A q-analog of the campbell-baker-hausdorff formula

The C-B-H Formula of Lie theory guarantees that exp(x)exp(y)=exp(w), where w can be expressed as an infinite series of (iterated) commutators, [a, b]=ab-ba, when x and y are noncommuting variables. It is shown here in Section 1 that a q-analog of this formula exists with the exponential replaced by its q-analog and the commutator replaced by the q-bracket [a, b]=qab-ba. In fact, the q-brackets span the ring of power series in x and y. Section 2 contains a little history of this problem, and a class of related algebras called q-Lie algebras is introduced in Section 3.     

Overlaps help: Improved bounds for group testing with interval queries

Given a finite ordered set of items and an unknown distinguished subset P of up to p positive elements, identify the items in P by asking the least number of queries of the type ‘‘does the subset Q intersect P?”, where Q is a subset of consecutive elements of {1,2,…,n}. This problem arises, e.g., in computational biology, in a particular method for determining splice sites in genes. We consider time-efficient algorithms where queries are arranged in a fixed number s of stages: In each stage, queries are performed in parallel. In a recent bioinformatics paper, we proved optimality (subject to lower-order terms) with respect to the number of queries, of some strategies for the special cases p=1 or s=2. Exploiting new ideas, we are now able to provide improved lower bounds for any p?2 and s?3 and improved upper bounds for larger s. Most notably, our new bounds converge as s grows. Our new query scheme uses overlapping query intervals within a stage, which is effective for large enough s. This contrasts with our previous results for s?2 where optimal strategies were implemented by disjoint queries. The main open problem is whether overlaps help already in the case of small s?3. Anyway, the remaining gaps between the current upper and lower bounds for any fixed s?3 amount to small constant factors in the main term. The paper ends with a discussion of practical implications in the case that the positive elements are well separated.     

A study on the structure of planning behaviour

Although there is an abundance of literature on planning, it is largely either about how we actually plan, about how we should plan, or about how we should organize our planning, and little effort seems to have been contributed to investigate how we actually organize our planning, which should become the basis for the above three kinds of research. In this paper, we consider first the relationship between planning and decision-making with a simple example (Section 2), and then the benefits and costs of planning more specifically (Section 3). After these preparatory considerations, we postulate a hypothesis which describes planning behaviour of individuals and organizations, with necessary definitions and assumptions (Section 4). Some propositions attained from the hypothesis are also included. For testing the validity of the hypothesis, an organism model of organization which deals with an organization as an open system in the environment is introduced (Section 5.1). We make additional assumptions to let the model plan (Section 5.2), and the behaviour of the model is generally supported by the practitioners and researcher interviewed (Section 5.3). Finally we discussed briefly the methodology employed and the possible applications of the hypothesis (Section 6).     

Convergence and error estimates for (m,l,s)-splines

In some recent papers, Bouhamidi and Le Méhauté introduced L^m,l,s-splines as a natural extension of J. Duchon's (m,s)-splines. In the present work, some results on convergence and error estimates for smoothing and interpolating L^m,l,s-splines (called here (m,l,s)-splines) are given. These results extend those presented in several papers by Duchon, Arcangéli and López de Silances for (m,s)-splines and also for D^m-splines (i.e. (m,0)-splines).     

Dynamic lot-sizing with price changes and price-dependent holding costs

This paper proposes a new formulation of the dynamic lot-sizing problem with price changes which considers the unit inventory holding costs in a period as a function of the procurement decisions made in previous periods. In Section 1, the problem is defined and some of its fundamental properties are identified. A dynamic programming approach is developed to solve it when solutions are restricted to sequential extreme flows, and results from location theory are used to derive an O(T²) algorithm which provides a provably optimal solution of an integer linear programming formulation of the general problem. In Section 2, a heuristic is developed for the case where the inventory carrying rates and the order costs are constant, and where the item price can change once during the planning horizon. Permanent price increases, permanent price decreases and temporary price reductions are considered. In Section 3, extensive testing of the various optimal and heuristic algorithms is reported. Our results show that, in this context, the two following intuitive actions usually lead to near optimal solutions: accumulate stock at the lower price just prior to price increase and cut short on orders when a price decrease is imminent.     

Periodic distributions and non-Liouville numbers

In an earlier paper (Translation-invariant linear forms and a formula for the Dirac measure, J. Functional Analysis8 (1971), 173–188) the author proved that the ratio $\text{α}\text{β}$ of two real numbers is a non-Liouville number if and only if there exist two Schwartz distributions A and B with compact supports on the real line such that δ′ = A ? τ_αA + B ? τ_βB. The present paper presents (in Section 3) a completely new and more elementary proof of this result (stated fully as Theorems 1 and 2 at the end of Section 1) based on some fundamental properties of the mapping H₀ = ? Δ and H = H₀ + T, where T which are established in Section 2. Further connections with Diophantine approximation (badly approximable numbers and Roth's Theorem) are presented in Section 4 where it is proved that the orders of the distributions A and B are always ?2 (Theorem 3) and almost always =3 (Theorem 4). Section 5 contains some partial results (Theorem 5 and Corollary 2 of Theorem 6) on the analogous question (as yet unsettled) for the Banach space C(T) of all continuous functions on the circle group T, and connects this problem with Sidon sets of integers.     

Measures of fuzzy sets and measures of fuzziness

The measures presented in this paper are defined by using Weber's concept of decomposable measures m of crisp sets, having in particular the Archimedean decomposable operations in view (Section 2). Measures $\text{m}$ of fuzzy sets are introduced as integrals with respect to m. For the Archimedean cases, Weber's integral will be used as alternative to Sugeno's and Choquet's concepts (Section 3). What ‘fuzziness’ means will be described by functions of fuzziness F (another name: entropy N-functions) with respect to a negation. In addition to the types of functions of fuzziness which are induced by concave functions, we discuss also the ones which are induced by fuzzy connectives (Section 4). Now, using m for measuring the ‘importance of items’ and F for the ‘fuzziness’ of the possible values of a fuzzy set ?, m?(F ° ?) serves us as a measure of the fuzziness $\text{F}$? of ?. The concepts of De Luca and Termini, Capocelli and De Luca, Kaufmann, Knopfmacher, Loo, Gottwald, Dombi and, under the restriction to the Archimedean cases, also the concepts of Trillas and Riera and Yager turn out to be special cases (Section 5).     

p-Sidon sets

Let G be a compact Abelian group with character group X. Bo?ejko and Pytlik [Colloq. Math.25 (1972), 117–124] introduced and studied several special types of lacunary subsets of X. This paper is based upon a hitherto unpublished detailed study of those types that most resemble Sidon sets, which the present authors had independently introduced and studied under the name of p-Sidon sets. Some, but not all, aspects of the theory of Sidon (= 1-Sidon) sets carry over to the more general setting. In Section 1 some properties of sets equivalent to p-Sidonicity are given. Section 2 contains several useful consequences of p-Sidonicity; see Theorems 2.1 and 2.4 and Corollaries 2.6 and 2.7. In Section 3, it is shown that certain Λ_q sets also satisfy some of the consequences listed in Section 2. Nevertheless, Λ_q sets need not be p-Sidon sets; see Theorem 3.1. Examples of ($\text{4}\text{3}$)-Sidon sets that are not Sidon sets are given in Section 5. The proof that these sets are ($\text{4}\text{3}$)-Sidon sets requires a brief study of 4-norms in Varopoulos algebras; see Section 4. In Section 6, some special results for the circle group are deduced. Many of these results appear to be new even for p = 1.     

Room squares with sub-squares

Suppose there is a Room square of side r. Then there is a Room square of side 4r + 1 with a sub-square of side r which is isomorphic to the original square.No example was known previously of a Room square of side s with a subsquare of side r where s ? 6r.     

Equatorial sporadic E and cross-field instability

The occurrence of sporadic E at an equatorial station during magnetically quiet daytime conditions corresponds almost exactly to the time during which the horizontal component of the earth’s magnetic field is above the mean night time level. Any large decrease of H below the night time level is accompanied by the disappearance of equatorial E_s?q reflections precisely for the period when the value of H remains below its night time level. Such disappearance of E_s?q can be attributed to the reversal of the current equal to, or greater than, the normal eastward equatorial electrojet current. During magnetically disturbed conditions, however, the depressions in H are not always accompanied by the disappearance of E_s?q. Whenever the normal E and sporadic E reflections can be resolved on the equatorial ionograms, the minimum virtual height of the normal E is seen to be clearly greater than that of the sporadic E layer. The creation of E region irregularities at equatorial latitudes giving the appearance of an E_s?q layer in daytime ionograms is suggested to be due to cross-field (plasma gradient) instability. The horizontal magnetic field and the upward Hall polarisation (electric) field produce irregularities in the lower E-region where the rate of increase of ambient electron density is large and directed upward. A temporary reversal of the electrojet current indicated by a decrease in H below the night time level and the disappearance of E_s?q are due to the temporary reversal of the vertical Hall polarisation field making it downward instead of upward which being opposite to the direction of the gradient of plasma density inhibits the cross-field instabilities.     

（r, s）-STABILITY OF UNFOLDING OFF-EQUIVARIANT BIFURCATION PROBLEM

In this paper,the F-equivariant (s,t)-equivalence relation and F-equivariant infinitesimally (r,s)-stability of F-equivariant bifurcation problem are defined. The criterion for F-equivariant infinitesimally (r, s)-stability is proven when F is a compact finite Lie group . Transversality condition is used to characterize the stability.     

Cohomology of Horizontal Forms

The complex of s-horizontal forms of a smooth foliation F on a manifold M is proved to be exact for every s = 1, . . . , n = codim F, and the cohomology groups of the complex of its global sections, are introduced. They are then compared with other cohomology groups associated to a foliation, previously introduced. An explicit formula for an s-horizontal primitive of an s-horizontal closed form, is given. The problem of representing a de Rham cohomology class by means of a horizontal closed form is analysed. Applications of these cohomology groups are included and several specific examples of explicit computation of such groups??even for non-commutative structure groups??are also presented.     

Coupled surface and grain boundary motion: a travelling wave solution

Existence and uniqueness are proven for a travelling wave solution for a problem in which motion by mean curvature is coupled with surface diffusion. This problem pertains to a bicrystal in a “quarter-loop” geometry in which one grain grows at the expense of the other, and the internal grain boundary between the two crystals contacts the exterior surface at a “groove root” or “tri-junction” where various balance laws hold. Far in front and behind the groove root the overall height of the bicrystal is assumed to be unperturbed. Whereas in a previous paper (Acta Mater. 51 (2003) 1981) a partially linearized formulation was considered for which explicit solutions could be found, here we treat the fully nonlinear problem. Employing an angle formulation and a scaled arc-length parameterization, we reduce the problem to the solution of a third order ODE with a jump condition at the origin. Existence is proven if m, the ratio of the exterior surface energy to the surface energy of the grain boundary, is less than about $\approx 0.92$. Uniqueness of these solutions is demonstrated within the class of single-valued solutions. A numerical comparison is made with the solution of the partially linearized formulation found earlier for the sake of illustration.     

Resource allocation in light of a stochastic stream of opportunities

The problem of resource allocation over time in a world of uncertainty has proven to be formidable indeed. Efforts to date have either removed the essential dynamics from the formulation or have produced complex backward optimization expressions of questionable usefulness. We deal with the development of an optimal strategy for the situation where potential projects arrive in accordance with a specified stochastic process. A project, once revealed, possesses a nonstochastic cash flow stream. The investor seeks to maximize his long-run rate of return (μ). The resulting (implicit) expression involving μ requires numerical solution in general. However, under certain conditions, a direct solution is possible. An illustrative example is presented.     

Fundamental solutions of singular SPDEs

This paper deals with some models of mathematical physics, where random fluctuations are modeled by white noise or other singular Gaussian generalized processes. White noise, as the distributional derivative od Brownian motion, which is the most important case of a Lévy process, is defined in the framework of Hida distribution spaces. The Fourier transformation in the framework of singular generalized stochastic processes is introduced and its applications to solving stochastic differential equations involving Wick products and singularities such as the Dirac delta distribution are presented. Explicit solutions are obtained in form of a chaos expansion in the Kondratiev white noise space, while the coefficients of the expansion are tempered distributions. Stochastic differential equations of the form P(ω, D) ◊ u(x, ω) = A(x, ω) are considered, where A is a singular generalized stochastic process and P(ω, D) is a partial differential operator with random coefficients. We introduce the Wick-convolution operator which enables us to express the solution as u = sA ◊ I^◊(−1), where s denotes the fundamental solution and I is the unit random variable. In particular, the stochastic Helmholtz equation is solved, which in physical interpretation describes waves propagating with a random speed from randomly appearing point sources.     

Du discret au continu pour des modèles de réseaux aléatoires d'atomes

We study in this Note the continuum (macroscopic) limit for some atomistic models for crystals. The purpose is to derive densities of mechanical energies from microscopic models. In contrast to the setting of a previous study, where the microscopic structure was assumed to be periodic, it is modelled here by a stochastic lattice, which enjoys some stationarity and ergodicity properties, following notions previously introduced elsewhere. To cite this article: X. Blanc et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Variance reduction method based on sensitivity derivatives,Part 2

A previous paper introduced a sampling method (SDES) based on sensitivity derivatives to construct statistical moment estimates that are more efficient than standard Monte Carlo estimates. In this paper we sharpen previous theoretical results and introduce a criterion to guarantee that the variance of SDES estimates is smaller than the variance of the Monte Carlo estimate. Previous numerical experiments demonstrated, and here we prove analytically, that the first-order SDES and Monte Carlo estimates converge at the same rate. We illustrate the efficiency of the SDES method of order n, where n is fixed, to estimate statistical moments with a Korteweg–de Vries equation with uncertain initial conditions.     

Approximation Schemes for the Generalized Traveling Salesman Problem

A graph Γ is called a Deza graph if it is regular and the number of common neighbors of any two distinct vertices is one of two fixed values. A Deza graph is called a strictly Deza graph if it has diameter 2 and is not strongly regular. In 1992, Gardiner et al. proved that a strongly regular graph that contains a vertex with disconnected second neighborhood is a complete multipartite graph with parts of the same size greater than 2. In this paper, we study strictly Deza graphs with disconnected second neighborhoods of vertices. In Section 2, we prove that, if each vertex of a strictly Deza graph has disconnected second neighborhood, then the graph is either edge-regular or coedge-regular. In Sections 3 and 4, we consider strictly Deza graphs that contain at least one vertex with disconnected second neighborhood. In Section 3, we show that, if such a graph is edge-regular, then it is the s-coclique extension of a strongly regular graph with parameters (n, k, λ, μ), where s is an integer, s ≥ 2, and λ = μ. In Section 4, we show that, if such a graph is coedge-regular, then it is the 2-clique extension of a complete multipartite graph with parts of the same size greater than or equal to 3.     

Halton-type sequences from global function fields

For any prime power q and any dimension s≥1, a new construction of (t, s)-sequences in base q using global function fields is presented. The construction yields an analog of Halton sequences for global function fields. It is the first general construction of (t, s)-sequences that is not directly based on the digital method. The construction can also be put into the framework of the theory of (u, e, s)-sequences that was recently introduced by Tezuka and leads in this way to better discrepancy bounds for the constructed sequences.