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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]

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]

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]

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]

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]

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]

In this paper we investigate continuity properties of functions $${f : \mathbb {R}_+ \to \mathbb {R}_+}$$ that satisfy the ( p, q)-Jensen convexity inequality where H stands for the pth power (or Hölder) mean. One of the main results shows that there exist discontinuous multiplicative functions that are ( p, p)-Jensen convex for all positive rational numbers p. A counterpart of this result states that if f is ( p, p)-Jensen convex for all $${p \in P \subseteq \mathbb {R}_+}$$ , where P is a set of positive Lebesgue measure, then f must be continuous. [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]

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 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]

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]

For a fixed positive number γ, a real-valued function f defined on a convex subset D of a normed space X is said to be γ-convex if it satisfies the inequality f(x′0) + f (x′1) ⩽ f(x0) + f(x1), for x′i ∊ [x0, x1], for x′i ∊[x0, x1], ∥x′i - xi∥ = γ, i = 0, 1, whenever x0, x1 ∊ D and ∥x0 - x1∥ ➮ γ. This paper presents some results on the boundedness and continuity of γ-convex functions. For instance, (a) if there is some x* ∊ D such that f is bounded below on D ∩ B (x*, γ), then so it is on each bounded subset of D; (b) if f is bounded on some closed ball B(x*, γ/2) ⊂ D and D′ is a closed bounded subset of D, then f is bounded on D′ if it is bounded above on the boundary of D′; (c) if dim X > 1 and the interior of D contains a closed ball of radius γ, then f is either locally bounded or nowhere locally bounded in the interior of D; (d) if D contains some open ball B(x*, γ/2) in which f has at most countably many discontinuities, then the set of all points at which f is continuous is dense in D. [ABSTRACT FROM AUTHOR]

In this paper, we study two variables operation continuity of lattice effect algebra with respect to its order topology. [ABSTRACT FROM AUTHOR]

Noiri in 1984, Dontchev, Ganster and Reilly in recent years and Reilly and Vamanamurthy in 1983 introduced the notion of perfectly continuous, regular set-connected and clopen functions, respectively. The aim of this paper is to introduce the notion of a new class of functions which is called almost clopen functions including the classes of perfectly continuous, regular set-connected and clopen functions. Furthermore, properties of almost clopen functions are obtained and relationships among almost clopenness, perfect continuity, regular set-connectedness, clopenness and almost continuity are investigated. [ABSTRACT FROM AUTHOR]

The vivacity of mathematics results (partly) from the fact that mathematics is stretched between several poles, not being committed to any one. The paper presents the following polarities: realism - idealism, the finite - the infinite, the discrete - the continuous, the approximate - the exact, certitute - probability, simplicity - complexity, unity - multiplicity. [ABSTRACT FROM AUTHOR]