Related papers: The M\"{o}bius Function of a Restricted Compositio…

A poset is called a symmetric chain decomposition if the poset can be expressed as a disjoint union of symmetric chains. For positive integers $m$ and $n$, let $N(m,n)$ denote the set of all compositions $\alpha=(\alpha_1,\cdots,\alpha_m)$,…

The lattice of partitions of a set and its d-divisible generalization have been much studied for their combinatorial, topological, and representation-theoretic properties. An ordered set partition is a set partition where the subsets are…

Fix $\lambda>-1/2$ and $\lambda \not=0$. Consider the Bessel operator (introduced by Muckenhoupt--Stein) $\triangle_\lambda:=-\frac{d^2}{dx^2}-\frac{2\lambda}{x} \frac d{dx}$ on $\mathbb{R_+}:=(0,\infty)$ with…

We show that a composition operator induced by an analytic self-map of the unit disc in the complex plane is weakly compact on the space BMOA precisely when the operator is compact on BMOA. As a crucial step we simplify the compactness…

Partition of unities appear in many places in analysis. Typically they are generated by compactly supported functions with a certain regularity. In this paper we consider partition of unities obtained as integer-translates of entire…

Macdonald defined two-parameter Kostka functions K_{\lambda\mu}(q,t) where \lambda, \mu are partitions. The main purpose of this paper is to extend his definition to include all compositions as indices. Following Macdonald, we conjecture…

Suppose that for some unit vectors $b_1,\ldots b_n$ in $\mathbb C^d$ we have that for any $j\neq k$ $b_j$ is either orthogonal to $b_k$ or $|\langle b_j,b_k\rangle|^2 = 1/d$ (i.e. $b_j$ and $b_k$ are unbiased). We prove that if $n=d(d+1)$,…

For a simply connected connected simple algebraic group $G$, it is known that a variety $B_{w_0}^-:=B^-\cap U\overline{w_0}U$ has a geometric crystal structure with a positive structure…

The Euler numbers have been widely studied. A signed version of the Euler numbers of even subscript are given by the coefficients of the exponential generating function 1/(1+x^2/2!+x^4/4!+...). Leeming and MacLeod introduced a…

We give an order-theoretic characterization of the essential image of the forgetful functor from the category of real/complex unital C*-algebras to the category of real/complex unital operator systems. It is based on the characterization of…

Anderson established a connection between core partitions and order ideals of certain posets by mapping a partition to its $\beta$-set. In this paper, we give a characterization of the poset $P_{(s,s+1,s+2)}$ whose order ideals correspond…

In this article we describe the Auslander-Reiten quiver for some posets with an involution, that we call types $\mathfrak{U}_n$ and $\mathfrak{U}_\infty$. These posets appear in the differentiation III of Zavadskij [12]. We follow the…

We study the relation between zero loci of Bernstein-Sato ideals and roots of b-functions and obtain a criterion to guarantee that roots of b-functions of a reducible polynomial are determined by the zero locus of the associated…

The exponential generating functions of {n^(n+m)} for arbitrary integer m are expressed as rational functions of the e.g.f. of {n^(n-1)} [the tree function] and then of the e.g.f. of {n^n} [the endofunction function]. The coefficients in…

We prove that the M\"obius function is orthogonal to polynomials over $\mathbb{F}_q[x]$ (up to a characteristic condition). We use this orthogonality property to count prime solutions to affine-linear equations of bounded complexity in…

Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUB) for the space of operators, $M(d, \mathbb{C})$, acting on such Hilbert spaces. The notion of MUUB reflects the…

For a general dyadic grid, we give a Calder\'{o}n-Zygmund type decomposition, which is the principle fact about the multilinear maximal function $\mathfrak{M}$ on the upper half-spaces. Using the decomposition, we study the boundedness of…

We give explicit analytical expressions for the partition function of $U(N)_{k}\times U(N+M)_{-k}$ ABJ theory at weak coupling ($k\rightarrow \infty )$ for finite and arbitrary values of $N$ and $M$ (including the ABJM case and its…

We study the problem of checking whether an existential sentence (that is, a first-order sentence in prefix form built using existential quantifiers and all Boolean connectives) is true in a finite partially ordered set (in short, a poset).…

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…