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This paper deals with positive stationary solutions and spreading speeds of monostable equations with nonlocal dispersal in spatially periodic habitats. The existence and uniqueness of positive stationary solutions and the existence and characterization of spreading speeds of such equations with symmetric convolution kernels are established in the authors’ earlier work [41] for following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth. The above conditions guarantee the existence of principal eigenvalues of nonlocal dispersal operators associated to linearized equations at the trivial solution. In general, a nonlocal dispersal operator may not have a principal eigenvalue. In this paper, we extend the results in [41] to general spatially periodic nonlocal monostable equations. As a consequence, it is seen that the spatial spreading feature is generic for monostable equations with nonlocal dispersal.

Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. Motivated by the paper [H. T. Koelink, On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc., 124 (1996) 887-898], in this paper we introduce two new classes of orthogonal functions, which are respectively Fourier transforms of the generalized ultraspherical polynomials and generalized Hermite polynomials, and then obtain their orthogonality relations using Parseval’s identity.

In this paper we consider continuous, SL ( 2 , R ) \text {SL}(2,\mathbb {R}) -valued, Schrödinger cocycles over irrational rotations. We prove two generic results on the Lyapunov exponents which improve the corresponding ones contained in a paper by Bjerklöv, Damanik and Johnson.

In our earlier paper it was proved that the singular locus of A g A_{g} (coarse moduli space of principally polarized abelian varieties over C \mathbb {C} ) is expressed as the union of irreducible varieties A g ( p , α ) A_{g}(p,\alpha ) representing abelian varieties with an order p p automorphism of fixed entire representation. In this paper we prove that A g ( p , α ) A_{g}(p,\alpha ) is an irreducible component of Sing A g \text {Sing} A_{g} if and only if for a general element of this variety its automorphism group modulo { ± 1 } \{\pm 1\} , G + G_{+} , satisfies the equivalent conditions: G + = ⟨ α ⟩ G_{+}=\langle \alpha \rangle or N G + ( ⟨ α ⟩ ) = ⟨ α ⟩ N_{G_{+}}(\langle \alpha \rangle )=\langle \alpha \rangle . We illustrate how these results can be used by studying the case g = 4 g=4 and p = 5 p=5 .

In this paper we study the regularity of solutions of a transmission problem arising in studying a fiber-reinforced composite media. It is described through a divergence type equation div ( A ∇ u ) = h , \mbox {div}(A \nabla u) = h, in an open set D D , where h h is a bounded function and A A is a uniformly elliptic matrix, bounded and with piecewise Hölder continuous coefficients. The subdomains D i D_i where A A is of class C α C^{\alpha } have disjoint closure and are of class C 1 , α C^{1,\alpha } . Exploiting an idea contained in a paper by Li and Vogelius, we obtain the optimal regularity result for solutions, proving that they are of class C 1 , α ( D ¯ i ) C^{1,\alpha }(\bar D_i) .

In this paper we present a probabilistic construction of countable abelian p p -groups with prescribed Ulm-sequence. This result provides a different proof for the existence theorem of abelian p p -groups with any given countable Ulm-sequence due to Ulm, which is sometimes called Zippin’s theorem. The basic idea, applying probabilistic arguments, comes from a result by Erdős and Rényi. They gave an amazing probabilistic construction of countable graphs which, with probability 1 1 , produces the universal homogeneous graph, therefore also called the random graph. P. J. Cameron says about this in his book Oligomorphic Permutation Groups [Cambridge University Press, 1990]: In 1963, Erdős and Rényi proved the following paradoxical result. … It is my contention that mathematics is unique among academic pursuits in that such an apparently outrageous claim can be made completely convincing by a short argument. The algebraic tool in the present paper needs methods developed in the 1970s, the theory of valuated abelian p p -groups. Valuated abelian p p -groups are natural generalizations of abelian p p -groups with the height valuation, investigated in detail by F. Richman and E. Walker, and others. We have to establish extensions of finite valuated abelian p p -groups dominated by a given Ulm-sequence. Probabilistic results of a similar nature have been established by A. Blass and G. Braun, and by M. Droste and D. Kuske.

In this paper, we remove the restriction z 0 ( x ) ≥ 0 z_{0}(x) \geq 0 or w 0 ( x ) ≤ 0 w_{0}(x) \leq 0 in the paper “Existence of Solutions to Hyperbolic Conservation Laws with a Source” (Commun. Math. Phys., 187 (1997), 327–340) and obtain the existence of solutions for the resonant, isothermal system of isentropic gas dynamics.

In this paper, we deal with the problem of uniqueness of minimal system of binomial generators of a semigroup ideal. Concretely, we give different necessary and/or sufficient conditions for uniqueness of such minimal system of generators. These conditions come from the study and combinatorial description of the so-called indispensable binomials in the semigroup ideal., Comment: 11 pages. This paper was initially presented at the II Iberian Mathematical Meeting (http://imm2.unex.es). To appear in the Proc. Amer. Math. Soc

In this paper we present two-sided estimates of Chernoff type for the weighted L w 2 L_{w}^{2} -distance of a smooth function to the k k -dimensional space of all polynomials of degree less than k k , whenever the weight function w w solves the Pearson differential equation and generates a finite or infinite sequence of classical orthogonal polynomials. These inequalities are simple corollaries of a unified general theorem, which is the main result of the paper.

This paper complements a recent result of Dilworth, Ferenczi, Kutzarova and Odell regarding the ergodicity of strongly asymptotic ℓ p \ell _p spaces. We state this result in a more general form, involving domination relations, and we show that every asymptotically Hilbertian space which is not isomorphic to ℓ 2 \ell _2 is ergodic. In particular, every weak Hilbert space which is not isomorphic to ℓ 2 \ell _2 must be ergodic. Throughout the paper we construct explicitly the maps which establish the fact that the relation E 0 E_0 is Borel reducible to isomorphism between subspaces of the Banach spaces involved.

In this paper, all spectra are localized at an odd prime. Let T ( 1 ) T(1) be the Ravenel spectrum characterized by B P ∗ BP_{\ast } -homology as B P ∗ [ t 1 ] BP_{\ast }[t_1] , T ( 1 ) / ( v 1 ) T(1)/(v_1) be the cofiber of the self-map v 1 : Σ 2 p − 2 T ( 1 ) → T ( 1 ) v_1: \Sigma ^{2p-2}T(1)\rightarrow T(1) and L 2 L_2 denote the Bousfield localization functor with respect to v 2 − 1 B P ∗ v_2^{-1}BP_{\ast } . In this paper, we determine the homotopy groups of L 2 T ( 1 ) / ( v 1 ) L_2T(1)/(v_1) .

In an unpublished paper, Hsu gives a proof of the convergence of the fundamental solutions. Since we had a problem understanding Hsu’s paper, in this paper we give a detailed proof of the convergence of the minimal positive fundamental solutions of the conjugate heat equation on complete non-compact manifolds under the Cheeger-Gromov convergence of Ricci flows.

Let M be an invariant subspace of the Bergman space L 2 a (D) and S M be the compression of the coordinate multiplication operator M z to the co-invariant subspace L 2 a (D) ⊖ M. The present paper determines when the identity representation of C* (S M ) is a boundary representation for the Banach subalgebra B(S M ). The paper also considers boundary representations on the co-invariant subspaces of L 2 a (B n ).

Let k, n ∈ N with k < n and let g k (R n ) denote the Grassmann manifold consisting of all k-dimensional linear subspaces in R n . In an earlier paper the authors showed that if the projections of a nonconvex closed set C C R n are convex and proper for projection directions from some nonempty open set P ⊂ g k (R n ), then C contains a closed copy of an (n―k―1)-manifold. In this paper we improve on that result by showing that that result remains valid under the weaker assumption that P is somewhere dense in g k (R n ).

This paper extends two recent improvements in the Hilbert space setting of the well-known Katznelson-Tzafriri theorem by establishing both a version of the result valid for bounded representations of a large class of abelian semigroups and a quantified version for contractive representations. The paper concludes with an outline of an improved version of the Katznelson-Tzafriri theorem for individual orbits, whose validity extends even to certain unbounded representations., 13 pages, to appear in Proceedings of the American Mathematical Society

Let G G be a locally compact group and let K K be its closed subgroup. Write G ^ K \widehat {G}_{K} for the set of irreducible representations with non-zero K K -invariant vectors. We call a pair ( G , K ) (G,K) admissible if for each irreducible representation ( π , V π ) (\pi , V_{\pi }) in G ^ K \widehat {G}_{K} , its K K -invariant subspace V π K V_{\pi }^{K} is of finite dimension. For each π \pi in G ^ K \widehat {G}_{K} , let π v i , ξ ¯ j \pi _{v_{i}, \overline {\xi }_{j}} ’s ( π v i , ξ ¯ j ( g K ) ≔ ⟨ v i , π ( g ) ξ j ⟩ ) (\pi _{v_{i}, \overline {\xi }_{j}}(gK)≔\langle v_{i}, \pi (g)\xi _{j}\rangle ) be the matrix coefficeints on G / K G/K induced by fixed orthonormal bases { v i } \{v_{i}\} and { ξ j } \{\xi _{j}\} for V π V_{\pi } and V π K V_{\pi }^{K} respectively. A probability measure μ \mu on G / K G/K is called a spectral measure if there is a subset Γ \Gamma of G ^ K \widehat {G}_{K} such that the set of all such matrix coefficients π v i , ξ ¯ j , π ∈ Γ , \pi _{v_{i}, \overline {\xi }_{j}},\ \pi \in \Gamma , constitutes an orthonormal basis for L 2 ( G / K , μ ) L^{2}(G/K, \mu ) with some suitable normalization of these matrix coordinate functions. In this paper, we shall give a characterization of a spectral measure for an admissible pair ( G , K ) (G,K) by using the Fourier transform on G / K G/K . Also, from this we show that there is a “local translation” (we call it locally regular representation in the sequel) of G G on L 2 ( G / K , μ ) L^{2}(G/K, \mu ) under a mild condition. This will give us some necessary conditions for the existence of spectral measures. In particular, the atomic spectral measures of finite supports for Gelfand pairs are studied.

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

In this paper, we extend the classical Bohr's inequality to the setting of the non-commutative Hardy space $H^1$ associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr's inequality for operators in the von Neumann-Schatten class $\cl C_1$ and square matrices of any finite order. Interestingly, we establish that the optimal bound for $r$ in the above mentioned Bohr's inequality concerning von Neumann-Shcatten class is 1/3 whereas it is 1/2 in the case of $2\times 2$ matrices and reduces to $\sqrt{2}-1$ for the case of $3\times 3$ matrices. We also obtain a generalization of our above-mentioned Bohr's inequality for finite matrices where we show that the optimal bound for $r$, unlike above, remains 1/3 for every fixed order $n\times n,\ n\ge 2$.

In 2012, Chan proved a number of congruences for the coefficients of Ramanujan’s ϕ \phi function. In this paper, we prove some new congruences modulo powers of 2 and 3 for Ramanujan’s ϕ \phi function by employing Newman’s identities and theta function identities.

The λ \lambda -property and the uniform λ \lambda -property were first introduced by R. Aron and R. Lohman in 1987 as geometric properties of Banach spaces. In 1989, Th. Shura and D. Trautman showed that the Schreier space possesses the λ \lambda -property and asked if it has the uniform λ \lambda -property. In this paper, we show that Schreier space does not have the uniform λ \lambda -property. Furthermore, we show that the dual of the Schreier space does not have the uniform λ \lambda -property.

Let ω \omega be a weight on a right cancellative semigroup S S . Let ‖ ⋅ ‖ ω \|\cdot \|_{\omega } be the weighted norm on the weighted discrete semigroup algebra ℓ 1 ( S , ω ) \ell ^1(S, \omega ) . In this paper, we prove that the weight ω \omega satisfies F-property if and only if the operator norm ‖ ⋅ ‖ ω o p \| \cdot \|_{\omega op} of ‖ ⋅ ‖ ω \| \cdot \|_{\omega } is exactly equal to another weighted norm ‖ ⋅ ‖ ω ~ 1 \| \cdot \|_{\widetilde {\omega }_1} . Though its proof is elementary, the result is unexpectedly surprising. In particular, the operator norm ‖ ⋅ ‖ 1 o p \| \cdot \|_{1 op} is same as the ℓ 1 \ell ^1 - norm ‖ ⋅ ‖ 1 \| \cdot \|_1 on ℓ 1 ( S ) \ell ^1(S) . Moreover, various examples are discussed to understand the relation among ‖ ⋅ ‖ ω o p \| \cdot \|_{\omega op} , ‖ ⋅ ‖ ω \| \cdot \|_{\omega } , and ℓ 1 ( S , ω ) \ell ^1(S, \omega ) .

This paper is concerned with a parabolic-elliptic system, which was originally proposed to model the evolution of biological transport networks. Recent results show that the corresponding initial-boundary value problem possesses a global weak solution, which, in particular, is also classical in the one and two dimensional cases. In this work, we establish a Serrin-type blowup criterion for classical solutions in the three dimensional setting.

This paper focuses on the stationary problem of the diffusive logistic equation on a bounded interval. We consider the ratio of a population size of a species to a carrying capacity which denotes a spatial heterogeneity of an environment. In one-dimensional case, Wei-Ming Ni proposed a variational conjecture that the supremum of this ratio varying a diffusion coefficient and a carrying function is 3. Recently, Xueli Bai, Xiaoqing He, and Fang Li [Proc. Amer. Math. Soc. 144 (2016), pp. 2161–2170] settled the conjecture by finding a special sequence of diffusion coefficients and carrying functions. Our interest is to derive a profile of the solutions corresponding to this maximizing sequence. Among other things, we obtain the exact order of the maximum and the minimum of solutions of the sequence. The proof is based on separating the stationary problem into two ordinary differential equations and smoothly adjoining each solution.

In this paper, we study the population optimization problem in the logistic reaction-diffusion model. The issue of maximizing the total population in a heterogeneous environment has attracted the attention of many researchers. For the n n -dimensional box domain, it has recently been shown that resource fragmentation is better than concentration in order to maximize the total population when the diffusion rate is sufficiently small. As resource concentration is known to be beneficial for the survival of the species, this contrasting phenomenon is quite surprising. We proved that the fragmentation phenomenon occurs for any general bounded domains in R n \mathbb {R}^n if the diffusion rate is sufficiently small.

Recently P. Hubert, C. Mauduit and A. Sarkozy introduced and studied the notion of pseudorandomness of binary lattices and gave a pseudorandom binary lattice. Later in other papers C. Mauduit and A. Sarkozy constructed some large families of "good" binary lattices. In this paper a large family of pseudorandom binary lattices is presented by using the multiplicative inverse and the quadratic character of finite fields.

Derivatives of a solution of an ODE Cauchy problem can be computed inductively using the Faa di Bruno formula. In this paper, we exhibit a noninductive formula for these derivatives. At the heart of this formula is a combinatorial problem, which is solved in this paper. We also give a more tractable form of the Magnus expansion for the solution of a homogeneous linear ODE.

In this paper we give simple, sufficient conditions for the existence of the stochastic integral for vector-valued processes X with values in a Banach space E; namely, X is of class (LD), and the stochastic measure I X is bounded and strongly additive in L p E (in particular, if I X is bounded in L p E and c 0 ⊈ E) and has bounded semivariation. The result is then applied to martingales and processes with integrable variation or semivariation. For martingales the condition of being of class (LD) is superfluous. For a square-integrable martingale with values in a Hilbert space, all the conditions are superfluous. For processes with p-integrable semivariation or p-integrable variation, the conditions of I X to be bounded and have bounded semivariation are superfluous. For processes with 1-integrable variation, all conditions are superfluous. In a forthcoming paper, we shall extend these results to local summability. The extension needs additional nontrivial work.

In a former paper we studied transformations on the set of all n n -dimensional subspaces of a Hilbert space H H which preserve the principal angles. In the case when dim ⁡ H ≠ 2 n \dim H\neq 2n , we could determine the general form of all such maps. The aim of this paper is to complete our result by considering the problem in the remaining case dim ⁡ H = 2 n \dim H=2n .

It is well known that the exceptional set in a resolution of a rational surface singularity is a tree of rational curves. We generalize the combinatoric part of this statement to higher dimensions and show that the highest cohomologies of the dual complex associated to a resolution of an isolated rational singularity vanish. We also prove that the dual complex associated to a resolution of an isolated hypersurface singularity is simply connected. As a consequence, we show that the dual complex associated to a resolution of a 3-dimensional Gorenstein terminal singularity has the homotopy type of a point., 10 pages, the structure of the paper has been slightly revised and a mistake in the proof of degeneration of spectral sequence in Lemma 2.3 (Lemma 2.4 of the published version of the paper) has been corrected

In this paper we study the nonexpansivity of the so-called relatively nonexpansive mappings. A relatively nonexpansive mapping with respect to a pair of subsets ( A , B ) (A,B) of a Banach space X X is a mapping defined from A ∪ B A\cup B into X X such that ‖ T x − T y ‖ ≤ ‖ x − y ‖ \|Tx-Ty\|\le \|x-y\| for x ∈ A x\in A and y ∈ B y\in B . These mappings were recently considered in a paper by Eldred et al. (Proximinal normal structure and relatively nonexpansive mappings, Studia Math. 171 (3) (2005), 283-293) to obtain a generalization of Kirk’s Fixed Point Theorem. In this work we show that, for certain proximinal pairs ( A , B ) (A,B) , there exists a natural semimetric for which any relatively nonexpansive mapping with respect to ( A , B ) (A,B) is nonexpansive. This fact will be used to improve one of the two main results from the aforementioned paper by Eldred et al. At that time we will also obtain several consequences regarding the strong continuity properties of relatively nonexpansive mappings and the relation between the two main results from the same work.

This paper is concerned with the behaviour of the period function of the quadratic reversible centers. In this context the interesting stratum is the family of the so-called Loud's dehomogenized systems, namely { x = -y + xy, {y = x+Dx 2 +Fy 2 . In this paper we show that the bifurcation set of the period function of these centers is contained in the rectangle K = (-7, 2) x (0,4). More concretely, we prove that if (D, F) ∉ K, then the period function of the center is monotonically increasing.

Refinable functions have been widely investigated because of their importance in wavelet theory and multiresolution analysis, as well as because of intrinsic interest. Problems involving refinability can be challenging and interesting problems in mathematics. Several papers have investigated refinability of splines and other classes of functions. The purpose of this paper is to develop necessary and sufficient conditions for the refinability of the class of step functions on the real line taking complex values.

In this paper we study a nonlinear eigenvalue problem driven by the p-Laplacian. Assuming for the right-hand side nonlinearity only unilateral and sign conditions near zero, we prove the existence of three nontrivial solutions, two of which have constant sign (one is strictly positive and the other is strictly negative), while the third one belongs to the order interval formed by the two opposite constant sign solutions. The approach relies on a combination of variational and minimization methods coupled with the construction of upper-lower solutions. The framework of the paper incorporates problems with concave-convex nonlinearities.

We prove that ΩSp(n) is stably indecomposable at the prime 2 if n ≥ 2 or n = ∞. Hopkins [2] proved that ΩSp(2) and ΩSp(3) are stably indecomposable. Later Hubbuck [3] added ΩG2 and ΩF4 to the list of such spaces. We [4] also proved that ΩE6 and ΩE7 are stably indecomposable. In this paper we will show that ΩSp(n) are stably indecomposable at the prime 2 for n ≥ 2, which was conjectured by Hubbuck. Theorem. ΩSp(n) is stably indecomposable at the prime 2 if n ≥ 2 or n = ∞. In contrast to our result ΩSU(n) is stably decomposable [1]. From now on until the end of this paper all spaces are assumed to be localized at the prime 2 and H∗(X) stands for H∗(X;F2). First we recall the ring structure of H∗(ΩSp(n)) and the dual action of the Steenrod algebra on them [5]: H∗(ΩSp(n)) ∼= F2[z1, z3, . . . , z2n−1], where z2i−1 ∈ H4i−2(ΩSp(n)). To state the action of the Steenrod algebra on z2i−1, for a positive integer i we define zi = (zb(i)) 2a(i) , where a(i) and b(i) are unique non-negative integers such that i = 2b(i) and b(i) is odd. Then we have Sqz2i−1 = ( 2i− 2− j j ) z2i−1−j. In particular, Sqz2i−1 = z2i−2. We also have Sqzi = ( i− 1− j j ) zi−j for any positive integer i. 2000 Mathematics Subject Classification. Primary 55P35, 55P42, 55S10.

Let ρ: G → GL(n, F) be a faithful representation of a finite group G. In this paper we proceed with the study of the image of the associated Noether map η G G : F[V(G)] G → F[V] G . In our 2005 paper it has been shown that the Noether map is surjective if V is a projective FG-module. This paper deals with the converse. The converse is in general not true: we illustrate this with an example. However, for p-groups (where p is the characteristic of the ground field F) as well as for permutation representations of any group the surjectivity of the Noether map implies the projectivity of V.

In this paper, we prove a uniqueness theorem for a free boundary problem which is given in the form of a variational inequality. This free boundary problem arises as the limit of an equation that serves as a basic model in population biology. Apart from the interest in the problem itself, the techniques used in this paper, which are based on the regularity theory of variational inequalities and of harmonic functions, are of independent interest, and may have other applications.

The paper deals with asymptotic phase and invariant foliations near periodic orbits, extending for two-dimensional smooth vector fields results that have been obtained by Chicone and Liu (2004). The problem of the existence of asymptotic phase is completely solved for analytic vector fields and is reduced to a problem of infinite codimension for C ∞ C^{\infty } systems. Moreover it is proven that whenever asymptotic phase occurs, or in other words, when the periodic orbit is isochronous, then there also exists a C ∞ C^{\infty } foliation, with leaves transversally cutting the periodic orbit and invariant under the flow of the vector field. The paper also provides some results in three dimensions.

This paper provides a version of the transversality theorem for a class of Lipschitz functions of the form f : R n × C → R n f:\textbf {R}^n\times C\to \textbf {R}^n , where C C is a convex subset of a normed vector space Z Z indexing the parameters in the problem. The set C C may be infinite-dimensional, and the notion of generic used is the measure-theoretic notion of prevalence introduced by Hunt, Sauer and Yorke (1992) and Christensen (1974). This paper also provides some results on sensitivity analysis for solutions to locally Lipschitz equations.

An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results complement a recent work by Ismail and Stanton. Quite surprisingly, in some cases the Taylor polynomials converge to a function which differs from the original one. We provide explicit expressions for the integral remainder. As application, we obtain some summation formulas for basic hypergeometric series. As far as we know, one of them is new. We conclude by studying the different forms of the binomial theorem in this context. 1. Introduction and definitions The problem of expanding a function with respect to a given polynomial basis has many implications in analysis. The simplest example of this kind is the Taylor's expansion theorem. In this paper, we replace the classical derivative by a difference operator of Askey-Wilson type. Our results complement the paper (3) of Ismail and Stanton and are a natural continuation of the point of view presented in (5), where a new approach to the theory of classical hypergeometric polynomials is given. In contrast with (3), our aim is to find sufficient conditions for t he Taylor series to converge, but not necessarily to the original function. In this more general setting, we may consider non-necessarily entire functions and we give an explicit expression for the limit of the remainders in terms of a contour integral. Using this and a new estimate for the q-shifted factorials, which might be of independent interest, we obtain a summation formula which is new as far as we know. As we explain below, it can be regarded as a non-symmetrized version of the non-terminating q-Saalschutz sum. As applications, we also provide a new proof of the q-Gauss summation formula and a list of binomial type summation formulas in the same line than Ismail's paper (2). Now we give some definitions which will be used in what follows. The notions we are presenting were already introduced in (5) with the aim of studying some aspects of the theory of hypergeometric polynomials. The relevance of this approach is justified in (5), where a more detailed exposition is given.

In this paper, we introduce sequentially defined Besov spaces adapted to the scaling invariant property of the 3D incompressible Navier-Stokes equations. We first show that these spaces are Banach spaces. We then establish regularity conditions in these spaces.

In this paper, Lie group representations on Hilbert spaces are studied in relation with operator ranges. Let R \mathcal {R} be an operator range of a Hilbert space H \mathcal {H} . Given the set Λ \Lambda of R \mathcal {R} -invariant operators, and given a Lie group representation ρ : G → GL ( H ) \rho :G\rightarrow \text {GL}(\mathcal {H}) , we discuss the induced semigroup homomorphism ρ ~ : ρ − 1 ( Λ ) → B ( R ) \widetilde {\rho }: \rho ^{-1}(\Lambda ) \rightarrow \mathcal {B(R)} for the operator range topology on R \mathcal {R} . In one direction, we work under the assumption ρ − 1 ( Λ ) = G \rho ^{-1} (\Lambda ) = G , so ρ ~ : G → B ( R ) \widetilde {\rho }:G\rightarrow \mathcal {B}(\mathcal {R}) is in fact a group representation. In this setting, we prove that ρ ~ \widetilde {\rho } is continuous (and smooth) if and only if the tangent map d ρ d\rho is R \mathcal {R} -invariant. In another direction, we prove that for the tautological representations of unitary or invertible operators of an arbitrary infinite-dimensional Hilbert space H \mathcal {H} , the set ρ − 1 ( Λ ) \rho ^{-1}(\Lambda ) is neither a group for a large set of nonclosed operator ranges R \mathcal {R} nor closed for all nonclosed operator ranges R \mathcal {R} . Both results are proved by means of explicit counterexamples.

In this paper we will give a simple stopping time proof in the $\mathbb{R}^d$ setting of the matrix weighted Triebel-Lizorkin bounds proved by F. Nazarov/S. Treil and A. Volberg, respectively. Furthermore, we provide explicit matrix A${}_p$ characteristic dependence and also discuss some interesting open questions., 13 pages, no figures, submitted

A number of open problems in the field of several complex variables naturally lead to the study of bihomogeneous polynomials r ( z , z ¯ ) r(z,\bar {z}) on C n + 1 \mathbb {C}^{n+1} . In particular, both the Ebenfelt sum of squares conjecture and the degree estimate conjecture for proper rational mappings between balls in complex Euclidean spaces lead to the study of the rank of the bihomogeneous polynomial r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z}\right \rVert ^2 under certain additional hypotheses. When r r has a diagonal coefficient matrix, these questions reduce to questions about real homogeneous polynomials. More specifically, we are led to study the rank of P = S Q P=SQ when Q Q is a homogeneous polynomial and S ( x ) = ∑ j = 0 n x j S(x) = \sum _{j=0}^n x_j . In this paper, we use techniques from commutative algebra to estimate the minimum rank of P = S Q P=SQ under the additional hypothesis that Q Q has maximum rank. The problem has already been solved for n + 1 ≤ 3 n+1 \leq 3 , and so we consider n + 1 ≥ 4 n+1\geq 4 . We obtain a minimum rank estimate that is sharp when n + 1 = 4 n+1=4 , and we exhibit a family of polynomials having this minimum rank. We also prove an estimate for n + 1 > 4 n+1>4 that, while not sharp, is non-trivial.

For any prime number p p , and integer k ⩾ 1 k\geqslant 1 , let F p k \mathbb {F}_{p^k} be the finite field of p k p^k elements. A famous problem in the theory of polynomials over finite fields is the characterization of all nonconstant polynomials F ∈ F p k [ x ] F\in \mathbb {F}_{p^k}[x] for which the value set { F ( α ) : α ∈ F p k } \{F(\alpha ): \alpha \in \mathbb {F}_{p^k}\} has the minimum possible size ⌊ ( p k − 1 ) / deg ⁡ F ⌋ + 1 \left \lfloor (p^k-1)/\deg F \right \rfloor +1 . For k ⩽ 2 k\leqslant 2 , the problem was solved in the early 1960s by Carlitz, Lewis, Mills, and Straus. This paper solves the problem for k = 3 k=3 .

Let F \mathbb {F} be a finite field and T T a transcendental element over F \mathbb {F} . An imaginary function field is defined to be a function field such that the prime at infinity is inert or totally ramified. For the totally imaginary case, in a recent paper the second author constructed infinitely many function fields of any fixed degree over F ( T ) \mathbb {F}(T) in which the prime at infinity is totally ramified and with ideal class numbers divisible by any given positive integer greater than 1. In this paper, we complete the imaginary case by proving the corresponding result for function fields in which the prime at infinity is inert. Specifically, we show that for relatively prime integers m m and n n , there are infinitely many function fields K K of fixed degree m m such that the class group of K K contains a subgroup isomorphic to ( Z / n Z ) m − 1 (\mathbb {Z}/n\mathbb {Z})^{m-1} and the prime at infinity is inert.

From our previous paper, it is known that the Magnus representation of the Torelli group is not faithful. In this paper, we characterize the kernel of its representation for a certain kind of elements.

Let β > 1 \beta >1 be a Pisot unit. A family of sets { X i } 1 ≤ i ≤ q \{X_i\}_{1\leq i\leq q} defined by a β \beta -numeration system has been extensively studied as an atomic surface or Rauzy fractal. For the purpose of constructing a Markov partition, a domain X ^ = ⋃ i = 1 q X ^ i \hat X=\bigcup _{i=1}^q \hat X_i constructed by an atomic surface has appeared in several papers. In this paper we show that the domain X ^ \hat X completely characterizes the set of purely periodic β \beta -expansions.

In their paper “An Introduction to Finsler Geometry,” J. C. Alvarez and C. Duran asked if there are other Minkowski planes besides the Euclidean for which the ratio of the Minkowski length of the unit “circle” to the Holmes-Thompson area of the unit disc equals 2. In this paper we show that this ratio is greater than 2, and that the ratio 2 is achieved only for Minkowski planes that are affine equivalent to the Euclidean plane. In other words, the ratio is 2 only when the unit “circle” is an ellipse.

We consider the initial value problem for the fourth order nonlinear Schrodinger type equation (4NLS) related to the theory of vortex filament. In this paper we prove the time local well-posedness for (4NLS) in the Sobolev space, which is an improvement of our previous paper.

The infinitesimal earthquake theorem gives a one-to-one correspondence between Thurston bounded earthquake measures and normalized Zygmund bounded functions. In this paper, we provide an intrinsic proof of a theorem given in an earlier paper by the author; that is, we show that the cross-ratio norm of a Zygmund bounded function is equivalent to the Thurston norm of the earthquake measure in the correspondence.