Formula For Weights Research Articles

Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests

We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (Sel Math 7(1):57–81, 2001). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are not determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the 4×4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4\ imes 4$$\\end{document} problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size n≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 4$$\\end{document}, which appear new for n≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 5$$\\end{document}. By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (Nucl Phys B 536:704–732, 1998), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation.

A non-separable progressive multivariate WENO-2r point value

The weighted essentially non-oscillatory technique using a stencil of 2r points (WENO-2r) is an interpolatory method that consists in obtaining a higher approximation order from the non-linear combination of interpolants of r+1 nodes. The result is an interpolant of order 2r at the smooth parts and order r+1 when an isolated discontinuity falls at any grid interval of the large stencil except at the central one. Recently, a new WENO method based on Aitken-Neville's algorithm has been designed for interpolation of equally spaced data at the mid-points and presents progressive order of accuracy close to discontinuities. This paper is devoted to constructing a general progressive WENO method for non-necessarily uniformly spaced data and several variables interpolating in any point of the central interval. Also, we provide explicit formulas for linear and non-linear weights and prove the order obtained. Finally, some numerical experiments are presented to check the theoretical results.

Separation of variables for scalar-valued polynomials in the non-stable range

Any complex-valued polynomial on (Rn)k decomposes into an algebraic combination of O(n)-invariant polynomials and harmonic polynomials. This decomposition, separation of variables, is granted to be unique if n≥2k−1. We prove that the condition n≥2k−1 is not only sufficient, but also necessary for uniqueness of the separation. Moreover, we describe the structure of non-uniqueness of the separation in the boundary cases when n=2k−2 and n=2k−3.Formally, we study the kernel of a multiplication map ϕ carrying out separation of variables. We devise a general algorithmic procedure for describing Ker ϕ in the restricted non-stable range k≤n<2k−1. In the full non-stable range n<2k−1, we give formulas for highest weights of generators of the kernel as well as formulas for its Hilbert series. Using the developed methods, we obtain a list of highest weight vectors generating Ker ϕ.

Ranking voting systems and surrogate weights: Explicit formulas for centroid weights

One of the most important issues in the field of ranking voting systems is the choice of the weighting vector. This issue has been addressed in the literature from different approaches, and one of them has been to obtain the weighting vector as a solution to a linear programming problem. In this paper we analyze some models proposed in the literature and show that one of their main shortcomings is that they cannot guarantee the uniqueness of the solution, so the winner or the final ranking of the candidates may depend on the chosen weighting vector. An alternative to these models is the use of surrogate weights, among which rank order centroid (ROC) weights stand out as the centroid of a specific simplex. Following this idea, in this paper we show the explicit expression for the weights that form the centroid of diverse simplices utilized in ranking voting systems, and we also see that certain surrogate weights frequently employed in literature can be derived as extreme cases where the simplices collapse into a single vector. Moreover, we argue that averaging two weighting vectors can be a valid approach in some cases and, in this way, we can get weighting vectors that closely resemble those used in some sports competitions.

FedSW: Federated learning with adaptive sample weights

Federated Learning (FL) is a machine learning approach in which a cluster of clients collaboratively trains a model without sharing the data of any clients. As the datasets of each client's task are typically noisy and statistically diverse, they are not easily learned by a global model, especially deep networks that are prone to overfitting on biased training data. This study proposes an adaptive sample weighting algorithm based on self-paced learning (SPL). Instead of weighting all samples equally, the algorithm assigns weights to each sample based on its impact on the global model. To achieve this, the client objectives are defined as the sum of the weighted empirical risks and the regularizer with respect to the weight of each sample. The final global model is obtained by alternately optimizing the model parameters and sample weights. By applying the implicit SPL regularizer, we derive an analytic formula for the optimal sample weights used in the experiments. We demonstrate that the algorithm converges and that our method is more stable and accurate than the series of federated averaging algorithms. Specifically, when 30% of the training data is corrupted, the test accuracy in our method is up to 50% higher than that achieved by the federated averaging algorithm.

An Adaptive Beam Forming Algorithm for Reduction of Coexistence- Interference and Improved Performance in 5G Communication Systems

Abstract: A couple of Wi-Fi systems coexisting in a 5G community might produce interference within the same frequency band, degrading the obtained sign's overall performance. This paper proposes a singular set of rules in antenna array processing to address interference-coexistence verbal exchange. We adopt a linear filter which is referred to as a Linearly confined minimal Variance (LCMV) filter out. We impose a log-sum penalty at the coefficients and upload it to the price feature based on traditional singly linearly restrained least suggest square (LC-LMS). The iterative formula for filter weights is derived. We show that the brand-new approach's convergence price is quicker than the conventional one in the usage of simulations in an antenna environment with a signal of hobby, noise, and interference. Furthermore, the proposed approaches suggest rectangular errors (MSE) are shown. Our approach has a lower MSE than the traditional LC-LMS set of rules, in line with the findings of the experiments. The cautioned adaptive beam forming approach may be used in a 5G gadget to cope with sign and interference coexistence.

Reservoir Computing for Solving Ordinary Differential Equations

There is a wave of interest in using physics-informed neural networks for solving differential equations. Most of the existing methods are based on feed-forward networks, while recurrent neural networks solvers have not been extensively explored. We introduce a reservoir computing (RC) architecture, an echo-state recurrent neural network capable of discovering approximate solutions that satisfy ordinary differential equations (ODEs). We suggest an approach to calculate time derivatives of recurrent neural network outputs without using back-propagation. The internal weights of an RC are fixed, while only a linear output layer is trained, yielding efficient training. However, RC performance strongly depends on finding the optimal hyper-parameters, which is a computationally expensive process. We use Bayesian optimization to discover optimal sets in a high-dimensional hyper-parameter space efficiently and numerically show that one set is robust and can be transferred to solve an ODE for different initial conditions and time ranges. A closed-form formula for the optimal output weights is derived to solve first-order linear equations in a one-shot backpropagation-free learning process. We extend the RC approach by solving nonlinear systems of ODEs using a hybrid optimization method consisting of gradient descent and Bayesian optimization. Evaluation of linear and nonlinear systems of equations demonstrates the efficiency of the RC ODE solver.

AN IMPROVED ADAPTIVE BEAMFORMING ALGORITHM FOR 5G INTERFERENCE-COEXISTENCE COMMUNICATION

Multiple wireless systems coexisting in a 5G network might produce interference in the same frequency band, degrading the received signal's performance. In this paper, a novel algorithm is proposed in antenna array processing to handle interference-coexistence communication. We adopt a linear filter which is called Linearly Constrained Minimum Variance (LCMV) filter. We impose a log-sum penalty on the coefficients and add it to the cost function based on classic singly linearly constrained least mean square (LC-LMS). The iterative formula for filter weights is derived. We demonstrate that the new method's convergence rate is faster than the traditional one using simulations in an antenna environment with a signal of interest, noise, and interferences. Furthermore, the proposed method's mean-square-error (MSE) is confirmed. Our technique has a lower MSE than the classic LC-LMS algorithm, according to the findings of the experiments. The suggested adaptive beam forming approach can be used in a 5G system to deal with signal and interference coexistence.

Chevalley formula for anti-dominant minuscule fundamental weights in the equivariant quantum K-group of partial flag manifolds

In this paper, we give an explicit formula of Chevalley type, in terms of the Bruhat graph, for the quantum multiplication with the class of the line bundle associated to an anti-dominant minuscule fundamental weight −ϖk in the torus-equivariant quantum K-group of the partial flag manifold G/PJ (where J=I∖{k}) corresponding to the maximal (standard) parabolic subgroup PJ of minuscule type in type A, D, E, or B. This result is obtained by proving a similar formula in a torus-equivariant K-group of the semi-infinite partial flag manifold QJ of minuscule type, and then by making use of the isomorphism between the torus-equivariant quantum K-group of G/PJ and the torus-equivariant K-group of QJ, recently established by Kato.

The Katok-Sarnak formula for higher weights

We prove a Katok-Sarnak type formula for Maass forms of even weight k and odd level N≥1 that includes and extends the results of Waldspurger, Kohnen-Zagier, Katok-Sarnak, Baruch-Mao, and Biró.

Chevalley formula for anti-dominant weights in the equivariant K-theory of semi-infinite flag manifolds

We prove a Chevalley formula for anti-dominant weights in the torus-equivariant K-group of semi-infinite flag manifolds, which is described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or equivalently, quantum Lakshmibai-Seshadri paths); in contrast to the Chevalley formula for dominant weights in our previous paper [17], the formula for anti-dominant weights has a significant finiteness property. Based on geometric results established in [17], our proof is representation-theoretic, and the Chevalley formula for anti-dominant weights follows from a certain identity for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra; in the proof of this identity, we make use of the (combinatorial) standard monomial theory for semi-infinite Lakshmibai-Seshadri paths, and also a string property of Demazure-like subsets of the set of semi-infinite Lakshmibai-Seshadri paths of a fixed shape, which gives an explicit realization of the crystal basis of a level-zero extremal weight module.

On homogeneous arcs and linear codes over finite chain rings

We consider projective Hjelmslev geometries over finite chain rings of length 2 with residue field of order q. In these geometries we introduce and investigate the so-called homogeneous arcs defined as multisets of points with respect to a special weight function. These arcs are associated with linearly representable q-ary codes. We establish a relation between the parameters of a linearly representable code and the parameters of the associated arc. We prove an inequality for the largest homogeneous weight of an arc of given size N. We consider the $$\tau $$ -dual of a homogeneous arc and give an explicit formula for the homogeneous weights of the hyperplanes with respect to the dual arc. We prove that if a homogeneous arc is of constant weight, this weight is forced to be zero and the arc is a sum of neighbour classes of points. We prove various numerical conditions on the parameters of homogeneous two-weight arcs, as well as an interesting identity involving the parameters of a projective homogeneous two-weight arc and its dual, which isturns out to be also a homogeneous two-weight arc. Finally, we present examples of two-weight homogeneous arcs some of which are new.

InverseK-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type

AbstractWe prove an explicit inverse Chevalley formula in the equivariantK-theory of semi-infinite flag manifolds of simply laced type. By an ‘inverse Chevalley formula’ we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a$\mathbb {Z}\left [q^{\pm 1}\right ]$-linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars$e^{\lambda }$, where$\lambda $is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type$E_8$. The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetricq-Toda operators for minuscule weights in ADE type.

Through a Random Route to the Goal: Theoretical Background and Application of the Method in Tourism Surveying in Poland

Abstract Classic survey methods are ineffective when surveying a small or rare population. Several methods have been developed to address this issue, but often without providing a full mathematical justification. In this paper we propose estimators of parameters relating to Random Route Sampling and explore their basic properties. A formula for the Horvitz-Thompson estimator weights is presented. Finally, a case of a tourism-related survey conducted in Poland is discussed.

Risk Management and the Optimal Combination of Equity Market Factors

Managing the intertemporal risk of optimally constructed multifactor portfolios adds to performance. The increases in Sharpe ratios are in addition to the utility that investors gain from controlling how much active risk they are exposed to over time. We derive a simple closed-form formula for security weights in optimal multifactor portfolios with an active-risk target. We test the risk control of five well-known factors—value, momentum, small size, low beta, and profitability—and the optimal multifactor portfolio. Our empirical research was carried out on the large-capitalization US equity market for 1966 through 2019. We conclude that for the equity market, more active factors are better than fewer if each subportfolio is “pure” as to factor, anchored to the benchmark, and combined on the basis of forecastable risks. Our portfolio construction methodology allows for transparent performance attribution and replication of the process in other markets and time periods.

Dynamic Risk Management of Equity Market Factors

Managing the intertemporal risk of optimally constructed multifactor portfolios adds to performance. The increases in Sharpe ratios are in addition to the utility that investors gain from controlling how much active risk they are exposed to over time. We derive a simple closed-form formula for security weights in optimal multifactor portfolios with an active-risk target. We test the risk control of five well-known factors—value, momentum, small size, low beta, and profitability—and the optimal multifactor portfolio. Our empirical research was carried out on the large-capitalization US equity market for 1966 through 2019. We conclude that for the equity market, more active factors are better than fewer if each factor subportfolio is “pure” as to factor, anchored to the benchmark, and combined on the basis of forecastable risks. Our portfolio construction methodology allows for transparent performance attribution and replication of the process in other markets and time periods.

A novel method optimizing the normalization of cardiac parameters in small animal models: the importance of dimensional indexing.

For indexing cardiac measures in small animal models, tibia length (TL) is a recommended surrogate for body weight (BW) that aims to avoid biases because of disease-induced BW changes. However, we question if indexing by TL is mathematically correct. This study aimed to investigate the relation between TL and BW, heart weight, ventricular weights, and left ventricular diameter to optimize the current common practice of indexing cardiac parameters in small animal models. In 29 healthy Wistar rats (age 5-34 wk) and 116 healthy Black 6 mice (age 3-17 wk), BW appeared to scale nonlinearly to TL1 but linearly to TL3. Formulas for indexing cardiac weights were derived. To illustrate the effects of indexing, cardiac weights between the 50% with highest BW and the 50% with lowest BW were compared. The nonindexed cardiac weights differed significantly between groups, as could be expected (P < 0.001). However, after indexing by TL1, indexed cardiac weights remained significantly different between groups (P < 0.001). With the derived formulas for indexing, indexed cardiac weights were similar between groups. In healthy rats and mice, BW and heart weights scale linearly to TL3. This indicates that not TL1 but TL3 is the optimal surrogate for BW. New formulas for indexing heart weight and isolated ventricular weights are provided, and we propose a concept in which cardiac parameters should not all be indexed to the same measure but one-dimensional measures to BW1/3 or TL1, two-dimensional measures to BW2/3 or TL2, and three-dimensional measures to BW or TL3. NEW & NOTEWORTHY In healthy rats and mice, body weight (BW) scales linearly to tibia length (TL) to the power of three (TL3). This indicates that for indexing cardiac parameters, not TL1 but TL3 is the optimal surrogate for BW. New formulas for indexing heart weight and isolated ventricular weights are provided, and we propose a concept of dimensionally consistent indexing. This concept is proposed to be widely applied in small animal experiments.

A cautionary note on the finite sample behavior of maximal reliability.

Several calls have been made for replacing coefficient α with more contemporary model-based reliability coefficients in psychological research. Under the assumption of unidimensional measurement scales and independent measurement errors, two leading alternatives are composite reliability and maximal reliability. Of these two, the maximal reliability statistic, or equivalently Hancock's H, has received a significant amount of attention in recent years. The difference between composite reliability and maximal reliability is that the former is a reliability index for a scale mean (or unweighted sum), whereas the latter estimates the reliability of a scale score where indicators are weighted differently based on their estimated reliabilities. The formula for the maximal reliability weights has been derived using population quantities; however, their finite-sample behavior has not been extensively examined. Particularly, there are two types of bias when the maximal reliability statistic is calculated from sample data: (a) the sample maximal reliability estimator is a positively biased estimator of population maximal reliability, and (b) the true reliability of composites formed with maximal reliability weights calculated from sample data is on average less than the population reliability. Both effects are more pronounced in small-sample scenarios (e.g., <100). We also demonstrate that the composite reliability estimator for equally weighted composite exhibits substantially less bias, which makes it a more appropriate choice for the small-sample case. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

An explicit formula for the smoother weights of the Hodrick–Prescott filter

Abstract By applying the Sherman–Morrison–Woodbury (SMW) formula and a discrete cosine transformation matrix, De Jong and Sakarya [De Jong, R. M., and N. Sakarya. 2016. “The Econometrics of the Hodrick–Prescott Filter.” Review of Economics and Statistics 98 (2): 310–317] recently derived an explicit formula for the smoother weights of the Hodrick–Prescott filter. More recently, by applying the SMW formula and the spectral decomposition of a symmetric tridiagonal Toeplitz matrix, Cornea-Madeira [Cornea-Madeira, A. 2017. “The Explicit Formula for the Hodrick–Prescott Filter in Finite Sample.” Review of Economics and Statistics 99: 314–318] provided a simpler formula. This paper provides an alternative simpler formula for it and explains the reason why our approach leads to a simpler formula.

Error Analysis for Determination of Accuracy of Johnson’s Formula, Dare’s Formula and Mother's Opinion for the Estimation of Birth Weight: Results of an Iranian Cross‐Sectional Study

Abstract Accurate fetal weight estimation is important for the management of birth. According to the existing literature, none of the clinical techniques is superior to any other. This study aimed to compare error analysis for the determination of the accuracy of Johnson’s formula, Dare’s formula and mother's opinion for the estimation of birth weight. A cross-sectional study involving 147 full-term, cephalic, singleton pregnancies, was performed. The mother’s opinion was recorded and Johnson’s and Dare’s formulas were used to calculate the fetal weight. Accuracy was determined by percentage error, absolute percentage error, and proportion of estimates within 10% of actual birth-weight. Statistical analysis was done using the RM ANOVA, Friedman and Cochran. P &lt; 0.05 was considered significant. The birth weight was correctly estimated in 34.7%, 58.5%, 66% of the cases using the mother’s estimate, Dare’s and Johnson’s formulas. Respectively the proportion of the mother's estimate and Dare’s formula was significantly lower than that of Johnson’s formula for all birth weights. No significant difference was observed in all the measures of accuracy for the low birth-weight range. The mean error of Johnson's formula in both normal weight and macrosomic groups was less than those obtained by the Dare's formula and mother's opinion. The Johnson’s formula is more accurate in actual birth weight estimation than the Dare ’s one and mother's opinion. In the normal weight range, Johnson’s formula is more accurate of the two, while in the macrosomic group, Dare’s formula appears to be more accurate.