Real Functions Research Articles

Structure of the algebra of transition types, and the cut-and-join operator

Simple real Hurwitz numbers enumerate real meromorphic functions on real algebraic curves all finite critical values of which are simple. M. Kazarian, S. Lando, and S. Natanzon constructed algebras of transition types having these numbers as structure constants, and deduced cut-and-join type equations for generating functions for them. The subject of the present paper is the structure of the algebras of transition types and approaches to efficient computation of simple real Hurwitz numbers.

The existence of real nine-dimensional manifolds which include classical one-parameter families of triply periodic minimal surfaces

Triply periodic minimal surfaces have been studied in many fields of natural science, and in particular, many one-parameter families of triply periodic minimal surfaces of genus three have been considered. In 1990s, the moduli theory of triply periodic minimal surfaces established by C. Arezzo and G. P. Pirola [1], [14], and they studied a relationship between the nullity of a minimal surface and the differential of its real period map from the viewpoint of complex geometry. The present paper develops their theory in terms of a real differential geometric aspect, and, by applying the classical transversal property to the real period map, we obtain the numerical evidence for the existence of real nine-dimensional manifolds of triply periodic minimal surfaces which include such one-parameter families. For each case that the transversal property fails, we give values of parameters from which new one-parameter families of triply periodic minimal surfaces issue.

Metric lines in the jet space

Abstract In the realm of sub-Riemannian manifolds, a relevant question is: what are the metric lines (isometric embedding of the real line)? The space of k k -jets of a real function of one real variable x x , denoted by J k ( R , R ) {J}^{k}\left({\mathbb{R}},{\mathbb{R}}) , admits the structure of a Carnot group. Every Carnot group is sub-Riemannian manifold, so is J k ( R , R ) {J}^{k}\left({\mathbb{R}},{\mathbb{R}}) . This study aims to present a partial result about the classification of the metric lines within J k ( R , R ) {J}^{k}\left({\mathbb{R}},{\mathbb{R}}) . The method is to use an intermediate three-dimensional sub-Riemannian space R F 3 {{\mathbb{R}}}_{F}^{3} lying between the group J k ( R , R ) {J}^{k}\left({\mathbb{R}},{\mathbb{R}}) and the Euclidean space R 2 {{\mathbb{R}}}^{2} .

Existence and concentration of ground state solutions for a quasilinear Choquard equation with critical exponential growth in ℝ2

In this paper, we consider the following quasilinear Choquard equation − ϵ 2 Δ u + V ( x ) u − ϵ 2 Δ ( u 2 ) u = ϵ μ − 2 ( 1 | x | μ ∗ F ( u ) ) f ( u ) in R 2 , where ϵ > 0 is a parameter, 0 < μ < 2 , ∗ is the convolution product in R 2 , V ( x ) is a continuous real function in R 2 , F ( u ) is the primitive function of f ( u ) and f has critical exponential growth with respect to the Trudinger-Moser inequality. By employing a change of variables, the quasilinear equation can be reduced to a semilinear equation, whose associated functional is well defined in a nonstandard Orlicz space and exhibits a mountain pass geometry. Under suitable assumptions on V and f, we investigate the existence and concentration behavior of positive ground state solutions for the above problem by variational methods.

A note on the space of all Toeplitz operators

Abstract We study Toeplitz operators on the space of all real analytic functions on the real line and the space of all holomorphic functions on finitely connected domains in the complex plane. In both cases, we show that the space of all Toeplitz operators is isomorphic, when equipped with the topology of uniform convergence on bounded sets, with the symbol algebra. This is surprising in view of our previous results, since we showed that the symbol map is not continuous in this topology on the algebra generated by all Toeplitz operators. We also show that in the case of the Fréchet space of all holomorphic functions on a finitely connected domain in the complex plane, the commutator ideal is dense in the algebra generated by all Toeplitz operators in the topology of uniform convergence on bounded sets.

Emergent spacetime and the ergodic hierarchy

Various diagnostics of the emergence of an arrow of time in the bulk description of a holographic theory have been proposed, including the decay of some real time correlation functions and the appearance of type III1 von Neumann algebras carrying half-sided modular inclusions. This Letter puts forward a close parallel between these diagnostics and a quantum formulation of the ergodic hierarchy of dynamical systems. Theories with an emergent spacetime appear to sit near the top of this hierarchy. Published by the American Physical Society 2024

On Solutions of Some Functional Equations of Radical Type

Many authors have studied various functional equations of forms patterned on the equation expressed as ga2+b2=g(a)+g(b), which can be considered, e.g., for real functions. Such equations are usually referred to as radical functional equations or of the radical type. Authors mainly study the so-called Ulam stability of such equations, i.e., they investigate how much the mappings satisfying the equations approximately (in a sense) differ from the exact solutions of these equations. Quite often, information about the solutions of these equations is also provided, but unfortunately, sometimes, such information is given in a misleading or incomplete way. It seems, therefore, that there is a need for a publication containing simple descriptions of such solutions (with appropriate examples), which would help in easy correction of such information and avoidance of similar problems for future authors. This is the main motivation for this expository paper. We present a general approach to the topic and consider two general forms of such equations. Moreover, the results presented in this paper show significant symmetry between the solutions of numerous functional equations and the solutions of equations of the radical type that correspond to them. To make this publication accessible to a wider audience, we omit various related information, avoid advanced generalizations, and present several simple examples.

Strong convergence of the empirical spectral distribution of a unified matrix model

Let [Formula: see text] is a positive definite matrix and [Formula: see text] with [Formula: see text] being independent and identically distributed (i.i.d.) centered real random variables. Consider the matrix [Formula: see text] where [Formula: see text] and [Formula: see text] are two projection matrices (deterministic or random) satisfying [Formula: see text] and [Formula: see text]. Additionally, if [Formula: see text] and [Formula: see text] are random matrices, they are independent of [Formula: see text]. In this paper, we demonstrate that the empirical spectral distribution of [Formula: see text] converges almost surely to a non-random distribution when [Formula: see text] and [Formula: see text], assuming [Formula: see text] has finite second moment.

Eccentric p-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and Graphs

The extension of the concept of p-summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space M afforded by the associated Arens–Eells space, along with the duality between M and the metric dual space M# defined by the real-valued Lipschitz functions on M. However, alternative approaches to measuring distances between sequences of elements of metric spaces (essentially involved in the definition of p-summability) exist. One approach involves considering specific subsets of the unit ball of M# for computing the distances between sequences, such as the real Lipschitz functions derived from evaluating the difference in the values of the metric from two points to a fixed point. We introduce new notions of summability for Lipschitz operators involving such functions, which are characterized by integral dominations for those operators. To show the applicability of our results, in the last part of this paper, we use the theoretical tools obtained in the first part to analyze metric graphs. In particular, we show new results on the behavior of numerical indices defined on these graphs satisfying certain conditions of summability and symmetry.

The first derivative test and the classification of stationary points

Given a real differentiable function f we say that a point x0 is a stationary point of f if f′ (x0) = 0.In any standard single-variable calculus class, students learn how to determine the nature of a stationary point by checking the sign of f(x) in intervals to the left and to the right of the stationary point. In doing so, they are performing the first derivative test.

An ergodic theorem with weights and applications to random measures, homogenization and hydrodynamics

We prove a multidimensional ergodic theorem with weighted averages for the action of the group Zd on a probability space. At level n weights are of the form n−dψ(j/n), j∈Zd, for real functions ψ decaying suitably fast. We discuss applications to random measures and to quenched stochastic homogenization of random walks on simple point processes with long-range random jump rates, allowing to remove the technical Assumption (A9) from Faggionato (2023) [Theorem 4.4]. This last result concerns also some semigroup and resolvent convergence particularly relevant for the derivation of the quenched hydrodynamic limit of interacting particle systems via homogenization and duality. As a consequence we show that also the quenched hydrodynamic limit of the symmetric simple exclusion process on point processes stated in Faggionato (2022) [Theorem 4.1] remains valid when removing the above mentioned Assumption (A9).

Residue Theorem and Logarithmic Residue Theorem

This passage is written for researching the application of residue theorem and Logarithmic Residue theorem. The residue theorem is a powerful tool for the analysis of complex functions. It also can be used to calculate the integral of the real functions. Residue theorem is the extension of the Cauchy’s integral theorem and Cauchy integral formula. Passage researched two theorems by defining the residue in two ways. Each of them is defined by Laurent series and defined by integral. Then the residue theorem is introduced with its definition and applications. The passage points out two instances for the applications of the residue theorem. After the residue theorem, the Logarithmic Residue theorem is listed with its definition and applications. There are also three instances for its applications. The residue theorem and Logarithmic Residue theorem mainly contribute for the research of the complex functions’ integral calculations. The residue theorem provides powerful mathematical tools in certain special types of real integration problems.

Impact and consequences of the error of estimated GFR in patients with heart failure

Heart failure is a highly prevalent disease, which courses with frequent readmissions, mainly by Acute Heart Failure (AHF). Reduced renal function is associated with increased mortality in patients with HF. Therefore, an accurate and precise evaluation of renal function in patients with HF is crucial. The error of estimated GFR (eGFR) is wide and common, showing a ± 30% variability compared to measured GFR (mGFR). However, there is no evidence on the error of formulas in reflecting real renal function and particularly the consequences of this error in patients with AHF. This is a prospective study comparing the impact of mGFR versus eGFR in the onset of cardiovascular (CV) outcomes in patients with AHF. This was tested with cox survival analysis. Measured GFR was determined by the plasma clearance of iohexol-dbs and eGFR by Cockroft-Gould, MDRD, CKD-EPI creatinine, CKD-EPI cystatin-C and CKD-EPI creatinine + cystatin-C equations formulas. Also the agreement between mGFR and eGFR was analyzed. A total of 90 patients were included. Average age was 66 (± 12 years) and 52 (58%) were male. Of them 53 patients (59%) had a cardiovascular event during follow-up, 22 fatal (41%). The agreement between mGFR and eGFR indicated moderate precision and accuracy (concordance correlation coefficient of 0.77; CI = 0.73–0.82). In multiple cox survival analysis, mGFR was significantly associated with cardiovascular events together with NTproBNP, BMI, LVEF and previous coronary artery disease (p = 0.037; HR = 0.98, 95% CI = 0.95–0.99). Estimated GFR by formulas was not significant. In patients with AHF the error of formulas is large, frequent and random, also, mGFR and not eGFR predicted future CV events. The error of eGFR may have clinical consequences in specific subpopulations.

Variations of the zero degrees isotherm and environmental lapse rate recorded with ADSB and Mode S EHS messages

This study focuses on analyzing the altitude of the zero degrees isotherm and variations in the environmental lapse rate (ELR) using the publicly available data collected from Automatic Dependent Surveillance–Broadcast (ADS-B) and Mode S Enhanced Surveillance (EHS) messages emitted by airplanes over a five-month period in 2021. The data was gathered using a professional receiver stationed in Bucharest (Romania). The aviation messages were decoded and the air temperature and pressure were determined, at the location of the airplane. The method has the advantage of the continuous messages that are emitted by the aircrafts during flight that allow instantaneous determination of the meteorological parameters at no additional costs. It can also be extended to permit almost real time maps of the ELR. When data was analyzed and a standard ELR value of 6.5 K/km is employed it was observed that the mean altitude of the 0 degrees isotherm exhibits a seasonal increase during the summer months, with an average altitude of 2874.2 m. The highest recorded altitude of the 0 degrees isotherm was found to be 5346.8 m, near Alexandria city (Romania), on 22.07.2021. Using a standard Least Mean Square algorithm alongside the International Standard Atmosphere pressure formula, the ELR values were calculated from pressure measurements data. The resulting mean ELR for the five-month period was determined to be 5.1331 K/km, slightly lower than the standard value.

Orthogonal Laurent Polynomials of Two Real Variables

ABSTRACTIn this paper, we consider an appropriate ordering of the Laurent monomials , that allows us to study sequences of orthogonal Laurent polynomials of the real variables and with respect to a positive Borel measure defined on such that . This ordering is suitable for considering the multiplication plus inverse multiplication operator on each variable and , and as a result we obtain five‐term recurrence relations, Christoffel–Darboux and confluent formulas for the reproducing kernel and a related Favard's theorem. A connection with the one‐variable case is also presented, along with some applications for future research.

GRID-FAST: A Grid-based Intersection Detection for Fast Semantic Topometric Mapping

This article introduces a novel approach to constructing a topometric map that allows for efficient navigation and decision-making in mobile robotics applications. The method generates the topometric map from a 2D grid-based map. The topometric map segments areas of the input map into different structural-semantic classes: intersections, pathways, dead ends, and pathways leading to unexplored areas. This method is grounded in a new technique for intersection detection that identifies the area and the openings of intersections in a semantically meaningful way. The framework introduces two levels of pre-filtering with minimal computational cost to eliminate small openings and objects from the map which are unimportant in the context of high-level map segmentation and decision making. The topological map generated by GRID-FAST enables fast navigation in large-scale environments, and the structural semantics can aid in mission planning, autonomous exploration, and human-to-robot cooperation. The efficacy of the proposed method is demonstrated through validation on real maps gathered from robotic experiments: 1) a structured indoor environment, 2) an unstructured cave-like subterranean environment, and 3) a large-scale outdoor environment, which comprises pathways, buildings, and scattered objects. Additionally, the proposed framework has been compared with state-of-the-art topological mapping solutions and is able to produce a topometric and topological map with up to 92% fewer nodes than the next best solution. The method proposed in this article has been implemented in the robotics framework ROS and is open-sourced. The code is available at: https://github.com/LTU-RAI/GRID-FAST.

Strain effect on the electronic and optical characteristics of FAGeX3 (X=Cl, Br, and I) perovskite materials: DFT analysis

This study investigates the electronic and optical properties of a perovskite material known as Formamidinium Germanium Halide (FAGeX3), where X represents the elements Chlorine (Cl), Bromine (Br), and Iodine (I). We explore the bandgap, density of state (DOS), and partial density of state (PDOS) to understand their electronic properties. We use two methods, PBE and HSE-06, to determine the bandgap. Further, we investigate the optical properties by investigating the real and imaginary functions of the dielectric constant, refractive index, electron energy loss function, and absorption coefficient. Our research extends to the impact of biaxial strain, both tensile and compressive, in the −6% to +6 % range. Without strain, the materials exhibit direct bandgaps at the R point, with FAGeCl3 showing the highest bandgap (2.1359 eV), followed by FAGeBr3 (1.7325 eV), and FAGeI3 with the lowest (1.2581 eV). Our results reveal that applying tensile strain increases the bandgap and induces a blueshift, shifting the optical responses to shorter wavelengths, while compressive strain reduces the bandgap and causes a redshift, enhancing longer wavelength responses. Our findings demonstrate that FAGeX3 perovskites exhibit highly tunable electronic and optical properties under strain, making them exceptional candidates for advanced optoelectronic applications.

Comprehensive study on the optoelectronic properties of ZnSnP2 compound by DFT and simulation for the application in a photodetector

Abstract This article presents the density function theory (DFT) calculation of ZnSnP2 (ZTP) and its application as a photodetector. A DFT calculation has been performed to determine ZTP's optical and electronic characteristics. The direct bandgap of ZnSnP2 is found to be 1.0 eV which agrees well with the previously reported bandgap (0.95 eV). The total density of states (TDOS) of ZTP is determined to be 1.40 states/eV which is attributed to the 3p orbital of P, with minor impacts from the 3d orbital of Zn and the 5p orbital of Sn to TDOS. The real and imaginary dielectric functions and refractive indices ZTP have been determined to be 16.44 and 17.60, 4.07 and 2.92, respectively. The absorption coefficient and reflectivity of ZTP obtained from this investigation are 2.6×105 cm-1 and 57.5%, respectively. After calculating the electrionic and optical properties, ZTP-based n-CdS/p-ZnSnP2/p+-AlSb photodetector (PD) with CdS and AlSb as the window and back surface field (BSF) layers, respectively, has been computationally analyzed and optimized using solar cell capacitance simulator (SCAPS-1D). In a single heterojunction, the photocurrent, voltage, responsivity, and detectivity values have been obtained at 44.52 mA/cm2, 0.66 V, 0.73 A/W, and 6.81×1013 Jones, respectively. Insertion of a thin AlSb BSF layer improves the photocurrent, voltage, responsivity, and detectivity to 48.75 mA/cm2, 0.78 V, 0.86 A/W, and 8.22×1014 Jones, respectively. The outcomes are highly promising for the fabrication of a high performance ZTP-based PD in the future.

Scenario Generation Based on Ant Colony Optimization for Modelling Stochastic Variables in Power Systems

Uncertainty is an important subject in optimization problems due to the unpredictable nature of real variables in the power system area, which can condition the solution’s accuracy. The effective modelling of stochastic variables can contribute to the reduction in losses in the system under evaluation and facilitate the implementation of an effective response in advance. To model uncertainty variables, the most extended technique is the scenario generation (SG) method. This method evaluates possible combinations of complete curves. Classical scenario generation methods are founded on probability distributions or robust stochastic optimization. This paper proposes a novel approach for constructing scenarios using the Ant Colony Optimization (ACO) algorithm, referred to as ACO-SG. This methodology does not require a previous statistical study of uncertainty data to generate new scenarios. A historical dataset and the desired number of scenarios are the inputs inserted into the algorithm. In the case study, the algorithm used historical data from the Savona Campus Smart Polygeneration Microgrid of the University of Genoa. The approach was applied to generate scenarios of photovoltaic generation and building consumption. The low values of the Euclidean distance were used in order to check the validity of the scenarios. Moreover, the error deviation of the scenarios generated with the goal of daily power were 1.77% and 0.144% for the cases of PV generation and building consumption, respectively. The different results for both cases are explained by the characteristics of the specific cases. Despite these different results, both were significantly low, which indicates the capability of the algorithm to generate any kind of feature within curves and its adaptability to any case of SG.

Elastic Fields Around Multiple Stiff Prestressed Arcs Located on a Circle

Abstract The plane strain problem of an isotropic elastic matrix subjected to uniform far-field load and containing multiple stiff prestressed arcs located on the same circle is considered. The boundary conditions for the arcs are described by those of either Gurtin–Murdoch or Steigmann–Ogden theories in which the arcs are endowed with their own elastic energies. The material parameters for each arc can in general be different. The problem is reduced to the system of real variables hypersingular boundary integral equations in terms of two scalar unknowns expressed via the components of the stress tensors of the arcs. The unknowns are approximated by the series of trigonometric functions that are multiplied by the square root weight functions to allow for automatic incorporation of the tip conditions. The coefficients in series are found from the system of linear algebraic equations that are solved using the collocation method. The expressions for the stress intensity factors are derived and numerical examples are presented to illustrate the influence of governing dimensionless parameters.