Mathematics

The main intention of the paper is to investigate an osculating curve under the conformal map. We obtain a sufficient condition for the conformal invariance of an osculating curve. We also find an equivalent system of a geodesic curve under the conformal transformation(motion) and show its invariance under isometry and homothetic motion.

We have already seen simple representations of modular Lie algebras of A l -type and C l -type. We shall further investigate simple representations of B l type, which turn out to be very similar in methodology as those types except for roots. So we may consider some conjectures relating to the representations of classical type modular Lie algebras.

Let p be a large prime, and let k≪logp . A new proof of the existence of any pattern of k consecutive quadratic residues and quadratic nonresidues is introduced in this note. Further, an application to the least quadratic nonresidues n p modulo p shows that n p ≪(logp)(loglogp) .

We analyze Non-linear chaffee infatne equation by groups theoretical method to get its symmetries and conservation laws.

It is known that the Collatz Conjecture (and the study of similar maps, here called "Hydra maps") can be stated in terms of solution sets of functional equations; or, equivalently, the fixed points of linear operators on spaces of analytic functions. Rather than studying potential fixed points of such operators, we examine their effect on the singularities of functions. To that end, we introduce the notion of a "dreamcatcher", an object which encodes the location and "virtual residue" of the singularities of an analytic function of a specified growth rate along the boundary of the function's region of convergence. Dreamcatchers can be given rigorous footing as elements of the Hilbert space L2(Q/Z) of complex-valued functions on Q/Z which are square-integrable with respect to the counting measure on Q/Z. The aforementioned linear operators (called here "permutation operators") are shown to "conserve" the singularities of their fixed points, in the sense that the dreamcatcher of a fixed point is itself the fixed point of the operator induced on L2(Q/Z) (the "dreamcatcher operator") by the permutation operator. Using Pontryagin duality and p-adic analysis, the fixed points of the operators acting on dreamcatchers for simple-pole-like singularities are completely determined for a large class of Hydra maps in the case where the fixed points are finitely supported on Q/Z. This enables qualitative conclusions to be made about those Hydra maps' orbit classes in the non-negative integers.

In this expository article, the real numbers are defined as infinite decimals. After defining an ordering relation and the arithmetic operations, it is shown that the set of real numbers is a complete ordered field. It is further shown that any complete ordered field is isomorphic to the constructed set of real numbers.

The idea that data lies on a non-linear space has brought up the concept of manifold learning as a part of machine learning.

Elementary transformations of equations Aψ=λψ are considered. The invertibility condition (Theorem 1) is established and similar transformations of Riccati equations in the case of second order differential operator A are constructed (Theorem 2). Applications to continuous fractions for Bessel functions and Chebyshev polynomials are established. It is shown particularly that the elementary solutions of Bessel equations are related to a fixed point transformations of Riccati equations.

The existing fractional grey prediction models mainly use discrete fractional-order difference and accumulation, but in the actual modeling, continuous fractional-order calculus has been proved to have many excellent properties, such as hereditary. Now there are grey models established with continuous fractional-order calculus method, and they have achieved good results. However, the models are very complicated in the calculation and are not conducive to the actual application. In order to further simplify and improve the grey prediction models with continuous fractional-order derivative, we propose a simple and effective grey model based on conformable fractional derivatives in this paper, and two practical cases are used to demonstrate the validity of the proposed model.

In this paper, some important properties of the windowed offset linear canonical transform (WOLCT) such as shift, modulation and orthogonality relation are introduced. Based on these properties we derive the convolution and correlation theorems for the WOLCT.