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I investigate combinations of quasi-quanta expressions and how they yield alternatesolutions for expressions inMorphic Topology of Numeric Energy: A Fractal Morphism of Topological Counting Shows Real Differentiation of Numeric Energy. For Praising Jehovah, I do publish these mathematical gesturing forms from the infinity meaning of His word. Thanks mom! This quasi-quanta language package outlines methods for combining by topo- logical functor entanglement, symbolic, numeric-energy components. Methods, guidelines and algebraic rules for combining the quasi-quanta into the energy number equivalencies are also notated herein. The Quasi-Quanta Language Package is intended to show the symbolic pat- terns for configuring the quasi quanta symbology into the numeric energy ex- pressions. This should put to rest any doubt that Energy Numbers are indeed a real, logically configured phenomenon a priori to real or complex numbers, but optionally mappable to the real or complex plane. Pre-numeric energy symbol configurations offer a broad language of pat- tern detection and logical symbol operation delineated with particular solving methods herein. This hopefully provides a new way to looking at the branches of mathematics and their inter-operable analog functions. So, inevitably, we decompose the current perspective on numbers and prove a novel method for ordering and combining symbolic orientations.

We study the differentiability, integral, and calculus of linear fuzzy-number-valued functions. Special emphasis is placed on the linear fuzzy-number-valued function F ˜ ( t ) = u ˜ 1 f 1 ( t ) + u ˜ 2 f 2 ( t ) + ⋯ + u ˜ m f m ( t ) , where u ˜ 1 , u ˜ 2 , … , u ˜ m denote fuzzy n-cell numbers and f 1 , f 2 , … , f m represent real functions of a real variable. The concepts of the limit and continuity of fuzzy n-cell-number-valued functions are defined, which are the basis for studying the calculus of linear fuzzy-number-valued functions. Using the fuzzy generalized difference introduced by Gomes and Barros (2015) [8] , we define a generalized difference for fuzzy n-cell numbers. Then a generalized differentiability of fuzzy n-cell-number-valued functions is proposed, and a sufficient condition for generalized differentiability of linear fuzzy-number-valued functions is given by means of real function theory. Furthermore, a Riemann integral of fuzzy n-cell-number-valued functions is introduced, and some properties of the integral are discussed. Finally, the relationship between generalized differentiability and the Riemann integral of the linear fuzzy-number-valued function F ˜ ( t ) = u ˜ 1 f 1 ( t ) + u ˜ 2 f 2 ( t ) + ⋯ + u ˜ m f m ( t ) is derived.

One of the most fundamental concepts in fuzzy set theory is the extension principle. It gives a generic way of dealing with fuzzy quantities by extending non-fuzzy mathematical concepts. There are a few examples, including the concept of fuzzy distance between fuzzy sets. The extension approach is then methodically applied to real algebra, with considerable development of fuzzy number operations. These operations are computationally appealing and generalized interval analysis. Although the set of real fuzzy numbers with extended addition or multiplication is no longer a group, it retains many structural qualities. The extension concept is demonstrated to be particularly beneficial for defining set-theoretic operations for higher fuzzy sets. We need some definitions related to our properties before we can create the properties of integration of a crisp real-valued function over a fuzzy interval. It is our goal in this article to develop and demonstrate certain characteristics of a real-valued function over a fuzzy interval in order to broaden the scope of the notion of integration of a real-valued function over a fuzzy interval. Some of these characteristics are linked to the operations of extended addition and extended subtraction, while others are not.

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.

In this paper we consider the semilinear damped wave problem of the form { ( α ( t ) u t ) t − β ( t ) Δ u + γ ( t ) u t + δ ( t ) u = β ( t ) f ( u ) , x ∈ Ω , t > τ , u ( x , t ) = 0 , x ∈ ∂ Ω , t ⩾ τ , u ( x , τ ) = u τ ( x ) , u t ( x , τ ) = v τ ( x ) , x ∈ Ω , where Ω is a bounded smooth domain in R N , N ⩾ 3 , τ ∈ R , f is a real valued function of a real variable with some suitable conditions of growth, regularity and dissipativity, and α , β , γ and δ are continuous real valued functions of a real variable with some suitable conditions of growth, regularity and signs. Using rescaling of time we prove existence, regularity, gradient-like structure, upper and lower semicontinuity of the pullback attractors for the evolution processes associated with this boundary initial value problem in a suitable phase space.

The commutation principle of Ramirez et al. (SIAM J Optim 23:687–694, 2013) proved in the setting of Euclidean Jordan algebras says that when the sum of a real valued function h and a spectral function $$\Phi $$ is minimized/maximized over a spectral set E, any local optimizer a at which h is Frechet differentiable operator commutes with the derivative $$h^{\prime }(a)$$ . In this note, we describe some analogs of the above result by assuming the existence of a subgradient in place of the derivative (of h) and obtaining strong operator commutativity relations. We show, for example: if a solves the problem $$\underset{E}{\max }\,(h+\Phi )$$ , then a strongly operator commutes with every element in the subdifferential of h at a; If E and h are convex and a solves the problem $$\underset{E}{\min }\,h$$ , then a strongly operator commutes with the negative of some element in the subdifferential of h at a. These results improve known operator commutativity relations for linear h and for solutions of variational inequality problems. We establish these results via a geometric commutation principle that is valid not only in Euclidean Jordan algebras, but also in a broader setting.

This work establishes the existence of addition theorems and double-angle formulas for $${\mathcal {C}}^k$$ real scalar functions. Moreover, we determine necessary and sufficient conditions for a bivariate function to be an addition formula for a $${\mathcal {C}}^k$$ real function. The double-angle formulas allow us to generate a duplication algorithm, which can be used as an alternative to the classical numerical methods to obtain an approximation for the solution of an ordinary differential equation. We demonstrate that this algorithm converges uniformly in any compact domain contained in the maximal domain of that solution. Finally, we carry out some numerical simulations showing a good performance of the duplication algorithm when compared with standard numerical methods.

In the present work, we prove an existence and uniqueness result of solutions to a quasilinear elliptic problem with nonlinear singular terms in the weighted Sobolev space. The equation that we consider is the following $$\begin{aligned} -\Delta _{p(\cdot )}^{\omega } u+\beta (u)=\frac{f(x)}{u^{\alpha }}, \end{aligned}$$ where $$\alpha \ge 1$$ , $$\beta $$ is a continuous non decreasing surjective real function on $${\mathbb {R}}$$ , f is a nonnegative function belonging to the Lebesgue space $$L^{m}(\Omega )$$ and $$m\ge 1$$ .

Given a real function $f$, the rate function for the large deviations of the diffusion process of drift $\nabla f$ given by the Freidlin-Wentzell theorem coincides with the time integral of the energy dissipation for the gradient flow associated with $f$. This paper is concerned with the stability in the hilbertian framework of this common action functional when $f$ varies. More precisely, we show that if $(f_h)_h$ is uniformly $\lambda$-convex for some $\lambda \in \mathbb{R}$ and converges towards $f$ in the sense of Mosco convergence, then the related functionals $\Gamma$-converge in the strong topology of curves.

We explore the relationship between analytic behavior of a computable real valued function and the computability-theoretic complexity of the individual values of its derivative (the function’s slopes) almost-everywhere. Given a computable function f, the values of its derivative $$f'(x)$$ , where they are defined, are uniformly computable from $$x'$$ , the Turing jump of the input. It is known that when f is $${\mathcal {C}}^2$$ , the values of $$f'(x)$$ are actually computable from x. We construct a $${\mathcal {C}}^1$$ function f so that, almost everywhere, $$f'(x)\ge _T x'$$ . Although the values $$f'(x)$$ at each point x cannot uniformly compute the corresponding jumps $$x'$$ of the inputs x almost everywhere for any $${\mathcal {C}}^1$$ function f, we produce an example of a $${\mathcal {C}}^1$$ function f such that $$f(x)\ge _T \emptyset '$$ uniformly on subsets of arbitrarily large measure, effectively (using the notion of a Schnorr test). We also explore analogous questions for weaker smoothness conditions, such as for f differentiable everywhere, and f differentiable almost everywhere.

As the field of study expands, or as the number of observations in a sample increases, data observed at discrete points is generally assumed to be sampled from an underlying real function. As the number of observation points increases, those observations are likely to be sampled from a real-valued function. In this case, the derived data will be examples of a functional structure. We analyzed the daily average temperature data at 65 discrete points in 18 cities in Turkey's Black Sea Region. Fourier basis functions were used as a basis function approach because the temperature data had a periodic structure. The data were then transformed into a continuous function using the basis function and roughness penalty approach. Functional data were obtained using the roughness penalty approach. The generalized cross-validation method was used to determine the smoothing parameter of the variable (temperature variable). Finally, smoothed functional principal components analysis was applied to the functional data. In this way, changes in temperature functions, which seem hard to tackle, were evaluated on the same graph using the mean function generated for the principal component function and using the functions and the mean function obtained using the principal component function.

A famous result due to Ko and Friedman (1982) asserts that the problems of integration and maximisation of a univariate real function are computationally hard in a well-defined sense. Yet, both functionals are routinely computed at great speed in practice. We aim to resolve this apparent paradox by studying classes of functions which can be feasibly integrated and maximised, together with representations for these classes of functions which encode the information which is necessary to uniformly compute integral and maximum in polynomial time. The theoretical framework for this is the second-order complexity theory for operators in analysis which was introduced by Kawamura and Cook (2012). The representations we study are based on rigorous approximation by polynomials, piecewise polynomials, and rational functions. We compare these representations with respect to polytime reducibility as well as with respect to their ability to quickly evaluate symbolic expressions in a given language. We show that the representation based on rigorous approximation by piecewise polynomials is polytime equivalent to the representation based on rigorous approximation by rational functions. With this representation, all terms in a certain language, which is expressive enough to contain the maximum and integral of most functions of practical interest, can be evaluated in polynomial time. By contrast, both the representation based on polynomial approximation and the standard representation based on function evaluation, which implicitly underlies the Ko-Friedman result, require exponential time to evaluate certain terms in this language. We confirm our theoretical results by an implementation in Haskell, which provides some evidence that second-order polynomial time computability is similarly closely tied with practical feasibility as its first-order counterpart., 33 pages, 4 figures

This work proposes a new calculation algorithm of the Wentzel–Kramers–Brillouin (WKB) solution for slow linear time-varying (LTV) systems based on the recursive formulation. For a linear dynamic system with time-varying (TV) mass, damping or stiffness, the recursive relations in real function form among the components of the WKB solution are obtained. The integral terms within the sampling intervals that cannot be solved analytically are approximated by simple algebraic calculations with high precision. Thus, the conventional expression of the WKB solution involving complex numerical integral terms is reduced to an analytical recursive formulation. An explicit function relationship between the TV system parameters and the dynamic response can be obtained when the external excitation is given. In the premise of similar accuracy, the recursive formulation has much higher efficiency than the conventional calculation process, which is verified by applying the proposed method to a particular LTV dynamic system.

It is proved: if $\phi(\tau,\xi)$ is a scalar continuous real function of arguments $\tau\in [a_{(n-1)},\ b_{(n-1)}]\subset R^{n-1},$ $\xi \in [a,\ b]\subset R^{1}$ and $\phi(\tau,a)\phi(\tau,b)0$ there exists a continuous function $\phi_{0}(\tau,\xi)$ such that $|\phi(\tau,\xi)-\phi_{0}(\tau,\xi)

In this paper, we introduced a Henstock-type integral named N-integral of a real valued function f on a closed and bounded interval [a,b]. The set N-integrable functions lie entirely between Riemann integrable functions and Henstock-Kurzweil integrable functions. Furthermore, this new integral integrates all improper Riemann integrable functions even if they are not Lebesgue integrable. It was shown that for a Henstock-Kurzweil integrable function f on [a,b], the following are equivalent:The function f is N-integrable;There exists a null set S for which given epsilon >0 there exists a gauge delta such that for any delta-fine partial division D={(xi,[u,v])} of [a,b] we have [(phi_S(D) Gamma_epsilon) sum |f(v)-f(u)||v-u

The paper deals with the following Kirchhoff equations: − a + b ∫ R N | ∇ u | 2 d x △ u + V ( x ) u = K ( x ) f ( u ) i n R N , u ∈ H 1 ( R N ) , where N ≥ 3 , f ( u ) is C 1 real function sati...

In this paper, we study the probability distribution of eigenvalues of the n×n Euclidean random matrix whose (i,j)th entry is of the form f(∥xi−xj∥2)−unδij∑kf(∥xi−xk∥2) for some real function f. Ra...

Graph of real valued continuous function with special addition and multiplication has already proven that is isomorphic to real number system. Furthermore, the graph of continous real valued function forms a field. The aim of this research was to generalize such concept to its n-tuple Cartesian Product and to prove that interchange of basis still able to be executed. The result of this research is n-tuple Cartesian Product of graph function forms a vector space over and interchange of basis still able to be executed

An Archimedean copula is characterised by its generator. This is a real function whose inverse behaves as a survival function. We propose a semiparametric generator based on a quadratic spline. This is achieved by modelling the first derivative of a hazard rate function, in a survival analysis context, as a piecewise constant function. Convexity of our semiparametric generator is obtained by imposing some simple constraints. The induced semiparametric Archimedean copula produces Kendall’s tau association measure that covers the whole range ( − 1 , 1 ) . Inference on the model is done under a Bayesian approach and for some prior specifications we are able to perform an independence test. Properties of the model are illustrated with a simulation study as well as with a real dataset.

Basing on the notion of algebraic function in real function theory, we study the question of the syntactic definition of the functions on universal algebras whose graphs are algebraic or piecewise-algebraic sets.

A vertex (respectively, edge) cycle stochastic function of a graph G is a labeling of vertices (respectively, edges) by a non-negative real valued function (respectively, ) such that for every cycle of G, the sum of labels of its vertices (respectively, edges) is 1. The graphs where we can define such a function are called vertex cycle stochastic graphs (respectively, edge cycle stochastic graphs). In this paper, we provide a structure theorem for biconnected cycle stochastic graphs, which is extended to characterize edge cycle stochastic graphs. We also find a minimal forbidden graph characterization for biconnected vertex cycle stochastic graphs and its description for vertex cycle stochastic graphs.

Let $P$ be a second-order, symmetric, and nonnegative elliptic operator with real coefficients defined on noncompact Riemannian manifold $M$, and let $V$ be a real valued function which belongs to the class of {\em small perturbation potentials} with respect to the heat kernel of $P$ in $M$. We prove that under some further assumptions (satisfying by a large classes of $P$ and $M$) the positive minimal heat kernels of $P-V$ and of $P$ on $M$ are equivalent. Moreover, the parabolic Martin boundary is stable under such perturbations, and the cones of all nonnegative solutions of the corresponding parabolic equations are affine homeomorphic, Comment: Proposition 6.6 and example 7.14 are added. The proof of Theorem 6.2 is modified

We give an extension of Hua’s inequality for C*-valued norm in pre-Hilbert C*-modules setting. As its applications, some known and new generalizations of Hua’s inequality are deduced. In particular, we establish the connection between C*-valued norm triangle inequality and C*-valued norm Hua’s inequality. We also present some operator versions of Hua’s inequality on Hilbert spaces. As a consequence, we obtain various forms of Hua’s inequality. Moreover, we indicate that for our operator, Hua’s inequality is equivalent to operator convexity of a given continuous real function.

Grandi’s paradox, which was posed for a real function of the form 1/(1+ x), has been resolved and extended to complex valued functions. Resolution of this approximately three-hundred-year-old paradox is accomplished by the use of a consistent truncation approach that can be applied to all the series expansions of Grandi-type functions. Furthermore, a new technique for improving the convergence characteristics of power series with alternating signs is introduced. The technique works by successively averaging a series at different orders of truncation. A sound theoretical justification of the successive averaging method is demonstrated by two different series expansions of the function 1/(1+ ex ) . Grandi-type complex valued functions such as 1/(i + x) are expressed as consistently-truncated and convergence-improved forms and Fagnano’s formula is established from the series expansions of these functions. A Grandi-type general complex valued function is introduced and expanded to a consistently truncated and successively averaged series. Finally, an unorthodox application of the successive averaging method to polynomials is presented.

We are interested in the asymptotic behavior, as t tends to $$+\infty $$ , of finite energy solutions and entropy solutions $$u_{n}$$ of nonlinear parabolic problems whose model is 0.1 $$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}-\Delta _{p}u+g(u)|\nabla u|^{p}=\mu &{}\text { in }(0,T)\times \Omega ,\\ u(0,x)=u_{0}(x)&{}\text { in }\Omega ,\\ u(t,x)=0&{}\text { on }(0,T)\times \partial \Omega \end{array}\right. } \end{aligned}$$ where $$\Omega \subseteq \mathbb {R}^{N}$$ is a bounded open set, $$N\ge 3$$ , $$u_{0}\in L^{1}(\Omega )$$ is a nonnegative initial data, while $$g:\mathbb {R}\mapsto \mathbb {R}$$ is a real function in $$C^{1}(\mathbb {R})$$ which satisfies sign condition with positive derivative and $$\mu $$ is a nonnegative measure independent on time which does not charge sets of null p-capacity.

A soft set is a generalization of a fuzzy set and is an efficient tool for modelling imprecise and vague data. The mathematical structures partially ordered sets and lattices play an important role in mathematics as well as in other disciplines such as computer science, engineering, cryptography, etc. Here we apply soft set theory to lattice theory and introduce the concept of a soft partial ordering and more concepts related to it. Mainly we introduce the concept of a generalized soft lattice(gs lattice) and investigate some of its fundamental properties. Further, we define a soft real valued function called soft valuation on a gs lattice and study its major properties. Moreover, we discuss the notion of a soft distance function in terms of soft valuation and using that function we discuss the conditions under which a gs lattice becomes a soft metric lattice. We conclude this study by defining a soft topological lattice and describe that when a gs lattice with a soft valuation becomes a soft topological lattice.

The subject of this paper is C 1 sign, monotonicity and convexity preserving spline interpolation to a set of ordered points from a real function of one real variable. Two solutions are proposed constructing, respectively, a Hermite parametric polynomial Cubic Spline (CS), and a Hermite Cubic Rational polynomial Spline (CRS). Both curves are based on the shape preserving Hermite Variable Degree Spline (VDS) Gabrielides and Sapidis (2018) (first introduced in Kaklis and Pandelis (1990)) and they use the Bezier representation of polynomials. Since the CS curve is parametric, the present problem also requires calculation of the y -component of CS for any specific x -value; a robust solution to this problem is discussed in detail. The CRS is non-parametric and it does solve the given interpolation-problem with its weights (which play the role of tension parameters) being directly computed using the properties of the VDS segments.

The concepts of Choquet sets and Silov boundary for a real function algebra and real function space has been introduced and studied mainly by Kulkarni and Limaye. Here we study these concepts for the Cartesian product of real function algebras and function spaces.

Using analytic continuation, we derive a new analytical solution for a screw dislocation in an N-phase laminate composed of N−2 intermediate elastic layers of equal thickness encased in two semi-infinite elastic media. The number of phases comprising the laminate can be arbitrarily large. The solution can be interpreted in terms of image screw dislocations, the magnitudes of which are determined by solving an algebraic equation of order N−2. We also obtain an elementary expression for the image force acting on the screw dislocation in terms of a single real function. Several specific examples are presented to demonstrate the solution. Analytic continuation is also applied to solve the related problem of a screw dislocation interacting with a multicoated circular elastic inhomogeneity.

This study devotes to the uncertainty principle under the free metaplectic transformation (an abbreviation of the metaplectic operator with a free symplectic matrix) of a real function. Covariance matrices in time, frequency and time–frequency domains are defined, and a relationship between these matrices and the free metaplectic transformation domain covariance is proposed. We then obtain two versions of lower bounds on the uncertainty product of the covariances of a real function in two free metaplectic transformation domains. It is shown here that a multivariable square integrable real-valued function cannot be both two free metaplectic transformations band limited. It is also seen that these two lower bounds depend not only on the minimum singular value of the blocks $${\mathbf {A}}_j,{\mathbf {B}}_j$$, $$j=1,2$$ found in free symplectic matrices but also on the covariance in time domain or in frequency domain. We thus reduce them to a new one which does not contain the covariances in time and frequency domains. Sufficient conditions that reach the lower bounds are derived. Example and simulation results are provided to validate the theoretical analysis.

In this note, we study the n × n random kernel matrix whose (i,j)th entry is of form f(xi′xj)−unδij∑kf(xi′xk) for some real function f and the xi’s are i.i.d. random vectors generated from lp ellip...

In this paper, we prove the existence and uniqueness of entropy solutions for the following equations in Orlicz spaces: $$\begin{aligned} \left\{ \begin{array}{c} \frac{\partial u}{\partial t}-div\Big (a \left( x,\nabla u(x,t)\right) \Big )+ \beta (u)=f\text { in }Q_{T}= ] 0.T[ \times \Omega \\ u=0\text { on }\Sigma _{T}=] 0.T[ \times \partial \Omega \\ u(0,.)=0 \text { in }\Omega , \end{array} \right. \qquad (1) \end{aligned}$$ where $$\ f$$ is an element of $$L^{1}( Q_{T} )$$ , the term $$-\text{ div }\Big (a(x,\nabla u(x,t))\Big )$$ is a Leray-Lions operator on $$W_0^{1,x}L_M(\Omega )$$ , with M(.) does not satisfy the $$\Delta _2$$ condition and $$\beta$$ is a continuous non decreasing real function defined on $$\mathbb {R}$$ with $$\beta (0)=0$$ . The investigation is made by approximation of the Rothe method which is based on a semi-discretization of the given problem with respect to the time variable.

It is very interesting and important to determine the function model for remote-sensing data. The existing statistical and artificial intelligence models still have some defects. The statistical models rely heavily on prior knowledge and cannot objectively reflect the real function model contained in the remote-sensing data. In addition, the existing artificial intelligence models can very easily fall into the local optimum and have a low efficiency for high-dimensional remote-sensing data. In this paper, we first decrease the complexity of remote-sensing data by using rough sets and propose an attribute reduction algorithm based on rough sets for remote-sensing data (ARRS-RSD). On the basis of the algorithm, this paper presents a function mining algorithm for remote-sensing data by using gene expression programming and rough sets (FMRS-ARGEP). In FMRS-ARGEP, a dynamic population generation policy and a new mutation operation based on self-adaptive rate adjustment are introduced to improve the convergence of the algorithm. The experimental results show that the proposed algorithm outperforms traditional algorithms in terms of the average running time, the number of condition attributes after reduction, the attribute reduction ratio, the average convergence speed, the number of convergences, and the ${R^{2}}$ value of the model.

We consider the space $A(\mathbb{T}^d)$ of absolutely convergent Fourier series on the torus $\mathbb{T}^d$. The norm on $A(\mathbb{T}^d)$ is naturally defined by $\|f\|_{A}=\|\widehat{f}\|_{l^1}$, where $\widehat{f}$ is the Fourier transform of a function $f$. For real functions $\varphi$ of a certain special form on $\mathbb T^d, \,d\geq 2,$ we obtain lower bounds for the norms $\|e^{i\lambda\varphi}\|_A$ as $\lambda\rightarrow\infty$. In particular, we show that if $\varphi(x, y)=a(x)|y|$ for $|y|\leq\pi$, where $a\in A(\mathbb{T})$ is an arbitrary nonconstant real function, then $\|e^{i\lambda\varphi}\|_{A(\mathbb{T}^2)}\gtrsim |\lambda|$., Comment: The introduction and Remarks modified to improve clarity

This research paper concentrates on the Kakeya problem. After the introduction of historical issue, we provide a thorough presentation of the results of Kakeya problem with some examples of the early solutions as well as the proof of the final outcome of this problem, the solution of which is known as Besicovitch Set. We give 3 different construction of Besicovitch set as well as the intuition of construction, which is related to iterated integral of 2-variable real function. We also give the Cunningham construction in which the area of a simply connected Kakeya set can also tend to 0. Furthermore, we generalize the process of generating a Kakeya set into a Kakeya dynamic. The definition of multiplicity enables us to estimate the area of a Kakeya set. In following discussion we provided a conjecture related to the solution in particular range. Finally, the derivation of the Kakeya problem is presented.

For any two distinct vertices $x$ and $y$ of a graph $G$, let $S\{x, y\}$ denote the set of vertices $z$ such that either $x$ lies on a $y-z$ geodesic or $y$ lies on an $x-z$ geodesic. Let $g: V(G) \rightarrow [0,1]$ be a real valued function and, for any $U \subseteq V(G)$, let $g(U)=\sum_{v \in U}g(v)$. The function $g$ is a strong resolving function of $G$ if $g(S\{x, y\}) \ge 1$ for every pair of distinct vertices $x, y$ of $G$. The fractional strong metric dimension, $sdim_f(G)$, of a graph $G$ is $\min\{g(V(G)): g \mbox{ is a strong resolving function of }G\}$. In this paper, after obtaining some new results for all connected graphs, we focus on the study of the fractional strong metric dimension of the corona product, the lexicographic product, and the Cartesian product of graphs., 18 pages, 7 figures

It is a familiar fact that inequalities have become a very popular method using fractional integrals, and that this method has been the driving force behind many studies in recent years. Many forms of inequality have been studied, resulting in the introduction of new trend in inequality theory. The aim of this paper is to use a fuzzy order relation to introduce various types of inequalities. On the fuzzy interval space, this fuzzy order relation is defined level by level. With the help of this relation, firstly, we derive some discrete Jensen and Schur inequalities for convex fuzzy interval-valued functions (convex fuzzy-IVF), and then, we present Hermite–Hadamard inequalities (HH-inequalities) for convex fuzzy-IVF via fuzzy interval Riemann–Liouville fractional integrals. These outcomes are a generalization of a number of previously known results, and many new outcomes can be deduced as a result of appropriate parameter “γ” and real valued function “Ω” selections. We hope that our fuzzy order relations results can be used to evaluate a number of mathematical problems related to real-world applications.

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.

We consider the dressed energy $\varepsilon$ of the XXZ chain in the massless antiferromagnetic parameter regime at $0 < \Delta < 1$ and at finite magnetic field. This function is defined as a solution of a Fredholm integral equation of the second kind. Conceived as a real function over the real numbers it describes the energy of particle-hole excitations over the ground state at fixed magnetic field. The extension of the dressed energy to the complex plane determines the solutions to the Bethe Ansatz equations for the eigenvalue problem of the quantum transfer matrix of the model in the low-temperature limit. At low temperatures the Bethe roots that parametrize the dominant eigenvalue of the quantum transfer matrix come close to the curve ${\rm Re}\, \varepsilon (\lambda) = 0$. We describe this curve and give lower bounds to the function ${\rm Re}\, \varepsilon$ in regions of the complex plane, where it is positive., Comment: 21 pages, 2 figures

In [1] two-sided bounds were derived for the probability of ruin of an insurance company, whose premium income is represented by an arbitrary increasing real function, the claims are dependent, integer valued r.v.s. and their inter-occurrence times are exponentially, non-identically distributed. Here we give an explicit algorithm to compute that probability.

In this paper, we introduce a new class of contact pseudo framed (CPF)-manifoldsM,g,f,λ,ξby a real tensor fieldfof type1,1, a real functionλsuch thatf3=λ2fwhereξis its characteristic vector field. We prove in our main Theorem 2 thatMadmits a closed 2-formΩifλis constant. In 1976, Blair proved that the vector fieldξof a normal contact manifold is Killing. Contrary to this, we have shown in Theorem 2 that, in general,ξof 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.

Let (M, 〈 , 〉) be a complete Riemannian manifold of dimension m ≥ 2, and let Δφ be a quasilinear operator depending on a real function φ, to be detailed in the next chapter. For instance, the family Δφ includes the Laplace–Beltrami operator Δ, defined with the sign agreement that $$\displaystyle \Delta u = \mathrm {div}(\nabla u) = \sum _{j=1}^m \frac {\partial ^2u}{\partial x_j^2} \qquad \text{on } \, \mathbb R^m. $$

This paper focuses on the use of Gaussian autocorrelation functions (ACF) in civil engineering applications involving random processes and random fields. It aims at debunking misgivings, verifying facts and figures, and formulating practical conclusions. A large majority of civil engineers active in random field modelling and reliability analysis is quite content to point out that the routine use of Gaussian autocorrelation functions is part of standard practice and perfectly harmless. A common approach in 2D random field problems, for instance, is to estimate an appropriate correlation length on some physical or empirical basis, and then plug it into a multivariate ACF that is both isotropic, and separable into a product of univariate ACFs: if both of these objectives are to be met, the Gaussian ACF naturally stands out as it is in fact the only real function to possess both of these properties. But as early as the nineteen-sixties, a substantive piece of electrical engineering literature pointed to “issues” and “red flags”. The claim was that the Gaussian ACF produces unrealistic results, violates certain principles concerning both the modelling and the estimation of random properties, and runs into results that possibly defy common sense. Similarly, geostatisticians have been issuing warnings of hyper-predictability, super-smoothness, wildly underestimated estimation errors, and artificial results in applications such as spatial kriging using Gaussian ACFs, leading to the recommendation that the Gaussian model should never be used in practice. This paper revisits the use of the Gaussian ACF and presents a sober but principled look at the entire issue. Importantly, it also considers the pros and cons of replacement ACF models and adjusted ACF models. The paper includes examples and measurable outcomes with the aim of providing a fair assessment and justifiable recommendations.

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.

For a system of first order ordinary differential equations, the Poincare method reduces the search of its periodic solutions to the search of their initial conditions, a finite-dimensional problem. This approach is applied to bounded perturbations of linear problems, and when the Cauchy problem is not uniquely solvable, a variant due to Stampacchia is introduced. A theorem of Krasnosel’skii-Perov estimates the Brouwer degree of the Poincare operator in terms of the degree of the right-hand member of the differential system. It is applied to compute the Brouwer degree of some gradient mappings under various conditions. For the study of periodic solutions of first order differential systems, the method of guiding functions, introduced by Krasnosel’skii and its school, is the analogous of the Lyapunov method in stability. It reduces the existence of a periodic solution to the search of a suitable real function, whose gradient has a behavior akin to the one of the right-hand member of the differential system. Various situations are considered, depending upon the continuability or the non-continuability over the period of the solutions of the Cauchy problem. The method of guiding functions is also extended to a more general class of equations important in mechanics, the evolution complementarity systems, whose periodic solutions are searched in a suitable closed convex cone.

In this paper, we introduce a general procedure to construct the Taylor series development of the inverse of an analytical function, in other words, given y=f(x), we provide the power series that defines its inverse x=hf(y). We apply the obtained results to solve nonlinear equations in an analytic way, and generalize Catalan and Fuss&ndash, Catalan numbers.