Superefficient estimation of the marginals by exploiting knowledge on the copula

We consider the problem of estimating the marginals in the case where there is knowledge on the copula. If the copula is smooth, it is known that it is possible to improve on the empirical distribution functions: optimal estimators still have a rate of convergence n^−1/2, but a smaller asymptotic variance. In this paper we show that for non-smooth copulas it is sometimes possible to construct superefficient estimators of the marginals: we construct both a copula and, exploiting the information our copula provides, estimators of the marginals with the rate of convergence logn/n.     

Cut-free formulations for a quantified logic of here and there

A predicate extension SQHT⁼ of the logic of here-and-there was introduced by V. Lifschitz, D. Pearce, and A. Valverde to characterize strong equivalence of logic programs with variables and equality with respect to stable models. The semantics for this logic is determined by intuitionistic Kripke models with two worlds (here and there) with constant individual domain and decidable equality. Our sequent formulation has special rules for implication and for pushing negation inside formulas. The soundness proof allows us to establish that SQHT⁼ is a conservative extension of the logic of weak excluded middle with respect to sequents without positive occurrences of implication. The completeness proof uses a non-closed branch of a proof search tree. The interplay between rules for pushing negation inside and truth in the “there” (non-root) world of the resulting Kripke model can be of independent interest. We prove that existence is definable in terms of remaining connectives.     

Grauert- and Lax-Halmos-type theorems and extension of matrices with entries in H

In the paper we prove an extension theorem for matrices with entries in H^∞(U) for U a Riemann surface of a special type. One of the main components of the proof is a Grauert-type theorem for “holomorphic” vector bundles defined on maximal ideal spaces of certain Banach algebras.     

Qregularity and an extension of the Evans-Griffiths criterion to vector bundles on quadrics

Here we define the concept of Qregularity for coherent sheaves on a smooth quadric hypersurface Q_n⊂Pⁿ⁺¹. In this setting we prove analogs of some classical properties. We compare the Qregularity of coherent sheaves on Q_n with the Castelnuovo-Mumford regularity of their extension by zero in Pⁿ⁺¹. We also classify the coherent sheaves with Qregularity −∞. We use our notion of Qregularity in order to prove an extension of the Evans-Griffiths criterion to vector bundles on quadrics. In particular, we get a new and simple proof of Knörrer’s characterization of ACM bundles.     

Integer cells in convex sets

Every convex body K in Rⁿ has a coordinate projection PK that contains at least cells of the integer lattice PZⁿ, provided this volume is at least one. Our proof of this counterpart of Minkowski's theorem is based on an extension of the combinatorial density theorem of Sauer, Shelah and Vapnik-Chervonenkis to Zⁿ. This leads to a new approach to sections of convex bodies. In particular, fundamental results of the asymptotic convex geometry such as the Volume Ratio Theorem and Milman's duality of the diameters admit natural versions for coordinate sections.     

Operator-stable distributions and stable marginals

Sharpe has shown that full operator-stable distributions μ on Rⁿ are infinitely divisible and for a suitable automorphism B depending on μ satisfy the relation $\text{μ}{}^{\mathrm{t}}\text{= μt}{}^{\mathrm{?B}}\text{1 δ(b(t))}$ for all t > 0. B is called an exponent for μ. It is proved here that if an operator-stable distribution on Rⁿ has n linearly independent univariate stable marginals, then its exponents are semi-simple operators. In addition necessary and sufficient conditions are given for such a distribution on R² to have univariate stable marginals. The proofs use a hitherto unpublished result of Sharpe's that all full operator-stable distributions are absolutely continuous. His proof is provided here.     

A method for proving polynomial enumeration formulas

We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by “explaining” their zeros using an appropriate combinatorial extension of the objects under consideration to negative integer parameters. We apply this method to prove a new refinement of the Bender-Knuth (ex-)Conjecture, which easily implies the Bender-Knuth (ex-)Conjecture itself. This is probably the most elementary way to prove this result currently known. Furthermore we adapt our method to q-polynomials, which allows us to derive generating function results as well. Finally we use this method to give another proof for the enumeration of semistandard tableaux of a fixed shape which differs from our proof of the Bender-Knuth (ex-)Conjecture in that it is a multivariate application of our method.     

Lattices on simplicial partitions

In this paper, (d+1)-pencil lattices on simplicial partitions in R^d are studied. The barycentric approach naturally extends the lattice from a simplex to a simplicial partition, providing a continuous piecewise polynomial interpolant over the extended lattice. The number of degrees of freedom is equal to the number of vertices of the simplicial partition. The constructive proof of this fact leads to an efficient computer algorithm for the design of a lattice.     

Visualizing elements of order two in the Weil-Châtelet group

Let E be an elliptic curve over an infinite field K with characteristic ≠2, and σ∈H¹(G_K,E)[2] a two-torsion element of its Weil-Châtelet group. We prove that σ is always visible in infinitely many abelian surfaces up to isomorphism, in the sense put forward by Cremona and Mazur in their article (J. Exp. Math. 9(1) (2000) 13). Our argument is a variant of Mazur's proof, given in (Asian J. Math. 3(1) (1999) 221), for the analogous statement about three-torsion elements of the Shafarevich-Tate group in the setting where K is a number field. In particular, instead of the universal elliptic curve with full level-three-structure, our proof makes use of the universal elliptic curve with full level-two-structure and an invariant differential.     

A linear-time algorithm for edge-disjoint paths in planar graphs

In this paper we discuss the problem of finding edge-disjoint paths in a planar, undirected graph such that each path connects two specified vertices on the boundary of the graph. We will focus on the “classical” case where an instance additionally fulfills the so-calledevenness-condition. The fastest algorithm for this problem known from the literature requiresO (n ^5/3(loglogn)^1/3) time, wheren denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which results in anO(n) algorithm. The proof of correctness immediately yields an alternative proof of the Theorem of Okamura and Seymour, which states a necessary and sufficient condition for solvability.     

Biproportional scaling of matrices and the iterative proportional fitting procedure

A short proof is given of the necessary and sufficient conditions for the convergence of the Iterative Proportional Fitting procedure. The input consists of a nonnegative matrix and of positive target marginals for row sums and for column sums. The output is a sequence of scaled matrices to approximate the biproportional fit, that is, the scaling of the input matrix by means of row and column divisors in order to fit row and column sums to target marginals. Generally it is shown that certain structural properties of a biproportional scaling do not depend on the particular sequence used to approximate it. Specifically, the sequence that emerges from the Iterative Proportional Fitting procedure is analyzed by means of the L ₁-error that measures how current row and column sums compare to their target marginals. As a new result a formula for the limiting L ₁-error is obtained. The formula is in terms of partial sums of the target marginals, and easily yields the other well-known convergence characterizations.     

Functions operating on multivariate distribution and survival functions—With applications to classical mean-values and to copulas

Functions operating on multivariate distribution and survival functions are characterized, based on a theorem of Morillas, for which a new proof is presented. These results are applied to determine those classical mean values on [0,1]ⁿ which are distribution functions of probability measures on [0,1]ⁿ. As it turns out, the arithmetic mean plays a universal rôle for the characterization of distribution as well as survival functions. Another consequence is a far reaching generalization of Kimberling’s theorem, tightly connected to Archimedean copulas.     

On the relation of closed forms and Trotter's product formula

The aim of this paper is to give a characterization in Hilbert spaces of the generators of C₀-semigroups associated with closed, sectorial forms in terms of the convergence of a generalized Trotter's product formula. In the course of the proof of the main result we also present a similarity result which can be of independent interest: for any unbounded generator A of a C₀-semigroup e^tA it is possible to introduce an equivalent scalar product on the space, such that e^tA becomes non-quasi-contractive with respect to the new scalar product.     

A multivariate extension of Sarhan and Balakrishnan’s bivariate distribution and its ageing and dependence properties

A new class of bivariate distributions (NBD) was recently introduced by Sarhan and Balakrishnan [A.M. Sarhan, N. Balakrishnan, A new class of bivariate distributions and its mixture, J. Multivariate Anal. 98 (2007) 1508-1527]. In this note, we give the joint survival function of a multivariate extension of the NBD, which is not an absolutely continuous multivariate distribution, and its marginal and extreme order statistics distributions are also derived. The multivariate ageing and dependence properties of the proposed n-dimensional distribution are also discussed, and then we analyze the stochastic ageing of its marginals and its minimum and maximum order statistics.     

Potentially good reduction of Barsotti-Tate groups

Let R be a complete discrete valuation ring of mixed characteristic (0,p) with perfect residue field, K the fraction field of R. Suppose G is a Barsotti-Tate group (p-divisible group) defined over K which acquires good reduction over a finite extension K^′ of K. We prove that there exists a constant c?2 which depends on the absolute ramification index e(K^′/Qp) and the height of G such that G has good reduction over K if and only if G[pc] can be extended to a finite flat group scheme over R. For abelian varieties with potentially good reduction, this result generalizes Grothendieck's “p-adic Néron-Ogg-Shafarevich criterion” to finite level. We use methods that can be generalized to study semi-stable p-adic Galois representations with general Hodge-Tate weights, and in particular leads to a proof of a conjecture of Fontaine and gives a constant c as above that is independent of the height of G.     

Three general multivariate semi-Pareto distributions and their characterizations

Three general multivariate semi-Pareto distributions are developed in this paper. First one—GMP^(k)(III) has univariate Pareto (III) marginals, it is characterized by the minimum of two independent and identically distributed random vectors. Second one—GMSP has univariate semi-Pareto marginals and it is characterized by finite sample minima. Third one—MSP is characterized through a geometric minimization procedure. All these three characterizations are based on the general and the particular solutions of the Euler's functional equations of k-variates.     

Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities

The initial purpose of the present paper is to provide a combinatorial proof of the minor summation formula of Pfaffians in [Ishikawa, Wakayama, Minor summation formula of Pfaffians, Linear and Multilinear Algebra 39 (1995) 285-305] based on the lattice path method. The second aim is to study applications of the minor summation formula for obtaining several identities. Especially, a simple proof of Kawanaka's formula concerning a q-series identity involving the Schur functions [Kawanaka, A q-series identity involving Schur functions and related topics, Osaka J. Math. 36 (1999) 157-176] and of the identity in [Kawanaka, A q-Cauchy identity involving Schur functions and imprimitive complex reflection groups, Osaka J. Math. 38 (2001) 775-810] which is regarded as a determinant version of the previous one are given.     

Hausdorff's moment problem and expansions in Legendre polynomials

A new proof is given for Hausdorff's condition on a set of moments which determines when the function generating these moments is in L². The proof uses Legendre polynomials and their discrete extensions found by Tchebychef. Then an extension is given to a weighted L² space using Jacobi polynomials and their discrete extensions.     

Short effective intervals containing primes

We prove that every interval ]x(1−Δ⁻¹),x] contains a prime number with Δ=28314000 and provided x?10726905041. The proof combines analytical, sieve and algorithmical methods.     

On marginal distributions and isomorphisms of stationary processes

LetT be an invertible ergodic aperiodic measure preserving transformation of a Lebesgue space, letA be a finite alphabet, and let π be a probability measure onA ⁿ which admits a mixing shift-invariant measureμ _π onΩ=A ^? such that the marginals of anyn successive coordinates are π and the entropyh(T) ofT is smaller than the entropy of the shift in (Ω,μ _π). Then there exists a shift invariant measure ν_π in Ω which also has marginals π and for whichT is isomorphic to the shift in (Ω, ν_π). This contains Krieger's finite generator theorem and strengthens the measure theoretic part of his approximation theorem for shift-invariant measures by showing that the preassigned marginal π can not only be achieved up to an ε>0 but exactly. Our result also contains an as yet unpublished theorem of Krieger, which says thatT can be embedded in an arbitrary mixing subshift of finite type, as long as the entropy of the subshift under the measure with maximal entropy exceeds that ofT. In the final section we show that the method can be extended to yield also exact marginals for the generator in the Jewett-Krieger theorem, i.e.T is shown to be isomorphic to a shift in (Ω, ν_π) where ν_π has exact marginals π and the shift is uniquely ergodic on the support of ν_π.