Essential dimension of quadrics

Let X be an anisotropic projective quadric over a field F of characteristic not 2. The essential dimension dim_es(X) of X, as defined by Oleg Izhboldin, is dim_es(X)=dim(X)-i(X) +1, where i(X) is the first Witt index of X (i.e., the Witt index of X over its function field).Let Y be a complete (possibly singular) algebraic variety over F with all closed points of even degree and such that Y has a closed point of odd degree over F(X). Our main theorem states that dim_es(X)dim(Y) and that in the case dim_es(X)=dim(Y) the quadric X is isotropic over F(Y).Applying the main theorem to a projective quadric Y, we get a proof of Izhboldins conjecture stated as follows: if an anisotropic quadric Y becomes isotropic over F(X), then dim_es(X)dim_es(Y), and the equality holds if and only if X is isotropic over F(Y). We also solve Knebuschs problem by proving that the smallest transcendence degree of a generic splitting field of a quadric X is equal to dim_es(X). To the memory of Oleg Izhboldin     

The first Bianchi identity in synthetic differential geometry

We give a synthetic treatment of the first Bianchi identity both in the style of differential forms and in the style of tensor fields on the lines of Lavendhomme (Basic Concepts of Synthetic Differential Geometry, Kluwer, Dordrecht, 1996). The tensor-field version of the identity is derived from the corresponding one for microcubes, just as we did for the Jacobi identity of vector fields with respect to Lie brackets in our previous paper (J. Theoret. Phys. 36 (1997) 1099–1131). As a by-product we have found out an identity of microcubes corresponding to the classical identity

R(X,Y,Z)=_XYZ−_YXZ−_[X,Y]Z

of tensor fields, which has largely simplified Lavendhomme's lengthy proof (Basic Concepts of Synthetic Differential Geometry, Kluwer, Dordrecht, 1996, Section 5.3, Proposition 8, pp. 176–180).     

Subsets of a Hilbert space,having finite hausdorff dimension

Let X₂, X₂ be Hilbert spaces, X₂ X₁, X₂ is dense in X₁, the imbedding is compact,m X₂, dim_H ⁱ m and h⁽ⁱ⁾(m) are the Hausdorff dimension and the limit capacity (information dimension) of the setm with respect to the metrics of the spaces X_i (i=1, 2). Two examples are constructed. 1) An example of a setm bounded in X₂, such that: a) h⁽¹⁾(m) < (and, consequently, dim_H ¹ m); b)m cannot be covered by a countable collection of sets, compact in X₂ (and, consequently, dim_H ² m=). 2) an Example of a setm, compact in X₂, such that h⁽¹⁾(m) < and h⁽²⁾(m)=.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 154–165, 1987.     

Identification of parameters by the distribution of the maximum random variable: The general multivariate normal case

Summary LetX ₁,X ₂, ...,X _r ber independentn-dimensional random vectors each with a non-singular normal distribution with zero means and positive partial correlations. Suppose thatX _i =(X _i1 , ...,X _in ) and the random vectorY=(Y ₁, ...,Y _n ), their maximum, is defined byY _j =max{X _ij :1ir}. LetW be another randomn-vector which is the maximum of another such family of independentn-vectorsZ ₁,Z ₂, ...,Z _s . It is then shown in this paper that the distributions of theZ _i 's are simply a rearrangement of those of theZ _j 's (and of course,r=s), whenever their maximaY andW have the same distribution. This problem was initially studied by Anderson and Ghurye [2] in the univariate and bivariate cases and motivated by a supply-demand problem in econometrics.     

Numerical properties of Chern classes of certain covering manifolds

In this paper, we study numerical properties of Chern classes of certain covering manifolds. One of the main results is the following: Let ψ : X → Pⁿ be a finite covering of the n-dimensional complex projective space branched along a hypersurface with only simple normal crossings and suppose X is nonsingular. Let c_i(X) be the i-th Chern class of X. Then (i) if the canonical divisor K_X is numerically effective, then (−1)^kc_k(X) (k ≥ 2) is numerically positive, and (ii) if X is of general type, then (−1)ⁿcⁱ_l (X) cⁱ_r, (X) > 0, where i_l + … + i_r = n. Furthermore we show that the same properties hold for certain Kummer coverings.     

A LIL type result for the product limit estimator

Summary Let X ₁,X ₂,...,X _n be i.i.d. r.v.'-s with P(X>u)=F(u) and Y ₁,Y ₂,...,Y _n be i.i.d. P(Y>u)=G(u) where both F and G are unknown continuous survival functions. For i=1,2,...,n set _i=1 if X _i Y _i and 0 if X _i >y _i , and Z _i =min {itX_i, Y_i}. One way to estimate F from the observations (Z _i , _i ) i=l,...,n is by means of the product limit (P.L.) estimator F _n ^* (Kaplan-Meier, 1958 [6]).In this paper it is shown that F _n ^* is uniformly almost sure consistent with rate O(log logn/n), that is P(sup ¦F _n ^* (u)– F(u)¦=0(log log n/n)=1 –<u<+ if G(T _F )>0, where T _F =sup{x F(x)>0}.A similar result is proved for the Bayesian estimator [9] of F. Moreover a sharpening of the exponential bound of [3] is given.     

Ample vector bundles and Bordiga surfaces

Let X be a smooth complex projective variety and let Z ? X be a smooth surface, which is the zero locus of a section of an ample vector bundle ? of rank dimX – 2 ≥ 2 on X. Let H be an ample line bundle on X, whose restriction H _Z to Z is a very ample line bundle and assume that (Z, H _Z ) is a Bordiga surface, i.e., a rational surface having (?², ?? (4)) as its minimal adjunction theoretic reduction. Triplets (X, ?, H) as above are discussed and classified. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decomposing symmetric exchanges in matroid bases

LetB,B be bases of a matroid, withX B, X B. SetsX,X are asymmetric exchange if(B – X) X and(B – X) X are bases. SetsX,X are astrong serial B-exchange if there is a bijectionf: X X, where for any ordering of the elements ofX, sayx _i ,i = 1, , m, bases are formed by the sets B₀ = B, B_i = (B_i–1 – x_i) f(x _i), fori = 1, , m. Any symmetric exchangeX,X can be decomposed by partitioning X = _i=1 ^m Y_i, X = _i=1 ^m Y_i, X, where (1) bases are formed by the setsB ₀ =B, B _i = (B _i–1 –Y _i ) Y _i ; (2) setsY _i ,Y _i are a strong serialB _i–1 -exchange; (3) properties analogous to (1) and (2) hold for baseB and setsY _i ,Y _i .     

Ordered Compactifications of Products of Two Totally Ordered Spaces

We describe the semilattice of ordered compactifications of X × Y smaller than _o X × _o Y where X and Y are certain totally ordered topological spaces, and where _o Z denotes the Stone–ech ordered- or Nachbin-compactification of Z. These basic cases are used to illustrate techniques for describing the semilattice of ordered compactifications of X × Y smaller than _o X × _o Y for arbitrary totally ordered topological spaces X and Y. Such products X × Y provide many counterexamples in the theory of ordered compactifications.     

On regularly branched maps

Let be a perfect map between finite-dimensional metrizable spaces and p1. It is shown that the space of all bounded maps from X into with the source limitation topology contains a dense G_δ-subset consisting of f-regularly branched maps. Here, a map is f-regularly branched if, for every n1, the dimension of the set is n(dimf+dimY)−(n−1)(p+dimY). This is a parametric version of the Hurewicz theorem on regularly branched maps.     

Equilibrium states for factor maps between subshifts

Let π:X→Y be a factor map, where (X,σX) and (Y,σY) are subshifts over finite alphabets. Assume that X satisfies weak specification. Let a=(a₁,a₂)∈R² with a₁>0 and a₂?0. Let f be a continuous function on X with sufficient regularity (Hölder continuity, for instance). We show that there is a unique shift invariant measure μ on X that maximizes . In particular, taking f≡0 we see that there is a unique invariant measure μ on X that maximizes the weighted entropy a₁hμ(σX)+a₂hμ°π−1(σY), which answers an open question raised by Gatzouras and Peres (1996) in [15]. An extension is given to high dimensional cases. As an application, we show that for each compact invariant set K on the k-torus under a diagonal endomorphism, if the symbolic coding of K satisfies weak specification, then there is a unique invariant measure μ supported on K so that dimHμ=dimHK.     

A note on the product of random elements of a semigroup

LetX=(X ₀,X ₁, ...) be a Markov chain on the discrete semigroupS. X is assumed to have one essential classC such thatCK, whereK is the kernel ofS. We study the processY=(Y ₀,Y ₁,...) whereY _n =X ₀ X ₁ ...X _n using the auxiliary process which is a Markov chain onS×S. The essential classes and the limiting distribution of theZ-chain are determined. (These results were obtained earlier byH. Muthsam, Mh. Math.76, 43–54 (1972). However, his proofs contained an error restricting the validity of his results.Supported in part by the Danish Ministry of Education and the Toroch Ellida Ljungbergs fond.     

The intersection homology <Emphasis Type="Italic">D</Emphasis>–module in finite characteristic

For Y a closed normal subvariety of codimension c of a smooth –variety X, Brylinski and Kashiwara showed that the local cohomology module ^c_Y(X,_X) contains a unique simple _X–submodule, denoted by (Y,X). In this paper the analogous result is shown for X and Y defined over a perfect field of finite characteristic. Moreover, a local construction of (Y, X) is given, relating it to the theory of tight closure. From the construction one obtains a criterion for the _X–simplicity of ^c_Y(X,_X). Mathematics Subject Classification (2000):14B15, 13N10     

Asymptotic structure and the existence of noncompact operators between Banach spaces

We investigate the problem of the existence of a noncompact operator T:X₀X→Y in terms of the asymptotic structure of separable Banach spaces X and Y. More precisely, for and , let T_ξ,η be the linear map which sends each x_i to y_i. We prove that if for some then every T:X₀X→Y is compact. If for n=2 all such maps have norm 1 we show the existence of a noncompact T:X₀X→Y.     

Interval-dividing processes

Summary If and then P(n ^–1·[(Y ₁)++(Y _n )] converges to cnts. law on R ¹) = P(n ^–1·[(Y ₁)++(Y _n )] converges to a cnts. law on R ¹). Thus if ,n then n ^–1[(X ₁)+...+(X _n )] converges a.s. The main result here generalizes this: Let X ₍₁₎ ⁿ , X ₍₂₎ ⁿ ,..., X _(n) ⁿ be the order statistics associated with X ₁, X ₂,,X _n. Define random variables Z ₁,Z ₂, by {Z _n =i}={X _n =X _(i) ⁿ }. Then if Z ₁,Z ₂,Z ₃, are independent and P(Z_ni)i/n, and {X _i} is bounded, n ^–1·[(X ₁)++(X _n)] converges a.s.     

On sharp Burkholder–Rosenthal-type inequalities for infinite-degree U-statistics

In this paper, we present a method that allows one to obtain a number of sharp inequalities for expectations of functions of infinite-degree U-statistics. Using the approach, we prove, in particular, the following result: Let D be the class of functions f :R₊→R₊ such that the function f(x+z)−f(x) is concave in xR₊ for all zR₊. Then the following estimate holds: for all fD and all U-statistics ∑_1i1<<ilnY_i1,…,il(X_i1,…,X_il) with nonnegative kernels Y_i1,…,il :R^l→R₊, 1i_kn; i_r≠i_s, r≠s; k,r,s=1,…,l; l=0,…,m, in independent r.v.'s X₁,…,X_n. Similar inequality holds for sums of decoupled U-statistics. The class D is quite wide and includes all nonnegative twice differentiable functions f such that the function f″(x) is nonincreasing in x>0, and, in particular, the power functions f(x)=x^t, 1<t2; the power functions multiplied by logarithm f(x)= (x+x₀)^t ln(x+x₀), 1<t<2, x₀max(e^{(3t2−6t+2)/(t(t−1)(2−t))},1); and the entropy-type functions f(x)=(x+x₀)ln(x+x₀), x₀1. As an application of the results, we determine the best constants in Burkholder–Rosenthal-type inequalities for sums of U-statistics and prove new decoupling inequalities for those objects. The results obtained in the paper are, to our knowledge, the first known results on the best constants in sharp moment estimates for U-statistics of a general type.     

Continuous mappings on subspaces of products with the κ-box topology

Much of General Topology addresses this issue: Given a function fC(Y,Z) with YY^′ and ZZ^′, find , or at least , such that ; sometimes Z=Z^′ is demanded. In this spirit the authors prove several quite general theorems in the context Y^′=(X_I)_κ=∏_iIX_i in the κ-box topology (that is, with basic open sets of the form ∏_iIU_i with U_i open in X_i and with U_i≠X_i for <κ-many iI). A representative sample result, extending to the κ-box topology some results of Comfort and Negrepontis, of Noble and Ulmer, and of Hušek, is this. Theorem Let ωκα (that means: κ<α, and [β<α and λ<κ]β^λ<α) with α regular, be a set of non-empty spaces with each d(X_i)<α, π[Y]=X_J for each non-empty JI such that |J|<α, and the diagonal in Z be the intersection of <α-many regular-closed subsets of Z×Z. Then (a) Y is pseudo-(α,α)-compact, (b) for every fC(Y,Z) there is J[I]^<α such that f(x)=f(y) whenever x_J=y_J, and (c) every such f extends to .     

Polyhedral banach spaces and extensions of compact operators

LetX be a polyhedral Banach space whose dual is anL ₁(μ) space for some measureμ. Then for each Banach spacesY ⊆Z and each compact operatorT: Y →X there exists a norm preserving compact extension Z →X.     

Linear programming and operator means

The shorted operator, the geometric mean, and the cascade limit are all examples of operations that are of the form sup{X¦C + K X ≥ 0}, where K X denotes the Kronecker product of the matrix K with the matrix X, K is a given n by n self-adjoint matrix, and C is a given positive semidefinite matrix. The supremum is taken with respect to the partial order generated by the positive semidefinite matrices. In all of the above examples the matrix K has exactly one negative eigenvalue. We show by linear programming techniques that if K has this property, and X_max = sup{X¦C + K X ≥ 0}, then (X_maxc, c) = inf tr(AY), subject to: −∑_{i,j = 1}ⁿk_ijY_ij ≥ cc^*, Y = {Y_ij)_{i,j = 1}ⁿ ≥ 0} In the case of the geometric mean A#B of two positive semidefinite matrices, this implies the new result that (A#Bc, c) = inf{tr(AY₁₁ + BY₂₂¦Y₁₂ + Y₂₁ ≥ cc^*, Y ≥ 0}.     

Correlations of functions of normal variables

Let (X₁, X₂,…, X_k, Y₁, Y₂,…, Y_k) be multivariate normal and define a matrix C by C_ij = cov(X_i, Y_j). If (i) (X₁,…, X_k) = (Y₁,…, Y_k) and (ii) C is symmetric positive definite, then 0 < varf(X₁,…, X_k) < ∞ corr(f(X₁,…, X_k),f(Y₁,…, Y_k)) > 0. Condition (i) is necessary for the conclusion. The sufficiency of (i) and (ii) follows from an infinite-dimensional version, which can also be applied to a pair of jointly normal Brownian motions.