The depth of a reflexive polytope

Given arbitrary integers d and r with $$d \ge 4$$ and $$1 \le r \le d + 1$$ , a reflexive polytope $${\mathscr {P}}\subset {\mathbb R}^d$$ of dimension d with $$\mathrm{depth}\,K[{\mathscr {P}}] = r$$ for which its dual polytope $${\mathscr {P}}^\vee $$ is normal will be constructed, where $$K[{\mathscr {P}}]$$ is the toric ring of $${\mathscr {P}}$$ .     

A sufficient condition for that an operator map is of the form of functional calculus

Positivity - Let $${\mathscr {H}}$$ be a Hilbert space, J be an open interval and $$B_J({\mathscr {H}})$$ be the set of all self-adjoint operators on $${\mathscr {H}}$$ with spectra in J. Suppose...     

The monoid of $$2 \times 2$$2×2 triangular boolean matrices under skew transposition is non-finitely based

Semigroup Forum - Let $$\mathscr {T\!B}_n$$ be the involution semigroup of all upper triangular boolean $$n\times n$$ matrices under the ordinary matrix multiplication and the skew transposition....     

On conditional expectations in $$L^p(\mu ;L^q(\nu ;X))$$ L p ( μ ; L q ( ν ; X ) )

Positivity - Let $$(A,{\mathscr {A}},\mu )$$ and $$(B,{\mathscr {B}},\nu )$$ be probability spaces, let $${\mathscr {F}}$$ be a sub- $$\sigma $$ -algebra of the product $$\sigma $$ -algebra...     

Metrics on Projections of the Von Neumann Algebra Associated with Tracial Functionals

Siberian Mathematical Journal - Let φ be a positive functional on a von Neumann algebra $$\mathscr{A}$$ and let $$\mathscr{A}^{\rm{pr}}$$ be the projection lattice in $$\mathscr{A}$$. Given...     

r-log-concavity of partition functions

The Ramanujan Journal - Let $$\hat{\mathscr {L}}$$ be the operator given by $$\hat{\mathscr {L}} \{a_n\}_{n \ge 0} = \{a_{n+1}^2 - a_{n} a_{n+2} \}_{n \ge 0}$$ . A sequence $$\{ a_n \}_{n \ge 0}$$...     

Precoherent quantale completions of partially ordered semigroups

Just as complete lattices can be viewed as the completions of posets, quantales can also be treated as the completions of partially ordered semigroups. Motivated by the study on the well-known Frink completions of posets, it is natural to consider the “Frink” completions for the case of partially ordered semigroups. For this purpose, we firstly introduce the notion of precoherent quantale completions of partially ordered semigroups, and construct the concrete forms of three types of precoherent quantale completions of a partially ordered semigroup. Moreover, we obtain a sufficient and necessary condition of the Frink completion on a partially ordered semigroup being a precoherent quantale completion. Finally, we investigate the injectivity in the category $$\mathbf {APoSgr}_{\le }$$ of algebraic partially ordered semigroups and their submultiplicative directed-supremum-preserving maps, and show that the $$\mathscr {E}_{\le }$$-injective objects of algebraic partially ordered semigroups are precisely the precoherent quantales, here $$\mathscr {E}_{\le }$$ denote the class of morphisms $$h:A\longrightarrow B$$ that preserve the compact elements and satisfy that $$h(a_1)\cdots h(a_n)\le h(b)$$ always implies $$a_1\cdots a_n\le b$$.     

Property T for Actions

We study property T for an action α of a discrete group Γ on a unital C*-algebra $$\mathscr{A}$$ . Our main results improve some well-known results about property T for groups. Moreover, we introduce Hilbert $$\mathscr{A}$$ -module property T and show that the action α has property T if and only if the reduced crossed product $$\mathscr{A}\;{\rtimes_{\alpha, r}}$$ Γ has Hilbert $$\mathscr{A}$$ -module property T.     

Infinitely Many Solutions of Nonlocal Kirchhoff-Type Equations via Perturbation Methods

Mathematical Notes - We study the multiplicity of weak solutions to the boundary-value problem $$\begin{alignedat}{2} - M\biggl(\iint_{\mathbb R^{2N}}|u(x)-u(y)|^2 K(x-y)\,d x\,d y\biggr)\mathscr...     

Bergman空间上Cowen-Douglas算子的换位

令H~∞(D)表示单位圆盘D上的有界解析函数全体构成的代数,对于■∈H~∞(D),在某种条件下,证明T=M■是一个指标为n的Cowen-Douglaus算子,并且给出了■(T)/rad■(T)可交换的一个充分条件.当n=1时,刻画了T的换位.     

A finite field approach to the Carlitz–Riordan q-Catalan numbers

Journal of Algebraic Combinatorics - Let q be a prime power and $${\mathbb {F}}_q$$ be the finite field with q elements. Suppose that $$n\ge 1$$ and $${{\mathscr {F}}}=\{E_1,\ldots ,E_{2n-1}\}$$ is...     

Existence,uniqueness and regularity of the projection onto differentiable manifolds

Annals of Global Analysis and Geometry - We investigate the maximal open domain $${\mathscr {E}}(M)$$ on which the orthogonal projection map p onto a subset $$M\subseteq {{\mathbb {R}}}^d$$ can be...     

单圈图的极小能量

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let Undenote the set of all connected unicyclic graphs with order n, and Ur n= {G ∈ Un| d(x) = r for any vertex x ∈ V(Cl)}, where r ≥ 2 and Cl is the unique cycle in G. Every unicyclic graph in Ur nis said to be a cycle-r-regular graph.In this paper, we completely characterize that C39(2, 2, 2) ο Sn-8is the unique graph having minimal energy in U4 n. Moreover, the graph with minimal energy is uniquely determined in Ur nfor r = 3, 4.     

Martingale inequalities for spline sequences

Positivity - We show that D. Lépingle’s $$L_1(\ell _2)$$-inequality $$\begin{aligned} \left\| \left( \sum _n {\mathbb {E}}[f_n | {\mathscr {F}}_{n-1}]^2 \right) ^{1/2}\right\|...     

左三角范畴的稳定化和幂等完备

Let (CΩ△) be a left triangulated category with a fully faithful endofunctor.We show a triangle-equivalence (S(C),△) ～=(S(C),△),where (S(C)△,) denotes the stabilization of the idempotent completion of (C,△) and (S(C),△) denotes the idempotent completion of the stabilization of (C,△).     

Tropical Duality in $$(d+2)$$ ( d + 2 ) -Angulated Categories

Applied Categorical Structures - Let $$\mathscr {C}$$ be a 2-Calabi–Yau triangulated category with two cluster tilting subcategories $$\mathscr {T}$$ and $$\mathscr {U}$$ . A result from...     

ON THE DIOPHANTINE EQUATION $\[\sum\limits_{i = 0}^k {\frac{1}{{{x_i}}}} = \frac{a}{n}\]$

In this paper,, the author proves the following result: Let $\[{E_{a,k}}(N)\]$ denote the number of natural numbers $\[n \le N\]$ for which equation $$\[\sum\limits_{i = 0}^k {\frac{1}{{{x_i}}}} = \frac{a}{n}\]$$ is insolable in positive integers $\[{x_i}(i = 0,1, \cdots ,k)\]$.Then $$\[{E_{a,k}}(N) \ll N\exp \{ - C{(\log N)^{1 - \frac{1}{{k + 1}}}}\} \]$$ where the implied constant depends on a and K.     

含有割点的图的最小特征值

Let ψ be a certain set of graphs.A graph is called a minimizing graph in the set ψ if its least eigenvalue attains the minimum among all graphs in ψ.In this paper,we determine the unique minimizing graph in ψn,where ψn denotes the set of connected graphs of order n with cut vertices.     

Sidon Basis in Polynomial Rings over Finite Fields

Czechoslovak Mathematical Journal - Let $${\mathbb{F}_q}\left[ t \right]$$ denote the polynomial ring over $${\mathbb{F}_q}$$ , the finite field of q elements. Suppose the characteristic of...