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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]

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.

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]

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 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]

Abstract Two open problems recently proposed by Xi and Luo (Adv. Differ. Equ. 2021:38, 2021) are resolved by evaluating explicitly three binomial sums involving harmonic numbers, that are realized mainly by utilizing the generating function method and symmetric functions.

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.

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.

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]

Given a Hopf algebra with distinguished bases indexed by combinatorial objects, along with a primitive generating set for this algebra, it is natural to consider how we can combinatorially interpret the primitivity of the elements in this generating set, by expanding these generators in terms of the distinguished bases of this algebra and then applying the comultiplication operation to these expansions and constructing sign-reversing involutions that determine the resultant cancellations that we obtain. In this article, we explore this idea, as applied to the power sum generators of the algebra of symmetric functions. [ABSTRACT FROM AUTHOR]

Two open problems recently proposed by Xi and Luo (Adv. Differ. Equ. 2021:38, 2021) are resolved by evaluating explicitly three binomial sums involving harmonic numbers, that are realized mainly by utilizing the generating function method and symmetric functions. [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.

It is known that every complex-valued homomorphism of the Fréchet algebra $H_{bs}(L_\infty)$ of all entire symmetric functions of bounded type on the complex Banach space $L_\infty$ is a point-evaluation functional $\delta_x$ (defined by $\delta_x(f) = f(x)$ for $f \in H_{bs}(L_\infty)$) at some point $x \in L_\infty.$ Therefore, the spectrum (the set of all continuous complex-valued homomorphisms) $M_{bs}$ of the algebra $H_{bs}(L_\infty)$ is one-to-one with the quotient set $L_\infty/_\sim,$ where an equivalence relation "$\sim$'' on $L_\infty$ is defined by $x\sim y \Leftrightarrow \delta_x = \delta_y.$ Consequently, $M_{bs}$ can be endowed with the quotient topology. On the other hand, $M_{bs}$ has a natural representation as a set of sequences which endowed with the coordinate-wise addition and the quotient topology forms an Abelian topological group. We show that the topology on $M_{bs}$ is metrizable and it is induced by the metric $d(\xi, \eta) = \sup_{n\in\mathbb{N}}\sqrt[n]{|\xi_n-\eta_n|},$ where $\xi = \{\xi_n\}_{n=1}^\infty,\eta = \{\eta_n\}_{n=1}^\infty \in M_{bs}.$

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.

A Boolean function f : { 0 , 1 } n → { 0 , 1 } is weighted symmetric if there exist a function g : Z → { 0 , 1 } and integers w 0 , w 1 , ... , w n such that f (x 1 , ... , x n) = g (w 0 + ∑ i = 1 n w i x i) holds. In this paper, we present algorithms for the circuit satisfiability problem of bounded depth circuits with AND, OR, NOT gates and a limited number of weighted symmetric gates. Our algorithms run in time super-polynomially faster than 2 n even when the number of gates is super-polynomial and the maximum weight of symmetric gates is nearly exponential. As a special case, we obtain an algorithm for the maximum satisfiability problem that runs in time poly (n t) ⋅ 2 n − n 1 / O (t) for instances with n variables and O (n t) clauses. Through the analysis of our algorithms, we show average-case lower bounds and compression algorithms for such circuits and worst-case lower bounds for majority votes of such circuits. [ABSTRACT FROM AUTHOR]

In this paper, we use (p,q)-integral to establish some Fejér type inequalities. In particular, we generalize and correct existing results of quantum Fejér type inequalities by using new techniques and showing some problematic parts of those results. Most of the inequalities presented in this paper are significant extensions of results which appear in existing literatures.

It is known that the so-called elementary symmetric polynomials $R_n(x) = \int_{[0,1]}(x(t))^n\,dt$ form an algebraic basis in the algebra of all symmetric continuous polynomials on the complex Banach space $L_\infty,$ which is dense in the Fr\'{e}chet algebra $H_{bs}(L_\infty)$ of all entire symmetric functions of bounded type on $L_\infty.$ Consequently, every continuous homomorphism $\varphi: H_{bs}(L_\infty) \to \mathbb{C}$ is uniquely determined by the sequence $\{\varphi(R_n)\}_{n=1}^\infty.$ By the continuity of the homomorphism $\varphi,$ the sequence $\{\sqrt[n]{|\varphi(R_n)|}\}_{n=1}^\infty$ is bounded. On the other hand, for every sequence $\{\xi_n\}_{n=1}^\infty \subset \mathbb{C},$ such that the sequence $\{\sqrt[n]{|\xi_n|}\}_{n=1}^\infty$ is bounded, there exists $x_\xi \in L_\infty$ such that $R_n(x_\xi) = \xi_n$ for every $n \in \mathbb{N}.$ Therefore, for the point-evaluation functional $\delta_{x_\xi}$ we have $\delta_{x_\xi}(R_n) = \xi_n$ for every $n \in \mathbb{N}.$ Thus, every continuous complex-valued homomorphism of $H_{bs}(L_\infty)$ is a point-evaluation functional at some point of $L_\infty.$ Note that such a point is not unique. We can consider an equivalence relation on $L_\infty,$ defined by $x\sim y \Leftrightarrow \delta_x = \delta_y.$ The spectrum (the set of all continuous complex-valued homomorphisms) $M_{bs}$ of the algebra $H_{bs}(L_\infty)$ is one-to-one with the quotient set $L_\infty/_\sim.$ Consequently, $M_{bs}$ can be endowed with the quotient topology. On the other hand, it is naturally to identify $M_{bs}$ with the set of all sequences $\{\xi_n\}_{n=1}^\infty \subset \mathbb{C}$ such that the sequence $\{\sqrt[n]{|\xi_n|}\}_{n=1}^\infty$ is bounded. We show that the quotient topology is Hausdorffand that $M_{bs}$ with the operation of coordinate-wise addition of sequences forms an abelian topological group.

Abstract We consider a certain quotient of a polynomial ring categorified by both the isomorphic Green rings of the symmetric groups and Schur algebras generated by the signed Young permutation modules and mixed powers respectively. They have bases parametrised by pairs of partitions whose second partitions are multiples of the odd prime p the characteristic of the underlying field. We provide an explicit formula rewriting a signed Young permutation module (respectively, mixed power) in terms of signed Young permutation modules (respectively, mixed powers) labelled by those pairs of partitions. As a result, for each partition λ , we discovered the number of compositions δ such that δ can be rearranged to λ and whose partial sums of δ are not divisible by p. [ABSTRACT FROM AUTHOR]

An ( r , k ) -parking function of length n may be defined as a sequence ( a 1 , … , a n ) of positive integers whose increasing rearrangement b 1 ≤ ⋯ ≤ b n satisfies b i ≤ k + ( i − 1 ) r . The case r = k = 1 corresponds to ordinary parking functions. We develop numerous properties of ( r , k ) -parking functions. In particular, if F n ( r , k ) denotes the Frobenius characteristic of the action of the symmetric group S n on the set of all ( r , k ) -parking functions of length n , then we find a combinatorial interpretation of the coefficients of the power series ( ∑ n ≥ 0 F n ( r , 1 ) t n ) k for any k ∈ Z . When k > 0 , this power series is just ∑ n ≥ 0 F n ( r , k ) t n ; when k < 0 , we obtain a dual to ( r , k ) -parking functions. We also give a q -analogue of this result. For fixed r , we can use the symmetric functions F n ( r , 1 ) to define a multiplicative basis for the ring Λ of symmetric functions. We investigate some of the properties of this basis. [ABSTRACT FROM AUTHOR]

Motivated by a recent paper, the author provides some new integral inequalities of Hermite–Hadamard type involving the product of an s-convex function and a symmetric function and applies these new established inequalities to construct inequalities for special means.

In the article, we prove that the symmetric function Fn(x1,x2,...,xn;r)= ∑ 1≤i12<...r≤n r∏j=1 (1+xij1−xij) 1/r is Schur convex, Schur multiplicatively convex and Schur harmonic convex on [0,1)n, and establish several new analytic inequalities by use of the theory of majorization, where r∈{1,2,...,n} and i1,i2,...in are integers. [ABSTRACT FROM AUTHOR]

The symmetric group S n acts on the polynomial ring Q [ x n ] = Q [ x 1 , … , x n ] by variable permutation. The invariant ideal I n is the ideal generated by all S n -invariant polynomials with vanishing constant term. The quotient R n = Q [ x n ] I n is called the coinvariant algebra . The coinvariant algebra R n has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization I n , k ⊆ Q [ x n ] of the ideal I n indexed by two positive integers k ≤ n . The corresponding quotient R n , k : = Q [ x n ] I n , k carries a graded action of S n and specializes to R n when k = n . We generalize many of the nice properties of R n to R n , k . In particular, we describe the Hilbert series of R n , k , give extensions of the Artin and Garsia–Stanton monomial bases of R n to R n , k , determine the reduced Gröbner basis for I n , k with respect to the lexicographic monomial order, and describe the graded Frobenius series of R n , k . Just as the combinatorics of R n are controlled by permutations in S n , we will show that the combinatorics of R n , k are controlled by ordered set partitions of { 1 , 2 , … , n } with k blocks. The Delta Conjecture of Haglund, Remmel, and Wilson is a generalization of the Shuffle Conjecture in the theory of diagonal coinvariants. We will show that the graded Frobenius series of R n , k is (up to a minor twist) the t = 0 specialization of the combinatorial side of the Delta Conjecture. It remains an open problem to give a bigraded S n -module V n , k whose Frobenius image is even conjecturally equal to any of the expressions in the Delta Conjecture; our module R n , k solves this problem in the specialization t = 0 . [ABSTRACT FROM AUTHOR]