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We establish Hadamard-type inequalities for a class of symmetric matrices called $k$-positive matrices for which the $m$-th elementary symmetric functions of their eigenvalues are positive for all $m\leq k$. These matrices arise naturally in the study of $k$-Hessian equations in Partial Differential Equations. For each $k$-positive matrix, we show that the sum of its principal minors of size $k$ is not larger than the $k$-th elementary symmetric function of their diagonal entries. The case $k=n$ corresponds to the classical Hadamard inequality for positive definite matrices. Some consequences are also obtained., Comment: v2: The assumption in Lemma 2.2 of the published version in Linear Algebra Appl. was modified to make it invariant under conjugation with orthogonal matrices. All arguments and results remain unchanged

In this paper, we prove the strong log-concavity and the unimodality of the various sequences of an extension of elementary symmetric function. The principal technique used is a combinatorial interpretation of determinants using lattice paths due to Gessel and Viennot. As applications, we establish the strong q-log-concavity and the unimodality of q-bi $$^{s}$$ nomial coefficients and their sequences lying on rays of the associated triangle.

The star transform is a generalized Radon transform mapping a function of two variables to its integrals along “star-shaped” trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in mathematical models of various imaging modalities based on scattering of elementary particles. The paper presents a comprehensive study of the inversion of the star transform. We describe the necessary and sufficient conditions for invertibility of the star transform, introduce a new inversion formula and discuss its stability properties. As an unexpected bonus of our approach, we prove a conjecture from algebraic geometry about the zero sets of elementary symmetric polynomials.

We derive a formula of Chern-Gauss-Bonnet type for the Euler characteristic of a four dimensional manifold-with-boundary in terms of the geometry of the Loewner-Nirenberg singular Yamabe metric in a prescribed conformal class. The formula involves the renormalized volume and a boundary integral. It is shown that if the boundary is umbilic, then the sum of the renormalized volume and the boundary integral is a conformal invariant. Analogous results are proved for asymptotically hyperbolic metrics in dimension four for which the second elementary symmetric function of the eigenvalues of the Schouten tensor is constant. Extensions and generalizations of these results are discussed. Finally, a general result is proved identifying the infinitesimal anomaly of the renormalized volume of an asymptotically hyperbolic metric in terms of its renormalized volume coefficients, and used to outline alternate proofs of the conformal invariance of the renormalized volume plus boundary integral.

Our aim in the present work is to identify all the possible standing wave configurations involving few vortices of different charges in an atomic Bose-Einstein condensate (BEC). In this effort, we deploy the use of a computational algebra approach in order to identify stationary multi-vortex states with up to 6 vortices. The use of invariants and symmetries enables deducing a set of equations in elementary symmetric polynomials, which can then be fully solved via computational algebra packages within Maple. We retrieve a number of previously identified configurations, including collinear ones and polygonal (e.g. quadrupolar and hexagonal) ones. However, importantly, we also retrieve a configuration with 4 positive charges and 2 negative ones which is unprecedented, to the best of our knowledge, in BEC studies. We corroborate these predictions via numerical computations in the fully two-dimensional PDE system of the Gross-Pitaevskii type which characterizes the BEC at the mean-field level.

Let n, k and a be positive integers. The Stirling numbers of the first kind, denoted by s(n, k), count the number of permutations of n elements with k disjoint cycles. Let p be a prime. Lengyel, Komatsu and Young, Leonetti and Sanna, Adelberg, Hong and Qiu made some progress in the study of the p-adic valuations of s(n, k). In this paper, by using Washington’s congruence on the generalized harmonic number and the n-th Bernoulli number Bn and the properties of m-th Stirling numbers of the first kind obtained recently by the authors, we arrive at an exact expression or a lower bound on vp(s(ap, k)) witha and k being integers such that $$1\le a\le p-1$$ and $$1\le k\le ap$$. This infers that for any regular prime $$p\ge 7$$ and for arbitrary integers a and k with $$5\le a\le p-1$$ and $$a-2\le k\le ap-1$$, one has $$v_{p}(H(ap-1,k)) < -\frac{\log{(ap-1)}}{2\log p}$$ with $$H(ap-1, k)$$ being the k-th elementary symmetric function of $$1, \frac{1}{2}, \ldots , \frac{1}{ap-1}$$ . This gives a partial support to a conjecture of Leonetti and Sanna. We also present results on $$v_p(s(ap^{n},ap^{n}-k))$$ from which one can derive that under certain condition, for any prime $$p\ge 5$$, any odd number $$k\ge 3$$ and any sufficiently large integer n, if $$(a,p)=1$$, then $$v_p(s(ap^{n+1},ap^{n+1}-k))=v_p(s(ap^{n},ap^{n}-k))+2$$. It confirms partially Lengyel’s conjecture.

In this paper, we investigate the contracting curvature flow of closed, strictly convex axially symmetric hypersurfaces in $$\mathbb {R}^{n+1}$$ and $$\mathbb {S}^{n+1}$$ by $$\sigma _k^\alpha $$ , where $$\sigma _k$$ is the k-th elementary symmetric function of the principal curvatures and $$\alpha \ge 1/k$$ . We prove that for any $$n\ge 3$$ and any fixed k with $$1\le k\le n$$ , there exists a constant $$c(n,k)>1/k$$ such that if $$\alpha $$ lies in the interval [1/k, c(n, k)], then we have a nice curvature pinching estimate involving the ratio of the biggest principal curvature to the smallest principal curvature at every point of the flow hypersurface, and we prove that the properly rescaled hypersurfaces converge exponentially to the unit sphere. In the case $$1

It is known that every continuous symmetric (invariant under the composition of its argument with each Lebesgue measurable bijection of $[0,1]$ that preserve the Lebesgue measure of measurable sets) polynomial on the Cartesian power of the complex Banach space $L_\infty$ of all Lebesgue measurable essentially bounded complex-valued functions on $[0,1]$ can be uniquely represented as an algebraic combination, i.e., a linear combination of products, of the so-called elementary symmetric polynomials. Consequently, every continuous complex-valued linear multiplicative functional (character) of an arbitrary topological algebra of the functions on the Cartesian power of $L_\infty,$ which contains the algebra of continuous symmetric polynomials on the Cartesian power of $L_\infty$ as a dense subalgebra, is uniquely determined by its values on elementary symmetric polynomials. Therefore, the problem of the description of the spectrum (the set of all characters) of such an algebra is equivalent to the problem of the description of sets of the above-mentioned values of characters on elementary symmetric polynomials. In this work, the problem of the description of sets of values of characters, which are point-evaluation functionals, on elementary symmetric polynomials on the Cartesian square of $L_\infty$ is completely solved. We show that sets of values of point-evaluation functionals on elementary symmetric polynomials satisfy some natural condition. Also, we show that for any set $c$ of complex numbers, which satisfies the above-mentioned condition, there exists an element $x$ of the Cartesian square of $L_\infty$ such that values of the point-evaluation functional at $x$ on elementary symmetric polynomials coincide with the respective elements of the set $c.$

In this paper, we construct two classes of permutation polynomials over finite fields. First, by one well-known lemma of Zieve, we characterize one class permutation polynomials of the finite field, which generalizes the result of Marcos. Second, by using the onto property of functions related to the elementary symmetric polynomial in multivariable and the general trace function, we construct another class permutation polynomials of the finite field. This extends the results of Marcos, Zieve, Qin and Hong to the more general cases. Particularly, the latter result gives a rather more general answer to an open problem raised by Zieve in 2010.

Let [Formula: see text] and [Formula: see text] be positive integers. We denote by [Formula: see text] the 2-adic valuation of [Formula: see text]. The Stirling numbers of the first kind, denoted by [Formula: see text], count the number of permutations of [Formula: see text] elements with [Formula: see text] disjoint cycles. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the study of the [Formula: see text]-adic valuations of [Formula: see text]. In this paper, by introducing the concept of [Formula: see text]th Stirling numbers of the first kind and providing a detailed 2-adic analysis, we show an explicit formula on the 2-adic valuation of [Formula: see text]. We also prove that [Formula: see text] holds for all integers [Formula: see text] between 1 and [Formula: see text]. As a corollary, we show that [Formula: see text] if [Formula: see text] is odd and [Formula: see text]. This confirms partially a conjecture of Lengyel raised in 2015. Furthermore, we show that if [Formula: see text], then [Formula: see text] and [Formula: see text], where [Formula: see text] stands for the [Formula: see text]th elementary symmetric functions of [Formula: see text]. The latter one supports the conjecture of Leonetti and Sanna suggested in 2017.

Conjectures on permanents are well-known unsettled conjectures in linear algebra. Let A be an n×n matrix and Sn be the symmetric group on n element set. The permanent of A is defined as perA=∑σ∈Sn∏i=1naiσ(i). The Merris conjectured that for all n×n doubly stochastic matrices (denoted by Ωn), nperA≥min1≤i≤n∑j=1nperA(j|i), where A(j|i) denotes the matrix obtained from A by deleting the jth row and ith column. Foregger raised a question whether per(tJn+(1−t)A)≤perA for 0≤t≤nn−1 and for all A∈Ωn, where Jn is a doubly stochastic matrix with each entry 1n. The Merris conjecture is one of the well-known conjectures on permanents. This conjecture is still open for n≥4. In this paper, we prove the Merris inequality for some classes of matrices. We use the sub permanent inequalities to prove our results. Foregger’s inequality is also one of the well-known inequalities on permanents, and it is not yet proved for n≥5. Using the concepts of elementary symmetric function and subpermanents, we prove the Foregger’s inequality for n=5 in [0.25, 0.6248]. Let σk(A) be the sum of all subpermanents of order k. Holens and Dokovic proposed a conjecture (Holen–Dokovic conjecture), which states that if A∈Ωn,A≠Jn and k is an integer, 1≤k≤n, then σk(A)≥(n−k+1)2nkσk−1(A). In this paper, we disprove the conjecture for n=k=4.

We show that for any permutation $w$ that avoids a certain set of 13 patterns of lengths 5 and 6, the Schubert polynomial $\mathfrak S_w$ can be expressed as the determinant of a matrix of elementary symmetric polynomials in a manner similar to the Jacobi-Trudi identity. For such $w$, this determinantal formula is equivalent to a (signed) subtraction-free expansion of $\mathfrak S_w$ in the basis of standard elementary monomials.

Four classes of multiple series, which can be considered as multifold counterparts of four classical summation formulae related to the Riemann zeta series, are evaluated in closed form.

The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized ( max ⁡ , + ) , Izhakian's extended and Izhakian–Rowen's supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. We show that in the case of idempotent semirings we can do this precisely when the Frobenius property is satisfied, that in the case of supertropical semirings this is always possible, and that in non-trivial symmetrized semirings this is never possible. Our results allow us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent.

In Haglund et al. (Trans. Amer. Math. Soc. 370(6):4029–4057, 2018), Haglund, Remmel and Wilson introduce a conjecture which gives a combinatorial prediction for the result of applying a certain operator to an elementary symmetric function. This operator, defined in terms of its action on the modified Macdonald basis, has played a role in work of Garsia and Haiman on diagonal harmonics, the Hilbert scheme, and Macdonald polynomials (Garsia and Haiman in J. Algebraic Combin. 5:191–244, 1996; Haiman in Invent. Math. 149:371–407, 2002). The Delta Conjecture involves two parameters q, t; in this article we give the first proof that the Delta Conjecture is true when $$q=0$$ or $$t=0$$ .

In this paper, we study a class of contracting flows of closed, convex hypersurfaces in the Euclidean space $${\mathbb {R}}^{n+1}$$ with speed $$r^{\alpha } \sigma _k$$, where $$\sigma _k$$ is the k-th elementary symmetric polynomial of the principal curvatures, $$\alpha \in {\mathbb {R}}^1$$, and r is the distance from the hypersurface to the origin. If $$\alpha \ge k+1$$, we prove that the flow exists for all time, preserves the convexity and converges smoothly after normalisation to a sphere centred at the origin. If $$\alpha

Fix a bounded domain Ω ⊆ C n and a positive definite kernel K on Ω, both invariant under S n , the permutation group on n symbols. Let H ⊆ Hol ( Ω ) be the Hilbert module determined by K. We show that H splits into orthogonal direct sum of subspaces P p H indexed by the partitions p of n. We prove that each submodule P p H is a locally free Hilbert module of rank equal to square of the dimension χ p ( 1 ) of the irreducible representation corresponding to p. Given two partitions p and q, we show that if χ p ( 1 ) ≠ χ q ( 1 ) , then the sub-modules P p H and P q H are not unitarily equivalent. We prove that if H is a contractive analytic Hilbert module on Ω, then the Taylor joint spectrum of the n-tuple of multiplication operators by elementary symmetric polynomials on P p H is clos ( s ( Ω ) ) , where s : C n → C n is the symmetrization map. It is then shown that this commuting tuple of operators defines a contractive homomorphism of the ring of symmetric polynomials C [ z ] S n in n variables, equipped with the sup norm on clos ( s ( Ω ) ) .

We construct the element of the Cartesian square of the complex Banach space L ∞ [ 0,1 ] of all Lebesgue measurable essentially bounded functions on [ 0,1 ] by the predefined values of elementary symmetric polynomials on this element. Results of this work can be applied to the investigation of an algebraic basis of the algebra of continuous symmetric polynomials on the Cartesian square of the complex Banach space L ∞ [ 0,1 ] .

The k-Hessian operator σ k is the k-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the k-Hessian equation σ k ( D 2 u ) = f with Dirichlet boundary condition u = 0 is variational; indeed, this problem can be studied by means of the k-Hessian energy − ∫ u σ k ( D 2 u ) . We construct a natural boundary functional which, when added to the k-Hessian energy, yields as its critical points solutions of k-Hessian equations with general non-vanishing boundary data. As a consequence, we establish a Dirichlet's principle for k-admissible functions with prescribed Dirichlet boundary data.

A generalized k-Yamabe problem is considered in this paper. Denoting Ric and R the Ricci tensor and the scalar curvature of a Riemannian space ( M , g ) respectively, we consider the σ k -type equation σ k ( λ s t ) = c o n s t . , where λ s t are the eigenvalues of the symmetric tensor s R i c − t R ⋅ g and σ k is the k − t h elementary symmetric polynomial. We show that the equation is solvable in a conformal class if s R i c − t R ⋅ g is in the convex cone Γ k + and 2 t > s > 0 .

We generalize the notion of Davenport constants to a `higher degree' and obtain various lower and upper bounds, which are sometimes exact as is the case for certain finite commutative rings of prime power cardinality. Two simple examples that capture the essence of these higher degree Davenport constants are the following. 1) Suppose $n = 2^k$, then every sequence of integers $S$ of length $2n$ contains a subsequence $S'$ of length at least two such that $\sum_{a_i,a_j \in S'} a_ia_j \equiv 0 \pmod{n}$ and the bound is sharp. 2) Suppose $n \equiv1 \pmod{2}$, then every sequence of integers $S$ of length $2n -1$ contains a subsequence $S'$ of length at least two such that $\sum_{a_i,a_j \in S'} a_ia_j \equiv 0 \pmod{n}$. These examples illustrate that if a sequence of elements from a finite commutative ring is long enough, certain symmetric expressions have to vanish on the elements of a subsequence., Comment: 16 pages

Partition functions of a canonical ensemble of non-interacting bound electrons are a key ingredient of the super-transition-array approach to the computation of radiative opacity. A few years ago, we published a robust and stable recursion relation for the calculation of such partition functions. In this paper, we propose an optimization of the latter method and explain how to implement it in practice. The formalism relies on the evaluation of elementary symmetric polynomials, which opens the way to further improvements., submitted to "High Energy Density Physics"

In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean n-space. This flow involves k-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz-Christoffel-Minkowski problem., 19 pages. arXiv admin note: text overlap with arXiv:2005.02376

In this paper, we first consider a class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space R n + 1 with speed u α f − β , where u is the support function of the hypersurface, f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. For α ⩽ 0 β ⩽ 1 − α , we prove that the flow has a unique smooth solution for all time, and converges smoothly after normalization, to a sphere centered at the origin. In particular, the results of Gerhardt [16] and Urbas [40] can be recovered by putting α = 0 and β = 1 in our first result. If the initial hypersurface is convex, this is our previous work [11] . If α ⩽ 0 β 1 − α and the ambient space is hyperbolic space H n + 1 , we prove that the flow ∂ X ∂ t = ( u α f − β − η u ) ν has a longtime existence and smooth convergence to a coordinate slice. The flow in H n + 1 is equivalent (up to an isomorphism) to a re-parametrization of the original flow in R n + 1 case. Finally, we find a family of monotone quantities along the flows in R n + 1 . As applications, we give a new proof of a family of inequalities involving the weighted integral of kth elementary symmetric function for k-convex, star-shaped hypersurfaces, which is an extension of the quermassintegral inequalities in [20] .

International audience; We present an observer for linear time-invariant (LTI) systems with measurement delay. Our design ensures that the observer error converges to zero within a prescribed terminal time. To achieve this, we employ time-varying output gains that approach infinity at the terminal time, which can be arbitrarily short but no shorter than the sensor delay time. We model the sensor delay as a transport partial differential equation (PDE) and build upon the cascade ODE-PDE setting while accounting for the infinite dimensionality of the sensor. To construct our time-varying gains, the observer design needs to be conducted in a particular system representation. For this reason, we employ a sequence of state transformations (and their inverses) mapping the original observer error model into (1) the observer form, (2) a sensor delay-compensated observer error form via backstepping, and (3) a particular diagonal form that is amenable to the selection of time-varying gains for prescribed-time stabilization. Our construction of the time-varying observer gains uses (a) generalized Laguerre polynomials, (b) elementary symmetric polynomials, and (c) polynomial-based Vandermonde matrices. A simulation illustrates the results.

Over the past decade there has been growing interest on characterizing which convex cones over Rn are spectrahedral, that is, are a linear section of the cone of positive semidefinite matrices. This interest is largely motivated by applications in control theory, optimization and combinatorics. One particular class of convex cones of interest is the class of hyperbolicity cones, where the (still open) Generalized Lax Conjecture states that every hyperbolicity cone is spectrahedral. Recent works [1, 2] have established that the hyperbolicity cones of the elementary symmetric polynomials and the homogeneous multivariate matching polynomial are spectrahedral, but the question of whether there exists an efficient spectrahedral representation for such cones remains open. Previous work [11] has provided exponential lower bounds on the spectrahedral representation of non-explicit hyperbolicity cones which are known to be spectrahedral. The current best lower unconditional bounds for explicit cones are the linear lower bounds proved by [7]. In this paper we establish the first superpolynomial hardness of the minimal spectrahedral representation for an explicit family of hyperbolicity cones, assuming Valiant's VP vs VNP conjecture is true, that is, that the permanent polynomial cannot be computed by algebraic circuits of polynomial size. More precisely, we prove that the hyperbolicity cone of Amini's homogeneous matching polynomial must require superpolynomial spectrahedral representations, assuming that Valiant's conjecture is true. This is the first work providing a (conditional) superpolynomial lower bound on the spectrahedral representation of an explicit hyperbolicity cone.

The Newton–Girard Formula allows one to write any elementary symmetric polynomial as a sum of products of power sum symmetric polynomials and elementary symmetric polynomials of lesser degree. It has numerous applications. We have generalized this identity by replacing the elementary symmetric polynomials with monomial symmetric polynomials. Our formula has an application in the field of Lie algebras.

The first linear code supporting a $4$-design was the $[11, 6, 5]$ ternary Golay code discovered in 1949 by Golay. In the past 71 years, sporadic linear codes holding $4$-designs or $5$-designs were discovered and many infinite families of linear codes supporting $3$-designs were constructed. However, the question as to whether there is an infinite family of linear codes holding an infinite family of $t$-designs for $t\geq 4$ remains open for 71 years. This paper settles this long-standing problem by presenting an infinite family of BCH codes of length $2^{2m+1}+1$ over $\mathrm{GF}(2^{2m+1})$ holding an infinite family of $4$-$(2^{2m+1}+1, 6, 2^{2m}-4)$ designs. Moreover, an infinite family of linear codes holding the spherical design $S(3, 5, 4^m+1)$ is presented., arXiv admin note: substantial text overlap with arXiv:1910.08265

We prove some "power" generalizations of Marcus-Lopes-style (including McLeod and Bullen) concavity inequalities for elementary symmetric polynomials, and convexity inequalities (of McLeod and Baston) for complete homogeneous symmetric polynomials. Finally, we present sundry concavity results for elementary symmetric polynomials, of which the main result is a concavity theorem that among other implies a well-known log-convexity result of Muir (1972/74) for positive definite matrices., 6 pages

Let $K[X_n]$ be the commutative polynomial algebra in the variables $X_n=\{x_1,\ldots,x_n\}$ over a field $K$ of characteristic zero. A theorem from undergraduate course of algebra states that the algebra $K[X_n]^{S_n}$ of symmetric polynomials is generated by the elementary symmetric polynomials which are algebraically independent over $K$. In the present paper we study a noncommutative and nonassociative analogue of the algebra $K[X_n]^{S_n}$ replacing $K[X_n]$ with the free metabelian Lie algebra $F_n$ of rank $n\geq 2$ over $K$. It is known that the algebra $F_n^{S_n}$ is not finitely generated but its ideal $(F_n')^{S_n}$ consisting of the elements of $F_n^{S_n}$ in the commutator ideal $F_n'$ of $F_n$ is a finitely generated $K[X_n]^{S_n}$-module. In our main result we describe the generators of the $K[X_n]^{S_n}$-module $(F_n')^{S_n}$ which gives the complete description of the algebra $F_n^{S_n}$., 9 pages

In this paper, we establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for Hessian quotient equations with prescribed asymptotic behavior at infinity. This extends the previous related results on the Monge–Ampere equations and on the Hessian equations, and rearranges them in a systematic way. Based on the Perron's method, the main ingredient of this paper is to construct some appropriate subsolutions of the Hessian quotient equation, which is realized by introducing some new quantities about the elementary symmetric polynomials and using them to analyze the corresponding ordinary differential equation related to the generalized radially symmetric subsolutions of the original equation.