Some properties concerning Milman's moduli

In this paper, some properties concerning Milman's moduli J(t,X) and S(t,X), including equivalent representations, upper and lower bounds as well as relationship between some geometric properties and these two moduli, are studied. In addition, some sufficient conditions for the fixed point property are presented.     

On a generalized James constant

We introduce a generalized James constant J(a,X) for a Banach space X, and prove that, if J(a,X)<(3+a)/2 for some a∈[0,1], then X has uniform normal structure. The class of spaces X with J(1,X)<2 is proved to contain all u-spaces and their generalizations. For the James constant J(X) itself, we show that X has uniform normal structure provided that , improving the previous known upper bound at 3/2. Finally, we establish the stability of uniform normal structure of Banach spaces.     

On a new geometric constant related to the von Neumann-Jordan constant

The von Neumann-Jordan constant CNJ(X) is computed for X being ?₂−?₁ and ?_∞−?₁ space by introducing a new geometric constant γX(t). These partly give an answer to an open question posed by Kato et al. Some basic properties of this new coefficient are investigated. Moreover, we obtain a new class of Banach spaces with uniform normal structure.     

On James and von Neumann-Jordan constants and sufficient conditions for the fixed point property

In this paper, we prove that a Banach space X and its dual space X^∗ have uniform normal structure if . The García-Falset coefficient R(X) is estimated by the CNJ(X)-constant and the weak orthogonality coefficient introduced by B. Sims. Finally, we present an affirmative answer to a conjecture by L. Maligranda concerning the relation between the James and CNJ(X)-constants for a Banach space.     

A note on Jordan-von Neumann constant and James constant

Let X be a non-trivial Banach space. L. Maligranda conjectured CNJ(X)?1+J²(X)/4 for James constant J(X) and von Neumann-Jordan constant CNJ(X) of X. Satit Saejung gave a proof of it in 2006. In this note, we show that the last step in Satit Saejung's proof is not valid. Using his proof, the result should be . On the other hand, we give a new proof of CNJ(X)?1+J²(X)/4. As an application, we give a relation between J(X) and J(lp(X)).     

Best-possible bounds for the distribution of a sum — a problem of Kolmogorov

Summary Recently, in answer to a question of Kolmogorov, G.D. Makarov obtained best-possible bounds for the distribution function of the sumX+Y of two random variables,X andY, whose individual distribution functions,F _X andF _Y, are fixed. We show that these bounds follow directly from an inequality which has been known for some time. The techniques we employ, which are based on copulas and their properties, yield an insightful proof of the fact that these bounds are best-possible, settle the question of equality, and are computationally manageable. Furthermore, they extend to binary operations other than addition and to higher dimensions.     

Answers to two questions posed by Farhi concerning additive bases

Let A be an asymptotic basis for N and X a finite subset of A such that A?X is still an asymptotic basis. Farhi recently proved a new batch of upper bounds for the order of A?X in terms of the order of A and a variety of parameters related to the set X. He posed two questions concerning possible improvements to his bounds. In this note, we answer both questions.     

Error analysis of large-time approximations for decay chains

Consider a radioactive decay chain X₁ → ? → X_n→ and let N_n(t) be the amount of X_n at time t. This paper establishes error bounds for large-time approximations to N_n(t) that include and generalize the transient equilibrium approximations and other known approximations. The error bounds allow one to find the range of t for which these approximations can be used with a given degree of precision.     

An Extension of a Simply Inequality Between von Neumann–Jordan and James Constants in Banach Spaces

For a non-trivial Banach space X, let J(X), CNJ(X), C_(NJ)~(p)(X) respectively stand for the James constant, the von Neumann–Jordan constant and the generalized von Neumann–Jordan constant recently inroduced by Cui et al. In this paper, we discuss the relation between the James and the generalized von Neumann–Jordan constants, and establish an inequality between them: C_(NJ)~(p)(X) ≤J(X) with p ≥ 2, which covers the well-known inequality CNJ(X) ≤ J(X). We also introduce a new constant, from which we establish another inequality that extends a result of Alonso et al.     

Upper bounds for the order of an additive basis obtained by removing a finite subset of a given basis

Let A be an additive basis of order h and X be a finite nonempty subset of A such that the set A?X is still a basis. In this article, we give several upper bounds for the order of A?X in function of the order h of A and some parameters related to X and A. If the parameter in question is the cardinality of X, Nathanson and Nash already obtained some of such upper bounds, which can be seen as polynomials in h with degree (|X|+1). Here, by taking instead of the cardinality of X the parameter defined by , we show that the order of A?X is bounded above by . As a consequence, we deduce that if X is an arithmetic progression of length ?3, then the upper bounds of Nathanson and Nash are considerably improved. Further, by considering more complex parameters related to both X and A, we get upper bounds which are polynomials in h with degree only 2.     

Large-scale linear programming: Geometry,working bases and factorizations

This paper is concerned with linear programming problems in which many of the constraints are handled implicitly by requiring that the vector of decision variables lie in a polyhedronX. It is shown that the simplex method can be implemented using a working basis whose size is the number of explicit constraints as long as the local structure ofX around the current point is known. Various ways of describing this local structure lead to known implementations whenX is defined by generalized or variable upper bounds or flow conservation constraints. In the general case a decomposition principle can be used to generate this local structure. We also show how to update factorizations of the working basis.     

On Fox?s m-dimensional category and theorems of Bochner type

We show that catm(X)=cat(jm), where catm(X) is Fox?s m-dimensional category, jm:X→X[m] is the mth Postnikov section of X and cat(X) is the Lusternik-Schnirelmann category of X. This characterization is used to give new “Bochner-type” bounds on the rank of the Gottlieb group and the first Betti number for manifolds of non-negative Ricci curvature. Finally, we apply these methods to obtain Bochner-type theorems for manifolds of almost non-negative sectional curvature.     

On upper bounds for the variance of functions of random variables

The upper bounds for the variance of a function g of a random variable X obtained in Cacoullos (1982) (for short CP) are improved in the case μ = E(X) ≠ 0. A main feature of these bounds is that they involve the second moment of the derivative or the difference of g. A multivariate extension for functions of independent random variables is also given.     

The minimal degree of plane models of double covers of smooth curves

If X is a smooth curve such that the minimal degree of its plane models is not too small compared with its genus, then X has been known to be a double cover of another smooth curve Y under some mild condition on the genera. However there are no results yet for the minimal degrees of plane models of double covers except some special cases. In this paper, we give upper and lower bounds for the minimal degree of plane models of the double cover X in terms of the gonality of the base curve Y and the genera of X and Y. In particular, the upper bound equals to the lower bound in case Y is hyperelliptic. We give an example of a double cover which has plane models of degree equal to the lower bound.     

Graph with given achromatic number

An irreducible (point-determining) graph is one in which distinct vertices have distinct neighbourhoods. Every graph X can be reduced to an irreducible graph X¹ by identifying all vertices with the same neighbourhood; the colourability properties of X¹ carry over to X. Hence irreducible graphs are instrumental in the study of achromatic number. We prove that there are only finitely many irreducible graphs with a given achromatic number, and describe all graphs with achromatic number less than four. We deduce certain bounds on the achromatic number X in terms of the number of vertices of X¹. In the course of the proofs we calculate the achromatic numbers of paths and cycles. Generalizations of the main theorem to homomorphisms other than colourings are discussed.     

On the normal approximation of a binomial random sum

We present upper bounds of the L _s norms of the normal approximation for random sums of independent identically distributed random variables X ₁ , X ₂ , . . . with finite absolute moments of order 2 + δ, 0 < δ ≤ 1, where the number of summands N is a binomial random variable independent of the summands X ₁ , X ₂ , . . . . The upper bounds obtained are of order (E N) ^?δ/2 for all 1 ≤ s ≤ ∞.     

Optimal designs for weighted approximation and integration of stochastic processes on [0,∞)

We study minimal errors and optimal designs for weighted L₂-approximation and weighted integration of Gaussian stochastic processes X defined on the half-line [0,∞). Under some regularity conditions, we obtain sharp bounds for the minimal errors for approximation and upper bounds for the minimal errors for integration. The upper bounds are proven constructively for approximation and non-constructively for integration. For integration of the r-fold integrated Brownian motion, the upper bound is proven constructively and we have a matching lower bound.     

Noetherian types of homogeneous compacta and dyadic compacta

The Noetherian type of a space is the least κ such that it has a base that is κ-like with respect to reverse inclusion. Just as all known homogeneous compacta have cellularity at most c, they satisfy similar upper bounds in terms of Noetherian type and related cardinal functions. We prove these and many other results about these cardinal functions. For example, every homogeneous dyadic compactum has Noetherian type ω. Assuming GCH, every point in a homogeneous compactum X has a local base that is c(X)-like with respect to containment. If every point in a compactum has a well-quasiordered local base, then some point has a countable local π-base.     

The Topology of Limits of Direct Systems Induced by Maps

Let Z, H be spaces. In previous work, we introduced the direct system X induced by the set of maps between the spaces Z and H. Now we will consider the case that X is induced by possibly a proper subset of the maps of Z to H. Our objective is to explore conditions under which X = dirlim X will be T₁, Hausdorff, regular, completely regular, pseudo-compact, normal, an absolute co-extensor for some space K, or will enjoy some combination of these properties.