Divisibility of certain $$\ell $$ -regular partitions by 2

The Ramanujan Journal - For a positive integer $$\ell $$ , let $$b_{\ell }(n)$$ denote the number of $$\ell $$ -regular partitions of a nonnegative integer n. Motivated by some recent conjectures...     

On $$\ell $$-regular bipartitions modulo $$\ell $$

Let \(B_\ell (n)\) denote the number of \(\ell \)-regular bipartitions of n. In this paper, we prove several infinite families of congruences satisfied by \(B_\ell (n)\) for \(\ell \in {\{5,7,13\}}\). For example, we show that for all \(\alpha >0\) and \(n\ge 0\),

$$\begin{aligned} B_5\left( 4^\alpha n+\frac{5\times 4^\alpha -2}{6}\right)\equiv & {} 0 \ (\text {mod}\ 5),\\ B_7\left( 5^{8\alpha }n+\displaystyle \frac{5^{8\alpha }-1}{2}\right)\equiv & {} 3^\alpha B_7(n)\ (\text {mod}\ 7) \end{aligned}$$

and

$$\begin{aligned} B_{13}\left( 5^{12\alpha }n+5^{12\alpha }-1\right) \equiv B_{13}(n)\ (\text {mod}\ 13). \end{aligned}$$

     

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...     

Saturated fusion systems over $$\mathcal {A}_2$$ A 2 -groups

Mathematische Zeitschrift - In this paper we study the saturated fusion systems $$\mathcal {F}$$ over $$\mathcal {A}_2$$ -groups S of order greater than $$p^4$$ . This study requires the analysis...     

Semisymmetric prime-valent graphs of order $$2p^3$$ 2 p 3

Journal of Algebraic Combinatorics - A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let p be a prime and $$\Gamma $$ a...     

Application of the $$\bar\partial$$ -dressing method to a $$(2+1)$$ -dimensional equation

Theoretical and Mathematical Physics - A remarkable method for investigating solutions of nonlinear soliton equation is the $$\bar\partial$$ -dressing method. Although there are other methods that...     

Weighted $$L^2$$ L 2 -norms of Gegenbauer polynomials

We study integrals of the form

$$\begin{aligned} \int _{-1}^1(C_n^{(\lambda )}(x))^2(1-x)^\alpha (1+x)^\beta {{\,\mathrm{\mathrm {d}}\,}}x, \end{aligned}$$

where \(C_n^{(\lambda )}\) denotes the Gegenbauer-polynomial of index \(\lambda >0\) and \(\alpha ,\beta >-1\). We give exact formulas for the integrals and their generating functions, and obtain asymptotic formulas as \(n\rightarrow \infty \).

     

Boundary functions in L2 H $$(\mathbb{B}^n )$$

We solve the Dirichlet problem for line integrals of holomorphic functions in the unit ball For a function u which is lower semi-continuous on we give necessary and sufficient conditions in order that there exists a holomorphic function f ∈ such that

Random embedding of $${\ell_p^n}$$ into $${\ell_r^N}$$

For any 0 < p < 2 and any natural numbers N > n, we give an explicit definition of a random operator \({S : \ell_p^n \to \mathbb{R}^N}\) such that for every 0 < r < p < 2 with r ≤ 1, the operator \({S_r = S : \ell_p^n \to \ell_r^N}\) satisfies with overwhelming probability that \({\|S_r\| \, \|(S_r)_{| {\rm Im}\, S}^{-1}\| \le C(p,r)^{n/(N-n)}}\), where C(p, r) > 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.     

Diagonal reduction algebra for $$\mathfrak{osp}(1|2)$$

Theoretical and Mathematical Physics - The problem of providing complete presentations of reduction algebras associated to a pair of Lie algebras $$( \mathfrak{G} , \mathfrak{g} )$$ has previously...     

On the family of elliptic curves $$\varvec{y^2=x^3-m^2x+p^2}$$

In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of \(E_{m,p} : y^2=x^3-m^2x+p^2\), where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of \(E_{m,p}(\mathbb {Q})\) is trivial for both the cases {\(m\ge 1\), \(m\not \equiv 0\pmod 3\)} and {\(m\ge 1\), \(m \equiv 0 \pmod 3\), with \(gcd(m,p)=1\)}. We also show that given any odd prime p and for any positive integer m with \(m\not \equiv 0\pmod 3\) and \(m\equiv 2\pmod {32}\), the lower bound for the rank of \(E_{m,p}(\mathbb {Q})\) is 2. Finally, we find curves of rank 9 in this family.     

The zero set $$\Lambda $$ of exponential polynomials and the Riesz property of its exponential system $$E_{\Lambda }$$ in $$L^2(0,T)$$

Archiv der Mathematik - Consider the class of exponential polynomials of the form $$\begin{aligned} f(z)=\sum _{n=0}^{N}\left( \sum _{k=0}^{m_n}c_{n,k}z^k\right) e^{h_n z}, \qquad 0=h_0&lt;...     

Matrix Gegenbauer Polynomials: The $$2\times 2$$ Fundamental Cases

In this paper, we exhibit explicitly a sequence of \(2 \times 2\) matrix valued orthogonal polynomials with respect to a weight \(W_{p,n}\), for any pair of real numbers p and n such that \(0<p<n\). The entries of these polynomiales are expressed in terms of the Gegenbauer polynomials \(C_k^\lambda \). The corresponding three-term recursion relations are also given, and we make some studies of the algebra of differential operators associated with the weight \(W_{p,n}\).     

Projectors on invariant subspaces of representations $$\mathrm{ad}^{\otimes2}$$ of Lie algebras $$so(N)$$ and $$sp(2r)$$ and Vogel parameterization

Theoretical and Mathematical Physics - Using the split Casimir operator, we find explicit formulas for the projectors onto invariant subspaces of the $$ \mathrm{ad} ^{ \otimes 2}$$ representation...     

Cubic edge-transitive graphs of order $$2^np$$ 2 n p

Journal of Algebraic Combinatorics - Let p be a prime. Cubic edge-transitive graphs of order $$2^np$$ have been classified for $$1\le n\le 4$$ in the literature. This paper is devoted to a general...     

On the quantum $$\mathfrak{osp}(1|2\ell)$$ Toda chain

Theoretical and Mathematical Physics - The orthosymplectic Lie superalgebra $$\mathfrak{osp}(1|\,2\ell)$$ is the closest superanalogue of standard Lie algebras. We demonstrate that the...     

On $$A_1^2$$ A 1 2 restrictions of Weyl arrangements

Journal of Algebraic Combinatorics - Let $$\mathcal {A}$$ be a Weyl arrangement in an $$\ell $$ -dimensional Euclidean space. The freeness of restrictions of $$\mathcal {A}$$ was first settled by a...     

On some permutation binomials and trinomials over $$\mathbb {F}_{2^n}$$

In this work, we completely characterize (1) permutation binomials of the form \(x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb {F}_{2^n}[x], n = 2^st, a \in \mathbb {F}_{2^{2t}}^{*}\), and (2) permutation trinomials of the form \(x^{2^s+1}+x^{2^{s-1}+1}+\alpha x \in \mathbb {F}_{2^t}[x]\), where s, t are positive integers. The first result, which was our primary motivation, is a consequence of the second result. The second result may be of independent interest.     

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

A note on the maximal expected local time of $${\text {L}}_2$$ -bounded martingales

Journal of Theoretical Probability - For an $${\text {L}}_2$$ -bounded martingale starting at 0 and having final variance $$\sigma ^2$$ , the expected local time at $$a \in \text {R}$$ is at most...