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In this paper, we first generalize to any hyper-Kahler manifold X with $$b_3(X)\not =0$$ results proved by O’Grady for hyper-Kahler manifolds of generalized Kummer type. In the second part, we restrict to hyper-Kahler manifolds of generalized Kummer type and prove, using results of Markman, that their Kuga–Satake correspondence is algebraic.

Very recently, I. Altun, M. Aslantas and H. Sahin [1] introduced the notion of $$p$$ -proximal contractions and surveyed the existence of best proximity points for such class of non-self mappings in metric spaces. In this note we show that this existence result is a straightforward consequence of the same conclusion in fixed point theory.

The main result of this paper states that if $${\mathcal {F}}$$ is a subgroup-closed saturated formation of full characteristic, then the $${\mathcal {F}}$$ -residual of a K- $${\mathcal {F}}$$ -subnormal subgroup S of a finite group G is a large subgroup of G provided that the $${\mathcal {F}}$$ -hypercentre of every subgroup X of G containing S is contained in the $${\mathcal {F}}$$ -residual of X. This extends a recent result of Aivazidis, Safonova and Skiba.

In this short note, we show that Theorem 4.3 of Liu and Yu (Monatshefte Math 195:173–176, 2021) is a consequence of Lemma 2 of Ballester-Bolinches and Esteban-Romero (J Aust Math Soc 75:181–191, 2003).

This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of Q were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can be violated near the boundary of these subdomains even for infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain Q by throwing out all possible shifts of the boundary Q by vectors of a certain group generated by shifts occurring in the difference operators. For the one dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. 2 Also there was obtained the smoothness (in Sobolev spaces W k ) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding -neighborhoods of certain points. However, the smoothness (in Ho lder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Ho lder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Ho lder space.

The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice $\Lambda_{K3}$, isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps $\pi_*$ and $\pi^*$ induced in cohomology by the rational quotient map $\pi:X\dashrightarrow Y$, where $X$ is a K3 surface admitting an order 3 symplectic automorphism $\sigma$ and $Y$ is the minimal resolution of the quotient $X/\sigma$; we deduce the relation between the N\'eron--Severi group of $X$ and the one of $Y$. Applying these results we describe explicit geometric examples and generalize the Shioda--Inose structures, relating Abelian surfaces admitting order 3 endomorphisms with certain specific K3 surfaces admitting particular order 3 symplectic automorphisms., Comment: 28 pages. Version 2: this is the published version of the paper. The last section of the previous version (v1) was erased (the results are only stated) and it is now contained in arXiv:2209.10141

Let R be an algebra. Given a noncommutative polynomial f, let f(R) stand for the additive subgroup of R generated by the image of f. For a unital or an affine algebra R, $$S_k(R)$$ is completely determined for any standard polynomial $$S_k$$ when R is generated by $$S_k(R)$$ as an ideal. Motivated by Bresar’s paper [Adv. Math. 374 (2020), 107346, 21 pp] and Robert’s paper [J. Oper. Theory 75 (2016), 387–408], under certain conditions we also prove that f(R) is equal to either [R, R] or the whole ring R. We obtain these results by studying the structure of Lie ideals L of a ring R whenever R is generated by [R, L] as an ideal.

In this paper, we study on the existence of transcendental entire solutions of certain non-linear generalized delay-differential equations. In this respect we have improved a recent result of Wang et al. (Turk J Math 43:941–954, 2019). Also at the time of investigating the solutions of shift equations we have improved as well as extended an earlier result due to Latreuch (Mediterr J Math 14:1–16, 2017). A handful number of examples have been exhibited by us relevant to the content of the paper to show that each case as demonstrated in the conclusions of the theorems actually occurs. At last we raise a question for future investigations.

The main purpose of this paper is to extend the result obtained by Beirao da Veiga (2000) from the whole-space case to slip boundary cases. Denote by u two components of the velocity u. To fix ideas set ū = (u1,u2, 0) (the half-space) or $${\boldsymbol{\bar u}} = {\hat u_1}{\hat e_1} + {\hat u_2}{\hat e_2}$$ (the general boundary case (see (7.1))). We show that there exists a constant K, which enjoys very simple and significant expressions such that if at some time τ ∈ (0,T) one has $$\lim {\sup _{t \to \tau - 0}}\left\| {{\boldsymbol{\bar u}}(t)} \right\|_{{L^3}(\Omega )}^3 < \left\| {{\boldsymbol{\bar u}}(\tau )} \right\|_{{L^3}(\Omega )}^3 + K$$ , then u is continuous at τ with values in L3(Ω). Roughly speaking, the above norm-discontinuity of merely two components of the velocity cannot occur for steps’ amplitudes smaller than K. In particular, if the above condition holds at each τ ∈ (0,T), the solution is smooth in (0,T) × Ω. Note that here there is no limitation on the width of the norms $$\left\| {{\boldsymbol{\bar u}}(t)} \right\|_{{L^3}(\Omega )}^3$$ . So K is independent of these quantities. Many other related results are discussed and compared among them. This is a second main aim of this paper. New results are proved in Sections 5–7.

In this paper, the author focuses on the joint effects of diffusion and advection on the dynamics of a classical two species Lotka-Volterra competition-diffusion-advection system, where the ratio of diffusion and advection rates are supposed to be a positive constant. For comparison purposes, the two species are assumed to have identical competition abilities throughout this paper. The results explore the condition on the diffusion and advection rates for the stability of former species. Meanwhile, an asymptotic behavior of the stable coexistence steady states is obtained.

In a recent paper, Ng, Tang and Tsai (Math. Ann. 2020) have found an explicit formula for the squeezing function of an annulus via the Loewner differential equation. Their result has led them to conjecture a corresponding formula for planar domains of any finite connectivity stating that the extremum in the squeezing function problem is achieved for a suitably chosen conformal mapping onto a circularly slit disk. In this paper we disprove this conjecture. We also give a conceptually simple potential-theoretic proof of the explicit formula for the squeezing function of an annulus which has the added advantage of identifying all extremal functions., Comment: Version 2: (1) a statement on the history of the notion of squeezing function has been corrected; (2) a new reference [5] (F. Deng: Levi's problem, convexity, and squeezing functions on bounded domains) has been added; (3) a small technical issue with numbering of equations has been resolved

This paper is devoted to investigate the possible ways of distinguishing regular and singular black holes (BHs) in modified gravity (MOG) called regular MOG (RMOG) and Schwarzschild MOG (SMOG) BHs through observational data from twin peak quasiperiodic oscillations (QPOs) which are generated by test particles in stable orbits around the BHs. The presence of MOG field causes to sufficiently the mpeak in effective potential for a radial motion of test particles. The effect of MOG parameter on specific angular momentum and energy has also studied. As a main part of the paper, we focus on investigations of QPOs around SMOG and RMOG BHs in RP model and the relations of upper and lower frequencies of twin peak QPOs in SMOG and RMOG BH models together with extreme rotating Kerr and Schwarzschild BH. Moreover, possible parameters for the central BHs of the objects GRS J1915 + 105 and XTE 1550 – 564 have also obtained numerically in the relativistic precession (RP) model. Finally, we provide comparisons of the innermost stable circular orbit (ISCO) and the orbits where twin peak QPOs with the ratio 3:2 taken place and show that QPOs can not be generated at/inside ISCO and there is a correlation between the radius of ISCO and QPO orbits.

This paper deals with a version of the two-timing method which describes various ‘slow’ effects caused by externally imposed ‘fast’ oscillations. Such small oscillations are often called vibrations and the research area can be referred as vibrodynamics. The governing equations represent a generic system of first-order ODEs containing a prescribed oscillating velocity $${\varvec{u}}$$ , given in a general form. Two basic small parameters stand in for the inverse frequency and the ratio of two time-scales; they appear in equations as regular perturbations. The proper connections between these parameters yield the distinguished limits, leading to the existence of closed systems of asymptotic equations. The aim of this paper is twofold: (i) to clarify (or to demystify) the choices of a slow variable, and (ii) to give a coherent exposition which is accessible for practical users in applied mathematics, sciences and engineering. We focus our study on the usually hidden aspects of the two-timing method such as the uniqueness or multiplicity of distinguished limits and universal structures of averaged equations. The main result is the demonstration that there are two (and only two) different distinguished limits. The explicit instruction for practically solving ODEs for different classes of $${\varvec{u}}$$ is presented. The key roles of drift velocity and the qualitatively new appearance of the linearized equations are discussed. To illustrate the broadness of our approach, two examples from mathematical biology are shown.

In this paper, in the light of weighted sharing of sets, we investigate the possible uniqueness of meromorphic function of restricted hyper order with its linear c-shift operator. Our first two theorems improve a number of earlier results. Our last theorem together with a corollary improves and extends a result due to Li, Lu and Xu [14]. Most importantly, our another corollary deducted from the last theorem not only provides an answer to an open question posed by Liu [16] but also noticeably improves two results of Chen and Chen [4]. A number of examples have been exhibited by us pertinent with the content of the paper. At the penultimate section which is also the application part of our result, we extend a recent result of Liu, Ma and Zhai [17]. Finally, presenting two examples, we conjecture that one of our result may hold for a larger class of functions and we place it as an open question for future investigations.

In this paper, extended H $$\ddot{o}$$ lder class $$H_\alpha ^{(w)}[0,\mu )$$ is considered. This class is the generalization of H $$\ddot{o}$$ lder class $$H_\alpha [0,\mu )$$ . Three new estimators $$E_N^{(1)}(f), E_N^{(2)}(f)$$ and $$E_N^{(3)}(f)$$ of functions of classes $$H_\alpha [0,\mu )$$ and $$H_\alpha ^{(w)}[0,\mu )$$ have been obtained. These estimators are best in approximation of functions by wavelet methods. The estimators obtained in this paper and the solution of ordinary and partial differential equations by extended Haar wavelet operational matrix method in the interval $$[0,\mu )$$ and its comparison with exact solution for different values of $$\mu$$ are the significant achievement of this research paper in approximation theory as well as Wavelet Analysis.

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Holder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form $$ {\partial}_{\overline{z}}f+ af+b\overline{f}=c, $$ where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that $$ {\partial}_{\overline{z}}f=g $$ with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincare problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type $$ {\partial}_{\overline{z}}f(z)=h(z)q\left(f(z)\right). $$ On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

This paper is a synopsis of the recent book [9]. The latter is dedicated to the stochastic Burgers equation as a model for 1d turbulence, and the paper discusses its content in relation to the Kolmogorov theory of turbulence.

This paper deals with the quantitative Schmidt's subspace theorem and the general from of the second main theorem, which are two correspondence objects in Diophantine approximation theory and Nevanlinna theory. In this paper, we give a new below bound for Chow weight of projective varieties defined over a number field. Then, we apply it to prove a quantitative version of Schmidt's subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety. Finally, we apply this new below bound for Chow weight to establish a general form of second main theorem in Nevanlinna theory for meromorphic mappings into projective varieties intersecting hypersurfaces in subgeneral position with a short proof. Our results improve and generalize the previous results in these directions., Comment: 21 pages. arXiv admin note: text overlap with arXiv:math/0408381 by other authors

Generalizing wavelets by adding desired redundancy and flexibility, framelets (i.e., wavelet frames) are of interest and importance in many applications such as image processing and numerical algorithms. Several key properties of framelets are high vanishing moments for sparse multiscale representation, fast framelet transforms for numerical efficiency, and redundancy for robustness. However, it is a challenging problem to study and construct multivariate nonseparable framelets, mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices. Moreover, all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment, and framelets derived from refinable vector functions are barely studied yet in the literature. In this paper, we circumvent the above difficulties through the approach of quasi-tight framelets, which behave almost identically to tight framelets. Employing the popular oblique extension principle (OEP), from an arbitrary compactly supported M-refinable vector function ϕ with multiplicity greater than one, we prove that we can always derive from ϕ a compactly supported multivariate quasi-tight framelet such that: (i) all the framelet generators have the highest possible order of vanishing moments; (ii) its associated fast framelet transform has the highest balancing order and is compact. For a refinable scalar function ϕ (i.e., its multiplicity is one), the above item (ii) often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived from ϕ satisfying item (i). We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices. Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter, which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets. This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders. This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.

The paper is devoted to the construction of an “integral” on an infinite-dimensional space, combining the approaches proposed previously and at the same time the simplest. A new definition of the construction and study its properties on a special class of functionals is given. An introduction of a quasi-scalar product, an orthonormal system, and applications in physics (path integral, loop space, functional derivative) are proposed. In addition, the paper contains a discussion of generalized functionals.

In this paper, we propose a generalized self-adaptive method for solving the multiple-set split feasibility problem in the framework of certain Banach spaces. Under some suitable conditions, we prove the strong convergence of the sequence generated by our method with a new way to select the step-sizes without prior knowledge of the operator norm. Several numerical experiments to illustrate the convergence behavior are presented. The results presented in this paper improve and extend the corresponding results in the literature.

We consider a ϕ-mixing shift T on a sequence space Ω and study the number of returns {Tkω ∈ U} to a union U of cylinders of length n until the first return {Tkω ∈ V} to another union V of cylinder sets of length m. It turns out that if probabilities of the sets U and V are small and of the same order, then the above number of returns has approximately geometric distribution. Under appropriate conditions we extend this result for some dynamical systems to geometric balls and Young towers with integrable tails. This work is motivated by a number of papers on asymptotical behavior of numbers of returns to shrinking sets, as well as by the papers on open systems studying their behavior until an exit through a “hole”.

The purpose of this paper is to give, through the second degree character, new characterizations of a part of the family of quasi-symmetric forms. In fact, thanks to the Stieltjes function and also the moments, we give necessary and sufficient conditions for a regular form to be at the same time of the second degree, quasi-symmetric and semiclassical one of class two. We focus our attention not only on the link between all these forms and the Jacobi forms $${{\mathcal {T}}}_{p, q}={{\mathcal {J}}}(p-1/2, q-1/2), \; p, q\in {\mathbb {Z}},~p+q\ge 0$$ but also on their connection with the Tchebychev form of the first kind $${{\mathcal {T}}}={\mathcal J}\left( -1/2, -1/2\right) $$ . The paper concludes by explicitly giving their characteristic elements of the structure relation and of the second order differential equation, which leads to interesting electrostatic models.

The object of the present paper is to study the notion of quasi-bi-slant submanifolds of almost contact metric manifolds as a generalization of slant, semi-slant, hemi-slant, bi-slant, and quasi-hemi-slant submanifolds. We study and characterize quasi-bi-slant submanifolds of Sasakian manifolds and provide non-trivial examples to signify that the structure presented in this paper is valid. Furthermore, the integrability of distributions and geometry of foliations are researched. Moreover, we characterize quasi-bi-slant submanifolds with parallel canonical structures.

In this paper, we consider the Laplace operator in a flat band whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude.

This paper is devoted to the study of tangential properties of measures with density in the Heisenberg groups $\mathbb{H}^n$. Among other results we prove that measures with $(2n+1)$-density have only flat tangents and conclude the classification of uniform measures in $\mathbb{H}^1$., Correction of typos and very minor details throughout the paper with respect to the published version

In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of P^3 ramified along a surface of degree 2d with defect has at least d(2d-1) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in P^(2n+2) with defect for d sufficiently large., v2: A proof for the Ciliberto-Di Gennaro conjecture is added (Section 5); Some minor corrections in the other sections. v3: some minor corrections in the abstract v4: The proof for the Ciliberto-Di Gennaro conjecture has been modified; The paper is split into two parts, the complete intersection case will be discussed in a different paper

In this paper, we formulate mathematical foundations of applications of boundary integral equations with strongly singular integrals understood in the sense of finite Hadamard value to numerical solution of certain boundary-value problems. We describe numerical schemes for solving boundary strongly singular equations based on quadrature formulas and the collocation method. Also, we make references to known results on the mathematical justification of the numerical methods described in the paper.

The study of the Dirichlet problem in the unit disk 𝔻 with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua [48] has been devoted to boundary-value problems (only with Holder continuous data) for the generalized analytic functions, i.e., continuous complex valued functions h(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form 𝜕zh + ah + b $$ \overline{h} $$ = c ; where it was assumed that the complex valued functions a; b and c belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar´e and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called A−harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions h : D → ℂ with the sources g : 𝜕zh = g ∈ Lp, p > 2 , and to generalized harmonic functions U with sources G : △U = G ∈ Lp, p > 2. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar´e problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.

For a semisimple real Lie group $G$ with an irreducible representation $\rho$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $\rho$ for existence of a group of affine transformations of $V$ whose linear part is Zariski-dense in $\rho(G)$ and that is free, nonabelian and acts properly discontinuously on $V$. This new criterion is more general than the one given in the author's previous paper "Proper affine actions in non-swinging representations" (submitted; available at arXiv:1605.03833), insofar as it also deals with "swinging" representations. We conjecture that it is actually a necessary and sufficient criterion, applicable to all representations., Comment: This paper generalizes the author's previous papers arXiv:1406.5906 and arXiv:1605.03833 . The structure of the proof is similar; a few passages are borrowed from the earlier papers. This version differs from the previous one only by a reference that I added

Recently, two of the authors of this paper constructed cyclic cocycles on Harish-Chandra's Schwartz algebra of linear reductive Lie groups that detect all information in the $K$-theory of the corresponding group $C^*$-algebra. The main result in this paper is an index formula for the pairings of these cocycles with equivariant indices of elliptic operators for proper, cocompact actions. This index formula completely determines such equivariant indices via topological expressions., 40 pages; updates based on referee comments; expanded proof of Proposition 3.3

Ever since the famous Erdős-Ko-Rado theorem initiated the study of intersecting families of subsets, extremal problems regarding intersecting properties of families of various combinatorial objects have been extensively investigated. Among them, studies about families of subsets, vector spaces and permutations are of particular concerns. Recently, we proposed a new quantitative intersection problem for families of subsets: For $${\cal F} \subseteq \left({\matrix{{[n]} \cr k \cr}} \right)$$ , define its total intersection number as $${\cal I}({\cal F}) = \sum\nolimits_{{F_1},{F_2} \in {\cal F}} {\left| {{F_1} \cap {F_2}} \right|} $$ . Then, what is the structure of $${\cal F}$$ when it has the maximal total intersection number among all the families in $$\left({\matrix{{[n]} \cr k \cr}} \right)$$ with the same family size? In a recent paper, Kong and Ge (2020) studied this problem and characterized extremal structures of families maximizing the total intersection number of given sizes. In this paper, we consider the analogues of this problem for families of vector spaces and permutations. For certain ranges of family size, we provide structural characterizations for both families of subspaces and families of permutations having maximal total intersection numbers. To some extent, these results determine the unique structure of the optimal family for some certain values of $$\left| {\cal F} \right|$$ and characterize the relationship between having maximal total intersection number and being intersecting. Besides, we also show several upper bounds on the total intersection numbers for both families of subspaces and families of permutations of given sizes.

The aim of this paper is to fill in the gaps in the formulation and the proof of a theorem contained in the paper by K.Ozeki (Aequ Math 25:247–252, 1982) published in this journal. We also give a short proof of this theorem and use it to obtain certain information about the factorization of polynomials of the form $$f(x)-f(y)$$ .

Let $$\alpha = (1+\sqrt{5})/2$$ and define the lower and upper Wythoff sequences by $$a_i = \lfloor i \alpha \rfloor $$ , $$b_i = \lfloor i \alpha ^2 \rfloor $$ for $$i \ge 1$$ . In a recent interesting paper, Kawsumarng et al. proved a number of results about numbers representable as sums of the form $$a_i + a_j$$ , $$b_i + b_j$$ , $$a_i + b_j$$ , and so forth. In this paper I show how to derive all of their results, using one simple idea and existing free software called Walnut. The key idea is that for each of their sumsets, there is a relatively small automaton accepting the Fibonacci representation of the numbers represented. I also show how the automaton approach can easily prove other results.

In this paper, we characterize that the boundedness, compactness and spectral structure of dual Toeplitz operators acting on the orthogonal complement of the harmonic Bergman space. This generalizes the corresponding results for dual Toeplitz operators on the orthogonal complement of the Bergman space due to Stroethoff and Zheng’s paper [Trans. Amer. Math. Soc., 354, 2495–2520 (2002)].

In the present paper we describe the family of all closed convex plane curves of class $$C^1$$ C 1 which have circles as their isoptics. In the first part of the paper we give a certain characterization of all ellipses based on the notion of isoptic and we give a geometric characterization of curves whose orthoptics are circles. The second part of the paper contains considerations on curves which have circles as their isoptics and we show the form of support functions of all considered curves.

This paper deals with a class of singular problems involving the fractional $$p(x,\cdot )$$ -Laplace operator of the form $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}_{p(x,\cdot )}u(x)= \frac{\lambda }{u^{\gamma (x)}}+u^{q(x)-1} &{} \hbox {in }\Omega , \\ u>0, \;\;\text {in}\;\; \Omega &{} \hbox {} \\ u=0 \;\;\text {on}\;\;{\mathbb {R}}^N\setminus \Omega , &{} \hbox {} \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a smooth bounded domain in $${\mathbb {R}}^N$$ ( $$N\ge 3$$ ), $$00$$ small enough. To our best knowledge, this paper is one of the first attempts in the study of singular problems involving fractional $$p(x,\cdot )$$ -Laplace operators.

In the paper, we consider the limit behavior of partial-sum random field (r.f.) $$ \left.{S}_n\left({t}_1,{t}_2;\right)X\left(b\left(\mathbf{n}\right)\right)\right)={\sum}_{k=1}^{\left[{n}_1{t}_1\right]}{\sum}_{l=1}^{\left[{n}_2{t}_2\right]}{X}_{k,l}\left(b\left(\mathbf{n}\right)\right), $$ where $$ \left\{{X}_{k,l}\left(b\left(\mathbf{n}\right)\right)={\sum}_{i=0}^{\infty }{\sum}_{j=0}^{\infty }{c}_{i,j}{\upxi}_{k-i,l-j}\left(b\left(\mathbf{n}\right)\right),k,l\in \mathrm{\mathbb{Z}}\right\},n\ge 1, $$ is a family (indexed by n = (n1, n2), ni ≥ 1) of linear r.f.s with filter ci,j = aibj and innovations ξk,l(b(n)) having heavy-tailed tapered distributions with tapering parameter b(n) growing to infinity as n → ∞. In [V. Paulauskas, Limit theorems for linear random fields with tapered innovations. I: The Gaussian case, Lith. Math. J., 61(2):261–273, 2021], we considered the so-called hard tapering as b(n) grows relatively slowly and the limit r.f.s for appropriately normalized Sn(t1, t2;X(b(n))) are Gaussian. In this paper, we consider the case of soft tapering where b(n) grows more rapidly in comparison with the case of hard tapering and stable limit r.f.s.We consider cases where the sequences {ai} and {bj} are long-range, short-range, and negatively dependent.

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrodinger equation $$i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u$$ in $$H^1$$ . The well-posedness theory in $$H^1$$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $$\beta =(4-2\alpha )/(n-2)$$ . It is still an open problem. The main contribution of this paper is to develop the theory in this case.

Recently, fixed point results via E -contractions (also called P- contractions in some papers) for self-mappings in metric spaces have been investigated. Some results generalize the well - known Edelstein’s theorem. In this paper, we provide some new fixed point theorems for mappings satisfying conditions of Edelstein - Suzuki type involving E -contractions. We also present several illustrative examples to compare our finding with some know results in the literature.

Solid codes can be used in information transmission over a noisy channel because they have remarkable synchronization and error-detecting capabilities in the presence of noise. In this paper, we focus on combinatoric properties of solid codes. We begin by characterizing solid codes by means of infix codes and unbordered words. Then, we discuss the decomposition of solid codes (in particular, the maximal solid codes). And finally, we investigate several properties of the products of the solid codes and some other kinds of codes. Our results given in this paper significantly enrich the theory of solid codes.

This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension. In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that satisfy general positioned point and cross-ratio conditions. Moreover, graphical contributions are introduced which provide a novel and structured way of understanding multiplicities of floor decomposed curves in $\mathbb{R}^3$. Additionally, so-called condition flows on a tropical curve are used to reflect how conditions imposed on a tropical curve yield different types of edges. This concept is applicable in arbitrary dimension., 36 pages, 15 figures; fixed minor issues, added references

In this paper, we are interested in the dependence between lifetimes based on a joint survival model. This model is built using the bivariate Sarmanov distribution with Phase-Type marginal distributions. Capitalizing on these two classes of distributions’ mathematical properties, we drive some useful closed-form expressions of distributions and quantities of interest in the context of multiple-life insurance contracts. The dependence structure that we consider in this paper is based on a general form of kernel function for the Bivariate Sarmanov distribution. The introduction of this new kernel function allows us to improve the attainable correlation range.

UDC 517.9 Many problems in applied mathematics can be transformed and described by the differential inclusion $\dot x\in f(t, x)-N_Qx$ involving $N_Qx,$ which is a normal cone to a closed convex set $Q \in \mathbb R^n$ at $x\in Q.$ The Cauchy problem of this inclusion is studied in the paper. Since the change of $x$ leads to the change of $N_Qx,$ solving the inclusion becomes extremely complicated. In this paper, we consider an ordinary differential equation containing a control parameter $K.$ When $K$ is large enough, the studied equation gives a solution approximating to a solution of the inclusion above. The theorem about the approximation of these solutions with arbitrary small error (this error can be controlled by increasing $K$) is proved in this paper.

This paper is a survey of results on countable families with natural degree spectra. These results were obtained by a modification of the methodology proposed by Wechner, who first found a family of sets with the spectrum consisting precisely of nonzero Turing degrees. Based on this method, many researchers obtained examples of families with other natural spectra. In addition, in this paper we extend these results and present new examples of natural spectra. In particular, we construct a family of finite sets with the spectrum consisting of exactly non-K-trivial degrees and also we find new sufficient conditions on $$ {\Delta}_2^0 $$ -degree a, which guarantees that the class {x : x ≰ a} is the degree spectrum of some family. Finally, we give a survey of our recent results on the degree spectra of α-families, where α is an arbitrary computable ordinal.

Let $\mathcal X$ be a Banach space and let T be a bounded linear operator on $\mathcal {X}$ . We denote by S(T) the set of all complex $\lambda \in \mathcal {C}$ such that T does not have the single-valued extension property. In this paper it is shown that if MC is a 2 × 2 upper triangular operator matrix acting on the Banach space $\mathcal {X} \oplus \mathcal {Y}$ , then the passage from σLD(A) ∪ σLD(B) to σLD(MC) is accomplished by removing certain open subsets of σd(A) ∩ σLD(B) from the former, that is, there is the equality σLD(A) ∪ σLD(B) = σLD(MC) ∪ℵ, where ℵ is the union of certain of the holes in σLD(MC) which happen to be subsets of σd(A) ∩ σLD(B). Generalized Weyl’s theorem and generalized Browder’s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how generalized Weyl’ theorem, generalized Browder’s theorem, generalized a-Weyl’s theorem and generalized a-Browder’s theorem survive for 2 × 2 upper triangular operator matrices on the Banach space.

The aim of this paper is to apply trigonometric functions with functional equations of generating functions. Using the resulted new equations and formulas from this application, we obtain many special numbers and polynomials such as the Stirling numbers, Bernoulli and Euler type numbers, the array polynomials, the Catalan numbers, and the central factorial numbers. We then introduce combinatorial sums related to these special numbers and polynomials. Moreover, we gave some remarks that relates our new findings from this paper to the relations found in earlier studies.

Let us fix a prime $p$. The Erd\H{o}s-Ginzburg-Ziv problem asks for the minimum integer $s$ such that any collection of $s$ points in the lattice $\mathbb{Z}^n$ contains $p$ points whose centroid is also a lattice point in $\mathbb{Z}^n$. For large $n$, this is essentially equivalent to asking for the maximum size of a subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero. In this paper, we give a new upper bound for this problem for any fixed prime $p\geq 5$ and large $n$. In particular, we prove that any subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero has size at most $C_p\cdot \left(2\sqrt{p}\right)^n$, where $C_p$ is a constant only depending on $p$. For $p$ and $n$ going to infinity, our bound is of the form $p^{(1/2)\cdot (1+o(1))n}$, whereas all previously known upper bounds were of the form $p^{(1-o(1))n}$ (with $p^n$ being a trivial bound). Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method. This method and its consequences were already applied by Naslund as well as by Fox and the author to prove bounds for the problem studied in this paper. However, using some key new ideas, we significantly improve their bounds., Comment: 11 pages

Let G be a finite 2-group. In our recent papers, we proved that in a finite 2-group of almost maximal class, the set of all commuting automorphisms, $$\mathcal {A}(G)=\lbrace \alpha \in Aut(G) :x\alpha (x)=\alpha (x)x~~for~ all~ x\in G\rbrace $$ is equal to the group of all central automorphisms, $$Aut_{c}(G)$$ , except only for five ones. Also, we determined the structure of $$Aut_{c}(G)$$ and $$\mathcal {A}(G)$$ for these five groups. Using these results, in this paper, we find the structure of $$\mathcal {A}(G)=Aut_{c}(G)$$ for the remaining 2-groups of almost maximal class. Also, we prove the following results

The subject of the present paper is to study the stability of a class of laminated Timoshenko-type systems in the whole line $$\mathbb {R}$$ combined with a heat conduction given by Gurtin–Pipkin’s law and acting only on one equation of the laminated Timoshenko-type system. The main result of this paper shows that the thermoelastic dissipation generated by Gurtin–Pipkin’s law is strong enough to stabilize the system at least polynomially, even if only the second or the third equation of the laminated Timoshenko-type system is controlled and the two other ones are free. When only the first equation of the laminated Timoshenko-type system is controlled, we give a necessary and sufficient condition for the polynomial stability. The polynomial decays in the $$L^2$$ -norm of the solution, and its higher-order derivatives with respect to the space variable are specified in terms of the regularity of the initial data and some connections between the coefficients. An application to the particular case of Timoshenko-type systems is also given. The proofs are based on the energy method and Fourier analysis combined with some well-chosen weight functions.