Zeros Of Function Research Articles

Zeros of the Degree Zero Functions From the Extended Selberg Class

Zeros of the Degree Zero Functions From the Extended Selberg Class

Interconnected renormalization of Hubbard bands and Green's function zeros in Mott insulators induced by strong magnetic fluctuations

Interconnected renormalization of Hubbard bands and Green's function zeros in Mott insulators induced by strong magnetic fluctuations

Topological Green's Function Zeros in an Exactly Solved Model and Beyond.

The interplay of topological electronic band structures and strong interparticle interactions provides a promising path towards the constructive design of robust, long-range entangled many-body systems. As a prototype for such systems, we here study an exactly integrable, local model for a fractionalized topological insulator. Using a controlled perturbation theory about this limit, we demonstrate the existence of topological bands of zeros in the exact fermionic Green's function and show that in this model they do affect the topological invariant of the system, but not the quantized transport response. Close to (but prior to) the Higgs transition signaling the breakdown of fractionalization, the topological bands of zeros acquire a finite "lifetime." We also discuss the appearance of edge states and edge zeros at real space domain walls separating different phases of the system. This model provides a fertile ground for controlled studies of the phenomenology of Green's function zeros and the underlying exactly solvable lattice gauge theory illustrates the synergetic cross pollination between solid-state theory, high-energy physics, and quantum information science.

Exciton condensation driven by bound states of Green's function zeros

Exciton condensation driven by bound states of Green's function zeros

Electronic properties, correlated topology, and Green's function zeros

There is extensive current interest in electronic topology in correlated settings. In strongly correlated systems, contours of Green's function zeros may develop in frequency-momentum space, and their role in correlated topology has increasingly been recognized. However, whether and how the zeros contribute to electronic properties is a matter of uncertainty. Here we address the issue in an exactly solvable model for a Mott insulator. We show that the Green's function zeros contribute to several physically measurable correlation functions in a way that does not run into inconsistencies. In particular, the physical properties remain robust to chemical potential variations up to the Mott gap, as it should be based on general considerations. Our work sets the stage for further understandings of the rich interplay among topology, symmetry, and strong correlations. Published by the American Physical Society 2024

Symmetry constraints and spectral crossing in a Mott insulator with Green's function zeros

Lattice symmetries are central to the characterization of electronic topology. Recently, it was shown that Green's function eigenvectors form a representation of the space group. This formulation has allowed the identification of gapless topological states even when quasiparticles are absent. Here we demonstrate the profundity of the framework in the extreme case, when interactions lead to a Mott insulator, through a solvable model with long-range interactions. We find that both Mott poles and zeros are subject to the symmetry constraints, and relate the symmetry-enforced spectral crossings to degeneracies of the original noninteracting eigenstates. Our results lead to new understandings of topological quantum materials and highlight the utility of interacting Green's functions toward their symmetry-based design. Published by the American Physical Society 2024

Learn2Extend: Extending Sequences by Retaining their Statistical Properties with Mixture Models

This paper addresses the challenge of extending general finite sequences of real numbers within a subinterval of the real line, maintaining their inherent statistical properties by employing machine learning. Our focus lies on preserving the gap distribution and pair correlation function of these point sets. Leveraging advancements in deep learning applied to point processes, this paper explores the use of an auto-regressive Sequence Extension Mixture Model (SEMM) for extending finite sequences, by estimating directly the conditional density, instead of the intensity function. We perform comparative experiments on multiple types of point processes, including Poisson, locally attractive, and locally repelling sequences, and we perform a case study on the prediction of Riemann ζ function zeroes. The results indicate that the proposed mixture model outperforms traditional neural network architectures in sequence extension with the retention of statistical properties. Given this motivation, we showcase the capabilities of a mixture model to extend sequences, maintaining specific statistical properties, i.e. the gap distribution, and pair correlation indicators.

MULTIPLE SOLUTIONS FOR -LAPLACIAN EQUATIONS WITH NONLINEARITY SUBLINEAR AT ZERO

AbstractWe consider the Dirichlet problem for $p(x)$ -Laplacian equations of the form $$ \begin{align*} -\Delta_{p(x)}u+b(x)\vert u\vert ^{p(x)-2}u=f(x,u),\quad u\in W_{0}^{1,p(x)}(\Omega). \end{align*} $$ The odd nonlinearity $f(x,u)$ is $p(x)$ -sublinear at $u=0$ but the related limit need not be uniform for $x\in \Omega $ . Except being subcritical, no additional assumption is imposed on $f(x,u)$ for $|u|$ large. By applying Clark’s theorem and a truncation method, we obtain a sequence of solutions with negative energy and approaching the zero function $u=0$ .

On subspaces of Orlicz spaces spanned by independent copies of a mean zero function

We study the subspaces of the Orlicz spaces $L_M$ spanned by independent copies $f_k$, $k=1,2,…$, of a function $f\in L_M$, $\int_0^1 f(t) dt=0$. Any such a subspace $H$ is isomorphic to some Orlicz sequence space $\ell_\psi$. In terms of dilations of the function $f$, a description of strongly embedded subspaces of this type is obtained, and conditions guaranteeing that the unit ball of such a subspace consists of functions with equicontinuous norms in $L_M$ are found. In particular, we prove that there is a wide class of Orlicz spaces $L_M$ (containing the $L^p$-spaces, $1\le p< 2$), for which each of the above properties of $H$ holds if and only if the Matuszewska-Orlicz indices of the functions $M$ and $\psi$ satisfy $\alpha_\psi^0>\beta_M^\infty$.

The Bloch–Nevanlinna Problem Under Lineability

The original version of the celebrated Bloch–Nevanlinna problem asked whether there exists or not a holomorphic function in the unit disc of bounded characteristic whose derivative is not of bounded characteristic. This problem has been solved in the affirmative by a number of mathematicians. Starting from a result on topological genericity of this class of functions due to Hahn, our work intends—under an algebraic as well as a topological point of view—to contribute to this topic. Specifically, the family of solutions of the Bloch–Nevanlinna problem is proved to be residual in appropriate topological spaces, and it contains, except for the zero function, dense maximal dimensional vector subspaces, large linear algebras and infinite-dimensional Banach spaces.

Zeros Under Unitary Weighted Composition Operators in the Hardy and Bergman Spaces

Let mathbb {D} denote the unit disk of the complex plane and let mathcal {A}^2(mathbb {D}) be the Bergman space that consists of those analytic functions on mathbb {D} that are of integrable square modulus with respect to the normalized area measure. Let varphi : mathbb {D} rightarrow mathbb {D} be an automorphism of the disk and consider C_varphi f=f circ varphi the operator defined from mathcal {A}^2(mathbb {D}) onto itself. Consider the unitary operator U_varphi f = varphi ^prime f circ varphi . Then if f in mathcal {A}^2(mathbb {D}) is even and U_varphi f is odd, then f is the zero function. The same is true if f in mathcal {A}^2(mathbb {D}) is odd and U_varphi f is even. Similar results can be proved for the Hardy space of the unit disk, that is, the space of analytic functions on mathbb {D}, whose Taylor coefficients are of summable square modulus. The result remains true for the Dirichlet space, that is, the space of analytic functions on mathbb {D}, whose derivatives are in mathcal {A}^2(mathbb {D}).

Local Orthogonal Moments for Local Features.

By introducing parameters with local information, several types of orthogonal moments have recently been developed for the extraction of local features in an image. But with the existing orthogonal moments, local features cannot be well-controlled with these parameters. The reason lies in that zeros distribution of these moments' basis function cannot be well-adjusted by the introduced parameters. To overcome this obstacle, a new framework, transformed orthogonal moment (TOM), is set up. Most existing continuous orthogonal moments such as Zernike moments, fractional-order orthogonal moments (FOOMs), etc. are all special cases of TOM. To control the basis function's zeros distribution, a novel local constructor is designed, and local orthogonal moment (LOM) is proposed. Zeros distribution of LOM's basis function can be adjusted with parameters introduced by the designed local constructor. Consequently, locations, where local features extracted from by LOM, are more accurate than those by FOOMs. In comparison with Krawtchouk moments and Hahn moments etc., the range, where local features are extracted from by LOM, is order insensitive. Experimental results demonstrate that LOM can be utilized to extract local features in an image.

Character Networks, the Zero Function, and the Lost Character: Solving Three Anomalies in Plot Genotype Theory

Character Networks, the Zero Function, and the Lost Character: Solving Three Anomalies in Plot Genotype Theory

The stickiness property for antisymmetric nonlocal minimal graphs

<p style='text-indent:20px;'>We show that arbitrarily small antisymmetric perturbations of the zero function are sufficient to produce the stickiness phenomenon for planar nonlocal minimal graphs (with the same quantitative bounds obtained for the case of even symmetric perturbations, up to multiplicative constants).</p><p style='text-indent:20px;'>In proving this result, one also establishes an odd symmetric version of the maximum principle for nonlocal minimal graphs, according to which the odd symmetric minimizer is positive in the direction of the positive bump and negative in the direction of the negative bump.</p>

A solution to the multidimensional additive homological equation

We prove that, for a finite-dimensional real normed space $V$, every bounded mean zero function $f\in L_\infty([0,1];V)$ can be written in the form $f=g\circ T-g$ for some $g\in L_\infty([0,1];V)$ and some ergodic invertible measure preserving transformation $T$ of $[0,1]$. Our method moreover allows us to choose $g$, for any given $\varepsilon>0$, to be such that $\|g\|_\infty\leq (S_V+\varepsilon)\|f\|_\infty$, where $S_V$ is the Steinitz constant corresponding to $V$.

Lineability and additivity of almost injective functions

We discuss additivity and lineability of classes of functions generalizing the concept of injectivity. In particular, we prove that it is consistent with ZFC that the class of almost injective functions has maximal possible lineability, i.e., it contains (with the exception of the zero function) a vector space of cardinality $2^\cont$.

A Hilbert–Kunz function with a periodic term that has a given period

A result of Monsky states that the Hilbert–Kunz function of a one-dimensional local ring of prime characteristic has a term ϕ \phi that is eventually periodic. For example, in the case of a power series ring in one variable over a prime-characteristic field, ϕ \phi is the zero function and is therefore immediately periodic with period 1. In additional examples produced by Kunz [Amer. J. Math. 91 (1969), pp. 772–784] and Monsky [Math. Ann. 263 (1983), pp. 43–49], ϕ \phi is immediately periodic with period 2. We show that, for every positive integer π \pi , there exists a ring for which ϕ \phi is immediately periodic with period π \pi .

On the best constant in the Lp estimate for the sharp maximal function

Let I be a fixed subinterval of R and let 1≤p<∞ be a fixed exponent. Suppose that f∈Lp(I) is an arbitrary function of integral zero and let f# be the BLO-based sharp maximal function of f. The paper contains the identification of the best constant Cp in the estimate‖f‖Lp≤Cp‖f#‖Lp. The proof rests on the explicit evaluation of the Bellman function associated with the above estimate.

Theoretical Study on Non-Improvement of the Multi-Frequency Direct Sampling Method in Inverse Scattering Problems

Generally, it has been confirmed that applying multiple frequencies guarantees a successful imaging result for various non-iterative imaging algorithms in inverse scattering problems. However, the application of multiple frequencies does not yield good results for direct sampling methods (DSMs), which has been confirmed through simulation but not theoretically. This study proves this premise theoretically by showing that the indicator function with multi-frequency can be expressed by the Bessel and Struve functions and the propagation direction of the incident field. This is based on the fact that the indicator function with single frequency can be expressed by the exponential and Bessel function of order zero of the first kind. Various simulation outcomes are shown to support the theoretical result.

A Creative Design Framework for Math Exam Questions Concerning Inequalities and Zeros of Functions with an Unknown Parameter in China National College Entrance Examination

The annual exam questions in China National College Entrance Examination (CNCEE, commonly known as Gaokao) have attracted wide attention from the entire society in China. This paper analyzes an important math exam question related to always holding inequalities with an unknown parameter from the 2020 CNCEE and builds different solutions via parameter separation or classified discussion, and then gives a systematic and effective design framework for questions related to problems of always holding inequalities and zeros of functions with an unknown parameter for the CNCEE. The framework reveals the internal mechanism of designing the math exam questions in the CNCEE. According to the design framework, two simulated math exam questions with reference solutions will be proposed.