Some <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Rigidity Results for Complete Manifolds with Harmonic Curvature

Let (M ⁿ , g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R m? the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R m? goes to zero uniformly at infinity if for \(p\geq \frac n2\), the L ^p -norm of R m? is finite. Moreover, If R is positive, then (M ⁿ , g) is compact. As applications, we prove that (M ⁿ , g) is isometric to a spherical space form if for \(p\geq \frac n2\), R is positive and the L ^p -norm of R m? is pinched in [0, C ₁), where C ₁ is an explicit positive constant depending only on n, p, R and the Yamabe constant. We give an isolation theorem of the trace-free Ricci curvature tensor of compact locally conformally flat Riemannian n-manifolds with constant positive scalar curvature, which extends Theorem 1 of Hebey and M. Vaugon (J. Geom. Anal. 6, 531–553, 1996). This result is sharp, and we can precisely characterize the case of equality. In particular, when n = 4, we recover results by Gursky (Indiana Univ. Math. J. 43, 747–774, 1994; Ann. Math. 148, 315–337, 1998).     

Expansive homoclinic classes of generic <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-vector fields

Let γ be a hyperbolic closed orbit of a C ¹ vector field X on a compact C ^∞ manifold M of dimension n ≥ 3, and let H _X(γ) be the homoclinic class of X containing γ. In this paper, we prove that C ¹-generically, if H _X(γ) is expansive and isolated, then it is hyperbolic.     

Existence of linear Pfaffian systems whose lower characteristic set has positive measure in <Emphasis Type="Italic">R</Emphasis><Superscript>3</Superscript>

We prove the existence of a completely integrable Pfaffian system ?x/?t i = A _i (t)x, x ∈ R ⁿ , t = (t ₁, t ₂, t ₃) ∈ R ₊ ³ , i = 1, 2, 3, with infinitely differentiable bounded coefficients and with lower characteristic set of positive three-dimensional Lebesgue measure.     

Curvature identities for principle <Emphasis Type="Italic">T</Emphasis><Superscript>1</Superscript>-bundles over almost Hermitian manifolds

We show that the identities R ₁, R ₂ and R ₃ for an almost Hermitian structure S on the base of the canonical principal T ¹-bundle are equivalent to their contact analogs for the induced almost contact metric structure S ^# on the total space of this bundle. We prove that the canonical connection of the canonical principal T ¹-bundle over a Hermitian or a quasi-Kähler manifold of class R ₃ is normal. We also prove that that the canonical connection of the canonical principal T ¹-bundle over a Vaisman-Gray manifold M of class R ₃ is normal if and only if the Lee vector of M belongs to the center of the adjoint K-algebra.     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

Let G be a finite group. The main result of this paper is as follows: If G is a finite group, such that Γ(G) = Γ(²G₂(q)), where q = 3²ⁿ⁺¹ for some n ≥ 1, then G has a (unique) nonabelian composition factor isomorphic to ² G ₂(q). We infer that if G is a finite group satisfying |G| = |² G ₂(q)| and Γ(G) = Γ (² G ₂(q)) then G ? = ² G ₂(q). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications of this result are also considered to the problem of recognition by element orders of finite groups.     

The existence of eigenvalues for operators acting in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We present conditions that allow us to prove the existence of eigenvalues and characteristic values for operator F(D) ? C(λ): L ²(R ^m ) → L ²(R ^m ), where F(D) is a pseudo-differential operator with a symbol F(iξ) and C(λ): L ²(R ^m ) → L ²(R ^m ) is a linear continuous operator.     

The modular group and words in its two generators<Superscript>*</Superscript>

Consider the full modular group PSL₂(?) with presentation 〈U,?S|U ³,?S ²〉. Motivated by our investigations on quasi-modular forms and the Minkowski question mark function (so that this paper may be considered as a necessary appendix), we are led to the following natural question. Some words in the alphabet {U, S} are equal to the unity; for example, USU ³ SU ² is such a word of length 8, and USU ³ SUSU ³ S ³ U is such a word of length 15. We consider the following integer sequence. For each n ∈ ?₀, let t(n) be the number of words in the alphabet {U, S} that equal the identity in the group. This is the new entry A265434 into the Online Encyclopedia of Integer Sequences. We investigate the generating function of this sequence and prove that it is an algebraic function over ?(x) of degree 3. As an interesting generalization, we formulate the problem of describing all algebraic functions with a Fermat property.     

On unconditional basis property in the space <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">R</Emphasis>) of a system of Franklin functions with vanishing means

We define a general Franklin system of functions on R with vanishing means, generated by an admissible sequence T. A necessary and sufficient condition on T is found for the corresponding general Franklin system of functions on R with vanishing means to be an unconditional basis in the space H ¹(R).     

Asymptotic <Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">u</Emphasis></Subscript>-Toeplitzness of weighted composition operators on <Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript>

Given a unilateral forward shift S acting on a complex, separable, innite dimensional Hilbert space H, an asymptotically S-Toeplitz operator is a bounded linear operator T on H satisfying that {S* ⁿ TS ⁿ } is convergent with respect to one of the topologies commonly used in the algebra of bounded linear operators on H. In this paper, we study the asymptotic T _u -Toeplitzness of weighted composition operators on the Hardy space H², where u is a nonconstant inner function.     

The probability that <Emphasis Type="Italic">r</Emphasis> elements of a rank <Emphasis Type="Italic">n</Emphasis> free group generate a rank <Emphasis Type="Italic">r</Emphasis> subgroup

Granted the three integers n ≥ 2, r, and R, consider all ordered tuples of r elements of length at most R in the free group F _n . Calculate the number of those tuples that generate in F _n a rank r subgroup and divide it by the number of all tuples under study. As R → ∞, the limit of the ratio is known to exist and equal 1 (see [1]). We give a simple proof of this result.     

Asymptotics of the Codimensions <Emphasis Type="Italic">c</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> in the Algebra <Emphasis Type="Italic">F</Emphasis><Superscript>(7)</Superscript>

The paper studies the additive structure of the algebra F⁽⁷⁾, i.e., a relatively free associative countably generated algebra with the identity [x₁,..., x₇] = 0 over an infinite field of characteristic ≠ 2, 3. First, the space of proper multilinear polynomials in this algebra is investigated. As an application, estimates for the codimensions c_n = dimF_n⁽⁷⁾ are obtained, where F_n⁽⁷⁾ stands for the subspace of multilinear polynomials of degree n in the algebra F⁽⁷⁾.     

A boundary Schwarz lemma on the classical domain of type <Emphasis Type="Italic">I</Emphasis>

Let R _I (m, n) be the classical domain of type I in ? ^m×n with 1 ≤ m ≤ n. We obtain the optimal estimates of the eigenvalues of the Fréchet derivative Df(\(\mathop Z\limits^ \circ \)) at a smooth boundary fixed point \(\mathop Z\limits^ \circ \)of R _I (m, n) for a holomorphic self-mapping f of R _I (m, n). We provide a necessary and sufficient condition such that the boundary points of R _I (m, n) are smooth, and give some properties of the smooth boundary points of R _I (m, n). Our results extend the classical Schwarz lemma at the boundary of the unit disk Δ to R _I (m, n), which may be applied to get some optimal estimates in several complex variables.     

Divergent solutions to the <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-supercritical NLS equations

We investigate the nonlinear Schrödinger equation iu _t +Δu+|u| ^p?1 u = 0with 1+ 4/N < p < 1+ 4/N?2 (when N = 1, 2, 1 + 4/N < p < ∞) in energy space H ¹ and study the divergent property of infinite-variance and nonradial solutions. If \(M{\left( u \right)^{\frac{{1 - {s_C}}}{{{s_C}}}}}E\left( u \right) \prec M{\left( Q \right)^{\frac{{1 - {s_C}}}{{{s_C}}}}}E\left( Q \right)\) and \(\left\| {{u_0}} \right\|_2^{\frac{{1 - {s_c}}}{{{s_c}}}}\left\| {\nabla {u_0}} \right\|_2^{\frac{{1 - {s_c}}}{{{s_c}}}}{\left\| {\nabla Q} \right\|_2}\), then either u(t) blows up in finite forward time or u(t) exists globally for positive time and there exists a time sequence t _n → +∞ such that \({\left\| {\nabla u\left( {{t_n}} \right)} \right\|_2} \to + \infty \). Here Q is the ground state solution of ?(1?s _c )Q+ΔQ+Q ^p?1 Q = 0. A similar result holds for negative time. This extend the result of the 3D cubic Schrödinger equation obtained by Holmer to the general mass-supercritical and energy-subcritical case.     

Generalized <Emphasis Type="Italic">p</Emphasis>-Adic Fourier Transform and Estimates of Integral Modulus of Continuity in Terms of This Transform

We consider a new class of functions on the p-adic linear space ? _p ⁿ for which a Fourier transform can be defined.We prove equalities of Parseval type, an inversion formula and a sufficient condition for a function to be represented as this Fourier transform. Also we give a sharp estimate of the L²(? _p ⁿ ) modulus of continuity in terms of Fourier transform generalizing the result of S. S. Platonov in the case n = 1. Finally we prove a generalization of this result and its converse for L^q(? _p ⁿ ) with appropriate q.     

On Linear Preservers of Two-Sided Gut-Majorization On <Emphasis Type="Bold">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">m</Emphasis></Subscript>

For X, Y ∈ M_n,m it is said that X is gut-majorized by Y, and we write X ?_gutY, if there exists an n-by-n upper triangular g-row stochastic matrix R such that X = RY. Define the relation ~_gut as follows. X ~_gutY if X is gut-majorized by Y and Y is gut-majorized by X. The (strong) linear preservers of ?_gut on ?ⁿ and strong linear preservers of this relation on M_n,m have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of ~_gut on ?ⁿ and M_n,m.     

Higher Algebraic <Emphasis Type="Italic">K</Emphasis>-theory of Ring Epimorphisms

Given a homological ring epimorphism from a ring R to another ring S, we show that if the left R-module S has a finite-type resolution, then the algebraic K-group K _n (R) of R splits as the direct sum of the algebraic K-group K _n (S) of S and the algebraic K-group K _n (R) of a Waldhausen category R determined by the ring epimorphism. This result is then applied to endomorphism rings, matrix subrings, rings with idempotent ideals, and universal localizations which appear often in representation theory and algebraic topology.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-consistency of wavelet estimators

This paper deals with the L ^p -consistency of wavelet estimators for a density function based on size-biased random samples. More precisely, we firstly show the L ^p -consistency of wavelet estimators for independent and identically distributed random vectors in R ^d . Then a similar result is obtained for negatively associated samples under the additional assumptions d = 1 and the monotonicity of the weight function.     

A calculus of Fourier integral operators with inhomogeneous phase functions on R<Superscript> <Emphasis Type="Italic">d</Emphasis> </Superscript>

We construct a calculus for generalized SG Fourier integral operators, extending known results to a broader class of symbols of SG type. In particular, we do not require that the phase functions are homogeneous. An essential ingredient in the proofs is a general criterion for asymptotic expansions within the Weyl-Hörmander calculus. We also prove the L²(R^d)-boundedness of the generalized SG Fourier integral operators having regular phase functions and amplitudes uniformly bounded on R^2d.     

The dual space of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> of a vector measure

For a vector measure ν having values in a real or complex Banach space and \({p \in}\) [1, ∞), we consider L ^p (ν) and \({L_{w}^{p}(\nu)}\), the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E _q (μ) and show it is σ-order continuous when p > 1. In this case, for the associate spaces we prove that L ^p (ν) ^×  = E _q (μ) and \({E_q(\mu)^\times = L_w^p(\nu)}\). It follows that \({L_p (\nu) ^{**} = L_w^p (\nu)}\). We also show that L ¹ (ν) ^×  may be equal or not to E _∞ (μ).     

On (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-absorbing ideals of commutative rings

Let R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a ₁?a _m ∈I for a ₁,…, a _m ∈R?U(R), then there are n of the a _i ’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bézout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m ? 1 maximal ideals.