Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations

In this article, we propose a new estimation methodology to deal with PCA for high-dimension, low-sample-size (HDLSS) data. We first show that HDLSS datasets have different geometric representations depending on whether a ρ-mixing-type dependency appears in variables or not. When the ρ-mixing-type dependency appears in variables, the HDLSS data converge to an n-dimensional surface of unit sphere with increasing dimension. We pay special attention to this phenomenon. We propose a method called the noise-reduction methodology to estimate eigenvalues of a HDLSS dataset. We show that the eigenvalue estimator holds consistency properties along with its limiting distribution in HDLSS context. We consider consistency properties of PC directions. We apply the noise-reduction methodology to estimating PC scores. We also give an application in the discriminant analysis for HDLSS datasets by using the inverse covariance matrix estimator induced by the noise-reduction methodology.     

Nonlocal Interaction Equations in Environments with Heterogeneities and Boundaries

We study well-posedness of a class of nonlocal interaction equations with spatially dependent mobility. We also allow for the presence of boundaries and external potentials. Such systems lead to the study of nonlocal interaction equations on subsets ? of ? ^d endowed with a Riemannian metric g. We obtain conditions, relating the interaction potential and the geometry, which imply existence, uniqueness and stability of solutions. We study the equations in the setting of gradient flows in the space of probability measures on ? endowed with Riemannian 2-Wasserstein metric.     

Highly rotating fluids in rough domains

We consider a rotating fluid in a domain with rough horizontal boundaries. The Rossby number, kinematic viscosity and roughness are supposed of characteristic size ε. We prove a strong convergence theorem on solutions of Navier–Stokes–Coriolis equations, as ε goes to 0, in the well-prepared case. We show in particular that the limit system is a two-dimensional Euler equation with a nonlinear damping term due to boundary layers. We thus give a substantial refinement of the results obtained on flat boundaries with the classical Ekman layers.     

Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions

We study asharpinterface model for phase transitions which incorporates the interaction of the phase boundaries with the walls of a container Ω. In this model, the interfaces move by their mean curvature and are normal to δΩ. We first establish local-in-time existence and uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the relation of the aforementioned model with atransitionlayer model. We prove that if Ω isconvex, the transition-layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We conclude with a discussion of the difficulties that arise in establishing this result in nonconvex domains. Communicated by David Kinderlehrer     

On Lebesgue-type inequalities for greedy approximation

We study the efficiency of greedy algorithms with regard to redundant dictionaries in Hilbert spaces. We obtain upper estimates for the errors of the Pure Greedy Algorithm and the Orthogonal Greedy Algorithm in terms of the best m-term approximations. We call such estimates the Lebesgue-type inequalities. We prove the Lebesgue-type inequalities for dictionaries with special structure. We assume that the dictionary has a property of mutual incoherence (the coherence parameter of the dictionary is small). We develop a new technique that, in particular, allowed us to get rid of an extra factor m^1/2 in the Lebesgue-type inequality for the Orthogonal Greedy Algorithm.     

Capturing incomplete information in resource allocation problems through numerical patterns

We look at the problem of optimizing complex operations with incomplete information where the missing information is revealed indirectly and imperfectly through historical decisions. Incomplete information is characterized by missing data elements governing operational behavior and unknown cost parameters. We assume some of this information may be indirectly captured in historical databases through flows characterizing resource movements. We can use these flows or other quantities derived from these flows as “numerical patterns” in our optimization model to reflect some of the incomplete information. We develop our methodology for representing information in resource allocation models using the concept of pattern regression. We use a popular goodness-of-fit measure known as the Cramer–Von Mises metric as the foundation of our approach. We then use a hybrid approach of solving a cost model with a term known as the “pattern metric” that minimizes the deviations of model decisions from observed quantities in a historical database. We present a novel iterative method to solve this problem. Results with real-world data from a large freight railroad are presented.     

Two-edge connected subgraphs with bounded rings: Polyhedral results and Branch-and-Cut

We consider the network design problem which consists in determining at minimum cost a 2-edge connected network such that the shortest cycle (a “ring”) to which each edge belongs, does not exceed a given length K. We identify a class of inequalities, called cycle inequalities, valid for the problem and show that these inequalities together with the so-called cut inequalities yield an integer programming formulation of the problem in the space of the natural design variables. We then study the polytope associated with that problem and describe further classes of valid inequalities. We give necessary and sufficient conditions for these inequalities to be facet defining. We study the separation problem associated with these inequalities. In particular, we show that the cycle inequalities can be separated in polynomial time when K≤4. We develop a Branch-and-Cut algorithm based on these results and present extensive computational results.     

The structure of hemispaces in Rn

We study hemispaces (i,e., convex sets with convex complements) in Rⁿ. We give several geometric characterizations of hemispaces and several ways of representing them with the aid of linear operators and lexicographical order. We obtain a metric-affine classification of hemispaces, in terms of their “rank” and “type,” and a “decomposition theorem.” We also give some characterizations of affine transformations which preserve a hemispace.     

Partitions of graphs into cographs

Cographs form the minimal family of graphs containing K₁ that is closed with respect to complementation and disjoint union. We discuss vertex partitions of graphs into the smallest number of cographs. We introduce a new parameter, calling the minimum order of such a partition the c-chromatic number of the graph. We begin by axiomatizing several well-known graphical parameters as motivation for this function. We present several bounds on c-chromatic number in terms of well-known expressions. We show that if a graph is triangle-free, then its chromatic number is bounded between the c-chromatic number and twice this number. We show that both bounds are sharp for graphs with arbitrarily high girth. This provides an alternative proof to a result by Broere and Mynhardt; namely, there exist triangle-free graphs with arbitrarily large c-chromatic numbers. We show that any planar graph with girth at least 11 has a c-chromatic number at most two. We close with several remarks on computational complexity. In particular, we show that computing the c-chromatic number is NP-complete for planar graphs.     

Cutting planes for integer programs with general integer variables

We investigate the use of cutting planes for integer programs with general integer variables. We show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 0–1 variables. We also explore the use of Gomory's mixed-integer cuts. We address both theoretical and computational issues and show how to embed these cutting planes in a branch-and-bound framework. We compare results obtained by using our cut generation routines in two existing systems with a commercially available branch-and-bound code on a range of test problems arising from practical applications. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.This research was partly performed when the author was affiliated with CORE, Université Catholique de Louvain.     

Quasidiagonal Solutions of the Yang–Baxter Equation, Quantum Groups and Quantum Super Groups

This paper answers a few questions about algebraic aspects of bialgebras, associated with the family of solutions of the quantum Yang–Baxter equation in Acta Appl. Math. 41 (1995), pp. 57–98. We describe the relations of the bialgebras associated with these solutions and the standard deformations of GL_n and of the supergroup GL(m|n). We also show how the existence of zero divisors in some of these algebras are related to the combinatorics of their related matrix, providing a necessary and sufficient condition for the bialgebras to be a domain. We consider their Poincaré series, and we provide a Hopf algebra structure to quotients of these bialgebras in an explicit way. We discuss the problems involved with the lift of the Hopf algebra structure, working only by localization.     

Noncommutative stochastic integration through decoupling

We present an abstract approach to noncommutative stochastic integration in the context of a finite von Neumann algebra equipped with a normal, faithful, tracial state, with respect to processes with tensor or freely independent increments satisfying a stationarity condition, using a decoupling technique. We obtain necessary and sufficient conditions for stochastic integrability of Lp-processes with respect to such integrators. We apply the theory to stochastic integration with respect to Boson and free Brownian motion.     

Divided power algebras and distributive laws

We study divided power structures over a product of operads with distributive law. We give a systematic method to characterise the divided power algebras over such a product from the structures of divided power algebra coming from each of the factor operads. We characterise divided power algebras with operadic derivation, as well as divided power p-level algebras in characteristic p, and divided power Poisson algebras in characteristic 3.     

Mathematics teacher education advanced methods: an example in dynamic geometry

We present a research work about an innovative national teacher training program in France: the Pairform@nce program, designed to sustain ICT integration. We study here training for secondary school teachers, whose objective is to foster the development of an inquiry-based approach in the teaching of mathematics, using investigative potentialities of dynamic geometry environments. We adopt the theoretical background of the documentational approach to didactics. We focus on the interactions between teachers and resources: teachers’ professional knowledge influences these interactions, which at the same time yield knowledge evolutions, a twofold process that we conceptualise as a documentational genesis. We followed in particular the work of a team of trainees; drawing on the data collected, we analyse their professional development, related with the training. We observe intertwined evolutions and stabilities, consistent with ongoing geneses.     

Robust combinatorial optimization with variable cost uncertainty

We present in this paper a new model for robust combinatorial optimization with cost uncertainty that generalizes the classical budgeted uncertainty set. We suppose here that the budget of uncertainty is given by a function of the problem variables, yielding an uncertainty multifunction. The new model is less conservative than the classical model and approximates better Value-at-Risk objective functions, especially for vectors with few non-zero components. An example of budget function is constructed from the probabilistic bounds computed by Bertsimas and Sim. We provide an asymptotically tight bound for the cost reduction obtained with the new model. We turn then to the tractability of the resulting optimization problems. We show that when the budget function is affine, the resulting optimization problems can be solved by solving n+1$n+1$ deterministic problems. We propose combinatorial algorithms to handle problems with more general budget functions. We also adapt existing dynamic programming algorithms to solve faster the robust counterparts of optimization problems, which can be applied both to the traditional budgeted uncertainty model and to our new model. We evaluate numerically the reduction in the price of robustness obtained with the new model on the shortest path problem and on a survivable network design problem.     

An expanded mixed covolume method for elliptic problems

We consider the mixed covolume method combining with the expanded mixed element for a system of first‐order partial differential equations resulting from the mixed formulation of a general self‐adjoint elliptic problem with a full diffusion tensor. The system can be used to model the transport of a contaminant carried by a flow in porous media. We use the lowest order Raviart‐Thomas mixed element space. We show the first‐order error estimate for the approximate solution in L² norm. We show the superconvergence both for pressure and velocity in certain discrete norms. We also get a finite difference scheme by using proper approximate integration formulas. Finally we give some numerical examples. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Stability and localization of unbounded solutions of a nonlinear heat equation in a plane

The article investigates unbounded solutions of the equation u _t = div (u ^σgrad u) + u ^β in a plane. We numerically analyze the stability of two-dimensional self-similar solutions (structures) that increase with blowup. We confirm structural stability of the simple structure with a single maximum and metastability of complex structures. We prove structural stability of the radially symmetrical structure with a zero region at the center and investigate its attraction region. We study the effect of various perturbations of the initial function on the evolution of self-similar solutions. We further investigate how arbitrary compact-support initial distributions attain the self-similar mode, including distributions whose support is different from a disk. We show that the self-similar mode described by a simple radially symmetrical structure is achieved only in the central region, while the entire localization region does not have enough time to transform into a disk during blowup. We show for the first time that simple structures may merge into a complex structure, which evolves for a long time according to self-similar law.     

Danielewski–Fieseler surfaces

We study a class of normal affine surfaces, with additive group actions, which contains in particular the Danielewski surfaces in A³ given by the equations xⁿz = P(y), where P is a nonconstant polynomial with simple roots. We call them Danielewski--Fieseler surfaces. We reinterpret a construction of Fieseler to show that these surfaces appear as the total spaces of certain torsors under a line bundle over a curve with an r fold point. We classify Danielewski-Fieseler surfaces through labelled rooted trees attached to such a surface in a canonical way. Finally, we characterize those surfaces which have a trivial Makar-Limanov invariant in terms of the associated trees.     

Some problems of the theory of quantum statistical systems with an energy spectrum of the fractional-power type

We consider the problem of the effective interaction potential in a quantum many-particle system leading to the fractional-power dispersion law. We show that passing to fractional-order derivatives is equivalent to introducing a pair interparticle potential. We consider the case of a degenerate electron gas. Using the van der Waals equation, we study the equation of state for systems with a fractional-power spectrum. We obtain a relation between the van der Waals constant and the phenomenological parameter ??, the fractional-derivative order. We obtain a relation between energy, pressure, and volume for such systems: the coefficient of the thermal energy is a simple function of ??. We consider Bose??Einstein condensation in a system with a fractional-power spectrum. The critical condensation temperature for 1 < ?? < 2 is greater in the case under consideration than in the case of an ideal system, where ?? = 2.