Integration by parts for a Lie group valued Brownian motion

LetG be a Lie group ofd×d matrices and be theLLie algebra ofG. We choose some Euclidean norm on , and an orthonormal basis (D ₁,...D _m ) relative to it. Let be the corresponding left invariant vector fields onG. In this paper we derive an integration by parts formula for aG-valued Brownian motion corresponding to the Laplacian .     

A representation theorem for weak automorphisms of a universal algebra

Given a group G and a descending chainG ₀,G ₁,...,G _n, of normal subgroups ofG, we prove that there exists a universal algebra , such that the chain ...W_n( )...W₁( }) W₀( )W( ) is isomorphic to the chain ...G _n ...G ₁G ₀G, where W( ) is the group of weak automorphisms of , and W_n( ) is the group of weak automorphisms of that leaves alln-ary operations fixed.We also prove that there are an infinite number of non-isomorphic algebras that satisfy the above.These results are a generalization of those proved by J. Sichler, in the special case when G=G₀, and G₁=G₂=...=G_n=....Presented by J. Mycielski.This paper comprises part of the author's doctoral dissertation at the University of Notre Dame in 1983. The author wishes to express her deep gratitude to Professor Abraham Goetz for suggesting this problem, for being extremely generous with his time and experience, and for giving her his constant encouragement. The author also thanks the reviewer for his helpful comments.     

Translative covering by homothetic copies

We show that for any convex bodyKE ² there exists a triangleT such that , where T is a suitable homothetic copy ofT with ratio . As a corollary we show that if (K _i) are homothetic copies of a given convex bodyKE ² with areaV(K)=1, then the condition is sufficient for the existence of a translative covering ofK by (K _i).     

Chu-Dualität und zwei Klassen maximal fastperiodischer Gruppen

This is a continuation of the paper Zwei Klassen lokalkompakter maximal fastperiodischer Gruppen, [6]. In [6], the classes and were introduced. We give sufficient conditions to conclude thatG is in if one knows thatG/G ₀ is in . If a groupG is in and ifG satisfies the Chu-duality then all closed subgroups ofG satisfy the Chu-duality. The Chu-quasi-dual of the Heisenberg groupH with integral coefficients is computed. It is shown thatH does not satisfy the Chu-duality, thatH is in , and thatH is not in .     

Towards a model theory for 2-hyponormal operators

We introduce the notion ofweak subnormality, which generalizes subnormality in the sense that for the extension ofT we only require that hold forf ; in this case we call a partially normal extension ofT. After establishing some basic results about weak subnormality (including those dealing with the notion of minimal partially normal extension), we proceed to characterize weak subnormality for weighted shifts and to prove that 2-hyponormal weighted shifts are weakly subnormal. Let { _n } _n=0 be a weight sequence and letW denote the associated unilateral weighted shift on . IfW is 2-hyponormal thenW is weakly subnormal. Moreover, there exists a partially normal extension on such that (i) is hyponormal; (ii) ; and (iii) . In particular, if is strictly increasing then can be obtained as

Signed Total Domination Nnumber of a Graph

The signed total domination number of a graph is a certain variant of the domination number. If is a vertex of a graph G, then N() is its oper neighbourhood, i.e. the set of all vertices adjacent to in G. A mapping f: V(G)-1, 1, where V(G) is the vertex set of G, is called a signed total dominating function (STDF) on G, if for each V(G). The minimum of values , taken over all STDF's of G, is called the signed total domination number of G and denoted by _st(G). A theorem stating lower bounds for _st(G) is stated for the case of regular graphs. The values of this number are found for complete graphs, circuits, complete bipartite graphs and graphs on n-side prisms. At the end it is proved that _st(G) is not bounded from below in general.     

Initial and Internal Limits of Supertemperatures

Let D and E be open subsets of which meet , with E bounded and D, and let . Given a non-negative supertemperature u on D ₊, we express its restriction to E ₊ as the sum of a temperature that vanishes at time 0, and a potential on that vanishes for all negative time. This enables us to use initial limit theorems about such potentials to prove similar results for u.     

Limit theorems for the ratio of the empirical distribution function to the true distribution function

Summary We consider almost sure limit theorems for and where _n is the empirical distribution function of a random sample ofn uniform (0, 1) random variables anda _n 0. It is shown that (1) ifna _n /log₂ n then both and converge to 1 a.s.; (2) ifna _n /log₂ n=d>0 (d>1) then has an almost surely finite limit superior which is the solution of a certain transcendental equation; and (3) ifna _n /log₂ n0 then and have limit superior + almost surely. Similar results are established for the inverse function _n ^–1 .Supported by the National Science Foundation under MCS 77-02255     

The Cauchy-Neumann problem for a viscous Hamilton-Jacobi equation

We study the weak solvability of viscous Hamilton-Jacobi equation: \,0,\,x\,\in\,\Omega,$" align="middle" border="0"> with Neumann boundary condition and irregular initial data ₀. The domain is a bounded open set and p > 0. The last part deals with the case a convex set and the initial data ₀ = in a open set D such that and      

Subcoercivity and subelliptic operators on Lie groups II: The general case

Let (,G, U) be a continuous representation of a Lie groupG by bounded operatorsg U (g) on the Banach space and let (, ,dU) denote the representation of the Lie algebra obtained by differentiation. Ifa ₁, ...,a _d is a Lie algebra basis of ,A _i =dU (a _i ) and whenever =(i ₁, ...,i _k ) we reconsider the operators

Asymptotic behaviour of the solution of a semilinear parabolic equation

We study the asymptotic behaviour ast tends to + of the solution of (u/t)–Lu+(u)–0,u _|=0 whereL is a second order self-adjoint elliptic operator and a maximal monotone graph of . If |(r)|/|r|²L ¹ (-1, 1) and ₁ is the first eigenvalue ofL we prove that u(.,t) converges uniformly on to some element of Ker (L + ₁ I) and that the limit is nonzero if |(r)|/|r| is nondecreasing. We give also some properties of the limit (monotonicity, continuity, range).     

K-theory of twisted differential operators on flag varieties

Let be a semisimple Lie algebra overk, an algebraically closed field of characteristic zero, and let be a Cartan subalgebra inside a Borel subalgebra of . LetU be the enveloping algebra of . For letM() denote the corresponding Verma modúle and letU _u=U/AnnM(). LetW be the Weyl group and letW ⁰ be the stabiliser of inW. We prove the following theorem, which affirms a conjecture of T.J. Hodges.Oblatum 16-XII-1994     

The axisymmetric free laminar jet of a power-law fluid

Summary The solution of the boundary-layer equations for an axisymmetric free laminar jet of a power-law fluid is investigated. It is shown that similar solutions are possible only forn1/2, wheren is the exponent in the power-law model. Forn>1/2, the axial velocity decay is proportional to the (–1/2n–1) power of the distance downstream and the jet spreads at a rate proportional to the (1/2n–1) power of the distance. Forn=1/2, the velocity decay and the rate of spread are exponential. Experimental data are presented which demonstrate that the predicted exponential decay occurs in real fluids.

---

Zusammenfassung Ähnliche Lösungen der Grenzschichtgleichungen für einen achsensymmetrischen laminaren Strahl von Potenzgesetz-Flüssigkeiten werden gegeben. Diese Lösungen werden mit vorhandenen experimentellen Daten verglichen.

---

Nomenclature d orifice diameter - D characteristic length - f preliminary dependent variable - F final dependent variable - J momentum inx-direction - K coefficient in power law model - n exponent in power law model - Re generalized Reynolds number - u x-component of velocity;u _o=characteristic velocity;u _p =initial velocity;=u/u _o;u _m =velocity on centerline of jet. - v y-component of velocity; - x coordinate in axial direction; ; = distance from orifice in experimental jet - y coordinate in radial direction; - function of - function of - parameter in final similarity transformation - preliminary independent variable - final independent variable - fluid density - shear stress - stream function     

Equivariant K-Homology and Restriction to Finite Cyclic Subgroups

For a discrete group G, we prove that a G-map between proper G–CW-complexes induces an isomorphism in G-equivariant K-homology if it induces an isomorphism in C-equivariant K-homology for every finite cyclic subgroup C of G. As an application, we show that the source of the Baum–Connes assembly map, namely K _* ^G (E(G, in)), is isomorphic to K _* ^G (E(G, )), where E(G, ) denotes the classifying space for the family of finite cyclic subgroups of G. Letting be the family of virtually cyclic subgroups of G, we also establish that and related results.     

On the ascending star subgraph decomposition of star forests

LetG be a graph of size for some integern2. ThenG is said to have an ascending star subgraph decomposition ifG can be decomposed inton subgraphsG ₁,G ₂, ...,G _n such that eachG _i is a star of sizei with 1in. We shall prove in this paper that a star forest with size , possesses an ascending star subgraph decomposition if the size of each component is at leastn, which is stronger than the conjecture proposed by Y. Alavi, A. J. Boals, G. Chartrand, P. Erds and O. R. Oellermann.     

Elementary Abelian Covers of Graphs

Let _G (X) be the set of all (equivalence classes of) regular covering projections of a given connected graph X along which a given group G Aut X of automorphisms lifts. There is a natural lattice structure on _G (X), where _{1 2} whenever ₂ factors through ₁. The sublattice _G () of coverings which are below a given covering : X^~ X naturally corresponds to a lattice _G () of certain subgroups of the group of covering transformations. In order to study this correspondence, some general theorems regarding morphisms and decomposition of regular covering projections are proved. All theorems are stated and proved combinatorially in terms of voltage assignments, in order to facilitate computation in concrete applications.For a given prime p, let _G ^p (X) _G (X) denote the sublattice of all regular covering projections with an elementary abelian p-group of covering transformations. There is an algorithm which explicitly constructs _G ^p (X) in the sense that, for each member of _G ^p (X), a concrete voltage assignment on X which determines this covering up to equivalence, is generated. The algorithm uses the well known algebraic tools for finding invariant subspaces of a given linear representation of a group. To illustrate the method two nontrival examples are included.     

Canonical and Boundary Representations on a Hyperboloid of One Sheet

For the hyperboloid of one sheet X=G/H, G=SO₀(1,2), H=SO₀(1,1), canonical representations R _,, C, =0,1, are defined as the restrictions to G of representations of the overgroup =SO₀(2,2) associated with a cone. They act on the torus containing two copies of X as open G-orbits. We study boundary representations generated by R _,. For some , they contain Jordan blocks. The decomposition of R _, into irreducible constituents includes a finite number (depending on ) of irreducible parts of the boundary representations.     

Operators of fibered groups

Let (G, ) denote a finite groupG with fibration . The group Aut (G, ) of operators of (G, ) is closely related to the group of collineations of (G, ). In this paper we investigate the geometric properties imposed on (G, ) by requiring that Aut (G, )=AutG. We find that in many instances this algebraic property restricts the geometry to a very special form.Dedicated to Professor Helmut Karzel on the occasion of his 60th birthday.