Vertices of simple modules and normal subgroups ofp-solvable groups

Let π be a set of prime numbers andG a finite π-separable group. Let θ be an irreducible π′-partial character of a normal subgroupN ofG and denote by I_π′ (G‖θ), the set of all irreducible π′-partial characters Φ ofG such that θ is a constituent of Φ_N. In this paper, we obtain some information about the vertices of the elements in I_π′ (G‖θ). As a consequence, we establish an analogue of Fong's theorem on defect groups of covering blocks, for the vertices of the simple modules (in characteristicsp) of a finitep-solvable group lying over a fixed simple module of a normal subgroup.     

Solution of monotone complementarity problems with locally Lipschitzian functions

The paper deals with complementarity problems CP(F), where the underlying functionF is assumed to be locally Lipschitzian. Based on a special equivalent reformulation of CP(F) as a system of equationsφ(x)=0 or as the problem of minimizing the merit functionΘ=1/2∥Φ∥ ₂ ² , we extend results which hold for sufficiently smooth functionsF to the nonsmooth case. In particular, ifF is monotone in a neighbourhood ofx, it is proved that 0 εδθ(x) is necessary and sufficient forx to be a solution of CP(F). Moreover, for monotone functionsF, a simple derivative-free algorithm that reducesΘ is shown to possess global convergence properties. Finally, the local behaviour of a generalized Newton method is analyzed. To this end, the result by Mifflin that the composition of semismooth functions is again semismooth is extended top-order semismooth functions. Under a suitable regularity condition and ifF isp-order semismooth the generalized Newton method is shown to be locally well defined and superlinearly convergent with the order of 1+p.     

On the asymptotic efficiency of estimators in an autoregressive process

Summary Let {X _t } be defined recursively byX _t =θX _t−1+U _t (t=1,2, ...), whereX ₀=0 and {U _t } is a sequence of independent identically distributed real random variables having a density functionf with mean 0 and varianceσ ². We assume that |θ|<1. In the present paper we obtain the bound of the asymptotic distributions of asymptotically median unbiased (AMU) estimators of θ and the sufficient condition that an AMU estimator be asymptotically efficient in the sense that its distribution attains the above bound. It is also shown that the least squares estimator of θ is asymptotically efficient if and only iff is a normal density function. University of Electro-Communications     

Simultaneous estimation of location parameters of the distribution with finite support

Summary LetX _i ,i=1,..., p be theith component of thep×1 vectorX=(X ₁,X ₂,...,X _p )′. Suppose thatX ₁,X ₂,...,X _p are independent and thatX _i has a probability density which is positive on a finite interval, is symmetric about θ _i and has the same variance. In estimation of the location vector θ=(θ₁, θ₂,...,θ _p )′ under the squared error loss function explicit estimators which dominateX are obtained by using integration by parts to evaluate the risk function. Further, explicit dominating estimators are given when the distributions ofX _i ′s are mixture of two uniform distributions. For the loss function such an estimator is also given when the distributions ofX _i ′s are uniform distributions.     

Multiparametric estimating equations

Summary Let {p(x, θ): θ∈Θ} be a family of densities where θ=(θ₁,θ₂), being θ₁ ∈ Θ₁ ak-dimensional parameter of interest, θ₂ ∈ Θ₂ a nuisance parameter and Θ=Θ₁×Θ₂. To estimate θ₁, vector estimating equations g(x,θ₁)=(g₁(x,θ₁),...,g_k(x,θ₁))=0 are considered. The standardized form of g(x,θ₁) is defined as g_s=(E_θ(∂g/∂θ′₁))⁻¹g. Then, within the classG ₁ of unbiased equations (i.e. satisfying E_θ(g)=0 (θ∈Θ)), an equationg ^*=0 is said to be optimum if the covariance matrices ofg _s andg _s ^* are such that is non-negative definite for allg∈ G ₁ and θ∈Θ. Sufficient conditions for optimality are discussed and, in particular, conditions for the optimality of the maximum conditional likelihood equation are analyzed. Special attention is given to non-regular cases. In addition, measures of the information about θ₁ contained in an estimating equation are presented and a Rao-Blackwell theorem is given. CIENES     

Improved confidence set estimators of a multivariate normal mean and generalizations

Summary LetX be the observed vector of thep-variate (p≧3) normal distribution with mean θ and covariance matrix equal to the identity matrix. Denotey ₊=max{0,y} for any real numbery. We consider the confidence set estimator of θ of the formC _δa,φ={θ:|θ−δ^a,φ(X)}≦c}, whereδ ^a,φ=[1−aφ({X})/{X}²]₊X is the positive part of the Baranchik (1970,Ann. Math. Statist.,41, 642–645) estimator. We provide conditions on ϕ(•) anda which guarantee thatC _δa.φ has higher coverage probability than the usual one, {θ:|θ−X|≦c}. This dominance result will be shown to hold for spherically symmetric distributions, which include the normal distribution,t-distribution and double exponential distribution. The latter result generalizes that of Hwang and Chen (1983,Technical Report, Dept. of Math., Cornell University).     

On semigroups generated by restrictions of elliptic operators to invariant subspaces

Let A be the closed unbounded operator inL _p(G) that is associated with an elliptic boundary value problem for a bounded domainG. We prove the existence of a spectral projectionE determined by the set Γ = {λ;θ ₁≦argλ≦θ ₂} and show thatAE is the infinitesimal generator of an analytic semigroup provided that the following conditions hold: 1<p<∞; the boundary ϖΓ of Γ is contained in the resolvent setp(A) ofA;π/2≦θ<θ ₂≦3π/2 ; and there exists a constantc such that (I)││(λ-A)^-1││≦c/│λ│ for λ∈ϖΓ. The following consequence is obtained: Suppose that there exist constantsM andc such that λ∈p(A) and estimate (I) holds provided that |λ|≧M and Re λ=0. Then there exist bounded projectionE ⁻ andE ⁺ such thatA is completely reduced by the direct sum decompositionL _p(G)=E⁻L_p (G) ⊕E⁺L_p (G) and each of the operatorsAE ⁻ and—AE ⁺ is the infinitestimal generator of an analytic semigroup.     

On the growth of mass for a viscous Hamilton-Jacobi equation

We investigate the large time behavior of positive solutions with finite mass for the viscous Hamilton-Jacobi equationu _t = Δu + |Δu| ^p ,t>0,x ∈ ℝ ^N , wherep≥1 andu(0,.)=u ₀≥0,u ₀≢0,u ₀∈L ¹. DenotingI _∞=lim _t→∞‖u(t)‖₁≤∞, we show that the asymptotic behavior of the mass can be classified along three cases as follows:

Zero sets and multiplier theorems for star-invariant sub spaces

Given an inner function θ, let {Kskθ/p}:= H^p ∩θ {Hsk0/p} be the corresponding star-invariant subspace of the Hardy spaceH ^p. We show that, unless θ is a finite Blaschke product, the zero sets for K _θ ^p -spaces are different for different p’s. We also investigate the (non)stability of zero sets when passing from {Kskθ/p} to {Ksku/q}, whereq > p and u is an inner function divisible by θ. This problem is motivated by the Beurling-Malliavin multiplier theorem for entire functions, and we solve it (at least in a natural special case) by proving an appropriate multiplier theorem for K _θ ^p .     

Universal non-compact operators between super-reflexive Banach spaces and the existence of a complemented copy of Hilbert space

Suppose that 1<p≦2, 2≦q<∞. The formal identity operatorI:l _p→l _qfactorizes through any given non-compact operator from ap-smooth Banach space into aq-convex Banach space. It follows that ifX is a 2-convex space andY is an infinite dimensional subspace ofX which is isomorphic to a Hilbert space, thenY contains an isomorphic copy ofl ₂ which is complemented inX.     

Small fractional parts of additive forms

Letf(x)=θ₁ x ₁ ^k +...+θ _s x _s ^k be an additive form with real coefficients, and ∥α∥ = min {|α-u|:uεℤ} denote the distance fromα to the nearest integer. We show that ifθ ₁,…,θ _s , are algebraic ands = 4k then there are integersx ₁,…,x _s , satisfying l ≤x ₁,≤ N and ∥f(x)∥ ≤ N ^E , withE = − 1 + 2/e. Whens = λk, 1 ≤λ ≤ 2k, the exponentE may be replaced byλE/4, and if we drop the condition thatθ ₁,…,θ _s , be algebraic then the result holds for almost all values of θεℝ ^s . Whenk ≥ 6 is small a better exponent is obtained using Heath-Brown’s version of Weyl’s estimate.     

Additive summable processes and their stochastic integral

We define and study a class of summable processes, called additive summable processes, that is larger than the class used by Dinculeanu and Brooks [D-B]. We relax the definition of a summable processesX:Ω×ℝ₊→E⊂L(F, G) by asking for the associated measureI _X to have just an additive extension to the predictableσ-algebra ℘, such that each of the measures (I _X) _z , forz∈(L _G ^p )*, beingσ-additive, rather than having aσ-additive extension. We define a stochastic integral with respect to such a process and we prove several properties of the integral. After that we show that this class of summable processes contains all processesX:Ω×ℝ₊→E⊂L(F, G) with integrable semivariation ifc ₀ ∋G.     

? 1-Summability of d-Dimensional Fourier Transforms

A general summability method of more-dimensional Fourier transforms is given with the help of a continuous function θ. Under some weak conditions on θ we show that the maximal operator of the ℓ ₁–θ-means of a tempered distribution is bounded from H _p (ℝ ^d ) to L _p (ℝ ^d ) for all d/(d+α)<p≤∞ and, consequently, is of weak type (1,1), where 0<α≤1 depends only on θ. As a consequence we obtain a generalization of the one-dimensional summability result due to Lebesgue, more exactly, the ℓ ₁–θ-means of a function f∈L ₁(ℝ ^d ) converge a.e. to f. Moreover, we prove that the ℓ ₁–θ-means are uniformly bounded on the spaces H _p (ℝ ^d ), and so they converge in norm (d/(d+α)<p<∞). Similar results are shown for conjugate functions. Some special cases of the ℓ ₁–θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de La Vallée-Poussin, Rogosinski, and Riesz summations.     

Embedding (p,p−1) graphs in their complements

A graphG is embeddable in its complement ifG is isomorphic with a subgraph of . A complete characterization is given of those (p,p−1) graphs which are embeddable in their complements. In particular, letG be a (p,p−1) graph wherep≧6 ifp is even andp≧9 ifp is odd; thenG is embeddable in if and only ifG is neither the starK _1,p−1 norK _1,n ∪C ₃ withn≧4.     

Boundedness criteria for singular integrals in weighted grand lebesgue spaces

Boundedness criteria for the Calderón singular integral, Riesz transform and Cauchy singular integral in generalized weighted grand Lebesgue spaces L ^p),θ _w , 1 < p < ∞, are studied. It is shown that an operator K of this type is bounded in L ^p),θ _w if and only if the weight w satisfies the Muckenhoupt A _p condition. Bibliography: 15 titles.     

Restricted summability of Fourier transforms and local Hardy spaces

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.     

Regularity of ideals and their radicals

In this paper we compare the regularityreg I of a homogeneous idealI⊂K[x ₁,...,x _n] with that of its radical. We prove that regI ≥ reg ifR/I is a BuchsbaumR-module or ifI is a monomial ideal. We also prove the same result when defines a non-singular curve inP ³ under some additional hypotheses     

Representations of deformed preprojective algebras and quantum groups

Let (Γ,I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group Zd. In this paper, we list all indecomposable representations of (Γ,I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Πλ(Γ,I). It is shown that those representations given by extending indecomposable representations of (Γ,I) are all simple representations of Πλ(Γ,I). Therefore, it is concluded that all simple representa-tions of rest...     

New estimators of discriminant coefficients as the gradient of log-odds

We consider the problem of estimating the discriminant coefficients, η=∑^1-(θ⁽¹⁾-θ⁽²⁾) based on two independent normal samples fromN _p (θ⁽¹⁾,∑) andN _p (θ⁽²⁾,∑). We are concerned with the estimation of η as the gradient of log-odds between two extreme situations. A decision theoretic approach is taken with the quadratic loss function. We derive the unbiased estimator of the essential part of the risk which is applicable for general estimators. We propose two types of new estimators and prove their dominance over the traditional estimator using this unbiased estimator.     

Closures of finitely generated ideals in Hardy spaces

LetH ^∞ be the algebra of bounded analytic functions in the unit diskD. LetI=I(f ₁,...,f _N) be the ideal generated byf ₁,...,f _N∈H ^∞ andJ=J(f ₁,...,f _N) the ideal of the functionsf∈H ^∞ for which there exists a constantC=C(f) such that |f(z)|≤C(|f ₁ (z)|+...;+|f _N (z)|),z∈D. It is clear that , but an example due to J. Bourgain shows thatJ is not, in general, in the norm closure ofI. Our first result asserts thatJ is included in the norm closure ofI ifI contains a Carleson-Newman Blaschke product, or equivalently, if there existss>0 such that