Halving PGL (2, 2 f ), f odd: A series of cryptocodes

A set S of permutations of k objects is -uniform, t-homogeneous if for every pair A, B of t-subsets of the ground set, there are exactly permutations in S mapping A onto B.Our main result (Theorem 1.2) is the construction of a (q – 1)-uniform, 2-homogeneous set of permutations of q + 1 objects contained in the projective group PGL(2, q), where q is a power of 2 with odd exponent.The main ingredient of the proof is a lemma concerning cubic equations in characteristic 2 (Lemma 2.6).The result is useful in the framework of theoretical secrecy and authentication. By a theorem of D.R. Stinson (Stinson 1990) one obtains families of cryptocodes which achieve perfect 2-fold secrecy and are 1-fold secure against spoofing (Corollary 1.3).     

Measurable Functionals and Finitely Absolutely Continuous Measures on Banach Spaces

We consider the structure of orthogonal polynomials in the space L ₂(B, ) for a probability measure on a Banach space B. These polynomials are described in terms of Hilbert–Schmidt kernels on the space of square-integrable linear functionals. We study the properties of functionals of this sort. Certain probability measures are regarded as generalized functionals on the space (B, ).     

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.     

The decomposition theorem for functions satisfying the law of large numbers

LetB be a Banach space with the Radon-Nikodym property and (S, , ) a probability space. Then anf: SB satisfies the strong law of large numbers if and only if there exists a Bochner integrable functionf ₁ and a Pettis integrable functionf ₂,f ₂f ₂=0 in the Glivenko-Cantelli norm, such thatf=f ₁+f ₂. The composition is unique.     

Weakly compactly generated Banach spaces and the strong law of large numbers

IfB is a weakly compactly generated Banach space andf: (S,S, ) satisfies the strong law of large numbers, thenf=f ₁+f ₂, wheref ₁ is Bochner -integrable andf ₂ is Pettis -integrable with Pettis norm 0. The decomposition is unique.     

Die Verteilung der schlichten Funktionen in einem Funktionenraum

We consider the set of regular functions . We construct a Borel measure and a class of outer measures _h onH. With these and _h we show that: (HS)=0 and _h (HS)=0, (S is the set of normed univalent functions). From _h (HS)=0 follows—forh=t —that the Hausdorff—Billingsley-dimension ofHS is zero.     

Finitely subadditive and countably subadditive outer measures

Results are given comparing countably subadditive (csa) outer measures and finitely subadditive (fsa) outer measures, especially relating to regularity and measurability conditions such as (*) condition:A setE (of an arbitrary setX), is measurable ( an outer measure),ES (the collection of measurable sets) iff (X)=(E)+(E). Specific examples are given contrasting csa and fsa outer measures. In particular fsa and csa outer measures derived from finitely additive measures defined on an algebra of sets generated by a lattice of sets, are investigated in some detail.     

Convergence of non-ergodic dynamical systems

Summary Let {T _t} be a flow on a probability space (S,L,}) which describes the time evolution of a dynamical system with state space S, and interpret as the initial distribution of the system. Then the distribution of the system at time t is given by T _t ^–1 . Our aim is to study the asymptotic behavior of T _t ^–1 both in general and in the particular cases of random rate and almost periodic systems. The results seem to indicate that convergence or mean convergence is the normal behavior in the non-ergodic case.     

BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials

We consider 3-parametric polynomialsP ^* (x; q, t, s) which replace theA _n-series interpolation Macdonald polynomialsP ^* (x; q, t) for theBC _n-type root system. For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation. Ass the polynomialsP ^* (x; q, t, s) becomeP ^* (x; q, t). We also prove a binomial formula for 6-parametric Koornwinder polynomials.     

Effilement minimal en une singularité isolée de l'équation de Schrödinger et application au principe de Picard

In this paper, the generalized Schrödinger equation (–)u=0 on the punctured unit disk of ² is investigated. If is rotation free and satisfies the Picard principle at the origin, it is shown that if a setE is minimal thin relatively to an extremal harmonic functionh with zero boundary values at {|x|=1}, there exists a sequence (r _n ) converging to zero such that B(O,r _n ) C E. Lete be the -unit. It is proved that if a measure satisfies _\E e h d<, for a minimal thin, relatively toh , setE then the Picard principle is valid for the measure + .

     

Vector-valued stochastic processes. I. Vector measures and vector-valued stochastic processes with finite variation

In this paper we study the relationship _V (M)=E(1 _M dV _S ) between operatorvalued processesV with finite variation V and operator-valued stochastic measures _V with finite variation | _V |. The variations satisfy the inequality | _V | _|V|, which, under certain conditions, is an equality (for example, ifV is measurable).     

Some extensions of the Hewitt-Savage zero-one law

Summary Let denote the class of infinite product probability measures = ₁× ₂× defined on an infinite product of replications of a given measurable space (X, A), and let denote the subset of for which (A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member , of belongs to . In the present paper, the class is shown to possess several closure properties. E.g., if and _{0 n} for some n 1, then ₀× ₁× ₂×.... While the current results do not permit a complete characterization of they demonstrate conclusively that is a much larger subset of than previous results indicated. The interesting special case X={0,1} is discussed in detail.Research supported by the National Science Foundation under grant No. MCS75-07556     

Method of moments on semigroups

If ( _j ) is a sequence of measures onR ^k having momentss _n ( _j ) of all ordersnN ₀ ^k and if for eachnN ₀ ^k the sequence (s _{n j} )) _jN converges to somet _n R then some subsequence of ( _j ) converges weakly to a measure with moments of all orders satisfyings _n ()=t _n for allnN^0/k . Thisindeterminate method of moments and the continuity theorems in probability theory suggest a common generalization, dealing with a commutative semigroupS, with involution and a neutral element, and measures on the dual semigroupS ^* ofcharacters on S—hermitian multiplicative complex functions not identically zero. In this setting, a continuity theorem holds for measures on the set of bounded characters,⁽²⁾ and an indeterminate method of moments whenS is finitely generated.⁽²⁾ The latter result is generalized in the present paper to the case of arbitraryS. This leads to a generalization of Haviland's criterion for theK-moment problem, and to a continuity theorem for the so-called perfect semigroups.     

Generalizations of Tarski's fixed point theorem for order varieties of complete meet semilattices

A poset P is -conditionally complete ( a cardinal) if every set X P all of whose subsets of cardinality < have an upper bound has a least upper bound. For we characterize the subposets of a -complete poset which can occur as the set of fixed points of some montonic function on P. This yields a generalization of Tarski's fixed point theorem. We also show that for every the class of -conditionally complete posets forms an order variety and we exhibit a simple generating poset for each such class.Research supported in part by NSERC while the author was visiting Professor Ivo Rosenberg at the Université de Montreal.Research supported in part by NSF-grant DMS-8703743.     

A New Relaxation Space for Obstacles

The space of obstacles (i.e. p-quasi upper semicontinuous functions) is endowed with a distance which is topologically equivalent to the -convergence. We find the metric completion of this space and we give some application for minimization problems of cost functionals depending on obstacles via their level sets. An element of the completion is a decreasing and _p -continuous on the left mapping Rt _t , where _t are positive Borel measures vanishing on sets of zero p-capacity.     

Operator stable laws: Series representations and domains of normal attraction

LetX,X ₁,X ₂,... be i.i.d. random vectors in ^d. The limit laws that can arise by suitable affine normalizations of the partial sums,S _n=X ₁+...+X _n, are calledoperator-stable laws. These laws are a natural extension to ^d of the stable laws on. Thegeneralized domain of attraction of [GDOA()] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to . If the linear part of the affine transformation is restricted to take the formn ^–B for some exponent operatorB naturally associated to thenX is in thegeneralized domain of normal attraction of [GDONA()]. This paper extends the theory of operator-stable laws and their domains of attraction and normal attraction.     

Fluctuation theory of media with pronounced spacetime inhomogeneity

A nonlinear determinstic thermodynamics is constructed for media with pronounced (r,t) inhomogeneity of the intensive variables or their derivatives. The balance equations in the theory are taken to be either the equations of an ideal fluid or an ideal fluid with heat conduction. The basic variables are taken to be (r,t) andT(r,t). The hypothesis of local equilibrium is represented in the form of the Gibbs-Duhem relation, the conjugate coordinates are (r,t) and (r,t), and the local potential isP(,T). The velocity potentialv _i (r,t) enters through the substantial derivative. A variational principle is formulated; in the case of an ideal fluid with heat conduction there arises naturally a local decrease of the entropy production:z ₂(t)=z ₀ ² exp(–2t/t), there0074 is the relaxation time.In memory of D. N. ZubarevJoint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 1, pp. 53–67, October, 1993.     

Compatible additive permutations of finite integral bases

Let be an additive permutation of a finite integral base. It is shown that ifB is symmetric, then there is a unique additive permutation ofB which is compatible with in the sense that ^–1 is also an additive permutation; and that, further, ifB is asymmetric, then there is no additive permutation ofB which is compatible with. Thus, in the symmetric case, there are no additively compatible sets (of permutations) forB of size greater than 3. This contrasts with the situation for completely compatible sets (equivalently, additive sequences of permutations) where for certainB compatible sets of size (resp. length) 4 or less are known, but where nothing is known of sets of greater size (resp. length). It is also noted how this result restricts the possibility of a useful multiplication theorem for the additive analogue of perfect systems of difference sets and graceful graphs.     

On the existence of wave operators for the Klein-Gordon equation

The problem of existence of wave operators for the Klein-Gordon equation ( _t ² –+²+iV_1t+V₂)u(x,t)=0 (x R ⁿ,t R, n3, >0) is studied where V₁ and V₂ are symmetric operators in L₂(R ⁿ) and it is shown that conditions similar to those of Veseli-Weidmann (Journal Functional Analysis 17, 61–77 (1974)) for a different class of operators are also sufficient for the Klein-Gordon equation.     

Infinitely divisible measures onp-adic groups

We show that any infinitely divisible measure on ap-adic algebraic group (p a prime) has a translatex, by an elementx centralizing the support of , such thatx can be embedded in a continuous real one-parameter semigroup {V _t } _t >0, asx=v ₁.