The lattice of extensions of the minimal logic

In this article, we survey the results on the lattice of extensions of the minimal logic Lj, a paraconsistent analog of the intuitionistic logic Li. Unlike the well-studied classes of explosive logics, the class of extensions of the minimal logic has an interesting global structure. This class decomposes into the disjoint union of the class Int of intermediate logics, the class Neg of negative logics with a degenerate negation, and the class Par of properly paraconsistent extensions of the minimal logic. The classes Int and Neg are well studied, whereas the study of Par can be reduced to some extent to the classes Int and Neg.     

Truncation error estimates for generalized Hermite sampling

The generalized Hermite sampling uses samples from the function itself and its derivatives up to order r. In this paper, we investigate truncation error estimates for the generalized Hermite sampling series on a complex domain for functions from Bernstein space. We will extend some known techniques to derive those estimates and the bounds of Jagerman (SIAM J. Appl. Math. 14, 714–723 1966), Li (J. Approx. Theory 93, 100–113 1998), Annaby-Asharabi (J. Korean Math. Soc. 47, 1299–1316 2010), and Ye and Song (Appl. Math. J. Chinese Univ. 27, 412–418 2012) will be special cases for our results. Some examples with tables and figures are given at the end of the paper.     

Zooming from global to local: a multiscale RBF approach

Because physical phenomena on Earth’s surface occur on many different length scales, it makes sense when seeking an efficient approximation to start with a crude global approximation, and then make a sequence of corrections on finer and finer scales. It also makes sense eventually to seek fine scale features locally, rather than globally. In the present work, we start with a global multiscale radial basis function (RBF) approximation, based on a sequence of point sets with decreasing mesh norm, and a sequence of (spherical) radial basis functions with proportionally decreasing scale centered at the points. We then prove that we can “zoom in” on a region of particular interest, by carrying out further stages of multiscale refinement on a local region. The proof combines multiscale techniques for the sphere from Le Gia, Sloan and Wendland, SIAM J. Numer. Anal. 48 (2010) and Applied Comp. Harm. Anal. 32 (2012), with those for a bounded region in ? ^d from Wendland, Numer. Math. 116 (2010). The zooming in process can be continued indefinitely, since the condition numbers of matrices at the different scales remain bounded. A numerical example illustrates the process.     

A contribution to perturbation analysis for total least squares problems

In this note, we present perturbation analysis for the total least squares (Tls) problems under the genericity condition. We review the three condition numbers proposed respectively by Zhou et al. (Numer. Algorithm, 51 (2009), pp. 381–399), Baboulin and Gratton (SIAM J. Matrix Anal. Appl. 32 (2011), pp. 685–699), Li and Jia (Linear Algebra Appl. 435 (2011), pp. 674–686). We also derive new perturbation bounds.     

Fast convergence of an inexact interior point method for horizontal complementarity problems

In Andreani et al. (Numer. Algorithms 57:457–485, 2011), an interior point method for the horizontal nonlinear complementarity problem was introduced. This method was based on inexact Newton directions and safeguarding projected gradient iterations. Global convergence, in the sense that every cluster point is stationary, was proved in Andreani et al. (Numer. Algorithms 57:457–485, 2011). In Andreani et al. (Eur. J. Oper. Res. 249:41–54, 2016), local fast convergence was proved for the underdetermined problem in the case that the Newtonian directions are computed exactly. In the present paper, it will be proved that the method introduced in Andreani et al. (Numer. Algorithms 57:457–485, 2011) enjoys fast (linear, superlinear, or quadratic) convergence in the case of truly inexact Newton computations. Some numerical experiments will illustrate the accuracy of the convergence theory.     

The Rockafellar Conjecture and type (FPV)

In this note, using a technique of Verona and Verona, we show that a result announced in “All maximal monotone operators in a Banach space are of type FPV” by A. Eberhard and R. Wenczel, Set-Valued Var. Anal. 22, 597–615, (2014), implies the truth of the Rockafellar conjecture. We then show that there is a gap in the logic of the Eberhard–Wenczel result, which we tried unsuccessfully to close. We also discuss briefly the connection with maximally monotone multifunctions of type (FPV).     

A Quantum Analog of Generalized Cluster Algebras

We define a quantum analog of a class of generalized cluster algebras which can be viewed as a generalization of quantum cluster algebras defined in Berenstein and Zelevinsky (Adv. Math. 195(2), 405–455 2005). In the case of rank two, we extend some structural results from the classical theory of generalized cluster algebras obtained in Chekhov and Shapiro (Int. Math. Res. Notices 10, 2746–2772 2014) and Rupel (2013) to the quantum case.     

Hereditary,Additive and Divisible Classes in Epireflective Subcategories of Top

Hereditary coreflective subcategories of an epireflective subcategory A of Top such that I ₂ ? A (here I ₂ is the two-point indiscrete space) were studied in [4]. It was shown that a coreflective subcategory B of A is hereditary (closed under the formation subspaces) if and only if it is closed under the formation of prime factors. The main problem studied in this paper is the question whether this claim remains true if we study the (more general) subcategories of A which are closed under topological sums and quotients in A instead of the coreflective subcategories of A. We show that this is true if A ? Haus or under some reasonable conditions on B. E.g., this holds if B contains either a prime space, or a space which is not locally connected, or a totally disconnected space or a non-discrete Hausdorff space. We touch also other questions related to such subclasses of A. We introduce a method extending the results from the case of non-bireflective subcategories (which was studied in [4]) to arbitrary epireflective subcategories of Top. We also prove some new facts about the lattice of coreflective subcategories of Top and ZD.     

Preconditioning of spectral methods via Birkhoff interpolation

High-order differentiation matrices as calculated in spectral collocation methods usually include a large round-off error and have a large condition number (Baltensperger and Berrut Computers and Mathematics with Applications 37(1), 41–48 1999; Baltensperger and Trummer SIAM J. Sci. Comput. 24(5), 1465–1487 2003; Costa and Don Appl. Numer. Math. 33(1), 151–159 2000). Wang et al. (Wang et al. SIAM J. Sci. Comput. 36(3), A907–A929 2014) present a method to precondition these matrices using Birkhoff interpolation. We generalize this method for all orders and boundary conditions and allowing arbitrary rows of the system matrix to be replaced by the boundary conditions. The preconditioner is an exact inverse of the highest-order differentiation matrix in the equation; thus, its product with that matrix can be replaced by the identity matrix. We show the benefits of the method for high-order differential equations. These include improved condition number and, more importantly, higher accuracy of solutions compared to other methods.     

Positivity for Generalized Cluster Variables of Affine Quivers

It has been proved in Lee and Schiffler, Ann. of Math. 182(1) 73–125 2015 that cluster variables of all skew-symmetric cluster algebras are positive. i.e., every cluster variable as a Laurent polynomial in the cluster variables of any fixed cluster has positive coefficients. We prove that every regular generalized cluster variable of an affine quiver is positive. As a corollary, we obtain that generalized cluster variables of affine quivers are positive and we also construct various positive bases. This generalizes the results in Dupont, J. Algebra Appl. 11(4) 19 2012 and Ding et al. Algebr. Represent. Theory 16(2) 491–525 2013.     

Generalized sinc-Gaussian sampling involving derivatives

The generalized sampling expansion which uses samples from a bandlimited function f and its first r derivatives was first introduced by Linden and Abramson (Inform. Contr. 3, 26–31, 1960) and it was extended in different situations by some authors through the last fifty years. The use of the generalized sampling series in approximation theory is limited because of the slow convergence. In this paper, we derive a modification of a generalized sampling involving derivatives, which is studied by Shin (Commun. Korean Math. Soc. 17, 731–740, 2002), using a Gaussian multiplier. This modification is introduced for wider classes, the class of entire functions including unbounded functions on ? and the class of analytic functions in a strip. It highly improves the convergence rate of the generalized sampling which will be of exponential order. We will show that many known results included in Sampl. Theory Signal Image Process. 9, 199–221 (2007) and Numer. Funct. Anal. Optim. 36, 419–437 (2015) are special cases of our results. Numerical examples show a rightly good agreement with our theoretical analysis.     

Prediction of catastrophes in space over time

Predicting rare events, such as high level up-crossings, for spatio-temporal processes plays an important role in the analysis of the occurrence and impact of potential catastrophes in, for example, environmental settings. Designing a system which predicts these events with high probability, but with few false alarms, is clearly desirable. In this paper an optimal alarm system in space over time is introduced and studied in detail. These results generalize those obtained by de Maré (Ann. Probab. 8, 841–850, 1980) and Lindgren (Ann. Probab. 8, 775–792, 1980, Ann. Probab. 13, 804–824, 1985) for stationary stochastic processes evolving in continuous time and are applied here to stationary Gaussian random fields.     

Semisimple Representations of Alternating Cyclotomic Hecke Algebras

We define alternating cyclotomic Hecke algebras in higher levels as subalgebras of cyclotomic Hecke algebras under an analogue of Goldman’s hash involution. We compute the rank of these algebras and construct a full set of irreducible representations in the semisimple case, generalising Mitsuhashi’s results Mitsuhashi (J. Alg. 240 535–558 2001, J. Alg. 264 231–250 2003).     

Volume of Tubes,Non Polynomial Behavior

The behavior of the volume of the tube around a given compact subset in finite dimension, depending on the radius r, is an old and important question. It is related to many fields, like differential geometry, geometric measure theory, integral geometry, and also probability and statistics. Federer (Trans. Amer. Math. Soc. 93, 418–491, 1959), introduces the class of sets with positive reach, for which the volume is given by a polynomial in the radius r. For applications, in numerical analysis and statistics for example, an “almost” polynomial behavior is of equal interest. We exhibit an example showing how far to a polynomial can be the volume of the tube, when the radius r tends to 0, for the simplest extension of the class of sets with positive reach, namely the class of (locally finite) union of sets with positive reach -satisfying a tangency condition- as introduced by Zähle (I. Math. Nachr. 119, 327–339, 1984).     

A hybrid viscosity iterative method with averaged mappings for split equilibrium problems and fixed point problems

In this paper, with the help of averaged mappings, we introduce and study a hybrid iterative method to approximate a common solution of a split equilibrium problem and a fixed point problem of a finite collection of nonexpansive mappings. We prove that the sequences generated by the iterative scheme strongly converges to a common solution of the above-said problems. We give some numerical examples to ensure that our iterative scheme is more efficient than the methods of Plubtieng and Punpaeng (J. Math Anal. Appl. 336(1), 455–469, 15), Liu (Nonlinear Anal. 71(10), 4852–4861, 10) and Wen and Chen (Fixed Point Theory Appl. 2012(1), 1–15, 18). The results presented in this paper are the extension and improvement of the recent results in the literature.     

Computationally efficient modular nonlinear filter stabilization for high Reynolds number flows

The nonlinear filter based stabilization proposed in Layton et al. (J. Math. Fluid Mech. 14(2), 325–354 2012) allows to incorporate an eddy viscosity model into an existing laminar flow codes in a modular way. However, the proposed nonlinear filtering step requires the assembly of the associated matrix at each time step and solving a linear system with an indefinte matrix. We propose computationally efficient version of the filtering step that only requires the assembly once, and the solution of two symmetric, positive definite systems at each time step. We also test a new indicator function based on the entropy viscosity model of Guermond (Int. J. Numer. Meth. Fluids. 57(9), 1153–1170 2008); Guermond et al. (J. Sci. Comput. 49(1), 35–50 2011).     

Rigid Binary Relations on a 4-Element Domain

An n-ary relation ρ on a set U is strongly rigid if it is preserved only by trivial operations. It is projective if the only idempotent operations in P o l ρ are projections. Rosenberg, (Rocky Mt. J. Math. 3, 631–639, 1973) characterized all strongly rigid relations on a set with two elements and found a strongly rigid binary relation on every domain U of at least 3 elements. Larose and Tardif (Mult.-Valued Log. 7(5-6), 339–362, 2001) studied the projective and strongly rigid graphs and constructed large families of strongly rigid graphs. ?uczak and Ne?et?il (J. Graph Theory. 47, 81–86, 2004) settled in the affirmative a conjecture of Larose and Tardif that most graphs on a large set are projective, and characterized all homogenous graphs that are projective. ?uczak and Ne?et?il (SIAM J. Comput. 36(3), 835–843, 2006) confirmed a conjecture of Rosenberg that most relations on a big set are strongly rigid. In this paper, we characterize all strongly rigid relations on a set with at least three elements to answer an open question by Rosenberg, (Rocky Mt. J. Math. 3, 631–639, 1973) and we classify the binary relations on the 4-element domain by rigidity and demonstrate that there are merely 40 pairwise nonisomorphic rigid binary relations on the same domain (among them 25 are pairwise nonisomorphic strongly rigid).     

On Weighted Kernels of Two Posets

A recent result of Aharoni Berger and Gorelik (Order 31(1), 35–43, 2014) is a weighted generalization of the well-known theorem of Sands Sauer and Woodrow (Theory Ser. B 33(3), 271–275, 1982) on monochromatic paths. The authors prove the existence of a so called weighted kernel for any pair of weighted posets on the same ground set. In this work, we point out that this result is closely related to the stable marriage theorem of Gale and Shapley (Amer. Math. Monthly 69(1), 9–15, 1962), and we generalize Blair’s theorem by showing that weighted kernels form a lattice under a certain natural order. To illustrate the applicability of our approach, we prove further weighted generalizations of the Sands Sauer Woodrow result.     

New error bounds for linear complementarity problems of Nekrasov matrices and <Emphasis Type="Italic">B</Emphasis>-Nekrasov matrices

New error bounds for the linear complementarity problems are given respectively when the involved matrices are Nekrasov matrices and B-Nekrasov matrices. Numerical examples are given to show that the new bounds are better respectively than those provided by García-Esnaola and Peña (Numer. Algor. 67(3), 655–667, 2014 and Numer. Algor. 72(2), 435–445, 2016) in some cases.     

<Emphasis Type="Italic">l</Emphasis><Subscript>1</Subscript>-<Emphasis Type="Italic">l</Emphasis><Subscript>2</Subscript> regularization of split feasibility problems

Numerous problems in signal processing and imaging, statistical learning and data mining, or computer vision can be formulated as optimization problems which consist in minimizing a sum of convex functions, not necessarily differentiable, possibly composed with linear operators and that in turn can be transformed to split feasibility problems (SFP); see for example Censor and Elfving (Numer. Algorithms 8, 221–239 1994). Each function is typically either a data fidelity term or a regularization term enforcing some properties on the solution; see for example Chaux et al. (SIAM J. Imag. Sci. 2, 730–762 2009) and references therein. In this paper, we are interested in split feasibility problems which can be seen as a general form of Q-Lasso introduced in Alghamdi et al. (2013) that extended the well-known Lasso of Tibshirani (J. R. Stat. Soc. Ser. B 58, 267–288 1996). Q is a closed convex subset of a Euclidean m-space, for some integer m ≥ 1, that can be interpreted as the set of errors within given tolerance level when linear measurements are taken to recover a signal/image via the Lasso. Inspired by recent works by Lou and Yan (2016), Xu (IEEE Trans. Neural Netw. Learn. Syst. 23, 1013–1027 2012), we are interested in a nonconvex regularization of SFP and propose three split algorithms for solving this general case. The first one is based on the DC (difference of convex) algorithm (DCA) introduced by Pham Dinh Tao, the second one is nothing else than the celebrate forward-backward algorithm, and the third one uses a method introduced by Mine and Fukushima. It is worth mentioning that the SFP model a number of applied problems arising from signal/image processing and specially optimization problems for intensity-modulated radiation therapy (IMRT) treatment planning; see for example Censor et al. (Phys. Med. Biol. 51, 2353–2365, 2006).