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Abstract: This is the first of a series of papers where we develop a theory of total positivity for loop groups. In this paper, we completely describe the totally nonnegative part of the polynomial loop group , and for the formal loop group we describe the totally nonnegative points which are not totally positive. Furthermore, we make the connection with networks on the cylinder. Our approach involves the introduction of distinguished generators, called whirls and curls, and we describe the commutation relations amongst them. These matrices play the same role as the poles and zeros of the Edrei–Thoma theorem classifying totally positive functions (corresponding to our case ). We give a solution to the “factorization problem” using limits of ratios of minors. This is in a similar spirit to the Berenstein–Fomin–Zelevinsky Chamber Ansatz where ratios of minors are used. A birational symmetric group action arising in the commutation relation of curls appeared previously in Noumi–Yamada’s study of discrete Painlevé dynamical systems and Berenstein–Kazhdan’s study of geometric crystals. [Copyright &y& Elsevier]

Abstract: Let be i.i.d. random variables with partial sums , . The now classical Baum–Katz theorem provides necessary and sufficient moment conditions for the convergence of for fixed . An equally classical paper by Heyde in 1975 initiated what is now called precise asymptotics, namely asymptotics for the same sum (for the case and ) when, instead, . In this paper we extend a result due to Klesov (1994), in which he determined the convergence rate in Heydeʼs theorem. [Copyright &y& Elsevier]

Abstract: We review the non-trivial issue of the relativistic description of a quantum mechanical system that, contrary to a common belief, kept theoreticians busy from the end of 1920s to (at least) mid 1940s. Starting by the well-known works by Klein–Gordon and Dirac, we then give an account of the main results achieved by a variety of different authors, ranging from de Broglie to Proca, Majorana, Fierz–Pauli, Kemmer, Rarita–Schwinger and many others. A particular interest comes out for the general problem of the description of particles with arbitrary spin, introduced (and solved) by Majorana as early as 1932, and later reconsidered, within a different approach, by Dirac in 1936 and by Fierz–Pauli in 1939. The final settlement of the problem in 1945 by Bhabha, who came back to the general ideas introduced by Majorana in 1932, is discussed as well, and, by making recourse also to unpublished documents by Majorana, we are able to reconstruct the line of reasoning behind the Majorana and the Bhabha equations, as well as its evolution. Intriguingly enough, such an evolution was identical in the two authors, the difference being just the period of time required for that: probably few weeks in one case (Majorana), while more than ten years in the other one (Bhabha), with the contribution of several intermediate authors. The important unpublished contributions by Majorana anticipated later results obtained, in a more involved way, by de Broglie (1934) and by Duffin and Kemmer (1938-9), and testify the intermediate steps in the line of reasoning that led to the paper published in 1932 by Majorana, while Bhabha took benefit of the corresponding (later) published literature. Majorana’s paper of 1932, in fact, contrary to the more complicated Dirac–Fierz–Pauli formalism, resulted to be very difficult to fully understand (probably for its pregnant meaning and latent physical and mathematical content): as is clear from his letters, even Pauli (who suggested its reading to Bhabha) took about one year in 1940-1 to understand it. This just testifies for the difficulty of the problem, and for the depth of Majorana’s reasoning and results. The relevance for present day research of the issue here reviewed is outlined as well. [Copyright &y& Elsevier]

Abstract: In this paper, the multidimensional limiter for the second order finite volume schemes on the unstructured grid, namely the Weighted Biased Average procedure developed in our previous paper is extended to high order finite volume schemes solving hyperbolic conservation laws. This extension relies on two key techniques: the secondary reconstruction and the successive limiting procedure. These techniques are discussed in detail in the present paper. Numerical experiments shows that this limiting procedure is very effective in removing numerical oscillations in the vicinity of discontinuities. And furthermore this procedure is efficient, robust and accuracy preserving. [Copyright &y& Elsevier]

Abstract: In this paper, we study the stability of stationary solutions w for the Navier–Stokes flows in an exterior domain with zero velocity at infinity. With suitable assumptions of w, by the works of Chen (1993), Kozono–Ogawa (1994) and Borchers–Miyakawa (1995), if then one can obtain where is a solution of the Navier–Stokes equations with the initial condition . In this paper, we will prove that for any if belongs to then one has [Copyright &y& Elsevier]

Abstract: In this paper we prove for , where k is an integer in , the existence of an initial value ψ, odd with respect to the k first coordinates, and with , such that the resulting solution of is global. In the case and , it is known that the solution u with the initial value blows up in finite time if either sufficiently small or sufficiently large. The result in this paper extends a similar result of Cazenave, Dickstein, and Weissler in the case , i.e. with and . [Copyright &y& Elsevier]

Abstract: This paper deals with a generating function of the Jacobi polynomials that satisfies the following properties (I) and (II). (I) The generating function is the kernel of an integral operator that is unitary. (II) The image of the unitary operator is a reproducing kernel Hilbert space of analytic functions and the reproducing kernel is given as a special value of the generating function above. A generating function that satisfies (I) is given in Watanabe (1998) . The purpose of this paper is to give a generating function that satisfies (I) and (II). From a group theoretical point of view, a similar construction for zonal spherical functions is given in Watanabe (2006) . [Copyright &y& Elsevier]

Abstract: In this paper we characterize bounded differences of composition operators from standard weighted Bergman spaces into Lebesgue spaces when and . In the process we also find an interesting connection between the difference operator and certain weighted composition operators answering a question raised in Saukko (2011) . By combining the results of this paper with those of Saukko (2011) , we obtain a complete characterization of bounded and compact differences between standard weighted Bergman spaces. [Copyright &y& Elsevier]

Abstract: In the present paper we define Hardy spaces with variable exponents on by the grand maximal function, and then investigate their several properties. The present paper will connect harmonic analysis with function spaces with variable exponents. We obtain the atomic decomposition and the molecular decomposition. With these decomposition proved, we investigate the Littlewood–Paley characterization. Also, we specify the dual spaces of Hardy spaces with variable exponents. They will turn out to be Campanato spaces with variable growth conditions. The present paper covers local Hardy spaces with variable exponents. [Copyright &y& Elsevier]

Abstract: We investigate the problem of enhancing the stability of a coupled transport–diffusion system with Dirichlet actuation and Dirichlet measurement. In the recent paper [H. Sano, Neumann boundary control of a coupled transport–diffusion system with boundary observation, J. Math. Anal. Appl. 377 (2011) 807–816], we treated the stabilization problem for the case with Neumann actuation and Dirichlet measurement, where the variable transformation of the state is performed by using the fractional power of an unbounded operator. However, we cannot use the similar transformation for the case with Dirichlet actuation and Dirichlet measurement, since it brings an ill-posed expression of the system. So, we use an algebraic approach for the formulation of the system. In this paper, it is shown that a reduced-order model with a finite-dimensional state variable is controllable and observable. The fact enables us to construct a finite-dimensional stability-enhancing controller for the original infinite-dimensional system by using a residual mode filter (RMF) approach. The novelty of this paper is the structure that the controller contains the dynamics with respect to the control variable. As a result, the state vector of the resulting closed-loop system includes the control variable as its entry. [Copyright &y& Elsevier]

Abstract: This is the second in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin diagrams. This paper deals with dihedral groups for which all special CM modules have rank one, and we show that all but four of the relations on such a reconstruction algebra are given simply as the relations arising from a reconstruction algebra of type A. As a corollary, the reconstruction algebra reduces the problem of explicitly understanding the minimal resolution (=G-Hilb) to the same level of difficulty as the toric case. [Copyright &y& Elsevier]

Abstract: The commercial automotive mufflers are generally of a complicated shape with multiply connected parts and complex acoustic elements. The analysis of such complex mufflers has always been a great challenge. In this paper, an Integrated Transfer Matrix method has been developed to analyze complex mufflers. Integrated transfer matrix relates the state variables across the entire cross-section of the muffler shell, as one moves along the axis of the muffler, and can be partitioned appropriately in order to relate the state variables of different tubes constituting the cross-section. The paper presents a generalized one-dimensional (1-D) approach, using the transfer matrices of simple acoustic elements, which are available from the literature. The present approach is robust and flexible owing to its capability to construct an overall matrix of the muffler with the transfer matrices of individual acoustic elements and boundary conditions, which can then be used to evaluate the transmission loss, insertion loss, etc. Results from the present approach have been validated through comparisons with the available experimental and three-dimensional finite element method (FEM) based results. The results show good agreement with both measurements and FEM analysis up to the cut-off frequency. [Copyright &y& Elsevier]

Abstract: In a recent paper (Buium et al., 2011 ), Buium et al. proved that f is a locally analytic function from the p-adic integers, to itself if and only if it is written as a restricted power series over with finitely many iterates of the Fermat quotient operator. In this paper we establish the function field analogs of their result. In addition, we deduce Wagnerʼs result, which is the function field analog of Mahlerʼs result for continuous functions on . [Copyright &y& Elsevier]

Abstract: In recent papers we have studied refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows. The present paper is a first step towards extending these considerations to alternating sign matrices where in addition some left and right columns are fixed. The main result is a simple linear relation between the number of alternating sign matrices where the top row as well as the left and the right column is fixed and the number of alternating sign matrices where the two top rows and the bottom row are fixed. This may be seen as a first indication for the fact that the refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows as well as left and right columns can possibly be reduced to the refined enumerations where only some top and bottom rows are fixed. For the latter numbers we provide a system of linear equations that conjecturally determines them uniquely. [Copyright &y& Elsevier]

Abstract: Image inpainting has been widely used in practice to repair damaged/missing pixels of given images. Most of the existing inpainting techniques require knowing beforehand where those damaged pixels are, either given as a priori or detected by some pre-processing. However, in certain applications, such information neither is available nor can be reliably pre-detected, e.g. removing random-valued impulse noise from images or removing certain scratches from archived photographs. This paper introduces a blind inpainting model to solve this type of problems, i.e., a model of simultaneously identifying and recovering damaged pixels of the given image. A tight frame based regularization approach is developed in this paper for such blind inpainting problems, and the resulted minimization problem is solved by the split Bregman algorithm first proposed by Goldstein and Osher (2009) . The proposed blind inpainting method is applied to various challenging image restoration tasks, including recovering images that are blurry and damaged by scratches and removing image noise mixed with both Gaussian and random-valued impulse noise. The experiments show that our method is compared favorably against many available two-staged methods in these applications. [Copyright &y& Elsevier]

Abstract: Delay tolerant networks (DTNs) are a special type of wireless mobile networks which may lack continuous network connectivity. Multicast is an important routing function that supports the distribution of data to a group of users: a service needed for many potential DTN applications. While multicasting in the Internet and in mobile ad hoc networks has been studied extensively, efficient multicasting in DTNs is a considerably different and challenging problem due to the probabilistic nature of contact among nodes. This paper aims to provide a non-replication multicasting scheme in DTNs while keeping the number of forwardings low. The address of each destination is not replicated, but is assigned to a particular node based on its contact rate level and active level. Our scheme is based on a dynamic multicast tree where each leaf node corresponds to a destination. Each tree branch is generated at a contact based on the compare–split rule proposed in this paper. The compare part determines when a new search branch is needed, and the split part decides how the destination set should be partitioned. When only one destination is left in the destination set, we use either wait (no further relay) or focus (with further relay) to reach the final destination. The effectiveness of our approach is verified through extensive simulations. Ratio-based-split performs best in the compare–split step, both in synthetic and real traces. Using the wait scheme can reduce the number of forwardings, while using the focus scheme can reduce the latency. [Copyright &y& Elsevier]

Abstract: Text: Let be the set of all positive integers pqr, where p, q, r are primes such that at least two of them are not equal. Denote by the largest prime factor of n. For , define . In 2006, Wushi Goldring proved that for any , there exists an i such that , where and (). If , then m is called a parent of n. Let be the set of all positive integers of . In this paper, we study the function w extensively. For example, one of our results is that there exist infinitely many which have at least parents in . Several open problems are posed. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=8FWvR8_KoHA. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to study sustainability of the pseudo-linear superposition principle, that is, of the nonlinear superposition principle (NLSP) observed in the pseudo-analysisʼ framework. The focus of this paper is on the so-called generated partial differential equation that is, for the purpose of this study, based on the stochastic Burgerʼs equation. [Copyright &y& Elsevier]

Abstract: On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. In each case, there are three possible interpretations for the double series. In an earlier paper, two of the present authors proved the first identity under one possible interpretation. In the present paper, the second identity is proved under a similar interpretation, with one additional assumption. Moreover, under a second interpretation, entirely different proofs of both identities, depending on weighted (or twisted) divisor sums, are offered. The two identities are intimately connected with the classical circle and divisor problems, respectively. [Copyright &y& Elsevier]

Abstract: Let denote the maximal number of limit cycles of polynomial systems of degree m. It is called the Hilbert number. The main part of Hilbertʼs 16th problem posed in 1900 is to find its value. The problem is still open even for . However, there have been many interesting results on the lower bound of it for . In this paper, we give some new lower bounds of this number. The results obtained in this paper improve all existing results for all based on some known results for . In particular, we obtain that grows at least as rapidly as for all large m. [Copyright &y& Elsevier]

Abstract: Let be a metric measure space satisfying the upper doubling and the geometrically doubling conditions in the sense of T. Hytönen. In this paper, the authors prove that the boundedness of a Calderón–Zygmund operator T on is equivalent to either of the boundedness of T from the atomic Hardy space to or from to . To this end, the authors first establish an interpolation result that a sublinear operator which is bounded from to and from to for some is also bounded on for all . A main tool used in this paper is the Calderón–Zygmund decomposition in this setting established by B.T. Anh and X.T. Duong. [Copyright &y& Elsevier]

Abstract: In a recent paper, we proved that assuming some initial data lead to a singularity for the 3D Navier–Stokes equations, there are also initial data with the minimal -norm which produce a singularity and the set of such data is compact up to translations and the natural scaling of the equation. The purpose of this paper is to analyze a more general case where the set of initial data potentially leading to a singularity is on a sphere centered at non-zero initial data leading to a global solution. [Copyright &y& Elsevier]

Abstract: Text: In this paper we investigate higher order dimensional Dedekind–Rademacher sums given by the expression where is a positive integer, are positive integers prime to and are non-negative integers. We study arithmetical properties of these sums. For instance, we prove that these sums are rational numbers and we explicit good bounds for their denominators. A reciprocity law is given generalizing a theorem of Rademacher for the classical Dedekind sums and a theorem of Zagier for higher dimensional Dedekind–Rademacher sums. On the other hand, our reciprocity results can be viewed as complements to the Beck reciprocity theorem. Taking , we derive the reciprocity and rationality theorems of Zagier. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=J1_5H28fgAg. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence and stability of periodic pyramidal traveling fronts for reaction–diffusion equations with bistable time-periodic nonlinearity in with . It is well known that two-dimensional periodic traveling curved fronts exist and are stable. In this paper, by constructing various of supersolutions and subsolutions, we first show that there exist three-dimensional periodic pyramidal traveling fronts, and then we prove that such periodic pyramidal traveling fronts are asymptotically stable. Finally, we further prove that our existence result holds for with . [Copyright &y& Elsevier]

Abstract: Stability in distribution of stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching have been studied by several authors and this kind of stability is an important property for stochastic systems. There are several papers which study this stability for stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching technically. In our paper, we are concerned with the general neutral stochastic functional differential equations with Markovian switching and we derive the sufficient conditions for stability in distribution. At the end of our paper, one example is established to illustrate the theory of our work. [Copyright &y& Elsevier]

Abstract: This paper extends recent developments on spectral sequence approach on the simplest normal form theory to parametric cases. In this paper, the method is first applied to two simple examples (parametric single zero and parametric resonant saddle singularities) and then, to parametric generalized Hopf singularity. We provide the simplest parametric normal form for these cases. [Copyright &y& Elsevier]

Abstract: In this paper, we will present a generalization for a minimization problem from I. Daubechies, M. Defrise, and C. DeMol (2004) . This generalization is useful for solving many practical problems in which more than one constraint are involved. In this regard, we will conclude the findings of many papers (most of which are on image processing) from this generalization. It is hoped that the approach proposed in this paper will be a suitable reference for some applied works where multi-frames, multi-wavelets, or multi-constraints are present in linear inverse problems. [Copyright &y& Elsevier]

Abstract: In a recent paper (Cucker et al., 2008 ) we analyzed a numerical algorithm for computing the number of real zeros of a polynomial system. The analysis relied on a condition number for the input system f. In this paper we look at as a random variable derived from imposing a probability measure on the space of polynomial systems and give bounds for both the tail and the expected value . [Copyright &y& Elsevier]

Abstract: For a 2-connected matroid M, Cunningham and Edmonds gave a tree decomposition that displays all of its 2-separations. When M is 3-connected, two 3-separations are equivalent if one can be obtained from the other by passing through a sequence of 3-separations each of which is obtained from its predecessor by moving a single element from one side of the 3-separation to the other. Oxley, Semple, and Whittle gave a tree decomposition that displays, up to this equivalence, all non-trivial 3-separations of M. Now let M be 4-connected. In this paper, we define two 4-separations of M to be 2-equivalent if one can be obtained from the other by passing through a sequence of 4-separations each obtained from its predecessor by moving at most two elements from one side of the 4-separation to the other. The main result of the paper proves that M has a tree decomposition that displays, up to 2-equivalence, all non-trivial 4-separations of M. [Copyright &y& Elsevier]

Abstract: Let n and k be positive integers, and let F be an alphabet of size n. A sequence over F of length m is a k-radius sequence if any two distinct elements of F occur within distance k of each other somewhere in the sequence. These sequences were introduced by Jaromczyk and Lonc in 2004, in order to produce an efficient caching strategy when computing certain functions on large data sets such as medical images. Let be the length of the shortest n-ary k-radius sequence. The paper shows, using a probabilistic argument, that whenever k is fixed and The paper observes that the same argument generalises to the situation when we require the following stronger property for some integer t such that : any t distinct elements of F must simultaneously occur within a distance k of each other somewhere in the sequence. [Copyright &y& Elsevier]

In this paper, we calculate the numbers of irreducible ordinary characters and irreducible Brauer characters in a block of a finite group G , whose associated fusion system over a 2-subgroup P of G (which is a defect group of the block) has an abelian hyperfocal subgroup of rank 3. [ABSTRACT FROM AUTHOR]

Abstract This paper classifies the factorizations of almost simple groups with a factor having at least two nonsolvable composition factors. This together with a previous classification result of the authors reduces the factorization problem of almost simple groups to the case where both factors have a unique nonsolvable composition factor. [ABSTRACT FROM AUTHOR]

Abstract In a recent paper, [14] , S. Doty and J. M. Douglass consider the degree r component L r (E) of the free Lie algebra L (E) on an n -dimensional vector space E over a suitable field k , of characteristic not dividing r. The authors discuss, amongst other things, the question of when this space, regarded as a module for the general linear group of degree n (equivalently the Schur algebra S (n , r)) has the double centraliser property. Some general classes are described but the question in general is left open. Here we show that at most one block component of L r (E) can fail to have the double centraliser property and give some examples of the failure of the property for that block (and therefore for the Lie module). [ABSTRACT FROM AUTHOR]

Abstract This paper is devoted to studying the modified b-family of equations with cubic nonlinearity, called the b -family of FORQ/MCH equations, which includes the cubic Camassa–Holm equation (also called Fokas–Olver–Rosenau–Qiao equation) as a special case. We first study the local well-posedness for the Cauchy problem of the equation, and then make good use of fine structure of the equation, we derive the precise blow-up scenario and a new blow-up result with respect to initial data. Finally, peakon solutions are derived. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is to establish Bogoliubov averaging principle of stochastic reaction–diffusion equation with a stochastic process and a small parameter. The solutions to stochastic reaction–diffusion equation can be approximated by solutions to averaged stochastic reaction–diffusion equation in the sense of convergence in probability and in distribution. Namely, we establish a weak law of large numbers for the solution of stochastic reaction–diffusion equation. [ABSTRACT FROM AUTHOR]

Abstract Introduced by Anglès, Pellarin, and Tavares Ribeiro, Drinfeld modules over Tate algebras are closely connected to Anderson log-algebraicity identities, Pellarin L -series, and Taelman class modules. In the present paper we define the de Rham map for Drinfeld modules over Tate algebras, and we prove that it is an isomorphism under natural hypotheses. As part of this investigation we determine further criteria for the uniformizability and rigid analytic triviality of Drinfeld modules over Tate algebras. [ABSTRACT FROM AUTHOR]

Abstract Given a commutative ring R with unity, we prove in this paper that every coordinate of R [ 2 ] is stably tame. [ABSTRACT FROM AUTHOR]

Highlights • We propose a variable weight test for both rare and common variants association based on affected sib pairs. • The variable weight structure makes it possible to analyze rare and common variants simultaneously. • The new method can improve the power of simultaneously detecting rare and common variants. • The proposed method is more powerful than the existing and related methods for detecting effects of rare and common variants. Abstract Current genome-wide association analysis has identified a great number of rare and common variants associated with common complex traits, however, more effective approaches for detecting associations between rare and common variants with common diseases are still demanded. Approaches for detecting rare variant association analysis will compromise the power when detecting the effects of rare and common variants simultaneously. In this paper, we extend an existing method of testing for rare variant association based on affected sib pairs (TOW-sib) and propose a variable weight test for rare and common variants association based on affected sib pairs (abbreviated as VW-TOWsib). The VW-TOWsib can be used to achieve the purpose of detecting the association of rare and common variants with complex diseases. Simulation results in various scenarios show that our proposed method is more powerful than existing methods for detecting effects of rare and common variants. At the same time, the VW-TOWsib also performs well as a method for rare variant association analysis. [ABSTRACT FROM AUTHOR]

Abstract We discuss a purely variational approach to the study of a wide class of second order nonhomogeneous dissipative hyperbolic PDEs. Precisely, we focus on the wave-like equations that present also a nonzero source term and a first-order-in-time linear term. The paper carries on the research program initiated in [14] , and developed in [15,21] , on the De Giorgi approach to hyperbolic equations. [ABSTRACT FROM AUTHOR]

Abstract Let n ≥ 2 and let A ∈ L (C n) be such that m (A) > 0. In this paper, we use a variational result for A -normalized univalent subordination chains, to deduce that every normalized univalent mapping which has A -parametric representation on B n can be approximated locally uniformly on B n by mappings which have A -parametric representation on B n and admit extensions, on one hand, to automorphisms of C n and, on the other hand, to quasiconformal diffeomorphisms of class C ∞ from C n onto C n , which are equal to the identity mapping in a neighborhood of ∞. Finally, we obtain approximation results of A -spirallike, starlike, and convex mappings by automorphisms of C n and quasiconformal diffeomorphisms of class C ∞ from C n onto itself, which are equal to the identity mapping in a neighborhood of ∞ , whose restrictions to B n have the same geometric property. We remark that even though these extensions exist for the same mapping on B n , they are not necessarily equal on C n ∖ B n ¯. [ABSTRACT FROM AUTHOR]

Abstract Let F (z) be a Hecke–Maass form for S L (3 , Z) (or S L (2 , Z)), A F (n , 1) (or A F (n)) be its Fourier coefficients, and L (s , F) the Godement–Jacquet L -function attached to F. And let μ F (n) be the coefficients of the reciprocal of L (s , F) in the absolutely convergent region ℜ s > 1. In this paper we prove that μ F (n) is strongly orthogonal to 1-step nilsequence. As applications, we prove that the Möbius function μ (n) is strongly orthogonal to A F (n , 1) e (n α) and A F (n) 2 e (n α). More precisely, we establish ∑ n ≤ N μ (n) A F (n , 1) e (n α) ≪ c N exp ⁡ (− c log ⁡ N) , and ∑ n ≤ N μ (n) A F (n) 2 e (n α) ≪ A N (log ⁡ N) − A. Moreover, we also show that μ F (n) is asymptotically orthogonal to any A C 0 (d) function, and all MMA and MMLS flows. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we seek to evaluate the most general form of moments of L (1 , χ) twisted by character value χ (c) as χ ranges over non-principal Dirichlet characters mod f. And we use this result to model the joint distribution for χ (c) and L (1 , χ). [ABSTRACT FROM AUTHOR]

Abstract In this paper, we discuss the variation of the numbers of the isomorphic classes of stable lattices when the weight and the level vary in a Hida deformation by using the Kubota–Leopoldt p -adic L -function. Then in Corollary 1.7 , we give a sufficient condition for the numbers of the isomorphic classes of stable lattices in Hida deformation to be infinite. [ABSTRACT FROM AUTHOR]

Abstract Let F be a family of 3-uniform linear hypergraphs. The linear Turán number of F is the maximum possible number of edges in a 3-uniform linear hypergraph on n vertices which contains no member of F as a subhypergraph. In this paper we show that the linear Turán number of the five cycle C 5 (in the Berge sense) is 1 3 3 n 3 / 2 asymptotically. We also show that the linear Turán number of the four cycle C 4 and { C 3 , C 4 } are equal asymptotically, which is a strengthening of a theorem of Lazebnik and Verstraëte [16]. We establish a connection between the linear Turán number of the linear cycle of length 2 k + 1 and the extremal number of edges in a graph of girth more than 2 k − 2. Combining our result and a theorem of Collier-Cartaino, Graber and Jiang [8] , we obtain that the linear Turán number of the linear cycle of length 2 k + 1 is Θ (n 1 + 1 k ) for k = 2 , 3 , 4 , 6. [ABSTRACT FROM AUTHOR]

Abstract The 'full' edge isoperimetric inequality for the discrete cube { 0 , 1 } n (due to Harper, Lindsey, Berstein and Hart) specifies the minimum size of the edge boundary ∂ A of a set A ⊂ { 0 , 1 } n , as function of | A |. A weaker (but more widely-used) lower bound is | ∂ A | ≥ | A | log ⁡ (2 n / | A |) , where equality holds whenever A is a subcube. In 2011, the first author obtained a sharp 'stability' version of the latter result, proving that if | ∂ A | ≤ | A | (log ⁡ (2 n / | A |) + ϵ) , then there exists a subcube C such that | A Δ C | / | A | = O (ϵ / log ⁡ (1 / ϵ)). The 'weak' version of the edge isoperimetric inequality has the following well-known generalization for the ' p -biased' measure μ p on the discrete cube: if p ≤ 1 / 2 , or if 0 < p < 1 and A is monotone increasing, then p μ p (∂ A) ≥ μ p (A) log p ⁡ (μ p (A)). In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if p μ p (∂ A) ≤ μ p (A) (log p ⁡ (μ p (A)) + ϵ) , then there exists a subcube C such that μ p (A Δ C) / μ p (A) = O (ϵ ′ / log ⁡ (1 / ϵ ′)) , where ϵ ′ : = ϵ ln ⁡ (1 / p). This result is a central component in recent work of the authors proving sharp stability versions of a number of Erdős–Ko–Rado type theorems in extremal combinatorics, including the seminal 'complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the 'full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the 'full' edge isoperimetric inequality, one relying on the Kruskal–Katona theorem. [ABSTRACT FROM AUTHOR]

Abstract The size ‖ σ ‖ of a split σ (a bipartition) of the leaf set of a tree is the cardinality of the smaller part. There exist many studies of properties of the phylogenetic tree shapes (trees with no vertex of degree 2) and general trees related to the split sizes. A general function Φ f (T) = ∑ σ ∈ Σ ⁎ (T) f (‖ σ ‖) was proposed for an (strictly) increasing function f , where the sum is over all non-trivial splits induced by internal edges. There are many interesting applications of this important concept. In particular, Φ f (⋅) was used to measure the balance of phylogenetic tree shapes. Extremal problems for Φ f (⋅) have been considered and partially answered for binary trees. In this paper, we study the extremal trees with a given degree sequence that maximize or minimize Φ f (⋅). These results can also be directly applied to phylogenetic tree shapes and their degree sequences. When the number of leaves is fixed, we can also compare the extremal trees of different degree sequences. This comparison is conducted for degree sequences of phylogenetic tree shapes with given number of leaves. Immediate consequences are shown. Related problems are also proposed for potential future work. [ABSTRACT FROM AUTHOR]

Abstract In this paper we give a new characterization of the dual polar graphs, extending the work of Brouwer and Wilbrink on regular near polygons. Also as a consequence of our characterization we confirm a conjecture of the authors on non-bipartite distance-regular graphs with smallest eigenvalue at most − k / 2 , where k is the valency of the distance-regular graph, in the case where c 2 ≥ 3 and a 1 = 1. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we describe the K -module H H 1 (L K (Γ)) of outer derivations of the Leavitt path algebra L K (Γ) of a row-finite graph Γ with coefficients in an associative commutative ring K with unit. We explicitly describe a set of generators of H H 1 (L K (Γ)) and relations among them. We also describe a Lie algebra structure of outer derivation algebra of the Toeplitz algebra. We prove that every derivation of a Leavitt path algebra can be extended to a derivation of the corresponding C ⁎ -algebra. [ABSTRACT FROM AUTHOR]

Abstract Olivieri, del Río and Simón defined strongly monomial groups and a significant result proved by them is the explicit description of the simple components of the rational group algebra Q G of a strongly monomial group G. In this paper, generalized strongly monomial groups are defined, which is a generalization of strongly monomial groups. Beside strongly monomial groups, an extensive list of important families of groups which are generalized strongly monomial is produced. A description of the simple components of Q G , when G is a generalized strongly monomial group, is also provided. Furthermore, the work by Jespers, Olteanu, del Río and Van Gelder on the construction of a subgroup of finite index in the group of central units of integral group ring of a strongly monomial group has also been generalized. [ABSTRACT FROM AUTHOR]

Abstract Let l be a prime number. In the present paper, we discuss a pro-l version of the congruence subgroup problem for mapping class groups of genus one. Our main result is that the pro-2 version has an affirmative answer, but the pro- l version for l ≥ 11 has a negative answer. [ABSTRACT FROM AUTHOR]