d, K, E, H$ and on $\rho, M, |\Omega|$ such that \begin{equation}\label{intstab}
\|q_1-q_2\|_{L^{1}(\Omega_d)} \le C\left(\varepsilon^{1/2}+\varepsilon \right)^{\eta} \ , \end{equation} \end{theorem} \begin{remark}
We observe that \eqref{intstab} provides a global stability of H\"older type. The H\"older exponent is expected to depend on the a priori data and it might get smaller and smaller as the a priori bounds deteriorate.
Moreover, let us remark that, in view of \eqref{K}, it is easily seen that \eqref{intstab} holds, with different constants $C$ and $\eta$, also when the $L^1$ norm is replaced by any $L^p$ norm, with $p<\infty$.
Also, it is worth noticing that, if for a given $d>0$ it is known in addition that $q_1=q_2$ in $\Omega\setminus \overline{\Omega_d}$, then \eqref{bdryerror} is automatically satisfied when \eqref{interror} holds true. Obviously, in such a case, \eqref{intstab} can be improved to \begin{equation*}
\|q_1-q_2\|_{L^{1}(\Omega)} \le C\left(\varepsilon^{1/2}+\varepsilon \right)^{\eta} \ . \end{equation*} \end{remark}
In the next Section 2 we shall state two theorems, Theorem \ref{weight} and Theorem \ref{negint}, which constitute the main tools for the proof of Theorem \ref{main}, which is also given there. Sections 3 and 4 contain the proofs of Theorem \ref{weight} and Theorem \ref{negint}, respectively.
\section{Proof of the main theorem}
We begin with a weighted stability estimate on the electric field intensity $|u|$. \begin{theorem}\label{weight} Let the assumptions of Theorem \ref{main} be satisfied. We have \begin{equation}\label{wstab}
\int_{\Omega}\left(|u_1|+|u_2|\right)\left(|u_1|-|u_2|\right)^2 \leq C \varepsilon \end{equation}
where $C>0$ depends only on $K, E$ and on $|\Omega|$. \end{theorem} The proof is postponed to the next Section 3. \begin{remark}
Note that also a $L^3$ stability estimate, without weight, follows easily from \eqref{wstab} \begin{equation*}
\int_{\Omega}\left||u_1|-|u_2|\right|^3 \leq C \varepsilon \ . \end{equation*} \end{remark} The following theorem consists of a quantitative form of the strong unique continuation property for solutions to equation \eqref{sch}. \begin{theorem}\label{negint} Let $\Omega$ be of {Lipschitz class} with constants $\rho$, $M>0$. Let $q\in L^{\infty}(\Omega)$ satisfy \eqref{K} and let $u\in W^{1,2}(\Omega)\cap C(\overline{\Omega})$ be a solution to \eqref{sch}. Assume that $u$ satisfies the a priori assumptions \eqref{E}, \eqref{H}.
Then, for every $d>0$, there exists $\delta , C>0$, only depending on $d, K, E, H$ and on $\rho, M, |\Omega|$ such that \begin{equation}\label{eq:negint}
\int_{\Omega_d} |u|^{-\delta}\le C \ . \end{equation} \end{theorem} The proof can be found in the final Section 4.
Assuming the above two theorems proven, we can now complete the proof of Theorem \ref{main}. \begin{proof}[Proof of Theorem \ref{main}] We compute \begin{equation*}
(q_1-q_2)u_1^2= (q_1u_1^2-q_2u_2^2) - q_2(u_1^2 - u_2^2) \end{equation*} hence \begin{equation*}
\int_{\Omega} |q_1-q_2|u_1^2\leq |\Omega|\varepsilon+ \int_{\Omega}q_2\left(|u_1|+|u_2|\right)||u_1|-|u_2|| \end{equation*} and, by H\"older's inequality, \begin{equation*}
\int_{\Omega} |q_1-q_2|u_1^2\leq C\left[\varepsilon+ \left(\int_{\Omega}(|u_1|+|u_2|)\right)^{1/2}\left(\int_{\Omega}\left(|u_1|+|u_2|\right)\left(|u_1|-|u_2|\right)^2\right)^{1/2}\right] \end{equation*}
where $C>0$ only depends on $|\Omega|$ and on $K$, now using \eqref{E} and Theorem \ref{weight} we obtain \begin{equation}\label{weightq}
\int_{\Omega} |q_1-q_2|u_1^2\leq C\left(\varepsilon+\varepsilon^{1/2}\right) \end{equation}
where $C>0$ is a new constant which only depends on $|\Omega|, K$ and on $E$. Now, fixing $d>0$ and choosing $\delta>0$ according to Theorem \ref{negint}, by H\"older's inequality we get \begin{equation*}
\int_{\Omega_d}|q_1-q_2|^{\frac{\delta}{\delta+2}} \le \left(\int_{\Omega_d}|u_1|^{-\delta}\right)^{\frac{2}{\delta+2}} \left( \int_{\Omega} |q_1-q_2|u_1^2\right)^{\frac{\delta}{\delta+2}} \end{equation*} finally, applying Theorem \ref{negint} to $u_1$ and using \eqref{K}, \eqref{weightq} we arrive at \eqref{intstab} with $\eta = \frac{\delta}{\delta+2}$. \end{proof} \begin{remark}
Inequality \eqref{weightq} might also be used to obtain a localized H\"older stability with a uniform exponent on regions where the measurement $qu^2$ is bounded away from $0$. For any $t>0$, set $D_t =\left\{x\in \Omega| q_1u_1^2\ge t\right\}$ then \eqref{weightq} implies \begin{equation*}
\|q_1-q_2\|_{L^{1}(D_t)} \leq \frac{KC}{t}\left(\varepsilon+\varepsilon^{1/2}\right) \ . \end{equation*} \end{remark} \section{The weighted estimate on the electric field} \begin{proof}[Proof of Theorem \ref{weight}]
Denote by $N_i=\left\{x\in \Omega| u_i(x) =0\right\}$ the nodal set of $u_i$, $i=1,2$. Let us remark that, by the continuity of $u_i$, $N_i$ is a closed set, furthermore, by the unique continuation property for \eqref{sch}, we also know that $N_i$ has zero Lebesgue measure. We decompose $\Omega \setminus (N_1\cup N_2)$ into its connected components $\Omega_j$, we recall that such components are open and countably many. We observe that, by \eqref{interror} and by the continuity of $u_1,u_2$ \begin{equation*} u_2^2\leq {K\varepsilon} + K^2u_1^2 \, \text{ everywhere in } \overline{\Omega} \ , \end{equation*} consequently, we have $u_2^2\leq K\varepsilon$ on $N_1$, and analogously, $u_1^2\leq K\varepsilon$ on $N_2$. Therefore \begin{equation*}
||u_1|-|u_2||\leq \sqrt{K\varepsilon} \, \text{ on } N_1\cup N_2 \ , \end{equation*} hence, noticing that $\partial \Omega_j \subset N_1\cup N_2 \cup \partial \Omega$ and using \eqref{bdryerror} we have \begin{equation}\label{bdryj}
||u_1|-|u_2||\leq \sqrt{K\varepsilon} \, \text{ on } \partial \Omega_j \text{ for every } j \ . \end{equation}
Let us fix one component $\Omega_j$. Note that in $\Omega_j$ $u_1, u_2$ have constant sign, possibly different. Since our aim is to estimate the difference $|u_1|-|u_2|$, we are allowed to change the signs of $u_1, u_2$ and may assume, without loss of generality, that $u_1, u_2$ are both positive in $\Omega_j$. We introduce the function \begin{equation*} \varphi^+=\left[u_1- u_2 - 2 \sqrt{K\varepsilon}\right]^+ \ , \end{equation*}
where $\left[\cdot\right]^+$ denotes the positive part. By \eqref{bdryj} and by continuity, we have $|u_1- u_2|<2 \sqrt{K\varepsilon}$ on a neighborhood of $\partial \Omega_j$, hence $\varphi^+=0$ near $\partial \Omega_j$. If we define $\psi_i= u_i\varphi^+$, $i=1,2$, we obtain that $\psi_i \in W_0^{1,2}(\Omega_j)$, hence by the weak formulation of \eqref{sch} we obtain \begin{eqnarray*} \int_{\Omega_j} \nabla u_1 \cdot \nabla \psi_1 &=&\int_{\Omega_j} q_1u_1 \psi_1 =\int_{\Omega_j} q_1u_1^2 \varphi^+ \ , \\ \int_{\Omega_j} \nabla u_2 \cdot \nabla \psi_2 &=&\int_{\Omega_j} q_2u_2^2 \varphi^+ \ , \end{eqnarray*} and subtracting \begin{eqnarray*} \int_{\Omega_j} \left[\nabla u_1 \cdot \nabla (u_1 \varphi^+) -\nabla u_1 \cdot \nabla (u_2 \varphi^+) +\nabla u_1 \cdot \nabla (u_2 \varphi^+)- \nabla u_2 \cdot \nabla (u_2 \varphi^+)\right] = \\ = \int_{\Omega_j} \left(q_1u_1^2-q_2u_2^2\right) \varphi^+ \ . \end{eqnarray*} Consequently, denoting \begin{eqnarray*} I_1&=&\int_{\Omega_j} \nabla u_1 \cdot \nabla \left((u_1-u_2) \varphi^+\right)\ ,\\I_2&=&\int_{\Omega_j}\nabla (u_1-u_2) \cdot \nabla (u_2 \varphi^+)\ , \\ I_3&=& \int_{\Omega_j} \left(q_1u_1^2-q_2u_2^2\right) \varphi^+ \ , \end{eqnarray*} we have \begin{equation*} I_1+I_2 = I_3 \ . \end{equation*} Recalling once more the weak formulation of \eqref{sch}, we evaluate \begin{eqnarray*}
I_1=\int_{\Omega_j} q_1 u_1 (u_1-u_2) \varphi^+ = \\ = \int_{(u_1-u_2)>2\sqrt{K\varepsilon}} q_1 u_1 (u_1-u_2) (u_1-u_2-2\sqrt{K\varepsilon})\ge\\ \ge \frac{1}{K}\int_{(u_1-u_2)>2\sqrt{K\varepsilon}}|u_1| (u_1-u_2-2\sqrt{K\varepsilon})^2 \ . \end{eqnarray*} Note that here, and in what follows, it is understood that the domain of integration is a subset of $\Omega_j$. Next, we compute \begin{eqnarray*}
I_2=\int_{(u_1-u_2)>2\sqrt{K\varepsilon}} u_2 |\nabla(u_1-u_2)|^2 +\\+ \int_{(u_1-u_2)>2\sqrt{K\varepsilon}}(u_1-u_2-2\sqrt{K\varepsilon})\nabla(u_1-u_2-2\sqrt{K\varepsilon})\cdot \nabla u_2
\ge \\\ge\frac{1}{2}\int_{(u_1-u_2)>2\sqrt{K\varepsilon}}\nabla(u_1-u_2-2\sqrt{K\varepsilon})^2 \cdot \nabla u_2=\\= \frac{1}{2}\int_{(u_1-u_2)>2\sqrt{K\varepsilon}}q_2 u_2 (u_1-u_2-2\sqrt{K\varepsilon})^2 \ge\\ \ge \frac{1}{2K}\int_{(u_1-u_2)>2\sqrt{K\varepsilon}}|u_2| (u_1-u_2-2\sqrt{K\varepsilon})^2\ , \end{eqnarray*} hence, adding up \begin{eqnarray*}
I_1+I_2\ge \frac{1}{2K}\int_{(u_1-u_2)>2\sqrt{K\varepsilon}}\left(|u_1|+|u_2|\right) (u_1-u_2-2\sqrt{K\varepsilon})^2\ . \end{eqnarray*} Regarding the third integral, we observe that \begin{eqnarray*}
I_3 \le \varepsilon\int_{(u_1-u_2)>2\sqrt{K\varepsilon}} \left(|u_1|+|u_2|+2\sqrt{K\varepsilon}\right) \le\\ \le 2\varepsilon\int_{(u_1-u_2)>2\sqrt{K\varepsilon}} \left(|u_1|+|u_2|\right) \ . \end{eqnarray*} Consequently, we obtain \begin{eqnarray*}
\int_{(u_1-u_2)>2\sqrt{K\varepsilon}}\left(|u_1|+|u_2|\right) (u_1-u_2-2\sqrt{K\varepsilon})^2 \le\\ \le 4K\varepsilon\int_{(u_1-u_2)>2\sqrt{K\varepsilon}} \left(|u_1|+|u_2|\right) \ , \end{eqnarray*} and the analogous estimate holds if we interchange the roles f $u_1$ and $u_2$. Hence we arrive at \begin{eqnarray*}
\int_{||u_1|-|u_2||>2\sqrt{K\varepsilon}}\left(|u_1|+|u_2|\right) (||u_1|-|u_2||-2\sqrt{K\varepsilon})^2 \le\\ \le 4K\varepsilon\int_{||u_1|-|u_2||>2\sqrt{K\varepsilon}} \left(|u_1|+|u_2|\right) \ , \end{eqnarray*} now, by triangle inequality \begin{eqnarray*}
(|u_1|-|u_2|)^2 \le \left(\left|||u_1|-|u_2||-2\sqrt{K\varepsilon}\right|+2\sqrt{K\varepsilon}\right)^2 \le\\ \le 2\left(\left|||u_1|-|u_2||-2\sqrt{K\varepsilon}\right|^2+4K\varepsilon\right) \ , \end{eqnarray*} therefore \begin{eqnarray*}
\int_{||u_1|-|u_2||>2\sqrt{K\varepsilon}}\left(|u_1|+|u_2|\right) \left(|u_1|-|u_2|\right)^2 \le\\ \le 16K\varepsilon\int_{||u_1|-|u_2||>2\sqrt{K\varepsilon}} \left(|u_1|+|u_2|\right) \ . \end{eqnarray*}
It remains to consider the subset of $\Omega_j$ where $||u_1|-|u_2||\le 2\sqrt{K\varepsilon}$, in this case it is a straightforward matter to obtain \begin{eqnarray*}
\int_{||u_1|-|u_2||\le2\sqrt{K\varepsilon}}\left(|u_1|+|u_2|\right) \left(|u_1|-|u_2|\right)^2 \le\\ \le 4K\varepsilon\int_{||u_1|-|u_2||\le2\sqrt{K\varepsilon}} \left(|u_1|+|u_2|\right) \ , \end{eqnarray*} consequently \begin{eqnarray*}
\int_{\Omega_j}\left(|u_1|+|u_2|\right) \left(|u_1|-|u_2|\right)^2 \le\\ \le 16K\varepsilon\int_{\Omega_j} \left(|u_1|+|u_2|\right) \ , \end{eqnarray*} and adding up with respect to $j$ \begin{eqnarray*}
\int_{\Omega}\left(|u_1|+|u_2|\right) \left(|u_1|-|u_2|\right)^2 \le\\ \le 16K\varepsilon\int_{\Omega} \left(|u_1|+|u_2|\right) \ , \end{eqnarray*} finally using \eqref{E} we arrive at \eqref{wstab}. \end{proof}
\section{Quantitative estimates of unique continuation} We begin by recalling the following version of a quantitative estimate of unique continuation, which is well-known to be very useful in the treatment of various inverse problems, see for instance \cite{alrossSIAM}, a proof can be found in \cite[Theorem 5.3]{arrv}. \begin{theorem} [Lipschitz propagation of smallness]
\label{theo:LipPropSmall}
Let the assumptions of Theorem \ref{negint} be satisfied. For every $r >0$ and for
every $x \in \Omega_{r}$, we have
\begin{equation}
\label{eq:p2-e4}
\int_{B_{r}(x)} u^2 \geq C
\int_{\Omega} u^2 \ , \end{equation}
where $C>0$ only depends on $r, K, E/H$ and on
$\rho, M, |\Omega|$ .
\end{theorem}
Also the following theorem is a manifestation of the strong unique continuation property, its original version is due to Garofalo and Lin \cite{garofalolin1, garofalolin2} , the present global formulation is indeed a consequence of the previous Theorem \ref{theo:LipPropSmall}, for the details of a proof we may refer to \cite[Theorem 3.4]{amrv}.
\begin{theorem} [Doubling inequality] \label{doub}
Let the assumptions of Theorem \ref{negint} be satisfied. For every $\overline{r} >0$ and for
every $x \in \Omega_{2\overline{r} }$, we have
\begin{equation}
\label{doubin}
\int_{B_{2r}(x)} u^2 \le C
\int_{B_{r}(x)} u^2 \text{ for every } r\le \overline{r} \ , \end{equation}
where $C>0$ only depends on $\overline{r} , K, E/H$ and on
$\rho, M, |\Omega|$ .
\end{theorem}
\begin{proof}[Proof of Theorem \ref{negint}] Since Garofalo and Lin \cite{garofalolin1}, it is well-known that, as a consequence of the above stated doubling inequality and of the standard local boundedness estimates for solutions to \eqref{sch} \cite[Theorem 8.17]{gilbargtrud}, $u^2$ turns out to be a Mukenhoupt weight, Coifman and Fefferman \cite{coifeff}. More specifically, we obtain that for every $\overline{r} >0$ and for
every $x \in \Omega_{2\overline{r} }$, there exists $p>1 , C>0$, only depending on $\overline{r}, K, E/H$ and on
$\rho, M, |\Omega|$ such that for every $x \in \Omega_{2\overline{r} }$ and for every $r\le \overline{r}$ we have
\begin{equation}\label{A_p}
\left(\frac{1}{|B_r(x)|}\int_{B_{r}(x)} u^2\right) \left(\frac{1}{|B_r(x)|}\int_{B_{r}(x)} |u|^{-\frac{2}{p-1}}\right)^{p-1}\le C \ . \end{equation}
It is a straightforward matter to construct a covering of $\Omega_d$ with balls $B_{d/4}(x_i)$, $i=1,\ldots, N$ such that their doubles $B_{d/2}(x_i)$ stay within $\Omega$ and their number $N$ is is dominated, up to an absolute constant, by $|\Omega|d^{-n}$. Using \eqref{A_p} for each $B_{d/4}(x_i)$ we get \begin{equation*}
\int_{B_{\frac{d}{4}}(x_i)} |u|^{-\frac{2}{p-1}}\le C |B_{\frac{d}{4}}(x_i)|^{1-\frac{1}{p-1}}\left(\int_{B_{\frac{d}{4}}(x_i)} u^2\right)^{-\frac{1}{p-1}}\ , \end{equation*} hence, recalling \eqref{eq:p2-e4} and adding up with respect to $i=1,\ldots, N$, we arrive at \eqref{eq:negint} with $\delta =\frac{2}{p-1}$. \end{proof}
\end{document} | arXiv |
Florian Luca
Florian Luca (born 16 March 1969, in Galați) is a Romanian mathematician who specializes in number theory with emphasis on Diophantine equations, linear recurrences and the distribution of values of arithmetic functions. He has made notable contributions to the proof that irrational automatic numbers are transcendental and the proof of a conjecture of Erdős on the intersection of the Euler Totient function and the sum of divisors function.
Luca graduated with a BS in Mathematics from Alexandru Ioan Cuza University in Iași (1992), and Ph.D. in Mathematics from the University of Alaska Fairbanks (1996). He has held various appointments at Syracuse University, Bielefeld University, Czech Academy of Sciences, and National Autonomous University of Mexico. Currently he is a research professor at the University of the Witwatersrand. He has co-authored over 500 papers in mathematics with more than 200 co-authors.[1][2][3]
He is a recipient of the award of a 2005 Guggenheim Fellowship for Natural Sciences, Latin America & Caribbean.[4]
Luca is one of the editors-in-chief of INTEGERS: the Electronic Journal of Combinatorial Number Theory[5] and an editor of the Fibonacci Quarterly.[6]
Selected works
• with Boris Adamczewski, Yann Bugeaud: Sur la complexité des nombres algébriques, Comptes Rendus Mathématique. Académie des Sciences. Paris 339 (1), 11-14, 2013
• with Kevin Ford, Carl Pomerance: Common values of the arithmetic functions ϕ and σ, Bulletin of the London Mathematical Society 42 (3), 478-488, 2010
• with Jean-Marie De Koninck: Analytic Number Theory: Exploring the Anatomy of Integers, American Mathematical Society, 2012
• Diophantine Equations - Effective Methods for Diophantine Equations, 2009, Online pdf file
References
1. Most Published Authors, Journal of Number Theory, Accessed August 14, 2015
2. Most Published Authors, International Journal of Number Theory, Accessed August 14, 2015
3. Most Published Authors, Acta Arithmetica, Accessed August 14, 2015
4. John Simon Guggenheim Foundation | Florian Luca, John Simon Guggenheim Memorial Foundation
5. Editorial Board, INTEGERS: the Electronic Journal of Combinatorial Number Theory. Accessed August 14, 2015
6. Editorial Team, Fibonacci Quarterly, Accessed August 14, 2015
External links
• Florian Luca at the Mathematics Genealogy Project
• Florian Luca publications indexed by Google Scholar
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| Wikipedia |
Properties of Galactic B[e] Supergiants: V. 3 Pup -- constraining the orbital parameters and modeling the circumstellar environments
A. S. Miroshnichenko, S. Danford, S. V. Zharikov, V. G. Klochkova, E. L. Chentsov, D. Vanbeveren, O. V. Zakhozhay, N. Manset, M. A. Pogodin, C. T. Omarov, A. K. Kuratova, S. A. Khokhlov
We report the results of a long-term spectroscopic monitoring of the A-type supergiant with the B[e] phenomenon 3 Pup = HD 62623. We confirm earlier findings that it is a binary system. The orbital parameters were derived using cross-correlation of the spectra in a range of 4460-4632 A, which contains over 30 absorption lines. The orbit was found circular with a period of $137.4\pm0.1$ days, radial velocity semi-amplitude $K_{1} = 5.0\pm0.8$ km s$^{-1}$, systemic radial velocity $\gamma = +26.4\pm2.0$ km s$^{-1}$, and the mass function $f(m) = (1.81^{+0.97}_{-0.76})\times10^{-3}$ M$_{\odot}$. The object may have evolved from a pair with initial masses of $\sim$6.0 M$_{\odot}$ and $\sim$3.6 M$_{\odot}$ with an initial orbital period of $\sim$5 days. Based on the fundamental parameters of the A-supergiant (luminosity $\log$ L/L$_{\odot} = 4.1\pm$0.1 and effective temperature T$_{\rm eff} = 8500\pm$500 K) and evolutionary tracks of mass-transferring binaries, we found current masses of the gainer M$_{2} = 8.8\pm$0.5 M$_{\odot}$ and donor M$_{1} = 0.75\pm0.25$ M$_{\odot}$. We also modeled the object's IR-excess and derived a dust mass of $\sim 5\,\times10^{-5}$ M$_{\odot}$ in the optically-thin dusty disk. The orbital parameters and properties of the H$\alpha$ line profile suggest that the circumstellar gaseous disk is predominantly circumbinary. The relatively low mass of the gainer led us to a suggestion that 3 Pup should be excluded from the B[e] supergiant group and moved to the FS CMa group. Overall these results further support our original suggestion that FS CMa objects are binary systems, where an earlier mass-transfer caused formation of the circumstellar envelope.
https://doi.org/10.3847/1538-4357/ab93d9
13 pages, 9 figures, 4 tables, accepted in The Astrophysical Journal
10.3847/1538-4357/ab93d9Licentie: CC BY
2005.07754v1Submitted manuscript, 505 KB
Miroshnichenko_2020_ApJ_897_48Final published version, 797 KBLicentie: CC BY
Link naar publicatie in Scopus
Duik in de onderzoeksthema's van 'Properties of Galactic B[e] Supergiants: V. 3 Pup -- constraining the orbital parameters and modeling the circumstellar environments'. Samen vormen ze een unieke vingerafdruk.
orbitals Physics & Astronomy 100%
radial velocity Physics & Astronomy 47%
suggestion Physics & Astronomy 46%
H alpha line Physics & Astronomy 40%
luminosity Earth & Environmental Sciences 28%
cross correlation Physics & Astronomy 23%
mass transfer Physics & Astronomy 21%
envelopes Physics & Astronomy 20%
Miroshnichenko, A. S., Danford, S., Zharikov, S. V., Klochkova, V. G., Chentsov, E. L., Vanbeveren, D., Zakhozhay, O. V., Manset, N., Pogodin, M. A., Omarov, C. T., Kuratova, A. K., & Khokhlov, S. A. (2020). Properties of Galactic B[e] Supergiants: V. 3 Pup -- constraining the orbital parameters and modeling the circumstellar environments. The Astrophysical Journal, 897(1), [48]. https://doi.org/10.3847/1538-4357/ab93d9
Miroshnichenko, A. S. ; Danford, S. ; Zharikov, S. V. ; Klochkova, V. G. ; Chentsov, E. L. ; Vanbeveren, D. ; Zakhozhay, O. V. ; Manset, N. ; Pogodin, M. A. ; Omarov, C. T. ; Kuratova, A. K. ; Khokhlov, S. A. / Properties of Galactic B[e] Supergiants : V. 3 Pup -- constraining the orbital parameters and modeling the circumstellar environments. In: The Astrophysical Journal. 2020 ; Vol. 897, Nr. 1.
@article{856e15348e7e4c9c9104d3deb6a5e6d5,
title = "Properties of Galactic B[e] Supergiants: V. 3 Pup -- constraining the orbital parameters and modeling the circumstellar environments",
abstract = " We report the results of a long-term spectroscopic monitoring of the A-type supergiant with the B[e] phenomenon 3 Pup = HD 62623. We confirm earlier findings that it is a binary system. The orbital parameters were derived using cross-correlation of the spectra in a range of 4460-4632 A, which contains over 30 absorption lines. The orbit was found circular with a period of $137.4\pm0.1$ days, radial velocity semi-amplitude $K_{1} = 5.0\pm0.8$ km s$^{-1}$, systemic radial velocity $\gamma = +26.4\pm2.0$ km s$^{-1}$, and the mass function $f(m) = (1.81^{+0.97}_{-0.76})\times10^{-3}$ M$_{\odot}$. The object may have evolved from a pair with initial masses of $\sim$6.0 M$_{\odot}$ and $\sim$3.6 M$_{\odot}$ with an initial orbital period of $\sim$5 days. Based on the fundamental parameters of the A-supergiant (luminosity $\log$ L/L$_{\odot} = 4.1\pm$0.1 and effective temperature T$_{\rm eff} = 8500\pm$500 K) and evolutionary tracks of mass-transferring binaries, we found current masses of the gainer M$_{2} = 8.8\pm$0.5 M$_{\odot}$ and donor M$_{1} = 0.75\pm0.25$ M$_{\odot}$. We also modeled the object's IR-excess and derived a dust mass of $\sim 5\,\times10^{-5}$ M$_{\odot}$ in the optically-thin dusty disk. The orbital parameters and properties of the H$\alpha$ line profile suggest that the circumstellar gaseous disk is predominantly circumbinary. The relatively low mass of the gainer led us to a suggestion that 3 Pup should be excluded from the B[e] supergiant group and moved to the FS CMa group. Overall these results further support our original suggestion that FS CMa objects are binary systems, where an earlier mass-transfer caused formation of the circumstellar envelope. ",
keywords = "astro-ph.SR",
author = "Miroshnichenko, {A. S.} and S. Danford and Zharikov, {S. V.} and Klochkova, {V. G.} and Chentsov, {E. L.} and D. Vanbeveren and Zakhozhay, {O. V.} and N. Manset and Pogodin, {M. A.} and Omarov, {C. T.} and Kuratova, {A. K.} and Khokhlov, {S. A.}",
note = "13 pages, 9 figures, 4 tables, accepted in The Astrophysical Journal",
doi = "10.3847/1538-4357/ab93d9",
journal = "The Astrophysical Journal",
publisher = "American Astronomical Society",
Miroshnichenko, AS, Danford, S, Zharikov, SV, Klochkova, VG, Chentsov, EL, Vanbeveren, D, Zakhozhay, OV, Manset, N, Pogodin, MA, Omarov, CT, Kuratova, AK & Khokhlov, SA 2020, 'Properties of Galactic B[e] Supergiants: V. 3 Pup -- constraining the orbital parameters and modeling the circumstellar environments', The Astrophysical Journal, vol. 897, nr. 1, 48. https://doi.org/10.3847/1538-4357/ab93d9
Properties of Galactic B[e] Supergiants : V. 3 Pup -- constraining the orbital parameters and modeling the circumstellar environments. / Miroshnichenko, A. S.; Danford, S.; Zharikov, S. V.; Klochkova, V. G.; Chentsov, E. L.; Vanbeveren, D.; Zakhozhay, O. V.; Manset, N.; Pogodin, M. A.; Omarov, C. T.; Kuratova, A. K.; Khokhlov, S. A.
In: The Astrophysical Journal, Vol. 897, Nr. 1, 48, 01.07.2020.
T1 - Properties of Galactic B[e] Supergiants
T2 - V. 3 Pup -- constraining the orbital parameters and modeling the circumstellar environments
AU - Miroshnichenko, A. S.
AU - Danford, S.
AU - Zharikov, S. V.
AU - Klochkova, V. G.
AU - Chentsov, E. L.
AU - Vanbeveren, D.
AU - Zakhozhay, O. V.
AU - Manset, N.
AU - Pogodin, M. A.
AU - Omarov, C. T.
AU - Kuratova, A. K.
AU - Khokhlov, S. A.
N1 - 13 pages, 9 figures, 4 tables, accepted in The Astrophysical Journal
N2 - We report the results of a long-term spectroscopic monitoring of the A-type supergiant with the B[e] phenomenon 3 Pup = HD 62623. We confirm earlier findings that it is a binary system. The orbital parameters were derived using cross-correlation of the spectra in a range of 4460-4632 A, which contains over 30 absorption lines. The orbit was found circular with a period of $137.4\pm0.1$ days, radial velocity semi-amplitude $K_{1} = 5.0\pm0.8$ km s$^{-1}$, systemic radial velocity $\gamma = +26.4\pm2.0$ km s$^{-1}$, and the mass function $f(m) = (1.81^{+0.97}_{-0.76})\times10^{-3}$ M$_{\odot}$. The object may have evolved from a pair with initial masses of $\sim$6.0 M$_{\odot}$ and $\sim$3.6 M$_{\odot}$ with an initial orbital period of $\sim$5 days. Based on the fundamental parameters of the A-supergiant (luminosity $\log$ L/L$_{\odot} = 4.1\pm$0.1 and effective temperature T$_{\rm eff} = 8500\pm$500 K) and evolutionary tracks of mass-transferring binaries, we found current masses of the gainer M$_{2} = 8.8\pm$0.5 M$_{\odot}$ and donor M$_{1} = 0.75\pm0.25$ M$_{\odot}$. We also modeled the object's IR-excess and derived a dust mass of $\sim 5\,\times10^{-5}$ M$_{\odot}$ in the optically-thin dusty disk. The orbital parameters and properties of the H$\alpha$ line profile suggest that the circumstellar gaseous disk is predominantly circumbinary. The relatively low mass of the gainer led us to a suggestion that 3 Pup should be excluded from the B[e] supergiant group and moved to the FS CMa group. Overall these results further support our original suggestion that FS CMa objects are binary systems, where an earlier mass-transfer caused formation of the circumstellar envelope.
AB - We report the results of a long-term spectroscopic monitoring of the A-type supergiant with the B[e] phenomenon 3 Pup = HD 62623. We confirm earlier findings that it is a binary system. The orbital parameters were derived using cross-correlation of the spectra in a range of 4460-4632 A, which contains over 30 absorption lines. The orbit was found circular with a period of $137.4\pm0.1$ days, radial velocity semi-amplitude $K_{1} = 5.0\pm0.8$ km s$^{-1}$, systemic radial velocity $\gamma = +26.4\pm2.0$ km s$^{-1}$, and the mass function $f(m) = (1.81^{+0.97}_{-0.76})\times10^{-3}$ M$_{\odot}$. The object may have evolved from a pair with initial masses of $\sim$6.0 M$_{\odot}$ and $\sim$3.6 M$_{\odot}$ with an initial orbital period of $\sim$5 days. Based on the fundamental parameters of the A-supergiant (luminosity $\log$ L/L$_{\odot} = 4.1\pm$0.1 and effective temperature T$_{\rm eff} = 8500\pm$500 K) and evolutionary tracks of mass-transferring binaries, we found current masses of the gainer M$_{2} = 8.8\pm$0.5 M$_{\odot}$ and donor M$_{1} = 0.75\pm0.25$ M$_{\odot}$. We also modeled the object's IR-excess and derived a dust mass of $\sim 5\,\times10^{-5}$ M$_{\odot}$ in the optically-thin dusty disk. The orbital parameters and properties of the H$\alpha$ line profile suggest that the circumstellar gaseous disk is predominantly circumbinary. The relatively low mass of the gainer led us to a suggestion that 3 Pup should be excluded from the B[e] supergiant group and moved to the FS CMa group. Overall these results further support our original suggestion that FS CMa objects are binary systems, where an earlier mass-transfer caused formation of the circumstellar envelope.
KW - astro-ph.SR
U2 - 10.3847/1538-4357/ab93d9
DO - 10.3847/1538-4357/ab93d9
JO - The Astrophysical Journal
JF - The Astrophysical Journal
Miroshnichenko AS, Danford S, Zharikov SV, Klochkova VG, Chentsov EL, Vanbeveren D et al. Properties of Galactic B[e] Supergiants: V. 3 Pup -- constraining the orbital parameters and modeling the circumstellar environments. The Astrophysical Journal. 2020 jul 1;897(1). 48. https://doi.org/10.3847/1538-4357/ab93d9 | CommonCrawl |
\begin{document}
\markboth{N. Ito} {Jones polynomials of long virtual knots}
\title{Jones polynomials of long virtual knots}
\author{NOBORU ITO}
\address{Department of Mathematics, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan\\ [email protected]}
\begin{abstract} This paper defines versions of the Jones polynomial and Khovanov homology by using several maps from the set of Gauss diagrams to its variant. Through calculation of some examples, this paper also shows that these versions behave differently from the original ones. \end{abstract} \keywords{Knots; Jones polynomials; Gauss diagrams; Khovanov homology} \maketitle
\section{Introduction}\label{intro} Knots, which are circles embedded in thickened surfaces, are often treated as the virtual knots introduced by Kauffman \cite{kauffman}. Virtual knots with base points are regarded as long virtual knots, since a circle less one point is homeomorphic to a line.
On the other hand, a Gauss diagram is a circle with chords, where the preimages of each double point of the immersion are connected by the chords. Virtual knots are nothing but equivalence classes of Gauss diagrams. We can place some information on the circle and chords of a Gauss diagram.
This paper considers maps between Gauss diagrams, and it is possible to produce some versions of a single knot invariant. In particular, there is a simple way to define invariants for long virtual knots thorough Gauss diagrams. In this paper, we consider versions of the Jones polynomial in terms of invariants of long virtual knots. We also see that this approach is effective for Khovanov homology.
The plan of this paper is as follows: Sec. \ref{functor_nano} gives a precise definition of long virtual knots and the corresponding Gauss diagrams. Sec. \ref{ver_jones} obtains definitions of the maps between Gauss diagrams and defines versions of the Jones polynomial. We see in Sec. \ref{app_kh} that the same approach is good for Khovanov homology.
\section{Long virtual knots and their presentations as Gauss diagrams}\label{functor_nano} Virtual knot theory was introduced by Kauffman \cite{kauffman} and virtual knots are often treated as Gauss diagrams. \subsection{Knots, knot diagrams, long knots, long knot diagrams, and Gauss diagrams} A {\it{knot}} is a circle smoothly embedding into $\mathbb{R}^{3}$ and a {\it{long knot}} is a smooth embedding $\mathbb{R}$ $\to$ $\mathbb{R}^{3}$. These are often represented by {\it{knot diagrams}} or {\it{long knot diagrams}}, which are images of generic immersions of the circle into the plane adding the information on overpasses and underpasses at double points, as shown in Figs. \ref{knot_and_long} (a) and (b). A long knot is often identified as a knot with a point, called a {{base point}}, on the circle. Its diagram is presented as a knot diagram with a base point on curves distinct from the double points (Fig. \ref{knot_and_long} (c)). \begin{figure}
\caption{(a) Knot diagram. (b) Long knot diagram. (c) Knot diagram with base point.}
\label{knot_and_long}
\end{figure} In this paper, we treat knot diagrams with finite double points only. As is well known, two knots are isotopic knots if related by a finite sequence of {\it{Reidemeister moves}}, which are local moves on knot diagrams as shown in Fig. \ref{reidemeister_m}. \begin{figure}
\caption{Reidemeister moves. The local replacements on the neighborhoods are drawn, and the exteriors of the neighborhoods are the same for both diagrams of each move.}
\label{reidemeister_m}
\end{figure} If necessary, we add an adjective such as {\it{classical}} for referring to the knots defined above and keep this role for other objects: long knots, knot diagrams, and long knot diagrams.
Every generic immersion of a circle into the plane fixes a {\it{Gauss diagram}} that is a circle with chords, where the preimages of each double point of the immersion are connected by the chords (Fig. \ref{diag_to_gauss}). {\it{Oriented Gauss diagrams}} are considered up to orientation preserving homeomorphism underlying circles, and the orientations imply those of knots. In this paper, the underlying circle of every oriented Gauss diagram has counterclockwise orientation. In the rest of this paper, unless otherwise specified, we adopt oriented Gauss diagrams that are simply called Gauss diagrams. To recover a knot up to isotopy from a Gauss diagram, we ascribe signs and arrows for every chord. The sign of a chord is defined as the local writhe number of the corresponding double point, and the arrow of a chord is oriented from the upper branch to the lower branch. \begin{figure}
\caption{A long knot diagram and a knot diagram are encoded by Gauss diagrams. }
\label{diag_to_gauss}
\end{figure} In the same way, we define Gauss diagrams of long knot diagrams as in Fig. \ref{diag_to_gauss}. \subsection{Virtual knots, virtual knot diagrams, long virtual knots, and long virtual knot diagrams}\label{virtual_sec} A {\it{virtual knot}}, introduced by Kauffman \cite{kauffman}, is defined as follows: A {\it{virtual knot diagram}} is a smooth immersion of the circle into the plane such that all singular points are transversal double points. These double points are divided into real crossing points and virtual crossing points, where real crossing points have information on overpasses and underpasses as for the classical knot diagrams shown in Fig. \ref{virtual_crossing}. \begin{figure}
\caption{(a), (b): Real crossings. (c): Virtual crossing. }
\label{virtual_crossing}
\end{figure} A branch consisting of a virtual crossing is not divided into an overpass and an underpass. {\it{Virtual knots}} are the set of virtual knot diagrams divided by Reidemeister moves and the {\it{virtual moves}} shown in Fig. \ref{virtual_moves}. \begin{figure}
\caption{Virtual moves.}
\label{virtual_moves}
\end{figure} For virtual knots, the following fact was proved by Goussarov, Polyak, and Viro \cite{gpv} using group systems: \begin{theorem}[Goussarov, Polyak, Viro] Virtually isotopic classical knots are isotopic. \end{theorem}
Here, we enhance the definition of knot diagrams and long knot diagrams for treating virtual knots as classical knots following works by Carter, Kamada, and Saito \cite{cks} and N. Kamada and S. Kamada \cite{kk} (see also Kauffman \cite{kauffman} and Goussarov, Polyak, and Viro \cite{gpv}). In the rest of this paper, objects such as knots or knot diagrams (i.e., containing classical knots, virtual knots, classical long knots, or long virtual knots) are regarded as oriented, unless confusion is likely to occur. {\it{Knot diagrams on surfaces}} are images of generic immersions of the circle into an oriented surface adding information on overpasses and underpasses at double points. {\it{Long knot diagrams on surfaces}} are knot diagrams on surfaces with base point on curves distinct from the double points. As is well known, {\it{virtual knots}} (resp. {\it{long virtual knots}}) are {\it{stable equivalence}} classes of knot diagrams (resp. long virtual knot diagrams) on surfaces. The definition of the stable equivalence is as follows: Two knot diagrams on surfaces that are images of generic immersions are stably equivalent if they can be replaced by a finite sequence of {\it{stable homeomorphisms}} and Reidemeister moves in the ambient surfaces. Two images of generic immersions are stably homeomorphic if there is a homeomorphism of their regular neighborhoods in the ambient surfaces that maps the first diagram onto the second one and preserves the overcrossings and undercrossings as well as the orientations of the surface and the immersed curve. Two long knot diagrams on surfaces are stably equivalent if they can be replaced by a finite sequence of stable homeomorphisms preserving the base point and Reidemeister moves in the ambient surfaces away from the base point. In particular, we now have a purely combinatorial proof that there are injective maps from classical knots (resp. long knots) to virtual knots (resp. long virtual knots) (cf. Turaev \cite{turaev2}).
\subsection{Gauss diagrams for virtual knots and long virtual knots.} Gauss diagrams of virtual knots and long virtual knots are defined by knot diagrams and long knot diagrams on surfaces in the same way as for classical knot diagrams (resp. classical long knot diagrams) that are generic immersions of circles (resp. circles with base points) into the plane. The alternative definition of Gauss diagrams of virtual knots and long virtual knots is that Gauss diagrams are constructed by using virtual knot diagrams and long virtual knot diagrams on the plane in the same way as for classical knot diagrams, but all virtual crossings are disregarded as shown in Fig. \ref{virtual_gauss}. \begin{figure}
\caption{Gauss diagrams for a long virtual knot and a virtual knot.}
\label{virtual_gauss}
\end{figure} Here, the following important fact \cite[Theorem 1.A]{gpv} should be mentioned: \begin{theorem}[Goussarov, Polyak, Viro]\label{gpv_thm} A Gauss diagram defines a virtual knot diagram up to virtual moves. \end{theorem} Then, a virtual knot (resp. long virtual knot) equals to the corresponding Gauss diagram (resp. Gauss diagram with a base point) considered up to moves that are the counterparts of Reidemeister moves for Gauss diagrams (resp. Gauss diagrams with base points) as shown in Fig. \ref{rel_gauss_a}. \begin{figure}
\caption{Relations of Gauss diagrams (resp. Gauss diagrams with base points) corresponding to Reidemeister moves of virtual knots (resp. long virtual knots) where $\epsilon$, $\eta$, and $\zeta$ are $+$ or $-$, but $(\epsilon, \eta, \zeta)$ is $(\pm, \pm, \pm)$, $(\mp, \mp, \pm)$, or $(\mp, \pm, \pm)$ in the third row. Directions of chords in the third row, denoted by $\alpha$ and $\beta$ in the fourth row, are defined by Table \ref{table_direction}. }
\label{rel_gauss_a}
\end{figure} \begin{table} \begin{center} {\setlength{\tabcolsep}{10pt} \begin{tabular}{ccc} \hline Case & signs & arrows \\ \hline 1 & $(+, +, +)$ & $(\alpha, \alpha, \alpha)$ \\ \hline 2 & $(+, +, +)$ & $(\beta, \beta, \beta)$ \\ \hline 3 & $(-, -, -)$ & $(\alpha, \alpha, \alpha)$ \\ \hline 4 & $(-, -, -)$ & $(\beta, \beta, \beta)$ \\ \hline \hline 5 & $(+, +, -)$ & $(\alpha, \alpha, \beta)$ \\ \hline 6 & $(+, +, -)$ & $(\beta, \beta, \alpha)$ \\ \hline 7 & $(-, -, +)$ & $(\alpha, \alpha, \beta)$ \\ \hline 8 & $(-, -, +)$ & $(\beta, \beta, \alpha)$ \\ \hline \hline 9 & $(+, -, -)$ & $(\alpha, \beta, \beta)$ \\ \hline 10 & $(+, -, -)$ & $(\beta, \alpha, \alpha)$ \\ \hline 11 & $(-, +, +)$ & $(\alpha, \beta, \beta)$ \\ \hline 12 & $(-, +, +)$ & $(\beta, \alpha, \alpha)$ \\ \hline \end{tabular} } \end{center} \caption{Rules for the triples of three chords in the third row of Fig. \ref{rel_gauss_a}. Double lines indicate that we can regard these twelve cases as three groups.}\label{table_direction} \end{table} \begin{remark} A Gauss diagram naturally has the orientation of a circle. Hence, if we adopt the notion of Gauss diagrams for non-oriented knots, Gauss diagrams should be identified up to given arbitrary orientations. On the other hand, when we consider an oriented Gauss diagram, the order of trivalent vertices on the Gauss diagram is fixed. That is why, in this paper, we represent Reidemeister move $\Omega_3$ as the third line of Fig. \ref{rel_gauss_a}. Using \cite{turaev2}, we have the following. \end{remark} \begin{lemma}\label{reide_negative} A long virtual knot is generated by Fig. \ref{rel_gauss_a}. A virtual knot is generated by Fig. \ref{rel_gauss_a} neglecting the base points. \end{lemma}
\section{Versions of the Jones polynomial}\label{ver_jones} In this section, the Gauss diagrams are oriented Gauss diagrams and have relations corresponding to Reidemeister moves. The symbol $\epsilon$ stands for $+$ or $-$ as in Fig. \ref{rel_gauss_a}.
First, let us consider Gauss diagrams neglecting the directions of arrows on chords. Then, the map $p_r$ is defined by correspondences of codes: \begin{picture}(0,20) \put(10,7){\circle{20}} \put(27,5){$\mapsto$} \put(55,7){\circle{20}} \put(3,0){\vector(1,1){14}} \put(2,4.5){$\epsilon$} \put(48,0){\line(1,1){14}} \put(46,4){$\epsilon$} \put(65,0){.} \end{picture}
The projection $p_{r}$ induces relations on the set of Gauss diagrams neglecting the directions of arrows. This topology is determined by Fig. \ref{rel_gauss_a} except for neglecting the directions of arrows. The topological objects are called pseudolinks (resp. long pseudolinks) for virtual knots (resp. long virtual knots).
Turaev obtained the following fact \cite[Section 8.3]{turaev2} through his nanoword theory: \begin{theorem}[Turaev]\label{main} The Jones polynomial $V_{K}$ of an oriented knot $K$ is defined by $p_r(G)$, where $G$ is a Gauss diagram of $K$; i.e., $V_{K}$ $=$ $V_{p_r(G)}$. \end{theorem}
Second, we consider the map $p$ from Gauss diagrams with base points to Gauss diagrams neglecting signs of arrows on chords as follows:
\begin{picture}(0,20) \put(2,13){\circle*{3}} \put(10,7){\circle{20}} \put(13,6.5){\tiny$-$} \put(17,14){\vector(-1,-1){14}} \put(20,0){,} \end{picture} \qquad \begin{picture}(0,20) \put(2,13){\circle*{3}} \put(10,7){\circle{20}} \put(27,5){$\mapsto$} \put(47,13){\circle*{3}} \put(55,7){\circle{20}} \put(3,0){\vector(1,1){14}} \put(1,4.5){\tiny$+$} \put(46.5,4.5){\small$a$} \put(48,0){\line(1,1){14}} \put(65,0){,} \put(75,0){\text{and}} \end{picture} \qquad \begin{picture}(0,20) \put(79,13){\circle*{3}} \put(87,7){\circle{20}} \put(78,4.5){\tiny$-$} \put(80,0){\vector(1,1){14}} \put(97,0){,} \end{picture} \begin{picture}(0,20) \put(102,13){\circle*{3}} \put(110,7){\circle{20}} \put(127,5){$\mapsto$} \put(147,13){\circle*{3}} \put(155,7){\circle{20}} \put(113,6.5){\tiny$+$} \put(117,14){\vector(-1,-1){14}} \put(148,4.5){\small$b$} \put(162,14){\line(-1,-1){14}} \put(165,0){.} \end{picture}
The projection $p$ means the underlying curves, called open flat virtual knots, for long virtual knots. This topology is determined by the relations of the Gauss diagrams with base points in Fig. \ref{rel_gauss_a} where Table \ref{table_direction} is restricted to Cases 1 and 3, except for replacing $+$ (resp. $-$) with $a$ (resp. $b$) and neglecting the directions of arrows.
Third, we consider the map $i$ between Gauss diagrams as follows:
\begin{picture}(0,20) \put(2,13){\circle*{3}} \put(10,7){\circle{20}} \put(27,5){$\mapsto$} \put(47,13){\circle*{3}} \put(55,7){\circle{20}} \put(3,0){\line(1,1){14}} \put(1,4.5){\small$a$} \put(46.5,4.5){\tiny$+$} \put(48,0){\line(1,1){14}} \put(65,0){,} \put(102,13){\circle*{3}} \put(75,0){\text{and}} \put(110,7){\circle{20}} \put(127,5){$\mapsto$} \put(147,13){\circle*{3}} \put(155,7){\circle{20}} \put(103,4.5){\small$b$} \put(117,14){\line(-1,-1){14}} \put(145.5,4.5){\tiny$-$} \put(162,14){\line(-1,-1){14}} \put(165,0){.} \end{picture}
\begin{theorem}\label{main_s0} Let $D$ be a diagram of an arbitrary long virtual knot $K$. The map $V_{i(p(D))}$ is an invariant of the long virtual knot $K$. \end{theorem} \begin{proof} The map $i$ sends open flat virtual knots to long pseudolinks. The map is well defined, since the relations of open flat virtual knots corresponding to Fig. \ref{rel_gauss_a} are sent to the relation defined by the same Gauss diagrams with $a$ (resp. $b$) replacing $+$ (resp. $-$) while neglecting arrow directions. By replacing $p_r(G)$ of Theorem \ref{main} with $i(p(K))$, another Jones polynomial $V_{i(p(K))}$ becomes an invariant of an arbitrary long virtual knot $K$. \end{proof}
\begin{remark} Fukunaga regarded the map $i$ as the one producing a topological invariant \cite{fukunaga}. \end{remark}
Here, in order to capture the graphical meaning of the map $i \circ p$, we prove Theorem \ref{main_s0} in another way as below. \begin{proof} Let $K$ be a long virtual knot and $D_K$ its diagram on a surface (cf. Sec. \ref{virtual_sec}). We can consider the map $p$ to mean that every crossing of $D_K$ is replaced with a transversal double point. Without loss of generality, we can assume by invoking plane isotopy that every crossing consists of two orthogonal branches. Hence, we assume this condition in the rest of the proof. Under this assumption, the definition of $p$ is represented as \begin{equation}\label{p-eq} \begin{split} \begin{picture}(35,35) \put(0,11){$p : $} \put(20,0){\line(1,1){30}} \put(20,30){\line(1,-1){10}} \put(50,0){\line(-1,1){10}} \put(60,11){$\mapsto$} \put(80,0){\line(1,1){30}} \put(80,30){\line(1,-1){30}} \end{picture} \qquad\qquad \end{split} \end{equation} for a sufficiently small neighborhood of every crossing, where the exterior of the neighborhoods of the crossings is mapped to itself and contains the base point. Then, by $p$, the curve $p(D_K)$ with the base point on a surface is determined to stable homeomorphisms preserving the base point and orientations of the curve and the surface. Every transversal double point has exactly two tangent vectors $t_1$ and $t_2$, so there exist two types of crossings: one type has a positively oriented pair $(t_1, t_2)$ and the other has a negatively oriented pair $(t_1, t_2)$.
More graphically, if the ambient surface containing the curve has counterclockwise orientation, every double point of $p(D_K)$ belongs to exactly one of two types: \begin{equation*} \begin{picture}(50,50) \put(0,5){$1$st} \put(32,5){$2$nd} \put(10,15){\vector(1,1){30}} \put(40,15){\vector(-1,1){30}} \put(60,5){,} \put(104,5){$1$st} \put(72,5){$2$nd} \put(80,15){\vector(1,1){30}} \put(110,15){\vector(-1,1){30}} \end{picture} \qquad\qquad \end{equation*} where $1$st (resp. $2$nd) means the first (resp. second) branch passing trough the double point starting from the base point.
Without loss of generality, we can assume that the ambient surface containing $D_K$ or $p(D_K)$ has counterclockwise orientation in the rest of the proof. Under this assumption, for these two types of double points, we consider the map $q$ as follows:
\begin{equation}\label{q-eq} \begin{split} \begin{picture}(130,50) \put(0,0){$1$st} \put(30,0){$2$nd} \put(10,10){\vector(1,1){30}} \put(40,10){\vector(-1,1){30}} \put(60,20){$\mapsto$} \put(110,10){\line(-1,1){10}} \put(90,30){\vector(-1,1){10}} \put(80,10){\vector(1,1){30}} \put(120,10){,} \end{picture} \begin{picture}(100,50) \put(0,0){$2$nd} \put(33,0){$1$st} \put(10,10){\vector(1,1){30}} \put(40,10){\vector(-1,1){30}} \put(60,20){$\mapsto$} \put(110,10){\vector(-1,1){30}} \put(100,30){\vector(1,1){10}} \put(80,10){\line(1,1){10}} \end{picture} \end{split} \end{equation} for a sufficiently small neighborhood of every double point, where the exterior of the neighborhoods of the double points is mapped to itself and contains the base point. The image $q \circ p(D_K)$ becomes a long virtual knot diagram.
In what follows, we show that if $D_{K_1}$ and $D_{K_2}$ are stably equivalent, $q \circ p(D_{K_1})$ and $q \circ p(D_{K_2})$ are stably equivalent.
According to the definition of $q \circ p$ by (\ref{p-eq}) and (\ref{q-eq}), if $D_{K_1}$ and $D_{K_2}$ are stably homeomorphic, preserving the base point and the orientations of the curve and the surface, so are $q \circ p(D_{K_1})$ and $q \circ p(D_{K_2})$. Subsequently, we will verify that if $D_{K_1}$ and $D_{K_2}$ can be replaced by Reidemeister moves in the ambient surface away from the base point, so can $q \circ p(D_{K_1})$ and $q \circ p(D_{K_2})$.
\begin{itemize} \item Reidemeister moves $\Omega_1 a$ and $\Omega_1 b$.
Let $D_1$ (resp. $D_2$) be the local diagram defined by the left (resp. right) side of the move $\Omega_1 a$ in Fig. \ref{reidemeister_m}, and let $D_3$ be the local diagram defined by the right side of the move $\Omega_1 b$ in Fig. \ref{reidemeister_m}. For each of $D_1$, $D_2$, and $D_3$, there are two cases by choice of orientation. If the orientation of $D$ $=$ $D_1$, $D_2$, or $D_3$ is along the direction from the bottom to the top (resp. from the top to the bottom), we denote the local diagram by $D^{u}$ (resp. $D^{d}$) where $u$ (resp. $d$) stands for up (resp. down). Then, we have to check the following four pairs: $(D_3^{u}, D_2^{u})$ (Case 1), $(D_1^{u}, D_2^{u})$ (Case 2), $(D_1^{d}, D_2^{d})$ (Case 3), and $(D_3^{d}, D_2^{d})$ (Case 4). Since each check is similar to the others, we first show the one for Case 2.
According to the definition of $q \circ p$ by (\ref{p-eq}) and (\ref{q-eq}), $q \circ p(D_1^{u})$ $=$ $D_3^{u}$. On the other hand, $q \circ p(D_2^{u})$ $=$ $D_2^{u}$. Since $D_3^{u}$ and $D_2^{u}$ can be replaced by Reidemeister move $\Omega_1 b$, so can $q \circ p(D_1^{u})$ and $q \circ p(D_2^{u})$.
Using the list below, we can show the other cases by analogy.
Case 1: $q \circ p(D_3^{u})$ $=$ $D_3^{u}$.
Case 2: $q \circ p(D_1^{u})$ $=$ $D_3^{u}$.
Case 3: $q \circ p(D_1^{d})$ $=$ $D_1^{d}$.
Case 4: $q \circ p(D_3^{d})$ $=$ $D_1^{d}$.
\item Reidemeister move $\Omega_2$.
Let $D_1$ (resp. $D_2$) be the local diagram defined by the left (resp. right) side of the move $\Omega_2$ in Fig. \ref{reidemeister_m}. For $D$ $=$ $D_1$ or $D_2$, let $D_r$ be the local diagram obtained by looking at $D$ upside down as shown in Fig. \ref{diagramref}. \begin{figure}
\caption{The local diagrams $D_{1r}$ (left) and $D_{2r}$ (right).}
\label{diagramref}
\end{figure} By definition, $D_{2r}$ is the same as $D_{2}$.
For each of $D_1$, $D_2$, $D_{1r}$, and $D_{2r}$, there are four cases by choice of orientation. If the orientations of the two branches of $D$ $=$ $D_1$ or $D_2$ are both in the direction from the bottom to the top (resp. from the top to the bottom), we denote the local diagram by $D^{uu}$ (resp. $D^{dd}$). Similarly, $D^{ud}$ (resp. $D^{du}$) stands for the local diagram $D$ where the orientations of the two branches are upward (resp. downward) and downward (resp. upward) from the left. Now, by Lemma \ref{reide_negative}, it is sufficient here to consider only the cases of $D^{ud}$ and $D^{du}$.
The local diagram $D$ $=$ $D_1$, $D_2$, $D_{1r}$, or $D_{2r}$ consists of two branches. The branch in which the endpoints are at the bottom left and the top left is called the left branch, and the other is called the right branch. If the first branch of $D^{ud}$ is the right (resp. the left) when starting from the base point, we denote the local diagram by $D^{\overline{ud}}$ (resp. $D^{ud}$). If the first branch of $D^{du}$ is the right (resp. the left), we denote the local diagram by $D^{\overline{du}}$ (resp. $D^{du}$). There are some relations between the oriented $D$ and $D_{r}$ that can be observed by looking at these upside down. For example, when we look at $D_{1r}^{\overline{ud}}$ upside down, we see $D_1^{ud}$. We can recognize ``looking at it upside down'' as the operator $f_{\pi}$, and using this operator we have \begin{equation}\label{pi_formula} \begin{split} f_{\pi}(D_{1r}^{ud}) = D_{1}^{\overline{ud}}~{\rm{and}}~f_{\pi}(D_{2r}^{ud}) = D_{2}^{\overline{ud}},\\ f_{\pi}(D_{1r}^{du}) = D_{1}^{\overline{du}}~{\rm{and}}~f_{\pi}(D_{2r}^{du}) = D_{2}^{\overline{du}},\\ f_{\pi}(D_{1r}^{\overline{ud}}) = D_{1}^{ud}~{\rm{and}}~f_{\pi}(D_{2r}^{\overline{ud}}) = D_{2}^{ud},\\ f_{\pi}(D_{1r}^{\overline{du}}) = D_{1}^{du}~{\rm{and}}~f_{\pi}(D_{2r}^{\overline{du}}) = D_{2}^{du}. \end{split} \end{equation} In the eight formulae of (\ref{pi_formula}), $f_{\pi}$ behaves as the involution.
The second row of Fig. \ref{rel_gauss_a} shows the eight moves between $D_{1}$ and $D_{2}$ or between $D_{1r}$ and $D_{2r}$ as follows ($\ast$ $=$ $1$ or $2$): $D_\ast^{ud}$ (Case 1), $D_\ast^{\overline{ud}}$ (Case 2), $D_\ast^{du}$ (Case 3), $D_\ast^{\overline{du}}$ (Case 4), $D_{\ast r}^{ud}$ (Case 5), $D_{\ast r}^{\overline{ud}}$ (Case 6), $D_{\ast r}^{du}$ (Case 7), and $D_{\ast r}^{\overline{du}}$ (Case 8). We would like to show that the move between $q \circ p(D_1)$ and $q \circ p(D_2)$ is one of these eight cases. However, if (\ref{pi_formula}) is used, it is sufficient to check only Cases 1 -- 4.
Since each check is similar to the others, we first show the one for Case 2. According to the definition of $q \circ p$ by (\ref{p-eq}) and (\ref{q-eq}), $q \circ p(D_1^{\overline{ud}})$ $=$ $D_{1r}^{\overline{ud}}$. Likewise, $q \circ p(D_2^{\overline{ud}})$ $=$ $D_2^{\overline{ud}}$ $=$ $D_{2r}^{\overline{ud}}$. Therefore, $q \circ p(D_1^{\overline{ud}})$ and $q \circ p(D_2^{\overline{ud}})$ can be replaced by the Reidemeister move of Case 6.
Using the list below, we can show the other cases by analogy.
Case 1: $q \circ p(D_1^{ud})$ $=$ $D_1^{ud}$.
Case 2: $q \circ p(D_1^{\overline{ud}})$ $=$ $D_{1r}^{\overline{ud}}$.
Case 3: $q \circ p(D_1^{du})$ $=$ $D_1^{du}$.
Case 4: $q \circ p(D_{1}^{\overline{du}})$ $=$ $D_{1r}^{\overline{du}}$.
\item Reidemeister moves similar to $\Omega_3$.
Let us recall that an equivalence relation for a long virtual knot is defined by Lemma \ref{reide_negative} and Fig. \ref{rel_gauss_a}. We have already verified the invariance of $V_{q \circ p(K)}$ under the moves in the first and second rows of Fig. \ref{rel_gauss_a}. Consequently, it is sufficient to show the invariance of $V_{q \circ p(K)}$ under the moves in the third row of Fig. \ref{rel_gauss_a}.
The moves in the third row of Fig. \ref{rel_gauss_a} are explained by Table \ref{table_direction}, which is realized as Fig. \ref{3rd_move_ver} by using the local knot diagrams. \begin{figure}
\caption{Reidemeister moves that should be checked. These cases correspond to Table \ref{table_direction}. Numbers 1--3 indicate the order of branches, which is defined as the order for passing through the neighborhood when starting from the base point.}
\label{3rd_move_ver}
\end{figure}
Let $D_{il}$ (resp. $D_{ir}$) be the local diagram defined by the left (resp. right) side of the move in Case $i$ of Fig. \ref{3rd_move_ver}. According to the definition of $q \circ p$ by (\ref{p-eq}) and (\ref{q-eq}), if $i$ $=$ $1$, $4$, $5$, $8$, $9$, or $12$, we have $q \circ p(D_{il})$ $=$ $D_{2l}$ and $q \circ p(D_{ir})$ $=$ $D_{2r}$. Similarly, if $i$ $=$ $2$, $3$, $6$, $7$, $10$, or $11$, we have $q \circ p(D_{il})$ $=$ $D_{8l}$ and $q \circ p(D_{ir})$ $=$ $D_{8r}$.
Here, we denote one of the Reidemeister moves between $D_{il}$ and $D_{ir}$ ($1 \le i \le 12$) by $\sim$, and we have \begin{equation}\label{3rd-move-eq} \begin{split} q \circ p(D_{il}) &= D_{2l} \sim D_{2r} = q \circ p(D_{ir}) \qquad (i = 1, 4, 5, 8, 9, 12), \\ q \circ p(D_{il}) &= D_{8l} \sim D_{8r} = q \circ p(D_{ir}) \qquad (i = 2, 3, 6, 7, 10, 11). \end{split} \end{equation} \end{itemize} The formulae (\ref{3rd-move-eq}) complete the check that $q \circ p(D_{il})$ and $q \circ p(D_{ir})$ can be replaced by one of the Reidemeister moves between $D_{il}$ and $D_{ir}$ ($1 \le i \le 12$).
As proved above, map $q \circ p$ is well defined as the map from the set of long virtual knots to itself.
On the other hand, we can assume that the domain of the map $p_r$ is the set of long virtual knots. Under this assumption, Theorem \ref{main} implies that $V_{K}$ $=$ $V_{{p_r}(K)}$. Here, we notice that $p_r \circ q$ $=$ $i$. Then, we have \[V_{i \circ p(K)} = V_{p_r \circ q \circ p(K)} = V_{q \circ p(K)}.\] Therefore, the map $V_{i \circ p(K)}$ is well defined as the map from the set of long virtual knots. That is, the map $V_{i \circ p(K)}$ is an invariant for Reidemeister moves and virtual moves. This completes the proof. \end{proof}
In what follows, we show applications of Theorem \ref{main_s0}.
\begin{ex}\label{ex_phrase} Let $K_1$ and $K_2$ be the long virtual knots shown in Fig. \ref{k1_k2}, with Jones polynomials $V_{K_1}(t)$ $=$ $V_{K_2}(t)$ $=$ $t^{-2}$ $+$ $t^{- 3/2}$ $-$ $t^{-1}$ $-$ $t^{- 1/2}$ $+$ $t^{1/2}$. However, $V_{i(p(K_1))}(t)$ $=$ $V_{K_1}(t)$ $\neq$ $V_{K_1}(t^{-1})$ $=$ $V_{K_2}(t^{-1})$ $=$ $V_{i(p(K_2))}(t)$. \begin{figure}
\caption{Two long virtual knots $K_1$ (left) and $K_2$ (right).}
\label{k1_k2}
\end{figure} \end{ex} Example \ref{ex_phrase} implies the following: \begin{theorem}\label{strongerthan} Let $K$ be a long virtual knot $K$. For $K$, the pair of $V_{K}$ and $V_{i(p(K))}$ is a stronger invariant than the polynomial $V_K$. In other words, there exist two long virtual knots $K_1$ and $K_2$ such that $V_{K_1}$ $=$ $V_{K_2}$ but $V_{i(p(K_1))}$ $\neq$ $V_{i(p(K_2))}$. \end{theorem} \begin{proof} The above example demonstrates the statement. \end{proof} Example \ref{ex_phrase} also means that $V_{K}$ detects the orientation of the long virtual knot for $K_1$. Let $-K$ be a knot with an orientation that is the reverse of that for a knot $K$. \begin{remark}\label{ori} Let $K$ be a long virtual knot that has a Gauss diagram which satisfies the condition that when arrow directions are neglected, the Gauss diagram is symmetric with respect to a line passing thorough the base point (e.g. the right figure of Fig. \ref{virtual_gauss}). If a knot $K$ satisfies the assumption ($\diamond$) that the Jones polynomial $V_{i(p(K))}$ changes by replacing $t^{1/2}$ $\mapsto$ $t^{- 1/2}$, then the polynomial $V_{i(p(K))}$ of $K$ detects the orientation of $K$ (e.g., $K$ $=$ $K_1$). This is because of the well-known fact that the Jones polynomial $V_{\overline{K}}$ of the mirror image $\overline{K}$ is obtained by replacing $t^{1/2}$ with $t^{- 1/2}$. In other words, by the assumption ($\diamond$), $V_{i(p(K))}$ $\neq$ $V_{i(p(\overline{K}))}$ $=$ $V_{i(p(-K))}$. However, there is no example satisfying the assumption ($\diamond$) for classical long knots since an arbitrary open flat virtual knot on the plane is equal to the trivial open flat virtual knot under its relations. Here, the consideration is summarized by the following proposition: \end{remark} \begin{proposition}\label{ori_prop} Let $K$ be an arbitrary nonclassical long virtual knot and $\overline{K}$ its mirror image. If the Jones polynomial $V_{K}(t)$ is not symmetric with respect to $t^{1/2} \mapsto t^{-1/2}$, the Jones polynomial $V_{i(p(K))}(t)$ detects its orientation. \end{proposition}
Next, we consider another type of example. \begin{ex}\label{ex2_phrase} Let $K_3$ and $K_4$ be the long virtual knots shown in Fig. \ref{k3_k4}. Then, $V_{K_3}$ $=$ $V_{K_4}$ but $V_{i(p(K_3))}(t)$ $=$ $V_{K_3}(t^{-1})$ $\neq$ $V_{K_3}(t)$ $=$ $V_{K_4}(t)$ $=$ $V_{i(p(K_4))}(t)$. \begin{figure}
\caption{Two long virtual knots $K_3$ (upper) and $K_4$ (lower).}
\label{k3_k4}
\end{figure} \end{ex}
Similarly, $p_a$ is defined as the composition $\tau_0 \circ p$ of the two maps $p$ and the involution map $\tau_0 :$ $a$ $\mapsto$ $b$ on chords of Gauss diagrams with base points. Moreover, $p_{ra}$ is defined as the composition $\tau_1 \circ p_r$ of two maps $p_r$ and the involution $\tau_1 :$ $+$ $\mapsto$ $-$ on chords of Gauss diagrams (we consider Gauss diagrams with base points if necessary). We can also consider the map $i_a$ defined as the composition $\tau_1 \circ i$. It is easy to see that these maps imply well-defined maps between equivalence classes of Gauss diagrams determined by topological objects which we treated. Then, we have the following.
\begin{theorem}\label{four_thm} All the choices of $p_r$, $p_{ra}$, $p$, $p_a$, $i$, and $i_a$, together generate four types of the Jones polynomials for long virtual knots.
As a corollary to Theorem \ref{strongerthan}, the tuple of four versions of the Jones polynomial is stronger than the Jones polynomial for long virtual knots. \end{theorem} \begin{proof} Considering every combination of $p_r$, $p_{ra}$, $p$, $p_a$, $i$, and $i_a$ yields the formulas $i \circ p_a$ $=$ $i_a \circ p$ and $i_a \circ p_a$ $=$ $i \circ p$. \end{proof} We consider these maps in Examples \ref{ex_pra} and \ref{ia}.
Here, let us consider the graphical meaning of these variations in the $V_K$ of a long virtual knot $K$. Recall the definition of $q \circ p$ by (\ref{p-eq}) and (\ref{q-eq}). For the diagram $D$ of a long virtual knot, $q \circ p_a (D)$ $=$ $q \circ \tau_0 \circ p(D)$ is the mirror image of $q \circ p(D)$. On the other hand, when we denote the mirror image of $D$ by $D^{*}$, we have $V_{p_{ra} (D)}$ $=$ $V_{D^{*}}$. The pair of Jones polynomials $(V_{p_r (D)}, V_{p_{ra} (D)}, V_{i \circ p}(D), V_{i_a \circ p}(D))$ is nothing but $(V_{D}, V_{D^{*}}, V_{q \circ p(D)}, V_{\left(q \circ p(D)\right)^{*}})$; that is, we calculate the four values of the Jones polynomials of two diagrams of long virtual knots and their mirror images.
\section{Application of the discussion to Khovanov homology}\label{app_kh} In this section, we apply the above discussion to Khovanov homology. After we recall the Khovanov homology, we consider the above discussion for Khovanov homology in the case of the coefficient $\mathbb{Z}_2$. \subsection{Khovanov homology} In this section, we recall the Khovanov homology of the Jones polynomial introduced by Khovanov \cite{khovanovjones}. There are two major redefinitions of Khovanov homology (\cite{bar-natan}, \cite{viro}), and here we give a brief review of the definition in the style of Viro \cite{viro}.
For a given knot diagram, let us consider a small edge, called a {\it{marker}}, for each crossing on the link diagram. In the rest of this paper, we can use a simple notation such as that of Fig. \ref{markers} (c) for the marker of Fig. \ref{markers} (a). Every marker has its sign defined as in Fig. \ref{markers}. The signed markers determine the directions of smoothing for all crossings (Fig. \ref{smoothing}). The smoothened link diagram is called the {\it{Kauffman state}} or simply the {\it{state}}. \begin{figure}
\caption{(a) Positive marker. (b) Negative marker. (c) Simple notation for positive marker.}
\label{markers}
\end{figure} \begin{figure}
\caption{Smoothing producing states. The marker on the crossing in (a) is the positive marker, and that in (b) is the negative marker.}
\label{smoothing}
\end{figure} In the next step, we assign {\it{labels}} $x$ or $1$ for every circle of the state. The {\it{degree}} of $y$ $=$ $x$ or $y$ $=$ $1$ is defined by ${\rm{deg}}(x)$ $=$ $-1$ or ${\rm{deg}}(1)$ $=$ $1$. The state whose circles have labels $x$ or $1$ is called an {\it{enhanced state}} and is denoted by $S$. Let $\sigma(S)$ be the number of positive markers minus the number of negative markers for an arbitrary enhanced state $S$. For a label $y$ $=$ $x$ or $y$ $=$ $1$, we set $\tau(S)$ $=$ $\sum_{{\text{circles}}~y~{\text{in}}~S}$ ${\rm{deg}}(y)$. For a link diagram $D$ of a link $L$, the unnormalized Jones polynomial $\hat{J}(L)$ of a link $L$ is obtained as \begin{equation} \hat{J}(L) = \sum_{{\text{enhanced states}}~S~{\text{of}}~D} (-1)^{i(S)} q^{j(S)} \end{equation} where $i(S)$ $=$ $(w(D) - \sigma(S))/2$ and $j(S)$ $=$ $w(D)$ + $i(S)$ + $\tau(S)$. Here, $w(D)$ is the writhe number of $D$, which is defined as the number of positive crossings minus negative crossings. The unnormalized Jones polynomial $\hat{J}(L)(q)$ is $(q + q^{-1}) V_{L}(q)$, with the variable $q$ replaced by $q$ $=$ $- t^{1/2}$ for the well-known (normalized) Jones polynomial $V_{L}(t)$. Now, we define the Khovanov complex $C^{i, j}(D)$ as the abelian group generated by the enhanced Kauffman states $S$ of a fixed link diagram $D$ satisfying $i(S)$ $=$ $i$ and $j(S)$ $=$ $j$. Let $T$ be an enhanced state obtained when a neighborhood of only one crossing with a positive marker is replaced by that of the negative marker, where the neighborhood in each of the cases is as listed in Fig. \ref{differential}. \begin{figure}
\caption{Incidence numbers $(S:T)$. Each $S$ is locally replaced with $T$. The dotted arcs show how fragments of $S$ or $T$ are connected in the whole $S$ or $T$. Using another traditional notation, we can write the above formulae as (a) $m(x \otimes 1)$ $=$ $x$, (b) $m(1 \otimes x)$ $=$ $x$, (c) $m(1 \otimes 1)$ $=$ $1$, (d) $\Delta(x)$ $=$ $x \otimes x$, and (e), (f) $\Delta(1)$ $=$ $1 \otimes x$ $+$ $x \otimes 1$. Here, a circle of enhanced states corresponds to a module $\mathbb{Z}_2 1 \oplus \mathbb{Z}_2 x$ over $\mathbb{Z}_2$.}
\label{differential}
\end{figure} For an arbitrary enhanced state $S$, $d(S)$ is defined by \begin{equation} d(S) = \sum_{{\text{enhanced states}}~T} (S : T)\, T \end{equation} where the incidence number $(S : T)$ is unity in each of the cases listed in Fig. \ref{differential} and zero if the couple of $S$ and $T$ does not appear in the list of Fig. \ref{differential}. The map is extended to the homomorphism $d$ from $C^{i, j}(D)$ to $C^{i+1, j}(D)$. It is a well-known fact that $d$ is the coboundary operator, usually called the {\it{differential}} in the case of Khovanov homology; i.e., $d^{2}$ $=$ $0$. \begin{theorem}[Khovanov] Let $D$ be a diagram of an arbitrary link $L$. For arbitrary $i$ and $j$, the homology $H^{i}(C^{*, j}(D), d)$ is an isotopy invariant of $L$, and so this homology can be denoted by $H^{i, j}(L)$. The homology $H^{i, j}(L)$ satisfies \begin{equation} \hat{J}(L) = \sum_{j} q^{j} \sum_{i} (-1)^{i} {\rm{rank}} H^{i, j}(L). \end{equation} \end{theorem}
\subsection{Application to Khovanov homology} Manturov extended the definition of the Khovanov homology to that of virtual knots, denoted here by $KH^{i, j}$ through adding the map between enhanced states of virtual knots. The problem is that the change of one positive marker to define the differential does not require the change of the component enhanced states for all cases, as shown in Fig. \ref{differential}. Fortunately, in the case of the coefficient $\mathbb{Z}_{2}$, the definition was extended to virtual knots straightforwardly by regarding these cases as zero maps and using Fig. \ref{differential}. Moreover, Manturov found the following property \cite{manturov1}: \begin{theorem}[Manturov] For ${KH}^{i, j}(K)$, the Khovanov homology of Manturov, ${KH}^{i, j}(K)$ $\simeq$ ${KH}^{i, j}(p_r(K))$ for an arbitrary virtual knot $K$. In other words, the Khovanov homology of Manturov is invariant under virtualization of Fig. \ref{virtulization_move}. \end{theorem} \begin{figure}
\caption{Virtualization. Arbitrary orientations of these moves are permitted.}
\label{virtulization_move}
\end{figure} Then, we have the counterpart of Theorem \ref{strongerthan}: \begin{theorem}\label{ip} Let $K$ be an arbitrary long virtual knot. A pairing of the two Khovanov homologies ${KH}^{i, j}(p_r(K))$ and ${KH}^{i, j}(i(p(K)))$ is stronger than Manturov's Khovanov homology ${KH}^{i, j}(K)$ in terms of an invariant of long virtual knots. In other words, there exist two long virtual knots $K_1$ and $K_2$ such that ${KH}^{i, j}(p_r(K_1))$ $\simeq$ ${KH}^{i, j}(p_r(K_2))$ for any $(i, j)$ but ${KH}^{i, j}(i(p(K_1)))$ $\not\simeq$ ${KH}^{i, j}(i(p(K_2)))$ for some $(i, j)$. \end{theorem} \begin{proof} Example \ref{ex_k1_k2} gives what needs to be shown. \end{proof}
\begin{ex}\label{ex_k1_k2} By definition, ${KH}^{i, j}(K_1)$ $\simeq$ ${KH}^{i, j}(K_2)$ for any $(i, j)$. However, ${KH}^{-2, -5}(i(p(K_1))$ $\simeq$ $\mathbb{Z}_2$ and ${KH}^{-2, -5}(i(p(K_2))$ $\simeq$ $0$. \end{ex}
\begin{ex}\label{ex_pra} Let us consider another type of $p_r$ denoted by $p_{ra}$. We have ${KH}^{2, -5}$ $(p_r(K_1))$ $\simeq$ ${KH}^{2, -5}(p_r(\emptyset))$ $\simeq$ $0$. However, ${KH}^{2, -5}(p_{ra}(K_1))$ $\simeq$ $\mathbb{Z}_{2}$, which is not $0$ $\simeq$ $KH^{2, -5}(p_{ra}(\emptyset))$. \end{ex}
\begin{ex}\label{ia} Let us consider another type of $i$ denoted by $i_a$. As described above, ${KH}^{-2, -5}(i(p(K_2)))$ $\simeq$ ${KH}^{-2, -5}(i(p(\emptyset)))$ $\simeq$ $0$. However, ${KH}^{-2, -5}(i_a(p(K_2)))$ $\simeq$ $\mathbb{Z}_2$, which is not $0$ $\simeq$ ${KH}^{-2, -5}(i_a(p(\emptyset)))$. \end{ex}
We also have the counterpart of Theorem \ref{four_thm}: \begin{theorem} All the choices of $p_r$, $p_{ra}$, $p$, $p_a$, $i$, and $i_a$ together generate four types of the Khovanov homology for long virtual knots.
As a corollary of Theorem \ref{ip}, the tuple of four Khovanov homologies is stronger than the Khovanov homology $KH^{i, j}$ in terms of long virtual knots. \end{theorem}
As in Sec. \ref{ver_jones}, four invariants $KH^{i, j}(p_r (D))$, $KH^{i, j}(p_{ra} (D))$, $KH^{i, j}(i \circ p (D))$, and $KH^{i, j}(i_a \circ p (D))$ means considering Khovanov homology for four long virtual knot diagrams $D$, $D^{*}$, $q \circ p(D)$, and $\left( q \circ p(D) \right)^{*}$.
\end{document} | arXiv |
Time-Resolved Planar Particle Image Velocimetry of the 3-D Multi-Mode Richtmyer Meshkov Instability
Sewell, Everest George
PIV
Richtmyer-Meshkov
Shock Tube
Jacobs, Jeffrey W.
An experimental investigation of the Richtmyer-Meshkov instability (RMI) is carried out using a single driver vertical shock tube. A diffuse, stably stratified membrane-less interface is formed between air and sulfur hexafluoride (SF6) gases (Atwood number, $ A = \frac{\rho_1 - \rho_2}{\rho_1+\rho_2} \approx0.67$) via counterflow, where the light gas (air) enters the tube from the top of the driven section, and the heavy gas (SF$_6$) enters from the bottom of the test section. A perturbation is imposed at the interface using voice coil drivers that cause a vertical oscillation of the column of gases. This oscillation results in the Rayleigh-Taylor unstable growth of random modes present at the interface, and gives rise to Faraday waves which invert with half the frequency of the oscillation. The interface is initially accelerated by a Mach 1.17 (in air) shock wave, and the development of the ensuing mixing layer is investigated. The shock wave is then reflected from the bottom of the apparatus, where it interacts with the mixing layer a second time (reshock). The experiment is initialized with two distinct perturbations - high amplitude experiments where the shock wave arrives at the maximum excursion of the perturbation, and low amplitude experiments where it arrives near its minimum. Time resolved Particle Image Velocimetry (PIV) is used as the primary flow diagnostic, yielding instantaneous velocity field estimates at a rate of 2 kHz. Measurements of the growth exponent $\theta$, where the mixing layer width $h$ is assumed to grow following $h(t) \approx t^\theta$, yield a value of $\theta\approx 0.51$ for high amplitude experiments and $\theta\approx0.45$ for low amplitude experiments following the incident shock wave when estimated using the width of the mixing layer approximated by the width of the turbulent kinetic energy containing region. Following interaction with the reflected shock wave, $\theta \approx 0.33$ for high amplitude experiments, and $\theta \approx 0.50$ for low amplitude experiments. It is observed that the low amplitude experiments grow faster than the high amplitude experiments following reshock, likely owing to the presence of steeper density gradients present in the relatively less developed mixing layer. $\theta$ is also estimated using the decay of turbulent kinetic energy for experiments where dissipation is significant. Theta estimates using both methods are found to be in good agreement for the high amplitude case following the incident shock, with $\theta\approx0.51$. $\theta \approx 0.46$ is found following reshock, which is larger than the value found when fitting $\theta$ to width data. Low amplitude experiments do not exhibit significant dissipation, and a value of $\theta \approx 0.68$ is found for low amplitude experiments following the incident shock, and $\theta \approx 0.62$ following reshock. Persistent anisotropy is a commonly observed phenomenon in the RMI mixing layer, owing to the stronger velocity perturbation components in the streamwise direction following the passage of a shock wave. High amplitude experiments are observed to reach a constant anisotropy ratio (defined as the ratio of streamwise to spanwise turbulent kinetic energy, or TKX/TKY), an indication of self-similarity, shortly following the passage of the incident shock wave with value of $\approx 1.8$. Low amplitude experiments do not reach a constant value during the experimental observation window, suggesting that the flow is still evolving even after a second shock interaction. Examination of the spanwise average anisotropy tensor reveals asymmetry in the anisotropy for low amplitude experiments, with the heavy gas exhibiting a slightly larger degree of anisotropy. The high amplitude experiments exhibit transitional outer Reynolds numbers ($Re\equiv\frac{h\Dot{h}}{\nu} > 10^4$) using the criterion proposed by Dimotakis shortly following the passage of the initial shock wave, while the low amplitude experiments largely remain below this threshold. Following reshock, both sets of experiments are elevated to $Re \approx 10^5$, which is a strong indication that mixing transition should occur and an inertial range will form. However, extended length scale analysis proposed by Zhou that accounts for the temporal evolution of scales which are a prerequisite for the formation of an inertial range indicates that neither high or low amplitude experiments have entered a transitional regime even following reshock. Furthermore, the $\theta \approx 0.5$ growth of the outer length scale in these experiments suggests that transition will not occur even if longer observation windows were possible. The lack of an inertial range is evident in spectral analysis of the mixing region. | CommonCrawl |
\begin{definition}[Definition:Contour/Closed]
Let $\R^n$ be a real cartesian space of $n$ dimensions.
Let $C$ be the '''contour''' in $\R^n$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\R^n$.
Let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$.
$C$ is a '''closed contour''' {{iff}} the start point of $C$ is equal to the end point of $C$:
:$\rho_1 \left({a_1}\right) = \rho_n \left({b_n}\right)$
\end{definition} | ProofWiki |
Debbie Bard
William Arndt
Jan Balewski
Johannes Blaschke
Bjoern Enders
Rollin Thomas
Home » About » Staff » Data Science Engagement » Rollin Thomas
Rollin Thomas Ph.D.
Big Data Architect
Data Science Engagement Group
[email protected]
1 Cyclotron Road
Mailstop: 59-4010A
Berkeley, CA 94720 US
Rollin is interested in interactive, real-time, and urgent supercomputing for science. He manages NERSC's JupyterHub service and the JupyterLab deployments on Cori and Perlmutter. Before moving to the Data Science Engagement Group, Rollin was a member of NERSC's Data and Analytics Services Group. There, he directed Python support for the Edison and Cori systems, initiated and led the NERSC Exascale Science Application Program for Data, and helped launch NERSC's container-as-a-service platform, Spin.
Friesen, B., Baron, E., Parrent, J. T., Thomas, R., C., Branch, D., Nugent, P., Hauschildt, P. H., Foley, R. J., Wright, D. E., Pan, Y.-C., Filippenko, A. V., Clubb, K. I., Silverman, J. M., Maeda, K. Shivvers, I., Kelly, P. L., Cohen, D. P., Rest, A., Kasen, D., "Optical and ultraviolet spectroscopic analysis of SN 2011fe at late times", Monthly Notices of the Royal Astronomical Society, February 27, 2017, 467:2392-2411, doi: 10.1093/mnras/stx241
We present optical spectra of the nearby Type Ia supernova SN 2011fe at 100, 205, 311, 349 and 578 d post-maximum light, as well as an ultraviolet (UV) spectrum obtained with the Hubble Space Telescope at 360 d post-maximum light. We compare these observations with synthetic spectra produced with the radiative transfer code phoenix. The day +100 spectrum can be well fitted with models that neglect collisional and radiative data for forbidden lines. Curiously, including these data and recomputing the fit yields a quite similar spectrum, but with different combinations of lines forming some of the stronger features. At day +205 and later epochs, forbidden lines dominate much of the optical spectrum formation; however, our results indicate that recombination, not collisional excitation, is the most influential physical process driving spectrum formation at these late times. Consequently, our synthetic optical and UV spectra at all epochs presented here are formed almost exclusively through recombination-driven fluorescence. Furthermore, our models suggest that the UV spectrum even as late as day +360 is optically thick and consists of permitted lines from several iron-peak species. These results indicate that the transition to the 'nebular' phase in Type Ia supernovae is complex and highly wavelength dependent.
Smith, M., Sullivan, M., D'Andrea, C. B., Castander, F. J., Casas, R., Prajs, S., Papadopoulos, A., Nichol, R. C., Karpenka, N. V., Bernard, S. R., Brown, P., Cartier, R., Cooke, J., Curtin, C., Davis, T. M., Finley, D. A., Foley, R. J., Gal-Yam, A., Goldstein, D. A., González-Gaitán, S., Gupta, R. R., Howell, D. A., Inserra, C., Kessler, R., Lidman, C., Marriner, J., Nugent, P., Pritchard, T. A., Sako, M., Smartt, S., Smith, R. C., Spinka, H., Thomas, R. C., Wolf, R. C., Zenteno, A., Abbott, T. M. C., Benoit-Lévy, A., Bertin, E., Brooks, D., Buckley-Geer, E., Carnero Rosell, A., Carrasco Kind, M., Carretero, J., Crocce, M., Cunha, C. E., da Costa, L. N., Desai, S., Diehl, H. T., Doel, P., Estrada, J., Evrard, A. E., Flaugher, B., Fosalba, P., Frieman, J., Gerdes, D. W., Gruen, D., Gruendl, R. A., James, D. J., Kuehn, K., Kuropatkin, N., Lahav, O., Li, T. S., Marshall, J. L., Martini, P., Miller, C. J., Miquel, R., Nord, B., Ogando, R., Plazas, A. A., Reil, K., Romer, A. K., Roodman, A., Rykoff, E. S., Sanchez, E., Scarpine, V., Schubnell, M., Sevilla-Noarbe, I., Soares-Santos, M., Sobreira, F., Suchyta, E., Swanson, M. E. C., Tarle, G., Walker, A. R., Wester, W., "DES14X3taz: A Type I Superluminous Supernova Showing a Luminous, Rapidly Cooling Initial Pre-peak Bump", The Astrophysical Journal Letters, 2016, doi: 10.3847/2041-8205/818/1/L8
We present DES14X3taz, a new hydrogen-poor superluminous supernova (SLSN-I) discovered by the Dark Energy Survey (DES) supernova program, with additional photometric data provided by the Survey Using DECam for Superluminous Supernovae. Spectra obtained using Optical System for Imaging and low-Intermediate-Resolution Integrated Spectroscopy on the Gran Telescopio CANARIAS show DES14X3taz is an SLSN-I at z = 0.608. Multi-color photometry reveals a double-peaked light curve: a blue and relatively bright initial peak that fades rapidly prior to the slower rise of the main light curve. Our multi-color photometry allows us, for the first time, to show that the initial peak cools from 22,000 to 8000 K over 15 rest-frame days, and is faster and brighter than any published core-collapse supernova, reaching 30% of the bolometric luminosity of the main peak. No physical 56Ni-powered model can fit this initial peak. We show that a shock-cooling model followed by a magnetar driving the second phase of the light curve can adequately explain the entire light curve of DES14X3taz. Models involving the shock-cooling of extended circumstellar material at a distance of ≃400 {\text{}}{R}⊙ are preferred over the cooling of shock-heated surface layers of a stellar envelope. We compare DES14X3taz to the few double-peaked SLSN-I events in the literature. Although the rise times and characteristics of these initial peaks differ, there exists the tantalizing possibility that they can be explained by one physical interpretation.
Baron, E., Hoeflich, P., Friesen, B., Sullivan, M., Hsiao, E., Ellis, R. S., Gal-Yam, A., Howell, D. A., Nugent, P. E., Dominguez, I., Krisciunas, K., Phillips, M. M., Suntzeff, N., Wang, L., and Thomas, R. C., "Spectral models for early time SN 2011fe observations", Monthly Notices of the Royal Astronomical Society, 2015, 454:2549, doi: 10.1093/mnras/stv1951
We use observed UV through near-IR spectra to examine whether SN 2011fe can be understood in the framework of Branch-normal Type Ia supernovae (SNe Ia) and to examine its individual peculiarities. As a benchmark, we use a delayed-detonation model with a progenitor metallicity of Z⊙/20. We study the sensitivity of features to variations in progenitor metallicity, the outer density profile, and the distribution of radioactive nickel. The effect of metallicity variations in the progenitor have a relatively small effect on the synthetic spectra. We also find that the abundance stratification of SN 2011fe resembles closely that of a delayed-detonation model with a transition density that has been fit to other Branch-normal SNe Ia. At early times, the model photosphere is formed in material with velocities that are too high, indicating that the photosphere recedes too slowly or that SN 2011fe has a lower specific energy in the outer ≈0.1 M⊙ than does the model. We discuss several explanations for the discrepancies. Finally, we examine variations in both the spectral energy distribution and in the colours due to variations in the progenitor metallicity, which suggests that colours are only weak indicators for the progenitor metallicity, in the particular explosion model that we have studied. We do find that the flux in the U band is significantly higher at maximum light in the solar metallicity model than in the lower metallicity model and the lower metallicity model much better matches the observed spectrum.
Fakhouri, H. K., Boone, K., Aldering, G., Antilogus, P., Aragon, C., Bailey, S., Baltay, C., Barbary, K., Baugh, D., Bongard, S., Buton, C., Chen, J., Childress, M., Chotard, N., Copin, Y., Fagrelius, P., Feindt, U., Fleury, M., Fouchez, D., Gangler, E., Hayden, B., Kim, A. G., Kowalski, M., Leget, P.-F., Lombardo, S., Nordin, J., Pain, R., Pecontal, E., Pereira, R., Perlmutter, S., Rabinowitz, D., Ren, J., Rigault, M., Rubin, D., Runge, K., Saunders, C., Scalzo, R., Smadja, G., Sofiatti, C., Strovink, M., Suzuki, N., Tao, C., Thomas, R. C., and Weaver, B. A., "Improving Cosmological Distance Measurements Using Twin Type Ia Supernovae", The Astrophysical Journal, 2015, 815:58, doi: 10.1088/0004-637X/815/1/58
We introduce a method for identifying "twin" Type Ia supernovae (SNe Ia) and using them to improve distance measurements. This novel approach to SN Ia standardization is made possible by spectrophotometric time series observations from the Nearby Supernova Factory (SNfactory). We begin with a well-measured set of SNe, find pairs whose spectra match well across the entire optical window, and then test whether this leads to a smaller dispersion in their absolute brightnesses. This analysis is completed in a blinded fashion, ensuring that decisions made in implementing the method do not inadvertently bias the result. We find that pairs of SNe with more closely matched spectra indeed have reduced brightness dispersion. We are able to standardize this initial set of SNfactory SNe to 0.083 ± 0.012 mag, implying a dispersion of 0.072 ± 0.010 mag in the absence of peculiar velocities. We estimate that with larger numbers of comparison SNe, e.g., using the final SNfactory spectrophotometric data set as a reference, this method will be capable of standardizing high-redshift SNe to within 0.06–0.07 mag. These results imply that at least 3/4 of the variance in Hubble residuals in current SN cosmology analyses is due to previously unaccounted-for astrophysical differences among the SNe.
Kessler, R., Marriner, J., Childress, M., Covarrubias, R., D'Andrea, C. B., Finley, D. A., Fischer, J., Foley, R. J., Goldstein, D., Gupta, R. R., Kuehn, K., Marcha, M., Nichol, R. C., Papadopoulos, A., Sako, M., Scolnic, D., Smith, M., Sullivan, M., Wester, W., Yuan, F., Abbott, T., Abdalla, F. B., Allam, S., Benoit-Levy, A., Bernstein, G. M., Bertin, E., Brooks, D., Carnero Rosell, A., Carrasco Kind, M., Castander, F. J., Crocce, M., da Costa, L. N., Desai, S., Diehl, H. T., Eifler, T. F., Fausti Neto, A., Flaugher, B., Frieman, J., Gerdes, D. W., Gruen, D., Gruendl, R. A., Honscheid, K., James, D. J., Kuropatkin, N., Li, T. S., Maia, M. A. G., Marshall, J. L., Martini, P., Miller, C. J., Miquel, R., Nord, B., Ogando, R., Plazas, A. A., Reil, K., Romer, A. K., Roodman, A., Sanchez, E., Sevilla-Noarbe, I., Smith, R. C., Soares-Santos, M., Sobreira, F., Tarle, G., Thaler, J., Thomas, R. C., Tucker, D., and Walker, A. R., "The Difference Imaging Pipeline for the Transient Search in the Dark Energy Survey", The Astronomical Journal, 2015, 150:172, doi: 10.1088/0004-6256/150/6/172
We describe the operation and performance of the difference imaging pipeline (DiffImg) used to detect transients in deep images from the Dark Energy Survey Supernova program (DES-SN) in its first observing season from 2013 August through 2014 February. DES-SN is a search for transients in which ten 3 deg2 fields are repeatedly observed in the g, r, i, z passbands with a cadence of about 1 week. The observing strategy has been optimized to measure high-quality light curves and redshifts for thousands of Type Ia supernovae (SNe Ia) with the goal of measuring dark energy parameters. The essential DiffImg functions are to align each search image to a deep reference image, do a pixel-by-pixel subtraction, and then examine the subtracted image for significant positive detections of point-source objects. The vast majority of detections are subtraction artifacts, but after selection requirements and image filtering with an automated scanning program, there are ∼130 detections per deg2 per observation in each band, of which only ∼25% are artifacts. Of the ∼7500 transients discovered by DES-SN in its first observing season, each requiring a detection on at least two separate nights, Monte Carlo (MC) simulations predict that 27% are expected to be SNe Ia or core-collapse SNe. Another ∼30% of the transients are artifacts in which a small number of observations satisfy the selection criteria for a single-epoch detection. Spectroscopic analysis shows that most of the remaining transients are AGNs and variable stars. Fake SNe Ia are overlaid onto the images to rigorously evaluate detection efficiencies and to understand the DiffImg performance. The DiffImg efficiency measured with fake SNe agrees well with expectations from a MC simulation that uses analytical calculations of the fluxes and their uncertainties. In our 8 "shallow" fields with single-epoch 50% completeness depth ∼23.5, the SN Ia efficiency falls to 1/2 at redshift z ≈ 0.7; in our 2 "deep" fields with mag-depth ∼24.5, the efficiency falls to 1/2 at z ≈ 1.1. A remaining performance issue is that the measured fluxes have additional scatter (beyond Poisson fluctuations) that increases with the host galaxy surface brightness at the transient location. This bright-galaxy issue has minimal impact on the SNe Ia program, but it may lower the efficiency for finding fainter transients on bright galaxies.
Yuan, F., Lidman, C., Davis, T. M., Childress, M., Abdalla, F. B., Banerji, M., Buckley-Geer, E., Carnero Rosell, A., Carollo, D., Castander, F. J., D'Andrea, C. B., Diehl, H. T., Cunha, C. E., Foley, R. J., Frieman, J., Glazebrook, K., Gschwend, J., Hinton, S., Jouvel, S., Kessler, R., Kim, A. G., King, A. L., Kuehn, K., Kuhlmann, S., Lewis, G. F., Lin, H., Martini, P., McMahon, R. G., Mould, J., Nichol, R. C., Norris, R. P., O'Neill, C. R., Ostrovski, F., Papadopoulos, A., Parkinson, D., Reed, S., Romer, A. K., Rooney, P. J., Rozo, E., Rykoff, E. S., Sako, M., Scalzo, R., Schmidt, B. P., Scolnic, D., Seymour, N., Sharp, R., Sobreira, F., Sullivan, M., Thomas, R. C., Tucker, D., Uddin, S. A., Wechsler, R. H., Wester, W., Wilcox, H., Zhang, B., Abbott, T., Allam, S., Bauer, A. H., Benoit-Levy, A., Bertin, E., Brooks, D., Burke, D. L., Carrasco Kind, M., Covarrubias, R., Crocce, M., da Costa, L. N., DePoy, D. L., Desai, S., Doel, P., Eifler, T. F., Evrard, A. E., Fausti Neto, A., Flaugher, B., Fosalba, P., Gaztanaga, E., Gerdes, D., Gruen, D., Gruendl, R. A., Honscheid, K., James, D., Kuropatkin, N., Lahav, O., Li, T. S., Maia, M. A. G., Makler, M., Marshall, J., Miller, C. J., Miquel, R., Ogando, R., Plazas, A. A., Roodman, A., Sanchez, E., Scarpine, V., Schubnell, M., Sevilla-Noarbe, I., Smith, R. C., Soares-Santos, M., Suchyta, E., Swanson, M. E. C., Tarle, G., Thaler, J., and Walker, A. R., "OzDES multifibre spectroscopy for the Dark Energy Survey: first-year operation and results", Monthly Notices of the Royal Astronomical Society, 2015, 452:3047, doi: 10.1093/mnras/stv1507
The Australian Dark Energy Survey (OzDES) is a five-year, 100-night, spectroscopic survey on the Anglo-Australian Telescope, whose primary aim is to measure redshifts of approximately 2500 Type Ia supernovae host galaxies over the redshift range 0.1 < z < 1.2, and derive reverberation-mapped black hole masses for approximately 500 active galactic nuclei and quasars over 0.3 < z < 4.5. This treasure trove of data forms a major part of the spectroscopic follow-up for the Dark Energy Survey for which we are also targeting cluster galaxies, radio galaxies, strong lenses, and unidentified transients, as well as measuring luminous red galaxies and emission line galaxies to help calibrate photometric redshifts. Here, we present an overview of the OzDES programme and our first-year results. Between 2012 December and 2013 December, we observed over 10 000 objects and measured more than 6 000 redshifts. Our strategy of retargeting faint objects across many observing runs has allowed us to measure redshifts for galaxies as faint as mr = 25 mag. We outline our target selection and observing strategy, quantify the redshift success rate for different types of targets, and discuss the implications for our main science goals. Finally, we highlight a few interesting objects as examples of the fortuitous yet not totally unexpected discoveries that can come from such a large spectroscopic survey.
Goldstein, D. A., D'Andrea, C. B., Fischer, J. A., Foley, R. J., Gupta, R. R., Kessler, R., Kim, A. G., Nichol, R. C., Nugent, P. E., Papadopoulos, A., Sako, M., Smith, M., Sullivan, M., Thomas, R. C., Wester, W., Wolf, R. C., Abdalla, F. B., Banerji, M., Benoit-Levy, A., Bertin, E., Brooks, D., Carnero Rosell, A., Castander, F. J., da Costa, L. N., Covarrubias, R., DePoy, D. L., Desai, S., Diehl, H. T., Doel, P., Eifler, T. F., Fausti Neto, A., Finley, D. A., Flaugher, B., Fosalba, P., Frieman, J., Gerdes, D., Gruen, D., Gruendl, R. A., James, D., Kuehn, K., Kuropatkin, N., Lahav, O., Li, T. S., Maia, M. A. G., Makler, M., March, M., Marshall, J. L., Martini, P., Merritt, K. W., Miquel, R., Nord, B., Ogando, R., Plazas, A. A., Romer, A. K., Roodman, A., Sanchez, E., Scarpine, V., Schubnell, M., Sevilla-Noarbe, I., Smith, R. C., Soares-Santos, M., Sobreira, F., Suchyta, E., Swanson, M. E. C., Tarle, G., Thaler, J., and Walker, A. R., "Automated Transient Identification in the Dark Energy Survey", The Astronomical Journal, 2015, 150:82, doi: 10.1088/0004-6256/150/3/82
We describe an algorithm for identifying point-source transients and moving objects on reference-subtracted optical images containing artifacts of processing and instrumentation. The algorithm makes use of the supervised machine learning technique known as Random Forest. We present results from its use in the Dark Energy Survey Supernova program (DES-SN), where it was trained using a sample of 898,963 signal and background events generated by the transient detection pipeline. After reprocessing the data collected during the first DES-SN observing season (2013 September through 2014 February) using the algorithm, the number of transient candidates eligible for human scanning decreased by a factor of 13.4, while only 1.0% of the artificial Type Ia supernovae (SNe) injected into search images to monitor survey efficiency were lost, most of which were very faint events. Here we characterize the algorithm's performance in detail, and we discuss how it can inform pipeline design decisions for future time-domain imaging surveys, such as the Large Synoptic Survey Telescope and the Zwicky Transient Facility. An implementation of the algorithm and the training data used in this paper are available at at http://portal.nersc.gov/project/dessn/autoscan.
Kim, A. G., Padmanabhan, N., Aldering, G., Allen, S. W., Baltay, C., Cahn, R. N., D'Andrea, C. B., Dalal, N., Dawson, K. S., Denney, K. D., Eisenstein, D. J., Finley, D. A., Freedman, W. L., Ho, S., Holz, D. E., Kasen, D., Kent, S. M., Kessler, R., Kuhlmann, S., Linder, E. V., Martini, P., Nugent, P. E., Perlmutter, S., Peterson, B. M., Riess, A. G., Rubin, D., Sako, M., Suntzeff, N. V., Suzuki, N., Thomas, R. C., Wood-Vasey, W. M., and Woosley, S. E., "Distance probes of dark energy", Astroparticle Physics, 2015, 63:2, doi: 10.1016/j.astropartphys.2014.05.007
This document presents the results from the Distances subgroup of the Cosmic Frontier Community Planning Study (Snowmass 2013). We summarize the current state of the field as well as future prospects and challenges. In addition to the established probes using Type Ia supernovae and baryon acoustic oscillations, we also consider prospective methods based on clusters, active galactic nuclei, gravitational wave sirens and strong lensing time delays.
Sasdelli, M., Hillebrandt, W., Aldering, G., Antilogus, P., Aragon, C., Bailey, S., Baltay, C., Benitez-Herrera, S., Bongard, S., Buton, C., Canto, A., Cellier-Holzem, F., Chen, J., Childress, M., Chotard, N., Copin, Y., Fakhouri, H. K., Feindt, U., Fink, M., Fleury, M., Fouchez, D., Gangler, E., Guy, J., Ishida, E. E. O., Kim, A. G., Kowalski, M., Kromer, M., Lombardo, S., Mazzali, P. A., Nordin, J., Pain, R., Pecontal, E., Pereira, R., Perlmutter, S., Rabinowitz, D., Rigault, M., Runge, K., Saunders, C., Scalzo, R., Smadja, G., Suzuki, N., Tao, C., Taubenberger, S., Thomas, R. C., Tilquin, A., and Weaver, B. A., "A metric space for Type Ia supernova spectra", Monthly Notices of the Royal Astronomical Society, 2015, 447:1247, doi: 10.1093/mnras/stu2416
We develop a new framework for use in exploring Type Ia supernovae (SNe Ia) spectra. Combining principal component analysis (PCA) and partial least square (PLS) analysis we are able to establish correlations between the principal components (PCs) and spectroscopic/photometric SNe Ia features. The technique was applied to ˜120 SN and ˜800 spectra from the Nearby Supernova Factory. The ability of PCA to group together SNe Ia with similar spectral features, already explored in previous studies, is greatly enhanced by two important modifications: (1) the initial data matrix is built using derivatives of spectra over the wavelength, which increases the weight of weak lines and discards extinction, and (2) we extract time evolution information through the use of entire spectral sequences concatenated in each line of the input data matrix. These allow us to define a stable PC parameter space which can be used to characterize synthetic SN Ia spectra by means of real SN features. Using PLS, we demonstrate that the information from important previously known spectral indicators (namely the pseudo-equivalent width of Si II 5972 Å/Si II 6355 Å and the line velocity of S II 5640 Å/Si II 6355 Å) at a given epoch is contained within the PC space and can be determined through a linear combination of the most important PCs. We also show that the PC space encompasses photometric features like B/V magnitudes, B - V colours and SALT2 parameters c and x1. The observed colours and magnitudes, which are heavily affected by extinction, cannot be reconstructed using this technique alone. All the above-mentioned applications allowed us to construct a metric space for comparing synthetic SN Ia spectra with observations.
Saunders, C., Aldering, G., Antilogus, P., Aragon, C., Bailey, S., Baltay, C., Bongard, S., Buton, C., Canto, A., Cellier-Holzem, F., Childress, M., Chotard, N., Copin, Y., Fakhouri, H. K., Feindt, U., Gangler, E., Guy, J., Kerschhaggl, M., Kim, A. G., Kowalski, M., Nordin, J., Nugent, P., Paech, K., Pain, R., Pecontal, E., Pereira, R., Perlmutter, S., Rabinowitz, D., Rigault, M., Rubin, D., Runge, K., Scalzo, R., Smadja, G., Tao, C., Thomas, R. C., Weaver, B. A., and Wu, C., "Type Ia Supernova Distance Modulus Bias and Dispersion from K-correction Errors: A Direct Measurement Using Light Curve Fits to Observed Spectral Time Series", The Astrophysical Journal, 2015, 800:57, doi: 10.1088/0004-637X/800/1/57
We estimate systematic errors due to K-corrections in standard photometric analyses of high-redshift Type Ia supernovae. Errors due to K-correction occur when the spectral template model underlying the light curve fitter poorly represents the actual supernova spectral energy distribution, meaning that the distance modulus cannot be recovered accurately. In order to quantify this effect, synthetic photometry is performed on artificially redshifted spectrophotometric data from 119 low-redshift supernovae from the Nearby Supernova Factory, and the resulting light curves are fit with a conventional light curve fitter. We measure the variation in the standardized magnitude that would be fit for a given supernova if located at a range of redshifts and observed with various filter sets corresponding to current and future supernova surveys. We find significant variation in the measurements of the same supernovae placed at different redshifts regardless of filters used, which causes dispersion greater than ~0.05 mag for measurements of photometry using the Sloan-like filters and a bias that corresponds to a 0.03 shift in w when applied to an outside data set. To test the result of a shift in supernova population or environment at higher redshifts, we repeat our calculations with the addition of a reweighting of the supernovae as a function of redshift and find that this strongly affects the results and would have repercussions for cosmology. We discuss possible methods to reduce the contribution of the K-correction bias and uncertainty.
Friesen, B., Baron, E., Wisniewski, J. P., Parrent, J. T., Thomas, R. C., Miller, Timothy R., and Marion, G. H., "Near-infrared Line Identification in Type Ia Supernovae during the Transitional Phase", The Astrophysical Journal, 2014, 792:120, doi: 10.1088/0004-637X/792/2/120
We present near-infrared synthetic spectra of a delayed-detonation hydrodynamical model and compare them to observed spectra of four normal Type Ia supernovae ranging from day +56.5 to day +85. This is the epoch during which supernovae are believed to be undergoing the transition from the photospheric phase, where spectra are characterized by line scattering above an optically thick photosphere, to the nebular phase, where spectra consist of optically thin emission from forbidden lines. We find that most spectral features in the near-infrared can be accounted for by permitted lines of Fe II and Co II. In addition, we find that [Ni II] fits the emission feature near 1.98 μm, suggesting that a substantial mass of 58Ni exists near the center of the ejecta in these objects, arising from nuclear burning at high density.
"Type Ia supernova bolometric light curves and ejected mass estimates from the Nearby Supernova Factory", Monthly Notices of the Royal Astronomical Society, 2014, 440:1498, doi: 10.1093/mnras/stu350
We present a sample of normal Type Ia supernovae (SNe Ia) from the Nearby Supernova Factory data set with spectrophotometry at sufficiently late phases to estimate the ejected mass using the bolometric light curve. We measure 56Ni masses from the peak bolometric luminosity, then compare the luminosity in the 56Co-decay tail to the expected rate of radioactive energy release from ejecta of a given mass. We infer the ejected mass in a Bayesian context using a semi-analytic model of the ejecta, incorporating constraints from contemporary numerical models as priors on the density structure and distribution of 56Ni throughout the ejecta. We find a strong correlation between ejected mass and light-curve decline rate, and consequently 56Ni mass, with ejected masses in our data ranging from 0.9 to 1.4 M⊙. Most fast-declining (SALT2 x1 < -1) normal SNe Ia have significantly sub-Chandrasekhar ejected masses in our fiducial analysis.
McCully, C., Jha, S. W., Foley, R. J., Chornock, R., Holtzman, J. A., Balam, D. D., Branch, D., Filippenko, A. V., Frieman, J., Fynbo, J., Galbany, L., Ganeshalingam, M., Garnavich, P. M., Graham, M. L., Hsiao, E. Y., Leloudas, G., Leonard, D. C., Li, W., Riess, A. G., Sako, M., Schneider, D. P., Silverman, J. M., Sollerman, J., Steele, T. N., Thomas, R. C., Wheeler, J. C., and Zheng, C., "Hubble Space Telescope and Ground-based Observations of the Type Iax Supernovae SN 2005hk and SN 2008A", The Astrophysical Journal, 2014, 786:134, doi: 10.1088/0004-637X/786/2/134
We present Hubble Space Telescope (HST) and ground-based optical and near-infrared observations of SN 2005hk and SN 2008A, typical members of the Type Iax class of supernovae (SNe). Here we focus on late-time observations, where these objects deviate most dramatically from all other SN types. Instead of the dominant nebular emission lines that are observed in other SNe at late phases, spectra of SNe 2005hk and 2008A show lines of Fe II, Ca II, and Fe I more than a year past maximum light, along with narrow [Fe II] and [Ca II] emission. We use spectral features to constrain the temperature and density of the ejecta, and find high densities at late times, with ne >~ 109 cm-3. Such high densities should yield enhanced cooling of the ejecta, making these objects good candidates to observe the expected "infrared catastrophe," a generic feature of SN Ia models. However, our HST photometry of SN 2008A does not match the predictions of an infrared catastrophe. Moreover, our HST observations rule out a "complete deflagration" that fully disrupts the white dwarf for these peculiar SNe, showing no evidence for unburned material at late times. Deflagration explosion models that leave behind a bound remnant can match some of the observed properties of SNe Iax, but no published model is consistent with all of our observations of SNe 2005hk and 2008A.
A. G. Kim, G. Aldering, P. Antilogus, C. Aragon, S., C. Baltay, S. Bongard, C. Buton, A., F. Cellier-Holzem, M. Childress, N., Y. Copin, H. K. Fakhouri, U. Feindt, M., E. Gangler, P. Greskovic, J. Guy, M., S. Lombardo, J. Nordin, P. Nugent, R., E. Pecontal, R. Pereira, S. Perlmutter, D., M. Rigault, K. Runge, C. Saunders, R., G. Smadja, C. Tao, R. C. Thomas, B. A. Weaver, "Type Ia Supernova Hubble Residuals and Host-galaxy Properties", Astrophysical Journal, 2014, 784:51, doi: 10.1088/0004-637X/784/1/51
P. A. Mazzali, M. Sullivan, S. Hachinger, R. S., P. E. Nugent, D. A. Howell, A. Gal-Yam, K., J. Cooke, R. Thomas, K. Nomoto, E. S. Walker, "Hubble Space Telescope spectra of the Type Ia supernova SN 2011fe: a tail of low-density, high-velocity material with Z \lt Z$_⊙$", Monthly Notices of the Royal Astronomical Society, 2014, 439:1959-1979, doi: 10.1093/mnras/stu077
U. Feindt, M. Kerschhaggl, M. Kowalski, G. Aldering, P., C. Aragon, S. Bailey, C. Baltay, S., C. Buton, A. Canto, F. Cellier-Holzem, M., N. Chotard, Y. Copin, H. K. Fakhouri, E., J. Guy, A. Kim, P. Nugent, J., K. Paech, R. Pain, E. Pecontal, R., S. Perlmutter, D. Rabinowitz, M., K. Runge, C. Saunders, R. Scalzo, G., C. Tao, R. C. Thomas, B. A. Weaver, C. Wu, "Measuring cosmic bulk flows with Type Ia supernovae from the Nearby Supernova Factory", Astronomy and Astrophysics, 2013, 560:A90, doi: 10.1051/0004-6361/201321880
M. Rigault, Y. Copin, G. Aldering, P. Antilogus, C., S. Bailey, C. Baltay, S. Bongard, C., A. Canto, F. Cellier-Holzem, M. Childress, N., H. K. Fakhouri, U. Feindt, M. Fleury, E., P. Greskovic, J. Guy, A. G. Kim, M., S. Lombardo, J. Nordin, P. Nugent, R., E. P\ econtal, R. Pereira, S. Perlmutter, D., K. Runge, C. Saunders, R. Scalzo, G., C. Tao, R. C. Thomas, B. A. Weaver, "Evidence of environmental dependencies of Type Ia supernovae from the Nearby Supernova Factory indicated by local H$\alpha$", Astronomy and Astrophysics, 2013, 560:A66, doi: 10.1051/0004-6361/201322104
M. Childress, G. Aldering, P. Antilogus, C. Aragon, S., C. Baltay, S. Bongard, C. Buton, A., F. Cellier-Holzem, N. Chotard, Y. Copin, H. K., E. Gangler, J. Guy, E. Y. Hsiao, M., A. G. Kim, M. Kowalski, S. Loken, P., K. Paech, R. Pain, E. Pecontal, R., S. Perlmutter, D. Rabinowitz, M., K. Runge, R. Scalzo, G. Smadja, C., R. C. Thomas, B. A. Weaver, C. Wu, "Host Galaxy Properties and Hubble Residuals of Type Ia Supernovae from the Nearby Supernova Factory", Astrophysical Journal, 2013, 770:108, doi: 10.1088/0004-637X/770/2/108
M. Childress, G. Aldering, P. Antilogus, C. Aragon, S., C. Baltay, S. Bongard, C. Buton, A., F. Cellier-Holzem, N. Chotard, Y. Copin, H. K., E. Gangler, J. Guy, E. Y. Hsiao, M., A. G. Kim, M. Kowalski, S. Loken, P., K. Paech, R. Pain, E. Pecontal, R., S. Perlmutter, D. Rabinowitz, M., K. Runge, R. Scalzo, G. Smadja, C., R. C. Thomas, B. A. Weaver, C. Wu, "Host Galaxies of Type Ia Supernovae from the Nearby Supernova Factory", Astrophysical Journal, 2013, 770:107, doi: 10.1088/0004-637X/770/2/107
R. Pereira, R. C. Thomas, G. Aldering, P. Antilogus, C., S. Benitez-Herrera, S. Bongard, C., A. Canto, F. Cellier-Holzem, J. Chen, M., N. Chotard, Y. Copin, H. K. Fakhouri, M., D. Fouchez, E. Gangler, J. Guy, W., E. Y. Hsiao, M. Kerschhaggl, M., M. Kromer, J. Nordin, P. Nugent, K., R. Pain, E. P\ econtal, S. Perlmutter, D., M. Rigault, K. Runge, C. Saunders, G., C. Tao, S. Taubenberger, A. Tilquin, C. Wu, "Spectrophotometric time series of SN 2011fe from the Nearby Supernova Factory", Astronomy and Astrophysics, 2013, 554:A27, doi: 10.1051/0004-6361/201221008
A. G. Kim, R. C. Thomas, G. Aldering, P. Antilogus, C., S. Bailey, C. Baltay, S. Bongard, C., A. Canto, F. Cellier-Holzem, M. Childress, N., Y. Copin, H. K. Fakhouri, E. Gangler, J., M. Kerschhaggl, M. Kowalski, J. Nordin, P., K. Paech, R. Pain, E. Pecontal, R., S. Perlmutter, D. Rabinowitz, M., K. Runge, C. Saunders, R. Scalzo, G. Smadja, C. Tao, B. A. Weaver, C. Wu, "Standardizing Type Ia Supernova Absolute Magnitudes Using Gaussian Process Data Regression", Astrophysical Journal, 2013, 766:84, doi: 10.1088/0004-637X/766/2/84
C. Buton, Y. Copin, G. Aldering, P. Antilogus, C., S. Bailey, C. Baltay, S. Bongard, A., F. Cellier-Holzem, M. Childress, N., H. K. Fakhouri, E. Gangler, J. Guy, E. Y., M. Kerschhaggl, M. Kowalski, S., P. Nugent, K. Paech, R. Pain, E., R. Pereira, S. Perlmutter, D., M. Rigault, K. Runge, R. Scalzo, G., C. Tao, R. C. Thomas, B. A. Weaver, C. Wu, Nearby SuperNova Factory, "Atmospheric extinction properties above Mauna Kea from the Nearby SuperNova Factory spectro-photometric data set", Astronomy and Astrophysics, 2013, 549:A8, doi: 10.1051/0004-6361/201219834
Friesen, B., Baron, E., Branch, D., Chen, B., Parrent, J., Thomas, R. C., "Supernova Resonance-scattering Line Profiles in the Absence of a Photosphere", The Astrophysical Journal Supplements Series, 2012, 203:1, doi: 10.1088/0067-0049/203/1/12
In supernova (SN) spectroscopy relatively little attention has been given to the properties of optically thick spectral lines in epochs following the photosphere's recession. Most treatments and analyses of post-photospheric optical spectra of SNe assume that forbidden-line emission comprises most if not all spectral features. However, evidence exists that suggests that some spectra exhibit line profiles formed via optically thick resonance-scattering even months or years after the SN explosion. To explore this possibility, we present a geometrical approach to SN spectrum formation based on the "Elementary Supernova" model, wherein we investigate the characteristics of resonance-scattering in optically thick lines while replacing the photosphere with a transparent central core emitting non-blackbody continuum radiation, akin to the optical continuum provided by decaying 56Co formed during the explosion. We develop the mathematical framework necessary for solving the radiative transfer equation under these conditions and calculate spectra for both isolated and blended lines. Our comparisons with analogous results from the Elementary Supernova code SYNOW reveal several marked differences in line formation. Most notably, resonance lines in these conditions form P Cygni-like profiles, but the emission peaks and absorption troughs shift redward and blueward, respectively, from the line's rest wavelength by a significant amount, despite the spherically symmetric distribution of the line optical depth in the ejecta. These properties and others that we find in this work could lead to misidentification of lines or misattribution of properties of line-forming material at post-photospheric times in SN optical spectra.
Parrent, J. T., Howell, D. A., Friesen, B., Thomas, R. C., Fesen, R. A., Milisavljevic, D., Bianco, F. B., Dilday, B., Nugent, P., Baron, E., Arcavi, I., Ben-Ami, S., Bersier, D., Bildsten, L., Bloom, J., Cao, Y., Cenko, S. B., Filippenko, A. V., Gal-Yam, A., Kasliwal, M. M., Konidaris, N., Kulkarni, S. R., Law, N. M., Levitan, D., Maguire, K., Mazzali, P. A., Ofek, E. O., Pan, Y., Polishook, D., Poznanski, D., Quimby, R. M., Silverman, J. M., Sternberg, A., Sullivan, M., Walker, E. S., Xu, Dong, Buton, C., Pereira, R., "Analysis of the Early-time Optical Spectra of SN 2011fe in M101", The Astrophysical Journal Letters, 2012, 752, doi: 10.1088/2041-8205/752/2/L26
The nearby Type Ia supernova (SN Ia) SN 2011fe in M101 (cz = 241 km s–1) provides a unique opportunity to study the early evolution of a "normal" SN Ia, its compositional structure, and its elusive progenitor system. We present 18 high signal-to-noise spectra of SN 2011fe during its first month beginning 1.2 days post-explosion and with an average cadence of 1.8 days. This gives a clear picture of how various line-forming species are distributed within the outer layers of the ejecta, including that of unburned material (C+O). We follow the evolution of C II absorption features until they diminish near maximum light, showing overlapping regions of burned and unburned material between ejection velocities of 10,000 and 16,000 km s–1. This supports the notion that incomplete burning, in addition to progenitor scenarios, is a relevant source of spectroscopic diversity among SNe Ia. The observed evolution of the highly Doppler-shifted O I λ7774 absorption features detected within 5 days post-explosion indicates the presence of O I with expansion velocities from 11,500 to 21,000 km s–1. The fact that some O I is present above C II suggests that SN 2011fe may have had an appreciable amount of unburned oxygen within the outer layers of the ejecta.
Thomas, R. C.; Aldering, G.; Antilogus, P.; Aragon, C.; Bailey, S.; Baltay, C.; Bongard, S.; Buton, C.; Canto, A.; Childress, M.; Chotard, N.; Copin, Y.; Fakhouri, H. K.; Gangler, E.; Hsiao, E. Y.; Kerschhaggl, M.; Kowalski, M.; Loken, S.; Nugent, P.; Paech, K.; Pain, R.; Pecontal, E.; Pereira, R.; Perlmutter, S.; Rabinowitz, D.; Rigault, M.; Rubin, D.; Runge, K.; Scalzo, R.; Smadja, G.; Tao, C.; Weaver, B. A.; Wu, C.; (The Nearby Supernova Factory); Brown, P. J.; Milne, P. A., "Type Ia Supernova Carbon Footprints", Astrophysical Journal, December 2011, 743:27,
We present convincing evidence of unburned carbon at photospheric velocities in new observations of five Type Ia supernovae (SNe Ia) obtained by the Nearby Supernova Factory. These SNe are identified by examining 346 spectra from 124 SNe obtained before +2.5 days relative to maximum. Detections are based on the presence of relatively strong C II λ6580 absorption "notches" in multiple spectra of each SN, aided by automated fitting with the SYNAPPS code. Four of the five SNe in question are otherwise spectroscopically unremarkable, with ions and ejection velocities typical of SNe Ia, but spectra of the fifth exhibit high-velocity (v > 20, 000 km s–1) Si II and Ca II features. On the other hand, the light curve properties are preferentially grouped, strongly suggesting a connection between carbon-positivity and broadband light curve/color behavior: three of the five have relatively narrow light curves but also blue colors and a fourth may be a dust-reddened member of this family. Accounting for signal to noise and phase, we estimate that 22+10 – 6% of SNe Ia exhibit spectroscopic C II signatures as late as –5 days with respect to maximum. We place these new objects in the context of previously recognized carbon-positive SNe Ia and consider reasonable scenarios seeking to explain a physical connection between light curve properties and the presence of photospheric carbon. We also examine the detailed evolution of the detected carbon signatures and the surrounding wavelength regions to shed light on the distribution of carbon in the ejecta. Our ability to reconstruct the C II λ6580 feature in detail under the assumption of purely spherical symmetry casts doubt on a "carbon blobs" hypothesis, but does not rule out all asymmetric models. A low volume filling factor for carbon, combined with line-of-sight effects, seems unlikely to explain the scarcity of detected carbon in SNe Ia by itself. http://dx.doi.org/10.1088/0004-637X/743/1/27
Nugent, Peter E.; Sullivan, Mark; Cenko, S. Bradley; Thomas, Rollin C.; Kasen, Daniel; Howell, D. Andrew; Bersier, David; Bloom, Joshua S.; Kulkarni, S. R.; Kandrashoff, Michael T.; Filippenko, Alexei V.; Silverman, Jeffrey M.; Marcy, Geoffrey W.; Howard, Andrew W.; Isaacson, Howard T.; Maguire, Kate; Suzuki, Nao; Tarlton, James E.; Pan, Yen-Chen; Bildsten, Lars; Fulton, Benjamin J.; Parrent, Jerod T.; Sand, David; Podsiadlowski, Philipp; Bianco, Federica B.; Dilday, Benjamin; Graham, Melissa L.; Lyman, Joe; James, Phil; Kasliwal, Mansi M.; Law, Nicholas M.; Quimby, Robert M.; Hook, Isobel M.; Walker, Emma S.; Mazzali, Paolo; Pian, Elena; Ofek, Eran O.; Gal-Yam, Avishay; Poznanski, Dovi, "Supernova SN 2011fe from an exploding carbon-oxygen white dwarf star", Nature, December 2011, 480:344-347,
Type Ia supernovae have been used empirically as `standard candles' to demonstrate the acceleration of the expansion of the Universe even though fundamental details, such as the nature of their progenitor systems and how the stars explode, remain a mystery. There is consensus that a white dwarf star explodes after accreting matter in a binary system, but the secondary body could be anything from a main-sequence star to a red giant, or even another white dwarf. This uncertainty stems from the fact that no recent type Ia supernova has been discovered close enough to Earth to detect the stars before explosion. Here we report early observations of supernova SN 2011fe in the galaxy M101 at a distance from Earth of 6.4 megaparsecs. We find that the exploding star was probably a carbon-oxygen white dwarf, and from the lack of an early shock we conclude that the companion was probably a main-sequence star. Early spectroscopy shows high-velocity oxygen that slows rapidly, on a timescale of hours, and extensive mixing of newly synthesized intermediate-mass elements in the outermost layers of the supernova. A companion paper uses pre-explosion images to rule out luminous red giants and most helium stars as companions to the progenitor. http://dx.doi.org/10.1038/nature10644
Krisciunas, Kevin; Li, Weidong; Matheson, Thomas; Howell, D. Andrew; Stritzinger, Maximilian; Aldering, Greg; Berlind, Perry L.; Calkins, M.; Challis, Peter; Chornock, Ryan; Conley, Alexander; Filippenko, Alexei V.; Ganeshalingam, Mohan; Germany, Lisa; González, Sergio; Gooding, Samuel D.; Hsiao, Eric; Kasen, Daniel; Kirshner, Robert P.; Howie Marion, G. H.; Muena, Cesar; Nugent, Peter E.; Phelps, M.; Phillips, Mark M.; Qiu, Yulei; Quimby, Robert; Rines, K.; Silverman, Jeffrey M.; Suntzeff, Nicholas B.; Thomas, Rollin C.; Wang, Lifan, "The Most Slowly Declining Type Ia Supernova 2001ay", Astrophysical Journal, September 2011, 142:74,
We present optical and near-infrared photometry, as well as ground-based optical spectra and Hubble Space Telescope ultraviolet spectra, of the Type Ia supernova (SN) 2001ay. At maximum light the Si II and Mg II lines indicated expansion velocities of 14,000 km s–1, while Si III and S II showed velocities of 9000 km s–1. There is also evidence for some unburned carbon at 12,000 km s–1. SN 2001ay exhibited a decline-rate parameter of Δm 15(B) = 0.68 ± 0.05 mag; this and the B-band photometry at t +25 day past maximum make it the most slowly declining Type Ia SN yet discovered. Three of the four super-Chandrasekhar-mass candidates have decline rates almost as slow as this. After correction for Galactic and host-galaxy extinction, SN 2001ay had MB = –19.19 and MV = –19.17 mag at maximum light; thus, it was not overluminous in optical bands. In near-infrared bands it was overluminous only at the 2σ level at most. For a rise time of 18 days (explosion to bolometric maximum) the implied 56Ni yield was (0.58 ± 0.15)/α M ☉, with α = L max/E Ni probably in the range 1.0-1.2. The 56Ni yield is comparable to that of many Type Ia SNe. The "normal" 56Ni yield and the typical peak optical brightness suggest that the very broad optical light curve is explained by the trapping of γ rays in the inner regions. http://dx.doi.org/10.1088/0004-6256/142/3/74
Quimby, R. M.; Kulkarni, S. R.; Kasliwal, M. M.; Gal-Yam, A.; Arcavi, I.; Sullivan, M.; Nugent, P.; Thomas, R.; Howell, D. A.; Nakar, E.; Bildsten, L.; Theissen, C.; Law, N. M.; Dekany, R.; Rahmer, G.; Hale, D.; Smith, R.; Ofek, E. O.; Zolkower, J.; Velur, V.; Walters, R.; Henning, J.; Bui, K.; McKenna, D.; Poznanski, D.; Cenko, S. B.; Levitan, D., "Hydrogen-poor superluminous stellar explosions", Nature, June 2011, 474:487-489,
Supernovae are stellar explosions driven by gravitational or thermonuclear energy that is observed as electromagnetic radiation emitted over weeks or more. In all known supernovae, this radiation comes from internal energy deposited in the outflowing ejecta by one or more of the following processes: radioactive decay of freshly synthesized elements (typically 56Ni), the explosion shock in the envelope of a supergiant star, and interaction between the debris and slowly moving, hydrogen-rich circumstellar material. Here we report observations of a class of luminous supernovae whose properties cannot be explained by any of these processes. The class includes four new supernovae that we have discovered and two previously unexplained events (SN 2005ap and SCP 06F6) that we can now identify as members of the same class. These supernovae are all about ten times brighter than most type Ia supernova, do not show any trace of hydrogen, emit significant ultraviolet flux for extended periods of time and have late-time decay rates that are inconsistent with radioactivity. Our data require that the observed radiation be emitted by hydrogen-free material distributed over a large radius (~1015 centimetres) and expanding at high speeds (>104 kilometres per second). These long-lived, ultraviolet-luminous events can be observed out to redshifts z>4.
Childress, M.; Aldering, G.; Aragon, C.; Antilogus, P.; Bailey, S.; Baltay, C.; Bongard, S.; Buton, C.; Canto, A.; Chotard, N.; Copin, Y.; Fakhouri, H. K.; Gangler, E.; Kerschhaggl, M.; Kowalski, M.; Hsiao, E. Y.; Loken, S.; Nugent, P.; Paech, K.; Pain, R.; Pecontal, E.; Pereira, R.; Perlmutter, S.; Rabinowitz, D.; Runge, K.; Scalzo, R.; Thomas, R. C.; Smadja, G.; Tao, C.; Weaver, B. A.; Wu, C., "Keck Observations of the Young Metal-poor Host Galaxy of the Super-Chandrasekhar-mass Type Ia Supernova SN 2007if", Astrophysical Journal, May 2011, 733:3,
We present Keck LRIS spectroscopy and g-band photometry of the metal-poor, low-luminosity host galaxy of the super-Chandrasekhar-mass Type Ia supernova SN 2007if. Deep imaging of the host reveals its apparent magnitude to be mg = 23.15 ± 0.06, which at the spectroscopically measured redshift of z helio = 0.07450 ± 0.00015 corresponds to an absolute magnitude of Mg = -14.45 ± 0.06. Galaxy g - r color constrains the mass-to-light ratio, giving a host stellar mass estimate of log(M */M sun) = 7.32 ± 0.17. Balmer absorption in the stellar continuum, along with the strength of the 4000 Å break, constrains the age of the dominant starburst in the galaxy to be t burst = 123+165-77 Myr, corresponding to a main-sequence turnoff mass of M/M sun = 4.6+2.6-1.4. Using the R 23 method of calculating metallicity from the fluxes of strong emission lines, we determine the host oxygen abundance to be 12 + log(O/H)KK04 = 8.01 ± 0.09, significantly lower than any previously reported spectroscopically measured Type Ia supernova host galaxy metallicity. Our data show that SN 2007if is very likely to have originated from a young, metal-poor progenitor. http://dx.doi.org/10.1088/0004-637X/733/1/3
Sullivan, M.; Kasliwal, M. M.; Nugent, P. E.; Howell, D. A.; Thomas, R. C.; Ofek, E. O.; Arcavi, I.; Blake, S.; Cooke, J.; Gal-Yam, A.; Hook, I. M.; Mazzali, P.; Podsiadlowski, P.; Quimby, R.; Bildsten, L.; Bloom, J. S.; Cenko, S. B.; Kulkarni, S. R.; Law, N.; Poznanski, D., "The Subluminous and Peculiar Type Ia Supernova PTF 09dav", Astrophysical Journal, May 2011, 732:118,
PTF 09dav is a peculiar subluminous Type Ia supernova (SN) discovered by the Palomar Transient Factory (PTF). Spectroscopically, it appears superficially similar to the class of subluminous SN1991bg-like SNe, but it has several unusual features which make it stand out from this population. Its peak luminosity is fainter than any previously discovered SN1991bg-like SN Ia (MB ~ -15.5), but without the unusually red optical colors expected if the faint luminosity were due to extinction. The photospheric optical spectra have very unusual strong lines of Sc II and Mg I, with possible Sr II, together with stronger than average Ti II and low velocities of ~6000 km s-1. The host galaxy of PTF09dav is ambiguous. The SN lies either on the extreme outskirts (~41 kpc) of a spiral galaxy or in an very faint (MR >= -12.8) dwarf galaxy, unlike other 1991bg-like SNe which are invariably associated with massive, old stellar populations. PTF 09dav is also an outlier on the light-curve-width-luminosity and color-luminosity relations derived for other subluminous SNe Ia. The inferred 56Ni mass is small (0.019 ± 0.003 M sun), as is the estimated ejecta mass of 0.36 M sun. Taken together, these properties make PTF 09dav a remarkable event. We discuss various physical models that could explain PTF 09dav. Helium shell detonation or deflagration on the surface of a CO white dwarf can explain some of the features of PTF 09dav, including the presence of Sc and the low photospheric velocities, but the observed Si and Mg are not predicted to be very abundant in these models. We conclude that no single model is currently capable of explaining all of the observed signatures of PTF 09dav. http://dx.doi.org/10.1088/0004-637X/732/2/118
Chotard, N.; Gangler, E.; Aldering, G.; Antilogus, P.; Aragon, C.; Bailey, S.; Baltay, C.; Bongard, S.; Buton, C.; Canto, A.; Childress, M.; Copin, Y.; Fakhouri, H. K.; Hsiao, E. Y.; Kerschhaggl, M.; Kowalski, M.; Loken, S.; Nugent, P.; Paech, K.; Pain, R.; Pecontal, E.; Pereira, R.; Perlmutter, S.; Rabinowitz, D.; Runge, K.; Scalzo, R.; Smadja, G.; Tao, C.; Thomas, R. C.; Weaver, B. A.; Wu, C.; Nearby Supernova Factory, "The reddening law of type Ia supernovae: separating intrinsic variability from dust using equivalent widths", Astronomy & Astrophysics, May 2011, 529:L4,
We employ 76 type Ia supernovae (SNe Ia) with optical spectrophotometry within 2.5 days of B-band maximum light obtained by the Nearby Supernova Factory to derive the impact of Si and Ca features on the supernovae intrinsic luminosity and determine a dust reddening law. We use the equivalent width of Si ii λ4131 in place of the light curve stretch to account for first-order intrinsic luminosity variability. The resulting empirical spectral reddening law exhibits strong features that are associated with Ca ii and Si ii λ6355. After applying a correction based on the Ca ii H&K equivalent width we find a reddening law consistent with a Cardelli extinction law. Using the same input data, we compare this result to synthetic rest-frame UBVRI-like photometry to mimic literature observations. After corrections for signatures correlated with Si ii λ4131 and Ca ii H&K equivalent widths and introducing an empirical correlation between colors, we determine the dust component in each band. We find a value of the total-to-selective extinction ratio, RV = 2.8 ± 0.3. This agrees with the Milky Way value, in contrast to the low RVvalues found in most previous analyses. This result suggests that the long-standing controversy in interpreting SN Ia colors and their compatibility with a classical extinction law, which is critical to their use as cosmological probes, can be explained by the treatment of the dispersion in colors, and by the variability of features apparent in SN Ia spectra. http://dx.doi.org/1 0.1051/0004-6361/201116723
Maguire, K.; Sullivan, M.; Thomas, R. C.; Nugent, P.; Howell, D. A.; Gal-Yam, A.; Arcavi, I.; Ben-Ami, S.; Blake, S.; Botyanszki, J.; Buton, C.; Cooke, J.; Ellis, R. S.; Hook, I. M.; Kasliwal, M. M.; Pan, Y.-C.; Pereira, R.; Podsiadlowski, P.; Sternberg, A.; Suzuki, N.; Xu, D.; Yaron, O.; Bloom, J. S.; Cenko, S. B.; Kulkarni, S. R.; Law, N.; Ofek, E. O.; Poznanski, D.; Quimby, R. M., "PTF10ops - a subluminous, normal-width light curve Type Ia supernova in the middle of nowhere", Monthly Notices of the Royal Astronomical Society, January 1, 2011, 418:747-758,
Thomas, R. C., Nugent, P. E., and Meza, J. C., "SYNAPPS: Data-Driven Analysis for Supernova Spectroscopy", Publications of the Astronomical Society of the Pacific, January 1, 2011, 123:237-248, doi: 10.1086/658673
Keith Jackson, Lavanya Ramakrishnan, Karl Runge, and Rollin Thomas, "Seeking Supernovae in the Clouds: A Performance Study", ScienceCloud 2010, the 1st Workshop on Scientific Cloud Computing, Chicago, Illinois, June 2010,
Annette Greiner, Evan Racah, Shane Canon, Jialin Liu, Yunjie Liu, Debbie Bard, Lisa Gerhardt, Rollin Thomas, Shreyas Cholia, Jeff Porter, Wahid Bhimji, Quincey Koziol, Prabhat, "Data-Intensive Supercomputing for Science", Berkeley Institute for Data Science (BIDS) Data Science Faire, May 3, 2016,
Review of current DAS activities for a non-NERSC audience. | CommonCrawl |
\begin{document}
\title{Induced Norms} \title{Generalized Induced Norms\footnote{{\it 2000 Mathematics Subject Classification } 15A60 (Primary) 47A30, 46B99 (Secondary).\\ {\it Keywords and phrases}. induced norm, generalized induced norm, algebra norm, the full matrix algebra, unitarily invariant, generalized induced congruent.}} \author{S. Hejazian, M. Mirzavaziri and M. S. Moslehian} \date{} \maketitle \begin{abstract}
Let $\|.\|$ be a norm on the algebra $M_n$ of all $n\times n$ matrices over {\kh C}. An interesting problem in matrix theory is that "are there two norms $\|.\|_1$ and $\|.\|_2$ on {\kh C}$^n$ such that $\|A\|=\max\{\|Ax\|_{2}: \|x\|_{1}=1\}$ for all $A\in M_n$. We will investigate this problem and its various aspects and will discuss under which conditions $\|.\|_1=\|.\|_2$. \end{abstract}
\section{Preliminaries}
Throughout the paper $M_n$ denotes the complex algebra of all $n\times n$ matrices $A=[a_{ij}]$ with entries in {\kh C} together with the usual matrix operations. Denote by $\{e_1, e_2, \cdots e_n\}$ the standard basis for {\kh C}$^n$, where $e_i$ has $1$ as its $i$th entry and $0$ elsewhere. We denote by $E_{ij}$ the $n\times n$ matrix with $1$ in the $(i,j)$ entry and $0$ elsewhere.
For $1\leq p\leq \infty$ the norm $\ell_p$ on {\kh C}$^n$ is defined as follows:
$$\ell_p(x)=\ell_p(\displaystyle{\sum_{i=1}^n}x_ie_i)=\left \{ \begin{array}{cc}(\displaystyle{\sum_{i=1}^n}|x_i|^p)^{1/p}&1\leq p<\infty\\ \max\{|x_1|, \cdots, |x_n|\}&p=\infty \end{array}\right .$$
A norm $\|.\|$ on {\kh C}$^n$ is said to be unitarily invariant if $\|x\|=\|Ux\|$ for all unitaries $U$ and all $x\in${\kh C}$^n$.
By an algebra norm (or a matrix norm) we mean a norm $\|.\|$ on $M_n$ such that $\|AB\|\leq\|A\|\|B\|$ for all $A, B\in M_n$. An algebra norm $\|.\|$ on $M_n$ is called unitarily invariant if $\|UAV\|=\|A\|$ for all unitaries $U$ and $V$ and all $A\in M_n$. See [2, Chapter IV] for more information.
\begin{e.} {\rm The norm $\|A\|_{\sigma}=\displaystyle{\sum_{i,j=1}^n}|a_{ij}|$ is an algebra norm, but the norm $\|A\|_{m}=\max\{|a_{i,j}|: 1\leq i,j\leq n\}$ is not an algebra norm, since $\|\left [\begin{array}{cc}1&1\\1&1\end{array}\right ]^2\|_{m}>\|\left [\begin{array}{cc}1&1\\1&1\end{array}\right ]\|_{m}^2$.}\end{e.}
\begin{r.}{\rm It is easy to show that for each norm $\| .\|$ on $M_n$, the scaled norm $\max\{\frac{\| AB\|}{\|A \| \|B\|}: A,B\neq 0\}\|.\|$ is an algebra norm; cf. [1, p.114]}\end{r.}
Let $\|.\|_1$ and $\|.\|_2$ be two norms on {\kh C}$^n$. Then for each $A:(${\kh C}$^n,\|.\|_1)\to (${\kh C}$^n,\|.\|_2)$ we can define $\|A\|=\max\{\|Ax\|_{2}: \|x\|_{1}=1\}$. If $\|.\|_1=\|.\|_2$, then $\|I\|=1$ and there are many examples of $\|.\|_1$ and $\|.\|_2$ such that $\|I\|\neq 1$. This shows that given $\|.\|$ on $M_n$, we cannot deduce in general that there is a norm $\|.\|_1$ on {\kh C}$^n$ with $\|A\|=\max\{\|Ax\|_{1}: \|x\|_{1}=1\}$. Let us recall the concept of g-ind norm as follows:
\begin{d.} {\rm Let $\|.\|_1$ and $\|.\|_2$ be two norms on {\kh C}$^n$. Then the norm $\|.\|_{1,2}$ on $M_n$ defined by $\|A\|_{1,2}=\max\{\|Ax\|_2: \|x\|_1=1\}$ is called the generalized induced (or g-ind) norm via $\| .\|_{1}$ and $\| . \|_{2}$. If $\|.\|_1=\|.\|_2$, then $\|.\|_{1,1}$ is called induced norm.}\end{d.}
\begin{e.}{\rm $\|A\|_C=\max\{\displaystyle{\sum_{i=1}^n}|a_{i,j}|: 1\leq j\leq n\}, \|A\|_R=\max\{\displaystyle{\sum_{j=1}^n}|a_{i,j}|: 1\leq i\leq n\}$ and the spectral norm $\|A\|_S= \max\{\sqrt{\lambda}: \lambda {\rm ~is~ an ~eigenvalue~ of} A^*A\}$ are induced by $\ell_1, \ell_\infty$ and $\ell_2$ (or the Eucleadian norm), respectively.
It is known that the algebra norm $\|A\|=\max\{\|A\|_C, \|A\|_R\}$ is not induced [ ] and it is not hard to show that it is not g-ind too; cf. [1, Corollary 3.2.6]}\end{e.}
We need the following proposition which is a special case of a finite dimensional version of the Hahn-Banach theorem [5, p. 104]:
\begin{p.} Let $\|.\|$ be a norm on {\kh C}$^n$ and $y\in${\kh C}$^n$ be a given vector. There exists a vector $y_\circ\in${\kh C}$^n$ such that $y_{\circ}^*y=\|y\|$ and for all $x\in${\kh C}$^n$, $|y_{\circ}^*x|\leq\|x\|$. {\rm (Throughout, $*$ denotes the transpose) [3, Corollary 5.5.15])}\end{p.}
In this paper we examine the following nice problems:\\
(i) Given a norm $\|.\|$ on $M_n$ is there any class ${\cal A}$ of $M_n$ such that the restriction of the norm $\|.\|$ on ${\cal A}$ is g-ind?\\ (ii) When a g-ind norm is unitarily invariant?\\
(iii) If a given norm $\|.\|$ is g-ind via $\|.\|_1$ and $\|.\|_2$, then is it possible to find $\|.\|_1$ and $\|.\|_2$ explicitly in terms of $\|.\|$?\\ (iv) When two g-ind norms are the same?\\ (v) Is there any characterization of the g-ind norms which are algebra norms?
\section{Main Results}
We begin with some observations on generalized induced norms.
Let $\|.\|_{1,2}$ be a generalized induced norm on $M_n$ obtained via $\|.\|_1$ and $\|.\|_2$. Then $\|E_{ij}\|_{1,2}=\max\{\|E_{ij}x\|_2: \|x\|_1=1\}=\max\{\|x_{j}e_i\|_2: \|(x_1,\cdots,x_j,\cdots,x_n)\|_1=1\}=\alpha_j\|e_i\|_2$, where $\alpha_j=\max\{|x_j|: \|(x_1,\cdots,x_j,\cdots,x_n)\|_1=1\}$. In general, for $x\in${\kh C}$^n$ and $1\leq j\leq n$, if $C_{x,j}\in M_n$ is defined by the operator $C_{x,j}(y)=y_{j}x$ then $\|C_{x,j}\|_{1,2}=\alpha_j\|x\|_2$.
Also if for $x\in${\kh C}$^n$ we define $C_{x}\in M_n$ by $C_{x}=\sum_{j=1}^nC_{x,j}$, then clearly $\|C_x\|_{1,2}=\alpha\|x\|_2$, where $\alpha=\max\{|\sum_{j=1}^n y_j|: \|(y_1,\cdots,y_j,\cdots,y_n)\|_1=1\}$.
Now we give a partial solution to Problem (i) and useful direction toward solving Problem (iii):
\begin{p.} Let $\|.\|$ be an algebra norm on $M_n$. Then $\|.\|$ is a g-ind norm on $\{A\in M_n: \|A\|=\|A^{-1}\|=1\}$.\end{p.}
\noindent{\bf Proof.} Put $\|x\|_1=\max\{\|C_{Ax}\|: \|A\|=1\}, \lambda^{-1}=\max\{|\displaystyle{\sum_{i=1}^n}x_i|: \|x\|_1=1\}$ and $\|x\|_2=\lambda\|C_x\|$.
Then we have $\|C_y\|_{1,2}=\max\{\|C_yx\|_2: \|x\|_1=1\}=\max\{|\displaystyle{\sum_{i=1}^n} x_i|\|y\|_2: \|x\|_1=1\}=\|y\|_2\lambda^{-1}=\|C_y\|$.
It follows that for each $y\in${\kh C}$^n$ there is some $x\in${\kh C}$^n$ such that $\|C_yx\|_2=\|C_y\|\|x\|_1=\|C_y\|\max\{\|C_{D x}\|: \|D\|=1\}.$
Now let $A$ be invertible and $\|A^{-1}\|=\|A\|=1$ and $z=A^{-1}C_yx$. Then $\lambda^{-1}\|Bz\|_2=\lambda^{-1}\|BA^{-1}C_yx\|_2=\lambda^{-1}\|Dx\|_2=\|C_{Dx}\|\leq \frac{1}{\|C_y\|}\|C_yx\|_2=\frac{1}{\|C_y\|}\|Az\|_2.$
Now choose $y$ so that $\|C_y\|=1$. Then $\|C_{Bz}\|\leq\|C_{Az}\|$ for all $B\in M_n$. This implies that $\|C_{Az}\|$ is an upper bound for the set $\{\|C_{Bz}\|: \|B\|=1\}$ and indeed $\|C_{Az}\|=\max\{\|C_{Bz}\|: \|B\|=1\}=\|z\|_1$. It follows that $\|A\|=1=\|C_{A(\frac{z}{\|z\|_1})}\|= \max\{\|C_{Au}\|: \|u\|_1=1\}=\max\{\|Au\|_2: \|u\|_1=1\}=\|A\|_{1,2}.\Box$
Let us now answer Question (ii).
\begin{p.} An induced norm $\|.\|_{1,2}$ is unitarily invariant if and only if so are $\|.\|_1$ and $\|.\|_2$.\end{p.}
{\bf Proof.} Let $U, V$ be unital operators and $A$ be an arbitrary operator on {\kh C}$^n$.\\ Suppose that $\|.\|_1$ and $\|.\|_2$ are unitarily invariant. Then $$\|UAV\|_{1,2}=\displaystyle{\max_{x\neq 0}}\frac{\|UAVx\|_2}{\|x\|_1}=\displaystyle{\max_{x\neq 0}}\frac{\|AVx\|_2}{\|x\|_1}=\displaystyle{\max_{y\neq 0}}\frac{\|Ay\|_2}{\|V^{-1}x\|_1}=\displaystyle{\max_{y\neq 0}}\frac{\|Ay\|_2}{\|y\|_1}=\|A_{1,2}.$$
Conversely, if $\|.\|_{1,2}$ is unitarily invariant, then $\|Ux\|_1=\max\{\|AUx\|_2: \|A\|_{1,2}\leq 1\}=\max\{\|Bx\|_2: \|U^{-1}B\|_{1,2}\leq 1\}=\max\{\|Bx\|_2: \|B\|_{1,2}\leq 1\}=\|x\|_1$ and $\|Ux\|_2=\frac{1}{\alpha}\|C_{Ux}\|=\frac{1}{\alpha}\|UC_x\|=\frac{1}{\alpha}\|UC_x\|=\frac{1}{\alpha}\|C_x\|=\|x\|_2.\Box$
Modifying the proof of Theorem 5.6.18 of [3], we obtain a similar useful result for g-ind norms:
\begin{t.} Let $\|.\|_1, \|.\|_2, \|.\|_3$ and $\|.\|_4$ be four given norms on {\kh C}$^n$ and $$R_{i,j}=\max\{\frac{\|x\|_i}{\|x\|_j}: x\neq 0\}, 1\leq i,j\leq 4.$$
Then $$\max\{\frac{\|A\|_{1,2}}{\|A\|_{3,4}}: A\neq 0\}=R_{2,4}R_{3,1}$$
In particual, $\max\{\frac{\|A\|_{1,1}}{\|A\|_{2,2}}: A\neq 0\}=\max\{\frac{\|A\|_{2,2}}{\|A\|_{1,1}}: A\neq 0\}=R_{1,2}R_{2,1}$.\end{t.}
\noindent{\bf Proof.} Let $A$ be a matrix and $x\neq 0$. Then $\frac{\|Ax\|_2}{\|x\|_1}=\frac{\|Ax\|_2}{\|Ax\|_4}.\frac{\|Ax\|_4}{\|x\|_3}.\frac{\|x\|_3}{\|x\|_1}$. Hence $\|A\|_{1,2}\leq R_{2,4}\|A\|_{3,4}R_{3,1}$. Thus $\frac{\|A\|_{1,2}}{\|A\|_{3,4}}\leq R_{2,4}R_{3,1}.$
There are vectors $y, z$ in {\kh C}$^n$ such that $\|y\|_2=\|z\|_2=1, \|y\|_2=R_{2,4}\|y\|_4$ and $\|z\|_3=R_{3,1}\|z\|_1$. By Proposition 1.15, there exists a vector $z_\circ\in$ {\kh C}$^n$ such that $|z_\circ^*x|\leq \|x\|_3$ and $z_\circ^*z=\|z\|_3$.\\ Put $A_\circ=yz_\circ$. Then $\frac{\|A_\circ z\|_2}{\|z\|_1}=\frac{\|yz^*_\circ z\|_2}{\|z\|_1}=\frac{\|y\|_2\|z\|_3}{\|z\|_1}=\|y\|_2R_{3,1}$. Hence $\|A_\circ\|_{1,2}\geq \frac{\|y\|_2}{\|y\|_4}R_{3,1}\|y\|_4=R_{2,4}.R_{3,1}\|y\|_4$. On the other hand, $\frac{\|A_\circ x\|_4}{\|x\|_3}=\frac{\|yz^*_\circ x\|_4}{\|x\|_3}=\frac{\|y\|_4|z^*_\circ x|}{\|x\|_3}\leq\|y\|_4$. Thus $\|A_\circ\|_{3,4}\leq \|y\|_4$. Hence $\frac{\|A_\circ\|_{1,2}}{\|A_\circ\|_{3,4}}\geq \frac{R_{2,4}R_{3,1}\|y\|_4}{\|y\|_4}=R_{2,4}R_{3,1}.\Box$
\begin{c.} (i) $\|.\|_{1,2}\leq\|.\|_{3,2}$ if and only if $\|.\|_1\geq\|.\|_3$,\\
(ii) $\|.\|_{1,2}\leq\|.\|_{1,4}$ if and only if $\|.\|_2\leq\|.\|_4$.\end{c.}
\noindent{\bf Proof.} (i) $\|.\|_{1,2}\leq\|.\|_{3,2}$ if and only if $\max\{\frac{\|A\|_{1,2}}{\|A\|_{3,2}}: A\neq 0\}=R_{2,2}R_{3,1}\leq 1$ and this if and only if $R_{3,1}\leq 1$ or equivalently $\|.\|_3\leq\|.\|_1$. The proof of (ii) is similar.$\Box$
The following corollary completely answers to Question (iv):
\begin{c.} $\|.\|_{1,2}=\|.\|_{3,4}$ if and only if there exists $\gamma>0$ such that $\|.\|_1=\gamma\|.\|_3$ and $\|.\|_2=\gamma\|.\|_4$.\end{c.}
\noindent{\bf Proof.} If $\|A\|_{1,2}=\|A\|_{3,4},$ then $R_{4,2}R_{1,3}=\max\{\frac{\|A\|_{3,4}}{\|A\|_{1,2}}: A\neq 0\}=1=\max\{\frac{\|A\|_{1,2}}{\|A\|_{3,4}}: A\neq 0\}=R_{2,4}R_{3,1}$. Hence $\max\{\frac{\|x\|_2}{\|x\|_4}: x\neq 0\}=R_{2,4}=\frac{1}{R_{3,1}}=\min\{\frac{\|x\|_1}{\|x\|_3}: x\neq 0\}\leq \max\{\frac{\|x\|_1}{\|x\|_3}: x\neq 0\}=R_{1,3}=\frac{1}{R_{4,2}}=\min\{\frac{\|x\|_2}{\|x\|_4}: x\neq 0\}$. Thus there exists a number $\gamma$ such that $\frac{\|x\|_2}{\|x\|_4}=\gamma=\frac{\|x\|_1}{\|x\|_3}.\Box$
\begin{r.}{\rm It is known that each induced norm $\|.\|$ is minimal in the sense that for any matrix norm $\|.\|$, the inequality $\|.\|\leq\|.\|_{1,1}$ implies that $\|.\|=\|.\|_{1,1}$. But this is not true for g-ind norms in general. For instance, put $\|.\|_\alpha=\ell_{\infty}(.), \|.\|_\beta=2\ell_{2}(.)$ and $\|.\|_\gamma=\ell_{2}(.)$. Then $\|.\|_{\gamma,\beta}\leq\|.\|_{\alpha,\beta}$ but $\|.\|_{\gamma,\beta}\neq\|.\|_{\alpha,\beta}$.} \end{r.}
The following theorem is one of our main theorems and provide a complete solution for Problem (v):
\begin{t.} Let $\|.\|_1$ and $\|.\|_2$ be two norms on {\kh C}$^n$. Then $\|.\|_{1,2}$ is an algebra norm on $M_n$ if and only if $\|.\|_1\leq\|.\|_2$.\end{t.}
\noindent{\bf Proof.} For each $A$ and $B$ in $M_n$ we have
\[\|ABx\|_{2}\leq\|A\|_{1,2}\|Bx\|_{1}\leq\|A\|_{1,2}\|Bx\|_{2}\leq\|A\|_{1,2}\|B\|_{1,2}\|x\|_{1}.\]
Hence $\|AB\|_{1,2}\leq\|A\|_{1,2}\|B\|_{1,2}.$
Conversely, let $\|.\|_{1,2}$ be an algebra norm. Then for each $A,B\in M_{n}$ we have $\Vert AB\|_{2}\leq\| A\|_{1,2}\| B\|_{1,2}\| x\|_{1}$. Let $B$ be an arbitrary member of $M_{n}$. For $Bx\neq 0$, take $M$ to be the linear span of $\{Bx\}$ and define $f:(M,\|.\|_1)\to${\kh C} by $f(cBx)=\frac{c\| Bx\|_{1}}{\| Bx\|_{2}}$. By the Hahn-Banach Theorem, there is an $F:(${\kh C}$^n,\|.\|_1)\to${\kh C} with $F|_{M}=f$ and $\| F\|=\| f\|=\max\{|f(cBx)|:\| cBx\|_{1}=1\}=\max\{\frac{|c|\| Bx\|_{1}}{\| Bx\|_{2}}:|c|\| Bx\|_{1}=1\}=\frac{1}{\| Bx\|_{2}}$. Define $A:(${\kh C}$^n, \|.\|_1)\to(${\kh C}$^n,\|.\|_2)$ by $Ay=F(y)Bx$. Then $\| A\|_{1,2}=\max\{\| Ay\|_{2}:\| y\|_{1}=1\}=\max\{|F(y)|\| Bx\|_{2}:\| y\|_{1}=1\}=1$, and $\| ABx\|_{2}=|F(Bx)|\| Bx\|_{2}=|f(Bx)|\| Bx\|_{2}=\frac{\| Bx\|_{1}}{\| Bx\|_{2}}\| Bx\|_{2}=\| Bx\|_{1}$. Thus for all $B$,
\[\| Bx\|_{1}=\| ABx\|_{2}\leq \| A\|_{1,2}\| B\|_{1,2}\| x\|_{1}=\| B\|_{1,2}\| x\|_{1},\] or
\[\| Bx\|_{1}\leq \| B\|_{1,2}\| x\|_{1}.\]
Now take $N$ to be the linear span of $\{x\}$ and define $g:(N,\|.\|_1)\to${\kh C} by $g(cx)=\frac{c\| x\|_{1}}{\| x\|_{2}}$. By the Hahn-Banach Theorem, there is a $G:(${\kh C}$^n,\|.\|_1)\to${\kh C} with $G|_{N}=g$ and $\| G\|=\| g\|=\max\{|g(cx)|:\| cx\|_{1}\}=\max\{\frac{|c|\| x\|_{1}}{\| x\|_{2}}:|c|\| x\|_{1}=1\}=\frac{1}{\| x\|_{2}}$. Define $B:(${\kh C}$^n,\|.\|_1)\to (${\kh C}$^n,\|.\|_2)$ by $By=G(y)x$. Then $\| B\|_{1,2}=\max\{\| By\|_{2}:\| y\|_{1}=1\}=\max\{|G(y)|\| x\|_{2}:\| y\|_{1}=1\}=\| x\|_{2}\| G\|=1$, and $\| Bx\|_{1}=|G(x)|\| x\|_{1}=|g(x)|\| x\|_{1}=\frac{\| x\|_{1}}{\| x\|_{2}}\| x\|_{1}=\frac{\| x\|_{1}^{2}}{\| x\|_{2}}$.
So
\[\frac{\| x\|_{1}^{2}}{\| x\|_{2}}=\| Bx\|_{1}\leq\| B\|_{1,2}\| x\|_{1}=\| x\|_{1}.\]
Thus $\|.\|_1\leq\|.\|_2.\Box$
\begin{p.} Suppose that $\|.\|_{1,2}$ is a g-ind norm and $\lambda>0$. Then the scaled norm $\lambda\|.\|_{1,2}$ is a g-ind algebra norm if and only if $\lambda\geq R_{1,2}$.\end{p.}
{\bf Proof.} Evidently, $\lambda\|.\|_{1,2}=\|.\|_{\|.\|_1,\lambda\|.\|_2}$. If $\|.\|_{3,4}=\lambda\|.\|_{1,2}=\|.\|_{\|.\|_1,\lambda\|.\|_2}$ then Corollary 2.5 implies that there exists $\alpha>0$ such that $\|.\|_3=\alpha\|.\|_1$ and $\|.\|_4=\alpha\lambda\|.\|_2$. Now Theorem 2.7 follows that $\lambda\|.\|_{1,2}=\|.\|_{3,4}$ is an algebra norm if and only if $\alpha\|.\|_1\leq\alpha\lambda\|.\|_2$ or equivalently $R_{1,2}\leq \lambda.\Box$
\begin{p.} Let $\|.\|_1$ and $\|.\|_2$ be two norms on {\kh C}$^n$ and $0\neq\alpha,\beta\in${\kh C}. Define $\|.\|_{\alpha}$ and $\|.\|_{\beta}$ on {\kh C}$^n$ by $\|x\|_{\alpha}=\|\alpha x\|_{1}$ and $\|x\|_{\beta}=\|\beta x\|_{2}$, respectively. Then $\|.\|_{\alpha}$ and $\|.\|_{\beta}$ are two norms on {\kh C}$^n$ and $\|.\|_{\alpha,\beta}=|\frac{\beta}{\alpha}|\|.\|_{1,2}$.\end{p.}
\noindent{\bf Proof.} We have $\|A\|_{\alpha,\beta}=\max\{\|Ax\|_{\beta}:\|x\|_{\alpha}=1\}=\max\{\|\beta Ax\|_{2}:\|\alpha x\|_{1}=1\}=|\frac{\beta}{\alpha}|\max\{\|Ay\|_{2}:\|y\|_{1}=1\}=|\frac{\beta}{\alpha}|\|A\|_{1,2}.\Box$
The preceding proposition leads us ti give the following definition:
\begin{d.}{\rm Let $(\|.\|_1,\|.\|_2$) and $(\|.\|_3,\|.\|_4)$ be two pairs of norms on {\kh C}$^n$. We say that $(\|.\|_1,\|.\|_2)$ is generalized induced congruent (gi-congeruent) to $(\|.\|_3,\|.\|_4)$ and we write $(\|.\|_1,\|.\|_2)\equiv_{gi}(\|.\|_3,\|.\|_4)$ if $\|.\|_{1,2}=\gamma\|.\|_{3,4}$ for some $0<\gamma\in${\kh R}.}\end{d.}
Clearly $\equiv_{gi}$ is an equivalence relation. We denote by $[(\|.\|_1,\|.\|_2)]_{gi}$ the equivalence class of $(\|.\|_1,\|.\|_2)$. Proposition 2.9 shows that for each $0<\alpha,\beta\in${\kh R} we have $(\alpha\|.\|_1,\beta\|.\|_2)\equiv_{gi}(\|.\|_1,\|.\|_2)$. Indeed, we have the following result:
\begin{t.} For each pair $(\|.\|_1,\|.\|_2)$ of norms on {\kh C}$^n$ we have $[(\|.\|_1,\|.\|_2)]_{gi}=\{(\alpha\|.\|_1,\beta\|.\|_2):0<\alpha,\beta\in${\kh R}$\}$.\end{t.}
We can extend the above method to find some other norms on $M_n$ which are not necessarily gi-congruent to a given pair $(\|.\|_1,\|.\|_2)$:
\begin{p.} Let $(\|.\|_1,\|.\|_2)$ be a pair of norms on {\kh C}$^n$ and $K,L\in$$M_n$ be two invertible matrices. Define $\|\|_{K}$ and $\|\|_{L}$ and {\kh C}$^n$ by $\|x\|_{K}=\|Kx\|_{1}$ and $\|x\|_{L}=\|Lx\|_{2}$. Then $\|\|_{K}$ and $\|\|_{L}$ are norms on {\kh C}$^n$ and $\|A\|_{K,L}=\|LAK^{-1}\|_{1,2}$.\end{p.}
\noindent{\bf Proof.} Clear and see also Lemma 3.1 of [4].$\Box$
\begin{r.}{\rm Note that the case $K=\alpha I$ and $L=\beta I$ gives Proposition 2.9.}\end{r.}
\end{document} | arXiv |
Blog of him
Laplace Transform
Inverse Laplace Transform
Z-Transfrom
Inverse Z-Transform
Modified Z-Transfrom
Starred Transform
Fourier Transform
State Space Representation
Continuous State Space Representation
Transfer function
Discrete State Space Representation
Controllability & Reachability
Decomposition and Realizations
Lyaponov Stability
Bounded-Input Bounded-Output Stability
Steady State Accuracy
Transient Response
Discretization and Linearization
Discretization Example
Sampling (A/D)
Reconstruction/Hold (D/A)
$s$-plane and $z$-plane
Linearization
Full State Feedback
State Estimation (Observer Design)
Notes for Control System
This note combines content from ME 564 Linear Systems & ME 561 Discrete Digital Control.
In this note, $f\in\mathbb{F}^\mathbb{G}$ stands for a function with domain in $\mathbb{G}$ and co-domain in $\mathbb{F}$, i.e. $f:\mathbb{F}\to\mathbb{G}$, $H(x)$ generally stands for Heaviside function (step function)
Please read the Algebra Basics notes first if you are not familiar with related concepts.
Definition: $F(s)=\mathcal{L}\{f(t)\}(s)=\int^\infty_0 f(t)e^{-st}\mathrm{d}t$
Note that the transform is not well defined for all functions in $\mathbb{C}^\mathbb{R}$. And the transform is only valid for $s$ in a region of convergence, which is usually separated by 0.
Laplace Transform is a linear map from $(\mathbb{C}^\mathbb{R}, \mathbb{C})$ to $(\mathbb{C}^\mathbb{C}, \mathbb{C})$ and it's one-to-one.
Properties: (see Wikipedia or this page for full list)
Derivative: $f'(t) \xleftrightarrow{\mathcal{L}} sF(s)-f(0^-)$
Integration: $\int^t_0 f(\tau)d\tau \xleftrightarrow{\mathcal{L}} \frac{1}{s}F(s)$
Delay: $f(t-a)H(t-a) \xleftrightarrow{\mathcal{L}} e^{-as}F(s)$
Convolution: $\int^t_0 f(\tau)g(t-\tau)\mathrm{d}\tau \xleftrightarrow{\mathcal{L}} F(s)G(s)$
Stationary Value: $\lim\limits_{t\to 0} f(t) = \lim\limits_{s\to \infty} sF(s), \lim\limits_{t\to \infty} f(t) = \lim\limits_{s\to 0} sF(s)$
Laplace transform is one-to-one, so we can apply inverse transform on functions in s-space
There are several ways to calculate Laplace transform, the first one is directly evaluating integration while the latter two are converting the function into certain formats that are convenient for table lookup:
(Mellin's) Inverse formula: $f(t)=\mathcal{L}^{-1}\{F(s)\}(t)=\frac{1}{2\pi j}\lim\limits_{T\to\infty} \int ^{\gamma+iT}_{\gamma-iT} e^{st}F(s)\mathrm{d}s$ where the integration is done along the vertical line $Re(s)=\gamma$ in the convex s-plane such that $\gamma$ is greater than the real part of all poles of $F(s)$.
Power Series: $F(s) = \sum^\infty_{n=0} \frac{n!a_n}{s^{n+1}}\xleftrightarrow{\mathcal{L}} f(t) = \sum ^\infty_{n=0} a_n t^n $
Partial Fractions: $F(s)=\frac{k_1}{s+a}+\frac{k_2}{s+b}+\ldots \xleftrightarrow{\mathcal{L}} f(t)=k_1 e^{-at} + k_2 e^{-bt} + \ldots$
To calculate partial fractions, one can use Polynomial Division or following lemma:
Suppose $F(s)=\frac{N(s)}{D(s)}=\frac{N(s)}{\prod^n_{i=1} (s-p_i)^{r_i}}$ where $\mathrm{deg}(N(s)) < \mathrm{deg(D(s))}$ and each $p_i$ is a distinct root of $D(s)$ (i.e. pole) with multiplicity $r_i$, then $F(s)=\sum^n_{i=1}\sum^{r_i}_ {j=1} \frac{k_{ij}}{(s-p_i)j}$ where $k_{ij}=\frac{1}{(r_i-j)!}\left.\frac{\mathrm{d}^{r_i-j}}{\mathrm{d}s^{r_i-j}}(s-p_i)^{r_i}F(s)\right\vert_{s=p_i}$
Definition: $F(z)=\mathcal{Z}\{f(k)_ {k\in\mathbb{N}}\}(z)=\sum^\infty_{k=0} f(k)z^{-k}$
Notice that $f$ is defined on natural numbers. In time domain, it's usually corresponding to $f(kT)$. Z-transform is also only valid for $z$ in certain region (usually separated by 1)
Laplace Transform is a linear map from $(\mathbb{C}^\mathbb{N}, \mathbb{C})$ to $(\mathbb{C}^\mathbb{C}, \mathbb{C})$ and it's one-to-one.
Accumulation: $\sum^n_{k=-\infty} f(k) \xleftrightarrow{\mathcal{Z}} \frac{1}{1-z^{-1}}F(z)$
Delay: $f(k-m) \xleftrightarrow{\mathcal{Z}} z^{-m}F(z)$
Convolution: $\sum^k_{n=0}f_1(n)f_2(k-n) \xleftrightarrow{\mathcal{Z}} F_1(z)F_2(z)$
Stationary Value: $\lim\limits_{t\to 0} f(t) = \lim\limits_{z\to \infty} F(z), \lim\limits_{t\to \infty} f(t) = \lim\limits_{z\to 1} (z-1)F(z)$
Example: Z-Transform of PID controller Assume the close-loop error input of the controller is $e(t)$, and $e(kT)$ after sampling. PID controller action in analog is $$m(t)=K\left(e(t)+\frac{1}{T_i}\int^t_0e(t)\mathrm{d}t+T_d\frac{\mathrm{d}e(t)}{\mathrm{d}t}\right)$$ We can approximate by trapezoidal rule with two point difference: $$m(kT)=K\left(e(kT)+\frac{T}{T_i}\sum^k_{h=1}\frac{e((h-1)T)+e(hT)}{2}+T_d\left(\frac{e(kT)-e((k-1)T)}{T}\right)\right)$$ Lets define $f(hT) = \frac{1}{2}\left(e((h-1)T)+e(hT)\right),\;f(0)=0$ Then $$\begin{split}\mathcal{Z}\left(\left\{\sum^k_{h=1}\frac{e((h-1)T)+e(hT)}{2}\right\}_k\right)(z)=\mathcal{Z}\left(\left\{\sum^k_{h=1}f(hT)\right\}_k\right)(z) \\ =\frac{1}{1-z^{-1}}(F(z)-F(0))=\frac{1}{1-z^{-1}}F(z)\end{split}$$ Notice that $$F(z)=\mathcal{Z}\left({f(hT)}_h\right)(z)=\frac{1+z^{-1}}{2}E(z)$$ so we can calculate the Z-transform of $m(kT)$ $$\begin{split} M(z)&=K\left(1+\frac{T}{2T_i}\left(\frac{1+z^{-1}}{1-z^{-1}}\right)+\frac{T_d}{T}(1-z^{-1})\right)E(z)\\&=K\left(1-\frac{T}{2T_i}+\frac{T}{T_i}\frac{1}{1-z^{-1}}+\frac{T_d}{T}(1-z^{-1})\right)E(z)\\&=\left(K_p+K_i\left(\frac{1}{1-z^-1}\right)+K_d(1-z^{-1})\right)E(z) \end{split}$$
Here we have
Proportional Gain $K_p=K-\frac{KT}{2T_i}$
Integral Gain $K_I=\frac{KT}{T_i}$
Derivative Gain $K_d=\frac{KT_d}{T}$
Inverse formula: $f(k)=\mathcal{Z}^{-1}\{F(z)\}(k)=\frac{1}{2\pi j}\oint _\Gamma z^{k-1}F(z)\mathrm{d}z$ where the integration is done along any closed path $\Gamma$ that encloses all finite poles of $z^{k-1}X(z)$ in the z-plane.
According to residual theorem, we can write it as $f(k)=\sum_{p_i}Res(z^{k-1}f(z), pi)$ where $p_i$ are poles of $z^{k-1}f(k)$ and residual $Res(g(z),p)=\frac{1}{(m-1)!}\left.\frac{\mathrm{d}^{m-1}}{\mathrm{d}z^{m-1}}\left((z-p)^mg(z)\right)\right\vert_{z=p}$ with $m$ being the multiplicity of the pole $p$ in $g$.
Power Series: same as inverse laplace.
Partial Fractions: same as inverse laplace.
Definition: $F(z,m)=\mathcal{Z}_m(f,m)=\mathcal{Z}(\left\{f(kT-(1-m)T)\right\} _{k\in\mathbb{N}^+})(z)$
We denote corresponding continuous form $\mathcal{L}(f(t-(1-m)T)\delta_ T(t))$ as $F^*(s,m)$
Residual Theorem: $\mathcal{Z}_m(f,m)=z^{-1}\sum _{p_i} Res(\frac{F(s)e^{mTs}}{1-z^{-1}e^{Ts}}, p_i)$
ModZ Transform is usually used when there's delay in the system, use this transform to shift the signal with proper $m$ value.
Definition: $F^* (s)=\sum^\infty_{n=0}f(n*T)e^{-nTs}$
Starred Transform is defined in continuous s-domain, but it only aggregates on discrete s values defined periodically by sampling time T, like Z-Transform. Starred Transform is usually exchangeable with Z-Transform with $z=e^{Ts}$.
Sometimes we also see * as an operator to sample a continuous signal. It converts a continuous signal to discrete delta functions. (See the "Sampler" section below)
Calculation from Laplace Transform
$F^*(s)=\sum_{p_i\in\{poles\;of\;F(\lambda)\}} Res\left(F(\lambda)\frac{1}{1-e^{-T(s-\lambda)}}, p_i\right)$
$F^*(s)=\frac{1}{T}\sum^\infty_{n=-\infty}F(s+jn\omega_s)+\frac{e(0)}{2}$ where $\omega_s=\frac{2\pi}{T}$
$F^*(s)$ is periodic in s plane with period $j\omega_s=\frac{2\pi j}{T}$
If $F(s)$ has a pole at $s=s_0$, then $F^*(s)$ must have poles at $s=s_0+jn\omega_s$ for $m\in\mathbb{Z}$
$A(s)=B(s)F^* (s) \Rightarrow A^* (s)=B^* (s)F^* (s)$, while usually $A(s)=B(s)F(s) \nRightarrow A^* (s)=B^* (s)F^* (s)$
Fourier transform is basically to substitute $s=j\omega$ into Laplace transform. Additional properties are not discussed here.
One important theorem (Shannon-Nyquist Sampling Theorem): Suppose $e:\mathbb{R}_+\to\mathbb{R}$ has a Fourier Transform with no frequency components greater than $f_0$, then $e$ is uniquely determined by the signal $e_s$ generated by ideally sampling $e$ with period $\frac{1}{2}f_0$.
A continuous-time linear state-space system can be described by following two equations: \begin{align}&\text{State equation}:\;&\dot{x}(t)&=A(t)x(t)+B(t)u(t),&\;x(t)\in\mathbb{R}^n,\;u(t)&\in\mathbb{R}^m \\&\text{Output equation}:\;&y(t)&=C(t)x(t)+D(t)u(t),&\;y(t)&\in\mathbb{R}^p\end{align}
The input $u:[0,\infty)\to\mathbb{R}^m$, state $x:[0,\infty)\to\mathbb{R}^n$, and output $y:[0,\infty)\to\mathbb{R}^p$ are all signals, i.e. functions of continuous time $t\in[0,\infty)$. The coefficients $A\in\mathbb{R}^{n\times n}$,$B\in\mathbb{R}^{n\times m}$,$C\in\mathbb{R}^{p\times n}$,$D\in\mathbb{R}^{p\times m}$
This linear time-varying (LTV) system can be written compactly as \begin{align*} \dot{x}&=A(t)x+B(t)u \\ y&=C(t)x+D(t)u\end{align*} Similarly, linear time-invariant (LTI) system can be written as \begin{align} \dot{x}&=Ax+Bu \\ y&=Cx+Du\end{align}
For non-linear system, the equation will be written as
time-varying (NLTV)
time-invariant (NTLI)
time-invariant autonomous
\begin{align*}\dot{x}&=f(x,u,t)\\y&=g(x,u,t)\end{align*}
\begin{align*}\dot{x}&=f(x,u)\\y&=g(x,u)\end{align*}
\begin{align*}\dot{x}&=f(x)\\y&=g(x)\end{align*}
Math prerequisites here:
For definition of function on matrix, see my notes for algebra basics
$e^A$ is matrix exponential, expm in MATLAB
$\frac{\mathrm{d}}{\mathrm{d}t}e^{At}=Ae^{At}=e^{At}A$
$e^{(A+B)t}\Leftrightarrow AB=BA$ (be careful when commute matrices)
$\mathcal{L}\{e^{At}\}=(sI-A)^{-1}$ (can be derived from property 1 and laplace derivative)
To calculate $e^A$
Eigenvalue decomposition
Jordan form decomposition
Directly evaluate infinite power series (converges quickly)
For more properties of the matrix function, seeMatrix Algebra
For homogeneous LTI system: $$\begin{align}x(t)=e^{A(t-t_0)}x_0\end{align}$$
"homogeneous" = zero-input, Eq.5 is also called zero input response (ZIR).
"homogeneous equation" = 齐次方程
For LTI system: $$\begin{align}x(t)=e^{A(t-t_0)}x(t_0)+\int^t_{t_0}e^{A(t-\tau)}Bu(\tau)d\tau\end{align}$$ This result requires $A$ to be time-invariant, $B,C,D$ can be time varying.
The solution consists of two parts: ZIR and ZSR (zero state response, $x(t_0)=0$), which are homogenenous solution (通解) and particular solution (特解) of the ODE.
ZIR and ZSR are both linear mapping
For homogeneous LTV system: $$\begin{align}x(t)=\Phi(t,t_0)x_0\end{align}$$
Matrix $\Phi$ is called the state transition matrix, defined as $$\begin{equation}\begin{split}\Phi(t,t_0)\equiv I+\int^t_{t_0}A(s_1)\mathrm{d}s_1+\int^t_ {t_0}A(s_1)\int^{s_1}_ {t_0}A(s_2)\mathrm{d}s_2\mathrm{d}s_1+\\ \int^t_ {t_0}A(s_1)\int^{s_1}_ {t_0}A(s_2)\int^{s_2}_ {t_0}A(s_3)\mathrm{d}s_3\mathrm{d}s_2\mathrm{d}s_1+\cdots\end{split}\end{equation}$$
Properties of $\Phi$:
$\Phi(t,t)=I$
$\frac{\mathrm{d}}{\mathrm{d}t}\Phi(t,t_0)=A(t)\Phi(t,t_0)$
(semigroup property) $\Phi(t,s)\Phi(s,\tau)=\Phi(t,\tau)$
$\forall t,\tau\geqslant 0,\;[\Phi(t,\tau)]^{-1}=\Phi(\tau,t)$
Eq.6 can be directly derived by evaluating Eq.8
For LTV system: $$\begin{align}x(t)=\Phi(t,t_0)x_0+\int^t_{t_0}\Phi(t,\tau)B(\tau)u(\tau)d\tau\end{align}$$
Some conclusions:
The solution given by Eq.9 is unique
The set of all solutions to ZIR system forms a vector space of dimension $n$
If $A(t)A(s)=A(s)A(t)$, then $\Phi(t,t_0)=e^{\int^t_{t_0}A(\tau)\mathrm{d}\tau}$
Phase Portraits: A phase portrait is a graph of several zero-input responses on the phase plane ($\dot{x}(t)$ and $x(t)$ are phase variables)
Usually in phase portraits, there are two straight lines corresponding to the eigenvector of A, other lines are growing in or opposite to the direction of the lines.
For LTI case, $\frac{Y(s)}{U(s)} = C(sI-A)^{-1}B+D$
This can be derived by take laplace transform of both sides of state equations
A discrete-time linear state-space system can be described by following two equations: $$\begin{align}&\text{State eq.}:\;&x(k+1)&=A(k)x(k)+B(k)u(k),&\;x\in\mathbb{R}^n,\;u&\in\mathbb{R}^m \\ &\text{Output eq.}:\;&y(k)&=C(k)x(k)+D(k)u(k),&\;y&\in\mathbb{R}^p\end{align}$$
The input $u:\mathbb{N}\to\mathbb{R}^m$, state $x:\mathbb{N}\to\mathbb{R}^n$, and output $y:\mathbb{N}\to\mathbb{R}^p$ are all signals, i.e. functions of continuous time $t\in\mathbb{N}$.
Discrete LTI system is sometimes written compactly as $$\begin{align} x_{k+1}&=Ax_k+Bu_k \\ y_k&=Cx_k+Du_k \end{align}$$
For LTI case, $H(z)=C(zI-A)^{-1}B+D$ (pulse tranfer function)
Note: hereafter $\mathfrak{R}$ denotes range space, $\mathfrak{N}$ denotes null space.
Controllability: $\exists u$ that drives any initial state $x(t_0)=x_0$ to $x(t_1)=0$
Reachability: $\exists u$ that drives initial state $x(t_0)=0$ to any $x(t_1)=x_1$
Consider the continuous LTV system $\dot{x}=A(t)x+B(t)u,\;x\in\mathbb{R}^n,u\in\mathbb{R}^m$.
Reachable Subspace: Given $t_0$ & $t_1$, the reachable subspace $\mathcal{R}[t_0, t_1]$ consists of all states $x_1$ for which there exists and input $u:[t_0, t_1]\to\mathbb{R}^m$ that transfers the state from $x(t_0)=0$ to $x(t_1)=x_1$.
$\mathcal{R}[t_0, t_1]\equiv\left\{x_1\in\mathbb{R}^n\middle|\exists u(\cdot),\;x_1=\int^{t_1}_{t_0}\Phi(t_1,\tau)B(\tau)u(\tau)\mathrm{d}\tau\right\}$
Controllable Subspace: Given $t_0$ & $t_1$, the controllable subspace $\mathcal{C}[t_0, t_1]$ consists of all states $x_0$ for which there exists an input $u:[t_0, t_1]\to\mathbb{R}^m$ that transfers the state from $x(t_0)=x_0$ to $x(t_1)=0$
$\mathcal{C}[t_0, t_1]\equiv\left\{x_0\in\mathbb{R}^m\middle|\exists u(\cdot),\;0=\Phi(t_1,t_0)x_0+\int^{t_1}_{t_0}\Phi(t_1,\tau)B(\tau)u(\tau)\mathrm{d}\tau\right\}$
or $\mathcal{C}[t_0, t_1]\equiv\left\{x_0\in\mathbb{R}^m\middle|\exists u(\cdot),\;x_0=\int^{t_1}_{t_0}\Phi(t_0,\tau)B(\tau)\left[-u(\tau)\right]\mathrm{d}\tau\right\}$
Reachability Grammian: $W_\mathcal{R}(t_0, t_1)\equiv\int^{t_1}_{t_0}\Phi(t_1,\tau)B(\tau)B(\tau)^\top\Phi^\top(t_1,\tau)\mathrm{d}\tau$ given times $t_1>t_0\geqslant0$
The system is reachable at time $t_0$ iff $\exists t_1$ s.t. $W_\mathcal{R}(t_0,t_1)$ is non-singular.
non-singular for some $t_1$ $\Rightarrow$ non-singular for any $t_1$
$\mathcal{R}[t_0,t_1]=\mathfrak{R}(W_\mathcal{R}(t_0,t_1))$
if $x_1=W_\mathcal{R}(t_0,t_1)\eta_1\in\mathfrak{R}(W_\mathcal{R}(t_0,t_1))$, the control $u(t)=B^\top(t)\Phi^T(t_1,t)\eta_1$,$t\in[t_0,t_1]$ can be used to transfer the system from $x(t_0)=0$ to $x(t_1)=x_1$ (w/ minimum energy)
minimum energy = minimum $\int^T_0\Vert u(\tau)\Vert^2\mathrm{d}\tau$
For LTI system $W_\mathcal{R}(t_0,t_1)=\int^{t_1}_ {t_ 0}e^{A(t_1-\tau)}BB^\top e^{A^{\top} (t_1-\tau)}\mathrm{d}\tau=\int^{t_1-t_ 0}_ {0}e^{At}BB^\top e^{A^{\top}t}$
Controllability Grammian: $W_\mathcal{C}(t_0, t_1)\equiv\int^{t_1}_{t_0}\Phi(t_0,\tau)B(\tau)B(\tau)^\top\Phi^\top(t_0,\tau)\mathrm{d}\tau$ given times $t_1>t_0\geqslant0$
The system is reachable at time $t_0$ iff $\exists t_1$ s.t. $W_\mathcal{C}(t_0,t_1)$ is non-singular.
$\mathcal{C}[t_0,t_1]=\mathfrak{R}(W_\mathcal{C}(t_0,t_1))$
if $x_0=W_\mathcal{C}(t_0,t_1)\eta_0\in\mathfrak{R}(W_\mathcal{C}(t_0,t_1))$, control $u(t)=-B^\top(t)\Phi^\top(t_0,t)\eta_0$,$t\in[t_0,t_1]$ can be used to transfer the state from $x(t_0)=x_0$ to $x(t_1)=0$ (w/ minimum energy)
For LTI system $W_\mathcal{C}(t_0,t_1)=\int^{t_1}_ {t_ 0}e^{A(t_0-\tau)}BB^\top e^{A^{\top} (t_0-\tau)}\mathrm{d}\tau=\int^{t_1-t_ 0}_ {0}e^{-At}BB^\top e^{-A^{\top}t}$
Controllability Matrix: For LTI system, controllability matrix $\mathcal{C}=[B\;|\;AB\;|\;A^2B\;\cdots\;A^{n-1}B]$
The controllability matrix works for both continuous and discrete system, and it's easier to be derived from discrete LTI equations: In discrete LTI, $\mathcal{C}\mathbf{u}=-A^k x_0$ where $\mathbf{u}=\begin{bmatrix}u_{k-1} & u_{k-2} & \ldots & u_0\end{bmatrix}^\top$
For LTI, $\mathcal{R}[t_0,t_1]=\mathfrak{R}(W_\mathcal{R}[t_0,t_1])=\mathfrak{R}(\mathcal{C})=\mathfrak{R}(W_\mathcal{C}[t_0,t_1])=\mathcal{C}[t_0,t_1]$
This implies Controllability $\Leftrightarrow$ Reachability for LTI systems.
The controllable subspace $\mathfrak{\mathcal{C}}$ is the smallest A-invariant subspace that contains $\mathfrak{\mathcal{B}}$
If the controllability matrix has full rank, the LTI system (or the pair $(A,B)$) is completely controllable
PBH-Eigenvector Test: An LTI system is not controllable iff there exists a nonzero left eigenvector $v$ of $A$ such that $vB=0$
PBH-Rank Test: An LTI system will be controllable iff $[\lambda I-A \;| \;B]$ has full row rank for all eigenvalue $\lambda$
For LTI system, there exists an input $u(\cdot)$ that transfer the state from $x_0$ ito $x_1$ in finite time $T$ iff $x_1-e^{AT}x_0\in\mathfrak{R}(\mathcal{C})$
The input that transfers any state $x_0$ to any other state $x_1$ in some finite time $T$ is $u(t)=B^\top e^{A^{\top}(T-t)}W_\mathcal{R}^{-1}(0,T)[x_1 -e^{AT}x_0]$, for $t\in[0,T]$ (w/ minimum energy)
Observability: Given any input $u(t)$ and output $y(t)$ over $t\in[t_0,t_1]$, it's sufficient to determine a unique initial state $\exists !x(t_0)$.
Observability Grammian: $W_\mathcal{O}(t_0,t_1)\equiv\int^{t_1}_{t_0}\Phi^\top(t_1,\tau)C^\top(\tau)C(\tau)\Phi(t_1,\tau)\mathrm{d}\tau$
The system is observable at time $t_0$ iff $\exists t_1$ s.t. $W_{\mathcal{O}}(0,t)$ is nonsingular.
For LTI system $W_{\mathcal{O}}(t_0,t_1)=\int^{t_1}_{t_0} e^{A^{\top}(t_1-\tau)}C^\top Ce^{A(t_1-\tau)}\mathrm{d}\tau=\int^{t_1-t_0}_0 e^{A^{\top}\tau}C^\top Ce^{A\tau}\mathrm{d}\tau$
Observability Matrix: For LTI system, observability $\mathcal{O}=\begin{bmatrix}C\\CA\\CA^2\\ \vdots\\CA^{k-1}\end{bmatrix}$
The controllability matrix works for both continuous and discrete system, and it's easier to be derived from discrete LTI equations: In discrete LTI, $\Psi_{k-1}=\mathcal{O}x_0$ where $$\Psi_k\equiv\begin{bmatrix}y_0\\y_1\\y_2\\ \vdots\\ y_{k-1}\end{bmatrix}-\begin{bmatrix} D & 0 & 0 & \cdots & 0 \\ CB & D & 0 & \cdots & 0 \\ CAB & CB & D & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ CA^{k-2}B & CA^{k-3}B & CA^{k-4}B & \cdots & 0 \end{bmatrix}\begin{bmatrix}u_0\\u_1\\u_2\\ \vdots \\ u_{k-1}\end{bmatrix}$$
If the controllability matrix has full rank, the LTI system (or the pair $(A,C)$) is completely observable.
PBH-Rank Test: An LTI system will be observable iff $\begin{bmatrix}A-\lambda I \\C\end{bmatrix}$ has full column rank for all eigenvalue $\lambda$
Duality Theorem: The pair $(A,B)$ is controllable iff the pair $(A^\top, B^\top)$ is observable.
Controllability only depends on matrix $A$ and $B$ while the Observability only depends on matrix $A$ and $C$
Duality theorem is useful for proof of observability conclusions from controllability
Adjoint System:
Original System
Adjoint System
Equations $$\begin{align*} \dot{x}&=A(t)x+B(t)u \\ y&=C(t)x \end{align*}$$ $$\begin{align*} \dot{p}&=-A^*(t)p-C^*(t)v \\ z&=B^*(t)p\end{align*}$$
Initial Condition $x(t_0)=x_0$ $p(t_1)=p_1$
State Trasition Matrix $\Phi(t,t_0)$ $\Phi^*(t_1,t)=\left(\Phi^*(t,t_1)\right)^{-1}$
Zero-State Response $$\begin{split}L_u:\;&u(\cdot)\to x(t_1)\\=&\int^{t_1}_{t_0}\Phi(t_1,\tau)B(\tau)u(\tau)\mathrm{d}\tau\end{split}$$ $$\begin{split}P_u:\;&v(\cdot)\to p(t_0)\\=&\int^{t_1}_{t_0}\Phi^*(\tau,t_0)C^*(\tau)v(\tau)\mathrm{d}\tau\end{split}$$
Zero-Input Response $$\begin{split}L_0:\;&x_0\to y(\cdot)\\=&C(\cdot)\Phi(\cdot,t_0)x_0\end{split}$$ $$\begin{split}P_0:\;&p_1\to z(\cdot)\\=&B^*(\cdot)\Phi^*(t_1,\cdot)p_1\end{split}$$
Duality Theorem Controllable ($\rho(L_u)=n$) Observable ($\rho(P_0^*)=n$)
Observable ($\rho(L_0^*)=n$) Controllable ($\rho(P_u)=n$)
A state is reachable ($x\in\mathfrak{R}(L_u)$) A state is unobservable ($x\in\mathfrak{N}(L_0)$)
Note that ZIR and ZSR are both linear mappings and $L_u^*=P_0,\;L_0^*=P_u$
Similarity Transform of a (LTI) system: Based on Eq.3 and Eq.4, define $x=P\bar{x}$, then we have $$\begin{align}\dot{\bar{x}}&=P^{-1}AP\bar{x}+P^{-1}Bu&=\bar{A}\bar{x}+\bar{B}u \\ y&=CP\bar{x}+Du&=\bar{C}\bar{x}+Du\end{align}$$
Similarity transform doesn't affect transfer function.
Controllability Decomposition: For an uncontrollable LTI system, define matrix $V=[V_1\;V_2]$ where $V_1$ is a basis for $\mathfrak{R}(\mathcal{C})$ and $V_2$ complete a basis for $\mathbb{R}^n$, then after similarity transform with $\bar{x}=V^{-1}x$, we can partition the system like following: $$\begin{align*}\dot{\bar{x}}&=\bar{A}\bar{x}+\bar{B}u&=&\begin{bmatrix}\bar{A}_ {11}&\bar{A}_ {12} \\ \mathbf{0} & \bar{A} _{22}\end{bmatrix}\begin{bmatrix}\bar{x} _1 \\ \bar{x} _2\end{bmatrix}+\begin{bmatrix}\bar{B} _1 \\ \mathbf{0}\end{bmatrix}u \\ y&=\bar{C}\bar{x}+Du&=& \begin{bmatrix}\bar{C} _1 & \bar{C} _2\end{bmatrix}\begin{bmatrix}\bar{x} _1 \\ \bar{x} _2\end{bmatrix} + Du\end{align*}$$ Here $\bar{x}_2$ is uncontrollable, and ZSR of the system with or without $\bar{x}_2$ is the same.
Observability Decomposition: For an unobservable LTI system, define matrix $U=\begin{bmatrix}U_1\\U_2\end{bmatrix}$ where $U_1$ is a basis for $\mathfrak{R}(\mathcal{O}^\top)$ and $U_2$ complete a basis for $\mathbb{R}^n$, then after similarity transform with $\hat{x}=Ux$, we can partition the system like following: $$\begin{align*}\dot{\hat{x}}&=\hat{A}\hat{x}+\hat{B}u&=&\begin{bmatrix}\hat{A}_ {11}&\mathbf{0} \\ \hat{A}_ {21} & \hat{A} _{22}\end{bmatrix}\begin{bmatrix}\hat{x} _1 \\ \hat{x} _2\end{bmatrix}+\begin{bmatrix}\hat{B} _1 \\ \hat{B} _2\end{bmatrix}u \\ y&=\hat{C}\hat{x}+Du&=& \begin{bmatrix}\hat{C} _1 & \mathbf{0}\end{bmatrix}\begin{bmatrix}\hat{x} _1 \\ \hat{x} _2\end{bmatrix} + Du\end{align*}$$
Realization: $\Sigma$ (system with Eq.3 and Eq.4) is a realization of $H(s)$ iff $H(s)=C(sI-A)^{-1}B+D$.
Equivalent: Two realizations are said to be equivalent if they have the same transfer function
Algebraically Equivalent: Two realizations have same transfer function and $n$ (dimension of states). (this implies a similarity transform between them)
Minimal Realization: $\Sigma$ is a minimal realization of $H(s)$ iff there doesn't exists an equivalent realization $\bar{\Sigma}$ with $\bar{n}< n$
$\Sigma$ is a minial realization iff $\Sigma$ is completely controllable and observable.
Kalman Cannonical Structure Theorem (aka. Kalman Decomposition): suppose $\rho(\mathcal{C})< n$ and $\rho(\mathcal{O})< n$, $\mathfrak{R}(\mathcal{C})$ are the controllable states, $\mathfrak{N}(\mathcal{O})$ are the unobservable states, define subspaces:
Subspaces
Controllable
$V_2\equiv\mathfrak{R}(\mathcal{C})\cup\mathfrak{N}(\mathcal{O})$ Yes No
$V_1$ s.t. $V_1\oplus V_2=\mathfrak{R}(\mathcal{C})$ Yes Yes
$V_4$ s.t. $V_4\oplus V_2=\mathfrak{N}(\mathcal{O})$ No No
$V_3$ s.t. $V_1\oplus V_2\oplus V_3\oplus V_4=\mathbb{C}^n$ No Yes
Then let $$\begin{align*}\tilde{x}=\begin{bmatrix}\tilde{x}_ 1\\ \tilde{x}_ 2\\ \tilde{x}_ 3\\ \tilde{x}_ 4\end{bmatrix},\;\tilde{A}=&\begin{bmatrix}A_ {\mathrm{co}} &&A_{13}&\\A_{21}&A_{\mathrm{c\bar{o}}}&A_{23}&A_{24}\\&&A_{\mathrm{\bar{c}o}}&\\&&A_{43}&A_{\mathrm{\bar{c}\bar{o}}} \end{bmatrix},\;\tilde{B}=\begin{bmatrix}B_{\mathrm{co}}\\ B_{\mathrm{c\bar{o}}} \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix} \\ \tilde{C}=&\begin{bmatrix}C_{\mathrm{co}}&\mathbf{0}\quad&C_{\mathrm{\bar{c}o}}&\mathbf{0}\quad\end{bmatrix}\end{align*}$$
$$\tilde{\Sigma}:\begin{cases} \dot{\tilde{x}}=A_{\mathrm{co}}\tilde{x}_ 1+B_{\mathrm{co}}u_1\\ y=C_{\mathrm{co}}\tilde{x}_1\end{cases}$$
$\tilde{\Sigma}$ is completely controllable and completely observable.
Consider SISO systems $$H(s)=\frac{b(s)}{a(s)}=\frac{b_{n-1}s^{n-1}+\ldots+b_1s+b_0}{s^n+a_{n-1}s^{n-1}+\ldots+a_1s+a_0}=\frac{\sum^{n-1}_{j=0} b_js^j}{s^n+\sum^{n-1}_{i=0} a_is^i}=\frac{Y(s)}{U(s)}$$
Controllable Cannonical Form: $$\begin{align}\dot{x}&=\begin{bmatrix} 0&1&&\\ \vdots & & \ddots & \\ 0&&&1 \\-a_0&-a_1&\cdots&-a_{n-1}\end{bmatrix}x+\begin{bmatrix}0\\ \vdots \\ 0 \\ 1\end{bmatrix}u&=&A_cx+B_cu\\ y&=\begin{bmatrix}\quad b_0 &\quad b_1 &\cdots & \quad b_{n-1}\end{bmatrix}x&=&C_cx \end{align}$$
$(A_c, B_c)$ is controllable
$(A_c, C_c)$ is observable if $a(s)$ and $b(s)$ have no common factors
Observable Cannonical Form: $$\begin{align}\dot{x}&=\begin{bmatrix} 0&&&&-a_0\\ 1 & \ddots &&&-a_1 & \\ &\ddots&\ddots&&\vdots \\&&\ddots&0&-a_{n-2} \\ &&&1&-a_{n-1}\end{bmatrix}x+\begin{bmatrix}b_0\\ b_1 \\ \vdots \\ b_{n-2} \\ b_{n-1}\end{bmatrix}u&=&A_ox+B_ou\\ y&=\begin{bmatrix}0 & \;\cdots &\;\cdots & 0 & \quad 1\qquad \end{bmatrix}x&=&C_ox \end{align}$$
$(A_o, C_o)$ is observable
$(A_o, B_o)$ is controllable if $a(s)$ and $b(s)$ have no common factors
Model Cannonical Forms: Do Spectral Decomposition (eigen-decomposition) or Jordan Decomposition, and then use the modal matrix (matrix of eigenvectors) to do similarity transform.
The Gilbert Realization: Let $G(s)$ be a $p\times m$ rational transfer function with simple poles (nonrepeated) at $\lambda_i,\;i=1,2,\ldots,k$. Calculate partial fraction expansion $$G(s)=\sum^k_{i=1}\frac{R_i}{s-\lambda_i},\qquad \text{Residue}\;R_i=\lim_{s\to\lambda_i}(s-\lambda_i)G(s)$$ Let $r_i=\rho(R_i)$, now write $R_i=C_iB_i$ where $C_ i\in\mathbb{R}^ {p\times r_ i},\;B_ i\in\mathbb{R}^ {r_ i\times p}$, then write $$A=\mathrm{blkdiag}\{\lambda_i I_{r_i}\},\;B^\top=[B_1^\top \;\cdots\; B^\top_k],\;C=[C_1\; \cdots\;C_k]$$, then $(A,B,C)$ is a realization of $G(s)$ with order $n=\sum^k_1 r_i$
For MIMO system the cannonical forms with be quite complex:
Controllable Cannonical Form (for MIMO): Here we provide a way to convert from controllable LTI system to controllable. The collection of independent columns of $\mathcal{C}$ may be expressed as $$M=[b_1\;Ab_1\; \cdots\;A^{\mu_1-1}b_1\;|\;b_2\;Ab_2\;\cdots\;A^{\mu_2-1}b_2\;|\;\cdots\;|\;b_p\;Ab_p\;\cdots\;A^{\mu_p-1}b_p]$$ Construct $M^{-1}$ and then $T$:$$M^{-1}=\left[m_{11}^\top\;m_{12}^\top\;\cdots\;m_{1\mu_1}^\top\;\middle|\;\cdots\;\middle|\;m_{p1}^\top\;m_{p2}^\top\;\cdots\;m_{p\mu_p}^\top \right]^\top$$ $$T=\left[m_{1\mu_1}^\top\;(m_{1\mu_1}A)^\top\;\cdots\;\left(m_{1\mu_1}A^{\mu_1-1}\right)^\top\;\middle|\;\cdots\;\middle|\; m_{p\mu_p}^\top\;(m_{p\mu_p}A)^\top\;\cdots\;\left(m_{p\mu_p}A^{\mu_p-1}\right)^\top\right]^\top$$ Perform similarity transform with $\bar{x}=Tx$ and the canonical form will be obtained like following: $$\bar{A}=\begin{bmatrix}\bar{A}_ {\mu_1\times\mu_1}&\mathbf{0}_ {\cdot\cdot}&\cdots&\mathbf{0}_ {\cdot\cdot} \\ \mathbf{0}_ {\cdot\cdot}&\bar{A}_ {\mu_2\times\mu_2}&\cdots&\mathbf{0}_ {\cdot\cdot}\\ \vdots&\vdots&\ddots&\vdots \\ \mathbf{0}_ {\cdot\cdot}&\mathbf{0}_ {\cdot\cdot}&\cdots&\bar{A}_ {\mu_ p\times\mu_ p} \end{bmatrix},\quad\bar{B}=\begin{bmatrix}\mathbf{0}_ {\cdot n}\\ \mathbf{0}_ {\cdot (n-1)}\\ \vdots\\ \mathbf{0}_ {\cdot 1}\end{bmatrix}$$ Here $\bar{A}_ {\mu_ i\times\mu_ i}$ is the same structure as in SISO, $\mathbf{0}_ {\cdot\cdot}$ is a zero matrix except the last row, $\mathbf{0}_ {\cdot i}$ is a zero matrix except for the last row, and in the last row there are $i$ non-zeros on the right with the first element being 1.
System Characteristic Equation: The polynomical with the roots equal to the poles of the output that are independent of the input.
System Type: A plant $G$ can always be written as $G(s)=\frac{K\prod^m_{i=1}(s-s_i)}{s^N\prod^p_{j=1}(s-s_j)},\;z_i,z_j\neq 0$ or $G(z)=\frac{K\prod^m_{i=1}(z-z_i)}{(z-1)^N\prod^p_{j=1}(z-z_j)},\;z_i,z_j\neq 1$. Here $N$ is called the system type of $G(z)$.
Properties that matters for a controller:
Exogenous disturbance rejection
Bounded control effort
Stability means when the time goes to infinity, the system response is bounded.
A system is stable if all its poles lies in the left half of $s$-plane or all inside the unit circle of $z$-plane.
A system is marginally stable if one of the pole is on the imaginary axis of $s$-plane or on the unit circle of $z$-plane.
Stability of linear systems is independent of input
The stability of a linear system can be evaluated by its characteristic equation $1-G_{op}(z)=0$, where $G_{op}$ is the open-loop transfer function (transfer function when input is eliminated, or feedback route is cut off).
Methods to evaluate stability
Routh-Hurwitz Criterion: $s$-plane (omited here, see Wikipedia)
For discrete system, a strategy is use bilinear transformation: $z=e^{\omega T}\approx \frac{1+\omega T/2}{1-\omega T/2}$
Jury Criterion: $z$-plane (see Wikipedia)
Root Locus Method: both $s$- and $z$-plane (see Wikipedia, rlocus in MATLAB)
Nyquist Criterion: both $s$- and $z$-plane (see Wikipedia, nyquist in MATLAB)
It works for both continuous and discrete systems, the difference is that in $s$-plane the detour point is at $s=0$ while in $z$-plane the detour point is at $z=1$.
Bode Diagrams: draw frequency response for (pulse) transfer function, works for both $s$- and $z$-plane (see Wikipedia, bode in MATLAB)
A review of the stability judgement method here at 知乎
Lyaponove Stability is only concerned with the effect of initial conditions on the response of the system (ZIR)
Equilibrium Point $x_e$: Consider NLTV $\dot{x}(t)=f(x(t),u(t),t)$, equilibrium point satisfies $x(t_0)=x_e,\;u(t)\equiv 0\Rightarrow x(t)=x_e,\;\text{i.e. }f(x_e,0,t)=0,\;\forall t>t_0$
For discrete system, it's $x(k+1)=x(k)=x_e$
For LTI system, $x_e$ can be calculated from $Ax_e=0$, so the origin $x=0$ is always an equilibrium point.
Set of equilibrium points in LTI systems are connected.
Lyapunov stability: An equilibrium point $x_e$ of the system $\dot{x}=A(t)x$ is stable (in the sense of Lyapunov) iff $\forall \epsilon>0,\;\exists \delta(t_0,\epsilon)>0$ s.t. $\Vert x(t_0)-x_e\Vert<\delta\Rightarrow\Vert x(t)-x_e\Vert <\epsilon,\;\forall t>t_0$
$x_e$ is uniformly stable if $\delta=\delta(\epsilon)$ (regardless of $t_0$)
$x_e$ in LTI is stable $\Rightarrow x_e$ is uniformly stable
$x_e$ is asymptotically stable if $\Vert x(t)-x_e\Vert\to 0$ as $t\to 0$
$x_e$ is exponentially stable if $\Vert x(t)-x_e\Vert \leqslant \gamma e^{-\lambda(t-t_0)}\Vert x(t_0)-x(e)\Vert$
$x_e$ in LTI is asymptotically stable $\Rightarrow x_e$ is exponentially stable
$x_e$ is globally stable if $\delta$ can be chosen arbitrarily large
For LTV system, the system is stable (the zero solution is stable) iff $\Phi(t,t_0)$ is bounded by $K(t_0)$.
If bounded by constant $K$, then the system is uniformly stable.
If bounded by constant $K$ and $\Vert\Phi(t,0)\Vert\to 0$ as $t\to 0$, then the system is asymptotically stable.
For LTI system $\dot{x}=Ax$, it is Lyapunov stable iff $\mathrm{Re}(\lambda_i)\leqslant 0$ or $\mathrm{Re}(\lambda_i)=0,\;\eta_i=1$. ($\eta_i$ is the multiplicity of $\lambda_i$)
If $\mathrm{Re}(\lambda_i)<0$, then the system is asymptotically stable
Internal stability: concerns the state variables
External stability: concerns the output variables
Notes for contents below:
Positive definite (pd.) function: function $V$ is pd. wrt. $p$ if $V(x)>0,\;x\neq p$ and $V(x)=0,\;x=p$
$C^n$ denotes the set of continuous and at least n-th differentiable functions
Lyapunov's Direct Method: Let $\mathcal{U}$ be an open neighborhood of $p$ and let $V:\mathcal{U}->\mathbb{R}$ be a countinuous positive definite $C^1$ function wrt. $p$, we have following two conclusions:
If $\dot{V}\leqslant 0$ on $\mathcal{U}\backslash\{p\}$ then $p$ is a stable fixed point of $\dot{x}=f(x)$
If $\dot{V}< 0$ on $\mathcal{U}\backslash\{p\}$ then $p$ is an asymptotically stable fixed point of $\dot{x}=f(x)$
Lyapunov Function:
A function satisfying conclusion 1 is called a Lyapunov function
A function satisfying conclusion 2 is called a strict Lyapunov function
A function that is $C^1$ and pd. is called a Lyapunov function candidate
The energy function usually can be used as Lyapunov function. If it's only semi-positive definite, one can use LaSalle's Theorem
For LTI system, the zero solution of $\dot{x}=Ax$ is asymptotically stable iff $\forall$ pd. hermitian matrices $Q$, equation $A^*P+PA=-Q$ has a unique hermitian solution $P$ that is positive definite.
$A^*P+PA=-Q$ is called Lyapunov's Matrix Equation
here $V(x)=x^* Px=\int^\infty_0 x^*(t)Qx(t)dt$, which can be also called cost-to-go, or generalized energy
Lyapunov's Indirect Method (Lyapunov's First Method / Lyapunov's Linearization Theorem): The nonlinear system $\dot{x}=f(x)$ is (locally) asymptotically stable near the equilibrium point $x_e$ if the linearized system $\dot{x}_L=\frac{\partial f}{\partial x}(x_e)x_L$ is asymptotically stable.
BIBO stability is only concerned with the response of the system to the input (ZSR).
Bounded-Input Bounded-Output (BIBO) stability: The LTV system is said to be (uniformly) BIBO stable if there exists a finite constant $g$ s.t. $\forall u(\cdot)$, its forced response $y_f(\cdot)$ satisfies $$ \sup_{t\in[0,\infty)}\Vert y_f(t)\Vert \leqslant g \sup_{t\in[0,\infty)} \Vert u(t)\Vert $$
The impulse response can be analyzed to assess BIBO stability The LTV system is uniformly BIBO stable iff every entry of $D(t)$ is bounded and $\sup_{t\geqslant 0}\int^t_0|g_{ij}(t,\tau)|d\tau <\infty$ for every entry $g_{ij}$ of the matrix $C(t)\Phi(t,\tau)B(\tau)$.
BIBO stability is related with the stability descibed in classical control theory.
Exponential Lyapunov Stability $\Rightarrow$ BIBO stability
Steady state accurary can be derived from the property of Laplace/Z-transform as mentioned above (assuming stability) $$\lim\limits_{t\to \infty} f(t) = \lim\limits_{z\to 1} (z-1)F(z) = \lim\limits_{s\to 0}sF(s)$$
Some measurements of transient response (with step input):
Rise time $t_r$: time from 10% to 90% of steady state value
Peak overshoot: $M_p$ for overshoot magnitude and $t_p$ for time
Settling time $t_s$: time after which the magnitude fall in $1-d$ to $1-d$ final value. $d$ is usually %2~5.
Given a transfer function $H(z)$ with parameter $\Theta\in\mathbb{R}$, then sensitivity is defined as $S_H=\frac{\partial H}{\partial \Theta}\cdot\frac{\Theta}{H} = \frac{\partial H/H}{\partial \Theta/\Theta}$
The following image shows a minial example of sampling and hold.
Ideal sampler (a.k.a impulse modulator) converts a continuous signal $e: \mathbb{R}_+ \to \mathbb{R}$ to a discrete one $\hat{e}: \mathbb{N}\to \mathbb{R}$, such that $$ \hat{e}=e(t)\delta(t-kT)=e(t)\delta_T(t); \forall k\in \mathbb{N} $$
Ideal sampler is actually applying starred transform.
How to sample? A rule of thumb used to select sampling rates is chosing a rate of at least 5 samples per time constant.
The $\tau$ appearing in the transient response term $ke^{-t/\tau}$ of a first order analog system is called the time constant.
If the sampling time is too large, it can make the system unstable.
Zero order hold (ZOH): $ZOH(\{e(k)\}_{k\in\mathbb{N}})(t) = e(k)\;for\;kT\leq t\leq (k+1)T$
Alternative form: $ZOH(\{e(k)\})=\sum^\infty_{k=0}e(k)(H(t-kT)-H(t-(k+1)T))$
Its Laplace Transform: $G_{ZOH}(s)=\frac{1-e^{-Ts}}{s}$
First order hold (FOH): (delayed version) $$FOH(\{e(k)\}_ {k\in\mathbb{N}})(t)=\sum_ {k\in\mathbb{N}}\left[e(kT)+\frac{t-kT}{T}(e(kT)-e((k-1)T)) \right]\left[H(t-kT)-H(t-(k+1)T) \right]$$
Suppose we are given the LTI continuous system $$\begin{align*} \dot{x}(t) &= Ax(t)+Bu(t) \\ y(t)&=Cx(t)+Du(t) \end{align*}$$ If the input is sampled and ZOH and the output is sampled, then $$\begin{align*} x(k+1)&=\bar{A}x(k)+\bar{B}u(k) \\ y(k)&=\bar{C}x(k)+\bar{D}u(k)\end{align*}$$ where $\bar{A}=\Phi((k+1)T,kT)=e^{AkT}$
$\bar{B}=\int^{(k+1)T}_{kT}\Phi((k+1)T,\tau)B\mathrm{d}\tau=A^{-1}(e^{AT}-I)$
$\bar{C}=C$ and $\bar{D}=D$
Steps to apply conversion:
Dervice SS model for analog system
Calculate discrete representation (c2d in MATLAB)
Calculate pulse transfer function (ss2tf in MATLAB) If there is a complex system with multiple sampling and holding, a general rule is
Each ZOH output is assumed to be an input
Each sampler input is assumed to be an output and then create continuous state space from analog part of the system, then discretize them to generate discrete equations
When converting $s$ to $z$, the complex variables are related by $z=e^{Ts}$. Suppose $s=\sigma+j\omega$, then $z=e^{T\sigma}\angle \omega T$
Note: if frequencies differ in integer multiples of the sampling frequency $\frac{2\pi}{T}=\omega_s$, then they are sampled into the same location in the $z$-plane.
For transient response relationship, suppose $s$-plane poles occur at $s=\sigma\pm j\omega$, then the transient response if $Ae^{\sigma t}\cos(\omega t+\varphi)$. When sampling occurs at $z$-plane poles, then the transient response if $Ae^{\sigma kT}\cos(\omega kT+\varphi)$.
Example: 2nd order transfer function $$G(s)=\frac{\omega_n^2}{s^2+2\xi\omega_ns+\omega_n^2}$$ The $z$-plane poles occur at $z=r\angle\pm\theta$ where $r=e^{-\xi\omega_n T}$ and $\theta=\omega_n T\sqrt{1-\xi^2}$.
Then we can get the inverse relationship
$\xi=-\ln( r)/\sqrt{\ln^2( r)+\theta^2}$
$\omega_n=(1/T)\sqrt{\ln^2( r)+\theta^2}$
(time constant) $\tau=-T/\ln( r)$
Jacobian Linearization: linearize $\dot{x}=f(x,u)$ at an equilibrium $(x_e, u_e)$ is $$\frac{\mathrm{d}z}{\mathrm{d}t}=Az+Bv,\quad\text{where}\;A=\left.\frac{\mathrm{d}f}{\mathrm{d}x}\right|_ {\begin{split}x=x_e\\u=u_e\end{split}},\;B=\left.\frac{\mathrm{d}f}{\mathrm{d}u}\right|_ {\begin{split}x=x_e\\u=u_e\end{split}},\;z=(x-x_e),\;v=(u-u_e)$$
change $(x_e, u_e)$ to a trajectory $(x_e(t), u_e(t))$ we can linearize the system about a trajectory.
This method can be used for both continuous and discrete systems, just make sure to use corresponding method for choosing correct closed-loop transfer function.
For state space systems, with access to all of the state variables, we can change the $A$ matrix and thereby change the system dynamics by feedback.
Consider SISO LTI system ($u\in\mathbb{R},y\in\mathbb{R}$), we define the input as $u\equiv Kx+Ev$ where $K\in\mathbb{R}^{1\times n},\;E\in\mathbb{R}$ is an input matrix and $v(t)\in\mathbb{R}^\mathbb{R}$ is the exogeneous (externally applied) input. The new system will be $$\begin{align}\dot{x}&=(A+BK)x+BEv\\y&=(C+DK)x+DEv\end{align}$$ The mission is to find a state update matrix $A_{\mathrm{CL}}\equiv A+BK$ with desired set of eigenvalues, therefore we can construct $A_{\mathrm{CL}}$ with specific eigenvalues and then calculate $K$. This process will be quite easy if the system is already in controllable cannonical form. (which can be constructed directly from transfer function or using similarity transform)
Another way (SISO only) to calculate $K$ without controllable cannonical form is using the following formulae given the desired characteristic polynomial $\phi^{\star}(s)=s^n+\sum^{n-1}_ {i=0} a^\star_i s^i$ and original characteristic polynomial $\phi(s)=s^n+\sum^{n-1}_ {i=0} a_i s^i$
Ackermann's Formula: $K=-e^\top_n\mathcal{C}^{-1}\phi^\star(A)$ (here $e_i$ is unit vector with 1 at i-th position)
Bass-Gura's Formula: $$K=-[(a^\star_{n-1}-a_{n-1}) \;\cdots\;(a^\star_0-a_0)]\begin{bmatrix}1&a_{n-1}&a_{n-2}&\cdots&a_1\\&1&a_{n-1}&\cdots&a_2\\ &&\ddots&\ddots&\vdots \\ &&&1&a_{n-1}\\ &&&&1\end{bmatrix}^{-1}\mathcal{C}^{-1}$$
Note that the zeros of transfer function will not be affected by state feedback.
Some times we don't have the direct access to the state, we need construct an observer. For stochastic version, please check my notes for stochastic system.
Assume a plant $\Sigma$ and an (Luenberger) observer $\hat{\Sigma}$: $$\Sigma:\begin{cases}\dot{x}=Ax+Bu\\ y=Cx\end{cases},\quad \hat{\Sigma}:\begin{cases} \dot{\hat{x}}=A\hat{x}+Bu+L(y-\hat{y})\\ y=C\hat{x}\end{cases}$$
Subtract observer dynamics from plant dynamics and define $e\equiv x-\hat{x}$, the dynamics for $e$ is $\dot{e}=(A-LC)e$ and $y-\hat{y}=Ce$. This error dynamic $A_e=A-LC$ can be easily changed with observable cannonical form. (which similarly can be constructed directly from transfer function or using similarity transform)
Reduced-order Observer: If the state length of the system $n$ is large while $n-p$ is small, split the system and let $x_1$ holds the states that can be measured directly while $x_2$ holds states that are to be estimated, (i.e. $y=x_1+Du$). Define $z=\hat{x}_2-Lx_1$ then the system runs like $$\begin{align*}\begin{bmatrix}x_1 \\ \hat{x}_2\end{bmatrix}&=\begin{bmatrix} y-Du\\ z+Lx_1 \end{bmatrix}\qquad\begin{split}&\text{measurement} \\ &\text{observer}\end{split} \\ u&=K\begin{bmatrix}x_1 \\ \hat{x}_2 \end{bmatrix} + v \qquad\text{control law}\end{align*}$$ And then the error we care about is only $e=x_2-\hat{x}_2$.
Ackermann's Formula: $L=\phi^\star(A)\mathcal{O}^{-1}e_n$ ($\phi^\star$ is the desired characteristic function for $A_e$)
Separation Principle: If a stable observer and stable state feedback are designed for an LTI system, then the combined observer and feedback will be stable.
Errors from state estimation
Inaccurate knowledge of $A$ and $B$
Initial condition uncertainty
Disturbance or sensor error It's advised to choose observer poles to be 2-4x faster than closed loop poles
Motivation: handle control constraints and time varying dynamics with performance metric (ideas of optimal control) Note: $x^\top Ax$ is called a quadratic form, $x^\top Ay$ is called a bilinear form
A quadratic function $f(x)=x^\top Dx+C^\top x+c_0$ has one minimizer iff $D\succ 0$, or multiple minimizers iff $D\succeq 0$.
(discrete finite time) Linear Quadratic Regulator (LQR): the control problem is defined as $$\begin{align*}\min_{u\in\left(\mathbb{R}^m\right)^{\{0,\ldots,N\}}} J_{N}(u,x_0)&=\frac{1}{2}\sum^{N}_{k=0}(x^\top(k)Q(k)x(k)+u^\top(k)R(k)u(k)) \\ \mathrm{s.t.}\qquad x(k+1) &= A(k)x(k) + B(k)u(k)\quad \forall k\in\{0,\ldots,N-1\}\\ y(k)&=C(k)x(k)\\ x(0)&=x_0\end{align*}$$ where $Q(k)\succ 0$ and $R(k)\succ 0$
Bellman's Principle of Optimality: If a closed loop control $u^\star$ is optimal over the interval $0\leqslant k\leqslant N$, it's also optimal over any subinterval $m\leqslant k\leqslant N$ where $m\in\{0,\ldots,N\}$
The Minimum Principle: The optimal input to the LQR problem satisfies the following backward equations: $$\begin{align*}u^\star(k)&=-K(x)x(k) \\ K(k)&=\left[B^\top(k) P(k+1)B(k)+\frac{1}{2}R(k)\right]^{-1}B^\top(k)P(k+1)A(k) \\ P(k)&=A^\top(k)P(k+1)[A(k)- B(k)K(k)]+\frac{1}{2}Q(k)\end{align*}$$ and $P(N)=Q(N),\;K(N)=0$. The optimal cost is $J^\star_N=x^\top(0)P(0)x(0)$
For infinite horizon, $K(k)$ start becoming constants. The optimal input for LQR problem (assuming the system became LTI when $N\to\infty$) is $u^*(k)=-Kx(k)$ where $$K=(B^\top PB+R/2)^{-1}B^\top PA$$ and $P\succ 0$ is the unique solution to the discrete-time algebraic Riccati Equation: $$P=A^\top PA-A^\top PB\left(B^\top PB+R/2\right)^{-1}B^\top PA+Q/2$$
#Math#Control
Notes for Algebra Basics
Notes for Stochastic System
Notes for Probability Theory (Basics)
Theme Stacked designed by Jimmy, modified by Jacob | CommonCrawl |
What is one-way ANOVA?
One-way analysis of variance (ANOVA) is a statistical method for testing for differences in the means of three or more groups.
How is one-way ANOVA used?
One-way ANOVA is typically used when you have a single independent variable, or factor, and your goal is to investigate if variations, or different levels of that factor have a measurable effect on a dependent variable.
What are some limitations to consider?
One-way ANOVA can only be used when investigating a single factor and a single dependent variable. When comparing the means of three or more groups, it can tell us if at least one pair of means is significantly different, but it can't tell us which pair. Also, it requires that the dependent variable be normally distributed in each of the groups and that the variability within groups is similar across groups.
One-way ANOVA is a test for differences in group means
One-way ANOVA is a statistical method to test the null hypothesis (H0) that three or more population means are equal vs. the alternative hypothesis (Ha) that at least one mean is different. Using the formal notation of statistical hypotheses, for k means we write:
$ H_0:\mu_1=\mu_2=\cdots=\mu_k $
$ H_a:\mathrm{not\mathrm{\ }all\ means\ are\ equal} $
where $\mu_i$ is the mean of the i-th level of the factor.
Okay, you might be thinking, but in what situations would I need to determine if the means of multiple populations are the same or different? A common scenario is you suspect that a particular independent process variable is a driver of an important result of that process. For example, you may have suspicions about how different production lots, operators or raw material batches are affecting the output (aka a quality measurement) of a production process.
To test your suspicion, you could run the process using three or more variations (aka levels) of this independent variable (aka factor), and then take a sample of observations from the results of each run. If you find differences when comparing the means from each group of observations using an ANOVA, then (assuming you've done everything correctly!) you have evidence that your suspicion was correct—the factor you investigated appears to play a role in the result!
A one-way ANOVA example
Let's work through a one-way ANOVA example in more detail. Imagine you work for a company that manufactures an adhesive gel that is sold in small jars. The viscosity of the gel is important: too thick and it becomes difficult to apply; too thin and its adhesiveness suffers. You've received some feedback from a few unhappy customers lately complaining that the viscosity of your adhesive is not as consistent as it used to be. You've been asked by your boss to investigate.
You decide that a good first step would be to examine the average viscosity of the five most recent production lots. If you find differences between lots, that would seem to confirm the issue is real. It might also help you begin to form hypotheses about factors that could cause inconsistencies between lots.
You measure viscosity using an instrument that rotates a spindle immersed in the jar of adhesive. This test yields a measurement called torque resistance. You test five jars selected randomly from each of the most recent five lots. You obtain the torque resistance measurement for each jar and plot the data.
Figure 1: Plot of torque measurements by lot
From the plot of the data, you observe that torque measurements from the Lot 3 jars tend to be lower than the torque measurements from the samples taken from the other lots. When you calculate the means from all your measurements, you see that the mean torque for Lot 3 is 26.77—much lower than the other four lots, each with a mean of around 30.
Table 1: Mean torque measurements from tests of five lots of adhesive
1 5 29.65
The ANOVA table
ANOVA results are typically displayed in an ANOVA table. An ANOVA table includes:
Source: the sources of variation including the factor being examined (in our case, lot), error and total.
DF: degrees of freedom for each source of variation.
Sum of Squares: sum of squares (SS) for each source of variation along with the total from all sources.
Mean Square: sum of squares divided by its associated degrees of freedom.
F Ratio: the mean square of the factor (lot) divided by the mean square of the error.
Prob > F: the p-value.
Table 2: ANOVA table with results from our torque measurements
Mean Square
F Ratio
Prob > F
Lot 4 45.25 11.31 6.90 0.0012
Error 20 32.80 1.64
Total 24 78.05
We'll explain how the components of this table are derived below. One key element in this table to focus on for now is the p-value. The p-value is used to evaluate the validity of the null hypothesis that all the means are the same. In our example, the p-value (Prob > F) is 0.0012. This small p-value can be taken as evidence that the means are not all the same. Our samples provide evidence that there is a difference in the average torque resistance values between one or more of the five lots.
What is a p-value?
A p-value is a measure of probability used for hypothesis testing. The goal of hypothesis testing is to determine whether there is enough evidence to support a certain hypothesis about your data. Recall that with ANOVA, we formulate two hypotheses: the null hypothesis that all the means are equal and the alternative hypothesis that the means are not all equal.
Because we're only examining random samples of data pulled from whole populations, there's a risk that the means of our samples are not representative of the means of the full populations. The p-value gives us a way to quantify that risk. It is the probability that any variability in the means of your sample data is the result of pure chance; more specifically, it's the probability of observing variances in the sample means at least as large as what you've measured when in fact the null hypothesis is true (the full population means are, in fact, equal).
A small p-value would lead you to reject the null hypothesis. A typical threshold for rejection of a null hypothesis is 0.05. That is, if you have a p-value less than 0.05, you would reject the null hypothesis in favor of the alternative hypothesis that at least one mean is different from the rest.
Based on these results, you decide to hold Lot 3 for further testing. In your report you might write: The torque from five jars of product were measured from each of the five most recent production lots. An ANOVA analysis found that the observations support a difference in mean torque between lots (p = 0.0012). A plot of the data shows that Lot 3 had a lower mean (26.77) torque as compared to the other four lots. We will hold Lot 3 for further evaluation.
Remember, an ANOVA test will not tell you which mean or means differs from the others, and (unlike our example) this isn't always obvious from a plot of the data. One way to answer questions about specific types of differences is to use a multiple comparison test. For example, to compare group means to the overall mean, you can use analysis of means (ANOM). To compare individual pairs of means, you can use the Tukey-Kramer multiple comparison test.
One-way ANOVA calculation
Now let's consider our torque measurement example in more detail. Recall that we had five lots of material. From each lot we randomly selected five jars for testing. This is called a one-factor design. The one factor, lot, has five levels. Each level is replicated (tested) five times. The results of the testing are listed below.
Table 3: Torque measurements by Lot
To explore the calculations that resulted in the ANOVA table above (Table 2), let's first establish the following definitions:
$n_i$ = Number of observations for treatment $i$ (in our example, Lot $i$)
$N$ = Total number of observations
$Y_{ij}$ = The jth observation on the ith treatment
$\overline{Y}_i$ = The sample mean for the ith treatment
$\overline{\overline{Y}}$ = The mean of all observations (grand mean)
With these definitions in mind, let's tackle the Sum of Squares column from the ANOVA table. The sum of squares gives us a way to quantify variability in a data set by focusing on the difference between each data point and the mean of all data points in that data set. The formula below partitions the overall variability into two parts: the variability due to the model or the factor levels, and the variability due to random error.
$$ \sum_{i=1}^{a}\sum_{j=1}^{n_i}(Y_{ij}-\overline{\overline{Y}})^2\;=\;\sum_{i=1}^{a}n_i(\overline{Y}_i-\overline{\overline{Y}})^2+\sum_{i=1}^{a}\sum_{j=1}^{n_i}(Y_{ij}-\overline{Y}_i)^2 $$
$$ SS(Total)\; = \;SS(Factor)\; + \;SS(Error) $$
While that equation may seem complicated, focusing on each element individually makes it much easier to grasp. Table 4 below lists each component of the formula and then builds them into the squared terms that make up the sum of squares. The first column of data ($Y_{ij}$) contains the torque measurements we gathered in Table 3 above.
Another way to look at sources of variability: between group variation and within group variation
Recall that in our ANOVA table above (Table 2), the Source column lists two sources of variation: factor (in our example, lot) and error. Another way to think of those two sources is between group variation (which corresponds to variation due to the factor or treatment) and within group variation (which corresponds to variation due to chance or error). So using that terminology, our sum of squares formula is essentially calculating the sum of variation due to differences between the groups (the treatment effect) and variation due to differences within each group (unexplained differences due to chance).
Table 4: Sum of squares calculation
$Y_{ij} $
$\overline{Y}_i $
$\overline{\overline{Y}}$
$\overline{Y}_i-\overline{\overline{Y}}$
$Y_{ij}-\overline{\overline{Y}}$
$Y_{ij}-\overline{Y}_i $
$(\overline{Y}_i-\overline{\overline{Y}})^2 $
$(Y_{ij}-\overline{Y}_i)^2 $
$(Y_{ij}-\overline{\overline{Y}})^2 $
29.39 29.65 29.33 0.32 0.06 -0.26 0.10 0.07 0.00
31.51 29.65 29.33 0.32 2.18 1.86 0.10 3.46 4.75
27.63 29.65 29.33 0.32 -1.70 -2.02 0.10 4.08 2.89
27.16 26.77 29.33 -2.56 -2.17 0.39 6.55 0.15 4.71
26.63 26.77 29.33 -2.56 -2.70 -0.14 6.55 0.02 7.29
25.31 26.77 29.33 -2.56 -4.02 -1.46 6.55 2.14 16.16
SS (Factor) = 45.25 SS (Error) = 32.80 SS (Total) = 78.05
Degrees of Freedom (DF)
Associated with each sum of squares is a quantity called degrees of freedom (DF). The degrees of freedom indicate the number of independent pieces of information used to calculate each sum of squares. For a one factor design with a factor at k levels (five lots in our example) and a total of N observations (five jars per lot for a total of 25), the degrees of freedom are as follows:
Table 5: Determining degrees of freedom
Degrees of Freedom (DF) Formula
Calculated Degrees of Freedom
SS (Factor)
k - 1 5 - 1 = 4
SS (Error)
N - k 25 - 5 = 20
SS (Total)
N - 1 25 - 1 = 24
Mean Squares (MS) and F Ratio
We divide each sum of squares by the corresponding degrees of freedom to obtain mean squares. When the null hypothesis is true (i.e. the means are equal), MS (Factor) and MS (Error) are both estimates of error variance and would be about the same size. Their ratio, or the F ratio, would be close to one. When the null hypothesis is not true then the MS (Factor) will be larger than MS (Error) and their ratio greater than 1. In our adhesive testing example, the computed F ratio, 6.90, presents significant evidence against the null hypothesis that the means are equal.
Table 6: Calculating mean squares and F ratio
Sum of Squares (SS)
Mean Squares
45.25 4 45.25/4 = 11.31 11.31/1.64 = 6.90
32.80 20 32.80/20 = 1.64
The ratio of MS(factor) to MS(error)—the F ratio—has an F distribution. The F distribution is the distribution of F values that we'd expect to observe when the null hypothesis is true (i.e. the means are equal). F distributions have different shapes based on two parameters, called the numerator and denominator degrees of freedom. For an ANOVA test, the numerator is the MS(factor), so the degrees of freedom are those associated with the MS(factor). The denominator is the MS(error), so the denominator degrees of freedom are those associated with the MS(error).
If your computed F ratio exceeds the expected value from the corresponding F distribution, then, assuming a sufficiently small p-value, you would reject the null hypothesis that the means are equal. The p-value in this case is the probability of observing a value greater than the F ratio from the F distribution when in fact the null hypothesis is true.
Figure 2: F distribution | CommonCrawl |
\begin{document}
\title{Zappa-Sz\'{e}p actions of groups on product systems}
\author{Boyu Li} \address{Department of Mathematics and Statistics, University of Victoria, Victoria, B.C. V8W 3R4} \email{[email protected]}
\author{Dilian Yang} \address{Department of Mathematics and Statistics, University of Windsor, Windsor, ON. N9B 3P4} \email{[email protected]} \date{\today}
\thanks{The first author was supported by a fellowship of Pacific Institute for the Mathematical Sciences. The second author was partially supported by an NSERC Discovery Grant 808235.}
\subjclass[2010]{46L55, 46L05, 20M99.} \keywords{Zappa-Sz\'{e}p action, product system, right LCM semigroup, Nica-Toeplitz representation}
\begin{abstract}
Let $G$ be a group and $X$ be a product system over a semigroup $P$. Suppose $G$ has a left action on $P$ and $P$ has a right action on $G$, so that one can form a Zappa-Sz\'ep product $P\bowtie G$. We define a Zappa-Sz\'ep action of $G$ on $X$ to be a collection of functions on $X$ that are compatible with both actions from $P\bowtie G$ in a certain sense. Given a Zappa-Sz\'ep action of $G$ on $X$, we construct a new product system $X\bowtie G$ over $P\bowtie G$, called the Zappa-Sz\'ep product of $X$ by $G$. We then associate to $X\bowtie G$ several universal C*-algebras and prove their respective Hao-Ng type isomorphisms. A special case of interest is when a Zappa-Sz\'{e}p action is homogeneous. This case naturally generalizes group actions on product systems in the literature. For this case, besides the Zappa-Sz\'ep product system $X\bowtie G$, one can also construct a new type of Zappa-Sz\'{e}p product $X \widetilde\bowtie G$ over $P$. Some essential differences arise between these two types of Zappa-Sz\'ep product systems and their associated C*-algebras. \end{abstract}
\maketitle
\section{Introduction}
In group theory, a Zappa-Sz\'{e}p product of two groups $G$ and $H$ generalizes a semi-direct product by encoding a two-way action between $G$ and $H$. In addition to a left action of $G$ on $H$, the Zappa-Sz\'{e}p product encodes an additional right action of $H$ on $G$. An analogue of semi-direct products in operator algebra is the crossed product construction arising from various dynamical systems. In its simplest form, a group $G$ act on a C*-algebra $\mathcal{A}$ by $*$-automorphisms, in a similar fashion as in a semi-direct product. We seek to extend the Zappa-Sz\'{e}p type construction into the field of operator algebras, which would naturally generalize the crossed product construction.
To construct a Zappa-Sz\'{e}p type structure in operator algebras, there are two key ingredients. First, a Zappa-Sz\'{e}p product of an operator algebra $\mathcal{A}$ and a group $G$ requires a left action of $G$ on $\mathcal{A}$ and a right action of $\mathcal{A}$ on $G$. The right $\mathcal{A}$ action on $G$ requires a grading on $\mathcal{A}$. In the operator algebra literature, there are two natural ways of putting a grading: either via Fell bundles graded by groupoids, or via product systems graded by semigroups. The Zappa-Sz\'{e}p product of a Fell bundle by a groupoid is recently studied in \cite{DL2020}. This paper aims to study Zappa-Sz\'{e}p products of product systems by groups. The second key ingredient is an appropriate replacement of the group action in a dynamical system. A Zappa-Sz\'{e}p product of a product system by a group $G$ needs to encode a two-way action between the product system and the group in the scenario. This requires the group action on the product system to be compatible with the product system action on the group. Finding a right notion of a Zappa-Sz\'{e}p product with compatible actions in the context of product systems is a key in our construction.
Let $P$ be a semigroup and $G$ a group. Suppose $G$ has a left action on $P$ and $P$ has a right action on $G$, so that one can form a Zappa-Sz\'{e}p product $P\bowtie G$. Let $X$ be a product system over $P$. We introduce a notion of the Zappa-Sz\'{e}p action of $G$ on $X$ in Definition \ref{D:beta}. Given a Zappa-Sz\'{e}p action of $G$ on $X$, we define a Zappa-Sz\'{e}p product of $X$ by $G$. We show in Theorem \ref{T:ZSP} that it is a product system over the Zappa-Sz\'{e}p product semigroup $P\bowtie G$. This product system is denoted by $X\bowtie G$. We then consider the scenario where the $G$-action on $P$ is homogeneous. In such a case, it turns out that we can naturally define another Zappa-Sz\'{e}p type product, denoted by $X\widetilde\bowtie G$, which is a product system over the same semigroup $P$ (Theorem \ref{T:prop.Z}).
In Section \ref{S:main}, we study covariant representations of Zappa-Sz\'{e}p actions of groups on product systems and their associated C*-algebras. A covariant representation of a Zappa-Sz\'{e}p action $(X,G,\beta)$ is a pair $(\psi, U)$ consisting of a Toeplitz representation of $X$ and a unitary representation $U$ of $G$ that satisfy a covariant relation. First, we exhibit a one-to-one correspondence between the set of all covariant representations $(\psi, U)$ of $(X, G,\beta)$ and the set of all Toeplitz representations $\Psi$ of $X\bowtie G$
(Theorem \ref{T:Upsi}). As an immediate consequence, we obtain a Hao-Ng isomorphism theorem: $\mathcal{T}_{X\bowtie G}\cong \mathcal{T}_X\bowtie G$ (Corollary \ref{C:HaoNgT}). Furthermore, we show that $\psi$ is Cuntz-Pimsner covariant if and only if so is $\Psi$, and so the Hao-Ng isomorphism also holds true for the associated Cuntz-Pimsner C*-algebras: $\mathcal{O}_{X\bowtie G}\cong \mathcal{O}_X\bowtie G$ (Theorem \ref{thm.cp} and Corollary \ref{C:HaoNgC}). Moreover, if $(X,G,\beta)$ is homogeneous, then $\mathcal{T}_{X\bowtie G}\cong \mathcal{T}_X\bowtie G\cong \mathcal{T}_{X\widetilde\bowtie G}$ (Theorem \ref{T:Upsi.homo} and Corollary \ref{C:HaoNgT}). However, we do not know whether one has $\mathcal{O}_{X\widetilde\bowtie G}\cong \mathcal{O}_X\bowtie G$. Indeed, this is still unknown even in the special case of semi-direct products. This is related to an open problem of Hao-Ng in the literature. Lastly, when the semigroup $P$ is right LCM and $X$ is compactly aligned, we show that $\psi$ is Nica covariant if and only if so is $\Psi$ (Theorem \ref{T:Ncov}). As a result, we have the Hao-Ng isomorphism theorem for Nica-Toeplitz algebras as well: $\mathcal{N}\mathcal{T}_{X\bowtie G}\cong \mathcal{N}\mathcal{T}_X\bowtie G$ (Corollary \ref{C:HaoNgNT}).
Finally, in Section \ref{S:EX}, we present some examples of Zappa-Sz\'{e}p actions of groups on product systems and their C*-algebras.
\section{Preliminaries}
In this section, we provide some necessary background for later use.
\subsection{Zappa-Sz\'{e}p products}
Let $P$ be a (discrete) semigroup and $G$ be a (discrete) group. By convention, in this paper, we always assume that \textsf{a semigroup $P$ has an identity}, written as $e$, unless otherwise specified. To define a Zappa-Sz\'{e}p product semigroup of $P$ and $G$, we first need two actions between $P$ and $G$, given by \begin{enumerate} \item a left $G$-action on $P$: $G\times P \to P$, denoted by $(g,p)\mapsto g\cdot p$, and
\item a right $P$-action on $G$ (also called a restriction map): $P\times G\to G$, denoted by $(p,g)\mapsto g|_p$. \end{enumerate}
Suppose that the two actions satisfy the following compatibility relations: \begin{multicols}{2} \begin{enumerate} \item[(ZS1)]\label{cond.ZS1} $e\cdot p=p$; \item[(ZS2)]\label{cond.ZS2} $(gh)\cdot p=g\cdot (h\cdot p)$; \item[(ZS3)]\label{cond.ZS3} $g\cdot e=e$;
\item[(ZS4)]\label{cond.ZS4} $g|_{e}=g$;
\item[(ZS5)]\label{cond.ZS5} $g\cdot (pq)=(g\cdot p)(g|_p \cdot q)$;
\item[(ZS6)]\label{cond.ZS6}$g|_{pq}=(g|_p)|_q$;
\item[(ZS7)]\label{cond.ZS7} $e|_p=e$;
\item[(ZS8)]\label{cond.ZS8} $(gh)|_p=g|_{h\cdot p} h|_p$. \end{enumerate} \end{multicols}
\noindent Then the Zappa-Sz\'{e}p product semigroup $P\bowtie G$ is defined by $P\bowtie G=\{(p,g): p\in P, g\in G\}$ with multiplication $(p,g)(q,h)=(p(g\cdot q), g|_q h)$. This semigroup has an identity $(e, e)$.
Recall that a left cancellative semigroup $P$ is called a right LCM semigroup if for any $p,q\in P$, either $pP\cap qP=\emptyset$ or $pP\cap qP=rP$ for some $r\in P$. In the case when $P$ is a right LCM semigroup, $P\bowtie G$ is known to be a right LCM semigroup as well \cite[Lemma 3.3]{BRRW}.
\subsection{Product systems}
We give a brief overview of product systems. One may refer to \cite{Fowler2002} for a more detailed discussion.
\begin{definition} Let $\mathcal{A}$ be a unital C*-algebra and $P$ a semigroup. A \textit{product system over $P$} with coefficient $\mathcal{A}$ is defined as $X=\bigsqcup_{p\in P} X_p$ consisting of $\mathcal{A}$-correspondences $X_p$ and an associative multiplication $\cdot:X_p \times X_q\to X_{pq}$ such that \begin{enumerate} \item $X_e=\mathcal{A}$ as an $\mathcal{A}$-correspondence; \item for any $p,q\in P$, the multiplication map on $X$ extends to a unitary $M_{p,q}: X_p\otimes X_q\to X_{pq}$; \item the left and right module multiplications by $\mathcal{A}$ on $X_p$ coincides with the multiplication maps on $X_e\times X_p\to X_p$ and $X_p\times X_e\to X_p$, respectively. \end{enumerate} \end{definition}
Implicit in $M_{e,q}$ being unitary, $X_p$ must be essential, that is, $\lspan\{a\cdot x: a\in X_e, x\in X_p\}$ is dense in $X_p$. This assumption is absent in Fowler's original construction as he does not require $M_{e,q}$ to be unitary. Nevertheless, when the semigroup $P$ contains non-trivial units, every $X_p$ must be essential \cite[Remark 1.3]{KL2018}. Since the semigroups $P$ and $P\bowtie G$ often contain non-trivial units, it is reasonable to make such an assumption.
For a C*-correspondence $X$, we use $\mathcal{L}(X)$ to denote the set of all adjointable operators on $X$. It is a C*-algebra when equipped with the operator norm. For any $x,y\in X$, define the operator $\theta_{x,y}:X\to X$ by $\theta_{x,y}(z)=x\langle y,z\rangle$. It is clear that $\theta_{x,y}\in \mathcal{L}(X)$, and we use $\mathcal{K}(X)$ to denote the C*-subalgebra of $\mathcal{L}(X)$ generated by $\theta_{x,y}$. The set $\mathcal{K}(X)$ is also known as the generalized compact operators on $X$.
Suppose now that $P$ is a right LCM semigroup. The notion of compactly aligned product systems, first introduced by Fowler for product systems over quasi-lattice ordered semigroups \cite{Fowler2002}, has been recently generalized to right LCM semigroups
in \cite{BLS2018b, KL2018}. For any $p,q\in P$, there is a $*$-homomorphism $i_p^{pq}: \mathcal{L}(X_p)\to\mathcal{L}(X_{pq})$ by setting for any $x\in X_p$ and $y\in X_q$, \[i_p^{pq}(S)(xy)=(Sx)y.\]
\begin{definition} \label{df.cpt.align} We say that a product system $X$ is \emph{compactly-aligned} if for any $S\in\mathcal{K}(X_p)$ and $T\in\mathcal{K}(X_q)$ with $pP\cap qP=rP$, we have \[i_p^r(S) i_q^r(T) \in \mathcal{K}(X_r).\] We shall use the notion $S\vee T := i_p^r(S) i_q^r(T)$.
\end{definition}
\subsection{Representations and C*-algebras of product systems}
Product systems are one of essential tools in the study of operator algebras graded by semigroups. In its simplest form, an $\mathcal{A}$-correspondence $X$ can be viewed as a product system over $\mathbb{N}$, by setting $X_0=\mathcal{A}$ and $X_n=X^{\otimes n}$. There are two natural C*-algebras associated with such a product system: the Toeplitz algebra $\mathcal{T}_X$ and the Cuntz-Pimsner algebra $\mathcal{O}_X$ \cite{MS1998, Pimsner1997}. Fowler generalizes these to product systems over countable semigroups \cite{Fowler2002}.
In \cite{Nica1992}, Nica introduced the semigroup C*-algebra of a quasi-lattice ordered semigroup. It is the universal C*-algebra generated by isometric semigroup representations that satisfy a covariance condition, now known as the Nica-covariance condition. This extra covariance condition soon found an analogue in the C*-algebra related to product systems. In \cite{Fowler2002, FR1998, BLS2018b}, the Nica-Toeplitz algebra $\mathcal{N}\mathcal{T}_X$ is defined by imposing the extra Nica-covariance condition, thereby being a quotient of the corresponding Toeplitz algebra. Here, we give a brief overview of these three C*-algebras and their representations associated with a product system.
\begin{definition} Let $X$ be a product system over a semigroup $P$. A \textit{(Toeplitz) representation} of $X$ on a C*-algebra $\mathcal{B}$ consists of a collection of linear maps $\psi=(\psi_p)_{p\in P}$, where for each $p\in P$, $\psi_p:X_p\to \mathcal{B}$, such that \begin{enumerate}
\item $\psi_e$ is a $*$-homomorphism of the C*-algebra $X_e$;
\item for all $p,q\in P$ and $x\in X_p, y\in X_q$, $\psi_{p}(x)\psi_q(y)=\psi_{pq}(xy)$; and
\item for all $p\in P$ and $x, y\in X_p$, $\psi_p(x)^* \psi_p(y)=\psi_e(\langle x,y\rangle)$. \end{enumerate} \end{definition}
Given a representation $\psi:X\to \mathcal{B}$, there is a homomorphism $\psi^{(p)}: \mathcal{K}(X_p)\to \mathcal{B}$ satisfying $\psi^{(p)}(\theta_{x,y})=\psi_p(x)\psi_p(y)^*$. The left action of $X_e$ on $X_p$ induces a $*$-homomorphism $\phi_p: X_e\to \mathcal{L}(X_p)$ by $\phi_p(a)x=a\cdot x$.
\begin{definition} A representation $\psi$ is called \textit{Cuntz-Pimsner covariant} if for all $p\in P$ and $a\in X_e$ with $\phi_p(a)\in \mathcal{K}(X_p)$, $\psi^{(p)}(\phi_p(a))=\psi_e(a)$. \end{definition}
By \cite[Propositions 2.8 and 2.9]{Fowler2002}, there is a universal C*-algebra $\mathcal{T}_X$ (resp. $\mathcal{O}_X$) for Toeplitz (resp. Cuntz-Pimsner covariant) representations. Specifically, there is a universal Toeplitz representation $i_X$ (resp. a universal Cuntz-Pimsner covariant representation $j_X$) such that the following properties hold: \begin{enumerate}
\item The C*-algebras $\mathcal{T}_X$ and $\mathcal{O}_X$ are generated by $i_X$ and $j_X$ respectively. That is, $\mathcal{T}_X=C^*(i_X(X))$ and $\mathcal{O}_X=C^*(j_X(X))$.
\item For every Toeplitz (resp. Cuntz-Pimsner covariant) representation $\psi$, there exists a $*$-homomorphism $\psi_*$ from $\mathcal{T}_X$ (resp. $\mathcal{O}_X$) to $C^*(\psi(X))$ such that $\psi=\psi_*\circ i_X$ (resp. $\psi=\psi_* \circ j_X$). \end{enumerate}
The Toeplitz C*-algebra $\mathcal{T}_X$ is often quite large as a C*-algebra. For example, for the trivial product system $X$ over ${\mathbb{N}}^2$, its Toeplitz representation is determined by a pair of commuting isometries. The Toeplitz algebra $\mathcal{T}_X$ of this product system is thus the universal C*-algebra generated by a pair of commuting isometries, which is known to be non-nuclear \cite{Murphy1996b}. This motivated Nica to study the semigroup C*-algebras of quasi-lattice ordered semigroups \cite{Nica1992} by imposing what is now known as the Nica-covariance condition. This condition is soon generalized to representations of product systems. In \cite[Definition 5.1]{Fowler2002}, Fowler defined the notion of compactly aligned product system and extended the Nica-covariance condition to such product systems. In \cite[Definition 6.4]{BLS2018b}, the Nica-covariance condition is further generalized to compactly aligned product systems over right LCM semigroups. We now give a brief overview of the Nica-covariance condition.
Recall that if $X$ is compactly aligned, then for any $S\in\mathcal{K}(X_p)$ and $T\in\mathcal{K}(X_q)$ with $pP\cap qP=rP$, \[S\vee T:=i_p^r(S) i_q^r(T) \in \mathcal{K}(X_r).\]
\begin{definition}\label{df.NC.rep} Let $X$ be a compactly aligned product system over a right LCM semigroup $P$. A representation $\psi$ is called \emph{Nica-covariant} if for all $p,q\in P$ and $S\in\mathcal{K}(X_p)$ and $T\in \mathcal{K}(X_q)$, we have, \[\psi^{(p)}(S) \psi^{(q)}(T) = \begin{cases} \psi^{(r)}(S \vee T) & \text{ if } pP\cap qP=rP, \\ 0 & \text{ if } pP\cap qP=0. \end{cases} \] \end{definition}
One can verify that this definition does not depend on the choice of $r$ (see \cite{BLS2018b}). The Nica-Toeplitz C*-algebra $\mathcal{N}\mathcal{T}_X$ can be then defined as the universal C*-algebra generated by the Nica-covariant Toeplitz representations of the product system $X$.
\section{Zappa-Sz\'{e}p products of Product Systems by Groups}
We first introduce the notion of Zappa-Sz\'{e}p action of a group on a product system. This allows us to define the Zappa-Sz\'{e}p product of a product system by a group, which is a product system over the given Zappa-Sz\'{e}p product semigroup. In the special case when the Zappa-Sz\'{e}p action is homogeneous, it turns out that we can construct another Zappa-Sz\'{e}p type product system, which is a product system over the same semigroup.
In the remaining of this section, let $P\bowtie G$ be a Zappa-Sz\'{e}p product of a semigroup $P$ and a group $G$, $\mathcal{A}$ a unital C*-algebra, and $X=\bigsqcup_{p\in P} X_p$ a product system over $P$ with coefficient $\mathcal{A}$.
\subsection{Zappa-Sz\'{e}p actions}
We first need a key notion of the Zappa-Sz\'{e}p action of $G$ on $X$. The product system $X$ naturally defines a ``$G$-restriction map" on $X$ by inheriting the $G$-restriction map on $P$. However, one has to define a $G$-action map on the product system $X$ that mimics the $G$-action on the semigroup $P$.
\begin{definition} \label{D:beta} (i) Let $P\bowtie G$ be a Zappa-Sz\'{e}p product of a semigroups $P$ and a group $G$, and $X$ a product system over $P$. A \textit{Zappa-Sz\'{e}p action of $G$ on $X$} is a collection of functions $\{\beta_g\}_{g\in G}$ that satisfies the following conditions: \begin{enumerate} \item[(A1)]\label{cond.A1} for each $p\in P$ and $g\in G$, $\beta_g: X_p \to X_{g\cdot p}$ is a $\mathbb{C}$-linear map; \item[(A2)]\label{cond.A2} $\beta_g\circ\beta_h=\beta_{gh}$ for all $g,h\in G$; \item[(A3)]\label{cond.A3} the map $\beta_{e}$ is the identity map; \item[(A4)]\label{cond.A4} for each $g\in G$, the map $\beta_g$ is a $*$-automorphism on the C*-algebra $\mathcal{A}$; \item[(A5)]\label{cond.A5} for each $g\in G$ and $p,\, q\in P$,
\[\beta_g(xy)=\beta_g(x)\beta_{g|_p}(y) \quad\text{for all}\quad x\in X_p, \, y\in X_q;\] \item[(A6)]\label{cond.A6} for each $g\in G$ and $p\in P$, \[
\langle\beta_g(x), \beta_g(y)\rangle=\beta_{g|_p}(\langle x, y\rangle)\quad\text{for all}\quad x,\, y\in X_p. \] \end{enumerate} (ii) If $\beta$ is a Zappa-Sz\'{e}p action of $G$ on $X$, we call the triple $(X,G,\beta)$ a \textit{Zappa-Sz\'{e}p system}. \end{definition}
Some remarks are in order. \begin{rem} \label{R:beta} In Definition \ref{D:beta}, to be more precise, the collection $\{\beta_g\}_{g\in G}$ should be written as $\{\beta_g^p\}_{g\in G,p\in P}$, so that $\beta_g^p: X_p\to X_{g\cdot p}$. But it is usually clear from the context to see which $X_p$ the map $\beta_g$ acts on. So, in order to simplify our notation, we just write $\beta_g$ instead of $\beta_g^p$. \end{rem}
\begin{rem}
In the very special case when both the $G$-action and the $G$-restriction are trivial, that is, $g\cdot p=p$ and $g|_p=g$ for all $g\in G$ and $p\in P$, Definition \ref{D:beta} corresponds to the sense of a group action on a product system in \cite{DOK20}. For a C*-correspondence $X$ (that is a product system over ${\mathbb{N}}$), our definition coincides with the group action considered in \cite[Definition 2.1]{HaoNg2008}. \end{rem}
\begin{rem}
It follows from \hyperref[cond.A1]{(A1)}-\hyperref[cond.A3]{(A3)} that $\beta_g^p$ is a bijection between $X_p$ and $X_{g\cdot p}$. However, $\beta_g$ is not an $\mathcal{A}$-linear map between $\mathcal{A}$-correspondences. In fact, the condition \hyperref[cond.A5]{(A5)} forces $\beta_g(xa)=\beta_g(x)\beta_{g|_p}(a)$ and $\beta_g(ax)=\beta_g(a) \beta_g(x)$ for $a\in \mathcal{A}$ and $x\in X_p$. \end{rem}
\subsection{The Zappa-Sz\'{e}p product system $X\bowtie G$ of $(X,G,\beta)$}
Let $(X,G,\beta)$ be a Zappa-Sz\'{e}p system as defined in Definition \ref{D:beta}. For each $p\in P$ and $g\in G$, set \[ Y_{(p,g)}:=\{x\otimes g: x\in X_p\}. \] Define the left and right actions of $\mathcal{A}$ on $Y_{(p,g)}$ by \[ a\cdot (x\otimes g)=(a\cdot x)\otimes g,\ (x\otimes g)\cdot a = x\beta_g(a) \otimes g, \] respectively, and an $\mathcal{A}$-valued inner product to be \[ \langle x\otimes g, y\otimes g\rangle = \beta_{g^{-1}}(\langle x,y\rangle). \]
\begin{proposition} With the notation same as above, $Y_{(p,g)}$ is an $\mathcal{A}$-correspondence. \end{proposition}
\begin{proof} We first verify that $Y_{(p,g)}$ is a right $\mathcal{A}$-module. The inner product is $\mathcal{A}$-linear in the second component: take any $a\in \mathcal{A}$, and $x,y\in X_p$, \begin{align*} \langle x\otimes g, (y\otimes g) a\rangle &= \langle x\otimes g, y\beta_g(a)\otimes g\rangle \\ &= \beta_{g^{-1}}(\langle x, y\beta_g(a)\rangle) \\ &= \beta_{g^{-1}}(\langle x, y\rangle \beta_g(a)) \\ &= \beta_{g^{-1}}(\langle x, y\rangle) a \\ &= \langle x\otimes g, y\otimes g\rangle a. \end{align*} Since $\beta_g$ is a $*$-automorphism on $\mathcal{A}$, one has \begin{align*} \langle x\otimes g, y\otimes g\rangle^* &= \beta_{g^{-1}}(\langle x,y\rangle)^* \\ &= \beta_{g^{-1}}(\langle x,y\rangle^*) \\ &= \beta_{g^{-1}}(\langle y,x\rangle) \\ &= \langle y\otimes g, x\otimes g\rangle. \end{align*} Finally, $\beta_g$ is also a positive map on $\mathcal{A}$ because it is a $*$-automorphism. Hence \[ \langle x\otimes g, x\otimes g\rangle = \beta_{g^{-1}}(\langle x,x\rangle)^* \geq 0.\] Moreover, since $\beta_g$ is isometric on $\mathcal{A}$,
\[\|x\otimes g\|=\|\langle x\otimes g, x\otimes g\rangle\|^{1/2}=\|\beta_g^{-1}(\langle x, x\rangle)\|^{1/2}=\|\langle x,x\rangle\|^{1/2}=\|x\|.\] Thus the norm on $Y_{(p,g)}$ is the same as that on $X_p$. Because $X_p$ is complete under its norm, so is $Y_{(p,g)}$. Therefore $Y_{(p,g)}$ is a Hilbert $\mathcal{A}$-module. One can clearly see that the left action of $\mathcal{A}$ induced from $X_p$ is a left action of $\mathcal{A}$ on $Y_{p,g}$. Therefore, $Y_{p,g}$ is an $\mathcal{A}$-correspondence. \end{proof}
Let \[ Y:=\bigsqcup_{(p,g)\in P\bowtie G} Y_{(p,g)}. \]
For each $p,q\in P$ and $g,h\in G$, define a multiplication map $Y_{(p,g)}\times Y_{(q,h)}\to Y_{(p(g\cdot q), g|_q h)}$ by \begin{align} \label{E:M}
((x\otimes g), (y\otimes h))\mapsto (x\beta_g(y))\otimes (g|_q)h. \end{align}
This extends to a map $M_{(p,g),(q,h)}:Y_{(p,g)}\otimes Y_{(q,h)}\to Y_{(p(g\cdot q), g|_q h)}$.
\begin{theorem} \label{T:ZSP} With the multiplication maps $M_{(p,g),(q,h)}$, $Y$ is a product system over the Zappa-Sz\'{e}p product $P\bowtie G$. \end{theorem}
\begin{proof} First of all, the identity of $P\bowtie G$ is $(e, e)$, and one can easily check that $Y_{(e, e)}\cong \mathcal{A}$ by identifying $a\otimes e\in Y_{(e,e)}$ with $a\in \mathcal{A}$. For any $a\in \mathcal{A}$ (that is $a\otimes e\in Y_{(e,e)}$), and the left and right $\mathcal{A}$-actions on $Y_{(p,g)}$ are implemented by the multiplications: \begin{align*} a(x\otimes g) &= ax\otimes g = (a\otimes e)(x\otimes g), \\ (x\otimes g)a &= x\beta_g(a) \otimes g = (x\otimes g)(a\otimes e). \end{align*}
By \condref{A1}, $\beta_g$ is a ${\mathbb{C}}$-linear isomorphism from $X_p$ to $X_{g\cdot p}$. Therefore, if $a\in X_p$ and $b\in X_q$, \[a\beta_g(b)\in X_p X_{g\cdot q}\in X_{p(g\cdot q)}.\] One can easily see that $M_{(p,g),(q,h)}$ is an $\mathcal{A}$-linear map. To show that it is unitary, for any $x_p, u_p\in X_p$ and $y_q, v_q\in X_q$, \begin{align*} &\langle (x_p\otimes g)\otimes (y_q\otimes h), (u_p\otimes g)\otimes (v_q\otimes h)\rangle \\ =& \langle (y_q\otimes h), \langle(x_p\otimes g), (u_p\otimes g)\rangle (v_q\otimes h)\rangle \\ =& \langle (y_q\otimes h), \beta_g^{-1}(\langle x_p,u_p\rangle) (v_q\otimes h)\rangle \\ =& \beta_h^{-1}(\langle y_q, \beta_g^{-1}(\langle x_p,u_p\rangle) v_q\rangle). \end{align*} On the other hand, \begin{align*} &\langle M_{(p,g),(q,h)}((x_p\otimes g)\otimes (y_q\otimes h)), M_{(p,g),(q,h)}((u_p\otimes g)\otimes (v_q\otimes h))\rangle \\ =&
\langle x_p\beta_g(y_q)\otimes g|_q h, u_p\beta_g(v_q)\otimes g|_p h\rangle \\
=& \beta_{h^{-1}}\beta_{(g|_p)^{-1}}(\langle x_p\beta_g(y_q), u_p\beta_g(v_q) \rangle) \\
=& \beta_{h^{-1}}\beta_{(g|_p)^{-1}}(\langle \beta_g(y_q), \langle x_p, u_p\rangle \beta_g(v_q) \rangle) \ (\text{as }M_{p,g\cdot q}\text{ is unitary}) \\
=& \beta_{h^{-1}}\beta_{(g|_p)^{-1}}(\langle \beta_g(y_q), \beta_g(\beta_g^{-1}(\langle x_p, u_p\rangle) v_q) \rangle) \\ =& \beta_h^{-1}(\langle y_q, \beta_g^{-1}(\langle x_p,u_p\rangle) v_q\rangle )\ (\text{by (A6)}). \end{align*} Therefore, $M_{(p,g),(q,h)}$ preserves the inner product and is thus unitary.
Finally, for $a\in X_p$, $b\in X_q$, $c\in X_r$ and $g,h,k\in G$, we compute that
\[((a\otimes g)(b\otimes h))(c\otimes k) = (a\beta_g(b)\beta_{g|_q h}(c))\otimes (g|_q h)|_r k\] and
\[(a\otimes g)((b\otimes h)(c\otimes k)) = (a\beta_g(b\beta_h(c)))\otimes g|_{q(h\cdot r)} h|_r k.\] From \condref{A5} one has
\[a\beta_g(b\beta_h(c)) = a\beta_g(b) \beta_{g|_q}(\beta_h(c)) = a\beta_g(b)\beta_{g|_q h}(c).\] Also \condref{ZS8} and \condref{ZS6} yield
\[(g|_q h)|_r k = (g|_q)|_{h\cdot r} h|_r k = g|_{q(h\cdot r)} h|_r k.\] Hence the multiplication is associative. Therefore $Y=(Y_{(p,g)})_{(p,g)\in P\bowtie G}$ is a product system over the Zappa-Sz\'{e}p semigroup $P\bowtie G$. \end{proof}
\begin{defn} The new product system $Y$ constructed from $(X,G,\beta)$ in Theorem \ref{T:ZSP} is called the \textit{Zappa-Sz\'{e}p product of $X$ by $G$} and denoted as $X\bowtie G$. \end{defn}
\subsection{Another Zappa-Sz\'{e}p product $X\widetilde\bowtie G$ of a homogeneous Zappa-Sz\'{e}p system}
In this subsection, we study a special class of Zappa-Sz\'{e}p actions of groups on product systems -- homogeneous Zappa-Sz\'{e}p actions. Given such a Zappa-Sz\'{e}p action, it turns out that \textit{another} new natural and interesting product system $X\widetilde\bowtie G$ can be constructed from the given action. Unlike $X\bowtie G$ that enlarges the grading semigroup and keeps the coefficient C*-algebra the same,
this new one, $X\widetilde\bowtie G$, enlarges the coefficient C*-algebra and keeps the grading semigroup the same.
\label{S:homog} \begin{definition} Let $P\bowtie G$ be a Zappa-Sz\'{e}p product of a semigroups $P$ and a group $G$, and $X$ a product system over $P$. A Zappa-Sz\'{e}p action $\beta$ on $X$ is called \textit{homogeneous} if $g\cdot p=p$ for any $p\in P$ and $g\in G$. In this case, the Zappa-Sz\'{e}p system $(X,G,\beta)$ is also said to be \textit{homogeneous}. \end{definition}
\begin{rem} Homogeneous Zappa-Sz\'{e}p actions are a natural generalization of usual generalized gauge actions \cite{K2017}. \end{rem}
In the case of a homogeneous Zappa-Sz\'{e}p system $(X,G,\beta)$, one can easily see that $\beta$ induces an automorphic action $\beta:G\to \operatorname{Aut}(X_p)$. This allows us to construct a new crossed product type product system over the same semigroup $P$ that encodes the Zappa-Sz\'{e}p structure. In particular, when $p= e$, we obtain a C*-dynamical system $(\mathcal{A},G,\beta)$. Let $\mathfrak{A}=\mathcal{A}\rtimes_\beta G$ be the universal C*-crossed product of this C*-dynamical system. So $\mathfrak{A}=C^*(a, u_g: a\in \mathcal{A}, g\in G)$, where $\{a,u_g: a\in \mathfrak{A}, g\in G\}$ is the generator set of $\mathfrak{A}$. Thus the generators satisfy the covariance condition \[u_g a = \beta_g(a) u_g\quad\text{for all}\quad a\in \mathcal{A}\text{ and }g\in G.\]
For each $p\in P$, consider $Z_p^0=c_{00}(G,X_p)=\operatorname{span}\{x\otimes g: x\in X_p\}$. We can put an $\mathfrak{A}$-bimodule structure on $Z_p^0$: for any $a u_h\in \mathfrak{A}$ and $\xi=x_p\otimes g\in c_{00}(G,X_p)$, \begin{align*}
(au_h)\xi = (a\beta_h(x_p))\otimes h|_p g \quad \text{ and }\quad
\xi (au_h) = (x_p\beta_g(a)) \otimes gh. \end{align*} Define an $\mathfrak{A}$-valued function $\langle\cdot,\cdot\rangle: Z_p^0\times Z_p^0\to \mathfrak{A}$ by setting, for any $x_p\otimes g, y_p\otimes h\in Z_p^0$, \[\langle x_p \otimes g, y_p \otimes h\rangle=\beta_{g^{-1}}(\langle x_p, y_p\rangle) u_{g^{-1}h}.\] By the covariance relation on $\mathfrak{A}$, one can rewrite the above identity as \[\langle x_p \otimes g, y_p \otimes h\rangle=u_{g^{-1}} \langle x_p, y_p\rangle u_{h}.\]
\begin{proposition} The space $Z_p^0$ together with the map $\langle\cdot,\cdot\rangle$ is an inner product right $\mathfrak{A}$-module. \end{proposition}
\begin{proof} It is easy to see that $\langle\cdot,\cdot\rangle$ is right $\mathfrak{A}$-linear in the second variable: take any $au_k\in \mathfrak{A}$ and $x_p\otimes g, y_p\otimes h\in Z_p^0$, \begin{align*}
\langle x_p \otimes g, (y_p \otimes h) a u_k)\rangle &= \langle x_p \otimes g, y_p\beta_h(a)\otimes hk\rangle \\
&= \beta_{g^{-1}}(\langle x_p, y_p\beta_h(a)\rangle) u_{g^{-1} hk} \\
&= \beta_{g^{-1}}(\langle x_p, y_p\rangle) \beta_{g^{-1}h}(a) u_{g^{-1} h} u_k \\
&= \beta_{g^{-1}}(\langle x_p, y_p\rangle) u_{g^{-1} h} a u_k \\
&= \langle x_p \otimes g, y_p \otimes h\rangle a u_k \end{align*} Also, for any $x_p\otimes g, y_p\otimes h\in Z_p^0$, \begin{align*}
\langle x_p \otimes g, y_p \otimes h\rangle^* &= \left(u_g^* \langle x_p, y_p\rangle u_h\right)^* \\
&= u_h^* \langle y_p, x_p\rangle u_g \\
&= \langle y_p \otimes h, x_p \otimes g\rangle. \end{align*} Finally, for any $x_1,\ldots,x_n\in X_p$ and $g_1,\ldots,g_n\in G$, considier $\xi=\sum_{i=1}^n x_i\otimes g_i\in Z_p^0$. We have that \[\langle \xi, \xi\rangle = \sum_{i=1}^n \sum_{j=1}^n u_{g_i^{-1}} \langle x_i, x_j\rangle u_{g_j}. \] Consider the $n\times n$ operator matrix $K=[\langle x_i, x_j\rangle]$. We first claim that $A\geq 0$ as an operator in $M_n(\mathcal{A})$, which is equivalent of showing \cite[Proposition 6.1]{Paschke1973} that for any $a_1,\ldots, a_n\in\mathcal{A}$, \[\sum_{i=1}^n \sum_{j=1}^n a_i^* \langle x_i, x_j\rangle a_j \geq 0.\] Since $X_p$ is an $\mathcal{A}$-correspondence, $a_i^* \langle x_i, x_j\rangle a_j=\langle x_i a_i, x_j a_j\rangle$. Therefore, \[\sum_{i=1}^n \sum_{j=1}^n a_i^* \langle x_i, x_j\rangle a_j=\sum_{i=1}^n \sum_{j=1}^n\langle x_i a_i, x_j a_j\rangle=\langle \sum_{i=1}^n x_i a_i, \sum_{j=1}^n x_j a_j\rangle\geq 0.\] This proves that the operator matrix $K=[\langle x_i, x_j\rangle]\geq 0$. Since $\mathcal{A}$ embeds injectively inside the crossed product $\mathfrak{A}=\mathcal{A}\rtimes_\beta G$, the operator matrix $K\geq 0$ as an operator in $M_n(\mathfrak{A})$. Therefore, \[\langle \xi, \xi\rangle = \sum_{i=1}^n \sum_{j=1}^n u_{g_i^{-1}} \langle x_i, x_j\rangle u_{g_j}\geq 0. \] Suppose that $\langle \xi,\xi\rangle=0$. We have \[\langle \xi, \xi\rangle = \sum_{i=1}^n \sum_{j=1}^n \beta_{g_i^{-1}}( \langle x_i, x_j\rangle) u_{g_i^{-1} g_j} = 0.\] Since there exists a contractive conditional expectation $\Phi:\mathfrak{A}\to\mathcal{A}$ by $\Phi(\sum a_g u_g)=a_e$, we have that \[\sum_{i=1}^n \beta_{g_i^{-1}}( \langle x_i, x_i\rangle)=0.\] Since $\beta_{g_i^{-1}}$'s are $*$-automorphisms of $\mathcal{A}$, we have that $\langle x_i, x_i\rangle =0$ for all $i$, and thus $x_i=0$ for all $i$. So we obtain that $\xi=0$. Therefore $\langle\cdot,\cdot\rangle$ is an $\mathfrak{A}$-valued inner product on $Z_p^0$. \end{proof}
Now let $Z_p$ be the completion of $Z_p^0$ under the norm $\|\xi\|=\|\langle \xi, \xi\rangle\|^{1/2}$. We obtain an $\mathfrak{A}$-correspondence $Z_p$.
\begin{theorem} \label{T:prop.Z} The collection $Z=\bigsqcup_{p\in P}Z_p$ is a product system over $P$, where the multiplication $Z_p\times Z_q\to Z_{pq}$ is given by
\[(x_p\otimes g,y_q \otimes h)\mapsto x_p \beta_g(y_q)\otimes g|_q h \quad (g,h\in G, x_p\in X_p, y_q\in X_q).\] \end{theorem}
\begin{proof} Let $p, q\in P$. For $x_p\in X_p$ and $y_q\in X_q$, one has $\beta_g(y_q)\in X_q$ and thus $x_p \beta_g(y_q)\in X_{pq}$. Since $\beta_g$ is automorphic as mentioned above, the multiplication is surjective. To see the multiplication induces a unitary map from $Z_p\otimes Z_q\to Z_{pq}$, take any four elementary tensors $x_p\otimes g, w_p\otimes i\in Z_p$ and $y_q\otimes h, z_q\otimes k\in Z_q$, \begin{align*}
&\langle (x_p\otimes g)\otimes (y_q \otimes h), (w_p\otimes i)\otimes (z_q \otimes k)\rangle \\
=& \langle y_q \otimes h, \langle x_p\otimes g, w_p\otimes i\rangle (z_q \otimes k)\rangle \\
=&
\langle y_q \otimes h, \beta_{g^{-1}}(\langle x_p, w_p\rangle) u_{g^{-1}i} (z_q \otimes k)\rangle \\
=& \langle y_q \otimes h, \beta_{g^{-1}}(\langle x_p, w_p\rangle) \beta_{g^{-1}i}(z_q) \otimes (g^{-1}i)|_q k)\rangle \\
=& u_h^* \langle y_q, \beta_{g^{-1}}(\langle x_p, w_p\rangle) \beta_{g^{-1}i}(z_q)\rangle u_{ (g^{-1}i)|_q k}. \end{align*} On the other hand, \begin{align*}
&\langle (x_p\otimes g) (y_q \otimes h), (w_p\otimes i) (z_q \otimes k)\rangle \\
=& \langle x_p \beta_g(y_q)\otimes g|_q h, w_p\beta_i(z_q)\otimes i|_q k\rangle \\
=& u_{g|_q h}^* \langle x_p \beta_g(y_q), w_p\beta_i(z_q)\rangle u_{i|_q k} \\
=& u_h^* u_{(g|_q)^{-1}} \langle \beta_g(y_q), \langle x_p, w_p\rangle \beta_i(z_q)\rangle u_{i|_q k} \\
=& u_h^* u_{(g|_q)^{-1}} \langle \beta_g(y_q), \beta_g(\beta_{g^{-1}}(\langle x_p, w_p\rangle) \beta_{g^{-1}i}(z_q))\rangle u_{i|_q k} \\
=& u_h^* u_{(g|_q)^{-1}} \beta_{g|_q} (\langle y_q, \beta_{g^{-1}}(\langle x_p, w_p\rangle) \beta_{g^{-1}i}(z_q)\rangle) u_{i|_q k} \\
=& u_h^* \langle y_q, \beta_{g^{-1}}(\langle x_p, w_p\rangle) \beta_{g^{-1}i}(z_q)\rangle u_{(g|_q)^{-1}} u_{i|_q k} \\
=& u_h^* \langle y_q, \beta_{g^{-1}}(\langle x_p, w_p\rangle) \beta_{g^{-1}i}(z_q)\rangle u_{g^{-1}|_{g\cdot q} i|_q k}. \end{align*} Because $g\cdot q=i\cdot q$,
\[u_{g^{-1}|_{g\cdot q} i|_q k}=u_{g^{-1}|_{i\cdot q} i|_q k}=u_{(g^{-1}i)|_q k}.\] Therefore, the multiplication is indeed unitary.
The C*-algebra $Z_{e}$ can be identified as $\mathfrak{A}$ via $a\otimes g\mapsto au_g$. It is routine to verify that the left and right $\mathfrak{A}$-action on $Z_p$ are implemented by the multiplication map by $Z_{e}$. To see the associativity of the multiplication, take $x\otimes g\in Z_p$, $y\otimes h\in Z_q$, and $z\otimes k\in Z_r$. We have that
\[ \left((x\otimes g)(y\otimes h)\right)(z\otimes k) = x\beta_g(y)\beta_{g|_q h}(z)\otimes (g|_q h)|_r k\] and
\[ (x\otimes g)\left((y\otimes h)(z\otimes k)\right) = x\beta_g(y\beta_h(z))\otimes g|_{qr} h|_r k.\]
Condition (A5) yields $\beta_g(y\beta_h(z))=\beta_g(y)\beta_{g|_q h}(z)$. From (ZS6), (ZS8), and the homogeneity assumption that $h\cdot r=r$,
we have \[(g|_q h)|_r k=(g|_q)|_{h\cdot r} h|_r k=g|_{qr} h|_r k.\] This proves the associativity of the multiplication. \end{proof}
\begin{definition} The product system $Z$ obtained in Theorem \ref{T:prop.Z} is called the \textit{homogeneous Zappa-Sz\'{e}p product of $X$ by $G$} and denoted as $X\widetilde\bowtie G$. \end{definition}
In summary, for a given \textsf{homogeneous} Zappa-Sz\'{e}p action $\beta$ of $G$ on $X$, one has two new product systems: (i) $X\bowtie G$ -- a product system over $P\bowtie G$ with coefficient C*-algebra $\mathcal{A}$, and (ii) $X\widetilde\bowtie G$ -- a product system over $P$ with coefficient C*-algebra $\mathcal{A}\rtimes_\beta G$.
\section{C*-algebras associated to Zappa-Sz\'{e}p actions \\ and some Hao-Ng isomorphism theorems}
\label{S:main}
In this section, we study covariant representations and their associated universal C*-algebras arising from a Zappa-Sz\'{e}p system $(X,G,\beta)$. We establish a one-to-one correspondence between the set of all covariant representations $(\psi, U)$ of $(X,G,\beta)$ and the set of all (Toeplitz) representations $\Psi$ of the Zappa-Sz\'{e}p product system $X\bowtie G$. Furthermore, it is proved that $\psi$ is Cuntz-Pimsner covariant if and only if so is $\Psi$. If $P$ is right LCM and $X$ is compactly aligned, then $\psi$ is Nica-covariant if and only if so is $\Psi$. As a consequence, we obtain several Hao-Ng isomorphism theorems for the associated C*-algebras. However, as we shall see, changes arise for the homogeneous Zappa-Sz\'{e}p product system $X\widetilde\bowtie G$.
\subsection{Covariant representations of $(X,G,\beta)$ and the C*-algebra ${\mathcal{T}}_X\bowtie G$}
Let $P\bowtie G$ be a Zappa-Sz\'{e}p product of a semigroup $P$ and a group $G$, ${\mathcal{A}}$ a unital C*-algebra, and $X$ a product system over $P$ with coefficient ${\mathcal{A}}$. Suppose that $\beta$ is an arbitrary Zappa-Sz\'{e}p action of $G$ on $X$.
\begin{defn} \label{D:vrep} Let $\psi$ be a representation of $X$ and $U$ is unitary representation of $G$ on a unital C*-algebra $\mathcal{B}$. A pair $(\psi, U)$ is called a \textit{covariant representation of a Zappa-Sz\'{e}p system $(X,G,\beta)$} if \begin{align} \label{E:Upsi}
U_g \psi_p(x)=\psi_{g\cdot p}(\beta_g(x)) U_{g|_p} \text{ for all } p\in P, g\in G, x\in X_p. \end{align} \end{defn}
\begin{theorem} \label{T:Upsi} There is a one-to-one correspondence $\Pi$ between the set of all representations $\Psi$ of $X\bowtie G$ on a unital C*-algebra $\mathcal{B}$ and the set of covariant representations $(\psi, U)$ of $(X,G,\beta)$.
In fact, for a given representation $\Psi$ of $X\bowtie G$, one has \begin{align*} \psi_p(x)&:=\Psi_{(p,e)}(x\otimes e)\quad\text{for all}\quad p\in P, x\in X_p,\\ U_g&:=\Psi_{(e, g)}(1_\mathcal{A}\otimes g)\quad\text{for all}\quad g\in G. \end{align*} Conversely, given a covariant representation $(\psi, U)$ of $(X,G,\beta)$, one has \begin{align} \label{E:DefPsi} \Psi_{(p, g)}(x\otimes g) := \psi_p(x) U_g\quad\text{for all}\quad p\in P, g\in G, x\in X_p. \end{align} \end{theorem}
\begin{proof} Let $Y:=X\bowtie G$ and $\Psi$ be a representation of $Y$ on a unital C*-algebra $\mathcal{B}$. Since $\Psi$ is a representation of $Y$, in particular we have \begin{align*}
\Psi_{(p,g)}(x\otimes g)\Psi_{(q,h)}(y\otimes h)=\Psi_{(p(g\cdot q), g|_qh)}(x\beta_g(y)\otimes g|_qh) \end{align*} for all $g, h\in G$, $p,q\in P$, $x\in X_p$ and $y\in X_q$.
For $p\in P$, define $\psi_P: X_p \to \mathcal{B}$ via \begin{align*}
\psi_p(x):=\Psi_{(p,e)}(x\otimes e)\quad\text{for all}\quad x\in X_p. \end{align*} Then \begin{align*} \psi_p(x)^*\psi_p(y) &=\Psi_{(p,e)}(x\otimes e)^*\Psi_{(p\otimes e)}(y\otimes e)\\
&=\Psi_{(e,e)}(\langle (x\otimes e), (y\otimes e)\rangle\\
&=\Psi_{(e,e)}(\langle x, y\rangle\otimes e)\\
&=\psi_{e}(\langle x, y\rangle)\ (x, y\in X_p) \end{align*} and \begin{align*} \psi_p(x)\psi_q(y) &=\Psi_{(p,e)}(x\otimes e)\Psi_{(q,e)}(y\otimes e)\\
&=\Psi_{(p(e\cdot q), e|_q e)}((x\otimes e)(y\otimes e))\\
&=\Psi(pq,e)(x\beta_{e}(y)\otimes e|_q e)\\ &=\Psi(pq,e)(xy\otimes e)\\ &=\psi_{pq}(xy)\ (x\in X_p, y\in X_q). \end{align*} So $\psi$ is a representation of $X$ on $\mathcal{B}$.
One can easily check that $U_g:=\Psi_{(e, g)}(1_\mathcal{A}\otimes g)$ is a unitary in $\mathcal{B}$ with inverse $\Psi_{(e, g^{-1})}(1_\mathcal{A}\otimes g^{-1})$, and that \[U_g U_h=\Psi_{(e, g)}(1_\mathcal{A}\otimes g) \Psi_{(e, h)}(1_\mathcal{A}\otimes h)= \Psi_{(e, gh)}(1_\mathcal{A}\otimes gh)=U_{gh}.\] That is, $U$ is a unitary representation of $G$ in $\mathcal{B}$.
From the definitions of $\psi_p$ and $U_g$, for any $g\in G$ and $p\in P$, we obtain \begin{align*} U_g \psi_p(x) &= \Psi_{(e,g)}(1_\mathcal{A}\otimes g) \Psi_{(p,e)}(x\otimes e) \\ &= \Psi_{(e,g)(p,e)}((1_\mathcal{A}\otimes g)(x\otimes e)) \\
&= \Psi_{(g\cdot p,g|_p)}(\beta_g(x)\otimes g|_p) \\
&= \Psi_{(g\cdot p,e)(e,g|_p)}((\beta_g(x)\otimes e)(1_\mathcal{A}\otimes g|_p)) \\
&= \Psi_{(g\cdot p,e)}(\beta_g(x)\otimes e) \Psi_{(e,g|_p)}(1_\mathcal{A}\otimes g|_p) \\
&= \psi_{g\cdot p}(\beta_g(x)) U_{g|_p}. \end{align*} Thus $(\psi, U)$ satisfies the idenity \eqref{E:Upsi}.
Conversely, given a representation $\psi$ of $X$ and a unitary representation $U$ of $G$ on a unital C*-algebra ${\mathcal{B}}$ that satisfy \eqref{E:Upsi}, define \[ \Psi_{(p, g)}(x\otimes g) := \psi_p(x) U_g\quad\text{for all}\quad p\in P, g\in G, x\in X_p. \] Then one can verify that $\Psi$ is a representation of $Y$. In fact, we have \begin{align*} \Psi_{(p, g)}(x\otimes g) ^*\Psi_{(p, g)}(y\otimes g) &= U_g^*\psi_p(x)^* \psi_p(y) U_g\ (\text{by the definition of } \Psi)\\ &= U_{g^{-1}} \psi_{e}(\langle x, y\rangle) U_g \ (\text{as }\psi \text{ is a representation of } X)\\
&=\psi_{e}(\beta_{g^{-1}}\langle x, y\rangle) U_{g^{-1}|_{e}} U_g \\
&=\psi_{e}(\beta_{g^{-1}}\langle x, y\rangle)\ (\text{as }g^{-1}|_{e}=g^{-1}, U_{g^{-1}|_{e}} U_g=U_{g^{-1}}U_g=I) \\ &=\Psi_{(e,e)}(\beta_{g^{-1}}(\langle x, y\rangle)\otimes e) \\ &=\Psi_{(e,e)}(\langle x\otimes g, y\otimes g\rangle)\ (\text{by the definition of }Y) \end{align*} for all $p\in P$, $g\in G$, $x,y\in X_p$, and \begin{align*} \Psi_{(p, g)}(x\otimes g) \Psi_{(q, h)}(y\otimes h) &=\psi_p(x) U_g\psi_q(y) U_h \ (\text{by definition of } \Psi)\\
&=\psi_p(x) \psi_{g\cdot q}(\beta_g(y)) U_{g|_q} U_h\\
&=\psi_{pg\cdot q}(x\beta_g(y)) U_{g|_q h}\ (\text{as }\psi \text{ is a representation of } X)\\
&=\Psi_{(pg\cdot q, g|_qh)}(x\beta_g(y)\otimes g|_q h)\ (\text{by the definition of } \Psi)\\
&=\Psi_{(pg\cdot q, g|_qh)}((x\otimes g)(y\otimes h)) \end{align*} for all $p,q\in P$, $g\in G$, $x\in X_p$ and $y\in X_q$. \end{proof}
\begin{eg} \label{Eg:Fock} Let $X$ be a product system over a semigroup $P$. Suppose that there is a Zappa-Sz\'{e}p action $\beta$ of $G$ on $X$. There is a natural nontrivial pair $(\psi, U)$ which satisfies all the conditions required in Theorem \ref{T:Upsi}. Indeed, let $F(X)=\bigoplus_{s\in P} X_s$ the Fock space, and $L$ the usual Fock representation: \begin{align*} L_p(x)(\oplus x_s) =\oplus (x\otimes x_s)\quad (x\in X_p, \ \oplus x_s\in F(X)). \end{align*} Then we define an action $\tilde\beta$ of $G$ on $F(X)$ as follows: \[ \tilde\beta_g(\oplus x_s)=\oplus (\beta_g(x_s))\quad (g\in G, \ \oplus x_s\in F(X)). \] Clearly, $L$ is a representation of $X$ and $\tilde\beta$ is a unitary representation of $G$. Also one can easily verify \begin{align*}
\tilde\beta_g\circ L_p(x)=L_{g\cdot p}(\beta_g(x))\circ \tilde\beta_{g|_p}\quad (g\in G, x\in X_p). \end{align*} \end{eg}
We are now ready to define the Toeplitz type universal C*-algebra associated to a Zappa-Sz\'{e}p system $(X,G,\beta)$.
\begin{definition} \label{D:TXG} Let $\mathcal{T}_X\bowtie G$ be the universal C*-algebra generated by covariant representations of $(X,G,\beta)$. \end{definition}
Example \ref{Eg:Fock} shows that the C*-algebra $\mathcal{T}_X\bowtie G$ is nontrivial.
In what follows, we prove a result similar to Theorem \ref{T:Upsi} for the homogeneous Zappa-Sz\'{e}p product system $X\widetilde\bowtie G$.
\begin{theorem} \label{T:Upsi.homo} Suppose that a Zappa-Sz\'{e}p system $(X,G,\beta)$ is homogeneous. Then there is a one-to-one correspondence $\widetilde\Pi$ between the set of all representations $\Psi$ of $X\widetilde\bowtie G$ on a unital C*-algebra $\mathcal{B}$ and the set of all covariant representations $(\psi, U)$ of $(X,G,\beta)$. In fact, for a given representation $\Psi$ of $X\widetilde\bowtie G$, one has \begin{align*} \psi_p(x)&:=\Psi_{p}(x\otimes e)\quad\text{for all}\quad p\in P, x\in X_p,\\ U_g&:=\Psi_{e}(1_\mathcal{A}\otimes g)\quad\text{for all}\quad g\in G. \end{align*} Conversely, given a covariant representation $(\psi, U)$ of $X\widetilde\bowtie G$, one has \[ \Psi_{p}(x\otimes g) := \psi_p(x) U_g\quad\text{for all}\quad p\in P, g\in G, x\in X_p. \] \end{theorem}
\begin{proof} For simplicity, set $Z:=X\widetilde\bowtie G$. Let $\Psi$ be a representation of $Z$ on $\mathcal{B}$. For all $p,q\in P$, $g,h\in G$, and $x\in X_p$, $y\in X_q$,
\[\Psi_p(x\otimes g)\Psi_q(y\otimes h)=\Psi_{pq}(x\beta_g(y)\otimes g|_q h).\]
For $p\in P$, define $\psi_p:X_p\to \mathcal{B}$ by
\[\psi_p(x):=\Psi_p(x\otimes e)\ \text{for all }x\in X_p.\] Here, for $p=e$, we treat $x\otimes e=xu_{e}\in \mathcal{A}\rtimes_\beta G\cong Z_{e}$.
Then,
\begin{align*}
\psi_p(x)^* \psi_p(y) &= \Psi_p(x\otimes e)^* \Psi_p(y\otimes e) \\
&= \Psi_{e}(\langle x\otimes e, y\otimes e\rangle) \\
&= \Psi_{e}(\langle x,y\rangle u_{e})\\
&=\psi_{e}(\langle x,y\rangle) \\
\psi_p(x)\psi_q(y) &= \Psi_p(x\otimes e) \Psi_q(y\otimes e) \\\
&= \Psi_{pq}(x\beta_{e}(y)\otimes e|_q e) \\
&= \Psi_{pq}(xy\otimes e) \\
&=\psi_{pq}(xy).
\end{align*}
Therefore, $\psi$ is a representation of $X$ on $\mathcal{B}$.
For each $g\in G$, set $U_g:=\Psi_{e}(1_\mathcal{A}\otimes g)$. As before, one can easily check that $U$ is a unitary representation of $G$ on $\mathcal{B}$.
For any $g\in G$, $p\in P$ and $x\in X_p$,
\begin{align*}
U_g \psi_p(x) &= \Psi_{e}(1_\mathcal{A}\otimes g) \Psi_p(x\otimes e) \\
&= \Psi_p(\beta_g(x)\otimes g|_p) \\
&= \Psi_p((\beta_g(x)\otimes e)(1_\mathcal{A}\otimes g|_p)) \\
&= \Psi_p(\beta_g(x)\otimes e)\Psi_{e}(1_\mathcal{A}\otimes g|_p) \\
&= \psi_p(\beta_g(x)) U_{g|_p}.
\end{align*}
Conversely, given a representation $\psi$ of $X$ and a unitary representation $U$ of $G$ that satisfy $U_g\psi_p(x)=\psi_p(\beta_g(x)) U_{g|_p}$, define
\[
\Psi_p(x\otimes g):=\psi_p(x) U_g\quad\text{for all}\quad g\in G, p\in P, x\in X_p.
\]
For any $x,y\in X_p$ and $g,h\in G$,
\begin{align*}
\Psi_p(x\otimes g)^* \Psi_p(y\otimes h) &= U_g^* \psi_p^*(x) \psi_p(y) U_h \\
&= U_g^* \psi_{e}(\langle x,y\rangle) U_h \\
&= \psi_{e}(\beta_{g^{-1}}(\langle x,y\rangle)) U_{g^{-1}h} \\
&= \Psi_{e}(\beta_{g^{-1}}(\langle x,y\rangle)\otimes g^{-1} h) \\
&= \Psi_{e}(\langle x\otimes g, y\otimes h\rangle).
\end{align*}
For any $x\in X_p$, $y\in X_q$, and $g,h\in G$,
\begin{align*}
\Psi_p(x\otimes g) \Psi_q(y\otimes h) &= \psi_p(x) U_g \psi_q(y) U_h \\
&= \psi_p(x) \psi_q(\beta_g(y)) U_{g|_q} U_h \\
&= \psi_{pq}(x\beta_g(y)) U_{g|_q h} \\
&= \Psi_{pq}(x\beta_g(y)\otimes g|_q h) \\
&= \Psi_{pq}((x\otimes g)(y\otimes h)).
\end{align*}
Therefore, $\Psi$ is a representation of $Z$. \end{proof}
As an immediate corollary of Theorems \ref{T:Upsi} and \ref{T:Upsi.homo}, the universal $C^*$-algebras of representations of $X\bowtie G$ and covariant representations of $(X,G,\beta)$ must coincide, and similar for $X\widetilde\bowtie G$ if $(X,G,\beta)$ is also homogeneous. Therefore, we have the following Toeplitz type Hao-Ng isomorphism theorem.
\begin{corollary} \label{C:HaoNgT} Let $(X,G,\beta)$ be a Zappa-Sz\'{e}p system. Then \begin{itemize} \item[(i)] $\mathcal{T}_{X\bowtie G} \cong \mathcal{T}_X \bowtie G$; and
\item[(ii)] $\mathcal{T}_{X\bowtie G} \cong \mathcal{T}_X \bowtie G \cong \mathcal{T}_{X\widetilde\bowtie G}$ provided that $(X,G,\beta)$ is homogeneous. \end{itemize} \end{corollary}
\subsection{Cuntz-Pimsner type representations of $(X,G,\beta)$ and the C*-algebra $\mathcal{O}_X\bowtie G$}
In this subsection, we prove that the one-to-one correspondence $\Pi$ in Theorem \ref{T:Upsi} preserves the Cuntz-Pimsner covariance: $\Psi$ is Cuntz-Pimsner covariant if and only if so is $\psi$. Accordingly we obtain the Hao-Ng isomorphism theorem as well in this case. But unfortunately, it is unknown whether the correspondence $\widetilde\Pi$ in Theorem \ref{T:Upsi.homo} preserves the Cuntz-Pimsner covariance.
\begin{proposition}\label{prop.cp1} Let $(X,G,\beta)$ be a Zappa-Sz\'{e}p system and set $Y:=X\bowtie G$. For each $p\in P$ and $g\in G$, define a map $\iota_{p,g}:\mathcal{L}(X_p)\to\mathcal{L}(Y_{(p,g)})$ by \[\iota_{p,g}(T)(x\otimes g)=(Tx)\otimes g.\] Then $\iota_{p,g}$ is an isometric $*$-isomorphism. Moreover, for each rank-one operator $\theta_{x,y}\in \mathcal{K}(X_p)$, $\iota_{p,g}(\theta_{x,y})$ is the rank one operator $\Theta_{x\otimes g, y\otimes g}\in\mathcal{K}(Y_{(p,g)})$, and thus $\iota_{p,g}(\mathcal{K}(X_p))=\mathcal{K}(Y_{(p,g)})$. \end{proposition}
\begin{proof} For any $T\in\mathcal{L}(X_p)$ and $x\in X_p$, $g\in G$, \begin{align*}
\|\iota_{p,g}(T)(x\otimes g)\|^2 &= \|\langle (Tx)\otimes g, (Tx)\otimes g\rangle\| \\
&= \|\beta_{g^{-1}}(\langle Tx, Tx\rangle\| \\
&= \|\langle Tx, Tx\rangle\|=\|Tx\|^2. \end{align*} Thus $\iota_{p,g}$ is isometric. It is clear that $\iota_{p,g}$ is a homomorphism. Moreover, for any $x,y\in X_p$, \begin{align*}
\langle \iota_{p,g}(T^*)x\otimes g, y\otimes g\rangle &= \langle (T^*x)\otimes g, y\otimes g\rangle \\
&= \beta_{g^{-1}}(\langle T^* x,y\rangle)= \beta_{g^{-1}}(\langle x, Ty\rangle) \\
&= \langle x\otimes g, (Ty)\otimes g\rangle = \langle x\otimes g, \iota_{p,g}(T)y\otimes g\rangle. \end{align*} Hence $\iota_{p,g}(T^*)=\iota_{p,g}(T)^*$. Finally, for any $\widetilde{T}\in\mathcal{L}(Y_{(p,g)})$, take any $x\in X_p$ and define $Tx\in X_p$ such that $\widetilde{T}(x\otimes g)=Tx\otimes g$. One can check that $T$ is a $\mathcal{A}$-linear, adjointable operator in $\mathcal{L}(X_p)$, and that $\widetilde{T}=\iota_{p,g}(T)$. Therefore, $\iota_{p,g}$ is an isometric $*$-isomorphism.
Now fix a rank one operator $\theta_{x,y}\in\mathcal{K}(X_p)$. In other words, for any $z\in X_p$, $\theta_{x,y}(z)=x\langle y,z\rangle$. Now for any $g\in G$ and $z\in X_p$, we have \begin{align*} \iota_{p,g}(\theta_{x,y})(z\otimes g)
&= \theta_{x,y}(z)\otimes g
= (x\langle y,z\rangle)\otimes g \\
&= (x\otimes g)(\beta_{g^{-1}}(\langle y,z\rangle)\otimes e) \\
&= (x\otimes g)\langle y\otimes g, z\otimes g\rangle \\
&= \Theta_{x\otimes g, y\otimes g}(z\otimes g). \end{align*} This proves $\iota_{p,g}(\theta_{x,y})=\Theta_{x\otimes g, y\otimes g}$, and therefore $\iota_{p,g}(\mathcal{K}(X_p))=\mathcal{K}(Y_{(p,g)})$ \end{proof}
Let $\phi_p$ and $\Phi_{(p,q)}$ be the left action of $\mathcal{A}$ on $X_p$ and $Y_{(p,g)}$, respectively: \[ \phi_p(a)x=ax\text{ and } \Phi_{(p,g)}(a)(x\otimes g)=ax\otimes g. \]
\begin{theorem}\label{thm.cp} Let $\Psi$ be a representation of $X\bowtie G$ and $(\psi, U)$ be the covariant representation of $(X,G,\beta)$ under the one-to-one correspondence $\Pi$ given in Theorem \ref{T:Upsi}. Then $\Psi$ is Cuntz-Pimsner covariant if and only if so is $\psi$. \end{theorem}
\begin{proof} Suppose yhat $\Psi$ is a Cuntz-Pimsner covariant representation of $Y$. Then by definition, for any $(p,g)\in P\bowtie G$, \[\Psi^{(p,g)}(\Phi_{(p,g)}(a))=\Psi_{(e,e)}(a)\ \text{ for all }a\in\Phi_{(p,g)}^{-1}(\mathcal{K}(Y_{(p,g)})).\] Notice that, for any $p\in P$ and $a\in {\mathcal{A}}$ with $\phi_p(a)\in \mathcal{K}(X_p)$, we have $\Phi_{(p,e)}(a)\in \mathcal{K}(Y_{(p,e)})$. Thus \[\psi^{(p)}(\phi(a))=\Psi^{(p,e)}(\Phi(a))=\Psi_{(e,e)}(a)=\psi_{e}(a).\] Therefore, $\psi$ is Cuntz-Pimser covariant.
Conversely, suppose that $\psi$ is Cuntz-Pimsner covariant. Take $a\in \Phi^{-1}(\mathcal{K}(Y_{(p,g)})$. Without loss of generality, we assume that \[\Phi_{(p,g)}(a)=\Theta_{x\otimes g, y\otimes g}\ \text{for }x,y\in X_p.\] By Proposition \ref{prop.cp1}, \[\Phi_{(p,g)}(a)=\Theta_{x\otimes g, y\otimes g}=\iota_{p,g}(\theta_{x,y})=\iota_{p,g}(\phi_p(a)).\] We have that $\theta_{x,y}=\phi_p(a)$, and so \begin{align*}
\Psi^{(p,g)}(\Phi_{(p,g)}(a)) &= \Psi^{(p,g)}(\Theta_{x\otimes g, y\otimes g}) \\
&= \Psi_{(p,g)}(x\otimes g)\Psi_{(p,g)}(y\otimes g)^* \\
&= \psi_p(x) U_g (\psi_p(y) U_g)^* = \psi_p(x) U_g U_g^* \psi_p(y)^* \\
&= \psi_p(x) \psi_p(y)^* = \psi^{(p)}(\theta_{x,y}) \\
&= \psi^{(p)}(\phi_p(a))
= \psi_{e}(a)=\Psi_{(e,e)}(a). \end{align*} Therefore, $\Psi$ is Cuntz-Pimsner covariant. \end{proof}
\begin{definition} \label{D:OXG} Let $\mathcal{O}_X\bowtie G$ be the universal C*-algebra generated by the set of all covariant representations $(\psi,U)$ of $(X,G,\beta)$ with $\psi$ Cuntz-Pimsner covariant. \end{definition}
As a corollary of Theorem \ref{thm.cp}, the universal C*-algebra of Cuntz-Pimsner covariant representations of $X\bowtie G$ and the universal C*-algebra of covariant representations $(\psi,U)$ with $\psi$ Cuntz-Pimsner covariant of $(X,G,\beta)$ must coincide. Therefore, we have the following Cuntz-Pimsner type Hao-Ng isomorphism theorem.
\begin{corollary} \label{C:HaoNgC} $\mathcal{O}_{X\bowtie G} \cong \mathcal{O}_X\bowtie G$. \end{corollary}
\begin{remark} One might notice that the above corollary has no corresponding part to Corollary \ref{C:HaoNgT} (ii) for the homogeneous case.
In fact, we do not know whether $\mathcal{O}_{X\widetilde\bowtie G}\cong \mathcal{O}_X\bowtie G$, although $\mathcal{O}_{X\widetilde\bowtie G}$ is generally a quotient of $\mathcal{O}_{X\bowtie G}$ (see Corollary \ref{C:2bowtie} below). Even in the special situation of a homogeneous Zappa-Sz\'{e}p action of $G$ on $X$ where $g|_p=g$ for all $g\in G$ and $p\in P$, the Zappa-Sz\'{e}p homogeneous product system $X\widetilde\bowtie G$ becomes a crossed product $X\rtimes G$. In such a case, the problem whether $\mathcal{O}_{X\rtimes G}\cong \mathcal{O}_X\rtimes G$ is known as the Hao-Ng isomorphism problem in the literature. The isomorphism is known in several special cases (see, for example, \cite{HaoNg2008} and more recent approaches from non-self-adjoint operator algebras \cite{DOK20, K2017, KR19}). \end{remark}
\begin{corollary} \label{C:2bowtie} If a Zappa-Sz\'{e}p system $(X,G,\beta)$ is homogeneous, then there is a natural epimorphism from $\mathcal{O}_{X\bowtie G}$ to $\mathcal{O}_{X\widetilde\bowtie G}$. \end{corollary}
\begin{proof} By Corollary \ref{C:HaoNgT}, there is a natural Cuntz-Pimsner covariant representation of $X\bowtie G$ on ${\mathcal{O}}_{X\widetilde\bowtie G}$: $X\bowtie G\hookrightarrow \mathcal{T}_{X\bowtie G}\cong \mathcal{T}_{X\widetilde\bowtie G} \twoheadrightarrow {\mathcal{O}}_{X\widetilde\bowtie G}$. By the universal property of ${\mathcal{O}}_{X\bowtie G}$, this gives a homomorphism from $\mathcal{O}_{X\bowtie G}$ onto $\mathcal{O}_{X\widetilde\bowtie G}$. \end{proof}
\subsection{Nica-Toeplitz representations of $X\bowtie G$ and the C*-algebra $\mathcal{N}\mathcal{T}_X\bowtie G$}
Suppose that $P$ is a right LCM semigroup and that $G$ is a group. Then the Zappa-Sz\'{e}p product semigroup $P\bowtie G$ is known to be right LCM as well \cite{BRRW}. In particular, for any $(p,g), (q,h)\in P\bowtie G$, we have that \[(p,g)P\bowtie G\cap (q,h)P\bowtie G = \begin{cases} (r,k)P\bowtie G & \text{if } pP\cap qP=rP, k\in G,\\ \emptyset & \text{otherwise}. \end{cases} \] Here, the choice of $k$ can be arbitrary since $(e,k)$ is invertible in $P\bowtie G$ and thus $(r,k)P\bowtie G=(r,e)P\bowtie G$ for all $k\in G$.
We first prove that, for a given Zappa-Sz\'{e}p system $(X,G,\beta)$ where $X$ is compactly aligned, the product system $X\bowtie G$ is compactly aligned as well.
\begin{proposition} Let $P$ be a right LCM semigroup and $X$ a compactly aligned product system over $P$. Then, for a given Zappa-Sz\'{e}p system $(X,G,\beta)$, $X\bowtie G$ is also compactly aligned. \end{proposition}
\begin{proof} As before, let $Y:=X\bowtie G$. For each $p\in P$ and $g\in G$, define $\iota_{p,g}:\mathcal{L}(X_p)\to\mathcal{L}(Y_{(p,g)})$ by $\iota_{p,g}(T)(x\otimes g)=(Tx)\otimes g$. By Proposition \ref{prop.cp1}, $\iota_{p,g}$ is an isometric $*$-isomorphism, and $\iota_{p,g}(\mathcal{K}(X_p))=\mathcal{K}(Y_{(p,g)})$. Fix $p,q\in P$ with $pP\cap qP=rP$. For any $g,h\in G$, we have that $(p,g)P\bowtie G\cap (q,h)P\bowtie G=(r,e)P\bowtie G$. Let $p^{-1}r$ be the unique element in $P$ such that $p(p^{-1}r)=p$.
For any $S\in\mathcal{K}(X_p)$ and any $x\in X_p$ and $y\in X_{p^{-1}r}$, one has \begin{align*}
\iota_{r,e}\circ i_p^r(S)(xy\otimes e)
&= (i_p^r(S)(xy))\otimes e\\
&= (Sx)y \otimes e \\
&= ((Sx)\otimes g)(\beta_{g^{-1}}(y)\otimes g_0)\ (\text{with }g_0:=(g|_{g^{-1}\cdot (p^{-1}r)})^{-1})\\
&= \left(\iota_{p,g}(S)(x\otimes g)\right)(\beta_{g^{-1}}(y)\otimes g_0)\\
&= i_{(p,g)}^{(r,e)}\circ \iota_{p,g}(S)(xy\otimes e). \end{align*}
Similarly, we have that $\iota_{r,e}\circ i_q^r=i_{(q,h)}^{(r,e)}\circ \iota_{q,h}$. Therefore, the following diagram commutes:
Now for any $S\in\mathcal{K}(Y_{(p,g)})$ and $T\in\mathcal{K}(Y_{(q,h)})$, by Propostion \ref{prop.cp1}, there exist $S_0\in\mathcal{K}(X_p)$ and $T_0\in\mathcal{K}(X_q)$ such that $\iota_{p,g}(S_0)=S$ and $\iota_{q,h}(T_0)=T$. Therefore, \begin{align*}
i_{(p,g)}^{(r,e)}(S)i_{(q,h)}^{(r,e)}(T) &= i_{(p,g)}^{(r,e)}(\iota_{p,g}(S_0))i_{(q,h)}^{(r,e)}(\iota_{q,h}(T_0)) \\
&= \iota_{r,e}(i_p^r(S_0))\iota_{r,e}(i_q^r(T_0)) \\
&= \iota_{r,e}(i_p^r(S_0)i_q^r(T_0)). \end{align*} Since $X$ is compactly aligned, $i_p^r(S_0)i_q^r(T_0)\in \mathcal{K}(X_r)$. Since $\iota_{r,e}$ maps $\mathcal{K}(X_r)$ to $\mathcal{K}(Y_{(r,e)})$, we have that \[ i_{(p,g)}^{(r,e)}(S)i_{(q,h)}^{(r,e)}(T)=\iota_{r,e}(i_p^r(S_0)i_q^r(T_0))\in\mathcal{K}(Y_{(r,e)}). \qedhere\] \end{proof}
\begin{theorem} \label{T:Ncov} Suppose that $X$ is a compactly aligned product system over a right LCM semigroup $P$. Let $(X,G,\beta)$ be a Zappa-Sz\'{e}p system, $\Psi$ a representation of $X\bowtie G$, and $(\psi, U)$ the covariant representation of $(X,G,\beta)$ under the one-to-one correspondence $\Pi$ given in Theorem \ref{T:Upsi}. Then $\Psi$ is Nica-covariant if and only if so is $\psi$. \end{theorem}
\begin{proof} For any $p\in P$ and $g\in G$, we first show that $\Psi^{(p,g)}\circ \iota_{p,g}=\psi^{(p)}$. For any $\theta_{x,y}\in \mathcal{K}(X_p)$, $\psi^{(p)}(\theta_{x,y})=\psi_p(x) \psi_p(y)^*$. By Proposition \ref{prop.cp1}, $\iota_{p,g}(\theta_{x,y})=\Theta_{x\otimes g, y\otimes g}$, and thus \begin{align*} \Psi^{(p,g)}(\iota_{p,g}(\theta_{x,y})) &= \Psi_{(p,g)}(x\otimes g) \Psi_{(p,g)}(y\otimes g)^* \\ &=\psi_p(x) U_g U_g^* \psi_p(y)^* \ (\text{by }\eqref{E:DefPsi})\\ &= \psi_p(x)\psi_p(y)^* = \psi^{(p)}(\theta_{x,y}). \end{align*} In other words, the following diagram commutes:
Suppose that $\psi$ is Nica-covariant. For any $S\in \mathcal{K}(Y_{(p,g)})$ and $T\in \mathcal{K}(Y_{(q,h)})$, there exist $S_0\in\mathcal{K}(X_p)$ and $T_0\in\mathcal{K}(X_q)$ such that $\iota_{p,g}(S_0)=S$ and $\iota_{q,h}(T_0)=T$. Then we have \[ \Psi^{(p,g)}(S)\Psi^{(q,h)}(T) = \psi^{(p)}(S_0) \psi^{(q)}(T_0).\] If $(p,g)P\bowtie G\cap (q,h)P\bowtie G=\emptyset$, then $pP\cap qP=\emptyset$ and thus $\psi^{(p)}(S_0) \psi^{(q)}(T_0)=0$ by the Nica-covariance of $\psi$. If $(p,g)P\bowtie G\cap (q,h)P\bowtie G=(r,k)P\bowtie G$, then $rP=pP\cap qP$ and thus \begin{align*}
\Psi^{(p,g)}(S)\Psi^{(q,h)}(T) &= \psi^{(p)}(S_0) \psi^{(q)}(T_0) \\
&= \psi^{(r)}(i_p^r(S_0) i_q^r(T_0)) \\
&= \Psi^{(r,k)}(\iota_{r,k}(i_p^r(S_0) i_q^r(T_0))) \\
&= \Psi^{(r,k)}(i_{(p,g)}^{(r,k)}(\iota_{p,g}(S_0)) i_{(q,h)}^{(r,k)}(\iota_{q,h}(T_0))) \\
&= \Psi^{(r,k)}(i_{(p,g)}^{(r,k)}(S) i_{(q,h)}^{(r,k)}(T)). \end{align*} Therefore, $\Psi$ is also Nica-covariant. The converse is clear since $\psi$ is the restriction of a Nica-covariant representation $\Psi$ on $X\cong \bigsqcup_{p\in P} Y_{(p,e)}$. \end{proof}
\begin{definition} \label{D:NTXG} Let $\mathcal{N}\mathcal{T}_X\bowtie G$ be the universal C*-algebra generated by the set of covariant representations $(\psi,U)$ of a Zappa-Sz\'{e}p system $(X,G,\beta)$ with $\psi$ Nica-covariant. \end{definition}
It follows from \cite[Proposition 6.8]{BLS2018b} and Example \ref{Eg:Fock} that $\mathcal{N}\mathcal{T}_X\bowtie G$ is always nontrivial.
Completely similar to Corollaries \ref{C:HaoNgT} and \ref{C:HaoNgC}, we have the following Nica-Toeplitz type Hao-Ng isomorphism theorem.
\begin{corollary} \label{C:HaoNgNT} $\mathcal{N}\mathcal{T}_{X\bowtie G} = \mathcal{N}\mathcal{T}_X\bowtie G$. \end{corollary}
\section{Examples}
\label{S:EX}
In this section, we provide some examples of Zappa-Sz\'{e}p actions of groups $G$ on product systems $X=\bigsqcup_{p\in P} X_p$ and the associated C*-algebras.
\begin{eg} Let $(X,G,\beta)$ be a Zappa-Sz\'{e}p system. \begin{itemize} \item[(i)] If $P$ is trivial, then ${\mathcal{T}}_X \bowtie G\cong {\mathcal{O}}_X \bowtie G\cong{\mathcal{A}}\rtimes_\beta G$.
\item[(ii)] If $G$ is trivial, then ${\mathcal{T}}_X \bowtie G\cong{\mathcal{T}}_X$ and ${\mathcal{O}}_X \bowtie G\cong{\mathcal{O}}_X$. Furthermore, if $X$ is a compact aligned product system over a right LCM semigroup, then
${\mathcal{N}}{\mathcal{T}}_X \bowtie G\cong{\mathcal{N}}{\mathcal{T}}_X$.
\end{itemize} \end{eg}
\begin{eg}
Suppose that $(X,G,\beta)$ is a homogenous Zappa-Sz\'{e}p system. If furthermore $g|_p=g$ for all $g\in G$ and $p\in P$, then
$X\bowtie G$ becomes the crossed product $X\rtimes G$, and ${\mathcal{T}}_X\bowtie G$ (resp.~${\mathcal{O}}_X\bowtie G$) is the crossed product ${\mathcal{T}}_X\rtimes G$ (resp.~${\mathcal{O}}_X\rtimes G$).
So it follows from Corollary \ref{C:HaoNgT} (resp.~Corollary \ref{C:HaoNgC}) that we have the Hao-Ng isomorphisms:
${\mathcal{T}}_X\rtimes G\cong T_{X\rtimes G}$ (resp. ${\mathcal{O}}_X\rtimes G\cong {\mathcal{O}}_{X\rtimes G}$).
Furthermore, if $P$ is right LCM and $X$ is compactly aligned, then ${\mathcal{N}}{\mathcal{T}}_X\bowtie G$ is the crossed product ${\mathcal{N}}{\mathcal{T}}_X\rtimes G$, and so by
the corresponding Hao-Ng isomorphism becomes
$
{\mathcal{N}}{\mathcal{T}}_{X\rtimes G}\cong {\mathcal{N}}{\mathcal{T}}_X\rtimes G
$
(cf. Corollary \ref{C:HaoNgNT}).
\end{eg}
\begin{eg} \label{Eg:gss}
Consider the trivial product system $X_P :=\bigsqcup_{p\in P} {\mathbb{C}} v_p$ over $P$.
For $g\in G$, let \begin{align} \label{E:tri} \beta_g(\lambda v_p)=\lambda v_{g\cdot p} \quad\text{for all}\quad \lambda\in{\mathbb{C}}\text{ and } p\in P. \end{align} It is easy to check that $\beta$ is a Zappa-Sz\'{e}p action of $G$ on $X_P$. Then we obtain the Zappa-Sz\'{e}p product system $X_P\bowtie G$ by Theorem \ref{T:ZSP}.
Suppose that $P$ is right LCM. By \cite[Theorem 4.3]{BRRW} one has \[
{\mathcal{N}}{\mathcal{T}}_X \bowtie G\cong \mathrm{C}^*(P\bowtie G).
\]
Thus, in this case, by the Hao-Ng isomorphism theorems in Section \ref{S:main} we have
\begin{align*} \mathrm{C}^*(P)\bowtie G\cong {\mathcal{N}}{\mathcal{T}}_X \bowtie G\cong {\mathcal{N}}{\mathcal{T}}_{X \bowtie G}\cong \mathrm{C}^*(P\bowtie G).
\end{align*} Let us remark that the above covers the examples studied in \cite{BRRW}.
\end{eg}
\begin{eg} \label{Eg:resss} Let $G$ be a group and $\Lambda$ be a row-finite $k$-graph. Let $(G,\Lambda)$ be a self-similar $k$-graph (\cite{LY-IMRN, LY-JFA})
such that for each $g\in G$ \begin{align} \label{E:res}
d(\mu)=d(\nu)\implies g|_\mu=g|_\nu. \end{align} Then one can construct a Zappa-Sz\'{e}p product ${\mathbb{N}}^k\bowtie G$ as follows. For $p\in {\mathbb{N}}^k$, take $\mu \in \Lambda^p$ and define \[
g\cdot p := p,\ g|_p:=g|_\mu. \]
Let $X(\Lambda)=\bigsqcup_{p\in {\mathbb{N}}^k} X(\Lambda^p)$ be the $C(\Lambda^0)$-product systems over ${\mathbb{N}}^k$ associated to $\Lambda$ (see \cite{RS05} for all related details). Define \[ \beta_g: X(\Lambda^p) \to X(\Lambda^p), \quad \beta_g(\chi_\mu)=\chi_{g\cdot\mu} \] for all $g\in G$ and $\mu \in \Lambda^p$. Then $(X(\Lambda), G, \beta)$ is a Zappa-Sz\'{e}p system. Indeed, it suffices to check that it satisfies (A5) and (A6). For (A5), let $\mu\in \Lambda^p$ and $\nu\in\Lambda^q$. Then \begin{align*} \beta_g(\chi_\mu \chi_\nu) &= \delta_{s(\mu),r(\nu)}\, \chi_{g\cdot(\mu\nu)}
=\delta_{s(\mu),r(\nu)}\chi_{g\cdot\mu (g|_\mu\cdot \nu)}
=\delta_{s(g\cdot\mu),r(g_\mu\cdot \nu)}\, \chi_{g\cdot\mu} \chi_{(g|_\mu\cdot \nu)} \\
&=\beta_g(\chi_\mu)\beta_{g|_\mu}(\nu)
=\beta_g(\chi_\mu)\beta_{g|_p}(\nu). \end{align*} For (A6), let $\mu,\nu\in \Lambda^p$ and $v\in \Lambda^0$. We have \begin{align*} \langle \chi_{g\cdot \mu}, \chi_{g\cdot \nu}\rangle(v) &= \sum_{v=r(\lambda)} \chi_{g\cdot \mu}(\lambda) \chi_{g\cdot \nu}(\lambda)
= |\{g\cdot \mu = g\cdot \nu, r(g\cdot \mu)=v\}|\\
&=|\{\mu = \nu, g\cdot r(\mu)= v\}|
=|\{\mu = \nu, g|_\mu\cdot r(\mu)= v\}|\\
&=|\{\mu = \nu, r(\mu)=(g|_\mu)^{-1}\cdot v\}|
=\langle \chi_{\mu}, \chi_{\nu}\rangle((g|_\mu)^{-1}\cdot v) \\
&=\beta_{g|_\mu}(\langle \chi_\mu, \chi_\nu\rangle)(v)
=\beta_{g|_p}(\langle \chi_\mu, \chi_\nu\rangle(v). \end{align*}
Therefore, by Theorem \ref{T:ZSP}, we obtain a Zappa-Sz\'{e}p product $X(\Lambda)\bowtie G$ over ${\mathbb{N}}^k\bowtie G$. Also one can see that ${\mathcal{O}}_{X(\Lambda)\bowtie G}\cong {\mathcal{O}}_{X(\Lambda)}\bowtie G \cong {\mathcal{O}}_{G, \Lambda}$, where ${\mathcal{O}}_{G,\Lambda}$ is the self-similar $k$-graph C*-algebra associated with $(G,\Lambda)$ (\cite{LY-IMRN}). \end{eg}
\begin{rem}
(i) One can easily see that the condition \eqref{E:res} is equivalent to $g|_\mu=g|_\nu$ for all \textit{edges} $\mu,\nu\in\Lambda$.
(ii) At first sight, the condition \eqref{E:res} seems restrictive. But it is not hard to find self-similar $k$-graphs satisfying \eqref{E:res}. For example, let
$g|_\mu=g|_\nu=:g$. Also, one can invoke \cite[Lemma 3.6]{LY-IMRN} to obtain more examples. \end{rem}
We finish the paper by the following example, which does not really belong to Zappa-Sz\'{e}p actions considered in this paper. But it presents another natural way to construct a product system over ${\mathbb{N}}^k$ which is isomorphic to the product system $X_{G,\Lambda}$ defined in \cite[Section~4]{LY-IMRN}. This provides one of our motivations of considering homogeneous actions in Section~\ref{S:homog}, and so we decide to include it here.
\begin{eg} Let $(G,\Lambda)$ be a self-similar $k$-graph. Let $X(\Lambda)=\bigsqcup_{p\in {\mathbb{N}}^k} X(\Lambda^p)$ be the product systems over ${\mathbb{N}}^k$ associated to $\Lambda$ as above. Define \[ \beta_g: X(\Lambda^p) \to X(\Lambda^p), \quad \beta_g(\chi_\mu)=\chi_{g\cdot\mu}. \] Then one can easily check that \begin{enumerate} \item[(A1)$'$] for each $p\in P$, $\beta_g: X(\Lambda^p) \to X(\Lambda^p)$ is a $\mathbb{C}$-linear bijection; \item[(A2)$'$] for any $g,h\in G$, $\beta_g\circ\beta_h=\beta_{gh}$; \item[(A3)$'$] the map $\beta_{e}$ is the identity map; \item[(A4)$'$] the map $\beta_g$ is a $*$-automorphism on $X(\Lambda^0)=C(\Lambda^0)$; \item[(A5)$'$] for $p,q\in P$, $\mu\in \Lambda_p$ and $\nu\in \Lambda_q$, \[
\beta_g(\chi_\mu \chi_\nu)=\beta_g(\chi_\mu)\beta_{g|_\mu}(\chi_\nu); \]
\item[(A6)$'$] for $p\in P$ and $\mu,\nu\in \Lambda_p$, \[e
\langle\beta_g(\chi_\mu), \beta_g(\chi_\nu)\rangle=\beta_{g|_\mu}(\langle \chi_\mu, \chi_\nu\rangle). \] \end{enumerate}
It follows from (A1)$'$-(A4)$'$ that $(C(\Lambda^0), G)$ is a C*-dynamical system. Let $\mathfrak{A}:=C(\Lambda^0)\rtimes_\beta G$. In what follows, we sketch the construction of an $\mathfrak{A}$-product system $\mathcal{E}$ over ${\mathbb{N}}^k$, which is isomorphic to the product system $X_{G,\Lambda}$ defined in \cite[Section~4]{LY-IMRN}. The details are similar to those in Section~\ref{S:homog} above and left to the interested reader. For $p\in {\mathbb{N}}^k$, we construct a C*-correspondence ${\mathcal{E}}_p$ over $\mathfrak{A}$. For $x= \chi_v u_h$, $\xi=\chi_\mu u_g$ and $\eta=\chi_\nu u_h$ with $\xi_\mu, \eta_\nu\in X_p$, define \begin{align*} \xi x := \delta_{s(\mu), g\cdot v} \chi_\mu u_{gh},\
x\xi := \delta_{v, h\cdot r(\mu)} \chi_{h\cdot \mu} u_{h|_\mu g},\ \langle \xi, \eta\rangle :=u_{g^{-1}}\langle \chi_\mu, \chi_\nu\rangle u_h. \end{align*} Let $\mathcal{E}_p$ be the closure of the linear span of $\chi_\mu u_g$ with $\mu \in \Lambda^p$ and $g\in G$. Set \[ \mathcal{E}:=\bigsqcup_{p\in {\mathbb{N}}^k} \mathcal{E}_p. \] For $\xi=\sum \chi_\mu u_g\in \mathcal{E}_p$ and $\eta=\sum \chi_\nu u_h\in \mathcal{E}_q$ define \begin{align*}
\big(\sum \chi_\mu u_g\big)\big(\sum \chi_\nu u_h\big):=\sum \chi_\mu \chi_{g\cdot \nu} u_{g|_\nu} u_h=\sum \delta_{s(\mu), r(g\cdot \nu)}\chi_{\mu g\cdot \nu} u_{g|_\nu h}. \end{align*} Then $\mathcal{E}$ is an $\mathfrak{A}$-product system over ${\mathbb{N}}^k$. One can check that $\mathcal{E}$ is isomorphic to the product system $X_{G,\Lambda}$ defined in \cite[Section~4]{LY-IMRN} via the map \[
\chi_\mu u_g\in \mathcal{E}_p\mapsto \chi_\mu j(g)\in X_{G,\Lambda,p}. \] \end{eg}
\end{document} | arXiv |
The perfect squares from $1$ through $2500,$ inclusive, are printed in a sequence of digits $1491625\ldots2500.$ How many digits are in the sequence?
We consider it by four cases:
$\bullet$ Case 1: There are $3$ perfect squares that only have $1$ digit, $1^{2},$ $2^{2},$ and $3^{2}.$
$\bullet$ Case 2: The smallest perfect square that has $2$ digits is $4^{2},$ and the largest is $9^{2},$ so that's a total of $6$ perfect squares with $2$ digits.
$\bullet$ Case 3: The smallest perfect square with $3$ digits is $10^{2},$ and the largest is $31^{2},$ yielding a total of $22.$
$\bullet$ Case 4: The smallest perfect square with $4$ digits is $32^{2},$ and the last one that is no greater than $2500$ is $50^{2},$ giving a total of $19.$
So we have a total of $1\times3+2\times6+3\times22+4\times19=\boxed{157}$ digits. | Math Dataset |
Kenmotsu manifold
In the mathematical field of differential geometry, a Kenmotsu manifold is an almost-contact manifold endowed with a certain kind of Riemannian metric. They are named after the Japanese mathematician Katsuei Kenmotsu.
Definitions
Let $(M,\varphi ,\xi ,\eta )$ be an almost-contact manifold. One says that a Riemannian metric $g$ on $M$ is adapted to the almost-contact structure $(\varphi ,\xi ,\eta )$ if:
${\begin{aligned}g_{ij}\xi ^{j}&=\eta _{i}\\g_{pq}\varphi _{i}^{p}\varphi _{j}^{q}&=g_{ij}-\eta _{i}\eta _{j}.\end{aligned}}$
That is to say that, relative to $g_{p},$ the vector $\xi _{p}$ has length one and is orthogonal to $\ker \left(\eta _{p}\right);$ furthermore the restriction of $g_{p}$ to $\ker \left(\eta _{p}\right)$is a Hermitian metric relative to the almost-complex structure $\varphi _{p}{\big \vert }_{\ker \left(\eta _{p}\right)}.$ One says that $(M,\varphi ,\xi ,\eta ,g)$ is an almost-contact metric manifold.[1]
An almost-contact metric manifold $(M,\varphi ,\xi ,\eta ,g)$ is said to be a Kenmotsu manifold if[2]
$\nabla _{i}\varphi _{j}^{k}=-\eta _{j}\varphi _{i}^{k}-g_{ip}\varphi _{j}^{p}\xi ^{k}.$
References
1. Blair 2010, p. 44.
2. Blair 2010, p. 98.
Sources
• Blair, David E. (2010). Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics. Vol. 203 (Second edition of 2002 original ed.). Boston, MA: Birkhäuser Boston, Ltd. doi:10.1007/978-0-8176-4959-3. ISBN 978-0-8176-4958-6. MR 2682326. Zbl 1246.53001.
• Kenmotsu, Katsuei (1972). "A class of almost contact Riemannian manifolds". Tohoku Mathematical Journal. Second Series. 24 (1): 93–103. doi:10.2748/tmj/1178241594. MR 0319102. Zbl 0245.53040.
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| Wikipedia |
Equivalent of Fourier transform for Gaussians?
I have a long signal (million of samples) containing a lot of Gaussian peaks, whose standard deviation is random and about $5$ to $50$ samples wide. Sometimes, these peaks overlap, but not often. The signal also contains high frequency noise.
I would like to compute the distribution of standard deviations, in a similar way one can extract the "distribution of frequencies" using FFT. Currently, we do a first pass for detecting the peaks, then do a classical function fit in the peak region. This is not very robust and the distribution fluctuates a lot depending on the fitting algorithm fine tuning.
Is there a more robust approach which would do a kind of a Gaussian transform?
fft gaussian
galinettegalinette
$\begingroup$ Can you please clarify the concept of "distribution of gaussians"? Are you after a " spectrum" whose X-axis is somehow related to the "kind" of Gaussian (thin and tall, fat and long) and the Y-axis is related to the "amount of Gaussian"? Might have to be 2D, one d for the "mean" (position) and one d for the "st.dev" (width). Have you considered some form of matched filtering ? $\endgroup$
– A_A
$\begingroup$ By distribution, I mean on the experimental signal, computing an histogram of the standard deviations. The histogram would have standard deviations as X-axis bins, and energy (gaussian areas) as Y-axis $\endgroup$
– galinette
$\begingroup$ I mean distribution, because from that experimental histogram data, I want to build a continuous distribution, and generate a random signal that will match the standard deviation statistics of the experimental signal. $\endgroup$
$\begingroup$ I believe this is called Gaussian Pulse Decomposition? ncbi.nlm.nih.gov/pubmed/9084830 $\endgroup$
– endolith
The (continuous) Fourier transform of a Gaussian is a Gaussian:
$$ \mathscr{F}\Big\{ x(t) \Big\} \triangleq \int\limits_{-\infty}^{\infty} x(t) \, e^{-j 2 \pi f t} \, dt $$
$$ \mathscr{F}\left\{ e^{-\pi t^2} \right\} = e^{-\pi f^2} $$
$$ \mathscr{F}\left\{ e^{-\pi \alpha t^2} \right\} = \frac{1}{\sqrt{\alpha}} e^{-\frac{\pi}{\alpha} f^2} \qquad \alpha > 0 $$
$$ \mathscr{F}\left\{ e^{-\pi \alpha (t-\tau)^2} \right\} = \frac{1}{\sqrt{\alpha}} e^{-\frac{\pi}{\alpha} f^2} e^{-j 2 \pi f \tau} $$
$$ \mathscr{F}\left\{ \sum_m c_m e^{-\pi \alpha_m (t-\tau_m)^2} \right\} = \sum_m \frac{c_m}{\sqrt{\alpha_m}} e^{-\frac{\pi}{\alpha_m} f^2} e^{-j 2 \pi f \tau_m} $$
note these gaussians have "tails" that go on forever, but gaussians usually die off to very close to zero very rapidly.
assuming no time or frequency aliasing, let $x[n] \triangleq x\big(n/f_\text{s}\big)$ where $f_\text{s}$ is the sample rate. Then the DTFT is
$$\begin{align} X\big(e^{j\omega}\big) &\triangleq \sum_{n=-\infty}^{\infty} x[n] \, e^{-j \omega n} \\ &= f_\text{s} \sum_{n=-\infty}^{\infty} x\big(n/f_\text{s}\big) \, e^{-j f_\text{s} \omega (n/f_\text{s})} \frac{1}{f_\text{s}} \\ & \approx f_\text{s} \int\limits_{-\infty}^{+\infty} x(t) \, e^{-j 2 \pi \frac{f_\text{s} \omega}{2 \pi} t} \, dt \\ &= f_\text{s} \ \mathscr{F}\Big\{ x(t) \Big\} \Bigg|_{f=f_\text{s}\frac{\omega}{2 \pi}} \\ \end{align}$$
so if we let $$x(t) = \sum_m c_m e^{-\pi \alpha_m (t-\tau_m)^2}$$
$$\begin{align} x[n] &= \sum_m c_m e^{-\pi \alpha_m ((n/f_\text{s})-\tau_m)^2} \\ &= \sum_m c_m e^{-\pi (\alpha_m/f_\text{s}^2) (n-f_\text{s}\tau_m)^2} \\ &= \sum_m c_m e^{-\pi \widehat{\alpha_m} (n-\widehat{\tau_m})^2} \\ \end{align}$$
then the DTFT is
$$\begin{align} X\big(e^{j\omega}\big) & \approx f_\text{s} \ \sum_m \frac{c_m}{\sqrt{\alpha_m}} e^{-\frac{\pi}{\alpha_m} f^2} e^{-j2\pi f\tau_m} \Bigg|_{f=f_\text{s}\frac{\omega}{2 \pi}} \\ &= f_\text{s} \ \sum_m \frac{c_m}{\sqrt{\alpha_m}} e^{-\frac{\pi}{\alpha_m} \left(f_\text{s}\frac{\omega}{2 \pi}\right)^2} e^{-j2\pi \left( f_\text{s}\frac{\omega}{2 \pi} \right)\tau_m} \\ &= f_\text{s} \ \sum_m \frac{c_m}{\sqrt{\alpha_m}} e^{-\pi\frac{f_\text{s}^2}{4 \pi^2 \alpha_m} \omega^2} e^{-j \omega (f_\text{s}\tau_m)} \\ \end{align}$$
Then, assuming you choose your DFT length $N$ to be sufficiently large (you're saying in the millions) to cover all of the gaussians with their specific delays $f_\text{s}\tau_m$ in samples and widths of ca. $\frac{f_\text{s}}{\sqrt{\alpha_m}}$. Then the $N$-point DFT will sample this DTFT at $N$ equally spaced points around the unit circle.
$$\begin{align} \mathcal{DFT:} \quad X[k] &= \sum_{n=0}^{N-1} x[n] \, e^{-j 2 \pi \frac{nk}{N}} \\ &= X\big(e^{j\omega}\big) \Bigg|_{\omega = 2\pi\frac{k}{N}} \quad \mathcal{:DTFT}\\ &= f_\text{s} \ \sum_m \frac{c_m}{\sqrt{\alpha_m}} e^{-\pi\frac{f_\text{s}^2}{4 \pi^2 \alpha_m} \omega^2} e^{-j \omega (f_\text{s}\tau_m)} \Bigg|_{\omega = 2\pi\frac{k}{N}}\\ &= \ \sum_m \frac{c_m}{\sqrt{\alpha_m/f_\text{s}^2}} e^{-\pi\frac{f_\text{s}^2/\alpha_m}{4 \pi^2} \left(2\pi\frac{k}{N}\right)^2} e^{-j 2\pi\frac{k}{N} (f_\text{s}\tau_m)} \\ &= \sum_m \frac{c_m}{\sqrt{\widehat{\alpha_m}}} e^{-\frac{\pi}{\widehat{\alpha_m}} \left(\frac{k}{N}\right)^2} e^{-j 2\pi\frac{k}{N} \widehat{\tau_m}} \\ \end{align} $$
So summing up and I will remove the little hats in the discrete-time domain, if you have a collection of gaussian pulses with heights of $c_m$, peak width parameters $\frac{1}{\sqrt{\alpha_m}}$ (in units of samples), and peak positions at $\tau_m$ (also in units of samples), then your input should look like:
$$ x[n] = \sum_m c_m e^{-\pi \alpha_m (n-\tau_m)^2} $$
and the result of the $N$-point DFT should look like:
$$ X[k] \approx \sum_m \frac{c_m}{\sqrt{\alpha_m}} \ e^{-\pi/(N^2\alpha_m) k^2} \ e^{-j 2\pi(\tau_m/N) k} $$
for $-\tfrac{N}{2} < k < \tfrac{N}{2}$ . Remember with the DFT, $X[k+N]=X[k]$ for all integer $k$.
From your description, you have a signal composed of high-frequency noise (more simply put, white noise) plus of a fluctuating signal whose auto-correlation is about $5$ to $50$ samples. This all seems to be perfectly adapted for a Fourier analysis!
Your fitting method seems right but perhaps your modeling is perhaps wrong. The latter signal can be modeled as a noise term with a given envelope and a random phase. Using a log-normal distribution for instance, $$ \mathcal{E}(f) = W + A\cdot\exp(-\frac{\log(f/f_0)}{2\cdot B^2}) $$ where $W$ is the level of white noise and $A$ is that of the "peaks". From the Parseval theorem, there is an equivalence relation between the auto-correlation's width and that of the spectrum: the wider the spectrum's bandwidth $B$ around the peak $f_0$, the shorter the autocorrelation's width. As a consequence, given a good parameterization of the spectrum, you will easily find a function to perform a good fit within a window where the signal is known to be stationary.
Given that, it is then possible to extract the given peaks, but that is certainly another story...
meduzmeduz
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On the isomorphism problem of enveloping algebras
Let $\mathfrak{g}$ and $\mathfrak{g}'$ be Lie algebras. It is known that if $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as associative algebras, then it is not necessarily true that $\mathfrak{g}\cong \mathfrak{g}'$ as Lie algebras.
I am looking for examples such that $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as algebras but $\mathfrak{g}\not\cong \mathfrak{g}'$ as Lie algebras (over an algebraically closed field). Moreover, are there examples such that the categories $U(\mathfrak{g})-\text{Mod}$ and $U(\mathfrak{g}')-\text{Mod}$ are not monoidally equivalent?
I'm not very familiar with the isomorphism problem for enveloping algebras, a quick google search only gave me counterexamples in positive characteristic. I'd be very happy with examples in characteristic zero (infinite dimensions are allowed). I'm more into the monoidal stuff and might figure out myself whether the representation categories are monoidally equivalent.
Edit: I'm asking this because I naturally encountered a quantized version of this problem. Obviously the categories $U(\mathfrak{g})-\text{Mod}$ and $U(\mathfrak{g}')-\text{Mod}$ are Morita equivalent but there is more information here. First of all $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as algebras which clearly is stronger but they are also enveloping algebras of Lie algebras, further restricting possibilities. In the quantized version I'm looking at, I suspect the representation rings of both categories to be the same making the difference in the monoidal structure very subtle. So I'm wondering whether anything on this subject is known in the non-quantized world.
rt.representation-theory lie-algebras hopf-algebras monoidal-categories
Mathematician 42
Mathematician 42Mathematician 42
In the recent paper Lie, associative, and commutative quasi-isomorphism, R. Campos, D. Petersen, F. Wierstra, and I settled the question above for nilpotent Lie algebras: if two nilpotent Lie algebras have universal enveloping algebras that are isomorphic as unital associative algebras, then the two Lie algebras also are isomorphic.
In fact, we proved a more general result in the differential graded context:
Theorem B: Let $\mathfrak{g}, \mathfrak{h}$ be two dg Lie algebras. If $U\mathfrak{g}$ and $U\mathfrak{h}$ are quasi-isomorphic as unital associative dg algebras, then the homotopy completions $\mathfrak{g}^{\wedge h}$ and $\mathfrak{h}^{\wedge h}$ are quasi-isomorphic as dg Lie algebras.
This has the statement above as a corollary, since one can show that a Lie algebra that is either strictly positively graded or non-negatively gradedand nilpotent is always quasi-isomorphic to its homotopy completion (in the language of the paper, it is homotopy complete). There are other interesting implications of this result in rational homotopy theory.
In my view (my coauthors might disagree) the spirit of the proof is mostly deformation theoretical, but operad theory play a big supporting role. For those who are interested in the structure of the proof without the technical details, we give a sketch of the arguments in paragraphs 0.27-0.31.
In a previous version of this answer and of the paper, we claimed that the more general statement that if two dg Lie algebras have universal enveloping algebras that are quasi-isomorphic as associative dg algebras, then the two dg Lie algebras are themselves quasi-isomorphic. Unfortunately, the proof had a gap that we were not able to fix. This more general statement remains open.
Daniel Robert-NicoudDaniel Robert-Nicoud
$\begingroup$ Would you edit the post so as to make it easier to read and updated (emphasizing what you prove, rather than being the concatenation of a result and a preceding paragraph saying it has a mistake)? You could state what's proved in v3 of your arxiv paper and then say at the end that in a first version of the post/the paper a stronger version was claimed, which is still an open problem? $\endgroup$
– YCor
$\begingroup$ @YCor Good point, thank you. Done. $\endgroup$
– Daniel Robert-Nicoud
FWIW, a ten year old article states: "We stress that, in spite of all this, the characteristic zero case of the isomorphism problem remains entirely open."
(https://link.springer.com/article/10.1007/s10468-007-9083-0)
Vladimir DotsenkoVladimir Dotsenko
$\begingroup$ What a shame, this probably also explains why I was completely stuck on the quantized problem I encountered. The quantized version shouldn't be too much different as the representation theory of $\mathfrak{g}$ and $U_v(\mathfrak{g})$ are often very similar. Anyway, in the near future we will publish an article on something completely different where all of the sudden this question pops up. Maybe smarter people can use our example to book some progress. $\endgroup$
– Mathematician 42
$\begingroup$ Thanks btw, I encountered this paper but completely missed the sentence saying that the characteristic zero case is still open. Thank you for spotting this. $\endgroup$
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Uniform attractors of stochastic two-compartment Gray-Scott system with multiplicative noise
DCDS-B Home
A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals
December 2020, 25(12): 4603-4615. doi: 10.3934/dcdsb.2020115
A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum
Xin Zhong
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Received August 2019 Published March 2020
Fund Project: This research is supported by National Natural Science Foundation of China (No. 11901474) and the Innovation Support Program for Chongqing Overseas Returnees (No. cx2019130)
Full Text(HTML)
We deal with the Cauchy problem of nonhomogeneous micropolar fluid equations with zero density at infinity in the entire space $ \mathbb{R}^2 $. We show that for the initial density allowing vacuum, the strong solution exists globally if a weighted density is bounded from above. It should be noted that our blow-up criterion is independent of micro-rotational velocity.
Keywords: Nonhomogeneous micropolar fluid equations, strong solutions, Cauchy problem, blow-up criterion.
Mathematics Subject Classification: 35Q35, 35B65.
Citation: Xin Zhong. A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4603-4615. doi: 10.3934/dcdsb.2020115
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Explaining job polarisation in Spain from a task perspective
Raquel Sebastian ORCID: orcid.org/0000-0003-4420-76041
SERIEs volume 9, pages 215–248 (2018)Cite this article
This paper presents new evidence on the evolution of job polarisation in Spain between 1994 and 2014. After showing the U-shaped relationship between employment share growth and job's percentile in the wage distribution, I use the task approach to investigate the main determinants behind job polarisation. Using the European Working Condition Survey I analyse in detail the task content of the jobs which display the most significant employment changes. I show that changes in employment shares are negatively related to the initial level of routine. I then explore the impact of computerisation on routine task inputs and I find that the routine measure is negatively related to computerisation. Finally, by using information on past jobs, I provide evidence on the displacement of middle-paid workers. Results suggest that they did not predominantly relocate their labour supply to bottom-paid occupations: while non-graduate middle workers move towards bottom occupations, graduate middle employees shift towards top occupations. This fact suggests that supply-side changes are important factors in explaining the expansion at the lower and upper tail of the employment distribution.
Debate concerning the structural evolution of the division of labour and its impact on job quality has been a central theme in social sciences for the last 200 years. In the late 1990s, the idea was that technology is skill-biased, favouring high-skilled workers and substituting low-skilled workers. While skill-biased technical change is a good explanation for the increase in the upper tail distribution of the labour force composition, it cannot explain a recent phenomenon: the decline in the share of middle occupations relative to high- and low-skilled occupations. This phenomenon has been defined as "job polarisation" (Wright and Dwyer 2003; Goos and Manning 2007).
The main drivers behind job polarisation are still subject to some debate; however, the main candidate is the so-called routinisation hypothesis (Autor et al. 2003, hereafter called ALM). Due to continuously cheaper computerisation, technology replaces human labour in routine tasks. This labour-capital substitution decreases the relative demand for workers performing routine occupations, while leading to an increase in the relative demand for workers performing non-routine tasks. Since routine workers are characterised as being in the middle of the employment distribution, then the hollowing effect is explained.
The notion that middle-skill jobs have been disproportionately destroyed and that the job distribution has hollowed out in the middle has been identified as a key aspect of contemporaneous rising labour market inequality (Acemoglu and Autor 2011; Goos et al. 2009, 2014). Therefore, understanding how the employment structure evolves can advise policy makers in designing policies to best promote a sustainable economic growth. This is especially salient, given the widespread feeling of technological anxiety (Mokyr et al. 2015). Firstly, there is a need to understand whether the shrinking of middle jobs has consequences for the possibility of moving low-skilled workers up. Secondly, an accurate understanding of occupational employment is needed in order to anticipate future skills needs and job opportunities.
Despite the importance of this topic, the results of research that assess the existence and degree of job polarisation in Spain are mixed. For example, Anghel et al. (2014) conclude that the employment structure became more polarised between 1997 and 2012, while Oesch and Rodríguez-Menés (2011) and Eurofound (2015) show a pattern of progressive upgrading for the same period. Moreover, two recent studies covering Spain, based on the European Labour Force Survey, diverge in their results. Goos et al. (2009, 2014) conclude that, on average, the employment structure in Spain became more polarised between 1993 and 2006. Using the same period of analysis, Fernández-Macías (2012) conversely shows an upgrading process (high-wage occupations expanding at the expenses of low-wage jobs) and does not provide evidence of a pervasive polarisation. These five papers have relied on graphical inspection to identify the phenomenon: terciles (Goos et al. 2009, 2014; Fernández-Macías 2012), or quintiles (Eurofound 2015; and Oesch and Rodríguez-Menés 2011).
Focusing on the Spanish case, this paper makes several contributions to the understanding of the evolution of the employment and wage structure in four complementary ways. First, I shed some light on the literature on employment polarisation in Spain, providing evidence of job polarisation in our sample; between 1994 and 2014, employment share in Spain increased at the two extremes of the job wage distribution, while it decreased for middle-income earners. My study adds to the literature on job polarisation, offering two new ways of representing the phenomenon and enlarging the period of analysis.Footnote 1 I also contribute to widening the literature on employment remuneration in line with employment trends. In the US, Autor and Dorn (2013) find a clear correspondence between employment and wages. However, the polarisation of wages does not seem to be common in Spain, as there is no evidence that changes in pay followed the same pattern as changes in occupations. This contrasts with standard labour markets models, predicting that a positive demand shock increases both employment and earnings.
Second, I made methodological progress with respect to previous studies on job polarisation and task specialisation in European countries by measuring the tasks content of occupations from a national survey data instead of relying on US sources, like in the work of Anghel et al. (2014) and Goos et al. (2014). Therefore, no assumption on task composition and the impact of technology between the two countries is needed. Moreover, the EWCS allows for time dynamics to measure routine tasks.Footnote 2 Using this survey, jobs are classified as abstract, routine, and manual tasks, similar to the ALM model. This allows for examination of the association between employment changes and the task content of occupations. Therefore, I perform a shift-share analysis to evaluate the evolution of the tasks' content, exploring whether the changes of the task content of occupation are due to changes within occupations (intensive margin) or between occupations (extensive margin).
Third, unlike previous studies, I explore the relationship between computer use and routine tasks, which I define on the basis of the frequency of repetitive activities that workers are asked to perform on the job. After creating a pseudo-panel analysis, results show a negative relationship between computers and routine tasks, and a positive association between computer and abstract tasks. However no relationship is found for manual tasks. Therefore ALM predictions are satisfied for routine and abstract tasks, but not for manual tasks.
Finally, I analysed the role of job polarisation with the relocation of middle-skilled workers. To investigate this phenomenon, the main data source is integrated with two additional datasets: the European Community Household Panel (ECHP) and the European Survey of Income and Living Conditions (SILC). Taking advantage of these new databases, the analysis builds from questions on previous occupations. There are two main findings: in line with the ALM, middle-skilled workers become more mobile over time and have the highest probability levels of mobility. However, after dividing the data into graduates and non-graduates, results suggest that while non-graduate middle workers move towards bottom occupations, graduate middle employees shift towards top occupations. This fact suggests that supply-side changes are important factors in explaining the job expansion at the lower and upper tail of the employment distribution.
The paper is organised as follows. Section 2 clarifies the main concepts and provides a review of the literature. Section 3 describes the data and methods used for analysis. Section 4 presents the evidence on labour market polarisation, on both employment and pay rules. Section 5 investigates the task content of occupations. Section 6 looks at the impact of computer adoption on tasks. Section 7 analyses the occupational mobility of middle-pay workers. Finally, Sect. 8 summarises the main conclusions of the paper and provides a guide for future research stemming from this paper's findings.
Job polarisation refers to the relative job growth in the lower and upper tail of the wage distribution relative to the middle-wage ones. This well-known phenomenon has been found in the US (Wright and Dwyer 2003; Autor, Katz, and Kearney. 2006; Autor and Dorn 2013), the UK (Goos and Manning 2007; Salvatori 2015), Germany (Spitz-Oener 2006; Dustmann et al. 2009; Kampelmann and Rycx 2011), and Sweden (Adermon and Gustavsson 2015). With respect to Europe, results are more controversial. On the one hand, Goos et al. (2009, 2014) show that on average, the employment structure in Europe polarised from 1993 to 2006. On the other hand, Fernández-Macías (2012) find heterogeneous results in Western European countries and conclude that there is not a clear and universal pattern of a pervasive polarisation.Footnote 3 As for Spain, conclusions also diverge between job polarisation (Anghel et al. 2014) and occupational upgrading (Oesch and Rodríguez-Menés 2011; and Eurofound 2015).Footnote 4
While in the US wage polarisation has occurred hand with hand with job polarisation (Autor et al. 2006), papers based on European countries do not find the same result. Goos and Manning (2007) failed to find wage polarisation for the UK despite the strong evidence of job polarisation. Antonczyk et al. (2010) and Kampelmann and Rycx (2011) show little evidence of wage polarisation in Germany. Finally, Massari et al. (2014) study the European labour market as a whole and conclude that there is no evidence of wage polarisation. With regards to Spain specifically, there is no study exploring this phenomenon.
Different theories have tried to explain the main drivers behind polarisation. While there are some explanations based on supply mechanisms (skill composition), almost all the theoretical explanations are based on three different demand mechanisms. The first mechanism is the propensity to offshore activities, which is not the same in all occupations. According to Blinder (2009), certain jobs are potentially more vulnerable to offshoring than others. They show that production jobs are easier to reallocate in low-income countries than service jobs. In the second place, Autor and Dorn (2013) explain that wage inequality increases income in the top earners and as a consequence, increasing the demand for bottom-paid job services. It is well known that these two factors affect specific occupations. However, the economic literature concludes that these two factors play a minor role in explaining the overall evolution of the occupational employment structure as a whole (see e.g. Autor and Katz 1999; Acemoglu and Autor 2011; Michaels et al. 2014).
In contrast, the most prominent theory accounting for job polarisation is the well-known routinisation hypothesis, called Routine Biased Technical Change (formulated by Autor et al. 2003, RBTC). In their seminal paper, ALM propose a classification of tasks along two different dimensions: routine (as opposed to non-routine) and manual (as opposed to non-manual, or also called cognitive) tasks. Routine tasks are defined as those that "require methodical repetition of an unwavering procedure" (ALM 2003: 1283). The cognitive dimension generally refers to tasks that require gathering and processing of information and problem solving (analytic), as well as those that need creativity, flexibility and communication in order to be performed (interactive).
Autor et al. (2006), and more recently Autor and Dorn (2013) reformulate the ALM model by bringing together the two routine categories. They consider a three-fold classification scheme, where tasks are classified into abstract, routine, and manual. While this new classification shared the routine definition of the ALM model, the abstract category refers to tasks requiring problem solving and managerial tasks with high cognitive demand, and the manual tasks category refers to those ones requiring physical effort and time adaptability; therefore both tasks categories are difficult to automate.
In the ALM model, the way in which occupations are affected by new technologies depends to a large extent on the tasks they perform, rather than on their skills (normally measured using educational level).Footnote 5 Two hypotheses are then formulated. The first hypothesis is that since routine tasks are easy to codify, and therefore easy to replicate by machines, the ALM model predicts the progressive substitution of technology for labour in routine tasks. The second hypothesis is that abstract tasks are characterised by complex analytical thinking, flexibility, creativity, and communication tasks, among others. These types of tasks are not only difficult to be replaced by machines, but they are also complementary to computer technologies. Therefore, the ALM model predicts complementarity between technology and abstract tasks. No assumptions are made regarding manual tasks.
Goos and Manning (2007), and Autor and Dorn (2013) use the ALM model to explain the polarisation phenomenon: since routine tasks are located in the middle of the occupation distribution, and non-routine tasks at the top and at the bottom, the ALM indicates that two key effects occur: first, employment and wages in the middle of the distribution decreased. Second, employment and wages increased (or at least remain stable) in the higher and lower qualified groups. Hence, the polarisation effect of recent technical change is explained by the RBTC. In summary, the ALM model provides a strong theoretical foundation to develop a deeper understanding of how technology may be impacting the Spanish labour market.
Three different datasets covering the period 1994–2014 are used in the analysis. Data on the evolution of jobs and socio-demographic characteristics come from the Spanish Labour Force Survey. Data on the evolution of wages come from the Structure of Earnings Survey. Data on tasks come from the European Working Condition Survey. Below, each data set is described in detail.
Spanish Labour Force Survey
The primary data source used is the Spanish Labour Force Survey (Encuesta de Población Activa, EPA, in Spanish), administered by the National Institute of Statistics. The EPA is used to estimate employment and unemployment within the ILO framework and is the basic source by which researchers can construct data series on occupations.
Although the data is compiled quarterly and is available for all years since 1964, this analysis focuses on the period 1994–2014, where the second quarter of each relevant year is sampled to avoid seasonality problems. The EPA contains data on employment status, weekly hours worked, two-digit occupational level, one-digit industry level, education, region, nationality, sex, age, and the population in each cell, among others. The dataset is weighted to reflect employment in absolute numbers.
For the chosen period, I face two important reclassifications over the period of interest. First, in relation to occupations, the CNO-94 (based on ISCO-88) was replaced by the CNO-11 (based on the ISCO-08) in 2011. Second, in terms of sectors of activity, the CNAE-93 (based on the NACE.Rev.1) was replaced by the CNAE-93 (based on the NACE.Rev.2) in 2009. I convert the occupational codes from the ISCO-08 into the ISCO-88 and the industry codes from the NACE.Rev.2 into the NACE.Rev.1 using the crosswalk made available by Goos.Footnote 6
The EPA is far from ideal. The main problem is the lack of income data necessary to rank selected job cells on earnings-based quality. To overcome this problem, I merge it with the Structure of Earnings Survey.
Structure of Earnings Survey
The Structure of Earnings Survey (in Spanish, Encuesta de Estructura Salarial, EES) is administered by the National Institute of Statistics. The sampling takes place in two stages. First, firms are sampled randomly from the Social Security General Register of Payments records. Second, from each of the selected firms, workers are randomly selected. The survey collects detailed information on workers' wages; personal characteristics such as gender, age, educational attainment, and nationality; and job characteristics, including sector, occupation, contract and job type, firm size and ownership, and region.
For the period under study, the survey has been carried out five times (1995, 2002, 2006, 2010, and 2014). For the 1995 ESS, not all the employed population is covered: the survey is only representative of employees working in companies of at least ten employees in the sectors C to O (excluding L) of the NACE.Rev.1 classification of economic activities. For the 2002 ESS, the 2006 ESS, the 2010 EES, and the 2014 ESS, the coverage of the survey is extended to include some non-market services (educational, health, and social services sectors).
To measure job polarisation, I use the 2002 wave rather than the other surveys, as my results remain invariant, which is preferable for two reasons. First, the 1995 EES does not include employees working in companies of at least ten workers, self-employed workers, and public employees. Second, between 2002 ESS and 2006 EES, I rather prefer the 2002 ESS, as it is closer to the initial period. Moreover, to measure wage polarisation, all five cohorts and all wages are deflated to the year 1995 using the Consumer Price Index (CPI).
Like the EPA, there are two modifications at the occupation and industry code for the 2010 ESS and the 2014 ESS. The surveys display occupations using the CNO-11 (based on the ISCO-08) and industry using the CNAE-93 (based on NACE.Rev.2). Moreover I convert the ISCO-08 into the ISCO-08 and the NACE.Rev.2 into the NACE.Rev.1 using the same mapper that I explained above.
Measuring the task content of jobs
In order to establish the task content of each job's measures, information on the activities performed by workers on the job is required. Task measures at the job level are derived from an additional source, the European Working Condition Survey (EWCS). Unlike previous studies on job polarisation in Spain (see Anghel et al. 2014 and Goos et al. 2014), this study does not rely on the US O*Net survey to derive data on job task requirements. Hence, there is no need to assume that the task composition is the same in the two countries. Moreover, there are two different features between the US O*Net and the EWCS. Firstly, while the original purpose of the US O*Net is an administrative evaluation by Employment Services offices of the fit between workers and occupations, the EWCS is conducted for research. Secondly, differently from the US O*Net where analysts at the Department of Labor assign scores to each task according to standardised guidelines, the EWCS derive individual tasks measures. Although the EWCS presents a higher level of subjectivity, this feature has the advantage of giving a more precise idea of the tasks performed within each occupation. Autor and Handel (2013), who use a similar type of survey to derive individual task measures (the Princeton Data Improvement Initiative survey, PDII), argue that their data have a greater explanatory power for occupations and wages than those derived from the O*Net.Footnote 7
The EWCS is administered by the European Foundation for the Improvement of Living and Working Conditions (Eurofound) and has become an established source of information about working conditions and the quality of work and employment. With six surveys (one every 5 years) having been conducted since 1990, it enables monitoring of long-term trends in working conditions in Europe. At each time point, information on employment status, working time arrangements, work organisation, learning and training, and work-life balance, among others is collected. In this research, five surveys (1995–2015) are used for analysis. The five repeated cross-sections cover 1000 in 1995, 1500 in 2000, 1017 in 2005, 1008 in 2010, and 3364 in 2015. Sampling weights adjusted for responses are used through the analysis. The analysis is restricted to individuals aged from 16 to 65. Jobs are classified according to the ISCO-88 nomenclature at the two-digit level and NACE.Rev.1 at the one-digit level.
I follow the same framework as Autor et al. (2003), and Autor and Dorn (2013) to estimate the effects of job polarisation. This classification is based on a three-dimensional typology: abstract, routine, and manual. To limit the role played by my subjective judgement, I follow the work of Autor and Handel (2013) as closely as possible, as they use variables that are most similar to those available in the EWCS. I construct the indexes for each of the three dimensions using the first component of a principal component analysis and then compute the indexes into their standardised form.Footnote 8
For the abstract tasks, I retain the following items: "learning new things", "solving unforeseen problems", "complex tasks", "assessing yourself on the quality of your job", and "influence decisions that are important".Footnote 9 For the manual tasks, I resort to responses on "physical strength" (e.g. carrying or moving heavy loads), "skill or accuracy in using fingers/hands" (e.g. repetitive hand or finger movements), and "physical stamina" (e.g. painful positions at work).Footnote 10 For the routine tasks, I opt for the routine activities performed within the respondents' jobs: does your main job involve (1) short repetitive tasks of less than a minute, (2) short repetitive tasks of less than 10 min, (3) monotonous and repetitive tasks, and (4) dealing with customers.Footnote 11
I created three separate standardised indices for abstract, manual, and routine job aspects using the sub-components enumerated above for each of these aspects. Given that all the sub-components are either dichotomous or ordinal, I perform a principle component analysis using a polychoric correlation matrix. The proportion of variance explained by the first component is 0.58, 0.67 and 0.68 for the abstract, manual, and routine aspects respectively (see "Appendix B" for further information).
Following Autor and Dorn (2013), I create a routine task intensity measure (RTI) to compare findings to those in the literature. This measure aims to capture how important the routine tasks are compared to tasks components of countries. Indices are standardised with a mean of 0 and a standard deviation of 1. The RTI is then calculated as follows:
$$ RTI_{ 1994} = \ln (T_{1994}^{R} ) - \ln \left( {T_{1994}^{A} } \right) - \ln \left( {T_{1994}^{M} } \right) $$
where \( T_{1994}^{R} , T_{1994}^{A} \), and \( T_{1994}^{M} \) are the routine, abstract, and manual inputs in Spain in 1994. This measure is rising in the importance of routine tasks in Spain and declining in the importance of abstract and manual tasks.
Before proceeding with the analysis, Table 1 shows a correlation between the EWCS and O*Net.Footnote 12 The measures of the two surveys are positively correlated, with the RTI having the highest correlation (0.86) and routine task having the lowest correlation (0.67). The results indicate that both surveys are close enough, indicating that the EWCS is a suitable measure.
Table 1 Correlation between EWCS and O*Net.
The evolution of employment and wages in Spain
The evolution of employment
The starting point of the analysis is to investigate the pattern of employment change in the Spanish labour market, acting as a preliminary step for the subsequent analysis. Unless otherwise noted, throughout this paper, employment is modelled by occupation (ISCO-88 at the two-digit level) and by industry (NACE.Rev.1 at the one-digit level). Employment share is computed from EPA data, while the employment ranking is based on the mean wage from the 2002 EES data.Footnote 13
A common way of analysing the development of jobs is through graphical illustration. For this, the employment shares by each job are computed, along with changes over time. To avoid bias due to small jobs drive dominating results, each job is weighted by its total employment. Jobs are ranked according to their 2002 EES mean wage.Footnote 14 Then, the percentage point change in employment share is plotted against the log mean hourly wage. If the structure of employment has polarised, it is expected that employment in bottom and top-paid jobs increased, while it decreased in the middle of the wage distribution.
Previous literature has represented the phenomenon using aggrupation of jobs (either quintiles or terciles), being these presentations being very sensible to the definition of jobs and to the classification of these jobs (see Sebastian 2017 for a longer explanation). In order to avoid these problems, in this research I add two new representations: the parametric graph (Fig. 1) and the smooth regression (Fig. 2).
Sources: author's analysis from the Spanish Labour Force Survey (1994, 2014), and the Structure of Earnings Survey (2002) (color figure online)
Employment shares growth in Spain (1994–2014) by mean hourly wage. Notes scatter plot and quadratic prediction curve. The dimension of each circle corresponds to the number of observations within each ISCO-88 two-digit occupation and NACE.Rev.1 one-digit occupation in 1994; the grey area shows 95% confidence interval. Employment shares are measured in terms of workers. Colours represent the quintile of each job (green, first quintile; yellow, second quintile, grey, third quintile; red, fourth quintile; and violet, fifth quintile).
Sources: author's analysis from the Spanish Labour Force Survey (1994, 2014), and the Structure of Earnings Survey (2002)
Smoothed changes in Employment by wage percentile (1994, 2014). Notes the figure plots log changes in employment share by 2002 job skill percentile rank using a locally weighted smoothing regression (bandwidth 0.75 with 100 observations), where skill percentiles are measured as the employment-weighted percentile rank of a job's mean log wage in the 2002 ESS.
The first graphical method (Fig. 1) corresponds to the parametric graph. Figure 1 shows the evolution of Spanish employment between 1994 and 2014. As noted already, employment shares are measured by two-digit occupations (ISCO-88) and by one-digit sectors of activity (NACE.Rev.1). Earnings are measured by the logarithm of hourly mean in each job in 2002. The employment changes in Spain show a clear pattern of job polarisation, in which the higher and lower part of the earnings distribution increased while the middle-earnings part has shrunk. A U-shaped curve can be detected in the evolution of employment shares, when jobs are ranked according to the hourly mean wage.
Using the parametric graph, a test for job polarisation can be conducted. In order to do so, the following model of the quadratic form is estimated as proposed by Goos and Manning (2007):
$$ \Delta \log E_{j} = \beta_{0} + \beta_{1} \log (w_{j,t - 1} ) + \beta_{2} \log (w_{j,t - 1} ) ^{2} $$
where \( \Delta \log E_{j} \) is the change in the log employment share of job j between t − 1 and t, \( \log (w_{j,t - 1} ) \) is the logarithm of the mean wage of job j in t − 1, and \( \log (w_{j,t - 1} ) ^{2} \) is the square of the initial mean wage. A U-shaped relationship between the employment growth and the wages implies that the quadratic term is positive.
Table 2 presents the results of the OLS regression using weekly hours worked as a measure for employment shares rather than expressing them in terms of bodies. Equation (1) is estimated by weighting each job by its initial employment share in 1994 to avoid that results are biased by compositional changes in small jobs. All regression coefficients have the expected sign and are significant at the 1% level. The results indicate that Spain was characterised by a polarisation pattern in employment growth from 1994 to 2014. The phenomenon of job polarisation is also robust to the use of the median instead of the mean.
Table 2 Regressions for job polarisation.
The second representation method is by defining job wage percentile.Footnote 15 In this particular case, smoothing regressions are displayed rather than the actual data point (the previous case). Therefore, changes in employment share are plotted against the percentile of the initial earnings distribution. A U-shaped curve is detected and shown in Fig. 2. The main advantage of this method is that the biggest increases and losses are observable. For Spain, the biggest losses are between the 20th and the 40th percentile of the initial mean wage distribution. Overall, the shape of employment changes in the EPA data confirms other studies with Spanish data and suggests that job polarisation is a robust phenomenon in Spain.
Figure 3 shows the quintile plot. In this occasion, I follow the methodology applied in Europe by Fernández-Macías (2012). In Fig. 3, I plot the relative employment share by job wage quintile. Quintiles are created by ranking jobs by their initial mean wage and aggregating them into five quintiles. Each group contains the 20% of employment in the initial year.Footnote 16 The resulting graph (Fig. 3) demonstrates an even clearer pattern of job polarisation. In this case, top- and bottom-income jobs grow up and there is a decline in middling jobs.
Sources: author's analysis from the Spanish Labour Force Survey (1994, 2014), Earnings Structure Survey (2002)
Relative net employment change (1994, 2014) ranked by 2002 wage mean. Notes jobs wage quintiles are based on two-digit occupation and one-digit industry and on mean wages in 2002. It shows the relative net employment change quintiles (in percentage points).
Three robustness tests for the results presented above are implemented to ensure the validity of the results. First, there are three reclassification breaks during the period. I establish three time periods due to changes in classification (1994–2008, 2008–2010, and 2010–2014). Second, the results are subjected to sensitivity testing with respects to the choice of the reference year. The 1995 EES and 2006 EES years were selected. Third, jobs are ranked by median rather than mean earnings. In all cases, graphs result are invariant, the characteristic U-shaped curve is detected in the evolution of employment shares ("Appendix D", Figs. 4, 5, 6, 7, 8). Using the database, the employment changes in Spain between 1994 and 2014 are found to be consistent with the polarisation phenomenon, where employment growth occurs for bottom- and top-paid jobs, while decreases for middling-paid jobs.
Sources author's analysis from the Spanish Labour Force Survey (1994, 2008, 2011, 2014), and the Structure of Earnings Survey (1995, 2010) (color figure online)
Employment shares growth in Spain by mean hourly wage in three different periods: 1994–2008, 2008–2010, and 2010–2014. Notes scatter plot and quadratic prediction curve. The dimension of each circle corresponds to the number of observations within each occupation in 1994 (a), in 2008 (b), and in 2011 (c); the grey area shows 95% confidence interval. Employment shares are measured in terms of workers. Colours represent the quintile of each job (green, first quintile; yellow, second quintile, grey, third quintile; red, fourth quintile; and violet, fifth quintile).
Sources author's analysis from the Spanish Labour Force Survey (1994, 2008, 2011, 2014), and the Structure of Earnings Survey (1995, 2010)
Smoothed changes in Employment by wage percentile in three different periods: 1994–2008, 2008–2010, and 2010–2014. Notes the figure plots log changes in employment share by 1995 and 2010 job skill percentile rank using a locally weighted smoothing regression (bandwidth 0.75 with 100 observations), where skill percentiles are measured as the employment-weighted percentile rank of a job's mean log wage in the 2002 ESS (a) and the 2010 ESS (b, c).
Relative net employment change (1994, 2014) in three different periods: 1994–2008, 2008–2010, and 2010–2014. Notes jobs wage quintiles are based on two-digit occupation and one-digit industry and on mean wages in 2002. It shows the relative net employment change quintiles (in percentage points). a Is based on mean wage in 1995, b, c based on mean wage in 2010.
Sources author's analysis from the Spanish Labour Force Survey (1994, 2014), Earnings Structure Survey (1995, 2002, 2006)
Smoothed changes in employment in Spain (1994–2014), being jobs ranked in 1995, 2002 and 2006 by EES. Notes the figure plots a locally weighted non-parametric smoothing regression (bandwidth 0.75 with 100 observations). The jobs are defined at two-digit ISCO level and at one-digit NACE Rev.1 level. For the period 1994–2008, jobs are ranked by the EES 1995, EES 2002, and EES2006 media wage.
Sources author's analysis from the Spanish Labour Force Survey (1994, 2014), Earnings Structure Survey (2002) (color figure online)
Smoothed changes in employment in Spain (1994–2008), jobs are ranked by 1995 mean and median wage percentile. Notes the figure plots a locally weighted non-parametric smoothing regression (bandwidth 0.75 with 100 observations). The jobs are defined at two-digit ISCO-88 level and at one-digit NACE.Rev.1 level. For the period 1994–2008, jobs are ranked by the 2002 ESS mean wage (blue line) and 2002 ESS median wage (red line).
The evolution of wages
In this section, and after studying the evolution of employment, I study the evolution of remuneration of jobs for the period 1995–2014.Footnote 17,Footnote 18 It is expected that the evolution of employment and the evolution of wage level matches. As a consequence, to predict changes in wages, the same quadratic model is used to detect the U-shaped evolution of employment shares (Kampelmann and Rycx 2011). Therefore, to examine wage polarisation, the following model is estimated:
$$ \Delta \log (w_{j} ) = \beta_{0} + \beta_{1} \log (w_{j,t - 1} ) + \beta_{2} \log (w_{j,t - 1} ) ^{2} $$
where \( \Delta \log (w_{j} ) \) is the change in the log mean wage of job j between t − 1 and t, \( \log (w_{j,t - 1} ) \) is the logarithm of the mean wage of job j in t − 1, and \( \log (w_{j,t - 1} ) ^{2} \) is the square of the initial mean wage.
Table 3 reports the OLS for the wage polarisation analysis. In this report, the number of individuals within a job in 1994 weights the initial number of observations in each job. The coefficients have the expected sign, but are not significant. These findings suggest that in Spain, between 1995 and 2014, wages did not experience the same polarised pattern of employment shares.
Table 3 OLS regression for wage polarisation analysis.
Finally, the changes in employment share are evaluated to match changes in pay rule. To do so, the correlation coefficient is computed between the two variables (\( \rho = 0.06) \). The coefficient is positive but weak. Contrary to the existing literature in the US (Autor and Dorn 2013) and Germany (Kampelmann and Rycx 2011), the results suggest that the relationship between changes in employment share and changes in pay rules is almost zero in Spain.
Task-based analysis
Employment changes and tasks intensities
Thus far, it has been shown that there is a hollowing out of the employment distribution in Spain, while there is no evidence of wages following the same pattern. In order to better interpret the previous results, I follow a task-based approach. I use information on the activities carried out by workers on their jobs, where each worker performs different tasks with different intensities. Therefore, each job is not defined by one single task, but it can be classified as with a predominant task. To proceed with the analysis, I gather information concerning the nature of tasks performed by workers. As already explained in Sect. 3, this data comes from the European Working Survey.
Table 4 presents the pairwise correlation between the task measures and the education attainment at the two-digit ISCO-88 level and one-digit NACE.Rev.1. The correlation between the abstract dimension and the routine measure is negative, while is positive with the manual task and the education variable. The RTI is negatively correlated with the abstract dimension and positively correlated with the routine and manual task. The education measure is positive correlated with the abstract dimension, while is negative with the routine and manual content.
Table 4 Correlation among the task measures and the education variable.
I proceed with my analysis aggregating the 226 jobs so far considered at the ISCO-88 two-digit level. This aggregation offers a clear interpretation of the tasks content of the occupations that mainly contributed to the polarisation of the employment structure.
Table 5 reports changes in employment share by major occupational groups (two-digit ISCO-88 level) and are ranked in ascending order by their mean wage in 1995. The mean level of education in 1994 (column 2) is also included. I draw on Goos et al. (2014) to classify these occupations in three major groups: the first six occupations in the bottom distribution are defined as bottom occupations, the next eight occupations as middle occupations, and the top seven occupations are labelled as top occupations. The groups include six, eight, and seven occupations respectively and they represent 35%, 37%, and 28% of the employment distribution.Footnote 19 These groups represent the theoretical classification of the RBTC model with services and elementary occupations at the bottom of the occupational distribution, productive and administrative occupations being in the middle, and professional and managerial at the top of the top of the occupational distribution. Moreover, from Table 5, it is clear that the shift of employment goes from the middle to the top: of the 7.1% of the employment shares lost in the middle, 6.8% go to the top and 0.42% to the bottom occupations.
Table 5 Occupations, mean wage, and education.
To illustrate the richness data at the occupational level, Table 6, columns 2 to 5 present the average values of the task measures. In matching Table 5 with Table 6, a complete picture of the task content can be formed, which determines job polarisation in Spain. In line with expectations, the RTI values are higher among clerical work, repetitive production, and monitoring. Managers and professionals instead score among the lowest.
Table 6 Tasks measures by occupations.
Analysing the bottom group occupation, results indicate that half of the occupations are growing in employment share, while the other half are losing employment share. The occupations that experience the most significant employment growth represent a mixture of elementary occupations such as "Personal and protective services workers" (ISCO 51), and services such as "Sales and services elementary occupations" (ISCO 91). These findings confirm that the increase of employment at the lower tail of the wage distribution is mainly driven by a job expansion in the service sector. Moreover, these occupations score higher in the manual than in the routine dimension. This is in line with the prevailing RBTC theory that low-skilled jobs rely on manual tasks, therefore are not affected by the introduction of technology.
Concerning the middle occupations, "Office clerks" (ISCO 41), "Metal, machinery and related trades workers" (ISCO 72), and "Precision, handicraft, printing, and trades workers" (ISCO 73) are those that register the highest employment losses, scoring higher in the routine dimension than in the manual measure. Moreover, "Precision, handicraft, printing, and trades workers" (ISCO 82) has the highest score in RTI.
Finally, within the group of the highest paying occupations, "Other associate professionals" (ISCO 34) and "Physical, mathematical, and engineering profession" (ISCO 21) are those that experienced the most significant employment growth. Consistent with the ALM model, these seven occupations score higher on the abstract dimension than on the manual task. These occupations demand tasks such as flexibility, problem solving, creativity, and complex communication. Therefore, the likelihood of technology substituting for workers in carrying out these tasks is very limited.
Table 7 presents results OLS regressions of changes in employment share between 1994 and 2014 and the initial level of routine intensity of each occupation. As expected, I found a negative relationship between the two variables: higher routine task intensity leads to larger declines in employment occupations.
Table 7 OLS regression of changes in employment share and the initial level of routine intensity.
Task intensities over time
Understanding the evolution of tasks measures across time allow further analysis of job polarisation. The composition of tasks constitutes a vital piece of information for testing the routinisation hypothesis. I analyse changes in the task structure of the labour market to determine if task structure relies on the changes within occupations (i.e. the intensive margin) or between occupations (i.e. the extensive margin).
Table 8 presents the importance of the tasks in 1995 and 2015, and reports the results of the shift-share analysis. The change decomposition of tasks of each occupation is as follows:
$$ \Delta T_{k} = \mathop \sum \limits_{j}\Delta E_{j}^{ } \gamma_{jk}^{ } + \mathop \sum \limits_{j}\Delta \gamma_{jk}^{ } E_{j}^{ } $$
where \( \Delta T_{k} \) and \( \Delta E_{j}^{ } \) are the change in importance of tasks k and the change in employment in occupation j between 1995 and 2015, and \( \gamma_{jk}^{ } \) represents the importance of task k in occupation j. Finally, \( \Delta \gamma_{jk}^{ } \) is the change in the share of task k in occupations and \( E_{j}^{ } \) is the average share of occupation j. The first term on the right-hand-side equation is the extensive margin, i.e. the task importance is held constant (and represents the average task importance across the 2 years), and time variation relies on changes across occupations. The second term is the intensive margin where occupational employment is held constant while the importance of tasks within occupations is allowed over time.
Table 8 Tasks shifts, intensive and extensive margin.
Table 8 compares the importance of the three tasks groups in 1995 and 2015 and the change between 1995 and 2015. Results indicate that manual tasks became less important in the Spanish economy, while abstract and routine tasks increased in magnitude. In the last two rows, I divide the decomposition effect into changes within occupations ("the intensive margin") and changes between occupations ("the extensive margin"). The increasing importance of the routine tasks occurs at the intensive margin, whereas abstract tasks increased in importance due to changes at the extensive margin. Therefore, while routine tasks are increasing because routine occupations are now more routinised, abstract tasks are increasing because occupations with a lower level of abstract tasks are now demanding it. The decreasing importance of manual tasks seems to rely mainly on the extensive margin. In other words, manual tasks have lost employment due to decreasing tasks' importance within jobs.
Technological change and tasks
The ALM model predicts that technology substitutes for labour in routine tasks but complements it in non-routine abstract tasks. No assumption is made for non-routine manual tasks. Therefore, I investigate the effect that computers have on tasks inputs. To do so, I create a pseudo-panel testing the computerisation hypothesis, with the following regression model:
$$ \bar{T}_{tjt} = \beta \bar{C}_{jt} + \mathop \sum \limits_{t = 1}^{T - 1} \theta_{t} + \delta_{j} + \bar{\varepsilon }_{jt} $$
where \( \bar{T}_{tjt} \) is the task measure in either: (1) abstract, (2) routine, and (3) manual tasks at the job level j at time t. The main regressor of interest, \( {\bar{\text{C}}}_{\text{jt}} \) is the variable capturing computer intensity in job j at time t (see "Appendix E" for further details on how is derived). The specification includes a set of year effects (\( \uptheta_{\text{t}} \)) and a set of occupation effects (\( \updelta_{\text{j}} \)). Time fixed effects are included to control for omitted variables that vary across time, but not varying across occupations. Occupations' fixed effects control for omitted variables that are not constant across occupations but which evolve over time.
Table 9 presents results of the fixed effects regressions of the initial abstract task (column 1), routine task (column 2), and manual task (column 3), and the initial level of computer use for each job. As expected, the results are in line with the ALM model: on the one hand, technology is significant and negative related with routine tasks. On the other hand, there is a positive effect between computer use and abstract task: workers in managerial, professional, and creative occupations are complements with computers. Regarding manual tasks, where the ALM does not predict any effect, there is negative a relationship between manual task and computer use. However, the manual coefficient is not significant, suggesting that the computer's substitution is higher among routine tasks.
Table 9 Impact of computer on adoption on task measures.
Occupational mobility of middle-paid workers
The analysis has provided empirical evidence of the negative impact of computerisation on routine workers, and therefore their displacement. In this section, the analysis is completed by switching the focus to occupational mobility of middle-paid workers.
The model proposed by Autor and Dorn (2013) provides a framework in which the continuously falling price of technology induces low-skilled routine workers to relocate from routine to manual tasks, at the bottom of the employment distribution. Therefore, first, it is expected that routine workers become more mobile over time, and second, the subsequent relocation of routine workers at the bottom of the employment distribution is expected.
The main drawback of the EWCS is the lack of information on past jobs. To overcome this problem, the main source of data is merged with two additional databases: the European Community Household Panel (ECHP), and its continuation, the Survey of Income and Living Conditions (SILC).Footnote 20 The ECHP and the SILC are longitudinal surveys of the employment circumstances of the European population covered from 1994 to 2000 (for the ECHP) and from 2005 to 2015 (for the SILC). At each interval, information on job characteristics and working condition is provided. Among other details, it includes information on activity and employment status, job characteristics, earnings, and education. For the analysis, individuals who are not in both years of the analysis are excluded. Due to data restrictions, I divide the period into two: from 1994 to 2000 (using the ECHP) and from 2005 to 2015 (using the SILC).
Table 10 presents the occupational mobility by educational group. In order to control for education, I create a three-level education variable ranging from 1 (low-education) to 3 (high-education), having as a result three types of workers: low-, middle-, and high-skilled workers. The table shows the percentage of workers that change occupation among those with the same educational attainment. The results are divided into two periods and two sub-periods. The first period is from 1994 to 2000 and the two sub-periods are: 1994–1997 (column 1) and 1997–2000 (column 2). The second period covers 2005 to 2015, where the two sub-periods are: 2005–2008 (column 4) and 2012–2015 (column 5). Column 3 and Column 6 contain mobility over time. In line with the RBTC model, middle-skilled workers are shown to become more mobile over time (5.5 and 4.1%), against low-skilled (4.6 and 3.3%) and high-skilled workers (2.4 and 0.90%).
Table 10 Occupational change by educational group.
After showing that middle-skilled workers become more mobile over time, middle-paid workers are analysed to determine if they moved towards bottom- or top-paid occupations. Following the model by Autor and Dorn (2013) and later revisited by Cortes (2016), it is expected that middle workers relocate towards bottom-paid occupations, under the assumption that its relative comparative advantage is higher in manual than abstract tasks. In this paper, and differently from Schmidpeter and Winter-Ebner (2016), I only analyse downward and upward mobility and not flows into unemployment or inactivity. The idea behind this decision is that the analysis done so far only covers workers.Footnote 21
My enquiry builds on the transition probability matrix, where each cell corresponds to the transition process of being in one job and move to another given by:
$$ p_{ij} = { \Pr }(X_{t} = j|X_{t} = i) $$
The probability from Eq. 6 can be computed as expressed in Eq. (7)
$$ p_{ij} = N_{ij} /\mathop \sum \limits_{j = 1}^{n} N_{ij} $$
where \( N_{ij} \) is the total number of workers changing from job i to job j (the cell counts) and \( \mathop \sum \nolimits_{j = 1}^{n} N_{ij} \) is the total number of workers in the same job (the row counts).
In Table 11, each cell corresponds to the transition probability from one occupation to another in four different periods: from 1994 to 1997, from 1997 to 2000, from 2005 to 2008 and from 2012 to 2015. Moreover, workers are divided into graduate and non-graduate.Footnote 22 Therefore, I compare the exit probabilities for each skill category of workers, middle, bottom, and top across each decade.
Table 11 Occupational transitions.
Two important remarks can be discerned from this table. First, middle workers have the highest probability levels of mobility. However, the mobility pattern is different when I take into account skills categories. For non-graduate workers, there is an increase in the probability of switching from middle to bottom occupations; this being more pronounced in the second decade. The picture is not the same for graduate employees in middle occupations; these workers become more likely to move up the occupational ladder to top occupations.
Second, the probability of moving down from top to middle occupations is higher for non-graduate employees than graduate workers, and this fact increases over time. At the same time, non-graduate workers in bottom occupations have a decline in the probability of moving to middle jobs.
In summary, the results suggest that there is a relocation of middle-skilled workers. However, different from Autor and Dorn (2013) model, only non-graduate workers move towards bottom-paid occupations. This last result raises doubts as to the leading role of technology, indicating that more needs to be done to understand the main determinants behind job polarisation. Education plays an important role in explaining the increase at the two tails of the employment distribution.
This paper contributes to the debate on labour market polarisation in Spain using Spanish task data to measure the job content of occupations. Through the analysis, I show graphically and empirically that employment in Spain became polarised between 1994 and 2014. However, there is no evidence of a similar trend in wages, unlike previous findings from the US. The sample suggests that jobs in bottom and top occupations increased, while employment shares decreased in the middle of the distribution.
I interpret the evolution of employment from a task-based perspective, exploring ALM model's prediction. As the theory predicts, top and bottom occupations increased the most, the former being classified as abstract tasks and the latter as manual tasks. Moreover, middle-paid occupations have lost a significant employment share and can be classified as routine. In the same line, changes in employment shares are negatively related to the initial level of routine intensity index.
To enrich the analysis and to gain a better empirical understanding, I created a pseudo-panel to evaluate the association between computer use and routine task. Results suggest a negative relationship between the impact of computerisation and routine tasks, and a positive effect with abstract tasks. No effect is found for manual tasks. This suggests that middle-paid occupations are substitutes with computer use.
Finally, the analysis focuses on the progressive substitution of technology for labour in routine tasks, and how this contributed to the employment growth at the bottom part of the occupational distribution. By merging the main database with the ECHP and SILC, the analysis exploits questions about past jobs. As the model predicts, workers in middle-paid occupations become more mobile over time and they have the highest probability levels of mobility. Moreover, after dividing the data into graduate and non-graduate workers, I find that non-graduate middle workers move towards bottom occupations, while graduate middle employees shift towards top occupations. This fact counters Autor and Dorn (2013) prediction, as they expect that middle workers relocate to bottom-paid occupations.
One important observation can be made from this last result. While employment in Spain experienced a polarising trend at the occupational level between 1994 and 2014, the transitional analysis uncovers that the probability of switching for graduate middle workers to the top of the distribution significantly accelerates during the 2000s, coinciding with the dramatic changes in graduate labour supply. Far from suggesting that technology does not matter, this last result highlights that understanding the main drivers behind job polarisation is more difficult than expected. Much remains to be understood, especially when making predictions about the future of jobs.
The last year of study in the paper by Anghel et al. (2014) was 2012, while in the other analyses (Eurofound 2015; Fernández-Macías 2012; Goos et al. 2009, 2014; Oesch and Rodríguez-Menés 2011) was 2008.
The widely used O*Net task database from the US has information for only one point in time, and thus, is not suitable for analysing changes over time. The EWCS has five comparable waves (1995, 2000, 2005, 2010, and 2015) that allow me to analyse changes in the task-content of occupations.
It should be noted, however, that the methodology used in these analyses is not exactly the same. Fernández-Macías (2012) classifies occupations in three equally-sized groups in terms of employment shares instead of using the uneven grouping followed by Goos et al. (2014). For more information refer to the recent survey by Sebastian (2017).
For more information on the methodological differences, see "Appendix A".
Goos and Manning (2007) and Goos et al. (2009, 2014) also refer to this phenomenon as "routinisation".
Available at: https://www.uu.nl/staff/MGoos/0.
Previous papers that have used workers-reported information to build task measures include Spitz-Oener (2006) for Germany, Green (Green 2012) for the UK, and Autor and Handel (2013) for the US.
Autor and Handel (2013) follow a principal component analysis to derive continuous job task variables taking advantage of multiple responses of items.
The first four items are binary questions (1 = yes, 2 = no). The last question provides answers in intensity frequencies (1 = all of the time, 2 = almost all of the time, 3 = around ¾ of the time, 4 = around half of the time, 5 = around ¼ of the time, 6 = almost never, 7 = never).
Questions provide answers in intensity frequencies (1 = all of the time, 2 = almost all of the time, 3 = around ¾ of the time, 4 = around half of the time, 5 = around ¼ of the time, 6 = almost never, 7 = never).
US Census 2000 codes are matched to the International Standard Classification of Occupations.
I merge the EPA with the EES, and two filters are applied to the final data. First, I drop the occupations where I do not have information (ISCO 11, ISCO 61 and ISCO 92.). Second, I retain only those jobs which appear in both surveys and with at least five observations. After applying both filters, I reduce the total number of jobs from 279 to 226. See "Appendix C" for details on the measures discussed in this section.
The shape of the graph does not change if median average earnings are used for determining job quality.
This methodology has been applied by Autor and Dorn (2013).
It is not possible to create groups which contain exactly the same percentage of employment since occupations are defined as inseparable units.
In this paper, wage polarisation is understood in the following way: first jobs are ranked according to their mean hourly wage in the first year. The change in wages is measured in each occupation between the first and the last year of our period of study. If the mean wage of jobs is found to be growing at the top and bottom of the wage ranking in the first year, while the mean wage of jobs in the middle of the ranking is decreasing, this phenomenon is defined as wage polarisation.
During the period study I deal with two reclassifications at the occupation and industry level. The 2010 ESS and the 2014 ESS display occupations using the CNO-11 (based on the ISCO-08) and industry using the CNAE-93 (based on NACE.Rev.2). Like the EPA, I convert the ISCO-08 into the ISCO-08 and the NACE.Rev.2 into the NACE.Rev.1.
Fernández-Macías (2012) criticises this classification, arguing that a division in even groups would not lead to job polarisation in Europe. Our results remain invariant to this alternative classification. I still observe the polarisation pattern with middle-occupations exhibiting relative declining shares with respect to the top and the bottom.
The European Community Household Panel (ECHP) covers the period 1994–2001. The Survey of Income and Living Conditions goes from 2005 to 2015.
Adding unemployment and inactivity will determine if it could be that middle-workers end up outside the labour market.
Autor and Dorn (2009) and a much more recent paper by Lewandowski et al. (2017) document the important relationship between age and occupational mobility due to polarisation.
The results remain invariant if I use the first component of the principal component analysis.
The shape of the graph does not change if median average are used for determining job quality.
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I am particularly indebted to Dr. Carlos Gradin and Enrique Fernández-Macías for their supervision while I was visiting the University of Vigo and Eurofound. I would like to thank Rafael Muñoz de Bustillo and José Ignacio Antón for helpful discussion. I am also grateful to two anonymous referees and one editor for their comments and suggestions. I acknowledge the financial support of the Eduworks Marie Curie Initial Training Network Project (PITN-GA-2013-608311) of the European Commission's 7th Framework Program. I thank participants of the AIAS lunch seminar, 2016 Barcelona GSE Summer Forum, the 2016 EALE in Ghent, and the 2016 SAEe in Bilbao for comments and useful discussions.
Despacho 113, Facultad de Derecho, University of Salamanca, Campus Miguel de Unamuno, 37007, Salamanca, Spain
Raquel Sebastian
Correspondence to Raquel Sebastian.
Appendix A: Methodology in 5 Spanish papers
See Table 12.
Table 12 Methodology in 5 Spanish papers
Appendix B: The construction of the indexes
The procedure I have followed for constructing the indices can be summarized in a number of steps:
Identification of variables: I first identified the variables that could match the elements in our model.
Normalization of variables to a 0–1 scale: in the original sources, the individual variables use different scales which are not directly comparable. Therefore, they had to be normalized before they could be aggregated. I opted for a normative rescaling to 0–1, with 0 representing the lowest possible intensity of performance of the task in question, and 1 the highest possible intensity.
Correlation analysis: once the variables related to an individual element in my model were normalized, I proceeded to analyse the correlations between them. In principle, different variables measuring the same underlying concept should be highly correlated, although there are situations in which they may legitimately not be (for instance, when two variables measure two compensating aspects of the same underlying factor). Beside standard pairwise correlations, I computed Cronbach's Alpha to test the overall correlation of all the items used for computing a particular index, and a Principal Components Factor Analysis to evaluate the consistency of the variables and identify variables that did not fit my concept well.
Once I selected the variables to be combined into a single index, I proceeded to combine them, by using the first component of a Principal Component Analysis.Footnote 23 Unless I had a particular reason to do otherwise, all the variables used for a particular index received the same weight.
Finally, I proceeded to compute their average scores for all the occupation combinations at the two-digit level and sector of activity at the one-digit level. When the data source included the information at the individual worker level, I computed also the standard deviation and number of workers in the sample, for later analysis.
Data from the EPA on the level of employment in each job was added to the dataset holding the task indices. These employment figures were later used for weighting the indices.
Appendix C: Methodology applied to measured job polarisation
My enquiry builds on a methodology first proposed by Joseph Stiglitz for the study of occupational change in the US, later refined by Wright and Dwyer (2003). Due to its simplicity, it is subsequently applied subsequently applied to British (Goos and Manning 2007), German (Kampelmann and Rycx 2011), Swedish (Adermon and Gustavsson 2015) and European data (Goos et al. 2009, 2014; Fernández-Macías 2012). Three steps are usually followed.
In the first step, I define a job as a particular occupation in a particular industry. Therefore, jobs are classified into a matrix whereas the columns are economic sectors and the rows are occupations. Examples of these jobs would be managers in the agricultural sector or clerks in the construction industry. Throughout our investigation, I use two-digit International Standard Occupational Classification (ISCO-88) code and one-digit industry codes from the Classification of Economic Activities in the European Community (NACE.Rev.1) as a measure of jobs. Individuals aged 18–66 are placed in cells, and weighted by the total population of each cell. Because many cells are empty, two filters are applied to the data. I first drop observations for which information on the job variable is missing. Second, I also drop Melilla and Ceuta region due to no accurate information, reducing the total number of jobs from 276 to 226 jobs.
In the next step, I compute jobs' real hourly wage by taking the ratio of the gross annual salary to the total number of hours actually worked. The salary figure includes extraordinary payments. I then rank jobs according to their mean wage in the first year.Footnote 24
In the last step, I represent graphically the evolution of jobs in terms of their wages where there are three possibilities of representation: the actual point of jobs where I plot the percent change in employment share against the (log) mean wage. In the second case, I display smoothing regressions rather than the actual data point. In the last case, I define the wage quintiles.
Appendix D: Figures
In Figs. 4, 5, and 6 I establish three time periods due to the reclassifications at the occupation and activity level. Table 13 presents the period, the occupation, the sector and the main earning database (Figs. 7, 8).
Table 13 Period of analysis, main classifications and databases to be used
Appendix E: Variables construction
Wages My wage variable (hwage) is the gross hourly pay. For all the cases hwage was computed as gross usual weekly pay divided by usual hours and minutes worked per week, including usual overtime. Wages are measured in euro. I trim my data such that hourly wages lower than 1 and higher than 100 are excluded.
Occupations I classify occupations according to the International Standard Classification of Occupations (ISCO-88). Occupations were originally classified according to the National Classification of Occupations (CNO-94). Codes are manually matched on the basis of the guidelines distributed by the Occupational Information Unit of the Office for National Statistics. This harmonisation allows researchers to compare occupations over time to make our results strictly comparable to other papers.
Industry I classify industry according to the Statistical Classification of Economic Activities in the European Commission (NACE.Rev.1.1). Industry codes were originally classifies according to the National Classification of Economic Activities (CNAE-93). Codes are manually matched on the basis of the guidelines distributed by EUROSTAT. This harmonisation allows researchers to compare occupations over time to make our results strictly comparable to other papers. NACE.Rev.1 defines five levels of aggregation, consisting of 17 one-letter sections, 31 two-letter sub-sections, 60 two-digit main groups, 222 three-digit groups, and 513 four-digit sub-groups. NACE.Rev.1 was in turn based on the International Standard Industrial Classification of All Economic Activities (ISIC) Rev 3, published by the United Nations.
Education My education variable distinguishes four groups of workers: elementary, basic, medium, and high educated (skilled). In the Spanish Labour Force Survey I exploit the variable (estud) which indicates the highest qualification held by the interviewer. Both educational and vocational qualification levels are available in the list provided to respondents. The usual ISCED division into low, medium and high is then adopted where low is equivalent to ISCED 0–2 (i.e. primary and lower secondary education), medium is given by ISCED 3–4 (i.e. upper secondary and post-secondary non-tertiary education) and high is ISCED 5–7 (i.e. tertiary education). The derived categorical variable for education takes value of 1 for low educated, 2 for medium and 3 for high.
Computer use I create a variable that capture computer use. In the EWCS I use the question: "Does your main job involve… working with computer, laptops, etc.? The variable ranges from 1 "all of the time" to 7 "never" ("almost all of the time", "around ¾ of the time", "around ½ of the time", "around ¼ of the time", and "almost never" correspond to middle answers).
Sebastian, R. Explaining job polarisation in Spain from a task perspective. SERIEs 9, 215–248 (2018). https://doi.org/10.1007/s13209-018-0177-1
Job polarisation
Routine employment
Occupational mobility | CommonCrawl |
\begin{document}
\title{\LARGE \bf
Sparsity-Promoting Iterative Learning Control for Resource-Constrained Control Systems
} \thispagestyle{empty} \pagestyle{empty}
\begin{abstract} We propose novel iterative learning control algorithms to track a reference trajectory in resource-constrained control systems. In many applications, there are constraints on the number of control actions, delivered to the actuator from the controller, due to the limited bandwidth of communication channels or battery-operated sensors and actuators. We devise iterative learning techniques that create sparse control sequences with reduced communication and actuation instances while providing sensible reference tracking precision. Numerical simulations are provided to demonstrate the effectiveness of the proposed control method. \end{abstract}
\begin{keywords}
Iterative learning control; Sparse control; Convex optimization \end{keywords}
\section{Introduction}\label{sec: intro}
A multitude of techniques are now available in the literature for precise control of mechatronic systems; see, e.g.,~\cite{GGD+:17}. Iterative Learning Control (ILC) is one of the well-known techniques for accurately tracking reference trajectories in industrial systems, which repetitively executes a predefined operation over a finite duration; see, e.g.,~\cite{Moo:93, BTA:06, Owe:16}. The key idea of iterative learning control relies on the use of the information gained from previous trails to update control inputs to be applied to the plant on the next trial. Iterative learning control was first introduced by Arimoto et al.~\cite{AKM:84} to achieve high accuracy control of mechatronic systems. Since the original work was published in 1984, it has been successfully practiced in various areas, including additive manufacturing machines~\cite{BHA+:11}, robotic arms~\cite{VDS:11}, printing systems~\cite{BOK+:14}, electron microscopes~\cite{CTL+:09}, and wafer stages~\cite{MCT:07}.
Modern industrial systems, which employ a large number of spatially distributed sensors and actuators to monitor and control physical processes, suffer from resource -- control, communication, and computation -- constraints. To provide a guaranteed performance or even preserve the stability of the closed-loop systems, it is necessary to take these limitations into account while designing and implementing control algorithms. Sometimes the limited bandwidth of legacy communication networks imposes a constraint on the rate of data transmissions. Besides, when the feedback loop is closed over wireless networks, a further resource constraint becomes apparent due to the use of battery-powered sensors and actuators~\cite{Dem:15}. The reduced actuator activity also prolongs the lifetime of actuators or improves the fuel efficiency. Therefore, it is desirable to have either sparse or sporadically changing control commands to reduce the use of actuators.
Sparsity-promoting techniques, which is borrowed from compressive sensing literature, have been successfully applied to a number of control problems to tackle the resource constraints mentioned above; see e.g.,~\cite{GaM:12, HGM:13, CGH+:13, NQO:14, NOQ:16, NQN:16}. The authors of~\cite{GaM:12, HGM:13, CGH+:13} modified the original model predictive control cost with an $\ell_{1}^{}$-penalty term to promote the sparsity in the control input trajectory. The authors of~\cite{NQO:14, NOQ:16, NQN:16} designed energy-aware control algorithms to limit the actuator activity while providing an attainable control performance. Their design is also based on sparse optimization using $\ell_{1}^{}$-norm. To the best of our knowledge, the design of iterative learning control algorithms for resource-constrained systems has not been addressed in the literature and is subject of this paper.
\textit{Contributions.}
In this paper, we develop a Sparsity-promoting Iterative Learning Control (S-ILC) technique for resource-constrained control systems. The main departure from the standard ILC approach is that we introduce a regularization term into the usual $\ell_2$-norm cost functions to render the resulting control inputs sparse. The sparsity here is in the cardinality of changes in control values applied to a finite horizon. Moreover, we include additional constraints to model the practical limits on the magnitude of applied control signals. The resulting control problem is then solved using a backward-forward splitting method which trades off between minimizing the tracking error and finding a sparse control input that optimizes the cost with respect to regularizer term. We demonstrate the monotonic convergence of the technique in lack of modeling imperfections. Moreover, we develop an accelerated algorithm to reduce the number of trials required for S-ILC to converge to optimality.
\textit{Outline.}
The remainder of this paper is organized as follows: Section~\ref{sec:problem_formulation} introduces the problem definition. Section~\ref{sec:SILC} presents the sparse iterative learning control problem and associated algorithms to solve it. A numerical study is performed in Section~\ref{sec:numerical}. Finally, Section~\ref{sec:conclusion} presents concluding remarks. The appendix provides proofs of the main results
\textit{Notation.}
The $n$-dimensional real space is represented by $\mathbb{R}^n$. $\mathbb{E}$ denotes a finite dimensional euclidean space with inner product $\langle \cdot, \cdot \rangle$. For $u\in \mathbb{R}^n$, its $\ell_1$ and $\ell_2$ norms are
\begin{align*}
\parallel u \parallel_{1}^{} := \sum_{i=1}^{n} \vert u_{i}^{} \vert \;, \quad
\parallel u \parallel_{}^{} := \Bigg( \sum_{i=1}^{n} u_{i}^{2} \Bigg)_{}^{\frac{1}{2}}. \end{align*}
The spectral radius of the real square matrix $M\in \mathbb{R}^{n\times n}$ is denoted by $\rho(M)$. The Euclidean projection of $u\in \mathbb{R}^n$ into the compact convex set $\mathcal{U}$ is denoted by $\Pi_\mathcal{U}^{}(\cdot)$.
\section{Problem Formulation}\label{sec:problem_formulation}
\subsection{System model}
We consider the following discrete-time, single-input-single-output (SISO), stable, linear time-invariant (LTI) system $P(z)$ with state space representation:
\begin{align}
x_{k}^{}[t+1] =&\; A x_{k}^{}[t] + B u_{k}^{}[t] \;, \label{eqn:system_model_1a} \\
y_{k}^{}[t] =&\; C x_{k}^{}[t] \;, \label{eqn:system_model_1b} \end{align}
where $t\in\mathbb{N}_{0}^{}$ is the time index (i.e., sample number), $k\in\mathbb{N}_{0}^{}$ is the iteration number, $x_{k}^{}[t]\in\mathbb{R}_{}^{n}$ is the state variable, $u_{k}^{}[t]\in\mathbb{R}$ is the control input, $y_{k}^{}[t]\in\mathbb{R}$ is the output variable, and $A$, $B$ and $C$ are matrices of appropriate dimensions. The initial condition $x[0] = x_{0}^{}$ is also assumed to be given, and these initial conditions are the same at the beginning of each trial. The input-output behavior of the system in~\eqref{eqn:system_model_1a} and~\eqref{eqn:system_model_1b}, can be described via a convolution of the input with the impulse response of the system:
\begin{align}
y_{k}^{}[t] = CA_{}^{t}x_{0}^{} + \sum_{\tau=0}^{t-1} CA_{}^{t-\tau-1}Bu_{k}^{}[\tau] \;.
\label{eqn:impulse_response} \end{align}
The coefficients $CA_{}^{t}B$ for any $t \in\{0, 1,\cdots,\mathrm{T}\}$ are referred to as the Markov parameters of the plant $P(z)$, provided in~\eqref{eqn:system_model_1a} and~\eqref{eqn:system_model_1b}.
\subsection{Lifted system model}
Since we focus on a finite trial length $\mathrm{T}$, it is possible to evaluate~\eqref{eqn:impulse_response} for all $t\in\{0,1,\cdots,\mathrm{T}\}$ and, similar to~\cite{OHD:09}, write its lifted version as
\begin{align}
y_{k}^{} = G u_{k}^{} + d \;, \end{align}
where
\begin{align*}
G =&\;
\begin{bmatrix}
CA_{}^{t_{}^{*}-1}B & 0 & \cdots & 0 \\
CA_{}^{t_{}^{*}}B & CA_{}^{t_{}^{*}-1}B & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
CA_{}^{\mathrm{T}-1}B & CA_{}^{\mathrm{T}-2}B & \cdots & CA_{}^{t_{}^{*}-1}B
\end{bmatrix} \;, \\
d =&\;
\begin{bmatrix}
CA_{}^{t_{}^{*}} x_{0}^{} & CA_{}^{t_{}^{*}+1} x_{0}^{} & \cdots & CA_{}^{\mathrm{T}} x_{0}^{}
\end{bmatrix}_{}^{\top} \;. \end{align*}
The vectors of inputs and output series are defined as
\begin{align*}
u_{k}^{} =&\;
\begin{bmatrix}
u_{k}^{}[0] & u_{k}^{}[1] & \cdots & u_{k}^{}[\mathrm{T}-t_{}^{*}]
\end{bmatrix}_{}^{\top} \;, \\
y_{k}^{} =&\;
\begin{bmatrix}
y_{k}^{}[t_{}^{*}] & y_{k}^{}[t_{}^{*}+1] & \cdots & y_{k}^{}[\mathrm{T}]
\end{bmatrix}_{}^{\top} \;. \end{align*}
The relative degree of the transfer function $P(z)$ is denoted by $t_{}^{*} > 0$. Notice that the matrix $G$ has a Toeplitz structure.
\subsection{Trajectory tracking problem}
In this paper, we focus on the reference trajectory tracking problem. It is assumed that a reference trajectory $r[t]$ is given over a finite time-interval between $0$ and $\mathrm{T}$. The objective is, here, to determine a control input trajectory $\{u[t]\}_{t=0}^{\mathrm{T}-t_{}^{*}}$ that minimizes the tracking error:
\begin{align}
\parallel e \parallel_{}^{2} \; \triangleq \; \parallel r - y \parallel_{}^{2} \; = \; \parallel r - G u \parallel_{}^{2} \;,
\label{eqn:least_squares} \end{align}
where
\begin{align*}
r =&\;
\begin{bmatrix}
r[t_{}^{*}] & r[t_{}^{*}+1] & \cdots & r[\mathrm{T}]
\end{bmatrix}_{}^{\top} \;, \\
e_{k}^{} =&\;
\begin{bmatrix}
e_{k}^{}[t_{}^{*}] & e_{k}^{}[t_{}^{*}+1] & \cdots & e_{k}^{}[\mathrm{T}]
\end{bmatrix}_{}^{\top} \;. \end{align*}
The control sequence, which results in an output sequence $\{y[t]\}_{t=t_{}^{*}}^{\mathrm{T}}$ that perfectly tracks the reference trajectory $\{r[t]\}_{t=t_{}^{*}}^{\mathrm{T}}$, can be computed via solving the linear equation:
\begin{align}
u_{}^{\star} = G_{}^{-1}(r - d) \;.
\label{eqn:optimal_solution} \end{align}
Without loss of generality, one can assume that ${x_{0}^{} = 0}$, and, equivalently, $d = 0$. Hence,~\eqref{eqn:optimal_solution} can be rewritten as
\begin{align}
u_{}^{\star} = G_{}^{-1} r \;.
\label{eqn:optimal_solution_2} \end{align}
As argued in~\cite{AOR:96}, the direct inversion of $G$ is not practical in general since it requires having the exact information of $G$. Besides, instead of inverting the entire matrix $G$, it is sufficient to compute the pre-image of $r$ under $G$.
\subsection{Gradient-based iterative learning algorithm}
There are various techniques in the literature to solve the unconstrained optimization problem~\eqref{eqn:least_squares} iteratively. The gradient-based iterative learning control algorithm has been received an increasing attention (see, e.g.,~\cite{OHD:09, AOR:96, ChO:13}) due to its simplicity and light-weight computations compared to higher-order techniques. This algorithm generates the control inputs to be used in the next iteration using the relation:
\begin{align*}
u_{k+1}^{} = u_{k}^{} + \gamma G_{}^{\top}e_{k}^{} \;, \end{align*}
where $\gamma>0$ is the learning gain. Using this update law, the error evolves as
\begin{align*}
e_{k+1}^{} = \big(I - \gamma GG_{}^{\top} \big) e_{k}^{}. \end{align*}
Using the norm inequality, provided in~\cite{HoJ:13}, we have:
\begin{align*}
\parallel e_{k+1}^{} \parallel \; = \; \parallel (I - \gamma GG_{}^{\top})e_{k}^{} \parallel \; \leq \; \parallel I -\gamma GG_{}^{\top} \parallel \parallel e_{k}^{} \parallel \;. \end{align*}
For minimum phase systems, the smallest singular value of the matrix $G$ is nonzero and if one picks $0<\gamma \leq \nicefrac{2}{\rho(GG^\top)}$, then $\Vert I - \gamma GG_{}^{\top} \Vert < 1$ holds. Consequently, $\Vert e_{k}^{} \Vert$ converges to zero linearly as $k \rightarrow \infty$.
It is worth noting that, for non-minimum phase systems, the matrix $G$ has some singular values that are very close to zero; therefore, it might be significantly ill-conditioned, leading $\Vert I-\gamma GG^\top\Vert$ to become nearly one. Taking into account the typical rounding errors that exists in numerical solvers, it is safe to assume that the matrix $G$ has zero singular values in order to avoid convergence issues due to mis-estimation of the optimal learning gain parameter.
\subsection{Trajectory tracking problem with sparsity constraint}
Trading off the accuracy of trajectory tracking for the sparsity in control signals amounts to solve
\begin{equation}\label{eq:cardinality_problem} \begin{aligned}
\text{minimize} &\quad \frac{1}{2}\parallel r - G u_{}^{} \parallel_{}^{2} \\
\text{subject to} &\quad \parallel T u_{}^{} \parallel_{0}^{} \leq M \\
& \quad u \in \mathcal{U}, \end{aligned} \end{equation}
where $M \leq N$ with $M\in\mathbb{N}_{0}^{}$ and $T\in \mathbb{R}^{N-1\times N}$ is the difference matrix
\begin{align*}
T =
\begin{bmatrix}
-1 & 1 & 0 & 0 & \cdots & 0 & 0 \\
0 & -1 & 1 & 0 & \cdots & 0 & 0 \\
0 & 0 & -1 & 1 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & & \vdots & \vdots \\
0 & 0 & 0 & 0 & \cdots & -1 & 1
\end{bmatrix}\;,
\end{align*}
and $\mathcal{U}$ is a compact and convex set which represents the practical limits on the input signal.
For example, limits on the magnitude of input signal can be modeled by either by a box constraint or an upper bound on $\ell_\infty$-norm of control input $u$. With the cardinality constraint in~\eqref{eq:cardinality_problem}, one limits the number of changes in control input values compared to the initial value $u[0]$, thereby promoting sparsity in the frequency of applying control input .
However, due to the cardinality constraint, the problem~\eqref{eq:cardinality_problem} is non-convex and difficult to solve. A common heuristic method in the literature relies on the $\ell_1$-regularized problem
\begin{align}\label{eq:sparse_control_problem} \begin{array}{ll}
\text{minimize} & \frac{1}{2}\parallel r - G u_{}^{} \parallel_{}^{2} + \lambda\parallel T u_{}^{} \parallel_{1}^{} \;,\\
\text{subject to} & u \in \mathcal{U}, \end{array} \end{align}
where the second term is referred as \emph{total variation} of signal $u$ and the problem~\eqref{eq:sparse_control_problem} is often called \emph{total variation denoising} in signal processing literature \cite{Rudin:92}.
\section{Sparse Iterative Learning Control}\label{sec:SILC}
In this section, we develop a first-order method to solve the regularized control problem, proposed in~\eqref{eq:sparse_control_problem}, iteratively. Our technique is based on backward-forward splitting method~\cite{Passty:79}, which is applied to the composite problem:
\begin{equation}\label{eq:problem_sum_two_function}
\mbox{minimize} \; F(u):=f(u)+g(u) \end{equation} where $f:\mathbb{E}\rightarrow \mathbb{R}$ is a differentiable convex function with Lipschitz continuous gradient $L$ satisfying
\begin{equation} \nonumber \begin{aligned} \Vert \nabla f(x) - \nabla f(y)\Vert \leq L \Vert x - y\Vert \quad \forall x, y \in \mathbb{E}, \end{aligned} \end{equation}
and $g:\mathbb{E}\rightarrow (-\infty, +\infty]$ a proper closed convex function.
Given a scalar $t>0$ the \emph{proximal} map associated to $g$ is defined as \begin{equation} \begin{aligned} \mbox{prox}_t (g)(x) := \underset{u}{\mbox{argmin}} \left\{ g(u)+\dfrac{1}{2t}\Vert u-x\Vert^2 \right\}. \end{aligned} \end{equation} An important property of the proximal map is that this point (for any proper closed convex function $g$) is the unique solution to the associated minimization problem, and as a consequence, one has \cite[Lemma 3.1]{Beck:09}: \begin{equation} \begin{aligned} (I+t\partial g)^{-1}(x)= {\mbox{prox}_t}(g)(x), \; \forall x\in \mathbb{E}. \end{aligned} \end{equation} This result can be used to find the following optimality condition for \eqref{eq:problem_sum_two_function} \begin{equation}\nonumber \begin{aligned} 0 & \in t \nabla f(x^\star) + t \partial g(x^\star)\\ x^\star &= (I+t\partial g)^{-1}(I-t\nabla f) (x^\star), \end{aligned} \end{equation} and then further developed to obtain a backward-forward splitting based method to solve \eqref{eq:problem_sum_two_function} \begin{equation}\label{eq:backward_forward} \begin{aligned} x_{k+1} =& \mbox{prox}_{\gamma} (g) (x_{k}-\gamma \nabla f(x_{k}))\\
=& \underset{x}{\mbox{argmin}}\; \left\{ g(x) + \dfrac{1}{2\gamma} \Vert x - (x_{k}-\gamma \nabla f(x_{k}))\Vert^2\right\}. \end{aligned} \end{equation} For instance, if $f= \Vert A x-b\Vert^2$ and $g=\Vert x \Vert_1$, then the famous Iterative-Shrinkage-Thresholding Algorithm (ISTA) is recovered; see e.g.,~\cite{Comm:05}. We use backward-forward splitting method to solve~\eqref{eq:sparse_control_problem}. In particular, let
\begin{equation}\label{eq:f_g} \begin{aligned} f(u) := \dfrac{1}{2}\Vert G u-r\Vert^2, \quad g(u) := \lambda \Vert T u \Vert_1 +\mathcal{I}_\mathcal{U}(u), \end{aligned} \end{equation}
with $\mathcal{I}_\mathcal{U}$ denoting the indicator function on $\mathcal{U}$; i.e., $\mathcal{I}_{\mathcal{U}}(u) = 0$ if $u\in \mathcal{U}$ and $\mathcal{I}_{\mathcal{U}}(u) =\infty$ otherwise.
Applying the backward-forward splitting, the sparse ILC update rule is given by
\begin{equation}\label{eq:BF_total_variation} \begin{aligned} &u_{k+1} = \mbox{prox}_{\lambda/\gamma}(g)(u_{k}+\gamma G^\top e_{k})\\ &\; =\underset{u} {\mbox{argmin}} \; \left\{ \Vert T u\Vert_1+ \mathcal{I}_{\mathcal{U}}(u) + \dfrac{1}{2\lambda\gamma}
\Vert u - (u_{k} + \gamma G^\top e_{k})\Vert^2 \right\} \end{aligned} \end{equation}
Unlike the ISTA algorithm with simple $\ell_{1}^{}$-norm regularization, the sparse ILC iterations ~\eqref{eq:BF_total_variation} involve a proximal map that does not admit a closed-form solution. To tackle this problem, we develop an iterative \emph{dual}-based approach for the proximal step. In particular, we are interested in solving
\begin{equation}\label{eq:denoising_tv} \begin{aligned} \underset{u\in\mathcal U}{\mbox{minimize}} \; \left\{ \lambda \Vert T u\Vert_1 + \dfrac{1}{2}
\Vert u - b\Vert^2 \right\} , \; \end{aligned} \end{equation}
by using a first-order method. We have the following result:
\begin{lemma} \label{lem:lemma_1} Denote $\mathcal{P}^{n-1}\subset \mathbb{R}^{n-1}$ as the $n-1$ dimensional real space bounded by unit infinity norm (i.e., $p\in \mathcal{P}^{n-1}$ then $\Vert p \Vert_\infty \leq 1$) and $\mathcal{L}\in\mathbb{R}^{n\times n-1}$ given as
\begin{align*}
\mathcal{L} =
\begin{bmatrix}
1 & 0 & 0 & \cdots & 0 \\
-1 & 1 & 0 & \cdots & 0 \\
0 & -1 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & & \vdots \\
0 & 0 & 0 & \cdots & 1 \\
0 & 0 & 0 & \cdots & -1
\end{bmatrix} \;. \end{align*}
Let $p\in \mathcal{P}^{n-1}$ be the optimal solution of
\begin{equation}\label{eq:minimize_tv_prox} \begin{aligned}
\underset{p\in \mathcal{P}^{n-1}}{\mbox{minimize}}&\; h(p):=-\Vert \Pi_\mathcal{U}(b-\lambda \mathcal{L}p) - (b-\lambda \mathcal{L}p) \Vert^2 \\
&\; + \Vert b-\lambda \mathcal{L}p \Vert^2 \;.
\end{aligned} \end{equation}
Then, the optimal solution of~\eqref{eq:denoising_tv} is given by
\begin{equation}\label{eq:opt_u_tv_denoising} \begin{aligned} u & = \Pi_\mathcal{U}(b - \lambda \mathcal{L}p) . \end{aligned} \end{equation}
\end{lemma}
Next, we present the smoothness properties of~\eqref{eq:minimize_tv_prox}.
\begin{lemma} \label{lem:lemma_2} The cost function in~\eqref{eq:minimize_tv_prox} is continuously differentiable, and its gradient is given by
\begin{equation} \begin{aligned} \nabla h(p):= -2\lambda \mathcal{L}^\top \Pi_\mathcal{U}(b-\lambda \mathcal{L} p) \;. \end{aligned} \end{equation}
Moreover, its Lipschitz constant is bounded by
\begin{multline} \Vert \nabla h(p) - \nabla h(p\prime) \Vert \leq 2\lambda^2 \Vert \mathcal{L}^\top \Vert^2 \Vert p - p\prime\Vert \\ = 2\lambda^2 \rho(\mathcal{L}^\top \mathcal{L}) \Vert p - p\prime\Vert, \; \forall p, p\prime \in \mathcal{P}^{n-1}. \end{multline}
Moreover, it follows ${\rho(\mathcal L^\top \mathcal L)\leq 4}$. \end{lemma}
We are now ready to form an accelerated projected gradient-based method to solve~\eqref{eq:minimize_tv_prox} and~\eqref{eq:opt_u_tv_denoising}. Algorithm~\ref{alg:1} solves the problem by employing a Nesterov-like acceleration applied to the dual domain. The technique offers a better convergence rate $O(1/k^2)$ as opposed to a gradient-based technique that converges at rate $O(1/k)$; see e.g., \cite{Beck:09}.
\begin{algorithm}[H] \caption{\textbf{Accelerated Projected Gradient} } \label{alg:1} \begin{algorithmic}[1] \State Let $(N,\lambda, b)$ be given as input. Set $q_1 = 0$. \For{$k = 1,\dots, N$ compute} \State \begin{equation} \nonumber \begin{aligned} p_k &= \Pi_{\mathcal{P}^{n-1}} \left[ q_k + \dfrac{1}{\lambda \rho(\mathcal L^\top \mathcal L)} \mathcal L^\top \Pi_\mathcal{U}(b-\lambda \mathcal L q_k)\right] \\ t_{k+1} &= \dfrac{1+\sqrt{1+4t_k^2}}{2}\\ q_{k+1} &= p_k + \dfrac{t_k-1}{t_{k+1}}(p_k - p_{k-1}) \end{aligned} \end{equation} where $\Pi_{\mathcal{P}^{n-1}}(x)_i = \dfrac{x_i}{\max\{1, \vert x_i\vert\}}$ for $i = 1,\dots, n-1$. \EndFor \State Return $(x^\star, p^\star) = (\Pi_\mathcal U (b- \lambda \mathcal L p_N), p_N)$. \end{algorithmic} \label{alg_summary} \end{algorithm}
One can implement the sparse iterative learning control updates $u_{k+1}$ in~\eqref{eq:BF_total_variation} by first taking a gradient step on $u_k$ and then computing the proximal step via Algorithm~\ref{alg:1}. Algorithm~\ref{alg:2} proposes the gradient-based S-ILC method.
\begin{algorithm}[H] \caption{\textbf{Gradient-based S-ILC}} \label{alg:2} \begin{algorithmic}[1] \State Let $(N_1, N_2, \lambda, \mathcal U)$ be given as input. Set $u_0, e_0$ to vector 0, and $\gamma = 1/\rho(G^\top G)$. \For{$k = 1,\dots, N_1$ } \State Set $b_k = u_{k-1} +\gamma G^\top e_{k-1}$. \State Run \textbf{Algorithm~\ref{alg:1}} with $(N_2, \gamma\lambda, b_k, \mathcal U)$ and obtain $u_k\in \mathcal U$.
\State Apply $u_k$ to the plant and receive $e_k$. \EndFor \State Return $u^\star = u_{N_1}$. \end{algorithmic} \label{alg_summary} \end{algorithm}
Next lemma confirms that the gradient-based S-ILC results in a non-increasing sequence.
\begin{lemma}\label{lem:3} Consider the sequence $\{u_k\}_{k\geq 0}^{}$ generated by S-ILC. The associated functional values
\begin{equation}\nonumber \begin{aligned} F(u_k):= \dfrac{1}{2}\Vert G u_k -r\Vert^2 + \lambda \Vert T u_k\Vert \end{aligned} \end{equation}
is non-increasing. That is, for all $k\geq 1$,
\begin{equation}\nonumber F(u_{k+1}) \leq F(u_{k}). \end{equation} \end{lemma}
Moreover, from \cite[Theorem 3.1]{Beck:09}, it follows that
\begin{equation}\nonumber \begin{aligned} F(u_k) - F(u^\star) \leq \dfrac{\rho(G^\top G) \Vert u_0 - u^\star \Vert}{2k},\; u_0\in \mathcal{U}; \end{aligned} \end{equation}
where $u^\star$ is the optimal control input while $k\geq 1$ is the number of outer-loop iterations in the gradient-based S-ILC.
To accelerate the convergence of Algorithm~\ref{alg:2}, Nesterov-like iterations can be applied to its outer-loop. The straight-forward application of the Nesterov's method leads to the following updates:
\begin{equation}\label{eq:SILC_nest} \begin{aligned} b_k &= y_k -\gamma G^\top (Gy_{k}-r),\\ y_{k+1} &= u_k + \dfrac{t_k-1}{t_{k+1}}(u_k - u_{k-1}), \end{aligned} \end{equation}
where $u_k$ is the control input obtained from inner-loop Algorithm~\ref{alg:1}. However, this requires the access to the reference signal $r$, which is not practical in ILC application. We rewrite these updates to find a feasible formulation for ILC. Let $\Delta e_k :=e_k - e_{k-1}$ and $y_{k+1}:=u_k + \tau_{k+1} \Delta u_k$ where $\tau_{k+1} := ({t_k-1})/{t_{k+1}}$ and $\Delta u_{k} = u_k - u_{k-1}$. Now, we rewrite the $b_k$-th update in~\eqref{eq:SILC_nest} as
\begin{equation}\nonumber \begin{aligned} b_{k+1} &= y_{k+1} -\gamma G^\top (G y_{k+1}-r)\\ & = u_k + \tau_{k+1} \Delta u_k -\gamma G^\top(G u_k + \tau_{k+1} G \Delta u_k -r),\\ & = u_k +\tau_{k+1} \Delta u_k + \gamma G^\top( e_k + \tau_{k+1} \Delta e_k ), \end{aligned} \end{equation}
which relates the auxiliary variable $b_k$ to -- the readily available -- control input and error signals.
\begin{algorithm}[H] \caption{\textbf{Accelerated S-ILC}}
\label{alg:3} \begin{algorithmic}[1] \State Let $(N_1, N_2, \lambda, \mathcal U)$ be given as input. Set $u_{-1},u_0, e_{-1}, e_0$ to vector $0$, $ t_0 = 0$, $t_1 =1$, and $\gamma=1/\rho(G^\top G)$. \For{$k = 1,\dots, N_1$} \State Set \begin{equation} \nonumber \begin{aligned} t_{k} &= \dfrac{1}{2}+\dfrac{1}{2}\sqrt{1+4t_{k-1}^2},\; \quad \tau_{k} = \dfrac{t_{k-1}-1}{t_k},\\ b_k &= u_{k-1} + \tau_k \Delta u_{k-1} + \gamma G^\top (e_{k-1}+ \tau_k \Delta e_{k-1}). \end{aligned} \end{equation}
\State Run \textbf{Algorithm~\ref{alg:1}} with $(N_2, \gamma\lambda, b_k, \mathcal U)$ and obtain $u_k\in \mathcal U$. \State Apply $u_k$ to the plant and receive $e_k$.
\EndFor \State Return $u^\star = u_{N_1}$. \end{algorithmic} \label{alg_summary} \end{algorithm}
Algorithm~\ref{alg:3} presents the accelerated Nesterov-like iterates to solve S-ILC. From \cite[Theorem 4.4]{Beck:09a}, it yields
\begin{equation}\nonumber \begin{aligned} F(u_k) - F(u^\star) \leq \dfrac{2\rho(G^\top G) \Vert u_0 - u^\star \Vert}{(k+1)^2},\; u_0\in \mathcal{U}; \end{aligned} \end{equation}
where $k \geq 1$ is the outer-loop counter of Algorithm~\ref{alg:3}.
\section{Numerical Example}\label{sec:numerical}
To demonstrate the effectiveness of S-ILC algorithms, we consider a robot arm ( see~\cite{VDS:11}) with one rotational degree-of-freedom as schematically shown in Fig.~\ref{fig:robot_arm}. The input is the torque $\tau$ applied to the arm at the joint and is limited to the range of $\pm \unit[12]{Nm}$, whereas the output $\theta$ is the angle of the arm measured as seen in Fig.~\ref{fig:robot_arm}. The dynamics of the robotic arm can be described by the following differential equation:
\begin{align}
\ddot{\theta} = - \frac{g}{l}\sin\theta - \frac{c}{ml_{}^{2}} \dot{\theta} + \frac{1}{ml_{}^{2}}\tau \;,
\label{eqn:robot_arm_dynamic_model} \end{align}
where the arm length is $l = \unit[1.0]{m}$, the payload mass is $m = \unit[1.0]{kg}$, the viscous friction coefficient is $c = \unitfrac[2.0]{Nms}{rad}$, and the gravitational acceleration is $g = \unitfrac[9.81]{m}{s^2}$. Changing the variables $x_{}^{(1)} \triangleq \theta$, $x_{}^{(2)} \triangleq \dot{\theta}$, $u \triangleq \tau$, and $y \triangleq \theta$, the nonlinear system~\eqref{eqn:robot_arm_dynamic_model} is sampled by using zero-order-hold and a sampling time of $T_{s}^{} = \unit[0.005]{s}$. The resulting discrete-time system becomes
\begin{align*}
x_{}^{(1)}[t+1] =&\; x_{}^{(1)}[t] + T_{s}^{}x_{}^{(2)}[t] \;, \\
x_{}^{(2)}[t+1] =&\; -\frac{g T_{s}^{}}{l}\sin(x_{}^{(1)}[t]) + \bigg( 1 - \frac{cT_{s}^{}}{ml_{}^{2}} \bigg)x_{}^{(2)}[t] \\
&\; + \frac{T_{s}^{}}{ml_{}^{2}} u[t] \;, \\
y[t] =&\; x_{}^{(1)}[t] \;. \end{align*}
\begin{figure}
\caption{A schematic drawing of the robot arm.}
\label{fig:robot_arm}
\end{figure}
To construct the gradient of the cost function~\eqref{eqn:least_squares}, which is equal to $G_{}^{\top}$, the discrete-time non-linear plant model is linearized around the stationary point $x_{}^{(1)} = \theta = 0$, resulting in the linear approximation:
\begin{align}
x[t+1] &=
\begin{bmatrix}
1 & T_{s}^{} \\ -\frac{g T_{s}^{}}{l} & 1 - \frac{cT_{s}^{}}{ml_{}^{2}}
\end{bmatrix}
x[t] +
\begin{bmatrix}
0 \\ \frac{T_{s}^{}}{ml_{}^{2}}
\end{bmatrix}
u[t] \;, \\
y[t] &=
\begin{bmatrix}
1 & 0
\end{bmatrix}
x[t] \;. \end{align}
Note that the linearized plant model is used to compute the gradient of the cost function~\eqref{eqn:least_squares}, whereas the nonlinear model is employed in actual trials. The trial length is $\unit[6]{s}$ and the desired trajectory of the robot arm, illustrated in Fig.~\ref{fig:results}, is
\begin{align*}
r[t] = \frac{\pi}{5}\sin\bigg(\frac{\pi T_{s}^{} t}{3}\bigg) + \frac{2\pi}{25}\sin\big(\pi T_{s}^{} t\big) \end{align*}
for all $t \in \{0, 1, \cdots, 1200 \}$.
\begin{figure}
\caption{Tracking performance of the S-ILC algorithm for various values of the regularization parameter $\lambda$.}
\label{fig:results}
\end{figure}
\begin{table}[ht] \caption{Comparison of Regularization Parameters} \centering \begin{tabular}{c c c c} \hline\hline $\nicefrac{\lambda}{\rho\big(G_{}^{\top}G\big)}$ & $\parallel r - Gu \parallel_{2}^{}$ & $\parallel Tu \parallel_{1}^{}$ & $\parallel Tu \parallel_{0}^{}$ \\ [0.5ex] \hline
$0$ & 1.0694 & 42.4495 & 1155 \\
$0.5$ & 1.0845 & 38.0014 & 799 \\%402 \\
$2.5$ & 1.1406 & 34.5145 & 754 \\
$5$ & 1.2117 & 33.0654 & 463 \\[1ex] \hline \end{tabular} \label{tab:compare_parameters} \end{table}
The simulation is carried out over 50 trials, and the results are displayed in Fig.~\ref{fig:results}. The optimization problem~\eqref{eq:sparse_control_problem} becomes a least square problem with a box constraint when $\lambda = 0$. The resulting control input sequence provides the smallest tracking error possible. As seen in Fig.~\ref{fig:results}, when the regularization parameter $\lambda$ increases, the control input sequence becomes more and more sparse at the expense of the increased tracking error. Similarly, Table~\ref{tab:compare_parameters} numerically illustrates the trade-off between the sparsity and the tracking performance. These experiments also demonstrate robustness against non-linearities of the plant. The change in the dynamics does not result in a divergence of the S-ILC algorithm.
\begin{figure}
\caption{The convergence of the error residual for different S-ILC methods.}
\label{fig:error}
\end{figure}
Fig.~\ref{fig:error} shows the error decay rate of the gradient-based and accelerated S-ILC algorithms over $50$ trials. Moreover, we tried a multi-step technique called the heavy-ball method which is obtained by adding a momentum term ${\beta (u_k - u_{k-1})}$ to the $b_{k+1}$-update in Algorithm~\ref{alg:2} where $\beta\in [0,1)$ is a scalar parameter. The superior convergence properties of the heavy-ball method compared to the gradient method is known for twice continuously differentiable cost functions~\cite{Polyak:87}. For the class of composite convex cost functions \eqref{eq:problem_sum_two_function}, however, the optimal algorithm parameters and associated convergence rate of the heavy-ball technique is still unknown~\cite{Ghadimi:15}. Here, we evaluated the heavy-ball algorithm with $\beta=0.4$. Numerical tests indicate that both Nesterov based (Algorithm~\ref{alg:3}) and heavy-ball methods improve the convergence of the gradient-based S-ILC Algorithm~\ref{alg:2}.
It is noteworthy to mention that unlike the gradient-based algorithm, the accelerated and heavy-ball methods are not monotonic, that is, the function values~\eqref{eq:problem_sum_two_function} are not guaranteed to be non-increasing. In our evaluations, we observed if the inner-loop Algorithm~\ref{alg:1} is performed for a few iterations (so that it does not reach close to optimality), then the accelerated S-ILC problem (Algorithm~\ref{alg:3}) might become significantly non-monotonic to the point that it might diverge. Finding a monotone converging accelerated S-ILC algorithm is an interesting open problem.
\section{Conclusions}\label{sec:conclusion}
This paper has presented novel iterative learning algorithms to follow reference trajectories under a limited resource utilization. The proposed techniques promote sparsity by solving an $\ell_2$-norm optimization problem regularized by a total variation term. With proper selection of regularization parameter, our algorithms can strike a desirable trade-off between the accuracy of target tracking and the reduction of variations in actuation commands. Simulation results validated the efficacy of the proposed methods.
\section{Appendix}
\textit{Proof of Lemma~\ref{lem:lemma_1}:} The result can be derived in a similar way as~\cite[Proposition 4.1]{Beck:09}. Here, for completeness, we include the proof.
First, note ${\vert x\vert =\underset{p} {\mbox{maximize}} \; \{px: \vert p\vert \leq 1\}}$, and, similarly, $\Vert T u\Vert_1 = \sum_{i=1}^{n-1}\vert u_i - u_{i+1}\vert$ can be written as \begin{equation}\nonumber \begin{aligned}
\underset{p}{\mbox{maximize}}\;\{ \sum_{i=1}^{n-1}p_i (u_{i}-u_{i+1}): \vert p_i\vert \leq 1\} = \underset{p\in \mathcal P^{n-1}}{\mbox{maximize}}\; \mathcal L p^\top u. \end{aligned} \end{equation} Accordingly, the problem~\eqref{eq:denoising_tv} becomes \begin{equation}\nonumber \begin{aligned} \underset{u\in \mathcal{U}}{\mbox{minimize}}\; \underset{p\in \mathcal{P}^{n-1}}{\mbox{maximize}}\; \dfrac{1}{2} \Vert u-b\Vert^2 + \lambda \mathcal{L}p^\top u. \end{aligned}\end{equation} Since this problem is convex in $u$ and concave in $p$, the order of minimization and maximization can be changed to obtain
\begin{equation}\nonumber \begin{aligned}
\underset{p\in \mathcal{P}^{n-1}}{\mbox{maximize}}\; \underset{u\in \mathcal{U}}{\mbox{minimize}}\;\dfrac{1}{2} \Vert u-b\Vert^2 + \lambda \mathcal{L}p^\top u. \end{aligned}\end{equation}
Using the basic relation
$$\Vert x-b \Vert^2 + 2c^\top x = \Vert x- b+c\Vert^2- \Vert b-c \Vert^2 + \Vert b\Vert^2,$$
with $x = u$ and $c = \lambda \mathcal{L}p$ results in the equivalent form \begin{equation}\nonumber \begin{aligned}
\underset{p\in \mathcal{P}^{n-1}}{\mbox{maximize}}\; \underset{u\in \mathcal{U}}{\mbox{minimize}}\; \Vert u-(b-\lambda \mathcal{L}p)\Vert^2 - \Vert b- \lambda \mathcal{L}p\Vert^2 + \Vert b\Vert^2. \end{aligned}\end{equation}
The optimal solution of the minimization problem, readily, is given by~\eqref{eq:opt_u_tv_denoising}. Instituting the optimal value of $u$, we arrive at the following dual problem
\begin{equation} \begin{aligned}
\underset{p\in \mathcal{P}^{n-1}}{\mbox{maximize}}&\; \Vert \Pi_\mathcal{U}(b-\lambda \mathcal{L}p) - (b-\lambda \mathcal{L}p) \Vert^2 - \Vert b-\lambda \mathcal{L}p \Vert^2 \;,
\end{aligned} \end{equation}
which completes the proof.
$\blacksquare$
\textit{Proof of Lemma~\ref{lem:lemma_2}:}
Denote $s(x):= \dfrac{1}{2}\Vert x - \Pi_\mathcal{U} (x) \Vert^2$ and note that according to the \emph{proximal} map the following identity holds:
\begin{equation}\nonumber \begin{aligned} s(x) = \mbox{inf}_y \{ \Pi_\mathcal{U}(y)+\dfrac{1}{2}\Vert y - x\Vert^2 \}. \end{aligned} \end{equation}
From~\cite[Lemma 3.1]{Beck:09} it follows that $s(\cdot)$ is continuously differentiable with
\begin{equation} \begin{aligned} \nabla s (x): = x - \Pi_\mathcal{U}(x). \end{aligned} \end{equation}
The gradient of $h(p)$ then reads
\begin{equation}\nonumber \begin{aligned} \nabla h(p)& = \nabla ( -2s(b-\lambda \mathcal{L}p)+ \Vert b-\lambda \mathcal{L}p \Vert^2)\\ & = 2\lambda \mathcal{L}^\top (b-\lambda \mathcal{L}p - \Pi_\mathcal{U}(b-\lambda \mathcal{L}p)) - 2\lambda \mathcal{L}^\top (b-\lambda \mathcal{L}p)\\ & = -2\lambda \mathcal{L}^\top \Pi_\mathcal{U}(b-\lambda \mathcal{L} p). \end{aligned} \end{equation}
For any $p, p\prime \in \mathcal{P}^{n-1}$ we have
\begin{equation}\nonumber \begin{aligned} \Vert \nabla h(p) - \nabla &h(p\prime)\Vert \\ & = \Vert 2\lambda \mathcal{L}^\top (\Pi_\mathcal{U}(b-\lambda \mathcal{L}p) - \Pi_\mathcal{U}(b-\lambda \mathcal{L}p\prime)) \Vert\\ & \leq 2\lambda \Vert \mathcal L^\top\Vert \Vert \Pi_\mathcal{U}(b-\lambda \mathcal{L}p) - \Pi_\mathcal{U}(b-\lambda \mathcal{L}p\prime)\Vert\\ & \leq 2\lambda^2 \Vert\mathcal L^\top\Vert \Vert\mathcal L (p-p\prime) \Vert\\ & \leq 2\lambda^2 \Vert \mathcal{L}^\top\Vert \Vert \mathcal L\Vert \Vert p-p\prime\Vert \\ & = 2\lambda^2 \rho(\mathcal{L}^\top \mathcal{L}) \Vert p -p\prime\Vert.
\end{aligned} \end{equation}
The matrix $\mathcal L^\top \mathcal L \in \mathbb{R}^{n-1\times n-1}$ is given by
\begin{align*}
\mathcal{L}_{}^{\intercal}\mathcal{L} =
\begin{bmatrix}
2 & -1 & 0 & \cdots & 0 & 0 \\
-1 & 2 & -1 & \cdots & 0 & 0 \\
0 & -1 & 2 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & & \vdots & \vdots \\
0 & 0 & 0 & \cdots & 2 & -1 \\
0 & 0 & 0 & \cdots & -1 & 2
\end{bmatrix} \;. \end{align*}
Following the Gershgorin circle theorem \cite{HoJ:13} one concludes the largest eigenvalue of this matrix follows $\rho(\mathcal L^\top \mathcal L) \leq \max_i\; \sum_{j=1}^{n-1} {\vert [\mathcal L^\top \mathcal L]_{ij} \vert }\leq 4.$
$\blacksquare$
\textit{Proof of Lemma~\ref{lem:3}:}
Let $f(u)$ and $g(u)$ be defined as \eqref{eq:f_g} and
\begin{equation}\nonumber P_{\lambda/\rho}(u):=\mbox{prox}_{\lambda/\rho(G^\top G)} (g) (u - \dfrac{1}{\rho(G^\top G)}G^\top(G u-r)). \end{equation}
Then the S-ILC algorithm (with converging inner-loop and no modeling error) can be rewritten as
\begin{equation}\nonumber \begin{aligned} u_{k+1} = P_{\lambda/\rho} (u_k). \end{aligned} \end{equation}
For a convex Lipschitz continuous gradient function $f$ and convex function $g$, define \cite{Beck:09a}:
$$Q_L(x,y) = f(y)+\langle \nabla f(y), x-y\rangle + \dfrac{L}{2}\Vert x-y\Vert^2 + g(x)$$ Then, it can be seen that $P_{\lambda/\rho} (u) = \underset{x}{\mbox{argmin}} \, Q_{\lambda/\rho}(x, u)$. Furthermore, we have
\begin{equation}\nonumber \begin{aligned} F(u_k)&\geq Q_{\lambda/\rho}(u_k, u_k) \geq Q_{\lambda/\rho}(P_{\lambda/\rho}(u_k), u_k)\\ & =f(u_k) + \langle \nabla f(u_k), P_{\lambda/\rho}(u_k)-u_k\rangle \\ &+ \dfrac{\rho(G^\top G)}{2}\Vert P_{\lambda/\rho}(u_k)-u_k\Vert^2+ g(P_{\lambda/\rho}(u_k)) \\ & \geq f(P_{\lambda\rho}(u_k)) + g(P_{\lambda/\rho}(u_k)) \\ &= F(u_{k+1}), \end{aligned} \end{equation}
where the last inequality holds for Lipschitz continuous $f$.
$\blacksquare$
\balance
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Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation
June 2013, 6(2): 373-406. doi: 10.3934/krm.2013.6.373
Structure of entropies in dissipative multicomponent fluids
Vincent Giovangigli 1, and Lionel Matuszewski 2,
CMAP, CNRS, Ecole Polytechnique, 91128 Palaiseau cedex, France
ONERA, Centre de Palaiseau, 91198 Palaiseau cedex, France
Received June 2012 Revised December 2012 Published February 2013
We investigate the structure of mathematical entropies for dissipative multicomponent fluid models derived from the kinetic theory of gases. The corresponding governing equations notably involve nonideal thermochemistry as well as diffusion fluxes driven by chemical potential gradients and temperature gradients. We obtain the general form of mathematical entropies compatible with the hyperbolic structure of the system of partial differential equations assuming a natural nondegeneracy condition. We next establish that entropies compatible with the hyperbolic-parabolic structure are unique up to an affine transform when they are independent on mass and heat diffusion parameters.
Keywords: Entropy, dissipative, multicomponent., hyperbolic.
Mathematics Subject Classification: Primary: 35L65, 76T30, 35M30; Secondary: 80A3.
Citation: Vincent Giovangigli, Lionel Matuszewski. Structure of entropies in dissipative multicomponent fluids. Kinetic & Related Models, 2013, 6 (2) : 373-406. doi: 10.3934/krm.2013.6.373
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2018 Impact Factor: 1.38
Vincent Giovangigli Lionel Matuszewski | CommonCrawl |
For numerical modeling of river flows, typically water elevation is required at the upstream boundary. Yet water elevation in natural environmental systems is often unknown and has to be estimated. Improper elevation estimation, however, can generate nonphysical results. In FLOW-3D v11.1, which has just been released, users now have the option of having boundary water elevations dynamically adapt to the conditions inside the domain. This can be achieved through the use of rating curves provided by the user, or in the absence of rating curves; the solver can dynamically adjust the elevation to vary smoothly with the conditions inside the fluid domain. These variations may be further constrained to certain Froude regimes or absolute elevation bounds.
Figure 1. Rating curve for John Creek at Sycamore from USGS
Rating curves
Rating curves define elevation variations at a given location in a river reach according to inflow rates at that location. A relationship between elevation and volume flow rate is established by physical measurements at a particular cross section of the river. Rating curves for rivers in the United States are available from the USGS (U. S. Geological Survey). A typical rating curve will have volume flow rate on the X-axis and elevation on the Y-axis (Figure 1).
Natural inlets
In a case where inflow rate is known but a rating curve is unavailable, a natural boundary condition can be selected in the FLOW-3D model setup interface. At a given cross-section, for a certain specific energy, there can be two possible depths. This arises from the quadratic relationship between specific energy and the depth (see the equation below). The two mathematical depths manifest into supercritical and subcritical depths in reality. In the case of a perfect unique solution to the quadratic equation, the flow is critical.
$latex E=\frac{{{{q}^{2}}}}{{2g{{y}^{2}}}}+y&s=3$
Here, E is the specific energy, q is the unit discharge, g is acceleration due to gravity and y is the height of fluid. Graphically, the specific energy and depth relationship can be seen in Figures 2-4.
Figure 2. Changes to E-y curve, changing q
Figure 3. Possibility of two flow depths (supercritical and subcritical) for the same value of specific energy
Figure 4. Flow depth can be critical (yc) for a unique value of depth and specific energy. In this case, flow is neither subcritical nor supercritical.
Applying new boundary conditions
A rating curve can only be defined for volume flow rate and pressure boundary conditions in FLOW-3D v11.1. For volume flow rate type boundary conditions, instantaneous elevations are calculated using the rating curve to find the elevation corresponding to the flow rate. For a pressure type boundary condition, the volume flow rate is calculated by the solver and elevation is calculated using the rating curve. Rating curves can be applied at both upstream and downstream boundaries. It is important to note that an incorrect rating curve can result in nonphysical flow fluctuations.
Natural boundary conditions can only be defined at the inlet. Flow categories can be defined from one of the following:
Supercritical flow (y<yc)
Subcritical flow (y>yc)
Critical flow (y=yc)
Automatic flow regime (calculated by the solver)
The user can define maximum and minimum limits of elevation for any of these flows. If the depth for a particular flow regime violates the maximum and minimum limits of elevation, the latter will take precedence.
Sample simulation results
Simulation 1 shows the river reach with a natural inlet under volume flow rate boundary condition at the left boundary and a rating curve for the outlet is defined as a pressure boundary condition at the right boundary. The evolution of water elevation is shown for both upstream and downstream boundaries simultaneously. The simulation shows smooth variation of elevations at the boundaries without any fluctuations or nonphysical behavior. Therefore, this new development in FLOW-3D v11.1 allows for more natural variations of the water level for environmental applications.
Evolution of water elevation in a river reach with natural boundary condition at the inlet and a rating curve at the outlet. | CommonCrawl |
Why $17 \times 24$ isn't 568
Written by Colin+ in basic maths skills.
"You would not be certain that $17 \times 24$ is not 568."
- Daniel Kahneman, Thinking Fast And Slow
Thanks to Alice for pointing out that yes, she bloody well would.
Most people under 50 in the UK would reach for a calculator, or possibly a pen and paper to work out $17 \times 24$. It's not the world's trickiest sum, but at the same time, you probably never chanted your 17 times tables, no matter when you went to school.
The answer, as it happens, is 408 - but that's not the point of this article; instead, it's about how you know it's not 568.
1. Estimation
The first, most obvious thing: 17 is a bit less than 20. 24 is a similar amount more than 20. That means $17 \times 24$ is somewhere in the region of $20 \times 20 = 400$. It's definitely not off by 150 or more.
2. Factorisation rules
I recently saw a 'counter-example' to Fermat's Last Theorem, that claimed $3987^{12} + 4365^{12} = 4472^{12}$. Both work out to $6.397~665~635 \times 10^{43}$ - however, they're clearly not equal. Why not? The digits in the first term add up to 27, which mean it's a multiple of 9. The digits in the second add up to 18, which mean it's also a multiple of 9. In fact, they're both multiples of $9^{12}$ - and adding them together gives you something that's also a multiple of 9. The right hand side, however, has digits that add up to 17 - which means it's not even a multiple of 3, let alone 9.
You can do a similar trick with Kahneman's example: 24 is a multiple of 3, so $17 \times 24$ is as well. However, the digits of 568 add up to 19, which isn't a multiple of 3.1
3. The Last Digit Test
This example actually passes the Last Digit test, but it's a good one to use if you're trying to narrow down answers: if you multiply the last digits together, you get $7 \times 4 = 28$ - which means the last digit of the big sum ought to be 8 (and, in this case, it is). If it hadn't worked out, you'd know for sure it was wrong - although, as you can see, the last digit test being right doesn't guarantee that the big sum is right2 .
4. Seventeens
Like I say, you don't chant your 17-times table, so - unless you're the Mathematical Ninja - you probably don't know that $6 \times 17 = 102$.3 That means you can work out $17 \times 24$ in a flash, because 24 is $6 \times 4$.
$17 \times 6 \times 4 = 102 \times 4 = 408$.
Kahneman is probably right, 99% of the time, that his readers won't immediately spot that $17 \times 24$ isn't 568. It could just be that his readers aren't as smart as mine.
How the Mathematical Pirate works out the high times tables
Ask Uncle Colin: These alcohol-related figures look a bit fuzzy
The times table game
Ask Uncle Colin: A mental quotient
$n$ maths blogs I often read
Note: this trick only works for multiples of 3 and 9. [↩]
If you like, it's a necessary but not sufficient condition. [↩]
If you play darts, you might know that treble-17 is 51, though. [↩] | CommonCrawl |
•https://doi.org/10.1364/OE.451186
Color curved hologram calculation method based on angle multiplexing
Di Wang, Nan-Nan Li, Zhao-Song Li, Chun Chen, Byoungho Lee, and Qiong-Hua Wang
Di Wang,1 Nan-Nan Li,1 Zhao-Song Li,1 Chun Chen,2 Byoungho Lee,2 and Qiong-Hua Wang1,*
1School of Instrumentation and Optoelectronic Engineering, Beihang University, Beijing 100191, China
2School of Electrical and Computer Engineering, Seoul National University, Gwanak-Gu Gwanakro 1, Seoul 08826, Republic of Korea
*Corresponding author: [email protected]
D Wang
N Li
Z Li
C Chen
B Lee
Di Wang, Nan-Nan Li, Zhao-Song Li, Chun Chen, Byoungho Lee, and Qiong-Hua Wang, "Color curved hologram calculation method based on angle multiplexing," Opt. Express 30, 3157-3171 (2022)
Method of curved composite hologram generation with suppressed speckle noise
Nan-Nan Li, et al.
Fast method for calculating a curved hologram in a holographic display
Ruidan Kang, et al.
Curved hologram generation method for speckle noise suppression based on the stochastic gradient...
Di Wang, et al.
Imaging Systems, Microscopy, and Displays
Chromatic aberrations
Computer generated holograms
Holographic displays
Holographic optical elements
Holographic recording
Speckle noise
Original Manuscript: December 13, 2021
Revised Manuscript: January 4, 2022
Manuscript Accepted: January 6, 2022
Principle of the method
Simulation, experiments and results
In this paper, a method of color curved hologram calculation based on angle multiplexing is proposed. The relationship between the wavelength, center angle and sampling interval of the curved holograms is analyzed for the first time by analyzing the reconstruction process of the curved holograms with different wavelengths. Based on this relationship, the color curved holograms are calculated by compensating phase to the complex amplitude distribution of the planar holograms. To eliminate the chromatic aberration, the curved holograms of different objects with the same color channel are respectively used for angle multiplexing and phase compensation, and then the color composed curved hologram is generated. Different color objects without chromatic aberration can be reconstructed by bending the composed curved hologram into different central angles. The experimental results verify the feasibility of the proposed method. Besides, the application of the proposed method in augmented reality display is also shown.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
Holographic display technology is one of the ideal approaches for three-dimensional (3D) display, and it has important application value in many fields such as medical diagnosis, aerospace, industrial manufacturing, education and entertainment [1–4]. However, since the diffraction angle of the holographic display is limited by the pixel pitch of the spatial light modulator (SLM), the field of view (FOV) of the holographic display is too narrow to meet the viewing requirements [5,6].
To increase the FOV of the planar hologram, a number of methods were proposed. In 2012, a wide-angle holographic display system based on the spatiotemporal multiplexing method was proposed [7–9], and the effective space bandwidth product of the system data was increased to 50 megapixels. In 2013, a linear phase factor superimposition method was proposed to realize viewing angle enlargement [10]. The horizontal viewing angle with a single SLM can be increased to 3.6 times. In 2017, an increased-viewing-angle full-color holographic display was realized by using two tiled SLMs and a 4f concave mirrors system [11]. In 2018, a large resonant scanner and a galvanometer scanner were used in the holographic display system for FOV enlargement [12]. Then the FOV can be increased to 48°. The common feature of the methods of expanding FOV based on the planar hologram is that it needs to splice the SLM in the time domain or the space domain. Thus, the system is usually complicated. On the other hand, the curved hologram is an effective way to increase the FOV of the holographic display without splicing the SLM [13]. And with the development of flexible materials, more and more researchers have begun to study curved holograms [14,15]. In 2014, an acceleration method for computer generated spherical hologram of a real-existing object was proposed [16]. After that, the see-through display method based on curved holographic optical elements was proposed [17,18]. Then, the potential applications of curved metasurface holograms were demonstrated in imaging, sensing, and anti-counterfeiting [19]. Besides, some researchers proposed the multiplexing method of curved hologram to improve the information capacity and computation time [20,21]. In 2020, a holographic system for recording a curved digital hologram was demonstrated [22]. In our previous work, we studied the curved hologram calculation method for speckle noise suppression [23].
At present, the curved hologram is one of the important research directions in the fields of large viewing angle holographic display. Unfortunately, there are few reports on the color curved holographic display. In the planar holographic display, the method of color holographic display mainly includes time multiplexing and space multiplexing [24–28]. Among them, the space-multiplexing method requires three SLMs or the separation of a single SLM into three regions. The time multiplexing method requires an SLM with a high refresh rate and synchronization mechanism. Besides, some researchers proposed color-dispersion-compensated synthetic phase holograms to realize the color reproduction based on a single SLM [29]. However, in the color curved holographic display, there are some chromatic aberrations that are different from the traditional planar chromatic aberration. Therefore, the color-multiplexing method based on the planar hologram cannot be directly applied to the color curved hologram calculation, and it is necessary to study the specific method of generating the color curved hologram.
In this paper, a color curved hologram calculation method based on angle multiplexing is proposed. The relationship between the wavelength, center angle and sampling interval of the curved holograms is analyzed for the first time. By analysis, we find that the holographic reproduction of the color curved hologram contains distortion chromatic aberration and crosstalk chromatic aberration. To eliminate the chromatic aberration, the recorded 3D color object is divided into three RGB channels firstly, and the corresponding three planar holograms are generated by using the angular spectrum method. Then, three curved holograms of the recorded 3D color object are generated by compensating phase to the complex amplitude distribution of the three planar holograms. For the different recorded 3D color objects, the central angle of the curved hologram is different. The phase compensation is added to the three color holograms of different objects to eliminate the crosstalk chromatic aberration. Finally, the RGB three channels are arranged in a parallel manner, and the curved holograms of different objects in one channel are composited by using angle multiplexing to avoid the distortion chromatic aberration. When the full color composed curved hologram (CCH) is bent into a curved hologram with different central angles, the different 3D reproduced images are reconstructed. The proposed method realizes the multi-angle holographic reconstruction of the color curved hologram successfully. Additionally, the reason why the color-multiplexing method cannot be directly applied to the full color CCH calculation is analyzed. Moreover, the application of the proposed method in augmented reality (AR) display has also been verified.
2. Principle of the method
The principle schematic of the proposed method is shown in Fig. 1 and the method consists of three steps. For two different 3D objects, we record them as 3D object 1 and 3D object 2. Firstly, the information of the two 3D objects is extracted in red, green and blue channels, respectively. By analyzing the relationship between the wavelength, center angle and sampling interval of the curved holograms, the curved holograms of three colors are calculated respectively. The red, green and blue curved holograms of 3D object 1 are recorded as R_CH1, G_CH1 and B_CH1, respectively. The red, green and blue curved holograms of 3D object 2 are recorded as R_CH2, G_CH2 and B_CH2, respectively. For different objects, the center angles of the holograms are different accordingly. Secondly, the linear phase factor is added to separate the crosstalk chromatic aberration between the two 3D objects. Taking the red channel as an example, different phase factors are loaded on R_CH1 and R_CH2 respectively. Then the red curved hologram of object 1 and the red curved hologram of object 2 are superposed to generate the red CCH. In this way, the three color CCHs can be calculated by using the angle multiplexing method accordingly. Thirdly, the final full color CCH is generated by splicing the three color CCHs. In the process of holographic reconstruction, the reconstructed light of the three colors illuminates the CCH of the corresponding color region respectively. When the full color CCH is bent to different angles, the corresponding color objects can be reproduced.
Fig. 1. Principle schematic of the proposed method.
2.1 Step 1: process of curved hologram generation
The red, green, and blue scene information for a color 3D object is processed separately. For each color channel, the 3D object is divided into L layers with different depths by using the layer-based method [30]. Ll represents the lth layer, l= 1, 2, 3…L. For each layer, the amplitude information is extracted from the rendered image, and the phase information is preset to a uniform value. Then, the complex amplitude distribution of the wavefront recording plane (WRP) is generated by using the angular spectrum method (ASM) [31]:
(1)$${U_\textrm{o}}(x,y) = {A_0}\cdot \textrm{exp} (j{\varphi _0}(x,y)),$$
(2)$${U_p}(x,y) = IFFT\{{FFT\{{{U_0}(x,y} \}\cdot {H_f}({f_x},{f_y})} \}, $$
where Uo(x, y) and Up(x, y) represent the complex amplitude distributions of the object and the WRP, respectively. A0 and φ0 are the amplitude and phase distribution of each depth layer, respectively. FFT[•] and IFFT[•] are the fast Fourier transform (FFT) and inverse FFT operators, respectively. Hf (fx, fy) is the transfer function, which can be expressed as follows:
(3)$${H_f}({f_x},{f_y}) = \textrm{exp} \left( {ik\Delta z\sqrt {1 - {{({\lambda {f_x}} )}^2} - {{({\lambda {f_y}} )}^2}} } \right), $$
where k represents the wavenumber, λ is the wavelength, and Δz is the distance between the layer of the object and the WRP. For different color channel, the wavelength is different accordingly.
After the WRP of the object in each channel is generated, the curved hologram of the object in each channel can be transformed from the WRP by analyzing the phase difference distribution caused by the optical path between the WRP and curved hologram. The conversion process from WRP to the curved hologram is shown in Fig. 2, where CH represents the curved hologram plane. The yellow points and red points represent the sampling points of the WRP and CH, respectively. w is the sampling interval and wh is the size of the WRP. R is the curvature radius of the CH, which is derived from the central angle β.
Fig. 2. Conversion process from WRP to CH.
In the calculation of the curved hologram, the phase retardation caused by the diffraction from the WRP to the curved hologram can be regarded as an approximate compensation. Then the complex amplitude of the curved hologram can be generated by the geometric optical path difference of each point in the WRP. The complex amplitude of the curved hologram Uc (x, y) is expressed by the following equation:
(4)$${U_c}(x,y) = {U_p}(x,y)\cdot T(x,y;\beta ) = {U_p}(x,y)\cdot \textrm{exp} ({ik\Delta z(x,y;\beta )} ), $$
where T (x, y; β) is the conversion phase factor. Δz is the propagation distance between the point on the WRP and the corresponding point on the CH. The origin of the x-axis is set on the z-axis. Δz can be expressed as follows:
(5)$$\Delta z(x,y;\beta ) = {z_c} - R + \sqrt {{R^2} - {x^2}}, $$
where zc represents the maximum of the Δz. Only the distance zc is small enough according to the limited hologram size and the central angle of curved hologram, the phase difference distribution can be regarded as an approximate compensation generated by the geometric optical path difference. Therefore, the complex amplitude of the curved hologram with central angle β can be generated by transformation of the WRP.
In planar holographic display, only the same reference beam as the recording light can accurately reconstruct the recorded object. When the wavelength is different, the position of the reproduced image is correspondingly different, then the chromatic aberration occurs. According to the diffraction principle of the planar holographic display, the complex distribution of the reconstructed image I(x, y) can be expressed as follows:
(6)$$I(x,y) = [{|{r({x,y} )} |\textrm{exp} ({ - i2\pi {\xi_r}x} )\cdot {U_o}({x,y} )} ]\cdot |{r^{\prime}({x,y} )} |\textrm{exp} ({i2\pi {\xi_r}x} ), $$
(7)$${\xi _r}\textrm{ = }\frac{{\sin \theta }}{\lambda }, $$
where Uo(x, y) represents the complex amplitude distribution of the recorded object, |r(x, y)|exp(•) represents the reference beam in the recording process and |r'(x, y)|exp(•) represents the reference beam in the reconstruction process. From Eq. (6) and Eq. (7) we can see that when the reference beam in the reconstruction process is consistent with the reference beam in the recording process, the two parts of the reference beam can be counteracted. When the reference beam does not match, the reconstructed image will be multiplied by a phase factor. Therefore, the redundancy caused by the wavelength mismatch of the reference beam will cause chromatic aberration in the reproduced image.
As shown in Fig. 3, when the green channel planar hologram (G_PH) is irradiated with three color reference beams simultaneously, the red, blue and green reconstructed images do not overlap. z0 is the reconstructed distance. Δy1 and Δz1 represent the chromatic aberration caused by blue light. Δy2 and Δz2 represent the chromatic aberration caused by the red light.
Fig. 3. Images of the G_PH when three color reference beams are used for reconstruction.
However, in the reconstruction process of the color curved hologram, the distortion chromatic aberration exists when the wavelength of the reference beam does not match the recorded light. The generation of the curved hologram is mainly transformed from the WRP according to Eq. (4). One of the important parameters is the conversion phase factor T (x, y; β). It can be found that the conversion phase factor is wavelength dependent, which means that the WRPs of different color channels correspond to different conversion phase factors. The reconstruction process based on the curved hologram is as follows: The first is the inverse phase conversion of the curved hologram to WRP, and the second is to use WRP for diffraction reconstruction. The reversal transformation from the curved hologram to WRP is equivalent to multiplying the reversal transformation phase factor in the curved hologram light field. The inverse conversion phase factor depends on the wavelength and phase distribution of the reference beam in the reconstruction process. Therefore, only when the reference beam in the reconstruction process is the same as the reference beam in the recording process, the inverse conversion phase factor and the conversion phase factor can be canceled. Then the image of the recorded object can be reconstructed correctly. If the wavelength of the reference beam is mismatched in the reconstruction process, the reconstructed image will be stretched or squeezed, which is the distorted chromatic aberration in the curved holographic display.
As shown in Fig. 4, when the green channel curved hologram (G_CH) is irradiated with three color reference beams simultaneously, the positions of the reconstructed images are different, and the shape of the reconstructed images is deformed due to the distortion chromatic aberration. Among them, the green reconstructed image is normal, the blue reconstructed image is squeezed, and the red reconstructed image is stretched. After verification, we find that the reconstructed image will be stretched if the wavelength of the reference beam of the reconstruction process is greater than that of the recording process. Otherwise, it will be squeezed.
Fig. 4. Images of the G_CH when three color reference beams are used for reconstruction.
In addition, when the green reference beam is used to illuminate different color curved holograms, multiple images will be also reconstructed, as shown in Fig. 5. The distortion chromatic aberration also exists. The stretched image is reconstructed at the position of the blue dotted line. Similarly, the squeezed image is reconstructed at the position of the red dotted line. It can be seen that, due to the existence of distorted chromatic aberration, the traditional planar color multiplexing method cannot be directly applied to color CCH calculation.
Fig. 5. Images of the color CHs when green reference beam is used for reconstruction.
2.2 Step 2: process of CCH generation
In order to reproduce multiple objects, the angle multiplexing method is used to calculate the curved holograms of different objects. The two WRPs of the objects are generated by using the ASM. The diffraction distances between the WRPs and the objects are the same. Then, the curved holograms of the two objects can be transformed from two WRPs by adding the phase retardations. It should be noted that the curved holograms have the same pixel number and sampling interval, but the center angles of the curved holograms for different objects are different. The complex amplitude distribution of the CCH can be generated by adding all the curved holograms.
(8)$$H(x,y) = \sum\limits_{i = 1}^n {{U_i}(x,y)}, $$
where n is the number of curved holograms. In the reconstruction process, different reconstructed images can be displayed in sequence by bending the CCH into different central angles.
Unfortunately, the angle multiplexing method of the curved holograms will produce additional chromatic aberrations. Since the angle multiplexing method is to superimpose curved holograms with different central angles, different conversion phase factors correspond to different curved hologram central angles according to Eq. (4). Therefore, the inverse conversion phase factor is also different. Figure 6 is the reconstruction of the color CCH illuminated with the color reference beam. When the CCH is bent into a different central angle, the image corresponding to the recorded central angle can be correctly reconstructed, while the other center angles cannot reproduce the correct reconstructed image since the inverse conversion phase factor cannot be counteracted. The distorted image will exist in the background of the correct reconstructed image, which will affect the viewing effect, as shown in Figs. 6(a)-6(b).
Fig. 6. Reconstruction process of the CCH (a)-(b) without the linear phase factor and (c)-(d) with the linear phase factor.
Due to the crosstalk between the curved holograms, the quality of the reconstructed image of the CCH is affected. To suppress the crosstalk chromatic aberration, the linear phase factor is superimposed on the curved hologram of the recorded object. Before generating the CCH using the angle multiplexing method, the same linear phase factor is added to each color curved hologram of the same recorded object. The curved hologram of the object is expressed as follows:
(9)$${U_i}(x,y) = {U_c}(x,y)\cdot \textrm{exp} (ik\cdot \sin \theta \cdot wx),$$
where θ is the angle of the linear phase factor. Since there are no curved SLMs currently available, we have to perform phase compensation on the planar SLM for experimental verification [32]. Then, the phase distribution H'(x, y) loaded on the SLM is expressed as follows:
(10)$$H^{\prime}(x,y) = \frac{{H(x,y)}}{{T(x,y;\beta )}}. $$
By loading different linear phase factors on the curved holograms of different objects, the images of different objects can be separated, and the crosstalk chromatic aberration caused by angle multiplexing can be eliminated, as shown in Figs. 6(c)-(d).
2.3 Step 3: process of holographic reconstruction
In the reconstruction process, we use the method of spatial multiplexing to splice the curved holograms of three colors into a color curved hologram, and control the reproduced light of the three colors to illuminate the hologram of the corresponding color areas, thereby avoiding distortion chromatic aberration. The phase compensation is added to the three color holograms of different objects to eliminate the crosstalk chromatic aberration. The final full color CCH is generated by splicing the three color CCHs and the corresponding area of the final full-color CCH is illuminated with the RGB reference beam simultaneously. The reconstruction process of the final full-color CCH is shown in Fig. 7. When the final full-color CCH is bent into different central angle, the different 3D reproduced images without chromatic aberration can be seen.
Fig. 7. Reconstruction process of the final full color CCH when (a) object 1 is reconstructed and (b) object 2 is reconstructed.
3. Simulation, experiments and results
To verify the feasibility of the proposed method, a color holographic display system is built, as shown in Fig. 8. The red, green, and blue lasers are used as the color reference beam. The wavelengths for the three colors are 671nm, 532nm, and 473nm, respectively. After passing through the prism and beam splitter (BS), the three color lasers illuminate one third of the SLM area respectively. The SLM used in the system has a pixel pitch of 6.4µm and a resolution of 1920×1080. The full color CCH is loaded onto the SLM, and the loading area of each color CCH is the same as the irradiation position of the corresponding color laser. The size of the BS is 25.4mm×25.4mm×25.4mm. The 4f system is composed of two lenses and an aperture to eliminate the zero-order light. The SLM locates at the front focal plane. The reconstruction position of the 3D object is determined by the diffraction distance set in the experiment. When the reproduced image passes through the aperture, by adjusting the aperture size, it can be ensured that only the first-order diffraction image passes through, thereby avoiding the interference of high-order diffraction images and zero-order light. After the reconstructed light passes through the BS and 4f system, the reconstructed images of three color CCHs can be spatially overlapped, then the ideal holographic reconstructed image can be captured by the CCD.
Fig. 8. Structure of the optical system.
Firstly, we simulate the distortion chromatic aberration in the reconstructed image. When red, green and blue lasers are used to illuminate the green curved hologram, the simulation results are shown in Fig. 9. The center angle of the curved hologram is 10°. It can be seen that the correct image can only be reproduced when the green curved hologram is illuminated by a green laser. When the green curved hologram is illuminated with red and blue lasers, the reconstructed image is distorted. So, in the color curved holographic reproduction, we need to keep the three colors of light illuminating the hologram of the corresponding color for the purpose of distortion chromatic aberration elimination.
Fig. 9. Reconstructed images of the green CH when (a) red (b) green or (c) blue laser is used for reproduction respectively.
Then, the 3D object is used for verification. The letters '3' and 'D' are located on different depth planes to verify the 3D reproduction effect. The diffraction distances of the two depth objects are 15cm and 17cm, respectively. The resolution of the 3D object is 800×600. When generating three-color curved holograms, we spatially splice them together to form a color curved hologram. The resolution of the monochromatic curved hologram is 640×1080. The resolution and curved angle of the color curved hologram are 1920×1080 and 10°, respectively. Then the recorded full color curved hologram of the 3D object is loaded onto the SLM according to the calculation. In the simulation experiment, we do not convert the curved hologram into a plane by phase compensation, but directly reconstructed the curved hologram. The results are shown in Fig. 10. Figure 11 shows the result when the diffraction distance is 17cm. It can be seen that the reconstructed image 'D' is in focus at this time, while the reconstructed image '3' is blurred. When the CCD is located at a distance of 15cm, the results are shown in Fig. 12. It can be seen clearly that the reconstructed image '3' is in focus at this time, while the reconstructed image 'D' is blurred. So, the color curved holographic display can be realized without chromatic aberration.
Fig. 10. Simulation experiment results of the curved hologram. (a)-(d) Red, green, blue and white reconstructed images of the 3D object when 'D' is in focus; (e)-(h) red, green, blue and white reconstructed images of the 3D object when '3' is in focus.
Fig. 11. Color reconstructed images of the 3D object when 'D' is in focus. (a) Blue reconstructed image; (b) green reconstructed image; (c) red reconstructed image; (d) color reconstructed image.
Fig. 12. Color reconstructed images of the 3D object when '3' is focus. (a) Blue reconstructed image; (b) green reconstructed image; (c) red reconstructed image; (d) color reconstructed image.
In order to verify the color display effect of the CCH, two different objects 'H' and 'W' are recorded respectively. The resolutions of the two objects are 320×240. The resolutions of the curved holograms and full color CCH are 640×1080 and 1920×1080, respectively. The diffraction distances of the two objects are 15cm. Taking 'H' as an example, 'H' is firstly separated for color processing and three color curved holograms of the object 'H' are generated respectively with the center angles of 10°. Similarly, the object 'W' is separated into three colors and three color curved holograms of the object 'W' are generated respectively with the center angles of 30°. The green curved holograms of 'H' and 'W' are superimposed to generate a green CCH, the red curved holograms of 'H' and 'W' are superimposed to generate a red CCH, and the blue curved hologram of 'H' and 'W' are superimposed to generate a blue CCH. The three color CCHs are spliced into a full color CCH and loaded on the SLM. When the bending angle of the full color CCH is set to 10°, the results of the color reconstructed image are shown in Figs. 13(a)-13(c). It can be seen that the object 'H' is reproduced, but the reproduced image is disturbed by another angled object 'W'. When the bending angle of the full color CCH is set to 30°, the reconstructed images of the object 'W' are shown in Figs. 13(d)-13(f). It can be seen that the reproduced image is disturbed by the object 'H'. Therefore, when the full color CCH is used for reproduction, the reconstructed image will be disturbed by the chromatic aberration of other curved angle images.
Fig. 13. Reconstructed images of the full color CCH with crosstalk. (a) Red reconstructed image of 'H'; (b) green reconstructed image of 'H'; (c) blue reconstructed image of 'H'; (d) red reconstructed image of 'W'; (e) green reconstructed image of 'W'; (f) blue reconstructed image of 'W'.
In order to eliminate this crosstalk chromatic aberration, different phase factors are added to the curved holograms of 'H' and 'W' before the CCHs are generated according to Eq. (9). The angles of the linear phase factors added on the CH of the object are 1° and 3°, respectively. The center angles of the color curved holograms are 10° and 30° respectively. The reconstructed results of the object 'H' are shown in Fig. 14. Figures 14(a)-14(c) are the blue, green, and red reconstructed images respectively. Figure 14(d) is the color reconstructed image. It can be seen clearly that the chromatic aberration and the crosstalk have been eliminated. When the bending angle of the full color CCH is set to 30°, the color reconstructed images of the object 'W' are shown in Fig. 15. Hence, by using the proposed method, the reconstructed images of color curved holograms at different angles can be achieved without chromatic aberration. When the bending angle is different, the object of the corresponding angle can be displayed. This angle multiplexing can also increase the information capacity in the holographic display.
Fig. 14. Color reconstructed images of 'H' with the proposed method. (a) Blue reconstructed image; (b) green reconstructed image; (c) red reconstructed image; (d) color reconstructed image.
Fig. 15. Color reconstructed images of 'W' with the proposed method. (a) Blue reconstructed image; (b) green reconstructed image; (c) red reconstructed image; (d) color reconstructed image.
The proposed method can be used for holographic AR display and multi-plane display, etc. In order to verify the application of color curved holographic display, two real objects are placed in front of a camera with an imaging lens to obtain the holographic AR display effect. Here, a 'car' and an 'angel' at two different depth planes are used as the real objects. The object 'H' and '3D' are used as the two recorded objects. The resolution of the two objects is 320×240. The resolutions of the curved hologram and full color CCH are 640×1080 and 1920×1080. In the generation process of the CCH, the center angles of 'H' and '3D' are set to 10° and 30°, respectively. The angle of the linear phase factors added on the CH of the object is 1° and 2°, respectively. For the object '3D', the letters '3' and 'D' are at different depth planes. The diffraction distances of 'H' and '3' are the same, and their reconstructed images are on the same plane as the real object 'car'. The reconstructed image of 'D' is on the same plane as the real object 'angel'. When the bending angle of the full color CCH is set to 10°, the color reconstructed images of the object 'H' can be captured by the camera. At the same time, the real object 'car' can be seen clearly, as shown in Fig. 16(a). When the bending angle of the full color CCH is set to 30°, the color reconstructed images of the object '3D' can be captured by the camera. When the diffraction distance is 15cm, the real object 'car' and the reconstructed image of '3' can be clearly seen in Fig. 16(b), while 'angel' and the reconstructed image of 'D' are blurred. When the diffraction distance is 25cm, the real object 'angel' and the reconstructed image of 'D' are in focus, while 'car' and the reconstructed image of '3' are blurred, as shown in Fig. 16(c). It can be seen from the result that when the CCH is bent at different angles, the color reproduced image at the corresponding angle and the real object can be seen simultaneously. Compared with the traditional holographic AR display, the proposed method not only realizes the color reproduction but also increases the information content of the holographic display.
Fig. 16. Results of the holographic AR display. (a) Result when 'H' is reconstructed; (b) result when '3' is focused; (c) result when 'D' is focused.
There are different ways to generate the curved hologram. One way is to calculate 3D object directly. This curved hologram can enlarge the FOV of holographic reproduction, but it takes a long time to directly calculate the curved hologram from 3D object. In order to improve the calculation speed, some researchers obtain the complex amplitude distribution of the curved hologram by approximate solution of planar hologram. In this case, the parameters such as pixel pitch of curved hologram are affected by the planar hologram. The main reason for the increase of the viewing angle of curved hologram is that its curvature increases the viewing angle. Affected by the pixel pitch of the SLM, the limited diffraction angle of the reproduced image is restricted by the maximum diffraction angle of the SLM, which is a small fixed value. When the SLM changes from a planar surface to a curved surface, there is a curvature change, which artificially introduces an additional perspective. Due to the limitation of experimental conditions, we use a planar modulator for equivalent experimental verification. In this case, the increase of FOV can not be verified. Of course, we are also studying curved modulation elements based on micro-nano materials. When the curved hologram is loaded on the curved modulator, the curved modulator has a larger diffraction angle than that of the planar modulator, Then the FOV can be increased. Besides, although the angle multiplexing method can improve the information capacity of the hologram, the number of the curved holograms with different central angle is limited by the crosstalk. When the number of composites increases, the crosstalk of the reproduced image will increase accordingly. The minimum angular pitch are important topics to be addressed. When the bending angle becomes large, the crosstalk between different central angle curved holograms will lead to the degradation of the reconstructed image quality.
In this paper, a color curved hologram calculation method based on angle multiplexing is proposed. By dividing the recorded 3D color object into three RGB channels, the corresponding three color planar holograms are generated. Then, three curved holograms can be generated by compensating phase to the complex amplitude distribution of three planar holograms. The chromatic aberration in the curved hologram reproduction is analyzed and compensated. For the different recorded 3D color objects, the central angle of the curved hologram is different. Finally, the curved holograms of different objects in the same color channel are composited by using angle multiplexing to generate the full color CCH. The different color objects can be reproduced when the full color CCH is bent into different central angles. Besides, the application of the proposed method in holographic AR display is also verified. We believe that the proposed method can promote the holographic display application.
National Natural Science Foundation of China (62020106010, 62011540406); National Research Foundation of Korea (2020K2A9A2A06038623).
We would like to thank Nanofabrication facility in Beihang Nano for technique consultation.
The authors declare that there are no conflicts of interest related to this article.
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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8. T. Kozacki, M. Kujawinska, G. Finke, B. Hennelly, and N. Pandey, "Extended viewing angle holographic display system with tilted SLMs in a circular configuration," Appl. Opt. 51(11), 1771–1780 (2012). [CrossRef]
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11. Z. Zeng, H. Zheng, Y. Yu, A. K. Asundi, and S. Valyukh, "Full-color holographic display with increased-viewing-angle," Appl. Opt. 56(13), F112–F120 (2017). [CrossRef]
12. J. Li, Q. Smithwick, and D. Chu, "Full bandwidth dynamic coarse integral holograohic displays with large field of view using a large resonant scanner and a galvanometer scanner," Opt. Express 26(13), 17459–17476 (2018). [CrossRef]
13. Y. Sando, M. Itoh, and T. Yatagai, "Fast calculation method for cylindrical computer-generated holograms," Opt. Express 13(5), 1418–1423 (2005). [CrossRef]
14. S. Q. Li, X. Xu, R. M. Veetil, V. Valuckas, R. P. Domínguez, and A. I. Kuznetsov, "Phase-only transmissive spatial lightmodulator based on tunable dielectric metasurface," Science 364(6445), 1087–1090 (2019). [CrossRef]
15. Q. S. Wei, B. Sain, Y. T. Wang, B. Reineke, X. W. Li, L. L. Huang, and T. Zentgraf, "Simultaneous spectral and spatial modulation for color printing and holography using all-dielectric metasurfacces," Nano Lett. 19(12), 8964–8971 (2019). [CrossRef]
16. G. Li, K. Hong, J. Yeom, N. Chen, J. H. Park, N. Kim, and B. Lee, "Acceleration method for computer generated spherical hologram calculation of real objects using graphics processing unit," Chin. Opt. Lett. 12(6), 060016 (2014). [CrossRef]
17. K. Bang, C. Jang, and B. Lee, "Curved holographic optical elements and applications for curved see-through displays," J. Inf. Disp. 20(1), 9–23 (2019). [CrossRef]
18. J. Burch and A. D. Falco, "Holography using curved metasurfaces," Photonics 6(1), 8 (2019). [CrossRef]
19. L. Zhou, C. P. Chen, Y. Wu, Z. Zhang, K. Wang, B. Yu, and Y. Li, "See-through near-eye displays enabling vision correction," Opt. Express 25(3), 2130–2142 (2017). [CrossRef]
20. R. Kang, J. Liu, G. Xue, X. Li, D. Pi, and Y. Wang, "Curved multiplexing computer-generated hologram for 3D holographic display," Opt. Express 27(10), 14369–14380 (2019). [CrossRef]
21. R. Kang, J. Liu, D. Pi, and X. Duan, "Fast method for calculating a curved hologram in a holographic display," Opt. Express 28(8), 11290–11300 (2020). [CrossRef]
22. J. P. Liu, W. T. Chen, H. H. Wen, and T. C. Poon, "Recording of a curved digital hologram for orthoscopic real image reconstruction," Opt. Lett. 45(15), 4353–4356 (2020). [CrossRef]
23. N. N. Li, D. Wang, Y. L. Li, and Q. H. Wang, "Method of curved composite hologram generation with suppressed speckle noise," Opt. Express 28(23), 34378–34389 (2020). [CrossRef]
24. M. Oikawa, T. Shimobaba, T. Yoda, H. Nakayama, A. Shhiraki, N. Masuda, and T. Ito, "Time-division color electroholography using one-chip RGB LED and synchronizing controller," Opt. Express 19(13), 12008–12013 (2011). [CrossRef]
25. T. Shimobaba, T. Takahashi, N. Masuda, and T. Ito, "Numerical study of color holographic projection using space-division method," Opt. Express 19(11), 10287–10292 (2011). [CrossRef]
26. Y. Zhao, K. C. Kwon, Y. L. Piao, S. H. Jeon, and N. Kim, "Depth-layer weighted prediction method for a full-color polygon-based holographic system with real objects," Opt. Lett. 42(13), 2599–2602 (2017). [CrossRef]
27. S. J. Liu, N. T. Ma, P. P. Li, and D. Wang, "Holographic near-eye 3D display method based on large-size hologram," Front. Mater. 8, 739449 (2021). [CrossRef]
28. S. F. Lin, P. Gentet, D. Wang, S. H. Lee, E. S. Kim, and Q. H. Wang, "Simply structured full-color holographic three-dimensional display using angular-compensating holographic optical element," Opt. Laser. Eng. 138, 106404 (2021). [CrossRef]
29. K. Choi, H. Kim, and B. Lee, "Full-color autostereoscopic 3D display system using color-dispersion-compensated synthetic phase holograms," Opt. Express 12(21), 5229–5236 (2004). [CrossRef]
30. Y. Zhao, L. Cao, H. Zhang, D. Kong, and G. Jin, "Accurate calculation of computer-generated holograms using angular-spectrum layer-oriented method," Opt. Express 23(20), 25440–25449 (2015). [CrossRef]
31. T. Shimobaba, K. Matsushim a, T. Kakue, N. Masuda, and T. Ito, "Scaled angular spectrum method," Opt. Lett. 37(19), 4128–4130 (2012). [CrossRef]
32. D. Wang, N. N. Li, Y. L. Li, Y. W. Zheng, and Q. H. Wang, "Curved hologram generation method for speckle noise suppression based on stochastic gradient descent algorithm," Opt. Express 29(26), 42650–42662 (2021). [CrossRef]
L. Shi, B. Li, C. Kim, P. Kellnhofer, and W. Matusik, "Towards real-time photorealistic 3D holography with deep neural networks," Nature 591(7849), 234–239 (2021).
T. Zhan, J. H. Xiong, J. Y. Zhou, and S. T. Wu, "Multifocal displays: review and prospect," PhotoniX 1(1), 10 (2020).
D. Wang, C. Liu, C. Shen, Y. Xing, and Q. H. Wang, "Holographic capture and projection system of real object based on tunable zoom lens," PhotoniX 1(1), 6 (2020).
K. Wakunami, P. Y. Hsieh, R. Oi, T. Senoh, H. Sasaki, Y. Ichihashi, M. Okui, Y. P. Huang, and K. Yamamoto, "Projection-type see-through holographic three-dimensional display," Nat. Commun. 7(1), 12954 (2016).
X. Ding, Z. Wang, G. Hu, J. Liu, K. Zhang, H. Li, B. Ratni, S. N. Burokur, Q. Wu, J. Tan, and C. W. Qiu, "Metasurface holographic image projection based on mathematical properties of Fourier transform," PhotoniX 1(1), 16 (2020).
Z. Wang, G. Lv, Q. Feng, A. Wang, and H. Ming, "Simple and fast calculation algorithm for computer-generated hologram based on integral imaging using look-up table," Opt. Express 26(10), 13322–13330 (2018).
T. Kozacki, G. Finke, P. Garbat, W. Zaperty, and M. Kujawińska, "Wide angle holographic display system with spatiotemporal multiplexing," Opt. Express 20(25), 27473–27481 (2012).
T. Kozacki, M. Kujawinska, G. Finke, B. Hennelly, and N. Pandey, "Extended viewing angle holographic display system with tilted SLMs in a circular configuration," Appl. Opt. 51(11), 1771–1780 (2012).
Z. Zhang, C. P. Chen, Y. Li, B. Yu, L. Zhou, and Y. Wu, "Angular multiplexing of holographic display using tunable multi-stage gratings," Mol. Cryst. Liq. Cryst. 657(1), 102–106 (2017).
Y. Z. Liu, X. N. Pang, S. Jiang, and J. W. Dong, "Viewing-angle enlargement in holographic augmented reality using time division and spatial tiling," Opt. Express 21(10), 12068–12076 (2013).
Z. Zeng, H. Zheng, Y. Yu, A. K. Asundi, and S. Valyukh, "Full-color holographic display with increased-viewing-angle," Appl. Opt. 56(13), F112–F120 (2017).
J. Li, Q. Smithwick, and D. Chu, "Full bandwidth dynamic coarse integral holograohic displays with large field of view using a large resonant scanner and a galvanometer scanner," Opt. Express 26(13), 17459–17476 (2018).
Y. Sando, M. Itoh, and T. Yatagai, "Fast calculation method for cylindrical computer-generated holograms," Opt. Express 13(5), 1418–1423 (2005).
S. Q. Li, X. Xu, R. M. Veetil, V. Valuckas, R. P. Domínguez, and A. I. Kuznetsov, "Phase-only transmissive spatial lightmodulator based on tunable dielectric metasurface," Science 364(6445), 1087–1090 (2019).
Q. S. Wei, B. Sain, Y. T. Wang, B. Reineke, X. W. Li, L. L. Huang, and T. Zentgraf, "Simultaneous spectral and spatial modulation for color printing and holography using all-dielectric metasurfacces," Nano Lett. 19(12), 8964–8971 (2019).
G. Li, K. Hong, J. Yeom, N. Chen, J. H. Park, N. Kim, and B. Lee, "Acceleration method for computer generated spherical hologram calculation of real objects using graphics processing unit," Chin. Opt. Lett. 12(6), 060016 (2014).
K. Bang, C. Jang, and B. Lee, "Curved holographic optical elements and applications for curved see-through displays," J. Inf. Disp. 20(1), 9–23 (2019).
J. Burch and A. D. Falco, "Holography using curved metasurfaces," Photonics 6(1), 8 (2019).
L. Zhou, C. P. Chen, Y. Wu, Z. Zhang, K. Wang, B. Yu, and Y. Li, "See-through near-eye displays enabling vision correction," Opt. Express 25(3), 2130–2142 (2017).
R. Kang, J. Liu, G. Xue, X. Li, D. Pi, and Y. Wang, "Curved multiplexing computer-generated hologram for 3D holographic display," Opt. Express 27(10), 14369–14380 (2019).
R. Kang, J. Liu, D. Pi, and X. Duan, "Fast method for calculating a curved hologram in a holographic display," Opt. Express 28(8), 11290–11300 (2020).
J. P. Liu, W. T. Chen, H. H. Wen, and T. C. Poon, "Recording of a curved digital hologram for orthoscopic real image reconstruction," Opt. Lett. 45(15), 4353–4356 (2020).
N. N. Li, D. Wang, Y. L. Li, and Q. H. Wang, "Method of curved composite hologram generation with suppressed speckle noise," Opt. Express 28(23), 34378–34389 (2020).
M. Oikawa, T. Shimobaba, T. Yoda, H. Nakayama, A. Shhiraki, N. Masuda, and T. Ito, "Time-division color electroholography using one-chip RGB LED and synchronizing controller," Opt. Express 19(13), 12008–12013 (2011).
T. Shimobaba, T. Takahashi, N. Masuda, and T. Ito, "Numerical study of color holographic projection using space-division method," Opt. Express 19(11), 10287–10292 (2011).
Y. Zhao, K. C. Kwon, Y. L. Piao, S. H. Jeon, and N. Kim, "Depth-layer weighted prediction method for a full-color polygon-based holographic system with real objects," Opt. Lett. 42(13), 2599–2602 (2017).
S. J. Liu, N. T. Ma, P. P. Li, and D. Wang, "Holographic near-eye 3D display method based on large-size hologram," Front. Mater. 8, 739449 (2021).
S. F. Lin, P. Gentet, D. Wang, S. H. Lee, E. S. Kim, and Q. H. Wang, "Simply structured full-color holographic three-dimensional display using angular-compensating holographic optical element," Opt. Laser. Eng. 138, 106404 (2021).
K. Choi, H. Kim, and B. Lee, "Full-color autostereoscopic 3D display system using color-dispersion-compensated synthetic phase holograms," Opt. Express 12(21), 5229–5236 (2004).
Y. Zhao, L. Cao, H. Zhang, D. Kong, and G. Jin, "Accurate calculation of computer-generated holograms using angular-spectrum layer-oriented method," Opt. Express 23(20), 25440–25449 (2015).
T. Shimobaba, K. Matsushim a, T. Kakue, N. Masuda, and T. Ito, "Scaled angular spectrum method," Opt. Lett. 37(19), 4128–4130 (2012).
D. Wang, N. N. Li, Y. L. Li, Y. W. Zheng, and Q. H. Wang, "Curved hologram generation method for speckle noise suppression based on stochastic gradient descent algorithm," Opt. Express 29(26), 42650–42662 (2021).
Asundi, A. K.
Bang, K.
Burch, J.
Burokur, S. N.
Cao, L.
Chen, C. P.
Chen, N.
Chen, W. T.
Choi, K.
Chu, D.
Ding, X.
Domínguez, R. P.
Dong, J. W.
Duan, X.
Falco, A. D.
Feng, Q.
Finke, G.
Garbat, P.
Gentet, P.
Hennelly, B.
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Chin. Opt. Lett. (1)
Front. Mater. (1)
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Mol. Cryst. Liq. Cryst. (1)
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(1) U o ( x , y ) = A 0 ⋅ exp ( j φ 0 ( x , y ) ) ,
(2) U p ( x , y ) = I F F T { F F T { U 0 ( x , y } ⋅ H f ( f x , f y ) } ,
(3) H f ( f x , f y ) = exp ( i k Δ z 1 − ( λ f x ) 2 − ( λ f y ) 2 ) ,
(4) U c ( x , y ) = U p ( x , y ) ⋅ T ( x , y ; β ) = U p ( x , y ) ⋅ exp ( i k Δ z ( x , y ; β ) ) ,
(5) Δ z ( x , y ; β ) = z c − R + R 2 − x 2 ,
(6) I ( x , y ) = [ | r ( x , y ) | exp ( − i 2 π ξ r x ) ⋅ U o ( x , y ) ] ⋅ | r ′ ( x , y ) | exp ( i 2 π ξ r x ) ,
(7) ξ r = sin θ λ ,
(8) H ( x , y ) = ∑ i = 1 n U i ( x , y ) ,
(9) U i ( x , y ) = U c ( x , y ) ⋅ exp ( i k ⋅ sin θ ⋅ w x ) ,
(10) H ′ ( x , y ) = H ( x , y ) T ( x , y ; β ) . | CommonCrawl |
\begin{definition}[Definition:Sigma-Algebra/Definition 3]
A '''$\sigma$-algebra''' $\Sigma$ is a $\sigma$-ring with a unit.
\end{definition} | ProofWiki |
\begin{definition}[Definition:Minkowski Functional of Open Convex Set]
Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $C$ be an open convex subset of $X$ with $0 \in C$.
We define the '''Minkowski functional''' of $C$, $p_C : X \to \hointr 0 \infty$ by:
:$\map {p_C} x = \inf \set {t > 0 : t^{-1} x \in C}$
for each $x \in X$.
\end{definition} | ProofWiki |
\begin{definition}[Definition:Summation/Finite Support]
Let $G$ be an abelian group.
Let $S$ be a set.
Let $f: S \to G$ be a mapping.
Let the support $\map \supp f$ be finite.
Let $g$ be the restriction of $f$ to $\map \supp f$.
The '''summation of $f$ over $S$''', denoted $\ds \sum_{s \mathop \in S} \map f s$, is the summation over the finite set $\map \supp f$ of $g$:
:$\ds \sum_{s \mathop \in S} \map f s = \sum_{s \mathop \in \map \supp f} \map g s$
Category:Definitions/Summations
\end{definition} | ProofWiki |
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''geodesic''
A geometric concept which is a generalization of the concept of a straight line (or a segment of a straight line) in Euclidean geometry to spaces of a more general type. The definitions of geodesic lines in various spaces depend on the particular structure (metric, line element, linear connection) on which the geometry of the particular space is based. In the geometry of spaces in which the metric is considered to be specified in advance, geodesic lines are defined as locally shortest. In spaces with a connection a geodesic line is defined as a curve for which the tangent vector field is parallel along this curve. In Riemannian and Finsler geometries, where the line element is given in advance (in other words, a metric in the tangent space at each point of the considered manifold is given), while the lengths of lines are obtained by subsequent integration, geodesic lines are defined as extremals of the length functional.
{{MSC|53}}
{{TEX|done}}
Geodesic lines were first studied by J. Bernoulli and L. Euler, who attempted to find the shortest lines on regular surfaces in Euclidean space. On such lines the [[Geodesic curvature|geodesic curvature]] vanishes, and the principal normal of such curves is parallel to the normal to the surface. Geodesic lines are preserved under isometric deformation. The motion of a conservative mechanical system with a finite number of degrees of freedom is described by a geodesic line in a suitably-chosen Riemannian space.
Geodesic lines in Riemannian spaces have been studied most thoroughly. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g0441201.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g0441202.png" />-dimensional Riemannian space with metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g0441203.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g0441204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g0441205.png" />. The definition of a geodesic line as an extremal makes it possible to write down its differential equation in arbitrary local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g0441206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g0441207.png" />, for any parametrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g0441208.png" />:
The notion of a ''geodesic line'' (also: ''geodesic'')
is a geometric concept which is a generalization of the concept of a
straight line (or a segment of a straight line) in Euclidean geometry
to spaces of a more general type. The definitions of geodesic lines in
various spaces depend on the particular structure (metric, line
element, linear connection) on which the geometry of the particular
space is based. In the geometry of spaces in which the metric is
considered to be specified in advance, geodesic lines are defined as
locally shortest. In spaces with a connection a geodesic line is
defined as a curve for which the tangent vector field is parallel
along this curve. In Riemannian and Finsler geometries, where the line
element is given in advance (in other words, a metric in the tangent
space at each point of the considered manifold is given), while the
lengths of lines are obtained by subsequent integration, geodesic
lines are defined as extremals of the length functional.
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g0441209.png" /></td> </tr></table>
Geodesic lines were first studied by J. Bernoulli and L. Euler, who
attempted to find the shortest lines on regular surfaces in Euclidean
space. On such lines the
[[Geodesic curvature|geodesic curvature]] vanishes, and the principal
normal of such curves is parallel to the normal to the
surface. Geodesic lines are preserved under isometric deformation. The
motion of a conservative mechanical system with a finite number of
degrees of freedom is described by a geodesic line in a
suitably-chosen Riemannian space.
Geodesic lines in Riemannian spaces have been studied most
thoroughly. Let $M^n$ be an $n$-dimensional Riemannian space with metric
tensor $g_{ij}$ of class $C^k$, $k\ge 2$. The definition of a geodesic line as an
extremal makes it possible to write down its differential equation in
arbitrary local coordinates $x^i$, $i=1,\dots,n$, for any parametrization $\def\g{\gamma} \g = (x^i(t))$:
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412010.png" /></td> </tr></table>
$$\def\d{\partial}\frac{d}{dt}\Big(\frac{\d F}{\d x^i}\Big) - \frac{\d F}{\d x^i} = 0,$$
$$\def\dx{\dot{x}} F = \sqrt{g_{ik} \dx^i\dx^k}, \quad \dx^i=\frac{dx^i}{dt}.$$
Another, equivalent, form of the equations of
geodesic lines is derived from the postulate of parallelity of the
tangent vector $\dot{\gamma}=(\dx^i)$ along $\gamma$. If $t$ is the arc length $s$ along a
geodesic, or a linear function of $s$, then
$$\frac{D}{dt}(\dot{\gamma})=0,\quad {\textrm or }\quad \ddot{x}^i + \Gamma_{jk}^i \dx^j\dx^k = 0.\label{1}$$
The definition of a
geodesic line by equation (1) also involves a canonical selection of a
parameter. In such a definition a geodesic line $\gamma=x(t,\xi)$, with initial
tangent vector $\xi$, $x(0,\xi) = x_0$, $\dx(0,\xi) = \xi$, passes through a given point $x_0$. The
mapping $\xi\mapsto x(1,\xi)$ of the tangent space at $x_0$ into the space under
consideration is the exponential mapping with pole $x_0$. Near the pole
$x_0$ it is a diffeomorphism, which introduces Riemannian coordinates
into the space under consideration.
Another, equivalent, form of the equations of geodesic lines is derived from the postulate of parallelity of the tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412011.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412013.png" /> is the arc length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412014.png" /> along a geodesic, or a linear function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412015.png" />, then
A number of properties of geodesic lines are preserved for curves
represented by second-order equations $\ddot{x}=F(x,\dot{x})$ if, like in (1), the
function $F$ is a homogeneous function of the second degree in
$(\dx^i)$. The search for such equations in terms of tangent bundles yields
the concepts of a
[[Spray|spray]] and their integral curves. Geodesic lines are a
special case of such curves
{{Cite|La}}.
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
The local behaviour of geodesic curves is similar to that of straight
lines in Euclidean space. A sufficiently short arc of a geodesic line
is the shortest among all rectifiable curves with the same ends. Only
one geodesic line passes through any point in a given direction. Each
point has a neighbourhood $U$ in which any two points can be connected
by a unique geodesic line lying in $U$
{{Cite|He}}.
The definition of a geodesic line by equation (1) also involves a canonical selection of a parameter. In such a definition a geodesic line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412017.png" />, with initial tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412020.png" />, passes through a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412021.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412022.png" /> of the tangent space at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412023.png" /> into the space under consideration is the exponential mapping with pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412024.png" />. Near the pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412025.png" /> it is a diffeomorphism, which introduces Riemannian coordinates into the space under consideration.
The question of the distance to which an arc of a geodesic line
issuing from a point $x_0$ may be extended so that it remains the
shortest among all curves in a neighbourhood of $x_0$ is one of the
problems of the calculus of variations. A comparison of a geodesic
line with nearby curves is based on the study of the second variation
of the length by considering particular fields of velocities (the
[[Jacobi vector field|Jacobi vector field]]) along the geodesic line
$\gamma(s)$ belonging to particular variations $\gamma(s,t)$. For any fixed $t$ the
curve $\gamma(s,t)$ remains a geodesic, while the parameter $s$ on it remains
canonical. If at the origin of $\gamma$ the velocity is zero, then the
points on $\gamma$ where some non-zero Jacobi vector field is zero are
called conjugate points. Geodesic lines remain the shortest of all
nearby curves up to the first conjugate point. For a geodesic arc
extended beyond the conjugate point there exists a shorter curve with
the same ends, which may be arbitrary near. A Jacobi vector field $\eta(s)$
satisfies the equation
$$\frac{D^2\eta}{ds^2} + R(\dot{\gamma},\eta)\dot{\gamma} = 0,$$
where $\dot{\gamma}$ is a tangent vector to the
geodesic $\gamma(s)$, while $R(\dot{\gamma},\eta)$ is the
[[Curvature transformation|curvature transformation]] or, in Fermi
coordinates $x^i$, $x^1 = s$:
$$\frac{d^2\eta^l}{ds^2} + R_{1k,1}^i\eta^k = 0,\label{2}$$
where $R^l_{ij,k}$ is the curvature tensor. The
connection between the Jacobi vector field and the curvature
determines the dependence of properties of the geodesics on the
curvature of the space. For instance, in a space of negative curvature
there are no conjugate points; if, in addition, the space is simply
connected, then any geodesic arc is shortest, and geodesic lines
issuing from a point diverge exponentially. These properties are of
importance in the theory of dynamical systems (cf.
[[Geodesic flow|Geodesic flow]]). The monotone nature of the effect of
the curvature forms the subject of several so-called comparison
theorems. In particular, the distance to the first conjugate point and
the lengths of the vectors of the Jacobi vector field on this segment
(normalized by the condition $\eta(0)=0$, $|D\eta/ds| = 1$) decrease as the curvature of
the space increases. Here, under a comparison of two geodesic lines it
is understood that all the curvatures of the second space majorize any
curvature of the first space at points corresponding to the same
{{Cite|GrKlMe}}.
A number of properties of geodesic lines are preserved for curves represented by second-order equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412026.png" /> if, like in (1), the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412027.png" /> is a homogeneous function of the second degree in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412028.png" />. The search for such equations in terms of tangent bundles yields the concepts of a [[Spray|spray]] and their integral curves. Geodesic lines are a special case of such curves [[#References|[2]]].
In general relativity theory equation (2) is the source of physical
interpretation of the space-time curvature by the behaviour of
geodesic lines
{{Cite|Sy}}.
The local behaviour of geodesic curves is similar to that of straight lines in Euclidean space. A sufficiently short arc of a geodesic line is the shortest among all rectifiable curves with the same ends. Only one geodesic line passes through any point in a given direction. Each point has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412029.png" /> in which any two points can be connected by a unique geodesic line lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412030.png" /> [[#References|[3]]].
If the comparison is not restricted to nearby curves, the arc of a
geodesic line may cease to be shortest before it has passed the
conjugate point. This is possible even in a simply-connected space,
i.e. the reasons for it may be both topological and metric.
The question of the distance to which an arc of a geodesic line issuing from a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412031.png" /> may be extended so that it remains the shortest among all curves in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412032.png" /> is one of the problems of the calculus of variations. A comparison of a geodesic line with nearby curves is based on the study of the second variation of the length by considering particular fields of velocities (the [[Jacobi vector field|Jacobi vector field]]) along the geodesic line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412033.png" /> belonging to particular variations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412034.png" />. For any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412035.png" /> the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412036.png" /> remains a geodesic, while the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412037.png" /> on it remains canonical. If at the origin of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412038.png" /> the velocity is zero, then the points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412039.png" /> where some non-zero Jacobi vector field is zero are called conjugate points. Geodesic lines remain the shortest of all nearby curves up to the first conjugate point. For a geodesic arc extended beyond the conjugate point there exists a shorter curve with the same ends, which may be arbitrary near. A Jacobi vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412040.png" /> satisfies the equation
The question of the effect of the curvature on the extension of the
arc between conjugate points on which the geodesic line remains
shortest is of importance in the study of the connection between the
curvature and the topological structure of the space. The dependence
of the number of closed geodesic lines or the number of different
geodesic lines connecting two points on the topological structure of
the space forms a subject of
[[Variational calculus in the large|variational calculus in the
large]]{{Cite|LySc}},
{{Cite|GrKlMe}},
{{Cite|Mi}}.
Families of geodesic lines, considered as possible trajectories of
motion, form a subject of the theory of dynamical systems and ergodic
theory.
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412042.png" /> is a tangent vector to the geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412043.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412044.png" /> is the [[Curvature transformation|curvature transformation]] or, in Fermi coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412046.png" />:
In spaces with an affine connection geodesic lines are defined by
equation (1). Local theorems on the existence and uniqueness of
geodesic lines connecting two points, and on the existence of a convex
neighbourhood, are preserved for such spaces.
Geodesic lines with similar properties are also defined in spaces with
a projective connection, and also in cases of more general connections
on manifolds.
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412048.png" /> is the curvature tensor. The connection between the Jacobi vector field and the curvature determines the dependence of properties of the geodesics on the curvature of the space. For instance, in a space of negative curvature there are no conjugate points; if, in addition, the space is simply connected, then any geodesic arc is shortest, and geodesic lines issuing from a point diverge exponentially. These properties are of importance in the theory of dynamical systems (cf. [[Geodesic flow|Geodesic flow]]). The monotone nature of the effect of the curvature forms the subject of several so-called comparison theorems. In particular, the distance to the first conjugate point and the lengths of the vectors of the Jacobi vector field on this segment (normalized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412050.png" />) decrease as the curvature of the space increases. Here, under a comparison of two geodesic lines it is understood that all the curvatures of the second space majorize any curvature of the first space at points corresponding to the same lengths [[#References|[4]]].
The geometrization of the problems of variational calculus for
functionals other than the length functional generated the concept of
[[Finsler space|Finsler space]] and the geodesic lines in that
space. Separation of basic geometrical properties of such spaces led
to the concept of
[[Geodesic geometry|geodesic geometry]], defined by the presence and
the extendability of geodesic lines.
In general relativity theory equation (2) is the source of physical interpretation of the space-time curvature by the behaviour of geodesic lines [[#References|[5]]].
The most intensively studied geodesic lines in metric spaces with an
irregular metric are those on a convex surface and in a
If the comparison is not restricted to nearby curves, the arc of a geodesic line may cease to be shortest before it has passed the conjugate point. This is possible even in a simply-connected space, i.e. the reasons for it may be both topological and metric.
[[Two-dimensional manifold of bounded curvature|two-dimensional
manifold of bounded curvature]]. Here, a geodesic line is not
The question of the effect of the curvature on the extension of the arc between conjugate points on which the geodesic line remains shortest is of importance in the study of the connection between the curvature and the topological structure of the space. The dependence of the number of closed geodesic lines or the number of different geodesic lines connecting two points on the topological structure of the space forms a subject of [[Variational calculus in the large|variational calculus in the large]] [[#References|[6]]], [[#References|[4]]], [[#References|[7]]].
necessarily a smooth curve; it need have no extension or — in a
two-dimensional manifold of bounded curvature — it may have a
Families of geodesic lines, considered as possible trajectories of motion, form a subject of the theory of dynamical systems and ergodic theory.
non-unique extension. A geodesic line on a convex surface invariably
has a semi-tangent; if it can be extended, the extension is unique;
In spaces with an affine connection geodesic lines are defined by equation (1). Local theorems on the existence and uniqueness of geodesic lines connecting two points, and on the existence of a convex neighbourhood, are preserved for such spaces.
geodesic lines issue from a point in almost-all directions. It was
found that in such spaces the class of quasi-geodesic lines (cf.
Geodesic lines with similar properties are also defined in spaces with a projective connection, and also in cases of more general connections on manifolds.
[[Quasi-geodesic line|Quasi-geodesic line]]), which is the closure of
the class of geodesic lines, is more natural than the class of
The geometrization of the problems of variational calculus for functionals other than the length functional generated the concept of a [[Finsler space|Finsler space]] and the geodesic lines in that space. Separation of basic geometrical properties of such spaces led to the concept of [[Geodesic geometry|geodesic geometry]], defined by the presence and the extendability of geodesic lines.
geodesic lines itself
{{Cite|Al}}.
The most intensively studied geodesic lines in metric spaces with an irregular metric are those on a convex surface and in a [[Two-dimensional manifold of bounded curvature|two-dimensional manifold of bounded curvature]]. Here, a geodesic line is not necessarily a smooth curve; it need have no extension or — in a two-dimensional manifold of bounded curvature — it may have a non-unique extension. A geodesic line on a convex surface invariably has a semi-tangent; if it can be extended, the extension is unique; geodesic lines issue from a point in almost-all directions. It was found that in such spaces the class of quasi-geodesic lines (cf. [[Quasi-geodesic line|Quasi-geodesic line]]), which is the closure of the class of geodesic lines, is more natural than the class of geodesic lines itself [[#References|[8]]].
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1962)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.L. Synge, "Relativity: the general theory" , North-Holland & Interscience (1960)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> L.A. Lyusternik, L.G. [L.G. Shnirel'man] Schnirelmann, "Méthode topologiques dans les problèmes variationelles" , Hermann (1934) (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.D. Aleksandrov, "Die innere Geometrie der konvexen Flächen" , Akademie Verlag (1955) (Translated from Russian)</TD></TR></table>
====Comments====
Riemannian coordinates are usually called normal
coordinates or geodesic polar coordinates, cf.
[[Geodesic coordinates|Geodesic coordinates]].
A convex neighbourhood, also called a normal neighbourhood, is a
neighbourhood $U$ in which any two points can be connected by a unique
geodesic in $U$.
Riemannian coordinates are usually called normal coordinates or geodesic polar coordinates, cf. [[Geodesic coordinates|Geodesic coordinates]].
A convex neighbourhood, also called a normal neighbourhood, is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412051.png" /> in which any two points can be connected by a unique geodesic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044120/g04412052.png" />.
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Klingenberg, "Lectures on closed geodesics" , Springer (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Busemann, "The geometry of geodesics" , Acad. Press (1955)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> B. O'Neill, "Semi-Riemannian geometry (with applications to relativity)" , Acad. Press (1983)</TD></TR></table>
{|
|valign="top"|{{Ref|Al}}||valign="top"| A.D. Aleksandrov, "Die innere Geometrie der konvexen Flächen", Akademie Verlag (1955) (Translated from Russian)
|valign="top"|{{Ref|BeGo}}||valign="top"| M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces", Springer (1988) (Translated from French) {{MR|0917479}} {{ZBL|0629.53001}}
|valign="top"|{{Ref|Bu}}||valign="top"| H. Busemann, "The geometry of geodesics", Acad. Press (1955) {{MR|0075623}} {{ZBL|0112.37002}}
|valign="top"|{{Ref|GrKlMe}}||valign="top"| D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen", Springer (1968) {{MR|0229177}} {{ZBL|0155.30701}}
|valign="top"|{{Ref|He}}||valign="top"| S. Helgason, "Differential geometry, Lie groups, and symmetric spaces", Acad. Press (1978) {{MR|0514561}} {{ZBL|0451.53038}}
|valign="top"|{{Ref|Kl}}||valign="top"| W. Klingenberg, "Lectures on closed geodesics", Springer (1978) {{MR|0478069}} {{ZBL|0397.58018}}
|valign="top"|{{Ref|Kl2}}||valign="top"| W. Klingenberg, "Riemannian geometry", de Gruyter (1982) (Translated from German) {{MR|0666697}} {{ZBL|0495.53036}}
|valign="top"|{{Ref|La}}||valign="top"| S. Lang, "Introduction to differentiable manifolds", Interscience (1962) pp. App. III {{MR|0155257}} {{ZBL|0103.15101}}
|valign="top"|{{Ref|LySc}}||valign="top"| L.A. Lyusternik, L.G. [L.G. Shnirel'man] Schnirelmann, "Méthode topologiques dans les problèmes variationelles", Hermann (1934) (Translated from Russian)
|valign="top"|{{Ref|Mi}}||valign="top"| J.W. Milnor, "Morse theory", Princeton Univ. Press (1963) {{MR|0163331}} {{ZBL|0108.10401}}
|valign="top"|{{Ref|Ra}}||valign="top"| P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse", Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
|valign="top"|{{Ref|Sy}}||valign="top"| J.L. Synge, "Relativity: the general theory", North-Holland & Interscience (1960) {{MR|0118457}} {{ZBL|0090.18504}}
Latest revision as of 21:59, 5 March 2012
2010 Mathematics Subject Classification: Primary: 53-XX [MSN][ZBL]
The notion of a geodesic line (also: geodesic) is a geometric concept which is a generalization of the concept of a straight line (or a segment of a straight line) in Euclidean geometry to spaces of a more general type. The definitions of geodesic lines in various spaces depend on the particular structure (metric, line element, linear connection) on which the geometry of the particular space is based. In the geometry of spaces in which the metric is considered to be specified in advance, geodesic lines are defined as locally shortest. In spaces with a connection a geodesic line is defined as a curve for which the tangent vector field is parallel along this curve. In Riemannian and Finsler geometries, where the line element is given in advance (in other words, a metric in the tangent space at each point of the considered manifold is given), while the lengths of lines are obtained by subsequent integration, geodesic lines are defined as extremals of the length functional.
Geodesic lines were first studied by J. Bernoulli and L. Euler, who attempted to find the shortest lines on regular surfaces in Euclidean space. On such lines the geodesic curvature vanishes, and the principal normal of such curves is parallel to the normal to the surface. Geodesic lines are preserved under isometric deformation. The motion of a conservative mechanical system with a finite number of degrees of freedom is described by a geodesic line in a suitably-chosen Riemannian space.
Geodesic lines in Riemannian spaces have been studied most thoroughly. Let $M^n$ be an $n$-dimensional Riemannian space with metric tensor $g_{ij}$ of class $C^k$, $k\ge 2$. The definition of a geodesic line as an extremal makes it possible to write down its differential equation in arbitrary local coordinates $x^i$, $i=1,\dots,n$, for any parametrization $\def\g{\gamma} \g = (x^i(t))$:
$$\def\d{\partial}\frac{d}{dt}\Big(\frac{\d F}{\d x^i}\Big) - \frac{\d F}{\d x^i} = 0,$$ where $$\def\dx{\dot{x}} F = \sqrt{g_{ik} \dx^i\dx^k}, \quad \dx^i=\frac{dx^i}{dt}.$$ Another, equivalent, form of the equations of geodesic lines is derived from the postulate of parallelity of the tangent vector $\dot{\gamma}=(\dx^i)$ along $\gamma$. If $t$ is the arc length $s$ along a geodesic, or a linear function of $s$, then $$\frac{D}{dt}(\dot{\gamma})=0,\quad {\textrm or }\quad \ddot{x}^i + \Gamma_{jk}^i \dx^j\dx^k = 0.\label{1}$$ The definition of a geodesic line by equation (1) also involves a canonical selection of a parameter. In such a definition a geodesic line $\gamma=x(t,\xi)$, with initial tangent vector $\xi$, $x(0,\xi) = x_0$, $\dx(0,\xi) = \xi$, passes through a given point $x_0$. The mapping $\xi\mapsto x(1,\xi)$ of the tangent space at $x_0$ into the space under consideration is the exponential mapping with pole $x_0$. Near the pole $x_0$ it is a diffeomorphism, which introduces Riemannian coordinates into the space under consideration.
A number of properties of geodesic lines are preserved for curves represented by second-order equations $\ddot{x}=F(x,\dot{x})$ if, like in (1), the function $F$ is a homogeneous function of the second degree in $(\dx^i)$. The search for such equations in terms of tangent bundles yields the concepts of a spray and their integral curves. Geodesic lines are a special case of such curves [La].
The local behaviour of geodesic curves is similar to that of straight lines in Euclidean space. A sufficiently short arc of a geodesic line is the shortest among all rectifiable curves with the same ends. Only one geodesic line passes through any point in a given direction. Each point has a neighbourhood $U$ in which any two points can be connected by a unique geodesic line lying in $U$ [He].
The question of the distance to which an arc of a geodesic line issuing from a point $x_0$ may be extended so that it remains the shortest among all curves in a neighbourhood of $x_0$ is one of the problems of the calculus of variations. A comparison of a geodesic line with nearby curves is based on the study of the second variation of the length by considering particular fields of velocities (the Jacobi vector field) along the geodesic line $\gamma(s)$ belonging to particular variations $\gamma(s,t)$. For any fixed $t$ the curve $\gamma(s,t)$ remains a geodesic, while the parameter $s$ on it remains canonical. If at the origin of $\gamma$ the velocity is zero, then the points on $\gamma$ where some non-zero Jacobi vector field is zero are called conjugate points. Geodesic lines remain the shortest of all nearby curves up to the first conjugate point. For a geodesic arc extended beyond the conjugate point there exists a shorter curve with the same ends, which may be arbitrary near. A Jacobi vector field $\eta(s)$ satisfies the equation $$\frac{D^2\eta}{ds^2} + R(\dot{\gamma},\eta)\dot{\gamma} = 0,$$ where $\dot{\gamma}$ is a tangent vector to the geodesic $\gamma(s)$, while $R(\dot{\gamma},\eta)$ is the curvature transformation or, in Fermi coordinates $x^i$, $x^1 = s$: $$\frac{d^2\eta^l}{ds^2} + R_{1k,1}^i\eta^k = 0,\label{2}$$ where $R^l_{ij,k}$ is the curvature tensor. The connection between the Jacobi vector field and the curvature determines the dependence of properties of the geodesics on the curvature of the space. For instance, in a space of negative curvature there are no conjugate points; if, in addition, the space is simply connected, then any geodesic arc is shortest, and geodesic lines issuing from a point diverge exponentially. These properties are of importance in the theory of dynamical systems (cf. Geodesic flow). The monotone nature of the effect of the curvature forms the subject of several so-called comparison theorems. In particular, the distance to the first conjugate point and the lengths of the vectors of the Jacobi vector field on this segment (normalized by the condition $\eta(0)=0$, $|D\eta/ds| = 1$) decrease as the curvature of the space increases. Here, under a comparison of two geodesic lines it is understood that all the curvatures of the second space majorize any curvature of the first space at points corresponding to the same lengths [GrKlMe].
In general relativity theory equation (2) is the source of physical interpretation of the space-time curvature by the behaviour of geodesic lines [Sy].
The question of the effect of the curvature on the extension of the arc between conjugate points on which the geodesic line remains shortest is of importance in the study of the connection between the curvature and the topological structure of the space. The dependence of the number of closed geodesic lines or the number of different geodesic lines connecting two points on the topological structure of the space forms a subject of variational calculus in the large[LySc], [GrKlMe], [Mi].
The geometrization of the problems of variational calculus for functionals other than the length functional generated the concept of a Finsler space and the geodesic lines in that space. Separation of basic geometrical properties of such spaces led to the concept of geodesic geometry, defined by the presence and the extendability of geodesic lines.
The most intensively studied geodesic lines in metric spaces with an irregular metric are those on a convex surface and in a two-dimensional manifold of bounded curvature. Here, a geodesic line is not necessarily a smooth curve; it need have no extension or — in a two-dimensional manifold of bounded curvature — it may have a non-unique extension. A geodesic line on a convex surface invariably has a semi-tangent; if it can be extended, the extension is unique; geodesic lines issue from a point in almost-all directions. It was found that in such spaces the class of quasi-geodesic lines (cf. Quasi-geodesic line), which is the closure of the class of geodesic lines, is more natural than the class of geodesic lines itself [Al].
Riemannian coordinates are usually called normal coordinates or geodesic polar coordinates, cf. Geodesic coordinates.
A convex neighbourhood, also called a normal neighbourhood, is a neighbourhood $U$ in which any two points can be connected by a unique geodesic in $U$.
[Al] A.D. Aleksandrov, "Die innere Geometrie der konvexen Flächen", Akademie Verlag (1955) (Translated from Russian)
[BeGo] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces", Springer (1988) (Translated from French) MR0917479 Zbl 0629.53001
[Bu] H. Busemann, "The geometry of geodesics", Acad. Press (1955) MR0075623 Zbl 0112.37002
[GrKlMe] D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen", Springer (1968) MR0229177 Zbl 0155.30701
[He] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces", Acad. Press (1978) MR0514561 Zbl 0451.53038
[Kl] W. Klingenberg, "Lectures on closed geodesics", Springer (1978) MR0478069 Zbl 0397.58018
[Kl2] W. Klingenberg, "Riemannian geometry", de Gruyter (1982) (Translated from German) MR0666697 Zbl 0495.53036
[La] S. Lang, "Introduction to differentiable manifolds", Interscience (1962) pp. App. III MR0155257 Zbl 0103.15101
[LySc] L.A. Lyusternik, L.G. [L.G. Shnirel'man] Schnirelmann, "Méthode topologiques dans les problèmes variationelles", Hermann (1934) (Translated from Russian)
[Mi] J.W. Milnor, "Morse theory", Princeton Univ. Press (1963) MR0163331 Zbl 0108.10401
[Ra] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse", Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[Sy] J.L. Synge, "Relativity: the general theory", North-Holland & Interscience (1960) MR0118457 Zbl 0090.18504
Geodesic line. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geodesic_line&oldid=15374
This article was adapted from an original article by Yu.A. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Geodesic_line&oldid=21569" | CommonCrawl |
# The Cauchy-Riemann equations
The Cauchy-Riemann equations are a set of partial differential equations that relate the real and imaginary parts of a complex-valued function. These equations are named after Augustin-Louis Cauchy and Carl Friedrich Gauss, who independently formulated them in the early 19th century. The Cauchy-Riemann equations have many applications in mathematics and physics, including in the study of complex analysis and the behavior of electric and magnetic fields.
The Cauchy-Riemann equations are given by:
$$
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}
$$
$$
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
$$
where $u$ and $v$ are the real and imaginary parts of a complex-valued function $f(x, y) = u + iv$.
Consider the function $f(x, y) = x^2 - y^2 + 2xyi$. To find the Cauchy-Riemann equations, we first find the partial derivatives of $u$ and $v$ with respect to $x$ and $y$:
$$
\frac{\partial u}{\partial x} = 2x + 2y
$$
$$
\frac{\partial u}{\partial y} = -2y
$$
$$
\frac{\partial v}{\partial x} = 2x
$$
$$
\frac{\partial v}{\partial y} = 2x + 2y
$$
Now, we can plug these values into the Cauchy-Riemann equations:
$$
2x + 2y = 2x
$$
$$
-2y = 2x + 2y
$$
The first equation implies that $2y = 0$, which means $y = 0$. The second equation implies that $-2y = 2x$, which means $x = 0$. Therefore, the only point at which the Cauchy-Riemann equations hold is the origin $(0, 0)$.
## Exercise
Find the Cauchy-Riemann equations for the function $f(x, y) = x^3 - 3xy^2 + x^2yi$.
The Cauchy-Riemann equations for this function are:
$$
\frac{\partial u}{\partial x} = 3x^2 - 3y^2
$$
$$
\frac{\partial u}{\partial y} = -6xy
$$
$$
\frac{\partial v}{\partial x} = 2x
$$
$$
\frac{\partial v}{\partial y} = 2x^2 - 6xy
$$
Now, we can plug these values into the Cauchy-Riemann equations:
$$
3x^2 - 3y^2 = 2x^2 - 6xy
$$
$$
-6xy = 2x^2 - 6xy
$$
The first equation implies that $3x^2 = 2x^2$, which means $x = 0$. The second equation implies that $6xy = 2x^2$, which means $y = \frac{1}{3}$. Therefore, the only point at which the Cauchy-Riemann equations hold is the origin $(0, 0)$.
# Using Green's theorem to solve line integrals
Green's theorem is a powerful tool in vector calculus that allows us to calculate line integrals in the plane. It is named after the mathematician George Green, who introduced it in the 19th century. Green's theorem is a generalization of the fundamental theorem of calculus and has many applications in physics, engineering, and mathematical modeling.
Green's theorem states that the line integral of a vector field $\vec{F}$ over a closed curve $C$ is equal to the double integral of the curl of $\vec{F}$ over the region enclosed by $C$:
$$
\oint_C \vec{F} \cdot d\vec{r} = \iint_R (\nabla \times \vec{F}) \cdot d\vec{A}
$$
## Exercise
Instructions:
Use Green's theorem to calculate the line integral of the vector field $\vec{F} = xi + yj$ over the curve $C$ given by $x = t^2 - 1$, $y = t^3 - t$.
### Solution
None
To apply Green's theorem, we first need to find the curl of $\vec{F}$:
$$
\nabla \times \vec{F} = \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)i + \left( \frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z} \right)j + \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right)k
$$
For our vector field $\vec{F} = xi + yj$, the curl is:
$$
\nabla \times \vec{F} = 0i + 0j + 0k
$$
Since the curl is zero, the line integral of $\vec{F}$ over $C$ is zero.
# Solving line integrals using JavaScript
JavaScript is a popular programming language that can be used to solve line integrals using Green's theorem. By defining a vector field and a curve, we can use JavaScript to numerically approximate the line integral of the vector field over the curve.
To solve a line integral using JavaScript, we can follow these steps:
1. Define the vector field $\vec{F}$ and the curve $C$.
2. Discretize the curve into a finite number of points.
3. Calculate the tangent vector at each point on the curve.
4. Calculate the dot product of $\vec{F}$ and the tangent vector at each point.
5. Sum the dot products to approximate the line integral.
## Exercise
Instructions:
Write a JavaScript function that calculates the line integral of the vector field $\vec{F} = xi + yj$ over the curve $C$ given by $x = t^2 - 1$, $y = t^3 - t$.
### Solution
None
Here is a JavaScript function that calculates the line integral of $\vec{F}$ over $C$:
```javascript
function lineIntegral(F, C, numPoints) {
let integral = 0;
for (let t = 0; t <= 1; t += 1 / numPoints) {
let x = t * t - 1;
let y = t * t * t - t;
let dx = 1 / numPoints;
let dy = (3 * t * t - 1) * dx;
let Fx = F(x, y)[0];
let Fy = F(x, y)[1];
integral += Fx * dx + Fy * dy;
}
return integral;
}
function F(x, y) {
return [x, y];
}
function C(t) {
return [t * t - 1, t * t * t - t];
}
console.log(lineIntegral(F, C, 1000));
```
This code defines a function `lineIntegral` that takes a vector field `F`, a curve `C`, and the number of points to discretize the curve. It then calculates the line integral of `F` over `C` using a numerical integration method.
# Applications of Green's theorem in computer graphics
Green's theorem has many applications in computer graphics, where it is used to calculate properties of shapes and curves. In computer graphics, the Cauchy-Riemann equations are often used to determine whether a curve is closed or not, and Green's theorem is used to calculate the area enclosed by the curve.
For example, in computer graphics, the area enclosed by a simple polygon can be calculated using Green's theorem. By defining a vector field with a non-zero curl, we can calculate the double integral of the curl over the polygon, which gives the area enclosed by the polygon.
## Exercise
Instructions:
Write a JavaScript function that calculates the area enclosed by a polygon using Green's theorem.
### Solution
None
Here is a JavaScript function that calculates the area enclosed by a polygon using Green's theorem:
```javascript
function polygonArea(polygon) {
let area = 0;
for (let i = 0; i < polygon.length; i++) {
let x1 = polygon[i][0];
let y1 = polygon[i][1];
let x2 = polygon[(i + 1) % polygon.length][0];
let y2 = polygon[(i + 1) % polygon.length][1];
area += (x1 * y2 - x2 * y1) / 2;
}
return Math.abs(area);
}
console.log(polygonArea([[0, 0], [1, 0], [1, 1], [0, 1]]));
```
This code defines a function `polygonArea` that takes a polygon as input and calculates its area using Green's theorem. The polygon is represented as an array of points, and the function calculates the area by summing the signed areas of the triangles formed by the polygon's vertices.
# Visualizing complex functions using JavaScript
JavaScript can be used to visualize complex functions in the plane. By defining a complex function and a grid of points in the plane, we can use JavaScript to calculate the values of the function at each point and visualize the function using a color map.
To visualize a complex function using JavaScript, we can follow these steps:
1. Define the complex function $f(z)$ and the grid of points in the plane.
2. Calculate the values of $f(z)$ at each point in the grid.
3. Create a color map that maps the values of $f(z)$ to colors.
4. Display the grid of points in the plane, with each point colored according to the value of $f(z)$ at that point.
## Exercise
Instructions:
Write a JavaScript function that visualizes the complex function $f(z) = z^2$ in the plane.
### Solution
None
Here is a JavaScript function that visualizes the complex function $f(z) = z^2$ in the plane:
```javascript
function complexFunction(z) {
let x = z[0];
let y = z[1];
return [x * x - y * y, 2 * x * y];
}
function grid(minX, maxX, minY, maxY, numPoints) {
let grid = [];
for (let x = minX; x <= maxX; x += (maxX - minX) / numPoints) {
for (let y = minY; y <= maxY; y += (maxY - minY) / numPoints) {
grid.push([x, y]);
}
}
return grid;
}
function visualizeComplexFunction() {
let canvas = document.getElementById("canvas");
let ctx = canvas.getContext("2d");
let points = grid(-5, 5, -5, 5, 100);
let maxAbs = 0;
for (let point of points) {
let value = complexFunction(point);
let absValue = Math.sqrt(value[0] * value[0] + value[1] * value[1]);
maxAbs = Math.max(maxAbs, absValue);
}
for (let point of points) {
let value = complexFunction(point);
let absValue = Math.sqrt(value[0] * value[0] + value[1] * value[1]);
let color = Math.floor((absValue / maxAbs) * 255);
ctx.fillStyle = "rgb(" + color + "," + color + "," + color + ")";
ctx.fillRect(point[0] * 100 + 300, -point[1] * 100 + 300, 1, 1);
}
}
visualizeComplexFunction();
```
This code defines a function `complexFunction` that represents the complex function $f(z) = z^2$, a function `grid` that generates a grid of points in the plane, and a function `visualizeComplexFunction` that visualizes the function using a color map.
# The relationship between partial derivatives and Green's theorem
The Cauchy-Riemann equations and Green's theorem are closely related to the concept of partial derivatives in calculus. The Cauchy-Riemann equations are a set of partial differential equations that relate the real and imaginary parts of a complex-valued function, while Green's theorem is a generalization of the fundamental theorem of calculus that allows us to calculate line integrals in the plane.
The relationship between partial derivatives and Green's theorem can be understood by considering the connection between the Cauchy-Riemann equations and the curl of a vector field. The Cauchy-Riemann equations are equivalent to the condition that the curl of a vector field is zero, which means that the vector field is conservative. In this case, the line integral of the vector field over a closed curve is equal to zero.
## Exercise
Instructions:
Show how the Cauchy-Riemann equations and Green's theorem are related to the concept of partial derivatives.
### Solution
None
The Cauchy-Riemann equations are a set of partial differential equations that relate the real and imaginary parts of a complex-valued function. These equations are given by:
$$
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}
$$
$$
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
$$
The curl of a vector field $\vec{F}$ is given by:
$$
\nabla \times \vec{F} = \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)i + \left( \frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z} \right)j + \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right)k
$$
If the vector field $\vec{F}$ satisfies the Cauchy-Riemann equations, then the curl of $\vec{F}$ is zero:
$$
\nabla \times \vec{F} = 0i + 0j + 0k
$$
This means that the vector field is conservative, and the line integral of $\vec{F}$ over a closed curve is equal to zero. This is a direct consequence of Green's theorem.
# Solving differential equations using Green's theorem
Green's theorem can also be used to solve differential equations in the plane. By defining a vector field that represents the differential equation and a curve in the plane, we can use Green's theorem to calculate the line integral of the vector field over the curve, which gives the value of the integral along the curve.
For example, the Cauchy-Riemann equations can be used to solve the differential equation $u_x + v_y = 0$ by defining a vector field $\vec{F} = (u_x - v_y)i + (u_y + v_x)j$ and using Green's theorem to calculate the line integral of $\vec{F}$ over a curve in the plane.
## Exercise
Instructions:
Write a JavaScript function that solves the differential equation $u_x + v_y = 0$ using Green's theorem.
### Solution
None
Here is a JavaScript function that solves the differential equation $u_x + v_y = 0$ using Green's theorem:
```javascript
function solveDifferentialEquation(F, C, numPoints) {
let integral = 0;
for (let t = 0; t <= 1; t += 1 / numPoints) {
let x = t * t - 1;
let y = t * t * t - t;
let dx = 1 / numPoints;
let dy = (3 * t * t - 1) * dx;
let Fx = F(x, y)[0];
let Fy = F(x, y)[1];
integral += Fx * dx + Fy * dy;
}
return integral;
}
function F(x, y) {
return [x - y, x + y];
}
function C(t) {
return [t * t - 1, t * t * t - t];
}
console.log(solveDifferentialEquation(F, C, 1000));
```
This code defines a function `solveDifferentialEquation` that takes a vector field `F` and a curve `C`, and calculates the line integral of `F` over `C` using Green's theorem. The function then returns the value of the integral, which is the value of the differential equation along the curve.
# The Stokes' theorem as a generalization of Green's theorem
Stokes' theorem is a generalization of Green's theorem that extends its application to three-dimensional space. It is named after George Gabriel Stokes, who introduced it in the 19th century. Stokes' theorem states that the line integral of a vector field $\vec{F}$ over a surface $S$ is equal to the surface integral of the curl of $\vec{F}$ over the region bounded by $S$:
$$
\oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot d\vec{S}
$$
Stokes' theorem has many applications in physics, engineering, and mathematical modeling, including in the study of fluid dynamics, electromagnetism, and differential geometry.
## Exercise
Instructions:
Write a JavaScript function that calculates the line integral of a vector field over a surface using Stokes' theorem.
### Solution
None
Here is a JavaScript function that calculates the line integral of a vector field over a surface using Stokes' theorem:
```javascript
function stokesTheorem(F, S, numPoints) {
let integral = 0;
for (let t = 0; t <= 1; t += 1 / numPoints) {
let x = t * t - 1;
let y = t * t * t - t;
let dx = 1 / numPoints;
let dy = (3 * t * t - 1) * dx;
let Fx = F(x, y)[0];
let Fy = F(x, y)[1];
integral += Fx * dx + Fy * dy;
}
return integral;
}
function F(x, y) {
return [x - y, x + y];
}
function S(x, y) {
return [x * x - y * y, x * y];
}
console.log(stokesTheorem(F, S, 1000));
```
This code defines a function `stokesTheorem` that takes a vector field `F` and a surface `S`, and calculates the line integral of `F` over `S` using Stokes' theorem. The function then returns the value of the integral, which is the value of the differential equation along the curve.
# Green's theorem in physics: electric and magnetic fields
Green's theorem has many applications in physics, including in the study of electric and magnetic fields. For example, Green's theorem can be used to calculate the electric potential $\phi$ at a point in the plane, given the charge distribution $\rho(x, y)$:
$$
\phi(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} \iint_R \frac{\rho(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} d\mathbf{A}'
$$
where $\epsilon_0$ is the vacuum permittivity constant.
## Exercise
Instructions:
Write a JavaScript function that calculates the electric potential at a point in the plane, given the charge distribution $\rho(x, y)$.
### Solution
None
Here is a JavaScript function that calculates the electric potential at a point in the plane, given the charge distribution $\rho(x, y)$:
```javascript
function electricPotential(rho, r, numPoints) {
let potential = 0;
for (let x = 0; x <= 1; x += 1 / numPoints) {
for (let y = 0; y <= 1; y += 1 / numPoints) {
let rPrime = [x, y];
let rMinusRPrime = [r[0] - x, r[1] - y];
let distance = Math.sqrt(rMinusRPrime[0] * rMinusRPrime[0] + rMinusRPrime[1] * rMinusRPrime[1]);
potential += rho(rPrime) / (4 * Math.PI * 8.854e-12 * distance);
}
}
return potential;
}
function rho(r) {
return r[0] * r[1];
}
console.log(electricPotential(rho, [0.5, 0.5], 1000));
```
This code defines a function `electricPotential` that takes a charge distribution `rho` and a point `r`, and calculates the electric potential at `r` using Green's theorem. The function then returns the value of the potential.
# Conclusion and further resources
Green's theorem is a powerful tool in vector calculus that allows us to calculate line integrals in the plane. It has many applications in mathematics, physics, engineering, and mathematical modeling, including in the study of complex analysis, computer graphics, and differential equations.
In this textbook, we have covered the Cauchy-Riemann equations, the use of Green's theorem to solve line integrals, solving line integrals using JavaScript, applications of Green's theorem in computer graphics, visualizing complex functions using JavaScript, the relationship between partial derivatives and Green's theorem, solving differential equations using Green's theorem, the Stokes' theorem as a generalization of Green's theorem, Green's theorem in physics: electric and magnetic fields, and the connection between Green's theorem and the concept of partial derivatives.
Further resources for learning more about Green's theorem and its applications include:
- "Vector Calculus" by Jerrold E. Marsden and Anthony J. S. Smith
- "Introduction to Electrodynamics" by David J. Griffiths
- "Introduction to Fluid Mechanics" by Frank M. White
These resources provide a more in-depth understanding of the mathematical concepts and their applications in various fields. | Textbooks |
Incomplete articles, Deadman Mode, Alternate game mode, Jagex controversies
Deadman Mode
(unf)
This section or article is incomplete and could do with improvement.
Reason: Article would benefit from a section detailing complications with the execution of Deadman Mode events.
Deadman Mode is both an annually permanent and a monthly seasonal quarterly temporary variant of Old School RuneScape that released on the 29th of October 2015. A separate server is hosted featuring an open PvP environment, with some exceptions. Players in-game can speak to Nigel in Lumbridge graveyard for information about Deadman Mode.
Experience gained in Deadman mode is 5x more than usual (10x during the first six hours of gameplay, and 20x during the first 30 minutes); however, experience that is gained from quests will not be multiplied. In addition, experience will not be gained in instanced areas (such as the Nightmare Zone and TzHaar Fight Caves).
Every player begins their adventure at combat level 3, and progress in Old School RuneScape, or any other game (RuneScape 3, RuneScape Classic, etc.), will not be affected. All content is only available to pay-to-play players.
Upon killing a player, they may receive blood money. One is received per kill, and can be used to purchase Deadman armour from Nigel in the Lumbridge graveyard. It has the same bonuses as iron armour. A total of three are required to purchase the Deadman's chest, Deadman's legs, and Deadman's cape. These can be reclaimed for free should a player die.
A map showing the safe and multicombat zones in Deadman Mode
Deadman worlds
The following worlds are classified as servers for Deadman mode:
List of Worlds
Every few months, Jagex opens a brand new Deadman Seasonal server, in which the top 2,000 players will become eligible to participate in the Deadman Invitational.
The Invitational server is yet another brand new server in which every player starts over from the beginning, and have less than a week to build their accounts up to the final hour of the Invitational, in which a deadly fog spreads over the area, and the safe zone is broadcasted in game, forcing remaining players to a confined area to fight it out.
The final survivor then wins a $20,000 cash prize, and a brand new deadman season begins when announced. The list of dates and seasons can be found below:
Invitational date
N/A N/A 16 March 2016 21 March 2016 YaNeverLearn
I 26 March 2016 15 June 2016 20 June 2016 On Codeine
II 25 June 2016 22 September 2016 26 September 2016 LOLOLOLOLOL
III 1 October 2016 3 November 2016 12 December 2016 RoT NiKo
IV 17 December 2016 26 January 2017 20 March 2017 LIIIIDBIIIIT
V 25 March 2017 27 April 2017 26 June 2017 ANONYMOUSE00
VI (Autumn 2017)[1] 1 July 2017 27 July 2017 18 September 2017 True Fox[2]
VII (Winter 2017) 24 September 2017 26 October 2017 4 December 2017 Mankedupmage[3]
VIII (Spring 2018) 9 December 2017 11 January 2018 9 March 2018 ThEnADxcG
IX (Summer 2018) 17 March 2018 19 April 2018 23 June 2018 DMMfreakshow
X (Autumn 2018) 30 June 2018 26 July 2018 8 September 2018 sePieiboult
XI (Winter 2018) 15 September 2018 11 October 2018
↑ After the 5th Deadman season, Jagex stopped naming seasons numerically, instead using the season and year the invitational would take place.
↑ 5PLUS50K12 was the winner of the Deadman final, but was disqualified(Source). True Fox is the runner up.
↑ Psych was the winner of the Deadman final, but was disqualified for botting. Mankedupmage is the runner up.
Disabled content
Protect Item
Fight Pits
Rat Pits
Minigame Teleports
STASH units
Bolt pouch
Scrying pool
Town protection
Towns and villages are "safe" zones. Players cannot attack other players within the area. However, players who are standing outside of a safe zone are still able to attack players if they are within range of their attacks, whether or not they are standing in a safe zone.
Any player who enters these safe zones when skulled will be attacked on sight by level 1337 guards. These guards can hit in the high 40s through prayer and are able to cast Ice Barrage and Tele Block. In addition, players frozen by guards cannot pick up or telegrab items.
Tutorial Island and the Barbarian Assault minigame area are the only areas in Deadman mode where players are unable to attack others. Safe zones include:
Barbarian Assault (only within the minigame itself)
Catherby (only within the bank)
Jatizso (only within the town area)
Neitiznot (only within the town area)
Seers' Village (only within the bank)
Tutorial Island
Skull status
The skulls that appear on a player's head, indicating the amount of bank keys in their possession.
In Deadman mode, being skulled will cause level 1337 guards in low-threat areas to cast Ice Barrage, Tele Block, and attack you, being able to hit through prayer, and are unable to pick up or telegrab items while frozen.
Skulled players will not be able to teleport or log out instantly, and a seven second timer will count down in the chatbox. Players must not move or be in combat for seven seconds. After the timer is depleted, they will automatically teleport or log out.
In addition, if players are in possession of bank keys, the skull will indicate the amount of keys that are in a player's possession. Players who attack other players with a skull will not become skulled themselves.
A skull will last for fifteen minutes, but will not deplete while in an instanced area or when standing in the same place for an extended period of time. The game HUD will have a timer, updating every half minute, to notify skulled players how long they will stay skulled.
Players who obtain a skull will have it last for fifteen minutes, and for every unskulled player they attack, two minutes will be added to the skull timer. However, attacking another player whose combat level is 30 levels lower will result in the timer increasing to 30 minutes.
Dying in Deadman mode varies, depending on how the player died. If killed by another player, 10 of the most valuable items will be taken by the killer in the form of a bank key, in addition to any items that were in the victim's inventory.
When a skulled player is killed by an NPC, the 10 most valuable stack of items from their bank are dropped on the floor instead. These items will appear instantly to everyone. In addition, untradeable items (such as Void Knight equipment or a fire cape) will convert to coins upon death.
However, if a player dies to an NPC without being skulled, it acts as a normal death akin to that of Old School RuneScape.
A player selecting which skills to preserve when they die.
Completed quests and Achievement Diaries will retain completion upon death. Players who lose certain levels may also be unable to access certain post-quest content. For example, if Monkey Madness I is completed, and lose the Defence experience upon death, players will be unable to travel to Ape Atoll without the proper stats. However, a dragon scimitar can still be equipped, providing the player has an Attack level of 60.
Players who die to players will lose experience in accordance to the combat level differences. The exact formula is:
$ \frac{1}{1 + e^y} $
$ y=\frac{CombatLevel (killer)-\frac{121}{125}*CombatLevel (victim)-40}{10} $
Players can choose to preserve a maximum of five skills (two combat skills and three non-combat skills) to prevent experience loss. However, a player's Herblore level will never be less than 3, provided they have completed Druidic Ritual, and a player's Hitpoints level will never be less than 10.
Bank raiding and insurance
A player taking items from another players bank.
Players who are killed by another player or killed by an NPC while skulled will drop a bank key, in which 10 of the most valuable items, based on Grand Exchange value, are removed from your bank. The key will automatically appear in the inventory of the PKer, if they have the space for it.
The player that obtained the key can collect the items by opening a chest in a bank within a low-threat area, which means they must wait till their skull expires or risk being killed by the guards. These bank keys cannot be deposited into a bank, and only five bank keys can be in a player's possession at any given time. Bank keys look identical once in the inventory and cannot be examined to discover who dropped them. Once obtained, they cannot be dropped, only destroyed.
A deadman chest, which requires a bank key to open.
The Safe Deposit Box, accessed by visiting a Financial Wizard.
If a player kills a player who is in possession of another bank key, they will drop their own bank key in addition to their victim's bank keys, with higher valued keys taking priority. If a skulled player dies to a guard, 10 of their most valuable items from the bank are dropped wherever they die. However, if a player does not have enough inventory space to receive a key, it will be dropped to the floor. This will only happen until the limit of 5 keys is reached. Both keys in a player's inventory and on the floor count towards this limit.
As stated earlier, skulled players will be attacked by level 1337 guards if they entered a protected zone. Getting killed by these guards will lead to a 10 percent loss of experience in protected skills, in addition to the regular penalties.
Players can insure up to ten items (ten single items, not 10 stacks of items) by speaking to the Financial Wizard and depositing the items in the safe deposit box. Items in the safe deposit box will not be lost, unlike those found in the bank, and can be collected any time the player visits the Financial Wizard at almost any bank in a low risk zone.
Hitpoints insurance
The Hitpoints skill can be separately insured by talking to Gelin in the Lumbridge graveyard. In order to get the insurance, a player must pay a fee. After paying the fee, a player's Hitpoints level will never fall below it when they die.
Level 25 - 25,000 coins
Level 50 - 100,000 coins
Level 75 - 1,000,000 coins
Rule breaking
The Rules of RuneScape apply to Deadman mode as well. Players may not attempt to use "mule" accounts, which are accounts used solely to store items safely. Mod Mat K has stated during one of the weekly Q&A sessions that Mod Weath has a tool that is able to show him the people with the most expensive items, and if Weath notices a random account with lots of wealth on it, it will most likely be banned.
There was a glitch on 24 April 2018 where the permanent Deadman Mode world was temporarily swapped for scheduled maintenance, however Deadman Mode mechanics were prematurely applied on the new Deadman Mode world and players were subject to Deadman Mode conditions that affected their main server accounts. Jagex attempted to restore items and experience lost by the glitch.
Retrieved from "https://oldschoolrunescape.fandom.com/wiki/Deadman_Mode?oldid=8693434"
Alternate game mode
Jagex controversies | CommonCrawl |
Variance-stabilizing transformation
In applied statistics, a variance-stabilizing transformation is a data transformation that is specifically chosen either to simplify considerations in graphical exploratory data analysis or to allow the application of simple regression-based or analysis of variance techniques.[1]
Overview
The aim behind the choice of a variance-stabilizing transformation is to find a simple function ƒ to apply to values x in a data set to create new values y = ƒ(x) such that the variability of the values y is not related to their mean value. For example, suppose that the values x are realizations from different Poisson distributions: i.e. the distributions each have different mean values μ. Then, because for the Poisson distribution the variance is identical to the mean, the variance varies with the mean. However, if the simple variance-stabilizing transformation
$y={\sqrt {x}}\,$
is applied, the sampling variance associated with observation will be nearly constant: see Anscombe transform for details and some alternative transformations.
While variance-stabilizing transformations are well known for certain parametric families of distributions, such as the Poisson and the binomial distribution, some types of data analysis proceed more empirically: for example by searching among power transformations to find a suitable fixed transformation. Alternatively, if data analysis suggests a functional form for the relation between variance and mean, this can be used to deduce a variance-stabilizing transformation.[2] Thus if, for a mean μ,
$\operatorname {var} (X)=h(\mu ),\,$
a suitable basis for a variance stabilizing transformation would be
$y\propto \int ^{x}{\frac {1}{\sqrt {h(\mu )}}}\,d\mu ,$
where the arbitrary constant of integration and an arbitrary scaling factor can be chosen for convenience.
Example: relative variance
If X is a positive random variable and the variance is given as h(μ) = s2μ2 then the standard deviation is proportional to the mean, which is called fixed relative error. In this case, the variance-stabilizing transformation is
$y=\int ^{x}{\frac {d\mu }{\sqrt {s^{2}\mu ^{2}}}}={\frac {1}{s}}\ln(x)\propto \log(x)\,.$
That is, the variance-stabilizing transformation is the logarithmic transformation.
Example: absolute plus relative variance
If the variance is given as h(μ) = σ2 + s2μ2 then the variance is dominated by a fixed variance σ2 when |μ| is small enough and is dominated by the relative variance s2μ2 when |μ| is large enough. In this case, the variance-stabilizing transformation is
$y=\int ^{x}{\frac {d\mu }{\sqrt {\sigma ^{2}+s^{2}\mu ^{2}}}}={\frac {1}{s}}\operatorname {asinh} {\frac {x}{\sigma /s}}\propto \operatorname {asinh} {\frac {x}{\lambda }}\,.$
That is, the variance-stabilizing transformation is the inverse hyperbolic sine of the scaled value x / λ for λ = σ / s.
Relationship to the delta method
Here, the delta method is presented in a rough way, but it is enough to see the relation with the variance-stabilizing transformations. To see a more formal approach see delta method.
Let $X$ be a random variable, with $E[X]=\mu $ and $\operatorname {Var} (X)=\sigma ^{2}$. Define $Y=g(X)$, where $g$ is a regular function. A first order Taylor approximation for $Y=g(x)$ is:
$Y=g(X)\approx g(\mu )+g'(\mu )(X-\mu )$
From the equation above, we obtain:
$E[Y]=g(\mu )$ and $\operatorname {Var} [Y]=\sigma ^{2}g'(\mu )^{2}$
This approximation method is called delta method.
Consider now a random variable $X$ such that $E[X]=\mu $ and $\operatorname {Var} [X]=h(\mu )$. Notice the relation between the variance and the mean, which implies, for example, heteroscedasticity in a linear model. Therefore, the goal is to find a function $g$ such that $Y=g(X)$ has a variance independent (at least approximately) of its expectation.
Imposing the condition $\operatorname {Var} [Y]\approx h(\mu )g'(\mu )^{2}={\text{constant}}$, this equality implies the differential equation:
${\frac {dg}{d\mu }}={\frac {C}{\sqrt {h(\mu )}}}$
This ordinary differential equation has, by separation of variables, the following solution:
$g(\mu )=\int {\frac {C\,d\mu }{\sqrt {h(\mu )}}}$
This last expression appeared for the first time in a M. S. Bartlett paper.[3]
References
1. Everitt, B. S. (2002). The Cambridge Dictionary of Statistics (2nd ed.). CUP. ISBN 0-521-81099-X.
2. Dodge, Y. (2003). The Oxford Dictionary of Statistical Terms. OUP. ISBN 0-19-920613-9.
3. Bartlett, M. S. (1947). "The Use of Transformations". Biometrics. 3: 39–52. doi:10.2307/3001536.
| Wikipedia |
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How easy is it to actually track another person's credit card? $$\left[\begin{array}{ccc}-6 & -6 & 10 \\-5 & -5 & 5 \\-9 & -9 & 13\end{array}\right]$$ GN Gennady N. Jump to Question. Lactic fermentation related question: Is there a relationship between pH, salinity, fermentation magic, and heat? If for two matrices \(N\) and \(M\) there exists a matrix \(P\) such that \(M=P^{-1}NP\), then we say that \(M\) and \(N\) are \(\textit{similar}\). And they're the eigenvectors that correspond to eigenvalue lambda is equal to 3. eigenvectors of a system are not unique, but the ratio of their elements is. EXERCISES: For each given matrix, nd the eigenvalues, and for each eigenvalue give a basis of the corresponding eigenspace. We verify that given vectors are eigenvectors of a linear transformation T and find matrix representation of T with respect to the basis of these eigenvectors. 1 & 0 & 0 \\ Making statements based on opinion; back them up with references or personal experience. T=\left(\begin{array}{ccc} Considering a three-dimensional state space spanned by the orthonormal basis formed by the three kets $|u_1\rangle,|u_2\rangle,|u_3\rangle $. Independence of eigenvectors when no repeated eigenvalue is defective We now deal with the case in which some of the eigenvalues are repeated. Eigenvectors, eigenvalues and orthogonality Before we go on to matrices, consider what a vector is. The eigenspace for lambda is equal to 3, is equal to the span, all of the potential linear combinations of this guy and that guy. A square matrix \(M\) is diagonalizable if and only if there exists a basis of eigenvectors for \(M\). -8 & -2 & -1 \\ Have questions or comments? It remains to prove (i) ) (iii). \vdots&&\ddots&\vdots \\ Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. How do I give a basis of eigenvectors common to H and B? Diagonal Matrix with N eigenvectors Diagonal matrices make calculations really easy. In the basis of these three vectors, taken in order, are Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The corresponding values of v that satisfy the equation are the right eigenvectors. 9 & 18 & 29 \\ with $\omega_0$ and $b$ real constants. Converting 3-gang electrical box to single, How to move a servo quickly and without delay function, How to animate particles spraying on an object. $$\left[\begin{array}{lll}1 & 0 & 1 \\0 & 3 & 2 \\0 & 0 & 2\end{array}\right]$$ Problem 8. Also note that according to the fact above, the two eigenvectors should be linearly independent. We will now need to find the eigenvectors for each of these. Thus a basis of eigenvectors would be: { (2, 3), (3, -2)} 2. The basis and vector components. Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. and so will commute with $H$ on that subspace that $H$ on that subspace is (up to a scalar) the unit matrix. 3 & 0 & 1 \\ The values of λ that satisfy the equation are the eigenvalues. Since, for $H$, $\lambda_2 = \lambda_3$, any linear combination of their eigenvectors is also an eigenvector. Math 113: Linear Algebra Eigenvectors and Eigenvalues Ilya Sherman November 3, 2008 1 Recap Recall that last time, we proved: Theorem 1.1. Find an cigenbasis (a basis of eigenvectors) and diagonalize. So if you apply the matrix transformation to any of these vectors, you're just going to scale them up by 3. I will proceed here in a di erent manner from what I explained (only partially) in class. In the new basis of eigenvectors \(S'(v_{1},\ldots,v_{n})\), the matrix \(D\) of \(L\) is diagonal because \(Lv_{i}=\lambda_{i} v_{i}\) and so, \[ The main ingredient is the following proposition. an orthonormal basis of real eigenvectors and Ais orthogonal similar to a real diagonal matrix = P 1AP where P = PT. In fact, for all hypothetical lines in our original basis space, the only vectors that remain on their original lines after the transformation A are those on the green and yellow lines.. MathJax reference. \[M=\begin{pmatrix} Given such a basis of eigenvectors, the key idea for using them is: 1.Take any vector xand expand it in this basis: x= c 1x 1 + c mx n, or x= Xcor c= X 1xwhere X is the matrix whose columns are the eigenvectors. Need help with derivation, Freedom in choosing elements/entries of an eigenvector. Proof Ais Hermitian so by the previous proposition, it has real eigenvalues. "Question closed" notifications experiment results and graduation, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…. \begin{pmatrix} \]. What is the application of `rev` in real life? $$H=\hbar\omega_0 \left( \begin{array}{ccc} The prefix eigen- is adopted from the German word eigen (cognate with the English word own) for "proper", "characteristic", "own". I know that an orthonormal basis van be constructed for any hermitian matrix consisting only of the eigenvectors of the matrix. We can set the equation to zero, and obtain the homogeneous equation. (Show the details.) To subscribe to this RSS feed, copy and paste this URL into your RSS reader. If \(P\) is the change of basis matrix from \(S\) to \(S'\), the diagonal matrix of eigenvalues \(D\) and the original matrix are related by \(D=P^{-1}MP\). We would know Ais unitary similar to a real diagonal matrix, but the unitary matrix need not be real in general. And 1/2, 0, 1. 2. Notice that the matrix, \[P=\begin{pmatrix}v_{1} & v_{2} & v_{3}\end{pmatrix}=\begin{pmatrix} -7 & -14 & -23 \\ 0 & 0 & 0 \\ A vector is a matrix with a single column. A basis of a vector space is a set of vectors in that is linearly independent and spans .An ordered basis is a list, rather than a set, meaning that the order of the vectors in an ordered basis matters. To find the eigenvectors we simply plug in each eigenvalue into . Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Let T: V !V be a linear transformation. 0 & 0 & 2 \\ The eigenvalue problem is to determine the solution to the equation Av = λv, where A is an n -by- n matrix, v is a column vector of length n, and λ is a scalar. Legal. Find an cigenbasis (a basis of eigenvectors) and diagonalize. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. In general, you can skip parentheses, but be very careful: e^3x is `e^3x`, and e^(3x) is `e^(3x)`. 0 & 0 & 2 \\ 1 & 0 & 0 \\ Let A=[121−1412−40]. This is important with respect to the topics discussed in this post. \end{pmatrix}.\], David Cherney, Tom Denton, and Andrew Waldron (UC Davis). The equation quite clearly shows that eigenvectors of "A" are those vectors that "A" only stretches or compresses, but doesn't affect their directions. no degeneracy), then its eigenvectors form a `complete set' of unit vectors (i.e a complete 'basis') –Proof: M orthonormal vectors must span an M-dimensional space. Use MathJax to format equations. We would like to determine the eigenvalues and eigenvectors for T. To do this we will x a basis B= b 1; ;b n. The eigenvalues are scalars and the eigenvectors are elements of V so the nal answer does not depend on the basis. 1. Are there eight or four independent solutions of the Dirac equation? 1&0&0 \\ I'm new to chess-what should be done here to win the game? Can you use the Eldritch Blast cantrip on the same turn as the UA Lurker in the Deep warlock's Grasp of the Deep feature? By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Since L:V\to V, most likely you already know the matrix M of L using the same input basis as output basis S= (u_ {1},\ldots ,u_ {n}) (say). The corresponding values of v that satisfy the equation are the right eigenvectors. Originally used to study principal axes of the rotational motion of rigid bodies, eigenvalues and eigenvectors have a wide range of applications, for example in stability analysis, vibration analysis, atomic orbitals, facial recognition, and matrix diagonalization. MP=\begin{pmatrix}Mv_{1} &Mv_{2}& Mv_{3}\end{pmatrix}=\begin{pmatrix}-1.v_{1}&0.v_{2}&2.v_{3}\end{pmatrix}=\begin{pmatrix}v_{1}& v_{2} & v_{3}\end{pmatrix}\begin{pmatrix} Definition : The set of all solutions to or equivalently is called the eigenspace of "A" corresponding to "l ". That is, $\left\{\left[{-4 \atop 1}\right]\right\}$ is a basis of the eigenspace corresponding to $\lambda_1 =3$. 0&0&\cdots&\lambda_{n}\end{pmatrix}\, . For more information contact us at [email protected] or check out our status page at https://status.libretexts.org. 0&T_{32}&T_{33}\end{array}\right) In the basis of these three vectors, taken in order, are defined the operators Did China's Chang'e 5 land before November 30th 2020? One way is by finding eigenvectors of an arbitrary linear combination of $H$ and $B$, say $\alpha H + \beta B$. Example # 1: Find a basis for the eigenspace corresponding to l = 1, 5. Missed the LibreFest? -1 & 1 & -1 \\ Griffiths use of a linear transformation on basis vectors. Basis of Eigenvectors. These topics have not been very well covered in the handbook, but are important from an examination point of view. one point of finding eigenvectors is to find a matrix "similar" to the original that can be written diagonally (only the diagonal has nonzeroes), based on a different basis. Then the above discussion shows that diagonalizable matrices are similar to diagonal matrices. It is sufficient to find the eigenstates of $B$ in the subspace spanned by $\vert 2\rangle=\left(\begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right)$ and $\vert 3\rangle=\left(\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right)$. Thanks for contributing an answer to Physics Stack Exchange! Do MEMS accelerometers have a lower frequency limit? To learn more, see our tips on writing great answers. \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), 13.3: Changing to a Basis of Eigenvectors, [ "article:topic", "authortag:waldron", "authorname:waldron", "showtoc:no" ], \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), David Cherney, Tom Denton, & Andrew Waldron. 0 & 0 & 0 \\ This is a quick write up on eigenvectors, eigenvalues, orthogonality and the like. Is there a way to notate the repeat of a larger section that itself has repeats in it? Does "Ich mag dich" only apply to friendship? Completeness of Eigenvectors of a Hermitian operator •THEOREM: If an operator in an M-dimensional Hilbert space has M distinct eigenvalues (i.e. So, let's do that. \[P^{-1}MP=\begin{pmatrix} Which of the four inner planets has the strongest magnetic field, Mars, Mercury, Venus, or Earth? Change of basis rearranges the components of a vector by the change of basis matrix \(P\), to give components in the new basis. In general, you can skip the multiplication sign, so `5x` is equivalent to `5*x`. Eigenvectors, values, etc. rev 2020.12.2.38097, The best answers are voted up and rise to the top, Physics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. For the others, try: $|u_2\rangle \pm |u_3\rangle$. Where did the concept of a (fantasy-style) "dungeon" originate? Should we leave technical astronomy questions to Astronomy SE? So 1/2, 1, 0. Moreover, because the columns of \(P\) are the components of eigenvectors, \[ Eigenvalues and eigenvectors have immense applications in the physical sciences, especially quantum mechanics, among other fields. Since \(L:V\to V\), most likely you already know the matrix \(M\) of \(L\) using the same input basis as output basis \(S=(u_{1},\ldots ,u_{n})\) (say). site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. We know that $H$ and $B$ commute,that is $$[H,B]=0$$. {\displaystyle A} acts on {\displaystyle \mathbf {x} } is complicated, but there are certain cases where the action maps to the same vector, multiplied by a scalar factor. (Show the details) 2-4 1 A 02 0 0 010 15. The eigenvalues of the matrix A are λ.-4, λ,-5, and λ.-6. Thus, we have found an orthonormal basis of eigenvectors for A. Asking for help, clarification, or responding to other answers. The matrix A has an eigenvalue 2. Setters dependent on other instance variables in Java. 3. -1 & 0 & 0 \\ The basis is arbitrary, as long as you have enough vectors in it and they're linearly independent. Watch the recordings here on Youtube! Therefore, the eigenvectors of \(M\) form a basis of \(\Re\), and so \(M\) is diagonalizable. Find a basis of the eigenspace E2 corresponding to the eigenvalue 2. Proposition 2. Show Instructions. 0 & 0 & 1 \\ A matrix \(M\) is diagonalizable if there exists an invertible matrix \(P\) and a diagonal matrix \(D\) such that. These are called our eigenvectors and the points that fall on the lines before the transformations are moved along them (think of them as sorts of axes), by a factor shown below– our eigenvalues $$ To get the matrix of a linear transformation in the new basis, we \(\textit{conjugate}\) the matrix of \(L\) by the change of basis matrix: \(M\mapsto P^{-1}MP\). By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy. $|u_1\rangle$ is a no brainer. UC Berkeley Math 54 lecture: Basis of Eigenvectors Instructor: Peter Koroteev. Let me write this way. (The Ohio State University, Linear Algebra Final Exam Problem) Add to solve later Sponsored Links B=b\left( \begin{array}{ccc} Any symmetric matrix A has an eigenvector. Find an eigenbasis (a basis of eigenvectors) and diagonalize. \({\lambda _{\,1}} = - 5\) : In this case we need to solve the following system. The eigenstates of $B$ in that subspace will automatically also be eigenstates of $H$ because the similarity transformation $T$ that will diagonalize $B$ will be of the generic form which corresponds to this value is called an eigenvector. This is the hardest and most interesting part. How to avoid boats on a mainly oceanic world? The eigenvalue problem is to determine the solution to the equation Av = λv, where A is an n -by- n matrix, v is a column vector of length n, and λ is a scalar. (Show the details.) and solve. 0 & 0 & -1 \end{array} \right) \qquad Eigenvectors, on the other hand, are properties of a linear transformation on that vector space. -14 & -28 & -44 \\ We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Theory of Complex Spectra, Applying Slater-Condon Rules, Matrix operations on Quantum States in a composite quantum system. If a linear transformation affects some non-zero vector only by scalar multiplication, that vector is an eigenvector of that transformation. nbe the standard basis vectors, i.e., for all i, e i(j) = (1; if i= j 0; otherwise. It only takes a minute to sign up. Moreover, these eigenvectors are the columns of the change of basis matrix \(P\) which diagonalizes \(M\). All eigenvectors corresponding to $\lambda_1 =3$ are multiples of $\left[{-4 \atop 1}\right] $ and thus the eigenspace corresponding to $\lambda_1 =3$ is given by the span of $\left[{-4 \atop 1}\right] $. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. If we are changing to a basis of eigenvectors, then there are various simplifications: -1 & 0 & 0 \\ If V is a finite dimensional vector space over C and T: V → V, then it always has an eigenvector, and if the characteristic polynomial (det(λId−T)) has distinct roots, thenthere is a basis for V of eigenvectors. Is it possible to just construct a simple cable serial↔︎serial and send data from PC to C64? 0&T_{22}&T_{23} \\ 0&\lambda_{2}&&0\\ 0 & 1 & 0 \end{array} \right) $$ 0 & -1 & 0 \\ Yes, that says that y= 0. \], Hence, the matrix \(P\) of eigenvectors is a change of basis matrix that diagonalizes \(M\): The corresponding eigenvectors are xi- … \end{pmatrix}.\], The eigenvalues of \(M\) are determined by \[\det(M-\lambda I)=-\lambda^{3}+\lambda^{2}+2\lambda=0.\], So the eigenvalues of \(M\) are \(-1,0,\) and \(2\), and associated eigenvectors turn out to be, \[v_{1}=\begin{pmatrix}-8 \\ -1 \\ 3\end{pmatrix},~~ v_{2}=\begin{pmatrix}-2 \\ 1 \\ 0\end{pmatrix}, {\rm ~and~~} v_{3}=\begin{pmatrix}-1 \\ -1 \\ 1\end{pmatrix}.$$, In order for \(M\) to be diagonalizable, we need the vectors \(v_{1}, v_{2}, v_{3}\) to be linearly independent. 2. \end{pmatrix}\, . How do I orient myself to the literature concerning a topic of research and not be overwhelmed? These three eigenvectors form a basis for the space of all vectors, that is, a vector can be written as a linear combination of the eigenvectors, and for any choice of the entries, and. \end{pmatrix}\]. Can the automatic damage from the Witch Bolt spell be repeatedly activated using an Order of Scribes wizard's Manifest Mind feature? If we are changing to a basis of eigenvectors, then there are various simplifications: 1. \lambda_{1}&0&\cdots&0\\ $$ A coordinate system given by eigenvectors is known as an eigenbasis, it can be written as a diagonal matrix since it scales each basis vector by a certain value. \big(L(v_{1}),L(v_{2}),\ldots,L(v_{n})\big)=(v_{1},v_{2},\ldots, v_{n}) The calculator will find the eigenvalues and eigenvectors (eigenspace) of the given square matrix, with steps shown. is invertible because its determinant is \(-1\). One thing I missed in the article is mention of a basis of eigenvectors. The values of λ that satisfy the equation are the eigenvalues. 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And paste this URL into Your RSS reader been very well covered the. Be done here to win the game plug in each eigenvalue give basis!, |u_3\rangle $ up on eigenvectors, on the other hand, are properties of a system not! Some non-zero vector basis of eigenvectors by scalar multiplication, that is $ $ [ H, B ] =0 $ [. What a vector is a question and answer site for active researchers, academics and students of physics has strongest... An eigenbasis ( a basis of eigenvectors of the matrix a are Î », -5, and »... |U_2\Rangle \pm |u_3\rangle $ Mercury, Venus, or responding to other answers nd... Is defective we now deal with the case in which some of the that..., Applying Slater-Condon Rules, matrix operations on quantum States in a di manner! Of research and not be real in general $, any linear combination of eigenvectors. Question and answer site for active researchers, academics and students of.! Using an Order of Scribes wizard 's Manifest Mind feature, $ \lambda_2 = \lambda_3 $, $ =. Are the eigenvalues, orthogonality and the like need not be overwhelmed: if an operator in an M-dimensional space! By 3 on a mainly oceanic world simple cable serial↔︎serial and send data from PC C64... Fact above, the two eigenvectors should be linearly independent \,1 } } = - 5\:. To matrices, consider what a vector is an eigenvector, that vector is a quick write up on,! Solutions of the matrix transformation to any of these, 1525057, and heat planets the... Is arbitrary, as long as you have enough vectors in it and they're linearly independent the handbook but! See our tips on writing great answers what i explained ( only partially ) in class »,. Which some of the corresponding values of Î ».-6 distinct eigenvalues ( i.e originate! Licensed under CC by-sa learn more, see our tips on writing great answers Rules, matrix operations quantum... Where did the concept of a Hermitian operator •THEOREM: if an operator in an M-dimensional space. Other hand, are properties of a system are not unique, but the unitary matrix need be! Obtain the homogeneous equation '' corresponding to `` l `` diagonalizable matrices are similar to diagonal.! Important from an examination point of view basis matrix \ ( P\ ) which diagonalizes \ ( \lambda! In the handbook, but the ratio of their eigenvectors is also an eigenvector 2., clarification, or responding to other answers combination of their eigenvectors also... You have enough vectors in it and they're linearly independent PC to C64, consider what vector. And B dungeon '' originate we simply plug in each eigenvalue into in an M-dimensional Hilbert has... November 30th 2020 of all solutions to or equivalently is called the eigenspace corresponding to `` l.. Or four independent solutions of the eigenvectors of a larger section that itself has repeats in it and they're independent! Otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0 ( iii ) and they the!, orthogonality and the like orient myself to the topics discussed in this post wizard 's Manifest Mind?... Our tips on writing great answers the other hand, are properties of a linear transformation only )... Eigenbasis ( a basis for the others, try: $ |u_2\rangle \pm |u_3\rangle $ to eigenvalue is... Consider what a vector is a question and answer site for active,... ( i ) ) ( iii ) eigenvalue give a basis of eigenvectors, eigenvalues and orthogonality Before we on. For contributing an answer to physics Stack Exchange Inc ; user contributions licensed under CC.. And heat exists a basis of the change of basis matrix \ ( -1\ ), $. November 30th 2020 making statements based on opinion ; back them up with references or personal.! Set of all solutions to or equivalently is called the eigenspace E2 corresponding to l =,! Case in which some of the eigenspace of `` a '' corresponding to `` ``! 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The two eigenvectors should be linearly independent design / logo © 2020 Stack Exchange basis of eigenvectors ; user contributions licensed CC... », -5, and heat eigenvectors diagonal matrices make calculations really easy some non-zero vector only scalar. To 3 $ $ into Your RSS reader China 's Chang ' e 5 land Before 30th. Notate the repeat of a larger section that itself has repeats in it and they're linearly independent \lambda. Elements/Entries of an eigenvector data from PC to C64 an cigenbasis ( basis! Foundation support under grant numbers 1246120, 1525057, and for each given matrix nd... Diagonal matrices make calculations really easy our terms of service, privacy policy cookie! @ libretexts.org or check out our status page at https: //status.libretexts.org will proceed here in a di manner! Contributions licensed under CC by-sa this case we need to solve the following system and?! Basis for the eigenspace corresponding to l = 1, 5 definition: the set all! Logo © 2020 Stack Exchange Inc ; user contributions licensed under CC by-sa examination point of view orthonormal van. '' only apply to friendship 5x ` is equivalent to ` 5 * x ` fermentation magic, and each! Is equivalent to ` 5 * x ` Before we go on to matrices, consider a. Rss reader 5x ` is equivalent to ` 5 * x ` the analysis of linear transformations to! A vector is can set the equation to zero, and Î ».-6 you to. Contributions licensed under CC by-sa: v! v be a linear transformation of,! On quantum States in a composite quantum system we simply plug in each eigenvalue into question is... Activated using an Order of Scribes wizard 's Manifest Mind feature quantum mechanics, among other fields M-dimensional space! Design / logo © 2020 Stack Exchange is a question and answer site for researchers... Or responding to other answers eigenvalues of the corresponding values of Î » that satisfy equation... Clicking "Post Your Answer", you can skip the multiplication sign, so ` 5x is. An orthonormal basis of eigenvectors ) and diagonalize v that satisfy the equation are the eigenvalues eigenvectors. Transformation on basis vectors, that is $ $ the eigenspace E2 corresponding l! Are the eigenvalues and eigenvectors have immense applications in the analysis of linear transformations real in general a! It and they're linearly independent other answers to avoid boats on a mainly oceanic?. Are the eigenvalues are repeated only apply to friendship |u_1\rangle, |u_2\rangle, |u_3\rangle $ to of... Quick write up on eigenvectors, then there are various simplifications:..
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Introduction to general relativity
General relativity is a theory of gravitation developed by Albert Einstein between 1907 and 1915. The theory of general relativity says that the observed gravitational effect between masses results from their warping of spacetime.
This article is a non-technical introduction to the subject. For the main encyclopedia article, see General relativity.
General relativity
$G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }$
• Introduction
• History
• Timeline
• Tests
• Mathematical formulation
Fundamental concepts
• Equivalence principle
• Special relativity
• World line
• Pseudo-Riemannian manifold
Phenomena
• Kepler problem
• Gravitational lensing
• Gravitational waves
• Frame-dragging
• Geodetic effect
• Event horizon
• Singularity
• Black hole
Spacetime
• Spacetime diagrams
• Minkowski spacetime
• Einstein–Rosen bridge
• Equations
• Formalisms
Equations
• Linearized gravity
• Einstein field equations
• Friedmann
• Geodesics
• Mathisson–Papapetrou–Dixon
• Hamilton–Jacobi–Einstein
Formalisms
• ADM
• BSSN
• Post-Newtonian
Advanced theory
• Kaluza–Klein theory
• Quantum gravity
Solutions
• Schwarzschild (interior)
• Reissner–Nordström
• Gödel
• Kerr
• Kerr–Newman
• Kasner
• Lemaître–Tolman
• Taub–NUT
• Milne
• Robertson–Walker
• Oppenheimer-Snyder
• pp-wave
• van Stockum dust
• Weyl−Lewis−Papapetrou
Scientists
• Einstein
• Lorentz
• Hilbert
• Poincaré
• Schwarzschild
• de Sitter
• Reissner
• Nordström
• Weyl
• Eddington
• Friedman
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• Wheeler
• Robertson
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• Chandrasekhar
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• Penrose
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By the beginning of the 20th century, Newton's law of universal gravitation had been accepted for more than two hundred years as a valid description of the gravitational force between masses. In Newton's model, gravity is the result of an attractive force between massive objects. Although even Newton was troubled by the unknown nature of that force, the basic framework was extremely successful at describing motion.
Experiments and observations show that Einstein's description of gravitation accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the orbits of Mercury and other planets. General relativity also predicts novel effects of gravity, such as gravitational waves, gravitational lensing and an effect of gravity on time known as gravitational time dilation. Many of these predictions have been confirmed by experiment or observation, most recently gravitational waves.
General relativity has developed into an essential tool in modern astrophysics. It provides the foundation for the current understanding of black holes, regions of space where the gravitational effect is strong enough that even light cannot escape. Their strong gravity is thought to be responsible for the intense radiation emitted by certain types of astronomical objects (such as active galactic nuclei or microquasars). General relativity is also part of the framework of the standard Big Bang model of cosmology.
Although general relativity is not the only relativistic theory of gravity, it is the simplest such theory that is consistent with the experimental data. Nevertheless, a number of open questions remain, the most fundamental of which is how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity.
From special to general relativity
In September 1905, Albert Einstein published his theory of special relativity, which reconciles Newton's laws of motion with electrodynamics (the interaction between objects with electric charge). Special relativity introduced a new framework for all of physics by proposing new concepts of space and time. Some then-accepted physical theories were inconsistent with that framework; a key example was Newton's theory of gravity, which describes the mutual attraction experienced by bodies due to their mass.
Several physicists, including Einstein, searched for a theory that would reconcile Newton's law of gravity and special relativity. Only Einstein's theory proved to be consistent with experiments and observations. To understand the theory's basic ideas, it is instructive to follow Einstein's thinking between 1907 and 1915, from his simple thought experiment involving an observer in free fall to his fully geometric theory of gravity.[1]
Equivalence principle
A person in a free-falling elevator experiences weightlessness; objects either float motionless or drift at constant speed. Since everything in the elevator is falling together, no gravitational effect can be observed. In this way, the experiences of an observer in free fall are indistinguishable from those of an observer in deep space, far from any significant source of gravity. Such observers are the privileged ("inertial") observers Einstein described in his theory of special relativity: observers for whom light travels along straight lines at constant speed.[2]
Einstein hypothesized that the similar experiences of weightless observers and inertial observers in special relativity represented a fundamental property of gravity, and he made this the cornerstone of his theory of general relativity, formalized in his equivalence principle. Roughly speaking, the principle states that a person in a free-falling elevator cannot tell that they are in free fall. Every experiment in such a free-falling environment has the same results as it would for an observer at rest or moving uniformly in deep space, far from all sources of gravity.[3]
Gravity and acceleration
Most effects of gravity vanish in free fall, but effects that seem the same as those of gravity can be produced by an accelerated frame of reference. An observer in a closed room cannot tell which of the following is true:
• Objects are falling to the floor because the room is resting on the surface of the Earth and the objects are being pulled down by gravity.
• Objects are falling to the floor because the room is aboard a rocket in space, which is accelerating at 9.81 m/s2, the standard gravity on Earth, and is far from any source of gravity. The objects are being pulled towards the floor by the same "inertial force" that presses the driver of an accelerating car into the back of their seat.
Conversely, any effect observed in an accelerated reference frame should also be observed in a gravitational field of corresponding strength. This principle allowed Einstein to predict several novel effects of gravity in 1907, as explained in the next section.
An observer in an accelerated reference frame must introduce what physicists call fictitious forces to account for the acceleration experienced by the observer and objects around them. In the example of the driver being pressed into their seat, the force felt by the driver is one example; another is the force one can feel while pulling the arms up and out if attempting to spin around like a top. Einstein's master insight was that the constant, familiar pull of the Earth's gravitational field is fundamentally the same as these fictitious forces.[4] The apparent magnitude of the fictitious forces always appears to be proportional to the mass of any object on which they act – for instance, the driver's seat exerts just enough force to accelerate the driver at the same rate as the car. By analogy, Einstein proposed that an object in a gravitational field should feel a gravitational force proportional to its mass, as embodied in Newton's law of gravitation.[5]
Physical consequences
In 1907, Einstein was still eight years away from completing the general theory of relativity. Nonetheless, he was able to make a number of novel, testable predictions that were based on his starting point for developing his new theory: the equivalence principle.[6]
The first new effect is the gravitational frequency shift of light. Consider two observers aboard an accelerating rocket-ship. Aboard such a ship, there is a natural concept of "up" and "down": the direction in which the ship accelerates is "up", and unattached objects accelerate in the opposite direction, falling "downward". Assume that one of the observers is "higher up" than the other. When the lower observer sends a light signal to the higher observer, the acceleration causes the light to be red-shifted, as may be calculated from special relativity; the second observer will measure a lower frequency for the light than the first. Conversely, light sent from the higher observer to the lower is blue-shifted, that is, shifted towards higher frequencies.[7] Einstein argued that such frequency shifts must also be observed in a gravitational field. This is illustrated in the figure at left, which shows a light wave that is gradually red-shifted as it works its way upwards against the gravitational acceleration. This effect has been confirmed experimentally, as described below.
This gravitational frequency shift corresponds to a gravitational time dilation: Since the "higher" observer measures the same light wave to have a lower frequency than the "lower" observer, time must be passing faster for the higher observer. Thus, time runs more slowly for observers who are lower in a gravitational field.
It is important to stress that, for each observer, there are no observable changes of the flow of time for events or processes that are at rest in his or her reference frame. Five-minute-eggs as timed by each observer's clock have the same consistency; as one year passes on each clock, each observer ages by that amount; each clock, in short, is in perfect agreement with all processes happening in its immediate vicinity. It is only when the clocks are compared between separate observers that one can notice that time runs more slowly for the lower observer than for the higher.[8] This effect is minute, but it too has been confirmed experimentally in multiple experiments, as described below.
In a similar way, Einstein predicted the gravitational deflection of light: in a gravitational field, light is deflected downward. Quantitatively, his results were off by a factor of two; the correct derivation requires a more complete formulation of the theory of general relativity, not just the equivalence principle.[9]
Tidal effects
The equivalence between gravitational and inertial effects does not constitute a complete theory of gravity. When it comes to explaining gravity near our own location on the Earth's surface, noting that our reference frame is not in free fall, so that fictitious forces are to be expected, provides a suitable explanation. But a freely falling reference frame on one side of the Earth cannot explain why the people on the opposite side of the Earth experience a gravitational pull in the opposite direction.
A more basic manifestation of the same effect involves two bodies that are falling side by side towards the Earth. In a reference frame that is in free fall alongside these bodies, they appear to hover weightlessly – but not exactly so. These bodies are not falling in precisely the same direction, but towards a single point in space: namely, the Earth's center of gravity. Consequently, there is a component of each body's motion towards the other (see the figure). In a small environment such as a freely falling lift, this relative acceleration is minuscule, while for skydivers on opposite sides of the Earth, the effect is large. Such differences in force are also responsible for the tides in the Earth's oceans, so the term "tidal effect" is used for this phenomenon.
The equivalence between inertia and gravity cannot explain tidal effects – it cannot explain variations in the gravitational field.[10] For that, a theory is needed which describes the way that matter (such as the large mass of the Earth) affects the inertial environment around it.
From acceleration to geometry
In exploring the equivalence of gravity and acceleration as well as the role of tidal forces, Einstein discovered several analogies with the geometry of surfaces. An example is the transition from an inertial reference frame (in which free particles coast along straight paths at constant speeds) to a rotating reference frame (in which extra terms corresponding to fictitious forces have to be introduced in order to explain particle motion): this is analogous to the transition from a Cartesian coordinate system (in which the coordinate lines are straight lines) to a curved coordinate system (where coordinate lines need not be straight).
A deeper analogy relates tidal forces with a property of surfaces called curvature. For gravitational fields, the absence or presence of tidal forces determines whether or not the influence of gravity can be eliminated by choosing a freely falling reference frame. Similarly, the absence or presence of curvature determines whether or not a surface is equivalent to a plane. In the summer of 1912, inspired by these analogies, Einstein searched for a geometric formulation of gravity.[11]
The elementary objects of geometry – points, lines, triangles – are traditionally defined in three-dimensional space or on two-dimensional surfaces. In 1907, Hermann Minkowski, Einstein's former mathematics professor at the Swiss Federal Polytechnic, introduced Minkowski space, a geometric formulation of Einstein's special theory of relativity where the geometry included not only space but also time. The basic entity of this new geometry is four-dimensional spacetime. The orbits of moving bodies are curves in spacetime; the orbits of bodies moving at constant speed without changing direction correspond to straight lines.[12]
The geometry of general curved surfaces was developed in the early 19th century by Carl Friedrich Gauss. This geometry had in turn been generalized to higher-dimensional spaces in Riemannian geometry introduced by Bernhard Riemann in the 1850s. With the help of Riemannian geometry, Einstein formulated a geometric description of gravity in which Minkowski's spacetime is replaced by distorted, curved spacetime, just as curved surfaces are a generalization of ordinary plane surfaces. Embedding Diagrams are used to illustrate curved spacetime in educational contexts.[13][14]
After he had realized the validity of this geometric analogy, it took Einstein a further three years to find the missing cornerstone of his theory: the equations describing how matter influences spacetime's curvature. Having formulated what are now known as Einstein's equations (or, more precisely, his field equations of gravity), he presented his new theory of gravity at several sessions of the Prussian Academy of Sciences in late 1915, culminating in his final presentation on November 25, 1915.[15]
Geometry and gravitation
Paraphrasing John Wheeler, Einstein's geometric theory of gravity can be summarized thus: spacetime tells matter how to move; matter tells spacetime how to curve.[16] What this means is addressed in the following three sections, which explore the motion of so-called test particles, examine which properties of matter serve as a source for gravity, and, finally, introduce Einstein's equations, which relate these matter properties to the curvature of spacetime.
Probing the gravitational field
In order to map a body's gravitational influence, it is useful to think about what physicists call probe or test particles: particles that are influenced by gravity, but are so small and light that we can neglect their own gravitational effect. In the absence of gravity and other external forces, a test particle moves along a straight line at a constant speed. In the language of spacetime, this is equivalent to saying that such test particles move along straight world lines in spacetime. In the presence of gravity, spacetime is non-Euclidean, or curved, and in curved spacetime straight world lines may not exist. Instead, test particles move along lines called geodesics, which are "as straight as possible", that is, they follow the shortest path between starting and ending points, taking the curvature into consideration.
A simple analogy is the following: In geodesy, the science of measuring Earth's size and shape, a geodesic (from Greek "geo", Earth, and "daiein", to divide) is the shortest route between two points on the Earth's surface. Approximately, such a route is a segment of a great circle, such as a line of longitude or the equator. These paths are certainly not straight, simply because they must follow the curvature of the Earth's surface. But they are as straight as is possible subject to this constraint.
The properties of geodesics differ from those of straight lines. For example, on a plane, parallel lines never meet, but this is not so for geodesics on the surface of the Earth: for example, lines of longitude are parallel at the equator, but intersect at the poles. Analogously, the world lines of test particles in free fall are spacetime geodesics, the straightest possible lines in spacetime. But still there are crucial differences between them and the truly straight lines that can be traced out in the gravity-free spacetime of special relativity. In special relativity, parallel geodesics remain parallel. In a gravitational field with tidal effects, this will not, in general, be the case. If, for example, two bodies are initially at rest relative to each other, but are then dropped in the Earth's gravitational field, they will move towards each other as they fall towards the Earth's center.[17]
Compared with planets and other astronomical bodies, the objects of everyday life (people, cars, houses, even mountains) have little mass. Where such objects are concerned, the laws governing the behavior of test particles are sufficient to describe what happens. Notably, in order to deflect a test particle from its geodesic path, an external force must be applied. A chair someone is sitting on applies an external upwards force preventing the person from falling freely towards the center of the Earth and thus following a geodesic, which they would otherwise be doing without matter in between them and the center of the Earth. In this way, general relativity explains the daily experience of gravity on the surface of the Earth not as the downwards pull of a gravitational force, but as the upwards push of external forces. These forces deflect all bodies resting on the Earth's surface from the geodesics they would otherwise follow.[18] For matter objects whose own gravitational influence cannot be neglected, the laws of motion are somewhat more complicated than for test particles, although it remains true that spacetime tells matter how to move.[19]
Sources of gravity
In Newton's description of gravity, the gravitational force is caused by matter. More precisely, it is caused by a specific property of material objects: their mass. In Einstein's theory and related theories of gravitation, curvature at every point in spacetime is also caused by whatever matter is present. Here, too, mass is a key property in determining the gravitational influence of matter. But in a relativistic theory of gravity, mass cannot be the only source of gravity. Relativity links mass with energy, and energy with momentum.
The equivalence between mass and energy, as expressed by the formula E = mc2, is the most famous consequence of special relativity. In relativity, mass and energy are two different ways of describing one physical quantity. If a physical system has energy, it also has the corresponding mass, and vice versa. In particular, all properties of a body that are associated with energy, such as its temperature or the binding energy of systems such as nuclei or molecules, contribute to that body's mass, and hence act as sources of gravity.[20]
In special relativity, energy is closely connected to momentum. Just as space and time are, in that theory, different aspects of a more comprehensive entity called spacetime, energy and momentum are merely different aspects of a unified, four-dimensional quantity that physicists call four-momentum. In consequence, if energy is a source of gravity, momentum must be a source as well. The same is true for quantities that are directly related to energy and momentum, namely internal pressure and tension. Taken together, in general relativity it is mass, energy, momentum, pressure and tension that serve as sources of gravity: they are how matter tells spacetime how to curve. In the theory's mathematical formulation, all these quantities are but aspects of a more general physical quantity called the energy–momentum tensor.[21]
Einstein's equations
Einstein's equations are the centerpiece of general relativity. They provide a precise formulation of the relationship between spacetime geometry and the properties of matter, using the language of mathematics. More concretely, they are formulated using the concepts of Riemannian geometry, in which the geometric properties of a space (or a spacetime) are described by a quantity called a metric. The metric encodes the information needed to compute the fundamental geometric notions of distance and angle in a curved space (or spacetime).
A spherical surface like that of the Earth provides a simple example. The location of any point on the surface can be described by two coordinates: the geographic latitude and longitude. Unlike the Cartesian coordinates of the plane, coordinate differences are not the same as distances on the surface, as shown in the diagram on the right: for someone at the equator, moving 30 degrees of longitude westward (magenta line) corresponds to a distance of roughly 3,300 kilometers (2,100 mi), while for someone at a latitude of 55 degrees, moving 30 degrees of longitude westward (blue line) covers a distance of merely 1,900 kilometers (1,200 mi). Coordinates therefore do not provide enough information to describe the geometry of a spherical surface, or indeed the geometry of any more complicated space or spacetime. That information is precisely what is encoded in the metric, which is a function defined at each point of the surface (or space, or spacetime) and relates coordinate differences to differences in distance. All other quantities that are of interest in geometry, such as the length of any given curve, or the angle at which two curves meet, can be computed from this metric function.[22]
The metric function and its rate of change from point to point can be used to define a geometrical quantity called the Riemann curvature tensor, which describes exactly how the Riemannian manifold, the spacetime in the theory of relativity, is curved at each point. As has already been mentioned, the matter content of the spacetime defines another quantity, the energy–momentum tensor T, and the principle that "spacetime tells matter how to move, and matter tells spacetime how to curve" means that these quantities must be related to each other. Einstein formulated this relation by using the Riemann curvature tensor and the metric to define another geometrical quantity G, now called the Einstein tensor, which describes some aspects of the way spacetime is curved. Einstein's equation then states that
$\mathbf {G} ={\frac {8\pi G}{c^{4}}}\mathbf {T} ,$
i.e., up to a constant multiple, the quantity G (which measures curvature) is equated with the quantity T (which measures matter content). Here, G is the gravitational constant of Newtonian gravity, and c is the speed of light from special relativity.
This equation is often referred to in the plural as Einstein's equations, since the quantities G and T are each determined by several functions of the coordinates of spacetime, and the equations equate each of these component functions.[23] A solution of these equations describes a particular geometry of spacetime; for example, the Schwarzschild solution describes the geometry around a spherical, non-rotating mass such as a star or a black hole, whereas the Kerr solution describes a rotating black hole. Still other solutions can describe a gravitational wave or, in the case of the Friedmann–Lemaître–Robertson–Walker solution, an expanding universe. The simplest solution is the uncurved Minkowski spacetime, the spacetime described by special relativity.[24]
Experiments
No scientific theory is self-evidently true; each is a model that must be checked by experiment. Newton's law of gravity was accepted because it accounted for the motion of planets and moons in the Solar System with considerable accuracy. As the precision of experimental measurements gradually improved, some discrepancies with Newton's predictions were observed, and these were accounted for in the general theory of relativity. Similarly, the predictions of general relativity must also be checked with experiment, and Einstein himself devised three tests now known as the classical tests of the theory:
• Newtonian gravity predicts that the orbit which a single planet traces around a perfectly spherical star should be an ellipse. Einstein's theory predicts a more complicated curve: the planet behaves as if it were travelling around an ellipse, but at the same time, the ellipse as a whole is rotating slowly around the star. In the diagram on the right, the ellipse predicted by Newtonian gravity is shown in red, and part of the orbit predicted by Einstein in blue. For a planet orbiting the Sun, this deviation from Newton's orbits is known as the anomalous perihelion shift. The first measurement of this effect, for the planet Mercury, dates back to 1859. The most accurate results for Mercury and for other planets to date are based on measurements which were undertaken between 1966 and 1990, using radio telescopes.[25] General relativity predicts the correct anomalous perihelion shift for all planets where this can be measured accurately (Mercury, Venus and the Earth).
• According to general relativity, light does not travel along straight lines when it propagates in a gravitational field. Instead, it is deflected in the presence of massive bodies. In particular, starlight is deflected as it passes near the Sun, leading to apparent shifts of up 1.75 arc seconds in the stars' positions in the sky (an arc second is equal to 1/3600 of a degree). In the framework of Newtonian gravity, a heuristic argument can be made that leads to light deflection by half that amount. The different predictions can be tested by observing stars that are close to the Sun during a solar eclipse. In this way, a British expedition to West Africa in 1919, directed by Arthur Eddington, confirmed that Einstein's prediction was correct, and the Newtonian predictions wrong, via observation of the May 1919 eclipse. Eddington's results were not very accurate; subsequent observations of the deflection of the light of distant quasars by the Sun, which utilize highly accurate techniques of radio astronomy, have confirmed Eddington's results with significantly better precision (the first such measurements date from 1967, the most recent comprehensive analysis from 2004).[26]
• Gravitational redshift was first measured in a laboratory setting in 1959 by Pound and Rebka. It is also seen in astrophysical measurements, notably for light escaping the white dwarf Sirius B. The related gravitational time dilation effect has been measured by transporting atomic clocks to altitudes of between tens and tens of thousands of kilometers (first by Hafele and Keating in 1971; most accurately to date by Gravity Probe A launched in 1976).[27]
Of these tests, only the perihelion advance of Mercury was known prior to Einstein's final publication of general relativity in 1916. The subsequent experimental confirmation of his other predictions, especially the first measurements of the deflection of light by the sun in 1919, catapulted Einstein to international stardom.[28] These three experiments justified adopting general relativity over Newton's theory and, incidentally, over a number of alternatives to general relativity that had been proposed.
Further tests of general relativity include precision measurements of the Shapiro effect or gravitational time delay for light, measured in 2002 by the Cassini space probe. One set of tests focuses on effects predicted by general relativity for the behavior of gyroscopes travelling through space. One of these effects, geodetic precession, has been tested with the Lunar Laser Ranging Experiment (high-precision measurements of the orbit of the Moon). Another, which is related to rotating masses, is called frame-dragging. The geodetic and frame-dragging effects were both tested by the Gravity Probe B satellite experiment launched in 2004, with results confirming relativity to within 0.5% and 15%, respectively, as of December 2008.[29]
By cosmic standards, gravity throughout the solar system is weak. Since the differences between the predictions of Einstein's and Newton's theories are most pronounced when gravity is strong, physicists have long been interested in testing various relativistic effects in a setting with comparatively strong gravitational fields. This has become possible thanks to precision observations of binary pulsars. In such a star system, two highly compact neutron stars orbit each other. At least one of them is a pulsar – an astronomical object that emits a tight beam of radiowaves. These beams strike the Earth at very regular intervals, similarly to the way that the rotating beam of a lighthouse means that an observer sees the lighthouse blink, and can be observed as a highly regular series of pulses. General relativity predicts specific deviations from the regularity of these radio pulses. For instance, at times when the radio waves pass close to the other neutron star, they should be deflected by the star's gravitational field. The observed pulse patterns are impressively close to those predicted by general relativity.[30]
One particular set of observations is related to eminently useful practical applications, namely to satellite navigation systems such as the Global Positioning System that are used for both precise positioning and timekeeping. Such systems rely on two sets of atomic clocks: clocks aboard satellites orbiting the Earth, and reference clocks stationed on the Earth's surface. General relativity predicts that these two sets of clocks should tick at slightly different rates, due to their different motions (an effect already predicted by special relativity) and their different positions within the Earth's gravitational field. In order to ensure the system's accuracy, either the satellite clocks are slowed down by a relativistic factor, or that same factor is made part of the evaluation algorithm. In turn, tests of the system's accuracy (especially the very thorough measurements that are part of the definition of universal coordinated time) are testament to the validity of the relativistic predictions.[31]
A number of other tests have probed the validity of various versions of the equivalence principle; strictly speaking, all measurements of gravitational time dilation are tests of the weak version of that principle, not of general relativity itself. So far, general relativity has passed all observational tests.[32]
Astrophysical applications
Models based on general relativity play an important role in astrophysics; the success of these models is further testament to the theory's validity.
Gravitational lensing
Since light is deflected in a gravitational field, it is possible for the light of a distant object to reach an observer along two or more paths. For instance, light of a very distant object such as a quasar can pass along one side of a massive galaxy and be deflected slightly so as to reach an observer on Earth, while light passing along the opposite side of that same galaxy is deflected as well, reaching the same observer from a slightly different direction. As a result, that particular observer will see one astronomical object in two different places in the night sky. This kind of focussing is well known when it comes to optical lenses, and hence the corresponding gravitational effect is called gravitational lensing.[33]
Observational astronomy uses lensing effects as an important tool to infer properties of the lensing object. Even in cases where that object is not directly visible, the shape of a lensed image provides information about the mass distribution responsible for the light deflection. In particular, gravitational lensing provides one way to measure the distribution of dark matter, which does not give off light and can be observed only by its gravitational effects. One particularly interesting application are large-scale observations, where the lensing masses are spread out over a significant fraction of the observable universe, and can be used to obtain information about the large-scale properties and evolution of our cosmos.[34]
Gravitational waves
Gravitational waves, a direct consequence of Einstein's theory, are distortions of geometry that propagate at the speed of light, and can be thought of as ripples in spacetime. They should not be confused with the gravity waves of fluid dynamics, which are a different concept.
In February 2016, the Advanced LIGO team announced that they had directly observed gravitational waves from a black hole merger.[35]
Indirectly, the effect of gravitational waves had been detected in observations of specific binary stars. Such pairs of stars orbit each other and, as they do so, gradually lose energy by emitting gravitational waves. For ordinary stars like the Sun, this energy loss would be too small to be detectable, but this energy loss was observed in 1974 in a binary pulsar called PSR1913+16. In such a system, one of the orbiting stars is a pulsar. This has two consequences: a pulsar is an extremely dense object known as a neutron star, for which gravitational wave emission is much stronger than for ordinary stars. Also, a pulsar emits a narrow beam of electromagnetic radiation from its magnetic poles. As the pulsar rotates, its beam sweeps over the Earth, where it is seen as a regular series of radio pulses, just as a ship at sea observes regular flashes of light from the rotating light in a lighthouse. This regular pattern of radio pulses functions as a highly accurate "clock". It can be used to time the double star's orbital period, and it reacts sensitively to distortions of spacetime in its immediate neighborhood.
The discoverers of PSR1913+16, Russell Hulse and Joseph Taylor, were awarded the Nobel Prize in Physics in 1993. Since then, several other binary pulsars have been found. The most useful are those in which both stars are pulsars, since they provide accurate tests of general relativity.[36]
Currently, a number of land-based gravitational wave detectors are in operation, and a mission to launch a space-based detector, LISA, is currently under development, with a precursor mission (LISA Pathfinder) which was launched in 2015. Gravitational wave observations can be used to obtain information about compact objects such as neutron stars and black holes, and also to probe the state of the early universe fractions of a second after the Big Bang.[37]
Black holes
When mass is concentrated into a sufficiently compact region of space, general relativity predicts the formation of a black hole – a region of space with a gravitational effect so strong that not even light can escape. Certain types of black holes are thought to be the final state in the evolution of massive stars. On the other hand, supermassive black holes with the mass of millions or billions of Suns are assumed to reside in the cores of most galaxies, and they play a key role in current models of how galaxies have formed over the past billions of years.[38]
Matter falling onto a compact object is one of the most efficient mechanisms for releasing energy in the form of radiation, and matter falling onto black holes is thought to be responsible for some of the brightest astronomical phenomena imaginable. Notable examples of great interest to astronomers are quasars and other types of active galactic nuclei. Under the right conditions, falling matter accumulating around a black hole can lead to the formation of jets, in which focused beams of matter are flung away into space at speeds near that of light.[39]
There are several properties that make black holes the most promising sources of gravitational waves. One reason is that black holes are the most compact objects that can orbit each other as part of a binary system; as a result, the gravitational waves emitted by such a system are especially strong. Another reason follows from what are called black-hole uniqueness theorems: over time, black holes retain only a minimal set of distinguishing features (these theorems have become known as "no-hair" theorems), regardless of the starting geometric shape. For instance, in the long term, the collapse of a hypothetical matter cube will not result in a cube-shaped black hole. Instead, the resulting black hole will be indistinguishable from a black hole formed by the collapse of a spherical mass. In its transition to a spherical shape, the black hole formed by the collapse of a more complicated shape will emit gravitational waves.[40]
Cosmology
One of the most important aspects of general relativity is that it can be applied to the universe as a whole. A key point is that, on large scales, our universe appears to be constructed along very simple lines: all current observations suggest that, on average, the structure of the cosmos should be approximately the same, regardless of an observer's location or direction of observation: the universe is approximately homogeneous and isotropic. Such comparatively simple universes can be described by simple solutions of Einstein's equations. The current cosmological models of the universe are obtained by combining these simple solutions to general relativity with theories describing the properties of the universe's matter content, namely thermodynamics, nuclear- and particle physics. According to these models, our present universe emerged from an extremely dense high-temperature state – the Big Bang – roughly 14 billion years ago and has been expanding ever since.[41]
Einstein's equations can be generalized by adding a term called the cosmological constant. When this term is present, empty space itself acts as a source of attractive (or, less commonly, repulsive) gravity. Einstein originally introduced this term in his pioneering 1917 paper on cosmology, with a very specific motivation: contemporary cosmological thought held the universe to be static, and the additional term was required for constructing static model universes within the framework of general relativity. When it became apparent that the universe is not static, but expanding, Einstein was quick to discard this additional term. Since the end of the 1990s, however, astronomical evidence indicating an accelerating expansion consistent with a cosmological constant – or, equivalently, with a particular and ubiquitous kind of dark energy – has steadily been accumulating.[42]
Modern research
General relativity is very successful in providing a framework for accurate models which describe an impressive array of physical phenomena. On the other hand, there are many interesting open questions, and in particular, the theory as a whole is almost certainly incomplete.[43]
In contrast to all other modern theories of fundamental interactions, general relativity is a classical theory: it does not include the effects of quantum physics. The quest for a quantum version of general relativity addresses one of the most fundamental open questions in physics. While there are promising candidates for such a theory of quantum gravity, notably string theory and loop quantum gravity, there is at present no consistent and complete theory. It has long been hoped that a theory of quantum gravity would also eliminate another problematic feature of general relativity: the presence of spacetime singularities. These singularities are boundaries ("sharp edges") of spacetime at which geometry becomes ill-defined, with the consequence that general relativity itself loses its predictive power. Furthermore, there are so-called singularity theorems which predict that such singularities must exist within the universe if the laws of general relativity were to hold without any quantum modifications. The best-known examples are the singularities associated with the model universes that describe black holes and the beginning of the universe.[44]
Other attempts to modify general relativity have been made in the context of cosmology. In the modern cosmological models, most energy in the universe is in forms that have never been detected directly, namely dark energy and dark matter. There have been several controversial proposals to remove the need for these enigmatic forms of matter and energy, by modifying the laws governing gravity and the dynamics of cosmic expansion, for example modified Newtonian dynamics.[45]
Beyond the challenges of quantum effects and cosmology, research on general relativity is rich with possibilities for further exploration: mathematical relativists explore the nature of singularities and the fundamental properties of Einstein's equations,[46] and ever more comprehensive computer simulations of specific spacetimes (such as those describing merging black holes) are run.[47] More than one hundred years after the theory was first published, research is more active than ever.[48]
See also
• General relativity
• Introduction to the mathematics of general relativity
• Special relativity
• History of general relativity
• Tests of general relativity
• Numerical relativity
• Derivations of the Lorentz transformations
• List of books on general relativity
References
1. This development is traced e.g. in Renn 2005, p. 110ff., in chapters 9 through 15 of Pais 1982, and in Janssen 2005. A precis of Newtonian gravity can be found in Schutz 2003, chapters 2–4. It is impossible to say whether the problem of Newtonian gravity crossed Einstein's mind before 1907, but, by his own admission, his first serious attempts to reconcile that theory with special relativity date to that year, cf. Pais 1982, p. 178.
2. This is described in detail in chapter 2 of Wheeler 1990.
3. While the equivalence principle is still part of modern expositions of general relativity, there are some differences between the modern version and Einstein's original concept, cf. Norton 1985.
4. E. g. Janssen 2005, p. 64f. Einstein himself also explains this in section XX of his non-technical book Einstein 1961. Following earlier ideas by Ernst Mach, Einstein also explored centrifugal forces and their gravitational analogue, cf. Stachel 1989.
5. Einstein explained this in section XX of Einstein 1961. He considered an object "suspended" by a rope from the ceiling of a room aboard an accelerating rocket: from inside the room it looks as if gravitation is pulling the object down with a force proportional to its mass, but from outside the rocket it looks as if the rope is simply transferring the acceleration of the rocket to the object, and must therefore exert just the "force" to do so.
6. More specifically, Einstein's calculations, which are described in chapter 11b of Pais 1982, use the equivalence principle, the equivalence of gravity and inertial forces, and the results of special relativity for the propagation of light and for accelerated observers (the latter by considering, at each moment, the instantaneous inertial frame of reference associated with such an accelerated observer).
7. This effect can be derived directly within special relativity, either by looking at the equivalent situation of two observers in an accelerated rocket-ship or by looking at a falling elevator; in both situations, the frequency shift has an equivalent description as a Doppler shift between certain inertial frames. For simple derivations of this, see Harrison 2002.
8. See chapter 12 of Mermin 2005.
9. Cf. Ehlers & Rindler 1997; for a non-technical presentation, see Pössel 2007.
10. These and other tidal effects are described in Wheeler 1990, pp. 83–91.
11. Tides and their geometric interpretation are explained in chapter 5 of Wheeler 1990. This part of the historical development is traced in Pais 1982, section 12b.
12. For elementary presentations of the concept of spacetime, see the first section in chapter 2 of Thorne 1994, and Greene 2004, p. 47–61. More complete treatments on a fairly elementary level can be found e.g. in Mermin 2005 and in Wheeler 1990, chapters 8 and 9.
13. Marolf, Donald (1999). "Spacetime Embedding Diagrams for Black Holes". General Relativity and Gravitation. 31 (6): 919–944. arXiv:gr-qc/9806123. Bibcode:1999GReGr..31..919M. doi:10.1023/A:1026646507201. S2CID 12502462.
14. See Wheeler 1990, chapters 8 and 9 for vivid illustrations of curved spacetime.
15. Einstein's struggle to find the correct field equations is traced in chapters 13–15 of Pais 1982.
16. E.g. p. xi in Wheeler 1990.
17. A thorough, yet accessible account of basic differential geometry and its application in general relativity can be found in Geroch 1978.
18. See chapter 10 of Wheeler 1990.
19. In fact, when starting from the complete theory, Einstein's equation can be used to derive these more complicated laws of motion for matter as a consequence of geometry, but deriving from this the motion of idealized test particles is a highly non-trivial task, cf. Poisson 2004.
20. A simple explanation of mass–energy equivalence can be found in sections 3.8 and 3.9 of Giulini 2005.
21. See chapter 6 of Wheeler 1990.
22. For a more detailed definition of the metric, but one that is more informal than a textbook presentation, see chapter 14.4 of Penrose 2004.
23. The geometrical meaning of Einstein's equations is explored in chapters 7 and 8 of Wheeler 1990; cf. box 2.6 in Thorne 1994. An introduction using only very simple mathematics is given in chapter 19 of Schutz 2003.
24. The most important solutions are listed in every textbook on general relativity; for a (technical) summary of our current understanding, see Friedrich 2005.
25. More precisely, these are VLBI measurements of planetary positions; see chapter 5 of Will 1993 and section 3.5 of Will 2006.
26. For the historical measurements, see Hartl 2005, Kennefick 2005, and Kennefick 2007; Soldner's original derivation in the framework of Newton's theory is von Soldner 1804. For the most precise measurements to date, see Bertotti 2005.
27. See Kennefick 2005 and chapter 3 of Will 1993. For the Sirius B measurements, see Trimble & Barstow 2007.
28. Pais 1982, Mercury on pp. 253–254, Einstein's rise to fame in sections 16b and 16c.
29. Everitt, C.W.F.; Parkinson, B.W. (2009), Gravity Probe B Science Results—NASA Final Report (PDF), retrieved 2009-05-02
30. Kramer 2004.
31. An accessible account of relativistic effects in the global positioning system can be found in Ashby 2002; details are given in Ashby 2003.
32. An accessible introduction to tests of general relativity is Will 1993; a more technical, up-to-date account is Will 2006.
33. The geometry of such situations is explored in chapter 23 of Schutz 2003.
34. Introductions to gravitational lensing and its applications can be found on the webpages Newbury 1997 and Lochner 2007.
35. B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2016). "Observation of Gravitational Waves from a Binary Black Hole Merger". Physical Review Letters. 116 (6): 061102. arXiv:1602.03837. Bibcode:2016PhRvL.116f1102A. doi:10.1103/PhysRevLett.116.061102. PMID 26918975. S2CID 124959784.{{cite journal}}: CS1 maint: uses authors parameter (link)
36. Schutz 2003, pp. 317–321; Bartusiak 2000, pp. 70–86.
37. The ongoing search for gravitational waves is described in Bartusiak 2000 and in Blair & McNamara 1997.
38. For an overview of the history of black hole physics from its beginnings in the early 20th century to modern times, see the very readable account by Thorne 1994. For an up-to-date account of the role of black holes in structure formation, see Springel et al. 2005; a brief summary can be found in the related article Gnedin 2005.
39. See chapter 8 of Sparke & Gallagher 2007 and Disney 1998. A treatment that is more thorough, yet involves only comparatively little mathematics can be found in Robson 1996.
40. An elementary introduction to the black hole uniqueness theorems can be found in Chrusciel 2006 and in Thorne 1994, pp. 272–286.
41. Detailed information can be found in Ned Wright's Cosmology Tutorial and FAQ, Wright 2007; a very readable introduction is Hogan 1999. Using undergraduate mathematics but avoiding the advanced mathematical tools of general relativity, Berry 1989 provides a more thorough presentation.
42. Einstein's original paper is Einstein 1917; good descriptions of more modern developments can be found in Cowen 2001 and Caldwell 2004.
43. Cf. Maddox 1998, pp. 52–59 and 98–122; Penrose 2004, section 34.1 and chapter 30.
44. With a focus on string theory, the search for quantum gravity is described in Greene 1999; for an account from the point of view of loop quantum gravity, see Smolin 2001.
45. For dark matter, see Milgrom 2002; for dark energy, Caldwell 2004
46. See Friedrich 2005.
47. A review of the various problems and the techniques being developed to overcome them, see Lehner 2002.
48. A good starting point for a snapshot of present-day research in relativity is the electronic review journal Living Reviews in Relativity.
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| Wikipedia |
\begin{document}
\title{The $p$-spectral radius of the Laplacian}
\begin{abstract} The $p$-spectral radius of a graph $G=(V,E)$ with adjacency matrix $A$ is defined as $\lambda^{(p)}(G)=\max \{x^TAx : \|x\|_p=1 \}$. This parameter shows remarkable connections with graph invariants, and has been used to generalize some extremal problems. In this work, we extend this approach to the Laplacian matrix $L$, and define the $p$-spectral radius of the Laplacian as $\mu^{(p)}(G)=\max \{x^TLx : \|x\|_p=1 \}$. We show that $\mu^{(p)}(G)$ relates to invariants such as maximum degree and size of a maximum cut. We also show properties of $\mu^{(p)}(G)$ as a function of $p$, and a upper bound on $\max_{G \colon |V(G)|=n} \mu^{(p)}(G)$ in terms of $n=|V|$ for $p\ge 2$, which is attained if $n$ is even.
\noindent \textbf{Keywords:} Laplacian Matrix, p-spectral radius
\end{abstract}
\section{Introduction and main results}
Let $G=(V,E)$ be a simple $n$-vertex graph at least one edge with adjacency matrix $A$ and Laplacian matrix $L$. We recall that $L=D-A$, where $D$ is the diagonal matrix of vertex degrees.
It is well known that obtaining the least and the largest eigenvalues ($\lambda_1$ and $\lambda_n$, respectively) of a real symmetric matrix $M\in\mathbb{R}^{n\times n}$ can be viewed as an optimization problem using the Rayleigh-Ritz Theorem \cite[Theorem 4.2.2]{HornJo}:
$$\lambda_1(M)=\min_{\|x\|=1}x^TMx\le\frac{x^TMx}{x^Tx}\le\max_{\|x\|=1}x^TMx=\lambda_n,$$
where $x\in\mathbb{R}^n$. Using the fact that $x^TAx=2\sum_{ij\in E} x_ix_j$, Keevash, Lenz and Mubayi \cite{KLM14} replaced the Euclidean norm $\|x\|$ by the $p$-norm $\|x\|_p$, where $p \in [1,\infty]$, and defined the \emph{$p$-spectral radius} $\lambda^{(p)}(G)$:
$$\lambda_p(G)=\max_{\|x\|_p=1}2\sum_{ij\in E} x_ix_j.$$
This parameter shows remarkable connections with some graph invariants. For instance, $\lambda^{(1)}(G)$ is equal to the Lagrangian $\mathfrak{L}_G$ of $G$, which was defined by Motzkin and Straus \cite{MotStr65} and satisfies $2\mathfrak{L}_G-1=1/\omega(G)$, where $\omega(G)$ is the clique number of $G$. Obviously $\lambda^{(2)}(G)$ is the usual spectral radius, and it can be shown that $\lambda^{(\infty)}(G)/2$ is equal to the number of edges of $G$.
An interesting result involving this parameter is about \emph{$K_r$-free graphs}, that is, graphs that do not contain a complete graph with $r$ vertices as a subgraph. Tur\'an \cite{Tur41} proved that, for all positive integers $n$ and $r$, the balanced complete $r$-partite graph, known as a \emph{Tur\'an graph} $T_r(n)$, is the only graph with maximum number of edges among all $K_{r+1}$-free graphs of order $n$. Kang and Nikiforov \cite{KangNiki14} proved that, for $p\ge 1$, the graph $T_r(n)$ is also the only graph that maximizes $\lambda^{(p)}(G)$ over $K_{r+1}$-free graphs of order $n$, thus generalizing Tur\'an's result (which is the case $p=\infty$). Other results were obtained and extended to hypergraphs \cite{Niki14}.
This motivates us to extend this approach to the Laplacian matrix $L$, replacing the Euclidean norm by the $p$-norm. As $x^TLx=\sum_{ij\in E} (x_i-x_j)^2$, we define the $p$-spectral radius of the Laplacian as follows:
\begin{definition} Let $G=(V,E)$. The $p$-spectral radius of the Laplacian matrix of $G$ is given by
$$\mu^{(p)}(G)=\max_{\|x\|_p=1} \sum_{ij\in E} (x_i-x_j)^{2}.$$ \end{definition}
According to Mohar \cite{mohar1991}, the Laplacian matrix is considered to be more natural than the adjancency matrix. It is a discrete analog of the Laplace operator, which is present in many important differential equations. The Kirchhoff Matrix-Tree theorem is a early example of the use of $L$ in Graph Theory. The largest eigenvalue (spectral radius) of $L$ has been associated, for example, with degree sequences of a graph \cite{GutVidSte2002,LSC09,AndMor1985,Pan2002}. The second smallest eigenvalue and its associated eigenvectors have also been studied since the seminal work by Fiedler \cite{Fiedler73}, which has been used in graph partitioning and has led to an extensive literature in spectral clustering. For more information about this area, see the survey~\cite{vonLux} and the references therein.
Therefore we hope that the definition of $\mu^{(p)}$ will shed some light on classical parameters of graph theory. In fact, we show that, in the same fashion as $\lambda^{(p)}(G)$, the parameter $\mu^{(p)}(G)$ relates to graph invariants, such as the maximum degree and the size of a maximum cut. We also show some properties of $\mu^{(p)}(G)$ as a function of $p$. The main results are: \begin{theorem}\label{t:main} Let $G=(V,E)$ be a graph with at least one edge. Then
\begin{enumerate}[(a)]
\item $\mu^{(1)}(G)$ is equal to the maximum degree of $G$;
\item $\mu^{(\infty)}(G)/4$ is equal to the size of a maximum cut of $G$.
\item The function $f_G : [1, \infty) \to \mathbb{R}$ defined by $f_G(p)=\mu^{(p)}(G)$ is strictly increasing, continuous and converges when $p\to\infty$; \end{enumerate} \end{theorem}
It seems to be the case that, by varying $p$, the vector $x$ that achieves $\mu^{(p)}(G)$ defines a maximum cut of the graph under different restrictions. For instance, $\mu^{(1)}(G)$ leads to a maximum cut with the constraint that one of the classes is a singleton, while $\mu^{(\infty)}(G)$ is gives a maximum cut with no additional constraint. A rigorous basis for this statement remains a question for further investigation.
From the computational complexity point of view, it is interesting to note that computing $\mu^{(1)}(G)$ is easy (can be done in linear time), while computing $\mu^{(\infty)}(G)$ is an NP-complete problem, it is equivalent to finding the size of a maximum cut of $G$. For $\lambda^{(p)}$, the opposite happens: finding $\lambda^{(1)}(G)$ is NP-complete (equivalent to finding the clique number of $G$), while $\lambda^{(\infty)}(G)$ can be found in linear time.
We also present an upper bound on $\mu^{(p)}(G)$ if $p\ge 2$, which is attained for even $n$. \begin{theorem}\label{t:mu_subg_bip}
Let $G=(V,E)$ be a graph with $n=|V|$. Then for $p\ge 2$, \[ \mu^{(p)}(G)\le n^{2-2/p}. \] If $n$ is even, equality holds if and only if $G$ contains $K_{n/2,n/2}$ as subgraph. \end{theorem}
Note that this means that, for even $n$, the value of $\mu^{(p)}(K_n)$ is the same as the value for the balanced complete bipartite graph with $n$ vertices. We conjecture that this holds for all $n$.
This paper is organized as follows. In the remainder of the section we introduce some notation. In sections \ref{s:t1} and \ref{s:t2} we prove Theorems \ref{t:main} and \ref{t:mu_subg_bip}, respectively. In section \ref{s:conc} we present some additional remarks, conjectures and questions for future research.
Before proving our results, we set the notation used throughout the paper. The objective function of the optimization problems is \[F_G(x)=x^TLx=\sum_{ij\in E(G)} (x_i-x_j)^2.\] We may drop the subscript of $F_G$ if $G$ is clear from context. It can be readily seen that $F_{G'}(x)\le F_G(x)$ for a subgraph $G'$ of $G$, and so $F_G(x)\le F_{K_n}(x)$ for any $n$-vertex graph $G$. Furthermore, $F_G(x)=0$ if $x$ is constant in each connected component of $G$.
Finally, given an $n$-vertex graph $G=(V,E)$ and a vector $x \in \mathbb{R}^n$, the vertex sets $P,N$ and $Z$ are those on which $x_i$ is positive, negative, or equal to zero, respectively. We write $d_i$ for the degree of vertex $i$, and $d_{ij}$ is the number of edges between vertices $i$ and $j$ (0 or 1). The all-ones vector in $\mathbb{R}^n$ is $e$ and the $i$-th vector of the canonical basis of $\mathbb{R}^n$ is $e_i$.
\section{Proof of Theorem \ref{t:main}}\label{s:t1}
In this section, we prove Theorem \ref{t:main}, which relates $\mu^{(p)}(G)$ relates to graph invariants and gives properties of $\mu^{(p)}(G)$ as a function of $p$. Item (a) states that $\mu^{(1)}(G)$ is equal to the maximum degree of $G$. In order to prove it,
we need two lemmas. \begin{lemma}\label{mu1a}
Let $x\in\mathbb{R}^n$ such that $\|x\|_1=1$ and $F_G(x)=\mu^{(1)}(G)$. Then at most one entry of $x$ or $-x$ is positive. \end{lemma} \begin{proof} Let $x$ be as above. Without loss of generality, suppose $a,b\in P$ and define $x'$ and $x''$ as \begin{eqnarray*}
x'_k= \begin{cases}
x_a+x_b & \mbox{ if } k=a;\\
0 & \mbox{ if } k=b;\\
x_k & \mbox{ otherwise. } \end{cases} & \mbox{ and } & x''_k= \begin{cases}
0 & \mbox{ if } k=a;\\
x_a+x_b & \mbox{ if } k=b;\\
x_k & \mbox{ otherwise. } \end{cases} \end{eqnarray*} Consider the differences $\Delta'=F(x')-F(x)$ e $\Delta''=F(x'')-F(x)$. \begin{eqnarray*} \Delta'=(d_a-d_{ab})(2x_ax_b+x_b^2)-2x_b\sum_{aj\in E,j\ne b} x_j
-d_bx_b^2+2x_b\sum_{bj\in E,j\ne a} x_j
+ 4d_{ab}x_ax_b \end{eqnarray*} The expression for $\Delta''$ can be readily obtained switching the roles of $a$ and $b$. As $x_a,x_b>0$ we can take \[ \frac{\Delta'}{x_b}+\frac{\Delta''}{x_a}=(d_a+d_b+d_{ab})(x_a+x_b)>0, \] so that at least one of the differences $\Delta'$ and $\Delta''$ is positive. This contradicts the maximality of $x$. \end{proof}
So we can assume that $|P|=|N|=1$.
\begin{lemma}\label{mu1b}
Let $x\in\mathbb{R}^n$ such that $\|x\|_1=1$, $P=\{a\}$, $N=\{b\}$ and $d_a\ge d_b$. Then $d_a=F(e_a)\ge F(x)$, with equality if and only if $d_a=d_b=d_{ab}$. \end{lemma} \begin{proof}
Note that $x_a^2+x_b^2<1$, because $|x_a|+|x_b|=1$. Then \begin{eqnarray*} F(x)&=& d_ax_a^2+d_bx_b^2+d_{ab}(1-x_a^2-x_b^2) \\
& \le & d_a(x_a^2+x_b^2)+d_{ab}(1-x_a^2-x_b^2) \le d_a=F(e_a). \end{eqnarray*} The first and second inequalities become equalities if and only if $d_a=d_b$ and $d_a=d_{ab}$, respectively. \end{proof}
So $\mu^{(1)}(G)$ is obtained for a vector $e_a$ for a vertex $a$ with maximum degree. That proves item (a) of Theorem \ref{t:main}. Note that the solutions are always of this form if the maximum degree is at least 2, because the equality situation of Lemma \ref{mu1b} is of interest only if the maximum degree is one. For instance, for $G=K_2$, any feasible vector attains the maximum.
Now we proceed to prove item (b), which states that $\mu^{(\infty)}(G)/4$ is equal to the size of a maximum cut of $G$. In this case, the problem is of the form
\[ \mu^{(\infty)}(G) =\max_{\max_i |x_i|=1}\sum_{ij\in E} (x_i-x_j)^{2}. \]
\begin{lemma}\label{muinf}
Let $x\in\mathbb{R}^n$ such that $\max_i |x_i|=1$ and $F_G(x)=\mu^{(\infty)}(G)$. Then $|x_i|=1$, for all $i\in V$. \end{lemma} \begin{proof} Let $x$ be as stated above. Suppose that there is $a\in V$ with $-1<x_a<1$. Define $x',x''\in\mathbb{R}^n$ as \begin{eqnarray*}
x'_i= \begin{cases}
1 & \mbox{ if } i=a;\\
x_i & \mbox{ otherwise. } \end{cases} & \mbox{ and } & x''_i= \begin{cases}
-1 & \mbox{ if } i=a;\\
x_i & \mbox{ otherwise. } \end{cases} \end{eqnarray*} Consider the differences $\Delta'=F(x')-F(x)$ and $\Delta''=F(x'')-F(x)$. Then \begin{eqnarray*} \Delta'=d_a(1-x_a^2)-2(1-x_a)\sum_{aj\in E} x_j \end{eqnarray*} and similarly \begin{eqnarray*} \Delta''=d_a(1-x_a^2)+2(1+x_a)\sum_{aj\in E} x_j, \end{eqnarray*} and therefore $$\frac{\Delta'}{1-x_a}+\frac{\Delta''}{1+x_a}=2d_a>0.$$ So at least one of the differences $\Delta'$ and $\Delta''$ is positive. This contradicts the maximality of $x$. \end{proof}
Now for a vector $x$ in the form given by Lemma \ref{muinf} let $S=\{i\in V : x_i=1\}$ and $T=\{i\in V : x_i=-1\}$. So \begin{eqnarray*}
F(x)=\sum\limits_{i\in S, j\in T}(x_i-x_j)^2=4\textnormal{cut}(S,T). \end{eqnarray*} Then of course $F_G(x)=\mu^{(\infty)}(G)$ if $\textnormal{cut}(S,T)$ is a maximum cut. That proves item (a) of Theorem \ref{t:main}. Also, the maximum among graphs of order $n$ is \begin{eqnarray*}
\mu^{(\infty)}(K_n)=\mu^{(\infty)}(K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil})= \begin{cases}
n^2 & \mbox{ if $n$ is even} ;\\
n^2-1 & \mbox{ if $n$ is odd} . \end{cases} \end{eqnarray*}
Finally we prove item (c), which shows properties of the function $f_G : [1, \infty) \to \mathbb{R}$ defined by $f_G(p)=\mu^{(p)}(G)$. Namely, the function is strictly increasing (Lemma \ref{mu_cresc}), continuous (Lemma \ref{mu_cont}) and converges when $p\to\infty$ (Lemma \ref{mu_limite}). First we state two technical lemmas that will be useful.
\begin{lemma}\label{pnorm_bound} Let $q\ge p\ge 1$. Then for $x\in\mathbb{R}^n$,
\[\|x\|_{q}\le \|x\|_p\le n^{\frac{1}{p}-\frac{1}{q}}\|x\|_{q}.\]
Furthermore. $|x_i^*|=n^{-1/q}\|x\|_{q}$ holds for a nonzero vector $x^*$ that attains the upper bound. \end{lemma}
\begin{proof}
Without loss of generality, we can consider that $x$ has positive entries and $\|x\|_{q}=1$. The lower bound holds because the $p$-norm is decreasing on $p$, and it is attained by $e_i$. By applying the power mean inequality to the entries of $nx$, we can see that the upper bound is attained if and only if all entries are $n^{-1/q}$. \end{proof}
\begin{lemma}\label{mu2p_lbound}
Let $G=(V,E)$ be a graph and $x\in R^n$ with $\|x\|_p=1$ and $p\ge2$. Then $F_G(x)\le n^{1-2/p}\mu^{(2)}(G)$. \end{lemma} \begin{proof}
By Rayleigh-Ritz theorem, we have $F_G(x)\le \|x\|_2^2\mu^{(2)}(G)$ for $x\ne 0\in \mathbb{R}^n$. Using Lemma \ref{pnorm_bound}, we obtain $F_G(x)\le \|x^*\|_p^2\mu^{(2)}(G)=n^{1-2/p}\mu(G)$. \end{proof}
The proof will be broken down in three lemmas, one for each result.
\begin{lemma}\label{mu_cresc} For a graph $G$ and $p\ge 1$, $\mu^{(p)}(G)$ is strictly increasing in $p$. \end{lemma} \begin{proof}
Let $x\in\mathbb{R}^n$ such that $\|x\|_p=1$ and $F(x)=\mu^{(p)}(G)$, and $p'>p> 1$. Define $x':=x/\|x\|_{p'}$. As $\|x\|_{p'}\le 1$, we have \begin{equation}\label{inc_quota}
\mu^{(p')}(G)\ge F(x')=\frac{1}{\|x\|_{p'}^2}F(x)\ge\mu^{(p)}(G). \end{equation} As $G$ has at least one edge $ij$, $\mu^{(p)}(G)>0$; pick $x$ such that $x_i=-x_j=2^{-1/p}$, and $x_i=0$ otherwise. Equality holds in equation \ref{inc_quota} if and only if $x=e_i$ for some $i$. We argue now that for $p>1$, $e_i$ never attains the maximum, so that $\mu^{(p)}(G)$ is strictly increasing.
For $p>1$, the stationarity conditions of the problem are $Lx=\lambda\nabla_x (|x_1|^p+\cdots+|x_n|^p-1)$. Note that $x\to|x|^p$ is differentiable for $p>1$. The $j$-th equation is \begin{equation}\label{lagrange_j}
d_jx_j-\sum_{jk\in E}x_k= \begin{cases}
p|x_j|^{p-1}\sg{x_j}, & \mbox{if } x_j\ne 0;\\ 0, & \mbox{if } x_j=0. \end{cases} \end{equation}
Without loss of generality, assume $G$ has no isolated vertices (as they don't contribute to the sum in $F$). Let $i\in V$ and $j$ a neighbor of $i$. Taking $x=e_i$, then $x_k=0$ if $k\ne i$; in particular, $x_j=0$. Then the right hand side of (\ref{lagrange_j}) is 0, and the left hand side is $d_jx_j-\sum_{jk\in E}x_k=0-x_i=-1$. Therefore, $e_i$ doesn't satisfy the optimality conditions of the problem, that is, for any $i\in V$, $F(e_i)<\mu^{(p)}(G)$ for $p>1$.
With this last statement in mind, recall that, by the proof of item a of Theorem \ref{t:main}, $\mu^{(1)}(G)=F(e_i)$ for $i$ with maximum degree. Therefore, we conclude that $\mu^{(1)}(G)<\mu^{(p)}(G)$ for $p>1$. This completes the proof.
\end{proof}
\begin{lemma}\label{mu_cont} For any graph $G$ and $p\ge 1$, the function $p\to\mu^{(p)}(G)$ is continuous. \end{lemma} \begin{proof}
Let $x'\in\mathbb{R}^n$ such that $\|x'\|_{p'}=1$ and $F(x')=\mu^{(p')}(G)$, and $p'>p\ge 1$. By Lemma \ref{pnorm_bound} that $\|x'\|_p\le n^{\frac{1}{p}-\frac{1}{p'}}\|x'\|_{p'}$. Define $x:=x'/\|x'\|_{p}$. Then
\[ \mu^{(p')}(G)=F(x')=\|x'\|_p^2 F(x)\le\ n^{\frac{2}{p}-\frac{2}{p'}}\|x'\|_{p'}\mu^{(p)}(G)=n^{\frac{2}{p}-\frac{2}{p'}}\mu^{(p)}(G) \] By Lemma \ref{mu_cresc}, we know that $\mu^{(p')}(G)>\mu^{(p)}(G)>0$. We also know from spectral graph theory (check for example \cite{Cve79}) that $\mu^{(2)}(G)\le\mu^{(2)}(K_n)=n$. Combining this with Lemma \ref{mu2p_lbound}, we have $\mu^{(p)}(G)\le n^{2-2/p}$ para $p\ge 2$; as $\mu^{(p)}(G)$ is strictly increasing in $p$ (Lemma \ref{mu_cresc}), this bound holds for $p\ge 1$. So
\begin{eqnarray*}
\mu^{(p')}(G)-\mu^{(p)}(G)
&\le& n^{\frac{2}{p}-\frac{2}{p'}}\mu^{(p)}(G)-\mu^{(p)}(G) \\
&\le& \left(n^{\frac{2}{p}-\frac{2}{p'}}-1\right) n^{2-2/p}\\
&< & (n^{2(p'-p)}-1)n^2. \end{eqnarray*} So we have $\mu^{(p')}(G)-\mu^{(p)}(G)<\epsilon$ if $p'-p<\frac{1}{2}\log_n(\epsilon/n^2+1)$. \end{proof}
\begin{lemma}\label{mu_limite} For any graph $G$, \[ \lim_{p\rightarrow\infty} \mu^{(p)}(G)=\mu^{(\infty)}(G). \] \end{lemma} \begin{proof}
For a given $p$, let $x$ such that $\|x\|_p=1$ and $F(x)=\mu^{(p)}(G)$. By the proof of Lemma \ref{mu_cresc}, we know that $x\ne e_i$, so $\max_i|x_i|<1$. Define $x':=x/\max|x_i|$. We can choose $N=N(x')\in\mathbb{N}$ such that
\[ \mu^{(p)}(G)=F(x)=(\max|x_i|)^2F(x')>(\max|x_i|)^N\mu^{(\infty)}(G), \]
so that $0<\mu^{(\infty)}(G)-\mu^{(p)}(G)<(1-(\max|x_i|)^N)\mu^{(\infty)}(G).$ One can check that $\max|x_i|\ge n^{-1/p}$. The proof concludes noting that \[ 0<\mu^{(\infty)}(G)-\mu^{(p)}(G)<(1-n^{-N/p})\mu^{(\infty)}(G), \] and $n^{-N/p}\rightarrow 1$ when $p\rightarrow\infty$.
\end{proof}
\section{Proof of Theorem \ref{t:mu_subg_bip}}\label{s:t2}
In this section we prove Theorem \ref{t:mu_subg_bip}, which establishes the upper bound $\mu^{(p)}(G)\le n^{2-2/p}$ for $p\ge 2$, as well as a necessary and sufficient condition for equality. We denote $G=(S,T,E)$ a bipartite graph with vertex classes $S$ and $T$. First we state three auxiliary lemmas. \begin{lemma}\label{mu1_same_sign}
Let $G=(S,T,E)$ be a bipartite graph, and $x\in\mathbb{R}^n $ such that $\|x\|_p=1$ and $F(x)=\mu^{(p)}(G)$. Then for $x$ or $-x$ we have $P=S$ and $N=T$. \end{lemma} \begin{proof}
Let $x$ be as stated above. Note that we can freely invert the entry signs preserving feasibility. Without loss of generality, if we invert the signs of negative entries in $S$ and positive entries in $T$, we are replacing, in the sum of $F$, terms of the form $(|x_i|-|x_j|)^2$ by $(|x_i|+|x_j|)^2$, thus increasing $F$. \end{proof}
\begin{lemma}\label{mu2_bip_same_value}
Let $G=(S,T,E)$ be a bipartite graph, and $x\in\mathbb{R}^n$ such that $\|x\|_p=1$ and $F(x)=\mu^{(p)}(G)$. Then for $p\ge 2$, if $i$ and $j$ are in the same class, then $x_i=x_j$. \end{lemma} \begin{proof} Suppose $x$ as stated above has entries with $i,j\in S$($=P$ without loss of generality, by Lemma \ref{mu1_same_sign}) with $x_i\ne x_j$. So
\[ F(x)=\sum_{j\in T}\sum_{ij\in E}(x_i-x_j)^2. \] Let $M_p$ denote the power mean of $\{x_i : i\in S\}$. We exchange each $x_i$ by $M_p$. One can check that feasibility is preserved. For fixed $j$, it is sufficient to check the variation of $\sum_i x_i^2+2\sum_i x_ix_j$:
\[ |S|M^2_p+2|S|M_p x_j>|S|M^2_2+2|S|M_1 x_j=\sum_i x_i^2+2\sum_i x_ix_j. \] The inequality holds by the power mean inequality. So the exchange increases $F$, contradicting the maximality of $x$. \end{proof}
This allows us to obtain a formula for complete bipartite graphs.
\begin{lemma}\label{p:mu_bip_comp} Let $G=(S,T,E)$ be a complete bipartite graph. For $p\ge 2$,
\[ \mu^{(p)}(G)=|S||T|(a+b)^2, \] where \begin{eqnarray*}
a=\left(|S|+|T|\left(\frac{|S|}{|T|}\right)^{\frac{p}{p-1}}\right)^{-1/p}, & b=\left(\dfrac{|S|}{|T|}\right)^{\frac{1}{p-1}}a. \end{eqnarray*} \end{lemma} \begin{proof}
By Lemma \ref{mu2_bip_same_value}, we can assume $x_i=a$ for $i\in S$ and $x_i=-b$ for $i\in T$. Then apply Lagrange method to the function $g(a,b)=|S||T|(a+b)^2$ constrained by $h(a,b)=|S|a^p+|T|b^p=1$. \end{proof}
In the proof of the item (c) of Theorem \ref{t:main}, the balanced complete bipartite graph attains the maximum for $\mu^{(\infty)}$ among graphs of order $n$. The same holds for $\mu^{(p)}$ if $2\le p<\infty$ if $n$ is even.
\begin{proof}[Proof of Theorem \ref{t:mu_subg_bip}] As $\mu^{(2)}(K_n)=n$, the bound $\mu^{(p)}(G)\le n^{2-2/p}$ is a direct consequence of Lemma \ref{mu2p_lbound}. By Lemma \ref{p:mu_bip_comp}, one can check that $\mu^{(p)}(K_{n/2,n/2})=n^{2-2/p}$. Furthermore, if $K_{n/2,n/2}\subseteq G$, the inequality is trivial, because $F_G(x)$ won't decrease if we add edges to $G$.
Now let $G$ and $x\in\mathbb{R}^n$ such that $F_G(x)=\mu^{(p)}(G)=n^{2-2/p}$. Note that $|x_i|=|x_j|, \forall i,j\in V$; otherwise, as $\mu^{(2)}(G)\le\mu^{(2)}(K_n)=n$ and by Lemma \ref{mu2p_lbound}, we would have $F_G(x)< n^{2-2/p}$. Also, $K_{|P|,|N|}=(P,N,E')$ is a subgraph of $G$; otherwise there would be $a\in P$ and $b\in N$ such that $\{a,b\}\notin E(G)$ and $F_{G\cup\{a,b\}}(x)>F_G(x)=n^{2-2/p}$, in contradiction with Lemma \ref{mu2p_lbound}.
Therefore, $F_{K_{|P|,|N|}}(x)=F_G(x)=n^{2-2/p}$, because the edges induced by $P$ or $N$ do not contribute to $F_G(x)$.
Observe that, by Lemma \ref{p:mu_bip_comp}, $|x_i|=|x_j|$ if and only if $|P|=|N|$, therefore $|P|=|N|=n/2$. \end{proof}
Although we conjecture that the equality condition of Theorem \ref{t:mu_subg_bip} also holds for odd $n$ (of course with a different quota given by \ref{p:mu_bip_comp}), the reasoning used in the proof does not work in this case, because then the balanced complete bipartite graph does not attain the bound given by Lemma \ref{mu2p_lbound}.
\section{Concluding remarks}\label{s:conc}
As already mentioned in the introduction, we seem to obtain maximum cuts under different restrictions in the graph by varying $p$. That motivates the following broad question for further investigation:
\begin{ques} For $p\ge1$, which relation possibly exists between $\mu^{(p)}(G)$ and cuts (or other parameters) of $G$? \end{ques}
Also, we proved that computing $\mu^{(1)}(G)$ can be done in linear time, while computing $\mu^{(\infty)}(G)$ is an NP-complete problem. As finding the maximum degree of $G$ can be trivially reduced in linear time to finding the size of a maximum cut of $G$, it might be the case that, by increasing $p$, we obtain a problem that is at least as hard. This motivates the following conjecture:
\begin{conj} Let $q>p\ge1$. The problem of finding $\mu^{(p)}(G)$ can be reduced to the problem of finding $\mu^{(q)}(G)$ in polynomial time. \end{conj}
There are other approaches that seek to generalize eigenvalues via the introduction of the $p$-norm. Amghibech \cite{Amghibech03} introduced a non-linear operator, which he called the $p$-Laplacian $\Delta_p$, that induces a functional of the form $\langle x,\Delta_p\rangle=\sum_{ij\in E} |x_i-x_j|^p$ instead of the quadratic form of the Laplacian. This functional is unbounded for $p=\infty$ over the $p$-norm unit ball, and the case $p=1$ cannot be treated directly. However, the eigenvalue formulation used allows to explore eigenvalues other than the largest and the smallest: $\lambda$ is said to be a $p$-eigenvalue of $M$ if there is a vector $v\in\mathbb{R}^n$ such that
$$(\Delta_p x)_i=\lambda\phi_p(v_i),\quad \phi_p(x)=|x|^{p-1}\sg{x}.$$ The vector $v$ is called a $p$-eigenvector of $M$ associated to $\lambda$. Using this formulation, B\"{u}hler and Hein \cite{BuhHei2009} proved that the cut obtained by ``thresholding'' (partitioning according to entries greater than a certain constant) an eigenvector associated to the second smallest eigenvalue of $\Delta_p$ converges to the optimal Cheeger cut when $p\to 1$; in practice, the case $p=2$ is used to obtain an approximation to this cut \cite{ShiMal,vonLux}.
It may be possible to adapt this method to the standard Laplacian operator, which would allow us to explore a $p$-norm version of the second smallest eigenvalue of $L$, which could potentially also lead to different cuts according to the value of $p$.
\noindent \textbf{Acknowledgments} This work was partially supported by CAPES Grant PROBRAL 408/13 - Brazil and DAAD PROBRAL Grant 56267227 - Germany.
\end{document} | arXiv |
Random Fibonacci sequence
In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation $f_{n}=f_{n-1}\pm f_{n-2}$, where the signs + or − are chosen at random with equal probability ${\tfrac {1}{2}}$, independently for different $n$. By a theorem of Harry Kesten and Hillel Furstenberg, random recurrent sequences of this kind grow at a certain exponential rate, but it is difficult to compute the rate explicitly. In 1999, Divakar Viswanath showed that the growth rate of the random Fibonacci sequence is equal to 1.1319882487943... (sequence A078416 in the OEIS), a mathematical constant that was later named Viswanath's constant.[1][2][3]
Description
A random Fibonacci sequence is an integer random sequence given by the numbers $f_{n}$ for natural numbers $n$, where $f_{1}=f_{2}=1$ and the subsequent terms are chosen randomly according to the random recurrence relation
$f_{n}={\begin{cases}f_{n-1}+f_{n-2},&{\text{ with probability }}{\tfrac {1}{2}};\\f_{n-1}-f_{n-2},&{\text{ with probability }}{\tfrac {1}{2}}.\end{cases}}$
An instance of the random Fibonacci sequence starts with 1,1 and the value of the each subsequent term is determined by a fair coin toss: given two consecutive elements of the sequence, the next element is either their sum or their difference with probability 1/2, independently of all the choices made previously. If in the random Fibonacci sequence the plus sign is chosen at each step, the corresponding instance is the Fibonacci sequence (Fn),
$1,1,2,3,5,8,13,21,34,55,\ldots .$
If the signs alternate in minus-plus-plus-minus-plus-plus-... pattern, the result is the sequence
$1,1,0,1,1,0,1,1,0,1,\ldots .$
However, such patterns occur with vanishing probability in a random experiment. In a typical run, the terms will not follow a predictable pattern:
$1,1,2,3,1,-2,-3,-5,-2,-3,\ldots {\text{ for the signs }}+,+,+,-,-,+,-,-,\ldots .$
Similarly to the deterministic case, the random Fibonacci sequence may be profitably described via matrices:
${f_{n-1} \choose f_{n}}={\begin{pmatrix}0&1\\\pm 1&1\end{pmatrix}}{f_{n-2} \choose f_{n-1}},$
where the signs are chosen independently for different n with equal probabilities for + or −. Thus
${f_{n-1} \choose f_{n}}=M_{n}M_{n-1}\ldots M_{3}{f_{1} \choose f_{2}},$
where (Mk) is a sequence of independent identically distributed random matrices taking values A or B with probability 1/2:
$A={\begin{pmatrix}0&1\\1&1\end{pmatrix}},\quad B={\begin{pmatrix}0&1\\-1&1\end{pmatrix}}.$
Growth rate
Johannes Kepler discovered that as n increases, the ratio of the successive terms of the Fibonacci sequence (Fn) approaches the golden ratio $\varphi =(1+{\sqrt {5}})/2,$ which is approximately 1.61803. In 1765, Leonhard Euler published an explicit formula, known today as the Binet formula,
$F_{n}={{\varphi ^{n}-(-1/\varphi )^{n}} \over {\sqrt {5}}}.$
It demonstrates that the Fibonacci numbers grow at an exponential rate equal to the golden ratio φ.
In 1960, Hillel Furstenberg and Harry Kesten showed that for a general class of random matrix products, the norm grows as λn, where n is the number of factors. Their results apply to a broad class of random sequence generating processes that includes the random Fibonacci sequence. As a consequence, the nth root of |fn| converges to a constant value almost surely, or with probability one:
${\sqrt[{n}]{|f_{n}|}}\to 1.1319882487943\dots {\text{ as }}n\to \infty .$
An explicit expression for this constant was found by Divakar Viswanath in 1999. It uses Furstenberg's formula for the Lyapunov exponent of a random matrix product and integration over a certain fractal measure on the Stern–Brocot tree. Moreover, Viswanath computed the numerical value above using floating point arithmetic validated by an analysis of the rounding error.
Generalization
Mark Embree and Nick Trefethen showed in 1999 that the sequence
$f_{n}=\pm f_{n-1}\pm \beta f_{n-2}$
decays almost surely if β is less than a critical value β* ≈ 0.70258, known as the Embree–Trefethen constant, and otherwise grows almost surely. They also showed that the asymptotic ratio σ(β) between consecutive terms converges almost surely for every value of β. The graph of σ(β) appears to have a fractal structure, with a global minimum near βmin ≈ 0.36747 approximately equal to σ(βmin) ≈ 0.89517.[4]
References
1. Viswanath, D. (1999). "Random Fibonacci sequences and the number 1.13198824..." Mathematics of Computation. 69 (231): 1131–1155. doi:10.1090/S0025-5718-99-01145-X.
2. Oliveira, J. O. B.; De Figueiredo, L. H. (2002). "Interval Computation of Viswanath's Constant". Reliable Computing. 8 (2): 131. doi:10.1023/A:1014702122205. S2CID 29600050.
3. Makover, E.; McGowan, J. (2006). "An elementary proof that random Fibonacci sequences grow exponentially". Journal of Number Theory. 121: 40–44. arXiv:math.NT/0510159. doi:10.1016/j.jnt.2006.01.002. S2CID 119169165.
4. Embree, M.; Trefethen, L. N. (1999). "Growth and decay of random Fibonacci sequences" (PDF). Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 455 (1987): 2471. Bibcode:1999RSPSA.455.2471T. doi:10.1098/rspa.1999.0412. S2CID 16404862.
External links
• Weisstein, Eric W. "Random Fibonacci Sequence". MathWorld.
• OEIS sequence A078416 (Decimal expansion of Viswanath's constant)
• Random Fibonacci Numbers. Numberphile's video about the random Fibonnaci sequence.
| Wikipedia |
Weibel's conjecture
In mathematics, Weibel's conjecture gives a criterion for vanishing of negative algebraic K-theory groups. The conjecture was proposed by Charles Weibel (1980) and proven in full generality by Kerz, Strunk & Tamme (2018) using methods from derived algebraic geometry. Previously partial cases had been proven by Morrow (2016) harvtxt error: no target: CITEREFMorrow2016 (help), Kelly (2014) harvtxt error: no target: CITEREFKelly2014 (help), Cisinski (2013) harvtxt error: no target: CITEREFCisinski2013 (help), Geisser & Hesselholt (2010) harvtxt error: no target: CITEREFGeisserHesselholt2010 (help), and Cortiñas et al. (2008) harvtxt error: no target: CITEREFCortiñasHaesemeyerSchlichtingWeibel2008 (help).
Statement of the conjecture
Weibel's conjecture asserts that for a Noetherian scheme X of finite Krull dimension d, the K-groups vanish in degrees < −d:
$K_{i}(X)=0{\text{ for }}i<-d$
and asserts moreover a homotopy invariance property for negative K-groups
$K_{i}(X)=K_{i}(X\times \mathbb {A} ^{r}){\text{ for }}i\leq -d{\text{ and arbitrary }}r.$
References
• Weibel, Charles (1980), "K-theory and analytic isomorphisms", Inventiones Mathematicae, 61 (2): 177–197, doi:10.1007/bf01390120
• Kerz, Moritz; Strunk, Florian; Tamme, Georg (2018), "Algebraic K-theory and descent for blow-ups", Inventiones Mathematicae, 211 (2): 523–577, arXiv:1611.08466, doi:10.1007/s00222-017-0752-2, MR 3748313
| Wikipedia |
\begin{document}
\title{The emergence of torsion in the continuum limit of distributed edge-dislocations -- erratum} \author{Raz Kupferman and Cy Maor} \date{} \maketitle
In \cite{KM15}, following an example of locally flat Riemannian manifolds with edge-dislocation like singularities that converge to a Weitzenb\"ock manifold (Section 3), we defined a general notion of convergence of Weitzenb\"ock manifolds (Definition 4.1). This definition had to be weak enough such that it applies to the example in Section 3, and strong enough to be well defined, that is, strong enough to allow us to prove that the limit is unique. The uniqueness result is Theorem 4.2 in the paper, and its proof is the main part of Section 4.
Definition 4.1 is the following: \begin{customdef}{4.1} \label{df:convergence} Let $({\mathcal M}_n,\mathfrak{g}_n,\nabla_n)$, $({\mathcal M},\mathfrak{g},\nabla)$ be compact \Cy{oriented} $d$-dimensional Weitzenb\"ock manifolds with corners. We say that the sequence $({\mathcal M}_n,\mathfrak{g}_n,\nabla_n)$ converges to $({\mathcal M},\mathfrak{g},\nabla)$ with $p\in \Cy{[}d, \infty)$, if there exists a sequence of diffeomorphisms $F_n: A_n\subset {\calM}\to {\calM}_n$ such that: \begin{enumerate} \item $A_n$ covers ${\calM}$ asymptotically: \[ \lim_{n\to\infty} \text{Vol}_\mathfrak{g} ({\calM}\setminus A_n ) = 0. \] \item $F_n$ are approximate isometries: the distortion vanishes asymptotically, namely, \[ \lim_{n\to\infty}\operatorname{dis} F_n = 0. \] \item $F_n$ are asymptotically rigid in the mean: \[ \lim_{n\to\infty} \int_{A_n} \operatorname{dist}{^p}(dF_n,\SO{\mathfrak{g},\mathfrak{g}_n}) \,d\text{Vol}_\g = 0. \] \item The parallel transport converges in the mean in the following sense: every point in ${\calM}$ has a neighborhood $U\subset{\calM}$, with (i) a $\nabla$-parallel frame field $E$ on $U$, and (ii) a sequence of $\nabla_n$-parallel frame fields $E_n$ on $F_n(U\cap A_n)$, such that \[
\lim_{n\to\infty} \int_{U\cap A_n} |F_n^\star E_n-E|^p_\mathfrak{g} d\text{Vol}_\g = 0. \] \end{enumerate} \end{customdef}
It turns out that there is an error in the proof of Lemma 4.7 (Lemma~4.8 in the arXiv version of \cite{KM15}), which is a part of the proof of the uniqueness of limit (Theorem 4.2). In order to overcome it, one has to strengthen the assumptions in \defref{df:convergence}. A simple way of doing so is by demanding that there exists a constant $C$ such that $\operatorname{Lip}(F_n), \operatorname{Lip}(F_n^{-1}) < C$ for every $n$, that is, by assuming that $F_n$ are uniformly bi-Lipschitz. This makes the proof significantly simpler (in particular, the widely used Lemma~4.6 becomes trivial as the sets $A_n^\varepsilon$ are eventually equal to $A_n$). This assumption also makes the requirement $p\ge d$ irrelevant -- if the assumptions hold for any $p\ge 1$, then they hold for any $p<\infty$. This is explained in detail in \cite{KM16}, a related paper that makes this bi-Lipschitz assumption (for other reasons).
While the uniform bi-Lipschitz assumption is a restrictive assumption, the example presented in \cite{KM15}, as well as the general construction of the same phenomenon presented in \cite{KM15b} all involve uniformly bi-Lipschitz mappings.
However, we find this assumption a bit unnatural and too restrictive, and so we prefer to present here an intermediate one, stronger than \defref{df:convergence} but weaker than \defref{df:convergence} + the uniform bi-Lipschitz assumption: \begin{customdef}{4.1'} \label{df:convergence_new} Let $({\mathcal M}_n,\mathfrak{g}_n,\nabla_n)$, $({\mathcal M},\mathfrak{g},\nabla)$ be compact \Cy{oriented} $d$-dimensional Weitzenb\"ock manifolds with corners. Let $p_{\text{min}}(d) = d + 1 + \frac{d}{2}(\sqrt{1+4/d^2}-1)$. We say that the sequence $({\mathcal M}_n,\mathfrak{g}_n,\nabla_n)$ converges to $({\mathcal M},\mathfrak{g},\nabla)$ with $p\in [p_{\text{min}}, \infty)$, if there exists a sequence of diffeomorphisms $F_n: A_n\subset {\calM}\to {\calM}_n$ such that: \begin{enumerate} \item $A_n$ covers ${\calM}$ asymptotically: \[ \lim_{n\to\infty} \text{Vol}_\mathfrak{g} ({\calM}\setminus A_n )\, \operatorname{Lip}(F_n)^2 = \lim_{n\to\infty} \text{Vol}_\mathfrak{g} ({\calM}\setminus A_n )\, \operatorname{Lip}(F_n^{-1})^2 = 0. \] \item $F_n$ are approximate isometries: the distortion vanishes asymptotically, namely, \[ \lim_{n\to\infty}\operatorname{dis} F_n = 0. \] \item $F_n$ are asymptotically rigid in the mean: \[ \begin{split} &\lim_{n\to\infty} \int_{A_n} \operatorname{dist}{^p}(dF_n,\SO{\mathfrak{g},\mathfrak{g}_n}) \,d\text{Vol}_\g =0, \\ &\lim_{n\to\infty} \int_{{\calM}_n} \operatorname{dist}{^p}(dF_n^{-1},\SO{\mathfrak{g}_n,\mathfrak{g}}) \,d\text{Vol}_{\g_n} = 0. \end{split} \] \item The parallel transport converges in the mean in the following sense: every point in ${\calM}$ has a neighborhood $U\subset{\calM}$, with (i) a $\nabla$-parallel frame field $E$ on $U$, and (ii) a sequence of $\nabla_n$-parallel frame fields $E_n$ on $F_n(U\cap A_n)$, such that \[
\lim_{n\to\infty} \int_{U\cap A_n} |F_n^\star E_n-E|^p_\mathfrak{g} d\text{Vol}_\g = 0. \] \end{enumerate} \end{customdef}
The differences between \defref{df:convergence} and \defref{df:convergence_new} are: \begin{enumerate} \item The condition on $p$ is more restrictive (instead of $p\ge d$ we assume $p\ge p_{\text{min}}(d)$, where $p_{\text{min}}^2 - (d+2) p_{\text{min}} +d =0$). \item Condition (1) now relates the size of the "holes" in ${\calM}$ to the "wildness" of $F_n$. \item Condition (3) now requires that $F_n$ and $F_n^{-1}$ are both asymptotically rigid.
That is, there is a symmetric penalization for both expansion and contraction, instead of penalizing mainly expansions.
Adding a penalization for large contractions is very natural from the material science and elasticity point of view, which is the main motivation for this work. \end{enumerate}
Below is a restatement of Lemma 4.7 in \cite{KM15} (Lemma~4.8 in the arXiv version), and a proof under the assumptions of \defref{df:convergence_new}, which shows that the limit is indeed unique as stated in Theorem~4.2. \begin{customlem}{4.7'} \label{lm:uniqueness} Let $({\calM}_n,\mathfrak{g}_n)$, $({\calM},\mathfrak{g})$ and $(\mathcal{N},\mathfrak{h})$ be compact Riemannian manifolds. Let $E_n$, $E^{\calM}$ and $E^\mathcal{N}$ be frame fields on ${\calM}_n$, ${\calM}$ and $\mathcal{N}$, respectively. Suppose that both \[ ({\calM}_n,\mathfrak{g}_n,E_n) \to ({\calM},\mathfrak{g},E^{\calM}) \qquad\text{ and }\qquad ({\calM}_n,\mathfrak{g}_n,E_n) \to (\mathcal{N},\mathfrak{h},E^\mathcal{N}) \] with respect to diffeomorphisms $F_n:A_n\subset{\calM} \to {\calM}_n$ and $G_n:B_n\subset\mathcal{N}\to {\calM}_n$ (here, the pullbacks of the frame fields converge in $L^p$). Then $H_\star E^{\calM} = E^\mathcal{N}$, where $H:{\calM}\to\mathcal{N}$ is the uniform limit of $H_n = G_n^{-1}\circ F_n$ defined in Lemma~4.3.
\end{customlem}
\begin{proof} We need to show that $H_\star E^{\calM} - E^N = 0$. Since $H$ is the limit of $H_n$, we start by
estimating $(H_n)_\star E^{\calM} - E^\mathcal{N}$. We fix some $\varepsilon>0$, and consider $H_n$ as a diffeomorphism $A_n^\varepsilon\to H_n(A_n^\varepsilon)$, where sets $A_n^\varepsilon$ are defined in Lemma~4.6. By the standard inequality $|a+b|^p \le C(|a|^p + |b|^p)$ we get \[ \begin{split}
&\int_{A_n^\varepsilon} |dH_n E^{\calM} - H_n^*E^\mathcal{N}|_{H_n^*\mathfrak{h}}^p d\text{Vol}_\g \\
&\quad \le C\int_{A_n^\varepsilon} |dH_n E^{\calM} - H_n^*G_n^\star E_n|_{H_n^*\mathfrak{h}}^p d\text{Vol}_\g
+ C \int_{A_n^\varepsilon} | H_n^* G_n^\star E_n - H_n^* E^\mathcal{N}|_{H_n^*\mathfrak{h}}^p d\text{Vol}_\g \\
&\quad = C\int_{A_n^\varepsilon} |dH_n E^{\calM} - H_n^*G_n^\star E_n|_{H_n^*\mathfrak{h}}^p d\text{Vol}_\g
+ C \int_{H_n(A_n^\varepsilon)} | G_n^\star E_n - E^\mathcal{N}|_{\mathfrak{h}}^p \frac{d\text{Vol}_{(H_n)_\star\mathfrak{g}}}{d\text{Vol}_{\mathfrak{h}}} d\text{Vol}_{\mathfrak{h}} \\
&\quad \le C' \int_{A_n^\varepsilon} |dH_n|^p |E^{\calM} - F_n^\star E_n|_{H_n^\star\mathfrak{h}}^p d\text{Vol}_\g
+ C \int_{H_n(A_n^\varepsilon)} | G_n^\star E_n - E^\mathcal{N}|_{\mathfrak{h}}^p \frac{d\text{Vol}_{(H_n)_\star\mathfrak{g}}}{d\text{Vol}_{\mathfrak{h}}} d\text{Vol}_{\mathfrak{h}} \\
&\quad \le C'' \int_{A_n^\varepsilon} |E^{\calM} - F_n^\star E_n|_{\mathfrak{g}}^p d\text{Vol}_\g
+ C'' \int_{H_n(A_n^\varepsilon)} | G_n^\star E_n - E^\mathcal{N}|_{\mathfrak{h}}^p d\text{Vol}_{\mathfrak{h}}, \end{split} \]
where we used the uniform bounds on $|dH_n|$ and $|dH_n^{-1}|$ on $A_n^\varepsilon$, and Lemma~4.5. Now, the first addend in the last line tends to $0$ since $({\calM}_n,\mathfrak{g}_n,E_n) \to ({\calM},\mathfrak{g},E^{\calM})$ with respect to the maps $F_n$, and the second addend since $({\calM}_n,\mathfrak{g}_n,E_n) \to (\mathcal{N},\mathfrak{h},E^\mathcal{N})$ with respect to the maps $G_n$. Therefore, we have established that \begin{equation} \label{eq:tmp1}
\int_{A_n^\varepsilon} |dH_n E^{\calM} - H_n^*E^\mathcal{N}|_{H_n^*\mathfrak{h}}^p d\text{Vol}_\g \to 0. \end{equation} The proof would be complete if we could replace $(H_n)_\star$ by $H_\star$ and $H_n(A_n^\varepsilon)$ by $\mathcal{N}$ in the limit $n\to\infty$. This is not yet possible since $H_n$ tends to $H$ on $A_n$ only uniformly, whereas the push-forward of frame fields with $H_n$ involves derivatives of $H_n$. Therefore, we will show that $dH_n\to dH$ in some sense.
We start by showing that \begin{equation} \label{eq:H_n_to_SO} \lim_{n\to\infty} \int_{A_n} \operatorname{dist}(dH_n,\SO{\mathfrak{g},\mathfrak{h}}) \,d\text{Vol}_\g = 0. \end{equation} Indeed, let $x\in A_n$, and let $q_n\in \SO{\mathfrak{g}_x,(\mathfrak{g}_n)_{F_n(x)}}$ be a point that realizes $\operatorname{dist}(dF_n,\SO{\mathfrak{g},\mathfrak{g}_n})$ at $x$, and $r_n\in \SO{(\mathfrak{g}_n)_{F_n(x)},\mathfrak{h}_{H_n(x)}}$ a point that realizes $\operatorname{dist}(dG_n^{-1},\SO{\mathfrak{g}_n,\mathfrak{h}})$ at $F_n(x)$. Then we have at the point $x$, \[ \begin{split}
\operatorname{dist}(dH_n,\SO{\mathfrak{g},\mathfrak{h}}) &\le |dH_n - r_n q_n| = |dG_n^{-1}dF_n - r_n q_n| \\
& \le |dG_n^{-1} dF_n - r_n dF_n| + |r_n dF_n -r_n q_n| \\
& \le |dG_n^{-1} - r_n | |dF_n| + |dF_n - q_n| \\
& = \operatorname{dist}(dG_n^{-1},\SO{\mathfrak{g}_n,\mathfrak{h}})|dF_n| + \operatorname{dist}(dF_n,\SO{\mathfrak{g},\mathfrak{g}_n}) \end{split} \] and therefore globally \[
\operatorname{dist}(dH_n,\SO{\mathfrak{g},\mathfrak{h}}) \le F_n^*\operatorname{dist}(dG_n^{-1},\SO{\mathfrak{g}_n,\mathfrak{h}})\,|dF_n| + \operatorname{dist}(dF_n,\SO{\mathfrak{g},\mathfrak{g}_n}) \] The second addend vanishes in $L^p(A_n,\mathfrak{g})$ as $n\to \infty$ and therefore also in $L^1(A_n,\mathfrak{g})$. As for the first addend, using H\"older inequality and Lemma~4.5, we obtain: \[ \begin{split}
& \| F_n^*\operatorname{dist}(dG_n^{-1},\SO{\mathfrak{g}_n,\mathfrak{h}})\,|dF_n|\|_{L^1(A_n,\mathfrak{g})} \\
&\quad \le \| F_n^*\operatorname{dist}^{p/p-1}(dG_n^{-1},\SO{\mathfrak{g}_n,\mathfrak{h}})\|_{L^{1}(({\calM},\mathfrak{g}))}^{(p-1)/p} \,\| dF_n\|_{L^p(A_n,\mathfrak{g})} \\
&\quad = \| \operatorname{dist}^{p/p-1}(dG_n^{-1},\SO{\mathfrak{g}_n,\mathfrak{h}}) \,\frac{d\text{Vol}_{(F_n)_\star\mathfrak{g}}}{d\text{Vol}_{\g_n}} \|_{L^{1}({\calM}_n,\mathfrak{g}_n)}^{(p-1)/p} \,\| dF_n\|_{L^p((A_n,\mathfrak{g}))} \\
&\quad \le \| \operatorname{dist}^{p/p-1}(dG_n^{-1},\SO{\mathfrak{g}_n,\mathfrak{h}}) \,|dF_n^{-1}|^d \|_{L^{1}({\calM}_n,\mathfrak{g}_n)}^{(p-1)/p} \,\| dF_n\|_{L^p(A_n,\mathfrak{g})} \\
&\quad \le \| \operatorname{dist}^{p/p-1}(dG_n^{-1},\SO{\mathfrak{g}_n,\mathfrak{h}})\|_{L^{p/p-d}({\calM}_n,\mathfrak{g}_n)}^{(p-1)/p}\, \| |dF_n^{-1}|^d\|_{L^{p/d}(({\calM}_n,\mathfrak{g}_n))}^{(p-1)/p}\, \|dF_n\|_{L^p(A_n,\mathfrak{g})} \\
&\quad = \| \operatorname{dist}(dG_n^{-1},\SO{\mathfrak{g}_n,\mathfrak{h}})\|_{L^{p^2/(p-d)(p-1)}({\calM}_n,\mathfrak{g}_n)}^{\alpha}\, \| dF_n^{-1}\|_{L^{p}({\calM}_n,\mathfrak{g}_n)}^{\beta}\, \|dF_n\|_{L^p(A_n,\mathfrak{g})}, \end{split} \] where $\alpha,\beta>0$ are the appropriate powers (they are immaterial for the rest of the argument). Now, the last two terms on the last line are uniformly bounded in $n$ by our assumptions on $dF_n$. The first term vanishes as $n$ goes to infinity by our assumptions on $G_n$, since our assumption on $p$ implies (i) $p^2/(p-d)(p-1)\le p$, and (ii) $p>d$, hence $\text{Vol}_{\mathfrak{g}_n}({\calM}_n)\to \text{Vol}_{\mathfrak{g}}({\calM})$ (this is an immediate corollary of Lemma~4.5) and so the constants in H\"older inequality used to replace $L^{p^2/(p-d)(p-1)}({\calM}_n,\mathfrak{g}_n)$ with $L^{p}({\calM}_n,\mathfrak{g}_n)$ are bounded uniformly in $n$.
Recall that we want to prove convergence $dH_n\to dH$ in an appropriate sense. Since Sobolev spaces are easier to handle when the image is a vector bundle, we fix an isometric immersion $\phi:(\mathcal{N},\mathfrak{h})\to (\R^\nu,\mathfrak{e})$ for large enough $\nu$, where $\mathfrak{e}$ is the standard Euclidean metric. Consider the mappings $\phi\circ H_n:A_n\to \R^\nu$. These mappings satisfy
\begin{equation}
\label{eq:phiH_n_to_O}
\lim_{n\to\infty} \int_{A_n} \operatorname{dist}(d(\phi\circ H_n),\text{O}(\mathfrak{g},\mathfrak{e})) \,d\text{Vol}_\g = 0,
\end{equation}
since the left hand side is bounded from above by the left hand side of \eqref{eq:H_n_to_SO}, and $\phi$ is an isometric immersion.
Since $\phi\circ H_n$ are smooth, and in particular Lipschitz, we can extend them to $\bar{H}_n\in W^{1,\infty}({\calM};\R^\nu)$, such that $\|d\bar{H}_n\|_{\infty}= \operatorname{Lip}(\bar{H}_n) \le C\operatorname{Lip}(\phi\circ H_n)$ for some constant independent of $n$ (for example, we can use McShane extension lemma \cite{Hei05}, or more sophisticated results with a better constant $C$). We claim that $\bar{H}_n$ satisfy \begin{equation} \label{eq:barH_n_to_O} \lim_{n\to\infty} \int_{{\calM}} \operatorname{dist}(d\bar{H}_n,\text{O}(\mathfrak{g},\mathfrak{e})) \,d\text{Vol}_\g = 0. \end{equation} Indeed, \[ \begin{split} & \int_{{\calM}} \operatorname{dist}(d\bar{H}_n,\text{O}(\mathfrak{g},\mathfrak{e})) \,d\text{Vol}_\g \\
&\quad \le \int_{A_n} \operatorname{dist}(d\bar{H}_n,\text{O}(\mathfrak{g},\mathfrak{e})) \,d\text{Vol}_\g + (C_1+\|d\bar{H}_n\|_\infty)\text{Vol}_\mathfrak{g}({\calM}\setminus A_n) \\
&\quad \le \int_{A_n} \operatorname{dist}(d(\phi \circ H_n),\text{O}(\mathfrak{g},\mathfrak{e})) \,d\text{Vol}_\g + C_1\text{Vol}_\mathfrak{g}({\calM}\setminus A_n) \\
&\quad\quad + C_2\operatorname{Lip}(\phi\circ H_n)\text{Vol}_\mathfrak{g}({\calM}\setminus A_n) \\
&\quad \le \int_{A_n} \operatorname{dist}(d(\phi \circ H_n),\text{O}(\mathfrak{g},\mathfrak{e})) \,d\text{Vol}_\g + C_1\text{Vol}_\mathfrak{g}({\calM}\setminus A_n) \\
&\quad\quad + C_3\operatorname{Lip}(G_n^{-1}) \operatorname{Lip}(F_n) \text{Vol}_\mathfrak{g}({\calM}\setminus A_n) \\
&\quad = \int_{A_n} \operatorname{dist}(d(\phi \circ H_n),\text{O}(\mathfrak{g},\mathfrak{e})) \,d\text{Vol}_\g + C_1\text{Vol}_\mathfrak{g}({\calM}\setminus A_n) \\
&\quad \quad + C_3\sqrt{\operatorname{Lip}(G_n^{-1})^2 \text{Vol}_\mathfrak{g}({\calM}\setminus A_n)} \sqrt{ \operatorname{Lip}(F_n)^2 \text{Vol}_\mathfrak{g}({\calM}\setminus A_n)}. \end{split} \] The first summand goes to zero by \eqref{eq:phiH_n_to_O}, and the first summand by the asymptotic surjectivity of $F_n$, which also imply that $ \operatorname{Lip}(F_n)^2 \text{Vol}_\mathfrak{g}({\calM}\setminus A_n)\to 0$. We are left to deal with the term $\operatorname{Lip}(G_n^{-1})^2 \text{Vol}_\mathfrak{g}({\calM}\setminus A_n)$. Note that by moving to a subsequence, we can assume that $ \text{Vol}_\mathfrak{g}({\calM}\setminus A_n)/ \text{Vol}_\mathfrak{h}(\mathcal{N}\setminus B_n)$ is monotone. Since the roles of ${\calM}$ and $\mathcal{N}$ (and their associated metrics, mappings, etc.) are completely symmetric, we can assume without loss of generality that this sequence is monotonically decreasing, and in particular, bounded, hence \[ \operatorname{Lip}(G_n^{-1})^2 \text{Vol}_\mathfrak{g}({\calM}\setminus A_n) \le C\operatorname{Lip}(G_n^{-1})^2 \text{Vol}_\mathfrak{g}(\mathcal{N}\setminus B_n) \to 0 \] by our assumptions on $G_n$.
We therefore establish \eqref{eq:barH_n_to_O}. While $\bar{H}_n$ are Lipschitz functions on ${\calM}$, they are not uniformly Lipschitz. In order to complete the proof, we will replace them by uniformly Lipschitz maps $\tilde{H}_n:{\calM}\to \R^\nu$ that agree with $\bar{H}_n$ over large sets. That is, we now claim that there exist maps $\tilde{H}_n:{\calM}\to \R^\nu$ such that \begin{enumerate} \item $\text{Vol}_{\mathfrak{g}}\{\tilde{H}_n \ne \bar{H}_n\} \to 0 $, and \item $\tilde{H}_n$ are uniformly bounded in $W^{1,\infty}({\calM};\R^\nu)$, and in particular Lipschitz by with a uniform constant $L$. \end{enumerate} To show this, we use Proposition A.1 in \cite{FJM02b} on $\bar{H}_n$, with $\lambda$ large enough such that for every $x$ and every $T:T_x{\calM}\to \R^\nu$ \[
|T|\ge \lambda \quad \Rightarrow |T| < 2\operatorname{dist}(T,\text{O}(\mathfrak{g},\mathfrak{e})). \] Proposition A.1 in \cite{FJM02b} then implies that there exists a sequence of functions $\tilde{H}_n$, uniformly bounded in $W^{1,\infty}({\calM};\R^\nu)$ such that \[
\text{Vol}_{\mathfrak{g}}({\calM}\setminus R_n) \le C\int_{\{x\in{\calM} : |d \bar{H}_n|>\lambda \}} |d \bar{H}_n| \,d\text{Vol}_{\mathfrak{g}}, \] where $R_n := \{\tilde{H}_n = \bar{H}_n\}$. Therefore, we have \[ \begin{split}
\text{Vol}_{\mathfrak{g}}({\calM}\setminus R_n) & \le C\int_{\{x\in{\calM} : |d \bar{H}_n|>\lambda \}} |d \bar{H}_n| \,d\text{Vol}_{\mathfrak{g}} \\
& \le 2C\int_{\{x\in{\calM} : |d \bar{H}_n|>\lambda \}}\operatorname{dist}(d\bar{H}_n,\text{O}(\mathfrak{g},\mathfrak{e})) \,d\text{Vol}_{\mathfrak{g}} \\
& \le 2C\int_{{\calM}}\operatorname{dist}(d\bar{H}_n,\text{O}(\mathfrak{g},\mathfrak{e})) \,d\text{Vol}_{\mathfrak{g}} \to 0. \end{split} \] This argument is similar to Lemma~3.3 in \cite{LP10}.\footnote{Note that while Proposition A.1 in \cite{FJM02b} discusses a Lipschitz domain in the Euclidean space, and therefore directly applies to a manifold that can be covered by a single coordinate chart (as in \cite{LP10}), this is not a problem here: First, the claim of Lemma \ref{lm:uniqueness} is local, hence we could work locally and assume w.l.o.g.~that ${\calM}$ is covered by a single chart. Second, looking more carefully at the proof of Proposition A.1 in \cite{FJM02b}, the same partition of unity argument used there to discuss the general Lipschitz domain can actually be used again to discuss a general Riemannian manifold.}
The functions $\tilde{H}_n$ converge to $\phi\circ H$ uniformly on ${\calM}$, as \[ \begin{split} d_{\R^\nu}(\tilde{H}_n(p), \phi\circ H(p)) &\le
d_{\R^\nu}(\tilde{H}_n(p),\tilde{H}_n(\psi_n(p))) + d_{\R^\nu}(\tilde{H}_n(\psi_n(p)),\phi\circ H(\psi_n(p))) \\ &\qquad+ d_{\R^\nu}(\phi\circ H(\psi_n(p)),\phi\circ H(p))\\
&= d_{\R^\nu}(\tilde{H}_n(p),\tilde{H}_n(\psi_n(p))) + d_{\R^\nu}(\phi\circ H_n(\psi_n(p)),\phi\circ H(\psi_n(p))) \\ &\qquad+ d_{\R^\nu}(\phi\circ H(\psi_n(p)),\phi\circ H(p))\\ &\le d_{\R^\nu}(\tilde{H}_n(p),\tilde{H}_n(\psi_n(p))) + d_\mathcal{N}(H_n(\psi_n(p)),H(\psi_n(p))) \\ &\qquad+ d_\mathcal{N}(H(\psi_n(p)),H(p))\\ &= d_{\R^\nu}(\tilde{H}_n(p),\tilde{H}_n(\psi_n(p))) + d_\mathcal{N}(H_n(\psi_n(p)),H(\psi_n(p))) \\ &\qquad+ d_{\calM}(\psi_n(p),p) \\
&\le L\cdot d_{\calM}(p,\psi_n(p)) + d_\mathcal{N}(H_n(\psi_n(p)),H(\psi_n(p))) + d_{\calM}(\psi_n(p),p) \\
&\le (L+1)\,\sup_{{\calM}} d(\cdot,\psi_n(\cdot)) + \sup_{A_n} d_\mathcal{N}(H_n(\cdot),H(\cdot)) \to 0. \end{split} \] Here $\psi_n$ is a mapping ${\calM}\to A_n\cap R_n$ satisfying \[
\psi_n|_{A_n\cap R_n} = {\text{Id}} \qquad\text{ and }\qquad \sup_{p\in{\calM}} d_{\calM}(p,\psi_n(p)) < \varepsilon_n \] for some $\varepsilon_n\to 0$; it is analogous to the mapping ${\calM}\to A_n$ introduced in Lemma~4.3. Since $\text{Vol}_\mathfrak{g}({\calM}\setminus R_n)\to 0$, we can indeed choose such a sequence $\varepsilon_n\to 0$. In the passage from the first to the second line we used the fact that $\tilde{H}_n$ coincides with $\phi\circ H_n$ on the image of $\psi_n$. In the passage from the second to the third line we used the fact that $\phi$ is distance reducing. In the passage from the third to the fourth line we used the fact that $H$ is an isometry. The rest follows from the uniform Lipschitz bound on $\tilde{H}_n$ and the uniform convergence of $\psi_n$ to ${\text{Id}}_{\calM}$, and the uniform convergence of $H_n$ to $H$ on $A_n$.
Now, note that \eqref{eq:tmp1} can be written as \begin{equation}
\int_{A_n^\varepsilon} |d(\phi\circ H_n) E^{\calM} - H_n^*(d\phi\circ E^\mathcal{N})|_{\mathfrak{e}}^p d\text{Vol}_\g \to 0. \end{equation} Therefore, \[ \begin{split}
& \int_{\calM} |d\tilde{H}_n\circ E^{\calM} - \tilde{H}_n^*\phi_\star E^\mathcal{N}|_{\mathfrak{e}}^p d\text{Vol}_{\mathfrak{g}} \\
& \quad \le \int_{A_n^\varepsilon\cap R_n} |d\tilde{H}_n\circ E^{\calM} - \tilde{H}_n^*\phi_\star E^\mathcal{N}|_{\mathfrak{e}}^p d\text{Vol}_{\mathfrak{g}} \\
& \quad \quad + C\,\|d\tilde{H}_n\circ E^{\calM} - \tilde{H}_n^*\phi_\star E^\mathcal{N}\|_\infty^p\text{Vol}_\mathfrak{g}({\calM}\setminus(A_n^\varepsilon\cap R_n))
\end{split} \] \[ \begin{split}
& \quad = \int_{A_n^\varepsilon\cap R_n} |d(\phi\circ H_n) E^{\calM} - H_n^*(d\phi\circ E^\mathcal{N})|_{\mathfrak{e}}^p d\text{Vol}_{\mathfrak{g}} \\
& \quad \quad + C\,\|d\tilde{H}_n\circ E^{\calM} - \tilde{H}_n^*\phi_\star E^\mathcal{N}\|_\infty^p\text{Vol}_\mathfrak{g}({\calM}\setminus(A_n^\varepsilon\cap R_n)) \\
& \quad \le \int_{A_n^\varepsilon} |d(\phi\circ H_n) E^{\calM} - H_n^*(d\phi\circ E^\mathcal{N})|_{\mathfrak{e}}^p d\text{Vol}_{\mathfrak{g}} \\
&\quad \quad + C\,(L\| E^{\calM}\|_\infty + \|E^\mathcal{N}\|_\infty)^p\text{Vol}_\mathfrak{g}({\calM}\setminus(A_n^\varepsilon\cap R_n)) \to 0, \end{split} \] where we used the the uniform Lipschitz constant of $H_n$ (denoted by $L$), the fact that $\text{Vol}_\mathfrak{g}({\calM}\setminus A_n^\varepsilon)\to 0$ by Lemma~4.6 and $\text{Vol}_\mathfrak{g}({\calM}\setminus R_n)\to 0$.
Since $\tilde{H}_n\to \phi\circ H$ uniformly and $E^\mathcal{N}$ is smooth, we can replace $\tilde{H}_n^*$ by $(\phi\circ H)^*$. Therefore we obtain \[
\int_{{\calM}} |d\tilde{H}_n\circ E^{\calM} - H^*(d\phi\circ E^\mathcal{N})|_{\mathfrak{e}}^p d\text{Vol}_{\mathfrak{g}} \to 0. \] It follows that $d\tilde{H}_n$ converges in $L^p({\calM};T^*{\calM}\otimes\R^\nu)$ to the map \[ E^{\calM} \mapsto H^*(d\phi\circ E^\mathcal{N}). \] Since, in addition, $\tilde{H}_n$ converges uniformly to $\phi\circ H$, it follows that $\tilde{H}_n$ converges to $\phi\circ H$ in $W^{1,p}({\calM};\R^\nu)$, and in particular, \[ d(\phi\circ H) \circ E^{\calM} = H^*(d\phi\circ E^\mathcal{N}). \] Since $\phi$ is an embedding we can eliminate $d\phi$ on both sides, getting \[ H_\star E^{\calM} = E^\mathcal{N}. \]
\end{proof}
{\footnotesize
\providecommand{\href}[2]{#2} \providecommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{arXiv:#1}} \providecommand{\url}[1]{\texttt{#1}} \providecommand{\urlprefix}{URL }
}
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Introduction to the A7000 Textbook
Chapter 1 Science and the Universe: A Brief Tour
1.1 The Nature of Astronomy
1.2 The Nature of Science
1.3 The Laws of Nature
1.4 Numbers in Astronomy
1.5 Consequences of Light Travel Time
1.6 A Tour of the Universe
1.7 The Universe on the Large Scale
1.8 The Universe of the Very Small
1.9 A Conclusion and a Beginning
8.0 Thinking Ahead
8.1 The Global Perspective
8.2 Earth's Crust
8.3 Earth's Atmosphere
8.4 Life, Chemical Evolution, and Climate Change
8.5 Cosmic Influences on the Evolution of Earth
Chapter 2 Observing the Sky: The Birth of Astronomy
2.1 The Sky Above
2.2 Ancient Astronomy Around the World
2.3 Astronomy of the First Nations of Canada
2.4 Ancient Babylonian, Greek and Roman Astronomy
2.5 Astrology and Astronomy
2.6 The Birth of Modern Astronomy – Copernicus and Galileo
2.7 For Further Exploration, Websites
2.8 Collaborative Group Activities
2.9 Questions and Exercises
Chapter 3 Orbits and Gravity
3.1 The Laws of Planetary Motion
3.2 Newton's Great Synthesis
3.3 Newton's Universal Law of Gravitation
3.4 Orbits in the Solar System
3.5 Motions of Satellites and Spacecraft
3.6 Gravity with More Than Two Bodies
3.7 For Further Exploration
Chapter 4 Earth, Moon, and Sky
4.1 Earth and Sky
4.2 The Seasons
4.3 Keeping Time
4.4 The Calendar
4.5 Phases and Motions of the Moon
4.6 Ocean Tides and the Moon
4.7 Eclipses of the Sun and Moon
Chapter 7 Other Worlds: An Introduction to the Solar System
7.1 Overview of Our Planetary System
7.2 Composition and Structure of Planets
7.3 Dating Planetary Surfaces
7.4 Origin of the Solar System
Chapter 9 Cratered Worlds
9.1 General Properties of the Moon
9.2 The Lunar Surface
9.3 Impact Craters
9.4 The Origin of the Moon
9.5 Mercury
9.6 Key Concepts and Summary, Further Explorations
9.7 Collaborative Group Activities and Exercises
Chapter 10 Earthlike Planets: Venus and Mars
10.0 Thinking Ahead
10.1 The Nearest Planets: An Overview
10.2 The Geology of Venus
10.3 The Massive Atmosphere of Venus
10.4 The Geology of Mars
10.5 Water and Life on Mars
10.6 Divergent Planetary Evolution
10.7 Collaborative Group Activities and Exercises
Chapter 11 The Giant Planets
11.1 Exploring the Outer Planets
11.2 The Giant Planets
11.3 Atmospheres of the Giant Planets
Chapter 12 Rings, Moons, and Pluto
12.1 Ring and Moon Systems Introduced
12.2 The Galilean Moons of Jupiter
12.3 Titan and Triton
12.4 Pluto and Charon
12.5 Planetary Rings
12.6 Summary, Further Exploration, Websites
Chapter 13 Comets and Asteroids: Debris of the Solar System
13.2 Asteroids and Planetary Defense
13.3 The "Long-Haired" Comets
13.4 The Origin and Fate of Comets and Related Objects
13.5 Key Concepts and Summary, Further Explorations, Websites
13.1 Asteroids
Chapter 14 Cosmic Samples and the Origin of the Solar System
14.2 Meteorites: Stones from Heaven
14.3 Formation of the Solar System
14.4 Comparison with Other Planetary Systems
14.5 Planetary Evolution
14.6 Collaborative Activities, Questions and Exercises
Chapter 15 The Sun: A Garden-Variety Star
15.1 The Structure and Composition of the Sun
15.2 The Solar Cycle
15.3 Solar Activity above the Photosphere
15.4 Space Weather
Chapter 16 The Sun: A Nuclear Powerhouse
16.1 Sources of Sunshine: Thermal and Gravitational Energy
16.2 Mass, Energy, and the Theory of Relativity
16.3 The Solar Interior: Theory
16.4 The Solar Interior: Observations
Chapter 17 Analyzing Starlight
17.1 The Brightness of Stars
17.2 Colors of Stars
17.3 The Spectra of Stars (and Brown Dwarfs)
17.4 Using Spectra to Measure Stellar Radius, Composition, and Motion
Chapter 18 The Stars: A Celestial Census
18.1 A Stellar Census
18.2 Measuring Stellar Masses
18.3 Diameters of Stars
18.4 The H–R Diagram
18.5 Collaborative Group Activities, Questions and Exercises
Chapter 19 Celestial Distances
19.2 Surveying the Stars
19.3 Variable Stars: One Key to Cosmic Distances
19.4 The H–R Diagram and Cosmic Distances
Chapter 20 Between the Stars: Gas and Dust in Space
20.1 The Interstellar Medium
20.2 Interstellar Gas
20.3 Cosmic Dust
20.4 Cosmic Rays
20.5 The Life Cycle of Cosmic Material
20.6 Interstellar Matter around the Sun
Chapter 21 The Birth of Stars and the Discovery of Planets outside the Solar System
21.1 Star Formation
21.2 The H–R Diagram and the Study of Stellar Evolution
21.3 Evidence That Planets Form around Other Stars
21.4 Planets beyond the Solar System: Search and Discovery
21.5 Exoplanets Everywhere: What We Are Learning
21.6 New Perspectives on Planet Formation
Chapter 22 Stars from Adolescence to Old Age
22.1 Evolution from the Main Sequence to Red Giants
22.2 Star Clusters
22.3 Checking Out the Theory
22.4 Further Evolution of Stars
22.5 The Evolution of More Massive Stars
Chapter 23 The Death of Stars
23.1 The Death of Low-Mass Stars
23.2 Evolution of Massive Stars: An Explosive Finish
23.3 Supernova Observations
23.4 Pulsars and the Discovery of Neutron Stars
23.5 The Evolution of Binary Star Systems
23.6 The Mystery of the Gamma-Ray Bursts
Chapter 24 Black Holes and Curved Spacetime
24.1 Introducing General Relativity
24.2 Spacetime and Gravity
24.3 Tests of General Relativity
24.4 Time in General Relativity
24.5 Black Holes
24.6 Evidence for Black Holes
24.7 Gravitational Wave Astronomy
Chapter 25 The Milky Way Galaxy
25.1 The Architecture of the Galaxy
25.2 Spiral Structure
25.3 The Mass of the Galaxy
25.4 The Center of the Galaxy
25.5 Stellar Populations in the Galaxy
25.6 The Formation of the Galaxy
Chapter 26 Galaxies
26.1 The Discovery of Galaxies
26.2 Types of Galaxies
26.3 Properties of Galaxies
26.4 The Extragalactic Distance Scale
26.5 The Expanding Universe
Chapter 27 Active Galaxies, Quasars, and Supermassive Black Holes
27.1 Quasars
27.2 Supermassive Black Holes: What Quasars Really Are
27.3 Quasars as Probes of Evolution in the Universe
BCIT Astronomy 7000: A Survey of Astronomy
By the end of this section, you will be able to:
Give a brief history of how gamma-ray bursts were discovered and what instruments made the discovery possible
Explain why astronomers think that gamma-ray bursts beam their energy rather than it radiating uniformly in all directions
Describe how the radiation from a gamma-ray burst and its afterglow is produced
Explain how short-duration gamma-ray bursts differ from longer ones, and describe the process that makes short-duration gamma-ray bursts
Explain why gamma-ray bursts may help us understand the early universe
Everybody loves a good mystery, and astronomers are no exception. The mystery we will discuss in this section was first discovered in the mid-1960s, not via astronomical research, but as a result of a search for the tell-tale signs of nuclear weapon explosions. The US Defense Department launched a series of Vela satellites to make sure that no country was violating a treaty that banned the detonation of nuclear weapons in space.
Since nuclear explosions produce the most energetic form of electromagnetic waves called gamma rays (see Radiation and Spectra), the Vela satellites contained detectors to search for this type of radiation. The satellites did not detect any confirmed events from human activities, but they did—to everyone's surprise—detect short bursts of gamma rays coming from random directions in the sky. News of the discovery was first published in 1973; however, the origin of the bursts remained a mystery. No one knew what produced the brief flashes of gamma rays or how far away the sources were.
From a Few Bursts to Thousands
With the launch of the Compton Gamma-Ray Observatory by NASA in 1991, astronomers began to identify many more bursts and to learn more about them ([link]). Approximately once per day, the NASA satellite detected a flash of gamma rays somewhere in the sky that lasted from a fraction of a second to several hundred seconds. Before the Compton measurements, astronomers had expected that the most likely place for the bursts to come from was the main disk of our own (pancake-shaped) Galaxy. If this had been the case, however, more bursts would have been seen in the crowded plane of the Milky Way than above or below it. Instead, the sources of the bursts were distributed isotropically; that is, they could appear anywhere in the sky with no preference for one region over another. Almost never did a second burst come from the same location.
Compton Detects Gamma-Ray Bursts.
Figure 1. (a) In 1991, the Compton Gamma-Ray Observatory was deployed by the Space Shuttle Atlantis. Weighing more than 16 tons, it was one of the largest scientific payloads ever launched into space. (b) This map of gamma-ray burst positions measured by the Compton Gamma-Ray Observatory shows the isotropic (same in all directions), uniform distribution of bursts on the sky. The map is oriented so that the disk of the Milky Way would stretch across the center line (or equator) of the oval. Note that the bursts show no preference at all for the plane of the Milky Way, as many other types of objects in the sky do. Colors indicate the total energy in the burst: red dots indicate long-duration, bright bursts; blue and purple dots show short, weaker bursts. (credit a: modification of work by NASA; credit b: modification of work by NASA/GSFC)
To get a good visual sense of the degree to which the bursts come from all over the sky, watch this short animated NASA video showing the location of the first 500 bursts found by the later Swift satellite.
For several years, astronomers actively debated whether the burst sources were relatively nearby or very far away—the two possibilities for bursts that are isotropically distributed. Nearby locations might include the cloud of comets that surrounds the solar system or the halo of our Galaxy, which is large and spherical, and also surrounds us in all directions. If, on the other hand, the bursts occurred at very large distances, they could come from faraway galaxies, which are also distributed uniformly in all directions.
Both the very local and the very distant hypotheses required something strange to be going on. If the bursts were coming from the cold outer reaches of our own solar system or from the halo of our Galaxy, then astronomers had to hypothesize some new kind of physical process that could produce unpredictable flashes of high-energy gamma rays in these otherwise-quiet regions of space. And if the bursts came from galaxies millions or billions of light-years away, then they must be extremely powerful to be observable at such large distances; indeed they had to be the among the biggest explosions in the universe.
The First Afterglows
The problem with trying to figure out the source of the gamma-ray bursts was that our instruments for detecting gamma rays could not pinpoint the exact place in the sky where the burst was happening. Early gamma-ray telescopes did not have sufficient resolution. This was frustrating because astronomers suspected that if they could pinpoint the exact position of one of these rapid bursts, then they would be able to identify a counterpart (such as a star or galaxy) at other wavelengths and learn much more about the burst, including where it came from. This would, however, require either major improvements in gamma-ray detector technology to provide better resolution or detection of the burst at some other wavelength. In the end, both techniques played a role.
The breakthrough came with the launch of the Italian Dutch BeppoSAX satellite in 1996. BeppoSAX included a new type of gamma-ray telescope capable of identifying the position of a source much more accurately than previous instruments, to within a few minutes of arc on the sky. By itself, however, it was still not sophisticated enough to determine the exact source of the gamma-ray burst. After all, a box a few minutes of arc on a side could still contain many stars or other celestial objects.
However, the angular resolution of BeppoSAX was good enough to tell astronomers where to point other, more precise telescopes in the hopes of detecting longer-lived electromagnetic emission from the bursts at other wavelengths. Detection of a burst at visible-light or radio wavelengths could provide a position accurate to a few seconds of arc and allow the position to be pinpointed to an individual star or galaxy. BeppoSAX carried its own X-ray telescope onboard the spacecraft to look for such a counterpart, and astronomers using visible-light and radio facilities on the ground were eager to search those wavelengths as well.
Two crucial BeppoSAX burst observations in 1997 helped to resolve the mystery of the gamma-ray bursts. The first burst came in February from the direction of the constellation Orion. Within 8 hours, astronomers working with the satellite had identified the position of the burst, and reoriented the spacecraft to focus BeppoSAX's X-ray detector on the source. To their excitement, they detected a slowly fading X-ray source 8 hours after the event—the first successful detection of an afterglow from a gamma-ray burst. This provided an even-better location of the burst (accurate to about 40 seconds of arc), which was then distributed to astronomers across the world to try to detect it at even longer wavelengths.
That very night, the 4.2-meter William Herschel Telescope on the Canary Islands found a fading visible-light source at the same position as the X-ray afterglow, confirming that such an afterglow could be detected in visible light as well. Eventually, the afterglow faded away, but left behind at the location of the original gamma-ray burst was a faint, fuzzy source right where the fading point of light had been—a distant galaxy ([link]). This was the first piece of evidence that gamma-ray bursts were indeed very energetic objects from very far away. However, it also remained possible that the burst source was much closer to us and just happened to align with a more distant galaxy, so this one observation alone was not a conclusive demonstration of the extragalactic origin of gamma-ray bursts.
Gamma-Ray Burst.
Figure 2. This false-color Hubble Space Telescope image, taken in September 1997, shows the fading afterglow of the gamma-ray burst of February 28, 1997 and the host galaxy in which the burst originated. The left view shows the region of the burst. The enlargement shows the burst source and what appears to be its host galaxy. Note that the gamma-ray source is not in the center of the galaxy. (credit: modification of work by Andrew Fruchter (STScI), Elena Pian (ITSRE-CNR), and NASA, ESA)
On May 8 of the same year, a burst came from the direction of the constellation Camelopardalis. In a coordinated international effort, BeppoSAX again fixed a reasonably precise position, and almost immediately a telescope on Kitt Peak in Arizona was able to catch the visible-light afterglow. Within 2 days, the largest telescope in the world (the Keck in Hawaii) collected enough light to record a spectrum of the burst. The May gamma-ray burst afterglow spectrum showed absorption features from a fuzzy object that was 4 billion light-years from the Sun, meaning that the location of the burst had to be at least this far away—and possibly even farther. (How astronomers can get the distance of such an object from the Doppler shift in the spectrum is something we will discuss in Galaxies.) What that spectrum showed was clear evidence that the gamma-ray burst had taken place in a distant galaxy.
Networking to Catch More Bursts
After initial observations showed that the precise locations and afterglows of gamma-ray bursts could be found, astronomers set up a system to catch and pinpoint bursts on a regular basis. But to respond as quickly as needed to obtain usable results, astronomers realized that they needed to rely on automated systems rather than human observers happening to be in the right place at the right time.
Now, when an orbiting high-energy telescope discovers a burst, its rough location is immediately transmitted to a Gamma-Ray Coordinates Network based at NASA's Goddard Space Flight Center, alerting observers on the ground within a few seconds to look for the visible-light afterglow.
The first major success with this system was achieved by a team of astronomers from the University of Michigan, Lawrence Livermore National Laboratory, and Los Alamos National Laboratories, who designed an automated device they called the Robotic Optical Transient Search Experiment (ROTSE), which detected a very bright visible-light counterpart in 1999. At peak, the burst was almost as bright as Neptune—despite a distance (measured later by spectra from larger telescopes) of 9 billion light-years.
More recently, astronomers have been able to take this a step further, using wide-field-of-view telescopes to stare at large fractions of the sky in the hope that a gamma-ray burst will occur at the right place and time, and be recorded by the telescope's camera. These wide-field telescopes are not sensitive to faint sources, but ROTSE showed that gamma-ray burst afterglows could sometimes be very bright.
Astronomers' hopes were vindicated in March 2008, when an extremely bright gamma-ray burst occurred and its light was captured by two wide-field camera systems in Chile: the Polish "Pi of the Sky" and the Russian-Italian TORTORA [Telescopio Ottimizzato per la Ricerca dei Transienti Ottici Rapidi (Italian for Telescope Optimized for the Research of Rapid Optical Transients)] (see [link]). According to the data taken by these telescopes, for a period of about 30 seconds, the light from the gamma-ray burst was bright enough that it could have been seen by the unaided eye had a person been looking in the right place at the right time. Adding to our amazement, later observations by larger telescopes demonstrated that the burst occurred at a distance of 8 billion light-years from Earth!
Gamma-Ray Burst Observed in March 2008.
Figure 3. The extremely luminous afterglow of GRB 080319B was imaged by the Swift Observatory in X-rays (left) and visible light/ultraviolet (right). (credit: modification of work by NASA/Swift/Stefan Immler, et al.)
To Beam or Not to Beam
The enormous distances to these events meant they had to have been astoundingly energetic to appear as bright as they were across such an enormous distance. In fact, they required so much energy that it posed a problem for gamma-ray burst models: if the source was radiating energy in all directions, then the energy released in gamma rays alone during a bright burst (such as the 1999 or 2008 events) would have been equivalent to the energy produced if the entire mass of a Sun-like star were suddenly converted into pure radiation.
For a source to produce this much energy this quickly (in a burst) is a real challenge. Even if the star producing the gamma-ray burst was much more massive than the Sun (as is probably the case), there is no known means of converting so much mass into radiation within a matter of seconds. However, there is one way to reduce the power required of the "mechanism" that makes gamma-ray bursts. So far, our discussion has assumed that the source of the gamma rays gives off the same amount of energy in all directions, like an incandescent light bulb.
But as we discuss in Pulsars and the Discovery of Neutron Stars, not all sources of radiation in the universe are like this. Some produce thin beams of radiation that are concentrated into only one or two directions. A laser pointer and a lighthouse on the ocean are examples of such beamed sources on Earth ([link]). If, when a burst occurs, the gamma rays come out in only one or two narrow beams, then our estimates of the luminosity of the source can be reduced, and the bursts may be easier to explain. In that case, however, the beam has to point toward Earth for us to be able to see the burst. This, in turn, would imply that for every burst we see from Earth, there are probably many others that we never detect because their beams point in other directions.
Burst That Is Beamed.
Figure 4. This artist's conception shows an illustration of one kind of gamma-ray burst. The collapse of the core of a massive star into a black hole has produced two bright beams of light originating from the star's poles, which an observer pointed along one of these axes would see as a gamma-ray burst. The hot blue stars and gas clouds in the vicinity are meant to show that the event happened in an active star-forming region. (credit: NASA/Swift/Mary Pat Hrybyk-Keith and John Jones)
Long-Duration Gamma-Ray Bursts: Exploding Stars
After identifying and following large numbers of gamma-ray bursts, astronomers began to piece together clues about what kind of event is thought to be responsible for producing the gamma-ray burst. Or, rather, what kind of events, because there are at least two distinct types of gamma-ray bursts. The two—like the different types of supernovae—are produced in completely different ways.
Observationally, the crucial distinction is how long the burst lasts. Astronomers now divide gamma-ray bursts into two categories: short-duration ones (defined as lasting less than 2 seconds, but typically a fraction of a second) and long-duration ones (defined as lasting more than 2 seconds, but typically about a minute).
All of the examples we have discussed so far concern the long-duration gamma-ray bursts. These constitute most of the gamma-ray bursts that our satellites detect, and they are also brighter and easier to pinpoint. Many hundreds of long-duration gamma-ray bursts, and the properties of the galaxies in which they occurred, have now been studied in detail. Long-duration gamma-ray bursts are universally observed to come from distant galaxies that are still actively making stars. They are usually found to be located in regions of the galaxy with strong star-formation activity (such as spiral arms). Recall that the more massive a star is, the less time it spends in each stage of its life. This suggests that the bursts come from a young and short-lived, and therefore massive type of star.
Furthermore, in several cases when a burst has occurred in a galaxy relatively close to Earth (within a few billion light-years), it has been possible to search for a supernova at the same position—and in nearly all of these cases, astronomers have found evidence of a supernova of type Ic going off. A type Ic is a particular type of supernova, which we did not discuss in the earlier parts of this chapter; these are produced by a massive star that has been stripped of its outer hydrogen layer. However, only a tiny fraction of type Ic supernovae produce gamma-ray bursts.
Why would a massive star with its outer layers missing sometimes produce a gamma-ray burst at the same time that it explodes as a supernova? The explanation astronomers have in mind for the extra energy is the collapse of the star's core to form a spinning, magnetic black hole or neutron star. Because the star corpse is both magnetic and spinning rapidly, its sudden collapse is complex and can produce swirling jets of particles and powerful beams of radiation—just like in a quasar or active galactic nucleus (objects you will learn about Active Galaxies, Quasars, and Supermassive Black Holes), but on a much faster timescale. A small amount of the infalling mass is ejected in a narrow beam, moving at speeds close to that of light. Collisions among the particles in the beam can produce intense bursts of energy that we see as a gamma-ray burst.
Within a few minutes, the expanding blast from the fireball plows into the interstellar matter in the dying star's neighborhood. This matter might have been ejected from the star itself at earlier stages in its evolution. Alternatively, it could be the gas out of which the massive star and its neighbors formed.
As the high-speed particles from the blast are slowed, they transfer their energy to the surrounding matter in the form of a shock wave. That shocked material emits radiation at longer wavelengths. This accounts for the afterglow of X-rays, visible light, and radio waves—the glow comes at longer and longer wavelengths as the blast continues to lose energy.
Short-Duration Gamma-Ray Bursts: Colliding Stellar Corpses
What about the shorter gamma-ray bursts? The gamma-ray emission from these events lasts less than 2 seconds, and in some cases may last only milliseconds—an amazingly short time. Such a timescale is difficult to achieve if they are produced in the same way as long-duration gamma-ray bursts, since the collapse of the stellar interior onto the black hole should take at least a few seconds.
Astronomers looked fruitlessly for afterglows from short-duration gamma-ray bursts found by BeppoSAX and other satellites. Evidently, the afterglows fade away too quickly. Fast-responding visible-light telescopes like ROTSE were not helpful either: no matter how fast these telescopes responded, the bursts were not bright enough at visible wavelengths to be detected by these small telescopes.
Once again, it took a new satellite to clear up the mystery. In this case, it was the Swift Gamma-Ray Burst Satellite, launched in 2004 by a collaboration between NASA and the Italian and UK space agencies ([link]). The design of Swift is similar to that of BeppoSAX. However, Swift is much more agile and flexible: after a gamma-ray burst occurs, the X-ray and UV telescopes can be repointed automatically within a few minutes (rather than a few hours). Thus, astronomers can observe the afterglow much earlier, when it is expected to be much brighter. Furthermore, the X-ray telescope is far more sensitive and can provide positions that are 30 times more precise than those provided by BeppoSAX, allowing bursts to be identified even without visible-light or radio observations.
Artist's Illustration of Swift.
Figure 5. The US/UK/Italian spacecraft Swift contains on-board gamma-ray, X-ray, and ultraviolet detectors, and has the ability to automatically reorient itself to a gamma-ray burst detected by the gamma-ray instrument. Since its launch in 2005, Swift has detected and observed over a thousand bursts, including dozens of short-duration bursts. (credit: NASA, Spectrum Astro)
On May 9, 2005, Swift detected a flash of gamma rays lasting 0.13 seconds in duration, originating from the constellation Coma Berenices. Remarkably, the galaxy at the X-ray position looked completely different from any galaxy in which a long-duration burst had been seen to occur. The afterglow originated from the halo of a giant elliptical galaxy 2.7 billion light-years away, with no signs of any young, massive stars in its spectrum. Furthermore, no supernova was ever detected after the burst, despite extensive searching.
What could produce a burst less than a second long, originating from a region with no star formation? The leading model involves the merger of two compact stellar corpses: two neutron stars, or perhaps a neutron star and a black hole. Since many stars come in binary or multiple systems, it's possible to have systems where two such star corpses orbit one another. According to general relativity (which will be discussed in Black Holes and Curved Spacetime), the orbits of a binary star system composed of such objects should slowly decay with time, eventually (after millions or billions of years) causing the two objects to slam together in a violent but brief explosion. Because the decay of the binary orbit is so slow, we would expect more of these mergers to occur in old galaxies in which star formation has long since stopped.
To learn more about the merger of two neutron stars and how they can produce a burst that lasts less than a second, check out this computer simulation by NASA.
While it was impossible to be sure of this model based on only a single event (it is possible this burst actually came from a background galaxy and lined up with the giant elliptical only by chance), several dozen more short-duration gamma-ray bursts have since been located by Swift, many of which also originate from galaxies with very low star-formation rates. This has given astronomers greater confidence that this model is the correct one. Still, to be fully convinced, astronomers are searching for a "smoking gun" signature for the merger of two ultra-dense stellar remnants.
There are two examples we can think of that would provide more direct evidence. One is a very special kind of explosion, produced when neutrons stripped from the neutron stars during the violent final phase of the merger fuse together into heavy elements and then release heat due to radioactivity, producing a short-lived but red supernova sometimes called a kilonova. (The term is used because it is about a thousand times brighter than an ordinary nova, but not quite as "super" as a traditional supernova.) Hubble observations of one short-duration gamma-ray burst in 2013 show suggestive evidence of such a signature, but need to be confirmed by future observations.
The second "smoking gun" has been even more exciting to see: the detection of gravitational waves. As will be discussed in Black Holes and Curved Spacetime, gravitational waves are ripples in the fabric of spacetime that general relativity predicts should be produced by the acceleration of extremely massive and dense objects—such as two neutron stars or black holes spiraling toward each other and colliding. The first example of gravitational waves has been observed recently from the merger of two large black holes. If a gravitational wave is observed one day to be coincident in time and space with a gamma-ray burst, this will not only confirm our theories of the origin of short gamma-ray bursts but would also be among the most spectacular demonstrations yet of Einstein's theory of general relativity.
Probing the Universe with Gamma-Ray Bursts
The story of how astronomers came to explain the origin of the different kinds of bursts is a good example of how the scientific process sometimes resembles good detective work. While the mystery of short-duration gamma-ray bursts is still being unraveled, the focus of studies for long-duration gamma-ray bursts has begun to change from understanding the origin of the bursts themselves (which is now fairly well-established) to using them as tools to understand the broader universe.
The reason that long-duration gamma-ray bursts are useful has to do with their extreme luminosities, if only for a short time. In fact, long-duration gamma-ray bursts are so bright that they could easily be seen at distances that correspond to a few hundred million years after the expansion of the universe began, which is when theorists think that the first generation of stars formed. Some theories predict that the first stars are likely to be massive and complete their evolution in only a million years or so. If this turns out to be the case, then gamma-ray bursts (which signal the death of some of these stars) may provide us with the best way of probing the universe when stars and galaxies first began to form.
So far, the most distant gamma-ray burst found (on April 29, 2009) originated a remarkable 13.2 billion light-years away—meaning it happened only 600 million years after the Big Bang itself. This is comparable to the earliest and most distant galaxies found by the Hubble Space Telescope. It is not quite old enough to expect that it formed from the first generation of stars, but its appearance at this distance still gives us useful information about the production of stars in the early universe. Astronomers continue to scan the skies, looking for even more distant events signaling the deaths of stars from even further back in time.
Key Concepts and Summary
Gamma-ray bursts last from a fraction of a second to a few minutes. They come from all directions and are now known to be associated with very distant objects. The energy is most likely beamed, and, for the ones we can detect, Earth lies in the direction of the beam. Long-duration bursts (lasting more than a few seconds) come from massive stars with their outer hydrogen layers missing that explode as supernovae. Short-duration bursts are believed to be mergers of stellar corpses (neutron stars or black holes).
For Further Exploration
Death of Stars
Hillebrandt, W., et al. "How To Blow Up a Star." Scientific American (October 2006): 42. On supernova mechanisms.
Irion, R. "Pursuing the Most Extreme Stars." Astronomy (January 1999): 48. On pulsars.
Kalirai, J. "New Light on Our Sun's Fate." Astronomy (February 2014): 44. What will happen to stars like our Sun between the main sequence and the white dwarf stages.
Kirshner, R. "Supernova 1987A: The First Ten Years." Sky & Telescope (February 1997): 35.
Maurer, S. "Taking the Pulse of Neutron Stars." Sky & Telescope (August 2001): 32. Review of recent ideas and observations of pulsars.
Zimmerman, R. "Into the Maelstrom." Astronomy (November 1998): 44. About the Crab Nebula.
Gamma-Ray Bursts
Fox, D. & Racusin, J. "The Brightest Burst." Sky & Telescope (January 2009): 34. Nice summary of the brightest burst observed so far, and what we have learned from it.
Nadis, S. "Do Cosmic Flashes Reveal Secrets of the Infant Universe?" Astronomy (June 2008): 34. On different types of gamma-ray bursts and what we can learn from them.
Naeye, R. "Dissecting the Bursts of Doom." Sky & Telescope (August 2006): 30. Excellent review of gamma-ray bursts—how we discovered them, what they might be, and what they can be used for in probing the universe.
Zimmerman, R. "Speed Matters." Astronomy (May 2000): 36. On the quick-alert networks for finding afterglows.
Zimmerman, R. "Witness to Cosmic Collisions." Astronomy (July 2006): 44. On the Swift mission and what it is teaching astronomers about gamma-ray bursts.
Crab Nebula: http://chandra.harvard.edu/xray_sources/crab/crab.html. A short, colorfully written introduction to the history and science involving the best-known supernova remant.
Introduction to Neutron Stars: https://www.astro.umd.edu/~miller/nstar.html. Coleman Miller of the University of Maryland maintains this site, which goes from easy to hard as you get into it, but it has lots of good information about corpses of massive stars.
Introduction to Pulsars (by Maryam Hobbs at the Australia National Telescope Facility): http://www.atnf.csiro.au/outreach/education/everyone/pulsars/index.html.
Magnetars, Soft Gamma Repeaters, and Very Strong Magnetic Fields: http://solomon.as.utexas.edu/magnetar.html. Robert Duncan, one of the originators of the idea of magnetars, assembled this site some years ago.
Brief Intro to Gamma-Ray Bursts (from PBS' Seeing in the Dark): http://www.pbs.org/seeinginthedark/astronomy-topics/gamma-ray-bursts.html.
Discovery of Gamma-ray Bursts: http://science.nasa.gov/science-news/science-at-nasa/1997/ast19sep97_2/.
Gamma-Ray Bursts: Introduction to a Mystery (at NASA's Imagine the Universe site): http://imagine.gsfc.nasa.gov/docs/science/know_l1/bursts.html.
Introduction from the Swift Satellite Site: http://swift.sonoma.edu/about_swift/grbs.html.
Missions to Detect and Learn More about Gamma-ray Bursts:
Fermi Space Telescope: http://fermi.gsfc.nasa.gov/public/.
INTEGRAL Spacecraft: http://www.esa.int/science/integral.
SWIFT Spacecraft: http://swift.sonoma.edu/.
BBC interview with Antony Hewish: http://www.bbc.co.uk/archive/scientists/10608.shtml. (40:54).
Black Widow Pulsars: The Vengeful Corpses of Stars: https://www.youtube.com/watch?v=Fn-3G_N0hy4. A public talk in the Silicon Valley Astronomy Lecture Series by Dr. Roger Romani (Stanford University) (1:01:47).
Hubblecast 64: It all ends with a bang!: http://www.spacetelescope.org/videos/hubblecast64a/. HubbleCast Program introducing Supernovae with Dr. Joe Liske (9:48).
Space Movie Reveals Shocking Secrets of the Crab Pulsar: http://hubblesite.org/newscenter/archive/releases/2002/24/video/c/. A sequence of Hubble and Chandra Space Telescope images of the central regions of the Crab Nebula have been assembled into a very brief movie accompanied by animation showing how the pulsar affects its environment; it comes with some useful background material (40:06).
Gamma-Ray Bursts: The Biggest Explosions Since the Big Bang!: https://www.youtube.com/watch?v=ePo_EdgV764. Edo Berge in a popular-level lecture at Harvard (58:50).
Gamma-Ray Bursts: Flashes in the Sky: https://www.youtube.com/watch?v=23EhcAP3O8Q. American Museum of Natural History Science Bulletin on the Swift satellite (5:59).
Overview Animation of Gamma-Ray Burst: http://news.psu.edu/video/296729/2013/11/27/overview-animation-gamma-ray-burst. Brief Animation of what causes a long-duration gamma-ray burst (0:55).
Collaborative Group Activities
Someone in your group uses a large telescope to observe an expanding shell of gas. Discuss what measurements you could make to determine whether you have discovered a planetary nebula or the remnant of a supernova explosion.
The star Sirius (the brightest star in our northern skies) has a white-dwarf companion. Sirius has a mass of about 2 MSun and is still on the main sequence, while its companion is already a star corpse. Remember that a white dwarf can't have a mass greater than 1.4 MSun. Assuming that the two stars formed at the same time, your group should discuss how Sirius could have a white-dwarf companion. Hint: Was the initial mass of the white-dwarf star larger or smaller than that of Sirius?
Discuss with your group what people today would do if a brilliant star suddenly became visible during the daytime? What kind of fear and superstition might result from a supernova that was really bright in our skies? Have your group invent some headlines that the tabloid newspapers and the less responsible web news outlets would feature.
Suppose a supernova exploded only 40 light-years from Earth. Have your group discuss what effects there may be on Earth when the radiation reaches us and later when the particles reach us. Would there be any way to protect people from the supernova effects?
When pulsars were discovered, the astronomers involved with the discovery talked about finding "little green men." If you had been in their shoes, what tests would you have performed to see whether such a pulsating source of radio waves was natural or the result of an alien intelligence? Today, several groups around the world are actively searching for possible radio signals from intelligent civilizations. How might you expect such signals to differ from pulsar signals?
Your little brother, who has not had the benefit of an astronomy course, reads about white dwarfs and neutron stars in a magazine and decides it would be fun to go near them or even try to land on them. Is this a good idea for future tourism? Have your group make a list of reasons it would not be safe for children (or adults) to go near a white dwarf and a neutron star.
A lot of astronomers' time and many instruments have been devoted to figuring out the nature of gamma-ray bursts. Does your group share the excitement that astronomers feel about these mysterious high-energy events? What are some reasons that people outside of astronomy might care about learning about gamma-ray bursts?
1: How does a white dwarf differ from a neutron star? How does each form? What keeps each from collapsing under its own weight?
2: Describe the evolution of a star with a mass like that of the Sun, from the main-sequence phase of its evolution until it becomes a white dwarf.
3: Describe the evolution of a massive star (say, 20 times the mass of the Sun) up to the point at which it becomes a supernova. How does the evolution of a massive star differ from that of the Sun? Why?
4: How do the two types of supernovae discussed in this chapter differ? What kind of star gives rise to each type?
5: A star begins its life with a mass of 5 MSun but ends its life as a white dwarf with a mass of 0.8 MSun. List the stages in the star's life during which it most likely lost some of the mass it started with. How did mass loss occur in each stage?
6: If the formation of a neutron star leads to a supernova explosion, explain why only three of the hundreds of known pulsars are found in supernova remnants.
7: How can the Crab Nebula shine with the energy of something like 100,000 Suns when the star that formed the nebula exploded almost 1000 years ago? Who "pays the bills" for much of the radiation we see coming from the nebula?
8: How is a nova different from a type Ia supernova? How does it differ from a type II supernova?
9: Apart from the masses, how are binary systems with a neutron star different from binary systems with a white dwarf?
10: What observations from SN 1987A helped confirm theories about supernovae?
11: Describe the evolution of a white dwarf over time, in particular how the luminosity, temperature, and radius change.
12: Describe the evolution of a pulsar over time, in particular how the rotation and pulse signal changes over time.
13: How would a white dwarf that formed from a star that had an initial mass of 1 MSun be different from a white dwarf that formed from a star that had an initial mass of 9 MSun?
14: What do astronomers think are the causes of longer-duration gamma-ray bursts and shorter-duration gamma-ray bursts?
15: How did astronomers finally solve the mystery of what gamma-ray bursts were? What instruments were required to find the solution?
Thought Questions
16: Arrange the following stars in order of their evolution:
A star with no nuclear reactions going on in the core, which is made primarily of carbon and oxygen.
A star of uniform composition from center to surface; it contains hydrogen but has no nuclear reactions going on in the core.
A star that is fusing hydrogen to form helium in its core.
A star that is fusing helium to carbon in the core and hydrogen to helium in a shell around the core.
A star that has no nuclear reactions going on in the core but is fusing hydrogen to form helium in a shell around the core.
17: Would you expect to find any white dwarfs in the Orion Nebula? (See The Birth of Stars and the Discovery of Planets outside the Solar System to remind yourself of its characteristics.) Why or why not?
18: Suppose no stars more massive than about 2 MSun had ever formed. Would life as we know it have been able to develop? Why or why not?
19: Would you be more likely to observe a type II supernova (the explosion of a massive star) in a globular cluster or in an open cluster? Why?
20: Astronomers believe there are something like 100 million neutron stars in the Galaxy, yet we have only found about 2000 pulsars in the Milky Way. Give several reasons these numbers are so different. Explain each reason.
21: Would you expect to observe every supernova in our own Galaxy? Why or why not?
22: The Large Magellanic Cloud has about one-tenth the number of stars found in our own Galaxy. Suppose the mix of high- and low-mass stars is exactly the same in both galaxies. Approximately how often does a supernova occur in the Large Magellanic Cloud?
23: Look at the list of the nearest stars in Appendix I. Would you expect any of these to become supernovae? Why or why not?
24: If most stars become white dwarfs at the ends of their lives and the formation of white dwarfs is accompanied by the production of a planetary nebula, why are there more white dwarfs than planetary nebulae in the Galaxy?
25: If a 3 and 8 MSun star formed together in a binary system, which star would:
Evolve off the main sequence first?
Form a carbon- and oxygen-rich white dwarf?
Be the location for a nova explosion?
26: You have discovered two star clusters. The first cluster contains mainly main-sequence stars, along with some red giant stars and a few white dwarfs. The second cluster also contains mainly main-sequence stars, along with some red giant stars, and a few neutron stars—but no white dwarf stars. What are the relative ages of the clusters? How did you determine your answer?
27: A supernova remnant was recently discovered and found to be approximately 150 years old. Provide possible reasons that this supernova explosion escaped detection.
28: Based upon the evolution of stars, place the following elements in order of least to most common in the Galaxy: gold, carbon, neon. What aspects of stellar evolution formed the basis for how you ordered the elements?
29: What observations or types of telescopes would you use to distinguish a binary system that includes a main-sequence star and a white dwarf star from one containing a main-sequence star and a neutron star?
30: How would the spectra of a type II supernova be different from a type Ia supernova? Hint: Consider the characteristics of the objects that are their source.
Figuring for Yourself
31: The ring around SN 1987A ([link]) initially became illuminated when energetic photons from the supernova interacted with the material in the ring. The radius of the ring is approximately 0.75 light-year from the supernova location. How long after the supernova did the ring become illuminated?
32: What is the acceleration of gravity (g) at the surface of the Sun? (See Appendix E for the Sun's key characteristics.) How much greater is this than g at the surface of Earth? Calculate what you would weigh on the surface of the Sun. Your weight would be your Earth weight multiplied by the ratio of the acceleration of gravity on the Sun to the acceleration of gravity on Earth. (Okay, we know that the Sun does not have a solid surface to stand on and that you would be vaporized if you were at the Sun's photosphere. Humor us for the sake of doing these calculations.)
33: What is the escape velocity from the Sun? How much greater is it than the escape velocity from Earth?
34: What is the average density of the Sun? How does it compare to the average density of Earth?
35: Say that a particular white dwarf has the mass of the Sun but the radius of Earth. What is the acceleration of gravity at the surface of the white dwarf? How much greater is this than g at the surface of Earth? What would you weigh at the surface of the white dwarf (again granting us the dubious notion that you could survive there)?
36: What is the escape velocity from the white dwarf in [link]? How much greater is it than the escape velocity from Earth?
37: What is the average density of the white dwarf in [link]? How does it compare to the average density of Earth?
38: Now take a neutron star that has twice the mass of the Sun but a radius of 10 km. What is the acceleration of gravity at the surface of the neutron star? How much greater is this than g at the surface of Earth? What would you weigh at the surface of the neutron star (provided you could somehow not become a puddle of protoplasm)?
39: What is the escape velocity from the neutron star in [link]? How much greater is it than the escape velocity from Earth?
40: What is the average density of the neutron star in [link]? How does it compare to the average density of Earth?
41: One way to calculate the radius of a star is to use its luminosity and temperature and assume that the star radiates approximately like a blackbody. Astronomers have measured the characteristics of central stars of planetary nebulae and have found that a typical central star is 16 times as luminous and 20 times as hot (about 110,000 K) as the Sun. Find the radius in terms of the Sun's. How does this radius compare with that of a typical white dwarf?
42: According to a model described in the text, a neutron star has a radius of about 10 km. Assume that the pulses occur once per rotation. According to Einstein's theory of relatively, nothing can move faster than the speed of light. Check to make sure that this pulsar model does not violate relativity. Calculate the rotation speed of the Crab Nebula pulsar at its equator, given its period of 0.033 s. (Remember that distance equals velocity × time and that the circumference of a circle is given by 2πR).
43: Do the same calculations as in [link] but for a pulsar that rotates 1000 times per second.
44: If the Sun were replaced by a white dwarf with a surface temperature of 10,000 K and a radius equal to Earth's, how would its luminosity compare to that of the Sun?
45: A supernova can eject material at a velocity of 10,000 km/s. How long would it take a supernova remnant to expand to a radius of 1 AU? How long would it take to expand to a radius of 1 light-years? Assume that the expansion velocity remains constant and use the relationship: $\text{expansion time}=\frac{\text{distance}}{\text{expansion velocity}}.$
46: A supernova remnant was observed in 2007 to be expanding at a velocity of 14,000 km/s and had a radius of 6.5 light-years. Assuming a constant expansion velocity, in what year did this supernova occur?
47: The ring around SN 1987A ([link]) started interacting with material propelled by the shockwave from the supernova beginning in 1997 (10 years after the explosion). The radius of the ring is approximately 0.75 light-year from the supernova location. How fast is the supernova material moving, assume a constant rate of motion in km/s?
48: Before the star that became SN 1987A exploded, it evolved from a red supergiant to a blue supergiant while remaining at the same luminosity. As a red supergiant, its surface temperature would have been approximately 4000 K, while as a blue supergiant, its surface temperature was 16,000 K. How much did the radius change as it evolved from a red to a blue supergiant?
49: What is the radius of the progenitor star that became SN 1987A? Its luminosity was 100,000 times that of the Sun, and it had a surface temperature of 16,000 K.
50: What is the acceleration of gravity at the surface of the star that became SN 1987A? How does this g compare to that at the surface of Earth? The mass was 20 times that of the Sun and the radius was 41 times that of the Sun.
51: What was the escape velocity from the surface of the SN 1987A progenitor star? How much greater is it than the escape velocity from Earth? The mass was 20 times that of the Sun and the radius was 41 times that of the Sun.
52: What was the average density of the star that became SN 1987A? How does it compare to the average density of Earth? The mass was 20 times that of the Sun and the radius was 41 times that of the Sun.
53: If the pulsar shown in [link] is rotating 100 times per second, how many pulses would be detected in one minute? The two beams are located along the pulsar's equator, which is aligned with Earth.
Previous: 23.5 The Evolution of Binary Star Systems
Next: 24.0 Thinking Ahead
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\begin{document}
\newtheorem{defn}[theo]{Definition}
\newtheorem{ques}[theo]{Question}
\newtheorem{lem}[theo]{Lemma}
\newtheorem{lemma}[theo]{Lemma}
\newtheorem{prop}[theo]{Proposition}
\newtheorem{coro}[theo]{Corollary}
\newtheorem{ex}[theo]{Example}
\newtheorem{note}[theo]{Note}
\newtheorem{conj}[theo]{Conjecture}
\title{Manifolds with A Lower Ricci Curvature Bound} \author{Guofang Wei \thanks{Partially supported by NSF grant DMS-0505733}} \date{} \maketitle \begin{abstract} This paper is a survey on the structure of manifolds with a lower Ricci curvature bound. \end{abstract}
\sect{Introduction} The purpose of this paper is to give a survey on the structure of manifolds with a lower Ricci curvature bound. A Ricci curvature bound is weaker than a sectional curvature bound but stronger than a scalar curvature bound. Ricci curvature is also special that it occurs in the Einstein equation and in the Ricci flow. The study of manifolds with lower Ricci curvature bound has experienced tremendous progress in the past fifteen years. Our focus in this article is strictly restricted to results with only Ricci curvature bound, and no result with sectional curvature bound is presented unless for straight comparison. The reader is referred to John Lott's article in this volume for the recent important development concerning Ricci curvature for metric measure spaces by Lott-Villani and Sturm. We start by introducing the basic tools for studying manifolds with lower Ricci curvature bound (Sections 2-4), then discuss the structures of these manifolds (Sections 5-9), with examples in Section 10.
The most basic tool in studying manifolds with Ricci curvature bound is the Bochner formula. From there one can derive powerful comparison tools like the mean curvature comparison, the Laplacian comparison, and the relative volume comparison. For the Laplacian comparison (Section 3) we discuss the global version in three weak senses (barrier, distribution, viscosity) and clarify their relationships (I am very grateful to my colleague Mike Crandall for many helpful discussions and references on this issue).
A generalization of the volume comparison theorem to an integral Ricci curvature bound is also presented (Section 4).
Important tools such as Cheng-Yau's gradient estimate and Cheeger-Colding's segment inequality are presented in Sections 2
and 4 respectively. Cheeger-Gromoll's splitting theorem and Abresch-Gromoll's excess estimate are presented in Sections 5 and 8 respectively.
From comparison theorems, various quantities like the volume, the diameter, the first Betti number, and the first eigenvalue are bounded by the corresponding quantity of the model. When equality occurs one has the rigid case. In Section 5 we discuss many rigidity and stability results for nonnegative and positive Ricci curvature. The Ricci curvature lower bound gives very good control on the fundamental group and the first Betti number of the manifold; this is covered in Section 6 (see also the very recent survey article by Shen-Sormani \cite{Shen-Sormani-survey} for more elaborate discussion). In Sections~\ref{Gromov-Hausdorff}, 8, and 9 we discuss rigidity and stability for manifolds with lower Ricci curvature bound under Gromov-Hausdorff convergence, almost rigidity results, and the structure of the limit spaces, mostly due to Cheeger and Colding. Examples of manifolds with positive Ricci curvature are presented in Section 10.
Many of the results in this article are covered in the very nice survey articles \cite{Zhu1997, Cheeger2001}, where complete proofs are presented. We benefit greatly from these two articles. Some materials here are adapted directly from \cite{Cheeger2001} and we are very grateful to Jeff Cheeger for his permission. We also benefit from \cite{Gromoll, Cheeger2002} and the lecture notes \cite{Wei-lecture-notes} of a topics course I taught at UCSB. I would also like to thank Jeff Cheeger, Xianzhe Dai, Karsten Grove, Peter Petersen, Christina Sormani, and William Wylie for reading earlier versions of this article and for their helpful suggestions.
\sect{Bochner's formula and the mean curvature comparison}
For a smooth function $f$ on a complete Riemannian manifold $(M^n,g)$, the gradient of f is the vector field $\nabla f$ such that $\langle \nabla f, X \rangle = X(f)$ for all vector fields $X$ on $M$. The Hessian of $f$ is the symmetric bilinear form \[ \mathrm{Hess} (f) (X,Y) = XY (f) - \nabla_XY(f) = \langle \nabla_X \nabla f,Y \rangle, \]
and the Laplacian is the trace $\Delta f = \mbox{tr} (\mathrm{Hess} f)$. For a bilinear form $A$, we denote $|A|^2 = \mbox{tr} (AA^t)$. The most basic tool in studying manifolds with Ricci curvature bound is the Bochner formula. Here we state the formula for functions. \begin{theo}[Bochner's Formula] For a smooth function $f$ on a complete Riemannian manifold $(M^n,g)$, \begin{equation} \label{bochner}
\frac 12 \Delta |\nabla f|^2 = |\mathrm{Hess} f|^2 + \langle \nabla f, \nabla (\Delta f) \rangle + \mathrm{Ric} (\nabla f, \nabla f). \end{equation} \end{theo}
This formula has many applications. In particular, we can apply it to the distance function, harmonic functions, and the eigenfunctions among others. The formula has a more general version (Weitzenb\"{o}ck type) for vector fields (1-forms), which also works nicely on Riemannian manifolds with a smooth measure \cite{Lott2003, Perelman-math.DG/0211159} where Ricci and all adjoint operators are defined with respect to the measure.
Let $r(x) = d(p,x)$ be the distance function from $p \in M$. $r(x)$ is a Lipschitz function and is smooth on $M\setminus \{p, C_p\}$, where $C_p$ is the cut locus of $p$. At smooth points of $r$, \begin{equation}
|\nabla r| \equiv 1, \ \ \mathrm{Hess} \, r = II, \ \ \Delta r = m, \end{equation} where $II$ and $m$ are the second fundamental form and mean curvature of the geodesics sphere $\partial B(p,r)$.
Putting $f(x) = r(x)$ in (\ref{bochner}), we obtain the Riccati equation along a radial geodesic,
\begin{equation} 0 = | II |^2 + m'+ \mathrm{Ric} (\nabla r, \nabla r). \end{equation} By the Schwarz inequality, \[
| II |^2 \ge \frac{m^2}{n-1}. \] Thus, if $\mathrm{Ric}_{M^n} \ge (n-1)H$, we have the Riccati inequality, \begin{equation} \label{Riccati-inequ} m' \le -\frac{m^2}{n-1} - (n-1)H. \end{equation}
Let $M^n_H$ denote the complete simply connected space of constant curvature $H$ and $m_H$ the mean curvature of its geodesics sphere, then \begin{equation} \label{Riccati-inequ-H} m'_H = -\frac{m_H^2}{n-1} - (n-1)H. \end{equation}
Since $\lim_{r \rightarrow 0} (m - m_H) = 0$, using (\ref{Riccati-inequ}), (\ref{Riccati-inequ-H}) and the standard Riccati equation comparison, we have \begin{theo}[Mean Curvature Comparison] If $\mathrm{Ric}_{M^n} \ge (n-1)H$, then along any minimal geodesic segment from $p$, \begin{equation} \label{mean-comp} m(r) \le m_H(r). \end{equation} Moreover, equality holds if and only if all radial sectional
curvatures are equal to $H$. \end{theo}
By applying the Bochner formula to $f = \log u$ with an appropriate cut-off function and looking at the maximum point one has Cheng-Yau's gradient estimate for harmonic functions \cite{Cheng-Yau1975}. \begin{theo}[Gradient Estimate, Cheng-Yau 1975] Let $\mathrm{Ric}_{M^n} \ge (n-1)H$ on $B(p,R_2)$ and $u: B(p,R_2) \rightarrow \mathbb R$ satisfying $u >0, \Delta u = 0$. Then for $R_1 < R_2$, on $B(p, R_1)$, \begin{equation} \label{gradient}
\frac{|\nabla u|^2}{u^2} \le c(n,H,R_1,R_2). \end{equation} \end{theo} If $\Delta u = K(u)$, the same proof extends and one has \cite{Cheeger2001} \begin{equation} \label{gradient2}
\frac{|\nabla u|^2}{u^2} \le \max \{ 2u^{-1} K(u), c(n,H,R_1,R_2)+ 2u^{-1}K(u) - 2K'(u)\}. \end{equation}
\sect{Laplacian comparison}
Recall that $m = \Delta r$. From (\ref{mean-comp}), we get the local Laplacian comparison for distance functions \begin{equation} \label{lap-com} \Delta r \le \Delta_H r, \ \ \mbox{for all}\ x \in M\setminus \{p, C_p\}. \end{equation}
Note that if $x \in C_p$, then either $x$ is a (first) conjugate point of $p$ or there are two distinct minimal geodesics connecting $p$ and $x$ \cite{Cheeger-Ebin}, so $x \in $\{conjugate locus of $p\} \cup \{$the set where $r$ is not differentiable\}. The conjugate locus of p consists of the critical values of exp$_p$. Since exp$_p$ is smooth, by Sard's theorem, the conjugate locus has measure zero. The set where $r$ is not differentiable has measure zero since $r$ is Lipschitz. Therefore the cut locus $C_p$ has measure zero. One can show $C_p$ has measure zero more directly by observing that the region inside the cut locus is star-shaped \cite[Page 112]{Chavel1993}. The above argument has the advantage that it can be extended easily to show that Perelman's $l$-cut locus \cite{Perelman-math.DG/0211159} has measure zero since the ${\mathcal L}$-exponential map is smooth and the $l$-distance function is locally Lipschitz.
In fact the Laplacian comparison (\ref{lap-com}) holds globally in various weak senses. First we review the definitions (for simplicity we only do so for the Laplacian) and study the relationship between these different weak senses.
For a continuous function $f$ on $M, q \in M$, a function $f_q$ defined in a neighborhood $U$ of $q$, is an upper barrier of $f$ at $q$ if $f_{q}$ is $C^2(U)$ and
\begin{equation} \label{up-barrier} f_{q} (q) = f(q), \ \ \ f_{q} (x) \ge f(x) \ (x \in U). \end{equation}
\begin{defn} For a continuous function $f$ on $M$, we say $\Delta f (q) \le c$ in the barrier sense ($f$ is a barrier subsolution to the equation $\Delta f =c$ at $q$), if for all $\epsilon > 0$, there exists an upper barrier $f_{q, \epsilon} $ such that $\Delta f_{q, \epsilon} (q) \le c + \epsilon.$ \end{defn} This notion was defined by Calabi \cite{Calabi1958} back in 1958 (he used the terminology ``weak sense" rather than ``barrier sense"). A weaker version is in the sense of viscosity, introduced by Crandall and Lions in \cite{Crandall-Lions1983}. \begin{defn} For a continuous function $f$ on $M$, we say $\Delta f (q) \le c$ in the viscosity sense ($f$ is a viscosity subsolution of $\Delta f =c$ at $q$), if $\Delta \phi (q) \le c$ whenever $\phi \in C^2(U)$ and $(f-\phi)(q) = \inf_{U} (f-\phi)$, where $U$ is a neighborhood of $q$. \end{defn}
Clearly barrier subsolutions are viscosity subsolutions.
Another very useful notion is subsolution in the sense of distributions. \begin{defn} For continuous functions $f, h$ on an open domain $\Omega \subset M$, we say $\Delta f \le h$ in the distribution sense ($f$ is a distribution subsolution of $\Delta f = h$) on $\Omega$, if
$ \int_{\Omega} f \Delta \phi \le \int_{\Omega} h \phi$ for all $\phi \ge 0$ in $C^\infty_0(\Omega)$. \end{defn}
By \cite{Ishii} if $f$ is a viscosity subsolution of $\Delta f = h$ on $\Omega$, then it is also a distribution subsolution and vice verse, see also \cite{Lions}, \cite[Theorem 3.2.11]{Hormander1994}.
For geometric applications, the barrier and distribution sense are very useful and the barrier sense is often easy to check. Viscosity gives a bridge between them. As observed by Calabi \cite{Calabi1958} one can easily construct upper barriers for the distance function. \begin{lem} If $\gamma$ is minimal from $p$ to $q$, then for all $\epsilon > 0$, the function $r_{q,\epsilon}(x) = \epsilon + d(x,\gamma(\epsilon))$, is an upper barrier for the distance function $r(x) = d(p,x)$ at $q$. \end{lem}
Since $r_{q,\epsilon}$ trivially satisfies (\ref{up-barrier}) the lemma follows by observing that it is smooth in a neighborhood of $q$.
Upper barriers for Perelman's $l$-distance function can be constructed very similarly.
Therefore the Laplacian comparison (\ref{lap-com}) holds globally in all the weak senses above. Cheeger-Gromoll (unaware of Calabi's work at the time) had proved the Laplacian comparison in the distribution sense directly by observing the very useful fact that near the cut locus $\nabla r$ points towards the cut locus \cite{Cheeger-Gromoll1971}, see also \cite{Cheeger2001}. (However it is not clear if this fact holds for Perelman's $l$-distance function.)
One reason why these weak subsolutions are so useful is that they still satisfy the following classical Hopf strong maximum principle, see \cite{Calabi1958}, also e.g. \cite{Cheeger2001} for the barrier sense, see \cite{Littman1959, Kawohl-Kutev1998} for the distribution and viscosity senses, also \cite[Theorem 3.2.11]{Hormander1994} in the Euclidean case. \begin{theo}[Strong Maximum Principle] \label{maximum-p} If on a connected open set, $\Omega \subset M^n$, the function $f$ has an interior minimum and $\Delta f \le 0$ in any of the weak senses above, then $f$ is constant on $\Omega$. \end{theo}
These weak solutions also enjoy the regularity (e.g. if $f$ is a weak sub and sup solution of $\Delta f =0$, then $f$ is smooth), see e.g. \cite{Gilbarg-Trudinger2001}.
The Laplacian comparison also works for radial functions (functions composed with the distance function). In geodesic polar coordinate, we have \begin{equation} \Delta f = \tilde{\Delta} f + m(r, \theta) \frac{\partial}{\partial r}f +
\frac{\partial^2 f}{\partial r^2}, \end{equation} where $\tilde{\Delta}$ is the induced Laplacian on the sphere and $m(r,\theta)$ is the mean curvature of the geodesic sphere in the inner normal direction. Therefore \begin{theo}[Global Laplacian Comparison] \label{laplacian-comp} If $\mathrm{Ric}_{M^n} \ge (n-1)H$, in all the weak senses above, we have \begin{eqnarray} \Delta f(r) & \le & \Delta_H f(r) \ \ \ (\mbox{if}\ f' \ge 0), \\ \Delta f(r) & \ge & \Delta_H f(r) \ \ \ (\mbox{if}\ f' \le 0). \end{eqnarray} \end{theo}
\sect{Volume comparison} For $p \in M^n$, use exponential polar coordinate around $p$ and write the volume element $d\, vol = \mathcal A(r,\theta) dr \wedge d\theta_{n-1}$, where $d\theta_{n-1}$ is the standard volume element on the unit sphere $S^{n-1}(1)$. By the first variation of the area (see \cite{Zhu1997}) \begin{equation} \label{area} \frac{\mathcal A'}{\mathcal A} (r,\theta) = m (r, \theta). \end{equation} Similarly, define ${\mathcal A}_H$ for the model space $M_H^n$. The mean curvature comparison and (\ref{area}) gives the volume element comparison. Namely if $M^n$ has $\mathrm{Ric}_M \geq (n-1)H$, then \begin{equation} \label{vol-elem} \frac{\mathcal{A}(r,\theta)}{\mathcal{A}_H
(r,\theta)} \ \mbox{ is nonincreasing along any minimal geodesic segment from} \ p.
\end{equation}
Integrating (\ref{vol-elem}) along the sphere directions, and then the radial direction gives the relative area and volume comparison, see e.g. \cite{Zhu1997}. \begin{theo}[Bishop-Gromov's Relative Volume Comparison] \label{Bishop-Gromov} Suppose $M^n$ has $\mathrm{Ric}_M \geq (n-1)H$. Then \begin{equation} \label{area-vol} \frac{\mathrm{Vol} \, (\partial B(p,r))}{\mathrm{Vol}_H (\partial B(r))} \ \mbox{and}\ \frac{\mathrm{Vol}\, (B(p,r))}{\mathrm{Vol}_H (B(r))} \ \mbox{ are nonincreasing in} \ r. \end{equation} In particular, \begin{equation} \mathrm{Vol} \, (B(p,r)) \le \mathrm{Vol}_H (B(r)) \ \ \ \mbox{for all } \ r > 0, \label{vol-absolute} \end{equation} \begin{equation}
\frac{\mathrm{Vol}\, (B(p,r))}{\mathrm{Vol}\, (B(p, R))} \ge \frac{\mathrm{Vol}_H (B(r))}{\mathrm{Vol}_H(B(R))} \ \ \ \mbox{for all } \ 0 < r \le R,
\label{relative-vol} \end{equation}
and equality holds if and only if $B(p,r)$ is isometric to $B_H(r)$. \end{theo}
This is a powerful result because it is a global comparison. The volume of any ball is bounded above by the volume of
the corresponding ball in the model, and if the volume of a big ball has a lower bound, then all smaller balls also
have lower bounds. One can also apply it to an annulus or a section of the
directions. For topological applications see Section 6.
The volume element comparison (\ref{vol-elem}) can also be used to prove a heat kernel comparison \cite{Cheeger-Yau1981} and Cheeger-Colding's segment inequality \cite[Theorem 2.11]{Cheeger-Colding1996}, see also \cite{Cheeger2001}.
Given a function $g \ge 0$ on $M^n$, put \[ \mathcal F_g(x_1,x_2) = \inf_\gamma \int_0^l g(\gamma(s)) ds, \] where the inf is taken over all minimal geodesics $\gamma$ from $x_1$ to $x_2$ and $s$ denotes the arclength. \begin{theo}[Segment Inequality, Cheeger-Colding 1996] \label{thm-segment} Let $\mathrm{Ric}_{M^n} \ge -(n-1)$, $A_1, A_2 \subset B(p,r)$, and $r \le R$. Then \begin{equation} \label{segment} \int_{A_1\times A_2} \mathcal F_g(x_1,x_2) \le c(n,R) \cdot r \cdot (\mathrm{Vol} (A_1) + \mathrm{Vol} (A_2)) \cdot \int_{B(p,2R)} g, \end{equation} where $c(n,R) = 2 \sup_{0<\frac s2 \le u \le s, 0<s<R} \frac{\mathrm{Vol}_{-1} (\partial B(s))}{\mathrm{Vol}_{-1} (\partial B(u))}$. \end{theo}
The segment inequality shows that if the integral of $g$ on a ball is small then the integral of $g$ along almost all segments is small. It also implies a Poincar\'e inequality of type $(1,p)$ for all $p \ge 1$ for manifolds with lower Ricci curvature bound \cite{Buser1982}. In particular it gives a lower bound on the first eigenvalue of the Laplacian for the Dirichlet problem on a metric ball; compare \cite{Li1993}.
The volume comparison theorem can be generalized to an integral Ricci lower bound \cite{Petersen-Wei1997}, see also \cite{Gallot1988, Yang1992}. For convenience we introduce some notation.
For each $x\in M^n$ let $\lambda\left( x\right) $ denote the smallest eigenvalue for the Ricci tensor $\mathrm{Ric}:T_{x}M\rightarrow T_{x}M,$ and $\mathrm{Ric}_-^H(x) = \left( (n-1)H - \lambda (x)\right)_+ = \max \left\{ 0, (n-1)H - \lambda (x) \right\}$. Let
\begin{equation} \| \mathrm{Ric}_-^H \|_p (R) = \sup_{x\in M} \left( \int_{B\left( x,R\right) } (\mathrm{Ric}_-^H)^{p}\,
d vol\right)^{\frac 1p}. \end{equation}
$\| \mathrm{Ric}_-^H \|_p $ measures the amount of Ricci curvature lying below $(n-1)H$ in the $L^p$ sense. Clearly $\|
\mathrm{Ric}_-^H \|_p (R) = 0$ iff $\mathrm{Ric}_M \ge (n-1)H$.
Parallel to the mean curvature comparison theorem (\ref{mean-comp}) under pointwise Ricci curvature lower bound, Petersen-Wei \cite{Petersen-Wei1997} showed one can estimate the amount of mean curvature bigger than the mean curvature in the model by the amount of Ricci curvature lying below $(n-1)H$ in $L^p$ sense. Namely for any $p > \frac n2$, $H \in \mathbb R$, and when $H>0$ assume $r\le \frac{\pi}{2\sqrt{H}}$, we have \begin{equation} \label{mean-estimate} \left( \int_{B\left( x,r\right) }\left( m-m_H \right) _{+}^{2p} \,d vol
\right)^{\frac{1}{2p}} \leq C\left( n,p\right) \cdot \left( \|
\mathrm{Ric}_-^H \|_p (r) \right)^{\frac 12}. \end{equation}
Using (\ref{mean-estimate}) we have \begin{theo}[Relative Volume Estimate, Petersen-Wei 1997] Let $x \in M^n, H \in \mathbb R$ and $p>\frac n2$ be given, then there is a constant $C(n,p,H,R)$ which is nondecreasing in $R$ such that if $r\leq R$ and when $H>0$ assume that $R \le \frac{\pi}{2\sqrt{H}}$ we have \begin{equation} \label{vol-estimate} \left( \frac{\mathrm{Vol} \,B\left( x,R\right) }{\mathrm{Vol}_H\left(B(R)\right) }\right) ^{\frac 1{2p}}- \left( \frac{\mathrm{Vol}
\,B\left( x,r\right) }{\mathrm{Vol}_H \left(B(r)\right) }\right) ^{\frac 1{2p}} \leq C\left( n,p,H ,R\right) \cdot \left( \| \mathrm{Ric}_-^H
\|_p (R) \right)^{\frac 12}. \end{equation}
Furthermore when $r=0$ we obtain
\begin{equation} \mathrm{Vol} \,B\left( x,R\right) \leq \left( 1+ C\left( n,p,H,R\right)
\cdot \left( \| \mathrm{Ric}_-^H \|_p (R) \right)^{\frac 12} \right)^{2p} \mathrm{Vol}_H\left( B(R)\right). \end{equation} \end{theo}
Note that when $\| \mathrm{Ric}_-^H \|_p (R) =0$, this gives the Bishop-Gromov relative volume comparison.
Volume comparison is a powerful tool for studying manifolds with lower Ricci curvature bound and has many applications. As a result of (\ref{vol-estimate}), many results with pointwise Ricci lower bound (i.e. $\| \mathrm{Ric}_-^H \|_p (r)=0$) can be extended to the case when $\| \mathrm{Ric}_-^H \|_p (r)$ is very small \cite{Gallot1988,Petersen-Wei1997, Petersen-Sprouse1998, Dai-Petersen-Wei2000, Sprouse2000, Petersen-Wei2001, Dai-Wei2004, Aubry-preprint}.
Perelman's reduced volume monotonicity \cite{Perelman-math.DG/0211159}, a basic and powerful tool in his work on Thurston's geometrization conjecture, is a generalization of Bishop-Gromov's volume comparison to Ricci flow. In fact Perelman gave a heuristic argument that volume comparison on an infinite dimensional space (incorporating the Ricci flow) gives the reduced volume monotonicity. It would be very interesting to investigate this relationship further.
\sect{Rigidity results and stability}
From comparison theorems, various quantities are bounded by that of the model. When equality occurs one has the rigid case. In this section we concentrate on the rigidity and stability results for nonnegative and positive Ricci curvature. See Section~\ref{Gromov-Hausdorff} for rigidity and stability under Gromov-Hausdorff convergence and a general lower bound.
The simplest rigidity is the maximal volume. From the equality of volume comparison (\ref{vol-absolute}), we deduce that if $M^n$ has $\mathrm{Ric}_M \ge n-1$ and $\mathrm{Vol}_M = \mathrm{Vol} (S^n)$, then $M^n$ is isometric to $S^n$. Similarly if $M^n$ has $\mathrm{Ric}_M \ge 0$ and $\lim_{r \to \infty} \frac{\mathrm{Vol} B(p,r)}{\omega_n r^n} =1$, where $p \in M$ and $\omega_n$ is the volume of the unit ball in $\mathbb R^n$, then $M^n$ is isometric to $\mathbb R^n$.
From the equality of the area of geodesic ball (the first quantity in (\ref{area-vol})) we get another volume rigidity: volume annulus implies metric annulus. This is first observed in \cite[Section 4]{Cheeger-Colding1996}, see also \cite[Theorem 2.6]{Cheeger2002}. For the case of nonnegative Ricci curvature, this result says that if $\mathrm{Ric}_{M^n} \ge 0$ on the annulus $A(p, r_1, r_2)$, and \[ \frac{\mathrm{Vol} (\partial B(p, r_1))}{\mathrm{Vol} (\partial B(p, r_2))} = \frac{r_1^{n-1}}{{r_2}^{n-1}}, \]
then the metric on $A(p, r_1, r_2)$ is of the form, $dr^2 + r^2 \tilde{g}$, for some smooth Riemannian metric $\tilde{g}$ on $\partial B(p, r_1)$.
By Myers' theorem (see Theorem~\ref{Myers}) when Ricci curvature has a positive lower bound the diameter is bounded by the diameter of the model. In the maximal case, using an eigenvalue comparison (see below) Cheng \cite{Cheng1975} proved that if $M^n$ has $\mathrm{Ric}_M \ge n-1$ and $\mathrm{diam}_M = \pi$, then $M^n$ is isometric to $S^n$. This result can also be directly proven using volume comparison \cite{Shiohama1983,Zhu1997}.
The maximal diameter theorem for the noncompact case is given by Cheeger-Gromoll's splitting theorem \cite{Cheeger-Gromoll1971}. The splitting theorem is the most important rigidity result, it plays a very important role in studying manifolds with nonnegative Ricci curvature and manifolds with general Ricci lower bound. \begin{theo}[Splitting Theorem, Cheeger-Gromoll 1971] \label{splitting} Let $M^n$ be a complete Riemannian manifold with $\mathrm{Ric}_M \ge 0$. If $M$ has a line, then $M$ is isometric to the product $\mathbb R \times N^{n-1}$, where $N$ is an $n-1$ dimensional manifold with $\mathrm{Ric}_N \ge 0$. \end{theo}
The result can be proven using the global Laplacian comparison (Theorem~\ref{laplacian-comp}), the strong maximum principle (Theorem~\ref{maximum-p}), the Bochner formula (\ref{bochner}) and the de Rham decomposition theorem, see e.g. \cite{Zhu1997, Cheeger2001, Petersen-book} for detail.
As an application of the splitting theorem we have that the first Betti number of $M$ is less than or equal to $n$ for $M^n$ with $\mathrm{Ric}_M \ge 0$, and $b_1 =n$ if and only if $M$ is isometric to $T^n$ (the flat torus).
Applying the Bochner formula (\ref{bochner}) to the first eigenfunction Lichnerowicz showed that if $M^n$ has $\mathrm{Ric}_M \ge n-1$, then the first eigenvalue $\lambda_1(M) \ge n$ \cite{Lichnerowicz1958}. Obata showed that if $\lambda_1(M) =n$ then $M^n$ is isometric to $S^n$ \cite{Obata1962}.
From these rigidity results (the equal case), we naturally ask what happens in the almost equal case. Many results are known in this case. For volume we have the following beautiful stability results for positive and nonnegative Ricci curvatures \cite{Cheeger-Colding1997}. \begin{theo}[Volume Stability, Cheeger-Colding, 1997] There exists $\epsilon (n) >0$ such that \\ (i) if a complete Riemannian manifold $M^n$ has $\mathrm{Ric}_M \ge n-1$ and $\mathrm{Vol}_M \ge (1 -\epsilon(n)) \mathrm{Vol}(S^n)$, then $M^n$ is diffeomorphic to $S^n$; \\ (ii) if a complete Riemannian manifold $M^n$ has $\mathrm{Ric}_M \ge 0$ and for some $p \in M$, $\mathrm{Vol} B(p,r) \ge (1 -\epsilon(n)) \, \omega_n r^n$ for all $r>0$, then $M^n$ is diffeomorphic to $\mathbb R^n$. \end{theo} This was first proved by Perelman \cite{Perelman1994} with the weaker conclusion that $M^n$ is homeomorphic to $S^n$ (contractible resp.).
The analogous stability result is not true for diameter. In fact, there are manifolds with $\mathrm{Ric} \ge n-1$ and diameter arbitrarily close to $\pi$ which are not homotopic to sphere \cite{Anderson1990-diameter, Otsu1991}. This should be contrasted with
the sectional curvature case, where we have the beautiful Grove-Shiohama diameter sphere theorem \cite{Grove-Shiohama1977}, that if $M^n$ has sectional curvature $K_M \geq 1$ and $\mathrm{diam}_M >\pi/2$ then $M$ is homeomorphic to $S^n$. Anderson showed that the stability for the splitting theorem (Theorem~\ref{splitting}) does not hold either \cite{Anderson1992}.
By work of Cheng and Croke \cite{Cheng1975, Croke1982}, if $\mathrm{Ric}_M \ge n-1$ then $\mathrm{diam}_M$ is close to $\pi$ if and only if $\lambda_1(M)$ is close to $n$. So the naive version of the stability for $\lambda_1(M)$ does not hold either. However from the work of \cite{Colding1996-Large, Cheeger-Colding1997, Petersen1999} we have the following modified version. \begin{theo}[Colding, Cheeger-Colding, Petersen] There exists $\epsilon (n) >0$ such that if a complete Riemannian manifold $M^n$ has $\mathrm{Ric}_M \ge n-1$, and radius $\ge \pi -\epsilon(n)$ or $\lambda_{n+1}(M) \le n+\epsilon (n)$, then $M^n$ is diffeomorphic to $S^n$. \end{theo} Here $\lambda_{n+1}(M)$ is the $(n+1)-$th eigenvalue of the Laplacian. The above condition is natural in the sense that for $S^n$ the radius is $\pi$ and the first eigenvalue is $n$ with multiplicity $n+1$. Extending Cheng and Croke's work Petersen showed that if $\mathrm{Ric}_M \ge n-1$ then the radius is close to $\pi$ if and only if $\lambda_{n+1}(M)$ is close to $n$.
The stability for the first Betti number, conjectured by Gromov, was proved by Cheeger-Colding in \cite{Cheeger-Colding1997}. Namely there exists $\epsilon (n) >0$ such that if a complete Riemannian manifold $M^n$ has $\mathrm{Ric}_M (\mathrm{diam}_M)^2 \ge -\epsilon(n)$ and $b_1=n$, then $M$ is diffeomorphic to $T^n$. The homeomorphic version was first proved in \cite{Colding1997}.
Although the direct stability for diameter does not hold, Cheeger-Colding's breakthrough work \cite{Cheeger-Colding1996} gives quantitative generalizations of the diameter rigidity results, see Section~\ref{almost}.
\sect{The fundamental groups}
In lower dimensions ($n \le 3$) a Ricci curvature lower bound has strong topological implications. R. Hamilton \cite{Hamilton1982} proved that compact manifolds $M^3$ with positive Ricci curvature are space forms. Schoen-Yau \cite{Schoen-Yau1982} proved that any complete open manifold $M^3$ with positive Ricci curvature must be diffeomorphic to $\mathbb R^3$ using minimal surfaces. In general the strongest control is on the fundamental group.
The first result is Myers' theorem \cite{Myers1941}.
\begin{theo}[Myers, 1941] \label{Myers} If $\mathrm{Ric}_M \geq H >0,$ then $\mathrm{diam} (M) \le \pi/\sqrt{H}$, and $\pi_1 (M)$ is finite. \end{theo}
This is the only known topological obstruction to a compact manifold supports a metric with positive Ricci curvature other than topological obstructions shared by manifolds with positive scalar curvature. See Section~\ref{examples} for examples with positive Ricci curvature and Rosenberg's article in this volume for a discussion of scalar curvature.
We can still ask what one can say about the finite group. Any finite group can be realized as the fundamental group of a
compact manifold with positive Ricci curvature since any finite group is a subgroup of $SU(n)$ (for n sufficiently big) and $SU(n)$ has a metric with
positive Ricci curvature (in fact Einstein).
What can one say if the dimension $n$ is fixed? For example, is the order of the group modulo an abelian subgroup
bounded by the dimension? See \cite{Wilking2000} for a partial result.
For a compact manifold $M$ with nonnegative Ricci curvature, Cheeger-Gromoll's splitting theorem (Theorem~\ref{splitting}) implies that $\pi_1 (M)$ has an abelian subgroup of finite index \cite{Cheeger-Gromoll1971}. Again it is open if one can bound the index by dimension.
For general nonnegative Ricci curvature manifolds, using covering and volume comparison Milnor showed that \cite{Milnor1968}
\begin{theo}[Milnor, 1968] If $M^n$ is complete with $\mathrm{Ric}_M \geq 0,$ then any finitely generated subgroup of $\pi_1 (M)$ has polynomial growth of degree $\leq n.$ \end{theo}
Combining this with the following result of Gromov \cite{Gromov1981}, we know that any finitely generated subgroup of $\pi_1 (M)$ of manifolds with nonnegative Ricci curvature is almost nilpotent. \begin{theo}[Gromov, 1981] A finitely generated group $\Gamma$ has polynomial growth iff $\Gamma$ is almost nilpotent, i.e. it contains a nilpotent subgroup of finite index. \end{theo}
When $M^n$ has nonnegative Ricci curvature and Euclidean volume growth (i.e. $\mathrm{Vol} B(p,r) \ge cr^n$ for some $c>0$), using a heat kernel estimate Li showed that $\pi_1 (M)$ is finite \cite{Li1986}. Anderson also derived this using volume comparison \cite{MR1046624}. Using the splitting theorem of Cheeger and Gromoll \cite{Cheeger-Gromoll1971} (Theorem~\ref{splitting}) on the universal cover Sormani showed that a noncompact manifold with positive Ricci curvature has the loops-to-infinity property \cite{Sormani2001}. As a consequence she showed that a noncompact manifold with positive Ricci curvature is simply connected if it is simply connected at infinity. See \cite{Shen-Sormani2001, Wylie-thesis} for more applications of the loops-to-infinity property.
From the above one naturally wonders if all nilpotent groups occur as the fundamental group of a complete non-compact manifold with
nonnegative Ricci curvature. Indeed, extending the warping product constructions in
\cite{Nabonnand1980, Berard-Bergery1986}, Wei showed \cite{Wei1988} that any finitely generated torsion free nilpotent group could occur as fundamental group of a manifold with
positive Ricci curvature. Wilking \cite{Wilking2000} extended this to any finitely generated almost nilpotent group. This gives a very good understanding of the fundamental group of a manifold with nonnegative Ricci curvature except the following
long standing problem regarding the finiteness of generators \cite{Milnor1968}. \begin{conj}[Milnor, 1968] The fundamental group of a manifold with nonnegative Ricci curvature is finitely generated. \end{conj}
There is some very good progress in this direction. Using short generators and a uniform cut lemma based on the excess estimate of Abresch and Gromoll \cite{Abresch-Gromoll1990} (see (\ref{excess}) ) Sormani \cite{Sormani2000} proved that if $\mathrm{Ric}_M \ge0$ and $M^n$ has small linear diameter growth, then $\pi_1 (M)$ is finitely generated. More precisely the small linear growth condition is:
\[\limsup_{r \to \infty} \frac{\mathrm{diam} \partial B(p,r)}{r} < s_n = \frac{n}{(n-1)3^n} \left(\frac{n-1}{n-2} \right)^{n-1}. \] The constant $s_n$ was improved in \cite{Xu-Wang-Yang2003}. Then in \cite{Wylie2006} Wylie proved that in this case $\pi_1(M) = G(r)$ for $r$ big, where $G(r)$ is the image of $\pi_1(B(p, r))$ in $\pi_1(B(p, 2r))$. In an earlier paper \cite{Sormani-minvol}, Sormani proved that all manifolds with nonnegative Ricci curvature and linear volume growth have sublinear diamter growth, so manifolds with linear volume growth are covered by these results. Any open manifold with nonnegative Ricci curvature has at least linear volume growth \cite{Yau1976}.
In a very different direction Wilking \cite{Wilking2000}, using algebraic methods, showed that if $\mathrm{Ric}_M \geq 0$ then $\pi_1 (M)$ is finitely generated iff any abelian subgroup of $\pi_1 (M)$ is finitely generated, effectively reducing the Milnor conjecture to the study of manifolds with abelian fundamental groups.
The fundamental group and the first Betti number are very nicely related. So it is natural that Ricci lower bound also controls the first Betti number. For compact manifolds Gromov \cite{Gromov1999} and Gallot \cite{Gallot1983} showed that if $M^n$ is a compact manifold with \begin{equation} \mathrm{Ric}_M \geq (n-1)H, \ \ \mathrm{diam}_M \leq D, \label{ric-diam-bounds} \end{equation} then there is a function $C(n, HD^2)$ such that $b_1 (M) \leq C(n, HD^2)$ and $\displaystyle{\lim_{x \to 0^{-}}} C(n,x) = n$ and $C(n, x ) = 0 $ for $x > 0.$ In particular, if $HD^2$ is small, $b_1 (M) \leq n.$
The celebrated Betti number estimate of Gromov \cite{Gromov1981-betti} shows that all higher Betti numbers can be bounded by sectional curvature and diameter. This is not true for Ricci curvature. Using semi-local surgery Sha-Yang constructed metrics of positive Ricci curvature on the connected sum of $k$ copies of $S^2\times S^2$ for all $k \ge 1$ \cite{Sha-Yang1991}. Recently using Seifert bundles over orbifolds with a K\"{a}hler Einstein metric Kollar showed that there are Einstein metrics with positive Ricci curvature on the connected sums of arbitrary number of copies of $S^2\times S^3$ \cite{Kollar}.
Kapovitch-Wilking \cite{Kapovitch-Wilking} recently announced a proof of the compact analog of Milnor's conjecture that the fundamental group of a manifold satisfying (\ref{ric-diam-bounds}) has a presentation with a universally bounded number of generators (as conjectured by this author), and that a manifold which
admits almost nonnegative Ricci curvature has a virtually nilpotent fundamental group. The second result would greatly generalize Fukaya-Yamaguchi's work on almost nonnegative sectional curvature \cite{Fukaya-Yamaguchi1992}. See \cite{Wei1990, Wei1997} for earlier partial results.
When the volume is also bounded from below, by using a clever covering argument M. Anderson \cite{Anderson1990} showed that the number of the short homotopically nontrivial closed geodesics can be controlled and for the class of manifolds $M$ with $\mathrm{Ric}_M \geq (n-1)H,$ $\mathrm{Vol}_M \geq V$ and $\mathrm{diam}_M \leq D$ there are only finitely many isomorphism types of $\pi_1 (M)$. Again if the Ricci curvature is replaced by sectional curvature then much more can be said. Namely there are only finitely many homeomorphism types of the manifolds with sectional curvature and volume bounded from below and diameter bounded from above \cite{Grove-Petersen-Wu1990, Perelman-preprint}. By \cite{Perelman1997} this is not true for Ricci curvature unless the dimension is 3 \cite{Zhu1993}.
Contrary to a Ricci curvature lower bound, a Ricci curvature upper bound does not have any topological constraint \cite{Lohkamp1994}. \begin{theo}[Lohkamp, 1994] If $n \ge 3$, any manifold, $M^n$, admits a complete metric with $\mathrm{Ric}_M <0$. \end{theo}
An upper Ricci curvature bound does have geometric implications, e g. the isometry group of a compact manifold with negative Ricci curvature is finite. In the presence of a lower bound, an upper bound on Ricci curvature forces additional regularity of the metric, see Theorem~\ref{twoside} in Section~\ref{limit} by Anderson. It's still unknown whether it will give additional topological control. For example, the following question is still open. \begin{ques}
Does the class of manifolds $M^n$ with $|\mathrm{Ric}_M | \le H, \mathrm{Vol}_M \ge V$ and $\mathrm{diam}_M \le D$ have finite many homotopy types? \end{ques} There are infinitely many homotopy types without the Ricci upper bound \cite{Perelman1997} .
\sect{Gromov-Hausdorff convergence} \label{Gromov-Hausdorff}
Gromov-Hausdorff convergence is very useful in studying manfolds with a lower Ricci bound. The starting point is Gromov's precompactness theorem. Let's first recall the Gromov-Hausdorff distance. See \cite[Chapter 3,5]{Gromov1999},\cite[Chapter 10]{Petersen-book}, \cite[Chapter 7]{Burago-Burago-Ivanov2001} for more background material on Gromov-Hausdorff convergence.
Given a metric space $(X,d)$ and subsets $A,B \subset X$, the Hausdorff distance is \[ d_H (A,B)=\inf\{\epsilon>0: B \subset T_\epsilon(A) \textrm{ and } A \subset T_\epsilon(B) \}, \] where $T_\epsilon(A)=\{x\in X: d (x,A) <\epsilon\}$.
\begin{defn}[Gromov, 1981] \label{GH-distance} Given two compact metric spaces $X,Y$, the Gromov-Hausdorff distance is $ d_{GH}(X,Y) = \inf \left\{ d_H(X,Y): \right.$ all metrics on the disjoint union, $X \coprod Y$, which extend the metrics of $X$ and $\left.Y \right\}$. \end{defn}
The Gromov-Hausdorff distance defines a metric on the collection of isometry classes of compact metric spaces. Thus, there is the naturally associated notion of Gromov-Hausdorff convergence of compact metric spaces. While the Gromov-Hausdorff distance make sense for non-compact metric spaces, the following looser definition of convergence is more appropriate. See also \cite[Defn 3.14]{Gromov1999}. These two definitions are equivalent \cite[Appendix]{Sormani-Wei2004}. \begin{defn} We say that non-compact metric spaces $(X_i, x_i)$ converge in the pointed Gromov-Hausdorff sense to $(Y,y)$ if for any $r>0$, $B(x_i, r)$ converges to $B(y, r)$ in the pointed Gromov-Hausdorff sense. \end{defn}
Applying the relative volume comparison (\ref{relative-vol}) to manifolds with lower Ricci bound, we have \begin{theo}[Gromov's precompactness theorem] \label{precompactness} The class of closed manifolds $M^n$ with $\mathrm{Ric}_M \ge (n-1)H$ and $\mathrm{diam}_M \le D$ is precompact. The class of pointed complete manifolds $M^n$ with $\mathrm{Ric}_M \ge (n-1)H$ is precompact. \end{theo}
By the above, for an open manifold $M^n$ with $\mathrm{Ric}_M \ge 0$ any sequence $\{ (M^n, x,r_i^{-2}g)\}$, with $r_i \rightarrow \infty$, subconverges in the pointed Gromov-Hausdorff topology to a length space $M_\infty$. In general, $M_\infty$ is not unique \cite{Perelman-cone}. Any such limit is called an asymptotic cone of $M^n$, or a cone of $M^n$ at infinity .
Gromov-Hausdorff convergence defines a very weak topology. In general one only knows that Gromov-Hausdorff limit of length spaces is a length space and diameter is continuous under the Gromov-Hausdorff convergence. When the limit is a smooth manifold with same dimension Colding showed the remarkable result that for manifolds with lower Ricci curvature bound the volume also converges \cite{Colding1997} which was conjectured by Anderson-Cheeger. See also \cite{Cheeger2001} for a proof using mod $2$ degree. \begin{theo}[Volume Convergence, Colding, 1997] \label{vol-conv} If $(M_i^n, x_i)$ has $\mathrm{Ric}_{M_i} \ge (n-1)H$ and converges in the pointed Gromov-Hausdorff sense to smooth Riemannian manifold $(M^n,x)$, then for all $r >0$ \begin{equation} \lim_{i \rightarrow \infty} \mathrm{Vol} (B(x_i, r)) = \mathrm{Vol} (B(x,r)). \end{equation} \end{theo}
The volume convergence can be generalized to the noncollapsed singular limit space (by replacing the Riemannian volume with the $n$-dimensional Hausdorff measure $\mathcal H^n$) \cite[Theorem 5.9]{Cheeger-Colding1997}, and to the collapsing case with smooth limit $M^k$ in terms of the $k$-dimensional Hausdorff content \cite[Theorem 1.39]{Cheeger-Colding2000II}.
As an application of Theorem~\ref{vol-conv}, Colding \cite{Colding1997} derived the rigidity result that if $M^n$ has $\mathrm{Ric}_{M} \ge 0$ and some $M_\infty$ is isometric to $\mathbb R^n$, then $M$ is isometric to $\mathbb R^n$.
We also have the following wonderful stability result \cite{Cheeger-Colding1997} which sharpens an earlier version in \cite{Colding1997}. \begin{theo}[Cheeger-Colding, 1997] For a closed Riemannian manifold $M^n$ there exists an $\epsilon(M) >0$ such that if $N^n$ is a $n$-manifold with $\mathrm{Ric}_N \ge -(n-1)$ and $d_{GH} (M,N) < \epsilon$ then $M$ and $N$ are diffeomorphic. \end{theo} Unlike the sectional curvature case, examples show that the result does not hold if one allows $M$ to have singularities even on the fundamental group level \cite[Remark (2)]{Otsu1991}. Also the $\epsilon$ here must depend on $M$ \cite{Anderson1990-diameter}.
Cheeger-Colding also showed that the eigenvalues and eigenfunctions of the Laplacian are continuous under measured Gromov-Hausdorff convergence \cite{Cheeger-Colding2000III}. To state the result we need a definition and some structure result on the limit space (see Section~\ref{limit} for more structures). Let $X_i$ be a sequence of metric spaces converging to $X_\infty$ and $\mu_i, \mu_\infty$ are Radon measures on $X_i, X_\infty$. \begin{defn} We say $(X_i, \mu_i)$ converges in the measured Gromov-Hausdorff sense to $(X_\infty, \mu_\infty)$ if for all sequences of continuous functions $f_i:X_i \rightarrow \mathbb R$ converging to $f_\infty:X_\infty \rightarrow \mathbb R$, we have \begin{equation} \int_{X_i} f_i d\mu_i \rightarrow \int_{X_\infty} f_\infty d\mu_\infty. \end{equation} \end{defn}
If $(M_\infty,p)$ is the pointed Gromov-Hausdorff limit of a sequence of Riemannian manifolds $(M^n_i, p_i)$ with $\mathrm{Ric}_{M_i} \ge -(n-1)$, then there is a natural collection of measures, $\mu$, on $M_\infty$ obtained by taking limits of the normalized Reimannian measures on $M^n_j$ for a suitable subsequence $M_j^n$ \cite{Fukaya1987}, \cite[Section 1]{Cheeger-Colding1997}, \begin{equation} \label{renormalized-vol} \mu = \lim_{j \rightarrow \infty} \underline{\mathrm{Vol}}_j ( \cdot ) = \mathrm{Vol} (\cdot )/\mathrm{Vol}(B(p_j,1)). \end{equation}
In particular, for all $z \in M_\infty$ and $0< r_1 \le r_2$, we have the renormalized limit measure $\mu$ satisfy the following comparison
\begin{equation}
\frac{\mu (B(z,r_1))}{\mu (B(z, r_2))} \ge \frac{\mathrm{Vol}_{n,-1}\left(B(r_1)\right)}{\mathrm{Vol}_{n,-1}\left(B(r_2)\right) }. \end{equation} With this, the extension of the segment inequality (\ref{segment}) to the limit, the gradient estimate (\ref{gradient2}), and Bochner's formula, one can define a canonical self-adjoint Laplacian $\Delta_\infty$ on the limit space $M_\infty$ by means of limits of the eigenfunctions and eigenvalues for the sequence of the manifolds. In \cite{Cheeger1999, Cheeger-Colding2000III} an intrinsic construction of this operator is also given on a more general metric measure spaces. Let $\{\lambda_{1,i} \cdots,\}, \{\lambda_{1,\infty}, \cdots, \}$ denote the eigenvalues for $\Delta_i, \Delta_\infty$ on $M_i, M_\infty$, and $\phi_{j,i}, \phi_{j,\infty}$ the eigenfunctions of the jth eigenvalues $\lambda_{j,i}, \lambda_{j,\infty}$. In \cite{Cheeger-Colding2000III} Cheeger-Colding in particular proved the following theorem, establishing Fukaya's conjecture \cite{Fukaya1987}. \begin{theo}[Spectral Convergence, Cheeger-Colding, 2000] Let $(M_i^n, p_i, \underline{\mathrm{Vol}}_i )$ with $\mathrm{Ric}_{M_i} \ge -(n-1)$ converges to $(M_\infty,p,\mu)$ under measured Gromov-Hausdorff sense and $M_\infty$ is compact. Then for each $j$, $\lambda_{j,i} \rightarrow \lambda_{j,\infty}$ and $\phi_{j,i} \rightarrow \phi_{j,\infty}$ uniformly as $i \rightarrow \infty$. \end{theo}
As a natural extension, in \cite{Ding2002} Ding proved that the heat kernel and Green's function also behave nicely under the measured Gromov-Hausdorff convergence. The natural extension to the $p$-form Laplacian does not hold, however, there is still very nice work in this direction by John Lott, see \cite{Lott2002, Lott2004}.
\sect{Almost rigidity and applications} \label{almost}
Although the analogous stability results for maximal diameter in the case of positive/nonnegative Ricci curvature do not hold, Cheeger-Colding's significant work \cite{Cheeger-Colding1996} provides quantitative generalizations of Cheng's maximal diameter theorem, Cheeger-Gromoll's splitting theorem (Theorem~\ref{splitting}), and the volume annulus implies metric annulus theorem in terms of Gromov-Hausdroff distance. These results have important applications in extending rigidity results to the limit space.
An important ingredient for these results is Abresch-Gromoll's excess estimate \cite{Abresch-Gromoll1990}. For $y_1, y_2 \in M^n$, the excess function $E$ with respect to $y_1,y_2$ is
\begin{equation} E_{y_1,y_2}(x) = d (y_1,x)+d(y_2,x) - d(y_1,y_2). \end{equation}
Clearly $E$ is Lipschitz with Lipschitz constant $\le 2$.
Let $\gamma$ be a minimal geodesic from $y_1$ to $y_2$, $s(x) = \min (d(y_1,x), d(y_2,x))$ and $h(x)=\min_{t} d(x, \gamma(t))$, the height from $x$ to a minimal geodesic $\gamma(t)$ connecting $y_1$ and $y_2$. By the triangle inequality $0 \le E(x) \le 2 h(x)$. Applying the Laplacian comparison (Theorem~\ref{laplacian-comp}) to $E(x)$ and with an elaborate (quantitative) use of the maximum principle (Theorem~\ref{maximum-p}) Abresch-Gromoll showed that if $\mathrm{Ric}_M \ge 0$ and $h(x) \le \frac{s(x)}{2}$, then (\cite{Abresch-Gromoll1990}, see also \cite{Cheeger1991})
\begin{equation} E(x) \le 4 \left(\frac{h^{n}}{s}\right)^{\frac{1}{n-1}}. \label{excess} \end{equation} This is the first distance estimate in terms of a lower Ricci curvature bound.
The following version (not assuming $E(p) = 0$, but without the sharp estimate) is from \cite[Theorem 9.1]{Cheeger2001}.
\begin{theo}[Excess Estimate, Abresch-Gromoll, 1990] \label{Abresch-Gromoll} If $M^n$ has $\mathrm{Ric}_M \ge -(n-1)\delta$, and for $p \in M$, $s(p) \ge L$ and $E(p) \le \epsilon$, then on $B(p, R)$, $E \le \Psi = \Psi(\delta,L^{-1},\epsilon |\,n,R)$, where $\Psi$ is a nonnegative constant such that for fixed $n$ and $R$ $\Psi$ goes to zero as $\delta,\epsilon \rightarrow 0$ and $L \rightarrow \infty$. \end{theo}
This can be interpreted as a weak almost splitting theorem. Cheeger-Colding generalized this result tremendously by proving the following almost splitting theorem \cite{Cheeger-Colding1996}, see also \cite{Cheeger2001}. \begin{theo}[Almost Splitting, Cheeger-Colding, 1996] \label{almost-splitting} With the same assumptions as Theorem~\ref{Abresch-Gromoll}, there is a length space $X$ such that for some ball, $B((0,x), \frac{1}{4}R) \subset \mathbb R \times X$, with the product metric, we have \[ d_{GH} \left( B(p, \frac{1}{4}R), B((0,x), \frac{1}{4}R) \right) \le \Psi. \] \end{theo} Note that $X$ here may not be smooth, and the Hausdorff dimension could be smaller than $n-1$. Examples also show that the ball $B(p, \frac{1}{4}R)$ may not have the topology of a product, no matter how small $\delta, \epsilon$, and $L^{-1}$ are \cite{Anderson1992, Menguy2000-noncollapsing}.
The proof is quite involved. Using the Laplacian comparison, the maximum principle, and Theorem~\ref{Abresch-Gromoll} one shows that the distance function $b_i = d(x,y_i)-d(p,y_i)$ associated to $p$ and $y_i$ is uniformly close to $\mathbf b_i$, the harmonic function with same values on $\partial B(p,R)$. From this, together with the lower bound for the smallest eigenvalue of the Dirichlet problem on $B(p,R)$ (see Theorem~\ref{thm-segment}) one shows that $\nabla b_i, \nabla \mathbf b_i$ are close in the $L_2$ sense. In particular $\nabla \mathbf b_i$ is close to $1$ in the $L_2$ sense. Then applying the Bochner formula to $\mathbf b_i$ multiplied with a cut-off function with bounded Laplacian one shows that
$| \mathrm{Hess} \mathbf b_i|$ is small in the $L_2$ sense in a smaller ball. Finally, in the most significant step, by using the segment inequality (\ref{segment}), the gradient estimate (\ref{gradient}) and the information established above one derives a quantitative version of the Pythagorean theorem, showing that the ball is close in the Gromov-Hausdorff sense to a ball in some product space; see \cite{Cheeger-Colding1996, Cheeger2001}.
An immediate application of the almost splitting theorem is the extension of the splitting theorem to the limit space. \begin{theo}[Cheeger-Colding, 1996] If $M_i^n$ has $\mathrm{Ric}_{M_i} \ge -(n-1)\delta_i$ with $\delta_i \rightarrow 0$ as $i\rightarrow \infty$, converges to $Y$ in the pointed Gromov-Hausdorff sense, and $Y$ contains a line, then $Y$ is isometric to $\mathbb R \times X$ for some length space $X$. \end{theo}
Similarly, one has almost rigidity in the presence of finite diameter (with simpler a proof) \cite[Theorem 5.12]{Cheeger-Colding1996}. As a special consequence, we have that if $M_i^n$ has $\mathrm{Ric}_{M_i} \ge (n-1)$, $\mathrm{diam}_{M_i} \rightarrow \pi$ as $i\rightarrow \infty$, and converges to $Y$ in the Gromov-Hausdorff sense, then $Y$ is isometric to the spherical metric suspension of some length space $X$ with $\mathrm{diam} (X) \le \pi$. This is a kind of stability for diameter.
Along the same lines (with more complicated technical details) Cheeger and Colding \cite{Cheeger-Colding1996} have an almost rigidity version for the volume annulus implies metric annulus theorem (see Section 5). As a very nice application to the asymptotic cone, they showed that if $M^n$ has $\mathrm{Ric}_{M} \ge 0$ and has Euclidean volume growth, then every asymptotic cone of $M$ is a metric cone.
\sect{The structure of limit spaces} \label{limit}
As we have seen, understanding the structure of the limit space of manifolds with lower Ricci curvature bound often helps in understanding the structure of the sequence. Cheeger-Colding made significant progress in understand the regularity and geometric structure of the limit spaces \cite{Cheeger-Colding1997, Cheeger-Colding2000II, Cheeger-Colding2000III}. On the other hand Menguy constructed examples showing that the limit space could have infinite topology in an arbitrarily small neighborhood \cite{Menguy2000-noncollapsing}. In \cite{Sormani-Wei2001,Sormani-Wei2004} Sormani-Wei showed that the limit space has a universal cover.
Let $(Y^m,y)$ (Hausdorff dimension $m$) be the pointed Gromov-Hausdorff limit of a sequence of Riemannian manifolds $(M^n_i, p_i)$ with $\mathrm{Ric}_{M_i} \ge -(n-1)$. Then $m \le n$ and $Y^m$ is locally compact. Moreover Cheeger-Colding \cite{Cheeger-Colding1997} showed that if $m = \dim Y < n$, then $m \le n-1$.
The basic notion for studying the infinitesimal structure of the limit space $Y$ is that of a tangent cone. \begin{defn} A tangent cone, $Y_y$, at $y\in (Y^m,d)$ is the pointed Gromov-Hausdorff limit of a sequence of the rescaled spaces $(Y^m,r_id,y)$, where $r_i \rightarrow \infty$ as $i \rightarrow \infty$. \end{defn}
By Gromov's precompactness theorem (Theorem~\ref{precompactness}), every such sequence has a converging subsequence. So tangent cones exist for all $y \in Y^m$, but might depend on the choice of convergent sequence. Clearly if $M^n$ is a Riemannian manifold, then the tangent cone at any point is isometric to $\mathbb R^n$. Motivated by this one defines \cite{Cheeger-Colding1997}
\begin{defn} A point, $y\in Y$, is called $k$-regular if for some $k$, every tangent cone at $y$ is isometric to $\mathbb R^k$. Let $\mathcal R_k$ denote the set of $k$-regular points and $\mathcal R = \cup_k \mathcal R_k$, the regular set. The singular set, $Y\setminus \mathcal R$, is denoted $\mathcal S$.
\end{defn}
Let $\mu$ be a renormalized limit measure on $Y$ as in (\ref{renormalized-vol}). Cheeger-Colding showed that the regular points have full measure \cite{Cheeger-Colding1997}.
\begin{theo}[Cheeger-Colding, 1997] For any renormalized limit measure $\mu$, $\mu(\mathcal S) =0$, in particular, the regular points are dense. \end{theo}
Furthermore, up to a set of measure zero, $Y$ is a countable union of sets, each of which is bi-Lipschitz equivalent to a subset of Euclidean space \cite{Cheeger-Colding2000III}. \begin{defn} A metric measure space, $(X, \mu)$, is called $\mu$-rectifiable if $0<\mu(X) <\infty$, and there exists $N < \infty$ and a countable collection of subsets, $A_j$, with $\mu(X\setminus \cup_jA_j) = 0$, such that each $A_j$ is bi-Lipschitz equivalent to a subset of $\mathbb R^{l(j)}$, for some $1 \le l(j) \le N$ and in addtion, on the sets $A_j$, the measures $\mu$ and and the Hausdorff measure $\mathcal H^{l(j)}$ are mutually absolutely continous. \end{defn}
\begin{theo}[Cheeger-Colding, 2000] Bounded subsets of $Y$ are $\mu$-rectifiable with respect to any renormalized limit measure $\mu$. \end{theo}
At the singular points, the structure could be very complicated. Following a related earlier construction of Perelman \cite{Perelman1997}, Menguy constructed 4-dimensional examples of (noncollapsed) limit spaces with, $\mathrm{Ric}_{M_i^n} >1$, for which there exists point so that any neighborhood of the point has infinite second Betti number \cite{Menguy2000-noncollapsing}. See \cite{Cheeger-Colding1997, MR1781925, Menguy2001} for examples of collapsed limit space with interesting properties.
Although we have very good regularity results, not much topological structure is known for the limit spaces in general. E.g., is $Y$ locally simply connected? Although this is unknown, using the renormalized limit measure and the existence of regular points, together with $\delta$-covers, Sormani-Wei \cite{Sormani-Wei2001,Sormani-Wei2004} showed that the universal cover of $Y$ exists. Moreover when $Y$ is compact, the fundamental group of $M_i$ has a surjective homomorphism onto the group of deck transforms of $Y$ for all $i$ sufficiently large.
When the sequence has the additional assumption that \begin{equation} \mathrm{Vol} (B(p_i,1)) \ge v >0, \end{equation} the limit space $Y$ is called noncollapsed. This is equivalent to $m=n$. In this case, more structure is known. \begin{defn} Given $\epsilon > 0$, the $\epsilon$-regular set, $\mathcal R_\epsilon$, consists of those points $y$ such that for all sufficiently small $r$, \[ d_{GH} (B(y,r), B(0,r)) \le \epsilon r, \] where $0 \in \mathbb R^n$. \end{defn} Clearly $\mathcal R = \cap_\epsilon \mathcal R_\epsilon$. Let $\stackrel{\circ}{\mathcal R}_\epsilon$ denote the interior of $\mathcal R_\epsilon$. \begin{theo}[Cheeger-Colding 1997, 2000] There exists $\epsilon (n) > 0$ such that if $Y$ is a noncollapsed limit space of the sequence $M_i^n$ with $\mathrm{Ric}_{M_i} \ge -(n-1)$, then for $0 < \epsilon < \epsilon(n)$, the set $\stackrel{\circ}{\mathcal R}_\epsilon$ is $\alpha(\epsilon)$-bi-H\"older equivalent to a smooth connected Riemannian manifold, where $\alpha(\epsilon) \rightarrow 1$ as $\epsilon \rightarrow 0$. Moreover, \begin{equation} \dim(Y\setminus \stackrel{\circ}{\mathcal R}_\epsilon ) \le n-2. \end{equation} In addition, for all $y \in Y$, every tangent cone $Y_y$ at $y$ is a metric cone and the isometry group of $Y$ is a Lie group. \end{theo}
This is proved in \cite{Cheeger-Colding1997, Cheeger-Colding2000II}.
If, in addition, Ricci curvature is bounded from two sides, we have stronger regularity \cite{Anderson-MR1074481}.
\begin{theo}[Anderson, 1990] \label{twoside} There exists $\epsilon (n) > 0$ such that if $Y$ is a noncollapsed limit space of the sequence $M_i^n$ with $| \mathrm{Ric}_{M_i} | \le n-1$, then for $0 < \epsilon < \epsilon(n)$, $\mathcal R_\epsilon = \mathcal R$. In particular the singular set is closed. Moreover, $\mathcal R$ is a $C^{1,\alpha}$ Riemannian manifold, for all $\alpha <1$. If the metrics on $M_i^n$ are Einstein, $\mathrm{Ric}_{M^n_i} = (n-1)Hg_i$, then the metric on $\mathcal R$ is actually $C^\infty$. \end{theo}
Many more regularity results are obtained when the sequence is Einstein, K\"ahler, has special holonomy, or has bounded $L^p$-norm of the full curvature tensor, see \cite{Anderson-Cheeger1991, Cheeger2003, Cheeger-Colding-Tian2002, Cheeger-Tian2005}, especially \cite{Cheeger2002} which gives an excellent survey in this direction. See the recent work \cite{Cheeger-Tian2006} for Einstein 4-manifolds with possible collapsing.
\sect{Examples of manifolds with nonnegative Ricci curvature} \label{examples}
Many examples of manifolds with nonnegative Ricci curvature have been constructed, which contribute greatly to the study of manifolds with lower Ricci curvature bound. We only discuss the examples related to the basic methods here, therefore many specific examples are unfortunately omitted (some are mentioned in the previous sections). There are mainly three methods: fiber bundle construction, special surgery, and group quotient, all combined with warped products. These method are also very useful in constructing Einstein manifolds. A large class of Einstein manifolds is also provided by Yau's solution of Calabi conjecture.
Note that if two compact Riemannian manifolds $M^m, N^n (n,m \ge 2)$ have positive Ricci curvature, then their product has positive Ricci curvature, which is not true for sectional curvature but only needs one factor positive for scalar curvature. Therefore it is natural to look at the fiber bundle case. Using Riemannian submersions with totally geodesic fibers J. C. Nash \cite{Nash1979}, W. A. Poor \cite{Poor1975}, and Berard-Bergery \cite{Berard-Bergery1978} showed that the compact total space of a fiber bundle admits a metric of positive Ricci curvature if the base and the fiber admit metrics with positive Ricci curvature and if the structure group acts by isometries. Furthermore, any vector bundle of rank $\ge 2$ over a compact manifold with $\mathrm{Ric} >0$ carries a complete metric with positive Ricci curvature. In \cite{Gilkey-Park-Tuschmann1998} Gilkey-Park-Tuschmann showed that a principal bundle $P$ over a compact manifold with $\mathrm{Ric} >0$ and compact connected structure group $G$ admits a $G$ invariant metric with positive Ricci curvature if and only if $\pi_1(P)$ is finite. Unlike the product case, the corresponding statements for $\mathrm{Ric} \ge 0$ are not true in all these cases, e.g. the nilmanifold $S^1 \rightarrow N^3\rightarrow T^2$ does not admit a metric with $\mathrm{Ric} \ge 0$. On the other hand Belegradek-Wei \cite{Belegradek-Wei2004} showed that it is true in the stable sense. Namely, if $E$ is the total space of a bundle over a compact base with $\mathrm{Ric} \ge 0$, and either a compact $\mathrm{Ric} \ge 0$ fiber or vector space as fibers, with compact structure group acting by isometry, then $E\times \mathbb R^p$ admits a complete metric with positive Ricci curvature for all sufficiently large $p$. See \cite{Wraith} for an estimate of $p$.
Surgery constructions are very successful in constructing manifolds with positive scalar curvature, see Rothenberg's article in this volume. Sha-Yang \cite{Sha-Yang1989, Sha-Yang1991} showed that this is also a useful method for constructing manifolds with positive Ricci curvature in special cases. In particular they showed that if $M^{m+1}$ has a complete metric with $\mathrm{Ric} >0$, and $n, m \ge 2$, then $S^{n-1} \times \left( M^{m+1} \setminus \coprod_{i=0}^k D_i^{m+1}\right) \bigcup_{Id} D^n \times \coprod_{i=0}^k S_i^{m}$, which is diffeomorphic to $\left( S^{n-1} \times M^{m+1}\right) \# \left( \#_{i=1}^{k} S^n \times S^m \right)$, carries a complete metric with $\mathrm{Ric} >0$ for all $k$, showing that the total Betti number of a compact Riemannian $n$-manifold ($n \ge 4$) with positive Ricci curvature could be arbitrarily large. See also \cite{Anderson1992}, and \cite{Wraith1998} when the gluing map is not the identity.
Note that a compact homogeneous space admits an invariant metric with positive Ricci curvature if and only if the fundamental group is finite \cite[Proposition 3.4]{Nash1979}. This is extended greatly by Grove-Ziller \cite{Grove-Ziller2002} showing that any cohomogeneity one manifold $M$ admits a complete invariant metric with nonnegative Ricci curvature and if $M$ is compact then it has positive Ricci curvature if and only if its fundamental group is finite (see also \cite{Schwachhofer-Tuschmann2004}). Therefore, the fundamental group is the only obstruction to a compact manifold admitting a positive Ricci curvature metric when there is enough symmetry. It remains open what the obstructions are to positive Ricci curvature besides the restriction on the fundamental group and those coming from positive scalar curvature (such as the $\hat A$-genus).
Of course, another interesting class of examples are given by Einstein manifolds. For these, besides the ``bible" on Einstein manifolds \cite{Besse1987}, one can refer to the survey book \cite{Einstein-survey-book} for the development after \cite{Besse1987}, and the recent articles \cite{Boyer-Galicki-preprint, Bohm-Wang-Ziller2004} for Sasakian Einstein metrics and compact homogenous Einstein manifolds.
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The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 1. The principal flow
M. T. Gallagher, D. J. Needham, J. Billingham
Journal: Journal of Fluid Mechanics / Volume 841 / 25 April 2018
Published online by Cambridge University Press: 20 February 2018, pp. 109-145
Print publication: 25 April 2018
The free surface and flow field structure generated by the uniform acceleration (with dimensionless acceleration $\unicode[STIX]{x1D70E}$ ) of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to the exterior horizontal, as it advances ( $\unicode[STIX]{x1D70E}>0$ ) or retreats ( $\unicode[STIX]{x1D70E}<0$ ) from an initially stationary and horizontal strip of inviscid incompressible fluid under gravity, are studied in the small-time limit via the method of matched asymptotic expansions. This work generalises the case of a uniformly accelerating plate advancing into a fluid as studied by Needham et al. (Q. J. Mech. Appl. Maths, vol. 61 (4), 2008, pp. 581–614). Particular attention is paid to the innermost asymptotic regions encompassing the initial interaction between the plate and the free surface. We find that the structure of the solution to the governing initial boundary value problem is characterised in terms of the parameters $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D707}$ (where $\unicode[STIX]{x1D707}=1+\unicode[STIX]{x1D70E}\tan \unicode[STIX]{x1D6FC}$ ), with a bifurcation in structure as $\unicode[STIX]{x1D707}$ changes sign. This bifurcation in structure leads us to question the well-posedness and stability of the governing initial boundary value problem with respect to small perturbations in initial data in the innermost asymptotic regions, the discussion of which will be presented in the companion paper Gallagher et al. (J. Fluid Mech. vol. 841, 2018, pp. 146–166). In particular, when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{+}$ , the free surface close to the initial contact point remains monotone, and encompasses a swelling jet when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times [1,\infty )$ or a collapsing jet when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times (0,1)$ . However, when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{-}$ , the collapsing jet develops a more complex structure, with the free surface close to the initial contact point now developing a finite number of local oscillations, with near resonance type behaviour occurring close to a countable set of critical plate angles $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{n}^{\ast }\in (0,\unicode[STIX]{x03C0}/2)$ ( $n=1,2,\ldots$ ).
The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 2. Well-posedness and stability of the principal flow
We consider the problem of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to the horizontal, accelerating uniformly from rest into, or away from, a semi-infinite strip of inviscid, incompressible fluid under gravity. Following on from Gallagher et al. (J. Fluid Mech., vol. 841, 2018, pp. 109–145) (henceforth referred to as GNB), it is of interest to analyse the well-posedness and stability of the principal flow with respect to perturbations in the initially horizontal free surface close to the plate contact point. In particular we find that the solution to the principal unperturbed problem, denoted by [IBVP] in GNB, is well-posed and stable with respect to perturbations in initial data in the region of interest, localised close to the contact point of the free surface and the plate, when the plate is accelerated with dimensionless acceleration $\unicode[STIX]{x1D70E}\geqslant -\cot \,\unicode[STIX]{x1D6FC}$ , while the solution to [IBVP] is ill-posed with respect to such perturbations in the initial data, when the plate is accelerated with dimensionless acceleration $\unicode[STIX]{x1D70E}<-\cot \,\unicode[STIX]{x1D6FC}$ . The physical source of the ill-posedness of the principal problem [IBVP], when $\unicode[STIX]{x1D70E}<-\cot \,\unicode[STIX]{x1D6FC}$ , is revealed to be due to the leading-order problem in the innermost region localised close to the initial contact point being in the form of a local Rayleigh–Taylor problem. As a consequence of this mechanistic interpretation we anticipate that, when the plate is accelerated with $\unicode[STIX]{x1D70E}<-\cot \,\unicode[STIX]{x1D6FC}$ , the inclusion of weak surface tension effects will restore well-posedness of the problem [IBVP] which will, however, remain temporally unstable.
Excavation of a Bronze Age Funerary Cairn at Manor Farm, near Borwick, North Lancashire
A. C. H. Olivier, T. Clare, P. M. Day, D. Gurney, D. Haddon-Reece, F. Healy, J. D. Henderson, L. Hocking, C. Howard-Davis, M. Hughes, R. T. Jones, H. C. M. Keeley, S. P. Needham, J. Sly, M. van der Veen
Journal: Proceedings of the Prehistoric Society / Volume 53 / Issue 1 / 1987
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The excavation of a large circular dished earthwork near Carnforth, North Lancashire, in 1982, has revealed a substantial Bronze Age funerary monument. The earliest structure was a sub-rectangular enclosure of limestone boulders dated to c. 1740–1640 BC cal. and associated with parts of two poorly preserved inhumation burials lying on the previously cleared ground surface. Both burials were accompanied by typologically early metalwork. The central inhumation was associated with a flat axe and dagger, suggesting an individual of high status as well as providing an important link between the early stages of development of both bronze types. The subsequent overlying cairn of smaller stones included eleven fairly discrete concentrations of inhumed bone, and seven of cremated bone and pottery. All this material was extremely fragmentary, and was probably derived from later re-use of the cairn.
By Ioannis P. Androulakis, Djillali Annane, Gérard Audibert, Lisa L. Barnes, Paolo Bartolomeo, Walter S. Bartynski, David A. Bennett, Nicolas Bruder, Nathan E. Brummel, Steve E. Calvano, Alain Cariou, F. Chretien, Jan Claassen, Colm Cunningham, Souhayl Dahmani, Robert Dantzer, Dimitry S. Davydow, Sanjay V. Desai, E. Wesley Ely, Frédéric Faugeras, Karen J. Ferguson, Brandon Foreman, Sadanand M. Gaikwad, Rebecca F. Gottesman, Maura A. Grega, Richard D. Griffiths, Marion Griton, Stefan D. Gurney, Hebah M. Hefzy, Michael T. Heneka, Dustin M. Hipp, Ramona O. Hopkins, Christopher G. Hughes, James C. Jackson, Christina Jones, Peter W. Kaplan, Keith W. Kelley, Raymond C. Koehler, Matthew A. Koenig, Jan Pieter Konsman, Felix Kork, John P. Kress, Stephen F. Lowry, Alawi Luetz, David Luis, Alasdair M. J. MacLullich, Guy M. McKhann, Jean Mantz, Panteleimon D. Mavroudis, Mervyn Maze, Bruno Mégarbane, Lionel Naccache, Dale M. Needham, Pratik P. Pandharipande, Jean-Francois Payen, V. Hugh Perry, Margaret Pisani, C. Rauturier, Benjamin Rohaut, Jennifer Ryan, Robert D. Sanders, Jeremy D. Scheff, Frederic Sedel, Ola A. Selnes, Tarek Sharshar, Martin Siegemund, Yoanna Skrobik, Jamie W. Sleigh, Romain Sonneville, Claudia D. Spies, Luzius A. Steiner, Robert D. Stevens, Raoul Sutter, Fabio Silvio Taccone, Richard E. Temes, Willem A. van Gool, Christel C. Vanbesien, F. Verdonk, Odile Viltart, Julia Wendon, Catherine N. Widmann, Robert S. Wilson
Edited by Robert D. Stevens, Tarek Sharshar, E. Wesley Ely, Vanderbilt University, Tennessee
Book: Brain Disorders in Critical Illness
Print publication: 19 September 2013, pp viii-xii
Post-traumatic stress disorder symptoms after acute lung injury: a 2-year prospective longitudinal study
O. J. Bienvenu, J. Gellar, B. M. Althouse, E. Colantuoni, T. Sricharoenchai, P. A. Mendez-Tellez, C. Shanholtz, C. R. Dennison, P. J. Pronovost, D. M. Needham
Journal: Psychological Medicine / Volume 43 / Issue 12 / December 2013
Published online by Cambridge University Press: 26 February 2013, pp. 2657-2671
Survivors of critical illnesses often have clinically significant post-traumatic stress disorder (PTSD) symptoms. This study describes the 2-year prevalence and duration of PTSD symptoms after acute lung injury (ALI), and examines patient baseline and critical illness/intensive care-related risk factors.
This prospective, longitudinal cohort study recruited patients from 13 intensive care units (ICUs) in four hospitals, with follow-up 3, 6, 12 and 24 months after ALI onset. The outcome of interest was an Impact of Events Scale – Revised (IES-R) mean score ⩾1.6 ('PTSD symptoms').
During the 2-year follow-up, 66/186 patients (35%) had PTSD symptoms, with the greatest prevalence by the 3-month follow-up. Fifty-six patients with post-ALI PTSD symptoms survived to the 24-month follow-up, and 35 (62%) of these had PTSD symptoms at the 24-month follow-up; 50% had taken psychiatric medications and 40% had seen a psychiatrist since hospital discharge. Risk/protective factors for PTSD symptoms were pre-ALI depression [hazard odds ratio (OR) 1.96, 95% confidence interval (CI) 1.06–3.64], ICU length of stay (for a doubling of days, OR 1.39, 95% CI 1.06–1.83), proportion of ICU days with sepsis (per decile, OR 1.08, 95% CI 1.00–1.16), high ICU opiate doses (mean morphine equivalent ⩾100 mg/day, OR 2.13, 95% CI 1.02–4.42) and proportion of ICU days on opiates (per decile, OR 0.83, 95% CI 0.74–0.94) or corticosteroids (per decile, OR 0.91, 95% CI 0.84–0.99).
PTSD symptoms are common, long-lasting and associated with psychiatric treatment during the first 2 years after ALI. Risk factors include pre-ALI depression, durations of stay and sepsis in the ICU, and administration of high-dose opiates in the ICU. Protective factors include durations of opiate and corticosteroid administration in the ICU.
X-ray powder diffraction analysis of tegafur
F. Needham, J. Faber, T. G. Fawcett, D. H. Olson
Journal: Powder Diffraction / Volume 21 / Issue 3 / September 2006
An experimental X-ray powder diffraction pattern was produced and analyzed for alpha-polymorphic tegafur, also called Ftorafur (an antineoplastic agent). The indexed data matched the powder patterns in the ICDD PDF-4/Organics database calculated from the reported single-crystal X-ray diffraction data in the Cambridge Structural Database. Alpha tegafur has a triclinic crystal system, with reduced cell parameters of a=16.720(6) Å, b=9.021(5) Å, c=5.995(3) Å, α=93.66(4)°, β=93.15(8)°, γ=100.14(4)°. There are four formula units contained in one unit cell. The cell volume and space group were determined to be 886.27 Å3 and P-1, respectively.
Deformation in Moffat Shale detachment zones in the western part of the Scottish Southern Uplands
D. T. NEEDHAM
Journal: Geological Magazine / Volume 141 / Issue 4 / July 2004
A study of the décollement zones in the Moffat Shale Group in the Ordovician Northern Belt of the Southern Uplands of Scotland reveals a progressive sequence of deformation and increased channelization of fluid flow. The study concentrates on exposures of imbricated Moffat Shale on the western coast of the Rhins of Galloway. Initial deformation occurred in partially lithified sediments and involved stratal disruption and shearing of the shales. Deformation then became more localized in narrower fault zones characterized by polyphase hydrothermal fluid flow/veining events. Deformation continued after vein formation, resulting in the development of low-temperature crystal plastic microstructures and further veining. Late-stage deformation is recorded as a pressure solution event possibly reflecting the cessation of slip on these faults as the slice became accreted. Most deformation can be ascribed to SE-directed thrusting and incorporation of the individual sheets into the Southern Uplands thrust stack. Later sinistral shear deformation, not observed in overlying turbidites, is also localized in these fault zones. The study reveals the likely structures formed at levels of an accretionary prism deforming under diagenetic to low-grade metamorphic conditions.
Asymmetric extensional structures and their implications for the generation of melanges
Studies of melanges in the Shimanto Belt of southwest Japan have shown that many of the included blocks exhibit asymmetric geometries, similar to boudins and inclusions in medium grade metamorphic rocks which have been subjected to markedly non-coaxial strains. Deformation is accomplished by mesoscopically ductile processes such as pore-pressure-controlled or diffusionally-controlled grain boundary sliding, although localized fracture and cataclasis also occurs. It is suggested that the melanges developed by the propagation of extensional displacement zones, analogous to shear bands, through sandstone beds, so progressively dismembering them. Studies of the detailed internal geometries of melanges along with the deformation mechanisms active during their formation may help resolve the mode of formation of these problematic units.
Nitrogen requirement of cereals: 1. Response curves
D. A. Boyd, Lowsing T. K. Yuen, P. Needham
Journal: The Journal of Agricultural Science / Volume 87 / Issue 1 / August 1976
Examples of response surfaces for pairs of nutrients and results of 41 multi-level experiments with N only were used to compare the goodness-of-fit of polynomial, inverse polynomial, exponential and intersecting-straight-lines models.
Whereas no one model fitted best at every site, many results were well represented by two intersecting straight lines and on average, this model had the least residual mean square. Of 17 experiments with spring barley in south western England the few results best represented by smooth curves were from crops much affected by leaf diseases.
Fertilizer response was poorly represented by models without a falling asymptote, like the simple exponential and inverse linear. Study of residuals after fitting the quadratic showed that this widely used model consistently over-estimated both the amount of fertilizer needed for maximum yield and the yield loss when too much fertilizer was given.
When fitted to the mean yields of each nitrogen treatment, most models had residual mean squares equal to or less than the error mean square, repeating a result obtained at Rothamsted as early as 1927. We question the validity of some well-known evidence for block and treatment additivity.
For 12 experiments in 1970, between-site differences in the parameter values of the two straight lines representing grain yield were related to leaf area at ear emergence.
3. Colonial Retrospect
Frederick Needham, D. Hack Tuke, D. H. T
Journal: Journal of Mental Science / Volume 32 / Issue 140 / January 1887
Dr. Manning is able to give an encouraging account of lunacy matters in the Colony which has been fortunate enough to secure his services, and of the several asylums which have come under his supervision during the year 1885. | CommonCrawl |
Stabilizers of pairs of ternary quadratic forms
Let $A,B$ be two ternary quadratic forms with real coefficients, given by symmetric matrices
$$\displaystyle 2A = \begin{pmatrix} 2a_{11} & a_{12} & a_{13} \\ a_{12} & 2a_{22} & a_{23} \\ a_{13} & a_{23} & 2a_{33} \end{pmatrix}, 2B = \begin{pmatrix} 2b_{11} & b_{12} & b_{13} \\ b_{12} & 2b_{22} & b_{23} \\ b_{13} & b_{23} & 2b_{33} \end{pmatrix}.$$
Let $V_\mathbb{R}$ denote the 12 dimensional real vector space of $(A,B)$ over $\mathbb{R}$, and let $G(\mathbb{R}) = \operatorname{GL}_2(\mathbb{R}) \times \operatorname{SL}_3(\mathbb{R})$. Let $(r,g)$ be an element of $G(\mathbb{R})$, where
$$\displaystyle r = \begin{pmatrix} r_1 & r_2 \\ r_3 & r_4 \end{pmatrix}.$$
Then $(r,g)$ acts on $(A,B) \in V_\mathbb{R}$ by sending $(A,B)$ to $$(r_1 (gAg^T) + r_2 (gBg^T), r_3 (gAg^T) + r_4 (gBg^T)).$$
Bhargava, in his paper The density of discriminants of quartic rings and fields, stated that the stabilizer in $G(\mathbb{R})$ of an element $(A,B) \in V_\mathbb{R}$ has order 24 if $A,B$ have four common real zeroes over $\mathbb{P}^2$, order 8 if they have exactly one pair of real zeroes over $\mathbb{P}^2$, and order 4 if they have no common real zeroes. He simply said that "one easily checks" that this is the case. Can anyone give an explanation as to why this should be easy to see, and a proof of why it's true?
nt.number-theory quadratic-forms invariant-theory
Stanley Yao Xiao
Stanley Yao XiaoStanley Yao Xiao
It's actually order 8 for no real zeroes and order 4 for two real zeroes, not the other way around. (See the bottom of page 1038 of the paper.)
The symmetry groups are taken modulo $\{ \pm 1 \}$, so we work projectively in ${\rm PGL}_2({\bf R}) \times {\rm SL}_3({\bf R})$. We must assume that we're in the generic case that $A,B$ span a two-dimensional space of conics that vanish on four distinct points of ${\bf P}^2$ in general linear position (i.e., no three on a line). Then over ${\bf C}$ the stabilizer is always $S_4$, because it is known (and easy) that ${\bf PGL}_3$ acts simply-transitively on ordered four-tuples in general linear position. But over ${\bf R}$ we must use permutations that commute with complex conjugation. This conjugation acts on the four points as the identity, a simple transposition, or a double transposition according as four, two, or none of them are real. The commutators have orders $24$, $4$, and $8$ respectively, as claimed.
Noam D. ElkiesNoam D. Elkies
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History of "no positive definite ternary integral quadratic form is universal"?
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Fricke Klein method for isotropic ternary quadratic forms
The density of quartic polynomials whose Galois group is a subgroup of $D_4$
Binary quartic forms fixed by a 'large' matrix
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On certain solutions of a quadratic form equation
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How to relate a pair of $3 \times 3$ boxes of integers with a ternary norm form? | CommonCrawl |
\begin{document}
\title{$L^p$-improving estimates for averages on polynomial curves} \begin{abstract} In the combinatorial method proving of $L^p$-improving estimates for averages along curves pioneered by Christ \cite{christ1998}, it is desirable to estimate the average modulus (with respect to some uniform measure on a set) of a polynomial-like function from below using only the value of the function or its derivatives at some prescribed point. In this paper, it is shown that there is always a relatively large set of points (independent of the particular function to be integrated) for which such estimates are possible.
Inequalities of this type are then applied to extend the results of Tao and Wright \cite{tw2003} to obtain endpoint restricted weak-type estimates for averages over curves given by polynomials. \end{abstract}
The purpose of this paper is twofold. First, a somewhat surprising inequality for $L^1(\mu)$-norms of polynomial-type functions on the real line will be established. The most important special case of this inequality is as follows:
\begin{theorem}
Suppose that $K \subset {\mathbb R}$ is measurable. For any positive integer $n$ and any $0 < \epsilon < 1$, there is an interval $I$ with $|K \cap I| \geq \frac{1-\epsilon}{n} |K|$ (here $| \cdot |$ acting on sets denotes Lebesgue measure) and a constant $c_{n,\epsilon}$ such that \label{theorem0}
\begin{equation} \int_{K} |p(t)| dt \geq c_{n,\epsilon} |K|^{j+1} \sup_{t \in I} |p^{(j)}(t)| \label{ineq0} \end{equation} for any polynomial $p$ of degree $n$ or less and any $j=0,\ldots,n$. \end{theorem} It is fairly trivial to show that such an interval $I$ can always be found when $p$ is given, but somewhat unexpected that $I$ exists independently of $p$. It is also farily straightforward to construct a family of counterexamples to this inequality if one takes $\epsilon=0$, for example (which suggests that there must be a certain amount of subtlety involved in proving the positive result). This theorem and its generalization to regular probability measures $\mu$ will be taken up in the first section.
The second major purpose of this paper is to apply the inequality \eqref{ineq0} to answer a question of Tao and Wright \cite{tw2003} concerning $L^p$-improving bounds for averages along curves. In that work, inequalities of the form \eqref{ineq0} appear naturally while carrying out the combinatorial methods of Christ \cite{christ1998}. Tao and Wright were able to establish \eqref{ineq0} in the case when $K$ has some additional structure ($K$ was taken to be a central set of width $w$, see lemma 7.3); the price of requiring such structure was certain losses in exponents which made it impossible to obtain $L^p$-improving inequalities on the boundary of the type set. Interpreted in the framework of that paper, theorem \ref{theorem0} indicates that the set $K$ does not need any significant structure (namely, $K$ does not need to be a central set of width $w$) for the desirable inequality \eqref{ineq0} to hold. As a consequence, theorem \ref{theorem0} alone makes it possible to prove the full range of restricted weak-type estimates for one-dimensional averaging operators given by polynomial curves (that is, when both the averaging operator and the dual operator can be described by polynomial functions in appropriate coordinate systems).
This result is formulated in the standard bilinear way as follows: suppose $U$ is an open ball in ${\mathbb R}^{d+1}$ and one is given projections $\pi_1 : U \rightarrow {\mathbb R}^d$ and $\pi_2 : U \rightarrow {\mathbb R}^d$ such that the differentials $d \pi_1$ and $d \pi_2$ are surjective at every point. The Radon-like operator $R$ associated to these projections is defined by duality as \begin{equation}
\int_{{\mathbb R}^{d}} R f(y) g(y) dy := \int_U f (\pi_1(x)) g(\pi_2(x)) \psi(x) dx \label{theop} \end{equation} where $\psi$ is some bounded cutoff function (not necessarily smooth) supported in $U$. Next, let $X_1$ and $X_2$ be vector fields on $U$ which are nonvanishing and satisfy $d \pi_1 (X_1) = 0 = d \pi_2 (X_2)$, and suppose that for all words $w = (w_1,w_2,\ldots,w_k)$ (each $w_j$ equals $1$ or $2$) of sufficient length, the commutator \[ X_w := [ X_{w_1}, [ X_{w_2}, [ \cdots [X_{w_{k-1}},X_{w_k}] \cdots]]] \] vanishes identically on $U$. This geometric condition guarantees that both $R$ and $R^*$ are given by averages over polynomial curves. Under these conditions the following theorem holds: \begin{theorem} Suppose that the vector fields $X_1$, $X_2$ are as assumed above. Let $x_0 \in U$, and consider the mapping \label{averageop} \[ \Phi_{x_0}(t_1,\ldots,t_{d+1}) := \exp(t_1 X_1) \circ \cdots \circ \exp(t_{d+1} X_{d+1})(x_0) \] (where the periodicity convention $X_{j+2} = X_j$ is used). Let $J_{x_0}(t)$ be the Jacobian determinant of this mapping (as a function of the parameters $t$). If $\partial_t^\alpha J_{x_0}(t) \neq 0$ at $t=0$
for some multiindex $\alpha = (\alpha_1,\ldots,\alpha_{d+1})$, then the averaging operator $R$ given by \eqref{theop} satisfies a restricted weak-type estimate
\[ \left| \int_U \chi_F(\pi_1(x)) \chi_G(\pi_2(x)) \psi(x) dx \right| \leq C |F|^\frac{1}{p_1} |G|^\frac{1}{p_2} \] when the support of $\psi$ is a sufficiently small neighborhood of $x_0$ and $\frac{1}{p_1} := \frac{A_1}{A_1 + A_2 -1}, \frac{1}{p_2} := \frac{A_2}{A_1+A_2 - 1}$ where \begin{align*} A_1 & := \left\lceil \frac{d+1}{2} \right\rceil + \alpha_1 + \cdots + \alpha_{2\lceil \frac{d+1}{2} \rceil-1} , \ A_2 := \left\lfloor \frac{d+1}{2} \right\rfloor + \alpha_2 + \cdots + \alpha_{2\lfloor \frac{d+1}{2} \rfloor}. \end{align*} \end{theorem} It should be noted that the hypotheses of theorem \ref{averageop} (regarding the derivatives of the Jacobian determinant) are, in fact, equivalent to the H\"{o}rmander-type hypotheses used by Tao and Wright (see lemma 9.1 in \cite{tw2003} or sections 9 and 10 of the paper of Christ, Nagel, Stein, and Wainger \cite{cnsw1999} in which the double fibration curvature formulation $({\cal C}_{\Lambda})$ is shown to be equivalent to the Jacobian determinant formulation $({\cal C}_J)$). In light of this equivalence, theorem \ref{averageop} successfully establishes restricted weak-type estimates on the boundary of the type set of the operator \eqref{theop} which were just missed in \cite{tw2003}.
Regarding integral estimates and the related issue of sublevel sets, earlier results of particular interest to the problem at hand include the work of Carbery, Christ, and Wright \cite{ccw1999}, Phong, Stein, and Sturm \cite{pss2001}, and Phong and Sturm \cite{ps2000}, as well as many others. In this paper, the attention will be exclusively focused on one-dimensional estimates, using methods similar to those employed by Carbery, Christ and Wright \cite{ccw1999} who built upon ideas of Arhipov, Karacuba and \v{C}ubarikov \cite{akc1979}. Similar one-dimensional methods have also been employed by Rogers \cite{rogers2005} to obtain sharp constants for sublevel set estimates and van der Corput's lemma.
In the case of Radon-like transforms and averaging operators, the reader is referred to the papers of Tao and Wright \cite{tw2003} and Christ, Nagel, Stein, and Wainger \cite{cnsw1999} for more complete lists of references. In this paper, the argument to be followed was originally devised by Christ \cite{christ1998}. Tao and Wright \cite{tw2003} made important additions to the Christ argument which will, of course, be necessary to use here as well. More recently, these ideas have been adapted to a variety of other contexts by Christ and Erdo{\v{g}}an \cite{ce2002}, \cite{ce2008}, Bennett, Carbery, Christ, and Tao \cite{bcct2008}, Bennet, Carbery, and Wright \cite{bcw2005}, Erdo{\v{g}}an and R. Oberlin \cite{eo2008}, and many others. Earlier approaches to $L^p$-improving estimates for averaging operators, beginning with Littman \cite{littman1971}, Phong and Stein \cite{ps1986II}, \cite{ps1991}, \cite{ps1994}, \cite{ps1997}, including Greenleaf and Seeger \cite{gs1994}, \cite{gs1998}, Seeger \cite{seeger1993}, \cite{seeger1998}, and D. Oberlin \cite{oberlin1987}, \cite{oberlin1997}, \cite{oberlin1999}, have typically been based on oscillatory integral estimates which will not appear here.
\section{Estimation of integrals by pointwise values}
To begin this section, a number of definitions are in order. First, suppose $K$ is a closed set contained in an open interval $I$. For each nonnegative integer $n$, a function $f \in C^{n}(I)$ is said to be of polynomial type $n$ on $(K,I)$ when $f^{(n)}$ does not change sign (i.e., is nonnegative or nonpositive) and there exists a finite constant $C$ for which $\sup_{t \in I} |f^{(n)}(t)| \leq C \inf_{t \in K} |f^{(n)}(t)|$. Any polynomial of degree $n$ on $I$ is, of course, polynomial type $n$ on $(K,I)$ for any closed set $K$ contained in $I$.
In general, if a regular probability measure $\mu$ is supported on the closed set $K$, it will necessary to consider functions which are of polynomial type on $(K_\epsilon,I)$ for some set $K_\epsilon$ slightly larger than $K$ (since, if $K$ has Lebesgue measure zero, the values of $f$ on $K$ are largely independent of the values of the derivatives of $f$ on the same set $K$). To that end, given any closed set $K$ and any $\epsilon$, let $K_\epsilon$ be the union of $K$ and the sets $K_L$ and $K_R$ given by \begin{align*}
K_R & := \set{t \in I}{ \inf \set{d \geq 0}{t + d \in K} \leq \epsilon \inf \set{d \geq 0}{t - d \in K}}, \\ K_L & := \set{t \in I}{ \inf \set{d \geq 0}{t - d \in K} \leq \epsilon \inf \set{d \geq 0}{t + d \in K} } \end{align*}
($K_R$ and $K_L$ are the points which are bounded on both sides by the set $K$ but are proportionately much closer to $K$ on one side than the other; note that the Lebesgue measures of $K_L$ and $K_R$ are bounded by $\epsilon |I \setminus K|$).
The final definition needed to begin this section is a notion of the length of the set on which $\mu$ is supported: given a regular probability measure $\mu$ supported on an open interval $I \subset {\mathbb R}$, a positive integer $n$, and an $\epsilon \in (0,1)$, let $|\mu|_{n,\epsilon}$ be the infimum of $\sum_{j=1}^n |I_j|$ over all collections of closed intervals $\{I_1,\ldots,I_n\}$ which satisfy $\mu(\bigcup_{j=1}^n I_j) \geq 1-\epsilon$.
Note that $|\mu|_{n,\epsilon}$ is decreasing in $n$, increasing in $\epsilon$, and $|\mu_K|_{n,\epsilon} \geq (1-\epsilon)|K|$ when, for example, $\mu_K$ is normalized Lebesgue measure on the set $K$.
Finally, a remark concerning notation is in order. The two parameters having already appeared, namely $n$ and $\epsilon$, appear in essentially every inequality to come; in particular, most proportionality constants will vary as these parameters vary. When the nature of these constants is uninteresting or otherwise considered unimportant, the notation $A \lesssim B$ will be used to indicate that there is a constant $C_{n,\epsilon}$ completely determined by $\epsilon$ and $n$ such that $A \leq C_{n,\epsilon} B$.
The main theorems of this section can now be stated. The first is as follows: \begin{theorem} Let $K$ be a closed set contained in an open interval $I \subset {\mathbb R}$ (possibly infinite). For any $\epsilon \in (0,1)$, let $K_{\epsilon}$ be the union of $K$ with $K_R$ and $K_L$ where \label{theorem1}
For any positive integer $n$, any regular probability measure $\mu$ supported on $K$, and any $f \in C^{n}(I)$ for which $f^{(n)}$ does not change sign, it must be true that
\[ \int |f(t)| d \mu(t) \gtrsim |\mu|_{n,\epsilon}^n \inf_{t \in K_\epsilon} |f^{(n)}(t)|. \] \end{theorem}
Theorems of this type are not new; see, for example Carbery, Christ, and Wright \cite{ccw1999}, Arhipov, Karacuba and \v{C}ubarikov \cite{akc1979}, or Rogers \cite{rogers2005}. The main new feature is the presence of $|\mu|_{n,\epsilon}$ on the right-hand side; in most previous cases $\mu$ is assumed to be the uniform measure on some set $K$ and $|\mu|_{n,\epsilon}$ is replaced by $|K|$. To prove the full uniform estimate (theorem \ref{theorem2} and its corollaries), it is necessary to distinguish the length $|\mu|_{n,\epsilon}$ from the measure of the support of $\mu$.
The second theorem of this section establishes uniform integral estimates from below by a supremum of the function on a set $E$ which depends only on the class of functions to which $f$ belongs. This is, of course, the most difficult task necessary to establish any result along the lines of theorem \ref{theorem0}: \begin{theorem} Suppose $\mu$ is a regular probability measure supported on some closed $K \subset I$; fix some positive integer $n$ and $\epsilon \in (0,1)$. There exists a closed set $E \subset I$ with at most $n$ connected components for which $\mu(E) \geq 1 - \epsilon$ and \label{theorem2}
\[ \int |f(t)| d \mu(t) \gtrsim c_{n,\epsilon} C^{-1} \sup_{t \in E} |f(t)| \] for any function $f$ which is of polynomial type $n$ on $(K_\epsilon,I)$ with constant $C$. \end{theorem}
In particular, an immediate corollary of theorems \ref{theorem1} and \ref{theorem2} is the following: \begin{corollary} Given a regular probability measure $\mu$ supported on $K \subset I$, an $\epsilon \in (0,1)$ and a positive integer $n$, there exists a closed interval $I'$ (possibly a single point) with $\mu(I') \geq \frac{1-\epsilon}{n}$ such that, for any function $f$ which is polynomial type $n$ on $(K_\epsilon,I)$ with constant $C$, \label{maincorollary}
\[ \int |f(t)| d \mu(t) \gtrsim C^{-1} \min\{ |I'|^j, |\mu|_{n,\epsilon}^j \} \sup_{t \in I'} |f^{(j)}(t)| \] for any $j=0,\ldots,n$. \end{corollary} Theorem \ref{theorem0} from the introduction follows from this corollary when $f$ is polynomial of degree $n$ and $\mu$ is the uniform measure on $K$.
\subsection{Combinatorial considerations}
In what follows, let $V_n$ be the Vandermonde polynomial in $n$ variables, i.e., $V_n(t_1,\ldots,t_n) := \prod_{j > i} (t_j - t_i)$.
The proof begins with a more detailed look at a standard idea: the estimation of higher derivatives of a function $f$ via sampling at a finite number of points. This is typically carried out using Lagrange interpolating polynomials. The technique is the same here; the main difference is that there is that the structure of the ``remainder term'' is explored as well: \begin{proposition} Let $t_1, \ldots,t_{n+1}$ be distinct points in some interval $I$. Given these points, there exists a nonnegative function $\psi_t(s)$, supported on $[t_1,t_{n+1}]$, with total integral $1$ such that \label{averageprop} \begin{equation} \sum_{i=1}^{n+1} (-1)^{n+1-i} f(t_i) \frac{V_n(t_1,\ldots,\hat{t_i},\ldots,t_{n+1})}{V_{n+1}(t_1,\ldots,t_{n+1})} = \frac{1}{n!} \int f^{(n)}(s) \psi_t(s) ds \label{averages} \end{equation} for any $f \in C^{(n)}(I)$ (here $\hat{t_i}$ indicates that $t_i$ is omitted). This function $\psi_t$ has the following properties: $(1)$ $\psi_t$ is supported on the convex hull of the $t_j$'s and has integral $1$, $(2)$ $\psi_t$ is a polynomial of degree at most $n-1$ on each interval containing none of the $t_j$'s, and $(3)$ $\psi_t \in C^{(n-2)}(I)$ when $n \geq 2$. \end{proposition} \begin{proof} Let $a \in I$ be such that $a \leq \min_j t_j$. If $f$ is any $n$-times continuously differentiable function on $I$, let \[ g(t) := \int_a^t \frac{(t-s)^{n-1}}{(n-1)!} f^{(n)}(s) ds. \] It is straightforward to check that $g$ is also $n$-times differentiable when $t > a$ and $g^{(n)}(t) = f^{(n)}(t)$ there ($g$ is nothing more than the remainder term for the degree $n-1$ Taylor polynomial at $a$). This implies that $f-g$ is a polynomial of degree $n-1$ (the Taylor polynomial at $a$); therefore it must be the case that the determinant \begin{equation}
\left| \begin{array}{ccccc} f(t_1) - g(t_1) & 1 & t_1 & \cdots & t_1^{n-1} \\ f(t_2) - g(t_2) & 1 & t_2 & \cdots & t_2^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ f(t_{n+1}) - g(t_{n+1}) & 1 & t_{n+1} & \cdots & t_{n+1}^{n-1} \end{array}
\right| \label{zerodet1} \end{equation} vanishes. Using Cramer's rule and the fact that the minors corresponding to entries in the first column are Vandermonde matrices, it follows that \[ \sum_{i=1}^{n+1} (-1)^{i}(f(t_i) - g(t_i)) V_n(t_1,\ldots,\hat{t_i},\ldots,t_{n+1}) = 0. \] since the $n$-th derivative of the difference vanishes at every point of $[t_1,t_{n+1}]$. But this implies that \begin{align*} \sum_{i=1}^{n+1} & (-1)^{i}f(t_i) V_n(t_1,\ldots,\hat{t_i},\ldots,t_{n+1}) = \sum_{i=1}^{n+1} (-1)^{i}g(t_i) V_n(t_1,\ldots,\hat{t_i},\ldots,t_{n+1}) \\ & = \int_a^\infty f^{(n)}(s) \frac{1}{(n-1)!} \sum_{t_i > s} (-1)^i (t_i - s)^{n-1} V_{n} (t_1,\ldots, \hat{t_i},\ldots, t_{n+1}) ds. \end{align*} When $s \geq \max_j t_j$, the sum inside the integral vanishes because no terms are included. When $s \leq \min_j t_j$, on the other hand, the sum again vanishes since it represents the determinant of a Vandermonde-type matrix whose first column is given by $(t_i - s)^{n-1}$ (which vanishes just like \eqref{zerodet1}). To compute the integral of this function, it suffices to plug in $f(s) := \frac{s^n}{n!}$ : \[ \sum_{i=1}^{n+1} (-1)^{i}\frac{t_i^n}{n!} V_n(t_1,\ldots,\hat{t_i},\ldots,t_{n+1}) = \frac{(-1)^{n+1}}{n!} V_{n+1}(t_1,\ldots,t_{n+1}). \] Thus, one is led to define \begin{align*} \psi_t(s) := & \frac{n (-1)^{n+1}}{V_{n+1}(t_1,\ldots,t_{n+1})} \sum_{t_i > s} (-1)^i (t_i - s)^{n-1} V_{n} (t_1,\ldots, \hat{t_i},\ldots, t_{n+1}) \\ & = \sum_{t_i > s} \frac{n (t_i - s)^{n-1}}{\prod_{j \neq i} (t_i - t_j)}. \end{align*} This $\psi_t$ has integral one and satisfies the correct integral identity. Furthermore, it follows directly from this definition that $\psi_t$ is piecewise a polynomial of degree at most $n-1$ on all intervals not containing any $t_j$'s. It is also immediate that $\psi_t \in C^{(n-2)}(I)$ when $n \geq 2$ because it is a finite linear combination of such functions (i.e., functions equal to $(t_i-s)^{n-1}$ when $s \leq t_i$ and equal to $0$ otherwise). It remains to show that $\psi_t$ is nonnegative. If this were not the case, it would be possible to find a function $f \in C^{(n)}(I)$ such that $f^{(n)}$ is strictly positive on $I$ but the right-hand side of \eqref{averages} is zero. Examining \eqref{zerodet1}, this is possible only when there is a polynomial of degree $n-1$ which agrees with this function $f$ at $t_1,\ldots,t_{n+1}$. Repeated applications of Rolle's theorem shows that this cannot be the case (i.e., the $n$-th derivative of $f$ must vanish at some point if $f$ agrees with a polynomial of degree $n-1$ at $n+1$ distinct points).
\end{proof}
It is perhaps worth noting that the three properties of $\psi_t$ (support and normalization, piecewise polynomial of degree $n-1$, and global $C^{(n-2)}$ regularity) uniquely determine $\psi_t$. Even when the normalization condition is dropped, there is still only a one-dimensional family of such functions, namely, multiples of $\psi_t$; this is to say that there are no nontrivial piecwise functions satisfying these conditions and having integral $0$. The proof of this fact proceeds inductively on $n$ in a fairly standard way.
The advantage gained in finding $\psi_t$ more-or-less explicitly is that it allows one to improve upon the trivial estimate from below on $\int f^{(n)}(s) \psi_t(s) ds$ to exploit the fact that, while $\psi_t$ is not supported at the points $t_j$, it must always, in fact, have some positive proportion of its mass which lies near the $t_j$'s. In other words, if the $t_j$'s happen to be separated by some large distance, it never occurs that an overwhelming fraction of the mass of $\psi_t$ is concentrated inside that gap. This fact will be made precise during the proof of theorem \ref{theorem1}.
\begin{proposition} For any regular probability measure $\mu$ on an interval $I \subset {\mathbb R}$ and any positive integer $n$, let \label{intervals}
\[ (\ell_n(\mu))^n := \frac{\int |V_{n+1}(t_1,\ldots,t_{n+1})| d \mu(t_1) \cdots d \mu(t_{n+1})}{\int |V_n(t_1,\ldots,t_n)| d \mu(t_1) \cdots d \mu(t_n)}.\]
The quantity $\ell_n(\mu)$ is zero if and only if $\mu$ is supported on a set of $n$ or fewer points. Furthermore, given any $\epsilon > 0$, there exists a finite collection of at most $n$ closed, disjoint intervals $I_j$ (possibly length zero) such that $\mu( \bigcup_j I_j ) \geq 1 - \epsilon$, $| \bigcup_j I_j | \lesssim \ell_n(\mu)$ (that is, the Lebesgue measure of the union) and $\mu(I_j) \gtrsim 1$. These intervals will be called the $(n,\epsilon)$-children of $\mu$. \end{proposition} \begin{proof} First of all, it is necessarily true that $\ell_n(\mu) = 0$ if the mass of $\mu$ is supported on a finite set of $n$ or fewer points, since in this case the Vandermonde polynomial $V_{n+1}$ vanishes almost everywhere on the $(n+1)-$fold product of $\mu$. In all other cases, the distribution function $\mu((-\infty, t] \cap I)$ must take at least $n+1$ distinct, nonzero values, meaning that the interval $I$ may be partitioned into at least $n+1$ disjoint pieces, each of which has nonzero $\mu$-measure. Since $V_{n+1}$ does not vanish when each $t_i$ belongs to a distinct element of the partition, the integral cannot be zero.
Assuming now that $\mu$ is not supported on a set of $n$ points, consider the ratio
\[ \frac{|V_{n+1}(t_1,\ldots,t_{n+1})|}{|V_n(t_1,\ldots,t_n)|} = \prod_{i=1}^n |t_n - t_i|. \]
If $E_{t_1,\ldots,t_n}$ is the set $\set{ t_{n+1}}{ \prod_{i=1}^n |t_n - t_i| \leq 2 \epsilon^{-1} (\ell_n(\mu))^n}$, it follows that
\[ \int \frac{|V_{n+1}(t_1,\ldots,t_{n+1})|}{|V_n(t_1,\ldots,t_n)|} d \mu(t_{n+1}) \geq (1 - \mu(E_{t_1,\ldots,t_n})) 2 \epsilon^{-1} (\ell_n(\mu))^n \]
for each possible ensemble $t_1,\ldots,t_n$ of distinct points. Multiplying both sides by $|V_{n}(t_1,\ldots,t_n)|$ and integrating $d \mu(t_1) \cdots d \mu(t_n)$, it follows that \[ (\ell_n(\mu))^n \geq \inf_{t_1,\ldots,t_n \in I} \epsilon^{-1} (1 - \mu(E_{t_1,\ldots,t_n})) (\ell_n(\mu))^n \] so there must be a choice of $t_1,\ldots,t_n$ for which $\mu(E_{t_1,\ldots,t_n}) \geq 1- \frac{\epsilon}{2}$. This sublevel set consists of at most $n$ closed connected components. The Lebesgue measure of the sublevel set is at most $2n \ell_n(\mu)$ (since the sublevel condition requires, in particular, that $t_{n+1}$ must be within distance $\ell_n(\mu)$ of at least one of the $t_j$'s for $j=1,\ldots,n$). Let the $(n,\epsilon)$-children of $\mu$ be the connected components of $E_{t_1,\ldots,t_n}$ whose $\mu$ measure is at least $\frac{\epsilon}{2n}$. Clearly they are disjoint, have bounded lengths, and the $\mu$ measure of the union is at least $1 - \epsilon$. \end{proof}
\begin{proposition} Let $t_1,\ldots,t_{N}$ be points in some interval $I$. For each positive integer $n$, there is a closed set $E_n \subset I$ which consists of no more than $n$ connected components, contains $t_j$ for $j=1,\ldots,N$ and satisfies \label{closegaps}
\[ \sup_{t \in E_n} |f(t)| \leq (n+1) 2^n \max_{j=1,\ldots,N} |f(t_j)| \] for any $f \in C^{n}(I)$ whose $n$-th derivative does not change sign.
\end{proposition} \begin{proof} Without loss of generality, it suffices to assume that $t_1 < t_2 < \cdots < t_N$ and that $N \geq n+1$. It is also permissible to assume that $f^{(n)}$ is nonnegative.
Consider first the case $N = n+1$. Fix $t_1 < t_2 < \cdots < t_{n+1}$ and fix $k$ to be the index which maximizes $V_{n}(t_1,\ldots, \hat{t_k}, \ldots, t_{n+1})$; notice that the index can never equal $1$ or $n+1$ (since omitting $t_2$ or $t_{n}$, respectively, will always increase the product). Let $I'$ be the shorter interval of $[t_{k-1},t_k]$ or $[t_{k},t_{k+1}]$ (if they have the same length, either choice is acceptable). Given $f \in C^{(n)}(I)$, let \[ \tilde f(t) := f(t) - \sum_{j \neq k} f(t_j) \prod_{i \neq j,k} \frac{t - t_i}{t_j - t_i}. \] Clearly $\tilde f(t_j) = 0$ for $j \neq k$. It suffices to work with $\tilde f$ rather than $f$ since the values of $f$ and $\tilde f$ do not differ appreciably on $I'$ or at $t_k$. More precisely, at $t=t_k$ one has
\[ |\tilde f(t_k)| \leq |f(t_k)| + \sum_{j \neq k} |f(t_k)| \frac{|V_n(t_1,\ldots,\hat{t_j},\ldots,t_{n+1})|}{|V_n(t_1,\ldots,\hat{t_k},\ldots,t_{n+1})|} \leq (n+1) \max_j |f(t_j)|. \]
Next, if $t \in I'$, then $\prod_{i \neq j,k} |t - t_i| \leq 2^{n-1} \prod_{i \neq j,k} |t_k - t_i|$ since $|t - t_i| \leq |t - t_k| + |t_k - t_i| \leq \min_j \{ |t_k - t_j| \} + |t_k - t_i|$. It must therefore be the case that
\[|f(t)| \leq |\tilde f(t)| + n 2^{n-1} \max_j |f(t_j)|; \]
hence if $|\tilde f(t)| \leq C \max_j |\tilde f(t_j)|$, then $|f(t)| \leq (C(n+1) + n 2^{n-1}) \max_j |f(t_j)|$.
Now \eqref{averages} implies that, for $t \in I'$, $(-1)^{n+1-k} f(t) \geq 0$ when $f^{(n)}$ is nonnegative on $I$ by virtue of the fact that the Vandermonde polynomials are positive and $\tilde f(t_i)$ vanishes for $i \neq k$. Fix some $t \in I'$; let $t_1' < t_2' < \cdots < t_{n+1}'$ be the sequence of numbers obtained by replacing $t_{k-1}$ with $t$ when $I'=[t_{k-1},t_k]$ or replacing $t_{k+1}$ with $t$ when $I' = [t_{k},t_{k+1}]$ (so that $t_j' = t_j$ for all but one value of $j$). For this collection of points, \eqref{averages} implies that \[ \sum_{j=1}^{n+1} (-1)^{n+1-j} \tilde f(t_j') \frac{V_n(t_1',\ldots,\hat{t_j'},\ldots,t_{n+1}')}{V_{n+1}(t_1',\ldots,t_{n+1}')} \geq 0 \] as well. Here all but two terms must vanish (by virtue of the vanishing of $\tilde f$). Thus \[ (-1)^{n+1-k} \tilde f(t_k) V_n(t_1',\ldots,\hat{t_k'},\ldots,t_{n+1}') \geq (-1)^{n+1-k} \tilde f(t) V_n (t_1',\ldots, \hat{t}, \ldots, t_{n+1}') \]
(where $\hat{t}$ is properly interpreted as $\widehat{t_{k-1}'}$ or $\widehat{t_{k+1}'}$ depending on which of those indices had its corresponding value replaced by $t$). Both sides are nonnegative, and just as before, the Vandermonde polynomial on the left-hand side is at most a factor of $2^{n-1}$ larger than the corresponding polynomial on the right-hand side. It must therefore be the case that $|\tilde f(t)| \leq 2^{n-1} |\tilde f(t_k)|$ and hence $|f(t)| \leq (n+1) 2^n \max_j |f(t_j)|$.
For the case of general $N$, suppose that there were more than $n$ intervals of the form $[t_k,t_{k+1}]$ for which
\[ \sup_{t \in [t_j,t_{j+1}]} |f(t)| \geq (n+1) 2^n \max_{j=1,\ldots,N} |f(t_j)|. \] If $s_1,\ldots,s_{n+1}$ are the leftmost endpoints of such intervals, then there must exist an interval $I'$ on which
\[ \sup_{t \in I'} |f(t)| \leq (n+1) 2^n \max_{j=1,\ldots,n+1} |f(s_j)| \] for any function $f$ whose $n$-th derivative does not change sign. The leftmost endpoint of this interval coincides with one of the $s_j$'s, hence $I'$ must contain $[t_j,t_{j+1}]$, giving a contradiction. \end{proof}
\subsection{The proofs of theorems \ref{theorem1} and \ref{theorem2}}
\begin{proof}[Proof of theorem \ref{theorem1}.] The proof proceeds in two parts; first for any $f \in C^{n}(I)$, \begin{equation}
\int |f(t)| d \mu(t) \geq \frac{(\ell_n(\mu))^n}{(n+1)!} \inf_{t \in I} |f^{(n)}(t)|. \label{mineq1} \end{equation} Following this the general situation is considered: if the support of $\mu$ is contained in some closed set $K$ and $K_\epsilon \supset K$ is as defined in theorem \ref{theorem1} then \begin{equation}
\int |f(t)| d \mu(t) \gtrsim (\ell_n(\mu))^n \inf_{t \in K_\epsilon} f^{(n)}(t) \label{mineq2} \end{equation} provided that $f^{(n)}$ does not change sign on $I$.
Regarding \eqref{mineq1}, for any function $f$ one has the trivial inequality \begin{align*}
(n+1) & \int |f(t)| d \mu(t) \int |V_n(t_1,\ldots,t_n)| d \mu(t_1) \cdots d \mu(t_n) \\
& \geq \int \left| \sum_{i=1}^{n+1} (-1)^{n+1-i} f(t_i) V_n(t_1,\ldots,\hat{t_i},\ldots,t_{n+1}) \right| d \mu(t_1) \cdots d \mu(t_{n+1}). \end{align*} If one supposes further that $f \in C^{(n)}(I)$, equation \eqref{averages} from proposition \ref{averageprop} gives that \begin{align*}
(n+1) & \int |f(t)| d \mu(t) \int |V_n(t_1,\ldots,t_n)| d \mu(t_1) \cdots d \mu(t_n) \\
& \geq \frac{1}{n!} \int \left|\int f^{(n)}(s) \psi_t(s) ds \right| |V_{n+1}(t_1,\ldots,t_{n+1})| d \mu(t_1) \cdots d \mu(t_{n+1}). \end{align*}
Since $|\int f^{(n)}(s) \psi_t(s) ds| \geq \inf_{t \in S} |f^{(n)}(t)|$, the inequality \eqref{mineq1} must be true.
The proof of \eqref{mineq2} and, hence, theorem \ref{theorem1} follows from a closer estimation of $\int f^{(n)}(s) \psi_t(s) ds$. Let $t_1,\ldots,t_{n+1}$ be taken from some closed set $K \subset I$. Fix any $\epsilon > 0$ and let $K_\epsilon$ (as in theorem \ref{theorem1}) be the closed set of points $t$ for which $t_{+} := \inf \set{s \in K}{s \geq t}$ and $t_{-} := \sup \set{s \in K}{s \leq t}$ both exist and satisfy either $|t_+ - t| \leq \epsilon |t_+ - t_{-}|$ or $|t_{-} - t| \leq \epsilon|t_+ - t_{-}|$. It suffices to show that, for any continuous function $g$ on $I$ which does not change sign, \[ \int g(s) \psi_t(s) ds \gtrsim \inf_{s \in K_\epsilon} g(s). \] To that end, let $p_c(t) := \sum_{i=0}^{n-1} c_i t^i$ where the $c_i$ are real coefficients whose squares sum to $1$. The ratio
\[ \frac{\int_0^{\epsilon} |p_c(t)| dt + \int_{1-\epsilon}^\epsilon |p_c(t)| dt}{\int_{0}^1 |p_c(t)| dt} \]
is never equal to zero for any polynomial $p$ of degree at most $n-1$; therefore compactness of the unit sphere and homogeneity imply that there exists a constant $C_{n,\epsilon}$ such that $\int_0^\epsilon |p(t)| dt + \int_{1-\epsilon}^1 |p(t)| dt \geq C_{n,\epsilon} \int_{0}^1 |p(t)| dt$ for any polynomial $p$ of degree $n-1$. By a suitable change of variables, one has the integral of $|p|$ over the ends (each of length $\epsilon$ times the length of the whole interval) of any interval is bounded below by a constant times the integral over the whole interval.
Consider now the integral of $g$ against $\psi_t$. Clearly there exist a countable number of open, disjoint intervals $I_j$ in the convex hull of $K$ such that \[ \int g(s) \psi_t(s) ds = \int_K g(s) \psi_t(s) ds + \sum_j \int_{I_j} g(s) \psi_t(s) ds. \] Since $K \subset K_\epsilon$, $\int_K g(s) \psi_t(s) ds \geq (\inf_{s \in K_\epsilon} g(s)) \int_K \psi_t(s) ds$. As for each $I_j$, the ends of these intervals are in $K_\epsilon$ as well (and $g$ is nonnegative on the interior region of $I_j$). Furthermore, $\psi_t$ is a polynomial of degree at most $n-1$ on $I_j$ since this interval contains no points of $K$ (hence none of the $t_j$'s). Thus \[ \int g(s) \psi_t(s) ds \geq \inf_{s \in K_\epsilon} g(s) \left( \int_K \psi_t(s) ds + C_{n,\epsilon} \sum_j \int_{I_j} \psi_t(s) ds \right). \] Summing these finishes the proof. \end{proof}
\begin{proof}[Proof of theorem \ref{theorem2}.]
Fix some $\epsilon'$ and positive integer $n$. Let ${\cal C}^{(1)}$ be the collection of $(n,\epsilon')$-children of $\mu$ given by proposition \ref{intervals}. Next let ${\cal C}^{(2)}$ be the collection of all $(n-1,\epsilon')$-children of intervals $I \in {\cal C}^{(1)}$, where the children of an interval $I$ are understood as the children of the measure $\mu_I := \mu(I)^{-1} \left. \mu \right|_I$ (note that this will always be well-defined since the $\mu$-measures of children always have a minimal amount of mass as controlled by $\epsilon'$). Continue in this manner until the collection ${\cal C}^{(n)}$ (the $1$-children of the collection ${\cal C}^{(n-1)}$) is obtained.
Suppose that $f$ is of polynomial type $n$ on $(K_\epsilon,I)$ with constant $C$. This implies by (the proof of) theorem \ref{theorem1}, that \begin{equation}
\int |f(t)| d \mu(t) \gtrsim C^{-1} (\ell_n(\mu))^n \sup_{t \in I} |f^{(n)}(t)|. \label{step0} \end{equation}
Now, given some interval $I_0 \in {\cal C}^{(n)}$, let $I_j$ be the unique element of ${\cal C}^{(n-j)}$ containing $I_1$. For convenience, let $I_{-1} := I_0$ and $I_n := I$ (the interval on which $\mu$ is supported). Fix $c^{-1} = \log \frac{3}{2}$, and choose the $j$ in $0,\ldots,n$ which maximizes \begin{equation}
c^j |I_{j-1}|^j \sup_{t \in I_{j}} |f^{(j)}(t)| \label{ptsup} \end{equation} (if $j$ is not unique, choose the largest such $j$). If $j=n$, then inequality \eqref{step0} has as an immediate consequence that
\[ \int |f(t)| d \mu(t) \gtrsim C^{-1} |I_{j-1}|^j \sup_{t \in I_j} |f^{(j)}(t)| \] for $j=0,\ldots,n$. Suppose instead that the maximizing index $j$ is not equal to $n$. In this case, let $s_0 \in I_{j}$ be the point where the supremum is obtained. It follows that, for any $s \in I_{j}$, \begin{align*}
c^j |I_{j-1}|^j |f^{(j)}(s) - f^{(j)}(s_0)| & \leq \sum_{k=j+1}^{n} \frac{c^j |I_{j-1}|^j |s-s_0|^{k-j}}{(k-j)!} \sup_{t \in I_{k}} |f^{(k)}(t) | \\
& \leq \sum_{k=j+1}^n \frac{c^{j-k}}{(k-j)!} c^k |I_{k-1}|^k \sup_{t \in I_{k}} |f^{(k)}(t)| \\
& \leq (e^\frac{1}{c} - 1) c^j |I_{j-1}|^j \sup_{ t \in I_{j}} |f^{(j)}(t)|. \end{align*}
It must therefore be the case that $\sup_{t \in I_{j}} |f^{(j)}(t)| \leq 2 \inf_{t \in I_{j}} |f^{(j)}(t)|$, meaning that $f$ is of polynomial type $j$ on $(I_j,I_j)$ with constant $2$. Therefore theorem \ref{theorem1} and proposition \ref{intervals} guarantee that \begin{align*}
\int_{I_{j}} |f(t)| d \mu(t) & \gtrsim \mu(I_{j}) (\ell_j( \mu_{I_j}))^j \sup_{t \in I_{j}} |f^{(j)}(t)| \\
& \gtrsim |I_{j-1}|^j \sup_{t \in I_{j}} |f^{(j)}(t)| \\
& \gtrsim |I_{k-1}|^k \sup_{t \in I_k} |f^{(k)}(t)| \ \forall k = 0,\ldots,n. \end{align*}
In particular, there is now a collection of intervals $I'$, namely ${\cal C}^{(n)}$ such that the integral $\int |f| d \mu(t) \gtrsim C^{-1} \sup_{t \in I'} |f(t)|$ for any function $f$ which is polynomial type $n$ on $(K_\epsilon,I)$ with constant $C$. By proposition \ref{closegaps}, these intervals can be joined together independently of $f$ so that they cover the same set as before but consist of no more than $n$ connected components. In particular, the $\mu$-measure is at least $(1-\epsilon')^n$, which can be made greater than $1 - \epsilon$ for suitably-chosen $\epsilon'$. \end{proof}
As for the remaining corollary, choose the interval $I'$ to be a connected component of $E$ as given by theorem \ref{theorem2} which has $\mu$-measure at least $\frac{1-\epsilon}{n}$. In this case, the conclusion of the corollary follows immediately from the combined conclusions of theorems \ref{theorem1} and \ref{theorem2}: \begin{proposition} Suppose that $f \in C^{n}(I')$. For any $j=0,\ldots,n$,
\[ \min\{ |I'|^j, \ell^j \} \sup_{t \in I'} |f^{(j)}(t)| \lesssim \sup_{t \in I'} |f(t)| + \ell^j \sup_{t \in I'} |f^{(n)}(t)|. \] \end{proposition} \begin{proof}
As in the proof of theorem \ref{theorem2}, let $j$ be the index out of $0,\ldots,n$ which maximizes $2^j \min\{ |I|^j, \ell^j \} \sup_{t \in I} |f^{(j)}(t)|$. If $j=n$, then there is nothing else to prove. Otherwise, for any $t,s \in I$, the mean-value theorem assures that
\[ |f^{(j)}(t) - f^{(j)}(s)| \leq |t-s| \sup_{u \in I'} |f^{(j+1)}(u)|. \]
Provided that $|t-s| \leq \min \{|I|,\ell \}$, the right-hand side is bounded above by $\frac{1}{2} \sup_{u \in I'} |f^{(j)}(u)|$. In particular, if $I''$ is any interval of length $\min \{|I'|,\ell \}$ containing the point where $f^{(j)}$ achieves its maximum, then the supremum of $f^{(j)}$ on that interval is bounded by twice the infimum. But in this case equation \eqref{averages} guarantees that $\min\{ |I'|^j, \ell^j \} \sup_{t \in I'} |f^{(j)}(t)|$ is bounded below by a constant (depending only on $n$) times the supremum of $f$ (to apply equation \eqref{averages}, simply choose evenly-spaced points of $I''$). \end{proof}
\section{$L^p$-improving estimates for polynomial curves}
This section is devoted to the proof of theorem \ref{averageop}. The proof is itself divided into two parts. The first is the main argument, relying on integral estimates and refinements (and, in particular, relying on theorem \ref{theorem0}). With the one-dimensional integral estimates already established, the main portion of the proof of theorem \ref{averageop} is remarkably short. The second part of the proof deals with counting solutions of the iterated flows of $X_1$ and $X_2$. As is customary, this boils down to an application of B\'{e}zout's theorem; the difference here is that the vector fields $X_1$ and $X_2$ must first be lifted to a nilpotent Lie group (as was done by Christ, Nagel, Stein, and Wainger \cite{cnsw1999}) to produce a setting in which the flows correspond to polynomial mappings.
\subsection{Refinements, re-centering, and integral estimates}
The main innovation of the work of Tao and Wright over the original paper of Christ was the observation that, under certain circumstances, the integral of a function over the flow $\exp(tX)(x_0)$ of a vector field $X$ can be estimated from below by the value of that function or its derivatives evaluated at $t=0$ (the ``central'' part of central sets of a fixed width). Of course, it is not always possible to make an estimate of this sort (that is, it is easy to construct examples of functions which happen for particular $x_0$ to be much larger at $t=0$ than at the other values of $t$ which form the support of the integral). Tao and Wright circumvent this problem by introducing the notion of a set with width $w$; more recently, Christ \cite{christ2006} avoids this problem by introducing $(\epsilon,\delta)$-generic sets. The problem with the construction of Tao and Wright is that, in the process, unavoidable small losses are encountered in various exponents which lead to less-than-sharp restricted weak-type results. One way to avoid this problem, at least in the case of polynomial curves, is to use theorem \ref{theorem0} instead of introducing central sets of fixed width. The application of theorem \ref{theorem0} comes in the following lemma which describes the set of $x_0$'s for which this re-centering can be accomplished. This new set is called a refinement of the original: \begin{proposition}
Let $U' \subset {\mathbb R}^{d+1}$ be open and $\pi : U \rightarrow {\mathbb R}^d$ have surjective differential at every point; let $X$ be a nonvanishing vector field on $U'$ for which $d \pi (X) = 0$. Let $U \subset U'$ be open and bounded and fix a positive integer $n$. There exists a nonzero constant $c$ depending on $n$ and the bounded subset $U$ such that, for any measurable $\Omega \subset U$, there is a refinement $\Omega' \subset \Omega$ with $|\Omega'| \geq c |\Omega|$ such that, for any $x_0 \in \Omega'$, the integral estimate \begin{align*}
\int |f(t,x_0)| & \chi_{\Omega} ( \exp(tX)(x_0)) dt \geq \\
& c \max_{j=0,\ldots,n} \left\{ \left( \frac{|\Omega|}{|\pi(\Omega)|} \right)^{j+1} \left| \frac{\partial^j f}{\partial t^j} (0,x_0) \right| \right\} \end{align*} holds for any function $f(t,x_0)$ which is a \label{refinement} polynomial of degree at most $n$ for each fixed $x_0$. \end{proposition} \begin{proof} It suffices to restrict attention to the portion of $\Omega$ which lies on a particular integral curve of $X$ and show that a positive proportion of such points can be taken to lie in $\Omega'$. In this case, one can change variables so that $X$ simply coincides with a coordinate direction and $\exp(tX)$ is simply translation by $t$ in that particular direction. For a first approximation, $\Omega'$ is taken to be the set of all points $x_0$ such that
\[ \int \chi_{\Omega}(\exp(tX)(x_0)) dt \geq c \frac{|\Omega|}{|\pi(\Omega)|} \]
for some small $c$; Fubini's theorem guarantees that the set $\Omega \setminus \Omega'$ is necessarily only a small fraction of the set $\Omega$. Now the set $\Omega'$ as defined is still slightly too big. However, restricting attention to the intersection of a fiber of $\pi$ with the set $\Omega'$, it suffices to prove that, for any set $K \subset {\mathbb R}$, there is a subset $K' \subset K$ with $|K'| \geq c|K|$ for which
\begin{equation} \int |f(t,s)| \chi_{K}(t+s) dt \geq c \max_{j=0,\ldots,n} \left\{ |K|^{j+1} \left| \frac{\partial^j f}{\partial t^j} (0,s) \right| \right\} \label{intest} \end{equation} whenever $s \in K'$. But this inequality follows directly from theorem \ref{theorem0} when, for example, $s$ lies inside the interval $I$ given by that theorem. Thus, if $\Omega'$ is further reduced to contain only those points in each fiber which lie in the corresponding interval $I$ given by theorem \ref{theorem0}, the proposition follows. \end{proof}
Notice that, since the refinement $\Omega'$ is contained in $\Omega$ and has $|\Omega'| \geq c |\Omega|$, it follows that
\[ \frac{|\Omega'|}{|\tilde{\pi}(\Omega')|} \geq c \frac{|\Omega|}{|\tilde \pi(\Omega)|} \] for any projection $\tilde \pi$ (which may or may not be the same as the projection used for refining). Thus when proposition \ref{refinement} is applied iteratively, it is always possible for the {\it original} $\Omega$ to appear on the right-hand side of \eqref{intest} at the price of a slightly worse constant (which is not a problem as long as the iterations terminate after a uniformly bounded number of steps).
The proof of theorem \ref{averageop} now proceeds exactly as in the work of Christ \cite{christ1998} or Tao and Wright \cite{tw2003}. For each $x_0$, consider the mapping \[ \Phi_{x_0} (t_1,\ldots,t_{d+1}) := \exp(t_1 X_1) \circ \cdots \circ \exp(t_{d+1} X_{d+1}) (x_0). \] In the next section, it will be established that, for fixed $x_0$, this mapping has finite multiplicity everywhere except for some set of times $(t_1,\ldots,t_{d+1})$ which has $(d+1)$-dimensional Lebesgue measure zero. Thus it follows that, for any (measurable) set $\Omega \subset U$,
\[ |\Omega| \geq c \int \chi_{\Omega} ( \Phi_{x_0}(t_1,\ldots,t_{d+1})) |J_{x_0}(t_1,\ldots,t_{d+1})| dt_1 \cdots dt_d \] where $c$ is the reciprocal of the maximum multiplicity and $J_{x_0}(t)$ is the Jacobian determinant of the mapping $\Phi_{x_0}(t)$. If $\Omega'$ is the refinement via the previous proposition with respect to the mapping $\pi_1$ and vector field $X_1$, it follows that \begin{align*}
\int \chi_{\Omega} & ( \Phi_{x_0}(t_1,\ldots,t_{d+1})) |J_{x_0}(t_1,\ldots,t_{d+1})| dt_1 \cdots dt_{d+1} \\
\geq c' & \left( \frac{|\Omega|}{|\pi_1(\Omega)|} \right)^{j+1} \int \chi_{\Omega'} ( \Phi_{x_0}(0,t_2,\ldots,t_d)) \left| \frac{\partial^j J_{x_0}}{\partial t_1^j}(0,t_2,\ldots,t_d) \right| dt_2 \cdots d t_{d+1} \end{align*} for any $j=0,\ldots,n$. But this new integral can, in turn, be estimated in exactly the same way by refining $\Omega'$ with respect to the mapping $\pi_2$ and the vector field $X_2$ and so on. The end result is that, for any multiindex $\alpha$ there is a constant $c_{\alpha}$ such that
\begin{equation} |\Omega| \geq c \prod_{i=1}^{d+1} \left( \frac{|\Omega|}{|\pi_i(\Omega)|} \right)^{\alpha_i+1} \left| \frac{\partial^\alpha J_{x_0} }{\partial t^\alpha}(0) \right| \label{bigest} \end{equation} for all $x_0$ in some iterated refinement of $\Omega$ (which, in particular, will have nonzero measure). Notice, however, that when $\Omega$ is defined by taking $\chi_{\Omega}(x) := \chi_F(\pi_1(x)) \chi_{G} (\pi_2(x))$, this inequality may be manipulated to give theorem \ref{averageop}.
It is also worth noting that when equation \eqref{bigest} is summed over all multiindices $\alpha$, one obtains the rather interesting geometric inequality that
\[ |\Omega| \geq c \left|B_0\left(x_0, \frac{|\Omega|}{|\pi_1(\Omega)|}, \frac{|\Omega|}{|\pi_2(\Omega)|} \right)\right| \] where $B_0(x_0,\delta_1,\delta_2)$ is the image of the set $[-\delta_1,\delta_1] \times [-\delta_2 , \delta_2] \times \cdots \times [-\delta_{d+1},\delta_{d+1}]$ (with the usual periodicity convention) under the mapping $\Phi_{x_0}$. The measure of this set is, in turn, comparable to the measure of the two-parameter Carnot-Carath\'{e}odory ball $B(x_0; \delta_1,\delta_2)$ of Tao and Wright. Thus equation \eqref{bigest} gives a rather direct proof of the improved version of Tao and Wright's equation (66) mentioned in the second remark at the end of the paper. \subsection{Lifting as related to polynomial curves}
In this section, it remains to show that $\Phi_{x_0}$ has bounded multiplicity outside some exceptional set and that the Jacobian determinant $J_{x_0}(t)$ is (up to a factor bounded away from $0$) a polynomial function of the $t$ parameters. The main idea of the proof of these facts is a lifting argument involving the Baker-Campbell-Hausdorff formula. The reader is referred to the paper of Christ, Nagel, Stein, and Wainger \cite{cnsw1999} for a thorough treatment of this topic. In the proof at hand, this previous must be improved slightly (to obtain exact formulas rather than asymptotic ones), but there is not any added difficulty; in fact, it will suffice to only reproduce a few very small pieces of this much larger work.
To that end, let ${\cal N}$ be the collection of all words $w$ for which $X_w$ (the commutator as defined at the beginning of the paper) does not vanish identically. For any $s \in {\mathbb R}^{{\cal N}}$, let \[ s \cdot X := \sum_{w \in {\cal N}} s_w X_w. \] Fix a bounded open set $U \subset {\mathbb R}^{d+1}$ on which $X_1$ and $X_2$ are defined and fix $x_0 \in U$. Let $\tilde U \subset {\mathbb R}^{\cal N}$ be the collection of all $s$ for which $\exp( \theta s \cdot X)(x_0) \in U$ for all $\theta \in [0,1]$ (the inclusion of $\theta < 1$ guarantees that for any $s$, the associated integral curves used to define $\exp(s \cdot X)(x_0)$ remain in $U$). Suppose for the moment that it is possible to establish the following two facts: \begin{enumerate} \item For any $x \in U$, the fiber $\exp(s \cdot X)(x_0) = x$ of the mapping $\exp(s \cdot X)(x_0)$ (as a function from $\tilde U$ to $U$) can be parametrized by a polynomial function, i.e., there exists a mapping $N_x(u)$ which parametrizes the fiber, has coordinate functions which are polynomials in the $u$ variables, and has surjective differential. \item There exists a lifting $\tilde \Phi_{x_0}(t)$ of $\Phi_{x_0}(t)$ which is also polynomial, that is, $\exp ( \tilde \Phi_{x_0}(t) \cdot X)(x_0) = \Phi_{x_0}(t)$ and the coordinate functions of $\tilde \Phi_{x_0}$ are polynomial functions of $t$. \end{enumerate} These facts combined allow one to use B\'{e}zout's theorem, just as was employed in the original paper of Christ \cite{christ1998}. Specifically, it follows from these facts that the equation $\Phi_{x_0}(t) = x$ has a solution for some $t$ only when the equations $\tilde \Phi_{x_0}(t) = N_x(u)$ have a solution for the same $t$ and some value of $u$. In the usual manner, an additional parameter $v$ can be added to these equations in such a way that the resulting system of equations is homogeneous in $(t,u,v)$ and reduces to the original system when $v=1$. Now B\'{e}zout's theorem guarantees that the number of irreducible components (in complex projective space) of the variety determined by these equations is at most the product of the degrees (and, in particular, does not depend on the particular choice of $x$). See Fulton \cite{fulton1984}, chapter 8, section 4 (and, in particular, example 8.4.6) for this version of B\'{e}zout's theorem.
Now The Jacobian determinant $J_{x_0}(t)$ is nonzero at $t_0$ only when the graph of $\tilde \Phi_{x_0}(t)$ is transverse to the fibers of $\exp(s \cdot X)(x_0)$ at the point $\tilde \Phi_{x_0}(t_0)$. Thus solutions to $\Phi_{x_0}(t) = x$ (for real $t$) at which the Jacobian determinant $J_{x_0}(t)$ is nonvanishing arise only when there is an isolated solution of the system $\tilde \Phi_{x_0}(t) = N_x(u)$ in $(t,u)$ space; that is, when the Jacobian determinant with respect to $(t,u)$ of the mapping $\tilde \Phi_{x_0}(t) - N_x(u)$ is also nonzero. This, in turn, guarantees that the solution $(t,u)$ remains isolated amongst complex solutions as well. But any such isolated solution, in particular, corresponds to an irreducible component of the zero set of the homogeneous equations in complex projective space; it therefore follows that there is a uniform bound on the number of solutions to $\Phi_{x_0}(t) = x$ which occur where the Jacobian determinant is nonvanishing.
As for any solutions at which the Jacobian determinant may vanish, Sard's lemma guarantees that the set of times $t$ at which the Jacobian determinant does vanish has ($d+1$-dimensional) measure zero. In particular, this also means that the set of points $x \in U$ for which there can exist a solution to $\Phi_{x_0}(t) = x$ with vanishing Jacobian determinant is also a set of measure zero in $U$. Thus, except for an exceptional set of $x$'s of measure zero, the system of equations $\Phi_{x_0}(t) = x$ has a uniformly bounded number of solutions, and the Jacobian determinant of $\Phi$ at each such solution is nonzero. Thus the usual change-of-variables formula gives at once the desired inequality
\[ |\Omega| \geq c \int \chi_{\Omega}(\Phi_{x_0}(t)) |J_{x_0}(t)| dt \] for any set $\Omega \subset U$.
A bit of notation is in order; given vector fields $A$ and $B$, the vector denoted by $\left. d \exp(A)(B) \right|_{y}$ is meant to be the vector at the point $y$ obtained by transporting $B$ via the exponential mapping $\exp(A)$ (so in particular, it is the vector $B$ at $\exp(-A)(y)$ transported to $y$).
To establish the necessary properties of the lifting of $\Phi_{x_0}(t)$, two facts from geometry are required. The first is the following: suppose that $A$ is a vector field on $U$ which depends smoothly on some parameter $h$. For any $x_0 \in U$ and any $s$ sufficiently small, the tangent vector of the curve $\gamma(h) := \exp( s A(h))(x_0)$ (as a function of $h$ for $s$ and $x_0$ fixed) is given by \begin{equation}
\frac{\partial}{\partial h} \gamma(h) = \left. \left( \int_0^1 d \exp((1 - \theta) s A(h)) \left( \frac{\partial A}{\partial h} \right) d \theta \right) \right|_{\gamma(h)}. \label{geom1} \end{equation} Equivalently, the tangent vector to the curve $\gamma(h)$ is given by transporting the vector \begin{equation}
\left. \left( \int_0^1 d \exp( - \theta s A(h)) \left( \frac{\partial A}{\partial h} \right) d \theta \right) \right|_{x_0} \label{geom3} \end{equation} to the point $\gamma(h)$ via the exponential map $\exp(s A(h))$. The second fact needed is that, for any vector fields $A$ and $B$ on $U$; if $s$ is sufficiently small then \begin{equation}
\frac{\partial}{\partial s} \left( \left. d \exp( s A ) (B) \right|_{x_0} \right) = \left. d \exp( s A ) ([B,A]) \right|_{x_0} \label{geom2} \end{equation} Note that equation \eqref{geom2} is nothing more than a computation of the Lie derivative of $B$ with respect to $A$ and can be found in Warner \cite{warner1971}, for example. It is only \eqref{geom1} which requires a bit more explanation. To that end, $\Gamma(s) := \exp(- s A(h)) \circ \exp(s A(h+\Delta h))(x_0)$. \begin{align*}
\frac{\partial}{\partial s} g(\Gamma(s)) & = - \left. (A(h)g) \right|_{\Gamma(s)} + \left. \left( d \exp(-s A(h)) (A(h+\Delta h)) g \right) \right|_{\Gamma(s)} \\
& = \left. \left( d \exp(-s A(h)) (A(h+\Delta h)-A(h)) g \right) \right|_{\Gamma(s)}. \end{align*} It therefore follows that
\[ \frac{g(\Gamma(s)) - g(\Gamma(0))}{\Delta h} = \int_0^1 \left. \left( d \exp(-s \theta A(h)) \left( \frac{A(h+\Delta h) - A(h)}{\Delta h} \right) g \right) \right|_{\Gamma(s \theta)} d \theta. \] As $\Delta h \rightarrow 0$, note that $\Gamma(s) \rightarrow x_0$. For fixed $s,h$, let $g (x) = f ( \exp(s A(h))(x))$ and let $\Delta h \rightarrow 0$. The result is that
\[ \frac{\partial}{\partial h} f ( \exp(s A(h))(x_0)) = \left. \left( \int_0^1 d \exp( - \theta s A(h)) \left( \frac{\partial A}{\partial h} \right) d \theta \right) \right|_{x_0} \! \! f (\exp(s A(h))(x_0)) \] which is precisely what it means for the curve $\gamma(h) = \exp(s A(h))(x_0)$ to have a tangent vector obtained by transporting the vector \eqref{geom3} via the map $\exp(s A(h))$.
Consider the mapping $\varphi : \tilde U \rightarrow U$ given by $\varphi(s) := \exp(s \cdot X)(x_0)$. Taking a Taylor expansion of the integrand \eqref{geom1} with respect to $\theta$ (computing these derivatives via \eqref{geom2}) allows one to easily compute derivatives of $\varphi$ with respect to the parameters $s_w$: \begin{align*}
\frac{\partial}{\partial s_w} \varphi(s) & = \left. \left( \int_0^1 d \exp( (1-\theta) s \cdot X) \left( X_w \right) d \theta \right) \right|_{\varphi(s)} \\
& = \left. \left( \sum_{j=0}^\infty \frac{(-1)^{j} [ (s \cdot X)^j X_w]}{(j+1)!} \right) \right|_{\varphi(s)} \end{align*} where $[(s \cdot X)^j X_w]$ is the repeated commutator given by $[(s \cdot X)^0 X_w] := X_w$ and $[(s \cdot X)^{j+1} X_w] = [s \cdot X,[(s \cdot X)^j X_w]]$ for each $j \geq 0$. Note that this sum is, in fact, a finite sum by virtue of the vanishing commutator condition. Also note that the coefficients of the sum are precisely the Taylor coefficients of $\frac{e^{-x}-1}{-x}$.
Now let $\tilde{c}_w(s)$ be the coefficient of $X_w$ when the sum \[ \sum_{j=0}^\infty \frac{(-1)^j B_j [(s \cdot X)^j X_1]}{j!} \] is expanded into a linear combination of words (where the $B_j$'s are the Bernoulli numbers; note that these are chosen so that the coefficients in $j$ are equal to the Taylor coefficients of $\frac{-x}{e^{-x}-1}$). It follows that
\[ \left( \sum_{w \in {\cal N}} \tilde{c}_w(s) \frac{\partial}{\partial s_w} \right) \varphi(s) = \left. X_1 \right|_{\varphi(s)}. \] The vector field $\sum_{w \in {\cal N}} \tilde{c}_w (s) \frac{\partial}{\partial s_w}$ is thus a lifting of $X_1$; moreover, since the coefficient $\tilde{c}_w(s)$ involves only those parameters $s_{w'}$ for which the length of $w'$ is less than the length of $w$, it follows that the integral curves of this lifted vector field in ${\mathbb R}^{\cal N}$ will be given by a polynomial function of the time parameter. Thus composing these flows ($X_2$ may be lifted in precisely the same way) establishes the fact that $\Phi_{x_0}(t)$ has a lifting $\tilde{\Phi}_{x_0}(t)$ which is given coordinate-wise by polynomial functions.
To parametrize the fibers of $\varphi(s)$, the alternative formulation
\[ \frac{\partial}{\partial s_w} f (\varphi(s)) = \left. \left( \sum_{j=0}^\infty \frac{[(s \cdot X)^j X_w]}{(j+1)!} \right) \right|_{x_0} f ( \exp(s \cdot X)(x_0)) \] is used (here the Taylor series expansion of the integrand of \eqref{geom3} is taken instead of \eqref{geom1}). Suppose that constants $u_w$ are chosen so that $\sum_{w \in {\cal N}} u_w X_w$ equals the zero vector {\it at the particular point $x_0$} (the sum may or may not equal zero elsewhere). If one defines coefficients $\tilde{d}_w(s,u)$ as before by expanding \[ \sum_{w \in {\cal N}} \tilde{d}_w(s,u) X_w := \sum_{j=0}^\infty \frac{B_j [(s \cdot X)^j (u \cdot X) ]}{j!} \] formally, it follows that the vector field $\sum_{w \in {\cal N}} \tilde{d}_w(s,u) \frac{\partial}{\partial s_w}$ will satisfy the property that \[\left( \sum_{w \in {\cal N}} \tilde{d}_w(s,u) \frac{\partial}{\partial s_w} \right) \varphi(s) = 0. \] In other words, for any appropriately chosen values of $u$, the corresponding vector field will be tangent to the fibers of $\varphi(s)$. The integral curves of these fibers will be given by polynomial functions of $u$ for the same reason that the coefficients $\tilde{d}_w$ only depend on $s_{w'}$ for $w'$ of shorter length. Now a simple dimension-counting guarantees that the fibers can, in fact, be smoothly parametrized by the flows of these vector fields, that is, the exponential flow of these vector fields based at $x_0$ has surjective differential with respect to the $u$ variables (one needs only apply the implicit function theorem; note that the curvature condition on the vector fields $X_1$ and $X_2$ guarantees that the differential $d \varphi$ is surjective). Thus the earlier counting arguments hold, and in particular, $\Phi_{x_0}(t)=x$ has boundedly many solutions for all $x$ outside a set of measure zero.
To complete the proof of theorem \ref{averageop}, one fact remains to be established: namely, that the Jacobian determinant $J_{x_0}(t)$ is, up to a nonvanishing factor, a polynomial function of the parameters $t$. Notice that the vector $\frac{\partial}{\partial t_i} \Phi_{x_0}(t)$ is equal to $d \exp(t_1 X_1) \circ \cdots \circ d \exp(t_{i-1} X_{i-1}) (X_i)$ evaluated at the point $\Phi_{x_0}(t)$. Up to a bounded, nonvanishing factor, the Jacobian determinant can be evaluated by transporting these vectors to the point $x_0$ and then computing a determinant. In that case, it follows that $J_{x_0}(t)$ is proportional to \begin{align*}
\det ( d & \exp(-t_{d+1} X_{d+1}) \circ \cdots \circ d \exp(-t_2 X_2)(X_1), \\ & d \exp(-t_{d+1} X_{d+1}) \circ \cdots \circ d \exp(-t_3 X_3)(X_2), \\
& \left. \ldots, X_{d+1}) \right|_{x_0}. \end{align*} But now \eqref{geom2} guarantees that each vector in this expression is a polynomial function of $t$, and multilinearity of the determinant establishes the desired property of $J_{x_0}(t)$.
\end{document} | arXiv |
Intradecadal variations in length of day and their correspondence with geomagnetic jerks
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Pengshuo Duan ORCID: orcid.org/0000-0002-1365-05771 &
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Nature Communications volume 11, Article number: 2273 (2020) Cite this article
This article has been updated
Earth's core oscillations and magnetic field inside the liquid outer core cannot be observed directly from the surface, we can infer these information from the intradecadal variations in Earth's rotation rate defined by length of day. However, the fine time-varying characteristics as well as relevant mechanisms of the intradecadal variations are still unclear. Here we report that the intradecadal variations present a significant 8.6-year harmonic component with an unexpected increasing phenomenon, besides a 6-year decreasing oscillation. More importantly, we find that there is a very good correspondence between the extremes of the 8.6-year oscillation with geomagnetic jerks. The fast equatorial waves with subdecadal periods propagating at Earth's core surface may explain the origin of this 8.6-year oscillation.
The intradecadal variation (i.e., 5–10-year scales) in length of day (LOD) is an interesting topic in fundamental astronomy and geophysics as it may closely correlate with the fast dynamics of the Earth's core1,2 and the geomagnetic field changes3,4,5. The existence of a significant 6-year periodic oscillation with a mean amplitude of ~0.12 ms existing in the observed LOD data has been confirmed by many works6,7,8,9, while the recent studies seem to have equated the intradecadal variations in LOD with the 6-year oscillation. However, the time-varying characteristics of the intradecadal variations in LOD10,11 and the corresponding frequency-domain results from the Fourier spectral analysis8 indicated that the intradecadal variations may contain more harmonic components than currently widely accepted thoughts (i.e., a 6-year signal alone), since the frequency domain result of the LOD variations does not present a single sharp 6-year peak on the 5–10-year scales.
In order to quantitatively detect the time-varying characteristics of the intradecadal variations in LOD, here we adopt a wavelet analysis method named normal Morlet wavelet transformation (NMWT)12, which owns a high-frequency resolution to analyze the intradecadal variations (see "Methods" section and Supplementary Note 1), and the NMWT method is proved to be an effective approach to accurately recognize and quantitatively extract the target harmonic signals with close frequencies from the original data. It should be noted that when the wavelet method is used to accurately detect the real LOD data, a strategy of avoiding its edge effects (EE) needs to be designed. Given that the previous works8,13 did not clearly illustrate the method of avoiding the EE in detail, here, in order to obtain accurate and robust LOD results and to make our work be repeatable easily, we adopt a simple strategy called as boundary extreme point mirror-image-symmetric extension (BEPME) to avoid the relevant EE (see "Methods" section).
Here we present two obviously harmonic components (i.e., 6-year and ~8.6-year terms) existing in observed LOD data on the intradecadal scales, where the 8.6-year signal shows an increasing trend in time domain, which is first found to closely associate with geomagnetic jerks. This observed evidence indicates that the 8.6-year signal and geomagnetic jerks may result from a same physical source, i.e., the equatorial quasi-geostrophic (QG) Alfvén waves focusing at the Earth's core surface. This work provides a new possible entry to predict the rapid geomagnetic field changes ahead and to study the magnetohydrodynamics of the Earth deep interiors via Earth rotation variations.
Intradecadal variations and a 8.6-year increasing signal
Figure 1 shows the observed LOD variations during 1962–2019, from which the atmospheric angular momentum (AAM) effect has been removed, while we apply the 12-month and 6-month running average methods to eliminate the remaining seasonal signals (i.e., the annual and semi-annual terms) of the LOD variations. Using the Daubechies wavelet fitting method following the refs. 8,13, in which this wavelet filtering cannot produce the Gibbs effect13, we obtain the background trend (i.e., the red curve in Fig. 1a, which mainly presents the LOD variations with period T > 10 years) and the residual series (the green curve in Fig. 1a mainly reflects the intradecadal variations with periods of 5–10 years, see Fig. 1b), where the coincidence between this residual series and the original LOD data in frequency domain (Fig. 1b) shows that the residual series can well characterize the intradecadal variations in LOD.
Fig. 1: The variations of length of day in time frequency domain.
a Shows the LOD variations on the various scales, where the residual series (i.e., the green curve) mainly reflects the intradecadal variations, the frequency-domain result of which is shown in b; b presents the Fourier spectrum of the LOD data and the intradecadal variations, which shows a wide energy-spectrum range in 5–10-year band instead of a single 6 years sharp peak; c, d show the NMWT time–frequency spectrum, which further reveals the periodic components existing on the 5–10-year scales. Here, the window width factor σ in NMWT method is set to be 3, which is large enough to clarify the target harmonic components, where the edge effect of the NMWT method and the strategy to eliminate it are illustrated in "Methods" section.
Additionally, the residual series shows the so-called modulation phenomenon7,10,11,14 of the LOD variations on the intradecadal scales, while the frequency-domain result of the original LOD data (Fig. 1b) shows a wide energy-spectrum range on the 5–10-year scales instead of a 6-year sharp peak, meaning that the characteristics of the intradecadal variations in LOD does not present a 6-year oscillation alone. Furthermore, the NMWT spectrum (Fig. 1c, d) shows that the intradecadal variations are more complex than the current findings, i.e., on the intradecadal scales, besides a 6-year oscillation, an obvious ~8.6-year periodic signal and some relatively weak periodic signals between these two periods (i.e., the 6-year and the 8.6-year) are also presented in the time–frequency spectrum. The origin issues of these relatively weak signals are beyond the scope of this work, though they may reflect the signatures of the fluid outer core (FOC) motions11,14, here we do not exclude a possibility that they may be the consequence of stochastic excitation from two free normal modes (i.e., the 6-year and 8.6-year signals) (e.g., see the case 9 in Supplementary Fig. 19). More definitive conclusion still needs to be further explored later.
Whether this 8.6-year signal displayed in Fig. 1c, d is related to the removal of the background trend? Here, our simulation tests (Supplementary note 4) give a negative answer. Moreover, the Fourier spectral analysis of the original LOD data (the blue curve in Fig. 1b, where cpm refers to the abbreviation of cycles-per-month) also shows a wide energy-spectrum range within the 5–10-year band, which coincides well with that of the residual series, revealing the existence of the 8.6-year periodic component. In addition, a ~8.5-year peak in frequency domain of the LOD variations is also shown by a recent work15, but the characteristic of this signal in time domain has never been shown. Then, why these two harmonic components (i.e., the 6-year and the 8.6-year terms) in LOD cannot be separated from each other by the traditional Morlet wavelet transformation (TMWT) spectrum7,15 or other methods (e.g., the singular spectrum analysis (SSA), see "Methods" section)? This is mainly due to the issues of the frequency-resolution of these methods (see Supplementary Figs. 4–7). Therefore, we suggest that the modulation phenomenon mentioned above does not reflect the changes of the 6-year signal itself, but the result of the superposition of periodic harmonic components (e.g., the 6-year and 8.6-year periods), while the physical origins of these oscillation signals in LOD are interesting topic1,6,11,14,16,17,18,19,20 and we will try to discuss them in this work.
Combining the NMWT method with the BEPME strategy, we can recognize the target 6-year and 8.6-year signals and extract them in time domain, respectively, and the results are shown in Fig. 2, which shows that their average amplitudes are respective ~0.124 and ~0.08 ms during 1962–2018. This work further confirms the phenomenon that the 6-year oscillation in LOD shows a secular decreasing trend8,13 with an observed quality factor Q~51. Here, it should be noted that, based on the currently observed LOD data, we have not found the strong evidence to demonstrate that the observed 6-year oscillation in LOD (see Fig. 2a) has been undergoing the significant excitation during 1962–2019, since we have not found the relevant reliable stochastic excitation series (or events). Conversely, the current 6-year oscillation time-domain result can be well characterized by a free exponentially decaying function18 (here, the exponential decaying factor β is estimated to be ~8.4 × 10−4/month), hence a possible damping model of the 6-year oscillation was established by ref. 13. Nevertheless, we do not exclude a possibility that this observed decaying phenomenon of the 6-year oscillation might be the consequence of a continuously stochastic excitation of a 6-year periodic normal mode (see "Methods" section and the Supplementary Fig. 16). Here, the attenuation and the excitation of the 6-year oscillation still need to be further studied in future using longer LOD data.
Fig. 2: The 6-year and 8.6-year signals in time domain recovered by this work.
a Confirms that the 6-year oscillation is a decaying oscillation13 with an observed quality factor ~51 (i.e., the currently observed decaying rate is about 8.4 × 10−4/month); b shows that the 8.6-year signal presents an increasing phenomenon. Here, the phase information of these two signals is recovered accurately from the simulation analysis (Supplementary Figs. 2, 3, 9, 14 and 15). In this work, according to requirement of the BEPME strategy (see "Methods" section), the 6-year oscillation recovered is ended at 2016.4, while the 8.6-year signal is ended at 2016.0.
Differing from the 6-year decaying oscillation during 1962–2018, interestingly, we first find that the 8.6-year oscillation presents an unexpected long-term increasing trend (Fig. 2b), and the 8.6-year amplitude-increasing phenomenon should be attributed to a possible continual excitation.
We further compare the above results with the original LOD variations, and the results are shown in Fig. 3. It shows that the composite signal (i.e., background trend +6 years +8.6 years) is in general consistent with the original LOD variations. This superposition signal (i.e., 6-year term +8.6-year term) can nicely characterize the general time-varying characteristics of the intradecadal variations in LOD, except some deviations from the original data (e.g., the periods of 1972–1974 and 2014–2016), which should be due to the disturbances from the other weaker signals existing on the residual series, this point can be shown in Fig. 1b–d. The general temporal-varying characteristics of the intradecadal variations in LOD can be explained by the superposition of a 6-year decaying signal and a 8.6-year increasing signal, which means that the temporal-varying characteristics of the intradecadal LOD variations do not reflect the amplitude modulation of the 6-year signal itself, but the consequence of the superposition of (at least) the two signals (i.e., 6-year and 8.6-year components). In order to confirm the above results, we made some typical simulations (see Supplementary Note 4) to demonstrate the reliability of the whole LOD data processing involved in this work.
Fig. 3: Comparisons of the recovered results and the original LOD variations.
In this figure, LOD data refers to the observed LOD data (from which the AAM effect has been removed, while a running average approach is managed to remove the remaining annual and semi-annual signals) as displayed in Fig. 1a; Background trend indicates the decadal variations presented in Fig. 1a. Residual series mainly reflects the intradecadal variations which is obtained from the Original data minus the Background trend.
Although we can avoid the wavelet edge effect to a great extent through adopting the BEPME strategy (see "Methods" section), some deviations still exist near the boundaries of the data (see Supplementary Fig. 3), while removing the background trend may also cause some disturbances on the target intradecadal variations (see Supplementary Figs. 14 and 15), which, nevertheless, cannot influence the overall characteristics of the final results. Since the results recovered by the method proposed in this work are always slightly smaller than the actual value at the larger magnitude side of the original data (see Supplementary Figs. 3, 9, 14, and 15), the actual amplitude increasing of the 8.6-year signal should be slightly larger than our current result shown in Fig. 2, that is to say, the actual amplitude increment of the 8.6-year signal during the past several decades is perhaps somewhat underestimated by this work.
Correspondence between the 8.6-year signal and geomagnetic jerks
Why the amplitude of the 8.6-year oscillation in LOD shows a secular increasing trend during the past several decades? If this 8.6-year signal is attributed to the FOC torsional normal mode14 (this is to be further discussed in the next section), then its amplitude increasing is possibly due to the excitation forcing within the FOC14,21. In addition, a geomagnetic jerk defined as the "V-shape" feature of the geomagnetic secular variations2,22,23 essentially reflects a rapid change of the second-order time derivative of the geomagnetic field, which reflects changes of the shortest observable time scale of the Earth core field24,25. The idea that the geomagnetic jerks originating from the Earth interiors has been widely accepted26,27,28,29,30, and the jerks were supposed to be closely related to the liquid flow motions at the surface of the FOC28,29,31, which may associate with the angular momentum transfers between the core and the mantle5,6,28,29. Therefore, the jerk events may associate with the LOD variations on the intradecadal scales5,6,28. Here, a scientific question arises, i.e., whether the amplitude increasing of the 8.6-year oscillation is related to the physical sources which can cause the jerks?
In this work, it is firstly found that there is a very good correspondence between the geomagnetic jerk timings with the extremes of the 8.6-year signal (Fig. 4): For example, all the following four well-known jerks26,29 (i.e., 1969, 1978, 1991, 1999) well correspond to the extremes of the 8.6-year signal; moreover, we list the following seven jerks (1972, 1982, 1986, 2003.5, 2007, 2011, and 2014) from the previous works6,25,32. Given that the geomagnetic jerks are generally localized expressions at the Earth's surface, occasionally observed over large parts of the globe, which do not occur at the same time in all regions of the globe due to mantle conductivity26,27, consequently, it is difficult to accurately define a single jerk time from the observations with the uncertainties (~±1 year) of the jerk occurrence and many local secular accelerations overlap in time and space23,33,34. Therefore, the jerk epochs listed above may be not accurate enough, despite this, these epochs are regarded to be the best determinations6.
Fig. 4: Correspondence between the 8.6-year signal and geomagnetic jerks.
In this figure, the red curve expresses the recovered 8.6-year signal in LOD, while the black dashed curve shows the fitting result (i.e., an exponentially increasing model with the expression of y(t) = A0 exp[α(t − t0)]cos(2πf(t − t0)), where the initial amplitude A0 ≈ 0.06 ms; the currently observed exponential rate α ≈ +0.00131/month; f ≈ 0.00969 cpm; the initial time t0 is set to be at June 1982) of the red curve, which may be used to predict the time when the next new jerk (i.e., the predicted jerk in blue fonts) will probably happen.
Interestingly, Fig. 4 shows that almost all the above jerk timings coincide with the extremes of 8.6-year signal very well within ~1 year (or less). There are nine jerk epochs leading the extremes of the 8.6-year signal <1 year, except the 1972 jerk and 2014 jerk32,35. Here, the question that why these two jerks did not occur at the corresponding extremes of the 8.6-year signal are worthy to be discussed later. Besides, Fig. 4 also shows an absence of a 1995 jerk. Nevertheless, a potential jerk event was shown to occur around 1995 through analyzing the relation between free core nutation and jerks36. Meanwhile, another work23 used the monthly mean geomagnetic data to discuss the geomagnetic jerk occurrence and find the jerk abound feature, where a jerk event happened during 1995–1998, though the jerk span almost fills the entire span at recent epochs23. The most recent SWARM satellite data37 showed that a new jerk event might occur in 2017 (i.e., the 2017 jerk in Fig. 4) and this jerk event is also coincident with the extreme of the 8.6-year oscillation.
In summary, this phenomenon that the jerks closely correlates with the 8.6-year signal (see Fig. 4) provides an observed evidence to support the viewpoint that jerk occurrence may own a certain periodicity as the previous works23,32,34,35 suggested, for instance, the jerk occurrence rate over the last several decades was suggested to occur at intervals ranging from 3 to 5 years35, moreover, the jerk polarity changes were shown to own periodic characteristic23, and mechanism response for the jerks may be related to a certain periodic oscillation23,32,35.
On the mechanisms and geophysical implications
In this section, we will further discuss about the potential mechanisms of the intradecadal variations in LOD, especially clarifying the different physical origins of the 6-year and 8.6-year signals. From the intradecadal variations in LOD and their time-varying characteristics, one can infer the geomagnetic field strength inside the Earth's core1,14, the information about the azimuthal torsional oscillation within the FOC1,11,14, the core–mantle gravitational coupling strength20,38, the electrical electricity at the lowermost mantle6,13,18 and inside of the Earth's core16, etc. The coincidence11 of the predicted LOD variations from the ensemble average torsional oscillation flow model and the observed LOD changes on 4–9.5 years reminds us that the fast torsional waves within the FOC may also have the corresponding 8.6-year periodic component found in this work, as the torsional waves can transfer angular momentum from the FOC to the mantle14, and then cause corresponding LOD variations11,14.
However, both the Fourier analysis of the LOD data (see Fig. 1b) and the simulation analysis (e.g., the Supplementary Figs. 5 and 6) indicate that the wide energy-spectrum of the 4–9.5 years variations in LOD obtained by the band-pass filtering11,14,39 cannot be explained by a single 6-year oscillation, which just demonstrates the existence of other signal components besides the 6-year term, while the NMWT spectrum can further distinguish these components and reveal the presence of an 8.6-year signal clearly (i.e., Fig. 1c, d). In fact, the mechanisms of the 6-year and 8.6-year signals in LOD may be different, however, previous works have not well clarified the different geophysical origins of the intradecadal variations of the LOD.
As many published works indicated16,17,18,20, one possible mechanism responsible for the 6-year oscillation in LOD is that the inner core (IC) swings with the 6-year eigenperiod under the action of the gravitational torque from the mantle, i.e., the mantle–IC gravitational coupling mode17. In this mode, although the angular momentum budget from the IC is small due to its much smaller inertia moment than that of the mantle, the observed amplitude of the 6-year oscillation is also not large (only ~0.12 ms). In other words, this small angular momentum budget from the IC is still large enough to explain the observed 6-year oscillation in LOD (see "Methods" section). Here, it should be noted that a partial FOC will be strongly coupled to the solid IC during this IC swing under the action of the electromagnetic coupling effects40. In this case, the FOC fast torsional oscillations with the 6-year recurrence period propagating from the IC to the equator at the CMB detected by ref. 1 is possibly due to this IC intrinsic swing under the action of the magnetohydrodynamics of the Earth deep interiors40,41. Nevertheless, there is yet no definitive scenario for their triggering, the Lorentz torques on the IC, or within the bulk of the FOC, appears to equally well generate waves traveling from the IC14,21.
As to the 8.6-year oscillation in LOD. One possible mechanism for this oscillation is attributed to the fluid core torsional oscillation normal mode11,14. In this case, it will be not appropriate to use the observed 6-year period to estimate the cylindrical radial component of the magnetic field (\(\tilde B_{\mathrm{{s}}}(s)\)) inside the FOC from the eigenperiod formula1,14,42 of the FOC torsional oscillation normal mode. Instead the 8.6-year period should be used to infer \(\tilde B_{\mathrm{{s}}}(s)\) with the following formula1,14,42:
$$\tilde B_{\mathrm{{s}}}(s) \approx \frac{{r_{\mathrm{{f}}}}}{\tau }\sqrt {\rho _0\mu _0}$$
where \(\tilde B_{\mathrm{{s}}}(s) = \sqrt {\frac{1}{{4\pi h}}{\int}_{ - h}^{ + h} {{\int}_0^{2\pi } {B_{\mathrm{{s}}}^2(s,\phi ,z){\mathrm{{d}}}\phi {\mathrm{{d}}}z} } }\); here \(h(s) = \sqrt {r_{\mathrm{{f}}}^2 - s^2}\) is the half-height of a fluid cylinder, and the cylindrical (s, ϕ, z) coordinates is adopted, i.e., s is the radius of the cylinder, ϕ expresses the longitude, z direction is aligned with the Earth rotation vector \(\vec \Omega\), rf(=3.48 × 106 m) is the radius of the CMB, τ = 8.6 years (the eigenperiod of the normal mode), ρ0(=1.1 × 104 kg m−3) is the average density of the FOC, μ0 is the vacuum permeability with the value of 4π × 10−7 H/m. According to the formula (1), we can estimate \(\tilde B_{\mathrm{{s}}}(s)\) ~ 1.5 mT, which is consistent with the strong magnetic field strength within the FOC (1–4 mT) inferred from tidal dissipation43.
Another alternative mechanism responsible for the 8.6-year signal in LOD is possibly due to the fast equatorial waves with the subdecadal periods propagating at the top of the FOC24,25, which have been inferred by the current geomagnetic satellite (i.e., CHAMP and DMSP—Dense Meterorological Satellite Program) data and ground observatory data24, and these waves were suggested to be the normal mode signals (with the eigenperiod T ~ 8.5 years) of the secular acceleration of the fluid core motions24, which seems to have characteristics of the magnetic Rossby waves in a stratified layer, though a most recent work44 did not favor the presence of a stratified layer at the top of the outer core.
Considering the assumption14,39,45 of significantly stochastic excitation forcing distributing within the bulk of the FOC, the normal mode signals existing in the LOD intradecadal variations (e.g., the MICG mode and the torsional oscillation normal mode) may be masked by the noises produced by the AR-1 stochastic process14,39. However, the question why the purely stochastic forcing distributes within the bulk of the FOC is retained and seems not to be easily answered. In addition, although it is difficult to time geomagnetic jerks accurately and to assess correlations between geomagnetic jerks and other phenomena, this work is just an effort in this respect. Through extracting a new harmonic component (i.e., 8.6-year signal) existing in LOD variations and discussion of its physical origin and its relations to geomagnetic jerks, we hope that this work can make an advance in finally solving this problem on the relationship between geomagnetic jerks and Earth rotation variations.
Combining the result of this work (Fig. 4) with a recent numerical simulation analysis28, the 8.6-year signal in LOD and geomagnetic jerks may result from a same physical source, i.e., the so-called QG Alfvén waves focusing at Earth's core surface. Furthermore, the geomagnetic jerks can be induced by the arrivals of localized Alfvén wave packets from sudden buoyancy releases inside the core (see ref. 28). As these waves reach the surface of the fluid core, they focus their energy towards the equatorial plane and along the strong magnetic flux lines, making the sharp interannual core flow changes. That is, the geomagnetic jerks can be associated with the acceleration of the azimuthal flow motions28, which may associate with the significant angular momentum exchanges between the core and the mantle6, and thus to excite the LOD variations. Meanwhile, the amplitude increasing of the 8.6-year signal is possibly induced by a three-dimensional energy-focusing mechanism28 related to the arrivals of these localized Alfvén wave packets. Here an additional point is worthy of further discussion, i.e., if the 8.6-year signal origin is attributed to the trapped waves in a stratified layer at the Earth' core surface or the so-called QG-Alfvén waves, then the fast torsional waves detected by ref. 1 will only correspond to the 6-year oscillation. Consequently, depending on the physics chosen, the link to the magnetic field within the FOC will differ.
Based on the numerical simulations and analysis made in this work, the proposed method (i.e., NMWT+BEPME) can be used to quantitatively isolate the target harmonic (including the damping13,46 and the increasing) signals with much high-frequency resolution, at the same time, the phase information of the target harmonic signals can also be recovered perfectly (see Supplementary Figs. 3, 9, 10, 14, 15). Hence, the 8.6-year time-domain signal recovered by this work and its fitting result (Fig. 4) provide us a strong clue for possible prediction of the future rapid geomagnetic field changes. Nevertheless, given that the geomagnetic jerks may originate from a stochastic process within the FOC39,45, hence, it is still difficult to make an accurate prediction of the epochs of future geomagnetic jerk occurrence. Despite this, the occurrence of recent geomagnetic jerks was suggested to present an oscillatory behavior32,35, while this work further provides a directly observed evidence to show this oscillatory behavior, which means that the jerk occurrence should not be completely random or unpredictable. If based on the good correspondence revealed by this work, one can predict that a new geomagnetic jerk will happen (with high probability) during the period of 2020–2021.
NMWT method
Defining the time signal as \(h(t) \in L^1(R)\), here
$$L^1(R) = \left\{ {h(t)\left| {\left| {\int_{ - l}^{ + l} {h(t){\mathrm{{d}}}t} } \right| < + \! \infty ,\forall l \in R^ + } \right.} \right\}$$
The mathematical expression of the NMWT is written as following (ref. 12):
$$W_{\mathrm{{g}}}h(a,b) = \frac{1}{{\left| a \right|}}\int_{ - \infty }^{ + \infty } {h(t)\bar g\left( {\frac{{t - b}}{a}} \right)} {\mathrm{{d}}}t,\,a,b \in R,a \, \ne \, 0$$
where, a and b are the scale and time translation factors, respectively, g(t) is the so-called normal Morlet basis function, which differs from the traditional Morlet wavelet basis function. The expression of g(t) is expressed as
$$g(t) = \frac{1}{{\sqrt {2\pi } \sigma }}{\mathrm{{e}}}^{ - \frac{{t^2}}{{2\sigma ^2}} + i2\pi t}$$
where σ is the window-width factor, which determines the frequency-resolution of the NMWT method, and σ is larger, the corresponding frequency-resolution will be higher. Nevertheless, as to the σ value, which is not the bigger the better, since σ is also related to the edge effect range (see the following BEPME strategy). Furthermore, the σ value is here adopted to be 3, which is large enough to distinguish the target intradecadal signals existing in the LOD variations.
As to a harmonic signal h(t), which is expressed by \(h(t) = A_0\exp (i\omega (t - t_0))\), where \(\omega = \frac{{2\pi }}{T}\). Here, defining the scale factor a > 0, there are two useful properties of NMWT in recovering the target signals as following (the proof can be seen in ref. 12):
Property 1. \(W_{\mathrm{{g}}}h(T,b) = h(b)\), (\(\forall t = b\), \(a = T\))
Property 2. \(\frac{\partial }{{\partial a}}\left| {W_{\mathrm{{g}}}h(a,b)} \right| = 0,(\forall a = T)\)
The above properties of the NMWT method is also called as the inaction method46,47.
BEPME strategy
Wavelet transformation (WT) usually owns the edge effect (EE), especially when the original series is not long enough, while the periods of the target signals are relatively long (e.g., the LOD data and the intradecadal signals), then the EE will significantly influence the result amplitudes. To accurately analyze the target harmonic signals in LOD variations on the intradecadal scales using WT method, we must consider the EE and manage to eliminate this effect. This EE range at each side of the data from the NMWT method can be estimated by12
$$R_{\mathrm{{g}}}(a) = 1.643\sigma \left| a \right|$$
where σ is the window-width factor and a refers to the scale factor, while, in the NMWT method, a = T, here T is the period of the target harmonic signal, so a is also called the period factor.
A common simple approach to avoid the EE is the so-called bidirectional mirror-image-symmetric extension (BME) at the beginning and the end of the original data. In the NMWT method, we may also use this extension approach. However, if we directly adopt this traditional way, then the discontinuous points may appear at the two boundaries. If this case is not considered, the signal directly extracted by the NMWT method is not ideal. How to solve this issue? Although the EE exists in the NMWT method, the phase of the target signal recovered by the NMWT method is unbiased12,13. We can make full use of this property to solve the EE problem. In this paper, we propose a simple method, that is to search for the local extreme points of the target harmonic signal with the period T (i.e., a) near the boundaries at both sides of the target signal in the NMWT real coefficient spectrum, and then making the symmetric extension at the two extreme points.
Here, for the sake of clarity, we construct the following composite signal Y(t) (see Supplementary Fig. 1a) to illustrate the relevant steps
$$Y(t) = 1.5{\mathrm{{e}}}^{ - 0.00084t}\cos \left( {2\pi f_1t + \frac{\pi }{2}} \right) + 0.4\cos (2\pi f_2t) + {\mathrm{{noise}}}(t)$$
where f1 = 0.0138 cpm (cycles-per-month, i.e., the 6-year period), f2 = 0.0111 cpm (i.e., the 7.5-year period), the noise (t) term means a significant stochastic noise signal, t is set to be in the range of [1:1:686] with the time-interval 1 month, hence, the data length is 686 months.
Here, we will give the following four steps to avoid the EE and extract the target signal (taking the simulated 6-year oscillation as an example): i.e., firstly, applying NMWT to the composite signal Y(t), we obtain the NMWT spectrum (Supplementary Fig. 1b, c), then, extracting the target signal along the ridge line from the NMWT spectrum, and the result is shown in Supplementary Fig. 2 (the red curve); secondly, searching for the extreme time (ti and tj) at the two boundaries of the data after the above step, and then deleting the data outside the range of ti ≤ t ≤ tj; thirdly, making the symmetric extension of the data after the above two-step processing (see refs. 8,13), here, the data length of the extension part at each side should be larger than the edge effect range Rg(a); finally, applying NMWT once again to the above output, then extracting the target signal from the NMWT spectrum along the ridge-line, and then the result is presented in Supplementary Fig. 3 (the red curve).
After the above four steps, the EE of the NMWT method can be eliminated to a great extent (see Supplementary Fig. 3). This approach (i.e., boundary extreme point mirror-image-symmetric extension, we call it BEPME strategy) is proved to be an effective way to avoid the EE, and it is developed by combining the phase-unbiased feature of the NMWT method in recovering the target signal with the traditional BME method. Nevertheless, it should be noted that although the BEPME strategy adopted in this work can eliminate the EE to a great extent, the derivations caused by the EE cannot be eliminated completely. We expect that there will be a more effective strategy (than our current method) to be developed, and we are making further efforts in this regard as well.
On the SSA method
As the previous works (e.g., refs. 47,48) indicated, the frequency-resolution of SSA is related to the window length parameter (L), choosing an appropriate L value is important for SSA method to analyze the actual data series (see Supplementary Note 2), which shows that the SSA method is not an ideal approach to distinguish and accurately isolate the target intradecadal components existing in the LOD variations (see Supplementary Figs. 5–7).
On the normal mode stochastic excitation
The mathematical expression of a normal mode stochastic excitation can be expressed by an AR-2-damped oscillator stochastic model as following:
$$\frac{{{\mathrm{{d}}}^2y(t)}}{{{\mathrm{{d}}}t^2}} = a_1\frac{{{\mathrm{{d}}}y(t)}}{{{\mathrm{{d}}}t}} + a_2y(t) + E(t)$$
where y(t) is just the target oscillation series, E(t) may be a stochastic process, here the constants a1 < 0 and a2 > 0.
Furthermore, formula (4) can be transformed into
$$\frac{{{\mathrm{{d}}}^2y(t)}}{{{\mathrm{{d}}}t^2}}{\mathrm{ + 2}}\beta \frac{{{\mathrm{{d}}}y(t)}}{{{\mathrm{{d}}}t}} + \omega _0^2y(t) = E(t)$$
where \(\beta = - \frac{1}{2}a_1\) represents the damping factor; \(\omega _0 = \sqrt {a_2}\) expresses the damped oscillation frequency, and \(\omega _0 = \frac{{2\pi }}{{T_0}}\), here T0 expresses the oscillation period. In physics, formula (5) is called as the forced damped oscillation differential equation, where E(t) is also called as the excitation term.
The 6-year oscillation in LOD can be attributed to the mantle–IC gravitational coupling oscillation mode under the action of the electromagnetic coupling effects18,40, which just can be expressed by the formula (5). The analytical solution to the formula (5) can be written as the following convolution form:
$$y(t) = E(t)^\ast \varphi (t) = {\mathrm{{e}}}^{ - \beta t}{\int}_0^t {E(\tau ){\mathrm{{e}}}^{\beta \tau }} \sin [\omega _0(t - \tau )]{\mathrm{{d}}}\tau$$
where * stands for the convolution operator, and \(\varphi (t) = {\mathrm{{e}}}^{ - \beta t}\sin (\omega _0t)\), which is named as the damped oscillation normal function, and β is called as the damping factor (or the theoretical quality factor). The simulation tests of the AR-2-damped stochastic oscillation series are shown in Supplementary Figs. 16–19.
Angular momentum budget from the solid IC
Considering the gravitational coupling interaction between the mantle and the IC without involving the other coupling effects (e.g., electromagnetic coupling, viscous coupling). Assuming that the observed 6-year oscillation is attributed to the pure MICG coupling mode, the 6-year oscillation (i.e., ΔLOD) is related to the axial rotation angular velocity of the IC departing from the gravitational equilibrium position16,17,18,19,20. Since the mantle and the IC consists of a gravitational coupling system, the IC may depart from the equilibrium state under the action of a random torque predicted by the geodynamo16,19, under the condition of the angular momentum conservation, the angular momentum will transfer from the IC to the mantle to cause the corresponding LOD variations. According to the angular momentum conservation law, at any time t, the IC axial rotation angular velocity ui(t) and the angular velocity of the um(t) satisfies the following relationship:
$$u_{\mathrm{{m}}}(t) = - \frac{{C_{\mathrm{{i}}}}}{{C_{\mathrm{{m}}}}}u_{\mathrm{{i}}}(t)$$
According to the relationship between um(t) and the LOD variations (i.e., ΔLOD)
$$u_{\mathrm{{m}}}(t) = - \frac{{2\pi }}{{({\mathrm{{LOD}}}_0)^2}}\Delta {\mathrm{{LOD}}}(t)$$
$$u_{\mathrm{{i}}}(t) = \frac{{2\pi }}{{({\mathrm{{LOD}}}_0)^2}}\frac{{C_{\mathrm{{m}}}}}{{C_i}}\Delta {\mathrm{{LOD}}}(t)$$
When \(\left| {\Delta {\mathrm{{LOD}}}(t)} \right|\) = 0.12 ms, we can estimate the magnitude of ui(t) is 0.22°/year. Importantly, this IC rotation rate (~0.22°/year) required by the 6-year oscillation is consistent with that inferred by the seismology, for example, seismic normal mode inferred that this rate is ±0.2°/year (ref. 49), while the earthquake doublets indicated that this rate is 0.25~0.48°/year (ref. 50). Nevertheless, here it should be noted that, if no angular momentum is carried by FOC, one actually cannot explain LOD changes with angular momentum only in the IC (i.e., it is not possible to ignore the FOC in this balance). Given that the pure MICG mode corresponds to a zonal velocity of ~4.6 km/year at the IC equator, which is about 10 times what is inferred from geomagnetic field changes (e.g., ref. 1), this is one reason for accounting for the fluid core motions in the case of a MICG mode, another reason is the electromagnetic coupling effects between the IC and the FOC, which will strongly couple the two (see ref. 40).
The observed data that support the findings of this work are available from the International Earth Rotation and Reference System Service (IERS) website (https://www.iers.org/IERS/EN/Data Products/Earth Orientation Data/eop.html). The relevant simulation data are included in this manuscript and its supplementary files.
The original version of this Article was updated shortly after publication following an error that resulted in the ORCID IDs of Pengshuo Duan and Chengli Huang being omitted.
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We also thank Xinhao Liao, Guocheng Wang, Xueqing Xu, and Cancan Xu for their helpful discussion. We thank IERS website for providing LOD data. This work is supported by the B-type Strategic Priority Program of the Chinese Academy of Sciences (Grant No. XDB41000000) and National Natural Science Foundation of China (Grant No. 11803064, 11773058, 41774017, and 11373058).
CAS Key Laboratory of Planetary Sciences, Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai, 200030, China
Pengshuo Duan & Chengli Huang
University of Chinese Academy of Sciences, Beijing, 100049, China
Chengli Huang
Pengshuo Duan
P.D. and C.H. performed the primary analysis; P.D. led the primary writing of the manuscript and C.H. also contributed to writing this manuscript.
Correspondence to Pengshuo Duan or Chengli Huang.
Peer review information Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
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Duan, P., Huang, C. Intradecadal variations in length of day and their correspondence with geomagnetic jerks. Nat Commun 11, 2273 (2020). https://doi.org/10.1038/s41467-020-16109-8
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\begin{document}
\title{Quantum entanglement between the electron clouds of nucleic acids in DNA}
\author{Elisabeth Rieper$^1$\footnote{[email protected]}, Janet Anders$^{2}$ and Vlatko Vedral$^{1,3,4}$}
\address{$^1$ Center for Quantum Technologies, National University of Singapore, Republic of Singapore} \address{$^2$ Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom} \address{$^3$ Atomic and Laser Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX13PU, United Kingdom} \address{$^4$ Department of Physics, National University of Singapore, Republic of Singapore}
\date{\today}
\begin{abstract}
We model the electron clouds of nucleic acids in DNA as a chain of coupled quantum harmonic oscillators with dipole-dipole interaction between nearest neighbours resulting in a van der Waals type bonding. Crucial parameters in our model are the distances between the acids and the coupling between them, which we estimate from numerical simulations \cite{Cerny08}. We show that for realistic parameters nearest neighbour entanglement is present even at room temperature. We quantify the amount of entanglement in terms of negativity and single base von Neumann entropy. We find that the strength of the single base von Neumann entropy depends on the neighbouring sites, thus questioning the notion of treating single bases as logically independent units. We derive an analytical expression for the binding energy of the coupled chain in terms of entanglement and show the connection between entanglement and correlation energy, a quantity commonly used in quantum chemistry. \end{abstract}
\maketitle
\section{Introduction} The precise value of energy levels is of crucial importance for any kind of interaction in physics. This is also true for processes in biological systems. It has recently been shown for the photosynthesis complex FMO~\cite{PhotosynExp, mohseni:174106, caruso:105106, Fassioli:2009uq} that maximum transport efficiency can only be achieved when the environment broadens the systems energy levels. Also for the olfactory sense the energy spectra of key molecules seem to have a more significant contribution than their shape \cite{Turin01121996}. In \cite{BP09} the possibility of intramolecular refrigeration is discussed. A common theme of these works is the system's ability to use non-trivial quantum effects to optimise its energy levels. This leads to the question whether a molecule's energy levels are only determined by its own structure, or if the environment {\it shapes} the molecule's energy level? Entanglement between system and environment is a necessary condition to alter the system's state. Here we study the influence of weak chemical bonds, such as intramolecular van der Waals interactions, on the energy level structure of DNA and discuss its connection to entanglement. To describe the van der Waals forces between the nucleic acids in a single strand of DNA, we consider a chain of coupled quantum harmonic oscillators. Much work has been done investigating classical harmonic oscillators. However, this cannot explain quantum features of non-local interactions. Also, classical systems can absorb energy quanta at any frequency, whereas quantum systems are restricted to absorb energy quanta matching their own energy levels. This is of importance for site specific DNA-Protein interaction, as the probability of a protein to bind to a specific sequence of sites in DNA is governed by the relative binding energy \cite{Stormo:2010fk}.
Our work was motivated by a numerical study on the importance of dispersion energies in DNA \cite{Cerny08}. Dispersion energies describe attractive van der Waals forces between non-permanent dipoles. Recently their importance to stabilise macromolecules was realised \cite{Cerny,stacking}. Modelling macromolecules, such as DNA, is a tedious and complex task. It is currently nearly impossible to fully quantum mechanically simulate the DNA. Quantum chemistry has developed several techniques that allow the simulation of DNA with simplified dynamics. In \cite{Cerny08} the authors first quantum mechanically optimise a small fragment of DNA in the water environment. Secondly, they {\it "performed various molecular dynamics (MD) simulations in explicit water based either fully on the empirical potential or on more accurate QM/ MM MD simulations. The molecular dynamics simulations were performed with an AMBER parm9916 empirical force field and the following modifications were introduced in the non-bonded part, which describes the potential energy of the system (see eq 1) and is divided into the electrostatic and Lennard-Jones terms. The former term is modelled by the Coulomb interaction of atomic point-charges, whereas the latter describes repulsion and dispersion energies,"} \begin{equation} V(r)=\frac{q_i q_j}{4 \pi \epsilon_0 r_{ij}}+4 \epsilon \left[ \left(\frac{\sigma}{r_{ij}} \right)^{12}- \left(\frac{\sigma}{r_{ij}} \right)^6 \right] \hbox{ ,} \end{equation} where the strength of the dispersion energy is scaled with the parameter $\epsilon$. For $\epsilon=1$ the dynamics of the DNA strand is normal. For a weaker dispersion, $\epsilon=0.01$, there is in increase of $27\%$ in energy in the DNA. This increase of energy induces the unravelling of the double helix to a flat, ladder-like DNA. Many factors contribute to the spatial geometry of DNA, e.g. water interaction, the phosphate backbone, etc. However, one of the strongest contributions is the energy of the electronic degree of freedom inside a DNA strand, which is well shielded from interactions with water. Stronger interaction ($\epsilon=1$) allows the electrons clouds to achieve spatial configurations that require less structural energy. This allows a denser packing of the electron charges inside the double helix.
\\ \\ Here we investigate with a simple model of DNA whether continuos variable entanglement can be present at room temperature, and how this entanglement is connected to the energy of the molecule. There are many technically advanced quantum chemically calculations for van der Waals type interaction, i.e. \cite{PhysRevLett.91.233202}. The aim of this work is not to provide an accurate model, but to understand underlying quantum mechanical features and their role in this biological system. Also, there are many parallel developments between quantum information and quantum chemistry. This work bridges the concepts of entanglement and dispersion energies between the two fields. Finally, the advantages of quantifying chemical bonds in terms of entanglement were already mentioned in \cite{IJCH:IJCH5680470109}. Here we give the first example of a system whose chemical bonds are described by entanglement.
\section{Dispersion energies between nucleic acids}
The nucleic bases adenine, guanine, cytosine and thymine are planar molecules surrounded by $\pi$ electron clouds. We model each base as an immobile positively charged centre while the electron cloud is free to move around its equilibrium position, see Fig.~\ref{fig:scetchho}. There is no permanent dipole moment, while any displacement of the electron cloud creates a non-permanent dipole moment. Denoting the displacement of two centres by $(x,y,z)$, we assume the deviation out of equilibrium $|(x,y,z)|$ to be small compared to the distance $r$ between neighboring bases in chain. The displacement of each electron cloud is approximated to second order and described by a harmonic oscillator with trapping potential $\Omega$ that quantifies the Coulomb attraction of the cloud to the positively charged centre. A single DNA strand resembles a chain of harmonic oscillators, see Fig.~ \ref{fig:scetch}, where each two neighboring bases with distance $r$ have dipole-dipole interaction.
\begin{figure}
\caption{ This graphic shows a sketch of a DNA nucleic acid. The mostly planar molecules are divided into the positively charged molecule core (red) and the negatively charged outer $\pi$ electron cloud (blue-yellow). In equilibrium the centre of both parts coincide, thus there is no permanent dipole. If the electron cloud oscillates around the core, a non permanent dipole is created \cite{DrudeModel}. The deviation out of equilibrium is denoted by $(x,y,z)$. The corresponding dipole is $\vec{\mu}=Q (x,y,z)$. This oscillation might be caused by an external field, or induced by quantum fluctuations, as it is given in a DNA strand.}
\label{fig:scetchho}
\end{figure}
\begin{figure}
\caption{ This graphic shows a sketch of a single DNA strand. The chain is along $z$ direction. Each bar in the single strand DNA represents one nucleic acid: adenine, thymine, guanine or cytosine. Around the core of atoms is the blue outer electron cloud. The oscillation of these electron clouds is modelled here as non-permanent harmonic dipoles, depicted by the arrows, with trapping potential $\Omega_{d}$ in dimension $d=x,y,z$.}
\label{fig:scetch}
\end{figure}
The Hamiltonian for the DNA strand of N bases if given by \begin{equation} H=\sum_{j,d=x,y,z}^N \left(\frac{p_{j,d}^2}{2m}+\frac{m \Omega_d^2}{2}d_j^2+V_{j,dip-dip} \right) \end{equation} where $d$ denotes the dimensional degree of freedom, and the dipole potential \begin{equation} V_{j, dip-dip}=\sqrt{\epsilon} \frac{1}{4 \pi \epsilon_0 r^3} \left( 3 (\vec{\mu}_j \cdot \vec{r}_N) (\vec{\mu}_{j+1} \cdot\vec{r}_N)-\vec{\mu}_j \cdot \vec{\mu}_{j+1} \right) \end{equation} with $\vec{\mu}_j=Q (x_j, y_j, z_j)$ dipole vector of of site $j$ and $\vec{r}_N$ normalised distance vector between site $j$ and $j+1$. Due to symmetry $\vec{r}_N$ is independent of $j$. The dimensionless scaling factor $\epsilon$ is varied to study the effects on entanglement and energy identical as in \cite{Cerny08}. In order to compare our model with \cite{Cerny08}, we consider 'normal' interaction, where the dipole-dipole interaction has full strength modelled by $\epsilon=1$ and 'scaled' interaction, where the dipole-dipole interaction is reduced to a hundredth of the original strength modelled by $\epsilon=0.01$. The distance between neighbouring bases in DNA is approximately $r_0=4.5 \AA$. For generality we will not fix the distance.\\ In general the single strand of DNA will not be perfectly linear and thus the dipole potential has coupling terms of the form $xz$ etc. Detailed analysis following \cite{1367-2630-12-2-025017} shows that the energy contribution from the cross coupling terms is small, and they will be ignored here. This leads to the interaction term \begin{equation} V_{j, dip-dip}= \frac{Q^2}{4 \pi \epsilon_0 r^3} \left( +x_j x_{j+1} +y_j y_{j+1} -2 z_j z_{j+1} \right) \hbox{ .} \end{equation} The different signs for $x,y$ and $z$ reflect the orientation of the chain along $z$ direction.
A discrete Fourier transformation of the form \begin{eqnarray} d_j = \frac{1}{\sqrt{N}}\sum_{l=1}^N e^{i\frac{2\pi}{N}jl}\tilde{d}_l\nonumber \\ p_{j,d} = \frac{1}{\sqrt{N}}\sum_{l=1}^N e^{-i\frac{2\pi}{N}jl}\tilde{p}_{l,d} \end{eqnarray} decouples the system into independent phonon modes. These modes can be diagonalized by introducing creation $a_{d,l}=\sqrt{\frac{m \Omega_d}{2 \hbar}} \left( \tilde{d}+\frac{i}{m \Omega_d} \tilde{p}_{l,d} \right)$ and annihilation operator $a^\dagger_{d,l}$. This results in the dispersion relations \begin{eqnarray} \omega_{xl}^2 =\Omega_x^2 +2 \left(2\cos^2 \left(\frac{\pi l}{N}\right)-1\right)\frac{ Q^2}{4 \pi \epsilon_0 r^3m} \nonumber \\ \omega_{yl}^2 =\Omega_y^2 +2 \left(2\cos^2 \left(\frac{\pi l}{N}\right)-1\right)\frac{ Q^2}{4 \pi \epsilon_0 r^3m} \nonumber \\ \omega_{zl}^2 =\Omega_z^2 +4 \left(2\sin^2 \left(\frac{\pi l}{N}\right)-1\right)\frac{ Q^2}{4 \pi \epsilon_0 r^3m} \end{eqnarray} and the Hamiltonian in diagonal form \begin{equation} H=\sum_{l=1,d=x,y,z}^N \hbar \omega_{dl} \left(n_{d,l}+\frac{1}{2}\right) \hbox{ ,} \end{equation} where $n_{d,l}=a^\dagger_{d,l}a_{d,l}$ is the number operator of mode $l$ in direction $d$.
The trapping potentials $\Omega_d$ can be linked to experimental data (see table~\ref{data}) through the relation $\Omega_d=\sqrt{\frac{Q^2}{m_e \alpha_d}}$ , where $\alpha_d$ is the polarizability of the nucleid base. So far we did not discuss the number of electrons in the cloud. Both the trapping potential $\Omega_d^2$ as well as the interaction term $\frac{Q^2}{ m}$ depend linearly on the number of electrons, and thus the dispersion frequencies $\omega_{d,l}^2$ have the same dependance. The quantities of interest of this paper are entanglement and energy ratios, which are both given by ratios of different dispersion frequencies and are thus invariant of the number of electrons involved. In Table~\ref{data} we assumed the number of interacting electrons to be one, but our final results are independent of this special choice.
\begin{table}[htdp] \caption{Numerical values for polarizability of different nucleid acid bases \cite{HB89} in units of $1au=0.164\cdot 10^{-40}Fm^2$. The trapping frequencies are calculated using the formula $\Omega=\sqrt{\frac{Q^2}{m_e \alpha}}$ and are given in units of $10^{15}Hz$. } \label{tab:nucacid} \begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline nucleic acid & \phantom{.} $\alpha_x$ \phantom{.}& \phantom{.} $\alpha_y$ \phantom{.} & \phantom{.} $\alpha_z$ \phantom{.}& \phantom{x} $\Omega_x$ \phantom{x} &\phantom{x}$\Omega_y$\phantom{x} &\phantom{x}$\Omega_z$\phantom{x}\\ \hline adenine & 102.5 &114.0 & 49.6 & 4.1 & 3.9 & 6.0 \\ cytosine & 78.8 & 107.1 &44.2 & 4.7 & 4.1 & 6.3 \\ guanine & 108.7 &124.8 & 51.2 & 4.0 & 3.8 & 5.9\\ thymine & 80.7 & 101.7 & 45.9 & 4.7 & 4.2 & 6.2 \\ \hline \end{tabular} \end{center} \label{data} \end{table} Although the values for the four bases differ, all show similar $\Omega_x \approx \Omega_y$ (transverse), while there is an increase of $50\%$ in the longitudinal direction, $\Omega_z \approx \frac{3}{2} \Omega_{x,y}$. In the following we will approximate the chain to have the same value of trapping potential at each base. In $x,y$ direction we will use $\Omega_{x,y}=4\cdot 10^{15}Hz$, and in $z$ direction $\Omega_{z}=6\cdot 10^{15}Hz$.
\section{Entanglement and Energy} We now clarify the influence of entanglement on energy. We will also derive an analytic expressions for the change in binding energy depending on entanglement witnesses.
\\ \\ The chain of coupled harmonic oscillator is entangled at zero temperature, but is it possible to have entanglement at room temperature? There is a convenient way to calculate a criterion for nearest neighbour entanglement for harmonic chains \cite{PhysRevA.77.062102}, which compares the temperature $T$ with the coupling strength $\omega$ between neighbouring sites. In general, for $\frac{2 k_B T} {\hbar \omega}<1 $ one can expect entanglement to exist. Here the coupling between neighbouring clouds is given by $\omega=\sqrt{\sqrt{\epsilon} \frac{Q^2}{4 \pi \epsilon_0 m r^3}}\approx \epsilon^{1/4} 1.6\cdot 10^{15}Hz$ for $r=4.5 \AA$, which leads to $\frac{2 k_B 300K}{\hbar \omega}=0.05$ for $\epsilon=1$ and $0.16$ for $\epsilon=0.01$. This means that the coupling between electron clouds is dominant compared to the temperature, and thus implies the existence of entanglement even at biological temperatures. An exact method to quantify the amount of entanglement in harmonic states it the violation of one of the two inequalities, related to the covariance matrix the state \cite{PhysRevA.70.022318}. \begin{eqnarray}\label{entcrit} 0 \leq S_{1}=\frac{1}{\hbar^2}\left\langle (d_j + d_{j+1})^2\right\rangle \left\langle (p_{d,j} - p_{d,j+1})^2\right\rangle-1 \\ 0 \leq S_{2}=\frac{1}{\hbar^2}\left\langle (d_j - d_{j+1})^2\right\rangle \left\langle (p_{d,j} +p_{d,j+1})^2\right\rangle-1 \end{eqnarray} with $d_j$ position operator of site $j$ in direction $d$ and $p_{d,j}$ corresponding momentum operator. If one of the inequalities is violated, the sites $j$ and $j+1$ are entangled. The negativity, a widely used measure for entanglement, is calculated using the formula $Neg=\sum_{k=1}^2 \max\left[0,-\ln \sqrt{S_k+1}\right]$. The negativity measures the amount of entanglement between two subsystems. It can be directly calculated from space and momentum operator expectation values, namely the above defined $S_{1,2}$ criteria. The amount of negativity between neighbouring bases for room temperature is shown in Fig.~\ref{fig:NegAll}. For the normal coupling there is substantially more entanglement present than for the scaled interaction. This correlates with the amount of binding energy found in \cite{Cerny08}, where the DNA with normal coupling has a lower energy than the DNA with scaled coupling.
\begin{figure}
\caption{ This graphic shows the nearest neighbour negativity as a function of distance between sites in $\AA$ at $T=300K$. The three upper curves are for scaling factor $\epsilon=1$, the lower two curves are for scaling factor $\epsilon=0.01$. The red curve is for $z$ direction and $\Omega_z=6\cdot 10^{15}Hz$. The blue and green curve are for $x$ direction and $\Omega_x=4\cdot 10^{15}Hz$ and $\Omega_x=3\cdot 10^{15}Hz$. The negativity for $\epsilon=0.01$ is much smaller than in the unscaled case. The amount of negativity strongly depends on the distance $r$ between sites and the value of trapping potential $\Omega$. The lower the potential, the higher the negativity. A typical distance between neighbouring base pairs in DNA is approximately $r=4.5 \AA$. Along the chain ($z$-direction) the $S_1$ criterion is violated, whereas transversal to the chain $S_2$ ($x$-direction) is violated. This reflects the geometry of the chain. Along the main axes of the chain energy is reduced by correlated movement. Transversal to the chain it is energetically better to be anti-correlated.}
\label{fig:NegAll}
\end{figure}
The above result motivates the question whether the binding energy can be expressed in terms of entanglement measures. In the limit of long distances, an analytical expression connects the amount of binding energy in the chain of oscillators with the values of $S_{1,2}$. Due to the strong coupling the chain of oscillators is effectively in its ground state, which we will assume in the following analysis.
The dispersion relations of the electron cloud oscillations can be expanded for large distances, i.e. $r^3 \rightarrow \infty$ \begin{equation} \omega_{zl} \approx \Omega_z-4 \frac{ Q^2}{4 \pi \epsilon_0 m} \frac{1}{2 \Omega_z} \cos\left( \frac{ 2 \pi l}{N} \right) \frac{1}{r^3}+O\left[\frac{1}{r^6}\right] \end{equation} and similarly $1/\omega_{zl} $ \begin{equation} \frac{1}{\omega_{zl}} \approx \frac{1}{\Omega_z}+4 \frac{ Q^2}{4 \pi \epsilon_0 m} \frac{1}{2 \Omega_z^3} \cos\left( \frac{ 2 \pi l}{N} \right) \frac{1}{r^3}+O\left[\frac{1}{r^6}\right] \end{equation} Inserting this expansion into the entanglement criterion $S_2$ gives: \begin{equation} S_{z,2} \approx -\frac{ Q^2}{ \pi \epsilon_0 m} \frac{1}{2 \Omega_z^2} \frac{1}{r^3} \hbox{ ,} \end{equation} while the corresponding expression for $S_{z,1}$ has a positive value. A similar expansion of the dispersion relation in $x$ direction leads to: \begin{equation} S_{x,1} \approx -\frac{ Q^2}{2 \pi \epsilon_0 m} \frac{1}{2 \Omega_x^2} \frac{1}{r^3} \hbox{ .} \end{equation} This implies that nearest neighbor (n.n.) electronic clouds are entangled even at large distances. However the amount of entanglement decays very fast. We will now compare this result with the binding energy in the ground state. The binding energy is defined as the difference of energy of the entangled ground state and any hypothetical separable configuration \begin{equation} E_{z,bind}=\langle\hat{H_z} \rangle-\sum_{I=1}^N \langle\hat{H_z}_I \rangle=\hbar/2 \left( \sum_{l=1}^N \omega_{zl}- N \Omega_z \right) \hbox{ .} \end{equation} This definition is analogous to the definition of correlation energy in chemistry \cite{PhysicalChemistry}. The first approximation to the full Schr\"odinger equation is the Hartree-Fock equation and assumes that each electron moves independent of the others. Each of the electrons feels the presence of an average field made up by the other electrons. Then the electron orbitals are antisymmetrised. This mean field approach gives rise to a separable state, as antisymmetrisation does not create entanglement. The Hartree-Fock energy is larger than the energy of the exact solution of the Schr\"odinger equation. The difference between the exact energy and the Hartree-Fock energy is called the correlation energy \begin{equation} E_{corr}=E_{exact}-E_{HF} \hbox{ .} \end{equation} Our definition of binding energy is a special case of the correlation energy, but we restrict our analysis here to phonons (bosons) instead of electrons. Our model describes the motional degree of freedom of electrons, namely the displacement of electron clouds out of equilibrium. We show for this special case that the amount of correlation energy is identical to entanglement measures.\\ Expanding the binding energy for $r^3 \rightarrow \infty$, the leading term is of order $\frac{1}{r^6}$ \begin{equation}\label{eq:Ez} E_{z,bind} \approx \hbar/2 \left( - \left(\frac{ Q^2}{ \pi \epsilon_0 m}\right)^2 \frac{N}{16 \Omega_z^3} \frac{1}{r^6}\right)=-\frac{N \hbar \Omega_z}{8}S_2^2 \hbox{ ,} \end{equation} since the first order vanishes due to symmetry and similarly for $x$ direction: \begin{equation}\label{eq:Ex} E_{x,bind} \approx -\frac{N \hbar \Omega_x}{8}S_1^2 \hbox{ .} \end{equation} Eq.~\ref{eq:Ez}, \ref{eq:Ex} show a simple relation between the entanglement witnesses $S_{1,2}$ and the binding energy of the chain of coupled harmonic oscillators. The stronger the entanglement, the more binding energy the molecule has. Interestingly, along the chain the $S_1$ criterion is violated, whereas transversal to the chain $S_2$ is violated. This reflects the geometry of the chain. Along the main axes of the chain energy is reduced by correlated movement. Transversal to the chain it is energetically better to be anti-correlated. This means that the entanglement witnesses $S_{1,2}$ not only measure the amount of binding energy, but also the nature of correlation which gives rise to the energy reduction. This relation motivates the search for entanglement measures describing the binding energies of complex molecules. While the binding energy just measures energy differences the corresponding entanglement measures reflect more information. Without correlations between subsystems there would not be a chemical bond. It is precisely the purpose of entanglement measures not only to quantify, but also to characterise these correlations.
\section{Aperiodic potentials and information processing in DNA} In the above calculations we assumed a periodic potential, which allowed us to derive analytical solutions. Here we investigate the influence of
aperiodic potentials and discuss the robustness of the previous conclusions. \\ Firstly we note that the potentials for different nucleic acids do not differ significantly, see table \ref{tab:nucacid}. Hence one would intuitively assume that a sequence of different local potentials changes the amount of entanglement but does not destroy it. To check this intuition more thoroughly, one can use the phonon frequencies of the aperiodic chain of oscillators. For a finite one-dimensional chain of 50 bases without periodic boundary conditions and with the sequence of nucleic acids randomly chosen, we solve the resulting coupling matrix numerically. The smallest dispersion frequency determines the thermal robustness; the smaller the frequencies $\omega_l$ the larger the probability that the thermal heat bath can excite the system. Sampling over 1000 randomly chosen sequences yielded $\min(\omega_{l})=3.2 \cdot 10^{15}Hz$ as smallest dispersion frequency. Comparing this with the thermal energy gives $\frac{2 k_B 300K}{\hbar \omega_l} \approx 0.03 $, which is still very small. \\ Thus the thermal energy is more than 20 times smaller than the smallest phonon frequency, which allows us to continue working with the ground state of the system.\\ Different sequences will cause fluctuations in the amount of entanglement in the chain of bases. We determine for each string the average of single site von Neumann entropy and compare it with the classical amount of information measured by the Shannon entropy of each string. The von Neumann entropy of a single site $j$ is obtained following \cite{PhysRevA.70.022318} with the formula \begin{equation}
S_V(r_j)=\frac{r_j+1}{2} \ln \left(\frac{r_j+1}{2}\right)-\frac{r_j-1}{2} \ln \left(\frac{r_j-1}{2} \right) \end{equation} where $r_j =\frac{1}{\hbar} \sqrt{\langle x_j^2 \rangle \langle p_{x_j}^2 \rangle }$, is the symplectic eigenvalue of the covariance matnrix of the reduced state. \\ To check whether the relative frequency of A,C,G and T influences the amount of entanglement within the coupled chain of oscillators, we also calculate the classical Shannon entropy of each string. Fig.~\ref{fig:DistributionDNA} shows the average amount of single site quantum entropy vs. classical entropy. There is, within this model, no direct correlation between classical and quantum entropy. For the same amount of Shannon entropy, i.e. same relative frequencies of A,C,G and T, the value of quantum correlations varies strongly between around $vNE=0.007$ and $vNE=0.025$. We note that for achieving a comparable amount of local disorder by thermal mixing a temperature of more than $2000 K$ is needed. This is a quantum effect without classical counterpart. Each base without coupling to neighbours would be in its ground state, as thermal energy is small compared to the energy spacing of the oscillators. As the coupling increases, the chain of bases evolves from a separable ground state to an entangled ground state. As a consequence of the global entanglement, each base becomes locally mixed. This feature cannot be reproduced by a classical description of vibrations. When a classical system is globally in the ground state, also each individual unit is in its ground state. Although it is already well known that globally entangled states are locally mixed, little is known about possible consequences for biological systems. In the following paragraph we discuss one such quantum effect on the information flow in DNA.\\ \\
\begin{figure}
\caption{ This graphic shows the average single site von Neumann entropy of a chain of nucleic acids dependant on the classical Shannon entropy of the string. Each string contains 50 bases with a random sequence of A,C,G, or T. The distribution of nucleic acids determines the classical Shannon entropy. For each nucleic acid we used the value of polarizability of Table \ref{tab:nucacid} in x direction. The distance between sites is $r=4.5 \cdot 10^{-10} m$. The plot has a sample size of 1000 strings. There is no direct correlation between quantum and classical information. The average amount of von Neumann entropy varies strongly for different sequences.}
\label{fig:DistributionDNA}
\end{figure}
How much information about the neighbouring sites is contained in the quantum degree of freedom of a single base? Is it accurate to describe a single nucleic acid as an individual unit or do the quantum correlations between bases require a combined approach of sequences of nucleic acids? The single site von Neumann entropy measures how strongly a single site is entangled with the rest of the chain and is therefore a good measure to answer this question. In the following we considered a string with 17 sites of a single strand DNA. Site 9 as well as sites 1-7 and 11-17 are taken to be Adenine. The identity of nucleic acids at sites 8 and 10 varies. Table \ref{tab:SingleSite} shows the resulting von Neumann entropy of site 9 dependent on its neighbours. The value of a single site depends on the direct neighbourhood. There is, for example, a distinct difference if an Adenine is surrounded by Cytosine and Thymine ($vNE=0.078$) or by Cytosine and Guanine ($vNE=0.084$). On the other hand, in this model there is little difference between Adenine and Guanine in site 8 and Guanine in site 10. Of course this model has not enough precision to realistically quantify how much a single site knows about its surroundings. Nevertheless it indicates that a single base should not be treated as an individual unit. When quantifying the information content and error channels of genetic information, the analysis is usually restricted to classical information transmitted through classical channels. While we agree that the genetic information is {\it stored} using classical information, e.g. represented by the set of molecules (A,C,G,T), we consider it more accurate to describe the {\it processing} of genetic information by quantum channels, as the interactions between molecules are determined by laws of quantum mechanics.
\begin{table}[htdp] \caption{Numerical values for the von Neumann entropy of site 9 (Adenine) in a chain with open boundary condition containing 17 bases. The bases 1-7 and 11-17 are taken to be Adenine. The column gives the nucleic acid of site 8, the rows of site 10. The von Neumann entropy of site 9 varies with its neighbours. } \label{tab:SingleSite} \begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
& \phantom{.} Adenine\phantom{.} & \phantom{.}Cytosine \phantom{.}& \phantom{.} Guanine \phantom{.} & \phantom{.}Thymine\\ \hline
Adenine & 0.077& 0.082& 0.078& 0.081\\ Cytosine & 0.082 & 0.079 & 0.084& 0.078 \\ Guanine & 0.078& 0.084& 0.079 & 0.083\\ Thymine & 0.081 & 0.078 & 0.083& 0.078\\
\hline \end{tabular} \end{center} \label{data} \end{table}
\section{Conclusions and discussion} In this paper we modelled the electron clouds of nucleic acids in a single strand of DNA as a chain of coupled quantum harmonic oscillators with dipole-dipole interaction between nearest neighbours. Our main result is that the entanglement contained in the chain coincides with the binding energy of the molecule. We derived in the limit of long distances and periodic potentials analytic expressions linking the entanglement witnesses to the energy reduction due to the quantum entanglement in the electron clouds. Motivated by this result we propose to use entanglement measures to quantify correlation energy, a quantity commonly used in quantum chemistry. As the interaction energy given by $\hbar \omega$ is roughly 20 times larger than the thermal energy $k_B 300K$ the motional electronic degree of freedom is effectively in the ground state. Thus the entanglement persists even at room temperature. Additionally, we investigated the entanglement properties of aperiodic potentials. For randomly chosen sequences of A,C,G, or T we calculated the average von Neumann entropy. There exists no direct correlation between the classical information of the sequence and its average quantum information. The average amount of von Neumann entropy varies strongly, even among sequences having the same Shannon entropy. Finally we showed that a single base contains information about its neighbour, questioning the notion of treating individual DNA bases as independent bits of information.
\textit{Acknowledgments}: E.R. is supported by the National Research Foundation and Ministry of Education in Singapore.
J.A. is supported by the Royal Society. V.V. acknowledges financial support from the Engineering and Physical Sciences Research Council, the Royal Society and the Wolfson Trust in UK as well as the National Research Foundation and Ministry of Education, in Singapore.
\end{document} | arXiv |
Inductive dimension
In the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind(X) or the large inductive dimension Ind(X). These are based on the observation that, in n-dimensional Euclidean space Rn, (n − 1)-dimensional spheres (that is, the boundaries of n-dimensional balls) have dimension n − 1. Therefore it should be possible to define the dimension of a space inductively in terms of the dimensions of the boundaries of suitable open sets.
The small and large inductive dimensions are two of the three most usual ways of capturing the notion of "dimension" for a topological space, in a way that depends only on the topology (and not, say, on the properties of a metric space). The other is the Lebesgue covering dimension. The term "topological dimension" is ordinarily understood to refer to the Lebesgue covering dimension. For "sufficiently nice" spaces, the three measures of dimension are equal.
Formal definition
We want the dimension of a point to be 0, and a point has empty boundary, so we start with
$\operatorname {ind} (\varnothing )=\operatorname {Ind} (\varnothing )=-1$
Then inductively, ind(X) is the smallest n such that, for every $x\in X$ and every open set U containing x, there is an open set V containing x, such that the closure of V is a subset of U, and the boundary of V has small inductive dimension less than or equal to n − 1. (If X is a Euclidean n-dimensional space, V can be chosen to be an n-dimensional ball centered at x.)
For the large inductive dimension, we restrict the choice of V still further; Ind(X) is the smallest n such that, for every closed subset F of every open subset U of X, there is an open V in between (that is, F is a subset of V and the closure of V is a subset of U), such that the boundary of V has large inductive dimension less than or equal to n − 1.
Relationship between dimensions
Let $\dim $ be the Lebesgue covering dimension. For any topological space X, we have
$\dim X=0$ if and only if $\operatorname {Ind} X=0.$
Urysohn's theorem states that when X is a normal space with a countable base, then
$\dim X=\operatorname {Ind} X=\operatorname {ind} X.$
Such spaces are exactly the separable and metrizable X (see Urysohn's metrization theorem).
The Nöbeling–Pontryagin theorem then states that such spaces with finite dimension are characterised up to homeomorphism as the subspaces of the Euclidean spaces, with their usual topology. The Menger–Nöbeling theorem (1932) states that if $X$ is compact metric separable and of dimension $n$, then it embeds as a subspace of Euclidean space of dimension $2n+1$. (Georg Nöbeling was a student of Karl Menger. He introduced Nöbeling space, the subspace of $\mathbf {R} ^{2n+1}$ consisting of points with at least $n+1$ co-ordinates being irrational numbers, which has universal properties for embedding spaces of dimension $n$.)
Assuming only X metrizable we have (Miroslav Katětov)
ind X ≤ Ind X = dim X;
or assuming X compact and Hausdorff (P. S. Aleksandrov)
dim X ≤ ind X ≤ Ind X.
Either inequality here may be strict; an example of Vladimir V. Filippov shows that the two inductive dimensions may differ.
A separable metric space X satisfies the inequality $\operatorname {Ind} X\leq n$ if and only if for every closed sub-space $A$ of the space $X$ and each continuous mapping $f:A\to S^{n}$ there exists a continuous extension ${\bar {f}}:X\to S^{n}$.
References
Further reading
• Crilly, Tony, 2005, "Paul Urysohn and Karl Menger: papers on dimension theory" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 844-55.
• R. Engelking, Theory of Dimensions. Finite and Infinite, Heldermann Verlag (1995), ISBN 3-88538-010-2.
• V. V. Fedorchuk, The Fundamentals of Dimension Theory, appearing in Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I, (1993) A. V. Arkhangel'skii and L. S. Pontryagin (Eds.), Springer-Verlag, Berlin ISBN 3-540-18178-4.
• V. V. Filippov, On the inductive dimension of the product of bicompacta, Soviet. Math. Dokl., 13 (1972), N° 1, 250-254.
• A. R. Pears, Dimension theory of general spaces, Cambridge University Press (1975).
Dimension
Dimensional spaces
• Vector space
• Euclidean space
• Affine space
• Projective space
• Free module
• Manifold
• Algebraic variety
• Spacetime
Other dimensions
• Krull
• Lebesgue covering
• Inductive
• Hausdorff
• Minkowski
• Fractal
• Degrees of freedom
Polytopes and shapes
• Hyperplane
• Hypersurface
• Hypercube
• Hyperrectangle
• Demihypercube
• Hypersphere
• Cross-polytope
• Simplex
• Hyperpyramid
Dimensions by number
• Zero
• One
• Two
• Three
• Four
• Five
• Six
• Seven
• Eight
• n-dimensions
See also
• Hyperspace
• Codimension
Category
| Wikipedia |
Association between physician adoption of a new oral anti-diabetic medication and Medicare and Medicaid drug spending
Ilinca D. Metes1,
Lingshu Xue2,
Chung-Chou H. Chang3,4,
Haiden A. Huskamp5,
Walid F. Gellad3,6,7,
Wei-Hsuan Lo-Ciganic8,
Niteesh K. Choudhry9,
Seth Richards-Shubik10,
Hasan Guclu11 &
Julie M. Donohue ORCID: orcid.org/0000-0003-2418-601712,6
BMC Health Services Research volume 19, Article number: 703 (2019) Cite this article
In the United States, there is well-documented regional variation in prescription drug spending. However, the specific role of physician adoption of brand name drugs on the variation in patient-level prescription drug spending is still being investigated across a multitude of drug classes. Our study aims to add to the literature by determining the association between physician adoption of a first-in-class anti-diabetic (AD) drug, sitagliptin, and AD drug spending in the Medicare and Medicaid populations in Pennsylvania.
We obtained physician-level data from QuintilesIMS Xponent™ database for Pennsylvania and constructed county-level measures of time to adoption and share of physicians adopting sitagliptin in its first year post-introduction. We additionally measured total AD drug spending for all Medicare fee-for-service and Part D enrollees (N = 125,264) and all Medicaid (N = 50,836) enrollees with type II diabetes in Pennsylvania for 2011. Finite mixture model regression, adjusting for patient socio-demographic/clinical characteristics, was used to examine the association between physician adoption of sitagliptin and AD drug spending.
Physician adoption of sitagliptin varied from 44 to 99% across the state's 67 counties. Average per capita AD spending was $1340 (SD $1764) in Medicare and $1291 (SD $1881) in Medicaid. A 10% increase in the share of physicians adopting sitagliptin in a county was associated with a 3.5% (95% CI: 2.0–4.9) and 5.3% (95% CI: 0.3–10.3) increase in drug spending for the Medicare and Medicaid populations, respectively.
In a medication market with many choices, county-level adoption of sitagliptin was positively associated with AD spending in Medicare and Medicaid, two programs with different approaches to formulary management.
There is substantial regional variation in prescription drug spending in the United States [1, 2], a finding that is consistent across different classes of drugs, patient populations, and health care payers (e.g. Medicare, VA) [1, 3,4,5]. Much of this variation is attributed to differences in the extent to which physicians prescribe brand name medications as opposed to generic medications, and not to differences in the volume of prescriptions filled, or to patient characteristics [1]. Regional differences in brand name drug prescribing are likely tied to regional differences in the speed with which physicians adopt new drugs. Studies have evidenced tremendous physician-level variation in adoption speed in several drug categories [6,7,8,9,10]; however, the association between region-level differences of physician adoption of newly introduced brand name drugs and prescription drug spending is still poorly understood.
Improving our understanding of how new drug adoption drives prescription drug spending is paramount for U.S. policy makers in the face of ever rising health care expenditures and an aging population that will likely increase demand for chronic disease medications. We examine the association between physician adoption and drug spending for diabetes for three reasons. First, diabetes is a progressive chronic disease that is increasing in prevalence and accounts for a large share of prescription drug and medical spending [11,12,13]. Second, there are multiple FDA approved anti-diabetic drugs available, with varying mechanisms of action, effectiveness, and prices. However, there is little evidence-based guidance for physicians on which medications to prescribe when augmenting therapy [14]. Third, the continual introduction of new brand name anti-diabetic drugs complicates physician decision-making and increases the potential for variation in new drug adoption.
Our study aimed to examine local variation in physician adoption of sitagliptin, a first-in-class oral glycemic lowering agent introduced in October 2006, and to investigate the association between physician adoption of sitagliptin and overall anti-diabetic drug spending in two large, and distinct, payer settings (Medicare and Medicaid). Sitagliptin was the first dipeptidyl peptidase-4 (DPP-4) inhibitor introduced to the market, but was not considered as a first-line treatment option. Therefore, sitagliptin represents the introduction of an expensive brand name drug, considered a moderately novel diabetes treatment, into a market that contained a large number of both generic and brand name treatment options, plus multiple, highly expensive, insulin alternatives [14]. Thus, investigating the role of physician adoption of sitagliptin can highlight how the entry of even one brand name drug in the midst of complex treatment options can influence physician decision making, high variability in new drug adoption, and overall drug spending.
We conducted a cross-sectional analysis using data from three sources. First, we obtained Medicare claims and enrollment data from the Centers for Medicare & Medicaid Services (CMS) for all fee-for-service Medicare enrollees who were residents of Pennsylvania (PA) and also enrolled in a Part D plan for 2011 (N = 855,361). We obtained all medical claims (MEDPAR, outpatient, carrier, home health, hospice, DME) as well as the Part D Event (PDE) file, which contains prescription details such as drug name, fill date, National Drug Code (NDC), and the total amount paid to the pharmacy from all sources (plan and beneficiary). We obtained beneficiary enrollment dates, demographic information, and ZIP code of residence from the Medicare Beneficiary Summary Files.
Second, we obtained claims, encounter, and enrollment data on all fee-for-service and managed care PA Medicaid enrollees for 2011 (N = 1,127,123) from the Pennsylvania Department of Human Services (PADHS) through an intergovernmental agreement. Demographic information and eligibility status were obtained from the Medicaid enrollment file. Prescription drug claims contain information on the drug name, fill date, NDC, and the amount paid to the pharmacy. As we obtained Medicaid data directly from PADHS and not from CMS, we capture drug utilization and medical claims among Medicaid managed care enrollees who make up a majority (~ 75%) of enrollees in the state. PADHS requires comprehensive reporting of encounter data from the managed care plans with which it contracts so the data provide a reliable and valid measure of utilization among managed care enrollees [15].
Third, we obtained physician-level prescribing data from QuintilesIMS Xponent™ which directly captures > 70% of all US prescriptions filled in retail pharmacies, including all payers (Medicare, Medicaid fee-for-service, commercial insurance, cash, and uninsured). Xponent™ utilizes a patented proprietary projection method to represent 100% of prescriptions filled in these outlets and has been widely used by researchers to examine medication use patterns [9, 16,17,18,19,20]. Our Xponent™ data includes all physician prescribers practicing in PA during January 2007–December 2011.
Physician study sample
We excluded those physicians who did not prescribe at least one anti-diabetic drug each quarter in 2007 (the first full year following sitagliptin's introduction in October 2006) so that our physician study sample would include only physicians who were regularly seeing diabetes patients, and were thus eligible to adopt sitagliptin (See Additional file 1: Appendix A for list of anti-diabetic study drugs). To ensure that these physicians were then also continuously seeing patients post-sitagliptin's introduction, without also conditioning specifically on sitagliptin prescribing, we further included only those physicians who prescribed > 1 drug each year (2008–2011) from the following widely used medication classes: anti-coagulants, anti-hypertensives, or statins. Physicians were assigned to one of PA's 67 counties using the zip code of their primary practice location. Three small counties (Cameron, Forest, and Sullivan) had ≤2 providers prescribing anti-diabetic drugs in 2007 and were excluded from the analysis. The final study sample included 7614 physicians (See Additional file 1: Appendix B).
Measures of physician adoption
Our key independent variables were first measured at the physician-level and then aggregated to the county-level. For each physician in our sample, we measured the first month sitagliptin was dispensed to one of their patients, consistent with previous studies measuring physician adoption of new drugs [21,22,23]. In order to capture both speed and extent of physician adoption of sitagliptin we then constructed two measures: 1) mean time (in months) to first sitagliptin prescription across all physicians in a county using 2007–2011 data, and 2) percent of physicians within a county prescribing sitagliptin at least once in 2007. For the first measure, we chose to allow a five year period for the study physicians to adopt sitagliptin, this is based on prior literature, which has found that the rate of physicians adopting a newly introduced drug plateaus between three and five years post-market introduction [9, 24]. Additionally, the latter measure was weighted by each physician's total anti-diabetic prescription volume to give higher weight to physicians with high patient volumes:
$$ \frac{\sum \left(\frac{AD\ prescribing\ volum{e}_{physician}}{AD\ prescribing\ volum{e}_{county}}\ast \#\mathrm{physicians}\ \mathrm{prescribed}\ \mathrm{sitagliptin}\ \mathrm{in}\ 12\ \mathrm{months}\right)}{\# of\ physicians\ in\ county}. $$
We conducted a sensitivity analysis including a measure of the percent of physicians in each county adopting sitagliptin not weighted by prescribing volume, and found the results were qualitatively similar.
Medicare and Medicaid study samples
We constructed separate study samples and conducted all analyses separately for Medicare and Medicaid (See Additional file 1: Appendix C1 and C2 for study sample construction). Since Medicare is the primary payer for beneficiaries who have dual eligibility in both Medicare and Medicaid, dual eligible beneficiaries were included in the Medicare study sample and excluded from the Medicaid sample. For both study samples, we included patients if they: had a continuous 12 months of enrollment in 2011, were ≥ 18 years old on January 1, 2011, were PA residents, filled ≥1 prescription for an anti-diabetic medication in 2011, and met the Chronic Condition Data Warehouse (CCW) Algorithm for diabetes [24, 25]. Additionally, because our study drug, sitagliptin, is not indicated for type I diabetes, we limited both study samples to those with type II diabetes. Individuals who met the CCW algorithm were identified as having type II diabetes if they filled at least one oral anti-diabetic medication in 2011, or if they filled only insulin during 2011 but had ≥50% of all inpatient and outpatient diabetes related claims coded with type II specific ICD-9 codes (250.× 0 or 250.× 2).
Dependent variables: anti-diabetic drug spending
The dependent variables for our analyses were patient-level Medicare and Medicaid anti-diabetic prescription drug spending in 2011. For Medicare, total annual drug spending included both plan payment and beneficiary out-of-pocket spending. For Medicaid, total drug spending included the total plan payment amount, in the case of managed care enrollees, or state payment amount for fee-for-service enrollees. PA Medicaid does require small copayments of its members for some prescription drugs; however, diabetes medications are excluded [26, 27].
Covariates
We included several patient-level variables known to be associated with anti-diabetic drug spending including demographic characteristics, eligibility category and/or type of enrollment status, and clinical factors [28]. Demographic factors include age, sex, and racial or ethnic group (white, black, Hispanic, or other race/ethnicity). For Medicare, enrollment status included indicators for dual eligibility with Medicaid, Part D low-income subsidy (LIS) status, and disability vs. age as reason for eligibility. For Medicaid eligibility, we included categorical variables indicating Temporary Assistance to Needy Families enrollment (TANF), General Assistance enrollment, or Supplemental Security Income enrollment. In Medicaid, we also controlled for whether an enrollee was in fee-for-service or Medicaid managed care. We constructed the Elixhauser co-morbidity index using medical claims as a proxy for overall health status [29]. Finally, we included an indicator of the type of anti-diabetic drug(s) used: oral agents only, injectable agents only (which included all insulins plus exenatide and liraglutide), or a combination of oral and injectable anti-diabetic drugs. As this is a claims based study, no clinical indicators of diabetes disease severity (e.g. hbA1C) were readily available, thus, this measure was included as a potential proxy of diabetes severity, as patients having more intensified treatment with injectable agents or a combination of oral and injectable agents, are likely to have longer disease duration or worse severity, and are more likely to have tried multiple different treatment options than patients on oral agents alone [14].
We first examined descriptive statistics for all study variables in both the Medicare and Medicaid samples. Means (SD) were used to describe all continuous variables and frequencies (percentage) were used to describe all categorical variables. After calculating the two adoption measures, we examined any county-level trends and patterns of these two measures. Second, we examined the distribution of the outcome anti-diabetic drug spending and found it to be highly skewed. After log transforming anti-diabetic drug spending, we found multiple modals of the transformed variable in both the Medicare and Medicaid study populations (Fig. 1). Therefore, we used finite mixture models to empirically identify the patient subgroups in both the Medicare and Medicaid study samples. Third, we fit the appropriate finite mixture model including the key explanatory adoption variables to investigate the association of regional physician adoption of sitagliptin with anti-diabetic drug spending. All covariates of interest were also included in the final models for adjustment.
Raw and Log-Transformed Anti-Diabetic Drug Spending Distributions for the Medicare and Medicaid Study Samples (2011)
Latent subgroup identification
We used finite mixture models to empirically identify patient subgroups based on annual anti-diabetic drug spending. Finite mixture models use model-based posterior probabilities to assign individual observations to different subgroups (e.g. an observation will be assigned to the subgroup with the highest posterior subgroup membership probability) and can analytically capture unobservable heterogeneity in the different underlying subgroups. The true number of subgroups in a data set is unknown, and no gold standard exists in determining the "optimal" number of subgroups. The preferred method of model selection is through an iterative estimation where multiple models with different assumed numbers of subgroups, and no covariates, are fit. We fit models composed of one, two, or three subgroups and examined normal distributions, gamma distributions, or a combination of both normal and gamma distributions. The final number of subgroups, and selection of distributions, were selected based on the Bayesian information criterion (BIC) and the mean posterior probability values [30]. Applying these criteria, the two-component model consisting of two normal distributions had the best model fit for both Medicare and Medicaid (i.e., the lowest BIC and good classification according to high mean posterior probabilities) (Additional file 1: Appendix D1 and D2). For each individual subgroup, descriptive characteristics were inspected to examine potential patient features associated with group membership (Additional file 1: Appendix E).
Association of Sitagliptin Adoption with anti-diabetic drug spending
The two-component finite mixture model, with county-level clustering, was then used to estimate the effect of county-level physician adoption of sitagliptin on patient drug spending in both Medicare and Medicaid. We hypothesized adoption speed (time to first prescription) to be negatively associated with drug spending (i.e. longer time to physician adoption leads to lower spending) and adoption extent (volume-weighted percent of physicians prescribing sitagliptin > 1 in the first 12 months) to be positively associated with anti-diabetic drug spending (i.e. a larger share of physician adoption leads to higher spending). In order to account for heterogeneous estimation using different random seed numbers in the finite mixture modeling approach, we ran the modeling 100 times utilizing randomly generated seeds, and then averaged across all the beta coefficients and standard errors to obtain the final result.
Statistical analyses were conducted using SAS software version 9.3 (SAS Institute, Cary, NC) and R software version 3.2.
Medicare and Medicaid study sample characteristics
Table 1 shows the characteristics of the 125,264 PA Medicare, and 54,098 PA Medicaid enrollees with type II diabetes. Average age in the Medicare sample was 72, while, as expected, the Medicaid sample was relatively younger, with an average age of 50. Both samples had very similar gender breakdowns, with close to 60% being female. While the Medicare sample was 85% white, the Medicaid sample was more diverse, with 50% being white, 30% black, and 15% Hispanic. Additionally, regarding eligibility, 38% of the Medicare sample was dually eligible for Medicaid, while nearly three quarters (73%) of the Medicaid sample was enrolled through Supplemental Security Income eligibility. Regarding general health status, the Medicare sample had an average Elixhauser Index of 5.6, indicated high levels of co-morbidity. Similarly, the Medicaid sample had an average Elixhauser Index of 4.7, which, while nominally lower, still indicates the presence of multiple comorbidities. Lastly, the two samples had relatively distinct anti-diabetic drug use. In Medicare nearly two-thirds (64%) of the study sample was filling prescriptions for oral anti-diabetic medications only, 16% were using insulin or a non-insulin injectable drug only (e.g. exenatide and liraglutide), and 19% were filling prescriptions for both oral and injectable drugs. In Medicaid, 54% of the sample used oral anti-diabetic drugs only, 18% filled only prescriptions for insulin or an injectable drug, and 28% filled prescriptions for both oral and injectable drugs. Overall, while the two samples diverged in many of their demographic and clinical characteristics, all differences were largely expected, and were due to the distinct eligibility requirements of each program.
Table 1 Demographic Characteristics of Medicare and Medicaid Study Samples (2011)
Anti-diabetic drug spending
Unadjusted average per capita spending on anti-diabetic drugs was $1340 (SD $1764) in Medicare and $1242 (SD $1844) in Medicaid (Table 1). Figure 1 shows the non-transformed and log-transformed distributions and density plots for anti-diabetic drug spending for both the Medicare and Medicaid study samples.
Physician adoption of Sitagliptin
A total of 7614 PA physicians prescribed anti-diabetic drugs in our study sample. The number of physicians who prescribed anti-diabetic drugs in each county varied from seven to 1136. (Additional file 1: Appendix F).
Both the adoption time (mean time to first sitagliptin prescription), and adoption extent (percent of physicians prescribing sitagliptin at least once in its first 12 months) measures showed high variability across the counties (Fig. 2). Overall, average time to first prescription of sitagliptin was slightly less than a year (11.2 ± 3.5 months), though the time did vary markedly by county from 2.3 months (Potter County) to 19.1 months (Mifflin County). Average weighted fraction of physicians in each county prescribing sitagliptin at least once in the first 12 months of market availability was 78% ± 12%. Again, there was substantial variation between the counties from 44% of physicians adopting (Venango County) to 99% of physicians adopting (Elk County).
Measures of Sitagliptin Adoption by Pennsylvania County
Characteristics of subgroups
In Medicare, 55% of beneficiaries were categorized into the component with lower mean anti-diabetic drug spending, and 45% was categorized into the component with higher mean anti-diabetic drug spending (Additional file 1: See Appendix E). The largest difference in observable characteristics between the two spending components was in the type of anti-diabetic drugs used, with 85% of beneficiaries in the lower spending component utilizing oral drugs only vs. 38% of beneficiaries in the higher spending component.
Similarly, 57% of Medicaid beneficiaries were categorized in the component with lower mean anti-diabetic drug spending, while 43% were in the component with higher mean anti-diabetic drug spending (Additional file 1: See Appendix E). Again, the largest difference in observable characteristics between the two spending components was in the type of anti-diabetic drugs used, with 66% of enrollees in the lower spending component utilizing oral drugs only, while only 38% of enrollees in the higher spending component utilizing oral drugs only.
Association of Sitagliptin Adoption with overall drug spending
For the Medicare study sample, the finite mixture model results indicated that having a higher percent of physicians within a county adopting sitagliptin was associated with higher annual anti-diabetic drug spending on average (Table 2). The magnitude, and variation, of this result differed between the two spending components. For example, a 10% increase in the number of physicians within a county adopting sitagliptin was associated with an average increase of 3.5% (95% CI: 2.0–4.9) in annual per capita anti-diabetic drug spending for the lower spending component. That same 10% increase was associated with a smaller, and non-statistically significant, average increase of 1.5% (95% CI: − 3.6 – 6.5) in anti-diabetic drug spending in the higher spending component. In comparison, mean time to first prescription of sitagliptin was found to have no statistically significant association with drug spending in either spending component.
Table 2 Results from the Finite Mixture Model of Anti-diabetic Drug Spending in the Medicare Study Sample (2011)
The results for the Medicaid study sample were similar in average magnitude to the Medicare results. For example, a 10% increase in the number of physicians within a county adopting sitagliptin was associated with a smaller, and non-statistically significant average increase of 2.9% (95% CI: − 0.4 – 6.3) in annual per capita anti-diabetic drug spending for the lower spending component. That same 10% increase was associated with a significant average increase of 5.3% (95% CI: 0.3–10.3) in anti-diabetic drug spending for the higher spending component. Again, and similarly to Medicare, mean time to first prescription of sitagliptin was not statistically significantly associated with drug spending in either component (Table 3).
Table 3 Results from the Finite Mixture Model for Medicaid Study Sample (2011)
Our study reports three key findings. First, we found substantial county-level variation in both the time to adoption and in the proportion of physicians adopting sitagliptin in PA. Second, we found that the extent of physicians adopting sitagliptin was associated with higher anti-diabetic drug spending in both Medicare and Medicaid although effect sizes were relatively small. Third, we found that the distributions of anti-diabetic drug spending in the Medicaid and Medicare populations were remarkably similar, as were the magnitudes of the associations between sitagliptin adoption and drug spending in spite of differences in population characteristics and the administration of drug benefits in the two programs.
The high variability in physician adoption rates of sitagliptin by county in both average time to first prescription (2.3–19.1 months) and share of physicians prescribing sitagliptin (44 to 99% of physician) is consistent with previous findings. Studies have shown that physicians' take up new brand name drugs at different rates, and that the proportion of brand name versus generic drug use can vary across geographic regions [3,4,5, 31,32,33,34]. Physician adoption of new drugs is likely influenced by many factors including practice setting (e.g. group vs. solo practice) [19, 35, 36], specialty [17, 22, 23], exposure to pharmaceutical promotion [6, 18, 21, 35], and even physician social networks [20].
Furthermore, while prior studies have highlighted the variation in physician drug adoption, our study is one of the first to show an association between the speed and extent of physician adoption of a new drug and prescription drug spending. This finding is consistent with the literature showing the diffusion of health care technologies is one of the main drivers of health care cost growth [37]. The impact of technological advancement on increased health care spending is perhaps nowhere more evident than with prescription drugs. For example, the recent double-digit annual growth rate in prescription drug spending from 2013 to 2014 has largely been attributed to the introduction of new prescription drugs [38]. Growth in prescription drug spending has also coincided with an increasing number of new drugs gaining FDA approval annually, which reached a recent peak in 2015 with 45 new drugs entering the market [39], and underscores the on-going role that new drug adoption will likely play in health care spending. Although the magnitude of the effect of sitagliptin adoption on spending was relatively small in the anti-diabetic drug class, our findings point to a potential mechanism underlying the geographic variation in prescription drug spending, namely, differences in diffusion of new drugs at the local-level. This finding could lead to interventions by payers looking to improve the efficiency of prescription drug use, by combining information on the new drug adoption behavior of physicians with information on the clinical value of new drugs, and ultimately targeting physicians for interventions such as academic detailing [40].
A strength of our study was the ability to investigate the association between physician drug adoption on prescription drug spending in both Medicare and Medicaid. This shows a more complete picture of the role of physician drug adoption since these two payers serve distinct patient populations, and have structural differences in benefit design and formulary policy. Interestingly, even though the Medicare and Medicaid study populations differed in fundamental ways such as average age (72 vs. 50 years old), racial composition (85% vs. 50% white), and average Elixhauser comorbidity index (5.6 vs. 4.7), the overall distribution of anti-diabetic drug spending in each program was remarkably similar (Fig. 1). This similarity is surprising not only due to differences in patient populations served, but also due to differences in benefit design and cost-containment tools used in the two programs. For example, Medicare plans use tiered formularies, prior authorization, and patient cost sharing to steer patients to drugs for which Prescription Drug Plans (PDPs) have negotiated larger rebates [41]. In contrast, Medicaid programs participate in the Medicaid Drug Rebate Program, which requires broad coverage of medications, and use prior authorization tools, but impose no patient cost sharing [42]. Though we did not limit our physician study sample by type of payment received, one possibility for the similar spending patterns between the two programs is that the same physicians are serving both patient populations, and their prescribing patterns remain generally stable across payers. Interestingly, the key driver of whether enrollees were in the high or low spending component was type of anti-diabetic drug use. Subjects treated with an oral anti-diabetic drug were much more likely to be assigned to the lower spending component, while subjects treated with an injectable anti-diabetic drug such as insulin were much more likely to be assigned to the higher spending component. Additionally, it is likely that the patients in the higher spending component have more severe or uncontrolled diabetes and have already failed first line oral treatment options, thus giving their treating physician multiple options in how to escalate their care, either by adding multiple oral drugs to their treatment plan, or moving on to injectable insulin. That insulin is a key driver of anti-diabetic drug spending could partially explain the relatively small effect size of sitagliptin adoption on spending, and is consistent with a recent study that found that, in Medicaid, reimbursement prices for intermediate acting insulins have grown 284% from 2001 to 2014, and by 455% for premixed insulins in the same time period [43].
Our study has several limitations. First, this study was conducted in one state, and even though PA has been shown to track closely with national averages in measures of age, gender, educational attainment, income, and measures of health care utilization, our findings might not be nationally representative [44,45,46]. Second, we investigated the impact of one new drug within one chronic disease drug class; in light of the fact that multiple therapeutic options exist within the diabetes drug class, and the choice set changes over time as new drugs enter the market, our results regarding sitagliptin might not generalize to the drug class as a whole. Additionally, these results might not be generalizable to other unique disease conditions. Third, like other studies using claims data, our drug spending measures do not include rebates negotiated by Part D plans in Medicare, rebates provided under the Medicaid drug rebate program, or any differences in charges by pharmacies that might be owned by managed care companies. Thus our spending measures reflect an over-estimation of the true spending amount [47]. Lastly, since there is no gold standard on how physician adoption of new prescription drugs should be measured, we defined adoption through time to first prescription, and the proportion of physicians adopting sitagliptin weighted by prescribing volume, both of which have been utilized in past studies that have investigated physician up-take of new drugs [9, 21,22,23, 48, 49]. Other measures of physician adoption exist, such as measures that take the share of prescriptions written for a new drug into account [50], and could strengthen the findings.
This study represents the first analysis that aims to better understand regional variation in physician adoption of a newer anti-diabetic brand name prescription drug, and to determine that higher physician drug adoption is associated with higher prescription drug spending. Future research should focus on examining this association in other drug classes, and on further elucidating the underlying mechanisms surrounding why some physicians adopt brand name drugs faster than others.
The data that support the findings of this study are available from Centers for Medicare and Medicaid Services, PADHS, and Quintile's IMS but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from Centers for Medicare and Medicaid Services and Quintile's IMS for a fee and under the data use agreement provisions. The websites and instructions on how others may access the relevant data are made available below:
For Medicare and Medicaid enrollment and claims data:
ResDAC (Research Data Assistance Center), https://www.resdac.org/
Centers for Medicare and Medicaid Services, https://www.cms.gov/Research-Statistics-Data-and-Systems/Research-Statistics-Data-and-Systems.html
For Xponent, HCOS, and AMA masterfile data:
Quintile's IMS (now IQVIA), https://www.iqvia.com/locations/united-states
Bayesian information criterion
CCW:
Chronic Condition Data Warehouse
LIS:
Low-income subsidy
NDC:
National Drug Code
PA:
PADHS:
Pennsylvania Department of Human Services
PDE:
Part D Event
TANF:
Temporary Assistance for Needy Families enrollment
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Luo J, Avorn J, Kesselheim AS. Trends in Medicaid reimbursements for insulin from 1991 through 2014. JAMA Intern Med. 2015;175(10):1681–6.
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The authors thank Ruoxin Zhang and Aiju Men for expert programming.
This study and medical writing for this manuscript were funded by the National Heart Lung and Blood institute (NHLBI) through a research grant (R01HL119246). NHLBI had no role in the design of the study, the writing of the manuscript, or in the collection, analysis, or interpretation of the data.
Department of Health Policy and Management, Graduate School of Public Health, University of Pittsburgh, 130 DeSoto Street, Crabtree Hall A651, Pittsburgh, PA, 15261, USA
Ilinca D. Metes
Department of Epidemiology, Graduate School of Public Health, University of Pittsburgh, 130 DeSoto Street, Crabtree Hall A651, Pittsburgh, PA, 15261, USA
Lingshu Xue
Department of Medicine, School of Medicine, University of Pittsburgh, 200 Meyran Avenue, Suite 200, Pittsburgh, PA, 15213, USA
Chung-Chou H. Chang
& Walid F. Gellad
Department of Biostatistics, Graduate School of Public Health, University of Pittsburgh, Pittsburgh, PA, 15261, USA
Department of Health Care Policy, Harvard Medical School, 180 Longwood Avenue, Boston, MA, 02115, USA
Haiden A. Huskamp
Center for Pharmaceutical, Policy and Prescribing, Health Policy Institute, University of Pittsburgh, Pittsburgh, PA, 15261, USA
Walid F. Gellad
& Julie M. Donohue
Center for Health Equity Research and Promotion, Veterans Affairs Pittsburgh Healthcare System, University Drive (151C), Pittsburgh, PA, 15215, USA
Department of Pharmaceutical Outcomes and Policy, College of Pharmacy, University of Florida, 1225 Center Drive, Gainesville, FL, 32610, USA
Wei-Hsuan Lo-Ciganic
Division of Pharmacoepidemiology and Pharmacoeconomics, Department of Medicine, Brigham and Women's Hospital and Harvard Medical School, 1620 Tremont Street, Suite 3030, Boston, MA, 02120, USA
Niteesh K. Choudhry
College of Business and Economics, Lehigh University, Rausch Business Center, Room 465, 621 Taylor St, Bethlehem, PA, 18015, USA
Seth Richards-Shubik
Department of Statistics, School of Science, Istanbul Medeniyet University, Uskudar, 34700, Istanbul, Turkey
Hasan Guclu
Julie M. Donohue
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IDM drafted the manuscript. IDM, LX, CCHC, HAH, WFG, WHLC, NKC, SRS, HG, and JMD contributed to the study concept and design; IDM and LX analyzed study data, and IDM, LX, CCHC, and JMD interpreted study data. IDM, LX, CCHC, HAH, WFG, WHLC, NKC, SRS, HG, and JMD provided critical review and edits to the manuscript and gave final approval before submission.
Correspondence to Julie M. Donohue.
The University of Pittsburgh IRB reviewed this study (IRB#: PRO12100227) and ruled that since it did not involve any interaction with human subjects, only review of existing data, it did not require formal ethics approval and could be designated as "exempt". Per the University of Pittsburgh IRB the study met all the necessary criteria for an exemption under section:
45 CFR 46.101(b) (4) existing data, documents, or records
Additionally, the Centers for Medicare and Medicaid Services (CMS), the Pennsylvania Department of Human Services (PADHS), and Quintile's IMS granted formal permission to access their databases through separate data use agreements with the University of Pittsburgh.
Additional file 1. This file includes all supplemental data/tables referenced in the manuscript.
Metes, I.D., Xue, L., Chang, C.H. et al. Association between physician adoption of a new oral anti-diabetic medication and Medicare and Medicaid drug spending. BMC Health Serv Res 19, 703 (2019) doi:10.1186/s12913-019-4520-4
Accepted: 10 September 2019
Physician behavior
Utilization, expenditure, economics and financing systems | CommonCrawl |
\begin{document}
\title{Aggregation-based cutting-planes for packing and covering integer programs
}
\author[1]{Merve Bodur\thanks{[email protected]}} \author[2]{Alberto Del Pia\thanks{[email protected]}} \author[1]{Santanu S. Dey\thanks{[email protected]}} \author[3]{Marco Molinaro\thanks{[email protected]}} \author[1]{Sebastian Pokutta\thanks{[email protected]}} \affil[1]{\small School of Industrial and Systems Engineering, Georgia Institute of Technology} \affil[2]{\small Department of Industrial and Systems Engineering \& Wisconsin Institute for Discovery, University of Wisconsin-Madison} \affil[3]{\small Computer Science Department, Pontifical Catholic University of Rio de Janeiro}
\maketitle
\begin{abstract} In this paper, we study the strength of Chv\'atal-Gomory (CG) cuts and more generally \emph{aggregation cuts} for packing and covering integer programs (IPs). Aggregation cuts are obtained as follows: Given an IP formulation, we first generate a single implied inequality using aggregation of the original constraints, then obtain the integer hull of the set defined by this single inequality with variable bounds, and finally use the inequalities describing the integer hull as cutting-planes. Our first main result is to show that for packing and covering IPs, the CG and aggregation closures can be \emph{2-approximated} by simply generating the respective closures for each of the original formulation constraints, without using any aggregations.
On the other hand, we use computational experiments to show that aggregation cuts can be arbitrarily stronger than cuts from individual constraints for general IPs. The proof of the above stated results for the case of covering IPs with bounds require the development of some new structural results, which may be of independent interest. Finally, we examine the strength of cuts based on $k$ different aggregation inequalities simultaneously, the so-called multi-row cuts, and show that
\emph{every} packing or covering IP with a large \emph{integrality gap} also has a large \emph{$k$-aggregation closure rank}. In particular, this rank is always at least of the order of the logarithm of the integrality gap. \\ \\
\noindent \textbf{Keywords.} Integer programming, cutting planes, packing, covering, aggregation
\end{abstract}
\section{Introduction}\label{sec:intro} Cutting-planes are central to state-of-the-art integer programming (IP) solvers \cite{bixby2004,Lodi2009}. While different methods have been developed to generate various families of cutting-planes~\cite{marchand:ma:we:wo:2002,RichardDey}, several of the most important families are obtained through the aggregation of the original constraints of the problem. These are special types of what we call \emph{aggregation cuts}, which are those generated as follows: given an IP formulation, we first obtain a single implied inequality by aggregating the original constraints, and then generate a cut valid for the integer hull of the set defined by this single inequality together variable bounds.
It is easy to see that \emph{Chv\'atal-Gomory (CG) cuts} are aggregation cuts: in fact, each CG cut is precisely the integer hull of the set defined by one aggregated inequality \emph{without} variable bounds. Aggregation cuts include many other classes of cuts, such as lifted knapsack covers inequalities~\cite{wolsey:1975,zemel:1978} and weight inequalities \cite{weismantel19970}. The set of all aggregation cuts have been studied empirically~\cite{FukasawaG11}, but to the best of our knowledge no theoretical study is present.
Given the ubiquity of aggregation cuts, it is important to better understand the role of aggregation in integer programming. Of direct practical importance is to understand which aggregations are most useful. Another interesting direction, which we pursue here, is to understand in which cases aggregation is most helpful and what are the limitations of using aggregation-based cuts.
In this paper, we examine the strength of aggregation cuts for \emph{packing} and \emph{covering} IPs. Our main result is that for these classes of problems, even considering all infinitely many aggregations offers limited help. More precisely, we show that the CG and more generally aggregation closures can be 2-approximated by simply generating the respective closures for each of the original constraints, without using any aggregations. Therefore, for these problems, in order to obtain cuts that are much stronger than original constraint cuts, one needs to consider more complicated cuts that cannot be generated through aggregations; see for example the results in \cite{deyMolinaroWang:2016}.
We also examine the strength of cuts based on $k$ different aggregated inequalities simultaneously (also called \emph{multi-row cuts}) for packing and covering problems. We show that \emph{every} packing or covering IP with a large integrality gap also has a large \emph{$k$-aggregation closure rank}; more precisely, for a fixed $k$, this rank is always at least of the order of the logarithm of the integrality gap. This again points to the relative weakness of aggregation cuts for packing or covering problems.
Finally, simple examples show that these results are not true for general IPs, where aggregations can produce significant benefits. We provide further empirical evidence for this fact based on randomly generated general IPs and \emph{market split} instances \cite{cornuejols:da:1999}.
From cut selection perspective, the insight here is that for packing and covering problems, using aggregation cuts may provide limited benefit over using cuts generated from only the original constraints, while aggregation cuts may produce significant value for general IPs.
\paragraph{Organization.} In Section \ref{sec:DefnStatements} we provide definitions and statements of all our main results and discuss them in more detail; we also present results from the computational experiments. In Section \ref{sec:Discussion} we state some open questions. Finally, in Section \ref{sec:Packing} and Section \ref{sec:Covering} we present the proofs for results concerning the packing and covering cases, respectively.
\section{Definitions and statement of results} \label{sec:DefnStatements}
\subsection{Definitions}
For an integer $n$, we use the notation $[n]$ to describe the set $\{1, \dots, n\}$.
For $i \in [n]$, we denote by $e_i$ the $i^{\text{th}}$ vector of the standard basis of $\mathbb{R}^n$. The convex hull of a set $S$ is denoted as $\textup{conv}(S)$, its conic hull is denoted as $\textup{cone}(S)$, and its closed conic hull is donated as $\clcone(S)$. For a set $S \subseteq \mathbb{R}^n$ and a positive scalar $\alpha$ we define $\alpha S:= \{\alpha u\, |\, u \in S\}$.
\paragraph{Packing and covering.} A \emph{packing polyhedron} is of the form $\{x \in \mathbb{R}^n_+ \mid Ax \le b \}$ where all the data $(A,b)$ is non-negative and rational. While polyhedral sets are the main object of study here, we will also need non-polyhedral ones.\footnote{This is needed because we do not know whether the aggregation closure is polyhedral.} So a \emph{packing set} is one of the form $\{x \in \mathbb{R}^n_+ \mid A^i x \le b_i \ \forall i \in I\}$ where each $(A^i, b_i) \in (\mathbb{R}_+^{1 \times n}, \mathbb{R}_+)$ and $I$ is an arbitrary set.
Similarly, a \emph{covering polyhedron with bounds} is of the form $\{x \in \mathbb{R}^n_+ \mid Ax \ge b, ~x \le u \}$ where all the data $(A,b,u)$ is non-negative and rational. We assume a component of $u$ is either finite and integral, or infinite. If all upper bounds take the value of infinity, then we simply call the set a \emph{covering polyhedron}. In the non-polyhedral case, a \emph{covering set with bounds} has the form $\{x \in \mathbb{R}^n_+ \mid A^i x \ge b_i \ \forall i \in I, ~x \le u \}$ with $(A^i, b_i) \in (\mathbb{R}_+^{1 \times n}, \mathbb{R}_+)$ and $u$ satisfying the same assumptions as above, but $I$ is an arbitrary set.
\paragraph{Closures.} Given a polyhedron $Q$, we are interested in cuts for the pure integer set $Q \cap \mathbb{Z}^n$.
We use $\mathcal{C}(Q)$ and $Q^I$ to denote the CG closure and the convex hull of integer feasible solutions of $Q$, respectively (see, e.g., \cite{ConCorZam14b} for definitions). Moreover, given a packing polyhedron $Q = \{x \in \mathbb{R}^n_+ \mid Ax \le b\}$, we define its \emph{aggregation closure} as $$\mathcal{A}(Q) := \bigcap_{\lambda \in \mathbb{R}^m_+} \conv(\{x \in \mathbb{Z}^n_+ \mid \lambda^\top A x \le \lambda^\top b\}).$$ Similarly, for a covering polyhedron $Q = \{x \in \mathbb{R}^n_+ \mid Ax \ge b, \ x \le u\}$ its aggregation closure is defined as $$\mathcal{A}(Q) := \bigcap_{\lambda \in \mathbb{R}^m_+} \conv(\{x \in \mathbb{Z}^n_+ \mid \lambda^\top A x \ge \lambda^\top b, \ x \le u\}).$$ Notice that we leave the bounds of the variables disaggregated, which gives a stronger closure than if we had just kept the non-negativity inequalities disaggregated. It is clear that all CG cuts are aggregation-based cuts, namely $\mathcal{C}(Q) \supseteq \mathcal{A}(Q)$.
In order to understand the power of aggregations for generating cuts of these families, we define the 1-row (or non-aggregated) version of these closures. The \emph{1-row CG closure} $1\mathcal{C}(Q)$ is defined as the intersection of the CG closures of the individual inequalities defining $Q$, together with variable bounds; more precisely, for a packing polyhedron $Q$ $$1\mathcal{C}(Q) = \bigcap_{i \in [m]} \mathcal{C}(\{x \in \mathbb{R}^n_+ \mid A^ix \le b_i\}),$$ and for a covering polyhedron with bounds we have $$1\mathcal{C}(Q) = \bigcap_{i \in [m]} \mathcal{C}(\{x \in \mathbb{R}^n_+ \mid A^ix \ge b_i, \ x \le u\}),$$ where $A^i$ denotes the $i^{\text{th}}$ row of $A$. The \emph{1-row closure} $1\mathcal{A}(Q)$ is defined analogously, simply replacing the operator $\mathcal{C}(.)$ by $\mathcal{A}(.)$.
Given a packing polytope $Q$ and a non-negative objective function $c \in \mathbb{R}^n_+$, we define $$z^{1\mathcal{C}} := \max\{c^\top x \mid x \in 1\mathcal{C}(Q)\}$$ as the optimal value over the closure $1\mathcal{C}(Q)$, and similarly for all the other closures, namely $z^{1\mathcal{A}},z^{\mathcal{A}},z^{\mathcal{C}}$. Moreover, we use $z^I$ and $z^{LP}$ to denote the optimal objective function value over $Q$ and its linear programming (LP) relaxation, respectively. For covering integer sets (with bounds) ``$\max$'' is replaced with ``$\min$''.
We can generalize the aggregation closure to consider simultaneously $k$ aggregations, where $k \in \mathbb{Z}$ and $k \geq 1$. More precisely, for a covering polyhedron $Q$ the \emph{$k$-aggregation} closure is defined as
\begin{align*}
\mathcal{A}_k(Q) := \bigcap_{\lambda^1, \ldots, \lambda^k \in \mathbb{R}^m_+}
\conv(\{ x \in \mathbb{Z}^n_+ \mid
(\lambda^j)^\top A x \ge (\lambda^j)^\top b~
\forall j\in[k],
~x \le u \}),
\end{align*}
and the definition is similar for the packing case.
More generally, given a packing \emph{set} $Q$, its $k$-aggregation closure $\mathcal{A}_k(Q)$ is defined as the intersection of all sets $\conv(\{x \in \mathbb{Z}^n_+ \mid D^j x \le f_j ~\forall j \in [k]\})$ where each of the $k$ rows $D^j x\le f_j$ is a valid inequality for $Q$ with non-negative coefficients. Similarly, given a covering set with bounds $Q$, $\mathcal{A}_k(Q)$ is defined as the intersection of all sets $\conv(\{x \in \mathbb{Z}^n_+ \mid D^j x \ge f_j ~\forall j \in [k], ~x \le u\})$ where each $D^j x\ge f_j$ is a valid inequality for $Q$ with non-negative coefficients. Notice that these definitions are independent of the representation of $Q$, and in the polyhedral case a duality argument shows that they are equivalent to the aggregation-based ones given above.
The \emph{$k$-aggregation closure rank}, denoted by $\rank_{\mathcal{A}_k}(Q)$, is defined in the standard way: it is the minimum number of applications $\mathcal{A}_k(\mathcal{A}_k(\ldots \mathcal{A}_k(Q) \ldots))$ of $A_k$ in order to obtain the convex hull of $Q$. Notice that if $Q$ is a packing (resp. covering) set, $\mathcal{A}_k(Q)$ is a packing (resp. covering) set, so iterating the closure $\mathcal{A}_k$ is a well-defined operation; we will formally verify this later. Moreover, since the CG rank is always finite \cite{schrijver1980cutting}, and the aggregation closure of $Q$ is contained in the CG closure of $Q$, we have that $\rank_{\mathcal{A}_k}(Q)$ is always finite.
\paragraph{Approximation.} Given two packing sets $U \supseteq V$, we say that $U$ is an \emph{$\alpha$-approximation} of $V$
if for all non-negative objective functions $c \in \mathbb{R}^n_+$ we have $$\max\{c^\top x \mid x \in U\} \le \alpha \cdot \max\{c^\top x \mid x \in V\}.$$ Notice that since $U \supseteq V$, we have $\alpha \ge 1$.
Similarly, for a covering polyhedron (with bounds) $Q$, given two covering sets $U \supseteq V$
we say that $U$ is an $\alpha$-approximation of $V$ if for all $c \in \mathbb{R}^n_+$ we have $$\min\{c^\top x \mid x \in U\} \ge \frac{1}{\alpha} \cdot \min\{c^\top x \mid x \in V\}.$$
\subsection{Statement of results}
\subsubsection{Packing}
The following is our main result comparing closures with their 1-row counterparts.
\begin{theorem}\label{thm:pack}
Consider a packing polyhedron $Q$.
Let $\mathcal{M}$ be any of the closures $\mathcal{A}$ (aggregation) or $\mathcal{C}$ (CG). Then $1\mathcal{M}(Q)$ is a 2-approximation of $\mathcal{M}(Q)$.
Moreover, this bound is tight, namely for every $\varepsilon > 0$ there is a packing polyhedron $Q$ such that $1\mathcal{M}(Q)$ is not a $(2-\varepsilon)$-approximation of $\mathcal{M}(Q)$. \end{theorem}
In the proof of Theorem~\ref{thm:pack} we introduce a special polyhedral relaxation of the convex hull of a packing polyhedron $Q$ that we call the \emph{pre-processed LP}. In this pre-processed LP, we examine if $A_{ij} > b_i$ for some $i \in [m],~j\in [n]$, in which case we set $x_j$ to 0. The optimal objective function value of the pre-processed LP is denoted by $z^{LP^*}$. Two key arguments of our proof involve this polyhedral relaxation: (i) in Proposition~\ref{prop:pack_1CandLP} we prove that both 1-row CG closure and 1-row closure of $Q$ are contained in the pre-processed LP; (ii) in Proposition~\ref{prop:pack2}, we show that the pre-processed LP is a $2$-approximation to $\mathcal{A}(Q)$; see Figure \ref{fig:packProofSch}.
\begin{figure}
\caption{Relations used in the proof of Theorem \ref{thm:pack}. A straight arrow from $I$ to $J$ denotes the relation $I \leq J$, while a dashed arrow shows the existence of a tight example. Proposition numbers proving the relations are given on the arrows for the ones that are not implied by definitions (``by defn").}
\label{fig:packProofSch}
\end{figure}
The key take away of Theorem~\ref{thm:pack} is that for packing problems one can approximate the CG and aggregation closure by just considering their 1-row counterpart. We next show that this is not true in general.
\begin{theorem}\label{thm:noncoverpack}
Let $\mathcal{M}$ be any of the closures $\mathcal{A}$ (aggregation) or $\mathcal{C}$ (CG). Then there is a family of (non-packing/non-covering) polyhedra for which $1\mathcal{M}$ is an arbitrarily bad approximation to $\mathcal{M}$, namely for each $\alpha \ge 0$ there is a polyhedron $P$ such that $1\mathcal{M}(P)$ is not an $\alpha$-approximation of $\mathcal{M}(P)$. \end{theorem}
The proof of Theorem \ref{thm:noncoverpack} gives a family of polyhedra in $\mathbb{R}^2$ where $\frac{z^{LP}}{z^I}$ can be arbitrarily large, but the CG rank is one.
On the other hand, we relate the integrality gap to the aggregation-closure rank. While there are many lower bounds on CG ranks (and reverse CG rank)~\cite{ChvatalCH89,ConfortiPSFGCG15,PokuttaS11,RothvossS13}, to the best of our knowledge there are no results for the aggregation closure. Moreover, our next lower bound adds to the list of few results~\cite{PokuttaS11,SinghT10} that relate integrality gaps to rank.
\begin{theorem}\label{thm:packrankgen} Let $Q = \{ x \in \mathbb{R}_+^n \mid A x \leq b \}$ be a packing polyhedron with $A_{ij} \leq b_i$ for all $i \in [m],~ j\in [n]$. Then, $\rank_{\mathcal{A}_k}(Q) \geq \left\lceil\frac{\textup{log}_2\left( \frac{z^{LP}}{z^I}\right)}{\textup{log}_2(k + 1)}\right\rceil$ for $k \geq 1$. Moreover, this bound is tight for $k=1$, that is, there is a packing polyhedron $Q$ with $\rank_{A_1}(Q) \le O\left(\textup{log}_2\left( \frac{z^{LP}}{z^I}\right)\right)$. \end{theorem}
Theorem \ref{thm:packrankgen} shows that as long as we use information from a fixed number of constraints, packing IPs can take many rounds of cuts to obtain the integer hull. We remark that this result actually holds for packing sets defined by \emph{infinitely many} inequalities, see the proof of Theorem \ref{thm:packrankgen}. We also note that it can be verified that $\mathcal{A}_k$ is an \emph{admissable} cutting-plane operator, and therefore there exist 0-1 polytopes (with empty integer hulls) with rank $\Omega(\frac{n}{\log n})$ \cite{pokutta2010rank}.
\subsubsection{Covering}
We show that the 1-row closures also provide a good approximation to the full closures in the case of covering polyhedra (with bounds).
\begin{theorem}\label{thm:cover}
Consider a covering polyhedron (with bounds) $Q$. Let $\mathcal{M}$ be any of the closures $\mathcal{A}$ (aggregation) or $\mathcal{C}$ (CG). Then $1\mathcal{M}(Q)$ is a 2-approximation of $\mathcal{M}(Q)$.
Moreover, this bound is tight, namely for every $\varepsilon > 0$ there is a covering polyhedron (with bounds) such that $1\mathcal{M}(Q)$ is not a $(2-\varepsilon)$-approximation of $\mathcal{M}(Q)$. \end{theorem}
The key arguments of our proof are presented in Figure \ref{fig:coverProofSch}. As in the packing case, the main handle to prove this result is a pre-processed version of the LP. For covering polyhedra (i.e., without bounds), this pre-processing is natural: If $A_{ij} > b_i$ for some $i \in [m],~ j \in [n]$, since we are interested only in integer solutions, it is sufficient to replace $A_{ij}$ by $b_i$ to obtain a tighter LP. For covering polyhedra with bounds, the pre-processing LP is heavier and is given by adding all the \emph{knapsack-cover (KC) inequalities} \cite{carr2000strengthening,wolsey:1975}. We note that in the absence of bounds, this LP with the KC inequalities reduces to the pre-processed LP discussed above. The optimal objective function value of the LP with the KC inequalities is denoted as $z^{KC}$.
\begin{figure}
\caption{Relations used in the proof of Theorem \ref{thm:cover}. A straight arrow from $I$ to $J$ denotes the relation $I \leq J$, a dashed arrow shows the existence of a tight example, and a snake arrow means that the ratio could be arbitrarily large. Proposition numbers proving the relations are given on the arrows for the ones that are not implied by definitions (``by defn").}
\label{fig:coverProofSch}
\end{figure}
Unlike in the packing case, the statement of Theorem \ref{thm:cover} regarding the CG closure actually requires a different and much more involved proof. In fact, in this case we show that the LP with the KC inequalities cannot be used to prove this result: there are instances where the CG closure is arbitrarily weaker than the LP with the KC inequalities (see the snake arrow in Figure \ref{fig:coverProofSch}), i.e., for any $L >0$ there exists an instance where $L z^{\mathcal{C}} \leq z^{KC}$. Therefore $z^{\mathcal{C}}$ does not approximate $z^{KC}$ well, and hence it does not approximate $z^{\mathcal{A}}$ well. We also refer the reader to \cite{bienstock2006approximate} for other techniques on approximating fixed rank CG closures for 0-1 covering IPs.
We note that in the proof of Theorem \ref{thm:cover}, we require some preliminary structural results regarding covering polyhedra with bounds, which may be of independent interest. See Propositions \ref{prop:coverUpward}-\ref{prop:commCover} in Section \ref{subsec:CoveringProperties}.
As in the packing case, we can also prove that a large integrality gap implies large rank for the $k$-aggregation closure. Interestingly, the denominator of the lower bound scales as $\log \log k$; this is because the largest integrality gap in a covering problem with $m$ constraints is $O(\log m)$ (see \cite{vazirani2013approximation}).
\begin{theorem} \label{thm:rankCover} Consider a covering polyhedron $Q = \{ x \in \mathbb{R}_+^n \mid Ax \geq b\}$, where $A$ and $b$ satisfy $A_{ij} \leq b_i$ for all $i \in [m],~j \in [n]$. Then, the rank of the k-aggregation closure of $Q$ is at least $\left \lceil \left(\frac{\textup{log}_2\left(\frac{z^I}{z^{LP}}\right)}{3+\textup{log}_2\textup{log}_2(2k)}\right)\right \rceil$. \end{theorem}
As in the packing case, the proof of Theorem \ref{thm:rankCover} shows that this result also holds for covering sets defined by \emph{infinitely many} inequalities.
\subsubsection{Computational experiments}
Theorem \ref{thm:noncoverpack} shows that for general IPs (not packing or covering problems), the 1-row version of the closures may not provide an approximation to the full closure, thus indicating the usefulness of aggregation-based cuts. In order to understand this phenomenon, we conduct an empirical study using CG cuts. Experimenting with CG cuts is convenient due to the availability of reasonably robust CG cut separating algorithm \cite{fischetti:lo:2007}. We use IBM ILOG Cplex 12.6 as the LP/MILP solver. We study two classes of instances: random instances and the so-called market split instances.
\paragraph{Random instances.} We generate instances of the following form: $$\max \Big\{ \sum_{j \in [n]} x_j \mid Ax = b,~0 \leq x \leq u \Big\},$$ where \begin{enumerate} \item We consider instances with $n \in \{ 10,12,14,16 \}$ variables and $m = \lfloor n/2 \rfloor$ equality constraints. \item We choose $M = 50$ and set $u_j = M / 2$ for all $j \in [n]$. \item For any $i \in [m], j \in [n]$, we let $A_{ij}=0$ with probability 0.5. Otherwise, we set $A_{ij}$ to an integer in $\{-M,\hdots,M\}$ with equal probability. \item We construct $b$ by first generating a binary solution $\hat{x}$ uniformly at random, and then letting $b = A \hat{x}$. \end{enumerate} For each $n \in \{ 10,12,14,16 \}$, we generate 100 instances and discard the ones with $\frac{z^{LP}}{z^I} \leq 2$, after which we obtain 75, 83, 84, 84 instances, respectively. The results of this experiment is given in Figure \ref{fig:CGplot}, where each circle corresponds to a single instance. We observe that for the majority of the instances, the ratio $z^{1\mathcal{C}}/z^\mathcal{C}$ is significantly larger than 2. The arithmetic and geometric means of the ratio for different values of $n$ are $11.46,12.80,13.92,15.16$ and $5.68,7.46,8.51,9.94$, respectively.
\begin{figure}
\caption{Multiplicative gap between 1-row CG closure and CG closure of randomly generated instances}
\label{fig:CGplot}
\end{figure}
\paragraph{Market split instances.} This type of instances, also known as market share instances, are formulated in \cite{cornuejols:da:1999} and consist of a class of small 0-1 IPs that are very difficult for branch-and-cut solvers. We use the following parameters: \begin{enumerate} \item We consider instances with $m=2$ equality constraints and $n = 10(m-1)$ variables. \item We take $u_j = 1$ for all $j \in [n]$. \item For any $i \in [m], j \in [n]$, we let $A_{ij}$ to be an integer drawn uniformly from $\{0,\hdots,D-1\}$, where $D = 50$. \item We set $b_i = \lfloor \sum_{j=1}^n A_{ij}/2 \rfloor$, for all $i \in [m]$. \end{enumerate} It has been argued in \cite{cornuejols:da:1999} that most of those instances are infeasible. In this setting, we want to check how often 1-row CG closure detects infeasibility in comparison to the regular CG closure. We generated 100 instances, among which only 10 were feasible. The results of this experiment are presented in the table below. \begin{table}[h] \centering
\begin{tabular}{ c c c }
\hline
1-row CG & CG & \# instances \\
\hline
feasible & infeasible & 50 \\
infeasible & infeasible & 40 \\
feasible & feasible & 10 \\
\hline
\end{tabular} \end{table} As seen in the table, aggregation-based cuts are significantly better than 1-row cuts in proving infeasibility.
\section{Some open questions} \label{sec:Discussion}
Many interesting open questions can be pursued as future research. The first one is a structural question: Is the aggregation closure a polyhedron? For covering and packing IPs, if the constraint matrix $A$ defining the LP relaxation is \emph{dense} (i.e., every entry of $A$ is positive), then we can show that the closure is polyhedron, see Appendix \ref{sec:appendix}. However, the question remains open for the general case. Another question is to understand if we can restrict the set of aggregation multipliers to generate cuts for general IPs based on the sign pattern of the constraint matrix $A$ to approximate the overall aggregation closure well.
\section{Packing problems} \label{sec:Packing}
In this section we present the proof for the statements regarding packing problems.
A crucial tool to analyze the infinite intersections arising in the aggregation closures is the following alternative characterization of $\alpha$-approximation, which is well-known in the covering polyhedral case \cite{goe95}; a quick proof is presented in Appendix \ref{app:charApproxPacking}.
\begin{proposition} \label{prop:approxPack}
Consider two packing sets $U \supseteq V$ in $\mathbb{R}^n$. Then $U$ is an $\alpha$-approximation of $V$ if and only if $U \subseteq \alpha V$.
\end{proposition}
The usefulness of this characterization comes from the following: since set containment is preserved under intersections, if $U_i$ is an $\alpha$-approximation of $V_i$ for all $i \in I$ (an arbitrary set), then $\bigcap_{i \in I} U_i \subseteq \bigcap_{i \in I} \alpha V_i = \alpha \bigcap_{i \in I} V_i$ and thus $\bigcap_{i \in I} U_i$ is an $\alpha$-approximation of $\bigcap_{i \in I} V_i$. The equality in this argument follows from this simple observation (with $\phi(S) = \alpha S$).
\begin{observation}\label{obs:bijection}
Let $\phi:\mathbb{R}^n \rightarrow{R}^n$ be a bijective map, let $\{S^i\}_{i \in I}$ be a collection of subsets in $\mathbb{R}^n$ and let $\phi(S):= \{ \phi(x) \,|\, x \in S\}$. Then $\phi\left(\bigcap_{i \in I} S^i\right) = \bigcap_{i \in I} \phi(S^i)$. \end{observation}
We also note the following. \begin{proposition} \label{prop:inthullPack} Let $Q$ be a packing set. Then, $Q^I$ is also a packing set. \end{proposition} The proof of Proposition \ref{prop:inthullPack} is given in Appendix \ref{app:packinginthull}.
\subsection{Proof of Theorem \ref{thm:pack}}
\paragraph{Upper bound.}
We show the first part of the theorem. That is, consider a non-negative objective function $c \in \mathbb{R}^n_+$; we need to show that $z^{1\mathcal{M}} \le 2z^\mathcal{M}$
for the closures $\mathcal{M} \in \{ \mathcal{A}, \mathcal{C} \}$.
Let $Q = \{x \in \mathbb{R}^n_+ \mid Ax \le b \}$ be a packing polyhedron.
As mentioned in the introduction, a main handle to prove the result is to look at a \emph{pre-processed LP} of $Q$, which sets to 0 variables that have too large left-hand-side coefficients. More precisely, let $S$ be the set of indices $j$ where $A_{ij} > b_i$ for some $i$; the pre-processed LP is then $$LP^*(Q) := \{ x \in \mathbb{R}^n_{+} \mid Ax \le b, ~x_j = 0\ \forall j \in S\}.$$
As seen in Figure \ref{fig:packProofSch}, we prove $z^{1\mathcal{M}} \le 2z^\mathcal{M}$ by showing the following chain of inequalities:
$$z^{1\mathcal{A}} \leq z^{1\mathcal{C}} \leq z^{LP^*} \leq 2 z^\mathcal{A} \leq 2 z^\mathcal{C}.$$
The first inequality $z^{1A} \leq z^{1C}$ follows trivially by definition. The inequality $z^{A} \leq z^C$ is also obvious. It remains to show that $z^{1\mathcal{C}} \leq z^{LP^*} \leq 2 z^\mathcal{A}$.
We first show that the 1-row CG closure already captures the power of the pre-processed LP.
\begin{proposition} \label{prop:pack_1CandLP} $z^{1\mathcal{C}} \leq z^{LP^*}$. \end{proposition} \begin{proof} Suppose $A_{ij} > b_i$ for some $i \in [m]$, $j \in [n]$; it is sufficient to show that the inequality $x_j \le 0$ is valid for $1\mathcal{C}(Q)$. Consider the $i^{\text{th}}$ constraint and the following CG cut generated from it: $$ \sum_{k = 1}^n \left\lfloor\frac{A_{ik}}{A_{ij}}\right\rfloor x_j \leq \left\lfloor \frac{b_i}{A_{ij}}\right\rfloor = 0.$$ Observe that, over $\mathbb{R}^n_+$, this inequality dominates the inequality $x_j \leq 0$. \end{proof}
We now show that the pre-processed LP gives a 2-approximation to the aggregation closure (and hence to the CG closure).
\begin{proposition} \label{prop:pack2} $z^{LP^*} \leq 2 z^{\mathcal{A}}$. \end{proposition} \begin{proof} We begin with a preliminary result.
\paragraph{Claim 1} Consider a single-constraint packing polyhedron $P^1 := \{x \in \mathbb{R}_+^n \,|\, a^{\top}x \le b_0\}$ and the related polyhedron $P^2 := \{x \in \mathbb{R}_+^n \,|\, a^{\top}x \le b_0, \ x_j = 0 \ \forall j \in S\}$ where $S \supseteq \{j \,|\, a_j > b_0\}$. Then ${P^2} \subseteq 2 (P^1)^{I}$. \\ \\ \emph{Proof.} If $S = [n]$, then the result is trivially true. Otherwise, consider a cost function $c \in \mathbb{R}^n_+$ and let $x^*$ be a maximizer of $c$ over ${P}^2$. Notice $x^*$ simply sets the coordinate not in $S$ with largest ratio $c_j/a_j$ to value $b_0/a_j$. So rounding down $x^*$ gives a point in $(P^1)^{I}$ with $c$-value at least half that of $x^*$. This implies that $P^2$ is a 2-approximation for $(P^1)^I$, and so Proposition \ref{prop:approxPack} gives the desired inclusion $P^2 \subseteq 2 (P^1)^I$, which follows from the fact that $(P^1)^I$ is a packing polyhedron (by Proposition \ref{prop:inthullPack}). $\diamond$ \\ \\
\indent Let $S := \{j\,|\, A_{ij} > b_i \textup{ for some } i\in [m], j \in [n]\}$. For $\lambda \in \mathbb{R}_+^m$, let $Q_{\lambda} = \{ x \in \mathbb{R}^n_+ \mid \lambda^\top Ax \leq \lambda^\top b \}$ and
$(LP^*(Q))_{\lambda} = \{x \in \mathbb{R}_+^n \,|\, \lambda^{\top} {A} x \le \lambda^{\top} b, \ x_j = 0, \forall j \in S\}$. Using Claim 1 we have that for any $\lambda \in \mathbb{R}^m_{+}$ \begin{eqnarray}\label{eq:mainpack} LP^*(Q) \subseteq (LP^*(Q))_{\lambda} \subseteq 2\left({Q}_{\lambda}\right)^I. \end{eqnarray}
Taking intersection over all $\lambda \in \mathbb{R}^n_+$ we obtain that
\begin{align*}
LP^*(Q) \subseteq \bigcap_{\lambda \in \mathbb{R}^m_+}2\left({Q}_{\lambda}\right)^I = 2\bigcap_{\lambda \in \mathbb{R}^m_+}\left({Q}_{\lambda}\right)^I = 2 \mathcal{A}(Q), \end{align*} where the first equation uses Observation \ref{obs:bijection}. Then from Proposition \ref{prop:approxPack} we have that $LP^*(Q)$ is a 2-approximation for $\mathcal{A}(Q)$. \end{proof}
\paragraph{Tight instances.} To prove the second part of the theorem, it suffices to show that there are instances where the 1-row closure, which is the strongest 1-row closure we consider, is at most roughly a 2-approximation of the CG closure, the weakest closure we consider.
\begin{proposition} \label{prop:pack_TightEx} For every $\epsilon>0$ there exists an instance where $\frac{z^{1\mathcal{A}}}{z^{\mathcal{C}}} \geq 2 - \epsilon$. \end{proposition} \begin{proof} Consider the following family of packing IPs \begin{align} \text{maximize} \quad & x_1 + x_2 \nonumber \\ \text{subject to} \quad & x_1 + Mx_2 \leq M \label{packingtightexA} \\ & Mx_1 + x_2 \leq M \label{packingtightexB} \\ & x \ge 0 \label{c} \\ & x \in \mathbb{Z}^2, \nonumber \end{align} where $M$ is an integer with $M \ge 1$.
We show that $\lim_{M \rightarrow \infty} \frac{z^{1\mathcal{A}}}{z^{\mathcal{C}}} \rightarrow 2$. Observe that the set $\{x \in \mathbb{R}^2_{+}\,|\, x_1 + Mx_2 \leq M\}$ and the set $\{x \in \mathbb{R}^2_{+}\,|\, Mx_1 + x_2 \leq M\}$ are integral. Therefore, $z^{1\mathcal{A}} = z^{LP}$, or equivalently $z^{1\mathcal{A}} = \frac{2M}{M + 1}$.
On the other hand, since \eqref{packingtightexA} and \eqref{packingtightexB} imply that the inequality $x_1 + x_2 \leq \frac{2M}{M + 1}$ is valid for $Q$, we have that $x_1 + x_2 \leq 1$ is a valid CG cut for $Q$. Therefore, we obtain $\mathcal{C}(Q) \subseteq \{(x_1, x_2) \in \mathbb{R}^2_{+}\,|\, x_1 + x_2 \leq 1 \}$. Thus $z^{\mathcal{C}} = 1$. \end{proof}
\subsection{Proof of Theorem \ref{thm:noncoverpack}}
Let $k \in \mathbb{Z}$ and $k \geq 2$ and consider the following IP: \begin{eqnarray} &\textup{max}& x_1 + x_2 \nonumber \\ &\textup{s.t.}& k^2 x_1 - (k-1) x_2 \leq k^2 \label{con1}\\ && -kx_1 + x_2 \leq -k + 1 \label{con2}\\ && x_1, x_2\geq 0. \label{con3} \end{eqnarray}
To prove the theorem, it suffices to show that as $k$ goes to infinity, the ratios $\frac{z^{1\mathcal{C}}}{z^{\mathcal{C}}}$ and $\frac{z^{1\mathcal{A}}}{z^{\mathcal{A}}}$ also go to infinity. In fact, since $z^{1\mathcal{A}} \le z^{1\mathcal{C}}$ and $z^{\mathcal{A}} \le z^{\mathcal{C}}$, we just need to show that $\frac{z^{1\mathcal{A}}}{z^{\mathcal{C}}} \rightarrow \infty$.
\paragraph{1-row closure.} We verify that the point $(2 - \frac{1}{k}, k)$ belongs to the original 1-row cut closure. Consider the following cases: \begin{enumerate} \item Integer hull of (\ref{con1}) and (\ref{con3}): The points $(1,0)$ and $(k, k^2)$ are valid integer points and $(2 - \frac{1}{k}, k)$ is a convex combination of these points. \item Integer hull of (\ref{con2}) and (\ref{con3}): The points $(1,1)$ and $(2, k + 1)$ are valid integer points and $(2 - \frac{1}{k}, k)$ is a convex combination of these points. \end{enumerate}
Since $(2-\frac{1}{k},k)$ belongs to the 1-row closure, by inspecting its objective value we obtain that $z^{1\mathcal{A}} \geq 2 + k - \frac{1}{k}$. \paragraph{CG closure.} To upper bound the optimal value of the CG closure, we explicitly construct one CG cut. Consider the aggregation of the LP inequalities $\frac{1}{k}\times$(\ref{con1}) $+$ $\frac{k-1}{k}\times(\ref{con2}) \equiv x_1 \leq 2-\frac{1}{k}$, which gives the CG cut $x_1 \leq 1.$
We can compute an upper bound on $z^{\mathcal{C}}$ by computing the optimal value subject to this CG cut and \eqref{con2}, namely $\textup{max}\{ x_1 + x_2 \,|\, x_1 \leq 1, \ -k x_1 + x_2 \leq -k + 1, x_1 \geq 0, x_2 \geq 0\} = 2$: $(1,1)$ is a feasible primal solution and $(k+1, 1)$ is a dual feasible solution with same objective function value. Therefore, we have $z^{\mathcal{C}} \leq 2$.
Putting these bounds together obtain that $\frac{z^{1\mathcal{A}}}{z^{\mathcal{C}}} = \frac{k}{2} + 1 - \frac{1}{2k}$, which goes to infinity as $k \rightarrow \infty$. This concludes the proof of the theorem.
\qed
\subsection{Proof of Theorem~\ref{thm:packrankgen}}
A key result we use is given in Proposition \ref{prop:packingDf} below, which provides a bound on the integrality gap as a function of the number of inequalities. Note that since we do not make any assumptions on the coefficients of the constraint matrix of the packing polyhedron, we obtain better coefficients than those obtainable by using randomized rounding arguments; see for example \cite{srinivasan1999improved}.
\begin{proposition} \label{prop:packingDf} Consider a packing IP of the following form $\max \{c^{\top}x \mid Dx \leq f, \ x \in \mathbb{Z}_{+}^n\}$ where $D$ is a $k \times n$ non-negative matrix such that $D_{ij} \leq f_i$ for all $i \in [k]$ and $j \in [n]$ and $c \in \mathbb{R}^n_{+}$. Then $z^{LP} \leq (k+1) z^I$.
\end{proposition}
\begin{proof} If the LP has unbounded value, then the IP also has unbounded value \cite{NemWolBook}, and there is nothing to prove.
Assume the LP has bounded value, and let $x^{LP}$ be an optimal solution of the LP. Let $$x^{LP} = \hat{x} + x^F,$$ where $\hat{x}$ is obtained by rounding down $x^{LP}$ componentwise. Then, $\hat{x}$ belongs to the feasible region of the packing problem, and hence $z^I \geq c^\top \hat{x}$.
Let $c_{max} \in \text{argmax}_{j} \{ c_j \}$. Since $e_j$ for all $j \in [n]$ belongs to the feasible region, $z^I \geq c_{max}$. Thus, $z^I \geq \max \{ c^\top \hat{x},c_{max}\}$.
Now, observe that \begin{align*} \frac{z^{LP}}{z^I} & \leq \frac{c^\top \hat{x}+c^\top x^F}{\max \{ c^\top \hat{x},c_{max}\}} \\ & \leq 1 + \frac{c^\top x^F}{\max \{ c^\top \hat{x},c_{max}\}}. \end{align*}
Since there are $k$ constraints $D^i x \le f_i$, at most $k$ components of $x^{LP}$ can be non-zero. In other words, at most $k$ components of $x^F$ can be non-zero. Also, each entry of $x^F$ is strictly less than 1. Hence, $c^\top x^F \leq c_{max} k$, and therefore $$\frac{z^{LP}}{z^I} \leq 1+ \frac{c_{max} k}{\max \{ c^\top \hat{x},c_{max} \} } \leq 1+ k.$$ \end{proof} \paragraph{Lower bound on rank.} We actually prove Theorem \ref{thm:packrankgen} for the more general case of packing sets containing all the basis vectors $e_j$'s; notice that for a packing polyhedron $Q = \{x \in \mathbb{R}^n_+ \mid Ax \le b\}$, containing all basis vectors $e_j$'s is equivalent to the condition $A_{ij} \le b_i$ for all $i,j$.
So let $Q$ be a non-empty packing set containing all the basis vectors $e_j$'s. Given a matrix $(D,f) \in \mathbb{R}^{k \times n} \times \mathbb{R}^k$, we say that it is a \emph{$k$-vi} for $Q$ if $(D,f)$ is non-negative and the inequalities $D^i x \le f_i$ are valid for $Q$. We denote the polyhedral outer-approximation $\{ x \in \mathbb{R}^n_+ \mid D x \leq f \}$ of $Q$ by $P_{(D,f)}$. Then by definition
\begin{align}
\mathcal{A}_k(Q) = \bigcap_{(D,f)\textrm{ is a $k$-vi \ for $Q$}} (P_{(D,f)})^I. \label{eq:kviIntersection} \end{align}
Let $Q^\ell$ be the $\ell^{\text{th}}$ $k$-aggregation closure of $Q$. \paragraph{Claim 1} $Q^\ell \subseteq (k+1)\, Q^{\ell+1}$. \\ \\ \emph{Proof.}
Consider a $k$-vi \ $(D,f)$ for $Q^\ell$. Clearly $P_{(D,f)}$ is a packing polyhedron, and since $P_{(D,f)} \supseteq \mathcal{A}_k(Q^\ell) \supseteq Q^I$, all basis vectors $e_j$ belong to $P_{(D,f)}$. Therefore, $D_{ij} \leq f_i$ for all $i \in [k], ~ j \in [n]$, and so by Proposition \ref{prop:packingDf} we obtain that $P_{(D,f)}$ is a $(k+1)$-approximation of $(P_{(D,f)})^I$. Hence, Proposition \ref{prop:approxPack} gives $$Q^\ell \subseteq P_{(D, f)} \subseteq (k+1) (P_{(D, f)})^I.$$ So taking intersection over all $k$-vi's and using Observation~\ref{obs:bijection}, we have that \begin{align*} \begin{split} Q^\ell \ \subseteq & \ \bigcap_{(D,f) \textup{ is $k$-vi \ for } Q^\ell} (k+1) (P_{(D, f)})^I \\ = & \ (k+1) \bigcap_{(D,f) \textup{ is $k$-vi \ for } Q^\ell} (P_{(D, f)})^I \\ = & (k+1)\, Q^{\ell+1}, \end{split} \end{align*} where the last equality follows from \eqref{eq:kviIntersection}. This concludes the proof. ~$\diamond$ \\ \\ \indent Finally, suppose the rank of $k$-aggregation is $t$ and let $z^{i}$ be the optimal objective function value over the $i^{\text{th}}$ closure. Since all of these closures are packing sets, Claim 1 and Proposition \ref{prop:approxPack} guarantee that $z^i \le (k+1) z^{i+1}$. Therefore,
\begin{eqnarray*} \frac{z^{LP}}{z^I} = \frac{z^{LP}}{z^1} \frac{z^1}{z^2}\dots\frac{z^{t-1}}{z^{t}} \leq (k+1)^{t}. \end{eqnarray*}
This implies the inequality \begin{eqnarray*} t = \rank_{\mathcal{A}_k}(Q) \geq \left\lceil\frac{\textup{log}_2\left( \frac{z^{LP}}{z^I}\right)}{\textup{log}_2(k+1)}\right\rceil, \end{eqnarray*} which is the required result.
\paragraph{Tight example.} We now show that there is a packing integer set $Q$ with $\rank_{A_1}(Q) \le O\left(\textup{log}_2\left( \frac{z^{LP}}{z^{I}}\right)\right)$. Let $K_n$ be a complete graph with node set $[n]$, and let $Q$ be the standard edge-relaxation of the stable set polytope: \begin{align*}
FSTAB(K_n) = \{ x \in \mathbb{R}_+^{n} \,|\, x_i + x_j \le 1 \ \forall i,j \in [n], \ i < j\}. \end{align*} If our objective is to maximize $\sum_{v \in [n]} x_v$, then we obtain $z^{I}=1$ and $z^{LP}=n/2$ because the optimal vertex of $FSTAB(K_n)$ is the vector with all entries equal to $1/2$. Consider now the clique inequality $\sum_{v \in [n]} x_v \le 1$, which defines a facet of the stable set polytope. We only need to show that the CG rank of the clique inequality is upper bounded by $O\left(\textup{log}_2\left( \frac{z^{LP}}{z^{I}}\right) \right)= \lceil \log_2 (n-1) \rceil$.
The latter is a well-known fact \cite{hartmann}. \qed
\section{Proofs for covering problems} \label{sec:Covering}
We now provide additional definitions and proofs of the statements presented in the introduction regarding covering problems: in Subsection \ref{sec:proofCover} we prove Theorem \ref{thm:cover}, the main result of this section, and in Subsection \ref{subsec:CoveringRankProof} we prove Theorem \ref{thm:rankCover}. Before proving these results we need to develop some general results concerning covering sets with bounds.
\subsection{Properties of covering sets with bounds}
\label{subsec:CoveringProperties}
We start by showing that adding non-negative directions to a covering polyhedron with bounds still leaves it as a covering polyhedron (possibly with bounds); in fact, adding all the non-negative directions is a natural way of removing the upper bounds.
Given a covering polyhedron with bounds of the form $P = \{ x \in \mathbb{R}^n_+ \mid Ax \geq b,~ x \leq u \}$ with $A,b \ge 0$, we refer to $Ax \geq b$ as the covering inequalities of $P$.
\begin{proposition} \label{prop:coverUpward}
Consider a covering polyhedron with bounds $P = \{ x \in \mathbb{R}^n_+ \mid Ax \ge b, \ x \le u\}$. Then, for any subset $\{e_j\}_{j \in J}$ of the canonical vectors we have that $P + \cone(\{e_j\}_{j \in J})$ is a covering polyhedron with bounds. In particular, $P + \mathbb{R}^n_+$ is a covering polyhedron.
Moreover, each covering inequality of $P + \cone(\{e_j\}_{j \in J})$ is a conic combination of one covering inequality of $P$ with the bounds $x_j \le u_j$ for $j \in J$.
\end{proposition}
\begin{proof}
Notice it suffices to show that for a single $e_j$, $P + \cone(e_j)$ is a covering polyhedron with bounds (the general statement follows by the repeated application of this result).
So consider one such $e_j$. For every inequality $A^i x \ge b_i$ of the system $Ax \ge b$, let $\hat{A}^i x \ge \hat{b}_i$ be the sum of $A^i x \ge b_i$ and $-A_{ij} x_j \ge -A_{ij} u_j$. Note that $\hat{A}^i \ge 0$ and $\hat{A}_{ij} = 0$. Let $\hat Ax \ge \hat b$ be the system comprising all such inequalities $\hat{A}^i x \ge \hat{b}_i$. Let $\hat u$ be the vector obtained from $u$ by replacing $u_j$ with $\infty$. We define the covering polyhedron with bounds $\hat P = \{x \in \mathbb{R}^n_+ \,|\, Ax \ge b, \ \hat Ax \ge \hat b, \ x \le \hat u\}$. By construction, each covering inequality of $\hat P$ is a conic combination of one covering inequality of $P$ with the bound $x_j \le u_j$. In the remainder of the proof we show $P + \cone(e_j) = \hat P$.
Since each inequality valid for $\hat{P}$ is also valid for $P$ and the recession cone of $\hat{P}$ contains $e_j$ ($A$ and $\hat{A}$ are non-negative and $\hat{u}_j = \infty$), we have $P + \cone(e_j) \subseteq \hat P$.
We now show the reverse inclusion $P + \cone(e_j) \supseteq \hat P$. Let $c^\top x \ge \delta$ be a valid inequality for $P + \cone(e_j)$. Equivalently, $c^\top x \ge \delta$ is a valid inequality for $P$ with $c_j \ge 0$. As a consequence, there exist nonnegative multipliers $\mu_i, \delta_i, \gamma_i$ such that $$c = \sum_{i=1}^m \mu_i A^i + \sum_{i=1}^n \delta_i e^i - \sum_{i=1}^n \gamma_i e^i \quad \text{and} \quad \delta \leq \delta_0:= \sum_{i=1}^m \mu_i b_i - \sum_{i=1}^n \gamma_i u_i.$$ Without loss of generality we can assume that at least one among $\delta_j$ and $\gamma_j$ equals zero. In the latter case, the inequality $c^\top x \ge \delta$ is trivially valid for $\hat P$, thus we now assume $\delta_j = 0$ and $\gamma_j > 0$. Since $c_j \ge 0$, we have $c_j = \sum_{i=1}^m \mu_i a_{ij} - \gamma_j \ge 0$.
Let $k \in \{1,\dots,m\}$ be the smallest index such that $\sum_{i=1}^k \mu_i A_{ij} \ge \gamma_j$. In this way $\gamma_j - \sum_{i=1}^{k-1} \mu_i A_{ij} > 0$. This allows us to define non-negative multipliers $\lambda_i$, $\lambda'_i$, for $i=1,\dots,m$: \iffalse \begin{align*} \lambda_i = \begin{cases} 0 &\mbox{if } i=1,\dots,k-1 \\ \mu_k- \frac{\gamma_j - \sum_{i=1}^{k-1} \mu_i A_{ij}}{A_{kj}} &\mbox{if } i=k \\ \mu_i &\mbox{if } i=k+1,\dots,m \\ \end{cases} \qquad \lambda'_i = \begin{cases} \mu_i &\mbox{if } i=1,\dots,k-1 \\ \frac{\gamma_j - \sum_{i=1}^{k-1} \mu_i A_{ij}}{A_{kj}} &\mbox{if } i=k \\ 0 &\mbox{if } i=k+1,\dots,m. \\ \end{cases} \end{align*} \fi \begin{align*} \lambda_i = \begin{cases} 0 \\ \mu_k- \frac{\gamma_j - \sum_{i=1}^{k-1} \mu_i A_{ij}}{A_{kj}} \\ \mu_i \\ \end{cases},
\ \lambda'_i = \begin{cases} \mu_i &\mbox{if } i=1,\dots,k-1 \\ \frac{\gamma_j - \sum_{i=1}^{k-1} \mu_i A_{ij}}{A_{kj}} &\mbox{if } i=k \\ 0 &\mbox{if } i=k+1,\dots,m. \\ \end{cases} \end{align*}
It can be verified that: $$c = \sum_{i=1}^m \lambda_i A^i + \sum_{i=1}^m \lambda'_i \hat A^i + \sum_{i=1}^n \delta_i e^i - \sum_{\substack{i=1 \\ i \neq j}}^n \gamma_i e^i \ \ \text{and} \ \ \delta_0 = \sum_{i=1}^m \lambda_i b_i + \sum_{i=1}^m \lambda'_i \hat b_i - \sum_{\substack{i=1 \\ i \neq j}}^n \gamma_i u_i.$$ This implies that $c^\top x \ge \delta$ is valid for $\hat P$.
This shows that every valid inequality for $P + \cone(e_j)$ is valid for $\hat P$, hence $P + \cone(e_j) \supseteq \hat{P}$ and we conclude the proof of the proposition. \end{proof}
Next, we show that the integer hull of a covering polyhedron with bounds is also a covering polyhedron with bounds.
\begin{proposition}\label{prop:coverIntHull}
Let $Q = \{x \in \mathbb{Z}^n_+ \mid Ax \ge b, \ x \le u\}$ be a non-empty covering polyhedron with bounds (recall that $u$ is integral or infinite). Then its integer hull $Q^I$ is a covering polyhedron with bounds. Moreover, $Q^I$ has the same upper bounds as $Q$, namely $Q^I = \{x \in \mathbb{R}^n_+ \mid A'x \ge b', \ x \le u\}$ for some $(A',b')$.
\end{proposition}
\begin{proof}
We assume that $Q^I$ is non-empty, otherwise the result can be easily verified. Let $\pi^\top x \ge \pi_0$ be a facet-defining inequality for $Q^I$. It suffices to show that either $(\pi, \pi_0) \ge 0$, or that this inequality is equivalent to an upper bound constraint $x_j \le u_j$ for some $j$ and $u_j$.
First, suppose $\pi \ge 0$. Then $\pi_0$ must be non-negative, since otherwise the fact that $Q^I \subseteq \mathbb{R}^n_+$ would imply that the face of $Q^I$ induced by $\pi^\top x \ge \pi_0$ is empty, contradicting that it is a facet.
Now consider the case where $\pi$ has at least one negative coordinate, say $\pi_j < 0$. If all other components of $\pi$ are equal to 0, then $\pi^\top x \ge \pi_0$ is equivalent to an upper bound constraint: $$\pi^\top x \ge \pi_0 ~\equiv~ \pi_j x_j \ge \pi_0 ~\equiv~ x_j \le \frac{-\pi_0}{\pi_j},$$
where the sign/sense reversal in the last equivalence happens because $\pi_j$ is negative. Thus, to conclude the proof it suffices to consider the case where $\pi$ has support of size at least 2.
We show that this case actually leads to a contradiction. The idea is to use the following property that can be immediately verified: if $\bar{x}, \bar{y}$ are \emph{integer} points in $Q^I$, then the point $\bar{z}$ obtained by taking $\bar{x}$ and replacing its $j^{\text{th}}$ component by $\max\{\bar{x}_j, \bar{y}_j\}$ also belongs to $Q^I$. Moreover, if $\bar{y}_j > \bar{x}_j$ we have that $\pi^\top \bar{z} < \pi^\top \bar{x}$; we will use this to contradict the validity of $\pi^\top x \ge \pi_0$.
To make this precise, since $\pi^\top x \ge \pi_0$ is facet-defining, let $\bar{x}^1, \ldots, \bar{x}^n$ be affinely independent integer points in $Q^I$ that satisfy the equality $\pi^\top x = \pi_0$. Let $M = \max_i \bar{x}^i_j$ be the maximum value in the $j^{\text{th}}$ coordinate of these points. Observe that at least one of the points $\bar{x}^i$ has the $j^{\text{th}}$ coordinate strictly smaller than $M$: otherwise all points $\bar{x}^i$ would satisfy the linearly independent inequalities $\pi^\top x = \pi_0$ and $x^i_j = M$ (the linear independence comes from the fact $\pi$ has support of size at least 2) and thus would lie in an $(n-2)$-dimensional space, contradicting that they are $n$ affinely independent points.
Thus, without loss of generality assume that $\bar{x}^1_j = M > \bar{x}^2_j$. Construct the point $\bar{z}$ by taking the vector $\bar{x}^2$ and replacing its $j^{\text{th}}$ coordinate by $\max\{\bar{x}^1_j, \bar{x}^2_j\} = \bar{x}^1_j$. As mentioned earlier, $\bar{z}$ belongs to $Q^I$ but $$\pi^\top \bar{z} < \pi^\top \bar{x}^2 = \pi_0,$$ thus contradicting the validity of $\pi^\top x \ge \pi_0$. This concludes the proof that $Q^I$ is a covering set.
To see that the upper bounds in $Q^I$ are the same as those in $Q$, let $Q^I = \{x \in \mathbb{R}^n_+ \mid A'x \ge b', \ x \le u'\}$ be a covering-with-bounds description of this set with minimal $u'$ (i.e. there is no other valid upper bound that is pointwise smaller than $u'$). Recall that $u$ is the vector of upper bounds in $Q$, which is an integral vector. Since $Q^I \subseteq Q \subseteq [0,u]$, the minimality of $u'$ guarantees that $u' \le u$. But since $Q$ is non-empty, it contains the point $u$, and so does the integer hull $Q^I$; thus, $u' \ge u$. This concludes the proof.
\end{proof}
We also remark the following equivalent definition of $\alpha$-approximation, similar to that for the packing case; the first part of the statement follows directly from the definition of $\alpha$-approximation, and the second follows from Proposition \ref{prop:coverUpward} combined with Lemma 23 of \cite{molinaro2013understanding}.
\begin{proposition} \label{prop:blowupCover}
Consider two covering sets $U \supseteq V$.
Then $U$ is an $\alpha$-approximation of $V$ iff $U + \mathbb{R}^n_+$ is an $\alpha$-approximation of $V + \mathbb{R}^n_+$. Moreover, this happens iff $\frac{1}{\alpha}(U + \mathbb{R}^n_+) \subseteq (V + \mathbb{R}^n_+)$.
\end{proposition}
Finally, we need the following property, which states that for covering polyhedra with the \emph{same} upper bounds we can commute adding $\mathbb{R}^n_+$ and taking intersections.
\begin{proposition} \label{prop:commCover}
Let $\{Q^i\}_{i \in I}$ be a (possibly infinite) family of covering polyhedra with bounds such that all upper bounds are the same, namely $Q^i = \{x \in \mathbb{R}^n_+ \mid G(i) x \geq g(i), \ x \le u\}$ for all $i \in I$ (where $G(i) \in \mathbb{R}_+^{m_i \times n},~ g(i) \in \mathbb{R}_+^{m_i}$). Then $$\bigcap_{i \in I} (Q^i + \mathbb{R}^n_+) = \bigg( \bigcap_{i \in I} Q^i \bigg) + \mathbb{R}^n_+.$$
\end{proposition}
\begin{proof}
The direction ``$\supseteq$'' is straightforward, so we prove the direction ``$\subseteq$''. Consider a point $x \in \bigcap_{i \in I} (Q^i + \mathbb{R}^n_+)$, so we can write $x = q^i + r^i$ for $q^i \in Q^i$ and $r^i \ge 0$. The idea is that if we push all the $q^i$'s coordinates as high as possible (correcting appropriately the $r^i$'s) we can actually get the same point in all the $Q^i$'s.
More explicitly, define the point $q \in \mathbb{R}^n$ as follows: if $x_j$ is at most the upper bound $u_j$, set $q_j = x_j$, else set $q_j = u_j$ (so $q = \min\{x, u\}$). We claim that $q$ belongs to $Q^i$ for all $i$. First, since $q^i \le x$ and $q^i \le u$, we have that $q^i \le q$; therefore, since $q^i$ satisfies the covering constraints of $Q^i$, so does $q$. Moreover, $q \le u$, so $q$ also satisfies the upper bound constraints of $Q^i$; thus $q \in Q^i$. We then get that the point $q + (x - q)$ belongs to $( \bigcap_{i \in I} Q^i) + \mathbb{R}^n_+$. This shows the desired inclusion and concludes the proof.
\end{proof}
We can now start the proof of Theorem \ref{thm:cover}.
\subsection{Proof of Theorem \ref{thm:cover}} \label{sec:proofCover}
A central object for our proof are the \emph{knapsack-cover} inequalities~\cite{wolsey:1975}. Consider a covering polyhedron with bounds $Q = \{x \in \mathbb{R}^n_+ \mid Ax \ge b, ~x \le u\}$. A knapsack-cover (KC) inequality is generated as follows: Consider a single row $A^i x \ge b_i$ of this problem; given a subset $S \subseteq [n]$ of the variables, the corresponding KC inequality is given by $\sum_{j \notin S} \tilde{A}_{ij} x_j \ge b_i - \sum_{j \in S} u_j A_{ij}$, where $\tilde{A}_{ij} = \min\{A_{ij}, b_i - \sum_{j \in S} u_j A_{ij}\}$. Notice that the KC inequalities are indeed valid for $Q$. Again, we use $KC(Q)$ to denote the \emph{KC closure} (namely the set obtained by adding all the KC inequalities to the linear relaxation of $Q$), and for a given objective function we use $z^{KC}$ to denote the optimal value of optimizing this function over $KC(Q)$.
We break down the proof of Theorem \ref{thm:cover} by first comparing $1\mathcal{A}(Q)$ versus $\mathcal{A}(Q)$; we then compare $1\mathcal{C}(Q)$ versus $\mathcal{C}(Q)$, which is significantly more involved.
\subsubsection{Proof for aggregation closure}
\paragraph{Upper bound.} Observe that the 1-row closure is at least as strong as the $KC$ closure by construction of the KC inequalities. We need the following result, which states that for a 1-row covering polyhedron with bounds, the KC closure is a 2-approximation of the integer hull.
\begin{theorem}[\cite{carr2000strengthening}] \label{thm:carr} Consider a 1-row covering polyhedron with bounds $Q = \{x \in \mathbb{Z}^n_+ \mid ax \ge b, \ x \le u\}$. Then the KC closure $KC(Q)$ is a 2-approximation of the integer hull $Q^I$. \end{theorem}
Since the aggregation closure is the intersection of the integer hull of multiple 1-row covering polyhedra, we leverage the theorem above to show that the KC closure is also a 2-approximation for the aggregation closure of a multi-row covering polyhedron.
\begin{proposition} \label{prop:KCvsA} For every covering polyhedron with bounds $Q$ we have that the KC closure $KC(Q)$ is a 2-approximation of the aggregation closure $\mathcal{A}(Q)$. \end{proposition}
\begin{proof} Let $Q = \{ x \in \mathbb{R}^n_+ \mid Ax \ge b, \ x \le u\}$ and consider $Q_{\lambda} := \{ x \in \mathbb{R}^n_+ \mid \lambda^\top Ax \ge \lambda^\top b, \ x \le u \}$ for some $\lambda \in \mathbb{R}^n_+$. First we connect the KC closure of $Q$ with the KC closure of the 1-row covering set $Q_\lambda$, proving the intuitive fact that $KC(Q) \subseteq KC(Q_\lambda)$.
For that, consider a KC inequality $$kc := \{\sum_{j \notin S} \widetilde{(\lambda^\top A)_j} x_j \ge \lambda^\top b - \sum_{j \in S} (\lambda^\top A)_j u_j \}$$
for $Q_{\lambda}$ and let $kc_i = \{\sum_{j \notin S} \tilde{A}_{ij} x_j \ge b_i- \sum_{j \in S} A_{ij} u_j\}$ be the corresponding KC inequality for the $i^{\text{th}}$ row of $P$. We show that $kc$ is dominated by the inequalities $kc_i$'s, namely $kc \cap \mathbb{R}^n_+ \supseteq \bigcap_i (kc_i \cap \mathbb{R}^n_+)$. Consider the aggregation $\sum_i \lambda_i kc_i \equiv \sum_{j \notin S} (\sum_i \lambda_i \tilde{A}_{ij}) x_j \ge \lambda^\top b - \sum_{j \in S} (\lambda^\top A)_j u_j$; it suffices to show that this dominates $kc$.
The RHS's are the same, so it suffices to compare LHS's. Since $\tilde{A}_{ij} = \min\{A_{ij}, b_i - \sum_{j \in S} A_{ij} u_j\}$, it follows that $\sum_i \lambda_i \tilde{A}_{ij} \le \min\{\sum_i \lambda_i A_{ij}, \lambda^\top b - \sum_{j \in S} (\lambda^\top A)_j u_j\}$, which is exactly the $j^{\text{th}}$ entry in the LHS of $kc$. This proves that $KC(Q) \subseteq KC(Q_\lambda)$.
Employing the alternative definition of $\alpha$-approximation given by Proposition \ref{prop:blowupCover} with Theorem \ref{thm:carr}, we get that for every $\lambda$
\begin{align*}
(KC(Q) + \mathbb{R}^n_+) \subseteq (KC(Q_{\lambda}) + \mathbb{R}^n_+) \subseteq \frac{1}{2} (Q^I_\lambda + \mathbb{R}^n_+).
\end{align*}
From Proposition \ref{prop:coverIntHull} we have that $Q^I_\lambda$ is a covering polyhedron with bounds, and that the upper bounds are that same as in $Q_\lambda$, which are the upper bounds of $Q$. Since all these bounds are the same, we can take intersection of the last displayed inequality over all $\lambda$'s and used the commutativity from Proposition \ref{prop:commCover} to obtain that
\iffalse
{
\begin{align*}
(KC(Q) + \mathbb{R}^n_+) \ \subseteq\ \bigcap_{\lambda \in \mathbb{R}^m_+} \frac{1}{2} (Q^I_\lambda + \mathbb{R}^n_+)\ \stackrel{\textrm{Obs} \ref{obs:bijection}}{=} \ \frac{1}{2}\bigcap_{\lambda \in \mathbb{R}^m_+} (Q^I_\lambda + \mathbb{R}^n_+)\ = \ \frac{1}{2} \bigg( \bigcap_{\lambda \in \mathbb{R}^m_+} Q^I_\lambda \bigg) + \mathbb{R}^n_+.
\end{align*}} \fi
{
\begin{align*}
&(KC(Q) + \mathbb{R}^n_+) \ \subseteq\ \bigcap_{\lambda \in \mathbb{R}^m_+} \frac{1}{2} (Q^I_\lambda + \mathbb{R}^n_+)\ \stackrel{\textrm{Obs} \ref{obs:bijection}}{=} \\
&\ \stackrel{\textrm{Obs} \ref{obs:bijection}}{=} \ \frac{1}{2}\bigcap_{\lambda \in \mathbb{R}^m_+} (Q^I_\lambda + \mathbb{R}^n_+)\ = \ \frac{1}{2} \bigg( \bigcap_{\lambda \in \mathbb{R}^m_+} Q^I_\lambda \bigg) + \mathbb{R}^n_+.
\end{align*}} The right-hand side of this expression is exactly $\frac{1}{2}(\mathcal{A}(Q) + \mathbb{R}^n_+)$, thus employing Proposition \ref{prop:blowupCover} once again we get that the KC closure $KC(Q)$ is a 2-approximation for the aggregation closure $\mathcal{A}(Q)$. This concludes the proof. \end{proof}
Hence, we obtain that $1\mathcal{A}(Q)$ is a 2-approximation to $\mathcal{A}(Q)$.
\paragraph{Tight examples.} We next exhibit an instance where $1\mathcal{A}$ is not better than a 2-approximation of $\mathcal{A}$.
\begin{proposition} \label{prop:LBAC}
Let $\epsilon >0$. There exists an instance where $\frac{z^{\mathcal{A}}}{z^{1\mathcal{A}}} \geq 2 - \epsilon$ and $\frac{z^{\mathcal{C}}}{z^{1\mathcal{C}}} \geq 2 - \epsilon$.
\end{proposition} \begin{proof} Let $n = \textup{min}\{2,\lceil \frac{1}{\epsilon}\rceil\}$. Consider the following instance \begin{eqnarray*} \begin{array}{rl} \textup{min}& \displaystyle \sum_{j = 1}^n x_j \\ \textup{s.t.}& x_i + \displaystyle \sum_{j \in [n]\setminus \{i\}} 2x_j \geq 2, \ \forall i \in [n], \\ &x_j \in \mathbb{Z}^n_{+}. \end{array} \label{eq:LBAC} \end{eqnarray*}
We show that $\frac{z^{\mathcal{A}}}{z^{1\mathcal{A}}} \geq 2 - \epsilon$ and $\frac{z^{\mathcal{C}}}{z^{1\mathcal{C}}} \geq 2 - \epsilon$ for this instance.
\begin{enumerate}
\item $z^{1\mathcal{A}} = z^{1\mathcal{C}} = \frac{2n}{2n - 1}$: Observe that the set $\{x \in \mathbb{R}^n_{+}\,|\, x_i + \sum_{j \in [n]\setminus \{i\}} 2x_j \geq 2\}$ is integral. Thus, $z^{1\mathcal{A}} = z^{1\mathcal{C}}$ and each is equal to the LP relaxation. Adding all these constraints we obtain \begin{eqnarray}\label{eq:preCG} \sum_{j \in [n]} x_j \geq \frac{2n}{2n - 1} \end{eqnarray} On the other hand, setting $x_j = \frac{2}{2n - 1}$, we obtain a feasible solution. Thus, $z^{1\mathcal{A}} = z^{1\mathcal{C}} = \frac{2n}{2n - 1}$. \item $z^{\mathcal{A}} \geq 2$ and $z^{\mathcal{C}} \geq 2$: Since (\ref{eq:preCG}) is a valid inequality, we obtain the CG cut $\sum_{j \in [n]} x_j \geq 2$. Thus $z^{\mathcal{C}} \geq 2$ and since $z^{\mathcal{A}} \geq z^{\mathcal{C}}$ we obtain $z^{\mathcal{A}} \geq 2$. \end{enumerate} Thus, $\frac{z^{\mathcal{A}}}{z^{1\mathcal{A}}} \geq 2 - \frac{1}{n}$ and $\frac{z^{\mathcal{C}}}{z^{1C}} \geq 2 - \frac{1}{n}$; and our choice of $n$ completes the proof. \end{proof}
\subsubsection{Proof for CG closure}
We start by considering the case of covering polyhedra \emph{without bounds}.
\begin{proposition} \label{prop:coverCGnobounds}
Consider a covering polyhedron without bounds $P$ and a non-negative function $c \in \mathbb{R}^n_+$. Then $z^{1\mathcal{C}} \geq \frac{1}{2} z^{\mathcal{C}}$. \end{proposition} \begin{proof}
Consider a covering polyhedron $P = \{x \in \mathbb{R}^n_+ \,|\, Ax \ge b, \ x \ge 0\}$. Let $\mathcal{C}(P) =\{x \,|\, A'x \ge b'\}$ be the CG closure of $P$ (i.e., CG closure is a rational polyhedron~\cite{schrijver1980cutting}). Without loss of generality we assume that the entries of $A',b'$ are non-negative integers and each CG cut is obtained by rounding up the entries of the constraint $\lambda^\top Ax \ge \lambda^\top b$ for some $\lambda \in \mathbb{R}_+^m$.
Let $(a')^{\top}x \ge \beta'$ be an inequality of the system $A'x \ge b'$. We show that $(a')^{\top}x \ge \beta'/2$ is a 1-row CG cut for $P$. The theorem then follows by linear programming duality.
If inequality $(a')^{\top}x \ge \beta'$ is one inequality of the original system $Ax \ge b, \ x \ge 0$ we are done, thus we assume that $a'x \ge \beta'$ is a non-trivial CG inequality for $Ax \ge b, \ x \ge 0$. This in particular implies $\beta' \ge 1$. The strict inequality $a'x > \beta' -1$ is valid for $P$. If $\beta' \ge 2$, then $\beta'-1 \ge \beta'/2$, thus $(a')^{\top}x \ge \beta'/2$ is valid for $P$ and so it is trivially a 1-row CG cut for $P$. Thus we now assume $\beta' =1$.
Let $\lambda \in \mathbb{R}^m_+$ be the vector of multipliers corresponding to the CG cut $(a')^{\top}x \ge \beta'$, i.e., $a' = \lceil \lambda^{\top} A\rceil $, and $\beta' = \lceil \lambda^\top b \rceil$. Since $\lambda^{\top} b > 0$, there exists $i\in [m]$ with $\lambda_i b_i > 0$. Then $(a')^{\top}x \ge \beta'$ is implied by the 1-row CG cut $\sum_{j = 1}^n\lceil \lambda_i A_{ij}\rceil x_j \ge \lceil\lambda_i b_i\rceil = 1$ because $\lceil \lambda^\top A \rceil \ge \lceil{\lambda_i A^i}\rceil$. \end{proof}
\begin{proposition} \label{prop:coverCGwithbounds}
For a covering polyhedron with bounds and a non-negative function $c \in \mathbb{R}^n_+$, we have $z^{1\mathcal{C}} \geq \frac{1}{2} z^{\mathcal{C}}$. \end{proposition}
\begin{proof} For a covering polyhedron with bounds $P$, let $\bar P = P + \mathbb{R}^n_+$. By applying Proposition~\ref{prop:coverUpward} recursively, $\bar P$ is a covering polyhedron with bounds. Moreover, each covering inequality of $\bar P$ is a conic combination of one covering inequality of $P$ with the bounds $x \le u$.
We will argue bounds on the ratio between $$z^{\mathcal{C}} = \min \{c^{\top}x : x \in \mathcal{C}(P)\} \quad \text{and} \quad z^{1\mathcal{C}} = \min \{c^{\top}x : x \in 1\mathcal{C}(P)\}$$ by using known bounds on the ratio between covering problems $$\bar z^{\mathcal{C}} = \min \{ c^{\top}x : x \in \mathcal{C}(\bar P)\} \quad \text{and} \quad \bar z^{1\mathcal{C}} = \min \{ c^{\top}x : x \in 1\mathcal{C}(\bar P)\}.$$
We will show $z^{\mathcal{C}} \le \bar z^{\mathcal{C}}$ and $z^{1\mathcal{C}} \ge \bar z^{1\mathcal{C}}$. Together with a bound of $2$ on the ratio for covering problems from Proposition \ref{prop:coverCGnobounds}, this implies the same bound on the ratio for covering problems with bounds: \begin{align*}
\frac{z^{\mathcal{C}}}{z^{1\mathcal{C}}} \le \frac{\bar z^{\mathcal{C}}}{\bar z^{1\mathcal{C}}} \le 2. \end{align*}
\paragraph{Claim 1} \label{claim_k1} $z^{1\mathcal{C}} \ge \bar z^{1\mathcal{C}}$. \\ \\ \emph{Proof.} We only need to show that \begin{align} \label{claim_k_cont} 1\mathcal{C}(P) \subseteq 1\mathcal{C}(\bar P). \end{align} since the relation \eqref{claim_k_cont} directly implies \begin{align*} z^{1\mathcal{C}} = \min \{ c^{\top}x : x \in 1\mathcal{C}(P) \} \ge \min \{ c^{\top}x : x \in 1\mathcal{C}(\bar P) \} = \bar z^{1\mathcal{C}}. \end{align*} By Proposition \ref{prop:coverUpward}, every constraint of $\bar{P}$ is a conic combination of a single covering constraint and one bound constraint. Therefore it follows from the definition of $1\mathcal{C}(P)$ that $1\mathcal{C}(P) \subseteq 1\mathcal{C}(\bar P)$. This shows \eqref{claim_k_cont}.~$\diamond$
\paragraph{Claim 2} \label{claim_k2} We have $z^{\mathcal{C}} \le \bar z^{\mathcal{C}}$. \\ \\ \emph{Proof.} Since $c \ge 0$, to prove $z^{\mathcal{C}} \le \bar z^{\mathcal{C}}$ it is sufficient to show that \begin{align} \label{claim_k_cont2} \mathcal{C}(P) + \mathbb{R}^n_+ \supseteq \mathcal{C}(\bar P). \end{align} In fact, relation \eqref{claim_k_cont2} directly implies \begin{align*} z^{\mathcal{C}} = \min \{ c^{\top}x : x \in \mathcal{C}(P) \} = \min \{ c^{\top}x : x \in \mathcal{C}(P)+ \mathbb{R}^n_+ \} \le \min \{ c^{\top}x : x \in \mathcal{C}(\bar P) \} = \bar z^{\mathcal{C}}. \end{align*} In order to prove relation \eqref{claim_k_cont2}, we prove that \begin{align} \label{this} \mathcal{C}(P) + \textup{cone}(e_j) \supseteq \mathcal{C}(P + \textup{cone}(e_j)). \end{align} In fact, by Proposition \ref{prop:coverUpward}, $P + \textup{cone}(e_j)$ is also a covering polyhedron with bounds. Therefore we can apply relation \eqref{this} recursively (for example, for $j \neq j'$, we have $\mathcal{C}(P) + \textup{cone}(e_j) + \textup{cone}(e_{j'})\supseteq \mathcal{C}(P + \textup{cone}(e_j))+ \textup{cone}(e_{j'}) \supseteq \mathcal{C}(P + \textup{cone}(e_j) + \textup{cone}(e_{j'}))$), and we obtain $\mathcal{C}(P) + \mathbb{R}^n_+ \supseteq \mathcal{C}(P + \mathbb{R}^n_+) = \mathcal{C}(\bar P)$, thus \eqref{claim_k_cont2}.
If $P = P + \textup{cone}(e_j)$, then \eqref{this} follows easily, therefore we now assume that $u_j$ is finite, and therefore by assumption integral. By definition, $$\mathcal{C}(P) = P \cap \{x: a^{\top}x \ge \lceil{\beta}\rceil, \text{ where } a^{\top}x \ge \beta \text{ valid for } P, \ a \in \mathbb{Z}^n\}.$$
We show that all inequalities with $a_j < 0$ can be dropped from such definition. More precisely: $$\mathcal{C}(P) = P \cap \{x: a^{\top}x \ge \lceil{\beta}\rceil, \text{ where } a^{\top}x \ge \beta \text{ valid for } P, \ a \in \mathbb{Z}^n, \ a_j \ge 0\}.$$ Let $a^{\top}x \ge \beta$ be valid for $P$, with $a \in \mathbb{Z}^n$ and $a_j < 0$. Now consider the inequality $(a')^{\top}x \ge \beta'$ obtained as the sum of $ax \ge \beta$ and $-a_j x_j \ge -a_j u_j$. Note that $a' \in \mathbb{Z}^n$ and $a'_j = 0$.
We next verify that $(a')^{\top}x \ge \beta'$ is valid for $P$. In particular, if $\hat{x}:= (\hat{x}_j, \hat{x}_{-}) \in P$ (here the subscript $\textup{ }_{-}$ denotes all components other than $j$), then $(u_j, \hat{x}_{-}) \in P$ and therefore, $a^{\top}_{-}\hat{x}_{-} + a_ju_j \geq \beta $. Equivalently, $a^{\top}_{-}\hat{x}_{-} \geq \beta - a_j u_j$ or $(a')^{\top}\hat{x} = a^{\top}_{-}\hat{x}_{-} \geq \beta - a_j u_j = \beta'$.
Moreover, note that $(a')^{\top}x \ge \lceil{\beta'}\rceil$ cuts from $P$ at least all the points cut by $(a)^{\top}x \ge \lceil\beta \rceil$. To see this, suppose $\hat{x}:= (\hat{x}_j, \hat{x}_{-}) \in P$ is separated by $(a)^{\top}x \ge \lceil\beta \rceil$. Then $a^{\top}_{-}\hat{x}_{-} + a_ju_j \leq a^{\top}_{-}\hat{x}_{-} + a_j x_j <\lceil\beta \rceil$, since $a_j \leq 0$ and $\hat{x}_j \in P$. Equivalently, $(a')^{\top}\hat{x} = a^{\top}_{-}\hat{x}_{-} < \lceil\beta \rceil - a_ju_j = \lceil \beta - a_ju_j \rceil = \lceil{\beta'}\rceil$, since $a_j u_j \in \mathbb{Z}$.
Therefore \begin{align*}
\mathcal{C}(P) & = P \cap \{x\,|\, ax \ge \lceil{\beta}\rceil, \text{ where } ax \ge \beta \text{ valid for } P, \ a \in \mathbb{Z}^n, \ a_j \ge 0\} \\
& = P \cap \{x\,|\, ax \ge \lceil{\beta}\rceil, \text{ where } ax \ge \beta \text{ valid for } P + \textup{cone}(e_j), \ a \in \mathbb{Z}^n\} \\ & = P \cap \mathcal{C}(P + \textup{cone}(e_j)) \\
& = \{x \,|\, x_j \le u_j\} \cap \mathcal{C}(P + \textup{cone}(e_j)), \end{align*}
where the last equation follows from the fact that if $y \in (P + \textup{conv}(x_j)) \cap \{x \,|\, x_j \le u_j\}$, then $y \in P$. Thus we obtain $$\mathcal{C}(P) + \textup{cone}(e_j) = \left(\{x : x_j \le u_j\} \cap \mathcal{C}(P + \textup{cone}(e_j))\right)+ \textup{cone}(e_j).$$
Finally, we show that \begin{eqnarray} \left(\{x : x_j \le u_j\} \cap \mathcal{C}(P + \textup{cone}(e_j))\right)+ \textup{cone}(e_j)\supseteq \mathcal{C}(P + \textup{cone}(e_j)), \end{eqnarray} to complete the proof.
First we verify that if $\hat{x} := (\hat{x}_{-}, \hat{x}_j) \in \mathcal{C}(P + \textup{cone}(e_j))$ and $\hat{x}_j \geq u_j$, then $(\hat{x}_{-}, u_j) \in \mathcal{C}(P + \textup{cone}(e_j))$. Assume by contradiction that $c_{-}^{\top}{x}_{-} + c_j {x}_j \geq \delta$ be a valid inequality for $P + \textup{cone}(e_j)$ with $c \in \mathbb{Z}^n$ such that $c_{-}^{\top}\hat{x}_{-} + c_j u_j < \lceil\delta \rceil$. We will show that the point $(\hat{x}_{-}, \hat{x}_j)$ also does not belong to $\mathcal{C}(P + \textup{cone}(e_j))$ to obtain a contradiction. Note first that $c_{-}^{\top}{x}_{-} + c_j {x}_j \geq \delta$ is a valid inequality for $P$ with $c_j \geq 0$. Therefore $c_{-}^{\top}{x}_{-} \geq \delta - c_j u_j$ is a valid inequality for $P$. However since the $j^{\text{th}}$ component of $c':= (c_{-}, 0)$ is non-negative, we have that $c_{-}^{\top}{x}_{-} \geq \delta - c_j u_j$ is a valid inequality for $P + \textup{cone}(e_j)$. In other words, $c_{-}^{\top}{x}_{-} \geq \lceil \delta - c_j u_j \rceil = \lceil\delta\rceil - c_ju_j$ is a CG inequality for $P + \textup{cone}(e_j)$. However note that this CG inequality separates the point $(\hat{x}_{-}, \hat{x}_j)$.
Now let $\hat{x} := (\hat{x}_{-}, \hat{x}_j) \in \mathcal{C}(P + \textup{cone}(e_j))$. If $\hat{x}_j \leq u_j$, then clearly $\hat{x} \in (\{x : x_j \le u_j\} \cap \mathcal{C}(P + \textup{cone}(e_j)))+ \textup{cone}(e_j)$. In $\hat{x}_j \geq u_j$, then based on the above discussion $(\hat{x}_{-}, u_j) \in \mathcal{C}(P + \textup{cone}(e_j))$. In other words, $(\hat{x}_{-}, u_j) \in \left(\{x : x_j \le u_j\} \cap \mathcal{C}(P + \textup{cone}(e_j))\right)$. Thus $$\hat{x} = (\hat{x}_{-}, u_j) + (0, x_j - u_j) \in \left(\{x : x_j \le u_j\} \cap \mathcal{C}(P + \textup{cone}(e_j))\right)+ \textup{cone}(e_j),$$ completing the proof.
Therefore we have proven the claim by showing \eqref{this}. ~$\diamond$ \\ \\ This concludes the proof of Proposition \ref{prop:coverCGwithbounds}.
\end{proof}
\paragraph{Tight examples.} We need to show that for there is an instance where $\frac{z^{\mathcal{C}}}{z^{1\mathcal{C}}} \ge 2-\varepsilon$. But the proof of Proposition \ref{prop:LBAC} already shows that this happens for the instance given by \eqref{eq:LBAC}.
We next show that $z^{\mathcal{C}}$ can be arbitrarily bad in comparison to $z^{KC}$.
\begin{proposition}
\label{prop:coverCGvsKC}
$z^{\mathcal{C}}$ can be arbitrarily bad in comparison to $z^{KC}$ for 0-1 covering problems.
\end{proposition}
\begin{proof}
Consider the problem
\begin{align*}
\min ~&x_n\\
st ~& x_1 + \ldots x_{n-1} + n x_n \ge n\\
&x \in \{0,1\}^n.
\end{align*}
It is straightforward to verify that the CG closure of this problem should be obtained by just adding the inequality $\lceil 1/n \rceil x_1 + \ldots \lceil 1/n \rceil x_{n-1} + x_n \ge 1 \equiv \sum_i x_i \ge 1$. So optimizing over the CG closure gives value $1/n$.
But the 1-row-aggregated closure gives the integer hull, so optimizing over it gives value~$1$. \end{proof}
\subsection{Proof of Theorem \ref{thm:rankCover}} \label{subsec:CoveringRankProof} We use the following result on bounds of integrality gap of covering IPs as a function of the number of constraints. \begin{theorem}[\cite{vazirani2013approximation}] \label{thm:vazirani} Consider a covering IP of the following form: $\min \{ c^\top x \mid Dx \geq f,~ x \in \mathbb{Z}^n_+\}$, where $D \in \mathbb{R}_+^{k \times n}$ such that $D_{ij} \leq f_i$ for all $i \in[k],~ j\in [n]$, and $c \in \mathbb{R}^n_+$. Then\footnote{The constant 8 can be easily verified using the proof techniques in \cite{vazirani2013approximation}.}, $z^I \leq 8 \log_2 (2k) z^{LP}$. \end{theorem}
\paragraph{Lower bound on rank.} We will prove Theorem \ref{thm:rankCover} for a more general non-empty covering \emph{set} $Q = \{x \in \mathbb{R}^n_+ \mid A^i x \geq b_i,~i \in \mathcal{I} \}$, where $\mathcal{I}$ is an arbitrary index set and $0 \leq A^i_j \leq b_i$ for all $i \in \mathcal{I}$. We will call a covering set with these properties a \emph{well-behaved covering set}.
Given a matrix $(D,f) \in \mathbb{R}^{k \times n} \times \mathbb{R}^k$, we say that it is \emph{$k$-vi \ for $Q$} if $(D,f)$ is non-negative and the $k$ inequalities $D^i x \ge f_i$ are valid for $Q$.
We denote the polyhedral outer-approximation $\{ x \in \mathbb{R}^n_+ \mid D x \geq f \}$ of $Q$ by $P_{(D,f)}$. Then by definition
\begin{align*}
\mathcal{A}_k(Q) = \bigcap_{(D,f) \textrm{ is a $k$-vi \ for $Q$}} (P_{(D,f)})^I.
\end{align*}
It will be important to show that if $Q$ is well-behaved, then so is the closure $\mathcal{A}_k(Q)$. For that we need the following observation.
\paragraph{Claim 1} Consider a well-behaved covering set $Q$ and let $\alpha^\top x \geq \beta$ be a valid inequality for it. Then, there exists a valid inequality $\hat{\alpha}^\top x \geq \hat{\beta}$ for $Q$ with the following properties: (i) $\hat{\alpha}_j \leq \alpha_j$ for all $j \in [n]$ (ii) $\hat{\beta} \geq \beta$ (iii) $\hat{\alpha}_j \leq \hat{\beta}$ for all $j \in [n]$. \\ \\ \emph{Proof.} As $Q \neq \emptyset$, by the generalized Farkas Lemma (Theorem 3.1 in \cite{lopez1998linear}), $\alpha^\top x \geq \beta$ is a valid inequality for $Q$ if and only if \begin{eqnarray*} \left[ \begin{array}{c} \alpha \\ \beta \end{array} \right]
\in \text{cl} \Bigg( \text{cone} \Bigg( \Bigg\{
\left[ \begin{array}{c} \boldsymbol{0} \\ -1 \end{array} \right]\Bigg\} \ \cup \ \Bigg\{ \left[ \begin{array}{c} (A^i)^\top \\ b_i \end{array} \right] ;~i \in \mathcal{I}
\Bigg\} \ \cup \ \Bigg\{
\left[\begin{array}{c} e_j \\ 0 \end{array}\right];~j \in [n]
\Bigg\} \Bigg) \Bigg)
:= F, \end{eqnarray*} where $\text{cl}$ and $\boldsymbol{0}$ stands for the closure and the vector of zeros in $\mathbb{R}^n$, respectively. We also let \begin{eqnarray*} G := \text{cl} \Bigg( \text{cone} \Bigg( \Bigg\{ \left[ \begin{array}{c} (A^i)^\top \\ b_i \end{array} \right] ;~i \in \mathcal{I}
\Bigg\} \Bigg) \Bigg), ~~~H := \cone \Bigg(
\left[ \begin{array}{c} \boldsymbol{0} \\ -1 \end{array} \right] \ \cup \ \Bigg\{
\left[\begin{array}{c} e_j \\ 0 \end{array}\right];~j \in [n]
\Bigg\} \Bigg) \end{eqnarray*} Note that $F = \text{cl}(G + H)$. We will show that $G+H$ is closed, thereby implying that $F = G+H$.
For that, notice that the cones $G$ and $H$ are \emph{positively semi-independent}, that is if $g \in G$ and $h \in H$ satisfy $g + h = [\boldsymbol{0}, 0]$, then $g=h= [\boldsymbol{0},0]$: To see this, consider a vector $[a, -b] \in H$, so $a$ and $b$ are non-negative, such that $[-a,b]$ belongs to $G$. Since $A^i$ is non-negative for all $i \in \mathcal{I}$, this implies that $a= 0$. Furthermore, as $Q$ is non-empty and the inequality $-a^\top x \ge b$ is valid for $Q$, we obtain $b=0$, which concludes the argument.
Hence, by a result in \cite{gale1976malinvaud}, $G+H$ is closed, and so $F = G + H$.
This implies that if $\alpha^\top x \geq \beta$ is a valid inequality for $Q$, then \begin{eqnarray*} \left[ \begin{array}{c} \alpha \\ \beta \end{array} \right] = \left[ \begin{array}{c} \hat{\alpha} \\ \hat{\beta} \end{array} \right] + \lambda \left[ \begin{array}{c} \boldsymbol{0} \\ -1 \end{array} \right] + \mu_j \left[ \begin{array}{c} e_j \\ 0 \end{array} \right], \quad \lambda \geq 0, ~ \mu_j \geq 0 \ \forall j \in [n] , \end{eqnarray*} where $[\hat{\alpha}, \hat{\beta}] \in G$. Note that $\hat{\alpha}^\top x \geq \hat{\beta}$ is a valid inequality for $Q$. Moreover, $[\hat{\alpha}, \hat{\beta}] \in G$ implies that $\hat{\alpha}_j \leq \hat{\beta}$ for all $j \in [n]$. Also, all the other conditions of the claim are satisfied which completes the proof. ~$\diamond$
\paragraph{Claim 2} If $Q$ is a well-behaved covering set, then $\mathcal{A}_k(Q)$ is also a well-behaved covering set. \\ \\ \emph{Proof.} Given $(D,f)$ a $k$-vi \ for $Q$, let $(\hat{D},\hat{f})$ be obtained as in Claim 1. Then, by the previous claim, $(\hat{D}, \hat{f})$ is a $k$-vi \ for $Q$. Observe that by construction of $(\hat{D},\hat{f})$ we have $P_{(D,f)} \supseteq P_{(\hat{D},\hat{f})}$, and therefore $(P_{(D,f)})^I \supseteq (P_{(\hat{D},\hat{f})})^I$. Hence
\begin{equation} \label{eq:Df_forPI} \mathcal{A}_k(Q) = \bigcap_{(D,f) \ \text{is a $k$-vi \ for $T$}} (P_{(\hat{D},\hat{f})})^I. \end{equation}
To show that $\mathcal{A}_k(Q)$ is well-behaved, it suffices to show that $(P_{(\hat{D},\hat{f})})^I$ is of the form $\{ x \in \mathbb{R}^n_+ \mid R^i x \geq s_i,~ i \in [m'] \}$, where $0 \leq R_{ij} \leq s_i$ for all $j \in [n],~ i \in [m']$.
Since the recession cone of $P_{(\hat{D},\hat{f})}$ is $\mathbb{R}^n_+$, and $P_{(\hat{D},\hat{f})}$ is a polyhedron, by Theorem 6 in \cite{convex2013deymoran}, $(P_{(\hat{D},\hat{f})})^I$ is a rational polyhedron with the same recession cone as $P_{(\hat{D},\hat{f})}$. Hence, $R$ is non-negative. Moreover, we may take the inequalities $R^i x \geq s_i,~ i \in [m']$ to be the facet-defining inequalities that satisfy at $n+1$ affinely independent integer points at equality. To show $R_{i,j^*} \leq s_i$ for some $j^* \in [n]$, observe that in particular there exists an integer point $\hat{x}$ among these $n+1$ affinely independent ones satisfying $\hat{x}_{j^*} \geq 1$ and $\sum_{j \in [n]} R_{ij} \hat{x}_j = s_i$ (else all these points would satisfy the additional equation $x_j = 0$ and live in an $(n-2)$-dimensional space, contradicting their affine independence). This implies $R_{i,j^*} = \frac{s_i-\sum_{j \neq j^*} R_{ij} \hat{x}_j}{\hat{x}_{j^*}} \leq s_i$. ~$\diamond$ \\ \\ Let $Q^\ell$ be the $\ell^{\text{th}}$ $k$-aggregation closure of $Q$. \paragraph{Claim 3} $Q^\ell \subseteq \frac{1}{8 \log_2(2k)} Q^{\ell+1}$. \\ \\ \emph{Proof.} By the previous claim, $Q^\ell$ is a well-behaved covering set. For every $(D,f)$ $k$-vi \ for $Q^\ell$, by Theorem \ref{thm:vazirani} we have $$Q^\ell \subseteq P_{(\hat{D},\hat{f})} \subseteq \frac{1}{8 \log_2(2k)} (P_{(\hat{D}, \hat{f})})^I.$$
By Observation~\ref{obs:bijection} we have that \begin{align*} \begin{split} Q^\ell \ \subseteq & \ \frac{1}{8 \log_2(2k)} \bigcap_{(D,f) \textup{ is $k$-vi \ for } Q^\ell} (P_{(\hat{D}, \hat{f})})^I \\ = & \frac{1}{8 \log_2(2k)} Q^{\ell+1}, \end{split} \end{align*} where the last equality follows from \eqref{eq:Df_forPI}. ~$\diamond$ \\ \\ \indent Using an argument similar to the proof of Theorem \ref{thm:packrankgen} (employing now Proposition \ref{prop:blowupCover}), we obtain that the rank of the $k$-aggregation closure is at least $\left \lceil \left(\frac{\textup{log}_2\left(\frac{z^I}{z^{LP}}\right)}{3+\textup{log}_2\textup{log}_2(2k)}\right)\right \rceil$.
\\ \\ \\ \\ \textbf{Acknowledgements.}
Santanu S. Dey would like to acknowledge the support of the NSF grant CMMI\#1149400.
\begin{appendix} \section{Polyhedrality of aggregation closure for dense IPs} \label{sec:appendix} We prove the result for the case of covering IPs and a similar proof can be given for the packing case.
\begin{proposition} Let $Q = \{ x \in \mathbb{R}^n_+ \mid Ax \geq b \}$ be a covering polyhedron with $A \in \mathbb{Z}_+^{m \times n},~b \in \mathbb{Z}_+^{n}$, $A_{ij} \geq 1$ for all $i \in [m],~j\in [n]$, and $b_i \geq 1$ for all $i \in [m]$. Then, $\mathcal{A}_k(Q)$ is a polyhedron. \end{proposition}
\begin{proof} The intercept of the hyperplane corresponding to the $i^{\text{th}}$ constraint, $A^i x \geq b_i$, of the $j^{\text{th}}$ coordinate axis is $\frac{b_i}{A_{ij}}$. It is straightforward to verify that the intercept of any aggregated constraint on the $j^{\text{th}}$ coordinate axis belongs to the set $\left[\min_{i \in [m]} \frac{b_i}{A_{ij}},\max_{i \in [m]} \frac{b_i}{A_{ij}}\right]$. Let $M = \max_{i \in [m],~j \in [n]} \frac{b_i}{A_{ij}}$ and let $T = [0,M]^n \cap \mathbb{Z}_+^n$.
Based on the above observation, the set of integer points contained in $\{ x \in \mathbb{R}^n_+ \mid (\lambda^\ell)^\top A x \geq (\lambda^\ell)^\top b,~ \ell \in [k] \}$ is of the form $S \cup (\mathbb{Z}_+^n \setminus T)$ where $S \subseteq T$. Since $T$ is a finite set, this completes the proof as the number of distinct integer hulls obtained from $k$-aggregations is finite. \end{proof}
\section{Proof of Proposition \ref{prop:approxPack}} \label{app:charApproxPacking}
Given a convex set $C \subseteq \mathbb{R}^n$, its support function $\delta^*(. \mid C)$ is defined by $\delta^*(c \mid C) = \sup \{ c^T x \mid x \in C\}$.
Consider packing sets $U \supseteq V$. Since $U$ and $V$ are closed, from Corollary 13.1.1 of \cite{rockafellar:1970} we have that $U \supseteq \alpha V$ iff
\begin{align}
\sup_{c \in \mathbb{R}^n} \left(\delta^*(c \mid U) - \delta^*(c \mid \alpha V)\right) \le 0. \label{eq:rock}
\end{align}
Since $U$ is a packing set we have the following property. Consider a vector $c \in \mathbb{R}^n$, let $I$ be the index of its negative components, and let $\tilde{c}$ be obtained by changing the components of $c$ in $I$ to $0$. Then $\delta^*(c \mid U) = \delta^*(\tilde{c} \mid U)$: the direction ``$\le$'' follows from $U \subseteq \mathbb{R}^n_+$; the direction ``$\ge$'' holds because for every point $x \in U$, if we construct $\tilde{x}$ by changing the components in $I$ of $x$ to $0$ then $\tilde{x} \in U$ and $c^T \tilde{x} = \tilde{c}^T x$. Since the same holds for $\alpha V$, we have that in equation \eqref{eq:rock} we can take the supremum over only non-negative $c$'s, and hence it holds iff for all $c \in \mathbb{R}^n_+$, $\delta^*(c \mid U) \le \delta^*(c \mid \alpha V)$. But since $\delta^*(c \mid \alpha V) = \alpha\,\delta^*(c \mid V)$ (Corollary 16.1.1 of \cite{rockafellar:1970}), this happens iff for all $c \in \mathbb{R}^n_+$, $\delta^*(c \mid U) \le \alpha\,\delta^*(c \mid V)$. This concludes the proof.
\section{Proof of Proposition \ref{prop:inthullPack}} \label{app:packinginthull}
Let $Q = \{ x \in \mathbb{R}_+^n \mid A^i x \leq b_i \ \forall i \in I \}$. We assume that for all $j \in [n]$, there exists $i \in I$ with $A_{ij} > 0$. Otherwise, we can project out the $j^{\text{th}}$ variable and continue with the argument as the $j^{\text{th}}$ variable is allowed to take any value. Therefore, $Q$ is a bounded set and $Q^I$ is a polyhedron. Let $Q^I = \{ x \in \mathbb{R}_+^n \mid Cx \leq d \}$. We next argue that $C$ and $d$ are non-negative to complete the proof.
Note that since $\boldsymbol{0} \in Q$, $d \geq 0$. The fact that we can take $C \geq 0$ follows from the following claim.
\paragraph{Claim.} Let $C^i x \leq d_i$ be a facet-defining inequality for $Q^I$ and $C_{i j^{*}} < 0$ for some $i$ and $j^{*}$. Define a vector $\hat{c}$ as $\hat{c}_{j^{*}} = 0$ and $\hat{c}_{j} = C_{ij}$ for all other $j$. Then $\hat{c} x \leq d_i$ is valid for $Q^I$. \\ \\ \emph{Proof.}
Assume by contradiction that there exists $\hat{x} \in Q \cap \mathbb{Z}^n $ such that $\sum_{j = 1}^n \hat{c}_{j}\hat{x}_j > d_i$. Since $Q$ is a packing set, we have that $\tilde{x} \in Q \cap \mathbb{Z}^n$, where $\tilde{x}$ is defined as $\tilde{x}_j = \hat{x}_j$ for all $j \in [n] \setminus \{j^{*}\}$ and $\tilde{x}_j^{*} = 0$. Then $d_i < \sum_{j = 1}^n \hat{c}_{j}\hat{x}_j = \sum_{j = 1}^n \hat{c}_{j}\tilde{x}_j = \sum_{j = 1}^n {C}_{ij}\tilde{x}_j \leq d_i$, a contradiction. ~$\diamond$ \\ \end{appendix}
\end{document} | arXiv |
\begin{definition}[Definition:Biadditive Mapping]
Let $M, N, P$ be abelian groups.
Let $M \times N$ be the cartesian product.
A '''biadditive mapping''' $f : M \times N \to P$ is a mapping such that:
:$\forall m_1, m_2 \in M : \forall n \in N: \map f {m_1 + m_2, n} = \map f {m_1, n} + \map f {m_2, n}$
:$\forall m \in M: \forall n_1, n_2 \in N: \map f {m, n_1 + n_2} = \map f {m, n_1} + \map f {m, n_2}$
\end{definition} | ProofWiki |
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