Thermodynamics and Hydrodynamics (Statistical Foundations): 3. Hydrodynamic Equations

We show that all the hydrodynamic equations can be obtained from the BBGKY hierarchy. The theory is constructed by expanding the distribution functions in series in a small parameter = R/L 10^–8, where R 10^–7cm is the radius of the correlation sphere and L is the characteristic macroscopic dimension. We also show that in the zeroth-order approximation with respect to this parameter, the BBGKY hierarchy implies the local equilibrium and the transport equations for the ideal Euler fluid; in the first-order approximation with respect to , the BBGKY hierarchy implies the hydrodynamic equations for viscous fluids. Moreover, we prove that the intrinsic energy flux must include both the kinetic energy flux proportional to the temperature gradient and the potential energy flux proportional to the density gradient. We show that the hydrodynamic equations hold for t 10^–12s and L R 10^–7cm.     

The transposition of locally compact,connected translation planes

This note deals with the transposition of translation planes in the topological context. We show that a topological congruenceC of the real vector space ²ⁿ has the property that every hyperplane of ²ⁿ contains a component ofC. This makes it possible to define the transposeP of the topological translation planeP associated withC; it is proved that the translation planeP is topological also. The relationship between collineation groups and the relationship between coordinatizing quasifields ofP andP are also discussed.     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

On the domain of attraction of an operator between supremum and sum

Summary Between the operations which produce partial maxima and partial sums of a sequenceY ₁,Y ₂, ..., lies the inductive operation:X _n =X _n-1(X _n-1+Y _n ),n1, for 0<<1. If theY _n are independent random variables with common distributionF, we show that the limiting behavior of normed sequences formed from {X _n ,n1}, is, for 0<<1, parallel to the extreme value case =0. ForFD() we give a full proof of the convergence, whereas forFD()D(), we only succeeded in proving tightness of the involved sequence. The processX _n is interesting for some applied probability models.     

Probability and expectation inequalities

Summary This paper introduces a mathematical framework within which a wide variety of known and new inequalities can be viewed from a common perspective. Probability and expectation inequalities of the following types are considered: (a)P(ZA) P(ZA) for some class of setsA, (b)k(Z)k(Z) for some class of functionsk, and (c)l(Z)k(Z) for some class of pairs of functionsl andk. It is shown, sometimes using explicit constructions ofZ andZ, that, in several cases, (a) (b) (c); included here are cases of normal and elliptically contoured distributions. A case where (a) (b) (c) is studied and is expressed in terms ofn monotone functions for (some of) which integral representations are obtained. Also, necessary and sufficient conditions for (c) are given.Research supported by the Air Force Office of Scientific Research under Grants AFOSR-75-2796 and AFOSR-80-0080Research supported by the National Science Foundation under Grants MCS78-01240 and MCS81-00748     

Measure ratio of equidecomposable sets

Let n 2. There are Lebesgue measurable sets A and B in ³ such that (B)/(A)=r and A _n B if and only if 2/n r n/2.     

Exact estimates for partially monotone approximation

f(x) — , - [–1,1], (f, ) — , as— f, . . (- ) (x _i,x _{i+ 1}) (i=0, 1, ...,s–1; =–1,x _s,=1), f(x) . , n=0,1,... _n() , [– 1,1] signf(x) sign _n(x) 0, ¦f(x)– _n(x)¦ C(s) (f, 1/n+1, C(s) s. , - , « » .     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

On the Integrability of Eigenfunctions of the Laplace-Beltrami Operator in the Unit Ball of C n

Let B denote the unit ball in C ⁿ , n1, and let , , and denote the volume measure, gradient, and Laplacian respectively, with respect to the Bergman metric on B. For R and 0<p<, we denote by L ^p the set of real, or complex-valued measurable functions f on B for which _B (1–|z|²)|f(z)| ^p d(z)<, and by D ^p the Dirichlet space of C ¹ functions f on B for which | f|L ^p . Also, for C, we denote by X the set of C ² real, or complex-valued functions f on B for which f=f. The main result of the paper is as follows: Let 0<p< and suppose R with –n ². Then L ^p X ={0}, and for 0, D ^p X ={0}(a) for all n+ when p1, and(b) for all when 0<p<1.By example it is shown that the result is best possible for all values of p with pn/(n+ .     

О сходимости метода совпадения

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On convex multipliers of convergence of some classes of bivariate fourier series

- ()N²,L _F ( ) — , 2- , {s _m() f} -L. — . (L _F( ),L _F( ) ={(k)} (kZ²) , fL_F( ) f , , L _F( ). - ={()} ={()} , n(())m()n(()+()) . R() , .. - . , . (L _F ( ),L _F ( )) , R(,)=O(1) (x).

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The author wishes to express his gratitude to S. A.Teljakovski for setting the problem and for his attention to this paper.     

On local oscillatory integrals with variable Calderón-Zygmund kernels, II

Let , be a real analytic function or a real-C function on ⁿ andk be a variable Calderón-Zygmund kernel. Define the oscillatory singular integral operatorT by

Approximation of the Distribution of the Supremum of a Centered Random Walk. Application to the Local Score

Let (X _n ) _{n 0} be a real random walk starting at 0, with centered increments bounded by a constant K. The main result of this study is: |P(S _n n x)–P( sup_{0 u 1} B _u x)| C(n,K) n/n, where x 0, ² is the variance of the increments, S _n is the supremum at time n of the random walk, (B _u ,u 0) is a standard linear Brownian motion and C(n,K) is an explicit constant. We also prove that in the previous inequality S _n can be replaced by the local score and sup₀ u 1 B _u by sup₀ u 1|B _u |.     

The Killing–Yano Tensor

We obtain a general solution of the equations determining the Killing–Yano tensor of rank p on an n-dimensional (1 p n – 1) pseudo-Riemannian manifold of constant curvature and discuss possible applications of the obtained result.     

Monotonic subsequences in permutations of n natural numbers

Let S_n be the set of all permutations of the numbers 1, 2,..., n, and letl _n() be the number of terms in the maximal monotonic subsequence contained in S_n. If M[l _n()] is the mean value ofl _n () on S_n, then, for all except a finite number of n, the bound M[l _n()] e n is valid.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 511–514, April, 1973.The author wishes to thank E. M. Nikishin for having posed the problem and for his constant interest in the work.     

Approximating derivations on ideals ofC *-algebras

For each^*-derivation of a separableC ^*-algebraA and each >0 there is an essential idealI ofA and a self-adjoint multiplierx ofI such that (–ad(ix))|I< and x.     

On a class of orthogonal series

, [0, 1], (n+1) n-. . [2]. — (. 5.4 5.6). . 6.4 2 [5]. , [4]. , , [6] [7]. [1].     

On the complexity of following the central path of linear programs by linear extrapolation II

A class of algorithms is proposed for solving linear programming problems (withm inequality constraints) by following the central path using linear extrapolation with a special adaptive choice of steplengths. The latter is based on explicit results concerning the convergence behaviour of Newton's method to compute points on the central pathx(r), r>0, and this allows to estimate the complexity, i.e. the total numberN = N(R, ) of steps needed to go from an initial pointx(R) to a final pointx(), R>>0, by an integral of the local weighted curvature of the (primal—dual) path. Here, the central curve is parametrized with the logarithmic penalty parameterr0. It is shown that for large classes of problems the complexity integral, i.e. the number of stepsN, is not greater than constm log(R/), where < 1/2 e.g. = 1/4 or = 3/8 (note that = 1/2 gives the complexity of zero order methods). We also provide a lower bound for the complexity showing that for some problems the above estimation can hold only for 1/3.As a byproduct, many analytical and structural properties of the primal—dual central path are obtained: there are, for instance, close relations between the weighted curvature and the logarithmic derivatives of the slack variables; the dependence of these quantities on the parameterr is described. Also, related results hold for a family of weighted trajectories, into which the central path can be embedded.On leave from the Institute of Mathematics, Eötvös University Budapest, H-1080 Budapest, Hungary.     

Convolution Powers of Probabilities on Stochastic Matrices

For any probability on the space of d×d stochastic matrices we associate a probability ; on a finite group—a subgroup of the permutation group—related to the kernel of the semigroup generated by the support of . We show that ⁿ converges iff ⁿ converges.     

Description of Hyperinvariant Subspaces of a Contraction in Terms of Its Characteristic Function

If T is a completely nonunitary contraction on a Hilbert space and L is its invariant subspace corresponding to a regular factorization of its characteristic function = , then L is hyperinvariant if and only if the following two conditions are fulfilled: (1) supp _* supp is of Lebesgue measure zero; (2) for every pair A H (E E) and A _* H (E _* E _*) intertwining by , i.e., such that A =A _*, there exists a function A _F H (F F) intertwining with A by and with A _* by , i.e., such that A = A _F and A _F = A _*. Bibliography: 4 titles.