O n ( m n,n, m n,m) relative differe nce sets with $$\text {gcd}( m,n)=1$$gcd( m,n)=1

There has been much research on \((p^{a},p^{b},p^{a},p^{a-b})\) relative difference sets with p a prime, while there are only a few results on (mn, n, mn, m) relative difference sets with \(\text {gcd}(m,n)=1\). The non-existence results on (mn, n, mn, m) relative difference sets with \(\text {gcd}(m,n)=1\) have only been obtained for the following five cases: (1) \(m=p,\ n=q,\ p>q\); (2) \(m=pq,\ n=3,\ p,q>3\); (3) \(m=4,\ n=p\); (4) \(m=2\) and (5) \(n=p\), where p, q are distinct odd primes. For the existence results, there are only four constructions of semi-regular relative difference sets in groups of size not a prime power with the forbidden subgroup having size larger than 2. In this paper, we present some more non-existence results on (mn, n, mn, m) relative difference sets with \(\text {gcd}(m,n)=1\). In particular, our result is a generalization of the main result of Hiramine’s work (J Comb Theory Ser A 117(7):996–1003, 2010). Meanwhile, we give a construction of non-abelian (16q, q, 16q, 16) relative difference sets, where q is a prime power with \(q\equiv 1\pmod {4}\) and \(q>4.2\times 10^{8}\). This is the third known infinite classes of non-abelian semi-regular relative difference sets.     

A simpler proof of a Katsurada’s theorem and rapidly converging series for $$\zeta {(2n+1)}$$ ζ ( 2 n + 1 ) and $$\beta {(2n)}$$ β ( 2 n )

Annali di Matematica Pura ed Applicata (1923 -) - In a recent work on Euler-type formulae for even Dirichlet beta values, i.e., $$\beta {(2n)}$$ , I have derived an exact closed-form expression for...     

Weighted limits in an $$(\infty ,1)$$ ( ∞ , 1 ) -category

We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal’s approach: we identify a meaningful construction for the quasi-category of weighted cones over a diagram in a quasi-category, whose terminal object is the weighted limit of the considered diagram. We then show that each weighted limit can be expressed as an ordinary limit. When the quasi-category arises as the homotopy coherent nerve of a category enriched over Kan complexes, we generalize an argument by Riehl-Verity to show that the weighted limit agrees with the homotopy weighted limit in the sense of enriched category theory, for which explicit constructions are available. When the quasi-category is complete, tensored and cotensored over the quasi-category of spaces, we discuss a possible comparison of our definition of weighted limit with the approach by Gepner-Haugseng-Nikolaus.

     

Répartitio n simulta née de S( n) et $$ S( n+1)$$ S( n+1) da ns les progressio ns arithmétiques

If \(q\ge 2\) is an integer, we denote by \(S_q(n)\) the sum of the digits in base q of the positive integer n and by \(v_q(n)\) its q-adic valuation. The goal of this work is to study exponential sums of the form \(\displaystyle \sum \nolimits _{n\le x}\exp \big (2i\pi \big (\frac{l}{m} S_q(n)+\frac{k}{m'}S_q(n+1)+\theta n\big )\big )\) in order to prove some statistical properties of integers n for which \(S_q(n)\) and \(S_q(n+1)\) belong to given arithmetic progressions. This extends the results obtained by Gelfond in 1968 and those obtained by Mauduit–Sárközy in 1996.     

Distortio n theorems o n the Lie ball $$ \mathcal{R}_{IV} $$ ( n) i n ℂ n

In this paper, we introduce the subfamilies H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) of holomorphic mappings defined on the Lie ball $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n) which reduce to the family of holomorphic mappings and the family of locally biholomorphic mappings when m = 1 and m → +∞, respectively. Various distortion theorems for holomophic mappings H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are established. The distortion theorems coincide with Liu and Minda’s as the special case of the unit disk. When m = 1 and m → +∞, the distortion theorems reduce to the results obtained by Gong for $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n), respectively. Moreover, our method is different. As an application, the bounds for Bloch constants of H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are given.     

Optimal portfolio selections via $$\ell _{1, 2}$$ ℓ 1 , 2 -norm regularization

Computational Optimization and Applications - There has been much research about regularizing optimal portfolio selections through $$\ell _1$$ norm and/or $$\ell _2$$ -norm squared. The common...     

On positive integers n with $$\sigma _l (2n+1)<\sigma _l (2n)$$ σ l ( 2 n + 1 ) < σ l ( 2 n )

Periodica Mathematica Hungarica - For any integer l and any positive integer n, let $$ \sigma _{l}(n)=\sum _{d\mid n}d^{l}$$ . In 1936, Erd?s proved that the set of positive integers n with...     

Schatten class Hankel and $$\overline{\partial }$$ ∂ ¯ -Neumann operators on pseudoconvex domains in $$\mathbb {C}^{n}$$ C n

Monatshefte für Mathematik - Let $$\Omega $$ be a $$C^2$$ -smooth bounded pseudoconvex domain in $$\mathbb {C}^n$$ for $$n\ge 2$$ and let $$\varphi $$ be a holomorphic function on $$\Omega $$...     

Series representations for the Apery constant $$\zeta (3)$$ ζ ( 3 ) involving the values $$\zeta (2n)$$ ζ ( 2 n )

The Ramanujan Journal - In this note, using the well-known series representation for the Clausen function, we also provide some new representations of Apery’s constant $$\zeta (3)$$ . In...     

The Intrinsic π-Operator on Domain Manifolds in $${\mathbb{C}^{n+1}}$$

The main aim of this article is to study the hypercomplex π-operator over \mathbbCⁿ⁺¹{\mathbb{C}^{n+1}} via real, compact, n + 1-dimensional manifolds called domain manifolds. We introduce an intrinsic Dirac operator for such types of domain manifolds and define an intrinsic π-operator, study its mapping properties and introduce a Clifford–Beltrami equation in this context.     

Heine’s method and $$A_{n}$$ A n to $$A_{m}$$ A m transformation formulas

The Ramanujan Journal - We apply Heine’s method—the key idea Heine used in 1846 to derive his famous transformation formula for $$_2\phi _1$$ series—to multiple basic series over...     

$${\overline{1}}$$ 1 ¯ -Evaluating linear functionals on function spaces

Positivity - Let C(X) denotes the Riesz algebra of all real valued continuous functions on a Tychonoff space X, and let L be a vector subspace of C(X). A linear functional $$H:L\rightarrow {\mathbb...     

On weighted absolute convergence of multiple Fourier series of a function of $${(p_{1} ,\ldots,p_{N})}$$–$${(\Lambda^1 ,\ldots ,\Lambda^N)}$$(Λ1,…,ΛN)– BV

We obtain a sufficient condition for the weighted β-absolute convergence of the multiple Fourier series

$$\sum_{k_{1} \in\mathbb{Z}} \cdots \sum_{k_{N} \in\mathbb{Z}} \gamma_{k_{1} ,\ldots ,k_{N}} |\hat{f}(k_1 ,\ldots ,k_N)|^\beta,$$

where \({0 < \beta < 2}\) and \({\gamma_{k_{1} ,\ldots ,k_{N}}}\) is a weight sequence, of a function f of the class \({(\Lambda^1 ,\ldots ,\Lambda^N)BV^{(p_{1} ,\ldots ,p_{N})}([0, 2\pi]^N)}\).

     

The [n 1,n 2, ...,n s]-th reduced KP hierarchy andW 1+∞ constraints

To every partition n=n₁+n₂+...+n_s one can associate a vertex operator realization of the Lie algebras a and gl_n. Using this construction, we obtain reductions of the s-component KP hierarchy, reductions which are related to these partitions. In this way we obtain matrix KdV-type equations. We show that the following two constraints on a KP -function are equivalent: (1) is a -function of the [n₁, n₂, ..., n_s]-th reduced KP hierarchy which satisfies the string equation L_–1=0; (2) satisfies the vacuum constraints of the W₁₊ algebra.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 32–42, July, 1995.     

从(n 1)/n×(n 1)=(n 1)/n (n 1)谈起

一问题的提出本刊2003年第5期刊载了《运用发现法解题》(以下简称《解题》)一文,文章在谈到“归纳发现法”时,提到这样一个例子: 1.观察下列各式     

Quantitative weakly compact sets and Banach-Saks sets in $$\ell _1$$ ℓ 1

Archiv der Mathematik - In this paper, we show a quantitative version of the theorem stating that relatively weakly compact sets in $$\ell _1$$ coincide with those having the Banach-Saks property....     

A Kähler structure for the $$\text {PU}(2, 1)$$ PU ( 2 , 1 ) configuration space of four points in $$S^3$$ S 3

Geometriae Dedicata - We show that an open subset $${\mathfrak {F}}_4''$$ of the $$\mathrm{PU}(2,1)$$ configuration space of four points in $$S^3$$ is in bijection with an open subset of...     

a_n={S_n-S_(n-1),n≥2,S_1,n=1的应用

设数列 {ａｎ}的前ｎ项和为Ｓｎ,则ａｎ 与Ｓｎ 间有 :ａｎ=Ｓｎ-Ｓｎ - 1 ,Ｓ1 ,ｎ≥ 2 ,ｎ=1.它是ａｎ 与Ｓｎ 之间相互转化的重要工具 ,应用它能较简洁地解出许多条件中含有Ｓｎ 的问题 .下面举例说明它的应用 .1　消去Ｓｎ,转化为ａｎ 来解例 1　设 {ａｎ}是正数组成的数列 ,其前ｎ项和为Ｓｎ.并且对所有自然数ｎ,ａｎ 与 2的等差中项等于Ｓｎ 与2的等比中项 .求数列 {ａｎ}的通项公式 .解　由题意有ａｎ 22 =2Ｓｎ (ｎ∈Ｎ ) ,整理得Ｓｎ=18(ａｎ 2 ) 2 ,由此得Ｓｎ 1 =18(ａｎ 1 2 ) 2 ,∴ａｎ 1=Ｓｎ 1 -Ｓｎ=18[(ａｎ 1 2 ) …