On the Infinitesimal Rigidity of Homogeneous Varieties

Let XP be a variety (respectively an open subset of an analytic submanifold) and let xX be a point where all integer valued differential invariants are locally constant. We show that if the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Segre P× P, n,m2, a Grassmaniann G(2,n+2), n4, or the Cayley plane OP², then X is the corresponding homogeneous variety (resp. an open subset of the corresponding homogeneous variety). The case of the Segre P²×P² had been conjectured by Griffiths and Harris in [GH]. If the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Veronese v₂(P) and the Fubini cubic form of X at x is zero, then X=v₂ (P) (resp. an open subset of v₂(P)). All these results are valid in the real or complex analytic categories and locally in the C category if one assumes the hypotheses hold in a neighborhood of any point x. As a byproduct, we show that the systems of quadrics I₂(P P) S²C, I₂(P¹× P) S²C and I₂(S₅) S²C¹⁶ are stable in the sense that if A S^* is an analytic family such that for t0,AA, then A₀A. We also make some observations related to the Fulton–:Hansen connectedness theorem.     

A remark on periodic Tchebyshev systems

The linear hull of a Tchebyshev system is called a Haarspace. Every Haar-space of periodic functions has odd dimension. It is shown that under certain conditions an n-dimensional Haar-space of periodic functions contains i-dimensional Haar-spaces U_i, i=1,3,...,n, with U₁U₃...U_n=U.     

Compact sets in the spaceL p (O,T; B)

Summary A characterization of compact sets in L^p (0, T; B) is given, where 1P and B is a Banach space. For the existence of solutions in nonlinear boundary value problems by the compactness method, the point is to obtain compactness in a space L^p (0,T; B) from estimates with values in some spaces X, Y or B where XBY with compact imbedding XB. Using the present characterization for this kind of situations, sufficient conditions for compactness are given with optimal parameters. As an example, it is proved that if {f_n} is bounded in L^q(0,T; B) and in L _loc ¹ (0, T; X) and if {f_n/t} is bounded in L _loc ¹ (0, T; Y) then {f_n} is relatively compact in L^p(0,T; B), p相似文献   

Stability of Completeness for Systems of Exponentials on Compact Convex Sets in ℂ

A sufficient condition for the stability of completeness of systems of exponentials with exponent shifts _n in spaces of functions continuous onb and holomorphic inside a compact convex set with nonempty interior is obtained. This condition is a generalization of an important special case of the corresponding result for the closed interval due to Redheffer and Alexander.     

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.     

Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

Complete convergence of weighted sums of martingale differences

LetF _oF ₁ ... be an increasing family of -algebras. For eachn1,X _n isF _n-measurable, andE(X _n|F_n–1) is zero almost surely, andE(|E_n|^p|F_n–1) is bounded by a finite constant almost surely for somep2. Leta _n1,...,a _nn be constants. Conditions are given to establish the complete convergence of (a _n1 X ₁+...+a _nnX_n)/n ^1/p , thereby obtaining an extension of Chow's (1966) result for the case of independent and identically distributed random variables. Whenp>2, the conditions are an improvement on existing results for the case of independence and identical distribution.     

Subfactors associated to compact Kac algebras

We construct inclusions of the form (B ₀P) ^G (B ₁P) ^G , whereG is a compact quantum group of Kac type acting on an inclusion of finite dimensional C^*-algebrasB ₀B ₁ and on aII ₁ factorP. Under suitable assumptions on the actions ofG, this is a subfactor, whose Jones tower and standard invariant can be computed by using techniques of A. Wassermann. The subfactors associated to subgroups of compact groups, to projective representations of compact groups, to finite quantum groups, to finitely generated discrete groups, to vertex models and to spin models are of this form.     

Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix

Let Agl(n, C) and let p be a positive integer. The Hessenberg variety of degree p for A is the subvariety Hess(p, A) of the complete flag manifold consisting of those flags S ₁ S _n–1 in ⁿ which satisfy the condition AS _i S _i+p ,for all i. We show that if A has distinct eigenvalues, then Hess(p, A) is smooth and connected. The odd Betti numbers of Hess(p, A) vanish, while the even Betti numbers are given by a natural generalization of the Eulerian numbers. In the case where the eigenvalues of A have distinct moduli, |₁|<<|₁|, these results are applied to determine the dimension and topology of the submanifold of U(n) consisting of those unitary matrices P for which A ₀=P ^-1 AP is in Hessenberg form and for which the diagonal entries of the QR-iteration initialized at A ₀ converge to a given permutation of ₁,_n.Research partially supported by the National Science Foundation under Grant ECS-8696108.     

Hankel operators and problems of best approximation of unbounded functions

For each function f, f VMO, there exists a unique function f₀, analytic in the circle and such that f–f₀=_f{gVMO_A}. We define the operator of best approximation (nonlinear) A, Af=f₀, fVMO, In the paper one considers the question of the preservation of a class under the action of the operator i.e. finding the classes X, X VMO, AX X. One investigates the classes X containing unbounded functions. It is proved that if P_X is the space of the symbols of the Hankel operators from a Banach space E of functions into the Hardy space H², then AX X. For E one can take almost any space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 5–17, 1985.     

Covering a convex body by smaller homothetic copies

Let a convex bodyAE ⁿ be covered bys smaller homothetic copies with coefficients ₁, ..., _s , respectively. It is conjectured that ₁ + ...+ _s n. This conjecture is confirmed in two cases:n is arbitrary ands=n+1;s is arbitrary andn=2.     

A solution of Hadwiger's covering problem for zonoids

For a convex body M ⁿ byb(M) the least integerp is denoted, such that there are bodiesM ₁, ...,M _p each of which is homothetic toM with a positive ratiok<1 andM ₁...M _p M. H. Martini has proved [7] thatb(M)<-3·2 ^n–2 for every zonotope M ⁿ , which is not a parallelotope.In the paper this Martini's result is extended to zonoids. In the proof some notions and facts of real functions theory are used (points of density, approximative continuity).     

Ein Kontinuitätssatz für k-holomorphe Mengen

Let k denote a non-trivial non-archimedean complete valuated field and X an irreducible k-affinoid space. We discuss the Hartog's domain H^*:=(X×Eⁿ) (U×Eⁿ) where øUX is an affinoid subdomain, Eⁿ is the n-dimensional unit-polydisc over k and Eⁿ is the ringdomain of all z==(z₁,...,z_n)Eⁿ with some coordinate |z_i|=1. The main result is the non-archimedean version of Rothstein's extensiontheorem for analytic subvarieties: Every k-holomorphic subvariety AH^* whose every branch has dimension (dim X + 1) can be extended to a k-holomorphic subvariety X×Eⁿ such that every branch of has dimension (dim X + l).     

Eigenvalue Estimates for Submanifolds with Locally Bounded Mean Curvature

We present a method to obtain lower bounds for firstDirichlet eigenvalue in terms of vector fields with positivedivergence. Applying this to the gradient of a distance functionwe obtain estimates of eigenvalue of balls inside the cut locus and of domains M B _N (p, r) in submanifolds M Nwith locally bounded mean curvature. Forsubmanifolds of Hadamard manifolds with bounded mean curvaturethese lower bounds depend only on the dimension of the submanifold and the bound on its mean curvature.     

Legitimate colorings of projective planes

For a projective plane _n of ordern, let( _n ) denote the minimum numberk, so that there is a coloring of the points of _n ink colors such that no two distinct lines contain precisely the same number of points of each color. Answering a question of A. Rosa, we show that for all sufficiently largen, 5 ( _n ) 8 for every projective plane _n of ordern. Research supported in part by Allon Fellowship and by a grant from the United States Israel Binational Science Foundation     

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

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.     

Bayesian estimation in the symmetric location problem

Summary Let X _i =+ _i for i=1, ..., n, where the _i's are i.i.d. F and F is symmetric about 0. F is assumed unknown or only partially known, and the problem is to estimate . Priors are put on the pair (F,). The priors on F are obtained from Doksum's neutral to the right priors, and include symmetrized Dirichlet priors. The marginal posterior distribution of given X ₁, ..., X _nis computed and its general properties studied. It is found that for certain classes of distributions of the _i's, the posterior distribution of is for all large n a point mass at the true value of . If the distribution of the _i's is not exactly symmetric, the Bayes estimates can behave very poorly.     

An embedding theorem for a limiting exponent

We consider the function space B _p ^l () of functionsf(x), defined on the domain of a certain class and characterized by specific differential-difference properties in L_p(). We prove a theorem on the embedding B _p,q ^l () L_q in the case whenl=n/p –n/q >0 and its generalization for vectorl, p, q.Translated from Matematicheski Zametki, Vol. 6, No. 2, pp. 129–138, August, 1969.     

Separation properties for Sturm-Liouville operators

Let q (x) be a positive function given on the interval I of the real axis; let P be the minimal operator generated in L₂(0, +) by the differential expression P[·]=-d²/dx²+q(x); let Q be the operator of multiplication by the function q(x). If D_p* Dq, then P [·] is said to be separated. In this note the separation of P[·] is proved for some growth regularity conditions on the fonction q (x), without assuming anything on its smoothness. One proves that if D_p* D_S, where S is the multiplication operator by the function s(x), satisfying some growth regularity condition, then D_Q D_S.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 349–359, September, 1973.