Common Zeros Research Articles

On the common zeros of quasi-modular forms for Γ+ 0(N) of level N = 1, 2, 3

Abstract In this article, we study common zeros of the iterated derivatives of the Eisenstein series for Γ 0 + ( N ) {\Gamma }_{0}^{+}\left(N) of level N = 1 , 2 N=1,2 , and 3, which are quasi-modular forms. More precisely, we investigate the common zeros of quasi-modular forms and prove that all the zeros of the iterated derivatives of the Eisenstein series θ m E k ( N ) {\theta }^{m}{E}_{k}^{\left(N)} of weight k = 2 , 4 , 6 k=2,4,6 for Γ 0 + ( N ) {\Gamma }_{0}^{+}\left(N) of level N = 2 , 3 N=2,3 are simple by generalizing the results of Meher and Gun-Oesterlé for SL 2 ( Z ) {{\rm{SL}}}_{2}\left({\mathbb{Z}}) .

Coarse nodal count and topological persistence

Courant’s theorem implies that the number of nodal domains of a Laplace eigenfunction is controlled by the corresponding eigenvalue. Over the years, there have been various attempts to find an appropriate generalization of this statement in different directions. We propose a new take on this problem using ideas from topological data analysis. We show that if one counts the nodal domains in a coarse way, basically ignoring small oscillations, Courant’s theorem extends to linear combinations of eigenfunctions, to their products, to other operators, and to higher topological invariants of nodal sets. We also obtain a coarse version of the Bézout estimate for common zeros of linear combinations of eigenfunctions. We show that our results are essentially sharp and that the coarse count is necessary, since these extensions fail in general for the standard count. Our approach combines multiscale polynomial approximation in Sobolev spaces with new results in the theory of persistence modules and barcodes.

Resultants of slice regular polynomials in two quaternionic variables

We introduce a non-commutative resultant, for slice regular polynomials in two quaternionic variables, defined in terms of a suitable Dieudonné determinant. We use this tool to investigate the existence of common zeros and common factors of slice regular polynomials and we give a kinematic interpretation of our results.

On the geometry of zero sets of central quaternionic polynomials

Let R be the ring H[x1,…,xn] of polynomials in n central variables over the real quaternion algebra H, and let I be a left ideal in R. We prove that if p∈R vanishes at all the common zeros of I in Hn with commuting coordinates, then as a slice regular quaternionic function, p vanishes at all common zeros of I in Hn. This confirms a conjecture of Gori, Sarfatti and Vlacci, who settled the two dimensional case.

A comparative analysis of several multivariate zero-inflated and zero-modified models with applications in insurance

. Given that insurance companies often operate across multiple lines of insurance business, where claim frequencies on different lines are often correlated, it often becomes advantageous to employ multivariate count modeling where dependence between lines of business can be included in the modeling. Due to the operation of bonus-malus systems where there is a reward to the insurance policyholder for not claiming, claims data in automobile insurance often exhibits an excess of common zeros, a characteristic known as multivariate zero-inflation. In this article, we propose two approaches to address this feature. The first approach involves utilizing a multivariate zero-inflated model, where we artificially enhance the probability of common zeros based on standard multivariate count distributions. The second approach applies a multivariate zero-modified model, which separately handles the common zeros and the number of claims incurred in each line given that at least one claim occurs. We present several models under these frameworks, along with detailed inference procedures. In the applications section, we conduct a comprehensive comparative analysis of these models using data from an automobile insurance portfolio.

Asymptotic Behavior of Generalized CR−iteration Algorithm and Application to Common Zeros of Accretive Operators

The purpose of this study is to provide a generalized CR−iteration algorithm for finding common fixed points (CFPs) for nonself quasi-nonexpansive mappings (QNEMs) in a uniformly convex Banach space. The suggested algorithm’s convergence analysis is analyzed in uniformly convex Banach spaces. Keywords: Fixed point, CR−iterative algorithm, nonself QNEMs.

Cyclic projection methods for solving the split common zero point problem in Banach spaces

Cyclic projection methods for solving the split common zero point problem in Banach spaces

COMMON ZEROS OF IRREDUCIBLE CHARACTERS

AbstractWe study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $\mathsf {S}_n$ , it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group G, with nonlinear irreducible characters of G as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by three. Lastly, the result for $\mathsf {S}_n$ is applied to prove the nonequivalence of the metrics on permutations induced from faithful irreducible characters of the group.

STRUCTURES OF THE SETS OF BOREL AND PICARD EXCEPTIONAL VECTORS OF ANALYTIC CURVES

Previously, we have described the structure of the sets of Picard and Borel exceptional vectors for a transcendental p-dimensional integer curve with linearly independent components without common zeros. It was proved that the set of Borel exceptional vectors together with the zero vector is a finite union of subspaces of dimension not higher than p-1 of a p-dimensional complex Euclidean space. In addition, the sum of the dimensions of all these subspaces does not exceed p and any pairwise intersection of these subspaces contains only the zero vector. The set of Picard exceptional vectors has the same structure. For an integer curve of non-integer or zero order, the set of Borel exceptional vectors together with the zero vector is a single subspace of dimension no higher than p-1, which is not the case for the set of Picard exceptional vectors of an integer curve of non-integer order. The sufficiency of these results is confirmed to some extent by the corresponding examples. In this paper, we show that the above results on the structure of the set of Borel and Picard exceptional vectors for integer curves can be transferred to the case of analytic curves in the circle. We have also constructed an analytic curve in the circle, which confirms, to some extent, the sufficiency of the results. However, it was not possible to find out whether the corresponding result is transferred to the case of analytic curves of non-integer or zero order.

Variational Inequalities Over the Intersection of Fixed Point Sets of Generalized Demimetric Mappings and Zero Point Sets of Maximal Monotone Mappings

In this paper, we consider a variational inequality problem which is defined over the intersection of the set of common fixed points of a finite family of generalized demimetric mappings and the set of common zero points of a finite family of maximal monotone mappings. We propose an iterative algorithm which combines the hybrid steepest descent method with the inertial technique to solve this variational inequality problem. Strong convergence theorem of the proposed method is established under standard and mild conditions. Moreover, we do not require any prior information regarding the Lipschitz and strongly monotone constants of the mapping. The results of this paper improve and extend several known results in the literature.

New Algorithms for Solving the Split Common Zero Point Problem in Hilbert Space

We introduce new self-adaptive algorithms for solving the split common zero point problem with multiple output sets in Hilbert space. We also apply our main results to solving split feasibility problems with multiple output sets.

A Halpern-type algorithm for a common solution of nonlinear problems in Banach spaces

Abstract In this article, we propose a Halpern-type subgradient extragradient algorithm for solving a common element of the set of solutions of variational inequality problems for continuous monotone mappings and the set of f-fixed points of continuous f-pseudocontractive mappings in reflexive real Banach spaces. In addition, we prove a strong convergence theorem for the sequence generated by the algorithm. As a consequence, we obtain a scheme that converges strongly to a common f-fixed point of continuous f-pseudocontractive mappings and a scheme that converges strongly to a common zero of continuous monotone mappings in Banach spaces. Furthermore, we provide a numerical example to illustrate the implementability of our algorithm.

Proportion of modular forms with transcendental zeros for general levels

Let $\Gamma$ be a congruence subgroup such that $\Gamma_{1}(N)\subset\Gamma\subset\Gamma_{0}(N)$ for some positive integer $N$. For a positive integer $k$, let $M_{k,\mathbf{Z}}(\Gamma)$ be the set of modular forms of weight $k$ on $\Gamma$ with integral Fourier coefficients. Let $R_{k}(\Gamma)$ be the set of common zeros in the upper half plane $\mathbf{H}$ of all the modular forms of weight $k$ on $\Gamma$. In this note, we prove that the density of modular forms in $M_{k,\mathbf{Z}}(\Gamma)$ with an algebraic zero $z \notin R_{k}(\Gamma)$ is zero.

Bergman kernels and equidistribution for sequences of line bundles on Kähler manifolds

Given a sequence of positive Hermitian holomorphic line bundles (Lp,hp) on a Kähler manifold X, we establish the asymptotic expansion of the Bergman kernel of the space of global holomorphic sections of Lp, under a natural convergence assumption on the sequence of curvatures c1(Lp,hp). We then apply this to study the asymptotic distribution of common zeros of random sequences of m-tuples of sections of Lp as p→+∞.

Bragg scattering of flexural-gravity waves by a series of polynyas in the context of blocking dynamics

Flexural-gravity wave scattering due to an array of polynyas is investigated from the perspective of the blocking dynamics. The canonical eigenfunction expansion method is generalized to account for multiple propagating wave modes within blocking frequencies. Bragg scattering occurs due to the presence of multiple gaps in the floating ice sheet, and the number of sub-harmonic peaks in wave reflection becomes one/two less than the number of gaps as the reflection coefficient varies with a change in gap/ice-sheet length. In addition, the amplitudes of harmonic peaks in wave reflection increase with an increase in the number of gaps. The variation of wave reflection with an increase in wavenumber/length of the ice sheet depicts that common zero minima occur for an even number of gaps, while common sub-harmonic maxima occur for an odd number of gaps. The scattering coefficients vary between zero and unity within the blocking frequencies, despite the individual amplitudes of the scattered waves becoming more than unity for certain frequencies. Noticeably, higher amplitudes of the scattered waves are associated with lower energy transfer rates and vice versa. Extrema in wave reflection occur for higher values of frequency within the primary and secondary blocking points. In addition, removable discontinuities are found in the scattering coefficient at the blocking frequencies, whereas a jump discontinuity is observed for certain frequencies within the blocking limits due to the incident wave mode conversion. Moreover, irregularities in the ice sheet's deflection are observed for any frequency within the blocking limit due to the superposition of three propagating wave modes.

Iterative manner involving sunny nonexpansive retractions for nonlinear operators from the perspective of convex programming as applicable to differential problems, image restoration and signal recovery

<abstract><p>In this paper, using sunny nonexpansive retractions which are different from the metric projection in Banach spaces, we develop the $ CR $-iteration algorithm in view of two quasi-nonexpansive nonself mappings and also give the convergence analysis for the proposed method in the setting of uniformly convex Banach spaces. Furthermore, our results can be applied for the purpose of finding common zeros of accretive operators, convexly constrained least square problems and convex minimization problems. Regarding application, some numerical experiments involving real-world problems are provided, with focus on differential problems, image restoration problems and signal recovery problems.</p></abstract>

Hilbert’s Nullstellensatz for analytic trigonometric polynomials

This paper proves such a new Hilbert’s Nullstellensatz for analytic trigonometric polynomials that if {fj}j=1n≥2 are analytic trigonometric polynomials without common zero in the finite complex plane ℂ then there are analytic trigonometric polynomials {gj}j=1n≥2 obeying ∑j=1n≥2fjgj=1 in ℂ, thereby not only strengthening Helmer’s Principal Ideal Theorem for entire functions, but also finding an intrinsic path from Hilbert’s Nullstellensatz for analytic polynomials to Pythagoras’ Identity on ℂ.

A Maple implementation of a modular algorithm for computing the common zeros of a polynomial and a regular chain

One of the core commands in the RegularChains library isTriangularize. The underlying decomposes the solution set of anpolynomial system into geometrically meaningful components representedby regular chains. This algorithm works by repeatedly calling aprocedure, called Intersect, which computes the common zeros of apolynomial p and a regular chain T.As the number of variables of p and T, as well as their degrees,increase, the call Intersect(p, T) becomes more and morecomputationally expensive. It was observed in (C. Chen an M. MorenoMaza, JSC 2012) that when the input polynomial system iszero-dimensional and T is one-dimensional then this cost can besubstantially reduced. The method proposed by the authors is aprobabilistic algorithm based on evaluation and interpolationtechniques. This is the type of method which is typically challengingto implement in a high-level language like Maple's language, as asharp control of computing resources (in particular memory) is needed.In this paper, we report on a successful Maple implementation of thisalgorithm. We take advantage of Maple's modp1 function which offersfast arithmetic for univariate polynomials over a prime field.The method avoids unlucky specialization and the probabilistic aspectonly comes from the fact that non-generic solutions are notcomputed.

New iterative methods for nonlinear operators as concerns convex programming applicable in differential problems, image deblurring, and signal recovering problems

In this paper, using sunny nonexpansive retractions, which are different from the metric projection in Banach spaces, a new type of study regarding the iterative methods in view of two quasi‐nonexpansive nonself mappings is presented. We also give the convergence analysis for the proposed method in the background of uniformly convex Banach spaces. Moreover, we apply our results to find solutions of common zeros of accretive operators, convexly constrained least square problems, and convex minimization problems. Furthermore, we also discuss novel applications of these methods to differential problems, image deblurring, and signal recovering problems.

A strong convergence theorem under a new shrinking projection method for nonlinear mappings in reflexive Banach spaces

In this paper, using a new shrinking projection method, we study the strong convergence theorem for finding a common point of the set of common fixed points of a finite family of Bregman demigeneralized mappings, common zero points of a finite family of Bregman inverse strongly monotone mappings and zero points of maximal monotone mapping in reflexive Banach space. Using this result, we get a new strong convergence theorem in Banach space. Our result extends and improves important recent results presented by many authors.