Elliptic Weight Research Articles

Elliptic Dimers on Minimal Graphs and Genus 1 Harnack Curves

This paper provides a comprehensive study of the dimer model on infinite minimal graphs with Fock’s elliptic weights (Fock, Inverse spectral problem for GK integrable system. arXiv e-prints arXiv:1503.00289 , 2015). Specific instances of such models were studied in Boutillier et al. (Invent Math 208(1):109–189, 2017), Boutillier et al. (Probability theory and related fields, 2018) and de Tilière (Electron J Probab 26:1–86, 2021); we now handle the general genus 1 case, thus proving a non-trivial extension of the genus 0 results of Kenyon (Invent Math 150(2):409–439, 2002) and Kenyon and Okounkov (Duke Math J 131(3):499–524, 2006) on isoradial critical models. We give an explicit local expression for a two-parameter family of inverses of the Kasteleyn operator with no periodicity assumption on the underlying graph. When the minimal graph satisfies a natural condition, we construct a family of dimer Gibbs measures from these inverses, and describe the phase diagram of the model by deriving asymptotics of correlations in each phase. In the $$\mathbb {Z}^2$$ -periodic case, this gives an alternative description of the full set of ergodic Gibbs measures constructed in Kenyon et al. (Ann Math 163(3):1019–1056, 2006). We also establish a correspondence between elliptic dimer models on periodic minimal graphs and Harnack curves of genus 1. Finally, we show that a bipartite dimer model is invariant under the shrinking/expanding of 2-valent vertices and spider moves if and only if the associated Kasteleyn coefficients are antisymmetric and satisfy Fay’s trisecant identity.

Elliptic and q-Analogs of the Fibonomial Numbers

In 2009, Sagan and Savage introduced a combinatorial model for the Fibonomial numbers, integer numbers that are obtained from the binomial coefficients by replacing each term by its corresponding Fibonacci number. In this paper, we present a combinatorial description for the $q$-analog and elliptic analog of the Fibonomial numbers. This is achieved by introducing some $q$-weights and elliptic weights to a slight modification of the combinatorial model of Sagan and Savage.

Elliptic classes of Schubert varieties

We introduce new notions in elliptic Schubert calculus: the (twisted) Borisov–Libgober classes of Schubert varieties in general homogeneous spaces G/P. While these classes do not depend on any choice, they depend on a set of new variables. For the definition of our classes we calculate multiplicities of some divisors in Schubert varieties, which were only known for full flag varieties before. Our approach leads to a simple recursions for the elliptic classes. Comparing this recursion with R-matrix recursions of the so-called elliptic weight functions of Rimanyi–Tarasov–Varchenko we prove that weight functions represent elliptic classes of Schubert varieties.

A class of partition functions associated with E τ,η gl3 by Izergin–Korepin analysis

Recently, a class of partition functions associated with higher rank rational and trigonometric integrable models were introduced by Foda and Manabe. We use the dynamical R-matrix of the elliptic quantum group Eτ,η(gl3) to introduce an elliptic analog of the partition functions associated with Eτ,η(gl3). We investigate the partition functions of Foda–Manabe type by developing a nested version of the elliptic Izergin–Korepin analysis and present the explicit forms as symmetrization of multivariable elliptic functions. We show that special cases are essentially the elliptic weight functions introduced in the works by Rimányi, Tarasov, and Varchenko; Konno; and Felder, Rimányi, and Varchenko.

Elliptic rook and file numbers

In this work, we construct elliptic analogues of the rook numbers and file numbers by attaching elliptic weights to the cells in a board. We show that our elliptic rook and file numbers satisfy elliptic extensions of corre- sponding factorization theorems which in the classical case were established by Goldman, Joichi and White and by Garsia and Remmel in the file number case. This factorization theorem can be used to define elliptic analogues of various kinds of Stirling numbers of the first and second kind as well as Abel numbers. We also give analogous results for matchings of graphs, elliptically extending the result of Haglund and Remmel.

ARTI (Adaptive Radio Tomographic Imaging): One New Adaptive Elliptical Weighting Model Combining with Tracking Estimates.

Imaging and tracking performance suffers from the mismatch between the model and the measurements in an adaptive radio tomographic imaging system. In this paper, a model-based approach is reviewed and a new adaptive elliptical weighting model is proposed, in which the coverage of ellipse and the voxels weightings can adaptively match the actual environments, and the Savitzky–Golay smoothing filter is presented to eliminate the influence of measurement noise and multipath interference. In our proposed model, the optimal coverage of ellipse and weightings can be obtained from voxel weightings distribution inside the ellipse and pseudo-position area and trailing phenomenon. Finally, the development efforts are evaluated and validated with real experiments conducted in indoor environments for a moving target. The results have shown that the proposed algorithm can improve the accuracy of image and location estimates compared with the normalized weight model and the const-eccentricity weight model.

Elliptic and K-theoretic stable envelopes and Newton polytopes

In this paper we consider the cotangent bundles of partial flag varieties. We construct the $$K$$ -theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes coincide with the elliptic stable envelopes defined by M. Aganagic and A. Okounkov. We give formulas for the $$K$$ -theoretic stable envelopes and our elliptic stable envelopes. We show that the $$K$$ -theoretic stable envelopes are suitable limits of our elliptic stable envelopes. That phenomenon was predicted by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms of the elliptic and trigonometric weight functions which originally appeared in the theory of integral representations of solutions of qKZ equations twenty years ago. (More precisely, the elliptic weight functions had appeared earlier only for the $${\mathfrak {gl}}_2$$ case.) We prove new properties of the trigonometric weight functions. Namely, we consider certain evaluations of the trigonometric weight functions, which are multivariable Laurent polynomials, and show that the Newton polytopes of the evaluations are embedded in the Newton polytopes of the corresponding diagonal evaluations. That property implies the fact that the trigonometric weight functions project to the $$K$$ -theoretic stable envelopes.

Probabilistic Diagnostic Algorithm-Based Damage Detection for Plates with Non-uniform Sections Using the Improved Weight Function

The current studies on damage detection based on guided wave mainly focus on uniform-section plate-like structures. However, plates with non-uniform sections are also essential structures in some engineering applications, such as ships, bridges, aircraft, and buildings. It is essential to identify damage in the non-uniform-section plates (NUSPs). An improvement is put forward to make reconstruction algorithm for the probability inspection of damage (RAPID) feasible on NUSPs. Reconstruction algorithm for the probability inspection of damage (RAPID) is a promising kind of tomography technique for detection and supervising of structures. The damage signatures based on the differences between the Lamb wave signals captured from the undamaged (reference) and damaged (present) states are used to assess the probability of the damage. The elliptical weight function commonly used in the RAPID algorithm limits the applications of the method in identifying damage in non-uniform-section plates. In this study, a novel method is proposed to choose the weight function that satisfies not only uniform-section plates but also NUSPs. A series of experiments on aluminum plates are carried out to validate the correctness of the method mentioned before. The results indicate that the improvement considerably increases the precision of the localization of damage in NUSPs.

Elliptically Distributed Lozenge Tilings of a Hexagon

We present a detailed study of a four parameter family of elliptic weights on tilings of a hexagon introduced by Borodin, Gorin and Rains, generalizing some of their results. In the process, we connect the combinatorics of the model with the theory of elliptic special functions. Using canonical coordinates for the hexagon we show how the $n$-point distribution function and transitional probabilities connect to the theory of $BC_n$-symmetric multivariate elliptic special functions and of elliptic difference operators introduced by Rains. In particular, the difference operators intrinsically capture all of the combinatorics. Based on quasi-commutation relations between the elliptic difference operators, we construct certain natural measure-preserving Markov chains on such tilings which we immediately use to obtain an exact sampling algorithm for these elliptic distributions. We present some simulated random samples exhibiting interesting and probably new arctic boundary phenomena. Finally, we show that the particle process associated to such tilings is determinantal with correlation kernel given in terms of the univariate elliptic biorthogonal functions of Spiridonov and Zhedanov.

Distance Attenuation-Based Elliptical Weighting-g Model in Radio Tomography Imaging

The selection of voxel weightings in the elliptical model is significant to determine the performance of the positioning in the radio tomography imaging system. In this paper, we propose an elliptical weighting model based on the distance of voxels within the scope of ellipse and line of sight path. To adaptively select the voxel weightings, a distance attenuation factor is introduced when the voxel weightings are calculated. Simulation results show that the proposed elliptical weight model improves the accuracy of the target position and also has fewer artifacts. Meanwhile, the mean square errors of the estimated three positions are decreased by 0.004, 0.03, and 0.01, respectively, in comparison with the classical elliptical weighting model.

Elliptic stable envelopes and finite-dimensional representations of elliptic quantum group

We construct a finite dimensional representation of the face type, i.e dynamical, elliptic quantum group associated with $sl_N$ on the Gelfand-Tsetlin basis of the tensor product of the $n$-vector representations. The result is described in a combinatorial way by using the partitions of $[1,n]$. We find that the change of basis matrix from the standard to the Gelfand-Tsetlin basis is given by a specialization of the elliptic weight function obtained in the previous paper[Konno17]. Identifying the elliptic weight functions with the elliptic stable envelopes obtained by Aganagic and Okounkov, we show a correspondence of the Gelfand-Tsetlin bases (resp. the standard bases) to the fixed point classes (resp. the stable classes) in the equivariant elliptic cohomology $E_T(X)$ of the cotangent bundle $X$ of the partial flag variety. As a result we obtain a geometric representation of the elliptic quantum group on $E_T(X)$.

Integral solutions to boundary quantum Knizhnik–Zamolodchikov equations

We construct integral representations of solutions to the boundary quantum Knizhnik–Zamolodchikov equations. These are difference equations taking values in tensor products of Verma modules of quantum affine sl2, with the K-operators acting diagonally. The integrands in question are products of scalar-valued elliptic weight functions with vector-valued trigonometric weight functions (boundary Bethe vectors). These integrals give rise to a basis of solutions of the boundary qKZ equations over the field of quasi-constant meromorphic functions in weight subspaces of the tensor product.

Elliptic Rook and File Numbers

Utilizing elliptic weights, we construct an elliptic analogue of rook numbers for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's $q$-rook numbers by two additional independent parameters $a$ and $b$, and a nome $p$. The elliptic rook numbers are shown to satisfy an elliptic extension of a factorization theorem which in the classical case was established by Goldman, Joichi and White and extended to the $q$-case by Garsia and Remmel. We obtain similar results for elliptic analogues of Garsia and Remmel's $q$-file numbers for skyline boards. We also provide an elliptic extension of the $j$-attacking model introduced by Remmel and Wachs. Various applications of our results include elliptic analogues of (generalized) Stirling numbers of the first and second kind, Lah numbers, Abel numbers, and $r$-restricted versions thereof.

Theta function solutions of the quantum Knizhnik–Zamolodchikov–Bernard equation for a face model

We consider the quantum Knizhnik–Zamolodchikov–Bernard equation for a face model with elliptic weights, the SOS model. We provide explicit solutions as theta functions. On the so-called combinatorial line, in which the model is equivalent to the three-colour model, these solutions are shown to be eigenvectors of the transfer matrix with periodic boundary conditions.

An improved approach for rainfall estimation over Indian summer monsoon region using Kalpana-1 data

In this paper, an improved Kalpana-1 infrared (IR) based rainfall estimation algorithm, specific to Indian summer monsoon region is presented. This algorithm comprises of two parts: (i) development of Kalpana-1 IR based rainfall estimation algorithm with improvement for orographic warm rain underestimation generally suffered by IR based rainfall estimation methods and (ii) cooling index to take care of the growth and decay of clouds and thereby improving the precipitation estimation.In the first part, a power-law based regression relationship between cloud top temperature from Kalpana-1 IR channel and rainfall from Tropical Rainfall Measuring Mission (TRMM) – precipitation radar specific to the Indian region is developed. This algorithm tries to overcome the inherent orographic issues of the IR based rainfall estimation techniques. Over the windward sides of the Western Ghats, Himalayas and Arakan Yoma mountain chains, separate regression coefficients are generated to take care of the orographically produced warm rainfall. Generally global rainfall retrieval methods fail to detect the warm rainfall over these regions. Rain estimated over the orographic region is suitably blended with the rain retrieved over the entire domain comprising of the Indian monsoon region and parts of the Indian Ocean using another regression relationship. While blending, a smoothening function is applied to avoid rainfall artefacts and an elliptical weighting function is introduced for the purpose.In the second part, a cooling index to distinguish rain/no-rain conditions is developed using Kalpana-1 IR data. The cooling index identifies the cloud growing/decaying regions using two consecutive half-hourly IR images of Kalpana-1 by assigning appropriate weights to growing and non-growing clouds. Intercomparison of estimated rainfall from the present algorithm with TRMM-3B42/3B43 precipitation products and Indian Meteorological Department (IMD) gridded rain gauge data are found to be encouraging. The advantages of the present algorithm are that it requires only two IR images as input without depending on other sources of information and simple to implement. The present algorithm performs better than the existing Kalpana-1 IR based rainfall estimation algorithm. Comparison with IMD rainfall data suggests that the underestimation of average rainfall has decreased by 30% for the present algorithm.

Benefits of Using Non-consolidated Domain Influence in Meshless Local Petrov-galerkin (Mlpg) Method for Solving Lefm Problems

This paper presents an efficient meshless method in the formulation of the weak form of local Petrov-Galerkin method MLPG. The formulation is carried out by using an elliptic domain rather than conventional isotropic domain of influence. Therefore, the method involves an MLPG formulation in conjunction with an anisotropic weight function. In the elliptic weight function, each node has three characteristic indicated that were major radius, inner radius, and the direction of the local domain. Furthermore, the space that will be covered by the elliptical domain will be less than the area of the circle (isotropic) at the same main diameter. This means leaving many points of integration are not necessary. Therefore, the computational cost will be decreased. MLPG method with the elliptical domain is used in solving problems of linear elastic fracture mechanism LEFM. MATLAB and Fortran codes are used for obtaining the results of this research .The results were compared with those presented in the literature which shows a reduction in the computational cost up to 15%, and an error criteria enhancement up to 25%.

ELLIPTICAL WEIGHTED HOLICs FOR WEAK LENSING SHEAR MEASUREMENT. III. THE EFFECT OF RANDOM COUNT NOISE ON IMAGE MOMENTS IN WEAK LENSING ANALYSIS

This is the third paper on the improvements of systematic errors in our weak lensing analysis using an elliptical weight function, called E-HOLICs. In the previous papers we have succeeded in avoiding error which depends on ellipticity of background image. In this paper, we investigate the systematic error which depends on signal to noise ratio of background image. We find that the origin of the error is the random count noise which comes from Poisson noise of sky counts. Random count noise makes additional moments and centroid shift error, and those 1st orders are canceled in averaging, but 2nd orders are not canceled. We derived the equations which corrects these effects in measuring moments and ellipticity of the image and test their validity using simulation image. We find that the systematic error becomes less than 1% in the measured ellipticity for objects with $S/N>3$.

ELLIPTICAL-WEIGHTED HOLICs FOR WEAK LENSING SHEAR MEASUREMENT. II. POINT-SPREAD FUNCTION CORRECTION AND APPLICATION TO A370

We have developed a new method(E-HOLICs) of estimating gravitational shear by adopting an elliptical weight function to measure background galaxy images in our previous paper. Following to the previous paper where isotropic Point Spread Function(PSF) correction is calculated, we consider an anisotropic PSF correction in this paper in order to apply E-HOLICs for real data. A an example, E-HOLICs is applied to Subaru data of massive and compact galaxy clusters A370, and is able to detect double peaks in the central region of the cluster consistent with the analysis of strong lensing. We also study the systematic error in E-HOLICs using STEP II simulation. In particular we consider the dependences of signal to noise ratio "S/N" of background galaxies in the shear estimation. Although E-HOLICs does improve systematic error due to the ellipticity dependence as shown in part 1, a systematic error due to the S/N dependence remains, namely E-HOLICs underestimates shear when background galaxies with low S/N objects are used. We discuss a possible improvement of the S/N dependence.

ELLIPTICALLY WEIGHTED HOLICs FOR WEAK-LENSING SHEAR MEASUREMENT. I. DEFINITIONS AND ISOTROPIC POINT-SPREAD FUNCTION CORRECTION

We develop a new method to estimate gravitational shear by adopting an elliptical weight function to measure background galaxy images. In doing so, we introduce a new concept of "zero plane" which is an imaginal source plane where shapes of all sources are perfect circles, and regard the intrinsic shear as the result of an imaginal lensing distortion. This makes the relation between the observed shear, the intrinsic shear and lensing distortion more simple and thus higher-order calculation more easy. The elliptical weight function allows us to measure the mutiplemoment of shape of background galaxies more precisely by weighting highly to brighter parts of image and moreover to reduce systematic error due to insufficient expansion of the weight function in the original approach of KSB. Point Spread Function(PSF) correction in E-HOLICs methods becomes more complicated than those in KSB methods. In this paper we studied isotropic PSF correction in detail. By adopting the lensing distortion as the ellipticity of the weight function, we are able to show that the shear estimation in E-HOLICs method reduces to solve a polynomial in the absolute magnitude of the distortion. We compare the systematic errors between our approach and KSB using STEP2 simulation. It is confirmed that KSB method overestimate the input shear for images with large ellipticities, and E-HOLICs correctly estimate the input shear even for such images. Anisotropic PSF correction and analysis of real data will be presented in forthcoming paper.

Q-Distributions on boxed plane partitions

We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon’s product formula for the number of plane partitions in a box. We then focus on the most general positive degenerations of these weights that are related to orthogonal polynomials; they form three 2-D families. For distributions from these families, we prove two types of results. First, we construct explicit Markov chains that preserve these distributions. In particular, this leads to a relatively simple exact sampling algorithm. Second, we consider a limit when all dimensions of the box grow and plane partitions become large and prove that the local correlations converge to those of ergodic translation invariant Gibbs measures. For fixed proportions of the box, the slopes of the limiting Gibbs measures (that can also be viewed as slopes of tangent planes to the hypothetical limit shape) are encoded by a single quadratic polynomial.