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.     

An inexact modified relaxed splitting preconditioner for the generalized saddle point problems from the incompressible Navier-Stokes equations

Based on the modified relaxed splitting (MRS) preconditioner proposed by Fan and Zhu (Appl. Math. Lett. 55, 18–26 2016), an inexact modified relaxed splitting (IMRS) preconditioner is proposed for the generalized saddle point problems arising from the incompressible Navier-Stokes equations. The eigenvalues and eigenvectors of the preconditioned matrix are analyzed, and the convergence property of the corresponding iteration method is also discussed. Numerical experiments are presented to show the effectiveness of the proposed preconditioner when it is used to accelerate the convergence rate of Krylov subspace methods such as GMRES.     

An hp finite element method for 4th order singularly perturbed problems

We present the analysis for the hp finite element approximation of the solution to singularly perturbed fourth order problems, using a balanced norm. In Panaseti et al. (2016) it was shown that the hp version of the Finite Element Method (FEM) on the so-called Spectral Boundary Layer Mesh yields robust exponential convergence when the error is measured in the natural energy norm associated with the problem. In the present article we sharpen the result by showing that the same hp-FEM on the Spectral Boundary Layer Mesh gives robust exponential convergence in a stronger, more balanced norm. As a corollary we also get robust exponential convergence in the maximum norm. The analysis is based on the ideas in Roos and Franz (Calcolo 51, 423–440, 2014) and Roos and Schopf (ZAMM 95, 551–565, 2015) and the recent results in Melenk and Xenophontos (2016). Numerical examples illustrating the theory are also presented.     

The Reich–Zaslavski property and fixed points of non-self multivalued mappings

Let (X, d) be a metric space, Y be a nonempty subset of X, and let \(T:Y \rightarrow P(X)\) be a non-self multivalued mapping. In this paper, by a new technique we study the fixed point theory of multivalued mappings under the assumption of the existence of a bounded sequence \((x_n)_n\) in Y such that \(T^nx_n\subseteq Y,\) for each \(n \in \mathbb {N}\). Our main result generalizes fixed point theorems due to Matkowski (Diss. Math. 127, 1975), W?grzyk (Diss. Math. (Rozprawy Mat.) 201, 1982), Reich and Zaslavski (Fixed Point Theory 8:303–307, 2007), Petru?el et al. (Set-Valued Var. Anal. 23:223–237, 2015) and provides a solution to the problems posed in Petru?el et al. (Set-Valued Var. Anal. 23:223–237, 2015) and Rus and ?erban (Miskolc Math. Notes 17:1021–1031, 2016).     

Spectral transfer morphisms for unipotent affine Hecke algebras

We classify the spectral transfer morphisms (cf. Opdam in Adv Math 286:912–957, 2016) between affine Hecke algebras associated to the unipotent types of the various inner forms of an unramified absolutely simple algebraic group G defined over a non-archimedean local field k. This turns out to characterize Lusztig’s classification (Lusztig in Int Math Res Not 11:517–589, 1995; in Represent Theory 6:243–289, 2002) of unipotent characters of G in terms of the Plancherel measure, up to diagram automorphisms. As an application of these results, the spectral correspondences associated with such morphisms (Opdam 2016), and some results of Ciubotaru, Kato and Kato [CKK] (also see Ciubotaru and Opdam in A uniform classification of the discrete series representations of affine Hecke algebras. arXiv:1510.07274) we prove a conjecture of Hiraga, Ichino and Ikeda [HII] on formal degrees and adjoint gamma factors in the special case of unipotent discrete series characters of inner forms of unramified simple groups of adjoint type defined over k.     

Semi-convergence analysis of preconditioned deteriorated PSS iteration method for singular saddle point problems

In this paper, we propose a two-parameter preconditioned variant of the deteriorated PSS iteration method (J. Comput. Appl. Math., 273, 41–60 (2015)) for solving singular saddle point problems. Semi-convergence analysis shows that the new iteration method is convergent unconditionally. The new iteration method can also be regarded as a preconditioner to accelerate the convergence of Krylov subspace methods. Eigenvalue distribution of the corresponding preconditioned matrix is presented, which is instructive for the Krylov subspace acceleration. Note that, when the leading block of the saddle point matrix is symmetric, the new iteration method will reduce to the preconditioned accelerated HSS iteration method (Numer. Algor., 63 (3), 521–535 2013), the semi-convergence conditions of which can be simplified by the results in this paper. To further improve the effectiveness of the new iteration method, a relaxed variant is given, which has much better convergence and spectral properties. Numerical experiments are presented to investigate the performance of the new iteration methods for solving singular saddle point problems.     

Hybridization of accelerated gradient descent method

We present a gradient descent algorithm with a line search procedure for solving unconstrained optimization problems which is defined as a result of applying Picard-Mann hybrid iterative process on accelerated gradient descent S M method described in Stanimirovi? and Miladinovi? (Numer. Algor. 54, 503–520, 2010). Using merged features of both analyzed models, we show that new accelerated gradient descent model converges linearly and faster then the starting S M method which is confirmed trough displayed numerical test results. Three main properties are tested: number of iterations, CPU time and number of function evaluations. The efficiency of the proposed iteration is examined for the several values of the correction parameter introduced in Khan (2013).     

<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).     

The <Emphasis Type="Italic">p</Emphasis>-Affine Capacity Redux

Continuing from Xiao (Adv Math 268:906–914, 2015; J Geom Anal 26:947–966, 2016), this note is devoted to the discovery of new geometric properties of the so-called \([1,n)\ni p\)-affine capacity in the Euclidean n-space.     

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     

Generalized growth and best approximation of entire functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-norm in several complex variables

The paper deals with the characterization of generalized order and generalized type of entire functions in several complex variables in terms of the coefficients of the development with respect to the sequence of extremal polynomials and the best L ^p -approximation and interpolation errors, 0 < p ≤ ∞, on a compact set K with respect to the set

$K_r = \left\{z \in \mathbb{C}^n, {\rm exp} (V_K (z)) \leq r\right\}$

where V _K is the Siciak extremal function of a L-regular compact set K or V _K is the pluricomplex Green function with a pole at infinity. It has been noticed that in the study of growth of entire functions, the set K _r has not been used so extensively in comparison to disk. Our results apply satisfactorily for slow growth in \({\mathbb{C}^n}\) , replacing the circle \({\{z \in \mathbb{C}; |z| = r\}}\) by the set K _r and improve and extend various results of Harfaoui (Int J Maths Math Sci 2010:1–15, 2010), Seremeta (Am Math Soc Transl 88(2):291–301, 1970), Shah (J Approx Theory 19:315–324, 1977) and Vakarchuk and Zhir (Ukr Math J 54(9):1393–1401, 2002).

     

O $$\varvec{}$$-orm preservers ad the Aleksadrov coservative $$\varvec{}$$-distace problem

The goal of this paper is to point out that the results obtained in the recent papers (Chen and Song in Nonlinear Anal 72:1895–1901, 2010; Chu in J Math Anal Appl 327:1041–1045, 2007; Chu et al. in Nonlinear Anal 59:1001–1011, 2004a, J. Math Anal Appl 289:666–672, 2004b) can be seriously strengthened in the sense that we can significantly relax the assumptions of the main results so that we still get the same conclusions. In order to do this first, we prove that for \(n \ge 3\) any transformation which preserves the n-norm of any n vectors is automatically plus-minus linear. This will give a re-proof of the well-known Mazur–Ulam-type result that every n-isometry is automatically affine (\(n \ge 2\)) which was proven in several papers, e.g. in Chu et al. (Nonlinear Anal 70:1068–1074, 2009). Second, following the work of Rassias and ?emrl (Proc Am Math Soc 118:919–925, 1993), we provide the solution of a natural Aleksandrov-type problem in n-normed spaces, namely, we show that every surjective transformation which preserves the unit n-distance in both directions (\(n\ge 2\)) is automatically an n-isometry.     

Congruences between word length statistics for the finitary alternating and symmetric groups

Bacher and de la Harpe (arxiv:1603.07943, 2016) study conjugacy growth series of infinite permutation groups and their relationships with p(n), the partition function, and \(p(n)_\mathbf{e }\), a generalized partition function. They prove identities for the conjugacy growth series of the finitary symmetric group and the finitary alternating group. The group theory due to Bacher and de la Harpe (arxiv:1603.07943, 2016) also motivates an investigation into congruence relationships between the finitary symmetric group and the finitary alternating group. Using the Ramanujan congruences for the partition function p(n) and Atkin’s generalization to the k-colored partition function \(p_{k}(n)\), we prove the existence of congruence relations between these two series modulo arbitrary powers of 5 and 7, which we systematically describe. Furthermore, we prove that such relationships exist modulo powers of all primes \(\ell \ge 5\).     

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.     

Every 3-connected $$\{K_{1,3},N_{1,2,3}\}$$-free graph is Hamilton-connected

A graph G is \(\{X,Y\}\)-free if it contains neither X nor Y as an induced subgraph. Pairs of connected graphs X, Y such that every 3-connected \(\{X,Y\}\)-free graph is Hamilton-connected have been investigated recently in (2002, 2000, 2012). In this paper, it is shown that every 3-connected \(\{K_{1,3},N_{1,2,3}\}\)-free graph is Hamilton-connected, where \(N_{1,2,3}\) is the graph obtained by identifying end vertices of three disjoint paths of lengths 1, 2, 3 to the vertices of a triangle.     

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.     

A brief note on the coarea formula

In this note we consider a special case of the famous Coarea Formula whose initial proof (for functions from any Riemannian manifold of dimension 2 into \({\mathbb {R}}\)) is due to Kronrod (Uspechi Matem Nauk 5(1):24–134, 1950) and whose general proof (for Lipschitz maps between two Riemannian manifolds of dimensions n and p) is due to Federer (Am Math Soc 93:418–491, 1959). See also Maly et al. (Trans Am Math Soc 355(2):477–492, 2002), Fleming and Rishel (Arch Math 11(1):218–222, 1960) and references therein for further generalizations to Sobolev mappings and BV functions respectively. We propose two counterexamples which prove that the coarea formula that we can find in many references (for example Bérard (Spectral geometry: direct and inverse problems, Springer, 1987), Berger et al. (Le Spectre d’une Variété Riemannienne, Springer, 1971) and Gallot (Astérisque 163(164):31–91, 1988), is not valid when applied to \(C^\infty \) functions. The gap appears only for the non generic set of non Morse functions.     

Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence

The Richardson variety X _α ^γ in the Grassmannian is defined to be the intersection of the Schubert variety X ^γ and opposite Schubert variety X _α . We give an explicit Gröbner basis for the ideal of the tangent cone at any T-fixed point of X _α ^γ , thus generalizing a result of Kodiyalam-Raghavan (J. Algebra 270(1):28–54, 2003) and Kreiman-Lakshmibai (Algebra, Arithmetic and Geometry with Applications, 2004). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the bounded RSK (BRSK). We use the Gröbner basis result to deduce a formula which computes the multiplicity of X _α ^γ at any T-fixed point by counting families of nonintersecting lattice paths, thus generalizing a result first proved by Krattenthaler (Sém. Lothar. Comb. 45:B45c, 2000/2001; J. Algebr. Comb. 22:273–288, 2005).     

On modified Gordon-Wixom interpolation schemes and their applications to nonlinear and exterior domain problems

We introduce and study extensions and modifications of the Gordon-Wixom transfinite barycentric interpolation scheme (Gordon and Wixom, SIAM J. Numer. Anal. 11(5), 909–933, 1974). We demonstrate that the modified Gordon-Wixom scheme proposed in Belyaev and Fayolle (Comput. Graph. 51, 74–80, 2015) reproduces harmonic quadratic polynomials in convex domains. We adapt the scheme for dealing with the exterior of a bounded domain and for the exterior of a disk, where we demonstrate that our interpolation formula reproduces harmonic functions. Finally, we show how to adapt the Gordon-Wixom approach for approximating p-harmonic functions and to derive computationally efficient approximations of the solutions to boundary value problems involving the p-Laplacian.