MICZ-Kepler: Dynamics on the cone over SO(n)

We show that the n-dimensional MICZ-Kepler system arises from symplectic reduction of the “Kepler problem” on the cone over the rotation group SO(n). As a corollary we derive an elementary formula for the general solution of the MICZ-Kepler problem. The heart of the computation is the observation that the additional MICZ-Kepler potential, |?|²/r ², agrees with the rotational part of the cone’s kinetic energy.     

To the question about the maximum principle for manifolds over local algebras

We consider manifolds over a local algebra A. We study basis functions of the canonical foliation which represent the real parts of A-differentiable functions. We prove that these are constant functions. We find the form of A-differentiable functions on some manifolds over local algebras, in particular, on compact manifolds. We obtain an estimate for the dimension of some spaces of 1-forms and analogs of the above results for the projective mappings of foliations.     

Relaxing support vectors for classification

We introduce a novel modification to standard support vector machine (SVM) formulations based on a limited amount of penalty-free slack to reduce the influence of misclassified samples or outliers. We show that free slack relaxes support vectors and pushes them towards their respective classes, hence we use the name relaxed support vector machines (RSVM) for our method. We present theoretical properties of the RSVM formulation and develop its dual formulation for nonlinear classification via kernels. We show the connection between the dual RSVM and the dual of the standard SVM formulations. We provide error bounds for RSVM and show it to be stable, universally consistent and tighter than error bounds for standard SVM. We also introduce a linear programming version of RSVM, which we call RSVMLP. We apply RSVM and RSVMLP to synthetic data and benchmark binary classification problems, and compare our results with standard SVM classification results. We show that relaxed influential support vectors may lead to better classification results. We develop a two-phase method called RSVM² for multiple instance classification (MIC) problems, where RSVM formulations are used as classifiers. We extend the two-phase method to the linear programming case and develop RSVMLP². We demonstrate the classification characteristics of RSVM² and RSVMLP², and report our classification results compared to results obtained by other SVM-based MIC methods on public benchmark datasets. We show that both RSVM² and RSVMLP² are faster and produce more accurate classification results.     

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.     

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.     

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.     

A cohomological construction of quantization functors of Lie bialgebras

We propose a variant to the Etingof-Kazhdan construction of quantization functors. We construct the twistor J_Φ associated to an associator Φ using cohomological techniques. We then introduce a criterion ensuring that the “left Hopf algebra” of a quasitriangular QUE algebra is flat. We prove that this criterion is satisfied at the universal level. This gives a construction of quantization functors, equivalent to the Etingof-Kazhdan construction.     

Weights in the cohomology of toric varieties

We describe the weight filtration in the cohomology of toric varieties. We present a role of the Frobenius automorphism in an elementary way. We prove that equivariant intersection homology of an arbitrary toric variety is pure. We obtain results concerning Koszul duality: nonequivariant intersection cohomology is equal to the cohomology of the Koszul complexIH _T ^* (X)⊗H*(T). We also describe the weight filtration inIH ^*(X). Supported by KBN 2P03A 00218 grant. I thank, Institute of Mathematics, Polish Academy of Science for hospitality.     

Complexity issues on bounded restrictive H-coloring

We consider a bounded version of the restrictive and the restrictive list H-coloring problem in which the number of pre-images of certain vertices of H is taken as parameter. We consider the decision and the counting versions, as well as, further variations of those problems. We provide complexity results identifying the cases when the problems are NP-complete or #P-complete or polynomial time solvable. We conclude stating some open problems.     

A class of semigroups regularized in space and time

We consider α-times integrated C-regularized semigroups, which are a hybrid between semigroups regularized in space (C-semigroups) and integrated semigroups regularized in time. We study the basic properties of these objects, also in absence of exponential boundedness. We discuss their generators and establish an equivalence theorem between existence of integrated regularized semigroups and well-posedness of certain Cauchy problems. We investigate the effect of smoothing regularized semigroups by fractional integration.     

The N-policy of a discrete time Geo/G/1 queue with disasters and its application to wireless sensor networks

We analyze an N-policy of a discrete time Geo/G/1 queue with disasters. We obtain the probability generating functions of the queue length, the sojourn time, and regeneration cycles such as the idle period and the busy period. We apply the queue to a power saving scheme in wireless sensor networks under unreliable network connections where data packets are lost by external attacks or shocks. We present various numerical experiments for application to power consumption control in wireless sensor networks. We investigate the characteristics of the optimal N-policy that minimizes power consumption and derive practical insights on the operation of the N-policy in wireless sensor networks.     

Aluthge transforms of unbounded weighted composition operators in L2-spaces

We describe the Aluthge transform of an unbounded weighted composition operator acting in an L²-space. We show that its closure is also a weighted composition operator with the same symbol and a modified weight function. We investigate its dense definiteness. We characterize p-hyponormality of unbounded weighted composition operators and provide results on how it is affected by the Aluthge transformation. We show that the only fixed points of the Aluthge transformation on weighted composition operators are quasinormal ones.     

Generalizing the modeling of fuzzy logic controllers by parameterized aggregation operators

We describe the type of reasoning used in the typical fuzzy logic controller, the Mamdani reasoning method. We point out the basic assumptions in this model. We discuss the S-OWA operators which provide families of parameterized “andlike” and “orlike” operators. We generalize the Mamdani model by introducing these operators. We introduce a method, which we call Direct Fuzzy Reasoning (DFR), which results from one choice of the parameters. We develop some learning algorithms for the new method. We show how the Takagi-Sugeno-Kang (TSK) method of reasoning is an example of this DFR method.     

Friedman-reflexivity

In the present paper, we explore an idea of Harvey Friedman to obtain a coordinate-free presentation of consistency. For some range of theories, Friedman's idea delivers actual consistency statements (modulo provable equivalence). For a wider range, it delivers consistency-like statements.We say that a sentence C is an interpreter of a finitely axiomatised A over U iff it is the weakest statement C over U, with respect to U-provability, such that $U+C$ interprets A. A theory U is Friedman-reflexive iff every finitely axiomatised A has an interpreter over U. Friedman shows that Peano Arithmetic, PA, is Friedman-reflexive.We study the question which theories are Friedman-reflexive. We show that a very weak theory, Peano Corto, is Friedman-reflexive. We do not get the usual consistency statements here, but bounded, cut-free, or Herbrand consistency statements. We illustrate that Peano Corto as a base theory has additional desirable properties.We prove a characterisation theorem for the Friedman-reflexivity of sequential theories. We provide an example of a Friedman-reflexive sequential theory that substantially differs from the paradigm cases of Peano Arithmetic and Peano Corto.Interpreters over a Friedman-reflexive U can be used to define a provability-like notion for any finitely axiomatised A that interprets U. We explore what modal logics this idea gives rise to. We call such logics interpreter logics. We show that, generally, these logics satisfy the Löb Conditions, aka K4. We provide conditions for when interpreter logics extend S4, K45, and Löb's Logic. We show that, if either U or A is sequential, then the condition for extending Löb's Logic is fulfilled. Moreover, if our base theory U is sequential and if, in addition, its interpreters can be effectively found, we prove Solovay's Theorem. This holds even if the provability-like operator is not necessarily representable by a predicate of Gödel numbers.At the end of the paper, we briefly discuss how successful the coordinate-free approach is.     

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.     

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.