On (strong) α-favorability of the Vietoris hyperspace

For a normal space X, α (i.e. the nonempty player) having a winning strategy (resp. winning tactic) in the strong Choquet game Ch(X) played on X is equivalent to α having a winning strategy (resp. winning tactic) in the strong Choquet game played on the hyperspace CL(X) of nonempty closed subsets endowed with the Vietoris topology τ _V . It is shown that for a non-normal X where α has a winning strategy (resp. winning tactic) in Ch(X), α may or may not have a winning strategy (resp. winning tactic) in the strong Choquet game played on the Vietoris hyperspace. If X is quasi-regular, then having a winning strategy (resp. winning tactic) for α in the Banach-Mazur game BM(X) played on X is sufficient for α having a winning strategy (resp. winning tactic) in BM(CL(X), τ _V ), but not necessary, not even for a separable metric X. In the absence of quasi-regularity of a space X where α has a winning strategy in BM(X), α may or may not have a winning strategy in the Banach-Mazur game played on the Vietoris hyperspace.     

Complete convergence in mean for double arrays of random variables with values in Banach spaces

The rate of moment convergence of sample sums was investigated by Chow (1988) (in case of real-valued random variables). In 2006, Rosalsky et al. introduced and investigated this concept for case random variable with Banach-valued (called complete convergence in mean of order p). In this paper, we give some new results of complete convergence in mean of order p and its applications to strong laws of large numbers for double arrays of random variables taking values in Banach spaces.     

Depth in bucket recursive trees with variable capacities of buckets

We consider bucket recursive trees of sizen consisting of all buckets with variable capacities1,2,...,b and with a specifc stochastic growth rule.This model can be considered as a generalization of random recursive trees like bucket recursive trees introduced by Mahmoud and Smythe where all buckets have the same capacities.In this work,we provide a combinatorial analysis of these trees where the generating function of the total weights satisfes an autonomous frst order diferential equation.We study the depth of the largest label（i.e.,the number of edges from the root node to the node containing label n）and give a closed formula for the probability distribution.Also we prove a limit law for this quantity which is a direct application of quasi power theorem and compute its mean and variance.Our results for b=1 reduce to the previous results for random recursive trees.     

Estimation of the trend function and auto-covariance for spatial models

We first establish, through a Berry–Esseen-type bound, the asymptotic normality of a local linear estimate of the regression function in a fixed design setting when the errors are stationary isotropic spatial random fields. On the other hand, we investigate the weak convergence of an empirical estimate of the variance of these errors in a general α-mixing setting.     

An efficient algorithm for mining high utility itemsets with negative item values in large databases

Utility itemsets typically consist of items with different values such as utilities, and the aim of utility mining is to identify the itemsets with highest utilities. In the past studies on utility mining, the values of utility itemsets were considered as positive. In some applications, however, an itemset may be associated with negative item values. Hence, discovery of high utility itemsets with negative item values is important for mining interesting patterns like association rules. In this paper, we propose a novel method, namely HUINIV (High Utility Itemsets with Negative Item Values)-Mine, for efficiently and effectively mining high utility itemsets from large databases with consideration of negative item values. To the best of our knowledge, this is the first work that considers the concept of negative item values in utility mining. The novel contribution of HUINIV-Mine is that it can effectively identify high utility itemsets by generating fewer high transaction-weighted utilization itemsets such that the execution time can be reduced substantially in mining the high utility itemsets. In this way, the process of discovering all high utility itemsets with consideration of negative item values can be accomplished effectively with less requirements on memory space and CPU I/O. This meets the critical requirements of temporal and spatial efficiency for mining high utility itemsets with negative item values. Through experimental evaluation, it is shown that HUINIV-Mine outperforms other methods substantially by generating much less candidate itemsets under different experimental conditions.     

The traces of Bessel potentials on regular subsets of Carnot groups

We prove the direct theorem on the traces of the Bessel potentials L _p ^α defined on a Carnot group, on the regular closed subsets called Ahlfors d-sets. The result is convertible for integer α, i.e., for the Sobolev spaces W _p ^α (the converse trace theorem was proven in [24]). This theorem generalizes A. Johnsson and H. Wallin’s results [13] for Sobolev functions and Bessel potentials on the Euclidean space.     

Process convergence of self-normalized sums of i.i.d. random variables coming from domain of attraction of stable distributions

In this paper we show that the continuous version of the self-normalized process Y _n,p (t)?=?S _n (t)/V _n,p ?+?(nt???[nt])X _[nt]?+?1/V _n,p ,0?<?t?≤?1; p?>?0 where $S_n(t)=\sum_{i=1}^{[nt]} X_i$ and $V_{(n,p)}=(\sum_{i=1}^{n}|X_i|^p)^{1/p}$ and X _i i.i.d. random variables belong to DA(α), has a non-trivial distribution iff p?=?α?=?2. The case for 2?>?p?>?α and p?≤?α?<?2 is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Csörg? et al. who showed Donsker’s theorem for Y _n,2(·), i.e., for p?=?2, holds iff α?=?2 and identified the limiting process as a standard Brownian motion in sup norm.     

Mathematical programming approaches for generating p-efficient points

Probabilistically constrained problems, in which the random variables are finitely distributed, are non-convex in general and hard to solve. The p-efficiency concept has been widely used to develop efficient methods to solve such problems. Those methods require the generation of p-efficient points (pLEPs) and use an enumeration scheme to identify pLEPs. In this paper, we consider a random vector characterized by a finite set of scenarios and generate pLEPs by solving a mixed-integer programming (MIP) problem. We solve this computationally challenging MIP problem with a new mathematical programming framework. It involves solving a series of increasingly tighter outer approximations and employs, as algorithmic techniques, a bundle preprocessing method, strengthening valid inequalities, and a fixing strategy. The method is exact (resp., heuristic) and ensures the generation of pLEPs (resp., quasi pLEPs) if the fixing strategy is not (resp., is) employed, and it can be used to generate multiple pLEPs. To the best of our knowledge, generating a set of pLEPs using an optimization-based approach and developing effective methods for the application of the p-efficiency concept to the random variables described by a finite set of scenarios are novel. We present extensive numerical results that highlight the computational efficiency and effectiveness of the overall framework and of each of the specific algorithmic techniques.     

Characterizations and closure under convolution of two classes of multivariate life distributions

Let ?(η) be the class of positive random vectors $\text{T}$ for which min_1?i?nα_iT_i is IFRA (NBU) for all α_i > 0, i = 1,…,n where n is an arbitrary positive integer. Characterizations of the classes ? and η are obtained and utilized to show that η is closed under convolution and that ? is closed under convolution provided one of the two convoluted vectors has independent components.     

Estimation améliorée explicite d'un degré de confiance conditionnel

We consider the problem of estimating the confidence statement of the usual confidence set, with confidence coefficient 1?α, of the mean of a p-variate normal distribution with identity covariance matrix. For p?5, we give an explicit sufficient condition for domination over the standard estimator 1?α by an estimator correcting it, that is, by 1?α+s where s is a suitable function. That condition mainly relies on a partial differential inequality of the form kΔs+s²?0 (for a certain constant k>0). It allows us to formally establish (with no recourse to simulations) this domination result. To cite this article: D. Fourdrinier, P. Lepelletier, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A framework for incremental generation of closed itemsets

Association rule mining from a transaction database (TDB) requires the detection of frequently occurring patterns, called frequent itemsets (FIs), whereby the number of FIs may be potentially huge. Recent approaches for FI mining use the closed itemset paradigm to limit the mining effort to a subset of the entire FI family, the frequent closed itemsets (FCIs). We show here how FCIs can be mined incrementally yet efficiently whenever a new transaction is added to a database whose mining results are available. Our approach for mining FIs in dynamic databases relies on recent results about lattice incremental restructuring and lattice construction. The fundamentals of the incremental FCI mining task are discussed and its reduction to the problem of lattice update, via the CI family, is made explicit. The related structural results underlie two algorithms for updating the set of FCIs of a given TDB upon the insertion of a new transaction. A straightforward method searches for necessary completions throughout the entire CI family, whereas a second method exploits lattice properties to limit the search to CIs which share at least one item with the new transaction. Efficient implementations of the parsimonious method is discussed in the paper together with a set of results from a preliminary study of the method's practical performances.     

Equilibrium in an ambiguity-averse mean–variance investors market

In a financial market composed of n risky assets and a riskless asset, where short sales are allowed and mean–variance investors can be ambiguity averse, i.e., diffident about mean return estimates where confidence is represented using ellipsoidal uncertainty sets, we derive a closed form portfolio rule based on a worst case max–min criterion. Then, in a market where all investors are ambiguity-averse mean–variance investors with access to given mean return and variance–covariance estimates, we investigate conditions regarding the existence of an equilibrium price system and give an explicit formula for the equilibrium prices. In addition to the usual equilibrium properties that continue to hold in our case, we show that the diffidence of investors in a homogeneously diffident (with bounded diffidence) mean–variance investors’ market has a deflationary effect on equilibrium prices with respect to a pure mean–variance investors’ market in equilibrium. Deflationary pressure on prices may also occur if one of the investors (in an ambiguity-neutral market) with no initial short position decides to adopt an ambiguity-averse attitude. We also establish a CAPM-like property that reduces to the classical CAPM in case all investors are ambiguity-neutral.     

The distribution-free newsboy problem under the worst-case and best-case scenarios

New theoretical foundations for analyzing the newsboy problem under incomplete information about the probability distribution of random demand are presented. Firstly, we reveal that the distribution-free newsboy problem under the worst-case and best-case demand scenarios actually reduces to the standard newsboy problem with demand distributions that bound the allowable distributions in the sense of increasing concave order. Secondly, we provide a theoretical tool for seeking the best-case and worst-case order quantities when merely the support and the first k moments of the demand are known. Using this tool we derive closed form formulas for such quantities in the case of known support, mean and variance, i.e. k = 2. Consequently, we generalize all results presented so far in literature for the worst-case and best-case scenarios, and present some new ones. Extensions of our findings to the cases of the known mode of a unimodal demand distribution, the known median, and to other stochastic inventory problems are indicated.     

Transformations of discrete closure systems

Discrete systems such as sets, monoids, groups are familiar categories. The internal structure of the latter two is defined by an algebraic operator. In this paper we concentrate on discrete systems that are characterized by unary operators; these include choice operators σ, encountered in economics and social theory, and closure operators φ, encountered in discrete geometry and data mining. Because, for many arbitrary operators α, it is easy to induce a closure structure on the base set, closure operators play a central role in discrete systems. Our primary interest is in functions f that map power sets 2 ^U into power sets 2 ^U′, which are called transformations. Functions over continuous domains are usually characterized in terms of open sets. When the domains are discrete, closed sets seem more appropriate. In particular, we consider monotone transformations which are “continuous”, or “closed”. These can be used to establish criteria for asserting that “the closure of a transformed image under f is equal to the transformed image of the closure”. Finally, we show that the categories MCont and MClo of closure systems with morphisms given by the monotone continuous transformations and monotone closed transformations respectively have concrete direct products. And the supercategory Clo of MClo whose morphisms are just the closed transformations is shown to be cartesian closed.     

Convergence of a continuation method under the Hölder condition of the first derivative

The aim of this paper is to describe a continuation method combining the Chebyshev’s method and the convex acceleration of Newton’s method to solve nonlinear equations in Banach spaces. The semilocal convergence analysis of the method is established using recurrence relations under the assumption that the first Fréchet derivative satisfies the Hölder continuity condition. This condition is milder than the usual Lipschitz condition. The computation of second Fréchet derivative is also avoided. Two real valued functions and a real sequence are used to establish a convergence criterion of R-order (1+p), where p∈(0,1] is the order of the Hölder condition. An existence and uniqueness theorem along with the closed form of error bounds is derived in terms of a real parameter α∈[0,1]. Two numerical examples are worked out to demonstrate the efficacy of our convergence analysis. For both the examples, the convergence conditions hold for the Chebyshev’s method (α=0). However, for the convex acceleration of Newton’s method (α=1), these convergence conditions hold for the first example but fail for the second example. For particular values of α, our method reduces to the Chebyshev’s method (α=0) and the convex acceleration of Newton’s method (α=1).     

A Generalization of Strassen’s Functional LIL

Let X ₁,X ₂,… be a sequence of i.i.d. mean zero random variables and let S _n denote the sum of the first n random variables. We show that whenever we have with probability one, lim?sup? _n→∞|S _n |/c _n =α ₀<∞ for a regular normalizing sequence {c _n }, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.     

The distribution of the size of the ancestor‐tree and of the induced spanning subtree for random trees

We consider two tree statistics that extend in a natural way the parameters depth of a node resp. distance between two nodes. The ancestor‐tree of p given nodes in a rooted tree T is the subtree of T, spanned by the root and these p nodes and generalizes the depth (ancestor‐tree of a single node), whereas the spanning subtree induced by p given nodes in a tree T generalizes the distance (induced spanning subtree of two nodes). We study the random variables size of the ancestor‐tree resp. spanning subtree size for two tree families, the simply generated trees and the recursive trees. We will assume here the random tree model and also that all () possibilities of selecting p nodes in a tree of size n are equally likely. For random simply generated trees we can then characterize for a fixed number p of chosen nodes the limiting distribution of both parameters as generalized Gamma distributions, where we prove the convergence of the moments too. For some specific simply generated tree families we can give exact formulæ for the first moments. In the instance of random recursive trees, we will show that the considered parameters are asymptotically normally distributed, where we can give also exact formulæ for the expectation and the variance. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Generalized congruence properties of the restricted partition function p(n,m)

Ramanujan-type congruences for the unrestricted partition function p(n) are well known and have been studied in great detail. The existence of Ramanujan-type congruences are virtually unknown for p(n,m), the closely related restricted partition function that enumerates the number of partitions of n into exactly m parts. Let ? be any odd prime. In this paper we establish explicit Ramanujan-type congruences for p(n,?) modulo any power of that prime ? ^α . In addition, we establish general congruence relations for p(n,?) modulo ? ^α for any n.     

Enumeration of power sums modulo a prime

We consider, for odd primes p, the function N(p, m, α) which equals the number of subsets S?{1,…,p ? 1} with the property that Σ_∞∈Sx^m ≡ α (mod p). We obtain a closed form expression for N(p, m, α). We give simple explicit formulas for N(p, 2, α) (which in some cases involve class numbers and fundamental units), and show that for a fixed m, the difference between N(p, m, α) and its average value p^?12^p?1 is of the order of $\text{exp}\text{(p}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ or less. Finally, we obtain the curious result that if p ? 1 does not divide m, then N(p, m, 0) > N(p, m, α) for all α ? 0 (mod p).     

Various disconnectivities of spaces and projectabilities of ?-groups

Arch denotes the category of archimedean ?-groups and ?-homomorphisms. Tych denotes the category of Tychonoff spaces with continuous maps, and α denotes an infinite cardinal or ∞. This work introduces the concept of an αcc-disconnected space and demonstrates that the class of αcc-disconnected spaces forms a covering class in Tych. On the algebraic side, we introduce the concept of an αcc-projectable ?-group and demonstrate that the class of αcc-projectable ?-groups forms a hull class in Arch. In addition, we characterize the αcc-projectable objects in W—the category of Arch-objects with designated weak unit and ?-homomorphisms that preserve the weak unit—and construct the αcc-hull for G in W. Lastly, we apply our results to negatively answer the question of whether every hull class (resp., covering class) is epireflective (resp., monocoreflective) in the category of W-objects with complete ?-homomorphisms (resp., the category of compact Hausdorff spaces with skeletal maps).