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The SL(2,C)-representation varieties of punctured surfaces form natural families parameterized by holonomies at the punctures. In this paper, we first compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauss-Manin connection on the natural family of the smooth SL(2,C)-representation variety of the one-holed torus., Comment: Minor stylistic revision from version 1, 21 pages

In this paper we establish a close connection between three notions attached to a modular subgroup, namely, the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup, and the set of elliptic zeta functions generalizing theWeierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.

For m, n ∈ , 1 < m ≤ n, we write n = n1 + … + nm where {n1 , … , nm} ⊂ . Let A1 , . . . , Am be n × n singular real matrices such thatwhere = {x : Ajx = 0}, dim() = n – nj, and A1 + … + Am is invertible. In this paper we study integral operators of the formn1 + … + nm = n, , 0 < r < 1, and the matrices Ai’s are as above. We obtain the Hp() − Lq() boundedness of Tr for 0 < p < and .

In this paper, we consider the following critical Kirchhoff type equation:By using variational methods that are constrained to the Nehari manifold, we prove that the above equation has a ground state solution for the case when 3 < q < 5. The relation between the number of maxima of Q and the number of positive solutions for the problem is also investigated.

In this paper we consider two natural notions of connectivity for hypergraphs: weak and strong. We prove that the strong vertex connectivity of a connected hypergraph is bounded by its weak edge connectivity, thereby extending a theorem of Whitney from graphs to hypergraphs. We find that, while determining a minimum weak vertex cut can be done in polynomial time and is equivalent to finding a minimum vertex cut in the 2-section of the hypergraph in question, determining a minimum strong vertex cut is NP-hard for general hypergraphs. Moreover, the problem of finding minimum strong vertex cuts remains NP-hard when restricted to hypergraphs with maximum edge size at most 3. We also discuss the relationship between strong vertex connectivity and the minimum transversal problem for hypergraphs, showing that there are classes of hypergraphs for which one of the problems is NP-hard, while the other can be solved in polynomial time.

In this paper, we classify all solutions ofwith the finite conditionsHere c is a positive number and β > −1.

This paper constitutes a comprehensive study of ten classes of self-maps on metric spaces ⟨X, d⟩ with the pointwise (i.e., local radial) and local contraction properties. Each of these classes appeared previously in the literature in the context of fixed point theorems.We begin with an overview of these fixed point results, including concise self contained sketches of their proofs. Then we proceed with a discussion of the relations among the ten classes of self-maps with domains ⟨X, d⟩ having various topological properties that often appear in the theory of fixed point theorems: completeness, compactness, (path) connectedness, rectifiable-path connectedness, and d-convexity. The bulk of the results presented in this part consists of examples of maps that show non-reversibility of the previously established inclusions between these classes. Among these examples, the most striking is a differentiable auto-homeomorphism f of a compact perfect subset X of ℝ with f′ ≡ 0, which constitutes also a minimal dynamical system. We finish by discussing a few remaining open problems on whether the maps with specific pointwise contraction properties must have the fixed points.

In this paper, we study a two-component Lotka–Volterra competition systemon a one-dimensional spatial lattice. By the comparison principle, together with the weighted energy, we prove that the traveling wavefronts with large speed are exponentially asymptotically stable, when the initial perturbation around the traveling wavefronts decays exponentially as j + ct → −∞, where j ∈ , t > 0, but the initial perturbation can be arbitrarily large on other locations. This partially answers an open problem by J.-S. Guo and C.-H.Wu.

The main result of this paper is the following: anyweightedRiemannian manifold$(M,g,\unicode[STIX]{x1D707})$,i.e., a Riemannian manifold$(M,g)$endowed with a generic non-negative Radon measure$\unicode[STIX]{x1D707}$, isinfinitesimally Hilbertian, which means that its associated Sobolev space$W^{1,2}(M,g,\unicode[STIX]{x1D707})$is a Hilbert space.We actually prove a stronger result: the abstract tangent module (à la Gigli) associated with any weighted reversible Finsler manifold$(M,F,\unicode[STIX]{x1D707})$can be isometrically embedded into the space of all measurable sections of the tangent bundle of$M$that are$2$-integrable with respect to$\unicode[STIX]{x1D707}$.By following the same approach, we also prove that all weighted (sub-Riemannian) Carnot groups are infinitesimally Hilbertian.

We consider three point positivemasses moving onS2andH2. An Eulerian-relative equilibrium is a relative equilibrium where the three masses are on the same geodesic. In this paper we analyze the spectral stability of these kind of orbits where the mass at the middle is arbitrary and the masses at the ends are equal and located at the same distance from the central mass. For the case of S2, we found a positive measure set in the set of parameters where the relative equilibria are spectrally stable, and we give a complete classiûcation of the spectral stability of these solutions, in the sense that, except on an algebraic curve in the space of parameters, we can determine if the corresponding relative equilibriumis spectrally stable or unstable. OnH2, in the elliptic case, we prove that generically all Eulerian-relative equilibria are unstable; in the particular degenerate case when the two equal masses are negligible, we get that the corresponding solutions are spectrally stable. For the hyperbolic case we consider the system where the mass in the middle is negligible; in this case the Eulerian-relative equilibria are unstable.

In this paper, we prove that the singular integral defined byis bounded on Lp() for 1 < p < ∞ and is of weak type (1,1), where Ω ∈ L log+ L() and , with a ∈ L∞() satisfying some restricted conditions.

In this paper, periodic steady-state of a liquid film flowing over a periodic uneven wall is investigated via a classical method. Specifically, we analyze a long-wave model that is valid at the near-critical Reynolds number. For the periodic wall surface, we construct an iteration scheme in terms of an integral form of the original steady-state problem. The uniform convergence of the scheme is proved so that we can derive the existence and the uniqueness as well as the asymptotic formula of the periodic solutions.

A 2-cell embedding of an Eulerian digraph D into a closed surface is said to be directed if the boundary of each face is a directed closed walk in D. In this paper, a method is developed with the purpose of enumerating unlabelled embeddings for an Eulerian digraph. As an application, we obtain explicit formulas for the number of unlabelled embeddings of directed bouquets of cycles Bn, directed dipoles OD2n and for a class of regular tournaments T2n+1.

This paper gives an equivalent form of Picard’s theorem via entire solutions of the functional equation f2 + g2 = 1 and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations.

In this paper we study the Hilbert transformations over L2() and L2() fromthe viewpoint of symmetry. For a linear operator over L2() commutative with the ax + b group, we show that the operator is of the form λI+ηH, where I and H are the identity operator and Hilbert transformation, respectively, and λ, η are complex numbers. In the related literature this result was proved by first invoking the boundedness result of the operator using some machinery. In our setting the boundedness is a consequence of the boundedness of the Hilbert transformation. The methodology that we use is the Gelfand–Naimark representation of the ax + b group. Furthermore, we prove a similar result on the unit circle. Although there does not exist a group like the ax + b group on the unit circle, we construct a semigroup that plays the same symmetry role for the Hilbert transformations over the circle L2().

The ER(2)-cohomology of Bℤ/(2q) and ℂℙn are computed along with the Atiyah–Hirzebruch spectral sequence for ER(2)*(ℂℙ∞). This, along with other papers in this series, gives us the ER(2)-cohomology of all Eilenberg–MacLane spaces.

This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. LetGbe a compact group andHa closed subgroup ofG. Letμbe the normalizedG-invariant measure over the compact homogeneous spaceG/Hassociated with Weil's formula and. We then present a structured class of abstract linear representations of the Banach convolution function algebrasLp(G/H,μ).

A theorem of Burgess and Stephenson asserts that in an exchange ring with central idempotents, every maximal left ideal is also a right ideal. The proof uses sheaf-theoretic techniques. In this paper, we give a short elementary proof of this important theorem.

It is known that every Toeplitz matrix T enjoys a circulant and skew circulant splitting (denoted by CSCS), i.e., T=C-S with C a circulant matrix and S a skew circulant matrix. Based on the variant of such a splitting (also referred to as CSCS), we first develop classical CSCS iterative methods and then introduce shifted CSCS iterative methods for solving hermitian positive definite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical CSCS iterative methods work slightly better than the Gauss-Seidel (GS) iterative methods if the CSCS is convergent and that there is always a constant $\alpha$ such that the shifted CSCS iteration converges much faster than the Gauss-Seidel iteration, no matter whether the CSCS itself is convergent or not., National Natural Science Foundation of China No. 11371075, The authors would like to thank the supports of the National Natural Science Foundation of China under Grant No. 11371075, the research innovation program of Hunan province for postgraduate students under Grant No. CX2015B374, the Portuguese Funds through FCT–Fundac˜ao para a Ciˆencia, within the Project UID/MAT/00013/2013., info:eu-repo/semantics/publishedVersion

In this paper, we show that the Möbius invariant function space Qpcan be generated by variant Dirichlet type spaces 𝒟μ,pinduced by finite positive Borel measures μ on the open unit disk. A criterion for the equality between the space 𝒟μ,pand the usual Dirichlet type space 𝒟pis given. We obtain a sufficient condition to construct different 𝒟μ,pspaces and provide examples. We establish decomposition theorems for 𝒟μ,pspaces and prove that the non-Hilbert space Qpis equal to the intersection of Hilbert spaces 𝒟μ,p. As an application of the relation between Qpand 𝒟μ,pspaces, we also obtain that there exist different 𝒟μ,pspaces; this is a trick to prove the existence without constructing examples.

In this paper we introduceHardy-Lorentz spaces with variable exponents associated with dilations in ℝn. We establishmaximal characterizations and atomic decompositions for our variable exponent anisotropic Hardy-Lorentz spaces.

AC*-algebra Ahas the ideal property if any ideal I of Ais generated as a closed two-sided ideal by the projections inside the ideal. Suppose that the limitC*-algebraAof inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has the ideal property. In this paper we will prove thatAcan be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension-drop interval algebras and matrix algebras over 2-dimensional spaces with torsionH2groups.

LetRbe a ring andb,c∊R. In this paper, we give some characterizations of the (b,c)-inverse in terms of the direct sum decomposition, the annihilator, and the invertible elements. Moreover, elements with equal (b,c)-idempotents related to their (b,c)-inverses are characterized, and the reverse order rule for the (b,c)-inverse is considered.

Suppose G is a connected complex Lie group and H is a closed complex subgroup. Then there exists a closed complex subgroup J of G containing H such that the fibration π:G/H ⟶ G/J is the holomorphic reduction of G/H; i.e., G/J is holomorphically separable and O(G/H)≅ π*O(G/J). In this paper we prove that if G/H is pseudoconvex, i.e., if G/H admits a continuous plurisubharmonic exhaustion function, then G/J is Stein and J/H has no non-constant holomorphic functions.

Fix an irreducible (finite) root system R and a choice of positive roots. For any algebraically closed field k consider the almost simple, simply connected algebraic group Gk over k with root system k. One associates with any dominant weight λ for R two Gk-modules with highest weight λ, the Weyl module V( λ)k and its simple quotient L(µ)k . Let λ and μ be dominant weights with μ < j such that μ is maximal with this property. Garibaldi, Guralnick, and Nakano have asked under which condition there exists k such that L(μ)k is a composition factor of V(j)k , and they exhibit an example in type E8 where this is not the case. The purpose of this paper is to to show that their example is the only one. It contains two proofs for this fact: one that uses a classiffication of the possible pairs (λ, μ), and another that relies only on the classiûcation of root systems.

In this paper, we construct two classes of rational function operators using the Poisson integrals of the function on the whole real axis. The convergence rates of the uniform and mean approximation of such rational function operators on the whole real axis are studied.

In this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple C* -algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.

In this paper, we prove that on a compact, n-dimensional Alexandrov space with curvature at least −1, the Wasserstein barycenter of Borel probability measures μ1 ,… , μm is absolutely continuous with respect to the n-dimensional Hausdorff measure if one of them is.

In this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy–Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on the vector-valued function with all ƒ j being BV-functions.

Let Ω ⊂ ℝn be a bounded Lipschitz domain and consider the energy functionalover the space of W1,2(Ω, ℝm) where the integrand is a smooth uniformly convex function with bounded second derivatives. In this paper we address the question of regularity for solutions of the corresponding system of Euler–Lagrange equations. In particular, we introduce a class of singularmaps referred to as traceless and examine themas a new counterexample to the regularity of minimizers of the energy functional ℱ[ ·, Ω] using a method based on null Lagrangians.

In this paper, the concept of the classical f -divergence (for a pair of measures) is extended to the mixed f divergence (for multiple pairs of measures). The mixed f -divergence provides a way to measure the difference between multiple pairs of (probability) measures. Properties for the mixed f -divergence are established, such as permutation invariance and symmetry in distributions. An Alexandrov-Fenchel type inequality and an isoperimetric type inequality for the mixed f -divergence will be proved and applications in the theory of convex bodies are given.

In 2012 Gubeladze (Adv.\ Math.\ 2012) introduced the notion of k-convex-normal polytopes to show that integral polytopes all of whose edges are longer than 4d(d+1) have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no difference between k- and (k+1)-convex-normality (for k >= 3) and improve the bound to 2d(d+1). In the second part we extend the definition to pairs of polytopes and show that for rational polytopes P and Q, where the normal fan of P is a refinement of the normal fan of Q, if every edge e_P of P is at least d times as long as the corresponding edge e_Q of Q, then (P+Q) \cap \Z^d = (P\cap \Z^d) + (Q \cap \Z^d)., 10 pages, 8 figures

Let $G$ denote a reductive algebraic group over $\mathbb{C}$ and $x$ a nilpotent element of its Lie algebra $\mathfrak{g}$. The Springer variety $\mathcal{B}_x$ is the closed subvariety of the flag variety $\mathcal{B}$ of $G$ parameterizing the Borel subalgebras of $\mathfrak{g}$ containing $x$. It has the remarkable property that the Weyl group $W$ of $G$ admits a representation on the cohomology of $\mathcal{B}_x$ even though $W$ rarely acts on $\mathcal{B}_x$ itself. Well-known constructions of this action due to Springer et al use technical machinery from algebraic geometry. The purpose of this note is to describe an elementary approach that gives this action when $x$ is what we call parabolic-surjective. The idea is to use localization to construct an action of $W$ on the equivariant cohomology algebra $H_S^*(\mathcal{B}_x)$, where $S$ is a certain algebraic subtorus of $G$. This action descends to $H^*(\mathcal{B}_x)$ via the forgetful map and gives the desired representation. The parabolic-surjective case includes all nilpotents of type $A$ and, more generally, all nilpotents for which it is known that $W$ acts on $H_S^*(\mathcal{B}_x)$ for some torus $S$. Our result is deduced from a general theorem describing when a group action on the cohomology of the fixed point set of a torus action on a space lifts to the full cohomology algebra of the space., 6 pages, title changed and made shorter, the presentation of the paper totally revised, final version to appear in the Canadian Mathematical Bulletin

This paper provides a constructive proof of the weak factorizations of the classical Hardy space $H^1(\mathbb{R}^n)$ in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of ${\rm BMO}(\mathbb{R}^n)$ (the dual of $H^1(\mathbb{R}^n)$) via commutators of the multilinear Riesz transforms., Comment: improved some typos

In a previous work, we associated a complete diòerential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we also have a realization functor fromthe category of complete diòerential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

In this paper we determine the L1 ⟶ L1 and L∞ ⟶ L∞ norms of an integral operator N related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.

We provide the differential equations that generalize the Newtonian N-body problem of celestial mechanics to spaces of constant Gaussian curvature, k, for all k real. In previous studies, the equations of motion made sense only for k nonzero. The system derived here does more than just include the Euclidean case in the limit when k tends to 0: it recovers the classical equations for k=0. This new expression of the laws of motion allows the study of the N-body problem in the context of constant curvature spaces and thus offers a natural generalization of the Newtonian equations that includes the classical case. We end the paper with remarks about the bifurcations of the first integrals., Comment: 19 pages, 1 figure

In this paper, double commutativity and the reverse order law for the core inverse are considered. Then, new characterizations of the Moore-Penrose inverse of a regular element are given by one-sided invertibilities in a ring. Furthermore, the characterizations and representations of the core and dual core inverses of a regular element are considered., 19 pages, http://dx.doi.org/10.4153/CMB-2016-045-7

In this paper we give the bifurcation diagram of the family of cubic vector fields z=, depending on the values of . The bifurcation diagram is in ℝ 4, but its conic structure allows describing it for parameter values in . There are two open simply connected regions of structurally stable vector fields separated by surfaces corresponding to bifurcations of homoclinic connections between two separatrices of the pole at infinity. These branch from the codimension 2 curve of double singular points. We also explain the bifurcation of homoclinic connection in terms of the description of Douady and Sentenac of polynomial vector fields.

Let R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

In this paper, we give a new definition of radicals of Green’s relations in an ordered semigroup and characterize left regular (right regular), intra regular ordered semigroups by radicals of Green’s relations. We also characterize the ordered semigroups that are unions and complete semilattices of t-simple ordered semigroups.

In this paper, a newnotion of scalar curvature for a Finslermetric F is introduced, and two conformal invariants Y(M, F) and C(M, F) are deûned. We prove that there exists a Finslermetric with constant scalar curvature in the conformal class of F if the Cartan torsion of F is suõciently small and Y(M, F)C(M, F) < Y(Sn) where Y(Sn) is the Yamabe constant of the standard sphere.

In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some information about the underlying groups by examining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similarly, we investigate complete representability as an operator algebra of deformed Fourier algebras on some ûnitely generated discrete groups with connection to the growth rate of the group.

LetRbe a commutative ring with identity. The co-annihilating-ideal graph ofR, denoted byAR, is a graph whose vertex set is the set of all non-zero proper ideals ofRand two distinct verticesIandJare adjacent whenever Ann(I) ∩ Ann(J) = {0}. In this paper we initiate the study of the co-annihilating ideal graph of a commutative ring and we investigate its properties.

In this paper, we study condition , which is a projective tensor product analogue of condition C'. We show that the finite-dimensional OLLP operator spaces have condition , and Mn (n > 2) does not have that property.

This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. LetGbe a compact group and letHbe a closed subgroup ofG. LetG/Hbe the left coset space ofHinGand letμbe the normalized G-invariant measure onG-Hassociated with Weil’s formula. Then we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert spaceL2(G/H,μ).

Let H be a group. The co-maximal graph of subgroups of H, denoted by Γ(H), is a graph whose vertices are non-trivial and proper subgroups of H and two distinct vertices L and K are adjacent in Γ(H) if and only ifH = LK. In this paper, we study the connectivity, diameter, clique number, and vertex chromatic number of Γ(H). For instance, we show that if Γ(H) has no isolated vertex, then Γ(H) is connectedwith diameter at most h. Also, we characterize all ûnite groupswhose co-maximal graphs are connected. Among other results, we show that if H is a ûnitely generated solvable group and Γ(H) is connected, and moreover, the degree of a maximal subgroup is finite, then (H) is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraically closed ûeld is zero or infinite.

Let G be a connected graph with vertex set V(G).The degree Kirchhoò index of G is defined as S'(G) = Σ{u,v}⊆V(G) d(u)d(v)R(u, v), where d(u) is the degree of vertex u, and R(u, v) denotes the resistance distance between vertices u and v. In this paper, we characterize the graphs having maximum and minimum degree Kirchhoò index among all n-vertex bicyclic graphs with exactly two cycles.

Let G be a finite group and let A(G) denote the Burnside ring of G. Then an inverse limit L(G) of the groups A(H) for proper subgroups H of G and a homomorphism res from A(G) to L(G) are obtained in a natural way. Let Q(G) denote the cokernel of res. For a prime p, let N(p) be the minimal normal subgroup of G such that the order of G/N(p) is a power of p, possibly 1. In this paper we prove that Q(G) is isomorphic to the cartesian product of the groups Q(G/N(p)), where p ranges over the primes dividing the order of G.

In an attempt to resolve a folklore conjecture of Erdös regarding the non-vanishing at s = 1 of the L-series attached to a periodic arithmetical function with period q and values in {−1, 1}, Livingston conjectured the -linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston’s conjecture for composite q ≥ 4, highlighting that a newapproach is required to settle Erdös conjecture. We also prove that the conjecture is true for prime q ≥ 3, and indicate that more ingredients will be needed to settle Erdös conjecture for prime q.