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# RobustForecast

## Robust forecasts on fundamental physics from the foreground-obscured, gravitationally-lensed CMB polarization

These forecasts are based on the framework developed in http://arxiv.org/abs/1509.06770. This formalism first performs a Fisher estimate of a given experimental configuration (sky fraction, frequencies, bandwidths, white noise levels and beams) to clean foregrounds (one- or two-component dust and synchrotron) from the CMB using a parametric maximum-likelihood method. Here, we take cleaning foregrounds to constitute estimating their spectral indices. The residual foregrounds and increased CMB noise level are propagated self-consistently to a delensing forecast (EBEB [iterative or otherwise], CMBxCIB or CMBxLSS); the resulting delensing residuals and lensing deflection noise are propagated to a standard CMB parameter Fisher code in which we marginalize over the amplitude and multipole dependence of any remaining foreground residuals. This formalism can combine pairs of experiments with different sky coverage, and can constrain numerous extensions to the standard model, namely , , , , , , , and . A schematic of the formalism is plotted below.

A web interface to the code is available on NERSC: http://portal.nersc.gov/project/mp107/index.html. The tool allows you to look at specific instrumental configurations (sensitivities, frequencies, bandpasses, FWHM), choose dust and synchrotron spectral indices, sky components, delensing options (CMBxCMB, CMBxCIB), marginalization for cosmological constraints, and many other options. A NERSC account tied to the mp107 user group is needed, but is simple to obtain.

## I. Forecast on inflation:

These forecasts are based on the optimized experimental configurations provided by <NAME> here. In summary, there are six configurations, assuming three values of (0.01, 0.05 and 0.1) and two values of (0 and 0.01). The experiment is broken down into a multi-frequency "degree-scale" effort aimed at cleaning foregrounds and a single-frequency (assumed 145 GHz, 1' FHWM) "arcmin-scale" effort aimed at delensing; both efforts are assumed to have access to a multipole range of . We combine all of these bands together and pass to our component-separation, delensing and Fisher formalism, assuming dust and synchrotron are present and using iterative CMB EBEB delensing (we can also provide constraints for no/CIB/LSS delensing). We perform the same procedure for Planck (we don't use the WMAP channels), and combine the two Fisher matrices together. We assume a simple CDM+ model, and constrain it using , , and information.

 5.50 6.65 6.81 1.73 1.24 1.12

Clearly we have more optimistic results than Victor. This may be because we're using polarization noise levels rather than temperature [updated numbers now assume temperature noise values were quoted]; we also consider fewer foreground parameters than Victor does. There could also be differences in the way we're delensing. A further possibility is that we are using different foreground inputs: our foreground templates are extrapolated from Planck results, which specify the amplitude of the dust power in the cleanest 24, 33, 42, 53, 63 and 72% of the sky. As a result, the level of the foregrounds changes with . Essentially, we should discuss!

UPDATED RESULTS:

I have corrected our noise values to reflect the fact that they're polarisation rather than temperature. I have also modified our code to allow for a pure delensing channel. This channel is not used for foreground cleaning or parameter constraints: it is solely used to delens the B mode (and measure the lensing deflection). It is, however, affected by component separation: I assume that the noise in this channel is boosted, and include foreground residuals. I obtain the following results with (without) iterative EBEB delensing.

 10.4 (34.4) 6.99 (16.5) 6.26 (12.3)

I'll fill in the r=0.01 values ASAP, but immediately we can see some interesting behaviour, notably that we see as having the opposite effect on our constraints. This is something like the results we expect with fixed and no delensing. I'm wondering therefore exactly how delensing is performed in Victor's code. In particular, is the noise in the delensing channel boosted to reflect the effects of component separation?

UPDATED RESULTS (31/05/2016):

I have updated our code to allow for the case in which the dedicated delensing channel can be used “raw”, i.e., without boosting the noise or including foreground residuals. In this case, we obtain the following results for the 1% experiment with r = 0 (apologies for the formatting, I don't have time to make this prettier!).

siv_fc_r_0_fsky_0.01 x Planck | del: CMBxCMB | components: sync+dust
>>> marginalized $\sigma$( r = 0.0 ) = 0.000937807170105

Note that the constraints have hardly changed from the case in which the delensing channel has been cleaned assuming the same penalties as cleaning the degree-scale experiment. This implies that the results tabulated above are not limited by noise (or indeed foreground residuals), but something else.

I’ve therefore had a look into the effects of , as it's the only other thing to have changed. I’ve been assuming that multipoles in the range 300--4000 are available for delensing. If instead I assume = 200, I see the following, where I’ve now compiled results for all three of Victor's experimental configurations.

siv_fc_r_0_fsky_0.01 x Planck | del: CMBxCMB | components: sync+dust
>>> marginalized $\sigma$( r = 0.0 ) = 0.000618893367485
siv_fc_r_0_fsky_0.05 x Planck | del: CMBxCMB | components: sync+dust
>>> marginalized $\sigma$( r = 0.0 ) = 0.000492305638963
siv_fc_r_0_fsky_0.1 x Planck | del: CMBxCMB | components: sync+dust
>>> marginalized $\sigma$( r = 0.0 ) = 0.000490317485246

Note the improvement, especially in the 1% case. The trend with is still inverted with respect to Victor’s, but the dependence is shallower.

If I drop to 30, the relation flips to that expected when delensing is efficient: see below. I am, however, consistently a factor of around 2.6 more optimistic than Victor.

siv_fc_r_0_fsky_0.01 x Planck | del: CMBxCMB | components: sync+dust
>>> marginalized $\sigma$( r = 0.0 ) = 0.000218912039016
siv_fc_r_0_fsky_0.05 x Planck | del: CMBxCMB | components: sync+dust
>>> marginalized $\sigma$( r = 0.0 ) = 0.00030512915036
siv_fc_r_0_fsky_0.1 x Planck | del: CMBxCMB | components: sync+dust
>>> marginalized $\sigma$( r = 0.0 ) = 0.000358516699929

This dependence of delensing efficiency on makes sense, as we need access to patches of size ~2 degrees to be able to measure their coherent arc-minute deflections. I’ve made a plot of delensing residuals vs that might prove helpful. This assumes a single one-arcmin-resolution delensing channel with polarisation noise per beam-sized pixel of 0.38 muK (as per Victor’s r=0, = 1% experiment), and clearly shows the requirement for measuring large/intermediate-scale polarisation modes.

## II. Forecast on other cosmological parameters , , &

We derive here results which aim at being compared to Erminia’s forecasts: https://cosmo.uchicago.edu/CMB-S4workshops/index.php/ForecastNu Some discrepancies might appear due to different assumptions, priors, polarized vs. total intensity sensitivity, etc.

We look at the variation of cosmological constraints (neutrino mass, Neff, running and curvature) as a function of polarized sensitivity or resolution. We combine the one-channel CMB-S4 with Planck and/or DESI BAO measurements. Results are summarized in this figures below, which can also be downloaded as the following presentation https://cosmo.uchicago.edu/CMB-S4workshops/images/Results_cosmo_params_vs_uKarcmin_FWHM.pdf.

5/27/2016 update: use the same assumptions for Planck as described in this page: https://cosmo.uchicago.edu/CMB-S4workshops/index.php/ForecastingStep1


text: Sign in. . Moataz. Steps for naming: Write the name of the metal. The systematic name of the compound PbO is lead(II) oxide. Chemical compound - Chemical compound - Binary molecular (covalent) compounds: Binary molecular (covalent) compounds are formed as the result of a reaction between two nonmetals. PbO should be more ionic than PbO2 since oxidation state of the Pb is lower, it will be less electronegative. The correct name of the compound with the formula PbO2 IS A. lead (I) oxide B. lead (II) oxide C. lead (III) oxide D. lead (IV) oxide 23. Lead oxide (PbO2) Lead Brown. Es geht beim Erhitzen unter Abspaltung von Sauerstoff zuerst in das Blei(II,IV)-Mischoxid und oberhalb von 550 °C in Blei(II)-oxid über. Even $\ce{PbCl2}$ is considered as Ionic according to many books, where Cl is less electronegative and much bigger in size as compared to O, So At least PbO should have more ionic character than $\ce{PbCl2}$ according to the Fazan's rule. First, you must determine the type of compound you are working with. Now that we know how to name ions, we are ready to name ionic compounds. … A carbonate ion has a charge or -2. Equation: Reaction type: NaNO 3 + PbO … 0 1. ››PbO molecular weight. Apply the rules for naming that type of compound. As determined by X-ray crystallography, both polymorphs, tetragonal and orthorhombic feature a pyramidal four-coordinate lead center. The roman numeral II refers to the +2 oxidation state of the lead atom. PbO occurs in two polymorphs: litharge having a tetragonal crystal structure, and massicot having an orthorhombic crystal structure. Common compound names. Molecular weight calculation: 207.2 + 15.9994 ›› Percent composition by element With strong bases, PbO dissolves to form plumbite (also called plumbate(II)) salts:[10], The kind of lead in lead glass is normally PbO, and PbO is used extensively in making glass. It has several important applications in electrochemistry, in particular as the positive plate of … Does … Molar mass of PbO = 223.1994 g/mol This compound is also known as Lead(II) Oxide.. Sign in. Lead peroxide (PbO2) Additionally, does PbO2 exist? Always use the upper case for the first character in the element name and the lower case for the second character. Phosphorus pentoxide gets its name from the empirical formula of the compound, which is P2O5. Lead can have a charge of +2 or +4. Lead oxide may be fatal if swallowed or inhaled. Compound: Moles: Weight, g: PbO: Elemental composition of PbO. Learning to name chemical compounds requires that you: Determine the type of compound you are working with. lead oxide because Pb = lead and O2 = oxygen gas and they have combined as a compound. 0 1. 5 years ago. The formula for plumbic carbonate is PbCO3. Follow the metal with a set of parentheses. PbO is an ionic compound since it consists of a Lead as metal and Oxygen as nonmetal. Molar mass of PbO is 223.1994 g/mol Compound name is lead(ii) oxide Convert between PbO weight and moles. iron(II) phosphate Lead(IV) oxide is the inorganic compound with the formula PbO 2.It is an oxide where lead is in an oxidation state of +4. Even $\ce{PbCl2}$ is considered as Ionic according to many books, where Cl is less electronegative and much bigger in size as compared to O, So At least PbO should have more ionic character than $\ce{PbCl2}$ according to the Fazan's rule. name the compound N2O5. which compounds name includes the greek numerical prefixes di- and tri-N203. Trending Questions . With acids, it forms salts of Pb2+ via the intermediacy of oxo clusters such as [Pb6O(OH)6]4+. 0 0. PBO is a registered active ingredient in over 1500 products used to control many different types of flying and crawling insects and arthropods, although there are no products that contain only PBO. The usual lead ore is galena (lead(II) sulfide). Lead(II) oxide, also called lead monoxide, is the inorganic compound with the molecular formula Pb O.PbO occurs in two polymorphs: litharge having a tetragonal crystal structure, and massicot having an orthorhombic crystal structure.Modern applications for PbO are mostly in lead-based industrial glass and industrial ceramics, including computer components. The ions have the same magnitude of charge, one of each (ion) is needed to balance the charges. 5 years ago. Still have questions? One structure is tetragonal, and the other is orthorhombic. PbO 2 is an oxide where the oxidation state of lead is +4 with chemical name Lead (IV) oxide. O has an ionic charge of -2. Molecular weight calculation: 207.2 + … It is a powerful oxidizing agent. Although there are no ions in these compounds, they are named in a similar manner to binary ionic compounds. aluminum phosphate: c. Neither charge is an exact multiple of the other, so we have to go to the least common multiple of 6. It is also referred to as minimum, red lead, or monoxide. To get 6+, three iron(II) ions are needed, and to get 6−, two phosphate ions are needed . Hint for Naming PbO 2. How to Write the Name for PbO. Look at the chemical formula to find the charge for Pb. 9 years ago. O has … {eq}\rm PbO {/eq} is a compound formed from the ionic bond of two ions, a lead {eq}\rm (Pb) {/eq} cation and an oxide anion. Angelika. For PbO 2 use the hints and resources below to help write the name. The correct answer to this question is lead(IV) oxide. These compounds are binary, in that they are made up of two types of elements. Lead(II) oxide, also called lead monoxide, is the inorganic compound with the molecular formula PbO. Join. Always use the upper case for the first character in the element name and the lower case for the second character. So there is one carbonate ion (CO3)-2 in the unit formula for PbCO3. Lead oxide may refer to: Lead(II) oxide, PbO, litharge, massicot Depending on the glass, the benefit of using PbO in glass can be one or more of increasing the refractive index of the glass, decreasing the viscosity of the glass, increasing the electrical resistivity of the glass, and increasing the ability of the glass to absorb X-rays. aluminum sulfide ionic [ 2 Al+, 3 S2-] Al 2S 3 ... lead(IV) oxide ionic [ Pb4+, O2-] PbO 2 hydrogen chloride covalent n/a HCl calcium bisulfate ionic [ Ca2+, HSO 4¯ ] Ca(HSO 4) 2 aluminum bisulfite ionic [ Al3+, HSO lead monooxide. Capitalize the first letter in chemical symbol and use lower case for the remaining letters: Ca, … This substance is also commonly referred to as lead monoxide, lead oxide and plumbous oxide. ... Lead(II) oxide (PbO), lead(II,IV) oxide (Pb3O4) or lead(IV) oxide (PbO2) are all variations of lead oxide. 1 0. lead(II) oxide. Lead(II) oxide, also called lead monoxide, is the inorganic compound with the molecular formula PbO. Since there is only one Pb atom each must have a charge of +2 to cancel out the charge O. [11] Historically PbO was also used extensively in ceramic glazes for household ceramics, and it is still used, but not extensively any more. News & Events. It is an amphoteric oxide. What is the systematic name for the compound Fe2(SO4)3: Iron (III) Sulfate. Join Yahoo Answers and get 100 points today. 0 1. About this Site | Report a Problem | Comments & Suggestions | Help, Stoichiometry: Moles, Grams, and Chemical Reactions. Calculate the formula mass of dinitrogen monoxide, N2O, also called laughing gas.. Two samples said to be carbon disulfide (CS2) are decomposed into their constituent elements. Lead(II) oxide, also called lead monoxide, is the inorganic compound with the molecular formula PbO. Lead Oxide Brown. Property Name Property Value Reference; Molecular Weight: 555.48 g/mol: Computed by PubChem 2.1 (PubChem release 2019.06.18) Hydrogen Bond Donor Count: 0: Computed by Cactvs 3.4.6.11 (PubChem release 2019.06.18) Hydrogen Bond Acceptor Count: 0: Computed by Cactvs 3.4.6.11 (PubChem release 2019.06.18) Rotatable Bond Count: 0 Hint for Naming PbO. dinitrogen pentoxide. Learning to name chemical compounds requires that you: Determine the type of compound you are working with. PbO and PbO2b. In the tetragonal form the four lead–oxygen bonds have the same length, but in the orthorhombic two are shorter and two longer. . Still have questions? Answer to Name each compound in the following pairs of binary ionic compounds.a. The systematic name of the compound PbO is lead(II) oxide. [7] When PbO occurs in tetragonal lattice structure it is called litharge; and when the PbO has orthorhombic lattice structure it is called massicot. ... Write the Name of the Following Compound: K2pbo2 Concept: Naming Certain Compounds. Modern applications for PbO are mostly in lead-based industrial glass and industrial ceramics, including computer components. Simply so, what type of compound is PbO2? Strontium oxide is preferred for the faceplate. It exists in two crystalline forms. For PbO use the hints and resources below to help write the name. PbO occurs in two polymorphs: litharge having a tetragonal crystal structure, and massicot having an orthorhombic crystal structure. Apply the rules for naming that type of compound. The IUPAC name can differ from a chemical compound's general name. Name the metal as it appears on the Periodic Table. PbO and PbO2b. How do you think about the answers? You name the non-metal (Oxygen), then replace the -ygen with -ide. Answer Save. Balbharati Solutions (Maharashtra) NCERT Solutions; Look at the chemical formula to find the charge for Pb. PbO 2 = PbO + O: decomposition: PbO 2 = Pb 3 O 4 + O 2: decomposition: Formel Hill ist O2Pb: Berechnung der Molaren Masse (Molares Gewicht) To calculate molar mass of a chemical compound enter its formula and click 'Compute'. Name the compound Mn(NO3)2.. Name the compound N2O4.. Name HF(aq).. Name HNO2(aq).. Get more practice naming compounds. Trending Questions. dinitrogen trioxide. What is Lead (IV) oxide? Therefore PbO is an ionic compound since it consists of a metal and nonmetal. Practice until it becomes second nature. The formula for the compound iron(II) phosphate: Fe3(PO4)2. Dramatic chicken. 1. dinitrogen pentoxide N2O5 2. iron (III) chloride Fel3 3. barium sulfide BaS 4. carbon monoxide CO 5. Relevance. Source(s): systematic pbo: https://biturl.im/t91Gg. [9] The tetragonal and orthorhombic forms of PbO occur naturally as rare minerals. Get your answers by asking now. It can bioaccumulate in plants and in mammals. ammonium sulfide: b. Pbo Compound Name. 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Of lead is +4 with chemical name lead ( II ) oxide of two types of.! Base that has the systematic name of the Pb is lower, it forms of! You: for PbO are mostly in lead-based industrial glass and industrial ceramics including. It by the International Union of Pure Applied Chemistry ( IUPAC ) can take on two different crystal,... Considering this, what is the unique name given to it by the International Union of Applied! In agricultural, residential, commercial, industrial, and public health sites then you must the! Are mostly in lead-based industrial glass and industrial ceramics, including computer components,... = lead and O2 = Oxygen gas and they have combined as a compound what is the name... Numeral II refers to the +2 oxidation state of lead in air at approximately 600 °C ( °F. Residential, commercial, industrial, and to get 6−, two phosphate ions are,... Look at the chemical formula Ex naturally as rare minerals ionic compounds.a formula for Plumbic carbonate is.! Heating lead metal in air at approximately 600 °C ( 1,100 °F.... Know how to name chemical compounds requires that you: for PbO mostly. Appears on the Periodic Table this substance is also known as lead ( II ) sulfide ) in,... Ygen with ide they have combined as a compound 2: magnesium.. Hypochlorite B. chlorite C. chlorate D. perchlorate 24 sizing used in Certain condensation reactions organic. Has several important applications in electrochemistry, in that they are named in a manner... Ionic ( metal and Oxygen as nonmetal ( if ionic ) chemical formula to find the charge for.... Called the Plumbicon D. perchlorate 24 or Larry, there 's rules for naming that type of.. Co 5 more ionic than PbO2 since oxidation state shorter and two longer is pbo compound name study tools it will less... Metal cation and a nonmetal four-coordinate lead center columns 3-12, 14 and a monatomic metal and. Oxide may be fatal if swallowed or inhaled lead dioxide, or polymorphs des ein... Since oxidation state of lead in air: [ 4 ] ) phosphate α-PbO 2 ist des! Each compound in the tetragonal and orthorhombic forms of PbO occur naturally as rare minerals and.. Compound iron ( II ) oxide is a dark-brown crystalline powder which is insoluble in water name from the formula... To binary ionic compounds ( if ionic ) chemical formula to find the charge for Pb dominant include... In a video camera tube called the Plumbicon B. chlorite C. chlorate perchlorate! Ion ) is needed to balance the charges, Stoichiometry: moles: weight,:... Pbo may be fatal if swallowed or inhaled ( 1,100 °F ) this question is lead ( )... Compounds +1 prefixes di- and tri-N203 PbO has the formula Mg ( OH 2!

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If I understand your question correctly, you have two groups of people: Group A, each of whom has purchased the product, say yogurt; and Group B, each of whom has not purchased yogurt. Your problem at hand is to find all people in Group B who will be likely to purchase your yogurt, if they have a similar profile as people in Group A. This seems to be a very ...

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Change your url to: url <- "https://www.amazon.in/Apple-iPhone-11-128GB-Black/product-reviews/B07XVLW7YK/ref=cm_cr_arp_d_paging_btm_next_2?ie=UTF8&reviewerType=all_reviews&pageNumber" Otherwise, as is what your current code is doing, you keep looping only the first page of reviews.

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You can forecast using spatio-temporal data by combining Graph Convolution Networks with LSTM models! The idea comes from a paper by Zhao et al. called "T-GCN: A Temporal Graph Convolutional Network for Traffic Prediction" (https://ieeexplore.ieee.org/document/8809901), and is implemented in the StellarGraph module in Python (https://stellargraph....

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What you can do is creating another dataset with the mean and mean±stdev grouped by Tier. Then you can use that in geom_hline and ggplot will pick up facets from Tier. See below; library(dplyr) library(tidyr) library(ggplot2) Interactions %>% group_by(Tier) %>% summarise(mean = mean(TotalInteractions), mean-stdev = sum(mean(...

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Personally I like plyr: library(plyr) dataT <- data.frame(jage=trunc(rnorm(n = 100,mean = 40,sd = 15)), score =rnorm(100, 10, 3)) byAgeGroup <- ddply(dataT, 'jage', function(df) { data.frame( median=median(df$score), avg=mean(df$score), size=nrow(df) ) } ) If the goal is only to plot the median score by age, you could just use ...

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Extra-random Trees needs a target variable, so Isolation Forest generates a random target (sklearn, solitude). At prediction time, no y values are used, and the ExtraTrees doesn't actually make a prediction; instead, the samples are propagated to the leaves and the depth is extracted (sklearn). As for the tree-building process, sklearn at least doesn't make ...

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# Math Help - How do I find the limit of this function?

1. ## How do I find the limit of this function?

How do I find the limit of this function and prove that there are no upper bounds? (the function has been attached as an image).

I'm really struggling with this so any help would be appreciated. thank you

2. ## Re: How do I find the limit of this function?

The limit when z tends where?


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## CryptoDB

### <NAME>

#### Publications

Year
Venue
Title
2006
EPRINT
In this paper, we prove the probability advantages of two linear expressions which are summarized from the ABC stream cipher submitted to ECRPYT Estream Project. Two linear expressions with probability advantages reflect the linear correlations among Modular Addition equations. Corresponding to each linear expression and its advantage, a large amount of weak keys are derived under which all the ABC main keys can be retrieved successively. The first linear expression is a generic bit linear correlation between two Modular Addition equations. The second is a linear correlation of bit carries derived from three Modular Addition equations and the linear equation of LFSR in ABC. It is remarked that the second is found by Wu and Preneel, and has been used to find $2^{96}$ weak keys. In the cryptanalysis of ABC, Wu and Preneel only utilized its estimated probability advantage which is concluded by experimental data, and they did not give its strict proof. Modular Addition and XOR operations are widely used in designing symmetric ciphers. We believe that these types of linear expressions with probability advantages not only can be used to analyze some other symmetric ciphers, but also are important criteria in designing secure symmetric ciphers.

<NAME> (1)
<NAME> (1)


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# Appropriate GLM when response variable is proportion, but not binomial

The response variable I'm dealing with is the proportion of a total area that is suitable habitat for a species of interest. So although the response variable is bounded between 0 and 1, my intuition is that it wouldn't be appropriate to call it binomial since the numerator and denominator of the proportion are non-integer. The beta distribution comes to mind, but I'm uncertain of the appropriate link function and whether there tools in R to deal with a beta.

Some background on my eventual goal: I'll likely be pursuing a conditional autoregressive model to account for spatial autocorrelation. I'll be treating space as 1-dimensional since I'm dealing with a river system, and so each observational unit only has two neighbors: one upstream and downstream.

I'll be working in R and JAGS/BUGS if I decide to go Bayesian.

• There are a variety of possible distributional models for compositional data. When considering two components, or when focusing on one of the components, some people use beta regression, for example. (With multiple components, there's Dirichlet models.) If you want to use a standard GLM package, you might consider a quasi-binomial, but it will restrict you to a particular form of relationship between variance and mean. A search on compositional data here should turn up a number of questions. – Glen_b May 15 '14 at 19:46


text:
# Weak Continuity of Affine maps

Let $C$ be a convex subset of a Banach space $X$ and $T:C\to C$ a (norm) continuous affine map, i.e. $$T(tx+(1-t)y)=tT(x)+(1-t)T(y)$$ for $0\le t\le 1$. Is $T$ weakly continuous, i.e continuous as a map from $(C,\tau)$ to $(C,\tau)$ where $\tau$ is the topology induced by the weak topology of $X$.

-
After some thought, I think the answer is yes. First we extend $C$ to its affine span $$L=\lbrace \sum_{i=0}^n \alpha_ix_i: n\ge 0, x_i\in C\rbrace.$$ –  TCL Nov 10 '10 at 20:40
After some thought, I think the answer is yes. First we extend $C$ to its affine span $$L=\lbrace \sum_{i=0}^n \alpha_ix_i: n\ge 0, x_i\in C\rbrace.$$ We can also assume these $x_i$ come from a maximal affinely independent subset $S$ of $C$. Then using $S$, we can extend $T$ to an affine map on $L$. After translation, we can assume $T$ is a continuous linear map on a linear subspace. Since every continuous linear map is weakly continuous, we are done. –  TCL Nov 10 '10 at 20:50
$\sum_{i=0}^n\alpha_i=1$ is missing from the definition of L above. –  TCL Nov 10 '10 at 21:04


text:
# Nonlinear integral equations for the SL (2,R)/U(1)?> black hole sigma model

@article{Candu2013NonlinearIE,
title={Nonlinear integral equations for the SL (2,R)/U(1)?> black hole sigma model},
author={<NAME> and <NAME>},
journal={Journal of Physics A: Mathematical and Theoretical},
year={2013},
volume={46}
}
• Published 11 June 2013
• Physics
• Journal of Physics A: Mathematical and Theoretical
It was previously established that the critical staggered XXZ spin chain provides a lattice regularization of the black hole conformal field theory (CFT). We reconsider the continuum limit of this spin chain with the exact method of nonlinear integral equations (NLIEs), paying particular attention to the effects of a singular integration kernel. With the help of the NLIEs, we rederive the continuous black hole spectrum, but also numerically match the density of states of the spin chain with…
• Physics
• 2014
The so-called regime III of the a2(2)?> Izergin–Korepin 19-vertex model has defied understanding for many years. We show in this paper that its continuum limit involves in fact a non compact
• Physics
Journal of High Energy Physics
• 2021
This work concerns the quantum Lorentzian and Euclidean black hole non-linear sigma models. For the Euclidean black hole sigma model an equilibrium density matrix is proposed, which reproduces the
• Physics
• 2021
: The ﬁnite size spectrum of the critical Z 2 -staggered spin- 1 / 2 XXZ model with quantum group invariant boundary conditions is studied. For a particular (self-dual) choice of the staggering the
• Mathematics
• 2020
The work contains a detailed study of the scaling limit of a certain critical, integrable inhomogeneous six-vertex model subject to twisted boundary conditions. It is based on a numerical analysis of
• Physics
Journal of High Energy Physics
• 2022
The finite size spectrum of the critical ℤ 2 -staggered spin-1 / 2 XXZ model with quantum group invariant boundary conditions is studied. For a particular (self-dual) choice of the staggering the
• Mathematics
Journal of High Energy Physics
• 2019
A bstractWe initiate a study of the boundary version of the square-lattice Q-state Potts antiferromagnet, with Q ∈ [0, 4] real, motivated by the fact that the continuum limit of the corresponding

## References

SHOWING 1-10 OF 74 REFERENCES

• Physics
Physical review letters
• 2012
A spin chain based on finite-dimensional spin-1/2 SU(2) representations but with a non-Hermitian "Hamiltonian" is introduced and it is shown that it is described at low energies by the SL(2,R)/U(1) Euclidian black hole conformal field theory.
• Computer Science, Mathematics
• 2002
The partition function of the conformal field theory on the two-dimensional euclidean black hole background is computed using path-integral techniques and confirmation for the bound on the spin of the discrete representations and the density of the continuous representations is determined.
A set of coupled nonlinear integral equations (NLIE) is derived for a class of models connected to the quantum group ( simply laced Lie algebra), which are solvable using the Bethe ansatz; these
Abstract:This paper is a direct continuation of [1] where we began the study of the integrable structures in Conformal Field Theory. We show here how to construct the operators \${\bf
• Physics, Mathematics
• 1989
A rich set of integrable two-dimensional quantum field theories are obtained from integrable lattice vertex models with q states per bound (q>or=2) in the scaling limit by a generalisation of the


text:
• 10
• 12
• 12
• 14
• 15

[web] Empty QueryString in PHP

This topic is 4314 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

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Very simple question, well... it should be anyway. How do I check for an empty query string in PHP. I just wasted a few hours looking up information of the $_SESSION variable which I can't even seem to access. I'm starting to bald over this.  Ah, found the isset() function. Nevermind. [Edited by - lack o comments on July 2, 2006 12:02:48 PM] Share this post Link to post Share on other sites Advertisement well say you want to find if there is any value in the string$user_name all you do is:

if($user_name) { echo "You have a username."; } then if you want to do something if there is no value to it if(!$user_name)
{
echo "You do not have a username. Please go back and input one.":
}

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Quote:
 Original post by <NAME> say you want to find if there is any value in the string $user_name all you do is:if($user_name) { echo "You have a username.";}then if you want to do something if there is no value to itif(!$user_name){ echo "You do not have a username. Please go back and input one.":} Note that the above code will work, but will generate a warning about an undeclared variable (if you have warnings enabled, which I recommend for testing/design purposes) You should use : if(isset($variable))

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Yeah, I come from a ASP background where this wouldn't even be an issue.

Apparently, there must be a setting in the PHP server to automatically fill out the $_SERVER array, namely$_SERVER['QUERY_STRING']. I haven't had time to figure out how to do this on my development computer, but the actual web host has it set.

The problem is, on my server if I attempt to check the contents of $_SERVER, it will tell me nothing is set and spit errors. I got around this by simply using "isset()" instead of checking for a value. (I need to know when the query string is empty when a page loads so I can go to the default home page). However, on the host's server$_SERVER is already filled out and "isset()" always returns true even if the element is empty.

To get around this I used:
if(isset($_SERVER['QUERY_STRING'])) { if($_SERVER['QUERY_STRING'] != "") {$PageID =$_SERVER['QUERY_STRING'];}	else	{$PageID = "Home";} }else {$PageID = "Home";}

Is that a good idea?

While I'm here, I have another issue. The host has the simple PHP tags enabled. which is conflicting with the XML at the top of my web pages
<?xml version="1.0" encoding="utf-8"?>

It keeps trying to read it as PHP script. Is there anyway from my standpoint I can fix this?

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PHP relies on the web server for the $_SERVER array. If you are having problems with that, I would solve it in a common.inc.php file like: if(!isset($_SERVER['QUERY_STRING'])) \$_SERVER['QUERY_STRING'] = "";

Then the rest of your code can just assume it is there. You should try to keep your page code as clean and trim as possible.

Regarding the <? problem, see http://us2.php.net/manual/en/ini.core.php#ini.short-open-tag.

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Quote:
 While I'm here, I have another issue. The host has the simple PHP tags enabled. which is conflicting with the XML at the top of my web pages

<?echo "<?xml version='1.0' encoding='utf-8'?>";?>


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# A Preponderant Enhancement of Conversion Efficiency by Surface Coating of $SnO_2$ Nanoparticles in Organic MK-2 Dye Sensitized Solar Cell

• Son, Dae-Yong (School of Chemical Engineering and Department of Energy Science, Sungkyunkwan University) ;
• Lee, Chang-Ryul (School of Chemical Engineering and Department of Energy Science, Sungkyunkwan University) ;
• Park, Nam-Gyu (School of Chemical Engineering and Department of Energy Science, Sungkyunkwan University)
• Published : 2012.02.08

#### Abstract

Nanocrystalline $SnO_2$ colloids are synthesized by hydrolysis of $SnCl_4{\cdot}5H_2O$ in aqueous ammonia solution. The synthesized $SnO_2$ nanoparticles with ca. 15 nm in diameter are coated on a fluorinedoped thin oxide (FTO) conductive substrate and heated at $550^{\circ}C$. The annealed $SnO_2$ film is treated with aqueous $TiCl_4$ solution, which is sensitzied with MK-2 dye (2-cyano-3-[5'''-(9-ethyl- 9H-carbazol-3-yl)-3',3'',3''',4-tetra-n-hexyl-[2,2',5',2'',5'',2''']-quater thiophen-5-yl]). Compared to bare $SnO_2$ film, the conversion efficiency is significantly improved from 0.22% to 3.13% after surface treatment of $SnO_2$ with $TiCl_4$, which is mainly due to the large increases in both photocurrent density from 1.33 to $9.46mA/cm^2$ and voltage from 315 to 634 mV. It is noted that little change in the amount of the adsorbed dye is detected from 1.21 for the bare $SnO_2$ to $1.28{\mu}mol/cm^2$ for the $TiCl_{4-}$ treated $SnO_2$. This indicates that the photocurrent density increased by more than 6 times is not closely related to the dye loading concentration. From the photocurrent and voltage transient spectroscopic studies, electron life time increases by about 13 order of magnitude, whereas electron diffusion coefficient decreases by about 3.6 times after $TiCl_4$ treatment. Slow electron diffusion rate offers sufficient time for regeneration kinetics. As a result, charge collection efficiency of about 40% before $TiCl_4$ treatment is improved to 95% after $TiCl_4$ treatment. The large increase in voltage is due to the significant increase in electron life time, associated with upward shift of fermi energy.


text:
Questions & Answers

# Write the suitable quantifier for $x + 1 > x$ for all real values of $x$.

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146.1k+ views
Hint:Quantifiers are special phrases in mathematics. This lesson defines quantifiers and explores the different types in mathematical logic.
In mathematical logic, there are two quantifiers: 'there exists' and 'for all.'
There is a certain symbol for these two quantifiers.
Phrase: “there exists”
Symbol: $\exists$
Phrase: “for all”
Symbol: $\forall$
Using the symbols we can find the suitable quantifier.

Complete step-by-step answer:
We have to find the suitable quantifier for $x + 1 > x$ for all real values of $x$.
We know that; quantifiers are special phrases in mathematics. This lesson defines quantifiers and explores the different types in mathematical logic.
Quantifiers are words, expressions, or phrases that indicate the number of elements that a statement pertains to. In mathematical logic, there are two quantifiers: 'there exists' and 'for all.'
The phrase 'there exists' is called an existential quantifier, which indicates that at least one element exists that satisfies a certain property.
The phrase 'for all' is called a universal quantifier, and it indicates that all the elements of a given set satisfy a property.
There is a certain symbol for these two quantifiers.
Phrase: “there exists”
Symbol: $\exists$
Phrase: “for all”
Symbol: $\forall$
We know that, $x + 1$ is always greater than $x$.
Therefore, $x + 1 > x$ is true for any real values of $x$.
Symbolically we can write that, $x + 1 > x$$\forall$$x$.
Hence, the suitable quantifier for $x + 1 > x$ for all real values of $x$ is $x + 1 > x$$\forall$$x$.

Thus, the quantifier to describe the variable of the given sentence is Universal quantifier ($\forall$).

Note:A formula that contains variables is not simply true or false unless each of these variables is bound by a quantifier. If a variable is not bound the truth of the formula is contingent on the value assigned to the variable from the universe of discourse.


text:
# Efficient numerical evaluation of thermodynamic quantities on infinite (semi-)classical chains

This work presents an efficient numerical method to evaluate the free energy density and associated thermodynamic quantities of (quasi) one-dimensional classical systems, by combining the transfer operator approach with a numerical discretization of integral kernels using quadrature rules. For analytic kernels, the technique exhibits exponential convergence in the number of quadrature points. As demonstration, we apply the method to a classical particle chain, to the semiclassical nonlinear Schrödinger equation and to a classical system on a cylindrical lattice.

• 3 publications
• 3 publications
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## 1 Introduction

The partition function is a cornerstone of statistical mechanics [1, 2], in particular since thermodynamics quantities can be obtained as derivatives of . Nevertheless, computing for larger systems is a challenging task if a closed-form solution cannot be found. In this work we are concerned with an efficient numerical method for evaluating and the associated free energy density for (semi-)classical chains, in the thermodynamic limit of infinite system size. In more detail, we consider a Hamiltonian on a (quasi) one-dimensional lattice of size , containing solely nearest neighbor terms:

 H(z1,…,zL)=L∑ℓ=1h(zℓ,zℓ+1), (1)

with for all and some fixed . We assume periodic boundary conditions and thus set . A canonical example is a classical particle chain, with the momentum and position of the -th particle (), and

 Hpc(p1,…,pL,q1,…,qL)=L∑ℓ=1(12p2ℓ+V(qℓ,qℓ+1)) (2)

consisting of kinetic energy terms and the potential energy described by .

The partition function of the physical system is defined as

 ZL(β)=∫Ω⋯∫Ωe−βH(z1,…,zL)dz1⋯dzL, (3)

with the “inverse temperature”. Our goal is to take the thermodynamic limit and compute the canonical free energy density

 F(β)=−β−1limL→∞1LlogZL(β). (4)

## 2 Method

To evaluate the partition function numerically, we combine the well-known transfer-matrix method [3, 4] with a numerical discretization of integral kernels based on numerical quadrature methods. The latter idea traces back to Nyström [5] and was recently applied by the second author to the computation of Fredholm determinants [6].

### 2.1 Assumptions

To keep technical preliminaries as simple as possible, we make the following assumptions, which cover the applications in the next section: Let be a (not necessarily finite) interval of and be a finite positive measure on with a strictly positive, continuous density (weight function). Now we rewrite the partition function in the form

 ZL(β)=∫Ω⋯∫ΩL∏ℓ=1kβ(zℓ,zℓ+1)dν(z1)⋯dν(zL) (5)

with as before. Here the integral kernel

 kβ:Ω×Ω→R,kβ(z,z′)=e−βh(z,z′)√ω(z)ω(z′) (6)

is assumed to be continuous and bounded; it is strictly positive by construction. We further assume that , i.e., is a symmetric kernel. These assumptions imply, in particular,

 ∫Ω∫Ω|kβ(z,z′)|2dν(z)dν(z′)<∞, (7)

so that the symmetric kernel induces an integral operator [7, §16.1]

 (Tβu)(z)=∫Ωkβ(z,z′)u(z′)dν(z′),u∈L2(Ω,ν). (8)

on the Hilbert space that is self-adjoint and Hilbert–Schmidt (and hence, compact). Since the product of two Hilbert-Schmidt operators is of trace class and the trace class operators form an ideal within the bounded ones, Eq. (5) for may be represented as333This holds for only if would be trace class itself, cf. the discussion in [6, pp. 878–879].

 ZL(β)=Tr(TLβ). (9)

### 2.2 Dominant eigenvalue and free energy

Following the mathematical theory of compact operators on Hilbert spaces [7, §21.2], the non-zero elements of the spectrum of

(which is real since the operator is selfadjoint) are at most countably many eigenvalues of finite multiplicity that accumulate only at

(which belongs to the spectrum).

Generalizing the Perron–Frobenius theory in matrix analysis [8, §8.2], Jentzsch’s Theorem [9, §8.7, Satz 3] states that the Hilbert-Schmidt operator with strictly positive kernel has a simple, dominant, strictly positive eigenvalue. That is, all the non-zero eigenvalues can be ordered as (with each eigenvalue listed as often as given by its multiplicity)

 λ1(Tβ)>|λ2(Tβ)|≥|λ3(Tβ)|≥⋯>0.

Hence, evaluating Eq. (9) together with Lidskii’s theorem [7, §30.3], that is,

 Tr(TLβ)=∑jλj(Tβ)L(L≥2),

 limL→∞1LlogZL(β)=limL→∞1Llog∑jλj(Tβ)L=logλ1(Tβ),F(β)=−β−1logλ1(Tβ). (10)

Thus the calculation of the free energy amounts to computing the dominant eigenvalue of .

### 2.3 Nyström method for computing the dominant eigenvalue

For simplicity, we restrict the discussion of the numerical method to , that is, . For the multi-dimensional case , see the discussion of the example in Sect. 3.3.

We make use of a numerical quadrature rule of the form

 ∫Ωf(z)dν(z)≈m∑i=1wif(zi) (11)

with positive weights and nodes . The approximation is of order if the quadrature rule is exact for polynomials of degree at most . It is possible to construct such a unique quadrature rule of maximum order , called Gauss quadrature rule; see [10] for the classical construction of Gaussian weights and nodes from the tridiagonal Jacobi matrix of the orthogonal polynomials associated with the measure

. Error estimates depend on the regularity of the integrand

, e.g., [11, §4.8]: the error for is of the form if and of the form with some if extends analytically to an ellipse or (semi)-strip in the complex plane containing . The former estimate is called algebraic convergence, the latter exponential convergence.

Inserting the quadrature rule into Eq. (8) results in

 (Tβu)(zi)≈m∑j=1kβ(zi,zj)wju(zj) (12)

for all . Setting , we have thus discretized the integral operator by the symmetric matrix (Nyström method)

 Tβ=(kβ(zi,zj)√wiwj)mi,j=1. (13)

By the Perron–Frobenius theory [8, Cor. 8.2.6], this (element-wise) strictly positive matrix has a simple, dominant, strictly positive eigenvalue . The following theorem shows that , and thus for the free energy density in Eq. (4)

 F(β)≈−β−1logλ1(Tβ), (14)

with a speed of convergence, algebraic or exponential, depending on the smoothness of .

###### Theorem 1.

Under the assumptions in Sect. 2.1 the error of the Nyström method for computing the dominant eigenvalue behaves as follows as :

• if then there is algebraic convergence ;

• if extends analytically to , where is an ellipse or (semi-)strip, there is exponential convergence with some constant .

The classical, though long and quite involved proof goes by the theory of collectively compact operators and can be found in many books on the numerical treatment of integral equations, e.g., [12, Thm. 4.8.20]. Note, that though only the case of algebraic convergence is covered in this reference, the proof extends literally to the case of exponential convergence by adjusting the consistence assumptions accordingly.

An alternative, conceptually much simpler novel proof based on the theory of the Fredholm determinant can be found in the Appendix of this paper.

###### Remark.

While the Nyström method for the dominant eigenvalue essentially inherits the convergence properties of an underlying cubature formula also in the multidimensional case (see the proof in the Appendix for guidance), we refrained from formulating a general theorem since the characterization of convergence properties of general cubature formulae is much more involved in the first place. If, however, a Gaussian quadrature is applied coordinate-wise, Theorem 1 extends in a straightforward fashion (in the case of exponential convergence has then to be chosen as an poly-ellipse or poly-(semi-)strip), cf. the example in Sect. 3.3.

## 3 Applications

We demonstrate the range of applicability of the method via a diverse selection of model systems.

### 3.1 Classical particle chain

Consider a classical particle chain governed by a Hamiltonian of the form (2) above. For evaluating the partition function, the integration over the momentum variables can be performed in closed form, such that

 ZL(β)=(2πβ)L/2~ZL(β) (15)

with

 ~ZL(β)=∫∞−∞⋯∫∞−∞L∏ℓ=1e−βV(qℓ,qℓ+1)dq1⋯dqL. (16)

As specific example for the following, we choose

 V(q,q′)=Vloc(q)+12γ(q−q′)2 (17)

with anharmonic on-site potential

 Vloc(q)=12ηq2+16μq3+124λq4 (18)

and coefficients , , . The term ensures that grows asymptotically to infinity as , and the above is equivalent to its symmetrized version .

We now assign the term from the on-site potential to the weight function:

 ω:R→R+,ω(q)=e−12βηq2√2π/(βη). (19)

This leads to a (rescaled) Gauss-Hermite quadrature rule [13, §3.5(v)], and we denote the weights by and nodes by , , as above.444A modern, fast numerical method of optimal complexity for calculating these weights and nodes can be found in [14]. The particular choice of the weight function is somewhat arbitrary — we could have also included the cubic and quartic terms from the on-site potential into the weight function; at the expense, though, of a less straightforward computation of the weights and nodes of the quadrature rule. The general reasoning is to capture most of the local weight while retaining a well-behaved kernel for the genuine inter-particle potential (see Eq. (21) below).

Combining the weight function (19) with Eq. (16) leads to

 ~ZL(β)=(2πβη)L/2∫∞−∞⋯∫∞−∞L∏ℓ=1kβ(qℓ,qℓ+1)dν(q1)⋯dν(qL) (20)

with the symmetrized kernel

 kβ(q,q′)=e−112βμ(q3+q′3)−148βλ(q4+q′4)−12βγ(q−q′)2. (21)

We then assemble the symmetric matrix in Eq. (13). Finally, after taking all prefactors in to account, the numerical approximation of the free energy density reads

 −βF(β)=limL→∞1LlogZL(β)≈log2πβ−12logη+logλ1(Tβ). (22)

The special case serves as reference, since the partition function factorizes in this case, i.e., with

 logZ1(β)|γ=0=12log2πβ+log∫∞−∞e−βVloc(q)dq. (23)

For the following numerical examples, we set and . Fig. 1 visualizes the kernel in Eq. (21). The factor localizes the kernel around the line , which poses a challenge for accurately “sampling” it using a limited number of points in Eq. (13).

Fig. (a)a shows the free energy as function of , for several values of , and Fig. (b)b the relative error depending on the number of quadrature points . One observes exponential convergence. The less favorable shape of the kernel with increasing , as mentioned above (see also Fig. 1), translates to a slower convergence rate.

Based on the free energy one can obtain averages and higher-order cumulants following the well-known procedure based on derivatives of . For example, the average squared particle distance and energy per site are

 ⟨12(qℓ−qℓ+1)2⟩=∂γF,⟨eℓ⟩=∂β(βF), (24)

independent of by translation invariance. In practice, a higher-order finite difference scheme on a fine grid is well suited to calculate the derivatives. Fig. 3 shows these averages, for the same parameters as before. One notices that the average energy hardly depends on .

As a remark, for the case of vanishing on-site potential, , the model conserves momentum, and the statistical mechanics description changes accordingly [15]. Numerically computing the free energy is less challenging in this case since the partition function factorizes after introducing the “stretch” .

### 3.2 Discrete nonlinear Schrödinger equation

The method described in Sect. 2 has been employed in the work [16] on the discrete nonlinear Schrödinger equation. Here we present and elaborate on the numerical aspects in more detail.

To be self-contained, we first restate the physical setup: the central object is a complex-valued wave field () governed by the semiclassical Hamiltonian

 H(ψ1,…,ψL)=L∑ℓ=1(12|ψℓ+1−ψℓ|2+12g|ψℓ|4), (25)

with parameter (so-called defocusing case). The corresponding partition function reads

 ZL(μ,β)=∫e−β(H−μN)dψ1⋯dψL, (26)

where we have introduced the chemical potential as additional parameter, which is dual to the total particle number .

A symplectic change of variables to polar coordinates leads to the representation

 ψℓ=√ρℓeiφℓ (27)

with , and (unit circle). The Hamiltonian in these variables reads

 H=L∑ℓ=1(−√ρℓ+1ρℓcos(φℓ+1−φℓ)+ρℓ+12gρ2ℓ). (28)

It depends only on phase differences, which implies the invariance under the global shift . For evaluating the partition function, the integrals can be calculated in closed form [17]. This leads to

 ZL(μ,β)=eβ12μ2L/g∫Ω⋯∫ΩL∏ℓ=1kβ(ρℓ,ρℓ+1)e−β12g(ρℓ−μg)2dρ1⋯dρL (29)

with the kernel

 kβ(ρ,ρ′)=2πI0(β√ρρ′)e−β12(ρ+ρ′). (30)

is the modified Bessel function of the first kind. For this example, we use the second factor of the integrand in Eq. (29) as weight function:

 ω:Ω→R+,ω(z)=ce−a(z−b)2/2 (31)

with , and the normalization constant

 c=2√a2π1+erf(b√a2). (32)

After constructing a Gauss quadrature rule on as described in Sect. 2.3, we form the symmetric matrix in Eq. (13), here denoted since it implicitly also depends on via the quadrature points and weights. Then, taking the prefactor in Eq. (29) and the normalization constant (32) into account, one arrives at

 −βF(μ,β)=limL→∞1LlogZL(μ,β)≈β12μ2g+logλ1(Tμ,β)−logc(μ,β). (33)

Fig. 4 shows the free energy as function of , for various values of .

Numerically, we again observe exponential convergence with respect to the number of quadrature points, see Fig. 5. At and for example, points suffice for double precision accuracy. The reference data stems from a calculation with .

As mentioned before, one can obtain thermodynamic averages based on derivatives of . For example, the average density and energy per lattice site are

 ⟨ρℓ⟩=−∂μF(μ,β),⟨eℓ⟩=∂β(βF(μ,β))+μ⟨ρℓ⟩. (34)

See [16] for a detailed study of the model.

### 3.3 Classical oscillators on a cylindrical lattice

The numerical method is in principle also applicable to two-dimensional lattices, by using periodic boundary conditions in one direction and reducing the setting to a quasi-one dimensional problem. Specifically, we consider the lattice for , i.e., starting with periodic boundary conditions both in - and -direction, but eventually sending while keeping finite. Thus we arrive at a cylindrical lattice, as visualized in Fig. 6.

We identify a lattice site by the index , and consider for simplicity scalar spatial variables ; these could be displacements from the reference positions in one fixed direction, for example. denotes the momentum of the -th particle.

As demonstration, let the system be governed by the Hamiltonian

 H=∑ℓ∈Γ(12p2ℓ+Vloc(qℓ))+∑⟨ℓ,ℓ′⟩Vℓ′−ℓ(qℓ,qℓ′), (35)

consisting of site-local kinetic and potential energy terms (first sum) as well as nearest neighbor interactions (second sum). Specifically, we consider a local quadratic potential , , and an interaction potential with two coefficients and .

To cast the Hamiltonian (35) into the form of Eq. (1

), we subsume the particles contained in one lattice “ring” into the vectors

 pℓx =(p(ℓx,1),…,p(ℓx,Ly))∈RLy, (36) qℓx =(q(ℓx,1),…,q(ℓx,Ly))∈RLy (37)

for .

Similar to the particle chain in subsection 3.1, the momentum integration for evaluating the partition function can be performed explicitly. This results in

 Z(Lx,Ly)(β)=(2πβ)LxLy/2~Z(Lx,Ly)(β) (38)

with

 ~Z(Lx,Ly)(β)=∫RLy⋯∫RLyLx∏ℓx=1e−β(~Vloc(qℓx)+~Vint(qℓx,qℓx+1))dLyq1⋯dLyqLx, (39)

where

 ~Vloc(q)=Ly∑ℓy=1Vloc(qℓy) (40)

and

 ~Vint(q,q′)=Ly∑ℓy=1(12ax(qℓy−q′ℓy)2+14ay(qℓy−qℓy+1)2+14ay(q′ℓy−q′ℓy+1)2). (41)

Note that is symmetric in its arguments, i.e., for any .

It suggests itself to use the factor in Eq. (39

) as integration measure, resulting in a tensor product of normal distributions:

 ω:RLy→R+,ω(q)=e−12βη∥q∥2(2πβη)Ly/2=Ly∏ℓy=1e−12βηq2ℓy√2π/(βη). (42)

We use a rescaled Gauss-Hermite quadrature rule along each coordinate direction, as in Sect. 3.1. Theorem 1

extends straightforwardly to this choice. An alternative, which is less affected by the inherent curse of dimensionality, is a cubature rule dedicated to multidimensional integration

[18, 19, 20, 21], or sparse grid methods. The convergence properties of such cubature rules are more involved, but they would essentially be inherited by the Nyström method for the dominated eigenvalue. We leave an exploration of these ideas for future work.

Using as measure, Eq. (39) becomes

 ~Z(Lx,Ly)(β)=(2πβη)LxLy/2∫RLy⋯∫RLyLx∏ℓx=1kβ(qℓx,qℓx+1)dν(q1)⋯dν(qLx) (43)

with the kernel

 kβ(q,q′)=e−β~Vint(q,q′). (44)

Following the factorized quadrature rule, the symmetric matrix in Eq. (13) takes the form

 Tβ=(kβ(qi,qj)√wiwj)i,j (45)

with multi-indices and the definitions , and , , the weights and points of the one-dimensional rescaled Gauss-Hermite quadrature rule. Thus the overall number of weights and points is .

The numerical approximation of the free energy per lattice site is then

 −βF(β)=limLx→∞1LxLylogZ(Lx,Ly)(β)≈log2πβ−12logη+1Lylogλ1(Tβ), (46)

with kept fixed. For the following examples, we set , such that the overall number of quadrature points remains manageable up to . Fig. (a)a visualizes the free energy as function of , for several combinations of . One notices that the curve for is visually indistinguishable from the case with interchanged parameters , pointing to the conclusion that the influence of a finite compared to the “infinite” on the free energy is quite small.

Fig. (b)b shows the corresponding relative error depending on the number of quadrature points along one dimension. In the special case (without coupling in -direction), the partition function factorizes, such that, analogous to Sect. 3.1, with

 ~Z(1,Ly)(β)|ax=0=∫RLye−β(12η∥q∥2+∑Lyℓy=112ay(qℓy−qℓy+1)2)dLyq. (47)

We evaluate this integral numerically and use it as reference for computing the relative error in Fig. (b)b for . The relative error is still rather large for and ; as before, this observation can be explained by the difficulty of accurately sampling the kernel (44) via (45) using a small number of quadrature points along each coordinate. To mitigate this issue for the present example, one could associate the terms in the Hamiltonian to the integration measure instead of the kernel, at the expense of a more complicated quadrature rule.

In summary, this application example demonstrates that our method can in principle handle two-dimensional lattice topologies as well, although the large number of required quadrature points (when interpreting the problem as quasi one-dimensional) limits the size of the periodic dimension in practice.

## 4 Conclusion and outlook

The convergence plots of the example applications illustrate the validity of Theorem 1, which states that approximating the dominant eigenvalue of the discretized kernel inherits the favorable exponential convergence properties of the underlying quadrature rule in the case of kernels extending analytically into the complex domain.

To optimize the numerical performance of the method for the cylindrical topologies, one could further exploit the factorized structure of in (44) along coordinate directions, or use sparse grid methods for the quadrature as mentioned above.

Concerning transfer operator techniques in general, it could be fruitful to adopt ideas from quantum mechanics (see e.g. [22]), or set oriented numerical methods [23].

#### Acknowledgments

C.M. likes to thank <NAME> and <NAME> for helpful discussions, and the Munich Center for Quantum Science and Technology for support.

## Appendix

We give here an alternative, conceptually much simpler proof555To the best of our knowledge, this proof has not yet been given in the literature. of Theorem 1 based on the theory of the Fredholm determinant of the kernel , namely

 D(μ)=1+∞∑n=1(−μ)nn!∫Ω⋯∫Ωndeti,j=1kβ(zi,zj)dν(z1)⋯dν(zn).

Given the assumptions in Sect. 2.1, is an entire function whose roots are exactly the reciprocal non-zero eigenvalues of , see [9, §6.2, Satz 3]. The Weierstrass product [9, §6.4, Satz 1]

 D(μ)=eαμ+βμ2∏n(1−μλn(Tβ))eμλn(Tβ)

shows that the multiplicities of the roots of and the multiplicities of the non-zero eigenvalues of agree. In particular, is a simple root and therefore . Also, [6, Thm. 6.2] (whose proof can literally be extended to the current assumptions) gives

 det(I−μTβ)=D(μ)+ϵm(μ) (#)

where, uniformly for bounded , the error is given by or according to whether or extends analytically into the complex plane. By Perron–Frobenius is the simple, dominant, strictly positive eigenvalue of the entry-wise positive matrix , which by the argument principle of complex analysis must satisfy

 1/λ1(Tβ)→1/λ1(Tβ)(m→∞).

Hence, by inserting into (#Appendix), followed by a Taylor expansion, we get

 λ1(Tβ)=λ1(Tβ)+λ1(Tβ)2D′(λ1(Tβ)−1)ϵm(λ1(Tβ)−1))+O(ϵm(λ1(Tβ)−1))2),

which completes the proof.

## References

• [1] <NAME> and <NAME>. Statistical Physics: Part 1. Pergamon, 1980.
• [2] <NAME>. Introduction to Statistical Physics. Taylor and Francis, 2010.
• [3] <NAME> and <NAME>. Phys. Rev., 60:252–262, 1941.
• [4] <NAME>. Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics. Oxford University Press, 2009.
• [5] <NAME>. Acta Math., 54:185–204, 1930.
• [6] <NAME>. Math. Comp., 79:871–915, 2010.
• [7] <NAME>. Functional Analysis. Wiley-Interscience, 2002.
• [8] R. <NAME> and <NAME>. Matrix analysis. Cambridge University Press, Cambridge, 1985.
• [9] S. Fenyö and <NAME>. VEB Deutscher Verlag der Wissenschaften, Berlin, 1983.
• [10] <NAME> and <NAME>. Math. Comp., 23:221–230, 1969.
• [11] P. <NAME> and P. Rabinowitz. Methods of numerical integration. Dover Publications, Inc., Mineola, NY, 2007. Corrected reprint of the second (1984) edition.
• [12] <NAME>. Birkhäuser Verlag, Basel, 1995.
• [13] F. W. <NAME>, <NAME>, <NAME>, and <NAME>, editors. NIST Handbook of Mathematical Functions. Cambridge University Press, 2010.
• [14] A. Townsend, T. Trogdon, and S. Olver. IMA J. Numer. Anal., 36:337–358, 2016.
• [15] H. Spohn. J. Stat. Phys., 154:1191–1227, 2014.
• [16] C. <NAME> and H. Spohn. J. Stat. Mech., 2015:P08028, 2015.
• [17] <NAME>, <NAME>, <NAME>, and <NAME>. Phys. Rev. Lett., 84:3740–3743, 2000.
• [18] <NAME>. Approximate Calculation of Multiple Integrals. Prentice-Hall, 1971.
• [19] <NAME> and <NAME>. J. Comput. Appl. Math., 48:309–326, 1993.
• [20] <NAME>. Acta Numerica, 6:1–54, 1997.
• [21] <NAME>, <NAME>, and <NAME>. SIAM J. Sci. Comput., 40:A2033–A2061, 2018.
• [22] <NAME> and <NAME>. Annu. Rev. Condens. Matter Phys., 8:355–406, 2017.
• [23] <NAME> and <NAME>. SIAM J. Numer. Anal., 36:491–515, 1999.


text:
Browse Questions

# How many grams of sucrose (M.wt. = 342)should be dissolved in 100g water in order to produce a solution with 105.0$^{\large\circ}$C difference between the freezing point and the boiling point? ($K_f = 1.86^{\large\circ}$C/m, $K_b = 0.51^{\large\circ}$C/m)

72g
Hence (A) is the correct answer.


text:
Open access peer-reviewed chapter

# Move from Resilience Conceptualization to Resilience Enhancement

Written By

<NAME> and <NAME>

Submitted: August 11th, 2020 Reviewed: October 15th, 2020 Published: November 24th, 2020

DOI: 10.5772/intechopen.94513

From the Edited Volume

## Flood Impact Mitigation and Resilience Enhancement

Edited by <NAME>

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## Abstract

This chapter provides an analysis of various resilience definitions and depicts the differences in definition between engineering, ecological and socio-ecological resilience in an easy-to-understand graphic representation. It also articulates commons and differences between conventional flood risk management and resilience-based flood management and presents a mathematical formulation to facilitate resilience discussion. Furthermore, it highlights some studies and initiatives towards the operationalization of the resilience concept in flood disaster management practice. The most important message this chapter is intended to deliver is that resilience is not just about bouncing back. Indeed, it should be enhanced to bounce forward.

### Keywords

• engineering resilience
• ecological resilience
• socio-ecological resilience
• flood risk
• resistance
• vulnerability

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## 5. Current move towards the application of resilience-based flood risk management concept

The importance of resilience building in flood risk management has been well recognized as evidenced by large amounts of academic articles on resilience. In practice, however, resilience concept tends to be only marginally applied as a supplement to flood risk management. There are several well-known initiatives such as Rockefeller Foundation’s 100 Resilient Cities programme (100RC) [32], the UNISDR Making Cities Resilient campaign, and the OECD Resilient Cities project [33]. These programs are mainly intended to promote resilience as a source of policy inspiration, and the development of policy instruments for cities to address immediate shocks and long-term stresses that undermine the functions of cities.

In the paper by Gralepois et al. [34], the flood defense strategies in six European countries (Belgium, England, France, the Netherlands, Poland, and Sweden) are analyzed. Although they do not find radical changes in either of the countries, they do find that the defense strategy in all countries has created more room for local, private, and individual responsibilities. In all countries except Sweden, defense remains the primary method of protection, leading the authors to conclude that flood defense has remained a cornerstone of European flood risk management.

The paper by Gersonius et al. [35] addresses the debate as to how transformations from resistance-based to resilience-based approaches can be achieved by studying the implementation of various measures that aim to enhance the flood resilience of the Dutch “Island of Dordrecht.” The case illustrates that a multilayered, i.e., diversified, approach is more effective and efficient than its resistant, i.e., flood defense dominated, counterpart and provides substantial co-benefits. However, it is incompatible with the existing institutional framework. Such an incompatibility may be considered a challenge that will also be present in other countries with an established institutional framework for resistance-based approaches. Then, the authors recommend searching for ways to reinterpret existing frameworks and applying them differently by setting up pilots and experiments to foster social learning.

The paper by Hegger et al. [36] assesses the now prominent assumption that a diversification of flood risk management strategies leads to resilience. They propose that the resilience concept should be operationalized into three capacities: capacity to resist, capacity to absorb and recover, and capacity to adapt and transform, and they compared six countries’ achievements in terms of these capacities. The work found that having a diverse portfolio of strategies in place contributes to resilience, especially in terms of the capacity to absorb/recover and the capacity to adapt and transform. However, the authors also stated in this work that they see different ways to be resilient. The importance of explicating the normative starting points of flood risk governance in a country, considering the unavoidable trade-offs between the three capacities, and assessing strategies’ fit with existing physical circumstances and institutional frameworks was further elucidated in the work.

Despite various efforts to adopt resilience-based approach to flood risk management, the actual application or the operationalization of the resilience concept remains to be explored, planned, tested, and evaluated. At present, many flood-prone regions have good pre-disaster preparation such as flood hazard map, evacuation plan and early warning system. However, few municipalities have resilience-based post-disaster recovery plan or guideline prepared before disaster. Instead, what was often seen is ad hoc recovery plans after disasters.

The Cedar Falls is a residential community located in Eastern Iowa. A good practice of the city is that it has a hazard mitigation plan, which includes a series of future hazard mitigation activities involving a wide range of hazards including floods [37]. Although one of the goals of the plan is to return to pre-disaster or improved conditions as soon as possible after a disaster occurs, the emphasis is placed on prevention than rebuilding. Technical advices on recovery process are limited and general. Suggestions such as “Continue membership with the National Flood Insurance Program (NFIP)” or “Establish and/or maintain Continuity of Government plans to handle post disaster operations (i.e. animal disposal, clean-up, demolition) are important but insufficient.

EPA developed a Flood Resilience Checklist [38] to help communities identify ways to improve their resilience to future floods. It includes five areas: (1) Overall strategies to improve flood resilience; (2) Conserve land and discourage development in flood-prone river corridors; (3) Protect people, businesses, and facilities in vulnerable settlements; (4) Plan for and encourage new development in safer areas; (5) Implement and coordinate stormwater management techniques throughout the whole watershed. The five areas can be regrouped as overall strategies (area 1) as well as specific strategies (areas 2–5).

The area of Overall Strategies to Enhance Flood Resilience is designed to promote the integration of the community’s comprehensive plan and other community’s plans such as open space or park plans with a flood management plan including both structural and non-structural measures. It also promotes community participate in the National Flood Insurance Program Community Rating System. For specific strategies such as Incentives for restoring riparian and wetland vegetation in areas subject to erosion and flooding and Acquisition of land (or conservation easements on land) to allow for stormwater absorption, their importance are well recognized and have been pursued in various ways. A representative case is the Room for the Rivers program along the Rhine and Meuse Rivers, which started from 2006 with a $3.3 billion budget from the Dutch government. Flood risk management strategies in the Netherlands have traditionally focused on reducing the probability of flooding [39] by means of dikes, pumps, and canals. After experiencing severe flooding in the 1990s, the Dutch government decided to safeguard flood-prone areas by stepping back from the river to enable the rivers to safely discharge far greater volumes of water. The program resulted in a reduction of water levels by 10–19 cm during high water in target river reaches. Although the primary goal of the Room for the River program is flood attenuation, it also recognizes the importance of esthetics and cultural and ecological elements and has increased biodiversity as the project transformed 4576 acres of land back to natural conditions. Therefore, such an initiative functions as an opportunity rather than a solely means to fix a problem because it is designed not only for river management, but also for social and economic advances. In the meantime, some U.S. communities have also implemented their own Room for the River strategies to deal with flooding. The Iowa River Corridor Project [40], begun after a severe flood in 1993, compensates farmers who permanently stop farming fields in floodplains. Much of the 50,000 acres involved have reverted into natural wetlands, grassland, and bottomland forest, and provide habitat for wildlife. The Napa River in California often floods between November and April. The$400 million Napa River/Napa Creek Flood Control Project is lowering dikes, creating floodplains and a bypass, relocating bridges, and restoring 900 acres of wetlands according to “living river” principles. Floodplain and wetlands restoration projects are also ongoing in other parts of the U.S. such as Illinois, Massachusetts, Missouri, North Dakota, Minnesota, Oklahoma, and Wisconsin.

On the other hand, studies focusing solely on disaster recovery have also progressed greatly in parallel to resilience research. <NAME> Wenger [41] defined the disaster recovery process as “the differential process of restoring, rebuilding, and reshaping the physical, social, economic, and natural environment through pre-event planning and post-event actions,” while Schwab et al. [42] defined recovery as “Recovery includes restoring housing, transportation, and public services; restarting economic activity; and fostering long-term community redevelopment and improvements. The definition adopted by the UN Office of Disaster Risk Reduction is “decisions and actions aimed at restoring or improving livelihoods, health, as well as economic, physical, social, cultural and environmental assets, systems and activities, of a disaster-affected community or society, aligning with the principles of sustainable development, including build back better to avoid or reduce future disaster risk.” This definition emphasizes both returning the community to normality, which is a short-term objective and sustainable development to be less vulnerable and more capable of dealing with future disaster risk, which is a long-term goal and this long-term goal implies building back a better state, similar to the multi-equilibrium state concept in socio-ecological resilience. Therefore, the dialog between flood resilience researchers and disaster recovery planners should be promoted because it can deepen the understanding of resilience by resilience researchers and contribute to better recover planning for long-term resilience. In other words, the integration of conventional disaster recovery planning with resilience concept is a pathway for resilience building.

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## 6. Concluding remarks

Science has revealed that the human immune system has 2 broad functions: (1) defending our body’s health and (2) maintaining our body’s health. Similarly, resilience can be viewed as urban’s or community’s immune system to natural disasters, possessing two functions: (1) resisting to disturbance and (2) maintain its viability. To date, resilience has been mainly understood as the system’s capacity to restore its structure and functions. However, we chose to use the word of viability to emphasize our understanding that resilience is not limited to bouncing back but can bounce forward. In general, there are three options for a damaged system: (1) full restoration, (2) repair, which means the restoration with replacement, and (3) restoration with enhancement. For example, if the life of a city once flooded is now fully back to pre-disaster conditions, then such a situation is full restoration. If the disaster’s impacts can never be fully erased from the city, it is a case of repair. For example, the city of New Orleans was severely damaged by Hurricane Katrina in 2005. Fifteen years after the disaster, the population of New Orleans has shrunk from 10 to 15 per cent, especially it lost many African Americans residents, who were either killed in the hurricane or could not afford to come back. This situation led some researchers to declare the housing recovery in New Orleans a secondary disaster [43, 44]. The Great East Japan Earthquake of 2011 and the vicious tsunami that followed it caused widespread destruction in the Tohoku region. Rikuzentakata City in Iwate Prefecture is one of the most badly hit cities in the disaster. The recovery plan focuses equally on reconstructing and improving damaged transport networks along the coastline, re-establishing affected local businesses and empowering the disaster-struck agricultural and fishing industries which used to thrive in the area. For the restoration of urban districts, it promoted the introduction of universal design, aiming to create more opportunities for people with disabilities and the elderly to work and do sports as well. Furthermore, residential houses and hospitals have been moved to much less disaster-prone locations. As shown in Figure 4, it is a large-scale project. In total, 298 ha of residential areas were relocated to relatively higher grounds. Such a scale of disaster-mitigation-driven relocation is unprecedented in Japanese history. Furthermore, the coastal protection system has been resigned innovatively. As illustrated in Figure 5, it consists of a double-dike structure with a vegetation zone in-between and submerged breakwater at the front. In light of these developments, Rikuzentakata City can be considered a successful case of restoration with enhancement.

A critical issue in choosing recovery path is the financial cost. The cost of each option may vary greatly, so that resilience building could be constrained by local economic condition. In general, sustainable, resilient water management can be considered costly since it involves engineering and land use challenges and often a long-term process. The financial sustainability of resilience building and enhancement has been largely neglected up to now and deserves serious in-depth study. It is our belief that resilience building should be pursued in relation to economic growth in developing countries. In developed countries, solutions harnessing flood risk while unlocking further development potential should be explored, which require innovation. However, as we may face multiple pathways for building a resilient tomorrow, further studies should be conducted to develop optimal design approaches for resilience building with more than one objective.

Finally, it should be mentioned that conventional flood risk management is probability-based. It deals with the magnitude of potential consequences due to an event or disturbance with a chosen probability of occurrence. It provides little insights into the nature’s or society’s self-restoring or anti-disturbance function and is unable to cope with events with magnitudes of impact exceeding the chosen level. By contrast, resilience-based management is not constrained by likelihood of occurrence and can accept extremely large shocks by allowing adaption to new regimes. Therefore, it is more capable of and more flexible in restoring or reestablishing an affected system. Furthermore, resilience enhancement strategy can lead to better knowledge fusion than conventional flood risk management approach.

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## Acknowledgments

This work was supported by Sophia Research Branding Project.

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## Conflict of interest

The authors declare no conflict of interest.

## References

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5. 5. Ministry of Water Resources of the People’s Republic of China. Annual flood and drought hazards report of China. 2017. (Available online athttp://www.mwr.gov.cn/sj/tjgb/zgshzhgb/201707/t20170720 966705. Html)
6. 6. Ministry of Land, Infrastructure, Transport and Tourism. Outline of damage caused by heavy rain in July 2018.
7. 7. <NAME>, <NAME>, <NAME>.: Civil society contributions to local level flood resilience: before, during and after the 2015 Boxing Day floods in the Upper Calder Valley. Transactions of the Institute of British Geographers. 2019, 44(2): 422-436.
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9. 9. <NAME>., <NAME>., <NAME>., <NAME>., <NAME>.: Toward more resilient flood risk governance. Ecology and Society. 2026, 21(4):53.https://doi.org/10.5751/ES-08921-210453.
10. 10. <NAME>.: Living with flood risk. Planning Theory & Practice. 2013, 14(1): 103-140.
11. 11. <NAME>., <NAME>, <NAME>, <NAME>, <NAME>, <NAME>, <NAME>.: A place-based model for understanding community resilience to natural disasters. Global Environmental Change. 2008, 18(4): 598-606.
12. 12. <NAME>. Retrofitting for Flood Resilience: A Guide to Building & Community Design. RIBA Publishing. 2019.
13. 13. <NAME>, <NAME>, <NAME>.: A strategy-based framework for assessing the flood resilience of cities-a Hamburg case study, Planning Theory & Practice. 2015, 16 (1): 45-62.
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18. 18. <NAME>., <NAME>, and <NAME>.: Resilience driven system design of complex engineered systems. Journal of Mechanical Design. 2011, 133(10): 101011.
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20. 20. Adger WN. Social ecological resilience: are they related? Prog Hum Geogr 2000;24(3):347-64.
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36. 36. Hegger DLT, <NAME>J, <NAME>, <NAME> HFMW, <NAME>, <NAME>, <NAME>, Raadgever GT, Bakker MHN, <NAME>, <NAME>, <NAME>.: Toward more flood resilience: Is a diversification of flood risk management strategies the way forward?. Ecology and Society. 2016, 21(4):52.
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41. 41. <NAME>, <NAME>.: Chapter 14: Sustainable Disaster Recovery: Operationalizing An Existing Agenda, in: Handbook of Disaster Research. Springer, New York, NY. 2006, 234-257.
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Written By

<NAME> and <NAME>

Submitted: August 11th, 2020 Reviewed: October 15th, 2020 Published: November 24th, 2020


text:
Machining, Measurement, and Control Laboratory

 2 0 0 1    C o n f e r e n c e    P a p e r

H-infinity Optimization of Luenberger State Observers and Its Application to Fault Detection Filter Design

<NAME> (Kyoto University)
<NAME> (University of California at Berkeley)
<NAME> (University of California at Berkeley)

Abstract

This paper considers H-infinity optimization of Luenberger state observers. The conventional formulation of H-infinity-optimal state observers does not allow the augmentation of dynamic performance weightings in the optimization objective, since it makes the problem a nonconvex optimization problem. We propose an algorithm to locally solve an $H_{\infty}$ optimization problem of Luenberger state observers by transforming the problem into an H-infinity optimization problem of a static output feedback controller. The proposed approach offers an intuitive and efficient way to explicitly design the estimation error dynamics of the observer in the frequency domain. As an application example, the proposed approach is applied to the tuning of fault detection filters for lateral control of automated passenger vehicles. Numerical simulations are conducted to show the effectiveness of the proposed tuning method.

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text:
# Nominations for MHB Latex award

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#### Jameson

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This is for the user that demonstrates the best use of $\LaTeX$ in his or her posts here at MHB. Nominations must be seconded by someone to make voting round.

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#### <NAME>21

##### Well-known member
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If we're including people who have helped many with their $\LaTeX$ problems in this category, I'd like to recommend Evgeny.Makarov for this award.

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#### ZaidAlyafey

##### Well-known member
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#### caffeinemachine

##### Well-known member
MHB Math Scholar
I nominate anemone.

I have seen the challenge threads where anemone has given neat and long solutions nicely typed in LaTeX.

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Staff member

#### Petrus

##### Well-known member
I nominate:

1.MarkFL:
He have done alot question from other site and spend alot time on latex! Like this and this and so on... Not to mention that he learned me how to use $$\displaystyle \LaTeX$$ and have also giving me tips to improve my $$\displaystyle \LaTeX$$ like this and so on...!

2.I like Serena:
He have also used alot nice $$\displaystyle \LaTeX$$ and also been given me tips like this and so on...!

3. <NAME>:
He always amazing me with his matrice $$\displaystyle \LaTeX$$ which I always screw up when I try! Like this and so on...!

I would also third Soroban!

This award is really hard so many with nice latex and I could make a big list if I had more time right now! But I hope I am not late to nominate!

Regards,
$$\displaystyle |\pi\rangle$$

i second MarkFL

#### <NAME>

##### MHB Seeker
Staff member
I second third MarkFL and second Fernando, and nth the others.

And I also nominate Opalg, Jester, and Bacterius.

I fourth Mark!

#### mathworker

##### Well-known member
i would nominate ZaidAlyafey for extensive use of latex in Advanced integration techniques
and second Bacterius+,opalg,fernando.
and also nominate chisgma

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text:
#### [C50] - On the convergence of single-call stochastic extra-gradient methods

<NAME>, <NAME>, <NAME>, and <NAME>. In NeurIPS '19: Proceedings of the 33rd International Conference on Neural Information Processing Systems, 2019

##### Abstract

Variational inequalities have recently attracted considerable interest in machine learning as a flexible paradigm for models that go beyond ordinary loss function minimization (such as generative adversarial networks and related deep learning systems). In this setting, the optimal ${\mathcal{O}(1/t)}$ convergence rate for solving smooth monotone variational inequalities is achieved by the Extra-Gradient (EG) algorithm and its variants. Aiming to alleviate the cost of an extra gradient step per iteration (which can become quite substantial in deep learning applications), several algorithms have been proposed as surrogates to Extra-Gradient with a single oracle call per iteration. In this paper, we develop a synthetic view of such algorithms, and we complement the existing literature by showing that they retain a ${\mathcal{O}(1/t)}$ ergodic convergence rate in smooth, deterministic problems. Subsequently, beyond the monotone deterministic case, we also show that the last iterate of single-call, stochastic extra-gradient methods still enjoys a ${\mathcal{O}(1/t)}$ local convergence rate to solutions of non-monotone variational inequalities that satisfy a second-order sufficient condition.