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# Real and Complex Analysis

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... We have that F N satisfies the following property when it is composed with a differential operator (see [33]): ...
... , N of the system (33) is a solution of the system (26) when N tends to infinity. As a consequence, to prove the existence of solutions of (26), it is sufficient to prove the existence of solutions for the system (33). ...
... we have thatū N nm , n, m = 0, . . . , N defined in (39) satisfies (33) for N ≥ K , and this concludes the proof. ...
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Peridynamics is a nonlocal generalization of continuum mechanics theory which addresses discontinuous problems without using partial derivatives and replacing them by an integral operator. As a consequence, it finds applications in the framework of the development and evolution of fractures and damages in elastic materials. In this paper we consider a one-dimensional nonlinear model of peridynamics and propose a suitable two-dimensional fast-convolution spectral method based on Chebyshev polynomials to solve the model. This choice allows us to gain the same accuracy both in space and time. We show the convergence of the method and perform several simulations to study the performance of the spectral scheme.
... for every x ∈ X ( [6,26,27,44,45]). Norm of the operator is defined by ...
... Finding norms of linear operators is one of the basic problems in operator theory. Many classical results can be found in books and surveys on functional analysis, operator theory and inequalities (see, for example, [6,7,9,10,16,23,26,27,44,45]; see also some of the original sources [13,14,28]). For some recent results in the topic, including some on multi-linear operators (for the definition and some examples see [52, p. 51-55]), see, for example, [4,9,11,20,[33][34][35][36]38,[40][41][42] and the related references therein. ...
... are non-increasing for each f ∈ L p w . Example 2. An example of such a space consists of all harmonic functions on R n [18,31], for which the integral means are nondecreasing functions (see, e.g., [17]; for one-dimensional case see [26]). Proof. ...
Article
We calculate the norms of several concrete operators, mostly of some integral-type ones between weighted-type spaces of continuous functions on several domains. We also calculate the norm of an integral-type operator on some subspaces of the weighted Lebesgue spaces.
... In a complex valued neural network, any regular analytic function cannot be bounded unless it reduces to a constant. This is known as Liouville's theorem [19]. That is to say, activation functions in complex-valued neural networks cannot be both bounded and analytic. ...
... Then, model (2.1) can be separated into three clusters C 1 = {1, 2, 3}, C 2 = {4, 5}, and C 3 = {6, 7}. With the above-mentioned settings, we can obtain A = diag(1, 1, 1), µ RR = µ RI = µ IR = µ II = θ RR = θ RI = θ IR = θ II = diag(1, 1, 1),C RR =C RI =C IR =C II = diag (14, 14.2, 14), (20,19,20), one can achievẽ ...
... For k = 1, 2, 3 in Figures 2-4, we have e k (t) → 0 when t → +∞, which implies that all states achieve CS asymptotically. Case 2 : LetΛ = diag (19,18,18), ...
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In cluster synchronization (CS), the constituents (i.e., multiple agents) are grouped into a number of clusters in accordance with a function of nodes pertaining to a network structure. By designing an appropriate algorithm, the cluster can be manipulated to attain synchronization with respect to a certain value or an isolated node. Moreover, the synchronization values among various clusters vary. The main aim of this study is to investigate the asymptotic and CS problem of coupled delayed complex-valued neural network (CCVNN) models along with leakage delay in finite-time (FT). In this paper, we describe several sufficient conditions for asymptotic synchronization by utilizing the Lyapunov theory for differential systems and the Filippov regularization framework for the realization of finite-time synchronization of CCVNNs with leakage delay. We also propose sufficient conditions for CS of the system under scrutiny. A synchronization algorithm is developed to indicate the usefulness of the theoretical results in case studies.
... One denotes by ( ) the vector space of all real valued continuous compactly supported functions defined on . For general results, terminology, and related background, see paragraphs from books and monographs [1][2][3][4][5][6][7][8][9][10]. Given a sequence ( ) ∈ℕ of real numbers, then one studies the existence, uniqueness, and construction of a linear positive form 1 defined on a function space 1 containing polynomials and continuous compactly supported real functions, such that the moment conditions T 1 (φ j ) = y j , j ∈ ℕ n , (2) are satisfied. ...
... The simplest cases are = ℝ and = ℝ + , regarded as closed subsets of ℝ. To obtain the equivalence from Example 3, one applies Corollary 6 to = ℝ, also using measure theory arguments [9] and properties of the Gamma function. ...
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First, this paper provides characterizing the existence and uniqueness of the linear operator solution 𝑇 for large classes of full Markov moment problems on closed subsets 𝐹 of ℝ𝑛. One uses approximation by special nonnegative polynomials. The case when 𝐹 is compact is studied. Then the cases when 𝐹 = ℝ𝑛 and 𝐹 = ℝ+n are under attention. Here, the main findings consist in proving and applying the density of special polynomials, which are sums of squares, in the positive cone of Lv1(ℝ𝑛), and respectively of Lv1(ℝ+n), for a large class of measures 𝜈. One solves the important difficulty created by the fact that on ℝ𝑛, 𝑛 ≥ 2, there exist nonnegative polynomials which are not expressible in terms of sums of squares. This is the second aim of the paper. On the other hand, two types of symmetry are outlined. Both these symmetry properties appear naturally from the thematic mentioned above. This is the third aim of the paper. They lead to new statements, illustrated in corollaries, and supported by a few examples.
... so that {θ n } n∈N ⊂ C(K × K ) is equicontinuous. Thus by the Arzelà-Ascoli theorem (see, e.g., [81,Theorem 11.28]) there exist θ ∈ C(K × K ) and a subsequence {θ n k } k∈N of {θ n } n∈N converging to θ in C(K × K ), so that {(θ n k , μ n k )} k∈N converges to (θ, μ) in C(K × K ) × P(K ). Then diam θ (K ) ∈ [C −1/2 , C 1/2 ] and for any x, y, z ∈ K we have (6.37) with θ in place of θ n , θ(x, x) = 0, θ(x, y) = θ(y, x) ≥ 0 and θ(x, y) ≤ θ(x, z) + θ(z, y) by the same properties of θ n k for k ∈ N, whence θ is a metric on K . ...
... Thus (3.3) holds with (d 1 , d 2 ) = (R E , θ), and (K , θ, μ) is VD by Proposition 6.10 and Theorem 4.5, which together imply that (K , R E , μ) is VD. Now since C(K ) is dense in L 1 (K , μ) (see, e.g., [81,Theorem 3.14]) and f ∈ L 1 (K , μ), the claim follows by Lebesgue's differentiation theorem [40, (2.8)] for (K , R E , μ), which requires (K , R E , μ) to be VD. ...
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We introduce the notion of conformal walk dimension, which serves as a bridge between elliptic and parabolic Harnack inequalities. The importance of this notion is due to the fact that, for a given strongly local, regular symmetric Dirichlet space in which every metric ball has compact closure (MMD space), the finiteness of the conformal walk dimension characterizes the conjunction of the metric doubling property and the elliptic Harnack inequality. Roughly speaking, the conformal walk dimension of an MMD space is defined as the infimum over all possible values of the walk dimension with which the parabolic Harnack inequality can be made to hold by a time change of the associated diffusion and by a quasisymmetric change of the metric. We show that the conformal walk dimension of any MMD space satisfying the metric doubling property and the elliptic Harnack inequality is two, and provide a necessary condition for a pair of such changes to attain the infimum defining the conformal walk dimension when it is attained by the original pair. We also prove a necessary condition for the existence of such a pair attaining the infimum in the setting of a self-similar Dirichlet form on a self-similar set, and apply it to show that the infimum fails to be attained for the Vicsek set and the N-dimensional Sierpiński gasket with N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 3$$\end{document}, in contrast to the attainment for the two-dimensional Sierpiński gasket due to Kigami (Math Ann 340(4):781–804, 2008).
... The existence of the solution (w * (x), a * ) implies that we can always choose a "maximizing sequence" u(w n (x)) = u(w * (x)), a n = a * . As a result, {u(w n (x))} is uniformly integrable under P(·|a n ) since u(w * (x)) is uniformly integrable under P(·|a * ) given that u(w * (x)) ∈ L 1 (X , P(·|a * )), which is followed by a standard result from real analysis (see, Rudin 1987, Chapter 1, Exercise 12). ...
... Thus, {x:v(x)≤0} v(x)d P(x|a * ) ≥ −C 0 , which further implies X v(x)d P(x|a * ) ≥ −C 0 . For the opposite direction, we claim that X v(x)d P(x|a * ) is bounded from above due to X v(x)d P(x|a * ) ≤ {x:v(x)≥0} v(x)d P(x|a * ) ≤ {x:v(x)≥0} (C 1 · |π(x)| +C 2 ) d P(x|a * ) < ∞, in which the last inequality follows the similar way in (45) under Assumption 3. Therefore, we show that v(x) ∈ L 1 (X , P(·|a * )), which implies the second term of Part 3 satisfies {x:v(x)<−K } v(x)d P(x|a * ) K →∞ − −−− → 0 by a standard result from real analysis (see, Rudin 1987, Chapter 1, Exercise 12). Hence, Part 3 in (43) is negative since the first term of Part 3 is negative for any K > 0. ...
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This paper proposes a new method for investigating the existence of a deterministic solution to pure moral hazard problems under a general setting without imposing a priori topological restriction on the contract space. Our method avoids the detour to show the existence of a random contract before showing the existence of a deterministic contract. We show the existence of a solution in the classical moral hazard setting wherein the agent’s utility is separable between money and effort, and the utilities of the principal and agent are concave in money. The proposed sufficient condition for the existence is comparable with the state-of-the-art results, and we use an easy-to-check approach. For example, we show the existence if the marginal incentive cost (per util given to the agent) is unbounded, or if the signal is finite. Also our approach can apply to multi-agent settings and the cases in which the agent utility is quasi-separable.
... In these points the value of | log(P A d )| is +∞. One can see for instance [Rud87, Chapter 1] as a reference. Lemma VII.3.8. ...
Thesis
In this thesis we investigate the sequence of Mahler measures of a family of bivariate regular exact polynomials, called Pd := P0≤i+j≤d xiyj , unbounded in both degree and the genus of the algebraic curve. We obtain a closed formula for the Mahler measure of Pd in termsof special values of the Bloch–Wigner dilogarithm. We approximate m(Pd), for 1 ≤ d ≤ 1000,with arbitrary precision using SageMath. Using 3 different methods we prove that the limitof the sequence of the Mahler measure of this family converges to 92π2 ζ(3). Moreover, we compute the asymptotic expansion of the Mahler measure of Pd which implies that the rate of the convergence is O(log dd2 ). We also prove a generalization of the theorem of the Boyd-Lawton which asserts that the multivariate Mahler measures can be approximated using the lower dimensional Mahler measures. Finally, we prove that the Mahler measure of Pd, for arbitrary d can be written as a linear combination of L-functions associated with an odd primitive Dirichlet character. In addition, we compute explicitly the representation of the Mahler measure of Pd in terms of L-functions, for 1 ≤ d ≤ 6.
... 20'li yaşların başında radyoda duymuş olduğu ney sesine 16 âşık olmuş ve bu sesin peşinde insan-ı kâmil olma yolunda adım atmıştır. Bu yol üzerinde ilerlerken Midhat Bahârî 17 (1875-1971), Abdülhay [Efendi] Öztoprak (1884-1961), Neyzen Gavsi Baykara 18 (1902-1967), Ladikli Ahmet Efendi (1887-1969), Sertarik Mesnevihan Şefik Can Dede , Cerrahi şeyhlerinden İbrahim Fahreddin Şevki Efendi (1885-1966), Muzaffer Ozak Efendi (1916-1985, Safer Dal Efendi gibi nüktedan ilim ve irfan sahibi kişilerle uzun yıllar meşk etme fırsatı bulmuştur. Bilhassa musiki yolunda kendisinin "Ayakta kütüphane" diye iltifat ettiği Hüseyin Tolon her daim meth ü sena ile yâd etmiştir. ...
Conference Paper
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The production of traditional building materials causes extensive resource and energy consumption. These materials cause global warming and habitat destruction by disrupting global ecosystems, causing climate change, and increasing greenhouse gas emission. In this context, research on developing durable, recyclable, sustainable, and innovative materials has been increasing in recent years to reduce the environmental impact of building materials. It is noteworthy that most of these studies focus on “living materials”. The increase in research on the subject will cause a change/transformation in the direction of increasing the use of recyclable materials in architecture. Living materials research is considered inspiring in creating the structures of 21st century architecture, and studies are carried out on the production of building elements such as walls, floors, roofs, and windows from living materials. The primary source of this studies is nature. Since in nature, an organism's decaying body or waste becomes a source of material and/or food for other organisms. This paper examines living materials produced as an alternative solution to the climate change crisis. The materials reached within the scope of the study were analyzed in terms of the living materials it contains and for what purpose they can be used in the future. According to the results obtained, if this research turns into a practice, they will be an environmentally friendly alternative source against cement-based materials, which are widely used and have a significant share in 〖CO〗_2 emissions.
... We now show that (UQD, ∂) is a principal PSL(2, R)-bundle over QC + . First recall from Theorem 14.19 of [25] and the subsequent remark that every α ∈ UQD extends to a homeomorphism α : H → α(H). In fact, since ∂α is a quasicircle, we have the following stronger result. ...
Preprint
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In [17], Labourie initiated the study of the dynamical properties of the space of $k$-surfaces, that is, suitably complete immersed surfaces of constant extrinsic curvature in $3$-dimensional manifolds, which he presented as a higher-dimensional analogue of the geodesic flow when the ambient manifold is negatively curved. In this paper, following the recent work [5] of Calegari--Marques--Neves, we study the asymptotic counting of surface subgroups in terms of areas of $k$-surfaces. We determine a lower bound, and we prove rigidity when this bound is achieved. Our work differs from that of [5] in two key respects. Firstly, we work with all quasi-Fuchsian subgroups as opposed to merely asymptotically Fuchsian ones. Secondly, as the proof of rigidity in [5] breaks down in the present case, we require a different approach. Following ideas outlined by Labourie in [19], we prove rigidity by solving a general foliated Plateau problem in Cartan--Hadamard manifolds. To this end, we build on Labourie's theory of $k$-surface dynamics, and propose a number of new constructions, conjectures and questions.
... Proof By[88, Theorem 1.34], we find that L r ω (R n ) has an absolutely continuous norm. Using the definition of L r ω (R n ), we know thatWe consider two cases based on the size of p. ...
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Let X be a ball Banach function space on $${\mathbb R}^n$$. In this article, under some mild assumptions about both X and the boundedness of the Hardy–Littlewood maximal operator on the associate space of the convexification of X, the authors prove that, for any locally integrable function f with $$\Vert \,|\nabla f|\,\Vert _{X}<\infty$$, \begin{aligned} \sup _{\lambda \in (0,\infty )}\lambda \left\| \left| \left\{ y\in {{\mathbb {R}}}^n:\ |f(\cdot )-f(y)| >\lambda |\cdot -y|^{\frac{n}{q}+1}\right\} \right| ^{\frac{1}{q}} \right\| _X\sim \Vert \,|\nabla f|\,\Vert _X \end{aligned}with the positive equivalence constants independent of f, where the index $$q\in (0,\infty )$$ is related to X and $$|\{y\in {\mathbb R}^n:\ |f(\cdot )-f(y)| >\lambda |\cdot -y|^{\frac{n}{q}+1}\}|$$ is the Lebesgue measure of the set under consideration. In particular, when $$X:=L^p({\mathbb R}^n)$$ with $$p\in [1,\infty )$$, the above formulae hold true for any given $$q\in (0,\infty )$$ with $$n(\frac{1}{p}-\frac{1}{q})<1$$, which when $$q=p$$ are exactly the recent surprising formulae of H. Brezis, J. Van Schaftingen, and P.-L. Yung, and which in other cases are new. This generalization has a wide range of applications and, particularly, enables the authors to establish new fractional Sobolev and new Gagliardo–Nirenberg inequalities in various function spaces, including Morrey spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, weighted Lebesgue spaces, Orlicz spaces, Orlicz-slice (generalized amalgam) spaces, and weak Morrey spaces, and, even in all these special cases, the obtained results are new. The proofs of these results strongly depend on the Poincaré inequality, the extrapolation, the exact operator norm on $$X'$$ of the Hardy–Littlewood maximal operator, and the exquisite geometry of $${\mathbb {R}}^n.$$
... where ⊂ \σ (S) is a finite set of closed rectifiable curves surrounding σ (S) = σ (A) (existence of such curves is shown in [24,Theorem 13.5]). Note that the integral ...
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We consider Gabor frames generated by a general lattice and a window function that belongs to one of the following spaces: the Sobolev space $$V_1 = H^1(\mathbb {R}^d)$$ V 1 = H 1 ( R d ) , the weighted $$L^2$$ L 2 -space $$V_2 = L_{1 + |x|}^2(\mathbb {R}^d)$$ V 2 = L 1 + | x | 2 ( R d ) , and the space $$V_3 = \mathbb {H}^1(\mathbb {R}^d) = V_1 \cap V_2$$ V 3 = H 1 ( R d ) = V 1 ∩ V 2 consisting of all functions with finite uncertainty product; all these spaces can be described as modulation spaces with respect to suitable weighted $$L^2$$ L 2 spaces. In all cases, we prove that the space of Bessel vectors in $$V_j$$ V j is mapped bijectively onto itself by the Gabor frame operator. As a consequence, if the window function belongs to one of the three spaces, then the canonical dual window also belongs to the same space. In fact, the result not only applies to frames, but also to frame sequences.
... As shown in Fig. 1, the linear target with a constant Sparameter leads to a holomorphic mapping, while the nonlinear target with a power-dependent scattering parameter leads to a non-holomorphic mapping. Mathematically, the nonholomorphic mapping should be expanded locally with both the perturbed term and its corresponding conjugation [33], [34], the latter of which is referred to as the localized conjugate component (LCC) in this paper. The situation, to some extent, resembles the cross-polarization phenomenon in polarization radar, where some targets could transform the incident righthand circular polarization to its left-hand counterpart [35], [36]. ...
Preprint
p>This is the preprint version of the paper "Characterization and Detection of RF Electronics by Localized Conjugate Component".</p
... We shall describe the situation in the simplest case in which g is assumed to be constant, say g ≡ 1. As shown in [2], or by simply invoking Weierstrass factorization theorem (see [11]), in the complex variable z = x + i y, when g ≡ 1, the complex gradient u x − i u y of u is uniquely determined by the Blaschke product ...
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We prove an existence result for the Backus interior problem in the Euclidean ball. The problem consists in determining a harmonic function in the ball from the knowledge of the modulus of its gradient on the boundary. The problem is severely nonlinear. From a physical point of view, the problem can be interpreted as the determination of the velocity potential of an incompressible and irrotational fluid inside the ball from measurements of the velocity field's modulus on the boundary. The linearized problem is an irregular oblique derivative problem, for which a phenomenon of loss of derivatives occurs. As a consequence, a solution by linearization of the Backus problem becomes problematic. Here, we linearize the problem around the vertical height solution and show that the loss of derivatives does not occur for solutions which are either (vertically) axially symmetric or oddly symmetric in the vertical direction. A standard fixed point argument is then feasible, based on ad hoc weighted estimates in H\"older spaces.
... Since p ≥ q, Jensen's inequality (see, e.g., Theorem 3.3 of [35]) implies that ...
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We study decay rates for bounded $C_0$-semigroups from the viewpoint of $L^p$-infinite-time admissibility and related resolvent estimates. In the Hilbert space setting, polynomial decay of semigroup orbits is characterized by the resolvent behavior in the open right half-plane. A similar characterization based on $L^p$-infinite-time admissibility is provided for multiplication semigroups on $L^q$-spaces with $1 \leq q \leq p < \infty$. We also give a sufficient condition for $L^2$-infinite-time admissibility for polynomially stable $C_0$-semigroups on Hilbert spaces.
... Let ε > 0. By a standard approximation result (see e.g. [8,Theorem 3.14]), there exists a continuous function g on [−π, π] such that f − g L 1 < ε/2. We can demand that m ≤ g ≤ M . ...
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We consider solutions $u_f$ to the one-dimensional Robin problem with the heat source $f\in L^1[-\pi,\pi]$ and Robin parameter $\alpha>0$. For given $m$, $M$, and $s$, $0\le m<s<M$, we identify the heat sources $f_0$, such that $u_{f_0}$ maximizes the temperature gap $\max_{[-\pi,\pi]}u_f -\min_{[-\pi,\pi]}u_f$ over all heat sources $f$ such that $m\le f\le M$ and $\|f\|_{L^1}=2\pi s$. In particular, this answers a question raised by J.~J.~Langford and P.~McDonald in \cite{LM}. We also identify heat sources, which maximize/minimize $u_f$ at a given point $x_0\in [-\pi,\pi]$ over the same class of heat sources as above and discuss a few related questions.
... The continuity follows by showing that there is always a subsequence ((s j k , ϕ j k )) k such that ∂ s B(s j k , ϕ j k ) converges to ∂ s B(s 0 , ϕ 0 ). Such a subsequence can be found recalling that we can always find a subsequence and a function σ ∈ L 2 (Ω) ( [32], Thm. 3.12) such that lim k→∞ ϕ j k (x) = ϕ 0 (x), a.e. ...
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In this paper we study an abstract framework for computing shape derivatives of functionals subject to PDE constraints in Banach spaces. We revisit the Lagrangian approach using the implicit function theorem in an abstract setting tailored for applications to shape optimization. This abstract framework yields practical formulae to compute the derivative of a shape functional, the material derivative of the state, and the adjoint state. Furthermore, it allows to gain insight on the duality between the material derivative of the state and the adjoint state. We show several applications of this method to the computation of distributed shape derivatives for problems involving linear elliptic, nonlinear elliptic, parabolic PDEs and distributions. We also compare our approach with other techniques for computing shape derivatives including the material derivative method and the averaged adjoint method.
... To do this, we first cite two theorems due to Harnack: [12,Ch. 11,Theorem 11.11]) Let G Ă R 2 be a bounded and connected domain. ...
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In his 1918 paper 'A General Form of Integral', Percy John Daniell developed a theory of integration capable of dealing with functions on arbitrary sets. Daniell's method differs from the measure-theoretic notion of integration. Linear functionals over vector lattices were considered as the fundamental objects on which he built the theory, rather than measures over sets. In this document, we explore Daniell's concept of integration and how his theory relates to the measure-theoretic notion of integration. We paint a picture of the historical context surrounding Daniell's ideas. Furthermore, we present examples due to Norbert Wiener, where the Daniell integral was employed on spaces too general for the standard integration techniques of the time.
... For instance, if F is the family of infinite sets, then the F -operators are precisely the operators that are topologically transitive. Suppose that F is the family of cofinite sets, then the F -operators are the operators that are mixing, see [19,23,31]. If F is the family of thick sets (those that contain arbitrarily long intervals), then the F -operators are the operators that are weak mixing, see [6,10,21]. ...
Article
A Furstenberg family F is a collection of infinite subsets of the set of positive integers such that if A⊂B and A∈F, then B∈F. For a Furstenberg family F, finitely many operators T1,...,TN acting on a common topological vector space X are said to be disjoint F-transitive if for every non-empty open subsets U0,...,UN of X the set {n∈N:U0∩T1-n(U1)∩...∩TN-n(UN)≠∅} belongs to F. In this paper, depending on the topological properties of Ω, we characterize the disjoint F-transitivity of N≥2 composition operators Cϕ1,…,CϕN acting on the space H(Ω) of holomorphic maps on a domain Ω⊂C by establishing a necessary and sufficient condition in terms of their symbols ϕ1,...,ϕN.
... Consequently, there is δ > 0 such that R n −B(ξ ,r) σ δ (y) y − ξ k dy δ ≤ γ + 1 2 , for all δ ∈ (0, δ], [42] where we note (γ + 1)/2 ∈ (γ, 1), and so lim δ→0 + S σ δ (y) y − ξ k dy ≤ lim δ→0 + γ + 1 2 1/δ = 0, [43] as desired. ...
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First-order optimization algorithms are widely used today. Two standard building blocks in these algorithms are proximal operators (proximals) and gradients. Although gradients can be computed for a wide array of functions, explicit proximal formulas are only known for limited classes of functions. We provide an algorithm, HJ-Prox, for accurately approximating such proximals. This is derived from a collection of relations between proximals, Moreau envelopes, Hamilton-Jacobi (HJ) equations, heat equations, and importance sampling. In particular, HJ-Prox smoothly approximates the Moreau envelope and its gradient. The smoothness can be adjusted to act as a denoiser. Our approach applies even when functions are only accessible by (possibly noisy) blackbox samples. We show HJ-Prox is effective numerically via several examples.
... Here we will explain boundedness [Khan & Talukder, 2021, Rudin, 1987, find out the equilibrium [Side et al. 2019] and basic reproduction number [Side et al., 2019] regarding the constant value of control variables 1 u and 2 u . ...
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Corruption is rapidly affecting any country’s economic, democratic, financial, social, and political stability. It has been a consistent social phenomenon that happens in all civilizations. Only in the last 20 years has this phenomenon been given serious attention. It has different forms and different impacts on the economy as well as society as a whole. Economic growth is slowed by corruption, which also has a detrimental effect on business operations, employment, and investment. Additionally, it has a detrimental effect on tax revenues as well as the effectiveness of various financial aid programs. So, it is necessary to reduce this global problem. For this reason, we propose a nonlinear deterministic model for the transmission dynamics of societal corruption in terms of optimal control problem, using two time-dependent controls namely the efforts aimed at preventing corruption through the use of social networks, media, and social organizations; including a strong and effective anti-corruption policy, and also the attempt to encourage the punishment of corrupt people to analyze the model. The goal of this study is to reduce the problem of corruption. The results show that our proposed model can help to alleviate this social issue. Our findings also reveal that by using both control strategies instead of just one we get a more effective result. Overall, this research suggests that the impact of corruption can be reduced by implementing anti-corruption media and advertising campaigns, as well as exposing corrupted people to jail and punishing them.
... So Y is a set of measure zero. As γ is absolutely continuous, it satisfies the Lusin N-property [13,Lemma 7.25], implying that γ(Y ) is also a set of measure zero. Let A = γ(X) and B = γ(Y ). ...
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This article studies a method of finding Lagrangian transformations , in the form of particle paths, for all scalar conservation laws having a smooth flux. These are found using the notion of weak diffeomorphisms. More precisely, from any given scalar conservation law, we derive a Temple system having one linearly degenerate and one genuinely nonlinear family. We modify the system to make it strictly hyperbolic and prove an existence result for it. Finally we establish that entropy admissible weak solutions to this system are equivalent to those of the scalar equation. This method also determines the associated weak diffeomorphism.
... For a locally compact Hausdorff space X we denote by M (X) the set of all Radon measures on X. With the usual definition of total variation |µ| of µ ∈ M (X) [59,Chapter 6], the space M (X) is a Banach space when endowed with the total variation norm given by µ T V := |µ|(X). Let C 0 (X) = (C 0 (X), · ∞ ) be the Banach space of all complex-valued, continuous functions on X which vanish at infinity, that is, for every ε > 0 there exists a compact set K ⊆ X such that |f (x)| ≤ ε for every x ∈ K c . ...
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This article bridges and contributes to two important research areas, namely the completeness problem for systems of translates in function spaces and the short-time Fourier transform (STFT) phase retrieval problem. As a first main contribution, we show that a complex-valued, compactly supported function can be uniquely recovered from samples of its spectrogram if certain density properties of an associated system of translates hold true. Secondly, we derive new completeness results for systems of discrete translates in spaces of continuous functions on compact sets. We finally combine these findings to deduce several novel recovery results from spectrogram samples. Our results hold for a large class of window functions, including Gaussians, all Hermite functions, as well as the practically highly relevant Airy disk function. To the best of our knowledge, our results constitute the first recovery guarantees for the sampled STFT phase retrieval problem with a non-Gaussian window.
... Note that t 0 λ t−s β(s)ds = t 0 λ η β(t − η)dη. Since α(t), β(t), λ t are all non-negative functions, from Tonelli's theorem [33] and by changing the order of the integrals, it follows that ...
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We introduce a class of distributed nonlinear control systems, termed as the flow-tracker dynamics, which capture phenomena where the average state is controlled by the average control input, with no individual agent has direct access to this average. The agents update their estimates of the average through a nonlinear observer. We prove that utilizing a proper gradient feedback for any distributed control system that satisfies these conditions will lead to a solution of the corresponding distributed optimization problem. We show that many of the existing algorithms for solving distributed optimization are instances of this dynamics and hence, their convergence properties can follow from its properties. In this sense, the proposed method establishes a unified framework for distributed optimization in continuous-time. Moreover, this formulation allows us to introduce a suit of new continuous-time distributed optimization algorithms by readily extending the graph-theoretic conditions under which such dynamics are convergent.
... Note that the sign is used in this proof to omit constant multipliers that may depend on r. Similarly to the previous lemma, it is sufficient to lower bound where we changed to polar coordinates in the last equality; here, S r−2 is the unit sphere in R r−1 , and σ r−2 is a measure on S r−2 such that, if A ⊆ S r−2 is a Borel set andÃ is the set of all points ru with 0 < r < 1 and u ∈ A, then σ r−2 (A) = (r − 1)m r−1 (Ã), where m r−1 is the Lebesgue measure on R r−1 (see Exercise 6, Chapter 8 of [27]). We continue: ...
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... Thus, the sequence |u n | q(x) , with 1 ≤ n < ∞, is equi-integrable in L 1 (Q T ). By Vitali's Convergence theorem (see [31]) ...
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