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A net $(x_\alpha)$ in a vector lattice $X$ is unbounded order convergent to $x \in X$ if $\lvert x_\alpha - x\rvert \wedge u$ converges to $0$ in order for all $u\in X_+$. This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net $(x_\alpha)$ in a Banach lattice $X$ is unbounded norm convergent to $x$ if $\lVert\lvert x_\alpha - x\rvert \wedge u\rVert\to 0$ for all $u\in X_+$. We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.

We revive the concept of Lambda-freeness of Mlotkowski, which describes a mixture of classical and free independence between algebras of random variables. In particular, we give a description of this in terms of cumulants; this will be instrumental in the subsequent paper [SW] where the quantum symmetries underlying these mixtures of classical and free independences will be considered., Comment: We rewrote and shortened the earlier version. The third version contains mainly the results which are new compared to the paper of Mlotkowski

This is the first in a series of two papers concerned with relative birational geometry of algebraic spaces. In this paper, we study Prufer spaces and Prufer pairs of algebraic spaces that generalize spectra of Prufer rings. As a particular case of Prufer spaces we introduce valuation algebraic spaces, and use them to establish valuative criteria of separatedness and properness that sharpen the standard criteria. In a sequel paper, we introduce a version of Riemann–Zariski spaces, and prove Nagata’s compactification theorem for algebraic spaces.

In this paper we assign, under reasonable hypothesis, to each pseudomanifold with isolated singularities a rational Poincar\'e duality space. These spaces are constructed with the formalism of intersections spaces defined by Markus Banagl and are indeed related to them in the even dimensional case., Comment: New version of the paper formerly known as "Spatialization of self dual complexes for spaces with isolated singularities". Final version to appear in manuscripta mathematica

The class of all locally quasi-convex (lqc) abelian groups contains all locally convex vector spaces (lcs) considered as topological groups. Therefore it is natural to extend classical properties of locally convex spaces to this larger class of abelian topological groups. In the present paper we consider the following well known property of lcs: “A metrizable locally convex space carries its Mackey topology ”. This claim cannot be extended to lqc-groups in the natural way, as we have recently proved with other coauthors (Ausenhofer and de la Barrera Mayoral in J Pure Appl Algebra 216(6):1340–1347, 2012; Diaz Nieto and Martin Peinador in Descriptive Topology and Functional Analysis, Springer Proceedings in Mathematics and Statistics, Vol 80 doi:10.1007/978-3-319-05224-3_7, 2014; Dikranjan et al. in Forum Math 26:723–757, 2014). We say that an abelian group G satisfies the Varopoulos paradigm (VP) if any metrizable locally quasi-convex topology on G is the Mackey topology. In the present paper we prove that in any unbounded group there exists a lqc metrizable topology that is not Mackey. This statement (Theorem C) allows us to show that the class of groups satisfying VP coincides with the class of finite exponent groups. Thus, a property of topological nature characterizes an algebraic feature of abelian groups.

This paper explores the consistency strength of The Proper Forcing Axiom ($\textsf{PFA}$) and the theory (T) which involves a variation of the Viale-Wei$\ss$ guessing hull principle. We show that (T) is consistent relative to a supercompact cardinal. The main result of the paper implies that the theory "$\sf{AD}$$_\mathbb{R} + \Theta$ is regular" is consistent relative to (T) and to $\textsf{PFA}$. This improves significantly the previous known best lower-bound for consistency strength for (T) and $\textsf{PFA}$, which is roughly "$\sf{AD}$$_\mathbb{R} + \textsf{DC}$".

This paper proves that the nearby cycles complexes on a certain family of PEL local models are central with respect to the convolution product of sheaves on the corresponding affine flag varieties. As a corollary, the semisimple trace functions defined using the action of Frobenius on those nearby cycles complexes are, via the sheaf-function dictionary, in the centers of the corresponding Iwahori–Hecke algebras. This is commonly referred to as Kottwitz’s conjecture. The reductive groups associated with the PEL local models under consideration are unramified unitary similitude groups with even dimension. The proof follows the method of Haines and Ngo (Compos Math 133:117–150, 2002). Upon completion of the first version of this paper, Pappas and Zhu released a preprint (now published as Pappas and Zhu in Invent Math 194(1):147–254, 2013) which contained within its scope the main theorem of this paper. However, the methods of Pappas and Zhu (2013) are very different, and some of the proofs from this paper have been useful in forthcoming work of Haines–Stroh.

In this paper, we introduce a cyclic subgradient extragradient algorithm and its modified form for finding a solution of a system of equilibrium problems for a class of pseudomonotone and Lipschitz-type continuous bifunctions. The main idea of these algorithms originates from several previously known results for variational inequalities. The proposed algorithms are extensions of the subgradient extragradient method for variational inequalities to equilibrium problems and the hybrid (outer approximation) method. The paper can help in the design and analysis of practical algorithms and gives us a generalization of the most convex feasibility problems.

In a previous paper by the authors, the existence of Haar projections with growing norms in Sobolev–Triebel–Lizorkin spaces has been shown via a probabilistic argument. This existence was sufficient to determine the precise range of Triebel–Lizorkin spaces for which the Haar system is an unconditional basis. The aim of the present paper is to give simple deterministic examples of Haar projections that show this growth behavior in the respective range of parameters.

This paper discusses Aspects and “Grundvorstellungen” in the development of the concepts of derivative and integral, which are considered central to the teaching of calculus in senior high school. We focus on perspectives that are relevant when these concepts are first introduced. In the context of a subject-matter didactical debate, the ideas are separated into two classes: firstly, mathematically motivated aspects, such as the limit of difference quotients or local linearization within the concept of derivative, as well as the product sum, antiderivative, and measure aspects of integration; secondly, the “Grundvorstellungen” associated with the concepts of derivative and integral. We regard finding a comprehensive description of Aspects and “Grundvorstellungen” an important objective of subject matter didactics. This description should clarify both the differences and the relationships between these perspectives, including mathematically motivated Aspects and “Grundvorstellungen” that are central to the students’ perspective. The primary objectives of this paper include a specification of the concepts of Aspects and “Grundvorstellungen” within the context of differentiation and integration, and a discussion of the relationships between the Aspects and “Grundvorstellungen” associated with the concepts of derivative and integral. We first present the characteristic properties of Aspects and “Grundvorstellungen”, including an account of related concepts and the current state of research. Aspects and “Grundvorstellungen” are analyzed, based on a subject-matter didactical analysis of the concepts of derivative and integral. We conclude with an account of how these insights can be beneficially exploited for introducing differentiation and integration in real-life environments, within the framework of a theory of concept understanding and subject matter didactics.

We study etale extensions of rings that have FIP., Comment: The paper entitled FIP and FCP products of ring morphisms (arXiv: 1312.1250 [math.AC]) is now split into three papers. The present paper contains the last section of the original paper and many other results on etale FIP extensions

In this paper we will show that certain universal homology classes which are fundamental in topology are algebraic. To be specific, the products of Eilenberg–MacLane spaces \({\mathcal K}_{2q}\equiv K(\mathbf Z,{2}) \times K(\mathbf Z,{4}) \times \cdots \times K(\mathbf Z,{2q}) \) have models which are limits of complex projective varieties. Precisely, we have \({\mathcal K}_{2q}= \varinjlim \nolimits _{d\rightarrow \infty }\mathcal C^{q}_{d}(\mathbf P^{n})\) where \(\mathcal C^{q}_{d}(\mathbf P^{n})\) denotes the Chow variety of effective cycles of codimension q and degree d on \(\mathbf P_{\mathbf C}^{n}\). It is natural to ask which elements in the homology of \({\mathcal K}_{2q}\) are represented by algebraic cycles in these approximations. In this paper we find such representations for the even dimensional classes which are known as Steenrod squares (as well as their Pontrjagin and join products). These classes are dual to the cohomology classes which correspond to the basic cohomology operations also known as the Steenrod squares.

In our preceding paper a generating set of the derived Picard group of a selfinjective Nakayama algebra was constructed combining some previous results for Brauer tree algebras and the technique of orbit categories developed there. In this paper we finish the computation of the derived Picard group of a selfinjective Nakayama algebra.

In a series of papers, Santilli and collaborators released various strong statements against the general theory of relativity (GTR) and the standard ΛCDM model of cosmology. In this paper we show that such claims are due to misunderstandings of basic concepts of gravitation and cosmology. In particular, we show that Santilli and collaborators demonstrated neither that the GTR is wrong, nor that the Universe is not expanding. We also show that the so-called iso-gravitation theory (IGT) of Santilli is in macroscopic contrast with geodesic motion and, in turn, with the Equivalence Principle (EP) and must therefore be ultimately rejected. Finally, we show that, although the so called iso-redshift could represent an interesting alternative (similar to the tired light theory historically proposed by Zwicky) to the Universe expansion from a qualitative point of view, it must be rejected from a quantitative point of view because the effect of iso-redshift is 10−6 smaller than the effect requested to achieve the cosmological redshift.

In this paper, we consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides if the region contains eigenvalue(s) or not. It is particularly suitable to test if zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples., Comment: The paper has been accepted for publication in SCIENCE CHINA Mathematics

The Farey sequence of order n consists of all reduced fractions a / b between 0 and 1 with positive denominator b less or equal to n. The sums of the inverse denominators 1 / b of the Farey fractions in prescribed intervals with rational bounds have a simple main term, but the deviations are determined by an interesting sequence of polygonal functions \(f_n\). In a former paper we obtained a limit function for \(n \rightarrow \infty \), which describes a scaling behaviour of the functions \(f_n\) in the vicinity of any fixed rational number a / b and which is independent of a / b. In this paper we prove that \(f_n(a/b)\) tends to zero for \(n \rightarrow \infty \) by using elementary representation formulas for \(f_n(a/b)\) as well as a variant of the prime number theorem. An application of this result immediately gives a global version of the scaling behaviour of the functions \(f_n\) around the rational numbers.

We investigate the possible values for geometric genera of subvarieties in a smooth projective variety. Values which are not attained are called gaps. For curves on a very general surface in $\mathbb{P}^3$, the initial gap interval was found by Xu (see [7] in References), and the next one in our previous paper (see [4] in References), where also the finiteness of the set of gaps was established and an asymptotic upper bound of this set was found. In the present paper we extend some of these results to smooth projective varieties of arbitrary dimension using a different approach., 9 pages, submitted preprint

Group algebras have been used in the context of Coding Theory since the beginning of the latter, but not in its full power. The article of Ferraz and Polcino Milies entitled Idempotents in group algebras and minimal abelian codes (Finite Fields Appl 13(2):382–393, 2007) gave origin to many thesis and papers linking these two subjects. In these works, the techniques of group algebras are mainly brought into play for the computing of the idempotents that generate the minimal codes and the minimum weight of such codes. In this paper I present a survey on the main results proceeding from applications of that seminal work.

Anamorphosis, as a drawing, represents shapes on a surface such that they appear in their natural form only under specific viewing conditions. Although anamorphoses are mainly studied in a historical context, they are currently experiencing a revival. Plane geometric anamorphoses are a specific sub-type of anamorphic drawings. Some practical problems may arise during the design and realization of plane geometric anamorphosis causing the 3D illusionistic effect to be impaired. The aim of this paper is to identify and analyze these problems. In the paper we use parametric analysis to quantify the distortion that may appear because the point of view is offset from the preferred point of view, and to simulate the deviations that can appear because of the errors in onsite realization. The analysis leads to a framework for the design of plane geometric anamorphosis that minimizes the impairment of the anamorphic illusion.

In this paper, an intersective polynomial is a monic polynomial in one variable with rational integer coefficients, with no rational root and having a root modulo m for all positive integers m. Let G be a finite noncyclic group and let r(G) be the smallest number of irreducible factors of an intersective polynomial with Galois group G over \(\mathbb {Q}\). Let s(G) be smallest number of proper subgroups of G having the property that the union of their conjugates is G and the intersection of all their conjugates is trivial. It is known that \(s(G)\le r(G)\). It is also known that if G is realizable as a Galois group over the rationals, then it is also realizable as the Galois group of an intersective polynomial. However it is not known, in general, whether there exists such a polynomial which is a product of the smallest feasible number s(G) of irreducible factors. In this paper, we study the case \(G=S_n\), the symmetric group on n letters. We prove that for every n, either \(r(S_n)=s(S_n)\) or \(r(S_n)=s(S_n)+1\) and that the optimal value \(s(S_n)\) is indeed attained for all odd n and for some even n. Moreover, we compute \(r(S_n)\) when n is the product of at most two odd primes and we give general upper and lower bounds for \(r(S_n)\).

In this paper we give an exponential lower bound for Cunningham's least recently considered (round-robin) rule as applied to parity games, Markov decision processes and linear programs. This improves a recent subexponential bound of Friedmann for this rule on these problems. The round-robin rule fixes a cyclical order of the variables and chooses the next pivot variable starting from the previously chosen variable and proceeding in the given circular order. It is perhaps the simplest example from the class of history-based pivot rules. Our results are based on a new lower bound construction for parity games. Due to the nature of the construction we are also able to obtain an exponential lower bound for the round-robin rule applied to acyclic unique sink orientations of hypercubes (AUSOs). Furthermore these AUSOs are realizable as polytopes. We believe these are the first such results for history based rules for AUSOs, realizable or not. The paper is self-contained and requires no previous knowledge of parity games.

A direct as well as iterative method (called the orthogonally accumulated projection method, or the OAP for short) for solving linear system of equations with symmetric coefficient matrix is introduced in this paper. With the Lanczos process the OAP creates a sequence of mutually orthogonal vectors, on the basis of which the projections of the unknown vectors are easily obtained, and thus the approximations to the unknown vectors can be simply constructed by a combination of these projections. This method is an application of the accumulated projection technique proposed recently by the authors of this paper, and can be regarded as a match of conjugate gradient method (CG) in its nature since both the CG and the OAP can be regarded as iterative methods, too. Unlike the CG method which can be only used to solve linear systems with symmetric positive definite coefficient matrices, the OAP can be used to handle systems with indefinite symmetric matrices. Unlike classical Krylov subspace methods which usually ignore the issue of loss of orthogonality, OAP uses an effective approach to detect the loss of orthogonality and a restart strategy is used to handle the loss of orthogonality. Numerical experiments are presented to demonstrate the efficiency of the OAP.

In this paper, we show that a generalized Sasakian space form of dimension >3 is either of constant sectional curvature, or a canal hypersurface in Euclidean or Minkowski spaces, or locally a certain type of twisted product of a real line and a flat almost Hermitian manifold, or locally a warped product of a real line and a generalized complex space form, or an $${\alpha}$$ -Sasakian space form, or it is of five dimension and admits an $${\alpha}$$ -Sasakian Einstein structure. In particular, a local classification for generalized Sasakian space forms of dimension >5 is obtained. A local classification of Riemannian manifolds of quasi constant sectional curvature of dimension >3 is also given in this paper.

The architecture of the pre-modern Islamic world broadly identifies itself with geometric design. In erecting buildings, architects-engineers of the Islamic world utilized distinct modes of geometric projections vital to the spatial conception of the building. These representations identify three modes of design drawings: plans, revetments/vertical surfaces, and reflected ceiling plans. This paper will discuss these modes of drawings and their unique role in relation to the architectural “design thinking” traditions. Much has been examined regarding two-dimensional Islamic geometric patterns (girih), but little exists in terms of a comprehensive framework investigating various modes of geometric drawings in relation to formal, spatial, and tectonic conceptions of the architectural space. This paper fills a critical gap in the literature about Islamic architecture and examines this topic through primary resources and original pamphlets.

This is an expository paper on Garland's vanishing theorem specialized to the case when the linear algebraic group is $\mathrm{SL}_n$. Garland's theorem can be stated as a vanishing of the cohomology groups of certain finite simplicial complexes. The method of the proof is quite interesting on its own. It relates the vanishing of cohomology to the assertion that the minimal positive eigenvalue of a certain combinatorial laplacian is sufficiently large. Since the 1970's, this idea has found applications in a variety of problems in representation theory, group theory, and combinatorics, so the paper might be of interest to a wide audience. The paper is intended for non-specialists and graduate students.

This is the second paper in a series following Tian and Xu (2015), on the construction of a mathematical theory of the gauged linear σ-model (GLSM). In this paper, assuming the existence of virtual moduli cycles and their certain properties, we define the correlation function of GLSM for a fixed smooth rigidified r-spin curve.

This paper has two main parts. First, we construct certain differential operators, which generalize operators studied by G. Shimura. Then, as an application of some of these differential operators, we construct certain p-adic families of automorphic forms. Building on the author’s earlier work, these differential operators map automorphic forms on a unitary group of signature (n, n) to (vector-valued) automorphic forms on the product \(U^\varphi \times U^{-\varphi }\) of two unitary groups, where \(U^\varphi \) denotes the unitary group associated to a Hermitian form \(\varphi \) of arbitrary signature on an n-dimensional vector space. These differential operators have both a p-adic and a \(C^{\infty }\) incarnation. In the scalar-weight, \(C^{\infty }\)-case, these operators agree with ones studied by Shimura. In the final section of the paper, we also discuss some generalizations to other groups and settings. The results from this paper apply to the author’s paper-in-preparation with J. Fintzen, E. Mantovan, and I. Varma and to her ongoing joint project with M. Harris, J. -S. Li, and C. Skinner; they also relate to her recent paper with X. Wan.

In a previous paper we showed that if one particular Fourier coefficient of the Siegel theta series of degree 4 for a 32-dimensional even unimodular extremal lattice is known then the other Fourier coefficients of the series are in principle determined. In this paper we choose the quaternary positive definite symmetric matrix (Formula presented.), and calculate the Fourier coefficient (Formula presented.) of the Siegel theta series of degree 4 associated with the five even unimodular extremal lattices which come from the five binary self-dual extremal [32,16,8] codes. As a result we can show that the five Siegel theta series of degree 4 associated with the five 32-dimensional even unimodular extremal lattices are distinct. © 2016 Mathematisches Seminar der Universität Hamburg and Springer-Verlag Berlin Heidelberg, Embargo Period 12 months

In a series of papers recently "checkerboard discrepancy" has been introduced, where a black-and-white checkerboard background induces a coloring on any curve, and thus a discrepancy, i.e., the difference of the length of the curve colored white and the length colored black. Mainly straight lines and circles have been studied and the general situation is that, no matter what the background coloring, there is always a curve in the family studied whose discrepancy is at least of the order of the square root of the length of the curve. In this paper we generalize the shape of the background, keeping the lattice structure. Our background now consists of lattice copies of any bounded fundamental domain of the lattice, and not necessarily of squares, as was the case in the previous papers. As the decay properties of the Fourier Transform of the indicator function of the square were strongly used before, we now have to use a quite different proof, in which the tiling and spectral properties of the fundamental domain play a role., Comment: 10 pages, 3 figures

In a recent paper, we introduced sine functions on commutative hypergroups. These functions are natural generalizations of those functions on groups which are products of additive and multiplicative homomorphisms. In this paper, we describe sine functions on different types of hypergroups, including polynomial hypergroups, Sturm–Liouville hypergroups, etc. A non-commutative hypergroup is also considered.

This paper gives simple proofs of Sylvester (` = 2) and Frobenius (` = 3) inequalities. Moreover, a new sufficient condition for the equality of the Frobenius inequality is provided. In addition, an extension for ` > 3 matrices and an application to generalized inverses are provided., This paper has been partially supported by Ministerio de Economia y Competitividad (Grant DGI MTM2013-43678P and Red de Excelencia MTM2015-68805-REDT). The author thanks the referees for their valuable suggestions.

In a recent paper with Thomas Hoffmann-Ostenhof, we proved that the number of critical points \(\ell _k\) in the boundary set of a minimal k-partition tends to \(+\infty \) as \(k\rightarrow +\infty \). In this note, we show that \(\ell _k\) increases linearly with k as suggested by a hexagonal conjecture about the asymptotic behavior of the energy of these minimal partitions. As in the original proof by Pleijel of his celebrated theorem, this involves Faber-Krahn’s inequality and Weyl’s formula, but this time, due to the magnetic characterization of the minimal partitions, we have to establish a Weyl’s formula for Aharonov-Bohm operator controlled with respect to a k-dependent number of poles. In a recent paper with Thomas Hoffmann-Ostenhof, we proved that the number of critical points \(\ell _k\) in the boundary set of a k-minimal partition tends to \(+\infty \) as \(k\rightarrow +\infty \). In this note, we show that \(\ell _k\) increases linearly with k as suggested by a hexagonal conjecture about the asymptotic behavior of the energy of these minimal partitions. As the original proof by Pleijel, this involves Faber-Krahn’s inequality and Weyl’s formula, but this time, due to the magnetic characterization of the minimal partitions, we have to establish a Weyl’s formula for Aharonov-Bohm operator controlled with respect to a k-dependent number of poles.

This paper is about the Fukaya category of a Fano hypersurface $X \subset \mathbb{CP}^n$. Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed-open string maps, weak proper Calabi-Yau structure, Abouzaid's split-generation criterion, and their analogues when weak bounding cochains are included. We then turn to computations of the Fukaya category of the hypersurface $X$: we construct a configuration of monotone Lagrangian spheres in $X$, and compute the associated disc potential. The result coincides with the Hori-Vafa superpotential for the mirror of $X$ (up to a constant shift in the Fano index $1$ case). As a consequence, we give a proof of Kontsevich's homological mirror symmetry conjecture for $X$. We also explain how to extract non-trivial information about Gromov-Witten invariants of $X$ from its Fukaya category., Comment: 127 pages, 7 figures. Minor changes in response to referee's comments, fixed small errors. Proposition 2.9 and Corollary 2.11 of the new version strengthen results from the previous version

In our previous paper, given an integral log smooth morphism $X\to S$ of fine log schemes of characteristic $p>0$, we studied the Azumaya nature of the sheaf of log differential operators of higher level and constructed a splitting module of it under an existence of a certain lifting modulo $p^{2}$. In this paper, under a certain liftability assumption which is stronger than our previous paper, we construct another splitting module of our Azumaya algebra over a scalar extension, which is smaller than our previous paper. As an application, we construct an equivalence, which we call the log local Cartier transform of higher level, between certain $\cal D$-modules and certain Higgs modules. We also discuss about the compatibility of the log Frobenius descent and the log local Cartier transform and the relation between the splitting module constructed in this paper and that constructed in the previous paper. Our result can be considered as a generalization of the result of Ogus-Vologodsky, Gros-Le Stum-Quir��s to the case of log schemes and that of Schepler to the case of higher level.

In [22] (Tong-Viet H P, Simple classical groups of Lie type are determined by their character degrees, J. Algebra, 357 (2012) 61–68), the following question arose: Which groups can be uniquely determined by the structure of their complex group algebras? The authors in [12] (Khosravi B et al., Some extensions of PSL(2,p 2) are uniquely determined by their complex group algebras, Comm. Algebra, 43(8) (2015) 3330–3341) proved that each extension of PSL(2,p 2) of order 2|PSL(2,p 2)| is uniquely determined by its complex group algebra. In this paper we continue this work. Let p be an odd prime number and q = p or q = p 3. Let M be a finite group such that |M| = h|PSL(2,q), where h is a divisor of |Out(PSL(2,q))|. Also suppose that M has an irreducible character of degree q and 2p does not divide the degree of any irreducible character of M. As the main result of this paper we prove that M has a unique nonabelian composition factor which is isomorphic to PSL(2,q). As a consequence of our result we prove that M is uniquely determined by its order and some information on its character degrees which implies that M is uniquely determined by the structure of its complex group algebra.

In this paper we consider the Martin compactification, associated with the operator $$\mathcal {L}= \Delta -1$$ , of a complete non-compact surface $$\Sigma ^2$$ with negative curvature. In particular, we investigate positive eigenfunctions with eigenvalue one of the Laplace operator $$\Delta $$ of $$\Sigma ^2$$ , and prove a uniqueness result: such eigenfunctions are unique up to a positive multiplicative constant, if they vanish on the part of the geometric boundary $$S_\infty (\Sigma ^2)$$ of $$\Sigma ^2$$ where the curvature is bounded above by a negative constant, and satisfy some growth estimate on the other part of $$S_\infty (\Sigma ^2)$$ where the curvature approaches zero. This uniqueness result plays an essential role in our recent paper (Cao and He, Infinitesimal rigidity of steady gradient Ricci soliton in dimension three. arXiv:1412.2714 , 2014, preprint), in which we prove an infinitesimal rigidity theorem for deformations of certain three-dimensional collapsed gradient steady Ricci soliton with a non-trivial Killing vector field.

In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over $A$ is isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra., Comment: Accepted by Science China Mathematics

In his seminal paper, Schoenberg (1942) characterized the class P(S^d) of continuous functions f:[-1,1] \to \R such that f(\cos \theta) is positive definite over the product space S^d \times S^d, with S^d being the unit sphere of \R^{d+1} and \theta being the great circle distance. In this paper, we consider the product space S^d \times G, for G a locally compact group, and define the class P(S^d, G) of continuous functions f:[-1,1]\times G \to \C such that f(\cos \theta, u^{-1}\cdot v) is positive definite on S^d \times S^d \times G \times G. This offers a natural extension of Schoenberg's Theorem. Schoenberg's second theorem corresponding to the Hilbert sphere S^\infty is also extended to this context. The case G=\R is of special importance for probability theory and stochastic processes, because it characterizes completely the class of space-time covariance functions where the space is the sphere, being an approximation of Planet Earth., Comment: 26 pages

For a given category KAC 2 , the present paper deals with the existence problem for the category DTC 2 (k), which is equivalent to KAC 2 , where DTC 2 (k) is the category whose objects are simple closed K-curves with even number l of elements in Z n , l ≠ 6, and morphisms are (digitally) K-continuous maps, and KAC 2 is a category whose objects are simple closed A-curves and morphisms are A-maps. To address this issue, the paper starts from the category denoted by KAC 1 whose objects are connected nD Khalimsky topological subspaces with Khalimsky adjacency and morphisms are A-maps in [S. E. Han and A. Sostak, Comput. Appl. Math., 32, 521–536 (2013)]. Based on this approach, in KAC 1 the paper proposes the notions of A-homotopy and A-homotopy equivalence and classifies the spaces in KAC 1 or KAC 2 in terms of the A-homotopy equivalence. Finally, the paper proves that, for Sa given category KAC 2 , there is DTC 2 (k), which is equivalent to KAC 2 .

This paper surveys recent nonlinear analyses of the motion of deformable solids to which are attached heavy rigid bodies. The deformable solids are described by geometrically exact equations of nonlinear viscoelasticity (of strain-rate type), which form quasilinear parabolic-hyperbolic systems. The main emphasis of this paper is on the treatment of problems in which the ratios of the inertias of the deformable body are small with respect to those of the rigid body. Some of these problems possess rigorous asymptotic expansions in a small inertia parameter with the leading terms of the reduced problems governed not by traditional ordinary differential equations for the rigid body, but by equations with memory. Some of these problems admit attractors with the dimensions of the attractors small when the inertia parameters are small. This paper describes a number of open problems.

In this paper we obtain quite general and definitive forms for Hardy-Littlewood type inequalities. Moreover, when restricted to the original particular cases, our approach provides much simpler and straightforward proofs and we are able to show that in most cases the exponents involved are optimal. The technique we used is a combination of probabilistic tools and of an interpolative approach; this former technique is also employed in this paper to improve the constants for vector-valued Bohnenblust-Hilletype inequalities.

This paper considers a class of nondissipative reaction–diffusion equations. We are particularly interested in globally well-posed equations exhibiting blow-up in infinite time. These are known as slowly nondissipative equations. We review the recently developed theory for this class of problems, where a characterization for the associated noncompact global attractor is obtained. In addition, we derive an extension for the permutation realization result that holds for dissipative equations. The outlined results are then illustrated with an example. A brief discussion on the similarities with the dissipative case closes the paper.

In this survey paper we illustrate a general strategy which consists on putting a module-theoretical result into a latticial frame (we call it latticization), in order to translate that result to Grothendieck categories (we call it absolutization) and module categories equipped with hereditary torsion theories (we call it relativization). The renowned Hopkins–Levitzki Theorem and Osofsky–Smith Theorem from Ring and Module Theory, we will discuss in the last two sections of the paper, are among the most relevant illustrations of the power of this strategy.

In a 1995 paper, Hof et al. obtain a sufficient combinatorial criterion on the subshift \(\Omega \) of the potential of a discrete Schrodinger operator which guarantees purely singular continuous spectrum on a generic subset of \(\Omega .\) In part, this condition requires that the subshift \(\Omega \) be palindromic, i.e., contains an infinite number of distinct palindromic factors. In the same paper, they introduce the class P of morphisms \(f:A^*\rightarrow B^*\) of the form \(a\mapsto pq_a\) with p and \(q_a\) palindromes, and ask whether every palindromic subshift generated by a primitive substitution arises from morphisms of class P or by morphisms of the form \(a\mapsto q_ap.\) In this paper we give a partial affirmative answer to the question of Hof et al.: we show that every rich primitive substitutive subshift is generated by at most two morphisms each of which is conjugate to a morphism of class P. More precisely, we show that every rich (or almost rich in the sense of finite defect) primitive morphic word \(y\in B^\omega \) is of the form \(y=f(x)\) where \(f:A^*\rightarrow B^*\) is conjugate to a morphism of class P, and where x is a rich word fixed by a primitive substitution \(g:A^*\rightarrow A^*\) conjugate to one in class P.

In a Hilbert space setting ${\mathcal{H}}$, in order to minimize by fast methods a general convex lower semicontinuous and proper function ${\Phi }: {\mathcal{H}} \rightarrow \mathbb {R} \cup \{+\infty \}$, we analyze the convergence rate of the inertial proximal algorithms. These algorithms involve both extrapolation coefficients (including Nesterov acceleration method) and proximal coefficients in a general form. They can be interpreted as the discrete time version of inertial continuous gradient systems with general damping and time scale coefficients. Based on the proper setting of these parameters, we show the fast convergence of values and the convergence of iterates. In doing so, we provide an overview of this class of algorithms. Our study complements the previous Attouch–Cabot paper (SIOPT, 2018) by introducing into the algorithm time scaling aspects, and sheds new light on the Guler seminal papers on the convergence rate of the accelerated proximal methods for convex optimization.

Professor Biagio Ricceri recently sent us a counterexample to a result in our paper [1]. After a revision of the paper, we have detected a non-correct use of Sard-Smale theorem in the proof of Section (aI-2-2) of Theorem 7. This leads us to add hypothesis (iv) to this theorem. In the particular case in which the operator B is linear, the hypothesis (iv) coincides with (iii) and it is non superfluous.

In this paper, we establish the approximate controllability results for a non-autonomous functional differential equation using the theory of linear evolution system, Schauder fixed point theorem, and by making use of resolvent operators. The results obtained in this paper, improve the existing ones in this direction, to a considerable extent. An example is also given to illustrate the abstract results.

In this paper we initiate the study of the maximal subalgebras of exceptional simple classical Lie algebras \g over algebraically closed fields k of positive characteristic p, such that the prime characteristic is good for \g. In this paper we deal with what is surely the most unnatural case; that is, where the maximal subalgebra in question is a simple subalgebra of non-classical type. We show that only the first Witt algebra can occur as a subalgebra of \g and give explicit details on when it may be maximal in \g., 32 pages; previous version missed two Hamiltonian cases

It is an immediate conclusion from Bavula's papers \cite{Bavula:GWA-def}, \cite{Bavula:GWA-tensor-product} that if a generalized Weyl algebra $A=\kk[z;\lambda,\eta,\varphi(z)]$ is homologically smooth, then the polynomial $\varphi(z)$ has no multiple roots. We prove in this paper that the converse is also true. Moreover, formal deformations of $A$ are studied when $\kk$ is of characteristic zero., Comment: Final version, 36 pages

In the previous paper \cite{L-Z}, for a characteristic problem with not necessarily small initial data given on a complete null cone decaying like that in the work \cite{Ch-K} of the stability of Minkowski spacetime by Christodoulou and Klainerman, we proved the local existence in retarded time, which means the solution to the vacuum Einstein equations exists in a uniform future neighborhood, while the global existence in retarded time is the weak cosmic censorship conjecture. In this paper, we prove that the local existence in retarded time still holds when the data is assumed to decay slower, like that in Bieri's work \cite{Bie} on the extension to the stability of Minkowski spacetime. Such decay guarantees the existence of the limit of the Hawking mass on the initial null cone, when approaching to infinity, in an optimal way., Comment: 30 pages. arXiv admin note: text overlap with arXiv:1406.0048