Unitary dilations and theC* algebra0 2

LetH be an infinite-dimensional separable Hilbert space, and letS=(S _ij)∈teB(H)⊗M ₂ be a unitary 2 × 2 matrix with operator entries. We study theC*-algebra generated by the operatorsS _ij, and show that the study of unitary dilations of isometriesT inH reduces to the special case whereS ₁₁ =T, andS ₂₁ = 0. We useC*-algebraic techniques to obtain detailed results about the set of all unitary dilations ofT. Work supported in part by NSF.     

Regularity of the attractor for 3-D complex Ginzburg-Landau equation

In this paper,the existence of global attractor for 3-D complex Ginzburg Landau equation is considered.By a decomposition of solution operator,it is shown that the global attractor A_i in H~i(Ω) is actually equal to a global attractor Aj in H~j(Ω)(i≠j,i,j = 1,2,…m).     

A remark on the unitary group of a tensor product of<Emphasis Type="Italic">n</Emphasis> finite-dimensional Hilbert spaces

LetH _i, 1 ≤ i ≤n be complex finite-dimensional Hilbert spaces of dimension d_i,1 ≤ i ≤n respectively withd _i ≥ 2 for everyi. By using the method of quantum circuits in the theory of quantum computing as outlined in Nielsen and Chuang [2] and using a key lemma of Jaikumar [1] we show that every unitary operator on the tensor productH =H ₁ ⊗H ₂ ⊗... ⊗H _n can be expressed as a composition of a finite number of unitary operators living on pair productsH _i ⊗H _j,1 ≤i,j ≤n. An estimate of the number of operators appearing in such a composition is obtained. Dedicated to Prof. A.K. Roy on his 62nd birthday.     

On the evolution of islands

Letn cells be arranged in a ring, or alternatively, in a row. Initially, all cells are unmarked. Sequentially, one of the unmarked cells is chosen at random and marked until, aftern steps, each cell is marked. After thekth cell has been marked the configuration of marked cells defines some number of islands: maximal sets of adjacent marked cells. Let ξ _k denote the random number of islands afterk cells have been marked. We give explicit expressions for moments of products of ξ _k ’s and for moments of products of 1/ξ _k ’s. These are used in a companion paper to prove that if a random graph on the natural number is made by drawing an edge betweeni≧1 andj>i with probabilityλ/j, then the graph is almost surely connected ifλ>1/4 and almost surely disconnected ifλ≦1/4.     

On a characterization of the exponential distribution by spacings

Summary LetX be a non-negative random variable with probability distribution functionF. SupposeX _i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X _i+1,n−X_i,n) and (n−j)(X _j+1,n−X_j,n) for somei, j andn, (1≦i<j<n). The work was partly completed when the author was at the Dept. of Statistics, University of Brasilia, Brazil.     

Direct products of cyclic semigroups an unions of groups

We consider direct productsS×U_eεE G _e=S ₁×…×S _n × U_eεE G _e of non-group finite cyclic semigroupsS _i, 1 ≤i ≤n, and finite unions of finite groups U_eεE G _e We prove that if such a semigroup is isomorphic to another of the same form, sayT×U _fεF H _f =T ₁×…×U _fεF H _f , whereT _j are non-group cyclic semigroups, 1≤j≤l, and U _fεF H _f is a union of groups, thenS is isomorphic toT and U_eεE G _e is isomorphic to U_fεF H _f . We then determine when a finite semigroup has such a decomposition and show how the direct factors can be found.     

On the Frattini normal embeddability of products ofp-groups

ItH _i is a finite non-abelianp-group with center of orderp, for 1≦j≦R, then the direct product of theH _i does not occur as a normal subgroup contained in the Frattini subgroup of any finitep-group. If the Frattini subgroup Φ of a finitep-groupG is cyclic or elementary abelian of orderp ², then the centralizer of Φ inG properly contains Φ. Non-embeddability properties of products of groups of order 16 are established.     

Balancing vectors in the max norm

Let v₁, ..., v _n be vectors inR ⁿ of max norm at most one. It is proven that there exists a choice of signs for which all partial sums have max norm at mostKn ^1/2. It is further shown that such a choice of signs must be anticipatory—there is no way to choose thei-th sign without knowledge of v _j forj>i.     

The riquier problem for a nonlinear equation unresolved with respect to the Lévy iterated laplacian

We present a method of solving for the nonlinear equationf(U(x),Δ _L ² U(x)) = Δ _L U(x) (Δ _L is an infinite-dimensional Laplacian) unresolved with respect to an iterated infinite-dimensional Laplacian and for the Riquier problem for this equation. Ukrainian NII MOD, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 423–427, March, 1999.     

A greedy algorithm for the optimal basis problem

The following problem is considered. Givenm+1 points {x _i }₀ ^m inR ⁿ which generate anm-dimensional linear manifold, construct for this manifold a maximally linearly independent basis that consists of vectors of the formx _i −x _j . This problem is present in, e.g., stable variants of the secant and interpolation methods, where it is required to approximate the Jacobian matrixf′ of a nonlinear mappingf by using values off computed atm+1 points. In this case, it is also desirable to have a combination of finite differences with maximal linear independence. As a natural measure of linear independence, we consider the hadamard condition number which is minimized to find an optimal combination ofm pairs {x _i ,x _j }. We show that the problem is not NP-hard, but can be reduced to the minimum spanning tree problem, which is solved by the greedy algorithm inO(m ²) time. The complexity of this reduction is equivalent to onem×n matrix-matrix multiplication, and according to the Coppersmith-Winograd estimate, is belowO(n ^2.376) form=n. Applications of the algorithm to interpolation methods are discussed. Part of the work was done while the author was visiting DIMACS, an NSF Science and Technology Center funded under contract STC-91-19999; DIMACS is a cooperative project of Rutgers University, Princeton University, AT&T Bell Laboratories and Bellcore, NJ, USA.     

Hamiltonian Cycles in Almost Claw-Free Graphs

 Some known results on claw-free graphs are generalized to the larger class of almost claw-free graphs. In this paper, we prove the following two results and conjecture that every 5-connected almost claw-free graph is hamiltonian. (1). Every 2-connected almost claw-free graph G∉J on n≤ 4 δ vertices is hamiltonian, where J is the set of all graphs defined as follows: any graph G in J can be decomposed into three disjoint connected subgraphs G ₁, G ₂ and G ₃ such that E _G (G _i , G _j ) = {u _i , u _j , v _i v _j } for i≠j and i,j = 1, 2, 3 (where u _i ≠v _i ∈V(G _i ) for i = 1, 2, 3). Moreover the bound 4δ is best possible, thereby fully generalizing several previous results. (2). Every 3-connected almost claw-free graph on at most 5δ−5 vertices is hamiltonian, hereby fully generalizing the corresponding result on claw-free graphs. Received: September 21, 1998 Final version received: August 18, 1999     

Arithmetical properties of the number of t-core partitions

Let Λ={λ ₁≥⋅⋅⋅≥λ _s ≥1} be a partition of an integer n. Then the Ferrers-Young diagram of Λ is an array of nodes with λ _i nodes in the ith row. Let λ _j ′ denote the number of nodes in column j in the Ferrers-Young diagram of Λ. The hook number of the (i,j) node in the Ferrers-Young diagram of Λ is denoted by H(i,j):=λ _i +λ _j ′−i−j+1. A partition of n is called a t-core partition of n if none of the hook numbers is a multiple of t. The number of t-core partitions of n is denoted by a(t;n). In the present paper, some congruences and distribution properties of the number of 2 ^t -core partitions of n are obtained. A simple convolution identity for t-cores is also given.      

Möbius Isoparametric Hypersurfaces in S n+1 with Two Distinct Principal Curvatures

A hypersurface x : M → S ⁿ⁺¹ without umbilic point is called a Möbius isoparametric hypersurface if its Möbius form Φ = ?ρ ^?2∑ _i (e _i (H) + ∑ _j (h _ij ?Hδ _ij )e _j (log ρ))θ _i vanishes and its Möbius shape operator $ {\Bbb {S}}A hypersurface x : M → S ^{n +1} without umbilic point is called a M?bius isoparametric hypersurface if its M?bius form Φ = −ρ⁻²∑ _i (e _i (H) + ∑ _j (h _ij −Hδ _ij )e _j (log ρ))θ _i vanishes and its M?bius shape operator ? = ρ⁻¹(S−Hid) has constant eigenvalues. Here {e _i } is a local orthonormal basis for I = dx·dx with dual basis {θ _i }, II = ∑ _ij h _ij θ _i ⊗θ _i is the second fundamental form, and S is the shape operator of x. It is clear that any conformal image of a (Euclidean) isoparametric hypersurface in S ^{n +1} is a M?bius isoparametric hypersurface, but the converse is not true. In this paper we classify all M?bius isoparametric hypersurfaces in S ^{n +1} with two distinct principal curvatures up to M?bius transformations. By using a theorem of Thorbergsson [1] we also show that the number of distinct principal curvatures of a compact M?bius isoparametric hypersurface embedded in S ^{n +1} can take only the values 2, 3, 4, 6. Received September 7, 2001, Accepted January 30, 2002     

Limiting Distributions for the Maximum of a Symmetric Function on a Random Point Set

Let U ₁, U ₂, ... be a sequence of independent random points taking values in a measurable space (S, Σ) according to a common probability P and let R be a symmetric, Borel/ -measurable function. Let H _n = max{h(U _i ,U _j ): 1≤ i < j≤ n} denote the maximum h-value over pairs of distinct points from U ₁,U ₂,...,U _n . Assumptions on the distribution of h(U ₁,·) are provided under which a continuous function of H _n converges in law to an extreme-value distribution upon suitable rescaling. The work is complementary to that appearing in Appel et al. (1999) J. Theor. Probab. 12, 27–47. on the almost-sure limiting behavior of H _n . In the first of two examples, the main result applied to the case of i.i.d. points distributed uniformly on the surface of a unit hypersphere in R ^d provides the limiting distribution of the maximum pairwise distance (chord length) among the first n of the points. The second example exhibits the limiting distribution of the minimum pair-wise distance among the first n of i.i.d. uniform points in a compact subset of R ^d .      

On Kendall's Process

LetZ₁, …, Z_nbe a random sample of sizen?2 from ad-variate continuous distribution functionH, and letV_i, nstand for the proportion of observationsZ_j,j≠i, such thatZ_j?Z_icomponentwise. The purpose of this paper is to examine the limiting behavior of the empirical distribution functionK_nderived from the (dependent) pseudo-observationsV_i, n. This random quantity is a natural nonparametric estimator ofK, the distribution function of the random variableV=H(Z), whose expectation is an affine transformation of the population version of Kendall's tau in the cased=2. Since the sample version ofτis related in the same way to the mean ofK_n, Genest and Rivest (1993,J. Amer. Statist. Assoc.) suggested that[formula]be referred to as Kendall's process. Weak regularity conditions onKandHare found under which this centered process is asymptotically Gaussian, and an explicit expression for its limiting covariance function is given. These conditions, which are fairly easy to check, are seen to apply to large classes of multivariate distributions.     

Order of magnitude bounds for expectations of Δ2-functions of generalized random bilinear forms

Let Φ be a symmetric function, nondecreasing on [0,∞) and satisfying a Δ₂ growth condition, (X ₁,Y ₁), (X ₂,Y ₂),…,(X _n ,Y _n ) be arbitrary independent random vectors such that for any given i either Y _i =X _i or Y _i is independent of all the other variates. The purpose of this paper is to develop an approximation of

Sums of almost equal prime cubes

We prove that almost all integers N satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 5; 6; 7; 8, i.e., N = p ₁³ + ... + p _j ³ with |p _i − (N/j)^1/3| ≦ $ N^{1/3 - \delta _j + \varepsilon } $ N^{1/3 - \delta _j + \varepsilon } (1 ≦ i ≦ j), for δ _j = 1/45; 1/30; 1/25; 2/45, respectively.     

Rook numbers and the normal ordering problem

For an element w in the Weyl algebra generated by D and U with relation DU=UD+1, the normally ordered form is w=∑c_i,jUⁱD^j. We demonstrate that the normal order coefficients c_i,j of a word w are rook numbers on a Ferrers board. We use this interpretation to give a new proof of the rook factorization theorem, which we use to provide an explicit formula for the coefficients c_i,j. We calculate the Weyl binomial coefficients: normal order coefficients of the element (D+U)ⁿ in the Weyl algebra. We extend these results to the q-analogue of the Weyl algebra. We discuss further generalizations using i-rook numbers.     

On simultaneous reduction of families of matrices to triangular or diagonal form by unitary congruences

Let {A_i }and {B_i } be two given families of n-by-n matrices. We give conditions under which there is a unitary U such that every matrix UA_iU ¹ is upper triangular. We give conditions, weaker than the classical conditions of commutativity of the whole family, under which there is a unitary U such that every matrix UA_jU ^? is upper triangular. We also give conditions under which there is one single unitary U such that every UA_iU ¹ and every UB_jU ^? is upper triangular. We give necessary and sufficient conditions for simultaneous unitary reduction to diagonal form in this way when all the A_j's are complex symmetric and all theB_j 's are Hermitian.     

Exact laws for sums of ratios of order statistics from the Pareto distribution

Consider independent and identically distributed random variables {X _nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X _n(i) ≤ X _n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables R _ij = X _n(j)/X _n(i).