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.     

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.     

Weight functions on the Kneser graph and the solution of an intersection problem of Sali

LetX, Y be finite sets and suppose thatF is a collection of pairs of sets (F, G),FX,GY satisfying |FF|s, |GG|t and |FF|+|GG|s+t+1 for all (F, G),F, GF. Extending a result of Sali, we determine the maximum ofF.     

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.     

First order properties of random posets

Let =(n,p) be a binary relation on the set [n]={1, 2, ..., n} such that (i,i) for every i and (i,j) with probability p, independently for each pair i,j [n], where i<j. Define as the transitive closure of and denote poset ([n], ) by R(n, p). We show that for any constant p probability of each first order property of R(n, p) converges as n .     

Stabilization of ill-posed Cauchy problems for parabolic equations

Summary In this paper we study the noncharacteristic Cauchy problem, u_t–(a(x)u_x)_x=0, x(0, l), t.(0, T], u(0, t)=(t), u_x(0,t)=0, 0tT, assuming only L for a. In the case of weak a priori bounds on u, we derive stability estimates on u of Hölder type in the interior and of logarithmic type at the boundary. Also the continuous dependence on a is considered.

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Sunto Nel presente lavoro consideriamo il problema di Cauchy non ben posto u_t= (a(x)u_x)_x, x(0, l), t(0, T), u(0, t)=(t), u_x(0, t)=0, 0tT. Supponiamo che a sia misurabile e limitato inferiormente e superiormente da constanti positive. Introduciamo delle limitazioni a priori su u e dimostriamo la dipendenza continua di u rispetto al dato sia in (0, l)×(0, T) (di tipo hölderiano) sia per x=l (di tipo logaritmico). Consideriamo, inoltre, la dipendenza continua di u da a.

     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. I

We prove a local limit theorem (LLT) on Cramer-type large deviations for sums S _V = _t V ( _t ), where _t , t Z , 1, is a Markov Gaussian random field, V Z , and is a bounded Borel function. We get an estimate from below for the variance of S _V and construct two classes of functions , for which the LLT of large deviations holds.     

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(×).     

On equivalence of rearrangements of the Haar system in dyadic Hardy and BMO spaces

H={h ₁,I } — , . : , I ¦(I)¦=¦I¦, ¦I¦ — I. H H ={h _(I),I} . , , . L ^p .

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Dedicated to Professor B. Szökefalvi-Nagy on his 75th birthday

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This research was supported in part by MTA-NSF Grants INT-8400708 and 8620153.     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

Zur Krümmungsverwandtschaft Von Zwangläufen in Affinen CK — Ebenen I

We study a map of osculating elements of an affine Cayley- Klein (CK-) plane into the Lie algebraA ₄(²) of the aequiform transformationsA ₄(²) of the given plane. If we use the real projective spaceP ₃() overA ₄(²) each osculating element defines a straight line inP ₃(). In the first part of this paper this map is studied in detail. In the second part we study second order properties of one- parameter motions and their corresponding properties in the Lie algebraA ₄(²). This is done by considering the analogen to the formula of EULERSAVARY in the image spaceP ³() overA ₄(²).     

On the Cesàro summability of orthogonal sequences

(C, ). , . 0<<1. 1) - ( _k ), _k =a _k , (C, ), . 2) , , (C, ) ; _k = =¦a _k ¦.     

A r (Ω)-Weighted Imbedding Inequalities for A-Harmonic Tensors

We prove the basic A _r ()-weighted imbedding inequalities for A-harmonic tensors. These results can be used to estimate the integrals for A-harmonic tensors and to study the integrability of A-harmonic tensors and the properties of the homotopy operator T: C (D, ^l )C (D, ^l–1).     

Estimate of the modulus of continuity of the generalized solutions of certain singular parabolic equations

One considers singular parabolic equations of the form (u)/t–u0,where sign u is a multivalued function, equal to -I for u<0, to 1 for u>0, and to the segment [-I,I] for u=0. Such a class of equations contains, in particular, the model for the two-phase Stefan problem, the porous medium equation, and the plasma equation. For the bounded generalized solutions u(x,t) of the indicated equations (without the assumption u/L²one has established a qualified local estimate of the modulus of continuity.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Ins'tituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 49–71, 1985.     

On the stability of the transport equation

We consider boundary value problems associated with the equation T=–A in a Hilbert Space, where T and A are bounded, self adjoint, injective, and A has a bounded inverse. We discuss the stability of the solution when A is perturbed by a self adjoint operator.     

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

Excesses of systems of exponential functions

Conditions on the closeness of real sequences {_n} and {_n} are studied which imply the equality of the excesses of the systems {exp(i_nx)} and {exp(i_nx)} in the space L²(–a, a). A theorem is formulated in terms of the difference of the sequences {_n} and {_n} enumerating the functions. In the corollaries of the theorem, conditions are given in terms of the behavior of the difference _n–_n0. An example is constructed showing that the condition _n–_n0 alone is not sufficient for equality of the excesses.Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 803–814, December, 1977.     

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.     

Rate of approximation by rectangular partial sums of double orthogonal series

( ) . .

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Dedicated to Professor K. Tandori on his seventieth birthday

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This research was supported in part by Grant # K41 100 of the Joint Fund of the Government of Ukraine and the International Science Foundation.