Analysis (Mathematics)

This book is intended as a text for a course in analysis, at the senior or
first-year graduate level.
A year-long course in real analysis is an essential part of the preparation
of any potential mathematician. For the first half of such a course , there
is substantial agreement as to what the syllabus should be. Standard topics
include: sequence and series , the topology of metric spaces, and the derivative
and the Riemannian integral for functions of a single variable. There are a
number of excellent texts for such a course, including books by Apostol [A] ,
Rudin [Ru] , Goldberg [Go] , and Royden [Ro] , among others.
There is no such universal agreement as to what the syllabus of the second
half of such a course should be . Part of the problem is that there are simply
too many topics that belong in such a course for one to be able to treat them
all within the confines of a single semester, at more than a superficial level.
At M.I.T., we have dealt with the problem by offering two independent
second-term courses in analysis. One of these deals with the derivative and
the Riemannian integral for functions of several variables, followed by a treatment
of differential forms and a proof of Stokes' theorem for manifolds in
euclidean space. The present book has resulted from my years of teaching
this course, The other deals with the Lebesque integral in euclidean space
and its applications to Fourier analysis.

It has never been accomplished to describe our behavior mathematically. Due to the fact that human behavior is highly erratic even the understanding of its causes are still sketchy. Assuming that we are all equal in our regulation of thought and behavior there are simply too many differences and partially inconsistencies, the attempts stopped in its onset. For having defined the five major groups of mankind, the sociopath, the artist, the median, the, and the psychopath［1］each group is related to each other but his decision making is hardwired differently and thou probably more easy to grasp than sticking to the consistency of all appearance for the regulation of behavior is quite similar in the five groups but different in its limits. Using the more formalistic tools of mathematics, this can open the review of the equations in verifying or falsifying the predictions on future behavior in an individual, at least after defining the group affiliation. Therefore a self-test has been established［2］to predetermine the group. In a hybrid-species ［3］, the eight main neuro-receptors in each group to have two optional origins. Measuring each by its own dominating patterns not only the amalgamation in each group can be defined but also the native patterns of the non-hybrid ancestry ［4］. Not only the variance of possible combination can distribute to the limits of brain-equations but also the time-axis of our memory, being rather different, illuminating the highly different decision making among offspring of hunters and farmers. A phenomenon probably explaining the variance in processing memory by peripheral distributed groups of ADHD and autism for ADHD memorizes in combining data with importance and such is been given an emotional response to the recall, while autism is mainly been given the exact time reference stored in a continuous frame of time-preference. The latter therefore have problems to distinguish between important and not important and the former lacks a passing timeframe, mirroring the primary form of acquiring resource, farming or hunting. With the boundaries set the graphs of the equation on resource-projection looks highly different by only changing its limits. Having not only focussed on the designation of human groups but also in behavioral shifts over time (from social to non-social) and the exact denomination of similar behavior, some rather simple equation could be defined to predict and consequently proof the predicate. Not only the proper use of words is necessary but also awareness that non-social individuals will often not answer truthfully. A topic that also can be mathematically predicted. The outcome will revolutionize our perception of mankind and ourselves, dawning more than one academic discipline, probably enabling us to go virtual and back again.

Introduced several new axiomatic systems, that are not less general than group theory, and discovered discontinuous analysis.

In this work I introduce and study in details the concepts of funcoids
which generalize proximity spaces and reloids which generalize uniform
spaces, and generalizations thereof. The concept of funcoid is generalized
concept of proximity, the concept of reloid is cleared from superfluous
details (generalized) concept of uniformity.

Also funcoids and reloids are generalizations of binary relations
whose domains and ranges are filters (instead of sets). Also funcoids
and reloids can be considered as a generalization of (oriented) graphs,
this provides us with a common generalization of calculus and discrete
mathematics.

I consider (generalized) limit of arbitrary (discontinuous) function, defined in terms of funcoids.
Definition of generalized limit makes it obvious to define such things as derivative of an arbitrary function, integral of an arbitrary function, etc. It is given a definition of non-dif\-fe\-re\-nti\-able solution of a (partial) differential equation. It's raised the question how do such solutions ``look like'' starting a possible big future research program.

The generalized solution of one simple example differential equation is also considered.

The generalized derivatives and integrals are linear operators. For example $\int_a^b f(x)dx - \int_a^b f(x)dx = 0$ is defined and true for \emph{every} function.

The concept of continuity is defined by an algebraic formula (instead
of old messy epsilon-delta notation) for arbitrary morphisms (including
funcoids and reloids) of a partially ordered category. In one formula
continuity, proximity continuity, and uniform continuity are generalized.

Also I define connectedness for funcoids and reloids.

Then I consider generalizations of funcoids: pointfree funcoids and
generalization of pointfree funcoids: staroids and multifuncoids.
Also I define several kinds of products of funcoids and other morphisms.

I define \emph{space} as an element of an ordered semigroup action, that is a semigroup action conforming to a partial order. Topological spaces, uniform spaces, proximity spaces, (directed) graphs, metric spaces, etc.\ all are spaces. It can be further generalized to ordered precategory actions (that I call interspaces). I build basic general topology (continuity, limit, openness, closedness, hausdorffness, compactness, etc.)\ in an arbitrary space. Now general topology is an algebraic theory.

Before going to topology, this book studies properties of co-brouwerian
lattices and filters.

Applied Mathematics and Sciences: An International Journal (MathSJ ) aims to publish original research papers and survey articles on all areas of pure mathematics, theoretical applied mathematics, mathematical physics, theoretical mechanics, probability and mathematical statistics, and theoretical biology. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics.

This paper deals with the symmetric space of functions and its subspace where continuous functions are dense is considered. Main properties of convolution which plays a vital role in harmonic analysis, as in other areas of mathematics are established in this space. Following the classical case, it is proved that the convolution can be approximated by linear combinations of shifts in a subspace of the considered space. An approximate identity for the convolution is also considered in that subspace.

The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, p-normal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed functions and preservation theorems.

Link: https://drive.google.com/file/d/1D6pCmLgEjdO9yAtWR7_UipBVM-ax8xrR/view?usp=sharing

Compactness is a generalization of the notion that a subset is both closed (it includes all its limit points) and bounded (All points lie within a finite distance).

En este artículo examino algunas especulaciones sobre la representación formal de biografías e historias vitales, tal como aparecen en el MANUSCRITO ENCONTRADO EN ZARAGOZA, novela-marco de Jan Potocki (1804, 1810). En la persona del geómetra Velázquez, Potocki a la vez satiriza las incipientes especulaciones de la Ilustración sobre una matematización de la psicología y comportamiento humanos, y participa sin embargo de ellas con contribuciones originales y sugerentes, que anticipan algunos elementos de narratologías formalistas posteriores.

"This is an important and ambitious book. It will probably be influential not only in colleges and universities but also in the mathematical community generally, It deserves careful study. because it is an unusual book, it is fitting to begin a review of it in an unusual way, viz.."--John Corcoran   

A Prologue, with Morris Kline
To past generations, mathematics was first and foremost man's finest creation for the investigation of nature. In the sixties of last century, the post WW-II generation was instructed to ask about any single mathematical expression of engineering applications, "What is its physical meaning?" which indicated the isolation of mathematics in the thought of its main future everyday users. M.Kline deduces that, "these are not the only grounds for distress. Grave misgivings and cause of dissension among mathematicians stems from the direction which research of the past one hundred years has taken. 

"The Late Morris Kline, 'A thinker who understood numbers better than anyone since Euclid', according to Omni, delivers a ringing indictment of modern mathematics." Nevertheless, he expresses his final review of the glorious achievements in the history of mathematics, recording its calamities in the loss of truth as a certain tragedy of the first magnitude. He laments with Rene' Descartes his cause of grief in the realization that, "the splendid show-case of human reasoning  exhibits a perfect structure, marred by shortcomings and vulnerability to the discovery of disastrous contradictions at any time as a blow to the stature of mathematics. 


Morris Concludes
"Talleyrand once remarked that an idealist cannot last long unless he is a realist and a realist cannot last long unless he is an idealist. When applied to mathematics , this observation speaks for the need to idealize real problems and to study them abstractly but it also says that the work  of the idealist who ignores reality will not survive. Mathematics should have its feet on the ground and its head in the clouds. It is the interplay between abstraction and concrete problems that produces live, vital , significant mathematics. Mathematics may like to soar into the clouds of abstract thought, but like birds they must return to earth for food! 

Joseph F Badir  amateurish comments
Demonstrations by derivations, or by paradox-ing?, since in orthodoxy Demonstration is by confessing, and like Socrates he may end a martyr. (This is really inspiring for a novice like me, to amazing philosophy of mathematics) Our professor (Ricci) once demonstrated saying; "For your convenience, I present to you, in a single paragraph, the essence of tensor analysis: Simply defined, "a tensor is a mathematical construction that 'eats' a bunch of vectors, and 'spits out' a scalar!"Paradox and Argumentation From Rhetoric to the Refinement of Mathematics
https://metode.cat/wp-content/uploads/2015/07/_86EN1_paradox_argumentation.pdf