L p Spectral Independence of Elliptic Operators via Commutator Estimates

Let {T _p:q ₁ p q ₂} be a family of consistent C ₀ semigroups on L ^p(), with q ₁,q ₂ [1,) and open. We show that certain commutator conditions on T _p and on the resolvent of its generator A _p ensure the p independence of the spectrum of A _p for p [q ₁,q ₂.Applications include the case of Petrovskij correct systems with Hölder continuous coefficients, Schrödinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coefficients.     

Systeme de poids sur une algebre de Lie nilpotente

Let g be anilpotent Lie algebra (of finite dimensionn over an algebraically closed field of characteristic zero) and let Der(g) be the algebra of derivations of g. Thesystem of weights of g is defined as being that of the standard representation of a maximal torus in Der(g) (see l.l). For a fixed integern, it is well-known that there are in general uncountably many isomorphism classes of nilpotent Lie algebra of dimensionn; but we show that there arefinitely many systems of weights, and each of them is explicitely constructed. The class of those Lie algebras having a given (arbitrary) system of weights is also studied.The first chapter is a setting for the study of nilpotent Lie algebras, used to prove some general theorems. In the second chapter, attention is restricted to a class of nilpotent Lie algebras for which our setting is particularly well adapted.

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Ce papier est extrait de mon travail de thèse [5] effectué sous la direction du Professeur Jean de Siebenthal que je remercie vivement.     

Representations indecomposables de dimension finie des algebres de Lie

Our problem is to determine which are the finite dimensional Lie algebras such that certain undercategories of the category of finite dimensional g-modules have only a finite number of indecomposable objects, up to isomorphism. As the study of the graph of g permits us to eliminate many Lie algebras, we construct it explicitely in the solvable case and indicate how to obtain it in the general case. For this, we give a characterization of the g-ideal, annihilator of the finite dimensional g-modules of height 2. Then it remains two types of Lie algebras which a supplementary study eliminates also. The result is: a Lie algebra g is solution of the problem if and only if its radical has dimension 1, and then g is uniserial.     

Factorization of One-Generated Formations

A large class of one-generated factorable foliated formations including composition, bicanonical, and other formations is described. V. A. Vedernikov suggested to study foliated formations with directions such that , where and are also some directions, from a unified point of view. We study one-generated factorable foliated formations with directions from the interval [,], where and are the directions of the bicanonical and composition formations, respectively. In particular, all irreducible factorizations of one-generated composition formations are studied.     

Generalized sequencing problem “Towers of Hanoi”

ForpN certain integer-valued functionsA _p (x), defined forx N {0}, are studied. These functions occur in a functional equation system corresponding to a generalized version of the transportation game Towers of Hanoi and their values may be interpreted as minimum numbers of moves. An explicit representation ofA _p (x) is given and so-called minimum partitions ofx with respect top are determined for allx N. The minimum partitions ofx are of interest concerning the realisation of the minimum number of moves by optimal policies.

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Zusammenfassung Es werden fürp N gewisse ganzzahlige für allex N {0} erklärte FunktionenA _p (x) untersucht, die bei einer Verallgemeinerung des unter dem Namen Türme zu Hanoi bekannten Transportspiels in einem entsprechenden Funktionalgleichungssystem auftreten und deren Funktionswerte sich als Mindestzugzahlen interpretieren. Es werden fürA _p (x) eine explizite Darstellung und sogenannte Minimalzerlegungen vonx bezüglichp für allex N bestimmt. Die Minimalzerlegungen vonx spielen eine besondere Rolle bei der Angabe von optimalen Strategien zur Realisierung der Mindestzugzahl.

     

On Dirichlet Processes Associated with Second Order Divergence Form Operators

Let be a d - dimensional Markov family corresponding to a uniformly elliptic second order divergence form operator. We show that for any quasi continuous in the Sobolev space the process (X) admits under P ^x a decomposition into a martingale additive functional (AF) M and a continuous AF A of zero quadratic variation for almost every starting point x if q=2, for quasi every x if q>2 and for every if is continuous, d=1 and or d>1 and q>d. Our decomposition enables us to show that in the case of symmetric operator the energy of A equals zero if q=2 and that the decomposition of (X) into the martingale AF M and the AF of zero energy A is strict if for some q>d. Moreover, our decomposition provides a probabilistic representation of A .     

On a coupled system of abstract differential equations similar to the thermoelasticity equations

The study of a model system of differential equations arising from the dynamical problems of thermo elasticity is continued. The case of shifts of the general type is investigated. We employ the commutant method based on the properties of the operator (A,B)=AB-BA.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 45, No. 8, pp. 1162–1165, August, 1993.     

Non-symmetric 2-designs modulo 2

Necessary conditions are obtained for the existence of a 2 – (v, k, ) design, for which the block intersection sizess ₁,s ₂, ...,s _n satisfys ₁ s ₂ ... s _n s (mod 2 ^e ), wheree is odd. These conditions are obtained by combining restrictions on the Smith Normal Form of the incidence matrix of the design with some well known properties of self-orthogonal binary codes with all weights divisible by 4.Research done at AT&T Bell Laboratories.     

Interpolation theorems for a family of spanning subgraphs

Let G be a graph with order p, size q and component number . For each i between p – and q, let be the family of spanning i-edge subgraphs of G with exactly components. For an integer-valued graphical invariant if H H is an adjacent edge transformation (AET) implies |(H)-(H^')|1 then is said to be continuous with respect to AET. Similarly define the continuity of with respect to simple edge transformation (SET). Let M _j() and m _j() be the invariants defined by . It is proved that both M _p–() and m _p–(;) interpolate over , if is continuous with respect to AET, and that M _j() and m _j() interpolate over , if is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.     

Semiregular Large Sets

We develop the notion of t-homogeneous, G-semiregular large sets of t-designs, show that there are infinitely many 3-homogeneous PSL(2, q)-semiregular large sets when q 3 mod 4, two sporadic 3-homogeneous AL(1,32)-semiregular large sets, and no other interesting t-homogeneous G-semiregular large sets for t 3.     

On the evaluation of multivariate polynomials and their derivatives

LetH be a polynomial inn>1 variables over the fields of real or complex numbers. An algorithm is presented here for the simultaneous evaluation ofH and its first and second (F-) derivativesH andH, or of any combination ofH,H,H. The evaluations ofH alone or ofH andH together are of the same order inn andd whered is the degree ofH, while the computation ofH,H, andH isd times this order. The process takes account of the sparsity pattern ofH by using a tree structure induced by the nonzero coefficients. It also allows for simultaneous operation with several polynomials with the same sparsity pattern. The data structure for the method is rather simple in nature and can be adapted easily to specific types of polynomials. Several possible implementations and their complexity are discussed.     

Phragmén-Lindelöf theorems for second-order quasilinear elliptic equations

Analogues are formulated of the well-known, in the theory of analytic functions, Phragmen-Lindelöf theorem for the gradients of solutions of a broad class of quasilinear equations of elliptic type. Examples are given illustrating the accuracy of the results obtained for the gradients of solutions of the equations of the form div(|U|^–2u)=f(x, u, u), where f(x, u, u) is a function locally bounded in ²ⁿ⁺¹. f(x, 0, u)=0, uf(x, u, u) c¦u¦1+q(1+ ¦u|), > 1, c > 0, q > 0, is an arbitrary real number, and n >- 2. The basic role in the technique employed in the paper is played by the apparatus of capacitary characteristics.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1376–1381, October, 1992.The author sincerely appreciates E. M. Landis's permanent attention and numerous useful discussions.     

Some inequalities about the covering radius of Reed-Muller codes

Let R(r, m) be the rth order Reed-Muller code of length 2 ^m , and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University.     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

Two-parameter Vilenkin multipliers and a square function

Two-parameter Vilenkin systems will be investigated. First we give a general sufficient condition for multipliers to be bounded between two-dimensional Hardy spaces H ^q(0<q1). By means of interpolation and duality argument, this theorem can be extended to other spaces. As a consequence, we can prove the (H ^q , L ^q)-boundedness of the Sunouchi operator U with respect to two-parameter Vilenkin systems for all 0 <q 1. Moreover, the equivalence f{H_q} ~ Uf_q (f H^q)follows for 1/2<q 1.     

On primitive ideals

We extend two well-known results on primitive ideals in enveloping algebras of semisimple Lie algebras, the Irreducibility theorem for associated varieties and Duflo theorem on primitive ideals, to much wider classes of algebras. Our general version of the Irreducibility Theorem says that if A is a positively filtered associative algebra such that gr A is a commutative Poisson algebra with finitely many symplectic leaves, then the associated variety of any primitive ideal in A is the closure of a single connected symplectic leaf. Our general version of the Duflo theorem says that if A is an algebra with a triangular structure, see § 2, then any primitive ideal in A is the annihilator of a simple highest weight module. Applications to symplectic reflection algebras and Cherednik algebras are discussed.     

The Structure of Hopf Algebras with a Weak Projection

We analyze the structure of a Hopf algebra H that has a Hopf subalgebra H " and a left H "-module coalgebra projection onto H ". In this situation H H " Q for Q = H / H "⁺ H, and the Hopf algebra structure on H can be recovered from suitable structures on Q, among others an in general nonassociative multiplication. The construction of H from H " and Q generalizes Radford biproduct, double crossproducts, and certain bicrossproducts. Further examples are Hopf algebras with a triangular decomposition, like all quantized enveloping algebras. In an appendix, we improve a standard criterion for a bicrossproduct A B of two Hopf algebras to be a Hopf algebra, and we show that in this case the antipode of the bicrossproduct is bijective if the antipodes of the factors are.     

Extremal Properties of Bases for Representations of Semisimple Lie Algebras

Let be a complex semisimple Lie algebra with specified Chevalley generators. Let V be a finite dimensional representation of with weight basis . The supporting graph P of is defined to be the directed graph whose vertices are the elements of and whose colored edges describe the supports of the actions of the Chevalley generators on V. Four properties of weight bases are introduced in this setting, and several families of representations are shown to have weight bases which have or are conjectured to have each of the four properties. The basis can be determined to be edge-minimizing (respectively, edge-minimal) by comparing P to the supporting graphs of other weight bases of V. The basis is solitary if it is the only basis (up to scalar changes) which has P as its supporting graph. The basis is a modular lattice basis if P is the Hasse diagram of a modular lattice. The Gelfand-Tsetlin bases for the irreducible representations of sl(n, ) serve as the prototypes for the weight bases sought in this paper. These bases, as well as weight bases for the fundamental representations of sp(2n, ) and the irreducible one-dimensional weight space representations of any semisimple Lie algebra, are shown to be solitary and edge-minimal and to have modular lattice supports. Tools developed here are used to construct uniformly the irreducible one-dimensional weight space representations. Similar results for certain irreducible representations of the odd orthogonal Lie algebra o(2n + 1, ), the exceptional Lie algebra G ₂, and for the adjoint and short adjoint representations of the simple Lie algebras are announced.     

On the asymptotic equivalence ofL pmetrics for convergence to normality

Summary This paper is concerned with the rate of convergence to zero of theL _pmetrics _np1p, constructed out of differences between distribution functions, for departure from normality for normed sums of independent and identically distributed random variables with zero mean and unit variance. It is shown that the _np are, under broad conditions, asymptotically equivalent in the strong sense that, for 1p, p, _np/_np is universally bounded away from zero and infinity asn.     

The tree number of a graph with a given girth

A family of subtrees of a graphG whose edge sets form a partition of the edge set ofG is called atree decomposition ofG. The minimum number of trees in a tree decomposition ofG is called thetree number ofG and is denoted by(G). It is known that ifG is connected then(G) |G|/2. In this paper we show that ifG is connected and has girthg 5 then(G) |G|/g + 1. Surprisingly, the case wheng = 4 seems to be more difficult. We conjecture that in this case(G) |G|/4 + 1 and show a wide class of graphs that satisfy it. Also, some special graphs like complete bipartite graphs andn-dimensional cubes, for which we determine their tree numbers, satisfy it. In the general case we prove the weaker inequality(G) (|G| – 1)/3 + 1.