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In this paper, we study fuzzy calculus in two main branches differential and integral. Some rules for finding limit and$gH$ -derivative of$gH$ -difference, constant multiple of two fuzzy-valued functions are obtained and we also present fuzzy chain rule for calculating$gH$ -derivative of a composite function. Two techniques namely, Leibniz's rule and integration by parts are introduced for fuzzy integrals. Furthermore, we prove three essential theorems such as a fuzzy intermediate value theorem, fuzzy mean value theorem for integral and mean value theorem for$gH$ -derivative. We derive a Bolzano's theorem, Rolle's theorem and some properties for$gH$ -differentiable functions. To illustrate and explain these rules and theorems, we have provided several examples in details.

In this paper, we prove that the intermediate value theorem remains true for the conformable fractional derivative and we prove some useful results using the definition of conformable fractional derivative given in R. Khalil, M. Al Horani, A. Yousef, M. Sababhehb [4].

A line splitting a triangle into two polygons (either two triangles or a triangle and a quadrilateral) of equal areas is called an area bisector line or simply an area bisector.

The quantization dimension of a probability measure on a metric compactum \\( X \\) does not exceed the box dimension of the support of the measure. We prove the following intermediate value theorem for the upper quantization dimension: If \\( X \\) is a metric compact space whose upper box dimension is equal to \\( aınfty \\) then for every real \\( b \\) such that \\( 0 b a \\) there exists a probability measure on \\( X \\) whose support is \\( X \\) and whose upper quantization dimension is \\( b \\).

Let f = I - k be a compact vector field of class C 1 on a real Hilbert space H . In the spirit of Bolzano’s Theorem on the existence of zeros in a bounded real interval, as well as the extensions due to Cauchy (in R 2 ) and Kronecker (in R k ), we prove an existence result for the zeros of f in the open unit ball B of H . Similarly to the classical finite dimensional results, the existence of zeros is deduced exclusively from the restriction f | S of f to the boundary S of B . As an extension of this, but not as a consequence, we obtain as well an Intermediate Value Theorem whose statement needs the topological degree. Such a result implies the following easily comprehensible, nontrivial, generalization of the classical Intermediate Value Theorem: If a half-line with extreme q ∉ f ( S ) intersects transversally the function f | S for only one point of  S , then any value of the connected component of H \\ f ( S ) containing q is assumed by f in B . In particular, such a component is bounded.

In this note, we obtain a simpler expression for the constant number given by Costin–Maz’ya on the sharp Hardy–Leray inequality for a class of solenoidal (namely divergence-free) vector fields, with respect to any radial power-weighted measure. The dependence of the constant on the weight exponent will be clear.

In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles such as co-finite choice, discrete choice, interval choice, compact choice and closed choice, which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also explore how existing classifications of the Hahn-Banach Theorem and Weak König's Lemma fit into this picture. Well-known omniscience principles from constructive mathematics such as LPO and LLPO can also naturally be considered as Weihrauch degrees and they play an important role in our classification. Based on this we compare the results of our classification with existing classifications in constructive and reverse mathematics and we claim that in a certain sense our classification is finer and sheds some new light on the computational content of the respective theorems. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. We develop a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example.

On taking the intermediate value theorem (IVT) and its converse as a point of departure, this paper connects the intermediate value property (IVP) to the continuity postulate typically assumed in mathematical economics, and to the solvability axiom typically assumed in mathematical psychology. This connection takes the form of four portmanteau theorems, two for functions and the other two for binary relations, that give a synthetic and novel overview of the subject. In supplementation, the paper also surveys the antecedent literature both on the IVT itself, as well as its applications in economic and decision theory. The work underscores how a humble theorem, when viewed in a broad historical frame, bears the weight of many far-reaching consequences; and testifies to a point of view that the apparently complicated can sometimes be under-girded by a most basic and simple execution.

We present a surprisingly short proof that for any continuous map f : ℝ n → ℝ m , if n > m, then there exists no bound on the diameter of fibers of f. Moreover, we show that when m = 1, the union of small fibers of f is bounded; when m > 1, the union of small fibers need not be bounded. Applications to data analysis are considered.

In this paper, we study some fundamental geometrical properties related to the \\[ S\\]-procedure. Given a pair of quadratic functions (g, f), it asks when “\\[g(x)=0 ~ f(x) 0\\]” can imply “(\\[ ın R\\]) (\\[ xın R^n\\]) \\[f(x) + g(x) 0.\\]” Although the question has been answered by Xia et al. (Math Program 156:513–547, 2016), we propose a neat geometric proof for it (see Theorem 2): the \\[ S\\]-procedure holds when, and only when, the level set \\[\\g=0\\\] cannot separate the sublevel set \\[\\f<0\\.\\] With such a separation property, we proceed to prove that, for two polynomials (g, f) both of degree 2, the image set of g over \\[\\f<0\\, g(\\f<0\\)\\], is always connected (see Theorem 4). It implies that the \\[ S\\]-procedure is a kind of the intermediate value theorem. As a consequence, we know not only the infimum of g over \\[\\f 0\\\], but the extended results when g over \\[\\f 0\\\] is unbounded from below or bounded but unattainable. The robustness and the sensitivity analysis of an optimization problem involving the pair (g, f) automatically follows. All the results in this paper are novel and fundamental in control theory and optimization.