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.     

Leaves in Representation Diagrams of Bipartite Distance-Regular Graphs

Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Let ₀ > ₁ ··· > _D denote the eigenvalues of and let q ^h _ij (0 h, i, j D) denote the Krein parameters of . Pick an integer h (1 h D – 1). The representation diagram = _h is an undirected graph with vertices 0,1,...,D. For 0 i, j D, vertices i, j are adjacent in whenever i j and q ^h _ij 0. It turns out that in , the vertex 0 is adjacent to h and no other vertices. Similarly, the vertex D is adjacent to D – h and no other vertices. We call 0, D the trivial vertices of . Let l denote a vertex of . It turns out that l is adjacent to at least one vertex of . We say l is a leaf whenever l is adjacent to exactly one vertex of . We show has a nontrivial leaf if and only if is the disjoint union of two paths.     

The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Let G denote a semisimple group, a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:(1) For any -quasi-invariant measure on B, and any probablity measure on , the norm of the operator () on L ²(B,) is equal to (), where is the unitary representation in L ²(X,), and is the regular representation of .(2) In particular this estimate holds when is Lebesgue measure on B, a Patterson–Sullivan measure, or a -stationary measure, and implies explicit lower bounds for the displacement and Margulis number of (w.r.t. a finite generating set), the dimension of the conformal density, the -entropy of the measure, and Lyapunov exponents of .(3) In particular, when G=PSL₂() and is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.We also prove that ()=_G() for any amenable action of G and L ¹(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the -entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when is any word-hyperbolic group, acting on their Poisson boundary, for example.     

The intuitionistic propositional calculus with quantifiers

Let L be the language of the intuitionistic propositional calculus J completed by the quantifiers and , and let calculus 2J in language L contain, besides the axioms of J, the axioms xB (x) B(y) and B(y) xB (x). A Kripke semantics is constructed for 2J and a completeness theorem is proven. A result of D. Gabbay is generalized concerning the undecidability of C2J⁺-extension of 2J by schemes x (x B) and x(A B(x))A xB (x) specificially: the undecidability is proven of each T theory in language L such that [2J]T [C2J+] ([2J] ([2J] denotes the set of all theorems of calculus 2J).Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 69–76, July, 1977.     

Onp-Helson sets in R n

p- . E R ⁿ -, f () _p(R ⁿ)., ER ⁿ 2nq ₀, E— - q ₀(q ₀-1). : q₀>2 n1 E R ⁿ 2nq ₀, p- p<₀. , f-[-, ]ⁿ, f A _p(R ⁿ) , _p([-, ]ⁿ) (1 << ).     

Embeddings of Lorentz—Marcinkiewicz spaces with mixed norms

X(Y) f -:X(Y)={fM(×): f_{X(Y)=f(x,.)YX<} . =(0, ), M (×) — , ×, X, Y, Z— . X(Y) Z(×).     

A note on rational approximation rate for Müntz system {x λn } with λ n ↘ 0

[Zho2] {x ⁿ } , _n 0 n .

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Supported in part by an NSERC Postdoctoral Fellowship and a CRF grant of University of Alberta.     

Estimation of the Intensity of the Flow of Nonmonotone Refusals in the Queuing System (≤ λ)/G/m

We consider a queuing system ()/G/m, where the symbol () means that, independently of prehistory, the probability of arrival of a call during the time interval dtdoes not exceed dt. The case where the queue length first attains the level r m+ 1 during a busy period is called the refusal of the system. We determine a bound for the intensity ₁(t) of the flow of homogeneous events associated with the monotone refusals of the system, namely, ₁(t) = O( ^{r+ 1}₁ ^{m– 1} _{r– m+ 1}), where _k is the kth moment of the service-time distribution.     

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.     

Local isometric immersions of two-dimensional metrics inE 4 with prescribed Gaussian torsion

Suppose that in a domain R(, B) of variables (r, ): (0 r , ₁ +B(r–r ₀ ) ₂–B(r–r₀), where > 0, B > 0, ₁ < 0 < ₂ are numbers) a metric ds² = dr² +G(r, )d ² and a function k(r, ) are given. The problem of isometrically immersing ds² in E ⁴ with prescribed Gaussian torsion is considered. The following is proved: The class C ⁵ metric ds ² is locally realized in the form of a class C ³ surface F ² whose Gaussian torsion is the prescribed class C ³ function (r, ).Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 38–47, 1992.     

Convergence of random products of compact contractions in Hilbert space

Let {T₁, ..., T_N} be a finite set of linear contraction mappings of a Hilbert space H into itself, and let r be a mapping from the natural numbers N to {1, ..., N}. One can form S_n=T_r(n)...T_r(1) which could be described as a random product of the T_i's. Roughly, the S_n converge strongly in the mean, but additional side conditions are necessary to ensure uniform, strong or weak convergence. We examine contractions with three such conditions. (W): x_n1, Tx_n1 implies (I-T)x_n0 weakly, (S): x_n1, Tx_n1 implies (I-T)x_n0 strongly, and (K): there exists a constant K>0 such that for all x, (I-T)x²K(x²–Tx²).We have three main results in the event that the T_i's are compact contractions. First, if r assumes each value infinitely often, then S_n converges uniformly to the projection Q on the subspace _i= ₁ ^N [x|T_ix=x]. Secondly we prove that for such compact contractions, the three conditions (W), (S), and (K) are equivalent. Finally if S=S(T₁, ..., T_N) denotes the algebraic semigroup generated by the T_i's, then there exists a fixed positive constant K such that each element in S satisfies (K) with that K.     

Eine Kennzeichnung der nicht fanoschen Rechtseitebenen durch Inzidenz,Parallelität und Kongruenz

Rectangular planes of characteristic 2 in the sense of H. KARZEL [7] will be characterized as incidence spaces with parallelism and congruence .     

Produktsätze für Verfahren zur Limitierung von Doppelfolgen

{p _mn } - ₀₀>0, (1, 1) (1.1) (1.2). {s _mn } J _p - ( bJ _p -lims _mn =), (1.3) 0<x,y<1 p _s (, )/p(x, y) x, y 1-. {r _mn } - , (1.5) 0<, <1. N _rp - , (1.6). , bJ _p -lims _mn = bJ _q -lim(N _rps) _mn =. J _p - . , .     

Index of exponential growth for a class of stochastic semigroups

We consider finite-dimensional homogeneous stochastic semigroups X _s ^t , 0 s t < assuming values in the space of real square matrices. For stochastic semigroups assuming values in the class of upper triangular matrices we compute the index of exponential growth , where · is the operator norm of a matrix. The answer is given in terms of the characteristic Y_t of the generating process Y_t of the semigroup X_s ^t:x=–(1/2), where is the smallest eigenvalue of the matrix B which defines the characteristic Y_t=Bt.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 78–84, 1988.     

Нормаляный поридок аддитявных арифметичыских функций на множестве дсдвинутых” пностых чясел

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Limiting behavior of weighted central paths in linear programming

We study the limiting behavior of the weighted central paths{(x(), s())} > 0 in linear programming at both = 0 and = . We establish the existence of a partition (B ,N ) of the index set { 1, ,n } such thatx _i() ands _j () as fori B , andj N , andx _N (),s _B () converge to weighted analytic centers of certain polytopes. For allk 1, we show that thekth order derivativesx ^(k) () ands ^(k) () converge when 0 and . Consequently, the derivatives of each order are bounded in the interval (0, ). We calculate the limiting derivatives explicitly, and establish the surprising result that all higher order derivatives (k 2) converge to zero when .     

On trees and tree dimension of ordered sets

We call an ordered set (X, ) a tree if no pair of incomparable elements ofX has an upper bound. It is shown that there is a natural way to associate a tree (T, ) with any ordered set (X, ), and (T, ) can be characterized by a universal property. We define the tree dimensiontd(X, ) of an ordered set as the minimal number of extensions of (X, ) which are trees such that the given order is the intersection of those tree orders. We give characterizations of the tree dimension, relations between dimension and tree dimension, and removal theorems.     

Hugoniót–Maslov Chains for Singular Vortical Solutions to Quasilinear Hyperbolic Systems and Typhoon Trajectory

According to Maslov, many 2D quasilinear systems of PDE possess only three algebras of singular solutions with properties of structural self-similarity and stability. They are the algebras of shock waves, narrow solitons, and square-root point singularities (solitary vortices). Their propagation is described by infinite chains of ODE (the Hugoniót–Maslov chains). We consider the Hugoniót-Maslov chain for the square-root point singularities of the shallow water equations. We discuss different related mathematical questions (in particular, unexpected integrability effects) as well as their possible application to the problem of typhoon dynamics.     

A two‐level queueing network model with blocking and non‐blocking messages

We analyze a novel twolevel queueing network with blocking, consisting of N level1 parallel queues linked to M level2 parallel queues. The processing of a customer by a level1 server requires additional services that are exclusively offered by level2 servers. These level2 servers are accessed through blocking and nonblocking messages issued by level1 servers. If a blocking message is issued, the level1 server gets blocked until the message is fully processed at the level2 server. The queueing network is analyzed approximately using a decomposition method, which can be viewed as a generalization of the wellknown twonode decomposition algorithm used to analyze tandem queueing networks with blocking. Numerical tests show that the algorithm has a good accuracy.