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.     

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.     

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.     

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.     

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.     

A Link Between the Log-Sobolev Inequality and Lyapunov Condition

We give an alternative look at the log-Sobolev inequality (LSI in short) for log-concave measures by semigroup tools. The similar idea yields a heat flow proof of LSI under some quadratic Lyapunov condition for symmetric diffusions on Riemannian manifolds provided the Bakry-Emery’s curvature is bounded from below. Let’s mention that, the general ?-Lyapunov conditions were introduced by Cattiaux et al. (J. Funct. Anal. 256(6), 1821–1841 2009) to study functional inequalities, and the above result on LSI was first proved subject to ?(?) = d²(?,x₀) by Cattiaux et al. (Proba. Theory Relat. Fields 148(1–2), 285–304 2010) through a combination of detective L² transportation-information inequality W₂I and the HWI inequality of Otto-Villani. Next, we assert a converse implication that the Lyapunov condition can be derived from LSI, which means their equivalence in the above setting.     

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.     

A generalization of the maximal-spacings in several dimensions and a convexity test.

The notion of maximal-spacing in several dimensions was introduced and studied by Deheuvels (Probab. Theory Related Fields 64(4), 411–424, 1983), for data uniformly distributed on the unit cube. Later on, Janson (Ann. Prob. 15, 274–280, 1987) extended the results to data uniformly distributed on any bounded set, and obtained a very fine result, namely, he derived the asymptotic distribution of different maximal-spacings notions. These results have been very useful in many statistical applications. We extend Janson’s results to the case where the data are generated from a Hölder continuous density that is bounded from below and whose support is bounded. As an application, we develop a convexity test for the support of a distribution.     

Central Limit Theorem for Lipschitz–Killing Curvatures of Excursion Sets of Gaussian Random Fields

Our interest in this paper is to explore limit theorems for various geometric functionals of excursion sets of isotropic Gaussian random fields. In the past, asymptotics of nonlinear functionals of Gaussian random fields have been studied [see Berman (Sojourns and extremes of stochastic processes, Wadsworth & Brooks, Monterey, 1991), Kratz and León (Extremes 3(1):57–86, 2000), Kratz and León (J Theor Probab 14(3):639–672, 2001), Meshenmoser and Shashkin (Stat Probab Lett 81(6):642–646, 2011), Pham (Stoch Proc Appl 123(6):2158–2174, 2013), Spodarev (Chapter in modern stochastics and applications, volume 90 of the series Springer optimization and its applications, pp 221–241, 2013) for a sample of works in such settings], the most recent addition being (Adler and Naitzat in Stoch Proc Appl 2016; Estrade and León in Ann Probab 2016) where a central limit theorem (CLT) for Euler integral and Euler–Poincaré characteristic, respectively, of the excursions set of a Gaussian random field is proven under some conditions. In this paper, we obtain a CLT for some global geometric functionals, called the Lipschitz–Killing curvatures of excursion sets of Gaussian random fields, in an appropriate setting.     

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.     

Multivariate Higher-Degree Stochastic Increasing Convexity

Building on the seminal work by Shaked and Shanthikumar (Adv Appl Probab 20:427–446, 1988a; Stoch Process Appl 27:1–20, 1988b), Denuit et al. (Eng Inf Sci 13:275–291, 1999; Methodol Comput Appl Probab 2:231–254, 2000; 2001) studied the stochastic s-increasing convexity properties of standard parametric families of distributions. However, the analysis is restricted there to a single parameter. As many standard families of distributions involve several parameters, multivariate higher-order stochastic convexity properties also deserve consideration for applications. This is precisely the topic of the present paper, devoted to stochastic \((s_1,s_2,\ldots ,s_d)\)-increasing convexity of distribution families indexed by a vector \((\theta _1,\theta _2,\ldots ,\theta _d)\) of parameters. This approach accounts for possible correlation in multivariate mixture models.     

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.     

The truncated Euler–Maruyama method for stochastic differential delay equations

The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao (Appl. Math. Comput. 217, 5512–5524 2011), and the theory there showed that the Euler–Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in L ^p ) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao (J. Comput. Appl. Math. 290, 370–384 2015) to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.     

Reflected Brownian Motion in a Convex Polyhedral Cone: Tail Estimates for the Stationary Distribution

Consider a multidimensional obliquely reflected Brownian motion in the positive orthant, or, more generally, in a convex polyhedral cone. We find sufficient conditions for existence of a stationary distribution and convergence to this distribution at an exponential rate, as time goes to infinity, complementing the results of Dupuis and Williams (Ann Probab 22(2):680–702, 1994) and Atar et al. (Ann Probab 29(2):979–1000, 2001). We also prove that certain exponential moments for this distribution are finite, thus providing a tail estimate for this distribution. Finally, we apply these results to systems of rank-based competing Brownian particles, introduced in Banner et al. (Ann Appl Probab 15(4):2296–2330, 2005).     

Exact Simulation for Discrete Time Spin Systems and Unilateral Fields

In this paper we generalize the technique presented by Häggström and Steif (Comb. Probab. Comput. 9:425–439, 2000) for the exact simulation of finite sections of infinite-volume Gibbs random fields, to a more general class of discrete time nearest neighbour spin systems. The main role is played by an auxiliary binary field, which indicates the sampling region. Percolation bounds can be used to prove that the algorithm terminates a.s. In the simplest case this field is Bernoulli; however blocking techniques can be used that destroy the independence property but extend the validity of the algorithm. Finally, the connection with stationary unilateral fields in the plane considered by Pickard (Adv. Appl. Probab. 12:655–671, 1980) and Galbraith and Walley (J. Appl. Probab. 19:332–343, 1982) is discussed.     

Efficient Computation of First Passage Times in Kou’s Jump-diffusion Model

S. G. Kou and H. Wang [First passage times of a jump diffusion process, Adv. Appl. Probab. 35 (2003) 504–531] give expressions of both the real Laplace transform of the distribution of first passage time and the real Laplace transform of the joint distribution of the first passage time and the running maxima of a jump-diffusion model called Kou model. These authors invert the former Laplace transform by using Gaver-Stehfest algorithm, and for the latter they need a large computing time with an algebra computer system. In the present paper, we give a much simpler expression of the Laplace transform of the joint distribution, and we also show, using Complex Analysis techniques, that both Laplace transforms can be extended to the complex plane. Hence, we can use inversion methods based on the complex inversion formula or Bromwich integral which are very efficent. The improvement in the computing times and accuracy is remarkable.     

No hexavalent half-arc-transitive graphs of order twice a prime square exist

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set and edge set, but not arc set. Let p be a prime. Wang and Feng (Discrete Math. 310 (2010) 1721–1724) proved that there exists no tetravalent half-arc-transitive graphs of order \(2p^2\). In this paper, we extend this result to prove that no hexavalent half-arc-transitive graphs of order \(2p^2\) exist.     

Improved local convergence analysis of the Gauss–Newton method under a majorant condition

We present a local convergence analysis of Gauss-Newton method for solving nonlinear least square problems. Using more precise majorant conditions than in earlier studies such as Chen (Comput Optim Appl 40:97–118, 2008), Chen and Li (Appl Math Comput 170:686–705, 2005), Chen and Li (Appl Math Comput 324:1381–1394, 2006), Ferreira (J Comput Appl Math 235:1515–1522, 2011), Ferreira and Gonçalves (Comput Optim Appl 48:1–21, 2011), Ferreira and Gonçalves (J Complex 27(1):111–125, 2011), Li et al. (J Complex 26:268–295, 2010), Li et al. (Comput Optim Appl 47:1057–1067, 2004), Proinov (J Complex 25:38–62, 2009), Ewing, Gross, Martin (eds.) (The merging of disciplines: new directions in pure, applied and computational mathematics 185–196, 1986), Traup (Iterative methods for the solution of equations, 1964), Wang (J Numer Anal 20:123–134, 2000), we provide a larger radius of convergence; tighter error estimates on the distances involved and a clearer relationship between the majorant function and the associated least squares problem. Moreover, these advantages are obtained under the same computational cost.     

Local convergence and the dynamics of a two-point four parameter Jarratt-like method under weak conditions

We present a local convergence analysis of a two-point four parameter Jarratt-like method of high convergence order in order to approximate a locally unique solution of a nonlinear equation. In contrast to earlier studies such us (Amat et al. Aequat. Math. 69(3), 212–223 2015; Amat et al. J. Math. Anal. Appl. 366(3), 24–32 2010; Behl, R. 2013; Bruns and Bailey Chem. Eng. Sci. 32, 257–264 1977; Candela and Marquina. Computing 44, 169–184 1990; Candela and Marquina. Computing 45(4), 355–367 1990; Chun. Appl. Math. Comput. 190(2), 1432–1437 2007; Cordero and Torregrosa. Appl. Math. Comput. 190, 686–698 2007; Deghan. Comput. Appl Math. 29(1), 19–30 2010; Deghan. Comput. Math. Math. Phys. 51(4), 513–519 2011; Deghan and Masoud. Eng. Comput. 29(4), 356–365 15; Cordero and Torregrosa. Appl. Math. Comput. 190, 686–698 2012; Deghan and Masoud. Eng. Comput. 29(4), 356–365 2012; Ezquerro and Hernández. Appl. Math. Optim. 41(2), 227–236 2000; Ezquerro and Hernández. BIT Numer. Math. 49, 325–342 2009; Ezquerro and Hernández. J. Math. Anal. Appl. 303, 591–601 2005; Gutiérrez and Hernández. Comput. Math. Appl. 36(7), 1–8 1998; Ganesh and Joshi. IMA J. Numer. Anal. 11, 21–31 1991; González-Crespo et al. Expert Syst. Appl. 40(18), 7381–7390 2013; Hernández. Comput. Math. Appl. 41(3-4), 433–455 2001; Hernández and Salanova. Southwest J. Pure Appl. Math. 1, 29–40 1999; Jarratt. Math. Comput. 20(95), 434–437 1966; Kou and Li. Appl. Math. Comput. 189, 1816–1821 2007; Kou and Wang. Numer. Algor. 60, 369–390 2012; Lorenzo et al. Int. J. Interact. Multimed. Artif. Intell. 1(3), 60–66 2010; Magreñán. Appl. Math. Comput. 233, 29–38 2014; Magreñán. Appl. Math. Comput. 248, 215–224 2014; Parhi and Gupta. J. Comput. Appl. Math. 206(2), 873–887 2007; Rall 1979; Ren et al. Numer. Algor. 52(4), 585–603 2009; Rheinboldt Pol. Acad. Sci. Banach Ctr. Publ. 3, 129–142 1978; Sicilia et al. J. Comput. Appl. Math. 291, 468–477 2016; Traub 1964; Wang et al. Numer. Algor. 57, 441–456 2011) using hypotheses up to the fifth derivative, our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. The dynamics of the family for choices of the parameters such that it is optimal is also shown. Numerical examples are also provided in this study     

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).