Functional Analytic Approach Research Articles

The functional analytic approach for quasi-periodic boundary value problems for the Helmholtz equation

The functional analytic approach for quasi-periodic boundary value problems for the Helmholtz equation

FunSpace: A functional and spatial analytic approach to cell imaging data using entropy measures.

Spatial heterogeneity in the tumor microenvironment (TME) plays a critical role in gaining insights into tumor development and progression. Conventional metrics typically capture the spatial differential between TME cellular patterns by either exploring the cell distributions in a pairwise fashion or aggregating the heterogeneity across multiple cell distributions without considering the spatial contribution. As such, none of the existing approaches has fully accounted for the simultaneous heterogeneity caused by both cellular diversity and spatial configurations of multiple cell categories. In this article, we propose an approach to leverage spatial entropy measures at multiple distance ranges to account for the spatial heterogeneity across different cellular organizations. Functional principal component analysis (FPCA) is applied to estimate FPC scores which are then served as predictors in a Cox regression model to investigate the impact of spatial heterogeneity in the TME on survival outcome, potentially adjusting for other confounders. Using a non-small cell lung cancer dataset (n = 153) as a case study, we found that the spatial heterogeneity in the TME cellular composition of CD14+ cells, CD19+ B cells, CD4+ and CD8+ T cells, and CK+ tumor cells, had a significant non-zero effect on the overall survival (p = 0.027). Furthermore, using a publicly available multiplexed ion beam imaging (MIBI) triple-negative breast cancer dataset (n = 33), our proposed method identified a significant impact of cellular interactions between tumor and immune cells on the overall survival (p = 0.046). In simulation studies under different spatial configurations, the proposed method demonstrated a high predictive power by accounting for both clinical effect and the impact of spatial heterogeneity.

Stochastic differential equations with critically irregular drift coefficients

This paper is concerned with stochastic differential equations (SDEs for short) with irregular coefficients. By utilising a functional analytic approximation approach, we establish the existence and uniqueness of strong solutions to a class of SDEs with critically irregular drift coefficients in a new critical Lebesgue space, where the element may be only weakly integrable in time. To be more precise, let b:[0,T]×Rd→Rd be Borel measurable, where T>0 is arbitrarily fixed and d⩾1. We consider the following SDEXt=x+∫0tb(s,Xs)ds+Wt,t∈[0,T],x∈Rd, where {Wt}t∈[0,T] is a d-dimensional standard Wiener process. For p,q∈[1,+∞), we denote by C[q]([0,T];Lp(Rd)) the space of all Borel measurable functions f such that t1qf(t)∈C([0,T];Lp(Rd)). If b=b1+b2 such that |b1(T−⋅)|∈C[q]([0,T];Lp(Rd)) with 2/q+d/p=1 and ‖b1(T−⋅)‖C[q]([0,T];Lp(Rd)) is sufficiently small, and that b2 is bounded and Borel measurable, then we show that there exists a weak solution to the above equation, and if in addition limt↓0⁡‖t1qb(T−t)‖Lp(Rd)=0, the pathwise uniqueness holds. Furthermore, we obtain the strong Feller property of the semi-group and the existence of density associated with the above SDE. Besides, we extend the classical results concerning partial differential equations (PDEs) of parabolic type with Lq(0,T;Lp(Rd)) coefficients to the case of parabolic PDEs with L[q]∞(0,T;Lp(Rd)) coefficients, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Theorem 3.1). Our results extend Krylov-Röckner and Krylov's profound results of SDEs with singular time dependent drift coefficients [20,23] to the critical case of SDEs with critically irregular drift coefficients in a new critical Lebesgue space.

Large time probability of failure in diffusive search with resetting for a random target in ℝ^{𝕕}–A functional analytic approach

We consider a stochastic search model with resetting for an unknown stationary target a ∈ R d , d ≥ 1 a\in \mathbb {R}^d,\ d\ge 1 , with known distribution μ \mu . The searcher begins at the origin and performs Brownian motion with diffusion coefficient D D . The searcher is also equipped with an exponential clock with rate r > 0 r>0 , so that if it has failed to locate the target by the time the clock rings, then its position is reset to the origin and it continues its search anew from there. In dimension one, the target is considered located when the process hits the point a a , while in dimensions two and higher, one chooses an ϵ 0 > 0 \epsilon _0>0 and the target is considered located when the process hits the ϵ 0 \epsilon _0 -ball centered at a a . Denote the position of the searcher at time t t by X ( t ) X(t) , let τ a \tau _a denote the time that a target at a a is located, and let P 0 d ; ( r , 0 ) P^{d;(r,0)}_0 denote probabilities for the process starting from 0. Taking a functional analytic point of view, and using the generator of the Markovian search process and its adjoint, we obtain precise estimates, with control on the dependence on a a , for the asymptotic behavior of P 0 d ; ( r , 0 ) ( τ a > t ) P^{d;(r,0)}_0(\tau _a>t) for large time, and then use this to obtain large time estimates on ∫ R d P 0 d ; ( r , 0 ) ( τ a > t ) d μ ( a ) \int _{\mathbb {R}^d}P^{d;(r,0)}_0(\tau _a>t)d\mu (a) , the probability that the searcher has failed up to time t t to locate the random target, for a variety of families of target distributions μ \mu . Specifically, for B , l > 0 B,l>0 and d ∈ N d\in \mathbb {N} , let μ B , l ( d ) ∈ P ( R d ) \mu ^{(d)}_{B,l}\in \mathcal {P}(\mathbb {R}^d) denote any target distribution with density μ B , l ( d ) ( a ) \mu _{B,l}^{(d)}(a) that satisfies lim | a | → ∞ log ⁡ μ B , l ( d ) ( a ) | a | l = − B . \begin{equation*} \lim _{|a|\to \infty }\frac {\log \mu _{B,l}^{(d)}(a)}{|a|^l}=-B. \end{equation*} Then we prove that lim t → ∞ 1 ( log ⁡ t ) l log ⁡ ∫ R d P 0 d ; ( r , 0 ) ( τ a > t ) μ B , l ( d ) ( d a ) = − B ( D 2 r ) l 2 . \begin{equation*} \lim _{t\to \infty }\frac 1{(\log t)^l}\log \int _{\mathbb {R}^d} P_0^{d;(r,0)}(\tau _a>t)\mu _{B,l}^{(d)}(da)=-B(\frac D{2r})^\frac l2. \end{equation*} The result is independent of the dimension. In particular, for example, if the target distribution is a centered Gaussian of any dimension with variance σ 2 \sigma ^2 , then for any δ > 0 \delta >0 , the probability of not locating the target by time t t falls in the interval ( e − ( 1 + δ ) D 4 r σ 2 ( log ⁡ t ) 2 , e − ( 1 − δ ) D 4 r σ 2 ( log ⁡ t ) 2 ) \big (e^{-(1+\delta )\frac {D}{4r\sigma ^2}(\log t)^2}, e^{-(1-\delta )\frac {D}{4r\sigma ^2}(\log t)^2}\big ) , for sufficiently large t t .

Optimal control of a two-dimensional contact problem with multiple unilateral constraints

In this article, we are concerned with optimal control of a frictionless contact problem with multiple unilateral constraints for a two-dimensional bar. The existence of an optimal trajectory-control pair is firstly proven under the framework of general cost functional. The Pontryagin maximum principle is then established for the investigational system equipped with many equality and inequality constraints in fixed final horizon case, owing to the Dubovitskii and Milyutin functional analytical approach. A remark concludes the article with the discussion, which address the utilization of obtained necessary optimality condition.

Normal approximation via non-linear exchangeable pairs

We propose a new functional analytic approach to Stein's method of exchangeable pairs that does not require the pair at hand to satisfy any approximate linear regression property. We make use of this theory in order to derive abstract bounds on the normal and Gamma approximation of certain functionals in the Wasserstein distance. Moreover, we illustrate the relevance of this approach by means of three instances of situations to which it can be applied: Functionals of independent random variables, finite population statistics and functionals on finite groups. In the independent case, and in particular for symmetric $U$-statistics, we demonstrate in which respect this approach yields fundamentally better bounds than those in the existing literature. Finally, we apply our results to provide Wasserstein bounds in a CLT for subgraph counts in geometric random graphs based on $n$ i.i.d. points in Euclidean space as well as to the normal approximation of Pearson's statistic.

Crafting Sequences of Sight and Sound: A Behavior Analysis of Filmmaking.

The discipline of film studies often engages in analyses of the functions of filmmakers' decisions in terms of their effects on viewers. Behavior analysis uses a similar, functional-analytic approach toward understanding the relationship between individuals' behavior and the environmental effects that maintain their behavior. Given converging similarities between the two disciplines, a functional analysis of filmmaking is provided, using Skinner (1957)'s Verbal Behavior as a guiding framework. Similar to behavioral conceptualizations of language and speaker-listener verbal episodes, the analysis prioritizes functional explanation of the controlling variables and conditions that underlie the meaning of filmmakers' behavior and behavioral products, rather than solely focusing on their topographical description. Viewers' responses to the audiovisual stimuli of the film are emphasized as key controlling variables, through rules specifying contingency relations as well as through contingency shaping, including when the filmmaker acts as a self-viewer who directly shapes their own behavior. Their responding as a self-viewer during the production and editing of a film is explored as a problem-solving process, similar to other artists who serve as their own audience when creating and editing their behavioral products.

The Availability of Critical Thinking Skills in the Arabic Language and Islamic Education Textbooks of the Third Grade Depending on their Teachers’ Point View in Jerash

This study aimed at investigating the availability of critical thinking skills in the Arabic language and Islamic education textbooks depending on their teachers’ point of view in Jerash Governorate. The researchers have used the functional analytical approach. They prepared a questionnaire that consisted of 36 paragraphs which have targeted four skills, and the sample consisted of 97 female teachers. After finding the arithmetic mean and the standard deviation of the four skills, the results revealed that the targeted skills are highly present in the Arabic language and Islamic education textbooks according to their teachers’ point of view. Accordingly, the researchers recommended conducting more studies that investigate critical thinking, and training the teachers to teach it, as well as to involve teachers in the process of composing school textbooks.

On functional analytic approach for Gleason’s problem in the theory of SCV

On functional analytic approach for Gleason’s problem in the theory of SCV

Magneto-transport and thermal properties of the Yukawa potential in cosmic string space-time

Yukawa potential has garnered remarkable attention since its proposition. It is applied in condensed matter physics, plasma physics, high energy physics and related areas. Based on this widespread applicability, this study focused on the analysis of the effects of topological defect on the thermal and magnetic properties of this system. Non-relativistic treatment of this potential in the presence of magnetic fields and Aharonov–Bohm is performed using the functional analytical approach (FAA). The energy equation and the wavefunction are analytically derived. The accessible energy levels are summed up to obtain to get the partition function that is used to derive the expressions for the thermo-magnetic properties of the Yukawa potential. These properties are analysed in detail by means of graphical representations. It is observed that in the various settings of the analysis, the system has a diamagnetic characteristic and the behaviour of the heat capacity is in accordance with the recognized law of Dulong–Petit, although some anomalies are observed. This irregular behaviour could be attributed to Schottky anomaly. The results of our research will be helpful in many fields of physics: condensed matter physics, high energy physics, etc.

SPF: A spatial and functional data analytic approach to cell imaging data.

The tumor microenvironment (TME), which characterizes the tumor and its surroundings, plays a critical role in understanding cancer development and progression. Recent advances in imaging techniques enable researchers to study spatial structure of the TME at a single-cell level. Investigating spatial patterns and interactions of cell subtypes within the TME provides useful insights into how cells with different biological purposes behave, which may consequentially impact a subject's clinical outcomes. We utilize a class of well-known spatial summary statistics, the K-function and its variants, to explore inter-cell dependence as a function of distances between cells. Using techniques from functional data analysis, we introduce an approach to model the association between these summary spatial functions and subject-level outcomes, while controlling for other clinical scalar predictors such as age and disease stage. In particular, we leverage the additive functional Cox regression model (AFCM) to study the nonlinear impact of spatial interaction between tumor and stromal cells on overall survival in patients with non-small cell lung cancer, using multiplex immunohistochemistry (mIHC) data. The applicability of our approach is further validated using a publicly available multiplexed ion beam imaging (MIBI) triple-negative breast cancer dataset.

On the value of non-Markovian Dynkin games with partial and asymmetric information

We prove that zero-sum Dynkin games in continuous time with partial and asymmetric information admit a value in randomised stopping times when the stopping payoffs of the players are general càdlàg measurable processes. As a by-product of our method of proof we also obtain existence of optimal strategies for both players. The main novelties are that we do not assume a Markovian nature of the game nor a particular structure of the information available to the players. This allows us to go beyond the variational methods (based on PDEs) developed in the literature on Dynkin games in continuous time with partial/asymmetric information. Instead, we focus on a probabilistic and functional analytic approach based on the general theory of stochastic processes and Sion’s min-max theorem (Pacific J. Math. 8 (1958) 171–176). Our framework encompasses examples found in the literature on continuous time Dynkin games with asymmetric information and we provide counterexamples to show that our assumptions cannot be further relaxed.

NUMERICAL COMPARISON OF ITERATIVE AND FUNCTIONAL-ANALYTICAL ALGORITHMS FOR INVERSE ACOUSTIC SCATTERING

Abstract In this work the numerical solution of acoustic tomography problem based on the iterative and functional-analytical algorithms is considered. The mathematical properties of these algorithms were previously described in works of R.G. Novikov for the case of the Schr ̈odinger equation. In the present work, for the case of two-dimensional scalar Helmholtz equation, the efficiency of the iterative algorithm in reconstruction of middle strength scat- terers and advantages of the functional-analytical approach in recovering strong scatterers are demonstrated. A filtering procedure is considered in the space of wave vectors, which additionally increases the convergence of the iterative algorithm. Reconstruction results of sound speed perturbations demonstrate the comparable noise immunity and resolution of the considered algorithms when reconstructing middle strength scatterers. A comparative nu- merical investigation of the iterative and functional-analytical algorithms in inverse acoustic scattering problems is implemented in this work for the first time.

Asymptotic behavior of the solutions of a transmission problem for the Helmholtz equation: A functional analytic approach

Let Ωi, Ωo be bounded open connected subsets of that contain the origin. Let for small ϵ > 0. Then, we consider a linear transmission problem for the Helmholtz equation in the pair of domains ϵΩi and Ω(ϵ) with Neumann boundary conditions on ∂Ωo. Under appropriate conditions on the wave numbers in ϵΩi and Ω(ϵ) and on the parameters involved in the transmission conditions on ϵ∂Ωi, the transmission problem has a unique solution (ui(ϵ, ·), uo(ϵ, ·)) for small values of ϵ > 0. Here, ui(ϵ, ·) and uo(ϵ, ·) solve the Helmholtz equation in ϵΩi and Ω(ϵ), respectively. Then, we prove that if x ∈ Ωo \ {0}, then uo(ϵ, x) can be expanded into a convergent power expansion of ϵ, for ϵ small enough. Here, if n is even and if n is odd, and δ2, 2 ≡ 1 and δ2, n ≡ 0 if n ≥ 3.

Continuous dependence of Lyapunov matrices with respect to perturbations for linear delay systems

AbstractIn this article, we consider general linear systems whose right‐hand sides are represented by the Riemann–Stieltjes integral with kernels given by a matrix function of bounded variation. We establish some minimal requirements on the convergence of the kernels that ensure the convergence of Lyapunov matrices of the perturbed systems. Using functional analytic approach, we analyze the continuity of the dependence of Lyapunov matrices on the right‐hand sides. Results obtained in this paper improve the previously known results for multiple delay systems and distributed delay systems. One significant advantage of our approach is that no assumption on the exponential stability is being made.

Black Bodies in America as the Metaphors for Oppression, Poverty, Violence, and Hate: Searching for Sustainable Solutions Beyond the Black-letter Law

Black people in America have often been labeled outlaws, deviants, and nonconformists who are disinterested in complying with the laid down rules. However, from a long range experience dating back to slavery, they recognize that rules in the American context whether the Slave Codes, Black Codes, Jim Crows, or the contemporary law, are machinations of the legal system to perpetuate oppression and violence against Blackness. Toward self-preservation, they have learned to radically resist acts of oppression such as wrongful arrests by the police and the functional denial of their rights to be presumed innocent, protest and bear arms, even if such resistance necessitates a stark disobedience to the law enforcement. The Stono Rebellion of 1739 and other slave uprisings used resistance to achieve the abolition of chattel slavery. In the contemporary times, the Black radical tradition pursues the eradication of criminal enslavement by promoting Black protests and resistance against wrongful arrests: wrongful arrests have been identified as the preliminary steps toward mass Black incarceration. In opposition to the mainstream perspective in American literature, this paper uses a functional-analytical approach to legal reasoning to analyze key legal, historical, and sociological issues surrounding the existence of Black people in America in order to show that slavery is still functionally alive: it argues positively for the legitimacy and appropriateness of the Black radical tradition as a reliable means of effectuating the myriad black-letter rights that started in 1865 under the Thirteenth Amendment.

Functional Approach for Solving Reduced Order of Index‐Four Hessenberg Differential‐Algebraic Control System

This research investigates differential‐algebraic equations with higher index (index four). Specifically, a functional analytic approach is proposed to find the solution of (index four) Hessenberg differential‐algebraic equations (DAEs). The approximate solution of the proposed functional approach for a suitable separable Hilbert space is obtained with the help of the direct optimisation technique and Ritz basis functions method. Illustrations have been ranked from an 8 × 8 test system of index‐four Hessenberg linear DAEs to a 4 × 4 DAE of rotating masses as well as an 8 × 8 differential‐algebraic generator model, where it reformulated index‐four linear Hessenberg DAEs. Their approximate solutions were obtained using the present approach with comparisons. The numerical results demonstrate the simplicity of the proposed approach and express suitable accuracy and efficiency.

ТЕОРЕТИЧНІ АСПЕКТИ ЕФЕКТИВНОСТІ ДІЯЛЬНОСТІ ФАРМАЦЕВТИЧНИХ ПІДПРИЄМСТВ

The article considered the essence of the category of efficiency in relation to the activities of pharmaceutical enterprises. The authors identified the genesis and development of this concept, its formation and the relationship between its characteristics and factors in terms of management. In particular, the essence and connections between the concepts of efficiency, effectiveness, productivity, profitability, and performance, which are used in the scientific and managerial literature, are established. Systematizing the existing definitions in the theory, the authors identified the five main approaches to understanding the efficiency of pharmaceutical enterprises: functional approach, target approach, problem approach, analytical approach, and complex approach. According to them, efficiency is classified by business functions, measurement, management objects and types. Besides, sixteen main types of the efficiency of pharmaceutical enterprises are identified, i.e.: financial efficiency, production efficiency, logistics efficiency, marketing efficiency, structural efficiency, dynamic efficiency, absolute efficiency, relative efficiency, organizational performance, performance of the enterprise division, efficiency of product (products, services, works) of the enterprise, employee performance, technological efficiency, economic efficiency, social efficiency, environmental efficiency. According to the results of the study, the authors provided their own definition of the efficiency of pharmaceutical enterprises as a complex and multifaceted management category that characterize the ability of the pharmaceutical company to achieve strategic and tactical goals, productively and economically use resources (financial, labor, material and technical, intangible), meet the needs of society (social), owners (economic) and the environment (environmental).

Optimal control of transverse vibration of a moving string with time-varying lengths

<p style='text-indent:20px;'>In this article, we are concerned with optimal control for the transverse vibration of a moving string with time-varying lengths. In the fixed final time horizon case, the Pontryagin maximum principle is established for the investigational system with a moving boundary, owing to the Dubovitskii and Milyutin functional analytical approach. A remark then follows for discussing the utilization of obtained necessary optimality condition.</p>

SOME math a`La Alexandrov

This is a short overview of the impact and current state of Alexandrov's functional analytical approach to extremal problems in the space of convex syrfaces.