Entire Functions Research Articles

Boundedness of the L-Index in the Direction of Composition of Slice Entire Functions and Slice Holomorphic Functions in the Unit Ball

Boundedness of the L-Index in the Direction of Composition of Slice Entire Functions and Slice Holomorphic Functions in the Unit Ball

REGIONÁLNE KLIMATICKÉ SCENÁRE FREKVENCIE VÝSKYTU DNÍ SO ZRÁŽKAMI NA SLOVENSKU

We present scenarios of how frequency of days with precipitation may change over the territory of Slovakia by 2050 and 2080 in the article. Climatic scenarios were created from three regional climate models EURO-CORDEX for the pessimistic emission scenario RCP8.5. We analysed the occurrence of the number of precipitation days for class intervals: 5 – 9.9 mm, 10 – 19.9 mm, 20 – 29.9 mm, 30 – 39.9 mm, as well as for open intervals ≥ 5 mm, ≥ 10 mm, ≥ 20 mm, ≥ 30 mm, and ≥ 40 mm. To correct systematic errors in climate models, we used data (daily precipitation totals) from the national network of rain gauges. The RCM models were corrected in two steps: first in terms of excessive occurrence of so-called “drizzle days”, and then in terms of systematic underestimation of daily sums across the entire distribution function (bias correction). The occurrence of days with daily rainfall exceeding 40 mm for the horizons of 2050 and 2080 is presented in the form of maps. As our analyses revealed, the annual frequency of days with daily rainfall exceeding 40 mm is expected to increase by approximately 20% by the middle of the 21st century and by up to 60% by the end of the century compared to the present. The paper also includes tables with relative changes and absolute frequencies of wet days for 75 selected locations in Slovakia.

Transcendental Julia sets of positive finite Lebesgue measure

We show that there exists a transcendental entire function whose Julia set has positive finite Lebesgue measure.

On the numerical radius of weighted shift operators with generalized geometric weights

In this paper, we give bounds on the numerical radius of the weighted shift operator T with generalized geometric weights(1,sq,q2,sq3,…,q2n−2,sq2n−1,…), where s>0 and 0<q<1. Also, we provide the entire function FT(z) whose minimal positive root gives the numerical radius of the weighted shift operator T. The purpose of this paper is to generalize the results of numerical radius for the weighted shift operator with geometric weights given in [5].

Arbitrary Random Variables and Wiman’s Inequality

We study the class of random entire functions given by power series, in which the coefficients are formed as the product of an arbitrary sequence of complex numbers and two sequences of random variables. One of them is the Rademacher sequence, and the other is an arbitrary complex-valued sequence from the class of sequences of random variables, determined by a certain restriction on the growth of absolute moments of a fixed degree from the maximum of the module of each finite subset of random variables. In the paper we prove sharp Wiman–Valiron’s type inequality for such random entire functions, which for given p∈(0;1) holds with a probability p outside some set of finite logarithmic measure. We also considered another class of random entire functions given by power series with coefficients, which, as above, are pairwise products of the elements of an arbitrary sequence of complex numbers and a sequence of complex-valued random variables described above. In this case, similar new statements about not improvable inequalities are also obtained.

Forensic Profiling of Road Traffic Accident Victims- A Retrospective Study

Background: Road traffic accidents (RTAs) have become a significant public health problem that requires a multi-disciplinary strategy to solve. Every year, 1.35 million people die because of RTAs. The effects of a traffic accident not only cause the sufferer to suffer physically, emotionally, and financially, but they also have a deadly impact on the way the entire family functions. Objectives: This study's objective was to assess the current RTA situation in Manikganj district, determine the injury pattern, explore the causes and frequency of accidents, and find out the sociodemographic characteristics of the victims. Methodology: This retrospective study was carried out in the Manikganj District hospital, Bangladesh, during the period of January, 2019 to December, 2020. Results: During the study period, there were 381 postmortems performed, of which 83 (21.78%) instances involved RTA. Of the victims, 49(59.04%) were men and 34(40.96%) women. Incidence of RTA (24, 28.91%) was the highest among people aged 21 to 30 years. The most accidents happened during the daytime (32, 38.55%). All victims sustained many abrasions and bruises (83, 100%), laceration was present in 46(55.42%), fractured ribs in 33(39.75%), fractured hipbones in 32(38.55%), skull bones fractured in 16(19.27%), head injury in 26 (31.32%) and intracranial haemorrhages in 26(31.32%) cases. Most victims (59, 71.08%) were pedestrians, followed by passengers and drivers. In 83 instances, head injuries accounted for most fatalities (22, 26.53%), followed by multiple traumas, haemorrhagic shock, spinal cord damage, septic shock and crush syndrome. Conclusion: Strict adherence to traffic laws, public awareness campaigns, and adequate driver training all reduce the likelihood of road accidents and lessen their severity. Journal of Monno Medical College December, 2023; 9 (2):53-57

Introduction to Nonlocal Field Theory Including Gravity

We give minireview of nonlocal field theory (infinite derivative field theory). We start with the discussion of the main peculiarities of nonlocal field theory on the example of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d = 4$$\\end{document} scalar \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\phi }^{4}}$$\\end{document}-model. The nonlocal \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\phi }^{4}}$$\\end{document}-model is ultraviolet finite, unitary, and macrocausal. One of the problems of nonlocal field theory is that the formfactor is an arbitrary entire function that makes the predictions extremely weak. We propose some additional principle that allows to fix the formfactor. Also we review the main results obtained in nonlocal quantum gravity, namely the nonlocal generalization of Einstein gravity leads to the superrenormalizable theory.

Unbounded operators and the uncertainty principle

We study a variant of the uncertainty principle in terms of the annihilation and creation operator on generalized Segal Bargmann spaces, which are used for the FBI-Bargmann transform. In addition, we compute the Berezin transform of these operators and indicate how to use spaces of entire functions in one variable to study the Szegő kernel for hypersurfaces in C 2 \mathbb C^2 .

Zero distribution of solutions of some linear differential equations

Assuming that A ( z ) = B 1 ( z ) e P ( z ) + B 2 ( z ) e ϱp ( z ) + Q ( z ) , where B 1 ( z ) , B 2 ( z ) , p ( z ) are nonzero polynomials, ϱ ∈ ( 0 , 1 ) , and Q ( z ) is an entire function of order less than deg ⁡ p , we study the oscillation of solutions of the linear differential equation f ( k ) + A ( z ) f = 0 . Some conditions on A ( z ) are given to guarantee that any non-trivial solution f satisfies λ ( f ) = ∞ or λ ( f ) ≥ σ ( A ) , where σ ( f ) and λ ( f ) denote respectively the order of f and the exponent of convergence of zeros of f. We also find some types of linear differential equations with zero-free solutions.

Duality, BMO and Hankel operators on Bernstein spaces

In this paper we deal with the problem of describing the dual space (Bκ1)⁎ of the Bernstein space Bκ1, that is the space of entire functions of exponential type (at most) κ>0 whose restriction to the real line is Lebesgue integrable. We provide several characterizations, showing that such dual space can be described as a quotient of the space of entire functions of exponential type κ whose restriction to the real line are in a suitable BMO-type space, or as the space of symbols b for which the Hankel operator Hb is bounded on the Paley–Wiener space Bκ/22. We also provide a characterization of (Bκ1)⁎ as the BMO space w.r.t. the Clark measures of the inner function ei2κz on the upper half-plane, in analogy with the known description of the dual of backward-shift invariant 1-spaces on the torus. Furthermore, we show that the orthogonal projection Pκ:L2(R)→Bκ2 induces a bounded operator from L∞(R) onto (Bκ1)⁎.Finally, we show that Bκ1 is the dual space of a suitable VMO-type space or as the space of symbols b for which the Hankel operator Hb on the Paley–Wiener space Bκ/22 is compact.

On the quadratic closeness of eigenfunctions of «unperturbed» and «perturbed» differentiation operators on an interval

The article considers the eigenvalue problem of the differentiation operator, when the spectral param- eter is also present in the boundary condition with an integral perturbation, where the integrand has the property of limited variation and has a value of unity at the ends of the segment [-1, 1]. The derivatives with respect to the time variable included in the boundary condition naturally arise when solving (by the Fourier method) initial boundary value problems for evolution equations. The conjugate operator is constructed. It is shown that the spectral questions of the conjugate operator have a similar structure. The characteristic determinant of the original direct spectral problem with an integral perturbation of the boundary condition and for the eigenvalue problem of a loaded first-order differential equation on a segment with a periodic boundary condition, which is an entire analytical function of the spectral parameter, is constructed. Based on the formula of the characteristic determinant, conclusions are drawn about the asymptotic behavior of the spectrum of the original «perturbed» differentiation operator and the loaded first-order differential equation on the segment. A special feature of the operator under consideration is the non-self-adjointness of the operator in L2(–1,1). The quadratic proximity of the systems of eigen functions of the «unperturbed» and «perturbed» differentiation operators and the Riesz basis property of these systems are proved. In this case, the system of eigenfunctions of the «perturbed» operator is not orthonormal.

Інтерполяційна задача у вагових просторах Пелі-Вінера

Yu. Lyubarskii and K. Seip K. (Revista Matematica Iberoamericana, 1997 (13), № 2) found the criterion for the existence of a uniqe solution of the interpolation problem f(λk) = bk in the Paley-Wiener spaces in terms of Makenhoupt’s (Ap) condition of entire functions of the exponential type at most π, where p is a real number more than 1, whose restriction to the real line coincides with the class of functions which module degree number p is integrable on R whith p-norm. These results make it possible to obtain the criterion of the unconditional basisness of the system of exponentials in the space of functions which module degree number p is an integrable on (–π; π). At the same time the sequence (λk) with a unique limit point at the infinity, for which mentioned interpolation problem has a uniqe solution is called a complete interpolating sequence in the Paley-Wiener spaces. For p = 2 those descriptions coincides with those given by Pavlov (1979), Nikolsky (1980), and Minkin (1982). We generalize these results to the weighted spaces of entire functions of the exponential type at most σ with the p-norm (there is a power function with exponent ω as a weight), where σ is a nonnegative real number, ω – real number more then –1. That is, we find conditions for the completeness of the interpolating sequence (λk) in the Paley-Wiener weighted space. Different forms of these conditions are considered. Among them there are Mackenhoupt’s conditions (continuous and discrete (Ap) conditions). There is proved that if (λk) is a complete interpolation sequence in the Paley-Wiener weighted space, then it is a relatively dense set in the space C. Also there constructed an example of a complete interpolating sequence for σ = π

Rational partitioning of spectral feature space for effective clustering of massive spectral image data

We have successfully proposed and demonstrated a clustering method that overcomes the “needle-in-a-haystack problem” (finding minuscule important regions from massive spectral image data sets). The needle-in-a-haystack problem is of central importance in the characterization of materials since in bulk materials, the properties of a very tiny region often dominate the entire function. To solve this problem, we propose that rational partitioning of the spectral feature space in which spectra are distributed, or defining of the decision boundaries for clustering, can be performed by focusing on the discrimination limit defined by the measurement noise and partitioning the space at intervals of this limit. We verified the proposed method, applied it to actual measurement data, and succeeded in detecting tiny (~ 0.5%) important regions that were difficult for human researchers and other machine learning methods to detect in discovering unknown phases. The ability to detect these crucial regions helps in understanding materials and designing more functional materials.

Hetero-oligomeric interaction as a new regulatory mechanism for protein arginine methyltransferases.

Protein arginine methylation is a versatile post-translational protein modification that has notable cellular roles such as transcriptional activation or repression, cell signaling, cell cycle regulation, and DNA damage response. However, in spite of their extensive significance in the biological system, there is still a significant gap in understanding of the entire function of the protein arginine methyltransferases (PRMTs). It has been well-established that PRMTs form homo-oligomeric complexes to be catalytically active, but in recent years, several studies have showcased evidence that different members of PRMTs can have cross-talk with one another to form hetero-oligomeric complexes. Additionally, these heteromeric complexes have distinct roles separate from their homomeric counterparts. Here, we review and highlight the discovery of the heterodimerization of PRMTs and discuss the biological implications of these hetero-oligomeric interactions.

Operational excellence in total productive maintenance: statistical reliability as support for planned maintenance pillar

PurposeTotal productive maintenance consists of strategies and procedures that aim to guarantee the entire functioning of machines in a production process so that production is not interrupted and no loss of quality in the final product occurs. Planned maintenance is one of the eight pillars of total productive maintenance, a set of tools considered essential to ensure equipment reliability and availability, reduce unplanned stoppage and increase productivity. This study aims to analyze the influence of statistical reliability on the performance of such a pillar.Design/methodology/approachIn this study, we utilized a multi-method approach to rigorously examine the impact of statistical reliability on the planned maintenance pillar within total productive maintenance. Our methodology combined a detailed statistical analysis of maintenance data with advanced reliability modeling, specifically employing Weibull distribution to analyze failure patterns. Additionally, we integrated qualitative insights gathered through semi-structured interviews with the maintenance team, enhancing the depth of our analysis. The case study, conducted in a fertilizer granulation plant, focused on a critical failure in the granulator pillow block bearing, providing a comprehensive perspective on the practical application of statistical reliability within total productive maintenance; and not presupposing statistical reliability is the solution over more effective methods for the case.FindingsOur findings reveal that the integration of statistical reliability within the planned maintenance pillar significantly enhances predictive maintenance capabilities, leading to more accurate forecasts of equipment failure modes. The Weibull analysis of the granulator pillow block bearing indicated a mean time between failures of 191.3 days, providing support for optimizing maintenance schedules. Moreover, the qualitative insights from the maintenance team highlighted the operational benefits of our approach, such as improved resource allocation and the need for specialized training. These results demonstrate the practical impact of statistical reliability in preventing unplanned downtimes and informing strategic decisions in maintenance planning, thereby emphasizing the importance of your work in the field.Originality/valueIn terms of the originality and practicality of this study, we emphasize the significant findings that underscore the positive influence of using statistical reliability in conjunction with the planned maintenance pillar. This approach can be instrumental in designing and enhancing component preventive maintenance plans. Furthermore, it can effectively manage equipment failure modes and monitor their useful life, providing valuable insights for professionals in total productive maintenance.

Asymptotic estimates for entire functions of minimal growth with given zeros

Let $\zeta=(\zeta_n)$ be an arbitrary complex sequence such that $0&lt;|\zeta_1|\le|\zeta_2|\le\dots$ and $\zeta_n\to\infty$ as $n\to\infty$, let $n_\zeta(r)$ and $N_\zeta(r)$ be the counting function and the integrated counting function of this sequence, respectively. By $\mathcal{E}_\zeta$ we denote the class of all entire functions whose zeros are precisely the $\zeta_n$, where a complex number that occurs $m$ times in the sequence $\zeta$ corresponds to a zero of multiplicity $m$. Suppose that $\Phi$ is a convex function on $\mathbb{R}$ such that $\Phi(\sigma)/\sigma\to+\infty$ as $\sigma\to+\infty$. It is proved that there exists an entire function $f\in\mathcal{E}_\zeta$ such that$$\varlimsup_{r\to+\infty}\frac{\ln\ln M_f(r)}{\Phi(\ln r)}\le\varlimsup_{r\to+\infty}\frac{\ln n_\zeta( r)}{\Phi(\ln r)},$$where $M_f(r)$ denotes the maximum modulus of the function $f$, and it is shown that the above inequality implies the inequality$$\varlimsup_{r\to+\infty}\frac{\ln\ln M_f(r)}{\Phi(\ln r)}\le\varlimsup_{r\to+\infty}\frac{\ln N_\zeta( r)}{\Phi(\ln r)}+\varlimsup_{\sigma\to+\infty}\frac{\ln\Phi'_+(\sigma)}{\Phi(\sigma)}.$$The formulated result is a consequence of the following more general statement: if the right-hand derivative $\Phi'_+$ of the function $\Phi$ assumes only integer values and $\sum_{n=1}^\infty e^{-\Phi(\ln|\zeta_n|)}&lt;+\infty$, then there exists an entire function $f\in\mathcal{E}_\zeta$ such that $\ln M_f(r)=o(e^{\Phi(\ln r)})$ as $r \to+\infty$.

Some inequalities for entire functions

Let $\mathcal{L}_p$ be the subspace of the space $L_p(\bR)$ consisting of the restriction to the real axis of all entire functions of exponential type $\le \pi$. In this paper, for any function $f\in \mathcal{L}_p \,\,(1\le p\le \infty)$, we obtain estimates for the norm of $f$ in terms of the sequence $(f(n/2))_{n\in\bZ},$ namely$$\frac12 \|f\|_{p,1}\le \|f\|_{\mathcal{L}_p}\le 2 \|f\|_{p,1},$$where $\|f\|_{p,1}:=\frac12(\|Jf\|_{\ell_p(\bZ)} +\|JT_{1/2}f\|_{\ell_p(\bZ)})$. Here $J:\mathcal{L}_p\to\ell_p(\bZ)$ is the linear operator given by the formula $$ (Jf)(n):=(-1)^nf(n), \quad n\in\bZ,$$ and $T_\tau$ is the shift by $\tau\in\bR$ of the function $f$,$$ (T_\tau f)(z):=f(z+\tau), \quad z\in\bC.$$

On Boundedness of the l-𝔐-Index of Entire Functions Represented by Series in a System of Functions

On Boundedness of the l-𝔐-Index of Entire Functions Represented by Series in a System of Functions

Point evaluation in Paley–Wiener spaces

We study the norm of point evaluation at the origin in the Paley–Wiener space PWp for 0 < p < ∞, i.e., we search for the smallest positive constant C, called Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document}, such that the inequality |f(0)|p≤C∥f∥pp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\vert f(0)\\vert ^{p}\\leq C \\Vert f\\Vert _{p}^{p}$$\\end{document} holds for every f in PWp. We present evidence and prove several results supporting the following monotonicity conjecture: The function p↦Cp/p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\mapsto {\\mathscr{C}}_{p}/p$$\\end{document} is strictly decreasing on the half-line (0, ∞). Our main result implies that Cp<p/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p} < p/2$$\\end{document} for 2 < p < ∞, and we verify numerically that Cp>p/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p} > p/2$$\\end{document} for 1 ≤ p < 2. We also estimate the asymptotic behavior of Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} as p → ∞ and as p → 0+. Our approach is based on expressing Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} as the solution of an extremal problem. Extremal functions exist for all 0 < p < ∞; they are real entire functions with only real zeros, and the extremal functions are known to be unique for 1 ≤ p < ∞. Following work of Hörmander and Bernhardsson, we rely on certain orthogonality relations associated with the zeros of extremal functions, along with certain integral formulas representing respectively extremal functions and general functions at the origin. We also use precise numerical estimates for the largest eigenvalue of the Landau–Pollak–Slepian operator of time-frequency concentration. A number of qualitative and quantitative results on the distribution of the zeros of extremal functions are established. In the range 1 < p < ∞, the orthogonality relations associated with the zeros of the extremal function are linked to a de Branges space. We state a number of conjectures and further open problems pertaining to Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} and the extremal functions.

On a difference equation with holomorphic coefficients generated by a hexagon

Let D be a hexagon with two straight and four right angles. We consider a sevenelement difference equation with holomorphic coefficients generated by this hexagon. A method for regularizing this equation is proposed. The solution is sought in the class of functions holomorphic outside the "half"of the boundary ᲛD and vanishing at infinity. It is represented as a Cauchy-type integral over the "half"of the boundary with an unknown density. Applications to the problem of moments for entire functions of exponential type (e.f.e.t.) are indicated.