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In this paper we consider the following question. When does there exist a square root of a probability measure supported on R+? This question is naturally related to subnormality of weighted shifts. The main result of this paper is that if μ is a finitely atomic probability measure having at most 4 atoms, then μ has a square root, i.e., there exists a measure ν such that μ=ν*ν (* means the convolution) if and only if the Aluthge transform of a subnormal weighted shift with Berger measure μ is subnormal. As an application of them, we give non-trivial, large classes of probability measures having a square root. We also prove that there are 6 and 7-atomic probability measures which don't have any square root. Our results have a connection to the following long-open problem in Operator Theory: characterize the subnormal operators having a square root.

In this paper we study a toy categorical version of Lusztig's induction and restriction functors for character sheaves, but in the abstract setting of multifusion categories. Let $\mathscr{C}$ be an indecomposable multifusion category and let $\mathscr{M}$ be an invertible $\mathscr{C}$-bimodule category. Then the center $\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$ of $\mathscr{M}$ with respect to $\mathscr{C}$ is an invertible module category over the Drinfeld center $\mathscr{Z}(\mathscr{C})$ which is a braided fusion category. Let $\zeta_{\mathscr{M}}:\mathscr{Z}_{\mathscr{C}}(\mathscr{M})\longrightarrow\mathscr{M}$ denote the forgetful functor and let $\chi_{\mathscr{M}}:\mathscr{M}\longrightarrow\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$ be its right adjoint functor. These functors can be considered as toy analogues of the restriction and induction functors used by Lusztig to define character sheaves on (possibly disconnected) reductive groups. In this paper we look at the relationship between the decomposition of the images of the simple objects under the above functors and the character tables of certain Grothendieck rings. In case $\mathscr{C}$ is equipped with a spherical structure and $\mathscr{M}$ is equipped with a $\mathscr{C}$-bimodule trace, we relate this to the notion of the crossed S-matrix associated with the $\mathscr{Z}(\mathscr{C})$-module category $\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$.

The general class of quadrilaterals where one diagonal is bisected by the other diagonal has appeared very rarely in the geometrical literature, but they have been named several times in connection with quadrilateral classifications. Günter Graumann strangely gave these objects two different names in [1, pp. 192, 194]: sloping-kite and sliding-kite. A. Ramachandran called them slant kites in [2, p. 54] and Michael de Villiers called them bisecting quadrilaterals in [3, pp. 19, 206]. The latter is a pretty good name, although a bit confusing: what exactly is bisected?We have found no papers and only two books where any theorems on such quadrilaterals are studied. In each of the books, one necessary and sufficient condition for such quadrilaterals is proved (see Theorem 1 and 2 in the next section). The purpose of this paper is to investigate basic properties of convex bisecting quadrilaterals, but we have chosen to give them a slightly different name. Let us first remind the reader that a quadrilateral whose diagonals have equal lengths is called an equidiagonal quadrilateral and one whose diagonals are perpendicular is called an orthodiagonal quadrilateral.

In this paper, the Taylor sequence space t p θ of non-absolute type is introduced which consists of all sequences whose Taylor transforms of order θ , ( 0 θ 1 ) , are in the space l p . It is shown that the space t p θ is a normed space which includes the space l p where 1 ≤ p ≤ ∞ . Moreover, in this paper a general upper estimate is obtained for the norm of Hausdorff matrices and Norlund matrices as operators from l p into the Taylor sequence space t p θ . Finally, the results are extended to the matrix domain of an arbitrary invertible matrix E in the sequence space l p .

Let 𝓜(Ω, μ) denote the algebra of all scalar-valued measurable functions on a measure space (Ω, μ). Let B ⊂ 𝓜(Ω, μ) be a set of finitely supported measurable functions such that the essential range of each f ∈ B is a subset of {0,1}. The main result of this paper shows that for any p ∈ (0, ∞), B has strict p-negative type when viewed as a metric subspace of L p (Ω, μ) if and only if B is an affinely independent subset of 𝓜(Ω, μ) (when 𝓜(Ω, μ) is considered as a real vector space). It follows that every two-valued (Schauder) basis of L p (Ω, μ) has strict p-negative type. For instance, for each p ∈ (0, ∞), the system of Walsh functions in L p [0,1] is seen to have strict p-negative type. The techniques developed in this paper also provide a systematic way to construct, for any p ∈ (2, ∞), subsets of L p (Ω, μ) that have p-negative type but not q-negative type for any q > p. Such sets preclude the existence of certain types of isometry into L p -spaces.

In a paper in 1962, Guido Zappa asked whether a non-trivial coset of a Sylow p-subgroup of a finite group could contain only elements whose orders are powers of p, and also in that case, at least one element of order p. The first question was raised again recently in a 2014 paper by Daniel Goldstein and Robert Guralnick, when generalising an answer by John Thompson in 1967 to a similar question by L.J. Paige. In this paper we give a positive answer to both questions of Zappa, showing somewhat surprisingly that in a number of non-abelian finite simple groups (including PSL ( 3 , 4 ) , PSU ( 5 , 2 ) and the Janko group J 3 ), some non-trivial coset of a Sylow 5-subgroup (of order 5) contains only elements of order 5. Also Zappa's first question is studied in more detail. Various possibilities for the group and its Sylow p-subgroup P are eliminated, and it then follows that | P | ≥ 5 and | P | ≠ 8 . It is an open question as to whether the order of the Sylow p-subgroup can be 7 or 9 or more.

In this paper, we examine embeddings of alternating and symmetric groups into almost simple groups of exceptional type. In particular, we prove that if the alternating or symmetric group has degree equal to 5, or 8 or more, then it cannot appear as the maximal subgroup of any almost simple exceptional group of Lie type. Furthermore, in the remaining open cases of degrees 6 and 7 we give considerable information about the possible embeddings. Note that no maximal alternating or symmetric subgroups are known in the remaining cases. This is the first in a sequence of papers aiming to substantially improve the state of knowledge about the maximal subgroups of exceptional groups of Lie type.

This is the first in our series of papers that concerns Hardy–Littlewood–Sobolev (HLS) type inequalities. In this paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space R n, $$\int {_{{R^n}}} \int {_{{R^n}}f\left( x \right)} {\left| {x - y} \right|^\lambda }g\left( y \right)dxdy \geqslant {\ell _{n,p,r}}{\left\| f \right\|_{{L^p}\left( {{R^n}} \right)}}{\left\| g \right\|_{{L^r}\left( {{R^n}} \right)}}$$ , for any non-negative functions f ∈ L p(R n), g ∈ L r(R n), and p, r ∈ (0, 1), λ > 0 such that 1/p+1/r −λ/n = 2. We will also explore some estimates for ln,p,r and the existence of optimal functions for the above inequality, which will shed light on some existing results in literature.

The aim of the paper is to present the binomial transformation formulae of Fibonacci numbers scaled by complex multipliers. Many of these new and nontrivial relations follow from the fundamental properties of the so-called delta-Fibonacci numbers defined by Wituła and Słota. The paper contains some original relations connecting the values of delta-Fibonacci numbers with the respective values of Chebyshev polynomials of the first and second kind.

This paper studies a higher dimensional generalization of Frieze's $\zeta(3)$-limit theorem in the Erd\"os-R\'enyi graph process. Frieze's theorem states that the expected weight of the minimum spanning tree converges to $\zeta(3)$ as the number of vertices goes to infinity. In this paper, we study the $d$-Linial-Meshulam process as a model for random simplicial complexes, where $d=1$ corresponds to the Erd\"os-R\'enyi graph process. First, we define spanning acycles as a higher dimensional analogue of spanning trees, and connect its minimum weight to persistent homology. Then, our main result shows that the expected weight of the minimum spanning acycle behaves in $O(n^{d-1})$., Comment: 24 pages

The k-core, defined as the maximal subgraph of minimum degree at least k, of the random graph G(n,p) has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [J Combin Theory Ser B 67 (1996), 111–151] determined the threshold dk for the appearance of an extensive k-core. The aim of the present paper is to describe how the k-core is “embedded” into the random graph in the following sense. Let k≥3 and fix d=np>dk. Colour each vertex that belongs to the k-core of G(n,p) in black and all remaining vertices in white. Here we derive a multi-type branching process that describes the local structure of this coloured random object as n tends to infinity. This generalises prior results on, e.g., the internal structure of the k-core. In the physics literature it was suggested to characterize the core by means of a message passing algorithm called Warning Propagation. Ibrahimi, Kanoria, Kraning and Montanari [Ann Appl Probab 25 (2015), 2743–2808] used this characterization to describe the 2-core of random hypergraphs. To derive our main result we use a similar approach. A key observation is that a bounded number of iterations of this algorithm is enough to give a good approximation of the k-core. Based on this the study of the k-core reduces to the analysis of Warning Propagation on a suitable Galton-Watson tree. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 00, 000–000, 2017

The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$., Comment: 22 pages; Minor corrections

This paper adjusts conditions on new defined given the name {a, d, c; r} ctype of contraction mappings on complete quasi metric space, confirms that there is only one fixed of any of these mappings with these adjustments, extends and generalizes results given in some previous research papers, and then builds a convergence theorem for a sequence of fixed points of {Sn}n∈N;{wn*}n∈N to the unique fixed of S; w, provided that limn→∞ Sn(w) = S(w).

The paper studies the algorithmic complexity of subproblems for satisfiability in positive integers of simultaneous divisibility of linear polynomials with nonnegative coefficients. In the general case, it is not known whether this problem is in the class NP, but that it is in NEXPTIME is known. The NP-completeness of two series of restricted versions of this problem such that a divisor of a linear polynomial is a number in the first case, and a linear polynomial is a divisor of a number in the second case is proved in the paper. The parameters providing the NP-completeness of these problems have been established. Their membership in the class P has been proven for smaller values of these parameters. For the general problem SIMULTANEOUS DIVISIBILITY OF LINEAR POLYNOMIALS, NP-hardness has been proven for its particular case, when the coefficients of the polynomials are only from the set {1, 2} and constant terms are only from the set {1, 5}.

Let N be a module in category $$\sigma [M]$$ . N is called generalized $$\delta$$ –M-small (briefly G– $$\delta$$ –M) if, $$N \subseteq \delta (L)$$ for some $$L \in \sigma [M]$$ . In this paper we characterize G– $$\delta$$ –M modules and get some suitable results related to this kind of modules. We will show that a module $$N \in \sigma [M]$$ is G– $$\delta$$ –M if and only if $$N \subseteq \delta (\hat{N})$$ , where $$\hat{N}$$ is the M-injective envelope of N in $$\sigma [M]$$ . Also we prove that if there is no non-zero G– $$\delta$$ –M module, then M is cosemisimple and by giving an example we show that the converse need not be true. The relation between G– $$\delta$$ –M modules and some other classes of modules would be investigated in this paper. The torsion theory cogenerated by this class of modules will be introduced and studied in this paper. For a module $$N \in \sigma [M]$$ we show that $$N = \mathrm{{Re}}_{GD[M]}(N)$$ iff for every nonzero homomorphism $$f : N \longrightarrow K$$ in $$\sigma [M],$$ $$\mathrm{{Im}}(f) \not \subseteq \delta (K)$$ iff $$\Delta _{\delta }(N,A) = 0,$$ for all $$A \in \sigma [M]$$ .

The paper considers the category ( $$ \mathcal{T} $$ , S, X) consisting of mappings S : $$ \mathcal{T} $$ −→ $$ \mathcal{T} $$ of spaces $$ \mathcal{T} $$ with distinguished subsets X ⊂ $$ \mathcal{T} $$ . Let rX (i, x0) be the distribution function of points of an S-orbit x0, x1 = S(x0), . . . , xi−1 = Si−1(x0) getting into X, and let δX (i, x0) be the deviation defined by the equation rX (i, x0) = aX i + δX (i, x0), where aX i is the average value. If δX (i, x0) = O(1), then such sets X are called bounded remainder sets. In the paper, bounded remainder sets X are constructed in the following cases: (1) the space $$ \mathcal{T} $$ is the circle, torus, or the Klein bottle; (2) the map S is a rotation of the circle, a shift or an exchange mapping of the torus; (3) X is a fixed subset X ⊂ $$ \mathcal{T} $$ or a sequence of subsets depending on the iteration number i = 0, 1, 2, . . .. Bibliography: 27 titles.

This paper concerns with the existence of solutions for the following fractional Kirchhoff problem with critical nonlinearity: $${\left( {\int {\int {_{{\mathbb{R}^{2N}}}\frac{{{{\left| {u\left( x \right) - u\left( y \right)} \right|}^2}}}{{{{\left| {x - y} \right|}^{N + 2s}}}}dxdy} } } \right)^{\theta - 1}}{\left( { - \Delta } \right)^s}u = \lambda h\left( x \right){u^{p - 1}} + {u^{2_s^* - 1}} in {\mathbb{R}^N},$$ where (−Δ) s is the fractional Laplacian operator with 0 < s < 1, 2 s * = 2N/(N − 2s), N > 2s, p ∈ (1, 2 s *), θ ∈ [1, 2 s */2), h is a nonnegative function and λ a real positive parameter. Using the Ekeland variational principle and the mountain pass theorem, we obtain the existence and multiplicity of solutions for the above problem for suitable parameter λ > 0. Furthermore, under some appropriate assumptions, our result can be extended to the setting of a class of nonlocal integro-differential equations. The remarkable feature of this paper is the fact that the coefficient of fractional Laplace operator could be zero at zero, which implies that the above Kirchhoff problem is degenerate. Hence our results are new even in the Laplacian case.

Let E be an elliptic curve defined over an algebraically closed field k whose characteristic is not 2 or 3. Let τ be a translation automorphism of E that is not of order 2. In a previous paper we studied an algebra A = A ( E , τ ) that depends on this data: A ( E , τ ) = ( S ( E , τ ) ⊗ M 2 ( k ) ) Γ where S ( E , τ ) is the 4-dimensional Sklyanin algebra associated to ( E , τ ) , M 2 ( k ) is the ring of 2 × 2 matrices over k, and Γ is ( Z / 2 ) × ( Z / 2 ) acting in a particular way as automorphisms of S and M 2 ( k ) . The action of Γ on S is compatible with the translation action of the 2-torsion subgroup E [ 2 ] on E. Following the ideas and results in papers of Artin–Tate–Van den Bergh, Smith–Stafford, and Levasseur–Smith, this paper examines the line modules, point modules, and fat point modules, over A, and their incidence relations. The right context for the results is non-commutative algebraic geometry: we view A as a homogeneous coordinate ring of a non-commutative analogue of P 3 that we denote by Proj n c ( A ) . Point modules and fat point modules determine “points” in Proj n c ( A ) . Line modules determine “lines” in Proj n c ( A ) . Line modules for A are in bijection with certain lines in P ( A 1 ⁎ ) ≅ P 3 and therefore correspond to the closed points of a certain subscheme L of the Grassmannian G ( 1 , 3 ) . Shelton–Vancliff call L the line scheme for A. We show that L is the union of 7 reduced and irreducible components, 3 quartic elliptic space curves and 4 plane conics in the ambient Plucker P 5 , and that deg ⁡ ( L ) = 20 . The union of the lines corresponding to the points on each elliptic curve is an elliptic scroll in P ( A 1 ⁎ ) . Thus, the lines on that elliptic scroll are in natural bijection with a corresponding family of line modules for A.

In this paper, we give a proof of the Gan–Gross–Prasad conjecture for the discrete series of U(p, q). There are three themes in this paper: branching laws of a small $$A_{\mathfrak {q}}(\lambda )$$ , branching laws of discrete series and inductive construction of discrete series. These themes are linked together by a reciprocity law and the notion of invariant tensor product.

Let D ∈ ℤ and let C D be the set of all monic cubic polynomials x 3 + ax 2 + bx + c ∈ ℤ[x] with the discriminant equal to D. Along the line of our preceding papers, the following Theorem has been proved: If D is square-free and 3 ∤ h(−3D) where h(−3D) is the class number of Q ( − 3 D ) , $\mathbb Q(\sqrt {-3D}),$ then all polynomials in C D have the same type of factorization over the Galois field 𝔽 p where p is a prime, p > 3. In this paper, we prove the validity of the above implication also for primes 2 and 3.

It is known that the monoid wreath product of any two semigroup varieties that are atoms in the lattice of all semigroup varieties may have a finite as well as an infinite lattice of subvarieties. If this lattice is finite, then as a rule it has at most eleven elements. This was proved in a paper of the author in 2007. The exclusion is the monoid wreath product Sl w N 2 of the variety of semilattices and the variety of semigroups with zero multiplication. The number of elements of the lattice L(Sl w N 2) of subvarieties of Sl w N 2 is still unknown. In our paper, we show that the lattice L(Sl w N 2) contains no less than 33 elements. In addition, we give some exponential upper bound of the cardinality of this lattice.

The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for σ \sigma -centered posets. In his 1973 paper he showed under this assumption that both ω 1 {\omega }_{1} and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for σ \sigma -centered posets implies that the Boolean algebra P ( ω ) / FIN \mathcal {P}(\omega ) / \operatorname {FIN} equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility.

This paper continues a recent resurgence of interest in combinatorial properties of a poset that are associated with graph properties of its cover graph and order diagram. The following two theorems appearing in a 1977 paper of Trotter and Moore have played important roles in motivating this more modern research: (1) The dimension of a poset is at most 3 when its cover graph is at tree; (2) The dimension of a poset is at most 3 when the poset has a zero and its order diagram is planar. Although the underlying ideas lay dormant for more than 30 years, the first of these two results has become the base case for recent results bounding the dimension of a poset in terms of (a) the tree-width of its cover graph, and (b) the maximum dimension of its blocks. The second result is the base case for bounding the dimension of a planar poset in terms of the number of minimal elements. Continuing with this line of research, we show that every poset whose cover graph is a tree is a circle order, i.e., it has a representation as a family of circular disks in the Euclidean plane partially ordered by inclusion.

Let k be a nonzero complex number. In this paper we consider k - circulant matrices with arithmetic sequence and investigate the eigenvalues, determinants and Euclidean norms of such matrices. Also, for k=1 , the inverses of such ( invertible ) matrices are obtained (in a way different from the way presented in the paper: M. Bahsi and S. Solak , On the Circulant Matrices with Arithmetic Sequence , Int. J. Contemp . Math. Sci. 5 (25) (2010), 1213-1222, and the Moore-Penrose inverses of such (singular) matrices are derived.

Let $H_1$, $H_2$, $H_3$ be real Hilbert spaces, let $A:H_1\rightarrow H_3$, $B:H_2\rightarrow H_3$ be two bounded linear operators. The general split common fixed-point problem under consideration in this paper is to $$\text{find}\ \ x \in \cap_{i=1}^p F(U_i),\ \ y \in \cap_{j=1}^r F(T_j)\ \ \text{such that}\ \ Ax = By,\eqno{(1)}$$ where $p$, $r\geq 1$ are integers, $U_i:H_1\rightarrow H_1$ $(1\leq i\leq p)$ and $T_j:H_2\rightarrow H_2$ $(1\leq j\leq r)$ are quasi-nonexpansive mappings with nonempty common fixed-point sets $\cap_{i=1}^pF(U_i)=\cap_{i=1}^p\{x\in H_1:U_ix=x\}$ and $\cap_{j=1}^rF(T_j)=\cap_{j=1}^r\{x\in H_2:T_jx=x\}$. Note that, the above problem (1) allows asymmetric and partial relations between the variables $x$ and $y$. If $H_2=H_3$ and $B=I$, then the general split common fixed-point problem (1) reduces to the general split common fixed-point problem proposed by Censor and Segal $\cite{C}$. In this paper, we introduce simultaneous parallel and cyclic algorithms for the general split common fixed-point problems (1). We introduce a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. We prove the weak convergence of the proposed algorithms and apply the proposed algorithms to the multiple-set split feasibility problems. Our results improve and extend the corresponding results announced by many others.

This is the last paper of a series of four papers in which we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number~$k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first two papers of this series, we decomposed the host graph $G$, and found a suitable combinatorial structure inside the decomposition. In the third paper, we refined this structure, and proved that any graph satisfying the conditions of the above approximate version of the Loebl-Komlos-Sos Conjecture contains one of ten specific configurations. In this paper we embed the tree $T$ in each of the ten configurations., Comment: 81 pages, 12 figures. A fix reflecting the change of Preconfiguration Clubs in Paper III, additional small changes

This is the third of a series of four papers in which we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first paper of the series, we gave a decomposition of the graph $G$ into several parts of different characteristics. In the second paper, we found a combinatorial structure inside the decomposition. In this paper, we will give a refinement of this structure. In the forthcoming fourth paper, the refined structure will be used for embedding the tree $T$., Comment: 59 pages, 4 figures; further comments by a referee incorporated; this includes a subtle but important fix to Lemma 5.1; as a consequence, Preconfiguration Clubs was changed

Suppose $R$ is an associative ring with identity 1. In this paper, We give the necessary and sufficient conditions for the existence and explicit representations of the group inverses of the following two classes block matrices. (i) $M=\left(\begin{array}{cc}AX+YBA (ii) $M=\left(\begin{array}{cc}A&B\\C&D\end{array}\right)$, where $A^{\sharp}$ exists, $A^{\pi}B=0$ and $(D-CA^{\sharp}B)^{\sharp}$ exists.\\ The paper's results generalize some relative results of Li et al. (J. Harbin Engineering Univ., 34, 658-661, 2013), Cao et al. (Appl. Math. Comput., 217, 10271-10277, 2011), Bu et al. (Appl. Math. Comput., 215, 132-139, 2009), Deng et al. (Linear Algebra Appl, 435, 2766-2783, 2011), and Liu et al. (J. Appl. Math. http://dx.doi.org/10.1155 /2013/247028). Some examples are given to illustrate our results.

A central question in geometric measure theory is whether geometric properties of a set translate into analytical ones. In 1960, E. R. Reifenberg proved that if an $n$-dimensional subset $M$ of $\mathbb{R}^{n+k}$ is well approximated by $n$-planes at every point and at every scale, then $M$ is a locally bi-H\"older image of an $n$-plane. Since then, Reifenberg's theorem has been refined in several ways in order to ensure that $M$ is a bi-Lipschitz image of an $n$-plane. In this paper, we show that a Carleson condition on the oscillation of the unit normal of an $n$-Ahlfors regular rectifiable subset $M$ of $\mathbb{R}^{n+1}$ satisfying a Poincar\'e-type inequality is sufficient to prove that $M$ is contained inside a bi-Lipschitz image of an $n$-dimensional affine subspace of $\mathbb{R}^{n+1}$. We also show that this Poincar\'e-type inequality encodes geometrical information about $M$, namely it implies that $M$ is quasiconvex., Comment: Throughout the paper, the parameters that the constants depend on are clearer. Theorem 4.8 is added in Section 4, and Remarks 5.9 and 5.10 added in Section 5

The purpose of this paper is to prove a general fixed point theorem for mappings involving almost altering distances and satisfying a new type of common limit range property which generalize the results from Theorem 2.9 [19]. In the last part of the paper, as applications, some fixed point results for mappings satisfying contractive conditions of integral type for almost contractive mappings for ?-contractive mappings and (?,?) - weak contractive mappings in metric spaces are obtained.

This is the second of a series of four papers in which we prove the following relaxation of the Loebl-Komlos--Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first paper of the series, we gave a decomposition of the graph $G$ into several parts of different characteristics; this decomposition might be viewed as an analogue of a regular partition for sparse graphs. In the present paper, we find a combinatorial structure inside this decomposition. In the last two papers, we refine the structure and use it for embedding the tree $T$., Comment: 38 pages, 4 figures; new is Section 5.1.1; accepted to SIDMA

Throughout this paper, we work over ${\mathbb C}$, and $n$ is an integer such that $n\geq 2$. For an Enriques surface $E$, let $E^{[n]}$ be the Hilbert scheme of $n$ points of $E$. By Oguiso and Schr\"oer, $E^{[n]}$ has a Calabi-Yau manifold $X$ as the universal covering space, $\pi :X\rightarrow E^{[n]}$ of degree $2$. The purpose of this paper is to investigate a relationship of the small deformation of $E^{[n]}$ and that of $X$ $({\rm Theorem}\ 1.1)$, the natural automorphism of $E^{[n]}$ $({\rm Theorem}\,1.2)$, and count the number of isomorphism classes of the Hilbert schemes of $n$ points of Enriques surfaces which has $X$ as the universal covering space when we fix one $X$ $({\rm Theorem}\,1.3)$.

In this paper we introduce a counterpart structure to the shamrocks studied in the paper A dual of Macmahon's theorem on plane partitions by M. Ciucu and C. Krattenthaler (2013) [5] , which, just like the latter, can be included at the center of a lattice hexagon on the triangular lattice so that the region obtained from the hexagon by removing it has its number of lozenge tilings given by a simple product formula. The new structure, called a fern, consists of an arbitrary number of equilateral triangles of alternating orientations lined up along a lattice line. The shamrock and the fern seem to be the only such connected structures with this property. It would be interesting to understand the reason for this.

In this paper, the complete moment convergence and the integrability of the supremum for weighted sums of negatively orthant dependent (NOD, in short) random variables are presented. As applications, the complete convergence and the Marcinkiewicz-Zygmund type strong law of large numbers for NODrandom variables are obtained. The results established in the paper generalize some corresponding ones for independent random variables and negatively associated random variables.

Let $V$ be a projective subvariety of $\mathbb P^n(\mathbb C)$. A family of hypersurfaces $\{Q_i\}_{i=1}^q$ in $\mathbb P^n(\mathbb C)$ is said to be in $N$-subgeneral position with respect to $V$ if for any $1\le i_1, Comment: This paper is a revised version of the manuscript "Second main theorem and unicity of meromorphic mappings for hypersurfaces of projective varieties in subgeneral position, arXiv:1302.1261" of the first author. This paper is accepted for publication in Acta Mathematica Vietnamica (2016)

We consider consecutive random subdivision of polygons described as follows. Given an initial convex polygon with $d\ge 3$ edges, we choose a point at random on each edge, such that the proportions in which these points divide edges are i.i.d. copies of some random variable $\xi$. These new points form a new (smaller) polygon. By repeatedly implementing this procedure we obtain a sequence of random polygons. The aim of this paper is to show that under very mild non-degenerateness conditions on $\xi$, the shapes of these polygons eventually become "flat" The convergence rate to flatness is also investigated; in particular, in the case of triangles ($d=3$), we show how to calculate the exact value of the rate of convergence, connected to Lyapunov exponents. Using the theory of products of random matrices our paper greatly generalizes the results of Volkov (2013) which are achieved mostly by using ad hoc methods., Comment: 35 pages, 2 figures

In this paper we clarify the quadratic irrationalities that can be admitted by an odd-degree complex irreducible character χ of an arbitrary finite group. Write Q ( χ ) to denote the field generated over the rational numbers by the values of χ, and let d > 1 be a square-free integer. We prove that if Q ( χ ) = Q ( d ) then d ≡ 1 (mod 4) and if Q ( χ ) = Q ( − d ) , then d ≡ 3 (mod 4). This follows from the main result of this paper: either i ∈ Q ( χ ) or Q ( χ ) ⊆ Q ( exp ⁡ ( 2 π i / m ) ) for some odd integer m ≥ 1 .

This paper studies Menon-type identities involving both multiplicative characters and additive characters. In the paper, we shall give the explicit formula of the following sum \[ \sum_{\substack{a \in \mathbb{Z}_n^{\ast} \\ b_1, \ldots, b_k \in \mathbb{Z}_n}} \gcd(a-1, b_1, \ldots, b_k, n) \chi(a) \lambda_1(b_1) \cdots \lambda_k(b_k), \] where for a positive integer $n$, $\mathbb{Z}_n^{\ast}$ is the group of units of the ring $\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}$, $\gcd$ represents the greatest common divisor, $\chi$ is a Dirichlet character modulo $n$, and for a nonnegative integer $k$, $\lambda_1, \ldots, \lambda_k$ are additive characters of $\mathbb{Z}_n$. Our formula further extends the previous results by Sury [13], Zhao-Cao [17] and Li-Hu-Kim [4].

Chen et al. (2004) strongly conjectured that R(Tn,Wm)=2n-1 if the maximum degree of Tn is small and m is even. Related to the conjecture, it is interesting to know for which tree Tn, we have R(Tn,Wm) > 2n-1 for even m. In this paper, we find the Ramsey number R(Tn,W8) for tree Tn with the maximum degree of Tn is at least n-3, namely R(Tn,W8) > 2n-1 for almost all such tree Tn. We also prove that if the maximum degree of Tn is large, then R(Tn,Wm) is a function of both m and n. In the end, we refine the conjecture of Chen et al. by giving the condition of small maximum degree. For a tree Tn with large maximum degree and even m, the R(Tn,Wm) is also unknown in general. In this paper, we shall determine the Ramsey number R(Tn,W8) for all trees Tn of order n with the maximum degree of Tn is at least n-3., 7 pages

Let $$\sigma $$ be an operator mean in the sense of Kubo and Ando. If the representation function $$f_\sigma $$ of $$\sigma $$ satisfies $$\begin{aligned} f_\sigma (t)^p\le f_\sigma (t^p) \text { for all } p>1, \end{aligned}$$then $$\sigma $$ is called a pmi mean. Our main interest is the class of pmi means (denoted by PMI). To study PMI, the operator means $$\sigma $$ satisfying $$\begin{aligned} f_\sigma (\sqrt{xy})\le \sqrt{f_\sigma (x)f_\sigma (y)}\quad (x,y>0) \end{aligned}$$are considered in this paper. The set of such means (denoted by GCV) includes certain significant examples and is included in PMI. The main result presented in this paper is that GCV is a proper subset of PMI. In addition, we investigate certain operator-mean classes including PMI.

Nielsen realization problem for the mapping class group $\text{Mod}(S_g)$ asks whether the natural projection $p_g: \text{Homeo}_+(S_g)\to \text{Mod}(S_g)$ has a section. While all the previous results use torsion elements in an essential way, in this paper, we focus on the much more difficult problem of realization of torsion-free subgroups of $\text{Mod}(S_g)$. The main result of this paper is that the Torelli group has no realization inside the area-preserving homeomorphisms., Comment: 22 pages, 5 figures

This paper counts the number of reduced quasigroup words of a particular length in a certain number of generators. Taking account of the relationship with the Catalan numbers, counting words in a free magma, we introduce the term peri-Catalan number for the free quasigroup word counts. The main result of this paper is an exact recursive formula for the peri-Catalan numbers, structured by the Euclidean Algorithm. The Euclidean Algorithm structure does not readily lend itself to standard techniques of asymptotic analysis. However, conjectures for the asymptotic behavior of the peri-Catalan numbers, substantiated by numerical data, are presented. A remarkable aspect of the observed asymptotic behavior is the so-called asymptotic irrelevance of quasigroup identities, whereby cancelation resulting from quasigroup identities has a negligible effect on the asymptotic behavior of the peri-Catalan numbers for long words in a large number of generators.

For graphs $G$ and $H$, a \emph{homomorphism} from $G$ to $H$ is an edge-preserving mapping from the vertex set of $G$ to the vertex set of $H$. For a fixed graph $H$, by \textsc{Hom($H$)} we denote the computational problem which asks whether a given graph $G$ admits a homomorphism to $H$. If $H$ is a complete graph with $k$ vertices, then \textsc{Hom($H$)} is equivalent to the $k$-\textsc{Coloring} problem, so graph homomorphisms can be seen as generalizations of colorings. It is known that \textsc{Hom($H$)} is polynomial-time solvable if $H$ is bipartite or has a vertex with a loop, and NP-complete otherwise [Hell and Ne\v{s}et\v{r}il, JCTB 1990]. In this paper we are interested in the complexity of the problem, parameterized by the treewidth of the input graph $G$. If $G$ has $n$ vertices and is given along with its tree decomposition of width $\mathrm{tw}(G)$, then the problem can be solved in time $|V(H)|^{\mathrm{tw}(G)} \cdot n^{\mathcal{O}(1)}$, using a straightforward dynamic programming. We explore whether this bound can be improved. We show that if $H$ is a \emph{projective core}, then the existence of such a faster algorithm is unlikely: assuming the Strong Exponential Time Hypothesis (SETH), the \textsc{Hom($H$)} problem cannot be solved in time $(|V(H)|-\epsilon)^{\mathrm{tw}(G)} \cdot n^{\mathcal{O}(1)}$, for any $\epsilon > 0$. This result provides a full complexity characterization for a large class of graphs $H$, as almost all graphs are projective cores. We also notice that the naive algorithm can be improved for some graphs $H$, and show a complexity classification for all graphs $H$, assuming two conjectures from algebraic graph theory. In particular, there are no known graphs $H$ which are not covered by our result. In order to prove our results, we bring together some tools and techniques from algebra and from fine-grained complexity., Comment: An extended abstract of this paper appeared on SODA 2020

© 2018, Springer Science+Business Media, LLC, part of Springer Nature. In this paper, we study properties of functions and sequences with a semi-heavy tail, that is, functions and sequences of the form w(x) = e−βxf(x), β > 0, resp., wn = cnfn, 0 < c < 1, where the function f(x), resp., the sequence (fn), is regularly varying. Among others, we give a representation theorem and study convolution properties. The paper includes several examples and applications in probability theory. ispartof: Lithuanian Mathematical Journal vol:58 issue:4 pages:480-499 status: published

Given a digital image (or digital object) (X,k),X⊂Zn, this paper initially establishes a group structure of the set of self-k-isomorphisms of (X,k) with the function composition, denoted by Isok(X) or Autk(X). In particular, let Ckn,l be a simple closed k-curve with l elements in Zn. Then, the group Isok(Ckn,l) is proved to be isomorphic to the standard dihedral group Dl with order l. The calculation of this quantity Isok(Ckn,l) is a key step for obtaining many new results. Indeed, it is essential for exploring many features of Isok(X). Furthermore, this quantity is proved to be a digital topological invariant. After proceeding with an Isok(X)-action on (X,k), we investigate some properties of fixed point sets by this action. Finally, we explore various features of fixed point sets by this action from the viewpoint of digital k-curve theory. This paper only deals with k-connected digital images (X,k) whose cardinality is equal to or greater than 2. Besides, we discuss some errors that have appeared in the lilterature.

Let q be a prime, P ⩾ 1 and let N q ( P ) denote the number of rational primes p ⩽ P that split in the imaginary quadratic field Q ( − q ) . The first part of this paper establishes various unconditional and conditional (under existence of a Siegel zero) lower bounds for N q ( P ) in the range q 1 / 4 + e ⩽ P ⩽ q , for any fixed e > 0 . This improves upon what is implied by work of Pollack and Benli–Pollack. The second part of this paper is dedicated to proving an estimate for a bilinear form involving Weyl sums for modular square roots (equivalently Salie sums). Our estimate has a power saving in the so-called Polya–Vinogradov range, and our methods involve studying an additive energy coming from quadratic residues in F q . This bilinear form is inspired by the recent automorphic motivation: the second moment for twisted L-functions attached to Kohnen newforms has recently been computed by the first and fourth authors. So the third part of this paper links the above two directions together and outlines the arithmetic applications of this bilinear form. These include the equidistribution of quadratic roots of primes, products of primes, and relaxations of a conjecture of Erdős–Odlyzko–Sarkozy.

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for [Formula: see text] knots (Tables A.1).

This paper is devoted to the study of groups from the category M of R-power groups. We examine problems on the commutation of the tensor completion with basic group operations and on the exactness of the tensor completion. Moreover, we introduce the notion of a variety and obtain a description of abelian varieties and some results on nilpotent varieties of A-groups. We prove the hypothesis on irreducible coordinate groups of algebraic sets for the nilpotent R-groups of nilpotency class 2, where R is a Euclidean ring. We state that the analog to the Lyndon result for the free groups (see [10]) holds in this case, whereas the analog to the Myasnikov–Kharlampovich result fails.The paper is dedicated to partial R-power groups which are embeddable to their A-tensor completions. The free R-groups and free R-products are described with usual group-theoretical free constructions.

Staircase tableaux are combinatorial objects that were first introduced due to a connection with the asymmetric simple exclusion process (ASEP) and Askey-Wilson polynomials. Since their introduction, staircase tableaux have been the object of study in many recent papers. Relevant to this paper, Hitczenko and Janson proved that distribution of parameters on the first diagonal is asymptotically normal. In addition, they conjectured that other diagonals would be asymptotically Poisson. Since then, only the second and the third diagonal were proven to follow the conjecture. This paper builds upon those results to prove the conjecture for the kth diagonal where k is fixed. In particular, we prove that the distribution of the number of α's (β's) on the kth diagonal, k > 1, is asymptotically Poisson with parameter 1/2. In addition, we prove that symbols on the kth diagonal are asymptotically independent and thus, collectively follow the Poisson distribution with parameter 1. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 795–818, 2016

In the present paper, we expand the domain of work on theconcept of semiderivations in 3-prime near-rings through the study ofstructure and commutativity of near-rings admitting semiderivations sat-isfying certain differential identities. Moreover, several examples havebeen provided at places which show that the assumptions in the hypothe-ses of various theorems are not altogether superfluous. 1. IntroductionThroughout this paper, N is a zero-symmetric left near ring. A near ringN is called zero symmetric if 0x= 0 for all x∈ N (recall that in a left nearring x0 = 0 for all x∈ N). N is called 3-prime if xNy = {0} implies x= 0or y = 0. The symbol Z(N) will represent the multiplicative center of N,that is, Z(N) = {x∈ N | xy= yxfor all y∈ N}.For any x,y∈ N; as usual[x,y] = xy−yxand x◦y= xy+yxwill denote the well-known Lie product andJordan product, respectively. Recall that N is called 2-torsion free if 2x= 0implies x= 0 for all x∈ N. For terminologies concerning near-rings we referto G. Pilz [7].An additive mapping d: N → N is said to be a derivation if d(xy) = xd(y)+d(x)yforall x,y∈ N, orequivalently, asnotedin [8], that d(xy) = d(x)y+xd(y)for all x,y ∈ N. An additive mapping d: N → N is called semiderivation ifthere exists a function g : N → N such that d(xy) = xd(y) + d(x)g(y) =g(x)d(y)+d(x)yand d(g(x)) = g(d(x)) for all x,y∈ N.Obviously, any deriva-tion is a semiderivation, but the converse is not true in general (see [6]). Therehas been a greatdeal of workconcerning derivations in near-rings(see [1, 2, 4, 5]where further references can be found). In this paper, we study the commuta-tivity of addition and multiplication of near-rings. Two well-known results forderivations in near-rings have been generalized for semiderivation. In fact, ourresults generalize some theorems obtained by the authors together with Rajiin [1].