[8] **viXra:1906.0569 [pdf]**
*submitted on 2019-06-30 18:51:51*

**Authors:** Saburou Saitoh

**Comments:** 14 Pages. The division by zero calculus requests the essential arrangements for equations and inequalities in analytic functions.

In this paper, we will examine the division by zero calculus from the viewpoints of equations and inequalities as a starting new idea.

**Category:** Functions and Analysis

[7] **viXra:1906.0509 [pdf]**
*submitted on 2019-06-27 03:36:56*

**Authors:** Galyna Kriukova

**Comments:** 6 Pages. Mohyla Mathematical Journal, Vol 1 (2018) http://mmj.ukma.edu.ua/article/view/152601

Hidden Markov models are a well-known probabilistic graphical model for time series of discrete, partially observable stochastic processes. We consider the method to extend the application of hidden Markov models to non-Gaussian continuous distributions by embedding a priori probability distribution of the state space into reproducing kernel Hilbert space. Corresponding regularization techniques are proposed to reduce the tendency to overfitting and computational complexity of the algorithm, i.e. Nystr¨om subsampling and the general regularization family for inversion of feature and kernel matrices. This method may be applied to various statistical inference and learning problems, including classification, prediction, identification, segmentation, and as an online algorithm it may be used for dynamic data mining and data stream mining. We investigate, both theoretically and empirically, the regularization and approximation bounds of the discrete regularization method. Furthermore, we discuss applications of the method to real-world problems, comparing the approach to several state-of-the-art algorithms.

**Category:** Functions and Analysis

[6] **viXra:1906.0329 [pdf]**
*replaced on 2019-06-21 07:53:29*

**Authors:** Pith Peishu Xie

**Comments:** 2 Pages.

The Operator axioms have deduced number systems. In this paper, we conjecture some inequalities in Operator axioms. The general inequalities show the value of Operator axioms.

**Category:** Functions and Analysis

[5] **viXra:1906.0237 [pdf]**
*submitted on 2019-06-13 14:10:23*

**Authors:** Jonathan W. Tooker

**Comments:** 66 Pages. This paper is undergoing some syntactical changes/improvements. The paper is fine as is, but there are some issues which are currently being improved. In the meantime, readers are directed to the finalized verisons of viXra:1811.0222 and viXra:1809.0234

Recent analysis has uncovered a broad swath of previously unconsidered real numbers called real numbers in the neighborhood of infinity. Here we extend the catalog of the rudimentary analytical properties of all real numbers by defining a set of fractional distance functions on the real number line and studying their behavior. The main results of this paper include (1) to prove that some real numbers are greater than any natural number, (2) to develop a technique for taking a limit at infinity via the ordinary Cauchy definition reliant on the classical epsilon-delta formalism, and (3) to demonstrate an infinite number of non-trivial zeros of the Riemann zeta function in the neighborhood of infinity. The methods used in this analysis include nothing other than basic arithmetic, a little trigonometry, and Euclidean geometry. In addition to the zeros used to disprove the Riemann hypothesis in earlier work, here we present yet more zeros which independently constitute the negation of the Riemann hypothesis.

**Category:** Functions and Analysis

[4] **viXra:1906.0236 [pdf]**
*replaced on 2019-07-06 10:52:26*

**Authors:** Jonathan W. Tooker

**Comments:** 5 Pages.

In this brief note, we propose a set of operations for the affinely extended real number called infinity. Under the terms of the proposition, we show that the Riemann zeta function has infinitely many non-trivial zeros on the complex plane.

**Category:** Functions and Analysis

[3] **viXra:1906.0185 [pdf]**
*submitted on 2019-06-11 20:12:46*

**Authors:** Saburou Saitoh

**Comments:** 11 Pages. We propose new problems in several complex analysis from the viewpoint of division by zero calculus.

In this paper, we will introduce the division by zero calculus in multiply dimensions in order to show some wide and new open problems as we see from one dimensional case.

**Category:** Functions and Analysis

[2] **viXra:1906.0163 [pdf]**
*replaced on 2019-06-20 06:50:03*

**Authors:** Andrej Liptaj

**Comments:** 8 Pages.

Derivative of a function can be expressed in terms of integration over a small neighborhood of the point of differentiation, so-called differentiation by integration method. In this text a maximal generalization of existing results which use one-dimensional integrals is presented together with some interesting non-analytic weight functions.

**Category:** Functions and Analysis

[1] **viXra:1906.0148 [pdf]**
*submitted on 2019-06-09 19:44:28*

**Authors:** Saburou Saitoh

**Comments:** 4 Pages. I gave a new type isoperimetric inequality and propose several fundamental open problems.

In this paper, as a direct application of Q. Guan's result on the conjugate analytic Hardy $H_2$ norm we will derive a new type isoperimetric inequality for Dirichlet integrals of analytic functions.

**Category:** Functions and Analysis