Clustering of high values in random fields

The asymptotic results that underlie applications of extreme random fields often assume that the variables are located on a regular discrete grid, identified with \(\mathbb {Z}^{2}\), and that they satisfy stationarity and isotropy conditions. Here we extend the existing theory, concerning the asymptotic behavior of the maximum and the extremal index, to non-stationary and anisotropic random fields, defined over discrete subsets of \(\mathbb {R}^{2}\). We show that, under a suitable coordinatewise mixing condition, the maximum may be regarded as the maximum of an approximately independent sequence of submaxima, although there may be high local dependence leading to clustering of high values. Under restrictions on the local path behavior of high values, criteria are given for the existence and value of the spatial extremal index which plays a key role in determining the cluster sizes and quantifying the strength of dependence between exceedances of high levels. The general theory is applied to the class of max-stable random fields, for which the extremal index is obtained as a function of well-known tail dependence measures found in the literature, leading to a simple estimation method for this parameter. The results are illustrated with non-stationary Gaussian and 1-dependent random fields. For the latter, a simulation and estimation study is performed.     

Averaging Fluctuations in Resolvents of Random Band Matrices

We consider a general class of random matrices whose entries are centred random variables, independent up to a symmetry constraint. We establish precise high-probability bounds on the averages of arbitrary monomials in the resolvent matrix entries. Our results generalize the previous results of Erd?s et al. (Ann Probab, arXiv:1103.1919, 2013; Commun Math Phys, arXiv:1103.3869, 2013; J Combin 1(2):15–85, 2011) which constituted a key step in the proof of the local semicircle law with optimal error bound in mean-field random matrix models. Our bounds apply to random band matrices and improve previous estimates from order 2 to order 4 in the cases relevant to applications. In particular, they lead to a proof of the diffusion approximation for the magnitude of the resolvent of random band matrices. This, in turn, implies new delocalization bounds on the eigenvectors. The applications are presented in a separate paper (Erd?s et al., arXiv:1205.5669, 2013).     

The divisible sandpile with heavy-tailed variables

This work deals with the divisible sandpile model when an initial configuration sampled from a heavy-tailed distribution. Extending results of Levine et al. (2015) and Cipriani et al. (2016) we determine sufficient conditions for stabilization and non-stabilization on infinite graphs. We determine furthermore that the scaling limit of the odometer on the torus is an $\alpha$-stable random distribution.     

Spectral estimation on the sphere with needlets: high frequency asymptotics

The angular power spectrum of a stationary random field on the sphere is estimated from the needlet coefficients of a single realization, observed with increasingly fine resolution. The estimator we consider is the one recently used in practice by (Faÿ et al. 2008) to estimate the power spectrum of the Cosmic Microwave Background. The consistency of the estimator, in the asymptotics of high frequencies, is proved for a model with a stationary Gaussian field corrupted by heteroscedastic noise and missing data.     

On a necessary condition for B-spline Gabor frames

In a previous note Gröchenig et al. prove that if g is a continuous function with compact support such that the translates of g form a partition of unity, then g cannot generate a Gabor frame for integer values of the frequency shift parameter b greater than 1 (Gröchenig et al. in IEEE Trans Inform Theory 49:3318–3320, 2003). We give a simpler proof of this result which applies also to windows g which are neither continuous nor with compact support. Our proof is based on a necessary condition for Gabor frames due to Heil and Walnut.     

Correction: Nearly Quartic Mappings in β-Homogeneous F-Spaces [Results Math. 63 (2013) 529–541]

In Chang et al. (Results Math. 63:529–541, 2013), Eshaghi Gordji et al. proved the Hyers-Ulam stability of a quartic functional equation in β-homogeneous F-spaces. In the main step of the proof of Chang et al. (Results Math. 63:529–541, 2013, Theorem 2.2), there is a fatal error. We correct the statement of Chang et al. (Results Math. 63:529–541, 2013, Theorem 2.2).     

An existence result for a class of quasilinear elliptic eigenvalue problems in unbounded domains

We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p–Laplacian operator and subcritical nonlinearities satisfying Ambrosetti–Rabinowitz type conditions. Using Morse theory and a cohomological local splitting as in Degiovanni et al. (Commun Contemp Math 12:475–486, 2010), we prove the existence of a nontrivial weak solution for all (real) values of the eigenvalue parameter. Our result is new even in the semilinear case p = 2 and complements some recent results obtained in Autuori et al. (Adv Anal Equ 18:1–48, 2013).     

Classification of α-Ricci flat graphs with girth at least five

Lin, Lu and Yau classified Ricci flat graphs with girth at least 5 in Lin et al., 2014 [7] and Cushing et al., 2018 [4,5]. In Lin et al., 2014, they defined $\alpha$-Ricci curvature for the definition of Ricci curvature. We will classify $\alpha$-Ricci flat graphs with girth at least 5 for all $\alpha \in (0,1)$.     

Symmetric multiple chessboard complexes and a new theorem of Tverberg type

We prove a new theorem of Tverberg–van Kampen–Flores type, which confirms a conjecture of Blagojevi? et al. about the existence of ‘balanced Tverberg partitions’ (Conjecture 6.6 in [Tverberg plus constraints, Bull. London Math. Soc. 46:953–967 (2014]). The conditions in this theorem are somewhat weaker than in the original conjecture, and we show that the theorem is optimal in the sense that the new (weakened) condition is also necessary. Among the consequences is a positive answer (Theorem 7.2) to the ‘balanced case’ of the question asking whether each admissible r-tuple is Tverberg prescribable (Blagojevi? et al. 2014, Question 6.9).     

Non-Stationary Abstract Friedrichs Systems

Inspired by the results of Ern et al. (Commun Partial Differ Equ 32:317–341, 2007) on the abstract theory for Friedrichs symmetric positive systems, we give the existence and uniqueness result for the initial- (boundary) value problem for the non-stationary abstract Friedrichs system. Despite the absence of the well-posedness result for such systems, there were already attempts for their numerical treatment by Burman et al. (SIAM J Numer Anal 48:2019–2042, 2010) and Bui-Thanh et al. (SIAM J Numer Anal 51:1933–1958, 2013). We use the semigroup theory approach and prove that the operator involved satisfies the conditions of the Hille–Yosida generation theorem. We also address the semilinear problem and apply the new results to a number of examples, such as the symmetric hyperbolic system, the unsteady div–grad problem, and the wave equation. Special attention was paid to the (generalised) unsteady Maxwell system.     

Fourier coefficients of GL(N) automorphic forms in arithmetic progressions

We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GL(N) Maass cusp forms for all \({N \geq 2}\) , satisfy a central limit theorem in a suitable range, generalizing the case N = 2 treated by Fouvry et al. (Commentarii Math Helvetici, 2014). Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums.     

Continuity of the solution mapping to parametric generalized vector equilibrium problems

In this paper, two kinds of parametric generalized vector equilibrium problems in normed spaces are studied. The sufficient conditions for the continuity of the solution mappings to the two kinds of parametric generalized vector equilibrium problems are established under suitable conditions. The results presented in this paper extend and improve some main results in Chen and Gong (Pac J Optim 3:511–520, 2010), Chen and Li (Pac J Optim 6:141–152, 2010), Chen et al. (J Glob Optim 45:309–318, 2009), Cheng and Zhu (J Glob Optim 32:543–550, 2005), Gong (J Optim Theory Appl 139:35–46, 2008), Li and Fang (J Optim Theory Appl 147:507–515, 2010), Li et al. (Bull Aust Math Soc 81:85–95, 2010) and Peng et al. (J Optim Theory Appl 152(1):256–264, 2011).     

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

“Arithmetic random waves” are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman in Annales de l’Insitute Henri Poincaré 9(1):109–130, 2008; Krishnapur et al. in Annals of Mathematics (2) 177(2):699–737, 2013). In this paper we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument has two main ingredients. An explicit derivation of the Wiener–Itô chaos expansion for the nodal length shows that it is dominated by its 4th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes). The rest of the argument relies on the precise analysis of the fourth order chaotic component.     

An iterative algorithm for a general system of variational inequalities and fixed point problems in q-uniformly smooth Banach spaces

In this paper, we introduce a new iterative algorithm for finding a common element of the set of common fixed points of an infinite family of notself strict pseudocontractions and the set of solutions of a general variational inequality problem for finite inverse-strongly accretive mappings in q-uniformly smooth Banach space. We obtain some strong convergence theorems under suitable conditions. Our results improve and extend the recent results announced by Qin et al. (J Comput Appl Math 233:231–240, 2009), Yao et al. (Acta Appl Math 110:1211–1244, 2010) and many others.     

An improved bit parallel exact maximum clique algorithm

This paper describes new improvements for BB-MaxClique (San Segundo et al. in Comput Oper Resour 38(2):571–581, 2011), a leading maximum clique algorithm which uses bit strings to efficiently compute basic operations during search by bit masking. Improvements include a recently described recoloring strategy in Tomita et al. (Proceedings of the 4th International Workshop on Algorithms and Computation. Lecture Notes in Computer Science, vol 5942. Springer, Berlin, pp 191–203, 2010), which is now integrated in the bit string framework, as well as different optimization strategies for fast bit scanning. Reported results over DIMACS and random graphs show that the new variants improve over previous BB-MaxClique for a vast majority of cases. It is also established that recoloring is mainly useful for graphs with high densities.     

The Jacobian Conjecture,a Reduction of the Degree to the Quadratic Case

The Jacobian Conjecture states that any locally invertible polynomial system in \({{\mathbb{C}}^n}\) is globally invertible with polynomial inverse. Bass et al. (Bull Am Math Soc 7(2):287–330, 1982) proved a reduction theorem stating that the conjecture is true for any degree of the polynomial system if it is true in degree three. This degree reduction is obtained with the price of increasing the dimension \({n}\). We prove here a theorem concerning partial elimination of variables, which implies a reduction of the generic case to the quadratic one. The price to pay is the introduction of a supplementary parameter \({0 \leq n' \leq n}\), parameter which represents the dimension of a linear subspace where some particular conditions on the system must hold. We first give a purely algebraic proof of this reduction result and we then expose a distinct proof, in a Quantum Field Theoretical formulation, using the intermediate field method.     

Exponentially Many Monochromatic <Emphasis Type="Italic">n</Emphasis>-Matchings in <Emphasis Type="Italic">K</Emphasis><Subscript>3<Emphasis Type="Italic">n</Emphasis>-1</Subscript>

Erdös et al and Gerencsér et al had shown that in any 2-edge-coloring of K _3n-1, there is a n-matching containing edges with the same color(we call such matching monochromatic matching). In this paper we show that for any 2-edge-coloring of K _3n-1 there exists a monochromatic subgraph H of K _3n-1 which contains exponentially many monochromatic n-matchings.     

On extremal indices greater than one for a scheme of series

The classic extremal index θ is an important parameter of asymptotic behavior of maxima of stationary random sequences. For applications, however, it is interesting to investigatemore complex structures. Recently, the extremal index was generalized to a scheme of series of random length. If the exact extremal index does not exist, then we consider partial indices. In contrast to the classic index, partial indices can be greater than one. In this paper, we consider a new model, where left and right partial indices can be greater than one and equal to each other, although the exact index does not exist.     

The Existence of Fixed Points via the Measure of Noncompactness and its Application to Functional-Integral Equations

In the present paper, using the concept of measure of non-compactness, we introduce the concept of a new contraction on a Banach space and obtain few generalizations of Darbo’s fixed-point theorem and extend some recent results of (Aghajani et al., J. Comput. Appl. Math. 260:68–77, 2014) and (Aghajani et al., Bull. Belg. Math. Soc. Simon Stevin 20:345–358, 2013). Also we show the applicability of obtained results to the theory of integral equations. A concrete example illustrating the mentioned applicability is also included.