Generalization of some results concerning Walsh series and the dyadic derivative

, , , . , . , , x(0,1),x2^–j ,j=1,2,..., 2 ⁿ . , ka _k 0 k k. , (0, 1) , , , , . , .     

The convolution equation of Choquet and Deny on nilpotent groups

G. Choquet and J. Deny have characterized the positive solutions of the convolution equation *= of measures on locally compact abelian groups, for a given positive measure . By elementary methods, we extend their characterization to locally compact nilpotent groups which complements the various existing results on the equation, and we work out the solutions explicitly for the Heisenberg groups and some nilpotent matrix groups, by finding all the exponential functions on these groups.     

Estimation of the Coefficient of Multiple Determination

Assume that we have iid observations on the random vector X = (X ,...,X ) following a multivariate normal distribution N (,) where both R and (p.d.) are unknown. Let denote the multiple correlation coefficient between X and (X ,...,X ). The parameter = , called the multiple coefficient of determination, indicates the proportion of variability in X explained by its best linear fit based on (X ,..., X ). In this paper we consider the point estimation of under the ordinary squared error loss function. The usual estimators (MLE, UMVUE) have complicated risk expressions and hence it is quite difficult to get exact decision-theoretic results. We therefore follow the asymptotic decision theoretic approach (as done by Ghosh and Sinha (1981, Ann. Statist., 9, 1334-1338)) and study Second Order Admissibility of various estimators including the usual ones.     

Regularization of the boundary-value problem for Sturm-Liouville equations with operator-valued coefficients

A regularization method is used to find an approximate solution to the inhomogeneous boundary-value problem –y(t)+Ay(t)=y(t)+h(t), t [0, b], cos CY–sin CY= where A is a self- adjoint unbounded operator on a Hilbert space H; C is an operator on H H; H H; h(t) L ₂ (H, (0, b)); Y, Y are regularizing boundary values of y (t).Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 23–28, 1989.     

On the interrelation of the generalized Besov-Nikol'skii and Weyl-Nikol'skii classes of functions

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Dedicated to Professors K. Tandori and L. Leindler on the occasion of their anniversaries

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This work was completed in support of the Russian Foundation of Fundamental Research (Project # 96-01-00094) and of the International Scientific Foundation (Grant # NCI-300).     

Right-ordered and free lattice-ordered groups

The free -groups over nilpotent groups are considered. The upper bound of -solvable length is found for an -group over a nilpotent group of finite rank; the lower bound of solvable length is given for a free -group over a finitely generated nilpotent group.Translated fromAlgebra i Logika, Vol. 34, No. 1, pp. 33–40, January–February, 1995.Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-1524.     

Sur la répartition des valeurs de la fonction d'Euler

Let (x) denote the number of those integers n with (n) x, where denotes the Euler function. Improving on a well-known estimate of Bateman (1972), we show that (x)-Ax R(x), where A=(2)(3)/(6) and R(x) is essentially of the size of the best available estimate for the remainder term in the prime number theorem.     

The best choice problem with an unknown number of objects

The secretary problem with a known prior distribution of the number of candidates is considered. Ifp(i)=p(N=i),i [, ] , where=inf{i :p(i) > 0} and=sup{i :p(i)0}, is the prior distribution of the numberN of candidates it will be shown that, if the optimal stopping rule is of the simple form, then the optimal stopping indexj=min satisfies asymptotically (as ) the equationj=exp .The probability of selecting the best object by the corresponding policy will be (j-1) p(i)/i. We also give an example of the distributionp for which the optimal stopping rule consists of a stopping set with two islands. We present an asymptotical solution for this example.     

Hypercentral Series and Paired Intersections of Sylow Subgroups of Chevalley Groups

Let G(k) be the Chevalley group of normal type associated with a root system G = , or of twisted type G = ^m,m = 2,3, over a field K. Its root subgroups X_s, for all possible s G⁺, generate a maximal unipotent subgroup U = UG(k) if p = charK < 0, U is a Sylow p-subgroup of G(K). We examine G and K for which there exists a paired intersection U U⁹, g G(K), which is not conjugate in G(K) to a normal subgroup of U. If K is a finite field, this is equivalent to a condition that the normalizer of U U⁹ in G(K)has a p-multiple index. Put p() = max(r,r)/(s,s) | r,s . We prove a statement (Theorem 1) saying the following. Let G(K) be a Chevalley group of Lie rank greater than 1 over a finite field K of characteristic p and U be its Sylow p-subgroup equal to UG(K); also, either G = and p() is distinct from p and 1, or G(K) is a twisted group. Then G(K) contains a monomial element n such that the normalizer U of Uⁿ in G(K) has a p-multiple index. Let K be an associative commutative ring with unity and (K,J) be a congruence subgroup of the Chevalley group (K) modulo a nilpotent ideal J. We examine an hypercentral series 1 Z₁ Z₂ ... Z_c-1 of the group U(K) (K,J). Theorem 2 shows that under an extra restriction on the quotient (J_t : J) of ideals, central series are related via Z_i = T_c-iC, 1 i < c, where C is a subgroup of central diagonal elements. Such a connection exists, in particular, if K = Z_pm and J = (p^d), 1 d < m, d| m.     

A Uniqueness Result for an Overdetermined Problem in Non-Linear Parabolic Potential Theory

In this paper we study the question of uniqueness for an inverse problem, arising in the (thermal) linear and/or non-linear potential theory. The overdetermined problem we shall study is represented by(div(|u| ^p–2u)–D _t u–+)u=0where supp()R ⁿ ×(0,), 1<p<, L and {t=} is bounded for >0.The problem has applications in shape-recognition in underground water/oil recovery, subject to shape-change during time intervals. The particular case u0, D _t u0, and p=2, is an example of the well-known Stefan.     

Canonical Semigroups of States and Cocycles for the Group of Automorphisms of a Homogeneous Tree

Let G=A ut(T) be the group of automorphisms of a homogeneous tree and let d(v,gv) denote the natural tree distance. Fix a base vertex e in T. The function (g)=exp(–d(e,ge)), being positive definte on G, gives rise to a semigroup of states on G whose infinitesimal generator d/d|₌₀=log() is conditionally positive definite but not positive definite. Hence, log() corresponds to a nontrivial cocycle (g): GH in some representation space H . In contrast with the case of PGL(2,), the representation is not irreducible.Let _o (g) be the derivative of the spherical function corresponding to the complementary series of A ut(T). We show that –d(e,ge) and _o (g) come from cohomologous cocycles. Moreover, _o is associated to one of the two (irreducible) special representations of A ut(T).     

Essential Spanning Forests and Electrical Networks on Groups

Let be a Cayley graph of a finitely generated group G. Subgraphs which contain all vertices of , have no cycles, and no finite connected components are called essential spanning forests. The set of all such subgraphs can be endowed with a compact topology, and G acts on by translations. We define a uniform G-invariant probability measure on show that is mixing, and give a sufficient condition for directional tail triviality. For non-cocompact Fuchsian groups we show how can be computed on cylinder sets. We obtain as a corollary, that the tail -algebra is trivial, and that the rate of convergence to mixing is exponential. The transfer-current function (an analogue to the Green function), is computed explicitly for the Modular and Hecke groups.     

Parabelscharen und grenzkreise in der isotropen ebene

It is known by H. Sachs [5] that the classical curve theorem of ABRAMESCU also holds in isotropic geometry. Generalising an idea due to O. Röschel [2] we regard all inscribed parabolas (s, t) of a triangle (t). This triangle is formed by the tangents of three neighbouring points of a C -curve k(t) in an isotropic plane. Let U((t)) be the circumcircle of (t) and I((t)) the incircle of the triangle (t) whose midpoints of the sides are the vertices of (t). The circle U((t)) is the locus of the isotropic focal points of (s, t) and the incircle I((T)) the envelope of the isotropic axes of (s, t). We prove that the ABRAMESU-circle — lim U((t)) — is identical with the locus of the focal points of lim (s, t) and the circle lim I((t)) with the envelope of the axes of lim (s, t). The characteristic points, different from k(t), of the circles lim U((t)) and lim I((t)) determine the direction of the affine-normal of k(t).Herrn Professor Helmut Mäurer zum 60. Geburtstag gewidmet     

Difference Sets with n = 2pm

Let D be a (v,k,) difference set over an abelian group G with even n = k - . Assume that t N satisfies the congruences t q _i ^fi (mod exp(G)) for each prime divisor q_i of n/2 and some integer f_i. In [4] it was shown that t is a multiplier of D provided that n > , (n/2, ) = 1 and (n/2, v) = 1. In this paper we show that the condition n > may be removed. As a corollary we obtain that in the case of n= 2p^a when p is a prime, p should be a multiplier of D. This answers an open question mentioned in [2].     

4-Dimensional compact projective planes with small nilradical

We consider 4-dimensional compact projective planes with a solvable 6-dimensional collineation group and with orbit type (2, 1), i.e. fixes a flagv W, acts transitively onL \{W} and fixes no point in the setW\{v}. We We prove a series of lemmas concerning the action of invariant subgroups of . These lemmas are applied to prove that the maximal connected nilpotent invariant subgroup of has dimension at least 4.Dedicated to Prof. H. Salzmann on the occasion of his 65th birthday     

Замечания о Сходкмости лагранжева интерполрования

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Semimodular lattices with isomorphic graphs

Two discrete modular lattice and have isomorphic graphs if and only if is of the form A × and is of the form A × for some lattices A and and . We prove that for discrete semimodular lattices and this latter condition holds if and only if and have isomorphic graphs and the isomorphism preserves the order on all cover-preserving sublattices of which are isomorphic to the seven-element, semimodular, nonmodular lattice (see Figure 1). This answers in the affirmative a question posed by J. Jakubik.     

On Ordering the Groups with Nilpotent Commutant

We prove that every group with nilpotent commutant, having an abelian normal subgroup such that the factor by this subgroup is nilpotent, is preorderable if and only if the group is -torsion-free. An example is exhibited of a nonorderable -torsion-free group with two-step nilpotent radical. This example demonstrates that for the variety of groups with nilpotent commutant the absence of -torsion in a group is not a sufficient condition for orderability.     

A density function and the structure of singularities of the mean curvature flow

We study singularity formation in the mean curvature flow of smooth, compact, embedded hypersurfaces of non-negative mean curvature in ⁿ⁺¹, primarily in the boundaryless setting. We concentrate on the so-called Type I case, studied by Huisken in [Hu 90], and extend and refine his results. In particular, we show that a certain restriction on the singular points covered by his analysis may be removed, and also establish results relating to the uniqueness of limit rescalings about singular points, and to the existence of slow-forming singularities of the flow.The main new ingredient introduced, to address these issues, is a certain density function, analogous to the usual density function in the study of harmonic maps in the stationary setting. The definition of this function is based on Huisken's important monotonicity formula for mean curvature flow.