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In this paper we shall introduce and study subclasses Rλ (α, s, t) and Pλ (α, s, t) of the class of λ - spirallike generalized Sakaguchi type function. Here we shall prove characterization and subordination properties for these subclasses and point out several interesting consequences of our results. [ABSTRACT FROM AUTHOR]

In this paper we give the definition of [λ ᵘ]-continuity of real-valued functions defined on an open interval, which is an example of path continuity. We give some properties of [λ, ᵘ]-continuous functions. The aim of the paper is to find the maximal additive class and the maximal multiplicative class for the family of [λ, ᵘ]-continuous functions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce two new classes of multifunctions called almost δ-semicontinuous multifunctions and weakly δ-semicontinuous multifunctions which are generalization of δ-semicontinuous multifunctions. In this paper we obtain several properties of such multifunctions. [ABSTRACT FROM AUTHOR]

In this paper, we study triples of the form (E, θ, ϕ) over a compact Riemann surface, where (E, θ) is a Higgs bundle and ϕ is a global holomorphic section of the Higgs bundle. Our main result is an description of a birational equivalence which relates geometrically the moduli space of Higgs bundles of rank r and degree d to the moduli space of Higgs bundles of rank r-1 and degree d. [ABSTRACT FROM AUTHOR]

An extremely simple proof of the K-K-M-S Theorem is given involving only Brouwer's fixed point theorem and some elementary calculus. A function is explicitly given such that a fixed point of it yields an intersection point of a balanced collection of sets together with balancing weights. Moreover, any intersection point of a balanced collection of sets together with balancing weights corresponds to a fixed point of the function. Furthermore, the proof can be used to show π-balanced versions of the K-K-M-S Theorem, with π-balancedness as introduced in Billera (1970). The proof makes clear that the conditions made with respect to π by Billera can be even weakened. [ABSTRACT FROM AUTHOR]

In this paper, we establish some inequalities of Fejér type for g-convex dominated functions. [ABSTRACT FROM AUTHOR]

The article reports on the nature of projectile motion in a uniform, downward gravitational field and resisted by a drag force due to the air through which it moves. It states that the nature of the projectile motion lies near the heart of the scientific revolution. The drag force is assumed to act in a direction opposite to that of the projectile's velocity and to depend only on its magnitude, therefore, Coriolis forces and Earth's curvature are ignored. The author notes that the projectile motion under quadratic drag was at the center of an episode in the calculus priority dispute.

Let Σ(p, k)(p, k ∈ N) be the class of functions f(z) = z-p+ak-p²k-p+... analytic in 0 < |z| < 1. A subclass M(p, k, α, λ) of Σ(p, k) is introduced. The object of the present paper is to investigate convolution properties for the class M(p, k, α, λ). Also a property of the partial sums of certain functions in this class is given. [ABSTRACT FROM AUTHOR]

This paper gives some characterizations of strongly θ-precontinuous functions. [ABSTRACT FROM AUTHOR]

Let L be a degree-2L-function associated to a Maass cusp form. We explore an algorithm that evaluates t values of L on the critical line in time O(t1+𝜖). We use this algorithm to rigorously compute an abundance of consecutive zeros and investigate their distribution. [ABSTRACT FROM AUTHOR]

A function $f:X\rightarrow X$ determines a topology $P(f)$ on $X$ by taking the closed sets to be those sets $A\subseteq X$ with $f(A)\subseteq A$. The topological space $(X,P(f))$ is called a functionally Alexandroff space. We completely characterise the homogeneous functionally Alexandroff spaces. [ABSTRACT FROM AUTHOR]

It has been shown that every n-person, perfect information game with no chance moves and bounded, lower semicontinuous payoffs has a subgame perfect-equilibrium in pure strategies. Here the same is proved when the payoffs are bounded and upper semicontinuous. [ABSTRACT FROM AUTHOR]

We obtain an exact estimate for a nonincreasing uniform rearrangement of a function of two variables from the Muckenhoupt class A. [ABSTRACT FROM AUTHOR]

Let f: ℝ+ → ℝ. The subject is the trace inequality Tr f( A) + Tr f( P2 AP2) ≦ Tr f( P12 AP12) + Tr f( P23 AP23), where A is a positive operator, P1; P2; P3 are orthogonal projections such that P1 + P2 + P3 = I, P12 = P1 + P2 and P23 = P2 + P3. There are several examples of functions f satisfying the inequality (called (SSA)) and the case of equality is described. [ABSTRACT FROM AUTHOR]

This study examines how students in the early stages of learning about the concept of functions, describe a curve and, in particular, evaluate the appropriateness of their argument about the representation of a function. Students are offered a message game which is related to a curve drawn on a coordinate system, representing an 'imaginary function'. In this message game, one half of the class must describe a curve to the other half who will then try to reproduce it as similarly as possible. Data analysis indicates that the idea of the curve that consists of series of points is very effective, while the idea of variation does not exist or is too limited among students at the beginning of learning functions. Thus, the concept of variation is not used as a criterion to describe a curve. More surprisingly, the students are working on the curve as if it was just a picture on a grid regardless of the representation of the function. [ABSTRACT FROM AUTHOR]

Abstract.  We summarize what we know about the clone lattice on an infinite set and formulate what we consider the most important open problems. [ABSTRACT FROM AUTHOR]

The article reports on the sharpening of Wielandt's characterization of the gamma function. It presents the two conditions that is satisfied with Euler's gamma function. According to the author, the two conditions by themselves do not characterize the gamma function and presents characterization of the gamma function through theorems that Wielandt discovered. He adds that a characterization in the real domain, found in 1922 by other scholars, is well known, while Wielandt's theorem is function-theoretic.

In Refs. [J. Math. Anal. Appl. 258:287-308, 2001; J. Math. Anal. Appl. 256:229-241, 2001], Yang and Li presented a characterization of preinvex functions and semistrictly preinvex functions under a certain set of conditions. In this note, we show that the same results or even more general ones can be obtained under weaker assumptions; we also give a characterization of strictly preinvex functions under mild conditions. [ABSTRACT FROM AUTHOR]

Choice functions play an important role in political science. We present structure results of choice functions following an algebraic approach. It is logical that rational political actors are not able to perceive every alternative in a set of alternatives X. They may also choose to exclude some alternatives from consideration. Let S be a subset of X that some set of political actors are not able to perceive. Then the complement X\S is the set comprising those alternatives considered in the decision process. If C is a choice function on X, then C(X\S) are the alternatives chosen and X\C(X\S) are those that are not. We are thus led to the concept of an upper choice function. We define upper choice functions in terms of choice functions and we give relationships between them. [ABSTRACT FROM AUTHOR]

The article analyzes the boundary asymptotic behavior of solutions for weighted p(x)-Laplacian equations that take infinite value on a bounded domain. It was found that the boundary asymptotic of solutions of the Laplacian function is continuous and positive. The p(x)-Laplacian possesses more complicated inhomogeneous nonlinearities, thus, some special techniques are needed. Its main difficulty arises from the lack of compactness. The uniqueness and asymptotic behavior of solutions for problem is a nonnegative function which is allowed to vanish on the boundary.

The article presents a research regarding the association between Jordan-von Neumann-type additive functional equations and functional inequalities. It proves the generalized Hyers-Ulam stability of the following functional inequalities: ρf(x)+f(y)+f(z)ρ ≤ ρ 2f((x+y+z)/2)ρ, ρf(x)+f(y)+f(z) ρ ≤ ρf(x+y+z)ρ, ρ f(x)+f(y)+2f(z)ρ ρ2f((x+y)/2+z)ρ in the spirit of the Rassias stability method for approximately homomorphisms. According to the report, the inequality that was brought in for the first time by Rassias provided a lot of influence in the development of a generalization relating to the Hyers-Ulam stability concept.

The article presents a discussion about the Koch Curve. It states that the mathematically popular Koch curve is usually used as an example to illustrate a continuous but not differentiable function and as an example of a non-rectifiable curve. However, it emphasizes that only the fact that it is not rectifiable is proved, and the proof that it is a curve and that no point does this curve have a tangent line is disregarded. It offers a mathematical and simple geometric proof that the Koch curve is indeed an arc and that it at no point has a tangent line.

The article presents a discussion about a summation formula that is used in certain transformations on generating algebraic and differential functions and the extraction and creation of coefficients. The goal of the article is to offer the method of coefficients. Mathematical functions are manipulated in order to arrive at a single and simple mathematical expression. It mentions that generating functions have emerged as one of the most popular approaches to combanitorial problems, which includes the analysis of algorithms.

The canonical function game is a game of length ω1 introduced by W. Hugh Woodin which falls inside a class of games known as Neeman games. Using large cardinals, we show that it is possible to force that the game is not determined. We also discuss the relationship between this result and Σ22 absoluteness, cardinality spectra and Π2 maximality for H( ω2) relative to the Continuum Hypothesis. [ABSTRACT FROM AUTHOR]

We investigate a variational approach to the Hele–Shaw flow ∂̸ t χ = &triangle; u + f χ, f ≥ 0 in &reals; N , where χ is the characteristic function of an open set Ω( t ) in &reals; N and solves - &triangle; u ( t , ·) = f in Ω( t ). By choosing a time step τ k = τ/2 k and iteratively solving a variational problem in &reals; N , we construct a staircase family of opens sets and a corresponding family of functions: as k → ∞, both sets and functions converge increasingly, at fixed time, to a weak solution of the problem. When the latter is not unique, the solution thus obtained is characterized by a minimality property, with respect to set inclusion, at fixed time. We also prove several monotonicity results of the solutions thus obtained, with respect to both the initial set and the forcing term f . In particular, these monotonicity properties imply that Ω( t ) has finite perimeter for every t , provided that Ω(0) has finite perimeter. Finally, under very mild assumptions, we prove that the number of connected components is nonincreasing, that Ω( t ) is connected for large t , and that it tends to fill the whole of &reals; N . [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for multipliers of a convolution of a smooth generalized integration operator to be topological automorphisms in the space of continuous functions. Moreover, an equivalent form of the conditions is found in terms of cyclic elements of the considered integration operator. [ABSTRACT FROM AUTHOR]

For z ∈ ∂Bn, the boundary of the unit ball in \Bbb {C}^n \ {\rm let} \ \Lambda(2)=\{ \lambda : \vert \lambda \vert \leqslan/math>. If f \in \Bbb{O}(B^n) then we call E(f)=\{ z \in \partial B^n: \int_{\Lambda(z)} \vert f(z) \vert ^2 d\Lambda(z)=\infty \} the exceptional set for f. In this note we give a tool for describing such sets. Moreover we prove that if E is a Gδ and Fσ subset of the projective (n − 1)-dimensional space \Bbb{P}^{n-1}=\Bbb{P}(\Bbb{C}^n) then there exists a holomorphic function f in the unit ball Bn so that E(f) = E. [ABSTRACT FROM AUTHOR]

It is shown that a function u satisfying |∂t+Δu|≦M(|u|+|∇u|), |u(x, t)|≦MeM|x|2 in (ℝn \ (BR) × [0, T] and u(x, 0) = 0 for x?ℝn \ BR must vanish identically in ℝn \ BR×[0, T]. [ABSTRACT FROM AUTHOR]

The article presents an elementary proof of Joris's Theorem. It utilizes Rolle's theorem, L'Hospital's rule, and Taylor's formula from elementary calculus. Joris considers the conditions under which a quotient of smooth functions is again smooth. Let g(x) = f(x)m and h(x) = f(x)n. It is assumed that, since m and n are relatively prime, one of m or n is odd. Function f is continuous as the composition of the continuous functions. Function f is continuous as the composition of the continuous functions n√x and h. Let K = f -1 ({0}). Then K is closed, and 0 = R \ K is an open subset of R, hence the disjoint union of countably many open intervals. On each of these intervals, both n√x and h are smooth on O.