Search

Between 1873 and 1876, Felix Klein published a series of papers that he later placed under the rubric anschauliche Geometrie in the second volume of his collected works (1922). The present study attempts not only to follow the course of this work, but also to place it in a larger historical context. Methodologically, Klein's approach had roots in Poncelet's principle of continuity, though the more immediate influences on him came from his teachers, Plücker and Clebsch. In the 1860s, Clebsch reworked some of the central ideas in Riemann's theory of Abelian functions to obtain complicated results for systems of algebraic curves, most published earlier by Hesse and Steiner. These findings played a major role in enumerative geometry, whereas Plücker's work had a strongly qualitative character that imbued Klein's early studies. A leitmotif in these works can be seen in the interplay between real curves and surfaces as reflected by their transformational properties. During the early 1870s, Klein and Zeuthen began to explore the possibility of deriving all possible forms for real cubic surfaces as well as quartic curves. They did so using continuity methods reminiscent of Poncelet's earlier approach. Both authors also relied on visual arguments, which Klein would later advance under the banner of intuitive geometry (anschauliche Geometrie). [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the sufficient conditions that guarantee the stability, continuity, and boundedness of solutions for a type of second-order stochastic delay integro-differential equation (SDIDE). To demonstrate the main results, we employ a Lyapunov functional. An example is provided to demonstrate the applicability of the obtained result, which includes the results of this paper and obtains better results than those available in the literature.

In this paper, through an intuitive vanil la proximal method perspective, we derive a concise unified acceleration framework (UAF) for minimizing a convex function that has H\"older continuous derivatives with respect to general (non-Euclidean) norms. The UAF reconciles two different high-order acceleration approaches, one by Nesterov [Math. Program., 112 (2008), pp. 159-181] and one by Monteiro and Svaiter [SIAM J. Optim., 23 (2013), pp. 1092-1125]. As a result, the UAF unifies the high-order acceleration instances of the two approaches by only two problem-related parameters and two additional parameters for framework design. Meanwhile, the UAF (and its analysis) is the first approach to make high-order methods applicable for high-order smoothness conditions with respect to non-Euclidean norms. Furthermore, the UAF is the first approach that can match the existing lower bound of iteration complexity for minimizing a convex function with H\"older continuous derivatives. For practical implementation, we introduce a new and effective heuristic that significantly simplifies the binary search procedure required by the framework. We use experiments on logistic regression to verify the effectiveness of the heuristic. Finally, the UAF is proposed directly in the general composite convex setting and shows that the existing high-order algorithms can be naturally extended to the general composite convex setting. [ABSTRACT FROM AUTHOR]

The weak L1 -space meets in many areas of mathematics. For example, the conjugate functions of Lebesgue integrable functions belong to the weak L1 -space. The difficulty of working with the weak L1 -space is that the weak L1 -space is not a normed space. Moreover, infinitely differentiable (even continuous) functions are not dense in this space. Due to this, the theory of approximation was not produced in this space. In the present paper, we introduced the concept of the modulus of continuity of the functions from the weak L1 -space, studied its properties, found a criterion for convergence to zero of the modulus of continuity of the function from the weak L1 -space, and proved in this space an analogue of the Jackson-type theorem. [ABSTRACT FROM AUTHOR]

In this paper we show that for a generalized Berger metric g ^ on S 3 close to the round metric, the conformally compact Einstein (CCE) manifold (M, g) with (S 3 , [ g ^ ]) as its conformal infinity is unique up to isometries. For the high-dimensional case, we show that if g ^ is an SU (k + 1) -invariant metric on S 2 k + 1 for k ≥ 1 , the non-positively curved CCE metric on the (2 k + 1) -ball B 1 (0) with (S 2 k + 1 , [ g ^ ]) as its conformal infinity is unique up to isometries. In particular, since in Li (Trans Amer Math Soc 369(6): 4385–4413, 2017), we proved that if the Yamabe constant of the conformal infinity Y (S 2 k + 1 , [ g ^ ]) is close to that of the round sphere then any CCE manifold filled in must be negatively curved and simply connected, therefore if g ^ is an SU (k + 1) -invariant metric on S 2 k + 1 which is close to the round metric, the CCE metric filled in is unique up to isometries. Using the continuity method, we prove an existence result of the non-positively curved CCE metric with prescribed conformal infinity (S 2 k + 1 , [ g ^ ]) when the metric g ^ is SU (k + 1) -invariant. [ABSTRACT FROM AUTHOR]

In this paper, we verify a relation between the symmetric continuity of lower semicontinuous extended seminorms on a locally convex cone with respect to the symmetric topology and barreledness of the locally convex cone. The assertion, whether a barreled cone is upper-barreled or not, was posed as a question in Ranjbari and Saiflu (J Math Anal Appl 332(2):1097–1108, 2007). We show that a barreled locally convex cone is not necessarily upper-barreled. [ABSTRACT FROM AUTHOR]

In this paper, we study heteroclinic bifurcation and the appearance of chaos in time-perturbed piecewise smooth hybrid systems with discontinuities on finitely many switching manifolds. The unperturbed system has a heteroclinic orbit connecting hyperbolic saddles of the unperturbed system that crosses every switching manifold transversally, possibly multiple times. By applying a functional analytical method, we obtain a set of Melnikov functions whose zeros correspond to the occurrence of chaos of the system. As an application, we present an example of quasiperiodically excited piecewise smooth system with impacts formed by two linked rocking blocks. [ABSTRACT FROM AUTHOR]

In a recent paper Engle et al. (Commun Math Phys 354:231–246, 2017) showed that there is a unique state on the reduced holonomy–flux ∗ -algebra of homogeneous isotropic loop quantum cosmology, that is invariant under residual diffeomorphims. This result has been claimed to be true both for the Ashtekar–Bojowald–Lewandowski framework and for that introduced by the present author. Unfortunately, the uniqueness proof relies on an incorrect argument which spoils the second case. In our short note, we are going to patch this issue, this way keeping the nice uniqueness result in both cases. Moreover, we will even extend the underlying operator algebraic statements as this might help later for studying higher-dimensional models. [ABSTRACT FROM AUTHOR]

Copyright of Revista Española de Filosofía Medieval is the property of Sociedad Espanola de Filosofia Medieval and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Abel et al. [U. Abel, M. Ivan, R. Pătănea, Appl. Math. Comput., 259 (2015), 116-123] introduced a Durrmeyer type integral variant of the Bernstein type operators based on two parameters defined by Stancu [D. D. Stancu, Calcolo, 35 (1998), 53-62]. Kajla [A. Kajla, Appl. Math. Comput., 316 (2018), 400-408] considered a Kantorovich modification of the Stancu operators wherein he studied some basic convergence theorems and also the rate of A-statistical convergence. In the present paper, we define a bivariate case of the operators proposed in [A. Kajla, Appl. Math. Comput., 316 (2018), 400-408] to study the degree of approximation for functions of two variables. We obtain the rate of convergence of these bivariate operators by means of the complete modulus of continuity, the partial moduli of continuity and the Peetre's K-functional. Voronovskaya and Gr¨ uss Voronovskaya type theorems are also established. We introduce the associated GBS (Generalized Boolean Sum) operators of the bivariate operators and discuss the approximation degree of these operators with the aid of the mixed modulus of smoothness for B¨ogel continuous and Bögel differentiable functions. [ABSTRACT FROM AUTHOR]

Considered in this paper are control systems of the form x = f (x, u). For such systems a number of related necessary conditions for various forms of stability and attractivity are presented. The paper starts by showing that Brockett's necessary condition for stabilizability via smooth feedback still persists if f is continuous and the class of allowable u increased to include continuous feedbacks. Using similar ideas to those used to prove the continuous Brockett result, again only assuming continuity of f and u, necessary conditions are then derived for global attractivity and for ultimate boundedness. [ABSTRACT FROM AUTHOR]

Let $ \mathcal{A} $ be a Banach algebra and $ n > 1 $, a fixed integer. The main objective of this paper is to talk about the commutativity of Banach algebras via its projections. Precisely, we prove that if $ \mathcal{A} $ is a prime Banach algebra admitting a continuous projection $ \mathcal{P} $ with image in $ \mathcal{Z}(\mathcal{A}) $ such that $ \mathcal{P}(a^n) = a^n\; \text{for all} \; a \in \mathcal{G} $, the nonvoid open subset of $ \mathcal{A} $, then $ \mathcal{A} $ is commutative and $ \mathcal{P} $ is the identity mapping on $ \mathcal{A} $. Apart from proving some other results, as an application we prove that, a normed algebra is commutative iff the interior of its center is non-empty. Furthermore, we provide some examples to show that the assumed restrictions cannot be relaxed. Finally, we conclude our paper with a direction for further research.

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class of operators: (n,k) -quasi class Q* operators, superclass of (n,k) -quasi- *-paranormal operators. An operator T is said to be (n,k) -quasi class Q* if it satisfies for all x ∈ H and for some nonnegative integers n and k. We will prove structural and spectral properties of this class of operators, and also prove the spectrum continuity of this class of operators. [ABSTRACT FROM AUTHOR]

This paper is a subsequent work of [Invent. Math., 2013, 191: 197–253]. The second fundamental theorem in Ahlfors covering surface theory is that, for each set Eq of q (⩾ 3) distinct points in the extended complex plane ℂ ¯ , there is a minimal positive constant H0(Eq) (called Ahlfors constant with respect to Eq), such that the inequality (q − 2) A (∑) − 4 π # (f − 1 (E q) ∩ U) ⩽ H 0 (E q) L (∂ ∑) holds for any simply-connected surface Σ = (f, Ū), where A(Σ) is the area of Σ, L(∂Σ) is the perimeter of Σ, and # denotes the cardinality. It is difficult to compute H0 (Eq) explicitly for general set Eq, and only a few properties of H0(Eq) are known. The goals of this paper are to prove the continuity and differentiability of H0 (Eq), to estimate H0 (Eq), and to discuss the minimum of H0(Eq) for fixed q. [ABSTRACT FROM AUTHOR]

Schramm–Loewner evolution ( SLE κ ) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by κ times Brownian motion. This yields a (half-plane) valued random field γ = γ (t , κ ; ω) . (Hölder) regularity of in γ (· , κ ; ω ), a.k.a. SLE trace, has been considered by many authors, starting with Rohde and Schramm (Ann Math (2) 161(2):883–924, 2005). Subsequently, Johansson Viklund et al. (Probab Theory Relat Fields 159(3–4):413–433, 2014) showed a.s. Hölder continuity of this random field for κ < 8 (2 - 3) . In this paper, we improve their result to joint Hölder continuity up to κ < 8 / 3 . Moreover, we show that the SLE κ trace γ (· , κ) (as a continuous path) is stochastically continuous in κ at all κ ≠ 8 . Our proofs rely on a novel variation of the Garsia–Rodemich–Rumsey inequality, which is of independent interest. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to define a q‐analogue of a parametric generalization of Baskakov operators introduced by Aral and Erbay (Math. Commun. 24(1) (2019), 119‐131). We establish some local direct results for these operators by means of the modulus of continuity and the Peetre's K‐functional. Weighted approximation properties are also established. Next, we construct a bivariate case of the above q‐operators and determine the rate of convergence in terms of the moduli of continuity. A Voronovskaja‐type theorem is derived too. Further, we study the rate of approximation of Bögel continuous and Bögel differentiable functions by the associated generalized Boolean sum (GBS) operators with the aid of mixed modulus of smoothness. We illustrate the rate of approximation of the q‐operators, their bivariate, and the GBS cases by means of graphics and tables and show that the GBS operators provide better rate of convergence than the bivariate operators. [ABSTRACT FROM AUTHOR]

In this paper, we develop an elementary and unified treatment, in the spirit of Rivière and Struwe (Comm. Pure. Appl. Math. 2008), to explore regularity of weak solutions of higher order geometric elliptic systems in critical dimensions without using conservation law. As a result, we obtain an interior Hölder continuity for solutions of the higher order elliptic system of de Longueville and Gastel in critical dimensions Δku = ∑i=0k−1Δi⟨Vi,du⟩ + ∑i=0 k−2Δiδ (wi, under critical regularity assumptions on the coefficient functions. This verifies an expectation of Rivière, and provides an affirmative answer to an open question of Struwe in dimension four when k = 2. The Hölder continuity is also an improvement of the continuity result of Lamm and Rivière and de Longueville and Gastel. [ABSTRACT FROM AUTHOR]2k, under critical regularity assumptions on the coefficient functions. This verifies an expectation of Rivière, and provides an affirmative answer to an open question of Struwe in dimension four when k = 2. The Hölder continuity is also an improvement of the continuity result of Lamm and Rivière and de Longueville and Gastel. [ABSTRACT FROM AUTHOR]

In this paper we study the dependence of solutions of uncertain initial value problems (UIVP) on the initial values. Introducing a contraction mapping and using Banach Fixed Point Theorem (BFPT), the existence and uniqueness (EaU) of solutions of the UIVP will be proven. We show that under appropriate assumptions, the solutions of UIVP are continues and differentiable with respect to initial conditions (ICs). The paper will be ended by proving a theorem about the existence of solutions of an autonomous UIVP under weaker conditions. This theorem is a generalization of PeanoTheorem to UDEs.

In this paper, we introduce some concepts of convexity and semicontinuity for real set-valued mappings similar to those of real single-valued mappings. Then, we obtain different results on the existence of solutions of set-valued equilibrium problems generalizing in a common way several old ones for both single-valued and set-valued equilibrium problems. Applications to Browder variational inclusions, with weakened conditions on the involved set-valued operator, are given. [ABSTRACT FROM AUTHOR]

Copyright of HUM: Journal of the Faculty of Arts at the University of Mostar is the property of University of Mostar, Faculty of Arts and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, the convergence for improved multi-point Chebyshev-like methods is considered. Compared to the results of Chebyshev method with second derivative in reference [M.A. Hernández, M.A. Salanova, J. Comput. Appl. Math. 126 (2000), pp. 131–143], the R-order is maximised and the Hölder continuity condition is also relaxed. Moreover, these improved methods do not need second derivatives, when the computation of second derivative is large, these improved methods will be more efficient than Chebyshev method. Under the relaxed condition, an existence-uniqueness theorem is proved. The R-order of convergence for these improved methods is also analysed. [ABSTRACT FROM AUTHOR]

This paper describes a study of picture fuzzy topological spaces. We prove some basic results related to picture fuzzy sets together with the introduction of new notions such as the rank, picture fuzzy base and picture fuzzy sub-base of picture fuzzy topological spaces. With the help of these notions, we present a method to design picture fuzzy topological spaces. Furthermore, we introduce the concept of continuity to picture fuzzy topological spaces and find a necessary and sufficient condition for a picture fuzzy continuous function between two picture fuzzy topological spaces. [ABSTRACT FROM AUTHOR]

We characterize the subdifferential of the supremum function of finitely and infinitely indexed families of convex functions. The main contribution of this paper consists of providing formulas for such a subdifferential under weak continuity assumptions. The resulting formulas are given in terms of the exact subdifferential of the data functions at the reference point, and not at nearby points as in [Valadier, C. R. Math. Acad. Sci. Paris, 268 (1969), pp. 39--42]. We also derive new Fritz John- and KKT-type optimality conditions for semi-infinite convex optimization, omitting the continuity/closedness assumptions in [Dinh et al., ESAIM Control Optim. Calc. Var., 13 (2007), pp. 580--597]. When the family of functions is finite, we use continuity conditions concerning only the active functions, and not all the data functions as in [Rockafellar, Proc. Lond. Math. Soc. (3), 39 (1979), pp. 331--355; Volle, Acta Math. Vietnam., 19 (1994), pp. 137--148]. [ABSTRACT FROM AUTHOR]

Knowledge or rule-based approaches are needed for quality assessment and assurance in professional or crowdsourced geographic data. Nevertheless, many types of geographic knowledge are statistical in nature and are therefore difficult to derive rules that are meaningful for this purpose. The rules of continuity and symmetry considered in this paper can be thought of as two concrete forms of the first law of geography, which may be used to formulate quality measures at the individual level without referring to ground truth. It is not clear, however, how much the rules can be faithful. Hence, the main objective is to test if the rules are consistent with street network data over the world. Specifically, for the rule of continuity we identify natural streets that connect smoothly in a network, and measure the spatial order of information (e.g. names, highway level, speed, etc.) along the streets. The measure is based on spatial auto-correlation indicators adapted for one dimension. For the rule of symmetry, we device an algorithm that recognize parallel road pairs (e.g. dual carriageways), and examine to what extent attributes in the pairs are identical. The two rules are tested against 28 cities selected from OpenStreetMap data worldwide; two professional data sets are used to show more insights. We found that the rules are consistent with street networks from a wide range of cities of different characteristics, and also noted cases with varying degrees of agreement. As a side-effect, we discussed possible limitations of the autocorrelation indicators used, where cautions are needed when interpreting the results. In addition, we present techniques that performed the tests automatically, which can be applied to new data to further verify (or falsify) our findings, or extended as quality assurance tools to detect data items that do not satisfy the rules and to suggest possible corrections according to the rules. [ABSTRACT FROM AUTHOR]

Sarasvati Amma published very few research papers. All her insights into the Indian mathematical (specifically, geometric) tradition are to be found in her book "Geometry in Ancient and Medieval India", published in 1979 but prepared as her thesis in the University of Madras 20 years earlier. The present article is, consequently, an evaluation of the mathematics described in the book and of the historiographic significance of its interpretation by her. The book pays specific attention to certain themes: e.g., the key ideas of the geometry of the Vedic period, cyclic quadrilaterals, geometric algebra etc. and, especially, the infinitesimal trigonometry of MÀdhava, all in a style designed to bring out the continuity in their evolution. The case is made in this article that Sarasvati Amma's work, along with the earlier book of B. Datta and A. N. Singh, marks the founding of an autonomous discipline of scholarship into India's mathematical past. [ABSTRACT FROM AUTHOR]

In the paper, two classes of involutive divisions related to admissible monomial orderings—≻-divisions and ≺-divisions—are considered. The latter may be viewed as an improvement of the class of induced divisions introduced in [1]. The continuity and constructivity of the ≺-divisions, as well as a number of additional properties of the ≻-divisions, are proved. Taking into account that the Janet division is a ≻-division associated with the lexicographical order, its “antipode�—a ≺-division associated with the same order—is separated in the class of the ≺-divisions. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce the concepts of η-sets, ηζ-sets,η-continuity and ηζ-continuity and to obtain a decomposition of continuity. [ABSTRACT FROM AUTHOR]

Let O be a subdomain in Rn and M<> be the local Hardy-Littlewood maximal function. In this paper, we show that both the commutator and the maximal commutator of Mg are bounded and continuous from the first order Sobolev spaces H"1,p' (O) to IV1,p(O) provided that 6 - I4z1'P2(-), 1 < pi,p2,p < oo and\/p = l/p1 + l/p2· These arc done by establishing several new pointwise estimates for the weak derivatives of the above commutators. As applications, the bounds of these operators on the Sobolev space with zero boundary values arc obtained. [ABSTRACT FROM AUTHOR]

For the infinite horizon cost function mixed constrained model predictive control, the largest possible stabilizable region for stable plants is the entire state space for both state and output feedback cases. However, for marginal or unstable cases, the largest possible stabilizable region is the constrained m-step stabilizable set for the state feedback case, and it is the region where the estimated state is in the constrained m-step stabilizable set throughout the trajectory for the output feedback case. Only attractivity over the largest stabilizable region is established for the state feedback case with stable or marginal plants and the output feedback case with stable plants [A. Zheng and M. Morari, IEEE Trans. Automat. Control, 40 (1995), pp. 1818-1823]. In this paper we show, for both state and output feedback cases, that the closed loop system with the mixed constrained model predictive controller possesses the exponential stability property, much stronger than the attractivity, on the largest possible stabilizable region. Here the exponential stability on the largest possible stabilizable region means that we can find the exponentially converging envelope for any initial condition in the region. Clearly this is much stronger than local exponential stability, for which the region for the envelope is not known and can be arbitrarily small. Moreover, the continuity properties of the mixed constrained model predictive control are also established. [ABSTRACT FROM AUTHOR]

The Problem of Continuity and Discreteness is the basic problem of philosophy and mathematics. For a long time, there is no clear understanding of this problem, which leads to the stagnation of the problem, because the essence of the problem is a problem of finity and infinity. The essence of the philosophical thought on which the mathematical definition of “line segment is composed of dots” is the idea of actual infinity, and geometric dot is equivalent to algebraic zero in terms of measure properties. This paper discusses the views of many philosophers on this problem, and on the basis of the view of dialectical infinity, gives two solutions that satisfy both philosophical and mathematical circles, so as to promote the healthy development of basic mathematics. Discrete the continuity, let the continuity become discreteness, but the discreteness itself is still a continuity, so as to realize the dialectical unity of continuity and discreteness. [ABSTRACT FROM AUTHOR]

This paper aimed to further introduce the finite element analysis of non-smooth data for semilinear stochastic subdiffusion problems driven by fractionally integrated additive noise. The mild solution of this stochastic model consisted of three different Mittag-Leffler functions. We analyzed the smoothness of the solution and utilized complex integration to approximate the error of the solution operator under non-smooth data. Consequently, optimal convergence estimates were obtained, and we also obtained the continuity conditions of the mild solution. Finally, the influence of the fractional parameters $ \alpha $ and $ \gamma $ on the convergence rates were accurately demonstrated through numerical examples.

The paper provides necessary and sufficient conditions for the existence of weak solutions for the Dirichlet problem within Bishop's constructive mathematics. It also considers the continuity of those weak solutions relative to the boundary data and the domain. [ABSTRACT FROM AUTHOR]

In this paper, we establish a relation-theoretic version of the results presented by Kim et al. (Journal of Nonlinear and Convex Analysis, 16 (9) (2015), 1779-1786). To showcase the versatility of our results, we furnish some illustrative examples. Furthermore, we exhibit an application of our results to establish sufficient conditions for the existence of a positive definite common solution to a pair of nonlinear matrix equations.

This paper studies the concept of fuzzy generalized topologies, which are generalizations of smooth topologies and Chang's fuzzy topologies. A basis of fuzzy generalized topological space will be defined as functions from the family of all fuzzy subsets of a non-empty set $ X $ to $ [0, 1] $ and some basic properties of their structure will be obtained. Some characterizations of the basis of fuzzy generalized topology, fuzzy generalized cotopology and the product of fuzzy generalized topological spaces will also be shown.

We obtain curvature estimates for long-time solutions of the continuity method on compact Kähler manifolds with semi-ample canonical line bundles. In this setting, initiated in [G. La Nave and G. Tian, A continuity method to construct canonical metrics, Math. Ann. 365(3) (2016) 911–921; Y. A. Rubinstein, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics, Adv. Math. 218(5) (2008) 1526–1565], we adapt arguments from [F. T.-H. Fong and Y. Zhang, Local curvature estimates of long-time solutions to the Kähler–Ricci flow, Adv. Math. 375 (2020) 107416] for the Kähler–Ricci flow to this setup. As an application, we derive curvature bounds for general metrics on product manifolds. [ABSTRACT FROM AUTHOR]

The paper discusses the problem of infinity with a special reference to mathematical truth. The roots of this problem, in a sense, go back to early efforts to provide a rational account of the phenomena of change, and the phenomenon of motion in particular, and, in connection with this, to the attempt to cast light on the complex relationship between continuity and discreteness. If mathematics is to serve as a tool of a rational description of the world, is it possible to construct a mathematical model of phenomenal continuity? This question was raised already by Zeno when the Pythagorean hypothesis of the essential ‘granularity’ of the world began to crumble as a result of the discovery of incommensurable magnitudes. The discovery of incommensurable magnitudes, and along with it the discovery of irrational numbers, produced well-known paradoxes and gave rise to the question to what extent, if at all, mathematics can deliver a ‘metaphysical’ description of the world. In more recent times, the idea of infinity was built into the infinitesimal calculus which became the foundation of the mathematical description of nature, but the metaphysical questions have not gone away. The issues surrounding actual and potential infinity continue to be the subject of intense debates. Whereas some recognise only potential infinities, others, like Georg Cantor, accept actually existent infinities. This is directly connected with the question of mathematical truth, which, according to the Platonist tradition exists as objectively given. The paper argues that the Cantorian transfinite set theory cannot withstand criticism, and that rather than providing a (metaphysical) description of reality mathematics merely offers possible models in terms of which the world might be better or more conveniently interpreted. The so-called ‘mathematical truth’ reduces to the question of ‘correctness’, viz. whether a given mathematical proposition does or does not follow from given premises.

In this paper, we introduce the concept of (r,s)-fuzzy minimal structure which is an extension of intuitionistic gradation of openness introduced by Samanta and Mondal (2002). Also, we introduce and study the concepts of (r,s)-fuzzy M continuity and several types of (r,s)-fuzzy minimal compactness on (r,s)-fuzzy minimal spaces. [ABSTRACT FROM AUTHOR]

In this paper we prove some coincidences and fixed point theorems for a pair of single-valued and multivalued mappings satisfying ϕ-contractive condition. Our results improve and extend some well known previous results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of pre α-irresolute and somewhat pre α-irresolute functions in topological spaces. We discuss and study some characterizations and basic properties of these classes of functions. Relations between this class and some other existing classes of functions are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the new class of weaker form of contra semi weakly g*-continuous functions in topological spaces and study some of their properties. [ABSTRACT FROM AUTHOR]

In this paper, we prove common fixed point theorems in fuzzy metric spaces for weakly compatible mappings along with property (E. A.) satisfying implicit relation. Property (E. A.) buys containment of ranges without any continuity requirement besides minimizing the commutativity conditions of the maps to commutativity at their point of coincidence. Moreover, property (E. A.) allows replacing the completeness requirement of the space with a more natural condition of closeness of the range. [ABSTRACT FROM AUTHOR]

In this paper, we define new classes of sets called J-g-open sets, J-gp-open sets, J-gs-open sets, J-αg-open sets, J- C(S)-sets, J-C*-sets and J-S*-sets. We investigate the relationships between these sets and with other sets in ideal topological spaces. Also, we obtain new decompositions of J-g-continuity in ideal topological spaces. [ABSTRACT FROM AUTHOR]

This paper aims to analyze the possible influences of the Solidarity Assimilation Pedagogy (SA) in the process of teaching and learning of continuity. The SA is a proposal that aims to break with traditional mathematics teaching, allowing students to receive bonus grades for group learning work unrelated to the assessment grade. The learning work of each class is based on a activity sheets composed of problems on the topic addressed. In this article, we analyze the development of the activity sheets about the continuity of a function, applied in a class of Calculus. The data analyzed through content analysis allowed us to infer that SA allows students to develop autonomy in learning, making them co-responsible for the whole process. [ABSTRACT FROM AUTHOR]

The concept of enlargement of a generalized topology was introduced by Á. Császár [4]. He also introduced the concepts of ( κ, λ)-continuity and ( κ, λ)-continuity on enlargements. In this paper, we characterize the ( κ, λ)-continuity and introduce the concept of strong ( κ, λ)-continuity on enlargements. In particular, we study characterizations for the strong ( κ, λ)-continuity and the relationships among ( μ,ν)-continuity, ( κ, λ)-continuity, strong ( κ, λ)-continuity and ( κ, λ)-continuity. [ABSTRACT FROM AUTHOR]

We introduce the path player game, a noncooperative network game with a continuum of mutually dependent set of strategies. This game models network flows from the point of view of competing network operators. The players are represented by paths in the network. They have to decide how much flow shall be routed along their paths. The competitive nature of the game is due to the following two aspects: First, a capacity bound on the overall network flow links the decisions of the players. Second, edges may be shared by several players which might have conflicting goals. In this paper, we prove the existence of feasible and pure-strategy equilibria in path player games, which is a nontrivial task due to noncontinuity of payoff functions and the infinite, mutually dependent strategy sets. We analyze different instances of path player games in more detail and present characterizations of equilibria for these cases. [ABSTRACT FROM AUTHOR]

In the paper, a generalization of a known theorem by Hardy and Young is obtained; a formula interrelating the integral of a 2π-periodic function over the period with the integral over the entire axis is established; new approximation characteristics for functions belonging to saturation classes of continuity modules of different orders for the spaces Lp of periodic functions are provided, and some issues concerning approximation, in the uniform metric, of continuous periodic functions even with respect to each of their variables and having nonnegative Fourier coefficients are considered. Bibliography: 17 titles. [ABSTRACT FROM AUTHOR]

In this paper, we study the regularity of solutions to the Monge-Ampère equation. We prove the log-Lipschitz continuity for the gradient under certain assumptions. We also give a unified treatment for the continuity estimates of the second derivatives. As an application we show the local existence of continuous solutions to the semigeostrophic equation arising in meteorology. [ABSTRACT FROM AUTHOR]

We give an elementary proof of the global well-posedness for the critical 2 D dissipative quasi-geostrophic equation. The argument is based on a non-local maximum principle involving appropriate moduli of continuity. [ABSTRACT FROM AUTHOR]

In 2004, Al-Hawary and Al-Omari introduced and explored the class of ωº--open sets which is strictly stronger than the class of ω--open sets and weaker than that of open sets. In this paper, we introduce what we call ωº--continuity and ωXº--continuity and we give several characterizations and two decompositions of ωº--continuity. Finally, new decompositions of continuity are provided. [ABSTRACT FROM AUTHOR]

In this paper, a necessary and sufficient condition for the inclusion of the class Λ BV in the class H is found. [ABSTRACT FROM AUTHOR]