Entire Functions Research Articles

Characterization of continuous homomorphisms on entire slice monogenic functions

Characterization of continuous homomorphisms on entire slice monogenic functions

A NEW PROGRAM FOR THE ENTIRE FUNCTIONS IN NUMBER THEORY

In this paper, we propose a new program for introducing the sign of the functional equation to present the entire functions of order one in number theory. We suggest some open problems for the zeros of these entire functions related to the completed Dedekind zeta function, completed quadratic Dirichlet L-functions, completed Ramanujan zeta function and completed automorphic L-function. These lead to the contribution to giving the deep understanding for diffusion equations associated with the entire functions of order one in number theory.

Completeness of the systems of Bessel functions of index $-5/2$

Let $L^2((0;1);x^4 dx)$ be the weighted Lebesgue space of all measurable functions $f:(0;1)\rightarrow\mathbb C$, satisfying $\int_{0}^1 t^4 |f(t)|^2\, dt<+\infty$. Let $J_{-5/2}$ be the Bessel function of the first kind of index $-5/2$ and $(\rho_k)_{k\in\mathbb N}$ be a sequence of distinct nonzero complex numbers. Necessary and sufficient conditions for the completeness of the system $\big\{\rho_k^2\sqrt{x\rho_k}J_{-5/2}(x\rho_k):k\in\mathbb N\big\}$ in the space $L^2((0;1);x^4 dx)$ are found in terms of an entire function with the set of zeros coinciding with the sequence $(\rho_k)_{k\in\mathbb N}$. In this case, we study an integral representation of some class $E_{4,+}$ of even entire functions of exponential type $\sigma\le 1$. This complements similar results on approximation properties of the systems of Bessel functions of negative half-integer index less than $-1$, due to B. Vynnyts'kyi, V. Dilnyi, O. Shavala and the author.

Wetzel families and the continuum

AbstractWe provide answers to a question brought up by Erdős about the construction of Wetzel families in the absence of the continuum hypothesis: A Wetzel family is a family of entire functions on the complex plane which pointwise assumes fewer than values. To be more precise, we show that the existence of a Wetzel family is consistent with all possible values of the continuum and, if is regular, also with Martin's Axiom. In the particular case of this answers the main open question asked by Kumar and Shelah [Fund. Math. 239 (2017) no. 3, 279–288]. In the buildup to this result, we are also solving an open question of Zapletal on strongly almost disjoint functions from Zapletal [Israel J. Math. 97 (1997) no. 1, 101–111]. We also study a strongly related notion of sets exhibiting a universality property via mappings by entire functions and show that these consistently exist while the continuum equals .

Amenability of bounded automata groups on infinite alphabets

AbstractWe study the action of groups generated by bounded activity automata with infinite alphabets on their orbital Schreier graphs. We introduce an amenability criterion for such groups based on the recurrence of the first‐level action. This criterion is a natural extension of the result that all groups generated by bounded activity automata with finite alphabets are amenable. Our motivation comes from the investigation of iterated monodromy groups of entire transcendental functions in holomorphic dynamics.

NEW CONJECTURES FOR THE ENTIRE FUNCTIONS ASSOCIATED WITH FRACTIONAL CALCULUS

In this paper, we address the entire Fourier sine and cosine integrals related to the Mittag-Leffler function. We guess that the entire functions have the real zeros in the entire complex plane. They can be connected with the well-known conjectures in analytic number theory. They are considered as the special solutions for the time-fractional diffusion equation within the Caputo fractional derivative.

Cauchy–Riemann beams in GRIN media

We investigate the propagation characteristics of Cauchy–Riemann beams in gradient-index media. Our study reveals two key findings: first, the preservation of their form during propagation; and second, the surprising feasibility of obtaining the two-dimensional fractional Fourier transform for any arbitrary entire function. We elucidate an explicit and straightforward expression for this transform. Additionally, our results contribute to a deeper understanding of the behavior of Cauchy–Riemann beams in inhomogeneous media, offering insights that may have implications for various applications in optics and wave propagation.

On the uniqueness problems of entire functions and their linear differential polynomials

On the uniqueness problems of entire functions and their linear differential polynomials

Growth of meromorphic solution related to linear difference equations and value sharing

In this paper, we consider the growth of meromorphic solutions of complex linear difference equations of certain types and derive some results, which improve the results of Chiang and Feng [On the Nevanlinna characteristic of f ( z + η ) and difference equations in the complex plane. Ramanujan J. 2008;16(1):105–129; On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans Am Math Soc. 2009;361:3767–3791] and Chen [Growth and zeros of meromorphic solution of some linear difference equations. J Math Anal Appl. 2011;373:235–241]. Also in this paper, we consider the sharing value problems of entire functions with their difference operators and obtain a result which improves the result of Li, Yang and Yi [Entire functions sharing an entire function of smaller order with their shifts. Proc Jpn Acad. 89 (A) (2013), 34–39]. Moreover, we show by illustrating a number of examples that our results are best possible in certain senses.

Intercropping improves maize yield and nitrogen uptake by regulating nitrogen transformation and functional microbial abundance in rhizosphere soil

Intercropping-driven changes in nitrogen (N)-acquiring microbial genomes and functional expression regulate soil N availability and plant N uptake. However, present data seem to be limited to a specific community, obscuring the viewpoint of entire N-acquiring microbiomes and functions. Taking maize intercropped with legumes (peanut and soybean) and non-legumes (gingelly and sweet potato) as models, we studied the effects of intercropping on N transformations and N-acquiring microbiomes in rhizosphere soil across four maize growth stages. Meanwhile, we compiled promising strategies such as random forest analysis and structural equation model for the exploitation of the associations between microbe-driven N dynamics and soil-plant N trade-offs and maize productivity. Compared with monoculture, maize intercropping significantly increased the denitrification rate of rhizosphere soils across four maize growth stages, net N mineralization in the elongation and flowering stages, and the nitrification rate in the seedling and mature stages. The abundance of most N-acquiring microbial populations was influenced significantly by intercropping patterns and maize growth stages. Soil available N components (NH4+-N, NO3−-N, and dissolved organic N content) showed a highly direct effect on plant N uptake, which mainly mediated by N transformations (denitrification rate) and N-acquiring populations (amoB, nirK3, and hzsB genes). Overall, the adaptation of N-acquiring microbiomes to changing rhizosphere micro-environments caused by intercropping patterns and maize development could promote soil N transformations and dynamics to meet demand of maize for N nutrient. This would offer another unique perspective to manage the benefits of the highly N-effective and production-effective intercropping ecosystems.

Reconstruction of the solution of inverse Sturm–Liouville problem

In this paper we are concerned with an inverse problem with Robin boundary conditions, which states that, when the potential on [0,1/2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$[0,1/2]$\\end{document} and the coefficient at the left end point are known a priori, a full spectrum uniquely determines its potential on the whole interval and the coefficient at the right end point. We shall give a new method for reconstructing the potential for this problem in terms of the Mittag-Leffler decomposition of entire functions associated with this problem. The new reconstructing method also deduces a necessary and sufficient condition for the existence issue.

Reproductive functions and pregnancy outcome in female patients with celiac disease.

Abnormalities in the reproductive functions are often ignored while evaluating a patient with celiac disease (CeD). We evaluated the entire reproductive functions in female patients with CeD. In a case control study between 2020 and 2021 using detailed questionnaire, we evaluated reproductive functions (age at menarche, menstrual pattern, fertility, pregnancy outcome and menopause) in biopsy-proven female patients with CeD of age >10years. The questionnaire was administered either in person or telephonically. Age-matched healthy female controls (twice the number) were also recruited. Of 1086 CeD patients, 470 were females and 288 were included. As compared with controls (n=586), females with CeD had higher age at menarche (14.6±2.0 vs 13.6±1.5years; P=0.001), delayed menarche (30.8% vs 11.4%; P=0.001), abnormal menstrual pattern (39.7% vs 25.8%; P<0.001), involuntary delay in conception at >1year (33.8% vs 11.8%; P=0.01), current infertility rate (10.5% vs 5.2%;P=0.028), and poorer overall pregnancy outcomes (abortion [23.5% vs 12.8%; P=0.001], pre-term birth [16.3% vs 3.7%; P=0.001]). Either one or more aspect of reproductive functions and pregnancy outcome is affected adversely in three-fourth female patients with CeD.

The Impact of the Limit q-Durrmeyer Operator on Continuous Functions

AbstractThe limit q-Durrmeyer operator, $$D_{\infty ,q}$$ D ∞ , q , was introduced and its approximation properties were investigated by Gupta (Appl. Math. Comput. 197(1):172–178, 2008) during a study of q-analogues for the Bernstein–Durrmeyer operator. In the present work, this operator is investigated from a different perspective. More precisely, the growth estimates are derived for the entire functions comprising the range of $$D_{\infty ,q}$$ D ∞ , q . The interrelation between the analytic properties of a function f and the rate of growth for $$D_{\infty ,q}f$$ D ∞ , q f are established, and the sharpness of the obtained results are demonstrated.

On Bloch’s “Principle of Topological Continuity”

AbstractWe discuss to what extent certain results about totally ramified values of entire and meromorphic functions remain valid if one relaxes the hypothesis that some value is totally ramified by assuming only that all islands over some Jordan domain are multiple. In particular, we prove a result suggested by Bloch which says that an entire function of order less than 1 has a simple island over at least one of two given Jordan domains with disjoint closures.

On a version of the Problem B of Mahler involving derivatives

In 1902, Paul St\"ackel constructed an analytic function $f(z)$ in a neighborhood of the origin, which was transcendental, and with the property that both $f(z)$ and its inverse, as well as its derivatives, assumed algebraic values at all algebraic points in this neighborhood. Inspired by this result, Mahler in 1976 questioned the existence of an transcendental entire function $f(z)$ such that $f(\overline{\mathbb{Q}})$ and $f^{-1}(\overline{\mathbb{Q}})$ are subsets of $\overline{\mathbb{Q}}.$ This problem was solved by Marques and Moreira in 2017. As St\"acklel's result involved derivatives, it is natural to question whether we have an analogous result for transcendental entire functions involving derivatives. In this article, we show that there are an uncountable amount of such functions.

Analyticity and supershift with irregular sampling

The notion of supershift generalizes that one of superoscillation and expresses the fact that the sampling of a function in an interval allows to compute the values of the function outside the interval. In a previous paper, we discussed the case in which the sampling of the function is regular and we are considering supershift in a bounded set, while here we investigate how irregularity in the sampling may affect the answer to the question of whether there is any relation between supershift and real analyticity on the whole real line. We show that the restriction to R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document} of any entire function displays supershift, whereas the converse is, in general, not true. We conjecture that the converse is true as long as the sampling is regular, we discuss examples in support and we prove that the conjecture is indeed true for periodic functions.

Power of an Entire Function Sharing One Value Partially with Its Derivative

Power of an Entire Function Sharing One Value Partially with Its Derivative

Exact Finite-Difference Calculus: Beyond Set of Entire Functions

In this paper, a short review of the calculus of exact finite-differences of integer order is proposed. The finite-difference operators are called the exact finite-differences of integer orders, if these operators satisfy the same characteristic algebraic relations as standard differential operators of the same order on some function space. In this paper, we prove theorem that this property of the exact finite-differences is satisfies for the space of simple entire functions on the real axis (i.e., functions that can be expanded into power series on the real axis). In addition, new results that describe the exact finite-differences beyond the set of entire functions are proposed. A generalized expression of exact finite-differences for non-entire functions is suggested. As an example, the exact finite-differences of the square root function is considered. The use of exact finite-differences for numerical and computer simulations is not discussed in this paper. Exact finite-differences are considered as an algebraic analog of standard derivatives of integer order.

Privacy Amplification by Iteration for ADMM with (Strongly) Convex Objective Functions

We examine a private ADMM variant for (strongly) convex objectives which is a primal-dual iterative method. Each iteration has a user with a private function used to update the primal variable, masked by Gaussian noise for local privacy, without directly adding noise to the dual variable. Privacy amplification by iteration explores if noises from later iterations can enhance the privacy guarantee when releasing final variables after the last iteration. Cyffers et al. explored privacy amplification by iteration for the proximal ADMM variant, where a user's entire private function is accessed and noise is added to the primal variable. In contrast, we examine a private ADMM variant requiring just one gradient access to a user's function, but both primal and dual variables must be passed between successive iterations. To apply Balle et al.'s coupling framework to the gradient ADMM variant, we tackle technical challenges with novel ideas. First, we address the non-expansive mapping issue in ADMM iterations by using a customized norm. Second, because the dual variables are not masked with any noise directly, their privacy guarantees are achieved by treating two consecutive noisy ADMM iterations as a Markov operator. Our main result is that the privacy guarantee for the gradient ADMM variant can be amplified proportionally to the number of iterations. For strongly convex objective functions, this amplification exponentially increases with the number of iterations. These amplification results align with the previously studied special case of stochastic gradient descent.

Abstract 1689: CRISPR functional genetic screening for immune enhancers of inhibitor receptors in human primary CD8 T cells

Abstract The restrained efficacy of T cell therapies in cancer treatment is associated with T cell exhaustion (TEX), an impaired effector function state that is marked by distinct epigenetic landscape. These unique epigenetic features promoted the progression of TEX, hindered reinvigoration of TEX state and controlled the abnormal expression of inhibitor receptors (IRs), which could serve as target sites in cancer treatment. Thus, deciphering the regulatory elements in primary human T cells will deeply reveal the mechanism controlling TEX and provide novel therapeutic strategies to promote clinical response. However, how enhancers can be effectively applied in anti-cancer treatment is still poorly understood because high throughput identifying the functional enhancers of inhibitor receptors at scale is technically limited in human T cells. Here, we first developed a pooled screening platform that employs a compact CRISPRi system and enabled efficient one-step delivery of pooled screening libraries to primary human T cells. Leveraging this platform, we screened large genomic regions (~300kb) at saturation around key TEX genes, two IRs—PDCD1, HAVCR2 and one transcription factor (TF)—TBX21 to interrogate functional enhancers in human primary CD8 T cells. We identified two enhancers for PDCD1, three enhancers for HAVCR2 and three enhancers for TBX21, all of which are located over 12kb away from each gene’s transcription start sites. Then, to parse the essential sequences that fulfill enhancer function, we employed Cas9-based in-situ saturation mutagenesis screening to the identified enhancers of PDCD1 and HAVCR2. We found disrupt essential sequences impaired the entire enhancer function, quantitatively phenocopied the effect of targeting the entire enhancer with CRISPRi. Transcription factor motifs analyses of these sequences revealed potential regulation of ZFX for controlling PDCD1 expression and STAT for HAVCR2 expression. Finally, we explored the functional consequences of applying enhancers to anti-cancer treatment. Indeed, targeting the identified enhancers of PDCD1 and HAVCR2 in CAR T cells lead to enhanced cancer cell killing efficiency in vitro. Together, our study provides a platform and paradigm to interrogate functional enhancers in human T cells, illuminated the great potential of manipulating enhancers in T cell therapies. Citation Format: Ping Wang, Josephine R. Giles, Hua Huang, Junwei Shi, E.John Wherry. CRISPR functional genetic screening for immune enhancers of inhibitor receptors in human primary CD8 T cells [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2024; Part 1 (Regular Abstracts); 2024 Apr 5-10; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2024;84(6_Suppl):Abstract nr 1689.