Related papers: $C-$Robin Functions and Applications

We study reduction schemes for functions of "many" variables into system of functions in one variable. Our setting includes infinite-dimensions. Following Cybenko-Kolmogorov, the outline for our results is as follows: We present explicit…

Let $V$ be a set of isolated points in $\RR^d$. Define a linear functional $\CL$ on the space of real polynomials restricted on $V$, $\CL f = \sum_{x \in V} f(x)\rho(x)$, where $\rho$ is a nonzero function on $V$. Polynomial subspaces that…

The functor that takes a ring to its category of modules has an adjoint if one remembers the forgetful functor to abelian groups: the endomorphism ring of linear natural transformations. This uses the self-enrichment of the category of…

In this paper, a new identity for convex functions is derived. A consequence of the identity is that we can derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in…

Combining the "method of restriction equations" of Rim\'anyi et al. with the techniques of symmetric functions, we establish the Schur function expansions of the Thom polynomials for the Morin singularities $A_3: ({\bf C}^{\bullet},0)\to…

The cyclic group labeled family of quasi-projection operators is used for investigation of decomposition of functions with respect to the cyclic group of order n . Series of new identities thus arising are demonstrated and new perspectives…

In this paper we study the coefficients of the powers of an ordinary generating function and their properties. A new class of functions based on compositions of an integer $n$ is introduced and is termed composita. We present theorems about…

Differential calculus is not a unique way to observe polynomial equations such as $a+b=c$. We propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials $a$, $b$ and $c$ satisfying the equation…

In this paper, we study functions of the roots of a univariate polynomial in which the roots have a given multiplicity structure $\mu$. Traditionally, root functions are studied via the theory of symmetric polynomials; we extend this theory…

We establish disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space, generalizing Lempert's formula for the convex case. This function is also known as the pluricomplex Green function…

We consider eigenfunctions of many-body system Hamiltonians associated with generalized (a-twisted) Cherednik operators used in construction of other Hamiltonians: those arising from commutative subalgebras of the Ding-Iohara-Miki (DIM)…

We generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota's theory of M\"obius inversion. The technique works for a large class of semigroups including: inverse…

We give a cohomological interpretation of both the Kac polynomial and the refined Donaldson-Thomas- invariants of quivers. This interpretation yields a proof of a conjecture of Kac from 1982 and gives a new perspective on recent work of…

We provide some conditions for the graph of a Hoelder-continuous function on \bar{D}, where \bar{D} is a closed disc in the complex plane, to be polynomially convex. Almost all sufficient conditions known to date --- provided the function…

Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda_0 + \sum_{k = 1}^d \lambda_k [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are…

Let $C$ be a subset of $\mathbb{R}^n$ (not necessarily convex), $f:C\to\mathbb{R}$ be a function, and $G:C\to\mathbb{R}^n$ be a uniformly continuous function, with modulus of continuity $\omega$. We provide a necessary and sufficient…

Functions like the exponential, Chebyshev polynomials, and monomial symmetric polynomials are preeminent among all special functions. They have simple definitions and can be expressed using easily specified integers like n!. Families of…

We associate convex bodies to a wide class of graded G-algebras where G is a connected reductive group. These convex bodies give information about the Hilbert function as well as multiplicities of irreducible representations appearing in…

Introduced by Polyak in 1966, the class of strongly quasiconvex functions includes some interesting nonconvex members, like the square root of the Euclidean norm or ratios with a nonnegative strongly convex numerator and a concave and…

In this paper we investigate some algebraic and geometric consequences which arise from an extremal bound on the Hilbert function of the general hyperplane section of a variety (Green's Hyperplane Restriction Theorem). These geometric…