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We discuss the dynamics beyond topological hyperbolicity considering homeomorphisms satisfying the shadowing property and generalizations of expansivity. It is proved that transitive countably expansive homeomorphisms satisfying the shadowing property are expansive in the set of transitive points. This is in contrast with pseudo-Anosov diffeomorphisms of the two-dimensional sphere that are transitive, cw-expansive, satisfy the shadowing property but the dynamical ball in each transitive point contains a Cantor subset. We exhibit examples of countably expansive homeomorphisms that are not finite expansive, satisfy the shadowing property and admits an infinite number of chain-recurrent classes. We further explore the relation between countable and entropy expansivity and prove that for surface homeomorphisms f : S → S f\colon S\to S satisfying the shadowing property and Ω ( f ) = S \Omega (f)=S , both countably expansive and entropy cw-expansive are equivalent to being topologically conjugate to an Anosov diffeomorphism.

We consider the problem of optimal partisan gerrymandering: a legislator in charge of redrawing the boundaries of equal-sized congressional districts wants to ensure the best electoral outcome for his own party. The so-called gerrymanderer faces two issues: the number of districts is finite and there is uncertainty at the level of each district. Solutions to this problem consists in cracking favorable voters in as many districts as possible to get tight majorities, and in packing unfavorable voters in the remaining districts. The optimal payoff of the gerrymanderer tends to increase as the uncertainty decreases and the number of districts is large. With an infinite number of districts, this problem boils down to concavifying a function, similarly to the optimal Bayesian persuasion problem. We introduce a measure of fairness and show that optimal gerrymandering is accordingly closer to uniform districting (full cracking), which is most unfair, than to community districting (full packing), which is very fair.

Let u be any positive integer. We construct infinite sequences of almost symmetric non-Weierstrass numerical semigroups whose conductors are the genera double minus $$2u-1$$ 2 u - 1 . Moreover, let v be any non-negative integer. We give an infinite number of non-Weierstrass numerical semigroups whose conductors are the genera double minus 2v.

This paper proposes an improved version of 3D pure proportional navigation (PPN) against a manoeuvring target. The main research hypothesis is that the performance of 3D PPN can be improved by properly selecting the direction of the guidance command as there exists an infinite number of potential directions complying with the PPN concept in 3D space. Analysis on the relative motion confirms the validity of the hypothesis and leads to the development of a new guidance algorithm. Unlike traditional 3D PPN, the guidance algorithm developed adapts the direction, but maintains the magnitude of the commanded acceleration proportional to only the line-of-sight (LOS) rate. The validity and performance of the proposed guidance algorithm are investigated through theoretical analysis and numerical simulations.

Junto a los órganos unipersonales, en las Administraciones españolas conviven infinidad de órganos colegiados, creados para las finalidades más diversas (coordinación, cooperación, participación…). La transparencia de estos órganos plantea múltiples cuestiones que afectan tanto a la existencia y vigencia misma de los órganos, como a su estructura y, sobre todo, funcionamiento, con especial relevancia a si se debe o no mantener la confidencialidad de los debates y deliberaciones previas a las votaciones de los acuerdos.

The paper introduces a method for the construction of bivariate copulas with the usage of specific values of the parameters α and β ( α , β transformation) and the parameters κ and λ in their domain. The produced bivariate copulas are defined in four subrectangles of the unit square. The bounds of the produced copulas are investigated, while a novel construction method for fuzzy copulas is introduced, with the usage of the produced copulas via α , β transformation in four subrectangles of the unit square. Following this construction procedure, the production of an infinite number of copulas and fuzzy copulas could be possibly achieved. Some applications of the proposed methods are presented.

The digitalized video footage of LeBron James’s slam-dunk commemorating Kobe Bryant is sold for 200,000 USD. How is that even possible, seeing as digital code can be copied and pasted an infinite number of times? By linking it to a unique code that is stored on blockchain, the video becomes an NFT, a nonfungible token. Here we have the avant-garde of virtual currencies, an almost mystical way of making digital content nonfungible, i.e. non-replicable, unique, perhaps even bestowed with an aur...

In this contribution to the special issue on Fakery in Africa, I examine the booming "fake essay" industry and draw on the role and perspectives increasingly occupied by of tens of thousands of young and highly-educated Kenyans. These so-called "Shadow Scholars" are part of a vast global online marketplace, an invisible knowledge production economy, where students and academics in the global North solicit and pay for their services in exchange for confidential and plagiarism-free essays, theses, dissertations, qualifications and publications. This article centres on descriptions of these writers as "shadows" as a means of complicating not only the most popular description of Africa in the global imagination - as existing in the shadow of an infinite number of different entities - but to challenge the notion of the shadow in relation to African knowledge production as being fake. It pays attention to the Kenyan writers' protestations that their knowledge, experiences and labour are all real and that analogies with shadows reduce them and the impact of their work to something that is non-existent and not alive. From their perspective the term shadow is pejorative because it further reduces the intellectual contribution of Africans, presenting them as derivative.Katika mchango huu kwa suala maalum kuhusu Udanganyifu barani Afrika, mimi huchunguza tasnia ya “insha bandia” na haswa jukumu na msimamo wa makumi ya maelfu ya vijana wakenya na wenye elimu ya kiwango cha juu. Hawa wanaoitwa “Vivuli vya Wasomi” ni sehemu ya soko kubwa la mkondoni ulimwenguni, uchumi wa maarifa usiojulikana, ambapo wanafunzi na wasomi katika kaskazini mwa Ulimwengu wanaomba na kulipia huduma zao kwa malipo ya insha za wizi na usiri, tasnifu, ujuzi wa kitaaluma na machapisho. Utafiti huu unaangazia ufafanuzi wa waandishi hawa kama “Vivuli” kama njia ya kutatanisha sio tu maelezo maarufu zaidi ya Afrika pia katika fikra za ulimwengu, kama ilivyo katika kivuli cha idadi isiyo na mwisho ya vyombo tofauti, lakini kupinga wazo la kivuli kuhusiana na uzao wa maarifa ya Kiafrika kama bandia. Inatilia maanani maandamano ya waandishi wa Kenya kwamba maarifa, uzoefu na kazi zao zote ni za kweli na kwamba mlinganisho na vivuli hupunguza na athari ya kazi yao kwa kitu ambacho hakipo na sio hai. Kulingana na maoni yao, neno kivuli ni la kudharaulisha kwa sababu linapunguza zaidi mchango wa kiakili wa Waafrika, kuwaonyesha kama waigaji.

In optical data transmission with using vortex laser beams, data can be encoded by the topological charge, which is theoretically unlimited. However, the topological charge of a single separate vortex (screw dislocation) is limited by possibilities of its generating. Therefore, we investigate here three examples of multivortex Gaussian light fields (two beams are form-invariant and one beam is astigmatic) with an unbounded (countable) set of screw dislocations. As a result, such fields have an infinite topological charge. The first beam has the complex amplitude of the Gaussian beam, but multiplied by the cosine function with a squared vortex argument. Phase singularity points of such a beam reside in the waist plane on the Cartesian axes and their density grows with increasing distance from the optical axis. The transverse intensity distribution of such a beam has a shape of a four-pointed star. All the optical vortices in this beam has the same topological charge of +1. The second beam also has the complex amplitude of the Gaussian beam, multiplied by the vortex-argument cosine function, but the cosine is raised to an arbitrary power. This beam has a countable number of the optical vortices, which reside in the waist plane uniformly on one Cartesian axis and the topological charge of each vortex equals to power, to which the cosine function is raised. The transverse intensity distribution of such beam consists of two light spots residing on a straight line, orthogonal to a straight line with the optical vortices. Finally, the third beam is similar to the first one in many properties, but it is generated with a tilted cylindrical lens from a 1D parabolic-argument cosine grating.

In optical data transmission with using vortex laser beams, data can be encoded by the topo-logical charge, which is theoretically unlimited. However, the topological charge of a single sepa-rate vortex is limited by possibilities of its generating. Therefore, in this work, we analyze light beams with an unbounded (countable) set of optical vortices. The summary topological charge of such beams is infinite. Phase singularities (isolated intensity nulls) in such beams typically have a unit topological charge and reside equidistantly (or not equidistantly) on a straight line in the beam cross section. Such beams are form-invariant and, on propagation in space, change only in scale and rotate. Orbital angular momentum of such multivortex beams is finite, since only a finite number of optical vortices fall into the area, where the Gaussian beam has a notable intensity. Other phase singularities are located in the periphery (and at the infinity), where the intensity is almost zero.

In this contribution, we consider the optimal control problem for a switched dynamical system. While such systems can exhibit rather complex behavior in the case of only one switch, the most interesting problem corresponds to the case, when the system undergoes an infinite number of switches. We study the limiting behavior of optimal solutions under such assumption and show that there are three types of solutions, two of which correspond to cyclic evolution of the system state and control.

This paper presents an approach for determining feasible separation regions for S-shaped distillation lines. There are three main assumptions: (1) the feed is a mixture of vapor and liquid in equilibrium; (2) the column has a partial or total condenser and a partial or total reboiler; and (3) the border of the feasible separation region is defined by the extreme operating modes of the distillation process. The separation region itself was constructed using a geometric model of the column, while accounting for the shape of the distillation lines. Contrary to well-known cases for C- and U-shaped distillation lines, the border of the feasible separation region for S-shaped lines is formed from a portion of a product composition multitude (i.e., the set of distillate and bottoms composition points for a column with an infinite number of stages operating at total reflux; part of a distillation limit, which demarcates the sloppy splits from the area that is not accessible by distillation; and additional segments). The feasible separation region includes additional product composition areas, which enable the design of distillation columns with fewer theoretical stages, operating under a lower reflux ratio, as demonstrated for a mixture of methyl ethyl ketone, benzene, and toluene.

In this paper, using variational method, we study the existence of an infinite number of solutions (some are positive, some are negative, and others are sign-changing) for a non-homogeneous elliptic Kirchhoff equation with a nonlinear reaction term.

This paper is concerned with the study of optimality conditions for minimax optimization problems with an infinite number of constraints, denoted by (MMOP). More precisely, we first establish necessary conditions for optimal solutions to the problem (MMOP) by means of employing some advanced tools of variational analysis and generalized differentiation. Then, sufficient conditions for the existence of such solutions to the problem (MMOP) are investigated with the help of generalized convexity functions defined in terms of the limiting subdifferential of locally Lipschitz functions. Finally, some of the obtained results are applied to formulating optimality conditions for weakly efficient solutions to a related multiobjective optimization problem with an infinite number of constraints, and a necessary optimality condition for a quasi e-solution to problem (MMOP).

In this paper, we apply the product-to-sum decomposition (PSD) E(s)E( − s) = Z(s) + Z( − s) to investigate the integral-square-error (ISE) computation for fractional-order time-delay (FOTD) feedback control systems. First, we show that the PSD approach to ISE evaluation is inapplicable to delay-free fractional systems. Then we show that, based on performing operations on polynomials in the delay operator z = e−τs, the PSD approach can be extended to FO systems with commensurate orders and delays. A key advantage of the PSD approach is that the Ip part of ISE contributed by the dynamics of infinite number of poles of E(s) can be computed from the finite number of poles of the decomposed function Z(s). Also, we present PSD-based formulae for ISE calculation. To demonstrate the PSD approach to ISE calculation for FOTD systems, illustrative examples are provided.

Quantum key distribution (QKD) offers a reliable solution to communication problems that require long-term data security. For its widespread use, however, the rate and reach of QKD systems must be improved. Twin-field (TF) QKD is a step forward toward this direction, with early demonstrations suggesting it can beat the current rate-versus-distance records. A recently introduced variant of TF-QKD is particularly suited for experimental implementation, and has been shown to offer a higher key rate than other variants in the asymptotic regime where users exchange an infinite number of signals. Here, we extend the security of this protocol to the finite-key regime, showing that it can overcome the fundamental bounds on point-to-point QKD with around $10^{10}$ transmitted signals. Within distance regimes of interest, our analysis offers higher key rates than those of alternative variants. Moreover, some of the techniques we develop are applicable to the finite-key analysis of other QKD protocols., 26 pages, 5 figures. v2: major changes on subsection "Concentration inequality for sums of dependent random variables", minor changes elsewhere. v3: Changes on subsection "Concentration inequality for sums of dependent random variables", to use the newest version of the inequality in Ref. [38]. Minor changes in citations elsewhere. v4: Fixed wrong citation numbers in graph legends

The article examines the infimum and supremum of an infinite number of fuzzy numbers. It is shown that familiar properties of these operations, which are valid for real numbers, may apply to fuzzy numbers only under certain conditions. А formula for computing the infimum and supremum of any set of fuzzy numbers is provided. Since the union and meet of type-2 fuzzy sets are defined via the infimum and supremum of fuzzy numbers, all the results obtained are applicable to these operations as well.

With the Internet of Things, an infinite number of devices must generate and share information flows that describe real life. this is why the IoT software platform is very complicated and composed of several entities that depend heavily on each other to connect the tangible world of objects to the virtual world and embeds an intelligence that offers multiple possibilities, stores and analyzes the measurement taken by a sensor, in order to monitor and control connected objects or to create a history allowing prediction and these strong dependencies which complicate the development and maintenance circuit and the evolution of this software, Not all projects adopt a formally identical architecture, however it is possible to schematize an optimal architecture with weak dependencies.

Fractal geometry can be applied in interior architecture by using certain shapes that depend on the principle of self-similarity and repeating the shape to an infinite number of times, and one of the most popular shapes that can be applied is the Cantor group that can be used as a basis in completing the facades and the locations of doors and windows. For snowflakes, which can be used in devising various forms of decoration, also the Serpinsky Triangle, which is considered one of the most important forms of fractal geometry, which can be applied in internal architecture. It is formed by following the molecular fractals of the components of a small molecule. Use it to generate multiple ideas to make shapes for horizontal projections, whether internal or external. يمکن تطبيق هندسة الفراکتال فى العمارة الداخلية عن طريق استخدام أشکال معينه اعتمدت على مبدأ التشابه الذاتى وتکرار الشکل الى عدد لانهائي من المرات , ومن أشهر الأشکال التى يمکن تطبيقها هى مجموعة کانتور التى يمکن استخدامها کأساس فى تصميم الواجهات ومواقع الأبواب والشبابيک , أيضا هناک مايسمى بمنحنى کوخ لرقائق الثلج والذى يمکن استخدامه فى استنباط أشکال متنوعه من الزخارف , أيضا مثلث سيربنسکى وهو يعتبر من أهم أشکال هندسة الفراکتال والتي يمکن تطبيقها فى العمارة الداخلية وهو يتکون عن طريق متابعة الفراکتالات الجزيئية لمکونات جزئ صغير , ومن أهم الأشکال لهندسة الفراکتال أيضا ما يسمى بالتکرارات الهندسية والتي يمکن استخدامها لإنتاج أفکار متعددة لعمل أشکال لمساقط أفقيه سواء داخليه أو خارجية .

From ancient times to the present, mathematicians have put forward many series expressions of the circular constant. Because of the importance of the circular constant to mathematical physics, the research on circular constant has never stopped. In this paper, the general function expression of the circular constant was given by studying the transient heat conduction equation. From the physical aspect of the derivation process of the circular constant expression, we can conclude that there is an infinite number of different series exist that can be used to express π.

Conformational variability and heterogeneity are crucial determinants of the function of biological macromolecules. The possibility of accessing this information experimentally suffers from severe under-determination of the problem, since there are a few experimental observables to be accounted for by a (potentially) infinite number of available conformational states. Several computational methods have been proposed over the years in order to circumvent this theoretically insurmountable obstacle. A large share of these strategies is based on reweighting an initial conformational ensemble which arises from, for example, molecular simulations of different qualities and levels of theory. In this work, we compare the outcome of three reweighting approaches based on radically different views of the conformational heterogeneity problem, namely Maximum Entropy, Maximum Parsimony and Maximum Occurrence, and we do so using the same experimental data. In this comparison we find both expected as well as unexpected similarities.