Symmetric Set Research Articles

The rank of sparse symmetric matrices over arbitrary fields

AbstractLet be an arbitrary field and be a sequence of sparse weighted Erdős–Rényi random graphs on vertices with edge probability , where weights from are assigned to the edges according to a matrix . We show that the normalized rank of the adjacency matrix of converges to a constant, and derive the limiting expression. Our result shows that for the general class of sparse symmetric matrices under consideration, the asymptotics of the normalized rank are independent of the edge weights and even the field, in the sense that the limiting constant for the general case coincides with the one previously established for adjacency matrices of sparse nonweighted Erdős–Rényi matrices over . Our proof, which is purely combinatorial in its nature, is based on an intricate extension of a novel perturbation approach to the symmetric setting.

Hybrid financing in a dual‐channel supply chain with asymmetric demand information

AbstractThis study delves into the interaction between hybrid financing and asymmetric demand information within a dual‐channel supply chain. In this setup, the supplier directly sells to customers and also through a capital‐constrained retailer. We investigate a unique financing approach involving a blend of bank loans and supplier equity investment to support the retailer's operational (procurement and marketing) activities. Analyzing the equilibrium strategies under both symmetric and asymmetric information settings yields intriguing insights. In the case of symmetric information, we find that the retailer's equilibrium order quantity decreases with the potential market size under hybrid financing, contrary to traditional notions. When asymmetric information is present, a higher acceptance of supplier equity investment by the retailer tends to lead to order quantity distortion downward, increasing signaling costs. Furthermore, a greater proportion of supplier equity investment prompts the retailer to order more products, ultimately boosting profits for both the retailer and supplier. This suggests that supplier equity investment can enhance supply chain efficiency and alleviate the double marginalization effect.

Математическое моделирование волнового поля методом конечных разностей

The system of differential equations of hydrodynamics is reduced to the inhomogeneous wave equation, for which a discrete model is constructed. The computational domain is a rectangle covered with a two-dimensional uniform calculation grid. Discrete analogues of the wave equation for Dirichlet and Neumann boundary conditions were developed. The paper describes the choice of calculation grids. Analytical expressions describing the approximation error in spatial coordinate directions for the second derivative operator based on the second and fourth order of accuracy schemes were obtained and, in this range of error, the sizes of grids were estimated. The computational experiments demonstrated that, to keep the calculation error in the specified limit 0.1- 1%, it is necessary to perform calculations on grids with the number of nodes per half wavelength in the range from 9 to 30 when using the second-order of accuracy scheme, and from 4 to 6 when using the fourth-order of accuracy scheme. It is important to choose the values of the time step and the weight parameter. At the weight parameter σ =1/12, the relative error is significantly less than at σ =0 (corresponding to an explicit scheme) and at σ =1/4 (corresponding to a symmetrical setting of coefficients in a weight scheme). The optimal values of the weight parameter were calculated in the context of maintaining the propagation frequency of oscillatory processes. The analysis of the discrete model gave a condition for the stability of the difference scheme, and an expression describing the approximation error in a time variable, depending on time steps and spatial coordinate directions, was determined. It was established that the approximation of initial conditions makes a smaller contribution to the total error than the approximation of the equation for subsequent time layers. Based on the proposed algorithms and approaches, a software package was designed to simulate the propagation of vibrations in a two-dimensional computational domain. A number of computational experiments were carried out to study, for example, the propagation of acoustic waves from antennas with different directional characteristics and the scattering of waves on the obstacles of different types. To determine the farfields of an acoustic antenna, it is proposed to make the calculation window movable and to find its location in space. This significantly reduces the calculation time for the propagation of sound waves over long distances.

Solving pick-up and delivery problems via deep reinforcement learning based symmetric neural optimization

Recently, there has been a growing trend to use deep reinforcement learning (DRL) to solve NP-hard combinatorial optimization problems such as routing problem, where a policy learned by a deep neural network guides the sequential construction of solutions. Despite their success, most of the previous studies omit the symmetry between the partitions of instances and between solutions, which makes them less effective in handling the more practical scenarios with symmetric features, such as pick-up and delivery problems (PDPs). PDPs are more challenging and practical than traditional routing problems because the correlation between pick-up nodes and delivery nodes in the former is multiple one-to-one and not simple as the correlation in the latter. Besides, most DRL-based methods cannot take both computational cost and the quality of solutions into account. To resolve this issue, we propose the Symmetric Neural Optimization for Pick-up and Delivery Problems (PD-SNO) to solve the pick-up and delivery traveling salesman problem (PDTSP) and the multi-commodity one-to-one pick-up and delivery traveling salesman problem (m-PDTSP), respectively. The PD-SNO leverages the symmetries between different partitions of instances to effectively capture the relationship between symmetric node sets (i.e., pick-up nodes and delivery nodes); the PD-SNO also leverages the symmetry between different trajectories of solution to guide the construction of solutions and training of policy. We evaluate the performance of our PD-SNO, and the results show that it outperforms various representative baselines, including traditional methods (both exact and heuristic algorithms) and state-of-the-art DRL-based methods (both construction and improvement types) on synthetic datasets. Then, we further assess the generalization performance of our PD-SNO on benchmark datasets. Finally, we research the independent effectiveness of our designs through ablation studies.

Rigidity for the perimeter inequality under Schwarz symmetrization

In this paper, we give necessary and sufficient conditions for the rigidity of the perimeter inequality under Schwarz symmetrization. The term rigidity refers to the situation in which the equality cases are only obtained by translations of the symmetric set. In particular, we prove that the sufficient conditions for rigidity provided in M. Barchiesi, F. Cagnetti and N. Fusco [Stability of the Steiner symmetrization of convex sets. J. Eur. Math. Soc. 15 (2013), 1245-1278.] are also necessary.

A neural circuit architecture for rapid learning in goal-directed navigation

Anchoring goals to spatial representations enables flexible navigation but is challenging in novel environments when both representations must be acquired simultaneously. We propose a framework for how Drosophila uses internal representations of head direction (HD) to build goal representations upon selective thermal reinforcement. We show that flies use stochastically generated fixations and directed saccades to express heading preferences in an operant visual learning paradigm and that HD neurons are required to modify these preferences based on reinforcement. We used a symmetric visual setting to expose how flies' HD and goal representations co-evolve and how the reliability of these interacting representations impacts behavior. Finally, we describe how rapid learning of new goal headings may rest on a behavioral policy whose parameters are flexible but whose form is genetically encoded in circuit architecture. Such evolutionarily structured architectures, which enable rapidly adaptive behavior driven by internal representations, may be relevant across species.

Exploring the role non-coding RNAs during myocardial cell fate.

Myocardial cell fate specification takes place during the early stages of heart development as the precardiac mesoderm is configured into two symmetrical sets of bilateral precursor cells. Molecular cues of the surrounding tissues specify and subsequently determine the early cardiomyocytes, that finally matured as the heart is completed at early postnatal stages. Over the last decade, we have greatly enhanced our understanding of the transcriptional regulation of cardiac development and thus of myocardial cell fate. The recent discovery of a novel layer of gene regulation by non-coding RNAs has flourished their implication in epigenetic, transcriptional and post-transcriptional regulation of cardiac development. In this review, we revised the current state-of-the-art knowledge on the functional role of non-coding RNAs during myocardial cell fate.

Calibrated EWMA estimators for time-scaled surveys with diverse applications

Recently, several memory-type mean estimators (including ratio, product, and logarithmic) have been developed. These estimators rely on exponentially weighted moving average (EWMA), which incorporate both historical and present sample data. In this article, we propose EWMA type calibrated estimators under single and double stratified random sampling (StRS). Because calibration method enhances the estimates by modifying the stratification weight, taking advantage of supplementary information. To evaluate the performance of estimators, various real-world time-scaled data sets pertaining to stock market and weather are taken into account. Additionally, we also conduct a simulation study using a bivariate symmetric data set. The numerical results show the superiority of proposed estimators (y¯TM,y¯TaM) over the adapted ones (y¯PM,y¯PaM).

Emergent modified gravity

A complete canonical formulation of general covariance makes it possible to construct new modified theories of gravity that are not of higher-curvature form, as shown here in a spherically symmetric setting. The usual uniqueness theorems are evaded by using a crucial and novel ingredient, allowing for fundamental fields of gravity distinct from an emergent space-time metric that provides a geometrical structure to all solutions. As specific examples, there are new expansion-shear couplings in cosmological models, a form of modified Newtonian dynamics can appear in a space-time covariant theory without introducing extra fields, and related effects help to make effective models of canonical quantum gravity fully consistent with general covariance.

An Improved Sorting Algorithm for Periodic PRI Signals Based on Congruence Transform

Recently, a signal sorting algorithm based on the congruence transform has been proposed, which is effective in dealing with the staggered Pulse Repetition Interval (PRI) signals. It can effectively sort the staggered PRI signals and obtain the sub-PRI sequence directly without sub-PRI ranking, and it is less affected by interfered pulses and pulse loss. Nevertheless, we find that the algorithm causes pseudo-peaks in the remainder histogram when sorting signals such as sliding PRI, sinusoidal PRI, etc. (collectively referred to as periodic PRI signal in this paper) and pseudo-peaks will cause errors in signal sorting. To solve the issue of pseudo-peaks when sorting periodic PRI signals, an improved sorting algorithm based on congruence transform is proposed. According to the analysis of the congruence characteristics of the periodic PRI signal, a novel method is proposed to identify pseudo-peaks based on the histogram peak amplitude and symmetric difference set. The signal sorting algorithm based on congruence transform is improved to achieve a good sorting effect on periodic PRI signals. Simulation experiments demonstrate that the novel algorithm can effectively sort periodic PRI signals and improve Precall, Pd, and Pf by 6.9%, 5.1%, and 3.2%, respectively, compared to the typical similar algorithms.

Frame set for Gabor systems with Haar window

We describe the full structure of the frame set for the Gabor system G(g;α,β):={e−2πimβ⋅g(⋅−nα):m,n∈Z} with the window being the Haar function g=−χ[−1/2,0)+χ[0,1/2). This is the first compactly supported window function for which the frame set is represented explicitly.The strategy of this paper is to introduce the piecewise linear transformation M on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish a necessary and sufficient condition for the Gabor system G(g;α,β) to be a frame, i.e., the symmetric invariant set of the transformation M is empty.

Conditions for equality in Anderson’s theorem

The classical Anderson theorem is a well-known result relevant to multivariate statistics and probability. It establishes an integral inequality for symmetric quasiconcave functions over a shifted symmetric convex set. The aim of this note is twofold. First, we provide necessary and sufficient conditions for equality in Anderson’s theorem, extending the result of Soms (1991). Second, we propose a set of tractable requirements that guarantee strict inequality in Anderson’s theorem. Our results are used to characterize the independence of Gaussian-distributed random variables.

Statistical inference for a novel distribution using ranked set sampling with applications

Working on symmetrical or asymmetrical data is complicated since each requires a different probability density function. Many statistical distributions can be used for these data types, where choosing one should be satisfied with the correct data type. So, we apply the ranked set sampling technique, which is essential in gaining data when dealing with units in a population is expensive. However, their classification is simple according to the variable of interest. The Unit Generalized Rayleigh distribution has recently played a crucial role in analyzing symmetrical or asymmetrical complex data sets specifically in modeling claim and risk data used in actuarial and financial studies, and its density can take different symmetric and asymmetric possible shapes. It is proposed in various areas, such as reliability, survival, economics, actuarial science, and insurance. We applied the ranked set sampling design in this article for gaining the model parameter estimations of the unit generalized Rayleigh model. Different estimation procedures and risk measures are computed. Moreover, the performance of these measures is illustrated via numerical simulation experiments. Under different proposed estimators, we conduct the validation of the suggested ranked set sampling design via numerous Monte Carlo simulation experiments by computing average bias and mean squared errors. At the end, we illustrated two real applications of the financial area for demonstrating the potential and the supremacy of the proposed ranked set sampling estimators.

ALGEBRAIC ADJUSTMENT OF MEASURED TRIANGLE ELEMENTS

It is difficult to align the measured elements of the triangle. A method of alignment using algebraic circles is proposed. If six elements are measured in a triangle: angles A, B, C, lengths of opposite sides a, b, c with mean square errors mA, mB, mC, ma, mb, mc, the values of the equalized elements must satisfy the requirements of the equations where D is the metric parameter of the triangle (the metric value through which the mathematical relationship between the sides and angles of the triangle is established), that is, the value of the diameter of the circle. If , which are found by formulas (1) and condition (2) is fulfilled, then this confirms the validity of the theorem of sines. In fact, it is a symmetrical set, because the number and composition of the upper indices corresponds to the number and composition of the lower indices, and this is a reflection of dimensions, since the triangle ABC has pairs of opposite elements: а- А; b- B; c- C. Let's construct the graph G Do of the diameter of the circle circumscribed around the aligned triangle, for which we will first construct the graphs G Di of the diameters of the circles circumscribed around the T-triangles, which are shown in fig. 1, and then combine them. Three vertices a, b, c form a "plane" - this is a metric field. The reflection of the metric field is the field of angular measurements - a "plane" defined by three points A, B, C. Note that for a measured triangle, the reflection of the metric field in the angular one is not unambiguous, so the first plane in fig. 1 is "inclined" to the second. After determining the most likely metric parameter Do of the triangle, it is possible to bring these planes into full correspondence. That is, Do is a "lens" [4],[8] with the help of which we obtain an unambiguous mapping of the metric field in the angular field and vice versa.

Strategyproof and fair matching mechanism for union of symmetric m-convex constraints.

We identify a new class of distributional constraints defined as a union of symmetric M-convex sets, which can represent a wide range of real-life constraints in two-sided matching settings. Since M-convexity is not closed under union, a union of symmetric M-convex sets does not belong to this well-behaved class of constraints. Consequently, devising a fair and strategyproof mechanism to handle this new class is challenging. We present a novel mechanism for it called Quota Reduction Deferred Acceptance (QRDA), which repeatedly applies the standard Deferred Acceptance mechanism by sequentially reducing artificially introduced maximum quotas. We show that QRDA is fair and strategyproof when handling a union of symmetric M-convex sets, which extends previous results obtained for a subclass of the union of symmetric M-convex sets: ratio constraints. QRDA always yields a weakly better matching for students than a baseline mechanism called Artificial Cap Deferred Acceptance (ACDA). We also experimentally show that QRDA outperforms ACDA in terms of nonwastefulness.

Improved results on normalized solutions for planar Schrödinger–Poisson equations with [formula omitted]-supercritical case

We prove the existence of normalized solutions for the following planar Schrödinger–Poisson equation −Δu+λu+12πln|⋅|∗|u|2u=|u|q−2u,x∈R2,∫R2u2dx=c, where c>0, q>4 and λ∈R arises as a Lagrange multiplier and is not a priori given. In the axially symmetric setting, new variational techniques and inequalities related with logarithmic convolution potential are developed to detect the geometry structure of the constrained functional, we believe, which can be applicable for more planar L2-constrained problems. Our study improves the one of Cingolani and Jeanjean (2019) from some small values of c to arbitrary c>0, which seems to be new in the context of normalized solutions.

Quotients of the curve complex

We consider three kinds of quotients of the curve complex, which are obtained by coning off uniformly quasiconvex subspaces: symmetric curve sets, non-maximal train track sets, and compression body disc sets. We show that the actions of the mapping class group on those quotients are strongly weakly properly discontinuously (WPD), which implies that the actions are non-elementary and those quotients are of infinite diameter.

Physical phase field model for phagocytosis

We propose and study a simple, physical model for phagocytosis, i.e. the active, actin-mediated uptake of micron-sized particles by biological cells. The cell is described by the phase field method and the driving mechanisms of uptake are actin ratcheting, modeled by a dynamic vector field, as well as cell-particle adhesion due to receptor-ligand binding. We first test the modeling framework for the symmetric situation of a spherical cell engulfing a fixed spherical particle. We then exemplify its versatility by studying various asymmetric situations like different particle shapes and orientations, as well as the simultaneous uptake of two particles. In addition, we perform a perturbation theory of a slightly modified model version in the symmetric setting, allowing to derive a reduced model, shedding light on the effective driving forces and being easier to solve. This work is meant as a first step in describing phagocytosis and we discuss several effects that are amenable to future modeling within the same framework.

Initial data on big bang singularities in symmetric settings

Initial data on big bang singularities in symmetric settings

Deep Diversity-Enhanced Feature Representation of Hyperspectral Images.

In this paper, we study the problem of efficiently and effectively embedding the high-dimensional spatio-spectral information of hyperspectral (HS) images, guided by feature diversity. Specifically, based on the theoretical formulation that feature diversity is correlated with the rank of the unfolded kernel matrix, we rectify 3D convolution by modifying its topology to enhance the rank upper-bound. This modification yields a rank-enhanced spatial-spectral symmetrical convolution set (ReS 3-ConvSet), which not only learns diverse and powerful feature representations but also saves network parameters. Additionally, we also propose a novel diversity-aware regularization (DA-Reg) term that directly acts on the feature maps to maximize independence among elements. To demonstrate the superiority of the proposed ReS 3-ConvSet and DA-Reg, we apply them to various HS image processing and analysis tasks, including denoising, spatial super-resolution, and classification. Extensive experiments show that the proposed approaches outperform state-of-the-art methods both quantitatively and qualitatively to a significant extent. The code is publicly available at https://github.com/jinnh/ReSSS-ConvSet.