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This question came to my mind thanks to this question which I found really interesting (and beautiful! Like the mathematician Philippe Caldero said in his book Histoires Hédonistes de Groupes et de Géométries (roughly translated) "Let us stop for a moment to contemplate the beauty of mathematics, that is after all the point of figures".).
It is also related to this other question.
Initialisation: You start on the point $(0,0)$ which correspond to the integer $n=0$, and you will walk from one point of $\mathbb Z^2$ to another. You start by walking on the right.
Each horizontal step you take increases the integer $n$ by $1$.
When $n$ is equal to a prime number, you take one step up, and you change the direction you were going to (if you were walking from left to right you will walk from right to left, and reciprocally).
Well now nothing stops us from going a little further, which we will do until $n=100$, and then until $n=1\,000$.
It seems that the walk is almost always on the right side of the $y$-axis. Though the walk is crossing the axis a few times.
Let us walk until $n=10\,000$.
Then we realise that we have completely changing the side of the axis we were walking on.
Will we cross the $y$-axis infinitely many times?
Will we walk out of any fixed vertical band centred on the $y$-axis?
Though any other result, or drawing (I did not succeed in drawing it for $n=10^5$), references about this walk would be of great interest.
Will we walk arbitrarily far off the y-axis?
Yes, because there are arbitrarily long gaps between consecutive primes.
In other words, there is no finite bound on the gap between two consecutive primes.
Firstly, Let's lay out the assumptions related to this.
The spacing between primes is Arbitrary.
Given that both of these are true, the answer is Yes for any point along the $x$ axis.
Given infinite time, and an infinite random walk left or right, any walk will eventually pass through any point.
Any walk which exists in a single axis, in this case, the $x$, will pass through any value of $x$ an infinite amount of times, given infinite time.
Assume the target $x=0$, where the walk is currently at $x=1$. The walk, has a $1/2$ chance to either, cross the $x=0$ within the next move.
But we need the chance that $x$ will ever cross $0$ again.
The percent chance for $x=2$ to reach $0$, within two turns, is $1/4$, which means that $x=1$ has, at least, a $1/2+1/4=3/4$ chance to successfully cross the $0$ in 3 turns.
This means that $x\to\inf$, $p\to0$ where $p$ is the chance that $x$ will equal $0$ in exactly $x$ moves.
$x$ cannot be $\inf$, so $p>0$. There is always a chance, and there are infinite tries. Even after failing,there's a 50% chance that your odds increase.
Thus, a Random Walk, given infinite time, will cross $x=0$, infinite times.
This Applies to Prime Gaps.
In this case, the Prime Gaps are just a random walk which changes direction at a Prime.
Below is a diagram showing how the Prime Walk (Blue) and Random Walk (Red) compare for values up to $y=500$.
Even though the Prime walk is strongly biased towards smaller numbers, there is no known maximum distance between prime gaps which means there is a chance, albeit an even smaller than normal, yet a non-zero chance that the next prime is far enough away to $x\le0$.
Of course, this proof is destroyed if Primes are either Proven non-infinite, against current proofs, or predictable.
Not the answer you're looking for? Browse other questions tagged prime-numbers random-walk visualization pattern-recognition or ask your own question.
Probability of maximum of a random walk?
Will a 2 dimensional random walk with random orientations almost certainly return near the origin infinitely often? | CommonCrawl |
A lot of arithmetic geometry has to do and took off with constructions or proofs of non-constructibility of specific figures only with ruler and compass. See e.g. the non-constructibility of the regular heptagon and the constructibility of the regular heptadecagon by Gauss.
I wonder what it means that there is a simple device that can be constructed with the help of a ruler and a compass that together with ruler and compass allows to construct arbitrary regular polygons very easily.
The device is nothing but a solid cone of finite height and a flexible string of length smaller than the circumference of the cone.
Now arrange $n$ dots on the string at equal distances and close the string such that also the first and the last dot have the same distance. I.e. you arrange $n$ dots equally on a closed string. This can be achieved with a compass alone.
After that you place the string on the cone, keeping all dots at same height until they all touch the cone. Projecting them vertically on the plane gives the corners of a regular $n$-gon.
Note that the cone is not a magic device like the angle trisector which may help to construct a rectangular heptagon, but adds nothing genuinely new to ruler and compass (because it can be constructed with the help of these). On the other hand, a flexible string is enough to create a ruler (as a physical device), and you need two rulers to make a compass. So a string might be enough for everything, in a specific sense? Note further, that angle trisectors - and more generally angle $k$-sectors - can easily be constructed for any angle $\alpha = 2\pi / n$: just place $kn$ dots on the string.
Is there something wrong with my device? Might it be not constructible with the help of ruler and compass alone? Do other parts of the described procedure add something else beyond ruler and compass (the cone itself is supposed not to)?
I guess, it's leaving the plane that "disqualifies" this kind of construction. Might this have to do with leaving the real number line and entering the complex plane when solving polynomial equations? ("Suddenly all polynomial equations can be solved!") On the other hand: The compass is a genuinely three-dimensional device, too!
The basic misunderstanding here is that "ruler and compass" is not really about which physical tools you're allowed to use.
To describe a circle with any center and radius.
(The fourth and fifth postulates are not basic constructions but are claims about what happens when certain other combinations of constructions are performed).
These three postulates are the definition of "ruler and compass". Often in English the alternative phrase "compass and straightedge" is preferred, to underscore the fact that there is none of the basic allowed constructions that depend depend on having measuring marks on your ruler.
In fact even without leaving the two-dimensional paper there are things you can imagine doing with a ruler that has measuring marks on it -- so-called neusis constructions. These are nevertheless not part of the basic operations. They can, for example, be used to trisect angles.
Mathematics is not about physical tools. Saying "ruler and compass" is merely supposed to remind you what the fundamental three constructions in Euclid are. You may, if you wish, object that it is a confusing or misleading shorthand -- that is ultimately a subjective judgment -- but this does not change the fact that what we really care about is not the tools named by the shorthand, but the three postulates it attempts to point to.
That's actually very clever! It shows that the ruler and compass model may actually be narrow-minded, since flexible strings are available, as is paper.
However, I believe that if flexible things are not allowed, then we still have unconstructability proofs. For instance, suppose everything we cut out is solid. Then we are only allowed to place it on the plane, and everything we get will have been constructed in the plane before cutting out the shape. So only if we are allowed to bend and stretch things, the third dimension gives additional possibilities.
Not the answer you're looking for? Browse other questions tagged euclidean-geometry math-history geometric-construction arithmetic-geometry or ask your own question.
With edge and compass construction, given cubes of volumes $a^3,b^3$, can one construct a cube of volume $a^3+b^3$?
Where is the hole in this argument asserting the constructibility of all regular polygons? | CommonCrawl |
The Rules of "Integrated Sums Sudoku"
Each column, each row, and each box (3$\times$3 subgrid) must have the numbers 1 to 9.
The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on the intersections between two diagonally adjacent squares.
Each small clue-number is the sum of two digits in the two squares that are horizontally or vertically or diagonally adjacent. The position of each pair of diagonally adjacent squares is indicated by either two forward slash marks // or two backward slash marks \\.
For example, the //14 on the intersection between the diagonally adjacent squares (2, 9) and (3, 8) means that possible pairs of numbers in the squares are: 5 and 9, 9 and 5; 6 and 8, or 8 and 6 respectively.
The \\5 on intersection between the diagonally adjacent squares (3, 4) and (4, 5) means that possible pairs of numbers in the squares are: 1 and 4, 4 and 1; 2 and 3, or 3 and 2 respectively.
Finally, the clue-number 12 on the border line betweeen the squares (4, 7) and (4, 8) means that possible pairs of numbers for these squares can be from the following combinations: 3 and 9, 9 and 3; 4 and 8, 8 and 4; 5 and 7, or 7and 5. | CommonCrawl |
Abstract. We consider the Navier-Stokes equations in the thin 3D domain $T^2\times(0,\epsilon)$, where $T^2$ is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick-force. We establish that, firstly, when $\epsilon\ll1$ the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges (as $\epsilon\to0$) to a unique stationary measure for the Navier-Stokes equation on $T^2$. Thus, the 2D Navier-Stokes equations on surfaces describe asymptotic in time and limiting in $\epsilon$ statistical properties of 3D solutions in thin 3D domains. | CommonCrawl |
While finally revisiting one of the geometry books on my shelf, Glen Bredon's Topology and Geometry, I encountered an exercise about showing that the projective space is homeomorphic to the mapping cone of a map that doubles a circle on itself (the complex squaring map $z \mapsto z^2$). The mapping cone has a nice visualization, first as a mapping cylinder, which takes a space $X$ and crosses it with the interval $I$ to form $X \times I$ (thus forming a "cylinder"), and then glues the bottom of it to another space $Y$ using a given continuous map $f : X \to Y$. Finally, to make the cone, it collapses the top to a single point. Of course, this can be visualized as deforming the bottom part of $X \times I$ through whatever contortion $f$ does, which might include self-intersection (and of course, it could be more gradual). So I used a good old friend, parametrizations, to help set up an explicit example. Take a look!
A cutaway view, now as a more solid surface, basically illustrating it now as a mapping cylinder (it is homeomorphic to projective space minus a disk, which is a Möbius strip). Anyone know a good glassblower so we can make vases that look like this?
with $0\leq u \leq 1$ or $0.97$, and $0 \leq v \leq 2\pi$. | CommonCrawl |
In this paper we study non-interactive correlation distillation (NICD), a generalization of noise sensitivity previously studied earlier. We extend the model to NICD on trees. In this model there is a fixed undirected tree with players at some of the nodes. One node is given a uniformly random string and this string is distributed throughout the network, with the edges of the tree acting as independent binary symmetric channels. The goal of the players is to agree on a shared random bit without communicating. Our new contributions include the following: (1). In the case of a $k$-leaf star graph (the model considered earlier by Mossel and O'Donnell), we resolve the open question of whether the success probability must go to zero as $k \to \infty$. We show that this is indeed the case and provide matching upper and lower bounds on the asymptotically optimal rate (a slowly-decaying polynomial). (2). In the case of the $k$-vertex path graph, we show that it is always optimal for all players to use the same 1-bit function. (3). In the general case we show that all players should use monotone functions. We also show, somewhat surprisingly, that for certain trees it is better if not all players use the same function. Our techniques include the use of the reverse Bonami-Beckner inequality. Although the usual Bonami-Beckner has been frequently used before, its reverse counterpart seems very little-known; To demonstrate its strength, we use it to prove a new isoperimetric inequality for the discrete cube and a new result on the mixing of short random walks on the cube. Another tool that we need is a tight bound on the probability that a Markov chain stays inside certain sets; we prove a new theorem generalizing and strengthening previous such bounds. On the probabilistic side, we use the ``reflection principle'' and the FKG and related inequalities in order to study the problem on general trees. | CommonCrawl |
Comment at (n,r)-category about the equivalence of fundamental categories. This is a coarser notion than equivalence of categories in the usual sense.
Format: MarkdownThanks. Do you have a reference for that notion? Is that in Grandis' work? We should have an entry on this. Recently, after I carried this question to the CatTheory mailing list I received a reply by Peter Bubenik who wrote that together with David Spivak they are in the process of proving the "directed homotopy hypothesis" relating (oo,1)-categories and some flavor of directed topological spaces. But even though I tried, I couldn't make him tell me what exactly it is they are proving and precisely which notions of equivalence etc they are using.
Thanks. Do you have a reference for that notion? Is that in Grandis' work? We should have an entry on this.
Recently, after I carried this question to the CatTheory mailing list I received a reply by Peter Bubenik who wrote that together with David Spivak they are in the process of proving the "directed homotopy hypothesis" relating (oo,1)-categories and some flavor of directed topological spaces.
But even though I tried, I couldn't make him tell me what exactly it is they are proving and precisely which notions of equivalence etc they are using.
Format: MarkdownAsked a question at [[(n,r)-category]].
Asked a question at (n,r)-category.
Format: MarkdownItexI have added to the Definition-section at [[(n,r)-category]] a precise definition: > In terms of the standard notion of [[(∞,n)-categories]] we can make this precise as follows: > For $-2 \leq n \leq \infty$, an **[[(n,0)-category]]** is an [[∞-groupoid]] that is [[n-truncated]]: an [[n-groupoid]]. > For $0 \leq r \lt \infty$, an **(n,r)-category** is an [[(∞,n)-category|(∞,r)-category]] $C$ such that for all [[object]]s $X,Y \in C$ the $(\infty,r-1)$-categorical [[hom-object]] $C(X,Y)$ is an $(n-1,r-1)$-category.
For −2≤q;n≤q;∞-2 \leq n \leq \infty, an (n,0)-category is an ∞-groupoid that is n-truncated: an n-groupoid.
For 0≤q;r<∞0 \leq r \lt \infty, an (n,r)-category is an (∞,r)-category CC such that for all objects X,Y∈CX,Y \in C the (∞,r−1)(\infty,r-1)-categorical hom-object C(X,Y)C(X,Y) is an (n−1,r−1)(n-1,r-1)-category.
Format: MarkdownItexIs there anything in those query boxes worth keeping?
Is there anything in those query boxes worth keeping?
Format: MarkdownItexI think their conclusions should be incorporated into the page.
I think their conclusions should be incorporated into the page. | CommonCrawl |
The method converges to a solution in a finite number of iterations.
Since $x_2 = x_0$, we will get $x_3 = x_1 = 0$.
So, $x_3 = 0$, and the method never converges. A choice.
Say $f(1) = 1.9, f(2) = 1.95, f(3) = 1.97, \ldots f(100) = 1.999, f(10000) = 1.999999, f(1000000) = 2$, here we say $f$ converges to the value 2. These things you have to think a lot as it is mathematics. But the above question is from Numerical Methods -- now not in GATE syllabus.
Which of the following statements is true in respect of the convergence of the Newton-Rephson procedure? It converges always under all circumstances. It does not converge to a tool where the second differential coefficient changes sign. It does not converge to a root where the second differential coefficient vanishes. None of the above. | CommonCrawl |
This list is based on what was entered into the 'organiser' field in a talk. It may not mean that [email protected] actually organised the talk, they may have been responsible only for entering the talk into the talks.cam system.
Long Term Electricity Market Design: Pricing Quality?
Optimal storage, investment and management under uncertainty - It is costly to avoid outages!
Effective behaviour of critical-contrast PDEs: micro-resonances, frequency conversion, and time dispersive properties.
Lower semicontinuity and relaxation of nonlocal $L^\infty$ functionals.
Lecture 2: Complexity results for integration.
Hurwitz stable and self-interlacing orthogonal polynomials.
Panel Discussion - can we/should we do 100% Renewables?
Dendroidal spaces and mapping spaces between little cubes operads.
Topological and dynamical obstructions to extending group actions.
Isomorphism theorems, random walks, and spin systems.
Generation of random dynamical systems for SPDE with nonlinear noise.
What can AI Contribute to Neuroscience?
Why New Computational Approaches are Needed?
Supersymmetry and Ward identities: an alternative approach to renormalization.
What does it mean to be continuous?
Where can AI add Value in Radiology?
To complete or make discrete? That is the question.
Modes, Ambidexterity and Chromatic homotopy.
Reflection Positivity which plays an important role in QFT and statistical mechanics.
A Whitehead theorem for periodic homotopy groups?
Tree complexes and obstructions to embeddings.
Rényi's Information Dimension Beyond I.I.D.
Have you seen this homology class??
Does data interpolation contradict statistical optimality?
Monte Carlo adjusted profile likelihood, with applications to spatiotemporal and phylodynamic inference.
UQ: does it require efficient linear algebra?
Stein Points: Efficient sampling from posterior distributions by minimising Stein Discrepancies.
Three of eleven topics on my mind : "Choose your own adventure"
Optimal Weighted Least Squares Methods for High Dimensional Approximation and Estimation.
Stochastic Galerkin mixed finite element approximation for parameter-dependent linear elasticity equations.
Multilevel Emulation and History Matching of EAGLE: an expensive hydrodynamical Galaxy formation simulation.
Studying black holes with gravitational waves: Why GW astronomy needs you!
Light Echoes in Art and Science: How far do the Two Constituencies Reflect Each Other in Theory and Practice?
Around unbalanced optimal transport: fluid dynamic, growth model, applications.
How to deform and shake images?
What can we learn from large deformation diffeomorphic metric mapping on spaces of rigid bodies?
How ought we to structure research so as to make progress in understanding ice interaction?
Antarctic Coastal Polynyas: Do Measurements of Winter Processes give clues to modeling Improvements and better model fidelity?
Discrete images, continuous world: A better basis for discussion?
What Sea Ice Physics is Missing from Models?
What do Climate Models need Sea Ice for?
Swimming of a simple vertebrate: Insights from computational and robotic models.
How much should we believe correlations between Arctic cyclones and sea ice extent?
When is all the sea ice gone?
A compilation of research and thoughts on the future of sea ice models.
Minimization of curvature dependent functional.
Small-amplitude steady water waves on flows with counter-currents.
Some thoughts on the role of the convection terms in the fluid mechanical PDEs.
How do human mathematicians avoid big searches?
Categorical structures for type theory in univalent foundations"
How does breaking detailed balance accelerate convergence to equilibrium?
Proof Assistants: From Symbolic Logic To Real Mathematics?
UniMath - its present and its future.
Transferability: as easy as ABC?
Are geodesic metric spaces determined by their Morse boundaries?
Unitary representations of reflection groups and their deformations.
Finiteness conditions for classifying spaces for the family of virtually cyclic subgroups.
Approximate groups: nilprogressions and the structure theorem.
Benjamini-Schramm convergence of arithmetic orbifolds.
Median spaces and spaces with thin triangles.
Topological finite generation of certain compact open subgroups of tree automorphisms.
Computing Kazhdan constants by computer.
A Banachic generalization of Shalom's property H_FD.
Geometry of finite quotients of groups.
Crossed-products by locally compact groups and intermediate subfactors.
Topics in Heegaard Floer homology IV, cont.
Random walks on random symmetric groups.
Topics in Heegaard Floer homology III, cont.
Topics in Heegaard Floer homology II, cont.
Convex subgroups of orderable groups.
Cubical Accessibility and bounds on curves on surfaces.
How should we assess whether a medical device is safe? What are the possible comparators?
Can we have medical privacy, cloud computing and genomics all at the same time?
Combining statistical disclosure limitation methods to preserve relationships and data-specific constraints in survey data.
Serial Killer Nurses: Is there an Epidemic?
The geometry of optimal experiment design for vector-valued Ornstein-Uhlenbeck processes.
What should a forensic scientist's likelihood ratio be?
How does Government surveillance affect perceived online privacy/security and online information disclosure?
'Data Safe Havens' as a framework to support record linkage in observational studies: evidence from the Project to Enhance ALSPAC through Record Linkage (PEARL).
How should we interpret Y-chromosome evidence?
Bayes & the Blame Game: How to ease the mutually felt frustration between law professionals and scientists.
Law, statistics and psychology, do they match?
Forensic Ecology: How do we get answers to questions? How do we present them to the court?
Forensic trace evidence – what are the questions we need to answer?
Probability and statistics – a criminal lawyer's perspective.
What COSTNET can do for and with you!
Partitioning Well-Clustered Graphs: Spectral Clustering Works!
Can you see the sound of a drum?
Measuring risk and utility in remote analysis and online data centres – why isn't this problem already solved?
Efficiency=Geometry? Decoding the DNA of prediction in gauge, gravity, and effective field theories.
Non-compactness of initial data sets in high dimensions.
Modelling magma migration through the continental lithosphere: the importance of multiple pulses and channelized flow.
Four arguments against the reaction-diffusion master equation (and one in its favour).
Lecture 14: (U. of Cambridge): Anomalous diffusion for a monomer, mean time for a polymer to loop.
Lecture 13: (U. of Cambridge): Stochastic biology: stochastic telomere model and Rouse polymer model.
Resolving genomes from metagenomic strain mixtures with 3C & Hi-C. Is it possible?
Lecture 11: (U. of Cambridge): statistics and analysis of super-resolution Single Particle trajectories.
Closure Scheme for Chemical Master Equations - Is the Gibbs entropy maximum for stochastic reaction networks at steady state?
PyURDME, MOLNs and StochSS — from new algorithms for spatial stochastic simulation to large-scale distributed computational experiments in "the cloud"
Lecture 5: (U. of Cambridge): Activation escape through a potential well.
Thermal convection in an heterogeneous mantle: plumes, piles, domes, and LLSVPs.
Rothschild Lecture: The Hawaiian Plume: What do Surface Observables Tell Us? | CommonCrawl |
Abstract : Simultaneous observations of PSR B0823+26 with ESA's XMM-Newton, the Giant Metrewave Radio Telescope and international stations of the Low Frequency Array revealed synchronous X-ray/radio switching between a radio-bright (B) mode and a radio-quiet (Q) mode. During the B mode we detected PSR B0823+26 in 0.2$-$2 keV X-rays and discovered pulsed emission with a broad sinusoidal pulse, lagging the radio main pulse by 0.208 $\pm$ 0.012 in phase, with high pulsed fraction of 70$-$80%. During the Q mode PSR B0823+26 was not detected in X-rays (2 $\sigma$ upper limit a factor ~9 below the B-mode flux). The total X-ray spectrum, pulse profile and pulsed fraction can globally be reproduced with a magnetized partially ionized hydrogen atmosphere model with three emission components: a primary small hot spot ($T$$\sim$3.6$\times10^6$ K, $R$$\sim$17 m), a larger cooler concentric ring ($T$$\sim$1.1$\times10^6$ K, $R$$\sim$280 m) and an antipodal hot spot ($T$$\sim$1.1$\times10^6 $ K, $R$$\sim$100 m), for the angle between the rotation axis and line of sight direction $\sim66^\circ$. The latter is in conflict with the radio derived value of $(84\pm0.7)^\circ$. The average X-ray flux within hours-long B-mode intervals varied by a factor $\pm$20%, possibly correlated with variations in the frequency and lengths of short radio nulls or short durations of weak emission. The correlated X-ray/radio moding of PSR B0823+26 is compared with the anti-correlated moding of PSR B0943+10, and the lack of X-ray moding of PSR B1822-09. We speculate that the X-ray/radio switches of PSR B0823+26 are due to variations in the rate of accretion of material from the interstellar medium through which it is passing. | CommonCrawl |
cvgmt: A stability result for nonlinear Neumann problems in Reifenberg flat domains in $\mathbb R^N$.
A stability result for nonlinear Neumann problems in Reifenberg flat domains in $\mathbb R^N$.
In this paper we prove that if $\Omega_k$ is a sequence of Reifenberg-flat domains in $\mathbb R^N$ that converges to $\Omega$ for the complementary Hausdorff metric and if in addition the sequence $\Omega_k$ has a ``uniform size of holes'', then the solutions $u_k$ of a Neumann problem of the divergence form converge to the solution $u$ of the same Neumann problem in $\Omega$. The result is obtained by proving the Mosco convergence of some Banach spaces. As an application, in the second part of the paper we prove a decay estimate on the gradient for solutions of nonlinear Neumann problems. The estimate is initially established when the boundary is flat and then a similar estimate for perturbed boundaries using the stability property is obtained. | CommonCrawl |
Abstract: A well known conjecture of Wigner, Dyson, and Mehta asserts that the (appropriately normalized) $k$-point correlation functions of the eigenvalues of random $n \times n$ Wigner matrices in the bulk of the spectrum converge (in various senses) to the $k$-point correlation function of the Dyson sine process in the asymptotic limit $n \to \infty$. There has been much recent progress on this conjecture, in particular it has been established under a wide variety of decay, regularity, and moment hypotheses on the underlying atom distribution of the Wigner ensemble, and using various notions of convergence. Building upon these previous results, we establish new instances of this conjecture with weaker hypotheses on the atom distribution and stronger notions of convergence. In particular, assuming only a finite moment condition on the atom distribution, we can obtain convergence in the vague sense, and assuming an additional regularity condition, we can upgrade this convergence to locally $L^1$ convergence. | CommonCrawl |
Abstract: Medical students and radiology trainees typically view thousands of images in order to "train their eye" to detect the subtle visual patterns necessary for diagnosis. Nevertheless, infrastructural and legal constraints often make it difficult to access and quickly query an abundance of images with a user-specified feature set. In this paper, we use a conditional generative adversarial network (GAN) to synthesize $1024\times1024$ pixel pelvic radiographs that can be queried with conditioning on fracture status. We demonstrate that the conditional GAN learns features that distinguish fractures from non-fractures by training a convolutional neural network exclusively on images sampled from the GAN and achieving an AUC of $>0.95$ on a held-out set of real images. We conduct additional analysis of the images sampled from the GAN and describe ongoing work to validate educational efficacy. | CommonCrawl |
On a rectangular $ 4 \times 2016 $ chessboard, a knight begins in the lower left corner, makes several knight moves, visits all squares of the chessboard at least once and in the end returns to the starting position.
Find the minimal number of squares that the knight must visit more than one time.
the knight must visit two additional squares twice.
The 2016 columns can be divided up into groups of three. The knight travels to to the far end (the red path), visiting half of the squares, then returns along the blue path to a square adjacent to the starting corner. The middle section can be repeated (flipped vertically each time) to extend the width of the board to any multiple of three, including 2016.
there are no circuits that don't repeat any squares on a $4 \times m$ board (Theorom 3.15). Two repeated squares must therefore be the minimum, because to return to the original square, the number of visited squares must be even, due to the way knights always move between the black and white squares on a chessboard, and repeating one square would result in an odd number of moves.
a closed tour that visits exactly two squares more than once on $4 \times m$ boards for all $m > 4$.
Not the answer you're looking for? Browse other questions tagged mathematics combinatorics checkerboard knight-moves or ask your own question. | CommonCrawl |
Bacterial diversity was studied in the rhizosphere of Suaeda japonica Makino, which is native to Suncheon Bay in South Korea. Soil samples from several sites were diluted serially, and pure isolation was performed by subculture using marine agar and tryptic soy agar media. Genomic DNA was extracted from 29 pure, isolated bacterial strains, after which their 16S rDNA sequences were amplified and analyzed. Phylogenetic analysis was performed to confirm their genetic relationship. The 29 bacterial strains were classified into five groups: phylum Firmicutes (44.8%), Gamma proteobacteria group (27.6%), Alpha proteobacteria group (10.3%), phylum Bacteriodetes (10.3%), and phylum Actinobacteria (6.8%). The most widely distributed genera were Bacillus (phylum Firmicutes), and Marinobacterium, Halomonas, and Vibrio (Gamma proteobacteria group). To confirm the bacterial diversity in rhizospheres of S. japonica, the diversity index was used at the genus level. The results show that bacterial diversity differed at each of the sampling sites. These 29 bacterial strains are thought to play a major role in material cycling at Suncheon Bay, in overcoming the sea/mud flat-specific environmental stress. Furthermore, some strains are assumed to be involved in a positive interaction with the halophyte S. japonica, as rhizospheric flora, with induction of growth promotion and plant defense mechanism.
Zhiyong, L., He, L. and Miao, X. 2007. Cultivable bacterial community from south China sea sponge as revealed by DGGE fingerprinting and 16S rDNA phylogenetic analysis. Curr. Microbiol. 55, 654-672.
Sung, H. R. and Ghim, S. Y. 2010. Bacterial diversity and distribution of cultivable bacteria isolated from Dokdo Island. Kor. J. Microbiol. Biotechnol. 38, 263-272.
Tang, Y. W., Von, G. A., Waddington, M. G., Hopkins, M. K., Smith, D. H., Li, H., Kolbert, C. P., Montgomery, S. O. and Persing, D. H. 2000. Identification of coryneform bacterial isolates by ribosomal DNA sequence analysis. J. Clin. Microbiol. 38, 1676-1678.
Whittaker, R. H. 1977. Evolution of species diversity in land communities. Evol. Biol. 10, 1-67.
Yoon, J. H., Kang, S. J., Lee, S. Y., Lee, M. H. and Oh, T. K. 2005. Virgibacillus dokdonensis sp. Nov., isolated from a Korean island, Dokdo, located at the edge of the Ease Sea in Korea. Int. J. Syst. Evol. Microbiol. 51, 1079-1086.
You, Y. H., Yoon, H., Kang, S. M., Shin, J. H., Choo, Y. S., Lee, I. J., Lee, J. M. and Kim, J. G. 2012. Fungal diversity and plant growth promotion of endophytic fungi from six halophytes in Suncheon Bay. J. Microbiol. Biotechnol. 22, 1550-1557.
Margalef, R. 1958. Information theory in ecology. Gen. Syst. 3, 36-71.
Pielou, E. C. 1975. Ecological diversity. John Wiley, New York. p 165.
GlÖckner, F. O., Fuchs, B. M. and Aman, R. 1999. Bacterioplankton compositions of lakesand oceans: a first comparison based on fluorescence- in situ-hybridization. Appl. Environ. Microbiol. 65, 3721-3726.
Gonzalez, J. M. and Moran, M. A. 1997. Numerical dominance of a group of marine bacteria in the $\alpha$-subclass of the class Proteobacteria in coastal seawater. Appl. Environ. Microbiol. 63, 4237-4242.
Ihm, B. S., Leem, J. S., Kim, J. W., Kim, H. S. and Ihm, H. B. 1998. Studies on the vegetation at the wetland of Suncheonman. Bull Inst Litt Envi Mokpo Nat. Univ. 15, 1-8.
Jang, S. K. and Cheong, C. J. 2010. Characteristics of grain size and organic matters in the tidal flat sediments of the Suncheon Bay. J. Kor. Soc. Mar. Environ. Eng. 13, 198-205.
Jeon, S. A., Sung, H. R., Park, Y. M., Park, J. H. and Ghim, S. Y. 2009. Analysis of endospore-forming bacteria or nitrogen-fixing bacteria community isolated from plants rhizo-sphere in Dokdo Island. Kor. J. Microbiol. Biotechnol. 37, 189-196.
Jeong, S. M. and Lee, M. B. 2004. Change of estuary landscape in Suncheon Bay, South Coast of Korea. J. Kor. Geomorphological Association 11, 127-139.
Kim, B. S., Oh, H. M., Kang, H., Park, S. and Chun, J. 2004. Remarkable bacterial diversity in the tidal flat sediment as revealed by 16S rDNA analysis. J. Microbiol. Biotechnol. 14, 205-211.
Chapman, V. J. 1974. Salt marshes and salt deserts of the world in Ecology of halophytes. Academic Press, New York. pp. 3-22.
Amann, R. I., Ludwig, W. and Schleifer, K. H. 1995. Phylogenetic identification and in situ detection of individual microbial cells without cultivation. Microbiol. Rev. 59, 143-169. | CommonCrawl |
Abstract: Let $g \geq 2$. A real number is said to be g-normal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let \phi denote Euler's totient function, let \sigma be the sum-of-divisors function, and let \lambda be Carmichael's lambda-function. We show that if f is any function formed by composing \phi, \sigma, or \lambda, then the number \[ 0. f(1) f(2) f(3) \dots \] obtained by concatenating the base g digits of successive f-values is g-normal. We also prove the same result if the inputs 1, 2, 3, \dots are replaced with the primes 2, 3, 5, \dots. The proof is an adaptation of a method introduced by Copeland and Erdos in 1946 to prove the 10-normality of 0.235711131719\ldots. | CommonCrawl |
You are trying to get highest total score you can reach with arranging the numbers by rotating (no reflection allowed) them without overlapping them on each other.
Since there are 8 lines of the grid are touched with each other, the total score would be $2\times8+3\times8=40$ which is the maximum score you can get with $2$ and $3$.
Note: I am very sorry to let you know there is better answer than 172. That's totally my mistake!
I have 172, with the correct tiles.
Otherwise I think the maximum cannot be found by a greedy approach (trying to maximize the contact point between numbers. | CommonCrawl |
The two-body potential of systems with long-range interactions decays at large distances as $V(r)\sim 1/r^\alpha$, with $\alpha\leq d$, where $d$ is the space dimension. Examples are: gravitational systems, two-dimensional hydrodynamics, two-dimensional elasticity, charged and dipolar systems. Although such systems can be made extensive, they are intrinsically non additive. Moreover, the space of accessible macroscopic thermodynamic parameters might be non convex. The violation of these two basic properties is at the origin of ensemble inequivalence, which implies that specific heat can be negative in the microcanonical ensemble and temperature jumps can appear at microcanonical first order phase transitions. The lack of convexity implies that ergodicity may be generically broken. We present here a comprehensive review of the recent advances on the statistical mechanics and out-of-equilibrium dynamics of systems with long-range interactions. The core of the review consists in the detailed presentation of the concept of ensemble inequivalence, as exemplified by the exact solution, in the microcanonical and canonical ensembles, of mean-field type models. Relaxation towards thermodynamic equilibrium can be extremely slow and quasi-stationary states may be present. The understanding of such unusual relaxation process is obtained by the introduction of an appropriate kinetic theory based on the Vlasov equation. | CommonCrawl |
Number of Boolean functions of n variables.
Is my understanding flawed, or are any of the statements above, in fact, semantically equivalent? Are the boolean operations used really $\lor$ and $\land$ (I feel like excluding $\lnot$ is quite odd)? Moreover, is there a formula for this sequence?
The reader may be interested to know that there is a certain equivalence of boolean functions that may be counted by Power Group Enumeration. This is the case where equivalence includes permutation of the inputs and / or simultaneous complementation of the inputs and possible complementation of the outputs. We do not repeat the details of the algorithm here as it is exactly the same as what was documented at this MSE link I or at this MSE link II. The algorithm can be described very straightforwardly as counting the number of ways we may cover the cycles of permutation $\alpha$ by cycles of a permutation $\beta$ where $\alpha$ is a permutation from the group permuting the slots and $\beta$ from the group permuting the repertoire. Hence we have it solved if we can compute the cycle indices of the two groups.
It remains to determine the length of the cycle these strings are on, which must be $d/2$ for one segment followed by its inverse since with four segments etc. the string would not be aperiodic. We now have all the ingredients to apply the construction from the first class without complementation, except that technically we have two kinds of contributions for $d$ where $\gamma_2(d)$ is not zero and $d$ is even, corresponding to source strings appearing and not appearing among the inverses. The computation remains the same -- compute the LCM of the length of all cycles for $\beta$ being generated by the cycles of $\tau$ and the number of strings having this profile in terms of divisors and inversion to determine the number of cycles of the length that was obtained. The contribution to the number of strings from the two cases is a factor of $\gamma_2(d)$ or $\gamma_1(d)-\gamma_2(d)$ accordingly. This concludes the documentation of the algorithm.
which is OEIS A299104. The Maple code for this computation goes as follows.
The sequence A000370 enumerates Boolean functions up to an equivalence relation. More precisely, a function with any of its inputs or output negated is equivalent to the original function. Also, if the inputs are permuted, it is equivalent as well. So the sequence enumerates the equivalence classes. In the case of $n=2$, for example, there are $4$ equivalence classes. $f(A,B)=T$ is one class with $2$ members, $f(A,B)=A$ is another with $4$ members, $f(A,B)=A\lor B$ with $8$ members, and $f(A,B)=A\oplus B$ with $2$ members. Given this, then $A000157(n) = A000370(n)/2$ as stated in the OEIS entry for A000157.
Not the answer you're looking for? Browse other questions tagged sequences-and-series functions boolean-algebra oeis or ask your own question.
how many semantically different boolean functions are there for n boolean variables?
Help at solving boolean function.
Why are $T_0, T_1, L, M, S$ the main five Boolean classes?
Are true and false each considered self-dual? | CommonCrawl |
My scientific interests are especially the study of complex charged and magnetic soft matter by means of computer simulations, and the development of simple theoretical models to describe them. More precisely I am currently working on the solution properties and association behavior of flexible and semi-flexible polyelectrolytes in various solvents and under various salt concentrations and salt types. In addition I am interested in the effective pair interactions of charged colloidal particles and their phase behavior. This includes simple DNA models, and DNA protein interactions, as well as developing coarse grained models for DNA-Histone complexes. We also investigate in depth polyelectrolyte hydrogels and magnetically interacting ferrogels, as well as pure ferrofluids, where special attention is given to the structure of the solution and the magnetic response functions. Another interest is the applicability of mean-field models for the description of models with long range interactions, and possible improvements beyond the mean-field approach. This include local density functional methods based on the Poisson-Boltzmann functional, as well as strong coupling theories such as Wigner-crystal methods. In addition I am interested in the development of fast methods for the computation of long range interactions. These include pure Coulomb as well as dipolar interactions in various geometries (3D-1D), and under various boundary conditions. And least but not last, we are interested in developing fast methods to deal with fluid-structure couplings using various coupling schemes of particles to a lattice-boltzmann fluid. These can be charged fluids, as well as fluids that undergo reactions at boundaries, such as needed for active Janus-Colloids,or for catalytic particles. We also are interested to apply machine learning algorithms for the development of force-fields with almost DFT precision.
Breitsprecher, Konrad and Szuttor, Kai and Holm, Christian.
"Electrode Models for Ionic Liquid-Based Capacitors".
The Journal of Chemical Physics C 119(22445–22451), 2015.
de Graaf, Joost and Peter, Toni and Fischer, Lukas P. and Holm, Christian.
"The Raspberry model for hydrodynamic interactions revisited. II. The effect of confinement".
The Journal of Chemical Physics 143(8)(084108), 2015.
de Graaf, Joost and Rempfer, Georg and Holm, Christian.
"Diffusiophoretic Self-Propulsion for Partially Catalytic Spherical Colloids".
IEEE Transactions on NanoBioscience 14(3)(272–288), 2015.
Holm, Christian and Gompper, Gerhard and Dill, Ken A..
"Preface: Special Topic on Coarse Graining of Macromolecules, Biopolymers, and Membranes".
The Journal of Chemical Physics 143(24)(242901), 2015.
Aleksandr V. Ryzhkov and Petr V. Melenev and Christian Holm and Yuriy L. Raikher.
"Coarse-grained molecular dynamics simulation of small ferrogel objects".
Journal of Magnetism and Magnetic Materials 383(277–280), 2015.
Fahrenberger, Florian and Xu, Zhenli and Holm, Christian.
"Simulation of electric double layers around charged colloids in aqueous solution of variable permittivity".
The Journal of Chemical Physics 141(6)(064902), 2014.
Marcia C. Barbosa and Markus Deserno and Christian Holm and René Messina.
"Screening of spherical colloids beyond mean field: A local density functional approach".
Physical Review E 69(051401), 2004.
"Efficient methods for long range interactions in periodic geometries plus one application".
Christian Holm and Matthias Rehahn and Wilhelm Oppermann and Matthias Ballauff.
Holm, Christian and Joanny, Jean-Fran\ccois and Kremer, Kurt and Netz, Roland R. and Reineker, Peter and Seidel, Christian and Vilgis, Thomas A. and Winkler, Roland G..
Springer Berlin Heidelberg, New York, 2004.
J. P. Huang and Christian Holm.
"Magnetization of polydisperse colloidal ferrofluids: Effect of magnetostriction".
Physical Review E 70(061404), 2004.
Ivanov, Alexey O. and Wang, Zuowei and Holm, Christian.
"Applying the Chain Formation Model to magnetic properties of aggregated ferrofluids".
Physical Review E 69(031206), 2004.
Hans Jörg Limbach and Mehmet Sayar and Christian Holm.
Journal of Physics: Condensed Matter 16(22)(2135–2144), 2004.
Hans Jörg Limbach and Christian Holm.
"Conformations and Solution Structure of Polyelectrolytes in Poor Solvent".
Polyelectrolytes , volume 9 of Dresdener Polymer Discussions 2003Editors: U. Scheler, , Dresden, Germany, 2004.
Vladimir Lobaskin and Burkhard Dünweg and Christian Holm.
"Electrophoretic mobility of a charged colloidal particle: A computer simulation study".
Journal of Physics: Condensed Matter 16(38)(S4063–S4073), 2004.
Bernward A. Mann and Ralf Everaers and Christian Holm and Kurt Kremer.
"Effect of image forces on polyelectrolyte adsorption at a charged surface".
Physical Review E 70(5)(051802), 2004.
Ali Naji and Axel Arnold and Christian Holm and Roland R. Netz.
"Attraction and unbinding of like–charged rods".
Holm, Christian and Limbach, Hans Jörg and Kremer, Kurt.
Journal of Physics: Condensed Matter 15(1)(S205–S211), 2003.
Jiménez-Ángeles, Felipe and Messina, René and Holm, Christian and Lozada-Cassou, Marcelo.
"Ion pairing in model electrolytes: A study via three-particle correlation functions".
The Journal of Chemical Physics 119(9)(4842-4856), 2003.
"Single-Chain Properties of Polyelectrolytes in Poor Solvent".
Journal of Physical Chemistry B 107(32)(8041–8055), 2003.
René Messina and Christian Holm and Kurt Kremer.
"Polyelectrolyte Multilayering in Spherical Geometry".
Zuowei Wang and Christian Holm and Hanns Walter Müller.
"Boundary condition effects in the simulation study of equilibrium properties of magnetic dipolar fluids".
The Journal of Chemical Physics 119(379), 2003.
"Computer simulation study of equilibrium properties of magnetic dipolar fluids".
Zuowei Wang and Christian Holm.
"Structure and magnetization properties of polydispersed ferrofluids: A molecular dynamics study".
Physical Review E 68(041401), 2003.
"MMM2D: A fast and accurate summation method for electrostatic interactions in 2D slab geometries".
Computer Physics Communications 148(3)(327–348), 2002.
"A novel method for calculating electrostatic interactions in 2D periodic slab geometries".
Chemical Physics Letters 354(324–330), 2002.
Arnold, Axel and de Joannis, Jason and Holm, Christian.
"Electrostatics in Periodic Slab Geometries I".
The Journal of Chemical Physics 117(2496–2502), 2002.
"Electrostatics in Periodic Slab Geometries II".
The Journal of Chemical Physics 117(2503–2512), 2002.
Jason de Joannis and Axel Arnold and Christian Holm.
Journal of Chemical Physics 117(2503–2512), 2002.
"Theory and simulations of rigid polyelectrolytes".
Holm, Christian and Kremer, Kurt and Deserno, Markus and Limbach, Hans Jörg.
"Computer Modeling of Charged Polymers".
John von Neumann Institute for Computing, Jülich, Germany, 2002.
"Conformational properties of poor solvent polyelectrolytes".
Computer Physics Commications 147(321–324), 2002.
Hans Jörg Limbach and Christian Holm and Kurt Kremer.
"Structure of polyelectrolytes in poor solvent".
"Conformation of a Polyelectrolyte Complexed to a Like-Charged Colloid".
Physical Review E 65(041805), 2002.
Journal of Chemical Physics 117(2947), 2002.
Messina, René and González Tovar, Enrique and Lozada-Cassou, Marcelo and Holm, Christian.
"Overcharging: The Crucial Role of Excluded Volume".
"Charge inversion in colloidal systems".
Computer Physics Communications 147(282–285), 2002.
Wang, Zuowei and Holm, Christian and Müller, Hanns Walter.
"Molecular dynamics study on the equilibrium magnetization properties and structure of ferrofluids".
Physical Review E 66(021405), 2002.
Markus Deserno and Christian Holm and Kurt Kremer.
"Molecular dynamics simulations of the cylindrical cell model".
Marcel Decker, New York, 2001.
Markus Deserno and Christian Holm and Jürgen Blaul and Matthias Ballauff and Matthias Rehahn.
"The Osmotic Coefficient of Rod-like Polyelectrolytes: Computer Simulation, Analytical Theory, and Experiment".
European Physical Journal E: Soft Matter 5(97–103), 2001.
Markus Deserno and Felipe Jiménez-Ángeles and Christian Holm and Marcelo Lozada-Cassou.
"Overcharging of DNA in the presence of salt: Theory and Simulation".
The Journal of Physical Chemistry B 105(44)(10983–10991), 2001.
"Cell-model and Poisson-Boltzmann-theory: A brief introduction".
Kluwer Academic Publishers, Dordrecht, Nl, 2001.
Christian Holm and Kurt Kremer.
"Computer Simulations of Charged Systems".
"What can Ising spins teach us about Quantum Gravity?".
"End-effects of strongly charged polyelectrolytes - a molecular dynamics study".
Journal of Chemical Physics 114(21)(9674–9682), 2001.
"Effect of colloidal charge discretization in the primitive model".
The European Physical Journal E 4(363–370), 2001.
"Strong electrostatic interactions in spherical collidal systems".
Physical Review E 64(021405), 2001.
Wang, Zuowei and Holm, Christian.
"Estimate of the Cutoff Errors in the Ewald Summation for Dipolar Systems".
The Journal of Chemical Physics 115(6277–6798), 2001.
Marcia C. Barbosa and Markus Deserno and Christian Holm.
"A stable local density functional approach to ion-ion correlations".
Markus Deserno and Christian Holm and Sylvio May.
"Fraction of Condensed Counterions around a Charged Rod: Comparison of Poisson-Boltzmann Theory and Computer Simulations".
"Strong attraction between charged spheres due to metastable ionized states".
Physical Review Letters 85(872–875), 2000.
"Ground state of two unlike charged colloids: An anology with ionic bonding".
"Polyelectrolytes in Solution - Recent Computer Simulations".
Proceedings of Yamada Conference ``Polyelectrolytes'', Inuyama, Japan , pages 27–36, Editors: I. Noda and E. Kokufuta, , Osaka, Japan, 1999.
Christian Holm and Wolfhard Janke.
"Simplicial Quantum Gravity on a Randomly Triangulated Sphere".
International Journal of Modern Physics A 14(24)(3885–3903), 1999.
Elmar Bittner and A. Hauke and Harald Markum and J. Riedler and Christian Holm and Wolfhard Janke.
"Lattice Models of 2D-Quantum Gravity".
Proceedings of the Eigth Marcel Grossmann Meeting , pages 769, Editors: Tsvi Piran and Remo Ruffini, , The Hebrew University, Jerusalem, Israel, June 22 - 27, 1997, 1999.
Elmar Bittner and A. Hauke and Christian Holm and Wolfhard Janke and Harald Markum and J. Riedler.
"$Z_2$-Regge versus Standard Regge Calculus in Two Dimensions".
Physical Review D 59(124018), 1999.
Uwe Micka and Christian Holm and Kurt Kremer.
"Strongly Charged, Flexible Polyelectrolytes in Poor Solvents - A Molecular Dynamics Study".
"Anwendung von FFT Ewald Gitter-Methoden in Simulationen von Polyelektrolyten".
Research Center Jülich, Jülich, 1998.
Holm, Christian and Kremer, Kurt and Vilgis, Thomas A..
"Polyelektrolyte: Grundlegende Probleme bei der Beschreibung weitverbreiteter Substanzen".
"Standard and $Z_2$-Regge Theory in two Dimensions".
Nuclear Physics B: Proceedings Supplements 63(1)(769-771), 1998.
"Lattice Models of Quantum Gravity".
"How to mesh up Ewald sums. II. An accurate error estimate for the Particle-Particle-Particle-Mesh algorithm".
Journal of Chemical Physics 109(7694), 1998.
"Critical Exponents of the Classical Heisenberg Ferromagnet".
Physical Review Letters 78(2265), 1997.
"Fixed Versus Random Triangulations in 2D Regge Calculus".
Physics Letters B 390(1)(59–63), 1997.
"Influence of the Path Integral Measure in Quantum Gravity".
Christian Holm and Wolfhard Janke and Tetsuo Matsui and Kazuhiko Sakakibara.
"Monte Carlo Study of Asymmetric 2D XY Model".
Physica A: Statistical Mechanics and its Applications 246(3)(633–645), 1997.
"Measuring the String Susceptibility in 2D Simplicial Quantum Gravity Using the Regge Approach".
Nuclear Physics B 477(465–488), 1996.
"Ising Spins on a Gravitating Sphere".
Physics Letters B 375(1)(69–74), 1996.
"2D Non-Perturbative Euclidean Quantum Gravity via Regge Calculus".
"Measure Dependence of 2D Simplicial Quantum Gravity".
Nuclear Physics B: Proceedings Supplements 42(1)(722–724), 1995.
"The Ising Transition in 2D Simplicial Quantum Gravity – Can Regge Calculus be Right?".
Nuclear Physics B: Proceedings Supplements 42(1)(725–727), 1995.
"Monte Carlo study of topological defects in the 3D Heisenberg model".
Journal of Physics A: Mathematical and General 27(7)(2553–2563), 1994.
"The critical behaviour of Ising spins on 2D Regge lattices".
Physics Letters B 335(2)(143–150), 1994.
"High Pecision Monte Carlo Determination of $\alpha/\nu$ in the 3D Classical Heisenberg Model".
International Journal of Modern Physics C 5(2)(267–270), 1994.
"High Precision Single-Cluster MC Measurement of the Critical Exponents of the Classical 3D Heisenberg Model".
Nuclear Physics B: Proceedings Supplements 30(846–849), 1993.
"Finite-size scaling study of the three-dimensional classical Heisenberg model".
Physics Letters A 173(1)(8–12), 1993.
"Critical exponents of the classical three-dimensional Heisenberg model: A single-cluster Monte Carlo study".
Physical Review B 48(2)(936–950), 1993.
Joan Adler and Christian Holm and Wolfhard Janke.
"High-Temperature Series Analysis of the Classical Heisenberg and XY Model".
Holm, Christian and Hennig, Jörg D..
"A Poincaré Gauge Theory on Regge Simplexes".
Classical and Quantum Systems- Foundations and Symmetries, Proceedings of the II. International Wigner Symposium , pages 723–726, Editors: Doebner, H. D. and Scherer, W. and Schroeck, Jr., F., , 1991.
"Differential Geometry, Gauge Theories, and Gravity".
International Journal of Theoretical Physics 29(1)(23–36), 1990.
Finkelstein, David and Holm, Christian.
"Neutrino-Neutrino Scattering in Quantum Network Dynamics".
Bulletin of the American Physical Society 34(114), 1989.
"The Hyperspin Structure of Unitary Groups".
Journal of Mathematical Physics 29(4)(978–986), 1988.
"Neutrino Spectrum of Einstein Universes".
Journal of Mathematical Physics 29(10)(2273–2279), 1988.
"The Hyperspin Structure of Einstein Universes and Their Neutrino Spectrum".
PhD thesis, Georgia Institute of Technology, 1987.
Finkelstein, David and Finkelstein, Shlomit Ritz and Holm, Christian.
Physical Review Letters 59(1265–1266), 1987.
"Christoffel Formula and Geodesic Motion in Hyperspin Manifolds".
International Journal of Theoretical Physics 25(11)(1209–1213), 1986.
John Eidson and Sarah Flynn and Christian Holm and D. Weeks and Ronald F. Fox.
"Elementary Explanation of Boundary Shading in Chaotic-Attractor Plots for the Feigenbaum Map and the Circle Map".
Physical Review A 33(2809–2812), 1986.
International Journal of Theoretical Physics 25(4)(441–463), 1986.
Master's thesis, Georgia Institute of Technology, feb, 1985.
This page was last modified on 16 April 2019, at 14:55. | CommonCrawl |
Who proved the modern form of the fundamental theorem of Galois theory?. Was it in the original Galois' manuscript?
Let the equation be given whose $m$ roots are $a,b,c,\ldots$. There will always be a group of permuations of the letters $a,b,c,\ldots$ which will have the following property: 1) that each function invariant under the substitutions of this group will be known rationally; 2) conversely, that every function of these roots which can be determined rationally will be invariant under these substitutions.
If an element of the splitting field of $K(a,b,c,\ldots)$ is left fixed by all the automorphisms of the Galois group then it is in $K$.
The fundamental theorem of Galois theory (i.e. the Galois correspondence) follows easily, though Edwards doesn't say who first stated it.
would be useful. This book about Galois theory is made from the historical point of view. In particular, it contains an English translation of Galois' memoir.
The modern form can be found in Galois Theory, by Emil Artin and Arthur N. Milgram page 46, published in 1944. I'm not expert in math history, but once, I heard Artin was the first person to wrote the modern account of Galois theory.
I am not particularly interested in mathematical history, but Peter Newmann is: http://www.ems-ph.org/books/book.php?proj_nr=137.
Not the answer you're looking for? Browse other questions tagged ho.history-overview or ask your own question.
What was Galois theory like before Emil Artin?
Did any new mathematics arise from Ruffini's work on the quintic equation?
Who first proved the fundamental theorem of projective geometry? | CommonCrawl |
Is it possible to shuffle a 3x3 Rubik's cube so that there's no more than 2 pieces of the same color in every face?
Can I take a standard $3 \times 3$ Rubik's Cube and shuffle it so that, for every face, there are no more than $2$ pieces with the same color?
Please answer if you have managed (or failed) to solve the question using an actual cube. No guessing here, thanks.
To find the solution just click "play".
Does anyone knows a shorter way?
How many different combinations of this setup exists?
I believe this works. Rotate diagonally opposite pairs of corners so that the front faces move to the sides. Rotate the vertical center slice by a half-turn. Interchange the four edge cubies around the equator.
Not the answer you're looking for? Browse other questions tagged combinatorics group-theory recreational-mathematics rubiks-cube or ask your own question.
Qubeks logic game - is it solvable in random initial state?
Say you had a Rubik's cube in a certain starting state and chose moves randomly and uniformly. How long on average would it take to solve it?
Diminishing upper limit on Rubik's Cube solutions - why so long?
A toroidal version of Rubik's cube: how many shuffles?
How can I swap opposite corners on the same face of a 2x2 Rubik's cube?
How many solvable configurations of the Rubik's cube have no two squares of the same color touching? | CommonCrawl |
This is the second half of 18.02A and can be taken only by students who took the first half in the fall term; it covers the remaining material in 18.02.
Abstract: We will discuss how the evolution of a random walker on the square grid leads to a second order partial differential equation known as the telegraph equation.
Abstract: We introduce the basics of quantum mechanics, show how it produces phenomena and probability distributions which cannot be simulated classically, and give some examples of interesting quantum protocols, including the Elitzur–Vaidman bomb tester.
Abstract: In this lecture, we will develop some calculus notions on a fractal set. More specifically, we will consider the Sierpinski gasket, and built on it analogs of the trigonometric and polynomial functions. What is remarkable, is that the tools we develop will come from the fractal structure of the Sierpinski gasket.
Abstract: We show how the layered neural net architecture of deep learning produces continuous piecewise linear functions as approximations to the unknown map from input to output. A combinatorial formula counts the number of linear pieces in a typical learning function.
Abstract: Surface tension is a property of fluid interfaces that leads to myriad subtle and striking effects in nature and technology. We describe a number of surface-tension-dominated systems and how to rationalize their behavior via mathematical modeling. Particular attention is given to the influence of surface tension on biological systems.
Abstract: Optimal transport is a mathematical tool that links probability to geometry. In this talk, we will show how transport can be brought from theory to practice, with applications in machine learning and computer graphics.
Abstract: I will discuss several games whose analysis involves interesting mathematics. First, in the mathematical version of tug of war, play begins at a game position $x_0$. At each turn a coin is tossed, and the winner gets to move the game position to any point within $\epsilon$ units of the current point. (One can imagine the two players are holding a rope, and the ``winner'' of the coin toss is the one who gets a foothold and then has the chance to pull one step in any desired direction.) Play ends when the game position reaches a boundary set, and player two pays player one the value of a "payoff function" defined on the boundary set.
So... what is the optimal strategy? How much does player one expect to win (in the $\epsilon \to 0$ limit) when both players play optimally? We will answer this question and also explain how this game is related to the "infinity Laplacian," to "optimal Lipschitz extension theory" and to a random turn version of a game called Hex.
Abstract: Biology and pharmacology are many times thought of as non-mathematical disciplines. In modern research practice that is hardly the case. In this talk we will discuss the applications of mathematics to the areas of systems biology and pharmacokinetic/pharmacodynamic modeling. Examples such as spatial pattern formulation (zebra stripes) via Turing instability of partial differential equations, controlling intrinsic biological randomness through stochastic differential equations, and individualized drug dosing and choice optimizations through nonlinear mixed effects models will be introduced. The student will leave with a new lens on how the mathematics they explored in the IAP can be applied to new disciplines which themselves uncover new mathematical problems.
Category theory is a relatively new branch of mathematics that has transformed much of pure math research. The technical advance is that category theory provides a framework in which to organize formal systems and by which to translate between them, allowing one to transfer knowledge from one field to another. But this same organizational framework also has many compelling examples outside of pure math.
In this course we provide an introductory tour of category theory, with a viewpoint toward modelling real-world phenomena. The course will begin with the notion of poset, and introduce central categorical ideas such as functor, natural transformation, (co)limit, adjunction, the adjoint functor theorem, and the Yoneda lemma in that context. We'll then move to enriched categories, profunctors, monoidal categories, operads, and toposes. Applications to resource theory, databases, codesign, signal flow graphs, and dynamical systems will help ground these notions, providing motivation and a touchstone for intuition. The aim of the course is to provide an overview of the breadth of research in applied category, so as to invite further study. | CommonCrawl |
Economies and societal structures in general are complex stochastic systems which may not lend themselves well to algebraic analysis. An addition of subjective value criteria to the mechanics of interacting agents will further complicate analysis. The purpose of this short study is to demonstrate capabilities of agent-based computational economics to be a platform for fairness or equity analysis in both a broad and practical sense.
In this paper, we study various new Hawkes processes. Specifically, we construct general compound Hawkes processes and investigate their properties in limit order books. With regards to these general compound Hawkes processes, we prove a Law of Large Numbers (LLN) and a Functional Central Limit Theorems (FCLT) for several specific variations. We apply several of these FCLTs to limit order books to study the link between price volatility and order flow, where the volatility in mid-price changes is expressed in terms of parameters describing the arrival rates and mid-price process.
This article aims to present an elementary analytical solution to the question of the formation of a structure of differentiation of rates of return in a classical gravitation model and in a model of the dynamics of price-wage spirals.
We introduce an affine extension of the Heston model where the instantaneous variance process contains a jump part driven by $\alpha$-stable processes with $\alpha\in(1,2]$. In this framework, we examine the implied volatility and its asymptotic behaviors for both asset and variance options. Furthermore, we examine the jump clustering phenomenon observed on the variance market and provide a jump cluster decomposition which allows to analyse the cluster processes. | CommonCrawl |
Finally, the MATLAB implementation of CEDD is available on-line.
The source code is quite simple and easy to be handled by all users. There is a main function that has the task of extracting the CEDD descriptor from a given image.
The descriptors, which include more than one features in a compact histogram, can be regarded that they belong to the family of Compact Composite Descriptors. A typical example of CCD is the CEDD descriptor. The structure of CEDD consists of 6 texture areas. In particular, each texture area is separated into 24 sub regions, with each sub region describing a color. CEDD's color information results from 2 fuzzy systems that map the colors of the image in a 24-color custom palette. To extract texture information, CEDD uses a fuzzy version of the five digital filters proposed by the MPEG-7 EHD. The CEDD extraction procedure is outlined as follows: when an image block (rectangular part of the image) interacts with the system that extracts a CCD, this section of the image simultaneously goes across 2 units. The first unit, the color unit, classifies the image block into one of the 24 shades used by the system. Let the classification be in the color $m, m \in [0,23]$. The second unit, the texture unit, classifies this section of the image in the texture area $a, a \in [0,5]$. The image block is classified in the bin $a \times 24 + m$. The process is repeated for all the image blocks of the image. On the completion of the process, the histogram is normalized within the interval [0,1] and quantized for binary representation in a three bits per bin quantization.
The most important attribute of CEDDs is the achievement of very good results that they bring up in various known benchmarking image databases. The following table shows the ANMRR results in 3 image databases. The ANMRR ranges from '0′ to '1′, and the smaller the value of this measure is, the better the matching quality of the query. ANMRR is the evaluation criterion used in all of the MPEG-7 color core experiments. | CommonCrawl |
Abstract: The current status of optical potentials employed in the prediction of thermonuclear reaction rates for astrophysics in the Hauser-Feshbach formalism is discussed. Special emphasis is put on $\alpha$+nucleus potentials. A novel approach for the prediction of $\alpha$+nucleus potentials is proposed. Further experimental efforts are motivated. | CommonCrawl |
Are the range proofs and pedersen commitments part of a transaction? Or are they not kept in a transaction at all?
where a is a random scalar.
Borromean ring signature is suitable here for reducing the signature size.
MRL's paper is confusing, it's better to read the source code.
There is a new output we want to make a "range proof".
$10$ is the amount, $a$ is the secret key. G and H are different base point.
because $2+8=10$. $a_i$ is random.
For the first line, we get $(C_0,C_0 - 1 \times 1H)$ these two points.
The first point's secret key is $a_0$.
We can't compute the second point's secret key.
The difference between the tow points is $1H$.
We sign a ring signature on these two points, a ring contains only two points.
$H()$ is a hash function to covert a point to scalar.
$s_1$ is random, $P_1$ is the second point.
The second line is similar but we should change the order of $(P_0,P_1)$ because we only know the second point's secret key.
At last we make four range proof.
In practice, the code is a little different for security.
Range proofs and and commitments are both kept in the transaction. Version 2 transactions (the ringct ones) now calculate transaction id a bit differently from v1 transactions, by hashing a set of hashes of several parts of the transaction, to allow future pruning of the range proofs. In that hypothetical future, range proofs will thus not be part of the (pruned) transaction anymore. Commitments will still be part of the transaction in that case.
Not the answer you're looking for? Browse other questions tagged ringct cryptography transaction-data range-proofs or ask your own question.
How will the range proof size reduction be accomplished? | CommonCrawl |
Department of Mathematics, University of Qom, Qom 3716146611, Iran.
In this paper, we study $L^p$-conjecture on locally compact hypergroups and by some technical proofs we give some sufficient and necessary conditions for a weighted Lebesgue space $L^p(K,w)$ to be a convolution Banach algebra, where $1<p<\infty$, $K$ is a locally compact hypergroup and $w$ is a weight function on $K$. Among the other things, we also show that if $K$ is a locally compact hypergroup and $p$ is greater than 2, $K$ is compact if and only if $m(K)$ is finite and $f\ast g$ exists for all $f,g\in L^p(K)$, where $m$ is a left Haar measure for $K$, and in particular, if $K$ is discrete, $K$ is finite if and only if the convolution of any two elements of $L^p(K)$ exists.
F. Abtahi, R. Nasr-Isfahani, and A. Rejali, On the $L^p$-conjecture for locally compact groups, Arch. Math., 89 (2007), pp. 237-242.
F. Abtahi, R. Nasr-Isfahani, and A. Rejali, Weighted $L^P$-conjecture for locally compact groups, Periodica Math. Hun., 60 (2010), pp. 1-11.
W.R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, De Gruyter, Berlin, 1995.
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If the lottery balls are being chosen at random, then the distribution of the number of times each ball comes up should follow the theoretical shape shown in white. Use the playback controls below the animation to restart, pause, or fast forward the draws.
Of course the actual distribution is more jagged, but the theoretical distribution allows us to see whether the 'leading' number is surprisingly far in front. Below we see the final observed distribution with an approximate theoretical distribution superimposed. The fit looks good, suggesting, as we would expect, that there is no systematic preference for particular numbers.
In Is the Lottery biased? we consider some of the mathematics behind the theoretical distribution of counts, and how we can check if the observed distribution is in conflict with the theoretical one.
Are the gaps what we would expect?
If you run the animation below, then if the lottery balls are being chosen at random, the distribution of the gaps should follow the theoretical shape in white when you click on 'Show histogram' and then 'Show theoretical'. This theoretical distribution is known as a Geometric distribution and is derived in Is the Lottery biased?.
After 1240 lottery draws, with 6 main balls being drawn each time, $6\times 1240 = 7440$ numbers have been drawn, and so there are 7440 gaps between two draws of the same number (the gaps until the first time each number is drawn are included in this total). The histogram below shows the distribution of all these 7440 gaps, with the theoretical geometric distribution superimposed. The gaps are divided into those below and above 40, so that the large gaps are clearly displayed: the theoretical distrbution seems to fit the observed distribution well, although there are inevitably some jagged bits in the tail.
The longest gap observed is 72, for number 17 , which appeared on draw 435 on 23rd February 2000, but did not appear again until draw 508 on 4th November 2000. How surprising is it to get a gap as large as this? After a specific occurrence of a particular number, this is extremely surprising, and there is only 8/100000 chance of such an extreme result. However, when we take into account that there were 7440 gaps observed and this was the largest one, it turns out that it is not surprising at all. In fact 72 is almost exactly the average maximum gap one would expect in a series of 1240 lottery draws!
Alternatively we can use the power of the computer to simulate 'fictional' lotteries, by picking 6 different numbers at random from 1 to 49, and then repeating this process as long as we want. The software contains 'random number generators' that should ensure that each number really does have an equal chance of being chosen. We simulated 1000 full lottery histories and found the longest gap in each history. These 1000 longest gaps had the distribution shown below: 420 out of 1000 were 72 or more.
As another example of using simulations, looking backwards from 20th October 2007, we saw that ball 14 was not drawn until the 53rd draw. The graph below shows the results of simulating 1000 lotteries until all the numbers had come up. In 60 of these simulations we had to wait until at least 53 draws before all the numbers had come up, showing the time we had to wait for ball 14 was not really very surprising.
In Is the Lottery biased? we consider the mathematics behind the theoretical distribution of gaps. | CommonCrawl |
Abstract: Various QCD correlators are calculated in the instanton liquid model in zeromode approximation and $1/N_c$ expansion. Previous works are extended by including dynamical quark loops. In contrast to the original "perturbative" $1/N_c$ expansion not all quark loops are suppressed. In the flavor singlet meson correlators a chain of quark bubbles survives the $N_c\to\infty$ limit causing a massive $\eta^\prime$ in the pseudoscalar correlator while keeping massless pions in the triplet correlator. The correlators are plotted and meson masses and couplings are obtained from a spectral fit. They are compared to the values obtained from numerical studies of the instanton liquid and to experimental results. | CommonCrawl |
Lemma 23.8.6. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $\mathfrak p \subset A$ be a prime ideal. If $A$ is a complete intersection, then $A_\mathfrak p$ is a complete intersection too.
Proof. Choose a prime $\mathfrak q$ of $A^\wedge $ lying over $\mathfrak p$ (this is possible as $A \to A^\wedge $ is faithfully flat by Algebra, Lemma 10.96.3). Then $A_\mathfrak p \to (A^\wedge )_\mathfrak q$ is a flat local ring homomorphism. Thus by Proposition 23.8.4 we see that $A_\mathfrak p$ is a complete intersection if and only if $(A^\wedge )_\mathfrak q$ is a complete intersection. Thus it suffices to prove the lemma in case $A$ is complete (this is the key step of the proof).
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09Q4. Beware of the difference between the letter 'O' and the digit '0'.
The tag you filled in for the captcha is wrong. You need to write 09Q4, in case you are confused. | CommonCrawl |
Here, $D$ is the number of dimensions. In our current case, $D=2$.
Note this looks a bit confusing with the use of two $\Sigma$ symbols. The one far to the right is stil used for summation, while the one on the left represents our covariance matrix.
Although we worked out the math for the case $D=2$, this generalizes to higher dimensions as well. Well, there you have it, the Gaussian in higher dimensions. And remember, this form assumes the $x_i$ random variables involved are all independent. | CommonCrawl |
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[N15] Nikulin, V. V., Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups, Izv. Ross. Akad. Nauk Ser. Mat., 79 (4) (2015), 103–158.
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Suppose you have a huge number of robots/vehicles and you want all of them to track some global value, maybe the average of the weight of the fuel that each contains.
One way to do this is to have a master server that takes in everyone's input and generates the output. So others can get it from the master. But this approach results in a single point of failure and a huge traffic to one server.
The other way is to let all robots talk to each other, so each robot will have information from others, which can then be used to compute the sum. Obviously this will incur a huge communication overhead. Especially if we need to generate the value frequently.
If we can tolerate approximate results, we have a third approach: consensus filters.
Here $x_i$ is the value obtained by robot $i$, $x_i(t)$ refers to the value of $x_i$ at time $t$ and $\dot x_i(t)$ is the change of value of $x_i$ at time $t$. Then $c_i$ is some initial value. We can think of the update as $x_i(t + 1) = x_i(t) + \sigma\dot x_i(t)$, where $\sigma$ is some small value.
And guess what, it can be proven that with these update rules, every $x_i$ converge to the average of the original $c_i$s, i.e.
The proof involve some kind of matrix/eigenvalue and other math blah blah so I will omit it here. You can refer to the reference below for the paper to it.
The update rule given above is one example of consensus filters. It's a filter that lets each robot compute the average of every other robots' initial values. This average value is unique so every robot will arrived at a 'consensus', asymptotically.
Notice that every robot only relies on the $x$ values of its neighbors, yet their initial values 'propagate' to the entire network, assuming that the network is connected.
The communication overhead is very low. You don't even need to set up routing tables etc. because agents are only doing single-hop communication.
In some case, the immediate, approximate value can be used, even when the network has not arrived at a consensus. i.e. if we do not require the consensus value to be very accurate, or we are OK with agents having values more related to nearby agents than all agents, we can let each agent use the approximate one very quickly.
The example given above is a static filter - static because it tracks only the initial value $c_i$. What if as the time goes, the value also changes? For example, the fuel each robot has may reduce over time, so the average value of such is also reducing.
That's where dynamic filters come in. Dynamic filters are similar to static ones, but they can track changing values - or signals.
The only change here is then when computing $\dot x$, we add $\dot u$, which is the change in the local value (in the example given, it will the change in weight of the fuel).
This filter also allows convergence to a consensus - even when the tracked value is changing! Of course, the change of the value cannnot be too drastic, otherwise there's no opportunity for it to 'propagate' to the network. But if the rate of change is more or less constant, the output is pretty accurate.
It's called 'highpass' because if you do some math to derive the Laplace transform, it apparently propagate high frequency signals, which most of the time are noises. There are other more complicated, but stable (less susceptible to drastic changes) filters, but they incur more math :P, and also other kinds of constraints.
Obviously tracking fuel isn't that fun, but you can apply consensus filters to other decentralized algorithms like Kalman filters and Sparse Gaussian Process to allow a lower communication overhead. Also, as mentioned above, these algorithms are OK with approximate average, so consensus filters work great. You can get more information in the references below.
That being said, this technique is still a hot research area, so it's not as mature yet.
Akyildiz, Ian F, Weilian Su, Yogesh Sankarasubramaniam, and Erdal Cayirci. 2002. "A Survey on Sensor Networks." Communications Magazine, IEEE 40 (8): 102–114.
Olfati-Saber, R., J.A. Fax, and R.M. Murray. 2007. "Consensus and Cooperation in Networked Multi-Agent Systems." Proceedings of the IEEE 95 (1): 215–233. doi:10.1109/JPROC.2006.887293.
Olfati-Saber, Reza. 2005. "Distributed Kalman Filter with Embedded Consensus Filters." In Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC '05. 44th IEEE Conference on, 8179–8184. doi:10.1109/CDC.2005.1583486.
Freeman, R.A., Peng Yang, and K.M. Lynch. 2006. "Stability and Convergence Properties of Dynamic Average Consensus Estimators." In Decision and Control, 2006 45th IEEE Conference on, 338–343. doi:10.1109/CDC.2006.377078. | CommonCrawl |
Proceedings of Machine Learning Research, PMLR 89:467-475, 2019.
We derive an algorithm that achieves the optimal (up to constants) pseudo-regret in both adversarial and stochastic multi-armed bandits without prior knowledge of the regime and time horizon. The algorithm is based on online mirror descent with Tsallis entropy regularizer. We provide a complete characterization of such algorithms and show that Tsallis entropy with power $\alpha = 1/2$ achieves the goal. In addition, the proposed algorithm enjoys improved regret guarantees in two intermediate regimes: the moderately contaminated stochastic regime defined by Seldin and Slivkins and the stochastically constrained adversary studied by Wei and Luo . The algorithm also obtains adversarial and stochastic optimality in the utility-based dueling bandit setting. We provide empirical evaluation of the algorithm demonstrating that it outperforms Ucb1 and Exp3 in stochastic environments. In certain adversarial regimes the algorithm significantly outperforms Ucb1 and Thompson Sampling, which exhibit close to linear regret.
%X We derive an algorithm that achieves the optimal (up to constants) pseudo-regret in both adversarial and stochastic multi-armed bandits without prior knowledge of the regime and time horizon. The algorithm is based on online mirror descent with Tsallis entropy regularizer. We provide a complete characterization of such algorithms and show that Tsallis entropy with power $\alpha = 1/2$ achieves the goal. In addition, the proposed algorithm enjoys improved regret guarantees in two intermediate regimes: the moderately contaminated stochastic regime defined by Seldin and Slivkins and the stochastically constrained adversary studied by Wei and Luo . The algorithm also obtains adversarial and stochastic optimality in the utility-based dueling bandit setting. We provide empirical evaluation of the algorithm demonstrating that it outperforms Ucb1 and Exp3 in stochastic environments. In certain adversarial regimes the algorithm significantly outperforms Ucb1 and Thompson Sampling, which exhibit close to linear regret. | CommonCrawl |
where the data term is reduced to known data from for the lossy case, and the first order differences are relaxed to a quadratic term for small distances. The weighghts can be also given as vectors for the dimensions with different weights, i.e. $\alpha_1$ as a weight for the differences in -direction Real values are interpreted as constant vectors.
When given as matrices, the upper triangular part of weights the diagonal differences in both positive directions (to direction, while the lower triangular denotes weights for .
The parameters lambda, lambdaIterate, and stoppingCriterion are explained in the optional parameters section of the CyclicProximalPoint this function internally uses.
The FixedMask sets pixel in to the value initially provided by and leaves them unchanged. the UnknownMask specifies unknown pixel, which are intialized and regularized during the algorithm.
% M : a manifold.
% either a number or one for each dimension.
% they are initialized in the cycles, when possible. | CommonCrawl |
This paper propose a novel way of tuning the LDA's hyper parameters $\alpha$, $\beta$ and topic_size $k$. The method used here is Differential Evolution Algorithm - a black box optimization method used widely outside machine learning domain, hence the name as LDADE. This paper also reviews multiple papers in this domain and points out the issues of running LDA without tuning has instability issues with topic formation. There are multiple tuning methods proposed before to address this topic instability issue eg; LDA-GA ( LDA with Genetic Algorithm). This paper claims LDADE method is more stable and converges faster.
$\alpha$ -> Dirichlet prior for Topic Distribution over document, initialized uniformly at starting and updated via Bayesian inference.
$\beta$ -> Dirichlet prior for Vocab distribution over Topic.
$k$ -> Number of topic. This affects the LDA performance.
Differential Evolution optimization can be used to minimize any function, it does search over parameter space to find the best fitting parameter that minimizes the target function. The convergence speed is much faster and DE method prunes down the possible search space pretty quickly. You can think this as - out of all the mutation happening over the population only the favouring mutations are being carry forward to the next generation. Similarly DE start with a population size over the parameter space and then nudge towards the parameters which was best in that generation to next, provided the new parameter improves the goal / function DE trying to minimize. Please refer bellow links for more explanations.
Searching for models close to data source behaviour. | CommonCrawl |
The TORCH concept is based on the detection of Cherenkov light produced in a quartz radiator plate. It is an evolution of the DIRC technique, extending the performance by the use of precise measurements of the emission angles and arrival times of detected photons. This allows dispersion in the quartz to be corrected for, and the time of photon emission to be determined with a target precision of $\rm 70~ps$ per photon. Combining the information from the 30 or so detected photons from each charged particle that traverses the plate, exceptional resolution on the time-of-flight of order $\rm 15~ps$ should be possible. The TORCH technique is a candidate for application in a future upgrade of the LHCb experiment, for low-momentum charged particle identification. Over a flight distance of $\rm 10~m$ it would provide clean pion-kaon separation up to $\rm 10~GeV$, in the busy environment of collisions at the LHC. Fast timing will also be crucial at higher luminosity for pile-up rejection.
A 5-year R&D program has been pursued with industry to develop suitable photon detectors with the required fast timing performance, fine spatial granularity (0.8 mm-wide pixels), long lifetime $\rm (5~C/cm^2$ integrated charge at the anode) and large active area (80% for a linear array). This is being achieved using $\rm 6 \times 6~cm^2$ micro-channel plate PMTs, and final prototype tubes are expected to be delivered early in 2017. Earlier prototype tubes have demonstrated most of the required features individually, using fast read-out electronics that has been developed based on NINO+HPTDC chips. A small-scale prototype of the optical arrangement has been tested in beam at CERN over the last year, and demonstrated close to nominal performance. Components for a large-scale prototype which will be read out using 10 MCP-PMTs, including a highly-polished synthetic quartz radiator plate of dimensions $\rm 125 \times 66 \times 1~cm^3$, are currently being manufactured for delivery on the same timescale. The status of the project will be reviewed, including the latest results from test beam analysis, and the progress towards the final prototype.
The TORCH detector is an evolution of the DIRC technique, for precision time-of-flight over large areas, being developed for a future upgrade of the LHCb experiment. The R&D project is delivering high-performance photon detectors and an optical system in synthetic quartz for a large-scale prototype. The status of the project will be reviewed, including the latest results from test beam analysis and progress towards the prototype. | CommonCrawl |
Use this tag for questions and problems involving vectors, e.g., in an Euclidean plane or space. More abstract questions, might better be tagged vector-spaces, linear-algebra, etc.
The commutator product as a replacement for the cross and wedge product?
How to prove $v_1\cdot v_2=|v_1||v_2|\cos(\theta)$ in n-dimensions?
It is easy to prove in 2D that $v_1\cdot v_2=|v_1||v_2|\cos(\theta)$ where $\theta$ is the angle between $v_1$ and $v_2$. But how to generalize? What is the proof in n-dimensions?
Why is the angle between vectors restricted?
Linear span of given vectors expresses a plane in $\mathbb R^3$?
Signed angle between higher-dimensional oriented vectors?
How to get a basis of $U \cap V$ where U and V are the column space of $A$ and $B$.?
How to find the rotation vector by deriving the final vector with respect to the displacement?
Deriving the scalar/dot product without using the dot product itself?
What's the point of the cross product?
How to see if a subspace is contained in another subspace?
Given a graph with a position vector and a vector with velocity, defining another vector. No idea where to start.
How is the following decomposition done? | CommonCrawl |
We investigate the dynamics, shape and stability of a thin viscous sheet subjected to an extensional flow under an imposed non-uniform temperature field. Using finite element simulations, we first solve for the stretching flow to determine the pre-buckling sheet thickness and in-plane flow velocities. Next, we use this solution as the base state and solve the linearized partial differential equation governing the out-of-plane deformation of the mid-surface as a function of two dimensionless operating parameters: the normalized stretching ratio $\alpha$ and a dimensionless width of the heating zone $\beta$. We show the sheet can become unstable via a buckling instability driven by the development of localized compressive stresses, and determine the global shape and growth rates of the most unstable mode. The growth rate is shown to exhibit a transition from stationary to oscillatory modes in region upstream of the heating zone. Finally, we investigate the effect of surface tension and present an operating diagram that indicates regions of the parameter space that minimizes or entirely suppresses the instability while achieving desired outlet sheet thickness. Therefore, our work is directly relevant to various industrial processes including the glass redraw \& float-glass method. | CommonCrawl |
Is there an example of a "almost-metric" that is not symmetric but satisfies the other axioms of a metric (positive-definiteness, triangle inequality)?
to hold. It is certainly interesting to find an example in $\mathbb R^n$ (even if just for a specific $n$) but I'm also interested in other "almost-metric-spaces".
Not the answer you're looking for? Browse other questions tagged geometry metric-spaces symmetry or ask your own question.
What values of $p$ make $d$ a metric?
is symmetric chi-squared distance "A" metric?
Is sum of two metrics a metric?
What do we call a metric that doesn't satisfy triangle inequality?
Proof that the triangle inequality holds in the following metric? | CommonCrawl |
Pasupathy, J (1997) PCAC and Modifications in Hadron Properties in Nuclear Medium. In: Modern Physics Letters A, 12 (26). pp. 1943-1949.
It is known that the Adler zero condition when imposed on pion amplitudes leads to several relations between hadron masses through the dual resonance formula of Lovelace, Shapiro and Veneziano. In particular the Lovelace quantization condition leads to the relation $\alpha^1(m^2_p-m^2_\pi)=1/2$ which is well satisfied experimentally. The Regge slope parameter α′ can be related to the gluon condensate. The latter is modified in the nuclear medium as compared to its value in the QCD vacuum. Combining the PCAC hypothesis with changes in gluon condensate leads to lowering of vector masses in the nuclear medium. It also leads to predictions regarding, nucleon isobar mass differences, pion decay constant in the medium, pion amplitudes and transverse momentum distribution of secondaries in heavy ion collisions. | CommonCrawl |
Hello all. Sage noob here. My friend recently told me about sage and convinced me to give it a whirl. (My background is MATLAB).
I would like to simply get the outer product of those two vectors. However it errors out.
I would like to think that I am boneheading something here, because this is a legitimate operation. I have multiple tutorial/documentation tabs open but nothing really elucidating why/how this doesnt work.
When multiplied to the right, a vector behaves like a column matrix, when multiplied to the left, a vector behaves like a row matrix. So your product is like multipliying a $2\times 1$ by another $2\times 1$ matrix, which is invalid.
Thanks tmonteil, however I am more confused now. If I try v*vC it works just find, but vC*v doesnt work - why the discrepancy? That is, why does it complain about the 'type' of data structure in one case, but does not complain about it in the other? Another thing: I tried to multiply v with a matrix A. It seems that it 'knows' how to interpret v regardless of which side I put it around A. Yet it does not know how to interpret v with A is substituted for vC? Not complaining, just trying to understand how it thinks. Thanks.
OK, I guess that makes sense... thanks!
That makes sense slelievre, but I am still confused as to why I can use "v" by itself. Why can I do v*A if A was a 2x2 matrix without having to tell it that v is a row vector? | CommonCrawl |
Consider writing a natural number as product of powers of natural numbers with given exponents, additionally requiring different base numbers for each power.
For example, $256$ can be written as a product of a square and a fourth power in three ways such that the base numbers are different.
Though $4^2$ and $2^4$ are both equal, we are concerned only about the base numbers in this problem. Note that permutations are not considered distinct, for example $16^2\times 1^4$ and $1^4 \times 16^2$ are considered to be the same.
Similarly, $10!$ can be written as a product of one natural number, two squares and three cubes in two ways ($10!=42\times5^2\times4^2\times3^3\times2^3\times1^3=21\times5^2\times2^2\times4^3\times3^3\times1^3$) whereas $20!$ can be given the same representation in $41680$ ways.
Let $F(n)$ denote the number of ways in which $n$ can be written as a product of one natural number, two squares, three cubes and four fourth powers. | CommonCrawl |
I describe work with Stefan Hollands that establishes a new criterion for the dynamical stability of black holes and black branes with respect to axisymmetric perturbations. Our analysis is done in vacuum general relativity without a cosmological constant in $D \geq 4$ spacetime dimensions, but our approach is applicable to much more general situations. We show that the positivity of the canonical energy, $\mathcal E$, on a subspace of linearized solutions that have vanishing linearized ADM mass and angular momentum implies mode stability. Conversely, failure of positivity of canonical energy on this subspace implies instability in the sense that there exist perturbations that cannot asymptotically approach a stationary perturbation. We further show that the canonical energy is related to the second order variations of mass, angular momentum, and horizon area by $\mathcal E = \delta^2 M - \sum_i \Omega_i \delta^2 J_i - (\kappa/8\pi) \delta^2 A$. This establishes that dynamic stability of a black hole is equivalent to its thermodynamic stability (i.e., its area, $A$, being a maximum at fixed ``state parameters'' $M$, $J_i$). For a black brane, we further show that a sufficient condition for instability is the failure of the Hessian of $A$ with respect to $M$, $J_i$ to be negative, thus proving a conjecture of Gubser and Mitra. We also prove that positivity of $\mathcal E$ is equivalent to the satisfaction of a ``local Penrose inequality,'' thus showing that satisfaction of this local Penrose inequality is necessary and sufficient for dynamical stability. | CommonCrawl |
where $y(t)$ is the output and $x(t)$ is the input given to the system.
$$y_1(t) + y_2(t) = x_1(e^t) + x_2(e^t)$$ How will system respond to this input?
Additivity requires a little more than direct addition of some $x_1$ and $x_2$. It should involve any linear combination of $x_1$ and $x_2$, $a_1 x_1 + a_2 x_2$ where $a_1$ and $a_2$ are "kind of scalars". This could make us dive into complicated stuff, like in Does scaling property imply superposition?
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Is the system represented by the equation $y(t) = x(2t)$ time invariant? | CommonCrawl |
Ideas for further contemplation from the blog post Complex Matrix Expansions.
Instead of expanding complex numbers into $2 \times 2$ real matrices, expand $m \times n$ complex matrices into $2m \times 2n$ real matrices.
Find analogues of every one of the above-mentioned properties under this generalized expansion. Warning: the analogue of (5) is not obvious!
Back then I had difficulty proving the analogue of (5) so I wrote a computer program to verify it by generating loads of randomized complex matrices and implementing arbitrary-precision arithmetic to multiply them. | CommonCrawl |
Little Ivica solves crossword puzzles every day. In case you haven't seen one, a crossword puzzle starts on a grid of $R \times C$ squares, each of which is either empty or blocked. The player's task is to write words in consecutive empty squares vertically (top down) or horizontally (left to right).
Ivica's sister has a strange habit of looking at crosswords Ivica has finished solving, and finding the lexicographically smallest word in it. She only considers words at least $2$ characters long.
Write a program that, given a crossword puzzle, finds that word.
The first line contains two integers $R$ and $C$ ($2 \le R, C \le 20$), the number of rows and columns in the crosswords.
Each of the following $R$ lines contains a string of $C$ characters. Each of those characters is either a lowercase letter of the English alphabet, or the character '#' representing a blocked square.
The input will be such that a solution will always exist.
Output the lexicographically smallest word in the crossword. | CommonCrawl |
The E791 collaboration Aitala, E.M. ; Amato, S. ; Anjos, J.C. ; et al.
Phys.Lett. B496 (2000) 9-18, 2000.
Using data from Fermilab fixed-target experiment E791, we have measured particle-antiparticle production asymmetries for lambda zero, cascade minus, and omega minus hyperons in pi minus-nucleon interactions at 500 GeV/c. The asymmetries are measured as functions of Feynman-x (x_F) and pt^2 over the ranges of -0.12 GE x_F LE 0.12 and 0 GE pt^2 LE 4 (GeV/c)^2. We find substantial asymmetries, even at x_F = 0. We also observe leading-particle- type asymmetries which qualitatively agree with theoretical predictions.
Phys.Lett. B495 (2000) 42-48, 2000.
We present a measurement of asymmetries in the production of $\Lambda_c^+$ and $\Lambda_c^-$ baryons in 500 GeV/c $\pi^-$--nucleon interactions from the E791 experiment at Fermilab. The asymmetries were measured as functions of Feynman x ($x_F$) and transverse momentum squared ($p_T^2$) using a sample of $1819 \pm 62$ $\Lambda_c$'s observed in the decay channel $\Lambda_c \to pK^-\pi^+$. We observe more $\Lambda_c^+$ than $\Lambda_c^-$ baryons, with an asymmetry of $(12.7\pm3.4\pm1.3) %$ independent of $x_F$ and $p_T^2$ in our kinematical range $(-0.1 < x_F < 0.6$ and $0.0 < p_T^2 < 8.0 (GeV/c)^2$). This $\Lambda_c$ asymmetry measurement is the first with data in both the positive and negative $x_F$ regions.
Phys.Lett. B411 (1997) 230-236, 1997.
This paper presents measurements of the production of Ds- mesons relative to Ds+ mesons as functions of x_F and square of p_t for a sample of 2445 Ds decays to phi pi. The Ds mesons were produced in Fermilab experiment E791 with 500 GeV/c pi- mesons incident on one platinum and four carbon foil targets. The acceptance-corrected integrated asymmetry in the x_F range -0.1 to 0.5 for Ds+- mesons is 0.032 +- 0.022 +- 0.022, consistent with no net asymmetry. The results, as functions of x_F and square of p_t, are compared to predictions and to the large production asymmetry observed for D+- mesons in the same experiment. These comparisons support the hypothesis that production asymmetries come from the fragmentation process and not from the charm quark production itself.
Phys.Lett. B371 (1996) 157-162, 1996.
We present asymmetries between the production of D+ and D- mesons in Fermilab experiment E791 as a function of xF and pt**2. The data used here consist of 74,000 fully-reconstructed charmed mesons produced by a 500 GeV/c pi- beam on C and Pt foils. The measurements are compared to results of models which predict differences between the production of heavy-quark mesons that have a light quark in common with the beam (leading particles) and those that do not (non-leading particles). While the default models do not agree with our data, we can reach agreement with one of them, PYTHIA, by making a limited number of changes to parameters used. | CommonCrawl |
then $R \to S$ is formally smooth in the $\mathfrak n$-adic topology.
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07EH. Beware of the difference between the letter 'O' and the digit '0'.
The tag you filled in for the captcha is wrong. You need to write 07EH, in case you are confused. | CommonCrawl |
"Resolution Of Conjectures Related To Lights Out! And Cartesian Products" by Bryan A. Curtis, Jonathan Earl et al.
Lights Out!\ is a game played on a $5 \times 5$ grid of lights, or more generally on a graph. Pressing lights on the grid allows the player to turn off neighboring lights. The goal of the game is to start with a given initial configuration of lit lights and reach a state where all lights are out. Two conjectures posed in a recently published paper about Lights Out!\ on Cartesian products of graphs are resolved.
Curtis, Bryan A.; Earl, Jonathan; Livingston, David; and Shader, Bryan L.. (2018), "Resolution Of Conjectures Related To Lights Out! And Cartesian Products", Electronic Journal of Linear Algebra, Volume 34, pp. 718-724. | CommonCrawl |
A universal construction in Goursat categories (Gran, Marino and Rodelo, Diana), In Cah. Topol. Géom. Différ. Catég., volume 49, 196-208, 2008.
Protolocalisations of exact Mal'cev categories (Borceux, Francis, Gran, Marino and Mantovani, Sandra), In Theory Appl. Categ., volume 21, 3-37, 2008.
On the number of subsets relatively prime to an integer (Ayad, Mohamed and Kihel, Omar), In J. Integer Seq., volume 11, 08-5, 2008.
On composite polynomials (Ayad, Mohamed and Ali, Nidal), In Int. J. Algebra, volume 2, 315-326, 2008.
Matsuki's double coset decomposition via gradient maps (Miebach, Christian), In J. Lie Theory, volume 18, 555-580, 2008.
Prediction via the conditional quantile for right censorship model (Ould-Saïd, Elias and Sadki, Ourida), In Far East J. Theor. Stat., volume 25, 145-179, 2008.
Asymptotic normality of the kernel estimator of conditional quantiles in a normed space (Ezzahrioui, M'hamed and Ould-Saïd, Elias), In Far East J. Theor. Stat., volume 25, 15-38, 2008.
Michel Kervaire 1927–2007 (S. Eliahou, P. de la Harpe, J.-C. Hausmann and C. Weber), In Notices Amer. Math. Soc., volume 55, 960-961, 2008.
Michel Kervaire (26 avril 1927–19 novembre 2007) (S. Eliahou, P. de la Harpe, J.-C. Hausmann and C. Weber), In Gaz. Math., 77-82, 2008.
Square packings in the flat torus (Gensane, Thierry and Ryckelynck, Philippe), In Geombinatorics, volume 17, 141-156, 2008.
Effective Dispersion Equations For Reactive Flows With Dominant Peclet and Damkohler Numbers (C. J. van Duijn, A. Mikelic, I. S. Pop and C. Rosier), In Advances in Chemical Engineering, volume 34, 1-45, 2008.
ADI Preconditioned Krylov Methods for Large Lyapunov Matrix Equations (K. Jbilou), Technical report 386, LMPA, 2008.
Stein high-order one step implicit Runge-Kutta methods for large-scale ordinary differential equations (A. Bouhamidi and K. Jbilou), Technical report 376, LMPA, 2008.
Vector Extrapolation Enhanced TSVD for Linear Discrete Ill-posed Problems (K. Jbilou, L. Reichel and H. Sadok), Technical report 370, LMPA, 2008.
The global Arnoldi process for solving the Sylvester-observer equation (M. Heyouni and K. Jbilou), Technical report 357, LMPA, 2008.
A note on the numerical approximate solutions for generalized Sylvester matrix equations with applications (A. Bouhamidi and K. Jbilou), Technical report 356, LMPA, 2008.
Résolution des équations matricielles de Sylvester par le processus 'Extended Block Arnoldi' (M. Heyouni and K. Jbilou), Technical report 355, LMPA, 2008.
Résolution des équations matricielles de Sylvester par le processus 'Extended Block Arnoldi' (M. Heyouni and K. Jbilou), Technical report, LMPA, 2008.
The global Arnoldi process for solving the Sylvester-observer equation (M. Heyouni and K. Jbilou), Technical report, LMPA, 2008.
Extended Arnoldi Methods for Large Sylvester Matrix Equations (M. Heyouni), Technical report, LMPA, 2008.
Infinite locally finite doubly-stochastic matrices (S. Rénier), Technical report, LMPA, 2008.
Three Paths to the Convergence Analysis of the Arnoldi Process for Eigenvalue Problems (M. Bellalij, Y. Saad and H. Sadok), Technical report, LMPA, 2008.
Annalysis of some Ktylov Subspace Methods for Nomrl Matrices via Approximation theory and Convex Optimization (M. Bellalij, Y. Saad and H. Sadok), Technical report, LMPA, 2008.
Nonparametric conditional density and conditional hazard estimation for dependent functional data (A. Laksaci, M. Lemdani and E. Ould-Saïd), Technical report, LMPA, 2008.
A Potential-Theoretic Approach to the Time-Dependent Oseen System (P. Deuring), Technical report, LMPA, 2008.
Spatial decay of time-dependent Oseen flows (P. Deuring), Technical report, LMPA, 2008.
Finite element error estimates for 3D exterior incompressible flow with nonzero velocity at infinity (P. Deuring), Technical report, LMPA, 2008.
A Weighted Eingenvalue Problem for the P-Laplacian Plus a Potential (M. Cuesta and H. Ramos Quoirin), Technical report, LMPA, 2008.
Generation of optimal packings from optimal packings (T. Gensane), Technical report, LMPA, 2008.
Strong uniform consistency of nonparametric estimation of the censored conditional quantile for functional regressors (M. Elbahi and E. Ould-Saïd), Technical report, LMPA, 2008.
Asymptotic normality of a kernel conditional quantile estimator for randomly left-truncation time series (E. Ould-Saïd and D. Yahia), Technical report, LMPA, 2008.
On the strong uniform consistency of the conditional mode estimator under $\alpha$- mixing condition and left-truncation (E. Ould-Saïd and A. Tatachak), Technical report, LMPA, 2008.
Asymptotic normality of a robust estimator of the regression function for functional time series data (M. Attouch, A. Laksaci and E. Ould-Saïd), Technical report, LMPA, 2008.
On the uniform strong convergence rate of a smooth conditional quantile kernel estimator for censored and dependent data (E. Ould-Saïd and O. Sadki), Technical report, LMPA, 2008.
On the central limit theorem of kernel conditional quantile estimator under random censorship (E. Ould-Saïd), Technical report, LMPA, 2008.
Asymptotic results for $L^1$- norm kernel estimator of conditional quantile for functional times series data (A. Laksaci, M. Lemdani and E. Ould-Saïd), Technical report, LMPA, 2008.
Asymptotic normality for a smooth kernel estimator of the conditional quantile for censored time series (E. Ould-Saïd and O. Sadki), Technical report, LMPA, 2008.
Asymptotic distribution of a nonparametric regression quantile with censored data and functional regressors (M. El Bahi and E. Ould-Saïd), Technical report, LMPA, 2008.
Matrices bistochastiques paires et impaires (Rénier, Simon), PhD thesis, Université du Littoral Côte d'Opale, 2008. | CommonCrawl |
Abstract: We consider nonparametric maximum-likelihood estimation of a log-concave density in case of interval-censored, right-censored and binned data. We allow for the possibility of a subprobability density with an additional mass at $+\infty$, which is estimated simultaneously. The existence of the estimator is proved under mild conditions and various theoretical aspects are given, such as certain shape and consistency properties. An EM algorithm is proposed for the approximate computation of the estimator and its performance is illustrated in two examples. | CommonCrawl |
The determination of the normalizer of the basis group in the group of units of the associated group ring is a question that naturally imposes by itself. In integral group rings, in particular, it has been observed that, for important classes of finite groups, this normalizer is minimal, in other words, $\mathcal N_\mathcal U(G)=G\cdot Z$. When this occurs, we say that the group in question and its integral group ring satisfy the normalizer property. This property, also known as (Nor), has recently gained great importance when Mazur, in [Ma95], noticed an interesting relation with the famous problem of isomorphism in integral group rings also known as (Iso). Exploring this connection, Hertweck in [He01] found an example of a finite group that does not satisfy (Nor), and indirectly, by the relation mentioned above, obtained a counterexample to (Iso). Given that the counter example of Hertweck to (Nor) consists of an extension given by a semidirect product, but [LPS99] proves that extensions given by direct products are solutions (Nor), it is important to investigate which other other extensions of finite groups answer the property. Recently, Petit Lobão e Sehgal in [PeS03] demonstrated the validity of (Nor) for the class of complete monomial groups; in other words, a wreath extension of a finite nilpotent group with the symmetric group on m letters. Zhengxing Li e Jinke Hai in a series of articles, among which we have [HL12], [HL12b], HL11], also obtained interesting solutions of this property. The purpose of this work is to verify the relation between (Nor) and extensions of groups, where such component groups are solutions (Nor), in order to obtain necessary and sufficient conditions to find positive solutions to the property in question.
Joint work with Thierry Petit Lobão (Universidade Federal da Bahia). | CommonCrawl |
On the Real Nullstellensatz for Global Analytic Functions.
An algebra of analytic functions on a complex analytic space can be endowed with a topology in such a way that closed ideals behave specially well. For instance, a closed ideal $\alpha$ admits a primary decomposition, $\alpha =\bigcap _i \alpha _i$. In fact, under some restrictions on the ideals $\alpha _i$, the Hilbert Nullstellensatz holds for the ideal $\alpha$. In this work we look for similar conditions in the real case.
Mathematics Subject Classification (2000): 14P99, 11E25, 32B10.
Keywords and Phrases: Nullstellensatz, sum of squares, positive semidefinite analytic function, compact set, analytic curve, normal analytic surface.
Full text, 24p.: pdf 270k. | CommonCrawl |
Time according to the gravity of Sagittarius A?
They found the factor of time on the varying distances from the SMBH, I am curious on how to turn those factors into relatable terms such as: At 1 km from the event horizon ____ time passes on earth in an hour.
I'm just not sure how to figure this out.
I am curious on how to turn those factors into relatable terms such as: At 1 km from the event horizon ____ time passes on earth in an hour.
The key here is to understand what the formula relates to. The (all important) observer is in this case a notional observer so distant that curvature (and hence gravitational effects) are negligible.
That observer is the "base clock" we reference. Everything is dilated relative to that clock.
Now in the specific case of Earth and Sagitarius A*, $r_e$ is huge so the dilation factor for Earth is as close to one as makes no difference in practice, so you don't really need to go to all that trouble and you can use $t_\infty \approx t_e$.
Not the answer you're looking for? Browse other questions tagged black-hole space-time space supermassive-black-hole time-dilation or ask your own question.
What is the optimal escape trajectory from near a black hole?
Does gravity bend light, and how much time does it take for light to cross gravity of a Black Hole?
How does gravity have an effect from the inside the event horizon of a black hole with the rest of the universe?
Will biological process be disrupted under strong gravitational time dilation? | CommonCrawl |
Abstract: The warped product $N_1\times_f N_2$ of two Riemannian manifolds $(N_1,g_1)$ and $(N_2,g_2)$ is the product manifold $N_1\times N_2$ equipped with the warped product metric $g=g_1+f^2 g_2$, where $f$ is a positive function on $N_1$. The notion of warped product manifolds is one of the most fruitful generalizations of Riemannian products. Such notion plays very important roles in differential geometry as well as in physics, especially in general relativity. Warped product manifolds have been studied for a long period of time. In contrast, the study of warped product submanifolds was only initiated around the beginning of this century.
In this article we survey important results on warped product submanifolds in various ambient manifolds. It is the author's hope that this survey article will provide a good introduction on the theory of warped product submanifolds as well as a useful reference for further research on this vibrant research subject. | CommonCrawl |
We study the shape of the Young diagram $\lambda$ associated via the Robinson–Schensted–Knuth algorithm to a random permutation in $S_n$ such that the length of the longest decreasing subsequence is not bigger than a fixed number $d$; in other words we study the restriction of the Plancherel measure to Young diagrams with at most $d$ rows. We prove that in the limit $n\to\infty$ the rows of $\lambda$ behave like the eigenvalues of a certain random matrix (namely the traceless Gaussian Unitary Ensemble random matrix) with $d$ rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asymptotically like the largest eigenvalue of the corresponding random matrix. | CommonCrawl |
We derive a sufficient condition and present explicit examples based on toric geometry for realizing meta-stable de Sitter vacua with small positive cosmological constant within type IIB string theory / F-theory flux compactifications with spontaneously broken supersymmetry. There are a number of `lamp post' constructions of de Sitter vacua in type IIB string theory and supergravity. We show that one of them -- the method of `K\"ahler uplifting' by F-terms from an interplay between non-perturbative effects and the leading $\alpha'$-correction -- allows for a more general parametric understanding of the existence of de Sitter vacua. The result is a condition on the values of the flux induced superpotential and the topological data of the Calabi-Yau compactification, which guarantees the existence of a meta-stable de Sitter vacuum if met. Our analysis explicitly includes the stabilization of all moduli, i.e. the K\"ahler, dilaton and complex structure modu li, by the interplay of the leading perturbative and non-perturbative effects at parametrically large volume. | CommonCrawl |
Abstract: The subject of this paper is the problem of estimating service time distribution of the $M/G/\infty$ queue from incomplete data on the queue. The goal is to estimate $G$ from observations of the queue--length process at the points of the regular grid on a fixed time interval. We propose an estimator and analyze its accuracy over a family of target service time distributions. The original $M/G/\infty$ problem is closely related to the problem of estimating derivatives of the covariance function of a stationary Gaussian process. We consider the latter problem and derive lower bounds on the minimax risk. The obtained results strongly suggest that the proposed estimator of the service time distribution is rate optimal. | CommonCrawl |
Abstract: We discuss the local geometry in the vicinity of a sphere $\mathbb P^1$ embedded with a negative normal bundle. We show that the behavior of the Kähler potential near a sphere embedded with a given normal bundle can be determined using the adjunction formula. As a by-product, we construct (asymptotically locally complex-hyperbolic) Kähler–Einstein metrics on the total spaces of the line bundles $\mathcal O(-m)$, $m\ge3$, over $\mathbb P^1$. | CommonCrawl |
We use integral field spectroscopy, from the SWIFT and Palm3K instruments, to perform a spatially-resolved spectroscopic analysis of four nearby highly star-forming `green pea' (GP) galaxies, that are likely analogues of star-forming systems at z~2.5-3. By studying emission-line maps in H$\alpha$, [NII]$\lambda \lambda$6548,6584 and [SII]$\lambda$$\lambda$6716,6731, we explore the kinematic morphology of these systems and constrain properties such as gas-phase metallicities, electron densities and gas-ionization mechanisms. Two of our GPs are rotationally-supported while the others are dispersion-dominated systems. The rotationally-supported galaxies both show evidence for recent or ongoing mergers. However, given that these systems have intact disks, these interactions are likely to have low mass ratios (i.e. minor mergers), suggesting that the minor-merger process may be partly responsible for the high SFRs seen in these GPs. Nevertheless, the fact that the other two GPs appear morphologically undisturbed suggests that mergers (including minor mergers) are not necessary for driving the high star formation rates in such galaxies. We show that the GPs are metal-poor systems (25-40 per cent of solar) and that the gas ionization is not driven by AGN in any of our systems, indicating that AGN activity is not co-eval with star formation in these starbursting galaxies. | CommonCrawl |
Legends often tell of great treasures. But you rarely get the chance to actually stumble upon those treasures. Most of them are lost in the sea, or hidden below mountains. But as you learned from one of your biggest idols, treasures do belong into a museum. And now you have the chance to make that happen.
On an expedition you found a large cave network. A native shaman has spoken about incredible values his ancestors have hidden in the caves. He even gave you an ancient map, depicting the cave network and the location of the treasures within. Sadly, the cave network is completely flooded. Since the trip out here takes forever, you decided to do a short dive and scout out the cave network. But on your arrival back at the entrance to the cave network you get the news$\ldots $ a volcano just erupted nearby. It is next to guaranteed that the lava will cover the entries to the cave network and the treasures will be lost forever.
That puts you on the spot. You only have a short time left, and only one lousy tank of air. So it is all on you. You only have time for a single dive. But how could you possible decide which route to take? The cave network is huge, and you should definitely try and rescue as much of the treasures as possible. You think back to your times as a computer scientist at the university$\ldots $ And then it hits you. You still have your laptop with you. You could write a program to help you figure out the best you can do in rescuing the treasures.
You may assume that neither locating or picking up a treasure within a cave nor the traversal of a cave consumes any air.
Each test set consists of multiple test cases. The file starts with a single number $t$ ($0 < t \le 2000$) on a single line, denoting the number of testcases in the file. Each test case starts with two integers $n$ and $m$ on a single line, where $n$ the number of caves and $m$ the number of the connecting tunnels in the network ($1 \leq n \leq 10\, 000$; $0 \leq m \leq 50\, 000$). This line is followed by $m$ lines, giving a description of the tunnels of the cave as three integers $a$ $b$ and $l$ with $a$, $b$ denoting caves and $l$ giving the amount of air necessary for diving through the tunnel ($0 \le a, b < n; 0 \le l \le 500$). After the tunnels follows an integer $i$ on a single line, giving the number of idols in the cave system ($0 \leq i \leq 8$). This line is followed by a single line containing $i$ integers $p_1, \ldots , p_ i$ giving the caves withing the network conaining an idol ($0 \le p_1,\ldots ,p_ i < n$). The input is concluded by a single number, giving the liters of air $a$ you have available ($0 \le a \le 1\, 000\, 000$). You will always start (and end) at the node with the label $0$.
For each test case print a number $X$ on a single line, where X is replaced by the maximal number of idols the diver can recover. | CommonCrawl |
Hamilton-Jacobi-Bellman approach for optimal control problems with discontinuous coefficients.
Abstract: This thesis deals with the Dynamical Programming and Hamilton-Jacobi-Bellman approach for a general class of deterministic optimal control problems with discontinuous coefficients.The tools essentially used in this work are based on the control theory, the viscosity theory forPartial Differential Equations, the nonsmooth analysis and the dynamical systems.
The first part of the thesis is concerned with the state constrained problem of discontinuous trajectories driven by impulsive dynamical systems. A characterization result of the value function of this problem has been obtained. Another contribution of this part consists of the extension of the HJB approach for the problems with time-measurable dynamical systems and in presence of time-dependent state constraints.
The second part is devoted to the problem on stratified domain, which consists of a union of subdomains separated by several interfaces. One of the motivations of this work comes from the hybrid control problems. Here new transmission conditions on the interfaces have been obtained to ensure the uniqueness and the characterization of the value function.
The third part investigates the homogenization of Hamilton-Jacobi equations in the framework of state-discontinuous Hamiltonians. This work considers the singular perturbation of optimal control problem on a periodic stratified structure. The limit problem has been analyzed and the associatedHamilton-Jacobi equation has been established. This equation describes the limit behavior of the value function of the perturbed problem when the scale of periodicity tends to 0.
Keywords: Optimal control problems, Hamilton-Jacobi-Bellman equations, viscosity solutions,nonsmooth analysis, impulsive differential equations, state constraints, multi-domains, stratified dynamical system, transmission conditions, singular perturbation.
Understanding large text corpora via sparse optimization.
Contrôle optimal d'équations différentielles avec – ou sans – mémoire.
Analyse de sensibilité pour des problèmes de commande optimale ; commande optimale stochastique sous contrainte en probabilité.
An approximation scheme for finding equilibria in energy markets with risk aversion.
Abstract: We discuss some recent results on convergence and rate of convergence of the classical augmented Lagrangian algorithm. First, we prove local primal-dual convergence under the sole assumption that the dual starting point is close to a multiplier satisfying the second-order sufficient optimality condition. In particular, no constraint qualifications of any kind are needed. Previous literature on the subject required, in addition, the linear independence constraint qualification and either the strict complementarity assumption or a stronger version of the second-order sufficient condition. Second, we show that when applied to optimization problems with complementarity constraints (MPCC), the augmented Lagrangian approach has theoretical global convergence guarantees that compare favorably to the alternatives, whether standard nonlinear programming solvers or those specially designed for this problem class. Moreover, we show that in practice the ALGENCAN implementation of the augmented Lagrangian method is a very good choice for MPCC if robustness and the quality of computed solutiuon is of primary concern.
Modelisation de la croissance metastatique d'un cancer et des traitements anti-cancereux.
Abstract: Le cancer est devenu la première cause de mortalité en France. L'utilisation de modèles mathématiques pour décrire cette maladie ainsi que les traitements administrés semble très prometeuse. Nous verrons dans cet exposé un certain nombre de difficultés auxquelles doivent faire face le médecin et les outils que les mathématiciens peuvent leur apporter.
(Sup + Bolza)-control problems as dynamic differential games.
Abstract: We consider a (L∞+Bolza)-control problem, namely a problem where the payoff is the sum of a sup (actually ess-sup) functional and a classical Bolza functional. Owing to the ⟨L1,L∞⟩ duality, the (L∞+Bolza)-control problem is rephrased in terms of a static differential game, where a new variable k plays the role of a maximizer. In this framework 1 − k is regarded as the available fuel for the maximizer. The relevant (and unusual) fact is that this static game is equivalent to the corresponding dynamic differential game, which allows the (upper) value function to verify a rather simple boundary value problem.
The whole approach will be hopefully utilized to get necessary conditions for a curve to be a minimizer.
Calcul d'ensemble de trajectoires de systèmes hybrides par des méthodes numériques garanties.
Abstract: Cet exposé s'intéresse au problème d'atteignabilité dans les systèmes hybrides, c'est-à-dire aux systèmes composés d'éléments évoluant continûment dans le temps et d'éléments évoluant de manière discrète dans le temps, pour lesquels nous voulons connaître l'ensemble des trajectoires pour un ensemble de conditions initiales. Cette problématique est abordée suivant une vision informaticienne issue du domaine de la vérification formelle. Nous présentons une approche fondée sur les méthodes d'intégration numérique garantie pour résoudre des équations différentielles ordinaires et sur l'interpolation polynomiale garantie pour calculer les instants d'apparition d'événements de zero-crosssing. Après avoir présenté le contexte de ce travail, nous aborderons les principales caractéristiques des outils utilisés qui seront illustrés à l'aide d'exemples.
An overview of scalable control and estimation using max-plus methods.
Abstract: Conventional approaches to solving optimal control problems using dynamic programming require the solution of Hamilton Jacobi equations. Numerical solution techniques suffer from the curse of dimensionality and are thus intractable in higher dimensions. This talk will provide a gentle introduction to a promising class of control synthesis methods which are more scalable than grid based methods. Further the application of these 'max-plus' methods to an intriguing range of problems in estimation and control of linear, non-linear and multi-sensor problems will be described.
Explicit solution for singular calculus of variations in infinite horizon.
Abstract: We consider a one dimensional infinite horizon calculus of variations problem (P), where the integrand is linear with respect to the velocity. The Euler-Lagrange equation, when defined, is not a differential equation as usual, but reduces to an algebraic (or transcendental) equation $C(x)=0 $. Thus this first order optimality condition is not informative for optimal solutions with initial condition $x_0 $ such that $C(x_0) \neq 0 $. To problem (P) we associate an auxiliary calculus of variations problem whose solutions connect as quickly as possible the initial conditions to some constant solutions. Then we deduce the optimality of these curves, called MRAPs (Most Rapid Approach Path), for (P). According to the optimality criterium we consider, we have to assume a classical transversality condition. We observe that (P) possesses the Turnpike property, the Turnpike set being given by the preceding particular constant solutions of the auxiliary problem.
Pursuit-Evasion Games with Multi-Pursuer: a decomposition approach.
Abstract: In a pursuit-evasion game, each pursuer attempts to minimize the distance between himself (P) and the evader (E) and capture it in the shortest time, whereas the evader tries to maximize the same distance to escape from being captured. In this talk, we deal with PE games with an evader and n pursuers. It is easy to imagine that the dimension of the space where the agents are moving increases with the number of pursuers, making the problem hard to solve. We will use the well-known PDE approach and an original decomposition technique to get over the big dimensionality contained in the problem. Simulation results will show the effectiveness of the proposed strategy of resolution as well as the limitations of it.
Optimal control of single species biological population.
Stabilité des arcs singuliers pour un problème de commande optimale perturbé.
Convergence of one-step methods for SDEs.
Modélisation de la production des globules rouges : Équations de transport non-linéaires et stabilité. | CommonCrawl |
Microscopic flows are almost always stable and laminar that allows precise control of chemical environment in micro-channels. We describe design and operation of several microfluidic devices, in which various types of environments are created for different experimental assays with live cells. In a microfluidic chemostat, colonies of non-adherent bacterial and yeast cells are trapped in micro-chambers with walls permeable for chemicals. Fast chemical exchange between the chambers and nearby flow-through channels creates essentially chemostatic medium conditions in the chambers and leads to exponential growth of the colonies up to very high cell densities. Another microfluidic device allows creation of linear concentration profiles of a pheromone ($\alpha$-factor) across channels with non-adherent yeast cells, without exposure of the cells to flow or other mechanical perturbation. The concentration profile remains stable for hours enabling studies of chemotropic response of the cells to the pheromone gradient. A third type of the microfluidic devices is used to study chemotaxis of human neutrophils exposed to gradients of a chemoattractant (fMLP). The devices generate concentration profiles of various shapes, with adjustable steepness and mean concentration. The ``gradient'' of the chemoattractant can be imposed and reversed within less than a second, allowing repeated quantitative experiments. | CommonCrawl |
Subject to: x_1 + x_2 + x_3 = 100, 2x_1 + x_2 + x_4 = 150, 5x_1 + 10x_2 + x_5 = 800, x_i \ge 0, i = 1, \dots, 5.
The vector x^T = [60, 30, 10, 0, 200] is a (nonbasic) feasible solution. Determine a vector y and a basic feasible solution of the form x+ty.
such that 80y_1 + 60y_2 > 0 and y_4 = 0 (because x_4 = 0), then a basic feasible solution x' can be found whose objective function value is at least as great as that of x.
We start by setting y_4 = 0 since x_4 = 0. Then choose y_1 = 1, y_2 = -2 to satisfy the resulting second equation 2y_1 + y_2 = 0. Substituting these values in turn into equations one and three yields y_3 = 1, y_5 = 15. Thus, we obtained the solution y^T = [1,-2,1,0,15]. Since 80\cdot 1 + 60\cdot (-2) < 0, we replace y with -y, so that y^T = [-1,2,-1,0,-15] and 80\cdot (-1) + 60\cdot 2 = 40 > 0.
Now we determine a scalar t such that x + ty is a basic feasible solution, and 80(x_1 + ty_1) + 60(x_2 + ty_2) \ge 80x_1 + 60x_2. Consider the vector (x+ty)^T = [60-t, 30+2t,10-t, 0, 200-15t]. Choosing the largest value of t that will make x+ty nonnegative, we get t = \min(60/1, 10/1, 200/15) = 10. Thus, the new candidate for a basic feasible solution is [50,50,0,0,50].
Example 2: Formulate the linear program to solve the following problem: The MF&S supplies castings to a variety of customers. It plans to devote the next week of production to just two, J and W. MF&S uses a combination of pure steel and scrap metal to fulfill its orders and has 400 pounds of pure steel and 360 pounds of scrap metal in stock. The pure steel costs MF&S $6 per pound and the scrap $3 per pound. Pure steel requires 3 hours per pound to process into a casting, while scrap requires only 2 hours per pound. Total available processing time in the week is 2,000 hours.
The castings for J each require 5 pounds of metal, with a quality control restriction limiting the ratio of scrap to pure steel to a maximum of 5/7. J has ordered 30 castings at a price of $50 each.
The castings for W each require 8 pounds of metal, with a quality restriction of maximum scrap to pure steel ratio of 2/3. W has ordered 40 castings at a price of $80 each.
Determine how MF&S should allocate their metal stocks to produce the castings ordered by these two customers if the objective is to maximize the value added to the metal, i.e., to maximize the selling price minus the cost of the metal.
Inventory contraints: x_1+y_1 \le 400, x_2 + y_2 \le 360.
Total processing time constraints: 3(x_1 + y_1)+2(x_2 + y_2) \le 2000.
Order constraints: x_1 + x_2 = 30 \times 5, y_1 + y_2 = 40\times 8.
Quality constraints: -5x_1 + 7x_2 \le 0 (equivalently, x_2 / x_1 \le 5 / 7), -2y_1 + 3y_2 \le 0 (equivalently, y_2 / y_1 \le 2/3).
Non-negativity constraints: x_1, x_2, y_1, y_2 \ge 0.
Example 3: RSC produces three imaginative lines of children's riding vehicles: ambulances, fire trucks, and roadsters. They sell exclusively by mail order and thus ship each vehicle individually.
RSC prides itself on the versatility of its employees. Each of the four employees can work in any of the three jobs, and each of the four works 40 hours a week.RSC is planning production for a 13-week quarter. Demand for their vehicles has shown a seasonal pattern, and they plan for the demand for fire tucks to be at least as great as the total demand for the other two lines. How should RSC allocate the time of their four employees, and how many of each vehicle should they make to maximize their profit during the period?
Maximize: 70x_1 + 80x_2 + 50x_3.
Subject to: (1/4 + 1/4 + 1/3)x_1 + (1/3 + 1/3 + 1/6)x_2 + (1/6+1/6+1/6)x_3 \le 40 \times 4 \times 13, x_1 - x_2 + x_3 \le 0 (equivalently, x_2 \ge x_1 + x_3), x_1, x_2, x_3 \ge 0.
It is clear that four employees should work on molding, dressing and packing & shipping subsequently. | CommonCrawl |
Your task is to find a minimum-price flight route from Syrjälä to Metsälä. You have one discount coupon, using which you can halve the price of any single flight during the route. However, you can use the coupon only once.
The first input line has two integers $n$ and $m$: the number of cities and flight connections. The cities are numbered $1,2,\ldots,n$. City 1 is Syrjälä, and city $n$ is Metsälä.
After this there are $m$ lines that describe the flights. Each line has three integers $a$, $b$, and $c$: a flight begins at city $a$, ends at city $b$, and its price is $c$. Each flight is unidirectional.
You can assume that it is always possible to get from Syrjälä to Metsälä.
Print one integer: the price of the cheapest route from Syrjälä to Metsälä.
When you use the discount coupon for a flight whose price is $x$, its price becomes $\lfloor x/2 \rfloor$ (it is rounded down to an integer). | CommonCrawl |
Is there an algorithm for solving the following problem: let $g_1,\ldots,g_n$ be permutations in some (large) symmetric group, and $g$ be a permutation that is known to be in the subgroup generated by $g_1,\ldots,g_n$, can we write $g$ explicitly as a product of the $g_i$'s?
My motivation is that I'm TAing an intro abstract algebra course, and would like to use the Rubik's cube to motivate a lot of things for my students, and would, in particular, like to show them an algorithm to solve it using group theory. (That is, I can write down what permutation of the cubes I have, and want to decompose it into basic rotations, which I then invert and do in the opposite order to get back to the solved state.) Though I'm interested in the more general case, not just for the Rubik(n) groups, if a solution works out.
Note: I don't really know what keywords to use for solving this problem, if someone can point me to the right search terms to google to get the results I'm looking for, I'll gladly close this.
Yes. The general rule of thumb is that groups described by permutations are computationally easy, groups described by generators and relations have computational problems that are generally undecidable, and matrix groups are somewhere in between.
There's a whole book "Permutation group algorithms" by Seress, Cambridge University Press, 2003.
The main technique for permutation groups is called the Schreier–Sims algorithm; there's a survey here, for instance. The rough idea is to stabilize the permuted elements one at a time. As Mitch already said, though, this doesn't find the shortest word in the generators that produces any particular group element, which is a more difficult problem.
Page 22 contains a survey of the problem you're interested in. Apparently, there is a polynomial time algorithm to check if a solution exists (it will also return some implicitly represented solution), but finding the solution with the fewest number of moves is, not surprisingly, PSPACE-complete.
Not the answer you're looking for? Browse other questions tagged co.combinatorics permutations gr.group-theory recreational-mathematics or ask your own question.
Which Turing machines accept the language of trivial words in a finitely presented group?
How hard is reconstructing a permutation from its differences sequence?
Modifying Dehn's algorithm to allow equal length replacements?
How hard is recognizing a permutation that is a square for the shift product? | CommonCrawl |
In this paper, we introduce the notion of a pre-Lie 2-algebra, which is the categorification of a pre-Lie algebra. We prove that the category of pre-Lie 2-algebras and the category of 2-term pre-Lie$_\infty$-algebras are equivalent. We classify skeletal pre-Lie 2-algebras by the third cohomology group of a pre-Lie algebra. We prove that crossed modules of pre-Lie algebras are in one-to-one correspondence with strict pre-Lie 2-algebras. O-operators on Lie 2-algebras are introduced, which can be used to construct pre-Lie 2-algebras. As an application, we give solutions of 2-graded classical Yang-Baxter equations in some semidirect product Lie 2-algebras.
Theory and Applications of Categories, Vol. 34, 2019, No. 11, pp 269-294. | CommonCrawl |
With the image class we created in previous posts, we are finally ready to have some fun with image processing. As a first step, let's try something simple.
Something simple huh? What can be simpler than linear transformation? I guess that's what we are targeting here: $$c_1 = \alpha \times c_0 + \beta$$ Simple as it is, we can still do something to make our images more appealing. Say we have somehow miscalculate the light and take a dark picture. A quick fix to it would be adjust the brightness of the image.
Since the brightness of a pixel is proportional to its RGB values, it is obvious that we can change the brightness of a pixel by simply changing its RGB value. The brightness filter adjust the overall brightness level of an input image by adding to or subtracting from every pixel values directly. This corresponds to having $\alpha=1$ and $\beta\ne 0$ in linear transformation equation. To maintain the color consistency, the amount of change must be the same for every channel in every pixel. | CommonCrawl |
Amanda L Wright and Bryce Vissel.
CAST your vote: is calpain inhibition the answer to ALS?. Journal of neurochemistry 137(2):140–1, April 2016.
Abstract A publication in the Journal of Neurochemistry by Rao et al. (2016) suggests that the overexpression of the calpain inhibitor, calpastatin (CAST) rescues neuron loss and increases survival of the amyotrophic lateral sclerosis (ALS) mouse model, hSOD1G93A. The findings of Rao et al. (2016) provide an insight into the mechanisms that lead to neuronal loss in ALS and suggest a cell loss pathway common to several neurodegenerative disorders that may be therapeutically targeted. Here, we highlight the findings of Rao et al. (2016) and discuss some key considerations required prior to assessing the potential use of calpain inhibitors in the clinic. Read the highlighted article 'Calpastatin inhibits motor neuron death and increases survival in hSOD1(G93A) mice' on page 253.
Mala V Rao, Jabbar Campbell, Arti Palaniappan, Asok Kumar and Ralph A Nixon.
Calpastatin inhibits motor neuron death and increases survival of hSOD1(G93A) mice.. Journal of neurochemistry 137(2):253–65, 2016.
Abstract Amyotrophic lateral sclerosis (ALS) is a progressive motor neuron disease with a poorly understood cause and no effective treatment. Given that calpains mediate neurodegeneration in other pathological states and are abnormally activated in ALS, we investigated the possible ameliorative effects of inhibiting calpain over-activation in hSOD1(G93A) transgenic (Tg) mice in vivo by neuron-specific over-expression of calpastatin (CAST), the highly selective endogenous inhibitor of calpains. Our data indicate that over-expression of CAST in hSOD1(G93A) mice, which lowered calpain activation to levels comparable to wild-type mice, inhibited the abnormal breakdown of cytoskeletal proteins (spectrin, MAP2 and neurofilaments), and ameliorated motor axon loss. Disease onset in hSOD1(G93A) /CAST mice compared to littermate hSOD1(G93A) mice is delayed, which accounts for their longer time of survival. We also find that neuronal over-expression of CAST in hSOD1(G93A) transgenic mice inhibited production of putative neurotoxic caspase-cleaved tau and activation of Cdk5, which have been implicated in neurodegeneration in ALS models, and also reduced the formation of SOD1 oligomers. Our data indicate that inhibition of calpain with CAST is neuroprotective in an ALS mouse model. CAST (encoding calpastatin) inhibits hyperactivated calpain to prevent motor neuron disease operating through a cascade of events as indicated in the schematic, with relevance to amyotrophic lateral sclerosis (ALS). We propose that over-expression of CAST in motor neurons of hSOD1(G93A) mice inhibits activation of CDK5, breakdown of cytoskeletal proteins (NFs, MAP2 and Tau) and regulatory molecules (Cam Kinase IV, Calcineurin A), and disease-causing proteins (TDP-43, $\alpha$-Synuclein and Huntingtin) to prevent neuronal loss and delay neurological deficits. In our experiments, CAST could also inhibit cleavage of Bid, Bax, AIF to prevent mitochondrial, ER and lysosome-mediated cell death mechanisms. Similarly, CAST over-expression in neurons attenuated pathological effects of TDP-43, $\alpha$-synuclein and Huntingtin. These results suggest a potential value of specific small molecule inhibitors of calpains in delaying the development of ALS. Read the Editorial Highlight for this article on page 140.
Takenari Yamashita, Sayaka Teramoto and Shin Kwak.
Phosphorylated TDP-43 becomes resistant to cleavage by calpain: A regulatory role for phosphorylation in TDP-43 pathology of ALS/FTLD. Neuroscience Research, December 2015.
Abstract TAR DNA-binding protein-43 (TDP-43) pathology, which includes the presence of abnormal TDP-43-containing inclusions with a loss of nuclear TDP-43 in affected neurons, is a pathological hallmark of amyotrophic lateral sclerosis (ALS) and/or frontotemporal lobar degeneration (FTLD). TDP-43 in the pathological brains and spinal cords of ALS/FTLD patients is abnormally fragmented and phosphorylated. It is believed that the generation of aggregation-prone TDP-43 fragments initiates TDP-43 pathology, and we previously reported that calpain has an important role in the generation of such aggregation-prone TDP-43 fragments. However, the role of phosphorylation in TDP-43 pathology has not been largely elucidated, despite previous observations that several kinases and their kinases are involved in TDP-43 phosphorylation. Here, we investigated the role of TDP-43 phosphorylation in the calpain-dependent cleavage of TDP-43 and found that phosphorylated, full-length TDP-43 and calpain-dependent TDP-43 fragments were more resistant to cleavage by calpain than endogenous full-length TDP-43 was. These results suggest that both phosphorylated and calpain-cleaved TDP-43 fragments persist intracellularly for a length of time that is sufficient for self-aggregation, thereby serving as seeds for inclusions.
R Stifanese, M Averna, R De Tullio, M Pedrazzi, M Milanese, T Bonifacino, G Bonanno, F Salamino, S Pontremoli and E Melloni.
Role of calpain-1 in the early phase of experimental ALS.. Archives of biochemistry and biophysics 562:1–8, November 2014.
Abstract Elevation in [Ca(2+)]i and activation of calpain-1 occur in central nervous system of SOD1(G93A) transgenic mice model of amyotrophic lateral sclerosis (ALS), but few data are available about the early stage of ALS. We here investigated the level of activation of the Ca(2+)-dependent protease calpain-1 in spinal cord of SOD1(G93A) mice to ascertain a possible role of the protease in the aetiology of ALS. Comparing the events occurring in the 120 day old mice, we found that [Ca(2+)]i and activation of calpain-1 were also increased in the spinal cord of 30 day old mice, as indicated by the digestion of some substrates of the protease such as nNOS, $\alpha$II-spectrin, and the NR2B subunit of NMDA-R. However, the digestion pattern of these proteins suggests that calpain-1 may play different roles depending on the phase of ALS. In fact, in spinal cord of 30 day old mice, activation of calpain-1 produces high amounts of nNOS active species, while in 120 day old mice enhanced-prolonged activation of calpain-1 inactivates nNOS and down-regulates NR2B. Our data reveal a critical role of calpain-1 in the early phase and during progression of ALS, suggesting new therapeutic approaches to counteract its onset and fatal course.
[Calpain plays a crucial role in TDP-43 pathology].. Rinshō shinkeigaku = Clinical neurology 54(12):1151–4, January 2014.
Abstract Amyotrophic lateral sclerosis (ALS) is the most common adult-onset motor neuron disease affecting healthy middle-aged individuals. Mislocalization of TAR DNA binding protein of 43 kDa (TDP-43) or TDP-43 pathology observed in the spinal motor neurons is the pathological hallmark of ALS. The mechanism generating TDP-43 pathology remained uncertain. Several reports suggested that cleavage of TDP-43 into aggregation-prone fragments might be the earliest event. Therefore, elucidation of the protease(s) that is responsible for TDP-43 cleavage in the motor neurons is awaited. ALS-specific molecular abnormalities other than TDP-43 pathology in the motor neurons of sporadic ALS patients include inefficient RNA editing at the GluA2 glutamine/arginine (Q/R) site, which is specifically catalyzed by adenosine deaminase acting on RNA 2 (ADAR2). We have developed the conditional ADAR2 knockout (AR2) mice, in which the ADAR2 gene is targeted in motor neurons. We found that Ca(2+)-dependent cysteine protease calpain cleaved TDP-43 into aggregation-prone fragments, which initiated TDP-43 mislocalization in the motor neurons expressing abnormally abundant Ca(2+)-permeable AMPA receptors. Here we summarized the molecular cascade leading to TDP-43 pathology observed in the motor neurons of AR2 mice and discussed possible roles of dysregulation of calpain-dependent cleavage of TDP-43 in TDP-43 pathology observed in neurological diseases in general.
Roberta De Tullio, Monica Averna, Marco Pedrazzi, Bianca Sparatore, Franca Salamino, Sandro Pontremoli and Edon Melloni.
Differential regulation of the calpain-calpastatin complex by the L-domain of calpastatin.. Biochimica et biophysica acta 1843(11):2583–91, 2014.
Abstract Here we demonstrate that the presence of the L-domain in calpastatins induces biphasic interaction with calpain. Competition experiments revealed that the L-domain is involved in positioning the first inhibitory unit in close and correct proximity to the calpain active site cleft, both in the closed and in the open conformation. At high concentrations of calpastatin, the multiple EF-hand structures in domains IV and VI of calpain can bind calpastatin, maintaining the active site accessible to substrate. Based on these observations, we hypothesize that two distinct calpain-calpastatin complexes may occur in which calpain can be either fully inhibited (I) or fully active (II). In complex II the accessible calpain active site can be occupied by an additional calpastatin molecule, now a cleavable substrate. The consequent proteolysis promotes the accumulation of calpastatin free inhibitory units which are able of improving the capacity of the cell to inhibit calpain. This process operates under conditions of prolonged [Ca(2+)] alteration, as seen for instance in Familial Amyotrophic Lateral Sclerosis (FALS) in which calpastatin levels are increased. Our findings show that the L-domain of calpastatin plays a crucial role in determining the formation of complexes with calpain in which calpain can be either inhibited or still active. Moreover, the presence of multiple inhibitory domains in native full-length calpastatin molecules provides a reservoir of potential inhibitory units to be used to counteract aberrant calpain activity.
A role for calpain-dependent cleavage of TDP-43 in amyotrophic lateral sclerosis pathology.. Nature communications 3:1307, 2012. | CommonCrawl |
Recent studies have demonstrated the power of recurrent neural networks for machine translation, image captioning and speech recognition. For the task of capturing temporal structure in video, however, there still remain numerous open research questions. Current research suggests using a simple temporal feature pooling strategy to take into account the temporal aspect of video. We demonstrate that this method is not sufficient for gesture recognition, where temporal information is more discriminative compared to general video classification tasks. We explore deep architectures for gesture recognition in video and propose a new end-to-end trainable neural network architecture incorporating temporal convolutions and bidirectional recurrence. Our main contributions are twofold; first, we show that recurrence is crucial for this task; second, we show that adding temporal convolutions leads to significant improvements. We evaluate the different approaches on the Montalbano gesture recognition dataset, where we achieve state-of-the-art results.
Realistic music generation is a challenging task. When building generative models of music that are learnt from data, typically high-level representations such as scores or MIDI are used that abstract away the idiosyncrasies of a particular performance. But these nuances are very important for our perception of musicality and realism, so in this work we embark on modelling music in the raw audio domain. It has been shown that autoregressive models excel at generating raw audio waveforms of speech, but when applied to music, we find them biased towards capturing local signal structure at the expense of modelling long-range correlations. This is problematic because music exhibits structure at many different timescales. In this work, we explore autoregressive discrete autoencoders (ADAs) as a means to enable autoregressive models to capture long-range correlations in waveforms. We find that they allow us to unconditionally generate piano music directly in the raw audio domain, which shows stylistic consistency across tens of seconds.
Learning useful representations without supervision remains a key challenge in machine learning. In this paper, we propose a simple yet powerful generative model that learns such discrete representations. Our model, the Vector Quantised-Variational AutoEncoder (VQ-VAE), differs from VAEs in two key ways: the encoder network outputs discrete, rather than continuous, codes; and the prior is learnt rather than static. In order to learn a discrete latent representation, we incorporate ideas from vector quantisation (VQ). Using the VQ method allows the model to circumvent issues of "posterior collapse" -- where the latents are ignored when they are paired with a powerful autoregressive decoder -- typically observed in the VAE framework. Pairing these representations with an autoregressive prior, the model can generate high quality images, videos, and speech as well as doing high quality speaker conversion and unsupervised learning of phonemes, providing further evidence of the utility of the learnt representations.
This paper introduces Associative Compression Networks (ACNs), a new framework for variational autoencoding with neural networks. The system differs from existing variational autoencoders (VAEs) in that the prior distribution used to model each code is conditioned on a similar code from the dataset. In compression terms this equates to sequentially transmitting the dataset using an ordering determined by proximity in latent space. Since the prior need only account for local, rather than global variations in the latent space, the coding cost is greatly reduced, leading to rich, informative codes. Crucially, the codes remain informative when powerful, autoregressive decoders are used, which we argue is fundamentally difficult with normal VAEs. Experimental results on MNIST, CIFAR-10, ImageNet and CelebA show that ACNs discover high-level latent features such as object class, writing style, pose and facial expression, which can be used to cluster and classify the data, as well as to generate diverse and convincing samples. We conclude that ACNs are a promising new direction for representation learning: one that steps away from IID modelling, and towards learning a structured description of the dataset as a whole.
Probabilistic generative models can be used for compression, denoising, inpainting, texture synthesis, semi-supervised learning, unsupervised feature learning, and other tasks. Given this wide range of applications, it is not surprising that a lot of heterogeneity exists in the way these models are formulated, trained, and evaluated. As a consequence, direct comparison between models is often difficult. This article reviews mostly known but often underappreciated properties relating to the evaluation and interpretation of generative models with a focus on image models. In particular, we show that three of the currently most commonly used criteria---average log-likelihood, Parzen window estimates, and visual fidelity of samples---are largely independent of each other when the data is high-dimensional. Good performance with respect to one criterion therefore need not imply good performance with respect to the other criteria. Our results show that extrapolation from one criterion to another is not warranted and generative models need to be evaluated directly with respect to the application(s) they were intended for. In addition, we provide examples demonstrating that Parzen window estimates should generally be avoided.
Due to the phenomenon of "posterior collapse," current latent variable generative models pose a challenging design choice that either weakens the capacity of the decoder or requires augmenting the objective so it does not only maximize the likelihood of the data. In this paper, we propose an alternative that utilizes the most powerful generative models as decoders, whilst optimising the variational lower bound all while ensuring that the latent variables preserve and encode useful information. Our proposed $\delta$-VAEs achieve this by constraining the variational family for the posterior to have a minimum distance to the prior. For sequential latent variable models, our approach resembles the classic representation learning approach of slow feature analysis. We demonstrate the efficacy of our approach at modeling text on LM1B and modeling images: learning representations, improving sample quality, and achieving state of the art log-likelihood on CIFAR-10 and ImageNet $32\times 32$.
This paper investigates recently proposed approaches for defending against adversarial examples and evaluating adversarial robustness. We motivate 'adversarial risk' as an objective for achieving models robust to worst-case inputs. We then frame commonly used attacks and evaluation metrics as defining a tractable surrogate objective to the true adversarial risk. This suggests that models may optimize this surrogate rather than the true adversarial risk. We formalize this notion as 'obscurity to an adversary,' and develop tools and heuristics for identifying obscured models and designing transparent models. We demonstrate that this is a significant problem in practice by repurposing gradient-free optimization techniques into adversarial attacks, which we use to decrease the accuracy of several recently proposed defenses to near zero. Our hope is that our formulations and results will help researchers to develop more powerful defenses.
We consider the task of unsupervised extraction of meaningful latent representations of speech by applying autoencoding neural networks to speech waveforms. The goal is to learn a representation able to capture high level semantic content from the signal, e.g. phoneme identities, while being invariant to confounding low level details in the signal such as the underlying pitch contour or background noise. The behavior of autoencoder models depends on the kind of constraint that is applied to the latent representation. We compare three variants: a simple dimensionality reduction bottleneck, a Gaussian Variational Autoencoder (VAE), and a discrete Vector Quantized VAE (VQ-VAE). We analyze the quality of learned representations in terms of speaker independence, the ability to predict phonetic content, and the ability to accurately reconstruct individual spectrogram frames. Moreover, for discrete encodings extracted using the VQ-VAE, we measure the ease of mapping them to phonemes. We introduce a regularization scheme that forces the representations to focus on the phonetic content of the utterance and report performance comparable with the top entries in the ZeroSpeech 2017 unsupervised acoustic unit discovery task.
Bellemare et al. (2016) introduced the notion of a pseudo-count, derived from a density model, to generalize count-based exploration to non-tabular reinforcement learning. This pseudo-count was used to generate an exploration bonus for a DQN agent and combined with a mixed Monte Carlo update was sufficient to achieve state of the art on the Atari 2600 game Montezuma's Revenge. We consider two questions left open by their work: First, how important is the quality of the density model for exploration? Second, what role does the Monte Carlo update play in exploration? We answer the first question by demonstrating the use of PixelCNN, an advanced neural density model for images, to supply a pseudo-count. In particular, we examine the intrinsic difficulties in adapting Bellemare et al.'s approach when assumptions about the model are violated. The result is a more practical and general algorithm requiring no special apparatus. We combine PixelCNN pseudo-counts with different agent architectures to dramatically improve the state of the art on several hard Atari games. One surprising finding is that the mixed Monte Carlo update is a powerful facilitator of exploration in the sparsest of settings, including Montezuma's Revenge.
We present a novel neural network for processing sequences. The ByteNet is a one-dimensional convolutional neural network that is composed of two parts, one to encode the source sequence and the other to decode the target sequence. The two network parts are connected by stacking the decoder on top of the encoder and preserving the temporal resolution of the sequences. To address the differing lengths of the source and the target, we introduce an efficient mechanism by which the decoder is dynamically unfolded over the representation of the encoder. The ByteNet uses dilation in the convolutional layers to increase its receptive field. The resulting network has two core properties: it runs in time that is linear in the length of the sequences and it sidesteps the need for excessive memorization. The ByteNet decoder attains state-of-the-art performance on character-level language modelling and outperforms the previous best results obtained with recurrent networks. The ByteNet also achieves state-of-the-art performance on character-to-character machine translation on the English-to-German WMT translation task, surpassing comparable neural translation models that are based on recurrent networks with attentional pooling and run in quadratic time. We find that the latent alignment structure contained in the representations reflects the expected alignment between the tokens.
This work explores conditional image generation with a new image density model based on the PixelCNN architecture. The model can be conditioned on any vector, including descriptive labels or tags, or latent embeddings created by other networks. When conditioned on class labels from the ImageNet database, the model is able to generate diverse, realistic scenes representing distinct animals, objects, landscapes and structures. When conditioned on an embedding produced by a convolutional network given a single image of an unseen face, it generates a variety of new portraits of the same person with different facial expressions, poses and lighting conditions. We also show that conditional PixelCNN can serve as a powerful decoder in an image autoencoder. Additionally, the gated convolutional layers in the proposed model improve the log-likelihood of PixelCNN to match the state-of-the-art performance of PixelRNN on ImageNet, with greatly reduced computational cost.
We propose a probabilistic video model, the Video Pixel Network (VPN), that estimates the discrete joint distribution of the raw pixel values in a video. The model and the neural architecture reflect the time, space and color structure of video tensors and encode it as a four-dimensional dependency chain. The VPN approaches the best possible performance on the Moving MNIST benchmark, a leap over the previous state of the art, and the generated videos show only minor deviations from the ground truth. The VPN also produces detailed samples on the action-conditional Robotic Pushing benchmark and generalizes to the motion of novel objects.
This paper introduces WaveNet, a deep neural network for generating raw audio waveforms. The model is fully probabilistic and autoregressive, with the predictive distribution for each audio sample conditioned on all previous ones; nonetheless we show that it can be efficiently trained on data with tens of thousands of samples per second of audio. When applied to text-to-speech, it yields state-of-the-art performance, with human listeners rating it as significantly more natural sounding than the best parametric and concatenative systems for both English and Mandarin. A single WaveNet can capture the characteristics of many different speakers with equal fidelity, and can switch between them by conditioning on the speaker identity. When trained to model music, we find that it generates novel and often highly realistic musical fragments. We also show that it can be employed as a discriminative model, returning promising results for phoneme recognition.
Deep autoregressive models have shown state-of-the-art performance in density estimation for natural images on large-scale datasets such as ImageNet. However, such models require many thousands of gradient-based weight updates and unique image examples for training. Ideally, the models would rapidly learn visual concepts from only a handful of examples, similar to the manner in which humans learns across many vision tasks. In this paper, we show how 1) neural attention and 2) meta learning techniques can be used in combination with autoregressive models to enable effective few-shot density estimation. Our proposed modifications to PixelCNN result in state-of-the art few-shot density estimation on the Omniglot dataset. Furthermore, we visualize the learned attention policy and find that it learns intuitive algorithms for simple tasks such as image mirroring on ImageNet and handwriting on Omniglot without supervision. Finally, we extend the model to natural images and demonstrate few-shot image generation on the Stanford Online Products dataset.
Sequential models achieve state-of-the-art results in audio, visual and textual domains with respect to both estimating the data distribution and generating high-quality samples. Efficient sampling for this class of models has however remained an elusive problem. With a focus on text-to-speech synthesis, we describe a set of general techniques for reducing sampling time while maintaining high output quality. We first describe a single-layer recurrent neural network, the WaveRNN, with a dual softmax layer that matches the quality of the state-of-the-art WaveNet model. The compact form of the network makes it possible to generate 24kHz 16-bit audio 4x faster than real time on a GPU. Second, we apply a weight pruning technique to reduce the number of weights in the WaveRNN. We find that, for a constant number of parameters, large sparse networks perform better than small dense networks and this relationship holds for sparsity levels beyond 96%. The small number of weights in a Sparse WaveRNN makes it possible to sample high-fidelity audio on a mobile CPU in real time. Finally, we propose a new generation scheme based on subscaling that folds a long sequence into a batch of shorter sequences and allows one to generate multiple samples at once. The Subscale WaveRNN produces 16 samples per step without loss of quality and offers an orthogonal method for increasing sampling efficiency.
We present a meta-learning approach for adaptive text-to-speech (TTS) with few data. During training, we learn a multi-speaker model using a shared conditional WaveNet core and independent learned embeddings for each speaker. The aim of training is not to produce a neural network with fixed weights, which is then deployed as a TTS system. Instead, the aim is to produce a network that requires few data at deployment time to rapidly adapt to new speakers. We introduce and benchmark three strategies: (i) learning the speaker embedding while keeping the WaveNet core fixed, (ii) fine-tuning the entire architecture with stochastic gradient descent, and (iii) predicting the speaker embedding with a trained neural network encoder. The experiments show that these approaches are successful at adapting the multi-speaker neural network to new speakers, obtaining state-of-the-art results in both sample naturalness and voice similarity with merely a few minutes of audio data from new speakers.
The recently-developed WaveNet architecture is the current state of the art in realistic speech synthesis, consistently rated as more natural sounding for many different languages than any previous system. However, because WaveNet relies on sequential generation of one audio sample at a time, it is poorly suited to today's massively parallel computers, and therefore hard to deploy in a real-time production setting. This paper introduces Probability Density Distillation, a new method for training a parallel feed-forward network from a trained WaveNet with no significant difference in quality. The resulting system is capable of generating high-fidelity speech samples at more than 20 times faster than real-time, and is deployed online by Google Assistant, including serving multiple English and Japanese voices. | CommonCrawl |
Prove the following corollary to the Fundamental Theorem of Galois Theory. Use only the FTGT statements to prove it. Let $E/F$ be a (finite) Galois extension, with Galois group $G=\gal_F(E)$. Let $L_1,L_2\in\sub_F(E)$ and $H_1,H_2\in\sub(G)$.
Let $f(x)\in\Q[x]$ be such that it has a non-real root. Let $E$ be the splitting field of $f(x)$ over $\Q$. Prove that $\gal_\Q(E)$ has even order.
Consider the polynomial $f(x)=x^3+2x^2+2x+2\in\Q[x]$, and $E$ its splitting field over $\Q$.
Show that $f(x)$ is irreducible over $\Q$.
Find $[E:\Q]$. Fully explain your calculation.
Let $E$ be a field, $G$ a subgroup of $\aut(E)$, $F=E_G$, and $L\in\sub_F(E)$. Show that $L^*=\aut_L(E)$, and it is a subgroup of $G$.
Let $E/L/F$ be a field tower.
Prove that if $E/F$ is a normal extension then so is $E/L$.
Prove that if $E/F$ is a Galois extension then so is $E/L$.
Let $F$ be a field, $\alpha_1,\dots,\alpha_n$ elements from some extension $E$ of $F$, and $R$ a commutative ring with unity. If $\varphi_1,\varphi_2:F(\alpha_1,\dots,\alpha_n)\to R$ are homomorphisms such that $\varphi_1(a)=\varphi_2(a)$ for all $a\in F$ and $\varphi_1(\alpha_i)=\varphi_2(\alpha_i)$ for $i=1,\dots,n$, then $\varphi_1=\varphi_2$.
Let $f(x)=x^5-2\in\Q[x]$, and $E$ the splitting field of $f(x)$. Consider the group $G=\aut_\Q(E)$.
What is the order of $G$?
What are the orders of elements in $G$?
Let $F=\F_p(t)$ be the field of rational functions on $t$ with coefficients in $\F_p$. Consider the polynomial $f(x)=x^p-t\in F[x]$.
Show that $f(x)$ has no root in $F$.
Show that the Frobeni\us endomorphism $\Phi:F\to F$ is not surjective.
Show that $f(x)$ has exactly one root, and that root has multiplicity $p$.
Show that $f(x)$ is irreducible over $F$. | CommonCrawl |
Jianghong Bao. Complex dynamics in the segmented disc dynamo. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3301-3314. doi: 10.3934\/dcdsb.2016098.
Cameron J. Browne, Sergei S. Pilyugin. Minimizing $\\mathcal R_0$ for in-host virus model with periodic combination antiviral therapy. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3315-3330. doi: 10.3934\/dcdsb.2016099.
Tan H. Cao, Boris S. Mordukhovich. Optimal control of a perturbed sweeping process via discrete approximations. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3331-3358. doi: 10.3934\/dcdsb.2016100.
Andrei Cozma, Christoph Reisinger. Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3359-3377. doi: 10.3934\/dcdsb.2016101.
Cristina Cross, Alysse Edwards, Dayna Mercadante, Jorge Rebaza. Dynamics of a networked connectivity model of epidemics. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3379-3390. doi: 10.3934\/dcdsb.2016102.
Ian H. Dinwoodie. Computational methods for asynchronous basins. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3391-3405. doi: 10.3934\/dcdsb.2016103.
Miao Du, Lixin Tian. Infinitely many solutions of the nonlinear fractional Schr\u00F6dinger equations. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3407-3428. doi: 10.3934\/dcdsb.2016104.
Nguyen Huu Du, Nguyen Thanh Dieu. Long-time behavior of an SIR model with perturbed disease transmission coefficient. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3429-3440. doi: 10.3934\/dcdsb.2016105.
Klemens Fellner, Evangelos Latos, Takashi Suzuki. Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3441-3462. doi: 10.3934\/dcdsb.2016106.
Hui Huang, Jian-Guo Liu. Error estimates of the aggregation-diffusion splitting algorithms for the Keller-Segel equations. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3463-3478. doi: 10.3934\/dcdsb.2016107.
Hyung Ju Hwang, Youngmin Oh, Marco Antonio Fontelos. The vanishing surface tension limit for the Hele-Shaw problem. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3479-3514. doi: 10.3934\/dcdsb.2016108.
Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3515-3550. doi: 10.3934\/dcdsb.2016109.
Kai Liu, Zhi Li. Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\\alpha$-stable processes. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3551-3573. doi: 10.3934\/dcdsb.2016110.
Jisun Lim, Seongwon Lee, Yangjin Kim. Hopf bifurcation in a model of TGF-$\\beta$ in regulation of the Th 17 phenotype. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3575-3602. doi: 10.3934\/dcdsb.2016111.
Rong Liu, Feng-Qin Zhang, Yuming Chen. Optimal contraception control for a nonlinear population model with size structure and a separable mortality. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3603-3618. doi: 10.3934\/dcdsb.2016112.
Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3619-3635. doi: 10.3934\/dcdsb.2016113.
Thanh-Anh Nguyen, Dinh-Ke Tran, Nhu-Quan Nguyen. Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3637-3654. doi: 10.3934\/dcdsb.2016114.
Tingting Su, Xinsong Yang. Finite-time synchronization of competitive neural networks with mixed delays. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3655-3667. doi: 10.3934\/dcdsb.2016115.
Yejuan Wang. On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3669-3708. doi: 10.3934\/dcdsb.2016116.
Chao Xing, Ping Zhou, Hong Luo. The steady state solutions to thermohaline circulation equations. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3709-3722. doi: 10.3934\/dcdsb.2016117.
Zhiting Xu. Traveling waves in an SEIR epidemic model with the variable total population. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3723-3742. doi: 10.3934\/dcdsb.2016118.
Hongfu Yang, Xiaoyue Li, George Yin. Permanence and ergodicity of stochastic Gilpin-Ayala population model with regime switching. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3743-3766. doi: 10.3934\/dcdsb.2016119.
Feng Zhou, Chunyou Sun. Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3767-3792. doi: 10.3934\/dcdsb.2016120.
Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21(10): 3793-3808. doi: 10.3934\/dcdsb.2016121. | CommonCrawl |
The integers are made up of positive numbers, negative numbers and zero. The positive numbers are like the naturals, but with a "plus" before: $$+1, +2, +3, +4, \ldots$$. Nevertheless, the "plus" of the positive numbers does not need to be be written. On the other hand, the negative numbers are like the naturals but with a "minus" before: $$-1, -2, -3, -4,\ldots$$ The number zero is special, because it is the only one that has neither a plus nor a minus, showing that it is neither positive nor negative.
Although they may seem a bit strange, the negative numbers are used every day.
For example, someone gets into an elevator on the ground floor. Nevertheless, he does not want to go up, rather he wants to go down because that is where the parking is. Then he pushes the button for the floor $$-1$$, the floor beneath the ground floor. If he had pushed the button for the first floor, he would have gone to the first floor: and this is not what he wanted!
A line is drawn and it is divided into equal segments.
The positive numbers are drawn on the right of the zero in order: first $$1$$, then $$2, 3$$, etc.
The negative numbers are drawn on the left of the zero as follows: first $$-1$$, then $$-2$$, $$-3$$, etc.
It is said that an integer is smaller than another one if when we draw it, it is placed on its left. In the previous drawing, we can see, for example, that: $$-2$$ is smaller than $$4$$, that $$-5$$ is smaller than $$-1$$, and that $$0$$ is smaller than $$3$$. To write this we will use the following symbol: $$<$$. This symbol means: the number that is on the left is smaller than the one that is on the right. In the previous example we would write: $$-2<4, -5<-1$$ and $$0<3$$.
$$5$$ is a natural number, therefore it is also an integer. Also, since it does not have a minus in front of it, it is positive. $$-31$$ is $$31$$ with a minus before it. As $$31$$ is natural, $$-31$$ is integer. And since it has a minus before, it is negative. $$-11.2$$ is $$11.2$$ with a minus before. But $$11.2$$ is not a natural number, therefore it is not an integer. $$80$$ is a natural number and therefore it is integer. Since it is not preceded by a minus, it is positive. $$6.2$$ is not natural, therefore it is not an integer.
The integers are: $$5, -31$$ and $$80$$. The only negative is $$-31$$, the other two are positive. | CommonCrawl |
How can we represent the state of the world and the goal? In a multi-task setting, enumerating all of the objects that the robot might need to pay attention to can become impractical: the number and types of objects might vary, and detecting them requires a dedicated vision pipeline. Instead, we can operate directly on the robot's sensors, representing the state as the image from the robot's camera and the goal as an image of the world as we would like it to be. To specify a new task, a user simply provides a goal image. We note that one could extend this work to more complex ways of specifying goals, such as through language or demonstrations, or by optimizing over goals as in this previous blog post.
The task: Make the world look like this image.
Reinforcement learning is a formalism for training agents to maximize the sum of rewards. For goal-conditioned reinforcement learning, one choice for the reward is the negative distance between the current state and the goal state, so that maximizing the reward corresponds to minimizing the distance to a goal state.
which effectively says, "choose the best action according to this Q function." By using this procedure, we obtain a policy that maximizes the sum of rewards, i.e. reaches various goals.
The nice thing about this goal resampling is that we can simultaneously learn how to reach multiple goals at once without needing more data from the environment. Overall, this simple modification can result in substantially faster learning.
The method outlined above makes two major assumptions: (1) you have access to a reward function and (2) you have access to a goal sampling distribution $p(g)$. Prior works that use this goal relabeling strategy ( Kaelbling '93 , Andrychowicz '17 , Pong '18 ) operate on ground truth state information (e.g., the Cartesian position of an object), where it is easy to manually design both the goal distribution $p(g)$ and reward function. However, when moving to vision-based tasks where goals are images, both of these assumptions introduce practical concerns. For one, it is not clear which reward function we should use, as pixel-wise distance to a goal image may not be semantically meaningful. Second, because our goals are images, we need a goal image distribution $p(g)$ from which we can sample goal images. Manually designing a distribution over goal images is a non-trivial task and image generation is still an active field of research. Instead, we would like our agent to autonomously imagine its own goals and learn how to reach them.
We can mitigate the challenges associated with goal-image conditioned Q learning by learning a representation for images and using this representation, rather than the images themselves, for RL. The key question becomes: what properties should our representation satisfy? To compute semantically meaningful rewards, we need a representation that captures the underlying factors of variations of images. Furthermore, we need a way to easily generate new goals.
We achieve these objectives by first training a generative latent variable model, which in our case is a variational autoencoder (VAE). This generative model converts high-dimensional observations $x$, like images, into low-dimensional latent variables $z$, and vice versa. The model is trained so that the latent variables capture the underlying factors of variation in an image, similar to the abstract representations a human may use to interpret the world and goals. Given a current image $x$ and goal image $x_g$, we convert them into latent variables $z$ and $z_g$ respectively. We then use these latent variables to representation the state and goal for our reinforcement learning algorithm. Learning Q functions and policies on top of this low-dimensional latent space rather than directly on images results in faster learning.
The agent encodes the current image ($x$) and goal image ($x_g$) into a latent space and use distances in that latent space for reward.
Using the latent variable representations for the images and goals also solves another problem: how to compute rewards. Rather than using pixel-wise error as our reward, we use the distance in the latent space for the reward to train our agent to reach a goal. In the full research paper describing our method, we show that this corresponds to maximizing the probability of reaching the goal and provides a much more effective learning signal.
This generative model is also important because it allows an agent to easily generate goals in the latent space. In particular, our generative model is designed so that sampling latent variables is trivial: we just sample latents from the VAE prior. We use this sampling mechanism for two reasons: First, it provides a mechanism for an agent set its own goals. The agent simply samples a value for the latent variable from our generative model, and tries to reach that latent goal. Second, this resampling mechanism is also used to relabel goals as mentioned above. Because our generative model is trained to encode real images into the prior, the samples from our latent variable prior correspond to meaningful latent goals.
Even without a human providing a goal, our agent can still generate its own goals, both for exploration and for goal relabeling.
All together, the latent variable representation of images (1) captures the underlying factors of a scene, (2) provides meaningful distances to optimize, and (3) provides an efficient goal sampling mechanism, allowing us to efficiently train a goal-conditioned reinforcement learning agent that operates directly on pixels. We call the overall method reinforcement learning with imagined goals (RIG).
We conducted experiments to test if we RIG would be sample-efficient enough to train a real world robot policy in a reasonable amount of time. We tested the robot's ability to reach user-specified positions and push objects to desired locations, as indicated by a goal image. The robot is trained with access only to 84x84 RGB images and without access to joint angles or object positions. The robot first learns by settings its own goals in the latent space. We can use the decoder to visualize the goals that the robot imagines for itself. In the GIF below, the top frame shows the decoded "imagined" goals, while the bottom frame shows the rollout of the actual policy.
The robot sets its own goals (top) and practices reaching them (bottom).
The human gives a goal image (top) and the robot reaches it (bottom).
Left: The Sawyer robot setup. Right: The human gives a goal image (top) and the robot reaches it (bottom).
Training a policy directly from images makes it easy to change tasks from reaching to object pushing. We simply added an object, added a table, and adjusted the camera. Lastly, despite working directly from pixels, these experiments did not take long to run. The reaching results took about an hour, while the pushing results took about 4.5 hours of real-robot interaction time. Many real-world robot reinforcement learning results use ground-truth state information like the position of an object. However, this usually requires additional machinery, like purchasing and setting up extra sensors or training an object-detection system. In contrast, our method only requires an RGB camera and works directly from the images.
For more results, including ablations and comparisons to baselines, we encourage readers to read the paper.
We've shown that we can train a real-world robot policy directly from images to achieve a variety of tasks in a sample-efficient way. There are a number of exciting next steps for this project. It might not be possible to represent all tasks with a goal image, and one could instead use other modalities, such as language and demonstrations, to represent goals. Also, while we provide a mechanism to sample goals for autonomous exploration, can we choose these goals in a more principled way to perform even better exploration? Incorporating ideas from intrinsic motivation would allow our policy to actively choose goals that will inform the policy to learn more quickly about what it can and cannot reach. Another future direction is to train our generative model so that it is aware of the dynamics. Encoding information about the environment dynamics could make the latent space even better suited for reinforcement learning, resulting in faster learning. Lastly, there are a variety of robot tasks whose state representation would be difficult to capture with sensors, such as manipulating deformable objects or handling scenes with variable number of objects. Scaling up RIG to solve these tasks would be an exciting next step.
The environment code is available here, and the algorithm code is available here.
We would like to thank Sergey Levine for his valuable feedback when preparing this blog post, as well as Deirdre Quillen and Kyle Hsu for feedback on later drafts of this post. | CommonCrawl |
By way of introduction, imagine an infinitely thin sheet of paper: it is two dimensional (with two sides. Two points on one side of that surface will be $\small x$ (arbitrary) units apart. If you introduce a small deflection on that surface between those points, the distance between those two points (in two dimensions) will increase. However, if you introduce a third dimension, the points could be represented (e.g.) on the surface of a sphere, where the direct connection between those points in the third dimension (through the volume determined by that topology) is shorter than the distance between those points on the surface of that surface (imagine any two points on a sphere).
arc length = $\small 2 \pi r \times (\theta \div 360)$ [$\small \theta$ in degrees].
2 * 3.14159 * 10 * (88/360) = 15.3589 cm.
arc length = $\small r \theta$ [$\small \theta$ in radians].
There are $\small (2 \pi\ rads)/360^\circ$, so in our previous example), so 88° = 88 x (2 x 3.14159)/360 = 1.535890 rad.
Therefore, the arc length is (10 cm)(1.535890 rad) = 15.3589 cm.
chord length c = 2(10)sin(88/2) = (20)sin(44) = 13.8932 cm.
Source!) incorrectly say to use radians in this formula ($\small c = 2 r sin(\theta/2)$), which gives the obviously wrong answer: 0.268 cm!
This website describes it correctly, and here are my handwritten calculations showing how it's done (note that in my example, I use a central angle of 90°, simplifying the calculations). | CommonCrawl |
Given two circles. It is required to find all their common tangents, i.e. all such lines that touch both circles simultaneously.
The described algorithm will also work in the case when one (or both) circles degenerate into points. Thus, this algorithm can also be used to find tangents to a circle passing through a given point.
The number of common tangents to two circles can be 0,1,2,3,4 and infinite. Look at the images for different cases.
Here, we won't be considering degenerate cases, i.e when the circles coincide (in this case they have infinitely many common tangents), or one circle lies inside the other (in this case they have no common tangents, or if the circles are tangent, there is one common tangent).
In most cases, two circles have four common tangents.
If the circles are tangent , then they will have three common tangents, but this can be understood as a degenerate case: as if the two tangents coincided.
Moreover, the algorithm described below will work in the case when one or both circles have zero radius: in this case there will be, respectively, two or one common tangent.
Summing up, we will always look for four tangents for all cases except infinite tangents case (The infinite tangents case needs to be handled separately and it is not discussed here). In degenerate cases, some of tangents will coincide, but nevertheless, these cases will also fit into the big picture.
Denote $r_1$ and $r_2$ the radii of the first and second circles, and by $(v_x,v_y)$ the coordinates of the center of the second circle and point $v$ different from origin. (Note: we are not considering the case in which both the circles are same).
Total we got eight solutions instead four. However, it is easy to understand where superfluous decisions arise: in fact, in the latter system, it is enough to take only one solution (for example, the first). In fact, the geometric meaning of what we take $\pm r_1$ and $\pm r_2$ is clear: we are actually sorting out which side of each circle there is a straight line. Therefore, the two methods that arise when solving the latter system are redundant: it is enough to choose one of the two solutions (only, of course, in all four cases, you must choose the same family of solutions).
The last thing that we have not yet considered is how to shift the straight lines in the case when the first circle was not originally located at the origin. However, everything is simple here: it follows from the linearity of the equation of a straight line that the value $a \cdot x_0 + b \cdot y_0$ (where $x_0$ and $y_0$ are the coordinates of the original center of the first circle) must be subtracted from the coefficient $c$. | CommonCrawl |
Abstract: A problem of index coding with side information was first considered by Y. Birk and T. Kol (IEEE INFOCOM, 1998). In the present work, a generalization of index coding scheme, where transmitted symbols are subject to errors, is studied. Error-correcting methods for such a scheme, and their parameters, are investigated. In particular, the following question is discussed: given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols $\delta$, what is the shortest length of a linear index code, such that every receiver is able to recover the required information? This question turns out to be a generalization of the problem of finding a shortest-length error-correcting code with a prescribed error-correcting capability in the classical coding theory. The Singleton bound and two other bounds, referred to as the $\alpha$-bound and the $\kappa$-bound, for the optimal length of a linear error-correcting index code (ECIC) are established. For large alphabets, a construction based on concatenation of an optimal index code with an MDS classical code, is shown to attain the Singleton bound. For smaller alphabets, however, this construction may not be optimal. A random construction is also analyzed. It yields another inexplicit bound on the length of an optimal linear ECIC. Finally, the decoding of linear ECIC's is discussed. The syndrome decoding is shown to output the exact message if the weight of the error vector is less or equal to the error-correcting capability of the corresponding ECIC. | CommonCrawl |
The analytic index of elliptic pseudodifferential operators on a singular foliation - Mathematics > Operator Algebras - Download this document for free, or read online. Document in PDF available to download.
Abstract: In previous papers arxiv:math-0612370 and arxiv:0909.1342 we defined theC*-algebra and the longitudinal pseudodifferential calculus of any singularfoliation M,F. Here we construct the analytic index of an elliptic operatoras a KK-theory element, and prove that the same element can be obtained from an-adiabatic foliation- TF on $M \times \R$, which we introduce here.
On JALT 2000-Towards the New Millennium. Proceedings of the JALT Annual International Conference on Language Teaching and Learning and Educational Materials Expo 26th, Shizuoka City, Japan, November 2-5, 2000. - Long, Robert, Ed.; van Troyer, Gene, Ed.; Lane, Keith, Ed.; Swanson, Malcom, Ed. | CommonCrawl |
I think this is primarily based on peoples' differing definitions, not any fundamental misunderstanding. There's no harm done when a student is learning grade school math.
then it has one element, as you might expect.
The natural numbers are those we use for counting objects. Some authors include $0$ as a 'natural' number of objects to have and some don't. There is no consensus in the mathematical community. We do know that the number zero came much later in human history than the rest of them, so some argue that it is less 'natural', but on the other hand it is perfectly normal to consider zero objects (not as abstract as considering -4 objects) so it may be a natural number. It depends on the author.
I think that modern definitions include zero as a natural number. But sometimes, expecially in analysis courses, it could be more convenient to exclude it.
in making limits, $0$ plays a role which is symmetric to $\infty$, and the latter is not a natural number.
I have seen children measure things with a ruler by aligning the $1$ mark instead of the $0$ mark. It is difficult to explain them why you have to start from $0$ when they are used to start counting from $1$. The marks in the rule identify the end of the centimeters, not the start, since the first centimeter goes from 0 to 1.
An example where counting from $1$ leads to somewhat wrong names is in the names of intervals between musical notes: the interval between C and F is called a fourth, because there are four notes: C, D, E, F. However the distance between C and F is actually three tones. This has the ugly consequence that a fifth above a fourth (4+3) is an octave (7) not a nineth! On the other hand if you put your first finger on the C note of a piano your fourth finger goes to the F note.
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Is $0$ a natural number?
What is the relationship between any natural number and two other natural numbers?
Is there any compelling reason why $0$ *shouldn't* be a natural number?
Is sum and product of $k$ natural numbers always different from that of some other $k$ natural numbers?
How many natural number this equation have?
Why natural set is an infinite set with each element a finite number? | CommonCrawl |
I wonder why this is? Is it just a coincidence?
Since the determinant represents the area of a parallelogram, the derivative at $x_0$ is the area of parallelogram formed by $ \langle f'(x_0),\ g'(x_0)\rangle $ and $ \langle f(x_0),\ g(x_0)\rangle $ where $ g(x_0) $ is scaled to 1. Still, this interpretation is kind of forced. What does it mean? | CommonCrawl |
dc.description.abstract In this paper we construct an "interior penalty" discontinuous Galerkin method to approximate the minimizer of a variational problem related to the $p(x)$-Laplacian. The function $p:\Omega\to [p_1,p_2]$ is log-Hölder continuous and $1<p_1\leq p_2<\infty$. We prove that the minimizers of the discrete functional converge to the solution. We also make some numerical experiments in dimension one to compare this method with the conforming Galerkin method, in the case where $p_1$ is close to one. This example is motivated by its applications to image processing. | CommonCrawl |
In a continuous time market model we consider the problem of existence of an equivalent martingale measure with density lying within given lower and upper bounds and we characterize a necessary and sufficient condition for this. In this sense our main result can be regarded as a version of the fundamental theorem of asset pricing. In our approach we suggest an axiomatic description of prices on $L_p$-spaces (with $p\in [1,\infty)$) and we rely on extension theorems for operators. | CommonCrawl |
6 Is it good to migrate my problem into MathOverflow?
4 Is MSE a good place for peer reviews? Is it appropriate to start a bounty for this type?
12 Connections in terms of tangent ($\infty$-)categories?
9 Is it pragmatically beneficial to pursue a master degree before applying for a math PhD in the U.S.?
8 Learning German from scratch in order to read mathematics? | CommonCrawl |
We can simply perform a greedy algorithm: stop at a rest stop which has the Maximal $C_i$ in this area you can arrived, and stay at this rest stop as long as possible. After that, find the next rest stop which also has the Maximal $C_j$ in next area. It's supposed $P_j > P_i$. And also stay at rest stop $J$ as long as possible, until Farmer John is coming.
It's easy to prove the greedy algorithm: If it didn't stay at the maximal $C_i$ rest stop, then changed to another rest stop $C_k$, that means $C_k$ must be less than $C_i$, so that $C_k\times T < C_i\times T$. So that can't be the best answer, so the greedy algorithm is correct. | CommonCrawl |
Nested quotations are great not only for writing literature with a complex narrative structure, but also in programming languages. While it may seem necessary to use different quotation marks at different nesting levels for clarity, there is an alternative. We can display various nesting levels using $k$-quotations, which are defined as follows.
A $1$-quotation is a string that begins with a quote character, ends with another quote character and contains no quote characters in-between. These are just the usual (unnested) quotations. For example, 'this is a string' is a $1$-quotation.
For $k > 1$, a $k$-quotation is a string that begins with $k$ quote characters, ends with another $k$ quote characters and contains a nested string in-between. The nested string is a non-empty sequence of $(k-1)$-quotations, which may be preceded, separated, and/or succeeded by any number of non-quote characters. For example, ''All 'work' and no 'play''' is a $2$-quotation.
Given a description of a string, you must determine its maximum possible nesting level.
The input consists of two lines. The first line contains an integer $n$ ($1 \le n \le 100$). The second line contains $n$ integers $a_1, a_2, \ldots , a_ n$ ($1 \le a_ i \le 100$), which describe a string as follows. The string starts with $a_1$ quote characters, which are followed by a positive number of non-quote characters, which are followed by $a_2$ quote characters, which are followed by a positive number of non-quote characters, and so on, until the string ends with $a_ n$ quote characters.
Display the largest number $k$ such that a string described by the input is a $k$-quotation. If there is no such $k$, display no quotation instead. | CommonCrawl |
Using pigeon hole principle show that among any $11$ integers, sum of $6$ of them is divisible by $6$.
I thought we can be sure that we have $6$ odd or $6$ even numbers among $11$ integers.Suppose that we have $6$ odd integers. Obviously sum of these $6$ odd numbers is divisible by $2$, so it remains to show that this sum is divisible by $3$ as well. But how?
one residue class must have 3 of these 5 integers belonging to it.
In the former case, let $x_0 \equiv 0 \mod 3$, $x_1 \equiv 1 \mod 3$ and $x_2 \equiv 2 \mod3$. Then summing $x_1, x_2, x_3$, we get, $x_1 + x_2 + x_3 \equiv 0 \mod 3$.
In the latter case, we have 3 integers among the 5, say $x_1, x_2, x_3$ such that, $x_1 \equiv x_2 \equiv x_3 \equiv k \mod 3$, again summing these three we get $x_1 + x_2 + x_3 \equiv 3k \equiv 0 \mod 3$.
This proves that among any 5 integers, sum of some 3 of them is divisible by 3.
Now, we have 11 integers. By the previous result, we can choose 3 of them such that there sum is divisible by 3. Denote this sum by $s_1$. Now, we are left with 8 integers, again, choose 3 of them such that there sum is divisible by 3. Denote this by $s_2$. Now, we are left with 5 integers. Choose $s_3$ similarly.
Thus we have 3 sums: $s_1, s_2, s_3$ (each of which are sums of 3 integers). These sums are divisible by 3. So, each of these sums are congruent to either 0 or 3 modulo 6.
Now, since there are 3 sums, and two residue clases ($, $), by Pigeonhole principle, one residue class must have two sums belonging to it. Let $s_i$ and $s_j$ be those sums. Either, $s_i \equiv s_j \equiv 0 \mod 6$ or $s_i \equiv s_j \equiv 3 \mod 6$. In both the cases, $s_i + s_j \equiv 0 \mod 6$.
Since, $s_i$ and $s_j$ are both sum of 3 integers, $s_i + s_j$ is a sum of 6 integers (which is divisible by 6). This completes the proof.
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Choose 38 different natural numbers less than 1000, Prove among these there exists at least two whose difference is at most 26.
Given 5 integers show that you can find two whose sum or difference is divisible by 6.
There are 31 houses on north street numbered from 1 to 57. Show at least two of them have consecutive numbers.
Pigeon hole principle: Prove that any set of six positive integers whose sum is 13 must contain at least one subset whose sum is three. | CommonCrawl |
Archive ouverte HAL - Particles approximations of Vlasov equations with singular forces : Propagation of chaos.
Abstract : We obtain the mean field limit and the propagation of chaos for a system of particles interacting with a singular interaction force of the type $1/|x|^\alpha$, with $\alpha <1$ in dimension $d \geq 3$. We also provide results for forces with singularity up to $\alpha < d-1$ but with large enough cut-off. This last result thus almost includes the most interesting case of Coulombian or gravitational interaction, but it is also interesting when the strength of the singularity $\alpha$ is larger but close to one, in which case it allows for very small cut-off.
Keywords : Derivation of kinetic equations. Particle methods. Vlasov equation. Propagation of chaos Derivation of kinetic equations. Particle methods. Vlasov equation. Propagation of chaos. | CommonCrawl |
From an email question "I am currently working on dosimeters. Lots of queries.. A broad overview of all dosimeters will help (if you can add when you get time)"
firstly why is larger volume of thimble ion chamber sometimes a drawback esp for steep dose gradients. I did understand this at some stage but I am confused after all this reading… is it because of electron disequilibrium?
The concept of spatial resolution is not clear and it says compromise between minimizing the volume to achieve best spatial resolution and maximising the volume to improve signal to noise ratio?
Dosimeters, obviously, measure dose. Either the measurement is direct or it is by some secondary/related phenomenon. So the ways it is done is to see how many electrons are liberated - not quite dose but close and 'precedes' dose. How to measure electrons, well they are charged, so a potential difference (voltage) will establish an electric field which will make the electrons flow to an electrode (oh my God! "electrons flow" that's a CURRENT!!!) where a current is measured. So liberated electrons are $\alpha$ dose.
What other things happen when dose is absorbed. Heat is generated (so you can use a calorimeter as a dosimeter. Bit coarse though, given the temperature change with 1 Gy! Liberated and absorbed electrons will cause chemical changes in redox states of coloured ions - so the changes of Fe3+ to Fe2+ (gain of an electron!). In some fluids, when electrons interact with chemicals there is an excitation change which releases energy as light when it 'relaxes', and if you have a light photon counter, you can then correlate scintillations counted with electron interactions.
So you see the dosimeter is usually based on a simple principle. Much of the complication comes in trying to calibrate the measurement with the dose, and noting in which conditions the relationship is true, and when it is not.
Thimbles - Does size matter?
Unfortunately, yes! The problem of measuring dose is that the larger the sample size of ionizations measured, the smaller the sampling error. This is true for everything. So a bigger thimble will give you a better, more accurate reading. If your very very small thimble measures 100 ionizations and then a stray cosmic electron enters as well, you measure 101 ionizations and have a 1% error. However if your very very large thimble measures 1,000,000 ionizations and another 10 stray electrons enter on a particularly active day for solar flares, you measure 1,000,010 ionizations and you have an error rate of 0.0001%. So bigger detectors lead to smaller errors. That's one part of the story!
The other part is the spatial resolution. The dosimeter's width can be thought of as a pixel in a picture. Here are three pictures of the same image with just the pixel size changed. In the first with large pixels, it is difficult to accurately identify the surface of the soft tissues, however with reducing size (i.e., increasing spatial resolution) the boundary becomes very easy to identify. With dosimeters it is the same. Recent work has been done on surface dose with film, gafchromic and mosfet dosimeters and the surface dose gets smaller each time. Is this because the dose changes with time or measurement, or because the dosimeter is measuring more accurately because its spatial resolution is better?
So you see, as pixel size (read dosimeter size!) gets smaller, spatial resolution gets better … BUT noise component of signal-to-noise ratio gets larger.
by email: Difference between absolute, reference and relative dosimetry?
My understanding - Absolute dosimeter is one which does not require calibration and it relies on its own accuracy instead of referring to a standard. They are the ones lying in NIST and other dosimeters around the world are calibrated (directly or indirectly) against these standard ones giving them a constant or calibration coefficient.
When we do yearly calibration of a MV beam (TRS-398 absorbed dose to water protocol) .. we use our good calibrated ionisation chamber.. get the reading, put in all the factors, coefficients and corrections) - get our absorbed dose to water at Dmax in Gy/MU — this would be under reference conditions ( reference dosimetry). We also calculate the PDD 20,10 and then put in the formula to know the TPR 20,10 (thats how we do it in our centre). All this using water tank.
The weekly QA are then done as a quick check to ensure we are getting the same output value or within tolerance for a particular linac with a particular beam energy using solid water phantoms. In the monthly QA - we also do beam energy check using PDD20/10 to ensure within tolerance value .. these weekly and monthly ones would be relative dosimetry. Does that all sound ok?
Or is it that absolute and reference dosimetry is same and all we do is relative dosimetry?
ABSOLUTE dosimetry is a direct measure of ionization or absorbed dose under standard conditions, which are things like calorimetry [measure energy deposited which eventually appears as heat], electrons released (in an ionization chamber where electronic charge is measured), or ion formation where the number of valence changes in a known amount of ions is directly related to the number of electrons (chemical dosimeter). Because of the need for accuracy, absolute dosimetry informs a standard and is usually tied to a single government based agency responsible for 'the standard'.
RELATIVE dosimetry refers to the use of a dosimeter which has a secondary standard, e.g., film dosimetry, is accurate but relies on the production of a known dose exposure somewhere on the film, so that relative opacity can be related to dose. This dose is actually 'measured' by a reference dosimeter (e.g., machine head parallel plate chambers).
REFERENCE dosimeters are produced as an STANDARD-defining exercise which will use an ABSOLUTE dosimeter. This implies that the dosimeter has sufficient accuracy, that it is measured under "Standard Conditions", i.e., fixed and reproducible, to established an absolute reference for other dosimeters. As an absolute standard, all other absolute dosimeters will have been handed in for checking and returned with a certificate that details accuracy and error. It also implies a management protocol for these dosimeters being referenced back to the Standard (or absolute) dosimeter periodically.
I think that what resides in the NZ Radiation Laboratory (and its Australian counterpart) is an ABSOLUTE DOSIMETER which forms the STANDARD OF DOSIMETRY measurement in that country [it's REFERENCE DOSIMETER]. The device that your department has which has been certificated by that body but which resides in your department is related this REFERENCE STANDARD OF DOSIMETRY. All the other devices for measuring dose that are internally calibrated within your department are then RELATIVE STANDARDS OF DOSIMETRY.
So you see that the word "RELATIVE" can have slightly different meanings (relative dose & relative standard), but reference is a supremacy issue. | CommonCrawl |
$K$ is quasi-isomorphic to a K-flat complex $E^\bullet $ whose terms are flat $R$-modules with $E^ i = 0$ for $i \not\in [a, \infty ]$.
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The tag you filled in for the captcha is wrong. You need to write 0BYL, in case you are confused. | CommonCrawl |
Abstract: A $k$-free like group is a $k$-generated group $G$ with a sequence of $k$-element generating sets $Z_n$ such that the girth of $G$ relative to $Z_n$ is unbounded and the Cheeger constant of $G$ relative to $Z_n$ is bounded away from 0. By a recent result of Benjamini-Nachmias-Peres, this implies that the critical bond percolation probability of the Cayley graph of $G$ relative to $Z_n$ tends to $1/(2k-1)$ as $n\to \infty$. Answering a question of Benjamini, we construct many non-free groups that are $k$-free like for all sufficiently large $k$. | CommonCrawl |
In algebraic topology we often encounter chain complexes with extra multiplicative structure. For example, the cochain complex of a topological space has what is called the $E_\infty$-algebra structure which comes from the cup product.
In this talk I present an idea for studying such chain complexes, $E_\infty$ differential graded algebras ($E_\infty$ DGAs), using stable homotopy theory. Namely, I discuss new equivalences between $E_\infty$ DGAS that are defined using commutative ring spectra.
ring spectra are equivalent. Quasi-isomorphic $E_\infty$ DGAs are $E_\infty$ topologically equivalent. However, the examples I am going to present show that the opposite is not true; there are $E_\infty$ DGAs that are $E_\infty$ topologically equivalent but not quasi-isomorphic. This says that between $E_\infty$ DGAs, we have more equivalences than just the quasi-isomorphisms.
I also discuss interaction of $E_\infty$ topological equivalences with the Dyer-Lashof operations and cases where $E_\infty$topological equivalences and quasi-isomorphisms agree. | CommonCrawl |
We present stellar evolution calculations of the remnant of the merger of two carbon-oxygen white dwarfs (CO WDs). We focus on cases that have a total mass in excess of the Chandrasekhar mass. After the merger, the remnant manifests as an $L \sim 3\times 10^4 L_\odot$ source for $\sim 10^4$ yr. A dusty wind may develop, leading these sources to be self-obscured and to appear similar to extreme AGB stars. Roughly $\sim 10$ such objects should exist in the Milky Way and M31 at any time. As found in previous work, off-center carbon fusion is ignited within the merger remnant and propagates inward via a carbon flame, converting the WD to an oxygen-neon (ONe) composition. By following the evolution for longer than previous calculations, we demonstrate that after carbon-burning reaches the center, neutrino-cooled Kelvin-Helmholtz contraction leads to off-center neon ignition in remnants with masses $\ge 1.35 M_\odot$. The resulting neon-oxygen flame converts the core to a silicon WD. Thus, super-Chandrasekhar WD merger remnants do not undergo electron-capture induced collapse as traditionally assumed. Instead, if the remnant mass remains above the Chandrasekhar mass, we expect that it will form a low-mass iron core and collapse to form a neutron star. Remnants that lose sufficient mass will end up as massive, isolated ONe or Si WDs. | CommonCrawl |
For more information, contact Richard Kent.
Abstract: The divergence of a pair of geodesic rays emanating from a point is a measure of how quickly they are moving away from each other. In Euclidean space divergence is linear, while in hyperbolic space divergence is exponential. Gersten used this idea to define a quasi-isometry invariant for groups, also called divergence, which has been investigated for classes of groups including fundamental groups of 3-manifolds, mapping class groups and right-angled Artin groups. I will discuss joint work with Pallavi Dani on divergence in right-angled Coxeter groups (RACGs). We characterise 2-dimensional RACGs with quadratic divergence, and prove that for every positive integer d, there is a RACG with divergence polynomial of degree d.
Abstract: I will discuss the distribution of critical values of a smooth random function on a compact m-dimensional Riemann manifold (M,g) described as a random superposition of eigenfunctions of the Laplacian. The notion of randomness that we use has a naturally built in small parameter $\varepsilon$, and we show that as $\varepsilon\to 0$ the distribution of critical values closely resemble the distribution of eigenvalues of certain random symmetric $(m+1)\times (m+1)$-matrices of the type introduced by E. Wigner in quantum mechanics. Additionally, I will explain how to recover the metric $g$ from statistical properties of the Hessians of the above random function.
Abstract: Swarup proved that every one-ended word hyperbolic group has a locally connected Gromov boundary. However for CAT(0) groups, non-locally connected boundaries are easy to construct. For instance the boundary of F_2 x Z is the suspension of a Cantor set.
In joint work with Kim Ruane, we have studied boundaries of CAT(0) spaces with isolated flats. If G acts properly, cocompactly on such a space X, we give a necessary and sufficient condition on G such that the boundary of X is locally connected. As a corollary, we deduce that such a group G is semistable at infinity.
The theory of fibered faces implies that pseudo-Anosov mapping classes with bounded normalized dilatation can be partitioned into a finite number of families with related dynamics. In this talk we discuss the problem of finding concrete description of the members of these families. One conjectural way generalizes a well-known sequence defined by Penner in '91. However, so far no known examples of this type come close to the smallest known accumulation point of normalized dilatations. In this talk we describe a different construction that uses mixed-sign Coxeter systems. A deformation of the simplest pseudo-Anosov braid monodromy can be obtained in this way, and hence this model does realize the smallest known accumulation point.
Strominger-Yau-Zaslow conjecture suggests that the Ricci-flat metric on Calabi-Yau manifolds might be related to holomorphic discs. In this talk, I will define a new open Gromov-Witten invariants on elliptic K3 surfaces trying to explain this conjecture. The new invariant satisfies certain wall-crossing formula and multiple cover formula. I will also establish a tropical-holomorphic correspondence. Moreover, this invariant is expected to be equivalent to the generalized Donaldson-Thomas invariants in the hyperK\"ahler metric constructed by Gaiotto-Moore-Neitzke. If time allowed, I will talk about the connection with disks counting on Calabi-Yau 3-folds.
Joint work with Ann Lemahieu (Lille). A notion of Milnor fibration for meromorphic functions and the corresponding concepts of monodromy and monodromy zeta function, introduced by Gussein-Zade, Luengo and Melle, invite to consider the notion of topological zeta function for meromorphic germs and the corresponding monodromy conjecture. We try to motive these notions and discuss the plane case. We show that the poles do not behave as in the holomorphic case but still do satisfy a generalization of the monodromy conjecture.
Inspired by Donaldson's program, we introduce the Kahler Ricci flow with conical singularities. The main part of this talk is to show that the conical Kahler Ricci flow exists for short time and for long time in a proper space. These existence results are highly related to heat kernel and Bessel functions. We will also discuss some easy applications of the conical Kahler Ricci flow in conical Kahler geometry.
In this talk, we explain the analysis of the following system of (degenerate) elliptic equation $$ \overline \partial^\pi w = 0, \, d(w^*\lambda \circ j) = 0 $$ associated for each given contact triad $(Q,\lambda,J)$ on a contact manifold $(Q,\xi)$. (Such an equation was first introduced by Hofer.) We directly work with this equation on the contact manifold without involving the symplectization process. We explain the basic analytical ingredients towards the construction of moduli space of solutions, which we call contact instantons. I will indicate how one can define contact homology type invariants using such a moduli space, which is still in progress. The talk is partially based on the joint work with Rui Wang.
Measurable group theory is the study of groups via their actions on measure spaces. While the classification for amenable groups was essentially complete by the early 1980's, progress for nonamenable groups has been slow to emerge. The last 15 years has seen a surge in the classification of ergodic actions of nonamenable groups, with methods coming from diverse areas. We will survey these new results, as well as, give an introduction to the operator algebra techniques that have been used.
We define the Pascal Triangle of a discrete (gray scale) image as a pyramidal ar- rangement of complex-valued moments and we explore its geometric significance. In particular, we show that the entries of row k of this triangle correspond to the Fourier series coefficients of the moment of order k of the Radon transform of the image. Group actions on the plane can be naturally prolonged onto the entries of the Pascal Triangle. We study the induced action of some common group actions, such as translation, rotations, and reflections, and we propose simple tests for equivalence and self- equivalence for these group actions. The motivating application of this work is the problem of recognizing "shapes" on images, for example characters, digits or simple graphics. Application to the MERGE project, in which we developed a fast method for segmenting hazardous material signs on a cellular phone, will be also discussed.
This is joint work with my graduate students Shanshan Huang and Andrew Haddad.
Three manifolds of constant vector curvature.
A Riemannian manifold M is said to have extremal curvature K if all sectional curvatures are bounded above by K or if all sectional curvatures are bounded below by K. A manifold with extremal curvature K has constant vector curvature K if every tangent vector to M belongs to a tangent plane of curvature K. For surfaces, having constant vector curvature is equivalent to having constant curvature. In dimension three, the eight Thurston geometries all have constant vector curvature. In this talk, I will discuss the classification of closed three manifolds with constant vector curvature. Based on joint work with Jon Wolfson.
We will describe an asymptotic relationship between the volume and the Betti numbers of certain locally symmetric spaces. The proof uses an exciting new tool: a synthesis of Gromov-Hausdorff convergence of Riemannian manifolds and Benjamini-Schramm convergence from graph theory.
String theory partition functions often have nice modular properties, which is well understood within the context of representation theory of (supersymmetric extensions) of Virasoro algebra. However, many questions of physical importance are preferrably addressed when string theory is formulated in terms of non-linear sigma model on a Riemann surface with a Riemannian manifold as target space. Traditionally, physicists have studied such sigma models within the realm of perturbation theory, overlooking a large class of very natural critical points of the path integral, namely, closed geodesics on the target space Riemannian manifold. We propose how to take into account the effect of these critical points on the path integral, and initiate its study on Ricci flat targe spaces, such as the K3 surface.
In this talk I will define all the concepts in the title, starting with what a contact manifold is. I will also explain how the heuristic arguments sketched in the literature since 1999 fail to define a homology theory and provide a foundation for a well-defined cylindrical contact homology, while still providing an invariant of the contact structure. A later talk will provide us with a large class of examples under which one can compute a well-defined version of cylindrical contact homology via a new approach the speaker developed for her thesis that is distinct and completely independent of previous specialized attempts.
The problem of computing the Reidemsieter number R(f) of a group automorphism f, that is, the number of f-twisted conjugacy classes, is related to questions in Lefschetz-Nielsen fixed point theory. We say a group has property R-infinity if every group automorphism has infinitely many twisted conjugacy classes. This property has been studied by Fel'shtyn, Gonzalves, Wong, Lustig, Levitt and others, and has applications outside of topology.
Twisted conjugacy classes in lamplighter groups are well understood both geometrically and algebraically. In particular the lamplighter group L_n does not have property R-infinity iff (n,6)=1. In this talk I will extend these results to Diestel-Leader groups with a surprisingly different conclusion. The family of Diestel-Leader groups provides a natural geometric generalization of the lamplighter groups. I will define these groups, as well as Diestel-Leader graphs and describe how these results include a computation of the automorphism group of this family. This is joint work with Melanie Stein and Peter Wong.
Earlier this year, Church, Ellenberg, and Farb developed a new framework for studying sequences of representations of the symmetric groups, using a concept they call an FI--module. I will give an overview of this theory, and describe how it generalizes to sequences of representations of the classical Weyl groups in Type B/C and D. The theory of FI--modules has provided a wealth of new results by numerous authors working in algebra, geometry, and topology. I will outline some of these results, including applications to configurations spaces and groups related to the braid group.
Following Thurston, certain classes of 3-manifolds yield holomorphic maps on the Teichmuller spaces of their boundary components. Inspired by numerical evidence of Kent and Dumas, we present a negative result about the regularity of such maps. Namely, we construct a path of deformations of the hyperbolic structure on a genus-2 handlebody, with two rank-1 cusps. The presence of some extra symmetry yields information about the convex core, which is used to conclude some inequalities involving the extremal length of a certain symmetric curve family. The existence of a critical point for the associated skinning map follows.
A complex projective structure is a certain geometric structure on a (real) surface, and it corresponds a representation from the fundamental group of the base surface into PSL(2,C). We discuss about a certain surgery operation, called a 2π–grafting, which produces a different projective structure, preserving its holonomy representation. This surgery is closely related to three-dimensional hyperbolic geometry.
Abstract: What is the relationship between manifolds and the structure of their diffeomorphism groups? On the positive side, a remarkable theorem of Filipkiewicz says that the group structure determines the manifold: if Diff(M) and Diff(N) are isomorphic, then M and N are diffeomorphic. On the negative side, we know little else. Could the group Diff(M) act by diffeomorphisms on M in nonstandard ways? Does the "size" of Diff(M) say anything about the complexity of M? Ghys asked if M and N are manifolds, and the group of compactly supported diffeomorphisms of N injects into the group of compactly supported diffeomorphisms of M, can the dimension of M be less than dim(N)? We'll discuss these and other questions, and answer these in the (already quite rich) case of dim(M)=1.
This page was last modified on 14 August 2013, at 08:28. | CommonCrawl |
This page presents a follow-up work on our CVPR'18 paper, we improved the proposed weakly-supervised 3D shape completion approach, referred to as amortized maximum likelihood (AML), as well as created two high-quality, challenging, synthetic benchmarks based on ShapeNet and ModelNet . We also presented extensive experiments on real data from KITTI and Kinect . The pre-print itself, a more detailed abstract as well as source code and data will be available below.
This work is based on our earlier CVPR'18 paper, but significantly improves results by utilizing an improved model and more detailed benchmarks. See below for a detailed description of the improvements over our CVPR'18 paper.
We address the problem of 3D shape completion from sparse and noisy point clouds, a fundamental problem in computer vision and robotics. Recent approaches are either data-driven or learning-based: Data-driven approaches rely on a shape model whose parameters are optimized to fit the observations; Learning-based approaches, in contrast, avoid the expensive optimization step by learning to directly predict complete shapes from incomplete observations in a fully-supervised setting. However, full supervision is often not available in practice. In this work, we propose a weakly-supervised learning-based approach to 3D shape completion which neither requires slow optimization nor direct supervision. While we also learn a shape prior on synthetic data, we amortize, i.e., learn, maximum likelihood fitting using deep neural networks resulting in efficient shape completion without sacrificing accuracy. On synthetic benchmarks based on ShapeNet and ModelNet as well as on real robotics data from KITTI and Kinect , we demonstrate that the proposed amortized maximum likelihood approach is able to compete with the fully supervised baseline of and outperforms the data-driven approach of , while requiring less supervision and being significantly faster.
We implemented a completely new pipeline for synthetic data generation to address shortcomings with respect to the visual quality of the predicted shapes. In particular, we used volumetric fusion to obtain detailed, watertight meshes, manually selected $220$ models from ShapeNet or ModelNet and computed both occupancy grids and signed distance functions (SDFs) as well as synthetic observations for experiments.
Additionally, we extended the proposed, weakly-supervised shape completion approach in order to enforce more variety and increase visual quality in the predicted, completed shapes.
We present extensive experiments on ModelNet considering both category-specific and category-agnostic experiments showing that our approach generalizes across object categories and the level of supervision can be reduced even further.
We also present experiments on Kinect , a real-world dataset of chairs and tables, showing that our approach is able to generalize form as few as $30$ training samples.
We increased the spatial resolution used for occupancy grids and SDFs to up to $64^3$ on ModelNet and Kinect and $48 \times 108 \times 48$ on ShapeNet and KITTI to obtain more detailed shape predictions.
We compare the proposed method with the work by Dai et al. as representative of a supervised, learning-based approach as well as an iterative closest point (ICP) baseline based on .
aml-shape-completion Shape completion implementations: amortized maximum likelihood (AML) (including the VAE shape prior), maximum likelihood (ML), Engelmann et al. , Dai et al. , iterative closest point (ICP) and the supervised baseline (Sup). Implementations are mostly in Torch and C++ (for ). Installation requirements and usage instructions are included. Note that the AML version in this repository obtains improved results over our CVPR'18 version at davidstutz/daml-shape-completion.
mesh-evaluation Efficient C++ implementation of mesh-to-mesh distance (accuracy and completeness) as well as mesh-to-point distance; this tool can be used for evaluation.
bpy-visualization-utils Python and Blender tools for visualization of meshes, occupancy grids and point clouds. These tools have been used for visualizations as presented in the paper.
mesh-voxelization Efficient C++ implementation for voxelizing watertight triangular meshes into occupancy grids and/or signed distance functions (SDFs). This tool was used to create the shape completion benchmarks as described below.
mesh-fusion This is a Python implementation of TSDF Fusion using and ; this approach was used to obtain simplified and watertight meshes for our synthetic benchmarks.
Except for ModelNet10 and Kinect, all downloads include benchmarks for three difference resolutions. On ShapeNet and KITTI, these are $24\times 54\times24$, $32\times72\times32$ and $48\times108\times48$. On ModelNet, these include $32^3$, $48^3$ and $64^3$.
SN-noisy: Amazon AWS MPI-INF The "clean" and "noisy" versions of our ShapeNet benchmark; which means that we synthetically generated observations without or with noise which can be used to benchmark shape completion methods. Note that this is not the same as for our CVPR'18 paper.
KITTI: Amazon AWS MPI-INF Our benchmark derived from KITTI; it uses the ground truth 3D bounding boxes to extract observations from the LiDAR point clouds. It does not include ground truth shapes; however, we tried to generate an alternative by considering the same bounding boxes in different timesteps. Note that this is not the same as for our CVPR'18 paper.
tables: Amazon AWS MPI-INF Single-category benchmarks derived from ModelNet's bathtubs, chairs, desks and tables.
ModelNet10: Amazon AWS MPI-INF Benchmark based on all ten categories from ModelNet10.
We also provide pre-trained models for the proposed approach and all baselines; the downloads include models on all datasets and for all resolutions.
AML Models (∼ 2.8GB): Amazon AWS MPI-INF Pre-trained Torch models for the proposed amortized maximum likelihood (AML) approach.
Dai et al. Models (∼ 11.6GB): Amazon AWS MPI-INF Pre-trained Torch models for the fully-supervised baseline of Dai et al. .
Supervised Baseline Models (∼ 1.6GB): Amazon AWS MPI-INF Pre-trained Torch models of our own fully-supervised baseline.
DVAE Shape Prior Models (∼ 2.3GB): Amazon AWS MPI-INF Pre-trained Torch models of our DVAE shape prior.
Nov 30, 2018. Data and models are now also available through a server provided by MPI-INF.
Oct 8, 2018. The paper has been accepted at IJCV: link.springer.com/article/10.1007/s11263-018-1126-y.
May 18, 2018. The pre-print is available on ArXiv.
June 7, 2018. Coda, data and models now available on GitHub.
Figure 1 (click to enlarge): Overview of the proposed, weakly-supervised 3D shape completion approach; see below or paper for details.
We propose an amortized maximum likelihood approach for 3D shape completion, see Figure 1, avoiding slow optimization as required by data-driven approaches and the required supervision of learning-based approaches. Specifically, we first learn a shape prior on synthetic shapes using a (denoising) variational auto-encoder. Subsequently, 3D shape completion can be formulated as a maximum likelihood problem. However, instead of maximizing the likelihood independently for distinct observations, we follow the idea of amortized inference and learn to predict the maximum likelihood solutions directly. Towards this goal, we train a new encoder which embeds the observations in the same latent space using an unsupervised maximum likelihood loss. This allows us to learn 3D shape completion in challenging real-world situations, e.g., on KITTI, and obtain sub-voxel accurate results using signed distance functions at resolutions up to $64^3$ voxels.
For experimental evaluation, we introduce two novel, synthetic shape completion benchmarks based on ShapeNet and ModelNet . We compare our approach to the data-driven approach by Engelmann et al. , a baseline inspired by Gupta et al (2015) and the fully-supervised learning-based approach by Dai et al. ; we additionally present experiments on real data from KITTI and Kinect . Experiments show that our approach outperforms data-driven techniques and rivals learning-based techniques while significantly reducing inference time and using only a fraction of supervision.
In the paper, we discuss various experiments on the four constructed benchmark datasets, i.e., ShapeNet, ModelNet, KITTI and Kinect. We consider the single-category case, the multi-category case as well as multiple resolutions. On KITTI and Kinect, we demonstrate that our approach is able to learn on data without ground truth and visually outperform related work. We also present insights regarding the learned latent space (of the shape prior) and the embedding learned during the inference step (see Figure 1).
Figure 1 (click to enlarge): Qualitative results of the proposed AML approach in comparison to related work by Engelmann et al. (Eng16) and Dai et al. (Dai17) on ShapeNet and ModelNet.
For example, Figure 2 presents experiments on ShapeNet and ModelNet, considering one object category at a time at low resolution (specifically, $24 \times 54 \times 24$ and $32^3$, respectively). As can be seen, our approach outperforms the data-driven approach — referred to as Eng16 — and is able to compete with — indicated as Dai17. This is remarkable, as we use up to $96\%$ less supervision compared to Dai17.
Figure 3 (click to enlarge): Multi-category results on ModelNet (left) and results on KITTI (right). For ModelNet, we used a resolution of $32^3$; on KITTI we use $32 \times 72 \times 32$ and additionally show partial ground truth in green.
In Figure 3, we also show some multi-category reuslts (left) as well as results on KITTI (right) — note that on KITTI, there is only partial ground truth available. The supervised baseline of Dai et al. (Dai17) was trained on ShapeNet, where we put significant effort into modeling KITTI's sensor and noise statistics. The experiments show that our approach performs well under even weaker supervision, i.e., when considering multiple object categories, and is able to learn on real data from KITTI, despite the noise and sparsity.
We presented a novel, weakly-supervised learning-based approach to 3D shape completion from sparse and noisy point cloud observations. We used a (denoising) variational auto-encoder to learn a latent space of shapes for one or multiple object categories using synthetic data. Based on the learned generative model, we formulated 3D shape completion as a maximum likelihood problem. In a second step, we then fixed the learned generative model and trained a new recognition model to amortize the maximum likelihood problem. Compared to related data-driven approaches, our approach offers fast inference at test time; in contrast to other learning-based approaches we do not require full supervision during training. On two newly created synthetic shape completion benchmarks, derived from ShapeNet's cars and ModelNet10, as well as on real data from KITTI and, we demonstrated that AML outperforms related data-driven approaches while being significantly faster. We further showed that AML is able to compete with fully-supervised approaches, both quantitatively and qualitatively, while using only $3-10\%$ supervision or less. We additionally showed that AML is able to generalize across object categories without category supervision during training.
Stutz D, and Geiger A (2018) Learning 3D Shape Completion from Laser Scan Data with Weak Supervision. In: Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR).
Dai A, Qi CR, Nießner M (2017) Shape completion using 3d-encoder-predictor cnns and shape synthesis. In: Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR).
Engelmann F, St¨uckler J, Leibe B (2016) Joint object pose estimation and shape reconstruction in urban street scenes using 3D shape priors. In: Proc. of the German Conference on Pattern Recognition (GCPR).
Gupta S, Arbel´aez PA, Girshick RB, Malik J (2015) Aligning 3D models to RGB-D images of cluttered scenes. In: Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR).
Wu Z, Song S, Khosla A, Yu F, Zhang L, Tang X, Xiao J (2015) 3d shapenets: A deep representation for volumetric shapes. In: Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR).
Chang AX, Funkhouser TA, Guibas LJ, Hanrahan P, Huang Q, Li Z, Savarese S, Savva M, Song S, Su H, Xiao J, Yi L, Yu F (2015) Shapenet: An information-rich 3d model repository. arXivorg 1512.03012.
Geiger A, Lenz P, Urtasun R (2012) Are we ready for autonomous driving? The KITTI vision benchmark suite. In: Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR).
Yang B, Rosa S, Markham A, Trigoni N, Wen H (2018) 3d object dense reconstruction from a single depth view. arXivorg abs/1802.00411. | CommonCrawl |
Let $p$ be a prime number, $V$ a complete discrete valuation ring of unequal caracteristics $(0,p)$, $G$ a smooth affine algebraic group over Spec $V$. Using partial divided powers techniques of Berthelot, we construct arithmetic distribution algebras, with level $m$, generalizing the classical construction of the distribution algebra. We also construct the weak completion of the classical distribution algebra over a finite extension $K$ of $\mathbf Q_p$. We then show that these distribution algebras can be identified with invariant arithmetic differential operators over $G$, and prove a coherence result when the ramification index of $K$ is $< p-1$. | CommonCrawl |
Learning a new skill by observing another individual, the ability to imitate, is a key part of intelligence in human and animals. Can we enable a robot to do the same, learning to manipulate a new object by simply watching a human manipulating the object just as in the video below?
The robot learns to place the peach into the red bowl after watching the human do so.
Such a capability would make it dramatically easier for us to communicate new goals to robots – we could simply show robots what we want them to do, rather than teleoperating the robot or engineering a reward function (an approach that is difficult as it requires a full-fledged perception system). Many prior works have investigated how well a robot can learn from an expert of its own kind (i.e. through teleoperation or kinesthetic teaching), which is usually called imitation learning. However, imitation learning of vision-based skills usually requires a huge number of demonstrations of an expert performing a skill. For example, a task like reaching toward a single fixed object using raw pixel input requires 200 demonstrations to achieve good performance according to this prior work. Hence a robot will struggle if there's only one demonstration presented.
Moreover, the problem becomes even more challenging when the robot needs to imitate a human showing a certain manipulation skill. First, the robot arm looks significantly different from the human arm. Second, engineering the right correspondence between human demonstrations and robot demonstrations is unfortunately extremely difficult. It's not enough simple to track and remap the motion: the task depends much more critically on how this motion affects objects in the world, and we need a correspondence that is centrally based on the interaction.
To enable the robot to imitate skills from one video of a human, we can allow it to incorporate prior experience, rather than learn each skill completely from scratch. By incorporating prior experience, the robot should also be able to quickly learn to manipulate new objects while being invariant to shifts in domain, such as a person providing a demonstration, a varying background scene, or different viewpoint. We aim to achieve both of these abilities, few-shot imitation and domain invariance, by learning to learn from demonstration data. The technique, also called meta-learning and discussed in this previous blog post, is the key to how we equip robots with the ability to imitate by observing a human.
So how can we use meta-learning to make a robot quickly adapt to many different objects? Our approach is to combine meta-learning with imitation learning to enable one-shot imitation learning. The core idea is that provided a single demonstration of a particular task, i.e. maneuvering a certain object, the robot can quickly identify what the task is and successfully solve it under different circumstances. A prior work on one-shot imitation learning achieves impressive results on simulated tasks such as block-stacking by learning to learn across tens of thousands of demonstrations. If we want a physical robot to able to emulate humans and manipulate a variety of novel objects, we need to develop a new system that can learn to learn from demonstrations in the form of videos using a dataset that can be practically collected in the real world. First, we'll discuss our approach for visual imitation of a single demonstration collected via teleoperation. Then, we'll show how it can be extended for learning from videos of humans.
In order to make robots able to learn from watching videos, we combine imitation learning with an efficient meta-learning algorithm, model-agnostic meta-learning (MAML). This previous blog post gives a nice overview of the MAML algorithm. In this approach, we use a standard convolutional neural network with parameters $\theta$ as our policy representation, mapping from an image $o_t$ from the robot's camera and the robot configuration $x_t$ (e.g. joint angles and joint velocities) to robot actions $a_t$ (e.g. the linear and angular velocity of the gripper) at time step $t$.
There are three main steps in this algorithm.
Three steps for our meta-learning algorithm.
Then, we optimize for the initial parameters $\theta$ by driving the updated policy to match the actions from another demonstration with the same object. After meta-training, we can ask the robot to manipulate completely unseen objects by computing gradient steps using a single demonstration of that task. This step is called meta-testing.
Placing items into novel containers using a single demonstration. Left: demo. Right: learned policy.
Learning to push a novel object by watching a human.
Learning to pick up a novel object and place it into a previously unseen bowl.
Learning to push a novel object by watching a human in a different environment from a different viewpoint.
Now that we've taught a robot to learn to manipulate new objects by watching a single video (which we also demonstrated at NIPS 2017), a natural next step is to further scale these approaches to the setting where different tasks correspond to entirely distinct motions and objectives, such as using a wide variety of tools or playing a wide variety of sports. By considering significantly more diversity in the underlying distribution of tasks, we hope that these models will be able to achieve broader generalization, allowing robots to quickly develop strategies for new situations. Further, the techniques we developed here are not specific to robotic manipulation or even control. For instance, both imitation learning and meta-learning have been used in the context of language (examples here and here respectively). In language and other sequential decision-making settings, learning to imitate from a few demonstrations is an interesting direction for future work.
We would like to thank Sergey Levine and Pieter Abbeel for valuable feedback when preparing this blog post. This article was initially published on the BAIR blog, and appears here with the authors' permission. | CommonCrawl |
every nonconstant polynomial over $F$ is a product of linear factors.
Proof. If $F$ is algebraically closed, then every irreducible polynomial is linear. Namely, if there exists an irreducible polynomial of degree $> 1$, then this generates a nontrivial finite (hence algebraic) field extension, see Example 9.7.6. Thus (1) implies (2). If every irreducible polynomial is linear, then every irreducible polynomial has a root, whence every nonconstant polynomial has a root. Thus (2) implies (3).
Assume every nonconstant polynomial has a root. Let $P \in F[x]$ be nonconstant. If $P(\alpha ) = 0$ with $\alpha \in F$, then we see that $P = (x - \alpha )Q$ for some $Q \in F[x]$ (by division with remainder). Thus we can argue by induction on the degree that any nonconstant polynomial can be written as a product $c \prod (x - \alpha _ i)$.
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Abstract: The static gluon-quark-antiquark interaction is investigated using lattice QCD techniques. A Wilson loop adequate to the static hybrid three-body system is developed and, using a $24^3 \times 48$ periodic lattice with $\beta = 6.2$, the potential energy of the system is measured for different geometries. For the medium range behaviour, when the quarks are far apart, we find a string tension which is compatible with two fundamental strings. On the other hand, when the quark and antiquark are nearby, the string tension is larger than two fundamental strings and is compatible with the Casimir scaling. | CommonCrawl |
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