Related papers: The symmetric function theorem via the Fa\`a di Br…

In one complex variable it is well known that if we consider the family of all holomorphic functions on the unit disc that fix the origin and with first derivative equal to 1 at the origin, then there exists a constant $\rho$, independent…

For each $a \in \mathbb{R}$, we define a Borel function $f_a : \mathbb{R} \to \mathbb{R}$ which encodes $a$ in a certain sense. We show that for each Borel $g : \mathbb{R} \to \mathbb{R}$, $f_a \cap g = \emptyset$ implies $a \in…

In any symmetric monoidal category, the $n$-th (co)equalizer symmetric power of an object $A$ is the (co)equalizer of all the permutations from $A^{\otimes n}$ to itself. If the symmetric monoidal category is $\mathbb{Q}_{\ge 0}$-linear,…

Harmonic functions of two variables are exactly those that admit a conjugate, namely a function whose gradient has the same length and is everywhere orthogonal to the gradient of the original function. We show that there are also partial…

We prove an equivalent of the Riemann hypothesis in terms of the functional equation (in its asymmetrical form) and the $a$-points of the zeta-function, i.e., the roots of the equation $\zeta(s)=a$, where $a$ is an arbitrary fixed complex…

Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem. In this paper we study the Schur-positivity of a family of symmetric functions. Given a partition \lambda, we denote by…

We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is…

We present a new method for proving Correa-Jofr\'e-Thibault theorem that monotonicity of subdifferential implies convexity of the function. This new method is based on barrier functions. Barrier functions help overcome some of the main…

We show that in supersymmetric theories, knowing the soft theorem for a single particle in a supermultiplet allows one to immediately determine soft theorems for the remainder of the supermultiplet. While soft theorems in supersymmetric…

We prove an implicit function theorem for non-commutative functions. We use this to show that if $p(X,Y)$ is a generic non-commuting polynomial in two variables, and $X$ is a generic matrix, then all solutions $Y$ of $p(X,Y)=0$ will commute…

We establish a general uniqueness theorem for subharmonic functions of several variables on a domain. A corollary from this uniqueness theorem for holomorphic functions is formulated in terms of the zero subset of holomorphic functions and…

We attempt to explain the ubiquity of tableaux and of Pieri and Cauchy formulae for combinatorially defined families of symmetric functions. We show that such formulae are to be expected from symmetric functions arising from representations…

In 2004, Rosas and Sagan developed the theory of symmetric functions in noncommuting variables, achieving results analogous to classical symmetric functions. On the other hand, in 2004, Desrosiers, Lapointe and Mathieu introduced the theory…

In combinatorics, P\'{o}lya's Enumeration Theorem is a powerful tool for solving a wide range of counting problems, including the enumeration of groups, graphs, and chemical compounds. In this paper, we present an extension of P\'{o}lya's…

According to symmetrization postulate for a system of identical particles, wave function has to be completely symmetric or completely anti-symmetric. In this paper we want to mathematically justify this postulate ignoring the spin part of…

The three invariants of C$^{1/2}$ are key to expressing this tensor and its inverse as a polynomial in C. Simple and symmetric expressions are presented connecting the two sets of invariants $I_1, I_2,I_3$ and $i_1, i_2,i_3 $ of C and…

We introduce a ring of noncommutative shifted symmetric functions based on an integer-indexed sequence of shift parameters. Using generating series and quasideterminants, this multiparameter approach produces deformations of the ring of…

In this paper, we provide some novel binomial convolution related to symmetric functions, as well as convolution sums without the binomial symbol. Moreover we give some new convolution sums of Bernoulli, Euler, and Genocchi numbers and…

In a previous work, both the constants of motion of a classical system and the symmetries of the corresponding quantum version have been computed with the help of factorizations. As their expressions were not polynomial, in this paper the…

We show that the Christoffel function (CF) factorizes (or can be disintegrated) as the product of two Christoffel functions, one associated with the marginal and the another related to the conditional distribution, in the spirit of "the CF…