Abstract We show that, under certain conditions, a strongly continuous semigroup admits an almost surely frequently hypercyclic random vector defined as a stochastic integral in Fréchet spaces with respect to the Brownian motion. Two criteria are given. We will apply the second criterion to three examples: translation semigroups on spaces of integrable functions, the exponential of weighted shifts, and the translation operators on the space of entire functions. This last example, with a stochastic approach, seems to be new in the literature. Some other examples are given.

Abstract Ecosystem management requires integrative tools for planning and decision-making at landscape scales. Vegetation water content (VWC) has been proposed as a holistic indicator of entire plant water status and ecosystem functioning. However, few studies have explored its relationships with ecological and environmental factors, especially in data-scarce regions like Patagonia. In this study, we used the SMAP dataset to analyzed VWC patterns across vegetation types, landscapes, and climate anomalies over the 2015–2016 to 2024–2025 growing seasons in Southern Patagonia, covering Santa Cruz and Tierra del Fuego provinces. The results show that VWC varies significantly among vegetation types. Forests and shrublands generally had higher VWC than grasslands. Among them, Nothofagus betuloides and evergreen mixed forests (NB) exhibited the highest VWC, with grasslands and dry shrublands being at the lowest level. Distinct seasonal VWC dynamics emerged along landscape gradients: a bell-shaped pattern in humid areas and an inverted bell-shaped pattern in arid ecosystems, primarily driven by winter snowmelt recharge and subsequent summer water stress. VWC responded differentially to El Niño-Southern Oscillation and Southern Annular Mode phases, with El Niño and La Niña exerting contrasting effects by province. To bridge science and application, we propose incorporating these metrics into adaptive management, specifically for: (i) defining conservation-priority zones for harvesting in water-limited forests, (ii) implementing dynamic fire risk thresholds based on seasonal depletion patterns, and (iii) adjusting silvopastoral stocking rates according to inter-annual VWC trends. These findings demonstrate that VWC is a robust indicator for guiding sustainable ecosystem management and conservation planning in Southern Patagonia.

Abstract This paper surveys the impact of Eremenko and Lyubich's paper “Examples of entire functions with pathological dynamics” , published in 1987 in the Journal of the London Mathematical Society . Through a clever extension and use of classical approximation theorems, the authors constructed examples exhibiting behaviours previously unseen in holomorphic dynamics. Their work laid foundational techniques and posed questions that have since guided a good part of the development of transcendental dynamics.

The Low-power Gigabit Transceiver (lpGBT) is a radiation-tolerant ASIC used in high-energy physics experiments for multipurpose high-speed bidirectional serial links. In 2024, almost 270,000 lpGBTs v1 were tested with a production test system that exercises the entire ASIC functionality to ensure its correct operation. Furthermore, qualification tests (Total Ionizing Dose, Single-Event Upsets, etc.) were done on lpGBTs from each production lot. Despite the thorough production and qualification tests, a design issue named “stuck at power-up” was discovered, affecting a maximum of 0.9% of delivered devices simultaneously. The test system setup developed for the characterisation of this behaviour and the results obtained are presented here.

Partial sharing and cross-sharing of entire function with its derivative

The article describes the tensor products of approximation spaces associated with regular elliptic operators on tensor products of Lebesgue spaces $L_2(\partial\Omega)$, where $\partial \Omega$ considers as smooth manifold that describes in the usual way by local system of local coordinates. We use the quasi-normed approximation spaces and subspaces of exponential type functions associated with such operators.} A connection between the tensor products of approximation spaces and interpolation spaces obtained by the real method of interpolation is showed. We prove the direct and inverse approximation theorems for Bernstein–Jackson type inequalities as well as we give the explicit dependence of constants on parameters of approximation spaces. Such constants are expressed via some normalization factor. Application to spectral approximations on tensor products of interpolation spaces associated with regular elliptic operators on compact manifolds is shown. In the article also consider the spectral approximations (Theorem 2), since the subspaces of entire functions of exponential type of regular elliptic operators on compact manifolds coincide with their spectral subspaces (Lemma 3).

Abstract In the paper, using the idea of normal family, we investigate the uniqueness problem of entire functions that share two values partially with their k -th derivatives. The obtained results improve the results of Lü, Xu and Yi (Ann. Polon. Math. 95(1) (2009), 67–75) in a large scale. Also, as an application of our results, we have settled the conjecture posed by Li and Yang (Illinois J. Math. 44(2) (2000), 349–362).

In connection to Brück conjecture we improve a uniqueness problem for entire functions that share a polynomial with linear differential polynomial.

Construction and approximation properties of exact neural network interpolation operators activated by entire functions

Abstract The escaping set of an entire function consists of the points in the complex plane that tend to infinity under iteration. This set plays a central role in the dynamics of transcendental entire functions. The goal of this survey is to explain this role, to summarise some of the main results in the area, and to identify a number of open questions.

Abstract In this paper, we discuss the existence of meromorphic solutions to the following system of difference equations $$\begin{cases}f(z+c)=e^{P}f-ae^{P}+a,\\ f(z+c)=e^{Q}f-be^{Q}+b,\end{cases}$$ where $$P$$ and $$Q$$ are entire functions, $$a$$ and $$b$$ are distinct complex numbers. Additionally, this work elucidates the application of these findings, thereby enhancing uniqueness results associated with meromorphic functions $$f(z)$$ and their shifts $$f(z+c)$$ .

Abstract Let be a transcendental entire map from the Eremenko–Lyubich class , and let be an attracting periodic point of period . We prove that the boundaries of components of the attracting basin of (the orbit of) have hyperbolic (and, consequently, Hausdorff) dimension larger than 1, provided has an infinite degree on an immediate component of the basin, and the singular set of is compactly contained in . The same holds for the boundaries of components of the basin of a parabolic ‐periodic point , under the additional assumption . We also prove that if an immediate component of an attracting basin of an arbitrary transcendental entire map is bounded, then the boundaries of components of the basin have hyperbolic dimension larger than 1. This enables us to show that the boundary of a component of an attracting basin of a transcendental entire function is never a smooth or rectifiable curve. The results provide a partial answer to a question from Hayman's list of problems in function theory.

The concept of proximate order is widely used in the theories of integer, meromorphic, subharmonic, and plurisubharmonic functions. In this paper, we provide a general interpretation of this concept as a proximate order relative to the model growth function. The classical proximate order in the sense of Valiron is the particular case of proximate order relative to the model growth function. The main result of this work is a lower estimate of the distance between the points at which the maximum modulus of the entire function and the set of zeros of this function is reached, using the concept of a proximate order relative to the model function.

This paper presents a generalization of the exponential base of special monogenic polynomials within the framework of Fr ́echet modules (F-modules). The study focuses on examining the convergence properties, specifically the effectiveness, of both the exponential simple base of special monogenic polynomials (ESBSMPs) and the exponential Cannon base of special monogenicpolynomials (ECBSMPs) in Fr ́echet modules. These properties are investigated on hyper-closed and open balls, in open regions surrounding hyper-closed balls, for all entire special monogenic functions, as well as at the origin. Furthermore, an explicit upper bound for the order of the exponential simple base is established and shown to be attainable. Finally, we extend the discussion to equivalent and similar bases, verifying that the derived results remain valid under such transformations, which confirms the robustness and general applicability of the findings.

Optoelectronic materials with proper charge storage play a pivotal role in the development of artificial neuromorphic devices aiming to mimic the visual, sensory, and memory functions of the human nervous system. This study presents the controllable charge storability in Indium Phosphide quantum dots through being capped with Zinc Selenide shells of different thicknesses. The organic transistors with the quantum dots integrated demonstrate shell‐thickness‐dependent optoelectronic memory characteristics, featuring optically programmable‐electrically erasable channel states. Analysis reveals that the optoelectronic performance of the device is ascribed to the photoexcitation and the following charge storage process in the quantum dots. The device of the thickest quantum‐dot‐shell performs well as an optoelectronic synapse to emulate the entire human visual sensory and memory function. The frequency‐dependent synaptic potentiation/depression, paired‐pulse facilitation, short/long‐term memory, and “learning‐experience” behavior are exhibited in the optoelectronic synaptic device through optical stimuli manipulation. Moreover, the optical sensory performance of the device can be enhanced by a positive gate bias. It enables a successful emulation of Pavlov's dog classical conditioning experiments, realizing the associative learning characteristic with optical and electric signals. This work provides an effective solution for a stable and controllable charge storage medium for optoelectronic synapse applications.

A classic definition of multisensory integration (MI) has been proposed as 'the presence of a (statistically) significant change in the response to a crossmodal stimulus complex compared to unimodal stimuli'. However, this general definition did not result in a broad consensus on how to quantify the amount of MI in the context of reaction time (RT). In this brief note, we argue that numeric measures of reaction times that only involve mean or median RTs do not uncover the information required to fully assess the effect of MI. We suggest instead novel measures that include the entire RT distributions functions. The central role is played by relative entropy (a.k.a. Kullback-Leibler divergence), a statistical concept in information theory, statistics, and machine learning to measure the (non-symmetric) distance between probability distributions. We provide a number of theoretical examples, but empirical applications and statistical testing are postponed to a later study.

For the differential equation $(1-z)^nw''+a(1-z)^mw' +bw=0$ (with $n>m\ge 0,\, a\in {\Bbb R},\, b\in {\Bbb R}$), the existence of analytical solutions in the unit disk of the form $f(z)=F(1/(1-z))$, where $F$ is an entire transcendental function, is studied. For such a function $F$, the growth, starlikeness, convexity, and close-to-convexity are investigated.

We study an integral transform—here called the Master Integral Transform—in which the kernel is an arbitrary entire function of finite order. When the nonzero Taylor coefficients of the kernel have positive Beurling–Malliavin density, we prove completeness and global injectivity in a Cauchy-weighted Hilbert space, and we furnish explicit Mellin–Fourier inversion formulae with exponentially decaying integrands. Classical Fourier, Laplace, and Mellin transforms appear only as strict special cases. Beyond these, we establish structural properties (multiplier/composition law, dilation covariance, parameter regularity) and present applications not captured by fixed-kernel frameworks, including inverse-kernel identification and hybrid boundary value models, e.g., the Poisson–Airy pair produces a closed-form transformed Green’s function and a solvable variable-coefficient PDE, illustrating capabilities unavailable to fixed-kernel frameworks.

The work is devoted to the study of complex-valued continuous symmetric polynomials on Cartesian products of complex Banach spaces of Lebesgue integrable functions. Let $L_p$, where $p\in [1;+\infty)$, be the complex Banach space of all complex-valued functions on $[0;1]$, the $p$th powers of absolute values of which are Lebesgue integrable. Let $\Xi_{[0;1]}$ be the set of all bijections $\sigma:[0;1] \to [0;1]$ such that both $\sigma$ and $\sigma^{-1}$ are measurable and preserve Lebesgue measure, i.e. $\mu(\sigma(E)) = \mu(\sigma^{-1}(E)) = \mu(E)$ for every Lebesgue measurable set $E\subset [0;1]$, where $\mu$ is Lebesgue measure. A function $f$ on the Cartesian product $L_{p_{1}} \times L_{ p_{2}} \times \ldots \times L_{p_{n}}$, where $p_1,p_2, \ldots, p_n \in [1;+\infty)$, is called symmetric if $f((x_1\circ\sigma;x_2\circ\sigma;\ldots;x_n\circ\sigma))=f((x_1;x_2;\ldots;x_n))$ for every $\sigma\in \Xi_{[0;1]}$ and $(x_1;x_2;\ldots;x_n)\in L_{p_{1}} \times L_{ p_{2}} \times \ldots \times L_{p_{n}}$. We construct an algebraic basis of the algebra of all complex-valued continuous symmetric polynomials on $L_{p_{1}} \times L_{ p_{2}} \times \ldots \times L_{p_{n}}$. Also we construct some isomorphisms of Fréchet algebras of complex-valued entire symmetric functions of bounded type on $L_{p_{1}} \times L_{ p_{2}} \times \ldots \times L_{p_{n}}$.

UDC 512.5 We introduce new classes of functions generalizing the well-known classes of functions of complex variable, such as the entire functions, meromorphic functions, rational functions, and polynomial functions, which take values in the set of circulant matrices with complex entries. For these new classes of functions, we extend some recently obtained characterization theorems presented in [B. Q. Li, Amer. Math. Monthly, 122, № 2, 169–172 (2015)] and [B. Q. Li, Amer. Math. Monthly, 132, № 3, 269–271 (2025)] to an algebraic structure of circulant matrices that includes several complex variables. Our characterization theorems generalize a recently established version of the fundamental theorem of algebra presented in [V. M. Abramov, Amer. Math. Monthly, 132, № 4, 356–360 (2025)].