Mathematical Monographs Research Articles

Duality, Vector Spaces and Absolutely Convex Modules

It is well-known (see Semadeni, Queen Pap. Pure Appl. Math., 33:1–98, 1973 and Pumplun and Rohrl, Commun. Algebra, 12(8):953–1019, 1984, 1985) that the embedding of vector spaces into the category of absolutely convex modules is reflective. As we will show, under a separatedness condition on these modules it is at the same time coreflective. This is a peculiar situation, see Kannan, Math. Ann., 195:168–174, (1972) and Hu\(\textrm {\u{s}}\)ek, Reflexive and coreflexive subcategories of unif and top, Seminar Uniform Spaces, Prague, 113–126, (1973), but we do find it also in the embedding \({\mathbf{Top}} \hookrightarrow {\mathbf{Ap}}\) (Lowen, Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad. Oxford Mathematical Monographs, Oxford University Press, London, UK, 1997) and, by extension, in the embedding \({\mathbf{lcTopVec}} \hookrightarrow {\mathbf{lcApVec}}\) (see Lowen and Verwulgen, Houst. J. Math, 30(4):1127–1142, 2004, and Sioen and Verwulgen, Appl. Gen. Topol., 4(2):263–279, 2003. We demonstrate that, in this setting, by duality arguments, absolutely convex modules are indeed the numerical counterpart of vector spaces. All these, at first sight unrelated facts, are comprised in the commutative scheme below with natural dualisation functors and their left adjoints.

Smoothness of Itô maps and diffusion processes on path spaces (I)

Let p ∈ [ 1 , 2 ) and α, ε > 0 be such that α ∈ ( p − 1 , 1 − ε ) . Let V, W be two Euclidean spaces. Let Ω p ( V ) be the space of continuous paths taking values in V and with finite p-variation. Let k ∈ N and f : W → Hom ( V , W ) be a Lip ( k + α + ε ) map in the sense of E.M. Stein [Stein E.M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, NJ, 1970]. In this paper we prove that the Itô map, defined by I ( x ) = y , is a local C k , ε 1 + ε map (in the sense of Fréchet) between Ω p ( V ) and Ω p ( W ) , where y is the solution to the differential equation d y t = f ( y t ) d x t , y 0 = a . This result strengthens the continuity results and Lipschitz continuity results in [Lyons T., Differential equations driven by rough signals. I. An extension of an inequality of L.C. Young, Math. Res. Lett. 1 (4) (1994) 451–464; Lyons T., Qian Z., System Control and Rough Paths, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2002] particularly to the non-integer case. It allows us to construct the fractional like Brownian motion and infinite dimensional Brownian motions on the space of paths with finite p-variation. As a corollary in the particular case where p = 1 , we obtain that the development from the space of finite 1-variation paths on R d to the space of finite 1-variation paths on a d-dimensional compact Riemannian manifold is a smooth bijection.

Distributed Digital Library of Mathematical Monographs

Distributed Digital Library of Mathematical Monographs

The place of exceptional covers among all diophantine relations

Let F q be the order q finite field. An F q cover φ : X → Y of absolutely irreducible normal varieties has a nonsingular locus. Then, φ is exceptional if it maps one–one on F q t points for ∞ -ly many t over this locus. Lenstra suggested a curve Y may have an Exceptional ( cover) Tower over F q Lenstra Jr. [Talk at Glasgow Conference, Finite Fields III, 1995]. We construct it, and its canonical limit group and permutation representation, in general. We know all one-variable tamely ramified rational function exceptional covers, and much on wildly ramified one variable polynomial exceptional covers, from Fried et al. [Schur covers and Carlitz's conjecture, Israel J. Math. 82 (1993) 157–225], Guralnick et al. [The rational function analogue of a question of Schur and exceptionality of permutations representations, Mem. Amer. Math. Soc. 162 (2003) 773, ISBN 0065-9266] and Lidl et al. [Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 65, Longman Scientific, New York, 1993]. We use exceptional towers to form subtowers from any exceptional cover collections. This gives us a language for separating known results from unsolved problems. We generalize exceptionality to p(ossibly)r(educible)-exceptional covers by dropping irreducibility of X. Davenport pairs (DPs) are significantly different covers of Y with the same ranges (where maps are nonsingular) on F q t points for ∞ -ly many t. If the range values have the same multiplicities, we have an iDP. We show how a pr-exceptional correspondence on F q covers characterizes a DP. You recognize exceptional covers and iDPs from their extension of constants series. Our topics include some of their dramatic effects • How they produce universal relations between Poincaré series. • How they relate to the Guralnick–Thompson genus 0 problem and to Serre's open image theorem. Historical sections capture Davenport's late 1960s desire to deepen ties between exceptional covers, their related cryptology, and the Weil conjectures.

The geometry of prior selection

This contribution is devoted to the selection of prior in a Bayesian learning framework. There is an extensive literature on the construction of non-informative priors and the subject seems far from a definite solution [Kass and Wasserman, Formal rules for selecting prior distributions: a review and annotated bibliography, Technical Report No. 583, Department of Statistics, Carnegie Mellon University, 1994]. We consider this problem in the light of the recent development of information geometric tools [Amari and Nagaoka, Methods of information geometry, in: Translations of Mathematical Monographs, AMS, vol. 191, Oxford University Press, Oxford, 2000]. The differential geometric analysis allows the formulation of the prior selection problem in a general manifold valued set of probability distributions. In order to construct the prior distribution, we propose a criteria expressing the trade off between decision error and uniformity constraint. The solution has an explicit expression obtained by variational calculus. In addition, it has two important invariance properties: invariance to the dominant measure of the data space and also invariance to the parametrization of a restricted parametric manifold. We show how the construction of a prior by projection is the best way to take into account the restriction to a particular family of parametric models. For instance, we apply this procedure to autoparallel restricted families. Two practical examples illustrate the proposed construction of prior. The first example deals with the learning of a mixture of multivariate Gaussians in a classification perspective. We show in this learning problem how the penalization of likelihood by the proposed prior eliminates the degeneracy occurring when approaching singularity points. The second example treats the blind source separation problem.

A hybrid algorithm for computing permanents of sparse matrices

The permanent of matrices has wide applications in many fields of science and engineering. It is, however, a #P-complete problem in counting. The best-known algorithm for computing the permanent, which is due to Ryser [Combinatorial Mathematics, The Carus Mathematical Monographs, vol. 14, Mathematical Association of America, Washington, DC, 1963], runs O(n2^n^-^1) in time. It is possible to speed up algorithms for matrices with special structures, which arise commonly in applications. Most algorithms discussed before focus on 0,1 matrix. In this paper, a hybrid algorithm is proposed. It is efficient for sparse matrices.

Logarithmic Combinatorial Structures: A Probabilistic Approach, by Richard Arratia, A. D. Barbour and Simon Tavaré, European Mathematical Society Monographs in Mathematics, 2003, 375 pp., 69

Logarithmic Combinatorial Structures: A Probabilistic Approach, by Richard Arratia, A. D. Barbour and Simon Tavaré, European Mathematical Society Monographs in Mathematics, 2003, 375 pp., 69

On the complexity of Schmüdgen's Positivstellensatz

Schmüdgen's Positivstellensatz roughly states that a polynomial f positive on a compact basic closed semialgebraic subset S of R n can be written as a sum of polynomials which are non-negative on S for certain obvious reasons. However, in general, you have to allow the degree of the summands to exceed largely the degree of f. Phenomena of this type are one of the main problems in the recently popular approximation of non-convex polynomial optimization problems by semidefinite programs. Prestel (Springer Monographs in Mathematics, Springer, Berlin, 2001) proved that there exists a bound on the degree of the summands computable from the following three parameters: The exact description of S, the degree of f and a measure of how close f is to having a zero on S. Roughly speaking, we make explicit the dependence on the second and third parameter. In doing so, the third parameter enters the bound only polynomially.

Parametric extensions of Shannon inequality and its reverse one in Hilbert space operators

We shall state the following parametric extensions of Shannon inequality and its reverse one in Hilbert space operators. Let p∈[0,1] and also let { A 1, A 2,…, A n } and { B 1, B 2,…, B n } be two sequences of strictly positive operators on a Hilbert space H such that ∑ j=1 n A j ♯ p B j ⩽ I . Then ∑ j=1 nS p+1(A j|B j)⩾ ∑ j=1 n(A j♮ p+1B j)+ I−∑ j=1 nA j♯ pB j × log ∑ j=1 n(A j♮ p+1B j)+ I−∑ j=1 nA j♯ pB j ⩾ log ∑ j=1 n(A j♮ p+1B j)+ I−∑ j=1 nA j♯ pB j ⩾∑ j=1 nS p(A j|B j) ⩾− log ∑ j=1 n(A j♮ p−1B j)+ I−∑ j=1 nA j♯ pB j ⩾− ∑ j=1 n(A j♮ p−1B j)+ I−∑ j=1 nA j♯ pB j × log ∑ j=1 n(A j♮ p−1B j)+ I−∑ j=1 nA j♯ pB j ⩾∑ j=1 nS p−1(A j|B j), where S q(A|B)=A 1 2 A −1 2 BA −1 2 q logA −1 2 BA −1 2 A 1 2 for A>0 , B>0 and any real number q and A♮ qB=A 1 2 A −1 2 BA −1 2 qA 1 2 for A>0 , B>0 and any real number q . In particular, if ∑ j=1 n A j =∑ j=1 n B j = I , then ∑ j=1 nS 2(A j|B j)⩾ ∑ j=1 nB jA j −1B j log ∑ j=1 nB jA j −1B j ⩾ log ∑ j=1 nB jA j −1B j ⩾∑ j=1 nS 1(A j|B j)⩾0 ⩾∑ j=1 nS(A j|B j)⩾− log ∑ j=1 nA jB j −1A j ⩾− ∑ j=1 nA jB j −1A j log ∑ j=1 nA jB j −1A j ⩾∑ j=1 nS −1(A j|B j), where S(A|B)=S 0(A|B)=A 1 2 logA −1 2 BA −1 2 A 1 2 which is the relative operator entropy of A>0 and B>0 . Our results can be considered as parametric extensions of the following celebrated Shannon inequality [Ann. Math. Statistics 22 (1951) 79; Bull. Syst. Tech. J. 27 (1948) 379; Pitman Monographs and Surveys in Pure and Applied Mathematics 97, Addison Wesley Longman, 1998, p. 233] which is very useful and so famous in information theory. Let { a 1, a 2,…, a n } and { b 1, b 2,…, b n } be two probability vectors. Then 0⩾∑ j=1 n a j log b j −∑ j=1 n a j log a j (see inequalities (2.4) of Corollary 2.4).

X-rays characterizing some classes of discrete sets

In this paper, we study the problem of determining discrete sets by means of their X-rays. An X-ray of a discrete set F in a direction u counts the number of points in F on each line parallel to u. A class F of discrete sets is characterized by the set U of directions if each element in F is determined by its X-rays in the directions of U. By using the concept of switching component introduced by Chang and Ryser [Comm. ACM 14 (1971) 21; Combinatorial Mathematics, The Carus Mathematical Monographs, No. 14, The Mathematical Association of America, Rahway, 1963] and extended in [Discrete Comput. Geom. 5 (1990) 223], we prove that there are some classes of discrete sets that satisfy some connectivity and convexity conditions and that cannot be characterized by any set of directions. Gardner and Gritzmann [Trans. Amer. Math. Soc. 349 (1997) 2271] show that any set U of four directions having cross ratio that does not belong to {4/3,3/2,2,3,4}, characterizes the class of convex sets. We prove the converse, that is, if U's cross ratio is in {4/3,3/2,2,3,4}, then the hv-convex sets cannot be characterized by U. We show that if the horizontal and vertical directions do not belong to U, Gardner and Gritzmann's result cannot be extended to hv-convex polyominoes. If the horizontal and vertical directions belong to U and U's cross ratio is not in {4/3,3/2,2,3,4}, we believe that U characterizes the class of hv-convex polyominoes. We give experimental evidence to support our conjecture. Moreover, we prove that there is no number δ such that, if | U|⩾ δ, then U characterizes the hv-convex polyominoes. This number exists for convex sets and is equal to 7 (see [Trans. Amer. Math. Soc. 349 (1997) 2271]).

Narkiewicz, W.The development of prime number theory from Euclid to Hardy and Littlewood (Monographs in Mathematics, Springer, 2000), xii+448 pp., 3 540 66289 8 (hardback), £58.50

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STRONG SHAPE AND HOMOLOGY (Springer Monographs in Mathematics)

By Sibe Mardešić: 489 pp., £55.00, isbn 3-540-66198-0 (Springer, Berlin, 2000).

DIFFERENTIAL EQUATIONS WITH OPERATOR COEFFICIENTS, WITH APPLICATIONS TO BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS (Springer Monographs in Mathematics)

DIFFERENTIAL EQUATIONS WITH OPERATOR COEFFICIENTS, WITH APPLICATIONS TO BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS (Springer Monographs in Mathematics)

D. Mitrović and D. Žubrinić Fundamentals of applied functional analysis (Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 91, Addison Wesley Longman, Harlow, 1998), 399 pp., 0 582 24694 6, £72.

D. Mitrović and D. Žubrinić Fundamentals of applied functional analysis (Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 91, Addison Wesley Longman, Harlow, 1998), 399 pp., 0 582 24694 6, £72.

THERMODYNAMICS OF CHAOS AND ORDER (Pitman Monographs and Surveys in Pure and Applied Mathematics 90)

By Victor L. Berdichevsky: 267 pp., £48.00, isbn 0 582 08734 1 (Longman, 1997).

Classical Solutions of Multidimensional Hele--Shaw Models

Existence and uniqueness of classical solutions for the multidimensional expanding Hele{Shaw problem are proved.

WREATH PRODUCTS OF GROUPS AND SEMIGROUPS (Pitman Monographs and Surveys in Pure and Applied Mathematics 74)

By J. D. P. Meldrum: 324 pp., £60.00, isbn 0 582 02693 8 (Longman, 1995).

Richard Dedekind. What are numbers and what should they be? (Was sind und was sollen die Zahlen?) Revised English translation of 70½ 1 with added notes by H. Pogorzelski, W. Ryan, and W. Snyder. RIM monographs in mathematics. Research Institute for Mathematics, Orono, Maine, 1995, viii + 91 pp.

Richard Dedekind. What are numbers and what should they be? (Was sind und was sollen die Zahlen?) Revised English translation of 70½ 1 with added notes by H. Pogorzelski, W. Ryan, and W. Snyder. RIM monographs in mathematics. Research Institute for Mathematics, Orono, Maine, 1995, viii + 91 pp. - Volume 61 Issue 2

EVOLUTIONARY INTEGRAL EQUATIONS AND APPLICATIONS (Monographs in Mathematics 87)

By Jan Prüss: 392 pp., £74.00, isbn 3 7643 2876 2 (Birkhäuser, 1993).

DIFFUSION EQUATIONS (Translations of Mathematical Monographs 111)

DIFFUSION EQUATIONS (Translations of Mathematical Monographs 111)