Invariant characters and coprime actions on finite nilpotent groups

Abstract. Let S be a subgroup of SLn(R), where R is a commutative ring with identity and n \geqq 3n \geqq 3. The order of S, o(S), is the R-ideal generated by x_ij, x_ii - x_jj (i 1 j)x_{ij},\ x_{ii} - x_{jj}\ (i \neq j), where (x_ij) ? S(x_{ij}) \in S. Let En(R) be the subgroup of SLn(R) generated by the elementary matrices. The level of S, l(S), is the largest R-ideal \frak q\frak {q} with the property that S contains all the \frak q\frak {q}-elementary matrices and all conjugates of these by elements of En(R). It is clear that l(S) \leqq o(S)l(S) \leqq o(S). Vaserstein has proved that, for all R and for all n \geqq 3n \geqq 3, the subgroup S is normalized by En(R) if and only if l(S) = o(S)     

The number of vertices of degree 5 in a contraction-critically 5-connected graph

An edge of a 5-connected graph is said to be 5-contractible if the contraction of the edge results in a 5-connected graph. A 5-connected graph with no 5-contractible edge is said to be contraction-critically 5-connected. Let V(G) and V₅(G) denote the vertex set of a graph G and the set of degree 5 vertices of G, respectively. We prove that each contraction-critically 5-connected graph G has at least |V(G)|/2 vertices of degree 5. We also show that there is a sequence of contraction-critically 5-connected graphs {G_i} such that lim_i→∞|V₅(G_i)|/|V(G_i)|=1/2.     

On Integral Graphs Which Belong to the Class

Let G be a simple graph and let ¯G denote its complement. A graph G is said to be integral if its spectrum contains integral values. If is integral with >1 and a>b, where mG denotes the m-fold union of the graph G, we show that it belongs to the class of integral graphs where (i) (kn ² s ²+km nst+ms,km ² t ²+kmnst+ms)= such that (m,)=1 and (s,)=1; (ii) k,m,n,s,t,z such that (k,m)=1, (k,s)=1, (m,n)=1, (m,s)=1, (n,t)=1 and (s,t)=1; and (iii) ns>m t. Acknowledgments.The author is very grateful to the referees for their valuable comments and suggestions concerning this paper.1991 Mathematics Subject Classification. 05 C 50     

Small embeddings for partial 5‐cycle systems

It has been conjectured that any partial 5‐cycle system of order u can be embedded in a 5‐cycle system of order v whenever v ≥ 3 u/ 2+1 and v ≡ 1 , 5 (mod 10) . The smallest known embeddings for any partial 5‐cycle system of order u is 10 u +5 . In this paper we significantly improve this result by proving that for any partial 5‐cycle system of order u ≥ 255 , there exists a 5‐cycle system of order at most (9 u +146) / 4 into which the partial 5‐cycle system of order u can be embedded. © 2011 Wiley Periodicals, Inc. J Combin Designs     

L r Estimates for Degenerate Elliptic Problems

We prove L ^r estimates for the Dirichlet problem –div(a(x,u,Du))=f with f in L ^q for 1q+, where the operator satisfies (|s|)|| ^p a(x,s,), with p>1. These estimates are obtained without symmetrization and are sharp in some cases.     

Two variable <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis> functions attached to eigenfamilies of positive slope

Consider a classical cusp eigenform f= _n=1 a _n (f)q ⁿ of weight k2 for ₀(N) with a Dirichlet character mod N, and let L _f (s,)= _n=1 (n)a _n (f)n ^-s denote the L-function of f twisted with an arbitrary Dirichlet character . For a prime number p5, consider a family of cusp eigenforms f _(k) of weight k , k {f _(k)= _n=1 a _n (f _(k))q ⁿ } containing f=f _(k), such that the Fourier coefficients a _n (f _(k)) are given by certain p-adic analytic functions k a _n (f _(k)). The purpose of this paper is to construct a two variable p-adic L function attached to Colemans family {f _(k)} of cusp eigenforms of a fixed positive slope =v _p ( _p )>0 where _p = _p (k ) is an eigenvalue (which depends on k ) of the Atkin operator U=U _p . Our p-adic L-function interpolates the special values L _f(k)(s,) at points (s,k ) with s=1,2,...,k -1. We give a construction using the Rankin-Selberg method and the theory of p-adic integration on a profinite group Y with values in an affinoid K-algebra A, where K is a fixed finite extension of Q _p . Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. In their turn, such measures come from Eisenstein distributions with values in certain Banach A-modules M =M (N;A) of families of overconvergent forms over A. To Robert Alexander Rankin in memoriam     

The short range expansion

Let V_i be short range potential and λ_i(ε) analytic functions. We show that the Hamiltonians H_ε = −Δ + ε⁻²∑_{i = l}ⁿλ_i(ε)V_i((· − x_i)/ε converge in the strong resolvent sense to the point interactions as ε → 0, and if V_i have compact support then the eigenvalues and resonances of H_ε, which remains bounded as ε → 0, are analytic in ε in a complex neighborhood of zero. We compute in closed form the eigenvalues and resonances of H_ε to the first order in ε.     

Meromorphic functions sharing small functions

Let f and g be meromorphic functions sharing four small functions a₁, a₂, a₃, a₄ a_1, a_2, a_3, a_4 ignoring multiplicities. If there is a small function a₅a_5 distinct from a_j, j=1, 2, 3, 4, a_j, j=1, 2, 3, 4, such that [`(N)](r,f=a<sub>5</sub>=g) 1 S(r,f) \overline {N}(r,f=a_5=g)\ne S(r,f) , then f=g f=g , where [`(N)](r,f=a₅=g) \overline{N}(r,f=a_5=g) is the counting function of those common zeros of f(z)-a₅(z) f(z)-a_5(z) and g(z)-a₅(z) g(z)-a_5(z) counted only once ignoring multiplicities.     

On the Volume of 5-Cycle Trades

 A 5-cycle trade of volume t is a graph G whose edge set can be partitioned into t 5-cycles in at least two ways, such that the two collections of 5-cycles have no 5-cycles in common. The foundation of a trade is the number of vertices in the graph G. This paper determines for which values of t and v there exists a 5-cycle trade of volume t and foundation v. Received: April 27, 1998 Final version received: February 15, 1999     

Any nilpotence degree occurs in mod-<Emphasis Type="Italic">p</Emphasis> cohomology rings of <Emphasis Type="Italic">p</Emphasis>-groups

Let p be an odd prime number and let n be an arbitrary positive integer. We prove that there exists a p-group whose mod-p cohomology ring has a nilpotent element H²() satisfying ⁿ0,^n+p–1=0. As a corollary, we exhibit a p-group whose mod-p cohomology ring contains an element of nilpotency degree n+1.Mathematical Subject Classification (2000): 20J06, 20D15, 55R40To Phuong and Nin     

Combined forced and free convection heat transfer from a horizontal flat plate

The problem of combined forced and free convection heat transfer in boundary-layer flow past a heated or cooled, horizontal flat plate is studied in the limits of large and small Prandtl number using the technique of matched asymptotic expansions. It is found that the effects of natural convection are confined to small changes in the basic forced convection velocity and temperature fields providedGr/Re ^5/2 Pr ^2/3 1 forPr1, orGr/Re ^5/2 Pr ^1/2 1 forPr1. The governing similarity equations at the first two orders in these small parameters are solved and the results compared with the numerical solutions of Sparrow and Minkowycz (1962).

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Zusammenfassung Das Problem der kombinierten erzwungenen und freien Konvektion in der Grenzschichtströmung entlang einer geheizten oder gekühlten horizontalen ebenen Platte wurde untersucht im Grenzfall von kleinen und grossen Prandtl-Zahlen, mit Hilfe der angepassten asymptotischen Entwicklungen. Der Effekt der natürlichen Konvektion ergab sich als kleine Korrektur der Geschwindigkeits-und Temperatur-Felder, fallsGr/Re ^5/2 Pr ^2/3 1 fürPr1 oderGr/Re ^5/2 Pr ^1/2 1 fürPr1. Die Ähnlichkeitslösungen der beiden ersten Ordnungen in diesen Parametern wurden bestimmt, und die Resultate wurden mit der numerischen Lösung von Sparrow und Minkowycz (1962) verglichen.

     

Subdivisions of K5 in Graphs Embedded on Surfaces With Face‐Width at Least 5

We prove that if G is a 5‐connected graph embedded on a surface Σ (other than the sphere) with face‐width at least 5, then G contains a subdivision of K₅. This is a special case of a conjecture of P. Seymour, that every 5‐connected nonplanar graph contains a subdivision of K₅. Moreover, we prove that if G is 6‐connected and embedded with face‐width at least 5, then for every v ∈ V(G), G contains a subdivision of K₅ whose branch vertices are v and four neighbors of v.     

A local structure theorem on 5‐connected graphs

An edge of a 5‐connected graph is said to be contractible if the contraction of the edge results in a 5‐connected graph. Let x be a vertex of a 5‐connected graph. We prove that if there are no contractible edges whose distance from x is two or less, then either there are two triangles with x in common each of which has a distinct degree five vertex other than x, or there is a specified structure called a K₄^?‐configuration with center x. As a corollary, we show that if a 5‐connected graph on n vertices has no contractible edges, then it has 2n/5 vertices of degree 5. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 99–129, 2009     

ON THE INCOMPLETENESS OF A DESCENDING CHAIN OF EXTENSIONS OF IMPLICATIONAL S5

C5.ω is obtained by adding, schematically, to the strict-implicational fragment C5 of S5 the axiom ((p → q) → (q → p)) → (q → p). This paper presents a fully general proof that neither C5.ω nor any of a descending chain of its extensions is complete with respect to any class of frames, correcting the garbled details of a version skeched in an earlier paper (same Zeitschrift 31 (1985), 201-208).     

Large Deviation Principle for Additive Functionals of Brownian Motion Corresponding to Kato Measures

Let (B _t ,P ^W _x ) be the Brownian motion. Let be a Radon measure in the Kato class and A _t the additive functional associated with . We prove that A _t /t obeys the large deviation principle.     

On problems with bounded state variables

A set of first-order necessary conditions is obtained for the general control problem of Bolza with bounded state constraints of the form (t, x)0, =1,...,m. With the solution required to satisfy the vector differential equationsx=f(t, x, u), whereu is control, an important feature of this paper is in relaxing the assumption on the rank of the matrix _x f _u generally made in attacking problems of this type. This is accomplished even though the solution may have an infinite number of intervals satisfying (t, x)=0 for various .The preparation of this paper was sponsored in part by the US Army Research Office, Grant No. DA-31-214-ARO(D)-355.     

Chern-simons theory in the SO(5)/U(2) harmonic superspace

We consider the superspace of D=3, N=5 supersymmetry using SO(5)/U(2) harmonic coordinates. Three analytic N=5 gauge superfields depend on three vector and six harmonic bosonic coordinates and also on six Grassmann coordinates. Decomposing these superfields in Grassmann and harmonic coordinates yields infinite-dimensional supermultiplets including a three-dimensional gauge Chern-Simons field and auxiliary bosonic and fermionic fields carrying SO(5) vector indices. The superfield action of this theory is invariant with respect to the D=3, N=6 conformal supersymmetry realized on N=5 superfields. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 2, pp. 217–234, November, 2008.     

Elliptic problems on ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript> with unbounded coefficients in classical Sobolev spaces

We discuss the existence and uniqueness in H¹(^N) and the H²(^N) regularity of the solutions of Au=f when f L²(^N) and A is a second-order linear elliptic operator whose first and zeroth order coefficients may be unbounded at infinity. We also investigate whether –A generates a C₀ or analytic semigroup on L². The approach in this nonweighted setting is based on a new and general method. The idea consists in embedding A into a suitable one-parameter family of operators (A_s)_s with A₀=A. The properties of A_s when s0 make it possible to prove that the boundary integrals arising from simple integration by parts over balls with increasing radius tend to 0 at infinity. This provides the needed estimates for uniqueness and regularity.Mathematics Subject Classification (1991): 35D05, 35D10, 35J15     

Equivalent Characterization of Entire Functions of Exponential Type

In this paper, we show that a necessary and sufficient condition for the fulfillment of the relation s _m ^(k) (f) – f^(k) _p 0 as m , 1 < p < , k 0,1,2,..., is that f B _,p , where B _,p = B L _p (R), and B denotes the subset of all entire functions of exponential type which are bounded on R, B _,p is usually called Paley-Wiener class, and s _m (f) is the unique cardinal spline of degree m – 1 interpolating f at the integers. Moreover, we obtain three equivalent forms for the characterization of the class B _,p .     

The lattice of extensions of the modal logic of two equivalence relations has the cardinality of the continuum

A continuum of different logics over S5⋇S5 is constructed. This proves that the cardinality of the lattice of all normal extensions of the logic of two equivalence relations Ext(S5⋇S5) is continuum.