Elementary Symmetric Functions Research Articles

Optimal unlabeled set partitioning with application to risk-based quarantine policies

We consider the problem of partitioning a set of items into unlabeled subsets so as to optimize an additive objective, i.e., the objective function value of a partition is equal to the sum of the contribution of each subset. Under an arbitrary objective function, this family of problems is known to be an -complete combinatorial optimization problem. We study this problem under a broad family of objective functions characterized by elementary symmetric polynomials, which are “building blocks” to symmetric functions. By analyzing a continuous relaxation of the problem, we identify conditions that enable the use of a reformulation technique in which the set partitioning problem is cast as a more tractable network flow problem solvable in polynomial-time. We show that a number of results from the literature arise as special cases of our proposed framework, highlighting its generality. We demonstrate the usefulness of the developed methodology through a novel and timely application of quarantining heterogeneous populations in an optimal manner. Our case study on real COVID-19 data reveals significant benefits over conventional measures in terms of both spread mitigation and economic impact, underscoring the importance of data-driven policies.

Newton’s identities and positivity of trace class integral operators

We provide a countable set of conditions based on elementary symmetric polynomials that are necessary and sufficient for a trace class integral operator to be positive semidefinite, which is an important cornerstone for quantum theory in phase-space representation. We also present a new, efficiently computable algorithm based on Newton’s identities. Our test of positivity is much more sensitive than the ones given by the linear entropy and Robertson-Schrödinger’s uncertainty relations; our first condition is equivalent to the non-negativity of the linear entropy.

Lattice Paths and Negatively Indexed Weight-Dependent Binomial Coefficients

In 1992, Loeb (Adv Math, 91:64–74, 1992) considered a natural extension of the binomial coefficients to negative entries and gave a combinatorial interpretation in terms of hybrid sets. He showed that many of the fundamental properties of binomial coefficients continue to hold in this extended setting. Recently, Formichella and Straub (Ann Comb, 23:725–748, 2019) showed that these results can be extended to the q-binomial coefficients with arbitrary integer values and extended the work of Loeb further by examining the arithmetic properties of the q-binomial coefficients. In this paper, we give an alternative combinatorial interpretation in terms of lattice paths and consider an extension of the more general weight-dependent binomial coefficients, first defined by Schlosser (Sém Lothar Combin, 81:24, 2020), to arbitrary integer values. Remarkably, many of the results of Loeb, Formichella and Straub continue to hold in the general weighted setting. We also examine important special cases of the weight-dependent binomial coefficients, including ordinary, q- and elliptic binomial coefficients as well as elementary and complete homogeneous symmetric functions.

Integral formulas for a foliated sub-Riemannian manifold

We apply the notion of foliation to a nonholonomic manifold, which was introduced for the geometric interpretation of constrained systems in mechanics. We prove a series of integral formulas for a foliated sub-Riemannian manifold, that is, a Riemannian manifold equipped with a distribution $${{\mathscr {D}}}$$ and a foliation $${{\mathscr {F}}}$$ whose tangent bundle is a subbundle of $${{\mathscr {D}}}$$ . Our integral formulas generalize some results for a foliated Riemannian manifold and involve the shape operators of $${\mathscr {F}}$$ with respect to normals in $${\mathscr {D}}$$ , the curvature tensor of induced connection on $${\mathscr {D}}$$ and arbitrary functions depending on elementary symmetric functions of eigenvalues of the shape operators. For a special choice of these functions, integral formulas with the Newton transformations of the shape operators of $${\mathscr {F}}$$ are obtained. Application to a foliated sub-Riemannian manifold with restrictions on the curvature and extrinsic geometry of $${\mathscr {F}}$$ and also to codimension-one foliations are given.

The Duality of the Volumes and the Numbers of Vertices of Random Polytopes

An identity due to Efron dating from 1965 relates the expected volume of the convex hull of n random points to the expected number of vertices of the convex hull of n+1 random points. Forty years later this identity was extended from expected values to higher moments. The generalized identity has attracted considerable interest. Whereas the left-hand side of the generalized identity—concerning the volume—has an immediate geometric interpretation, this is not the case for the right-hand side—concerning the number of vertices. A transformation of the right-hand side applying an identity for elementary symmetric polynomials overcomes the blemish. The arising formula reveals a duality between the volumes and the numbers of vertices of random polytopes.

From Symmetric Functions to Partition Identities

In this paper, we show that some classical results from q-analysis and partition theory are specializations of the fundamental relationships between complete and elementary symmetric functions.

Recurrences in Terms of Special Polynomials for Exponential Sums of Elementary Symmetric Polynomials over Finite Fields

Recurrences in Terms of Special Polynomials for Exponential Sums of Elementary Symmetric Polynomials over Finite Fields

A Proof of the Extended Delta Conjecture

Abstract We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $\Delta _{h_l}\Delta ' _{e_k} e_{n}$ , where $\Delta ' _{e_k}$ and $\Delta _{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $\operatorname {\mathrm {GL}}_m$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.

UWB Anchor Node Optimization Deployment Using MGWO for the Coal Mine Working Face End

Appropriate deployment of anchor nodes (ANs) is an essential prerequisite for achieving high-precision positioning. To ameliorate the comprehensive positioning performance, the weight position dilution of precision (WPDOP), that is utilized to evaluate the appropriateness of the ANs placement geometry is proposed by taking into account the different error variances for each AN. A simple closed-form calculation WPDOP method is theoretically derived for UWB positioning system by utilizing elementary symmetric polynomials and Newton’s identity, which is applied to evaluate the Ultra-wideband (UWB) deployment at the end of coal mine working face (CMWF). Several basic ANs layout strategies, based on the projection shape in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x-y</i> plane, are designed for the end of CMWF, and the modified grey wolf optimizer (MGWO) algorithm is executed to optimize the ANs deployment scheme in order to attain the better configuration and higher positioning accuracy. Simulation and experiments verification are simultaneously performed via the UWB positioning system. The experimental results sufficiently demonstrate that MGWO deployment scheme offers lower average WPDOP values than the obtained ones when applying random placement, and the lowest average improvement percentage was approximately equal to 37.6%. Furthermore, after implementing the MGWO algorithm, the V-shape deployment configuration is capable to provide the lowest average WPDOP values and outperforms the other arrangements, keeping the lowest localization error. This is regarded as the best deployment and it gives significant guidance for the ANs sensor deployment at the end of CMWF.

Absolutely k -incoherent quantum states and spectral inequalities for the factor width of a matrix

We investigate the set of quantum states that can be shown to be $k$-incoherent based only on their eigenvalues (equivalently, we explore which Hermitian matrices can be shown to have small factor width based only on their eigenvalues). In analogy with the absolute separability problem in quantum resource theory, we call these states "absolutely $k$-incoherent", and we derive several necessary and sufficient conditions for membership in this set. We obtain many of our results by making use of recent results concerning hyperbolicity cones associated with elementary symmetric polynomials.

On multivariate orthogonal polynomials and elementary symmetric functions

We study families of multivariate orthogonal polynomials with respect to the symmetric weight function in d variables Bγ(x)=∏i=1dω(xi)∏i<j|xi-xj|2γ+1,x∈(a,b)d,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} B_{\\gamma }(\\mathtt {x}) = \\prod \\limits _{i=1}^{d} \\omega (x_{i}) \\prod \\limits _{i<j} |x_{i}-x_{j}|^{2\\gamma +1}, \\quad \\mathtt {x}\\in (a,b)^{d}, \\end{aligned}$$\\end{document}for gamma >-1, where omega (t) is an univariate weight function in t in (a,b) and mathtt {x} = (x_{1},x_{2}, ldots , x_{d}) with x_{i} in (a,b). Applying the change of variables x_{i},i=1,2,ldots ,d, into u_{r},r=1,2,ldots ,d, where u_{r} is the r-th elementary symmetric function, we obtain the domain region in terms of the discriminant of the polynomials having x_{i},i=1,2,ldots ,d, as its zeros and in terms of the corresponding Sturm sequence. Choosing the univariate weight function as the Hermite, Laguerre, and Jacobi weight functions, we obtain the representation in terms of the variables u_{r} for the partial differential operators such that the respective Hermite, Laguerre, and Jacobi generalized multivariate orthogonal polynomials are the eigenfunctions. Finally, we present explicitly the partial differential operators for Hermite, Laguerre, and Jacobi generalized polynomials, for d=2 and d=3 variables.

Hessian equations of Krylov type on compact Hermitian manifolds

Abstract In this article, we are concerned with the equations of Krylov type on compact Hermitian manifolds, which are in the form of the linear combinations of the elementary symmetric functions of a Hermitian matrix. Under the assumption of the 𝒞-subsolution, we obtain a priori estimates in Γ k − 1 {\Gamma }_{k-1} cone. By using the method of continuity, we prove an existence theorem, which generalizes the relevant results. As an application, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.

Codes from symmetric polynomials

We define and study a class of Reed-Muller type error-correcting codes obtained from elementary symmetric functions in finitely many variables. We determine the code parameters and higher weight spectra in the simplest cases.

Certificate complexity of elementary symmetric Boolean functions

Boolean functions have important applications in information technology and computer science. Certificate complexity is an important combinatorial measure of Boolean function complexity. In this paper, we first introduce many concise and efficient notations then we study the elementary symmetric Boolean functions and obtain their certificate complexities when their degrees are odd or powers of 2. We show that both upper and lower bounds of certificate complexities can be attained.

Differential equations for the closed geometric crystal chains

We present two types of systems of differential equations that can be derived from a set of discrete integrable systems which we call the closed geometric crystal chains. One is an extended Lotka–Volterra type system, and the other seems to be generally new but reduces to a previously known system in a special case. Both equations have Lax representations associated with what are known as the loop elementary symmetric functions, which were originally introduced to describe products of affine type A geometric crystals for symmetric tensor representations. Examples of the derivations of the continuous time Lax equations from a discrete time one are described in detail, where a novel method of taking a continuum limit by assuming asymptotic behaviors of the eigenvalues of the Lax matrix in Puiseux series expansions is used.

Chromatic symmetric functions of Dyck paths and [formula omitted]-rook theory

The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian–Wachs q-analogue have important connections to Hessenberg varieties, diagonal harmonics and LLT polynomials. In the, so called, abelian case they are also curiously related to placements of non-attacking rooks by results of Stanley and Stembridge (1993) and Guay-Paquet (2013). For the q-analogue, these results have been generalized by Abreu and Nigro (2021) and Guay-Paquet (private communication), using q-hit numbers. Among our main results is a new proof of Guay-Paquet’s elegant identity expressing the q-CSFs in a CSF basis with q-hit coefficients. We further show its equivalence to the Abreu–Nigro identity expanding the q-CSF in the elementary symmetric functions. In the course of our work we establish that the q-hit numbers in these expansions differ from the originally assumed Garsia–Remmel q-hit numbers by certain powers of q. We prove new identities for these q-hit numbers, and establish connections between the three different variants.

On symmetric partial differential operators

Let \(\sigma _{1}, \dots , \sigma _{k}\) be the elementary symmetric functions of the complex variables \(x_{1}, \dots , x_{k}\). We say that \(F \in {\mathbb {C}}[\sigma _{1}, \dots , \sigma _{k}]\) is a trace function if their exists \(f \in {\mathbb {C}}[z]\) such that \(F(\sigma _{1}, \dots , \sigma _{k}) = \sum _{j=1}^{k} f(x_{j})\) for all \(\sigma \in {\mathbb {C}}^{k}\). We give an explicit finite family of second order differential operators in the Weyl algebra \(W_{2}:= {\mathbb {C}}[\sigma _{1}, \dots , \sigma _{k}]\langle \frac{\partial }{\partial \sigma _{1}}, \dots , \frac{\partial }{\partial \sigma _{k}}\rangle \) which generates the left ideal in \(W_{2}\) of partial differential operators killing all trace functions. The proof uses a theorem for symmetric differential operators analogous to the usual symmetric functions theorem and the corresponding map for symbols. As an application, we obtain for each integer k a holonomic system which is a quotient of \(W_{2}\) by an explicit left ideal whose local solutions are linear combinations of the branches of the multivalued root of the universal equation of degree k: \(z^{k} + \sum _{h=1}^{k} (-1)^{h}\sigma _{h}z^{k-h} = 0\).

Motion planning for a high order Dubins vehicle

AbstractIn this paper, we focus on the motion planning problem (MPP) of a generalized Dubins vehicle (GDV) with higher order dynamics, aiming at mitigating the drawbacks caused by the lack of smoothness of the Dubins curves. In the two‐dimensional case, the MPP is converted, by exploiting the Pontryagin maximum principle, into the problem of solving a set of algebraic equations (AEs). The solution of the set of AEs, which can also be applied to the MPP of chained‐form and flat systems, is found by systematically using the concept of elementary symmetric polynomial. Furthermore, in extending our results to the three‐dimensional case, the explicit expression of a new class of space curves is obtained using the Frenet frame. Simulation results show the superiority of the curves generated by the GDV over that generated by the conventional Dubins vehicle.

A Combinatorial Approach to the Groebner Bases for Ideals Generated by Elementary Symmetric Functions

Previous work by Mora and Sala provides the reduced Groebner basis of the ideal formed by the elementary symmetric polynomials in $n$ variables of degrees $k=1,\dots,n$, $\langle e_{1,n}(x), \dots, e_{n,n}(x) \rangle$. Haglund, Rhoades, and Shimonozo expand upon this, finding the reduced Groebner basis of the ideal of elementary symmetric polynomials in $n$ variables of degree $d$ for $d=n-k+1,\dots,n$ for $k\leq n$. In this paper, we further generalize their findings by using symbolic computation and experimentation to conjecture the reduced Groebner basis for the ideal generated by the elementary symmetric polynomials in $n$ variables of arbitrary degrees and prove that it is a basis of the ideal.

Positivity of Chromatic Symmetric Functions Associated with Hessenberg Functions of Bounce Number 3

We give a proof of the Stanley-Stembridge conjecture on chromatic symmetric functions for the class of all unit interval graphs with independence number 3. That is, we show that the chromatic symmetric function of the incomparability graph of a unit interval order in which the length of a chain is at most 3 is positively expanded as a linear sum of elementary symmetric functions.