Generalized Differentiability of Continuous Functions

Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated.

The Concentration Behavior of Ground States for a Class of Kirchhoff-type Problems with Hartree-type Nonlinearity

In this paper, we consider the Kirchhoff equation with Hartree-type nonlinearity\left\{\begin{aligned} \displaystyle-&\displaystyle\biggl{(}\varepsilon^{2}a+% \varepsilon b\int_{\mathbb{R}^{3}}\lvert\nabla u\rvert^{2}\mathop{}\!dx\biggr{% )}\Delta u+V(x)u=\varepsilon^{\mu-3}\biggl{(}\int_{\mathbb{R}^{3}}\frac{K(y)F(% u(y))}{\lvert x-y\rvert^{\mu}}\mathop{}\!dy\biggr{)}K(x)f(u),\\ &\displaystyle u\in H^{1}(\mathbb{R}^{3}),\end{aligned}\right.where {\varepsilon>0} is a small parameter, {a,b>0}, {\mu\in(0,3)}, {V,K} are two positive continuous function and F is the primitive function of f which is superlinear but subcritical at infinity in the sense of the Hardy–Littlewood–Sobolev inequality. We show that the equation admits a positive ground state solution for {\varepsilon>0} sufficiently small. Furthermore, we prove that these ground state solutions concentrate around such points which are both the minima points of the potential V and the maximum points of the potential K as {\varepsilon\to 0}.

The Discrete Analog of the Newton-Leibniz Formula in the Problem of Summation over Simplex Lattice Points

Definition of the discrete primitive function is introduced in the problem of summation over simplex lattice points. The discrete analog of the Newton-Leibniz formula is found

Dual Neural Network Method for Solving Multiple Definite Integrals

This study, which examines a calculation method on the basis of a dual neural network for solving multiple definite integrals, addresses the problems of inefficiency, inaccuracy, and difficulty in finding solutions. First, the method offers a dual neural network method to construct a primitive function of the integral problem; it can approximate the primitive function of any given integrand with any precision. On this basis, a neural network calculation method that can solve multiple definite integrals whose upper and lower bounds are arbitrarily given is obtained with repeated applications of the dual neural network to construction of the primitive function. Example simulations indicate that compared with traditional methods, the proposed algorithm is more efficient and precise in obtaining solutions for multiple integrals with unknown integrand, except for the finite input-output data points. The advantages of the proposed method include the following: (1) integral multiplicity shows no influence and restriction on the employment of the method; (2) only a finite set of known sample points is required without the need to know the exact analytical expression of the integrand; and (3) high calculation accuracy is obtained for multiple definite integrals whose integrand is expressed by sample data points.

JUDICIAL ACTIVISM – AN ASSET TO JUDICIARY BY INDIAN CONSTITUTION

Constitution means the structure of a body, organism or organization i.e. what constitutes it or of what it consists of. Constitution of a country spells out the basic fundamental principles or established precedents on which the state is organized. It lays down the structure of the political system under which its people are to be governed. It establishes the main organs of the State-the legislature, the executive and the judiciary, demarcates their responsibilities and regulates their relationships with each other and with the people. All authority in the hands of any organs, institutions or functionaries of the state flow from the Constitution. In a country like ours, adopting a written Constitution which mandates Judicial Review of the constitutionality of State activity in cases needing it and the laws enacted by legislature, the role of Judiciary cannot be restricted to the primitive function of dispensing justice. The role of judiciary in enforcing judicial review, must for all purposes keep the Government in good tune with the changing times and it should not be allowed to drift to become anachronistic or out of reasoning with the need of the day.

Semiclassical ground state solutions for a Choquard type equation in ℝ2 with critical exponential growth

In this paper we study a nonlocal singularly perturbed Choquard type equation $$-\varepsilon^2\Delta u +V(x)u =\vr^{\mu-2}\left[\frac{1}{|x|^{\mu}}\ast \big(P(x)G(u)\big)\right]P(x)g(u)$$ in ℝ2, where ε is a positive parameter, \hbox{$\frac{1}{|x|^\mu}$} with 0 < μ < 2 is the Riesz potential, ∗ is the convolution operator, V(x), P(x) are two continuous real functions and G(s) is the primitive function of g(s). Suppose that the nonlinearity g is of critical exponential growth in ℝ2 in the sense of the Trudinger-Moser inequality, we establish some existence and concentration results of the semiclassical solutions of the Choquard type equation in the whole plane. As a particular case, the concentration appears at the maximum point set of P(x) if V(x) is a constant.