On Differences of Semicubical Powers

For X,Y,>0, let and define I ₈(X,Y,) to be the cardinality of the set. In this paper it is shown that, for >0, Y ²/X ³=O(), =O(Y ³/X ³) and X=O (Y ²), one has I ₈(X,Y,)=O(X ² Y ²+X min (X ^{3/2} Y ³, X ^{11/2} Y ^{–1})+X min (^{1/3} X ² Y ³, X ^{14/3} Y ^{1/3})), with the implicit constant depending only on . There is a brief report on an application of this that leads, by way of the Bombieri-Iwaniec method for exponential sums, to some improvement of results on the mean squared modulus of a Dirichlet L-function along a short interval of its critical line.     

On the asymptotic and invariantσ-algebras of random walks on locally compact groups

Summary Consider a random walk of law on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG . We study the relation between the spacesL (G , ^a ,Q) andL (G , ⁱ ,Q) where ^a and ⁱ stand for the asymptotic and invariant -algebras, respectively. We obtain a factorizationL (G , ^a ,Q) L (G , ⁱ ,Q)L (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powers ⁿ.     

Barth type vanishing theorems

Let X be a smooth, projective, d-dimensional subvariety of ⁿ (). Barth's theorem says that H ^q (X, ^p _X )=0 when pq and q+p2d–n (if p=0 we must have q>0). It is very interesting to look for analogous vanishing theorems for H ^q (X, ^p _X (m)), m (see [S-S], [F], [S]). In this paper we prove some vanishing theorems for H ^q (X, ^p _X (1)), for H ^q (X, ^p _X (m)) when m–1, and, if dim(X)=n–2, for H ^q (X, ² _X (m)) and H ^q (X, S ^{k 1} _X (m)). We use standard techniques and some of our previous results.     

A Class of Iterative Algorithms and Solvability of Nonlinear Variational Inequalities Involving Multivalued Mappings

The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented.Determine an element x ^* K and u ^* T(x ^*) such that u ^*, x – x ^* 0 for all x K where T: K P(H) is a multivalued mapping from a real Hilbert space H into P(H), the power set of H, and K is a nonempty closed convex subset of H. The iterative procedure adopted here is represented by a nonlinear variational inequality: for arbitrarily chosen initial points x ⁰, y ⁰ K, u ⁰ T(y ⁰) and v ⁰ T(x ⁰), we have u ^k + x ^k+1 – y ^k , x – x ^k+1 0, x K, for u ^k T(y ^k ) and for k 0where v ^k + y ^k – x ^k , x – y ^k 0, x K and for v ^k T(x ^k ).     

GBRD's: Some new constructions for difference matrices,generalised hadamard matrices and balanced generalised weighing matrices

A new construction is given for difference matrices. The generalized Hadamard matrices GH(q(q – 1)²; EA(q)) are constructed whenq andq – 1 are both prime powers. Other generalised Hadamard matrices are also shown to exist. For example, there exist GH(n; G) forn = 5² 2 3, 2⁶ 3², 11² 2² 3, 17² 2 3², 53² 2 3³, 71² 2² 3², 107² 2² 3³, 149² 5² 2 3,.... Finally, a new construction for the BGW ((q ⁴ – 1)/(q – 1),q ³,q ²(q – 1);q _q-1), and a construction for the new BGW ((q ⁸ – 1)/(q ² – 1),q ⁶,q ⁴(q ² – 1);G) are given, wheneverq is a prime power, andG is a group of orderq + 1.     

Carleson measure characterization of Bloch functions

We give several equivalences of Bloch functions and little Bloch functions. Using these results we obtain the generalized Carleson measure characterization of Bloch functions and the generalized vanishing Carleson measure characterization of little Bloch functions, that is,f B if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized Carleson measure;f B ₀ if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized vanishing Carleson measure, whereD f( > 0) is the fractional derivative of analytic functionf of order, m denotes the normalised Lebesgue measure.Supported partly by the Young Teacher Natural Science Foundation of Shandong Province.     

Bessel Potentials with Logarithmic Components and Sobolev-Type Embeddings

In a recent paper Edmunds, Gurka, and Opic [5] showed that Sobolev spaces of order k, based on the Zygmund spaces L ^n/k (log L) (R ⁿ ), are continuously embedded into L (R ⁿ ) if > 1/p, p n/k. In this paper we replace L ^n/k (log L) (R ⁿ ) by the Lebesgue space L ^n/k (R ⁿ ) and increase the smoothness of the functions involved by a "logarithmic" order > 1/p to obtain the continuous embedding into L (R ⁿ ). Both approaches turn out to be equivalent. We also derive results of Trudinger-type [16] on embeddings into Orlicz spaces in the limit case k = n/p as well as results of Brézis-Wainger-type [2] on almost Lipschitz continuity in the superlimiting case k = n/p + 1.     

Existence of weak solutions of 2-D Euler equations with initial vorticityω 0∈L(log+ L) α (α>0)

It is proved that there exist global weak solutions of 2-D Euler equations inR ² under the assumption that the initial vorticity belongs to a kind of wider spaces,L ¹L(log⁺ L) (>0), which are Orlicz spaces containing spacesL ^p L ¹,L(log⁺ L) L (>1/2) and so on. This result improves on that of [2], [4], [11]. Moreover, these solutions are obtained by vanishing the viscosity term of Navier-Stokes equations.     

Integrals involving Gegenbauer and Hermite polynomials on the imaginary axis

Summary The integral _- [C _2n (it)]^–2(1+t ²)^–-1/2 dt is evaluated for > –1/2 whereC _2n is the Gegenbauer polynomial of degree 2n. Letting gives the value _- [H _2n (it)]^–2 e ^{1-1/2t 2} dt involving the Hermite polynomialH _2n of degree 2n. The result is obtained using Gegenbauer functions of the second kind.     

Sur les égalités du miroir et certaines formes faibles de la conjecture de Greenberg

Let p be an odd prime. For any CM number field K containing a primitive p^th-root of unity, class field theory and Kummer theory put together yield the well known reflection inequality ^+– between the plus and minus parts of the -invariant of K. Greenbergs conjecture asserts that ⁺ is always trivial. We study here a weak form of this conjecture, namely ⁺=^– if and only if ⁺=^–=0.     

An O(n 2) Active Set Algorithm for Solving Two Related Box Constrained Parametric Quadratic Programs

Recently, O(n ²) active set methods have been presented for minimizing the parametric quadratic functions (1/2)x Dx–a x+| x–c| and (1/2)x Dx–a x+(/2)( x–c)², respectively, subject to lxb, for all nonnegative values of the parameter . Here, D is a positive diagonal n×n matrix, and a are arbitrary n-vectors, c is an arbitrary scalar; l and b are arbitrary n-vectors such that lb. In this paper, we show that each one of these algorithms may be used to simultaneously solve both parametric programs withno additional computational cost.     

Separation of hypersurfaces

We give a generalization of results obtained in [15]. LetK ⁿ denote the set of embedded hypersurfaces in ⁿ⁺¹; for all xSⁿ and MK ⁿ we denote by C _x ^M the apparent contour ofM in the directionx. Then we give a sufficient condition on WSⁿ such that the map _W ^{K n}:K ⁿ P(T Sⁿ) , defined by _W ^{K n} (M)={C _w ^M ¦ wW}, is injective.     

Complexes whose boundaries cannot be pushed around

LetK ^{n n} be a triangulatedn-ball. Examples are given to show that unlike in the two-dimensional case, the following hold for alln3: (1) there are nonconvexK ⁿ with no convex simplexwise linear embeddingsK ^{n n} , even though there are strictly convex simplexwise linear embeddings K ^{n n} ; (2) there are convexK ⁿ , with no spanning simplices, such that not every simplexwise linear embeddingf: K ^{n n} with convex image can be extended to a simplexwise linear embedding ofK ⁿ ; (3) there are convexK ⁿ such that the space of simplexwise linear homeomorphisms ofK ⁿ , fixed on K ⁿ , is not path connected.Partially supported by NSF Contract DMS-8503388.     

Fano'sche Moufang-Ebenen mit Gauss-Figur sind pappos'sch

LetV be a quadrilateral in aMoufang-plane , in which theFano-proposition is valid. Take the pointsP,Q,R respectively in the diagonalsp,q,r ofV and construe the pointsP ^*,Q ^*,R ^* inp,q, r harmonic toP,Q,R with respect to pairs of edges ofV. IfP,Q,R are collinear, so areP ^*,Q ^*,R ^*, if and only if is aPappos-plane. Is V classical, the pointsP ₁ p,Qq,Rr and their harmonic conjugatesP ₁ ^* ,Q ^*,R ^* (construed as above mentioned) lay in a curve of 2^nd order.

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R. Artzy zum 70. Geburtstag zugeeignet     

A Criterion of Sign Definiteness for Quadratic and Quasi-Quadratic Forms in a Cone

A new criterion of sign definiteness is given for quadratic and quasi-quadratic forms in a cone in space ⁿ . The basic results are proved for the nonnegative cone K ₊ ⁿ : x 0, x ⁿ . A change of variables reduces the case of an arbitrary octant of space ⁿ to K ₊ ⁿ .     

Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations

Summary In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) _t v =v +f(x,v )+(x,v ) . Here is a strongly-elliptic second-order operator with constant coefficients, h:=DH _xx-h, and the space variablex takes values on the unit circleS ¹. The functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<mM wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv ² is aC ^k (S ¹)-valued Markov process for each 0<1/2, whereC (S ¹) is the Banach space of real-valued continuous functions onS ¹ which are Hölder-continuous of exponent . We prove, under some further natural assumptions onf and which imply that the zero element ofC (S ¹) is a globally exponentially stable critical point of the unperturbed equation _t ⁰ = ⁰ +f(x,⁰), that has a unique stationary distributionv ^K, on (C (S ¹), (C ^K (S ¹))) when the perturbation parameter is small enough. Some further calculations show that as tends to zero,v ^K, tends tov ^K,0, the point mass centered on the zero element ofC (S ¹). The main goal of this paper is to show that in factv ^K, is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv ^K, is the LDP for the process , which has been shown in an earlier paper. Our methods of deriving the LDP forv ^K, based on the LDP for are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC (S ¹) is inherently infinite-dimensional.This work was performed while the author was with the Department of Mathematics, University of Maryland, College Park, MD 20742, USA     

Conditions for separately subharmonic functions to be subharmonic

Letu be a function on ^m × ⁿ , wherem2 andn2, such thatu(x, .) is subharmonic on ⁿ for each fixedx in ^m andu(.,y) is subharmonic on ^m for each fixedy in ⁿ . We give a local integrability condition which ensures the subharmonicity ofu on ^m × ⁿ , and we show that this condition is close to being sharp. In particular, the local integrability of (log⁺ u ⁺) ^m+n–2+ is enough to secure the subharmonicity ofu if >0, but not if <0.     

Positive definite dot product kernels in learning theory

In the classical support vector machines, linear polynomials corresponding to the reproducing kernel K(x,y)=xy are used. In many models of learning theory, polynomial kernels K(x,y)=_l=0^Na_l(xy)^l generating polynomials of degree N, and dot product kernels K(x,y)=_l=0⁺a_l(xy)^l are involved. For corresponding learning algorithms, properties of these kernels need to be understood. In this paper, we consider their positive definiteness. A necessary and sufficient condition for the dot product kernel K to be positive definite is given. Generally, we present a characterization of a function f:RR such that the matrix [f(xⁱx^j)]_i,j=1^m is positive semi-definite for any x¹,x²,...,x^mRⁿ, n2. Supported by CERG Grant No. CityU 1144/01P and City University of Hong Kong Grant No. 7001342.AMS subject classification 42A82, 41A05     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.