Continuous Functions Research Articles

Extraction rates of algorithmically random continuous functionals

Extraction rates of algorithmically random continuous functionals

Discrete Pseudo-Quasi Overlap Functions and Their Applications in Fuzzy Multi-Attribute Group Decision-Making

The overlap function, a continuous aggregation function, is widely used in classification, decision-making, image processing, etc. Compared to applications, overlap functions have also achieved fruitful results in theory, such as studies on the fundamental properties of overlap functions, various generalizations of the concept, and the construction of additive and multiplicative generators based on overlap functions. However, most of the research studies on the overlap functions mentioned above assume commutativity and continuity, which can limit their practical applications. In this paper, we remove the symmetry and continuity from overlap functions and define discrete pseudo-quasi-overlap functions on finite chains. Meanwhile, we also discuss their related properties. Then, we introduce pseudo-quasi-overlap functions on sub-chains and construct discrete pseudo-quasi-overlap functions on finite chains using these sub-chain functions. Unlike quasi-overlap functions on finite chains generated by the ordinal sum, discrete pseudo-quasi-overlap functions on finite chains constructed through pseudo-quasi-overlap functions on different sub-chains are dissimilar. Eventually, we remove the continuity from pseudo-automorphisms and propose the concept of pseudo-quasi-automorphisms. Based on this, we utilize pseudo-overlap functions, pseudo-quasi-automorphisms, and integral functions to obtain discrete pseudo-quasi-overlap functions on finite chains; moreover, we apply them to fuzzy multi-attribute group decision-making. The results indicate that—compared to overlap functions and pseudo-overlap functions—discrete pseudo-quasi-overlap functions on finite chains have stronger flexibility and a wider range of practical applications.

HPB O04 - Human Splenic Ex-Vivo Perfusion Model: A Platform for Translational Basic Science Research

Abstract Background The function of the spleen has always been shrouded in mystery and since the Renaissance leading figures have attempted to understand its function. Vesalius questioned Galen and Marcello Malpighi (1628-1694) gave the first accurate description of the histology and the physiology. The spleen still poses unique challenges as invasive approaches may produce intraperitoneal bleeding, differences between human and animal spleens complicates comparative studies, and post-mortem examinations reveal only late-stage disease manifestations and functional studies are impossible. This study describes a human splenic ex-vivo perfusion model for translational research, using microbiological studies as markers of splenic physiological function. Method Patients with left-sided pancreatic cancers scheduled for distal pancreatic and splenic resection were identified from the Hepato-Pancreatic and Biliary MDT at the University Hospitals of Leicester NHS Trust. Patients were informed, consented, and the surgical approach remained standard, with splenic artery ligation deferred until specimen extraction. The artery was cannulated, and residual blood clots were cleared with Custodiol® HTK solution. The perfusion technique employs an extracorporeal circuit incorporateing anoxygenator and a heated water chamber maintaining a 37°C perfusate. The perfused spleen was infected with Streptococcus Pneumoniae strains, with regular biopsies and perfusate gas analysis performed. Results Since February 2023, nine spleens have been successfully perfused, with an average weight of 169.5 grams, cold ischemic time of 116 minutes, and warm ischemic time of 12.2 minutes. Pressure was maintained at 80 mmHg and hourly perfusate gas analysis demonstrated a mean pH of 7.1 and a gradual increase in lactate. Histopathological examination revealed clear distinctions between the lymphoid tissue and splenic red pulp, with intact central arterioles and healthy cell morphology at time 0 and after 6 hours of perfusion. Additionally, the perfused spleens cleared S. pneumoniae strains to below-detection levels. Conclusion Our work demonstrates that a stable human splenic ex-vivoperfusion model which maintains structural and physiological integrity can be achieved from tissue that would otherwise be discarded. The continued physiological function is confirmed by the clearance of S. pneumoniae. While currently employed for microbiological studies, this model has the potential to facilitate a wide range of different translational science studieswhile reducing the need for animal experimentation. The perfusion times employed for the present study were sufficient to demonstrate bacterial clearance which was the aim but there is capacity in the system to significantly prolong the duration.

Paired piecewise linear regression model with fixed nodes

In this paper we develop a method for constructing a paired regression model in the class of piecewise linear continuous functions with fixed nodes. The concept of linear division of the correlation field, its partial sections and its nodes is introduced into consideration. Using the linear division of the correlation field, a class of piecewise linear functions with fixed nodes is determined. The linear division is revealed during the visual analysis of the correlation field. Using the least squares method, a system of linear equations is compiled to find point estimates of the parameters of the approximating function. With the exception of two unknown angular coefficients (unknown) this system is reduced to a tridiagonal system of equations, the unknowns of which are the nodal values of the approximating function. The tridiagonal system is solved by the run-through method. An algorithm was constructed to demonstrate the operation of the developed method. Its initial data is arrays of the corresponding values of the explanatory and dependent variables, as well as an array of numbers of the right ends of the intervals defining the nodes. An array of fixed nodes is constructed from an array of values of the explanatory variable and an array of numbers of the right ends of the intervals. Next, an array of point estimates of the parameters of the approximating function is constructed. This algorithm is implemented in Python in the form of user-defined functions.

A Hierarchical Bayesian Model for Estimating Age-Specific COVID-19 Infection Fatality Rates in Developing Countries.

The COVID-19 infection fatality rate (IFR) is the proportion of individuals infected with SARS-CoV-2 who subsequently die. As COVID-19 disproportionately affects older individuals, age-specific IFR estimates are imperative to facilitate comparisons of the impact of COVID-19 between locations and prioritize distribution of scarce resources. However, there lacks a coherent method to synthesize available data to create estimates of IFR and seroprevalence that vary continuously with age and adequately reflect uncertainties inherent in the underlying data. In this article, we introduce a novel Bayesian hierarchical model to estimate IFR as a continuous function of age that acknowledges heterogeneity in population age structure across locations and accounts for uncertainty in the estimates due to seroprevalence sampling variability and the imperfect serology test assays. Our approach simultaneously models test assay characteristics, serology, and death data, where the serology and death data are often available only for binned age groups. Information is shared across locations through hierarchical modeling to improve estimation of the parameters with limited data. Modeling data from 26 developing country locations during the first year of the COVID-19 pandemic, we found seroprevalence did not change dramatically with age, and the IFR at age 60 was above the high-income country estimate for most locations.

Multi‐bump solutions for the nonlinear magnetic Schrödinger equation with logarithmic nonlinearity

AbstractIn this paper, we study the following nonlinear magnetic Schrödinger equation with logarithmic nonlinearity where the magnetic potential , is a parameter and the nonnegative continuous function has the deepening potential well. Using the variational methods, we obtain that the equation has at least multi‐bump solutions when is large enough.

Shape Preserving Approximation of Periodic Functions: Conclusion

AbstractWe give here the final results about the validity of Jackson-type estimates in comonotone approximation of $$2\pi $$ 2 π -periodic functions by trigonometric polynomials ($$q=1$$ q = 1 ). For coconvex ($$q=2$$ q = 2 ) and the so called co-q-monotone, $$q>2$$ q > 2 , approximations, everything is known by now. Thus, this paper concludes the research on Jackson type estimates of Shape Preserving Approximation of periodic functions by trigonometric polynomials. It is interesting to point out that the results for comonotone approximation of a periodic function are substantially different than the analogous results for comonotone approximation of a continuous function on a finite interval, by algebraic polynomials.

On Some Topological Properties of Ch(X) and of the Dual Operator

The hypo-topology on the algebra C(X) of real-valued continuous functions defined on a Tychonoff space 2X f ∈ C(X) with its hypograph, hypof = {(x,t) ∈ X x R:f(x) ≥ t}. This topology is very useful in the calculus of variations and in optimization theory (e.g. Maximization problems). We denote C(X) with the hypo-topology by Ch(X). Our study deals with fundamental properties of these function spaces, and with the linear operators on them and as well as the characterization of the topological properties of Ch(X) in terms of topological properties of the base space X. We are studying the linear operator between the functional algebras Ch(X) and the Ch(Y). We are primarily concerned with the continuity of the evaluation functional, the general evaluation and the continuity of the characters of Ch(X) before the investigation of the properties of the dual operator f* of a continuous function f:X→Y. This operator is defined by f*:Ch (Y)→Ch (X), where f*(g)=gof for all g ∈ C(Y). The continuity of f* enables us to characterize the continuity of an algebra homomorphism of the type φ:Ch (Y)→Ch (X) for a realcompact space Y. For such a space, we present a type of Riesz theorm with states that an algebra homomorphism φ: Ch(Y)→Ch(X) is continuous if and only if there exists a unique hypo-function f: X→Y such that φ = f*. There after, we give the equivalence between the properties of f and those of f*. The study of the continuous linear functional on Ch(X) helps us to compute the topological dual space of this algebra. We show here that this dual space is useful only when the set of isolated points of X is dense.

Gelfand Triplets, Ladder Operators and Coherent States

Inspired by a similar construction on Hermite functions, we construct two series of Gelfand triplets, each one spanned by Laguerre–Gauss functions with a fixed positive value of one parameter, considered as the fundamental one. We prove the continuity of different types of ladder operators on these triplets. Laguerre–Gauss functions with negative values of the fundamental parameter are proven to be continuous functionals on one of these triplets. Different sorts of coherent states are considered and proven to be in some spaces of test functions corresponding to Gelfand triplets.

Dominated and absolutely summing operators on the space Crc(X,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\,C_{rc}(X,E)$$\\end{document} of vector-valued continuous functions

Let X be a completely regular Hausdorff space and E and F be Banach spaces. Let Crc(X,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{rc}(X,E)$$\\end{document} denote the Banach space of all continuous functions f:X→E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:X\\rightarrow E$$\\end{document} such that f(X) is a relatively compact set in E, and βσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _\\sigma $$\\end{document} be the strict topology on Crc(X,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{rc}(X,E)$$\\end{document}. We characterize dominated and absolutely summing operators T:Crc(X,E)→F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T:C_{rc}(X,E)\\rightarrow F$$\\end{document} in terms of their representing operator-valued Baire measures. It is shown that every absolutely summing (βσ,‖·‖F)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\beta _\\sigma ,\\Vert \\cdot \\Vert _F)$$\\end{document}-continuous operator T:Crc(X,E)→F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T:C_{rc}(X,E)\\rightarrow F$$\\end{document} is dominated. Moreover, we obtain that every dominated operator T:Crc(X,E)→F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T:C_{rc}(X,E)\\rightarrow F$$\\end{document} is absolutely summing if and only if every bounded linear operator U:E→F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U:E\\rightarrow F$$\\end{document} is absolutely summing.

Time Scale Transformation in Bivariate Pearson Diffusions: A Shift from Light to Heavy Tails

Heavy-tailed Pearson diffusions provide a natural alternative to well-known Ornstein–Uhlenbeck and Cox–Ingersoll–Ross processes in applications that require addressing heavy-tailed behavior. In this paper, all three heavy-tailed Pearson diffusions, having inverse gamma, Fisher–Snedecor and Student stationary distributions, are constructed via an absolutely continuous time-change process employed in a specific functional transformation of CIR or OU. Moreover, time-change rates in stochastic clocks are continuous functionals of the CIR process.

On Ψω-factorizable groups

A topological group G is called Ψω-factorizable (resp. M-factorizable) if every continuous real-valued function on G admits a factorization via a continuous homomorphism onto a topological group H with ψ(H)≤ω (resp. a first-countable group). The first purpose of this article is to discuss some characterizations of Ψω-factorizable groups. It is shown that a topological group G is Ψω-factorizable if and only if every continuous real-valued function on G is Gδ-uniformly continuous, if and only if for every cozero-set U of G, there exists a Gδ-subgroup N of G such that UN=U. Sufficient conditions on the Ψω-factorizable group G to be M-factorizable are that G is τ-fine and τ-steady for a cardinal τ.

A note on the L1 discretization error for the Caputo derivative in Hölder spaces

We establish uniform error bounds of the L1 discretization of the Caputo derivative of Hölder continuous functions. The result can be understood as: error = degree of smoothness - order of the derivative. We present an elementary proof and illustrate its optimality with numerical examples.

Relatively functionally countable subsets of products

A subset A of a topological space X is called relatively functionally countable (RFC) in X, if for each continuous function f:X→R the set f[A] is countable. We prove that all RFC subsets of a product ∏n∈ωXn are countable, assuming that spaces Xn are Tychonoff and all RFC subsets of every Xn are countable. In particular, in a metrizable space every RFC subset is countable.The main tool in the proof is the following result: for every Tychonoff space X and any countable set Q⊆X there is a continuous function f:Xω→R2 such that the restriction of f to Qω is injective.

Evaluation of the cross sections and photoelectron angular distribution parameters following atomic photoionization under extreme conditions

In this manuscript, we report on a theoretical study of the atomic structures, cross sections, and photoelectron angular distribution parameters following the atomic photoionization under extreme conditions. To achieve this goal, a relativistic approach using the Dirac-Coulomb Hamiltonian within the framework of relativistic configuration interaction, taking advantage of independent particle basis wave functions, is proposed. To describe the interaction of charged particles in the ideal and non-ideal plasmas, the Debye potential and the pseudopotential are implemented, the latter being derived from a progressive resolution of the Bogolyubov chain equations. Both bound and continuous state wave functions, essential for a comprehensive understanding of quantum systems, are determined from the modified local central potential, which is obtained in a self-consistent manner to represent the electronic shielding effect on the nuclear potential. The photoionization processes are evaluated using the relativistic distorted wave approach, which is consistent with the principles of relativistic Dirac theory and thus provides an accurate description of the dynamics. As a test desk, the present method is applied to the evaluation of the energies, ionization potentials, wave functions, cross sections, and photoelectron angular distribution parameters, using the single photon ionization of the Li-like Fe XXIV ions as the basis for analysis. Our results demonstrate that the plasma environment effect not only decreases the ionization potentials and increases the cross sections but also affects the photoelectron angular distribution parameters across different shells, leading to a more balanced and symmetrical photoelectron distribution pattern. A detailed comparison is made between our results, and the available well-established theoretical predictions and experimental data of the unshielded case in the literature shows a good agreement. The present work provides novel insights and theoretical models that not only help us to better understand the fundamental properties of the complex systems but are also beneficial for innovative applications in the study of astrophysical and laboratory plasmas.

Simplicity of crossed products of the actions of totally disconnected locally compact groups on their boundaries

We prove that if a totally disconnected locally compact group admits a topologically free boundary, then the reduced crossed product of continuous functions on its Furstenberg boundary by the group is simple. We also prove a partial converse of this result.

$\delta(\tau_{1},\tau_{2})$-Continuous Functions

This paper introduces a new class of functions called $\delta(\tau_{1},\tau_{2})$-continuous functions. Several characterizations of $\delta(\tau_{1},\tau_{2})$-continuous functions are investigated. The relationships between $\delta(\tau_{1},\tau_{2})$-continuity and the other types of $\delta(\tau_{1},\tau_{2})$-continuity are also discussed.

Pythagorean Fuzzy Soft Somewhat Continuous Functions

In this work, we introduce the concept of Pythagorean fuzzy soft somewhat open sets utilizing the Pythagorean fuzzy soft interior operator, extending its application to Pythagorean fuzzy soft topological spaces. This study aims to enhance decision-making processes in future-assisted economies by addressing the limitations of existing fuzzy set theories. We investigate the distinctive properties of Pythagorean fuzzy soft somewhat open sets as a subclass of Pythagorean fuzzy soft somewhere dense sets. Additionally, we explore Pythagorean fuzzy soft somewhat metamorphism’s within the context of Pythagorean fuzzy soft somewhat continuous functions, offering new insights into their topological invariant. Through detailed analysis and examples, we demonstrate the applicability of these concepts in various scientific and engineering problems. This work provides a comprehensive framework for understanding and utilizing Pythagorean fuzzy soft sets in complex decision-making scenarios. Finally, we compare various relationships across some generalizations of Pythagorean fuzzy soft continuous functions.

Generalized Linear Differential Equation using Hyers - Ulam Stability Approach

In this paper, We demonstrate the Hyers - Ulam stability of linear differential equation of fourth order. We interact with the differential equation\begin{align*}\gamma^{iv} (\omega) + \rho_1 \gamma{'''} (\omega)+ \rho_2 \gamma{''} (\omega) + \rho_3 \gamma' (\omega) + \rho_4 \gamma(\omega) = \chi(\omega),\end{align*}where $\gamma \in c^4 [\alpha,\beta], \chi \in [\alpha,\beta]$. Hyers-Ulam stability concerns the robustness of solutions of functional equations under small perturbations, ensuring that a solution approximately satisfying the equation is close to an exact solution. We extend this concept to fourth-order linear differential equations and continuous functions. Using fixed-point methods and various norms, we establish conditions under which such equations exhibit Hyers-Ulam stability. Several illustrative examples are provided to demonstrate the application of these results in specific cases, contributing to the growing understanding of stability in higher-order differential equations. Our findings have implications in both theoretical research and practical applications in physics and engineering.

Upper and Lower Rarely m-I-Continuous Multifunctions

In 1979, Popa [24] first introduced rarely continuous functions. In this paper, we introduce upper and lower rarely m-continuous multifunctions. Moreover, we extend this concept to a multifunction F : (X, τ, I) → (Y, σ), where (X, τ, I) is an ideal topological space.