Interlineation of functions of 2 variables on M (M≥2) lines with the highest algebraic precision

A general algorithm is proposed for constructing interlineation operators , x=(x₁, x₂) with the properties

On the continuity of functions in connection with a problem of V. G. Krotov

В РАБОтЕ ДАЕтсь ОтВЕт НА ОДИН ВОпРОс, пОстАВ лЕННыИ В. г. кРОтОВыМ. УстАНОВлЕН О, ЧтО ЕслИ Ф(х) — МОНОтОННО ВО жРАстАУЩАь ФУНкцИь,Ф (0)=0, Ф(2х)≦кФ(х), х[0, ∞), тО $$\left\{ {f:\left\| {\sum\limits_{k = 1}^\infty {\mu _k \Phi (\lambda _k \left| {S_k - f} \right|)} } \right\|_c< \infty } \right\} \subseteqq C \Leftrightarrow \sum\limits_{k = 1}^\infty {\mu _k } \Phi (\lambda _k ) = \infty $$ Дль пРОИжВОльНых НЕО тРИцАтЕльНых ЧИслОВ ых пОслЕДОВАтЕльНОстЕ И {Μ_k} И {λ_k}. (жДЕсьS _k ОБОжНАЧАЕт ЧАстНУУ с УММУ пОРьДкАk РьДА ФУ РьЕ ФУНкцИИf). УстАНОВлЕН О тАкжЕ, ЧтО ВО МНОгИх слУЧАьх $$\left\{ {f:\left\| {\sum\limits_{k = 1}^\infty {\mu _k \Phi (\lambda _k \left| {\tilde S_k - \tilde f} \right|)} } \right\|_c< \infty } \right\} \subseteqq C \Leftrightarrow \sum\limits_{k = 1}^\infty {\frac{1}{{k\lambda _k }}} \Phi ^{ - 1} \left( {\frac{1}{{k\mu _k }}} \right)< \infty .$$      

Sums of the higher divisor function of diagonal homogeneous forms

The Ramanujan Journal - We give an asymptotic formula for $$\sum _{1\le n_1.n_2, \ldots ,n_l\le x^{1/r}}\tau _k(n^r_1+n^r_2+\ldots +n^r_l)$$ , where $$\tau _k(n)$$ represents the k-th divisor...     

Additive arithmetic functions meet the inclusion–exclusion principle: Asymptotic formulas concerning the GCD and LCM of several integers

Lithuanian Mathematical Journal - We obtain asymptotic formulas for the sums $$ {\sum}_{n_1,\dots, {n}_k\leqslant x} $$ f((n1, . . . , nk)) and $$ {\sum}_{n_1,\dots, {n}_k\leqslant x} $$ f([n1, . ....     

Interlineation of functions of two variables on M(M>-2) lines preserving the class Cr(R2)

An algorithm for constructing the operator O_Mn({_kx}; x, y) with the properties

Inequalities for the Inner Radii of Nonoverlapping Domains

Ukrainian Mathematical Journal - We consider the problem of maximum of a functional$$ {}^{r^{\gamma }}\left({B}_0,0\right)\coprod \limits_{k=1}^nr\left({B}_k,{a}_k\right), $$ where...     

Semigroup and Riesz transform for the Dunkl–Schrödinger operators

Semigroup Forum - Let $$L_k=-\Delta _k+V$$ be the Dunkl–Schrödinger operators, where $$\Delta _k=\sum _{j=1}^dT_j^2$$ is the Dunkl Laplace operator associated to the Dunkl operators...     

On the irreducibility of the Severi variety of nodal curves in a smooth surface

Archiv der Mathematik - Let X be a smooth projective surface and $$L\in \mathrm {Pic}(X)$$ . We prove that if L is $$(2k-1)$$ -spanned, then the set $${\tilde{V}}_k(L)$$ of all nodal and...     

Commutative Complex Algebras of the Second Rank with Unity and Some Cases of Plane Orthotropy. II

Ukrainian Mathematical Journal - For an algebra $$ {\mathbbm{B}}_0:= \left\{{c}_1e+{c}_2\omega :{c}_k\in \mathbb{C},k=1,2\right\} $$ , e2 = ω2 = e, eω = ωe = ω, over the field...     

Painless Breakups—Efficient Demixing of Low Rank Matrices

Journal of Fourier Analysis and Applications - Assume we are given a sum of linear measurements of s different rank-r matrices of the form $$\varvec{y}= \sum _{k=1}^{s} \mathcal {A}_k...     

THE GLOBAL SMOOTH SOLUTIONS OF SECOND ORDER QUASILINEAR HYPERBOLIC EQUATIONS WITH DISSIPATIVE BOUNDARY CONDITIONS

The paper deals with the following boundary problem of the second order quasilinear hyperbolic equation with a dissipative boundary condition on a part of the boundary:u_(tt)-sum from i,j=1 to n a_(ij)(Du)u_(x_ix_j)=0, in (0, ∞)×Ω,u|Γ_0=0,sum from i,j=1 to n, a_(ij)(Du)n_ju_x_i+b(Du)u_t|Γ_1=0,u|t=0=φ(x), u_t|t=0=ψ(x), in Ω, where Ω=Γ_0∪Γ_1, b(Du)≥b_0>0. Under some assumptions on the equation and domain, the author proves that there exists a global smooth solution for above problem with small data.     

On the sum of divisors of mixed powers in short intervals

The Ramanujan Journal - Let d(n) denote the Dirichlet divisor function. Define $$\begin{aligned} \mathcal {S}_k(x,y):= \sum _{\begin{array}{c} x-y     

The universal minimal one parameter system with quasi-discrete spectrum

The universal minimal one parameter system will be characterized as the space $$\Gamma ^{\infty }$$, in which $$\Gamma$$ is the Bohr compactification of the additive group $${\mathbb {R}}$$ of real numbers. In this way, we need to show that $$\Gamma ^\infty$$ is isomorphic to the spectrum of $$W({\mathbb {R}})$$, the norm closure of the invariant algebra generated by the maps $$\exp q(t)$$, where q(t) is a real polynomial on $${\mathbb {R}}$$.     

О кОЁФФИцИЕНтАх тРИг ОНОМЕтРИЧЕскИх НУль-РьДОВ

Let {? _k } _k=0 ^∞ be a numerical sequence satisfying the conditions $$\varrho _k \downarrow 0(k \to \infty )and\mathop \sum \limits_{k = 0}^\infty \varrho _k^2 = + \infty .$$ It is proved that there exists a trigonometric series $$\mathop \sum \limits_{k = 0}^\infty \varrho '_k \cos 2\pi (kx + \theta _k )$$ where ¦?′ _k ¦≦? _k ,k=0, 1, 2, ..., possessing the following property. For each measurable and a.e. finite functionF(x), x?[0, 1], the numbersδ _k =0 or 1,k=0, 1, ..., may be chosen in such a way that the series $$\mathop \sum \limits_{k = 0}^\infty \delta _k \varrho '_k \cos 2\pi (kx + \theta _k )$$ converges toF(x) a.e. on [0,1]. In addition, ifF(x)=0, then \(\delta _{k_0 } \varrho '_{k_0 } \ne 0\) for at least onek ₀≧0. Certain generalizations are discussed, too.     

The pure cactus group is residually nilpotent

We show that for each $$n\in {\mathbb {N}}$$ the pure cactus group $$\Gamma _{n+1}$$ embeds into a right-angled Coxeter group and, therefore, is residually nilpotent. In addition, we construct an explicit residually torsion-free nilpotent subgroup of finite index in $$\Gamma _{n+1}$$ .     

On triviality of the reduced Whitehead group over Henselian fields

Let F be a Henselian field of q-cohomological dimension 3, where q is a prime. Let $$\Gamma _F$$ be the totally ordered Abelian value group of F and let D be a central division algebra over F of index a power of q such that the characteristic of the residue field $$\overline{F}$$ is coprime to q. We show that when $$1\le \dim _{{\mathbb {F}}_q}(\Gamma _F/q\Gamma _F)\le 3$$ , the reduced Whitehead group of D is trivial.     

Eisenstein series twisted with non-expanding cusp monodromies

The Ramanujan Journal - Let $$\Gamma $$ be a geometrically finite Fuchsian group and suppose that $$\chi :\Gamma \rightarrow {{\,\mathrm{GL}\,}}(V)$$ is a finite-dimensional representation with...     

Smallest graphs with given automorphism group

Journal of Algebraic Combinatorics - For a finite group G, denote by $$\alpha (G)$$ the minimum number of vertices of any graph $$\Gamma $$ having $$\mathrm {Aut}(\Gamma )\cong G$$ . In this paper,...     

On the convergence and divergence of multiple Fourier series with respect to the Walsh—Paley system in the spacesC andL

Работа касается вопр осов сходимости и рас ходимости кратных рядов Фурье п о системе Уолша-Пэли в метрикахС и L. Из доказанных тео рем следует, в частности, ч то еишf∈Е (Е=С или E=L) и существует н атуральноеi ₀ (1≦i ₀ ≦N) так ое, что и $$\begin{gathered} \omega (\delta _{i_0 } ;f)_E = o\left( {\frac{1}{{1n^N \frac{1}{{\delta _{i_0 } }}}}} \right) (\delta _{i_0 } \to 0) \hfill \\ \omega (\delta _k ;f)_E = o\left( {\frac{1}{{1n^N \frac{1}{{\delta _k }}}}} \right), k \ne i_0 (\delta _k \to 0), k = 1,2, ...,N, \hfill \\ \end{gathered}$$ тоN-кратный ряд Фурье функцииf по системе У олша-Пэли сходится по Прингсхе йму в смысле метрики пространств аЕ. Доказано также, что вы шеотмеченное утверж дение неусиляемо в метрикеL не только для системы Уолша, но и для некоторого класс а ОНС, ограниченных в совок упности.