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.     

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.     

The embedding of linearly ordered sets

It is shown that if a linearly ordered set B does not contain as subsets sets of order type and ^* then B can be embedded in 2 . We construct an example of a set satisfying the above conditions which cannot be embedded in any 2 if < . Simultaneously we show that for any ordinal, 2 ⁺¹ cannot be embedded in 2 and that there exists at least ₊₁ distinct dense order types of cardinality 2 .Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 83–88, January, 1972.In conclusion, I wish to take the opportunity to thank Yu. L. Ershov for kindness and assistance in this work.     

Some Conditions for a C0-Semigroup to Be Asymptotically Finite-Dimensional

We study the class of bounded C ₀-semigroups T=(T _t ) _t0 on a Banach space X satisfying the asymptotic finite dimensionality condition: codim X ₀(T)<, where X ₀(T):={x X:lim_t T _t x=0}. We prove a theorem which provides some necessary and sufficient conditions for asymptotic finite dimensionality.     

Two results on exact boundary control of parabolic equations

(1) The known nullcontrollability result for boundary control ofu _t = u_xx + q(x)u is generalized to consider a time-dependent coefficientq. (2) For boundary control ofu _t =Au (where it is known thatC _T : (initial data) (optimal nullcontrol for timeT) exists for allT>0) it is shown that logC _T =145-1 asT 0.     

Black holes on the plane drawn by a Wiener process

Summary We say that the discD()R ², of radius , located around the origin isp-covered in timeT by a Wiener processW(·) if for anyzD() there exists a 0tT such thatW(t) is a point of the disc of radiusp, located aroundz. The supremum of those 's (0) is studied for which,D() isp-covered inT.     

On Bahadur asymptotic efficiency in a semiparametric regression model

ＯＮＢＡＨＡＤＵＲＡＳＹＭＰＴＯＴＩＣＥＦＦＩＣＩＥＮＣＹＩＮＡＳＥＭＩＰＡＲＡＭＥＴＲＩＣＲＥＧＲＥＳＳＩＯＮＭＯＤＥＬＬＩＡＮＧＨＵＡ（梁华）;ＣＨＥＮＧＰＩＮＧ（成平）（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｓＳｃｉｅｎｃｅ,ｔｈｅＣｈｉｎｅｓｅＡ...     

Study of a filtration expanded to include an honest time

Summary Let (,, P) be a measurable space, and { _t} be a filtration on (,). Then, given a fixed honest timeL a new filtrationG _t} is defined, the smallest containing { _t} and for whichL is a stopping time, and the martingales, semimartingales and stopping times of this new filtration are characterised.     

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.     

The existence of probability measures with specified projections

Let X and Y be locally compact-compact topological spaces, F X×Y is closed, and P(F) is the set of all Borel probability measures on F. For us to find, for the pair of probability measures (_x, _y P (X)×P(Y), a probability measure P(F) such that _X = _X ^–1 , _Y = _Y ^–1 it is necessary and sufficient that, for any pair of Borel sets A X, B Y for which (A× B) F=Ø, the condition _XA+ _YB 1 holds.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 573–576, October, 1973.     

Sensitivity analysis of the greatest eigenvalue of a symmetric matrix via the ɛ-subdifferential of the associated convex quadratic form

LetA(·) be ann × n symmetric affine matrix-valued function of a parameteruR ^m , and let (u) be the greatest eigenvalue ofA(u). Recently, there has been interest in calculating (u), the subdifferential of atu, which is useful for both the construction of efficient algorithms for the minimization of (u) and the sensitivity analysis of (u), namely, the perturbation theory of (u). In this paper, more generally, we investigate the Legendre-Fenchel conjugate function of (·) and the -subdifferential (u) of atu. Then, we discuss relations between the set (u) and some perturbation bounds for (u).The author is deeply indebted to Professor J. B. Hiriart-Urruty who suggested this study and provided helpful advice and constant encouragement. The author also thanks the referees and the editors for their substantial help in the improvement of this paper.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).     

The trace to subsets ofR n of Besov spaces in the general case

R ⁿ. , , , F R ⁿ, F , R ⁿ R ⁿ . ^p,q (Rⁿ), >0, 1, q, — ( ) Rⁿ. , ^p,q (Rⁿ) ^F Rⁿ. , q B ^p,q (F), = – (n–)/, >0, — « », ad — F, . , . : , F=R ^d,F— « » F— R ⁿ, « », F. .

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This work has been supported in part by the Swedish Natural Science Research Council.     

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(×).     

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. , - , « » .     

Galerkin methods for parabolic equations with nonlinear boundary conditions

A variety of Galerkin methods are studied for the parabolic equationu _t =(a(x) u),x ⁿ ,t (O,T], subject to the nonlinear boundary conditionu _v =g(x,t,u),x,t (O,T] and the usual initial condition. Optimal order error estimates are derived both inL ² () andH ¹ () norms for all methods treated, including several that produce linear computational procedures.The authors were partially supported by The National Science Foundation during the preparation of this paper.     

Analysis of a class of fractional programming problems

We propose a solution strategy for fractional programming problems of the form max _xx g(x)/ (u(x)), where the function satisfies certain convexity conditions. It is shown that subject to these conditions optimal solutions to this problem can be obtained from the solution of the problem max _xx g(x) + u(x), where is an exogenous parameter. The proposed strategy combines fractional programming andc-programming techniques. A maximal mean-standard deviation ratio problem is solved to illustrate the strategy in action.     

Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups

Let ( _t ) _t0 be a -semistable convolution semigroup of probability measures on a Lie groupG whose idempotent ₀ is the Haar measure on some compact subgroupK. Then all the measures ₁ are supported by theK-contraction groupC _K() of the topological automorphism ofG. We prove here the structure theoremC _K()=C()K, whereC() is the contraction group of . Then it turns out that it is sufficient to study semistable convolution semigroups on simply connected nilpotent Lie groups that have Lie algebras with a positive graduation.     

Optimal growth rates for a viscous Hamilton-Jacobi equation

The growth of the L^m-norm, m [1,], of non-negative solutions to the Cauchy problem _t u – u = |u| is studied for non-negative initial data decaying at infinity. More precisely, the function is shown to be bounded from above and from below by positive real numbers. This result indicates an asymptotic behaviour dominated by the hyperbolic Hamilton-Jacobi term of the equation. A one-sided estimate for ln u is also established.     

Formulating Measure Differential Inclusions in Infinite Dimensions

Measure differential inclusions were introduced by J. J. Moreau to study sweeping processes, and have since been used to study rigid body dynamics and impulsive control problems. The basic formulation of an MDI is d / d (t) K(t) where is a vector measure, an unsigned measure, and K() is a set-valued map with closed, convex values and is hemicontinuous. Note that need not be absolutely continuous with respect to . Stewart extended Moreau's original concept (which applied only to cone-valued K()) to general convex sets, and gave strong and weak formulations of d / d (t) K(t) where K(t) R ⁿ . Here the strong and weak formulations of Stewart are extended to infinite-dimensional problems where K(t) X where X is a separable reflexive Banach space; they are shown to be equivalent under mild assumptions on K().