On the Rédei zeta function

Let L be a locally finite lattice. An order function ν on L is a function defined on pairs of elements x, y (with x ≤ y) in L such that ν(x, y) = ν(x, z) ν(z, y). The Rédei zeta function of L is given by ?(s; L) = Σ_x∈Lμ(Ô, x) ν(Ô, x)^?s. It generalizes the following functions: the chromatic polynomial of a graph, the characteristic polynomial of a lattice, the inverse of the Dedekind zeta function of a number field, the inverse of the Weil zeta function for a variety over a finite field, Philip Hall's φ-function for a group and Rédei's zeta function for an abelian group. Moreover, the paradigmatic problem in all these areas can be stated in terms of the location of the zeroes of the Rédei zeta function.     

A construction for PBIBD(2)'s

This note gives a new construction for PBIBD(2)'s that generalizes a construction of Hall's for finite projective planes, and that leads to a new PBIBD(2) with parameters (v, b, k, r, λ₁, λ₂) = (36, 60, 10, 0, 2).     

A Bennett concentration inequality and its application to suprema of empirical processesUne inégalité de concentration de type Bennett et son application aux maxima de processus empiriques

We introduce new concentration inequalities for functions on product spaces. They allow to obtain a Bennett type deviation bound for suprema of empirical processes indexed by upper bounded functions. The result is an improvement on Rio's version [6] of Talagrand's inequality [7] for equidistributed variables. To cite this article: O. Bousquet, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 495–500.     

The 1-width of (0,1)-matrices having constant row sum 3

In this note we establish upper bounds for the 1-width of an m × n matrix of 0's and 1's having three 1's in every row and having a constant number, c, of 1's in every column. When c = 3, this upper bound is $\text{n}\text{2}$ and when c = 4 this estimate is $\text{5n}\text{9}$. In these cases the upper bound is best possible, in the sense that for every possible size there exist matrices with this maximal 1-width. The technique of proof is also used to improve the known bound for the 1-width of (0, 1)-matrices with constant line sum 4.     

On the complexity of growth of the number of distinct fuzzy switching functions

Several attempts have been made to enumerate fuzzy switching (FSF's) and to develop upper and lower bounds for the number of FSF's of n variables in an effort to better understand the properties and the complexity of FSF's. Previous upper bounds are 2⁴ⁿ [9] and 2^{2–3n—2n—1} [7].It has also been shown that the exact numbers of FSF's of n variables for n = 0, 1, 2, 3, and 4 are 2, 6, 8, 84, 43 918 and 160 297 985 276 respectively.This paper will give a brief overview of previous approaches to the problem, study some of the properties of fuzzy switching functions and give improved upper and lower bounds for a general n.     

A compactification problem of J. De Groot

Recently, De Groot's conjecture that cmp X = def X holds for every separable and metrizable space X has been negatively resolved by Pol. In previous efforts to resolve De Groot's conjecture various functions like cmp have been introduced. A new inequality between two of these functions is established. Many examples which have been constructed so far in relation with the conjecture are obtained by attaching a locally compact space to a compact space. An upper bound for the compactness deficiency def of the resulting space is given.     

An almost-prime sieve

We compare and contrast three methods for estimating the number of integers in an interval of length x which have fewer than k distinct prime factors less than z, with special attention to the case k = 2. An iterative method based on the case k = 1 is simplest. If z is sufficiently small compared to x one may use a kind of Brun sieve. Selberg's sieve method gives a good estimate for k = 2 but leads into technical difficulties as k increases.     

Inequalities induced by network connections II. Hybrid connections

It has been found that interesting mathematical relationships arise from a vectorial generalization of Kirchhoff's and Ohm's laws, in which the “resistors” become Hermitian positive semidefinite (PSD) linear operators. In analogy to the parallel connection of resistors Anderson and Duffin studied the parallel sum R : S of two PSD operators on a finite dimensional space, defined by $\text{R : S = R(R + S)}{}^2\text{S}$. Duffin and Trapp then studied the hybrid connection. This paper generalizes some of their results to a much broader class of electrical connections.     

Remarques sur le spectre de l'opérateur de Dirac

We describe a new family of examples of hypersurfaces in the sphere satisfying the limiting-case in C. Bär's extrinsic upper bound for the smallest eigenvalue of the Dirac operator. To cite this article: N. Ginoux, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Hermitian forms and the large and small sieves

The author observes that two Hermitian forms have the same largest eigenvalue. A large sieve result of Roth-Bombieri type and Selberg's upper bound sieve with a Montgomery type error term are derived.     

Finite common coverings of pairs of regular graphs

The graph G is a covering of the graph H if there exists a (projection) map p from the vertex set of G to the vertex set of H which induces a one-to-one correspondence between the vertices adjacent to v in G and the vertices adjacent to p(v) in H, for every vertex v of G. We show that for any two finite regular graphs G and H of the same degree, there exists a finite graph K that is simultaneously a covering both of G and H. The proof uses only Hall's theorem on 1-factors in regular bipartite graphs.     

Exponential asymptotics and adiabatic invariance of a simple oscillator

An alternative proof is provided for Littlewood's asymptotic expression arising from Lorentz's problem (1911) on the adiabatic invariance of a simple pendulum. Our approach is based on a standard WKB approximation. Our proof is simpler than those of both Littlewood (1963) and Wasow (1973). If the coefficient function in their differential equation is analytic, then Littlewood's asymptotic expression can even be replaced by an exponentially small term. To cite this article: C.H. Ou, R. Wong, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

k-dimensional skeletoids in Rn and the Hilbert cube

In their paper on pseudo-boundaries and pseudo-interiors R. Geoghegan and R.R. Summerhill construct k-dimensional pseudo-boundaries in $\text{R}$ⁿ, where they used West's notion of a pseudo-boundary, rather than Toruńczyk's. In this paper we construct pseudo-boundaries in the sense of Toruńczyk (skeletoids) in $\text{R}$ⁿ and use this result to find k-dimensional skeletoids in the Hilbert cube.     

Sur les théorèmes de Serre,Bass et Forster–Swan

This Note introduces a new approach to Serre's Splitting Off Theorem, Bass's Stable Range and Cancellation Theorems, and Forster–Swan's Theorem. A new dimension for commutative rings and some multilinear alternating maps give a means of getting unimodular vectors, without noetheriannity hypothesis. To cite this article: L. Ducos, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Jensen's inequality for g-expectation: part 1

Briand et al. (Electron. Comm. Probab. 5 (2000) 101–117) gave a counterexample and proposition to show that given g,g-expectations usually do not satisfy Jensen's inequality for most of convex functions. This yields a natural question, under which conditions on g, do g-expectations satisfy Jensen's inequality for convex functions? In this paper, we shall deal with this question in the case that g is convex and give a necessary and sufficient condition on g under which Jensen's inequality holds for convex functions. To cite this article: Z. Chen et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Upper eigenvalue bounds for pencils of matrices

Eigenvalue bounds are obtained for pencils of matrices A ? vB where A is a Stieltjes matrix and B is positive definite, under assumptions suitable for the estimation of asymptotic convergence rates of factorization iterative methods, where B represents the approximate factorization of A. The upper bounds obtained depend on the “connectivity” structure of the matrices involved, which enters through matrix graph considerations; in addition, a more classical argument is used to obtain a lower bound. Potential applications of these results include a partial confirmation of Gustafsson's conjecture concerning the nonnecessity of Axelsson's perturbations.     

Hall-conditions and the three-family problem

It is shown that Philip Hall's SDR theorem cannot be extended in a certain (natural) direction when more than two families are involved, except (possibly) when they form partitions.     

Über Primzahlen der Formn2 n +1 bzw.p2 p +1

Using Selberg's sieve upper bounds for the number of primes of the formn2 ⁿ +1 withn≤x and the number of primes of the formp2 ^p +1 withp≤x are derived.     

Chapter 1. Extension of the fundamental theorem of finite semigroups

This paper proves that some useful commutivity relations exist among semigroup wreath product factors that are either groups or combinatorial “units” U₁, U₂, or U₃. Using these results it then obtains some characterizations of each of the classes of semigroups buildable from U₁'s, U₂'s, and groups (“buildable” meaning “dividing a wreath product of”).We show that up to division U₁'s can be moved to the right and U₂'s, and groups to the left over other units and groups, if it is allowed that the factors involved be replaced by their direct products, or in the case of U₂, even by a wreath product. From this it is deduced that U₁'s and U₂'s do not affect group complexity, that any semigroup buildable from U₁'s, U₂'s, and groups has group complexity 0 or 1, and that all such semigroups can be represented, up to division, in a canonical form—namely, as a wreath product with all U₁'s on the right, all U₂'s on the left, and a group in the middle. This last fact is handy for developing charactérizations.An embedding theorem for semigroups with a unique 0-minimal ideal is introduced, and from this and the commutivity results and some constructions proved for RLM semigroups, there is obtained an algebraic characterization for each class of semigroups that is a wreath product-division closure of some combination of U₁'s, U₂'s, and the groups. In addition it is shown, for i = 1,2,3, that if the unit U_i does not divide a semigroup S, then S can be built using only groups and units not containing U_i. Thus, it can be deduced that any semigroup which does not contain U₃ must have group complexity either 0 or 1. This then establishes that indeed U₃ is the determinant of group complexity, since it is already proved that both U₁ and U₂ are transparent with regard to the group complexity function, and it is known that with U₃ (and groups) one can build semigroups with complexities arbitrarily large. Another conclusion is a combinatorial counterpart for the Krohn-Rhodes prime decomposition theorem, saying that any semigroups can be built from the set of units which divide it together with the set of those semigroups not having unit divisors. Further, one can now characterize those semigroups which commute over groups, showing a semigroup commutes to the left over groups iff it is “R₁” (i.e., does not contain U₁, i.e., is buildable form U₂'s and groups), and commutes to the right over groups iff it does not contain U₂ (i.e., is buildable from groups and U₁'s). Finally, from the characterizations and their proofs one sees some ways in which groups can do the work of combinatorials in building combinatorial semigroups.     

Santaló's inequality on Cn by complex interpolationInégalité de Santaló sur Cn par interpolation complexe

A new approach to Santaló's inequality on $\text{C}{}^{\mathrm{n}}$ is obtained by combining complex interpolation and Berndtsson's generalization of Prékopa's inequality. To cite this article: D. Cordero-Erausquin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 767–772.