Support theorems for Funk-type isodistant Radon transforms on constant curvature spaces

A connected maximal submanifold in a constant curvature space is called isodistant if its points are in equal distances from a totally geodesic of codimension 1. The isodistant Radon transform of a suitable real function f on a constant curvature space is the function on the set of the isodistants that gives the integrals of f over the isodistants using the canonical measure. Inverting the isodistant Radon transform is severely overdetermined because the totally geodesic Radon transform, which is a restriction of the isodistant Radon transform, is invertible on some large classes of functions. This raises the admissibility problem that is about finding reasonably small subsets of the set of the isodistants such that the associated restrictions of the isodistant Radon transform are injective on a reasonably large set of functions. One of the main results of this paper is that the Funk-type sets of isodistants are admissible, because the associated restrictions of the isodistant Radon transform, we call them Funk-type isodistant Radon transforms, satisfy appropriate support theorems on a large set of functions. This unifies and sharpens several earlier results for the sphere, and brings to light new results for every constant curvature space.

Error Bounds Related to Midpoint and Trapezoid Rules for the Monotonic Integral Transform of Positive Operators in Hilbert Spaces

For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the followingmonotonic integral transformwhere the integral is assumed to exist forT a positive operator on a complex Hilbert spaceH. We show among others that, if β ≥ A, B ≥ α > 0, and 0 < δ ≤ (B − A)2 ≤ Δ for some constants α, β, δ, Δ, thenandwhere is the second derivative of as a real function.Applications for power function and logarithm are also provided.

A New Class of Contact Pseudo Framed Manifolds with Applications

In this paper, we introduce a new class of contact pseudo framed (CPF)-manifolds M , g , f , λ , ξ by a real tensor field f of type 1,1 , a real function λ such that f 3 = λ 2 f where ξ is its characteristic vector field. We prove in our main Theorem 2 that M admits a closed 2-form Ω if λ is constant. In 1976, Blair proved that the vector field ξ of a normal contact manifold is Killing. Contrary to this, we have shown in Theorem 2 that, in general, ξ of a normal CPF-manifold is non-Killing. We also have established a link of CPF-hypersurfaces with curvature, affine, conformal collineations symmetries, and almost Ricci soliton manifolds, supported by three applications. Contrary to the odd-dimensional contact manifolds, we construct several examples of even- and odd-dimensional semi-Riemannian and lightlike CPF-manifolds and propose two problems for further consideration.

Directional Shift-Stable Functions

Recently, some new types of monotonicity—in particular, weak monotonicity and directional monotonicity of an n-ary real function—were introduced and successfully applied. Inspired by these generalizations of monotonicity, we introduce a new notion for n-ary functions acting on [0,1]n, namely, the directional shift stability. This new property extends the standard shift invariantness (difference scale invariantness), which can be seen as a particular directional shift stability. The newly proposed property can also be seen as a particular kind of local linearity. Several examples and a complete characterization for the case of n=2 of directionally shift-stable aggregation and pre-aggregation functions are also given.

Aristóteles Contra o Bem dos Economistas

Despite its origins within moral philosophy, economists think their science has nothing to do with the good. They appeal to some kind of Hume’s guillotine that divides the descriptive and the normative. With that in hand, they affirm the solely descriptive aspect of their discipline. I argue this is not the case. Economists have, as they need to, an all encompassing notion of the good. I suggest going back to Aristotelian arguments to show the shortcomings of this good of economists. Aristotle is helpful because of his analysis of chrematistics and the real function of money. Hence, the loosely utilitarian good of the economists is confronted with a robust sense of the good and the human form. The capability approach is the first to identify these weak points on economic theory and to propose a sort of Aristotelian comeback. However, I claim the capabilities approach itself doesn’t follow the Aristotelian arguments used to attack economists to its necessary conclusion. Therefore, I suggest that the recent advances in neo-Aristotelian ethical naturalism can be used to reformulate economics by dispossessing economists of their sumo bonnun.

Calculation of Two Types of Quaternion Step Derivatives of Elementary Functions

We aim to get the step derivative of a complex function, as it derives the step derivative in the imaginary direction of a real function. Given that the step derivative of a complex function cannot be derived using i, which is used to derive the step derivative of a real function, we intend to derive the complex function using the base direction of the quaternion. Because many analytical studies on quaternions have been conducted, various examples can be presented using the expression of the elementary function of a quaternion. In a previous study, the base direction of the quaternion was regarded as the base separate from the basis of the complex number. However, considering the properties of the quaternion, we propose two types of step derivatives in this study. The step derivative is first defined in the j direction, which includes a quaternion. Furthermore, the step derivative in the j+k2 direction is determined using the rule between bases i, j, and k defined in the quaternion. We present examples in which the definition of the j-step derivative and (j,k)-step derivative are applied to elementary functions ez, sinz, and cosz.

STRONG CONTINUITY OF FUNCTIONS FROM TWO VARIABLES

The concept of continuity in a strong sense for the case of functions with values in metric spaces is studied. The separate and joint properties of this concept are investigated, and several results by Russell are generalized. A function $f:X \times Y \to Z$ is strongly continuous with respect to $x$ /$y$/ at a point ${(x_0, y_0)\in X \times Y}$ provided for an arbitrary $\varepsilon> 0$ there are neighborhoods $U$ of $x_0$ in $X$ and $V$ of $y_0$ in $Y$ such that $d(f(x, y), f(x_0, y)) <\varepsilon$ /$d((x, y), f (x, y_0))<\varepsilon$/ for all $x \in U$ and $y \in V$. A function $f$ is said to be strongly continuous with respect to $x$ /$y$/ if it is so at every point $(x, y)\in X \times Y$. Note that, for a real function of two variables, the notion of continuity in the strong sense with respect to a given variable and the notion of strong continuity with respect to the same variable are equivalent. In 1998 Dzagnidze established that a real function of two variables is continuous over a set of variables if and only if it is continuous in the strong sense with respect to each of the variables. Here we transfer this result to the case of functions with values in a metric space: if $X$ and $Y$ are topological spaces, $Z$ a metric space and a function $f:X \times Y \to Z$ is strongly continuous with respect to $y$ at a point $(x_0, y_0) \in X \times Y$, then the function $f$ is jointly continuous if and only if $f_{y}$ is continuous for all $y\in Y$. It is obvious that every continuous function $f:X \times Y \to Z$ is strongly continuous with respect to $x$ and $y$, but not vice versa. On the other hand, the strong continuity of the function $f$ with respect to $x$ or $y$ implies the continuity of $f$ with respect to $x$ or $y$, respectively. Thus, strongly separately continuous functions are separately continuous. Also, it is established that for topological spaces $X$ and $Y$ and a metric space $Z$ a function $f:X \times Y \to Z$ is jointly continuous if and only if the function $f$ is strongly continuous with respect to $x$ and $y$.

Solovay reducibility and continuity

The objective of this study is a better understandingof the relationships between reduction and continuity. Solovay reduction is a variation of Turing reduction based on the distance of two real numbers. We characterize Solovay reduction by the existence of a certain real function that is computable (in the sense of computable analysis) and Lipschitz continuous. We ask whether thereexists a reducibility concept that corresponds to H¨older continuity. The answer is affirmative. We introduce quasi Solovay reduction and characterize this new reduction via H¨older continuity. In addition, we separate it from Solovay reduction and Turing reduction and investigate the relationships between complete sets and partial randomness.

Development of the Theory of the Functions of Real Variables in the First Decades of the Twentieth Century

In (Biacino 2018) the evolution of the concept of a real function of a real variable at the beginning of the twentieth century is outlined, reporting the discussions and the polemics, in which some young French mathematicians of those years as Baire, Borel and Lebesgue were involved, about what had to be considered a genuine real function. In this paper a technical survey of the arising function and measure theory is given with a particular regard to the contribution of the Italian mathematicians Vitali, Beppo Levi, Fubini, Severini, Tonelli etc … and also with the purpose of exposing the intermediate steps before the final formulation of Radom-Nicodym-Lebesgue Theorem and the Italian method of calculus of variations.