| viXra.org > Functions and Analysis > 2011 |
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| viXra.org > Functions and Analysis > 2011 |
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[12] viXra:2011.0202 [pdf] submitted on 2020-11-30 08:56:18
Authors: Zafar Turakulov
Comments: 10 Pages. A new method for solving linear ordinary differential equations is discovered. This material exposes examples of general solutions obtained this way.
We have worked out a new geometric approach to linear ordinary differential equations of second order which makes it possible to obtain general solutions to infinite number of equations of this sort. No need new families of special functions and their theories arose, solutions are composed straightforwardly. In this work we present a number of particular cases of equations with their general solutions. The solutions are divided into four groups the same way one encounters in any book on special functions.
Category: Functions and Analysis
[11] viXra:2011.0182 [pdf] submitted on 2020-11-26 11:12:58
Authors: Yu-Lin Chou
Comments: 3 Pages. expository article
It is a classical but relatively less well-known result that,
for every given measure space and every given $1 \leq p \leq +\infty$,
every sequence in $L^{p}$ that converges in $L^{p}$ has a subsequence converging almost everywhere.
The typical proof is a byproduct of proving the completeness of $L^{p}$ spaces, and hence is not necessarily ``application-friendly''.
We give a simple, perhaps more ``accessible'' proof of this result for all finite measure spaces.
Category: Functions and Analysis
[10] viXra:2011.0169 [pdf] submitted on 2020-11-24 08:40:11
Authors: Yu-Lin Chou
Comments: 2 Pages. expository article
Establishing on a finite measure space the implication for almost everywhere convergence to imply weak convergence of the corresponding induced measures (in particular to imply convergence in distribution) is usually indirect, convergence in measure being the transition. We give a simple, pedagogically informative proof for the implication.
Category: Functions and Analysis
[9] viXra:2011.0163 [pdf] replaced on 2021-02-19 00:35:24
Authors: Pranjal Jain
Comments: 18 Pages.
This paper proves inequalities among generalised f-means and provides formal conditions which a function of several inputs must satisfy in order to be a `meaningful' mean. The inequalities we prove are generalisations of classical inequalities including the Jensen inequality and the inequality among the Quadratic and Pythagorean means. We also show that it is possible to have meaningful means which do not fall into the general category of f-means.
Category: Functions and Analysis
[8] viXra:2011.0131 [pdf] replaced on 2020-11-25 01:38:50
Authors: Yu-Lin Chou
Comments: 6 Pages. Two minor changes: an example added; a sentence modified.
We prove the precompactness of a collection of Borel probability measures over an arbitrary metric space
precisely under a new legitimate notion,
which we term
\textit{topological stationarity},
regulating the sequential behavior
of Borel probability measures directly in terms of the open sets.
Thus the important direct part of Prokhorov's theorem,
which permeates the weak convergence theory,
admits a new version with the original and sole assumption --- tightness --- replaced by topological stationarity.
Since,
as will be justified,
our new condition is not vacuous and is logically independent of tightness,
our result deepens the understanding of the connection between precompactness of Borel probability measures and metric topologies.
Category: Functions and Analysis
[7] viXra:2011.0090 [pdf] submitted on 2020-11-12 10:26:50
Authors: Yu-Lin Chou
Comments: 6 Pages. expository article
Although the concept of Brownian motion or Wiener process is quite popular,
proving its existence via construction is a relatively deep work and would not be stressed outside mathematics.
Taking the existence of Brownian motion in $C([0,1], \R)$ ``for granted'' and following an existing implicit thread,
we intend to present an explicit, simple treatment of the existence of Brownian motion in the space $C([0, +\infty[, \R)$ of all continuous real-valued functions on the ray
$[0, +\infty[$ with moderate technical intensity.
In between the developments, some informative little results are proved.
Category: Functions and Analysis
[6] viXra:2011.0074 [pdf] submitted on 2020-11-10 10:06:17
Authors: Claude Michael Cassano
Comments: 9 Pages. [Corrections made by viXra Admin to conform with scholarly norm]
The Bessel linear homogeneous ordinary differential equation (LHODE) with real parameter may be solved exactly using my "Vector Space Transformation Technique" similarly to the usage to solve the "Exact Solution of All Half-Integer Bessel LHODE Formula" - using "Exact solution of ODEs - Vector Space Transformation Technique - Part 2", Theorem I.1.
Category: Functions and Analysis
[5] viXra:2011.0052 [pdf] submitted on 2020-11-08 11:22:28
Authors: Yu-Lin Chou
Comments: 2 Pages. expository article
To prove the equivalence between the $\eps$-$\delta$ characterization and the topological characterization of the continuity of maps acting between metric spaces, there are two typical approaches in, respectively, analysis and topology. We provide another proof that would be pedagogically informative, resembling the typical proof method --- principle of appropriate sets --- associated with sigma-algebras.
Category: Functions and Analysis
[4] viXra:2011.0044 [pdf] submitted on 2020-11-06 09:03:38
Authors: Yu-Lin Chou
Comments: 4 Pages. expository article
Given a countable collection of almost sure events, the event that at least one of the events occurs is ``evidently'' almost sure. It is, however, not so trivial to assert that the event for every event of the collection to occur is almost sure. Measure theory helps to furnish a simple, definite, and affirmative answer to the question stated in the title. This useful proposition seems to rarely, if not never,
occur in a teaching material regarding measure-theoretic probability; our proof in particular would help the beginning students in probability theory to get a feeling of almost sure events.
Category: Functions and Analysis
[3] viXra:2011.0030 [pdf] replaced on 2020-11-10 05:05:50
Authors: Yu-Lin Chou
Comments: 3 Pages. expository article; the first version is somewhat misleading.
With a new proof approach,
we show that the normal distribution with mean zero and variance $1/3$ is the distribution of the integrals $\int_{[0,1]}W_{t}\df t$ of the sample paths of Wiener process $W$ in $C([0,1], \R)$.
Category: Functions and Analysis
[2] viXra:2011.0029 [pdf] submitted on 2020-11-04 09:55:35
Authors: Yu-Lin Chou
Comments: 5 Pages.
Denote by $\CDF^{0,1}(\R)$ the class of all (cumulative) distribution functions on $\R$ with zero mean and unit variance;
if $F \in \CDF^{0,1}(\R)$,
we are interested in the asymptotic behavior of the function sequence
$(x \mapsto nF(x/\sqrt{n}))_{n \in \N}$.
We show that
$\inf_{F \in \CDF^{0,1}(\R)}\liminf_{n \to \infty}nF(x/\sqrt{n}) \geq \Phi(x)$ for all $x \in \R$,
which in particular would be a result obtained for the first time regarding the growth order of an arbitrary standardized distribution function on $\R$ near the origin.
Category: Functions and Analysis
[1] viXra:2011.0005 [pdf] submitted on 2020-11-01 18:23:20
Authors: Hiroshi Okumura, Saburou Saitoh
Comments: 10 Pages.
In this paper we introduce a very interesting property of the Laurent expansion in connection with the division by zero calculus and Euclid geometry by H. Okumura. The content may be related to analytic motion of figures. We will refer to some similar problems in the spectral theory of closed operators.
Category: Functions and Analysis
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