Finite-dimensionality and growth of algebras specified by polylinearly interrelated generators

We investigate the finite-dimensionality and growth of algebras specified by a system of polylinearly interrelated generators. The results obtained are formulated in terms of a function ρ. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1435–1440, October, 2005.     

An acceleration theorem for the ρ-algorithm

Summary. The ρ-algorithm of Wynn is an excellent device for accelerating the convergence of some logarithmically convergent sequences. Until now a convergence theorem and an acceleration theorem for the ρ-algorithm have not been obtained. The purpose of this paper is to give an acceleration theorem for the ρ-algorithm. Moreover, it is proved that the ρ-algorithm cannot accelerate linear convergence. Numerical examples are given. Received October 20, 1994 / Revised version received July 2, 1995     

On a class of stochastic partial differential equations related to turbulent transport

Summary. We consider the Cauchy problem for the mass density ρ of particles which diffuse in an incompressible fluid. The dynamical behaviour of ρ is modeled by a linear, uniformly parabolic differential equation containing a stochastic vector field. This vector field is interpreted as the velocity field of the fluid in a state of turbulence. Combining a contraction method with techniques from white noise analysis we prove an existence and uniqueness result for the solution ρ∈C ^1,2([0,T]×ℝ ^d ,(S)^*), which is a generalized random field. For a subclass of Cauchy problems we show that ρ actually is a classical random field, i.e. ρ(t,x) is an L ²-random variable for all time and space parameters (t,x)∈[0,T]×ℝ ^d . Received: 27 March 1995 / In revised form: 15 May 1997     

High-Dimensional Centrally Symmetric Polytopes with Neighborliness Proportional to Dimension

Let A be a d by n matrix, d < n. Let C be the regular cross polytope (octahedron) in Rⁿ. It has recently been shown that properties of the centrosymmetric polytope P = AC are of interest for finding sparse solutions to the underdetermined system of equations y = Ax [9]. In particular, it is valuable to know that P is centrally k-neighborly. We study the face numbers of randomly projected cross polytopes in the proportional-dimensional case where d ∼ δn, where the projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of Rⁿ. We derive ρ_N(δ) > 0 with the property that, for any ρ < ρ_N(δ), with overwhelming probability for large d, the number of k-dimensional faces of P = AC is the same as for C, for 0 ≤ k ≤ ρd. This implies that P is centrally ⌊ ρ d ⌋-neighborly, and its skeleton Skel_{⌊ ρ d ⌋}(P) is combinatorially equivalent to Skel_{⌊ ρ d⌋}(C). We display graphs of ρ_N. Two weaker notions of neighborliness are also important for understanding sparse solutions of linear equations: weak neighborliness and sectional neighborliness [9]; we study both. Weak (k,ε)-neighborliness asks if the k-faces are all simplicial and if the number of k-dimensional faces f_k(P) ≥ f_k(C)(1 – ε). We characterize and compute the critical proportion ρ_W(δ) > 0 such that weak (k,ε) neighborliness holds at k significantly smaller than ρ_W · d and fails for k significantly larger than ρ_W · d. Sectional (k,ε)-neighborliness asks whether all, except for a small fraction ε, of the k-dimensional intrinsic sections of P are k-dimensional cross polytopes. (Intrinsic sections intersect P with k-dimensional subspaces spanned by vertices of P.) We characterize and compute a proportion ρ_S(δ) > 0 guaranteeing this property for k/d ∼ ρ < ρ_S(δ). We display graphs of ρ_S and ρ_W.     

On tensor products of completely positive linear maps between pro-C *-algebras

In this paper we define the tensor products of completely positive linear maps between pro-C^*-algebras and discuss about connection between the KSGNS construction associated with the strict completely positive linear maps ρ and θ and the KSGNS construction associated with ρ ⊗ θ. This research was partially supported by CEEX grant -code PR-D11-PT00-48/2005 from The Romanian Ministry of Education and Research and partially by CNCSIS (Romanian National Council for Research in High Education) grant-code A 1065/2006.     

Univalent holomorphic mappings on a complex manifold with a C1 exhaustion function

Let M be a complex manifold with a exhaustion function ρ. Assume that ρ attains its minimum at a single point o and dρ≠ 0 on M\{o}. In this case, we will give a necessary condition and a sufficient condition for locally biholomorphic mappings on M to be biholomorphic and to have Φ-like images. Received: 15 May 1998 / Revised version: 8 March 1999     

Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems

ln) iterations, where ν is the parameter of a self-concordant barrier for the cone, ε is a relative accuracy and ρ_f is a feasibility measure. We also discuss the behavior of path-following methods as applied to infeasible problems. We prove that strict infeasibility (primal or dual) can be detected in O(ln) iterations, where ρ_· is a primal or dual infeasibility measure. Received April 25, 1996 / Revised version received March 4, 1998 Published online October 9, 1998     

Oligomorphic clones

A permutation group on a countably infinite domain is called oligomorphic if it has finitely many orbits of finitary tuples. We define a clone on a countable domain to be oligomorphic if its set of permutations forms an oligomorphic permutation group. There is a close relationship to ω-categorical structures, i.e., countably infinite structures with a first-order theory that has only one countable model, up to isomorphism. Every locally closed oligomorphic permutation group is the automorphism group of an ω-categorical structure, and conversely, the canonical structure of an oligomorphic permutation group is an ω-categorical structure that contains all first-order definable relations. There is a similar Galois connection between locally closed oligomorphic clones and ω-categorical structures containing all primitive positive definable relations. In this article we generalise some fundamental theorems of universal algebra from clones over a finite domain to oligomorphic clones. First, we define minimal oligomorphic clones, and present equivalent characterisations of minimality, and then generalise Rosenberg’s five types classification to minimal oligomorphic clones. We also present a generalisation of the theorem of Baker and Pixley to oligomorphic clones. Presented by A. Szendrei. Received July 12, 2005; accepted in final form August 29, 2006.     

Novikov’s inversion formula for the attenuated Radon transform—A new approach

We study the inversion of weighted Radon transforms in two dimensions, R_ρƒ(L)=ƒ_L =ƒ(·), where the weight function ρ(L, x), L a line and x ∈ L, has a special form. It was an important breakthrough when R.G. Novikov recently gave an explicit formula for the inverse of R_ρ when ρ has the form(1.2); in this case R_ρ is called the attenuated Radon transform. Here we prove similar results for a somewhat larger class of ρ using completely different and quite elementary methods.     

Asymptotic Elasticity in Atomic Monoids

Let M be a commutative atomic monoid (i.e. every nonzero nonunit of M can be factored as a product of irreducible elements). Let ρ(x) denote the elasticity of x ∈ M, R(M) = {ρ(x) | x ∈ M} the set of elasticities of elements in M, and ρ(M) = sup R(M) the elasticity of M. Define \overline{ρ}(x) = lim_n→∞ ρ(xⁿ) to be the asymptotic elasticity of x. We determine some basic properties of the function \overline{ρ} and determine its image for certain block monoids.     

Smolyak's algorithm for weighted <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-approximation of multivariate functions with bounded <Emphasis Type="Italic">r</Emphasis>th mixed derivatives over ℝ<Superscript>d</Superscript>

We are interested in numerical algorithms for weighted L₁ approximation of functions defined on . We consider the space ℱ_r,d which consists of multivariate functions whose all mixed derivatives of order r are bounded in L₁-norm. We approximate f∈ℱ_r,d by an algorithm which uses evaluations of the function. The error is measured in the weighted L₁-norm with a weight function ρ. We construct and analyze Smolyak's algorithm for solving this problem. The algorithm is based on one-dimensional piecewise polynomial interpolation of degree at most r−1, where the interpolation points are specially chosen dependently on the smoothness parameter r and the weight ρ. We show that, under some condition on the rate of decay of ρ, the error of the proposed algorithm asymptotically behaves as , where n denotes the number of function evaluations used. The asymptotic constant is known and it decreases to zero exponentially fast as d→∞.     

Asymptotic behavior of the critical probability for ρ-percolation in high dimensions

We consider oriented bond or site percolation on ℤ ^d ₊. In the case of bond percolation we denote by P _p the probability measure on configurations of open and closed bonds which makes all bonds of ℤ ^d ₊ independent, and for which P _p {e is open} = 1 −P _p e {is closed} = p for each fixed edge e of ℤ ^d ₊. We take X(e) = 1 (0) if e is open (respectively, closed). We say that ρ-percolation occurs for some given 0 < ρ≤ 1, if there exists an oriented infinite path v ₀ = 0, v ₁, v ₂, …, starting at the origin, such that lim inf _{n →∞} (1/n) ∑ _i=1 ⁿ X(e _i ) ≥ρ, where e _i is the edge {v _i−1 , v _i }. [MZ92] showed that there exists a critical probability p _c = p _c (ρ, d) = p _c (ρ, d, bond) such that there is a.s. no ρ-percolation for p < p _c and that P _p {ρ-percolation occurs} > 0 for p > p _c . Here we find lim _{d →∞} d ^1/ρ p _c (ρd, bond) = D ₁ , say. We also find the limit for the analogous quantity for site percolation, that is D ₂ = lim _{d →∞} d ^1/ρ p _c (ρ, d, site). It turns out that for ρ < 1, D ₁ < D ₂ , and neither of these limits equals the analogous limit for the regular d-ary trees. Received: 7 January 1999 / Published online: 14 June 2000     

Generalized V-univexity type-I for multiobjective programming with n-set functions

In the last time important results in multiobjective programming involving type-I functions were obtained (Yuan et al. in: Konnov et al. (eds) Lecture notes in economics and mathematical systems, 2007; Mishra et al. An Univ Bucureşti Ser Mat, LII(2): 207–224, 2003). Following one of these ways, we study optimality conditions and generalized Mond-Weir duality for multiobjective programming involving n-set functions which satisfy appropriate generalized univexity V-type-I conditions. We introduce classes of functions called (ρ, ρ′)-V-univex type-I, (ρ, ρ′)-quasi V-univex type-I, (ρ, ρ′)-pseudo V-univex type-I, (ρ, ρ′)-quasi pseudo V-univex type-I, and (ρ, ρ′)-pseudo quasi V-univex type-I. Finally, a general frame for constructing functions of these classes is considered. This research was supported by Grant PN II code ID No. 112/01.10.2007, CEEX code 1/2006 No. 1531/2006, and CNCSIS A No. 105 GR/2006.     

A Stochastic Analog of Bogolyubov's Second Theorem

We establish an estimate for the rate at which a solution of an ordinary differential equation subject to the action of an ergodic random process converges to a stationary solution of a deterministic averaged system on time intervals of order for some 0 < ρ < 1. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 879–894, July, 2005.     

Small sample properties of the power function ofF tests in two-way error component regression

In this paper we investigate properties of the power function of the generalized least squaresF test for linear hypotheses under regression models with two-way error component model. The covariance structure of the model depends on the correlation coefficients ρ1 and ρ2 corresponding to the random effects. This model has been frequently applied to the analysis of panel data. In general, we show that the power is a monotonically increasing function of ρ1(ρ2) in a region which is close to the ρ1(ρ2) axis, and a monotonically decreasing function of ρ1(ρ2) in a region close to the ρ2(ρ1) axis. This research is supported by the National Natural Science Foundation of China, the Natural Science Foundation of Beijing, a project of Science and Technology of Beijing Education Committee, the Academy of Finland, and the University of Tampere.     

Non-relativistic global limits of entropy solutions to the isentropic relativistic Euler equations

We consider in this paper the relativistic Euler equations in isentropic fluids with the equation of state p = κ²ρ, where κ, the sound speed, is a constant less than the speed of light c. We discuss the convergence of the entropy solutions as c→∞. The analysis is based on the geometric properties of nonlinear wave curves and the Glimm’s method.     

On global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities

We consider the Cauchy problem for the equations of selfgravitating motions of a barotropic gas with density-dependent viscosities μ(ρ), and λ(ρ) satisfying the Bresch–Desjardins condition, when the pressure P(ρ) is not necessarily a monotone function of the density. We prove that this problem admits a global weak solution provided that the adiabatic exponent γ associated with P(ρ) satisfies ${\gamma > \frac{4}{3}}${\gamma > \frac{4}{3}}.     

Erdős measures,sofic measures,and Markov chains

We consider the random variable ζ = ξ₁ρ+ξ₂ρ²+…, where ξ₁, ξ₂, … are independent identically distibuted random variables taking the values 0 and 1 with probabilities P(ξ_i = 0) = p₀, P(ξ_i = 1) = p₁, 0 < p₀ < 1. Let β = 1/ρ be the golden number. The Fibonacci expansion for a random point ρζ from [0, 1] is of the form η₁ρ + η₂ρ² + … where the random variables η_k are {0, 1}-valued and η_kη_k+1 = 0. The infinite random word η = η₁η₂ … η_n … takes values in the Fibonacci compactum and determines the so-called Erdős measure μ(A) = P(η ∈ A) on it. The invariant Erdős measure is the shift-invariant measure with respect to which the Erdős measure is absolutely continuous. We show that the Erdős measures are sofic. Recall that a sofic system is a symbolic system that is a continuous factor of a topological Markov chain. A sofic measure is a one-block (or symbol-to-symbol) factor of the measure corresponding to a homogeneous Markov chain. For the Erdős measures, the corresponding regular Markov chain has 5 states. This gives ergodic properties of the invariant Erdős measure. We give a new ergodic theory proof of the singularity of the distribution of the random variable ζ. Our method is also applicable when ξ₁, ξ₂, … is a stationary Markov chain with values 0, 1. In particular, we prove that the distribution of ζ is singular and that the Erdős measures appear as the result of gluing together states in a regular Markov chain with 7 states. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 28–47.     

Accuracy of approximation of subharmonic functions by logarithms of moduli of analytic functions in the Chebyshev metrics

It is known that a subharmonic function of finite order ρ can be approximated by the logarithm of the modulus of an entire function at a point z outside an exceptional set up to C log |z|. In this paper, we prove that if such an approximation becomes more precise, i.e., the constant C decreases, then, beginning with C = ρ/4, the size of the exceptional set enlarges substantially. Similar results are proved for subharmonic functions of infinite order and for functions that are subharmonic in the unit disk. These theorems improve and complement a result by Yulmukhametov. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 55–73.