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In this work we consider the notion of B-equivalence of pseudometrics. Two pseudometrics d1 and d2 on a set X are called B-equivalent, where B is a subgroup of the group of all bijections on X, if there exists an element b of B such that d1(x, y) = d2(b(x), b(y)) for every x, y ∈ X, that is, d1 can be obtained from d2 by permutating elements of X with the aid of the bijection b. The group B generates the group b B of transformations of the set of all pseudometrics on X, elements of which act as d(·, ·) 7→ d(b(·), b(·)), where d is a pseudometrics on X and b ∈ B. A function f on the set of all pseudometrics on X is called b B-symmetric if f is invariant under the action on its argument of elements of the group b B. If two pseudometrics d1 and d2 are B-equivalent, then f(d1) = f(d2) for every b B-symmetric function f. In general, the technique of symmetric functions is well-developed for the case of symmetric continuous polynomials and, in particular, for the case of symmetric continuous linear functionals on Banach spaces. To use this technique for the construction of b B-symmetric functions on sets of pseudometrics, we map these sets to some appropriate Banach space V, which is isometrically isomorphic to the Banach space ℓ1 of all absolutely summing real sequences. We investigate symmetric (with respect to an arbitrary group of symmetry, elements of which map the standard Schauder basis of ℓ1 into itself) linear continuous functionals on ℓ1. We obtain the complete description of the structure of these functionals. Also we establish analogical results for symmetric linear continuous functionals on the space V. These results are used for the construction of b B-symmetric functionals on the set of all pseudometrics on an arbitrary set X for the following case: the group B of bijections on X, that generates the group b B, is such that the set of all x ∈ X, for which there exists b ∈ B such that b(x) ̸= x, is finite. [ABSTRACT FROM AUTHOR]

Symmetry is often regarded as an integral aspect about aesthetics. This motivates the pursuit of interdisciplinary studies based on the use of subjects in mathematics concerned with symmetry in conjunction with aesthetics. What is referred to as a symmetric function in the field of algebraic combinatorics is an abstraction based on polynomials that exhibit a symmetric property, and this leads us to pursue an algebraic combinatorics-inspired exploration based on aesthetics. In particular, we use different bases and transitions between them to create aesthetically pleasing visualizations of symmetric functions. We see that these visualizations in turn raise new and interesting questions. [ABSTRACT FROM AUTHOR]

In this work, we investigate algebras of symmetric and block-symmetric polynomials and analytic functions on complex Banach spaces of Lebesgue measurable functions for which the pth power of the absolute value is Lebesgue integrable, where p ∈ [ 1, + ∞ ), and Lebesgue measurable essentially bounded functions on [ 0, 1 ]. We show that spectra of Fréchet algebras of block-symmetric entire functions of bounded type on these spaces consist only of point-evaluation functionals. Also we construct algebraic bases of algebras of continuous block-symmetric polynomials on these spaces. We generalize the above-mentioned results to a wide class of algebras of symmetric entire functions. [ABSTRACT FROM AUTHOR]

The present investigation focus on applying the theories of differential subordination, differential superordination and related sandwich-type results for the study of some subclasses of symmetric functions connected through a linear extended multiplier operator, which was previously defined by involving the $ q $-Choi-Saigo-Srivastava operator. The aim of the paper is to define a new class of analytic functions using the aforementioned linear extended multiplier operator and to obtain sharp differential subordinations and superordinations using functions from the new class. Certain subclasses are highlighted by specializing the parameters involved in the class definition, and corollaries are obtained as implementations of those new results using particular values for the parameters of the new subclasses. In order to show how the results apply to the functions from the recently introduced subclasses, numerical examples are also provided.

We prove that there are no 6-cycles of the form xn+3 = xif(xj, xk), with i, j, k ∈ {n, n+1, n+2} pairwise distinct, whenever f : (0,∞) x (0,∞) → (0,∞) is a continuous symmetric function, that is, f(x, y) = f(y, x), for all x, y > 0. Moreover, we obtain all the 6-cycles of potential form and present some open questions relative to the search of p-cycles whenever symmetry does not hold. [ABSTRACT FROM AUTHOR]

The present investigation focus on applying the theories of differential subordination, differential superordination and related sandwich-type results for the study of some subclasses of symmetric functions connected through a linear extended multiplier operator, which was previously defined by involving the q-Choi-Saigo-Srivastava operator. The aim of the paper is to define a new class of analytic functions using the aforementioned linear extended multiplier operator and to obtain sharp differential subordinations and superordinations using functions from the new class. Certain subclasses are highlighted by specializing the parameters involved in the class definition, and corollaries are obtained as implementations of those new results using particular values for the parameters of the new subclasses. In order to show how the results apply to the functions from the recently introduced subclasses, numerical examples are also provided. [ABSTRACT FROM AUTHOR]

We consider universal approximations of symmetric and anti-symmetric functions, which are important for applications in quantum physics, as well as other scientific and engineering computations. We give constructive approximations with explicit bounds on the number of parameters with respect to the dimension and the target accuracy ϵ. While the approximation still suffers from the curse of dimensionality, to the best of our knowledge, these are the first results in the literature with explicit error bounds for functions with symmetry or anti-symmetry constraints

In this paper, the relationship between r -convexity and Schur-convexity is investigated first. Moreover, the r -convexity and Schur-convexity of a class of functions are studied. As applications, some new inequalities on Minkowski's inequality are established. [ABSTRACT FROM AUTHOR]

In this work, we investigate algebras of block-symmetric and weakly symmetric polynomials and analytic functions on complex Banach spaces of Lebesgue measurable functions, for which the pth power of the absolute value is Lebesgue integrable, where p ∈ [1, +∞), and Lebesgue measurable essentially bounded functions on [0, 1]. We construct generating systems of algebras of all weakly symmetric continuous complex-valued polynomials on these spaces. Also we establish conditions under which sets of weakly symmetric analytic functions are algebras. [ABSTRACT FROM AUTHOR]

We describe an algebraic technique for operating with power series whose coefficients are represented by integrals of symmetric functions fn defined on the Cartesian powers Ωn of a set Ω with a measure μ. Moreover, each of the coefficient functions fn is obtained by means of a special mapping from graphs with n labeled vertices belonging to a fixed class. This technique has application to equilibrium statistical mechanics and to problems of enumeration of graphs. [ABSTRACT FROM AUTHOR]

In the plane of a triangle ABC, let L 1 as L 2 be a pair of nonparallel lines. A coordinate system having L 1 as x-axis and L 2 as y-axis is defined, and transformation formulas for representations of points and lines in (x, y) coordinates and barycentric coordinates are derived. The set of central lines consists of two types: even and odd, with consequences for the cases that L 1 and L 2 are both even, or both odd, or one even and the other odd. A mapping T(P, U) is defined for pairs (P, U) of points. This mapping maps lines to lines. A distinctive example of a bivariate coordinate system is introduced as the Steiner system, in which the axes are those of the Steiner circumellipse. In this system, the line y = x is the Euler line of ABC. [ABSTRACT FROM AUTHOR]

In this work, we present the notion of a weakly symmetric function. We show that the subset of all weakly symmetric elements of an arbitrary vector space of functions is a vector space. Moreover, the subset of all weakly symmetric elements of some algebra of functions is an algebra. Also we consider weakly symmetric functions on the complex Banach space $L_p[0,1]$ of all Lebesgue measurable complex-valued functions on $[0,1]$ for which the $p$th power of the absolute value is Lebesgue integrable. We show that every continuous linear functional on $L_p[0,1],$ where $p\in (1,+\infty),$ can be approximated by weakly symmetric continuous linear functionals.

Two classes of determinants with their matrix entries being polynomial differences are evaluated in closed product expressions. They extend with many free parameters the Vandermonde-like determinant identities associated with the classical root systems. [ABSTRACT FROM AUTHOR]

We study functions of the roots of an integer polynomial p = p (x) with m ≥ 2 distinct roots ρ = (ρ 1 , ... , ρ m) of multiplicity μ = (μ 1 , ... , μ m) , μ 1 ≥ ⋯ ≥ μ m ≥ 1. Traditionally, root functions are studied via the theory of symmetric polynomials; we generalize this theory to μ -symmetric polynomials. We initiate the study of the vector space of μ -symmetric polynomials of a given degree δ via the concepts of μ -gist and μ -ideal. In particular, we are interested in the root D μ + (r 1 , ... , r m) := ∏ 1 ≤ i < j ≤ m (r i − r j) μ i + μ j . The D-plus discriminant of p is D + (p) := D μ + (ρ 1 , ... , ρ m). This quantity appears in the complexity analysis of the root clustering algorithm of Becker et al. (ISSAC 2016). We conjecture that D μ + (r 1 , ... , r m) is μ -symmetric, which implies D + (p) is rational. To explore this conjecture experimentally, we introduce algorithms for checking if a given root function is μ -symmetric. We design three such algorithms: one based on Gröbner bases, another based on canonical bases and reduction, and the third based on solving linear equations. Each of these algorithms has variants that depend on the choice of a basis for the μ -symmetric functions. We implement these algorithms (and their variants) in Maple and experiments show that the latter two algorithms are significantly faster than the first. [ABSTRACT FROM AUTHOR]

In this paper, we discuss and introduce a new study on the connection between geometric function theory, especially sandwich theorems, and Viete's theorem in elementary algebra. We obtain some conclusions for differential subordination and superordination for a new formula of complete homogeneous symmetric functions class involving an ordered cyclic operator. In addition, certain sandwich theorems are found. [ABSTRACT FROM AUTHOR]

The enumeration of lattice paths is an important counting model in enumerative combinatorics. Because it can provide powerful methods and technical support in the study of discrete structural objects in different disciplines, it has attracted much attention and is a hot research field. In this paper, we summarize two kinds of the lattice path counting models that are single lattice paths and family of nonintersecting lattice paths and their applications in terms of the change of dimensions, steps, constrained conditions, the positions of starting and end points, and so on. (1) The progress of classical lattice path such as Dyck lattice is introduced. (2) A method to study the enumeration of lattice paths problem by generating function is introduced. (3) Some methods of studying the enumeration of lattice paths problem by matrix are introduced. (4) The family of lattice paths problem and some counting methods are introduced. (5) Some applications of family of lattice paths in symmetric function theory are introduced, and a related open problem is proposed. [ABSTRACT FROM AUTHOR]

We examine the coefficients in front of Chern numbers for complex genera, and pay special attention to the Td 1 2 -genus, the Γ-genus as well as the Todd genus. Some related geometric applications to hyper-Kähler and Calabi-Yau manifolds are discussed. Along this line and building on the work of Doubilet in 1970s, various Hoffman-type formulas for multiple-(star) zeta values and transition matrices among canonical bases of the ring of symmetric functions can be uniformly treated in a more general framework. [ABSTRACT FROM AUTHOR]

In this paper, we extend the chromatic symmetric function X to a chromatic k-multisymmetric function X k , defined for graphs equipped with a partition of their vertex set into k parts. We demonstrate that this new function retains the basic properties and basis expansions of X , and we give a method for systematically deriving new linear relationships for X from previous ones by passing them through X k. In particular, we show how to take advantage of homogeneous sets of G (those S ⊆ V (G) such that each vertex of V (G) ﹨ S is either adjacent to all of S or is nonadjacent to all of S) to relate the chromatic symmetric function of G to those of simpler graphs. Furthermore, we show how extending this idea to homogeneous pairs S 1 ⊔ S 2 ⊆ V (G) generalizes the process used by Guay-Paquet to reduce the Stanley-Stembridge conjecture to unit interval graphs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a method for finding the integral of symmetric multivariate function. We compute the number of nodes which the method use them, and also by using the Gauss-Legendre integrating, we obtain the approximate value the generalized symmetric function. Theoretical consideration has been discussed and some examples were presented to show the ability of the method for approximate value of integral of the symmetric functions. In this numerical integration approach, for a symmetric function that has the same calculations at a number of different node points, only calculations are performed for a node and the result is multiplied by the number of repetitions of similar cases. In addition to modulating errors due to rounding and expanded error during calculations, much less memory is used than the numerical integration method. Also in this approach, the time of numerical integration is reduced and the numerical results confirm this. [ABSTRACT FROM AUTHOR]

In this work, we present the notion of a weakly symmetric function. We show that the subset of all weakly symmetric elements of an arbitrary vector space of functions is a vector space. Moreover, the subset of all weakly symmetric elements of some algebra of functions is an algebra. Also we consider weakly symmetric functions on the complex Banach space Lp[0,1] of all Lebesgue measurable complex-valued functions on [0,1] for which the pth power of the absolute value is Lebesgue integrable. We show that every continuous linear functional on Lp[0,1], where p∈(1,+∞), can be approximated by weakly symmetric continuous linear functionals. [ABSTRACT FROM AUTHOR]

Symmetric functions play an important role in several subjects of mathematics, such as algebraic combinatorics, representation theory of finite groups and algebraic geometry. In this paper, we study the solutions of the following system of equations related to bases of symmetric functions: where , are natural numbers, and . Our main theorems generalize several known results. [ABSTRACT FROM AUTHOR]

Product fault diagnosis has always been the focus of quality and reliability research. However, a failure–rate curve of some products is a symmetrical function, the fault analysis result is not true because the failure period of the products cannot be judged accurately. In order to solve the problem of fault diagnosis, this paper proposes a new Takagi-Sugeno (T-S) dynamic fault tree analysis method based on a Bayesian network accompanying the Wiener process. Firstly, the top event, middle event, and bottom event of the product failure mode are determined, and the T-S dynamic fault tree is constructed. Secondly, in order to form the Bayesian network diagram of the T-S dynamic fault tree, the events in the fault tree are transformed into nodes, and the T-S dynamic gate is also transformed into directed edges. Then, the Wiener process is used to model the performance degradation process of the stationary independent increment of the symmetric function distribution, and the maximum likelihood estimation method is applied to estimate the unknown parameters of the degradation model. Next, the product residual life prediction model is established based on the concept of first arrival time, and a symmetric function of failure–rate curve is obtained by using the product failure probability density function. According to the fault density function derived from the Wiener process, the reverse reasoning algorithm of the Bayesian network is established. Combined with the prior probability of the bottom event, the posterior probability of the root node is calculated and sorted as well. Finally, taking the insufficient braking force of electromagnetic brakes as an example, the practicability and objectivity of the new method are proved. [ABSTRACT FROM AUTHOR]

Group Equivariant Operators (GEOs) are a fundamental tool in the research on neural networks, since they make available a new kind of geometric knowledge engineering for deep learning, which can exploit symmetries in artificial intelligence and reduce the number of parameters required in the learning process. In this paper we introduce a new method to build non-linear GEOs and non-linear Group Equivariant Non-Expansive Operators (GENEOs), based on the concepts of symmetric function and permutant. This method is particularly interesting because of the good theoretical properties of GENEOs and the ease of use of permutants to build equivariant operators, compared to the direct use of the equivariance groups we are interested in. In our paper, we prove that the technique we propose works for any symmetric function, and benefits from the approximability of continuous symmetric functions by symmetric polynomials. A possible use in Topological Data Analysis of the GENEOs obtained by this new method is illustrated.

It is well known that Heron's equality provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its edges. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges. Surprisingly, this has proved to be far from simple, and no explicit solutions exist for cyclic polygons having n > 4 edges. In this paper we investigate such a problem by following a new and elementary approach, based on the idea that the simple geometry underlying Heron's and Brahmagupta's equalities hides the real players of the game. In details, we propose to focus on the dissection of the edges determined by the incircles of a suitable triangulation of the cyclic polygon, showing that this approach leads to an explicit formula for the area as a symmetric function of the lengths of these segments. We also show that such a symmetry can be rediscovered in Heron's and Brahmagupta's results, which consequently represent special cases of the provided general equality. [ABSTRACT FROM AUTHOR]

Research in the area of secure multi-party computation using a deck of playing cards, often called card-based cryptography, started from the introduction of the five-card trick protocol to compute the logical AND function by den Boer in 1989. Since then, many card-based protocols to compute various functions have been developed. In this paper, we propose two new protocols that securely compute the n -variable equality function (determining whether all inputs are equal) E : { 0 , 1 } n → { 0 , 1 } using 2 n cards. The first protocol can be generalized to compute any doubly symmetric function f : { 0 , 1 } n → Z using 2 n cards, and any symmetric function f : { 0 , 1 } n → Z using 2 n + 2 cards. The second protocol can be generalized to compute the k -candidate n -variable equality function E : (Z / k Z) n → { 0 , 1 } using 2 ⌈ lg ⁡ k ⌉ n cards. [ABSTRACT FROM AUTHOR]

In this paper, we discuss and introduce a new study on the connection between geometric function theory, especially sandwich theorems, and Viete’s theorem in elementary algebra. We obtain some conclusions for differential subordination and superordination for a new formula of complete homogeneous symmetric functions class involving an ordered cyclic operator. In addition, certain sandwich theorems are found.

Motivated by a question in Schubert calculus, we study the interplay of quasisymmetric polynomials with the divided symmetrization operator, which was introduced by Postnikov in the context of volume polynomials of permutahedra. Divided symmetrization is a linear form which acts on the space of polynomials in n indeterminates of degree n - 1 . We first show that divided symmetrization applied to a quasisymmetric polynomial in m indeterminates can be easily determined. Several examples with a strong combinatorial flavor are given. Then, we prove that the divided symmetrization of any polynomial can be naturally computed with respect to a direct sum decomposition due to Aval–Bergeron–Bergeron, involving the ideal generated by positive degree quasisymmetric polynomials in n indeterminates. [ABSTRACT FROM AUTHOR]

VLSI technology makes the integration of multiple electronic components into a single electronic chip possible but Moore has already predicted that further miniaturization of VLSI chips will cause excess heat dissipation. Because of this impediment faced by this technology, the scientists have turned their focus to nano-technological alternatives to keep up the continuous device improvement and continue with the miniaturization procedure without facing any hurdle from the aspect of heat dissipation and space and time complexity. Memristor can be such a nano-alternative, which has the potential of very rapid execution, miniaturization and very low power consumption compared to transistor-based technology. The main objective of this paper is to show an application of symmetric function in cryptography. In order to do so, first, three-input and five-input symmetric functions are implemented in memristor-based nanocrossbar circuits. Then a dynamic circuit of symmetric functions is implemented where the input voltages convert the circuit to three-input or five-input symmetric function circuit as and when required. The simulation results show the satisfactory behavior of our proposed design. [ABSTRACT FROM AUTHOR]

We introduce a family of varieties Y n , λ , s , which we call the Δ -Springer varieties , that generalize the type A Springer fibers. We give an explicit presentation of the cohomology ring H ⁎ (Y n , λ , s) and show that there is a symmetric group action on this ring generalizing the Springer action on the cohomology of a Springer fiber. In particular, the top cohomology groups are induction products of Specht modules with trivial modules. The λ = (1 k) case of this construction gives a compact geometric realization for the expression in the Delta Conjecture at t = 0. Finally, we generalize results of De Concini and Procesi on the scheme of diagonal nilpotent matrices by constructing an ind-variety Y n , λ whose cohomology ring is isomorphic to the coordinate ring of the scheme-theoretic intersection of an Eisenbud–Saltman rank variety and diagonal matrices. [ABSTRACT FROM AUTHOR]

In this paper, the authors study new inequalities and generalizations for symmetric means and give new proofs for some known results by applying majorization theory.

Product fault diagnosis has always been the focus of quality and reliability research. However, a failure–rate curve of some products is a symmetrical function, the fault analysis result is not true because the failure period of the products cannot be judged accurately. In order to solve the problem of fault diagnosis, this paper proposes a new Takagi-Sugeno (T-S) dynamic fault tree analysis method based on a Bayesian network accompanying the Wiener process. Firstly, the top event, middle event, and bottom event of the product failure mode are determined, and the T-S dynamic fault tree is constructed. Secondly, in order to form the Bayesian network diagram of the T-S dynamic fault tree, the events in the fault tree are transformed into nodes, and the T-S dynamic gate is also transformed into directed edges. Then, the Wiener process is used to model the performance degradation process of the stationary independent increment of the symmetric function distribution, and the maximum likelihood estimation method is applied to estimate the unknown parameters of the degradation model. Next, the product residual life prediction model is established based on the concept of first arrival time, and a symmetric function of failure–rate curve is obtained by using the product failure probability density function. According to the fault density function derived from the Wiener process, the reverse reasoning algorithm of the Bayesian network is established. Combined with the prior probability of the bottom event, the posterior probability of the root node is calculated and sorted as well. Finally, taking the insufficient braking force of electromagnetic brakes as an example, the practicability and objectivity of the new method are proved.

This paper investigates the Schur-convexity, Schur-geometric convexity, and Schur-harmonic convexity for the elementary symmetric composite function and its dual form. The inverse problems are also considered. New inequalities on special means are established by using the theory of majorization.

Let (x 1 , ... , x n , y 1 , ... , y n) be a list of 2 n commuting variables, (θ 1 , ... , θ n , ξ 1 , ... , ξ n) be a list of 2 n anticommuting variables, and C [ x n , y n ] ⊗ ∧ { θ n , ξ n } be the algebra generated by these variables. D'Adderio, Iraci, and Vanden Wyngaerd introduced the Theta operators on the ring of symmetric functions and used them to conjecture a formula for the quadruply-graded S n -isomorphism type of C [ x n , y n ] ⊗ ∧ { θ n , ξ n } / I where I is the ideal generated by S n -invariants with vanishing constant term. We prove their conjecture in the 'purely fermionic setting' obtained by setting the commuting variables x i , y i equal to zero. [ABSTRACT FROM AUTHOR]