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The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $\alpha$ and a convex body $B$, $g_{\alpha}(B)$ is the infimum of $\alpha$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{\alpha}(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $\alpha$. In this paper, we prove that $g_{\alpha}(B)\leq h_{\alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{\alpha} (B) > 2^{d-\alpha}$ for almost all $\alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.

In 1993, Shub and Smale posed the problem of finding a sequence of univariate polynomials of degree $N$ with condition number bounded above by $N$. In a previous paper by C. Belt\'an, U. Etayo, J. Marzo and J. Ortega-Cerd\`a, it was proved that the optimal value of the condition number is of the form $O(\sqrt{N})$, and the sequence demanded by Shub and Smale was described by a closed formula (for large enough $N\geqslant N_0$ with $N_0$ unknown) and by a search algorithm for the rest of the cases. In this paper we find concrete estimates for the constant hidden in the $O(\sqrt{N})$ term and we describe a simple formula for a sequence of polynomials whose condition number is at most $N$, valid for all $N=4M^2$, with $M$ a positive integer., Comment: 21 pages

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 this paper, we study a problem posed by Furstenberg on intersections between × 2 , × 3 \times 2, \times 3 invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used to obtain further improvements. For example, we show that if A 2 , A 3 ⊂ [ 0 , 1 ] A_2,A_3\subset [0,1] are closed and × 2 , × 3 \times 2, \times 3 invariant respectively, assuming that dim ⁡ A 2 + dim ⁡ A 3 > 1 \dim A_2+\dim A_3>1 then A 2 ∩ ( u A 3 + v ) A_2\cap (uA_3+v) is sparse (defined in this paper) and has box dimension zero uniformly with respect to the real parameters u , v u,v such that u u and u − 1 u^{-1} are both bounded away from 0 0 .

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.

In a little-known paper Hurwitz gave an infinite series representation of the class number for positive definite binary quadratic forms. In this paper we give a similar formula in the indefinite case. We also give a simple proof of Hurwitz's formula and indicate some extensions., Journal of the European Mathematical Society, 23 (12), ISSN:1435-9855, ISSN:1435-9863

A misuse of terminology has occurred in the statement and proof of Theorem 4.1 in our paper [Proc. Amer. Math. Soc. 143 (2015), pp. 1981–1989] which caused some justifiable misinterpretation of this result. To recover this result we provide a new definition and give the statement of our result in terms of this definition. The proof of the new version is an improvement of the old proof. The effect of the new definition on further relevant results in our paper has been adopted in a remark.

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

In this paper a second order elliptic equation with nonsmooth coefficients is considered in grand-Sobolev classes $$W_{q)}^{2} \left( \varOmega \right) $$ on a bounded n-dimensional domain $$\varOmega \subset R^{n} $$ with a sufficiently smooth boundary $$\partial \varOmega $$ , generated by the norm of the grand-Lebesgue space $$L_{q)}\left( \varOmega \right) $$ . These spaces are non-separable and therefore the definition of a reasonable solution in them faces certain difficulties. For this purpose, a subspace $$N_{q)}^{2} \left( \varOmega \right) $$ is distinguished in which infinitely differentiable and finite functions are dense. The strict inclusion $$W_{q}^{2} \left( \varOmega \right) \subset N_{q)}^{2} \left( \varOmega \right) $$ holds, where $$W_{q}^{2} \left( \varOmega \right) $$ is the classical Sobolev space. This raises specific questions dictated by the theory of spaces $$W_{q}^{2} \left( \varOmega \right) $$ , for example, the characterization of the space of traces of functions from $$N_{q)}^{1} \left( \varOmega \right) $$ cannot be characterized following the classical case. In this paper, the corresponding theorems concerning traces, extensions, and compactness of a family of functions from $$N_{q)}^{k} \left( \varOmega \right) $$ are proved. These results are applied to obtain a Schauder-type estimate up to the boundary. Schauder-type estimates make it possible to establish the fredholmness of the Dirichlet problem for the considered equation in spaces $$N_{q)}^{2} \left( \varOmega \right) $$ with data from grand-Lebesgue type spaces that are different from Lebesgue spaces. Therefore, the results of this work cannot be directly obtained from the results of the $$L_{p}$$ -theory. This work is a continuation of the research carried out by the authors in articles (Bilalov and Sadigova in Complex Var Elliptic Equ, 2020. https://doi.org/10.1080/17476933.2020.1807965 ; Bilalov and Sadigova in Sahand Commun Math Anal, 2021. https://doi.org/10.22130/scma.2021.521544.893 .

In this paper, we will establish a formula that relates the product $$\prod _{\omega ^n=1} {\omega x-1 \atopwithdelims ()p-1}$$ to generalized and homogeneous multiple harmonic sums, this would allow us to derive new identities and congruences. The congruences considered in this paper are congruences in $$\mathbb {C}_p$$ which are also valid in $$\mathbb {Z}_p$$ and $$\mathbb {Z}$$ . In order to prove these congruences, we employ well-known theorems for symmetric functions and harmonic numbers.

In this paper, we are concerned with the following integral system [Formula: see text] where [Formula: see text], [Formula: see text], and [Formula: see text] ([Formula: see text]). When [Formula: see text], such an integral system is associated with the best constants of the Hardy–Littlewood–Sobolev inequality. Chen, Li and their cooperators obtained optimal integrability intervals of the finite energy solutions by an argument of contraction and shrinking operators. This result is helpful to well understand the classification of the extremal functions of the Hardy–Littlewood–Sobolev inequality. The critical condition plays a key role in their work. In this paper, we study optimal integrability intervals when the positive solutions have some initial integrability. Now, the critical condition is not necessary, and we apply a weaker condition, the Serrin-type condition, to establish some important relations of exponents which come into play to lift the regularity. In addition, we also generalize this result to the case of [Formula: see text].

This paper introduces self-similarity inherent in planar Milich-Jennings centered flip graphs derived from the Narayana sequence. We show that self-similarity found in a Narayana sequence yields a connected spanning subgraph with a centered flip. This paper has several main results (1) Every Narayana sequence constructs a flip graph, (2) Every Narayana sequence is self-similar and (3) Every Pascal 3-triangle has a free group presentation.

The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions. Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entrywise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at "opposite ends", and in both cases the preservers have to be absolutely monotonic. We complete the classification of positivity preservers that act entrywise except on specified "diagonal/principal blocks", in every case other than the two above. (In fact we achieve this in a more general framework.) This yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic., Minor edits in exposition, 19 pages. The paper now uses the style file of Trans. AMS (to appear)

This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator $$\overline{\partial }$$ ∂ ¯ on spaces $${\mathcal {E}}{\mathcal {V}}(\varOmega ,E)$$ E V ( Ω , E ) of $${\mathcal {C}}^{\infty }$$ C ∞ -smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights $${\mathcal {V}}$$ V . Vector-valued means that these functions have values in a locally convex Hausdorff space E over $${\mathbb {C}}$$ C . We derive a counterpart of the Grothendieck-Köthe-Silva duality $${\mathcal {O}}({\mathbb {C}}\setminus K)/{\mathcal {O}}({\mathbb {C}})\cong {\mathscr {A}}(K)$$ O ( C \ K ) / O ( C ) ≅ A ( K ) with non-empty compact $$K\subset {\mathbb {R}}$$ K ⊂ R for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of $$\overline{\partial }:{\mathcal {E}} {\mathcal {V}}(\varOmega ,E)\rightarrow {\mathcal {E}}{\mathcal {V}} (\varOmega ,E)$$ ∂ ¯ : E V ( Ω , E ) → E V ( Ω , E ) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on $${\mathcal {E}}{\mathcal {V}}(\varOmega ,{\mathbb {C}})$$ E V ( Ω , C ) .

For a fixed digraph G, the Constraint Satisfaction Problem with the template G, or CSP(G) for short, is the problem of deciding whether a given input digraph H admits a homomorphism to G. The dichotomy conjecture of Feder and Vardi states that CSP(G), for any choice of G, is solvable in polynomial time or NP-complete. This paper confirms the conjecture for a class of oriented trees called special triads. As a corollary we get the smallest known example of an oriented tree (with 33 vertices) defining an NP-complete CSP(G).

A special case of a fundamental result of Loewner and Horn [Trans. Amer. Math. Soc. 1969] says that given an integer $n \geq 1$, if the entrywise application of a smooth function $f : (0,\infty) \to \mathbb{R}$ preserves the set of $n \times n$ positive semidefinite matrices with positive entries, then $f$ and its first $n-1$ derivatives are non-negative on $(0,\infty)$. In a recent joint work with Belton-Guillot-Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and used it to strengthen the Schoenberg-Rudin characterization of dimension-free positivity preservers [Duke Math. J. 1942, 1959]. In recent works with Belton-Guillot-Putinar [Adv. Math. 2016] and with Tao [Amer. J. Math., in press] we used local, real-analytic versions at the origin of the Horn-Loewner condition, and discovered unexpected connections between entrywise polynomials preserving positivity and Schur polynomials. In this paper, we unify these two stories via a Master Theorem (Theorem A) which (i) simultaneously unifies and extends all of the aforementioned variants; and (ii) proves the positivity of the first $n$ nonzero Taylor coefficients at individual points rather than on all of $(0,\infty)$. A key step in the proof is a new determinantal / symmetric function calculation (Theorem B), which shows that Schur polynomials arise naturally from considering arbitrary entrywise maps that are sufficiently differentiable. Of independent interest may be the following application to symmetric function theory: we extend the Schur function expansion of Cauchy's (1841) determinant (whose matrix entries are geometric series $1 / (1 - u_j v_k)$), as well as of a determinant of Frobenius [J. reine angew. Math. 1882] (whose matrix entries are a sum of two geometric series), to arbitrary power series, and over all commutative rings., Comment: Final version, to appear in Transactions of the American Mathematical Society. 19 pages, no figures

We investigate criteria for circle packing(CP) types of disk triangulation graphs embedded into simply connected domains in $ \mathbb{C}$. In particular, by studying combinatorial curvature and the combinatorial Gauss-Bonnet theorem involving boundary turns, we show that a disk triangulation graph is CP parabolic if \[ \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)} = \infty, \] where $k_n$ is the degree excess sequence defined by \[ k_n = \sum_{v \in B_n} (\mbox{deg}\, v - 6) \] for combinatorial balls $B_n$ of radius $n$ and centered at a fixed vertex. It is also shown that the simple random walk on a disk triangulation graph is recurrent if \[ \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)+\sum_{j=0}^{n} (k_j +6)} = \infty. \] These criteria are sharp, and generalize a conjecture by He and Schramm in their paper from 1995, which was later proved by Repp in 2001. We also give several criteria for CP hyperbolicity, one of which generalizes a theorem of He and Schramm, and present a necessary and sufficient condition for CP types of layered circle packings generalizing and confirming a criterion given by Siders in 1998., Comment: 45 pages, 19 figures; to appear in TAMS

Let $$\ell $$ be a fixed odd positive integer. In this paper, using some classical results on the generalized Ramanujan-Nagell equation, we completely derive all solutions (p, x, m, n) of the equation $$x^2=4p^n-4p^m+\ell ^2$$ with $$\ell ^21$$ , where p is a prime, x, m, n are positive integers satisfying $$\gcd (x,\ell )=1$$ and $$m4p^m$$ . As an example of using this method, we find all solutions (p, x, m, n) of the equation for $$\ell \in \{5,7\}$$ .

Let C be an Abelian group. A class X of Abelian groups is called a CE• H-class if for any groups A, B ∈ X, it follows from the existence of isomorphisms E• (A) ≅ E• (B) and Hom(C,A) ≅ Hom(C,B) that there is an isomorphism A ≅ B. In this paper, conditions are studied under which the class $$ {\mathfrak{I}}_{\mathrm{cd}}^{\mathrm{ad}} $$ of completely decomposable almost divisible Abelian groups and class $$ {\mathfrak{I}}_{\mathrm{cd}}^{\ast } $$ of completely decomposable torsion-free Abelian groups A where Ω(A) contains only incomparable types are CE• H-classes, where C is a completely decomposable torsion-free Abelian group.

For fixed $r\geq 3$ and $n$ divisible by $r$, let ${\mathcal H}={\mathcal H}^r_{n,M}$ be the random $M$-edge $r$-graph on $V=\{1,\ldots ,n\}$; that is, ${\mathcal H}$ is chosen uniformly from the $M$-subsets of ${\mathcal K}:={V \choose r}$ ($:= \{\mbox{$r$-subsets of $V$}\}$). Shamir's Problem (circa 1980) asks, roughly, for what $M=M(n)$ is ${\mathcal H}$ likely to contain a perfect matching (that is, $n/r$ disjoint $r$-sets)? In 2008 Johansson, Vu and the author showed that this is true for $M>C_rn\log n$. More recently the author proved the asymptotically correct version of that result: for fixed $C> 1/r$ and $M> Cn\log n$, $P({\mathcal H} ~\mbox{contains a perfect matching})\rightarrow 1 \,\,\, \mbox{as $n\rightarrow\infty$}.$ The present work completes a proof, begun in that recent paper, of the definitive "hitting time" statement: $\mbox{Theorem.}$ If $A_1, \ldots ~$ is a uniform permutation of ${\mathcal K}$, ${\mathcal H}_t=\{A_1\dots A_t\}$, and \[ T=\min\{t:A_1\cup \cdots\cup A_t=V\}, \] then $P({\mathcal H}_T ~\mbox{contains a perfect matching})\rightarrow 1 \,\,\, \mbox{as $n\rightarrow\infty$}$.

In this paper, the authors define the homology of sets, which comes from and contains the ideas of path homology and embedded homology. Moreover, A Kunneth formula for sets associated to the homology of sets is given.

Lin-Lu-Yau introduced a notion of Ricci curvature for graphs and obtained a complete classification for all Ricci-flat graphs with girth at least five. In this paper, we characterize all Ricci-flat graphs of girth four with vertex-disjoint 4-cycles.

In this paper, we prove gradient estimates for the positive solutions of $$\cal{L}u=0$$ and $$\cal{L}u = {{\partial u} \over {\partial t}}$$ on conformal solitons, where $$\cal{L}\left( \cdot \right) = \Delta \left( \cdot \right) + \left\langle {X,\nabla \left( \cdot \right)} \right\rangle$$ . We also give some applications for them.

Treedepth is a well-studied graph invariant in the family of "width measures" that includes treewidth and pathwidth. Understanding these invariants in terms of excluded minors has been an active area of research. The recent Grid Minor Theorem of Chekuri and Chuzhoy [12] establishes that treewidth is polynomially approximated by the largest k x k grid minor. In this paper, we give a similar polynomial excluded-minor approximation for treedepth in terms of three basic obstructions: grids, tree, and paths. Specifically, we show that there is a constant c such that every graph of treedepth ≥ kc contains one of the following minors (each of treedepth ≥ k): • the k x k grid, • the complete binary tree of height k, • the path of order 2k. Let us point out that we cannot drop any of the above graphs for our purpose. Moreover, given a graph G we can, in randomized polynomial time, find either an embedding of one of these minors or conclude that treedepth of G is at most kc. This result has potential applications in a variety of settings where bounded treedepth plays a role. In addition to some graph structural applications, we describe a surprising application in circuit complexity and finite model theory from recent work of the second author [28].

Let $$[\, \cdot \,]$$ be the floor function. In this paper we show that whenever $$\eta $$ is real, the constants $$\lambda _i$$ satisfy some necessary conditions, then for any fixed $$1

The Littlewood-Richardson coefficients $c^{\lambda}_{\mu\nu}$ give the multiplicity of an irreducible polynomial ${\rm GL}_n$-representation $F^{\lambda}_n$ in the tensor product of polynomial representations $F^{\mu}_n\otimes F^{\nu}_n$. In this paper, we generalize these coefficients to an $r$-fold tensor product of rational representations, and give a new method for computing them using an analogue of statistical contingency tables. We demonstrate special cases in which our method reduces to counting statistical contingency tables with prescribed margins. Finally, we extend our result from the general linear group to both the orthogonal and symplectic groups.

In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic sequence along squares $(n^2)_{n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, we give for these subsequences a criterion to decide whether the densities exist, in which case they are also computable. In particular in the prime case these densities are all rational. We also deduce from a recent result of the third author and Lema\'nczyk that all subshifts generated by automatic sequences are orthogonal to any bounded multiplicative aperiodic function., Comment: 35 pages. We added an Appendix concerning upper densities of subsequences of automatic sequences

A graph $X$ is defined inductively to be $(a_0,\dots,a_{n-1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius $1$ around $v$ is an $(a_1,\dots,a_{n-1})$-regular graph. Such a graph $X$ is said to be highly regular (HR) of level $n$ if $a_{n-1}\neq 0$. Chapman, Linial and Peled studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders "globally and locally". They ask whether such HR-graphs of level 3 exist. In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can lead to such graphs. Given a Coxeter system $(W,S)$ and a subset $M$ of $S$, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope $\mathcal{P}_{W,M}$, which form an infinite family of expander graphs when $(W,S)$ is indefinite and $\mathcal{P}_{W,M}$ has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of $(W,S)$. The expansion stems from applying superapproximation to the congruence subgroups of the linear group $W$. This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked by Chapman, Linial and Peled., Comment: 22 pages, 2 tables. Dedicated to the memory of John Conway and Ernest Vinberg

Let X X be an affine spherical variety, possibly singular, and L + X \mathsf L^+X its arc space. The intersection complex of L + X \mathsf L^+X , or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified L L -functions. Such relationships were previously established in Braverman–Finkelberg–Gaitsgory–Mirković for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier–Ngô–Sakellaridis for toric varieties and L L -monoids. In this paper, we compute this intersection complex for the large class of those spherical G G -varieties whose dual group is equal to G ˇ \check G , and the stalks of its nearby cycles on the horospherical degeneration of X X . We formulate the answer in terms of a Kashiwara crystal, which conjecturally corresponds to a finite-dimensional G ˇ \check G -representation determined by the set of B B -invariant valuations on X X . We prove the latter conjecture in many cases. Under the sheaf–function dictionary, our calculations give a formula for the Plancherel density of the IC function of L + X \mathsf L^+X as a ratio of local L L -values for a large class of spherical varieties.

Let $${\mathbb{D}_r}: = \{ z = x + {\rm{i}}y \in \mathbb{C}:\left| z \right| < r\} ,\,r \le 1$$ . For a normalized analytic function f in the unit disk $$\mathbb{D}: = {\mathbb{D}_1}$$ , estimating the Dirichlet integral $$\Delta (r,f) = \int {\,\int_{{\mathbb{D}_r}} {{{\left| {f\prime (z)} \right|}^2}dxdy} } $$ , z = x +iy, is an important classical problem in complex analysis. Geometrically, Δ(r, f) represents the area of the image of $${\mathbb{D}_r}$$ under f counting multiplicities. In this paper, our main objective is to estimate areas of images of $${\mathbb{D}_r}$$ under non-vanishing analytic functions of the form (z/f)μ, μ > 0, in principal powers, when f ranges over certain classes of analytic and univalent functions in $$\mathbb{D}$$ .

In a 2015 article Cuenya and Ferreyra defined a class of functions in $$L^p$$ -spaces, denoted by $$c_n^p(x)$$ . The class $$c_n^p(x)$$ contains the class of $$L^p$$ -differentiability functions, denoted by $$t_n^p(x)$$ , introduced in a 1961 article by Calderon-Zygmund. A more recent paper by Acinas, Favier and Zo introduced a new class of functions in Orlicz spaces $$L^\Phi$$ , called $$L^\Phi$$ -differentiable functions in the present article. The class of $$L^\Phi$$ -differentiable functions is closely related to the class $$t_n^p(x)$$ . In this work, we define a class of functions in $$L^\Phi$$ , denoted by $$c_n^{\Phi }(x)$$ . The class $$c_n^{\Phi }(x)$$ is more general than the class of $$L^{\varPhi}$$ -differentiable functions. We prove the existence of the best local $$\Phi$$ -approximation for functions in $$c_n^{\varPhi }(x)$$ and study the convexity of the set of cluster points of the set of best $$\Phi$$ -approximations to a function on an interval when their measures tend to zero.

It is known that if a bivariate meanKis symmetric, continuous and strictly increasing in each variable, then for every meanMthere is a unique mean$$N\,$$Nsuch thatKis invariant with respect to the mean-type mapping$$\left( M,N\right) ,$$M,N,which means that$$K\circ \left( M,N\right) =K$$K∘M,N=KandNis called aK-complementary mean forM(Matkowski in Aequ Math 57(1):87–107, 1999). This paper extends this result for a large class of nonsymmetric means. As an application, the limits of the sequences of iterates of the related mean-type mappings are determined, which allows us to find all continuous solutions of some functional equations.

Let Z2 denote a cyclic group of 2 order and Z 2 2 = Z2 × Z2 the direct product of groups. Suppose that (M, Φ) is a closed and smooth manifold M with a smooth Z 2 2 -action whose fixed point set is the disjoint union of two real projective spaces with the same dimension. In this paper, the authors give a sufficient condition on the fixed data of the action for (M, Φ) bounding equivariantly.

This paper considers comparison problems of predictor and estimator in the context of seemingly unrelated regression models ( $$\mathrm{SURM}$$ s). $$\mathrm{SURM}$$ s are a class of multiple regression equations with correlated errors among the equations from linear regression models. Our aim is to establish a variety of equalities and inequalities for comparing covariance matrices of the best linear unbiased predictors ( $$\mathrm{BLUP}$$ s) and the ordinary least squares predictors ( $$\mathrm{OLSP}$$ s) of unknown vectors under $$\mathrm{SURM}$$ s by using various rank and inertia formulas of block matrices. The results for comparisons of the best linear unbiased estimators ( $$\mathrm{BLUE}$$ s) and the ordinary least squares estimators ( $$\mathrm{OLSE}$$ s) in the models are also considered.

Let λf (n) be the normalized n-th Fourier coefficient of holomorphic eigenform f for the full modular group and ℙc(x): = {p ≤ x ∣ [pc]prime}, c ∈ ℝ+. In this paper, we show that for all 0 < c < 1 the mean value of λf(n) in ℙc(x) is ≪x log−Ax assuming the Riemann Hypothesis. Unconditionally, in the sense of Lebesgue measure, it holds for almost all c ∈ (e, 1 − e).

Among the finitely generated groups of diffeomorphisms of the line and the circle, groups that act freely on the orbit of almost every point of the line (circle) are allocated. The paper is devoted to the study of the structure of the set of finitely generated groups of orientation-preserving diffeomorphisms of the line and the circle of C(1) smoothness with a given number of generators and the property noted above. It is shown that such a set contains a massive subset (contains a countable intersection of open everywhere dense subsets). Such a result for finitely generated groups of orientation-preserving diffeomorphisms of the circle, in the case of C(2) smoothness, was obtained by the author earlier.

In this paper, we consider the following two-coupled fractional Laplacian system with two or more isolated singularities $$\left\{ {\begin{array}{*{20}{c}} {{{( - \Delta )}^s}u}&{ = {\mu _1}{u^{2q + 1}} + \beta {u^{p1 - 1}}{v^{p2}},}&\; \\ {{{( - \Delta )}^s}u}&{ = {\mu _2}{v^{2q + 1}} + \beta {u^{p1}}{v^{p2 - 1}}}&{\text{in}\;{\mathbb{R}^n}\backslash \Lambda ,} \\ {u > 0,}&{v > 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&\; \end{array}} \right.$$ where s ∈ (0, 1), n > 2s and n ≥ 2. μ1, μ2 and β are all positive constants. p1, p2 > 1 and $${p_1} + {p_2} = 2q + 2 \in \left( {{{2n - 2s} \over {n - 2s}},\left. {{{2n} \over {n - 2s}}} \right]} \right.$$ . Λ ⊂ ℝn contains finitely many isolated points. By the method of moving plane, we obtain the symmetry results for positive solutions to above system.

This paper is mainly concerned with sets which do not contain four-term arithmetic progressions, but are still very rich in three term arithmetic progressions, in the sense that all sufficiently large subsets contain at least one such progression. We prove that there exists a positive constant c and a set A ⊂ F^n_q which does not contain a four-term arithmetic progression, with the property that for every subset A′ ⊂ A with |A′| ≥ |A|^(1−c), A′ contains a nontrivial three term arithmetic progression. We derive this from a more general quantitative Roth-type theorem in random subsets of F^n_q, which improves a result of Kohayakawa-Luczak-Rodl/Tao-Vu. We also discuss a similar phenomenon over the integers, where we show that for all ϵ > 0, and all sufficiently large N ∈ N, there exists a four-term progression-free set A of size N with the property that for every subset A′ ⊂ A with |A′| ≫ 1/(logN)^(1−ϵ)⋅ N contains a nontrivial three term arithmetic progression. Finally, we include another application of our methods, showing that for sets in F^n_q or Z the property of "having nontrivial three-term progressions in all large subsets" is almost entirely uncorrelated with the property of "having large additive energy".

The goal of this paper is to describe the relationship between generalized B-opers, generalized $$\mathrm {SO}(2n,{\mathbb {C}})$$ -opers and (G, P)-opers. In particular, we show that to each generalized B-oper there is a naturally associated (G, P)-oper, but there are some (G, P)-opers that do not arise as generalized B-opers or $$\mathrm {SO}(2n,{\mathbb {C}})$$ -opers.

All convex polygons whose envelopes of all area or perimeter-bisecting lines, their area or perimeter-bisecting deltoids, are tri-cusped and tri-concave-out (segment-free) closed curves were classified by the authors in a recent article. The present paper computes the area of each envelope as a fraction of the area of its corresponding polygon, thereby generalizing the formulas obtained for triangles by a group of students led by John Conway for the area-bisecting deltoid, and by the authors for the perimeter-bisecting deltoid.

Let $$G_n:=U(n)\ltimes {\mathbb {H}}_n$$ be the semidirect product of the unitary group acting by automorphisms on the Heisenberg group $${\mathbb {H}}_n.$$ It was pointed out in [5], that the unitary dual $$\widehat{G_n}$$ of $$G_n$$ is homeomorphic to the space of admissible coadjoint orbits $${\mathfrak {g}}_n^\ddagger /G_n$$ of $$G_n.$$ One of the important subset of $$\widehat{G_n}$$ is what is called the cortex of $$G_n,$$ cor $$(G_n),$$ which is the set of all $$\pi \in \widehat{G_n}$$ that cannot be Hausdorff separated from the identity representation $$1_{G_n}$$ of $$G_n.$$ In the present paper, we use the orbit method of Lipsman to determine the cortex of $$G_n.$$

Let G be a nonempty closed subset of a Banach space X. Let $${\cal B}(X)$$ be the family of nonempty bounded closed subsets of X endowed with the Hausdorff distance and $${{\cal B}_G}(X) = \overline {\{ A \in {\cal B}(X):A \cap G\emptyset \} } $$ , where the closure is taken in the metric space $$({\cal B}(X),H)$$ . For x ∈ X and $$F \in {{\cal B}_G}(X)$$ , we denote the nearest point problem inf{∥x − g∥: g ∈ G} by min(x, G) and the mutually nearest point problem inf{∥f − g∥: f ∈ F,g ∈ G} by min(F, G). In this paper, parallel to well-posedness of the problems min(x, G) and min(F, G) which are defined by De Blasi et al., we further introduce the weak well-posedness of the problems min(x, G) and min(F, G). Under the assumption that the Banach space X has some geometric properties, we prove a series of results on weak well-posedness of min(x, G) and min(F, G). We also give two sufficient conditions such that two classes of subsets of X are almost Chebyshev sets.

The equitable tree-coloring can formulate a structure decomposition problem on the communication network with some security considerations. Namely, an equitable tree-$k$-coloring of a graph is a vertex coloring using $k$ distinct colors such that every color class induces a forest and the sizes of any two color classes differ by at most one. In this paper, we show some theoretical results on the equitable tree-coloring of graphs by proving that every $d$-degenerate graph with maximum degree at most $\Delta$ is equitably tree-$k$-colorable for every integer $k\geq (\Delta+1)/2$ provided that $\Delta\geq 9.818d$, confirming the equitable vertex arboricity conjecture for graphs with low degeneracy., Comment: This is the final version acceptted for publication in Acta Mathematica Sinica, English Series, with an improvement of the lower bound for the maximum degree and with the title changed

Linear projections from $$\mathbb {P}^k$$ P k to $$\mathbb {P}^h$$ P h appear in computer vision as models of images of dynamic or segmented scenes. Given multiple projections of the same scene, the identification of sufficiently many correspondences between the images allows, in principle, to reconstruct the position of the projected objects. A critical locus for the reconstruction problem is a variety in $$\mathbb {P}^k$$ P k containing the set of points for which the reconstruction fails. Critical loci turn out to be determinantal varieties. In this paper we determine and classify all the smooth critical loci, showing that they are classical projective varieties.

In this paper, criteria for strict convexity of Orlicz sequence spaces equipped with the p-Amemiya norms $$(1\le p\le \infty )$$ are presented. The obtained results widen the characterization of strict convexity of Orlicz sequence spaces. From our results it follows that there are Orlicz sequence spaces which are strictly convex for p-Amemiya norms with $$1

Slater introduced the point-addition operation on graphs to characterize 4-connected graphs. The $$\varGamma $$ -extension operation on binary matroids is a generalization of the point-addition operation. In this paper, for a given binary matroid M we characterize the closure operator and the flats of the $$\varGamma $$ -extension binary matroid $$M^X$$ in terms of the closure operator of original binary matroid M.

The edge-connectivity of a connected hypergraph H is the minimum number of edges (named as edge-cut) whose removal makes H disconnected. It is known that the edge-connectivity of a hypergraph is bounded above by its minimum degree. H is super edge-connected, if every edge-cut consists of edges incident with a vertex of minimum degree. A hypergraph H is linear if any two edges of H share at most one vertex. We call H uniform if all edges of H have the same cardinality. Sufficient conditions for equality of edge-connectivity and minimum degree of graphs and super edge-connected graphs are known. In this paper, we present a generalization of some of these sufficient conditions to linear and/or uniform hypergraphs.

A finite group $${{{\mathbb {G}}}}$$ is said to have perfect order subsets if for every d, the number of elements of $$ {{{\mathbb {G}}}}$$ of order d (if there are any) divides $$|{{{\mathbb {G}}}}|.$$ In this paper, we make a modest improvement on the result of Ford, Konyagin and Luca [3] about perfect order subsets.

In this paper, we consider compositions of integers when only parts of size at most 3 are allowed in both composition and its conjugate composition. First we obtain a relation between the number of such compositions and the Fibonacci numbers. Then we provide combinatorial identities between these compositions and compositions into 1’s and 2’s, compositions into odd parts, and compositions into parts greater than 1. Further, several generalized results for these compositions are obtained.

We deal with an alienation problem for an Euler–Lagrange type functional equation $$\begin{aligned} f(\alpha x + \beta y) + f(\alpha x - \beta y) = 2\alpha ^2f(x) + 2\beta ^2f(y) \end{aligned}$$ f ( α x + β y ) + f ( α x - β y ) = 2 α 2 f ( x ) + 2 β 2 f ( y ) assumed for fixed nonzero real numbers $$\alpha ,\beta ,\, 1 \ne \alpha ^2 \ne \beta ^2$$ α , β , 1 ≠ α 2 ≠ β 2 , and the classic quadratic functional equation $$\begin{aligned} g(x+y) + g(x-y) = 2g(x) + 2g(y). \end{aligned}$$ g ( x + y ) + g ( x - y ) = 2 g ( x ) + 2 g ( y ) . We were inspired by papers of Kim et al. (Abstract and applied analysis, vol. 2013, Hindawi Publishing Corporation, 2013) and Gordji and Khodaei (Abstract and applied analysis, vol. 2009, Hindawi Publishing Corporation, 2009), where the special case $$g = \gamma f$$ g = γ f was examined.