Science topics: Mathematics

Science topic

# Mathematics - Science topic

Mathematics, Pure and Applied Math

Questions related to Mathematics

Members of the mathematics group:

I read that there are 147 members in the mathematics group here in ResearchGate. I therefore want to bring once again the issue I initiated some time ago-establishing a mathematical science journal here. I will appreciate if we participate in the discussion and put our efforts to bring the journal to existence. We all will appreciate later when the journal becomes an international hub for mathematicians who work in pure and or applied mathematics alike and post their results with a minumum hassel and make their works be read and see their contribution to Mathematics and thereby to society. I will once again approximately quote Lobachevsky : " No part of Mathematics however abstract it may look or be that will not be used to the good of mankind " . It is this spirit that makes almost all mathematicians need to participate ( not only as readers ) in contributing works and results how small they may be who will later produce larger results in their fields.

To give credit to the scientic network that we are using -ResearchGate, we can make copyrighted to it as :

©EJCS YYYY-ResearchGate

As you may recall, the title of the journal was: "Electronic Journal of Cross Sections in Mathematics" acronymed by EJCSM

I will appreciate your inputs.

Sincerely,

I have prepared a paper on Graph theory but I don't know how or where to publish it.

I know here f '(1) =0. I found in some texts if f ' (x) = 0 for some x then we can't apply N-R method. Is there any other technique to find first approximation for x^3-3x+1 = 0 taking initial approximation as 1? Which is correct ? a) 1 b) 0.5 c) 1.5 d) 0

I know parallel lines can't touch but what if they share all of the same points? Just wondering if there are any exceptions.

Moments Descriptors are invariant under RST ( Rotation, Scaling, Translation) in PR.

Is Moment Descriptor almost invariant under Optimization?

Is Moment Descriptor almost invariant under Guassian Noise?

Can the reduction of pixels to draw an optimal shape of a planar curve shape using mouse on computer affect recognition rate? If yes then upto which extent ?

Notice, that if f:K --> M is an injective map which can be defined by a finite statement, then

for every y in img(f) there is an x in K satisfying the relation y = f(x), which can be regarded as a definition for y. Thus, either both x and y can be defined by finite statements, or no one of them can be defined finitely.

May i know a book which gives a basic results or informations about Matrix theory?

How do I improve my English and my mathematical representation

My impression is that many people with the English
very bad deal.
Written submissions must be Formulated very exact.
I hope this does not fall back on me.
(It should therefore be possible to use the mathematical symbols
On this portal). See my earlier post.
Very helpful are:
For better English:
stardict-3
http://code.google.com/p/stardict-3/downloads/detail?name=stardict-3.0.3.exe&can=2&q =
LingoPad
http://www.ego4u.de/de/lingopad
For better mathematical representation:
Mathematics Microsoft Add-In for Word and OneNote
http://www.microsoft.com/download/en/details.aspx?id=17786
6.0 Unicode Character Code Charts
http://www.unicode.org/charts/
scroll to Symbols and Punctuation
scroll to Mathematical Symbols
For example, select Mathematical Operators
http://www.unicode.org/charts/PDF/U2200.pdf
The key combination must find themselves
On my laptop, for example, 2200 is universal quantifier then Alt + C. ⇒ ∀
Unicode BabelMap
http://www.babelstone.co.uk/Software/BabelMap.html
Microsoft Mathematics 4.0
http://www.microsoft.com/download/en/details.aspx?id=15702
Anyone can look around yourself.
Examples:
∀ ∈ x（ℝ) ∧ x (periodisch) ≝ ℚ
π√2
⇒

as I say vogels method is usually better but not always, have you seen a contradiction example?

thanks a lot

My question is : can we use adomian decomposition method for solving non linear equations in complex space ?

Please , if it is possible , send us your comments for me.

I am ready to cooperate about this idea in order to reasearch and writing scientific articles.

Thanks a lot

I would like to send my publications to this publisher. I would like to ask if someone has an experience with this publisher. Thank you!

There are mathematical objects or situations for which one needs an infinite number of definitions, statements or descriptions?

Mathematical objects or situations are special numbers, for example,
Quantities in general, functions .....

For instance, pi = 3.141592... can be defined as the ratio between the length and the diameter of a circle. Any integer can be defined by a finite sequence of figures, and so on.

Homotopy continuation method provide an useful aproach to find zeros of nonlinear equation systems in a globally convergent way. Homotopy methods transform a hard problem into a simpler one and solve it, then gradually deform this simpler problem into the original one. I usually solve equations from nonlinear circuits: diode, bipolar, MOS. Now, i want to solve other kind of equations with applications, specially if the equation is multivaluated. ¿Somebody want to collaborate with me?

If f(x) is an increasing real valued function, what about its inverse?

Maybe my English is poor.

So someone misunderstands the question.

Let me rewrite it in another words.

The problem I am interested in is as follows:

Given g(x).

We want to find out:

whether there exists a function f(x) that f(f(x))=g(x).

whether there exists a continuous function f(x) that f(f(x))=g(x).

and so on...

What's more,if we have a pair of such functions f1(x) and f2(x) that each allows f(f(x))=g(x),with what condition can we prove f1(x)=f2(x) for any x?

For example.

If g(x)=x.

There exists a continuous function f1(x)=x allows f1(f1(x))=x.

But we also have f2(x)=-x allows f2(f2(x))=x.

Here f1 is not always equals to f2.

However,if we add a condition that f must be monotone increasing,then we abandon f2.

We can prove only f1 suits for our demands.

What's the general situation?

===============================================================

The previous edition：

Where g(x) is already known,for example g(x)=exp(x)+x^2.

What can we say about the existence and uniqueness of f(x)?

If we require that f is continuous or analytic?

If there are some books about it,please give the names.

Commuting vector fields means that their Lie bracket equals zero.

Dear Srs,

First I would like to excuse my English, but is not

my native language.

I am a software developer with interest in Automaton Theory, I have a few related project that you can see here http://fsvieira.com/,

but I lack of math knowledge.

So, I made my own definition of a non-deterministic recursive automaton, that you can find here http://fsvieira.com/nar.pdf, and now I want to proof that my definition can accept the same languages that a non-deterministic pushdown automaton can.

But I don't have the skills to do it, so I just asking for some hints, what exercises should I do, what books can I read...

Thanks.

What are the isometries of the Hilbert Cube I^\infty = the set of all sequences (x_i) such that x_i \in [0, 1] endowed with the metric

d((x_i), (y_i)) = \sum_{i=0}^\infty 2^{-i} |x_i - y_i| ?

Here y can be assumed as a function of any independent variable ! I want the differentiate its present value of f(x(n)) with respect to its past value f(x(n-1)

Few examples are FFT,wavelet ect

y = 1=(x²+c) is a one-parameter family of solutions of the first-order Dierential Equation;
y+ 2xy² = 0. Find a solution of the first-order Initial Value Problem(IVP) consisting of
this differential equation and the given initial condition. Give the largest interval I over
which the solution is defined.
y(2) =1/3

We are working on coupled fibonacci sequences of higher orders.

I need this software to be interactive: allowing to enter a signal on real time of simulation. Maybe SimuLinks can do this, I could not say. I would appreciate all the information you can apport.

Can anybody suggest me the problems where research could be done in the field of Linear Integral equation.

Bessel Functions

Dear all, I've been searching for a "beginner's introduction" to Bessel Functions in its general form (and in electromagnetics in its specific form) except what is found on Wikipedia and youtube. The sources I found were not clear and misleading.
I would much appreciate your feedback.

I have a very interesting problem. I find that solving in a closed form the Rogers Ramanujan continued fraction

R=R(q), q=e^(-pi sqrt(r)) , r positive rational,

is equivalent to solve the equation

aX^2+bX+b^2/(20a)=CX^(5/3) : (1)

and X=[R(q^2)^(-5)-11-R(q^2)^5]b/(250a)

(the numbers C, a, b are related to the jr invariant with the relation jr= 250C^3/(a^2 b),

in order to generate the Rogers Ramanujan Continued fraction).

(1) is a six degree polynomial equation and of quite simple form. The program Mathematica (Version.6) for some values of a,b, and r, (say)

a=4,b=125 and r=1/5, 2/5, 3/5, 4/5, 6/5, 9/5, 12/5, 14/5, 17/5,

give the exact solution (solves the equation, for these values).

The problem is that I can not solve the polynomial equation and Im not aware of Galois Theory.

How Mathematica solves this equation?

Can anyone solve this polynomial equation?

See also the article on arXiv: "On a General Sextic Equation Solved by Rogers Ramanujan Continued Fraction". (by Nikos Bagis).

Note that I want to evaluate the RRCF not using the fifth degree modulus k_{25r}.

Anyone care to spare a thought on the idea that gravity has a magnetic-like component that is it's inverse (cosmological constant). I think the number zero is a purely human creation, and should be regarded as our latest step in evolution into the realm of intelligent life.

How the no of generator relate with the order of a cyclic group?

Can anybody suggest me some open problems in mathematics involving matrix theory

I basically work with matrices for analyzing Quantum systems. I would like to know if there are any open questions regarding matrix formulation. Please suggest some links to good papers on this topic

I feel weird without good looking math expressions

The following list includes free math software and tools together with the corresponding descriptions and download sites.

Operating systems:

Scientific Linux: A linux distribution put together by Fermilab and CERN. Freely available from

Ubuntu Linux: A Linux distribution, easy to install and freely available from

Debian: Perhaps the best Linux distribution.

DesktopBSD:

A freeBSD distribution easy to use which can be tested through a live DVD. Freely available from

BSD: Several Unix distributions.

Applications for symbolic calculus.

wxMaxima:

Calculus with a graphic interface. Freely available from

Axiom: Similar to the preceding one.

Euler: id.

Scilab: id.

octave: id.

Gap: Computational discrete algebra,

R: Statistics

PSPP: Statistics

haskell: Pure and lazy functional programming language with an interpreter.

Astronomy:

Stellarium: Free astronomy appl.

Star charts: Free star charts PDF files.

Math graphics:

Gnuplot: To build any graphic in 2D or 3D. Freely available from

DISLIN: A graphical library, easy to use.

Word processors:

TexMacs: WYSIWYG editor with a graphical interface, by means of which one can type scientific texts, and export them in PDF, PS, HTML, LaTeX formats. Freely available from

Lyx: Similar to the preceding one.

Miktex: A complete LaTeX distribution for Windows. Freely available from

TexMaker: A LaTeX editor: Freely available from

TeXniccenter: Another powerful LaTeX editor for Windows OS.

Kile: Another LaTeX editor.

TexShop: A LaTeX distribution and editor for Mac OS X. Freely available from

Texlive: A LaTeX distribution for Linux and Unix OS'.

Open Office: A package similar to Microsoft Office:

I have the following 3 non-linear equations:

(summation(i=1 to N)) (p(i)*k1(i)) <= Ith

(summation(i=1 to N)) (p(i)*k2(i)) <= Ith

(summation(i=1 to N)) p(i) <= pT

where p(i)=[delf/{{lambda1*k1(i)}+{lambda2*k2(i)}+{lambda3}}]-[{2*sigma^2}/{h^2}]

Here N,Ith,pT,sigma,h,delf are constants.All values of k(i) from 1 to N are known. The values of lambda1,lambda2,lambda3 need to be found.How can I solve these equations using fsolve in MATLAB?

I am a student of Computer Science and I'm just loving the story on Linear Algebra.

I wonder if there is a good indication of books regarding this subject?

Thank you in advance!

I have these equations with parameters. I have to plot graph for numerical solutions

I have three data sets,A,B and C.A is dependent on B and C therefore has (C*B number of data points).

C= inv( Trans(A) * inv(B) * A )

A is a rectangular matrix, and B is a large square matrix.

Assume the integral of "z*h(z,p,q)" over all values of the scalar z is equal to that of "z*h(z,p)", where both of the scalar-valued h(.) functions respectively integrate to 1 taken over all values of z. So then, is this true iff h(z,p,q) = h(z,p) for all z, p, and q? (Bonus points if you can also let me know the same for discrete z.) Thanks in advance!

Number theory and computer science question

Loop as only one vertices or more than one vertex?

Graph theory

When I read about Fourier transform, there are several definitions about Fourier transform. It is because there are several conventions about it. Which kind of definition should I refer to? Because it is a little bit confusing.

Anyone can help me in find solution to problems like

P * x + Q * y + ......... >= some constant

Capitals(P,Q) are constants while lower case letters(x,y) are variable

Problem states that we have to find solution to this problem while keeping solution minimized such that it should be greater than a given constant.

My interest is to solve nonlinear problems using HAM(homotopy analysis method) and prefect it in theory.

Dear friends, can somebody help me to know how I can simplify the equation (x-y)^0.5 as an approximation in terms of x and y?

Inverse matrix on PPU and on SPU using SIMD instructions.

This article will talk about how to convert some scalar code to SIMD code for the PPU and SPU using the inverse matrix as an example.

Most of the time in the video games, programmers are not doing a standard inverse matrix. It is too expensive. Instead, to inverse a matrix, they consider it as orthonormal and they just do a 3x3 transpose of the rotation part with a dot product for the translation. Sometimes the full inverse algorithm is necessary.

The main goal is to be able to do it as fast as possible. This is why the code should use SIMD instructions as much as possible.

A vector is an instruction operand containing a set of data elements packed into a one-dimensional array. The elements can be fixed-point or floating-point values. Most Vector/SIMD Multimedia Extension and SPU instructions operate on vector operands. Vectors are also called Single-Instruction, Multiple-Data (SIMD) operands, or packed operands.

SIMD processing exploits data-level parallelism. Data-level parallelism means that the operations required to transform a set of vector elements can be performed on all elements of the vector at the same time. That is, a single instruction can be applied to multiple data elements in parallel.

I am at present in need of help with the mathematical package bifurcation XPPAUTO

Actually, I have 3 diff. eqns and when I apply XPP I get results, some of which I cannot interpret. If anyone is interested I can give the eqns etc.

I m intrested in the solution of nonlinear hyperbolic partial differential equations with various techniques such as lie group theoritic method, vandyke and gutmaan technique etc,

Yang-Fourier transforms and Yang-Laplace transforms are new tools to deal with fractal differential equations and dynamical systems in fractal space.

For more detalis, see http://shootingcupoche.com/profile/Yang_Xiaojun/blog/22668_X_Yang_Local_Fractional_Inte

Does anyone know how to solve a system of equations where the number of equations is more than the number of unknowns?

Any help or references is highly appreciated.

Hallo,

there is a new proof of the 3n+1-Problem !

The paper ist available

Perhaps there is a flaw in the proof

What is your opinon?

The Collatz-conjecture (the famous 3n+1-problem):

we construct a sequence of integers starting with integer n = a_0

If a_j is even, the next number ist a_(j+1) = a_j/2.

If a_j is odd, the next number ist a_(j+1) = 3*a_j+1.

Example n = 6:

6, 3, 10, 5, 16, 8, 4, 2, 1

The Collatz-conjecture: the sequence with a every positive starting-integer ends always in the sequence 4,2 1

We usually expresed the numbers in a decimal base and define the irrationals how the numbers with non-periodic representation in the decimal base.

But, if a number have a non-periodic representation in a decimal base then this number have a non-periodic representation in any other base?

I try to solve this conjeture (maybe is easy) if someone have a idea is good receive!

Is there any skew symmetric matrix of odd order which is non singular over a finite field?

Hi I am an M.Sc graduate in mathematics and computer sciences. Suggest me some good universities in Asia to do P.hD

If we have: z = f (x,y) and z = f (t), could you please answer to my below questions:

1) Can I say: x = f (t) and y = f (t)?

2) How can I analyze dz/ dt?

Thanks in advance for your help.

Best Regards

Gholamreza Soleiman

graceful labeling

iam doingresearch in graph theory. will you help me in this. iam doing in graceful labeling.

There are five red balls and two green balls in a closed box. Two players consequently put a hand into the box and select a ball (without replacement). A player who first selects a green ball becomes the winner. Find the probability that the winner is the player who started the game....

Hi,

I am starting to explore better ways to use ResearchGate, not just for my benefit but to do something of value for others.

I will start posting some links to various papers, publications, presentations, and briefs, as well as joining in to different discussions.

I am looking for networking, and collaboration, and work (job(s)).

Certainly I am open to sharing ideas, critiques, comments, views, and helping others. To

me, everything does require an attitude of synergy and symbiosis in order for us to

succeed, as scientists, as people.

FYI, if anyone is interested, I just wrote up this summary:

These are URLs about me, some of what I am doing, including past, and also including

things that are "orthogonal" and obviously more directed at surviving in a "non-friendly

ecosystem" as far as science and especially exploratory and non-mainstream

("non-major-institutional/corporate") R&D.

---------------------------

---------------------------------

and just about my background, http://tetradyn.com/professormd

I can do some things pro bono and voluntary, as part of a team, etc., to help advance the

general interests and causes of good research, solid science, improved education, and

better public understanding.

However, I also seek (need) work: part-time, temporary, full-time of course, in US and/or

anywhere in the world.

Best regards,

Martin D

+1-757-847-5511

+1-202-415-7295 cell

Calculus (Latin, calculus, a small stone used for counting) is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change[1], in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.

Historically, calculus was called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) may refer to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus, and join calculus.

Colleagues, I am planning to change this thread to a category.

However, at the moment, I will post my second communication of the thread CMT.

Let us define another differential operator of infinite terms as :

e^{-D}:=∑_{j=0}^{∞}(((-1)^{j}D^{(j)})/(j!))

when j=0, we have the identity operator, and D:=(d/(dx))

Then as in my first communication post, we can question the following:

(∀ψεC^{∞}(I,ℝ))Λ(∀xεI), what will be

e^{-D}(ψ(x))=∑_{j=0}^{∞}(((-1)^{j}D^{(j)}ψ(x))/(j!))?

Consider the following example:

Example 1: Take ψ(x)=e^(x) the usual natural exponential function.

Claim: e^{-D}(ψ(x))=ψ(x-1)

Indeed,

e^{-D}(ψ(x))=∑_{j=0}^{∞}(((-1)^{j}D^{(j)}ψ(x))/(j!))

=∑_{j=0}^{∞}(((-1)^{j}D^{(j)}(e^{x}))/(j!))

=∑_{j=0}^{∞}(((-1)^{j}e^{x})/(j!))

=e^(x)∑_{j=0}^{∞}(((-1)^{j})/(j!))

=e^(x-1)=ψ(x-1)

∴ e^{-D}ψ(x)=ψ(x-1) ... which is a right translation of ψ by a unit.

One can extend this result further and write a corollary as :

Corollary: (∀kεℕ):(e^{-D})^{k}ψ(x)=ψ(x-k)-right translate of ψ by k-units.

Example 2. Let φ(x)=x³+x²+x+1.Then

e^{-D}φ(x)=∑_{j=0}^{∞}(((-1)^{j}D^{(j)}φ(x))/(j!))

=∑_{j=0}^{∞}(((-1)^{j}D^{(j)}(x³+x²+x+1))/(j!))

=x³-2x²+2x

But the expression we have at the end is precisely φ(x-1).

That is, once again we have a similar result :

e^{-D}φ(x)=φ(x-1)

Corollary: ∀ p(x) , e^{-D}p(x)=p(x-1)

Conjecture: ∀ψεC^{∞}(I,ℝ), e^{-D}ψ(x)=ψ(x-1)

Corollary to the conjecture: (∀kεℕ)(∀ψεC^{∞}(I,ℝ)),e^{-kD}ψ(x)=ψ(x-k)

Further communications will be posted on operators defined from combinations of both.

Hi, I am interested in fixed point theory in different spaces. Random fixed point theory is also my subject of interest. If you are interested, then we can start discuss.

Colleagues,

Recently we have a new group " International Professors " added to our group. It is therefore possible to create a new forum in order we share insights, new methods, interesting class encounters and new concepts introduced when teaching mathematics. This will create a plat form to share and know how curriculums are apart or close on global settings and might give a hint to education policy makers what they have to expect from mathematics curriculums in order to go at par with international standards.

I will therefore present my first communication.

It is on enlarging the usual differential operator D:=(d/(dx)) in variable x to something else. We know that the usual differentiation makes functions to loose their smoothness or regularity as we say it, by a degree (if they are not infinitely many times continuously differentiable ).

The types of questions I have, can therefore be given as extra exercises or new insights to students who take calculus on sequences, series and convergence, to engage them to think more about, not only single calculus operations but, combined of them and thereby do algebraic computations at the same time.

Let us define a new differential operator of infinite terms as :

∑_{j=0}^{∞}((D^{(j)})/(j!))=:e^{D} , for j=0, we have the identity operator.

Then for a real valued C^{∞}- function defined on some non-degenerate open interval I (or ℝ-for that matter ) we can question the following:

what will be the action of e^{D} on such functions.

That is, if ψεC^{∞}(I,ℝ), what will be ∑_{j=0}^{∞}((D^{(j)}ψ(x))/(j!))?

The very immediate question will be the question of summability of the series indicated?

But we take cases in which that condition works:

Example 1: Take ψ(x)=e^{x} the usual natural exponential function.

We see that e^{D}(e^{x}) converges to the sum : eψ(x)=ψ(x+1).

Indeed,

∑_{j=0}^{∞}((D^{(j)}ψ(x))/(j!))=∑_{j=0}^{∞}((D^{(j)}(e^{x}))/(j!))

=∑_{j=0}^{∞}((e^{x})/(j!))

=e^{x}∑_{j=0}^{∞}(1/(j!))

=e^{x+1}=ψ(x+1)

∴ e^{D}ψ(x)=ψ(x+1)-which is a left translation of ψ by a unit.

One can extend this result further and write a corollary as :

Corollary: (e^{D})^{k}ψ(x)=ψ(x+k)-left translate of ψ by k-units.

Example 2. Let φ(x)=x³+x²+x+1.Then

e^{D}φ(x)=∑_{j=0}^{∞}((D^{(j)}φ(x))/(j!))

=∑_{j=0}^{∞}((D^{(j)}(x³+x²+x+1))/(j!))

=x³+4x²+6x+4

But the expression we have at the end is φ(x+1).

Therefore once again we have :

e^{D}φ(x)=φ(x+1)

Claim: For a polynomial function p(x) , e^{D}p(x)=p(x+1)

Conjecture: ∀ψεC^{∞}(I,ℝ), e^{D}ψ(x)=ψ(x+1)

We can also define a similar operator that results in right translations of C^{∞}-functions by counts of units as:

e^{-D}:=∑_{j=0}^{∞}(((-1)^{j}D^{(j)})/(j!))

Further communications will be posted on the last operator and combinations of both.

analytic real-valued functions

Hey every body. I have a big question (at least for me!!), what we means by analytic real-valued functions on a closed interval or half closed interval. If any one can help me, realy I need this

y= -x/(a^2-x^2)

what is dy/dx?

where a is a constant

if there is any nice mathematicians , please help me out

Like hyperbolic and circular trigonometric functions can we able to generalize trigonometric ratios with respect to a general curve?

There are various implementations and variations of the LLL-algorithm, depending on the specific scope. Different "editions" have differet input variables and so on.. Has anyone experience of any of these implementations?

Given three vectors x,y,z., how do i plot the magnitude[sqrt(x^2+y^2+z^2)] and show it in 3D using matlab or mathematica?

If you have any other math package i can use and how-that would be great too.

Can we relate Grobner Bases for ideals to Computational Mathematics or Applied Mathematics?........thanks

I need a neat but detailed explanation on the introduction of a scavenger into a predator -prey lotka volterra model,i will ask that the assumptions made are clearly outlined as this explanation is given.

Thank you fellow mathematicians

What are the main differences between finsler spaces and riemann spaces

Hi,

looking for a way around the liar and logic contradictions

I have introduced a new logical dimension:

Statements are not absolutely true or false anymore

but true or false related to a viewing angel

or kind of logical layer or meta-level.

With this new dimension problems become solvable

that are unsolvable with classical logic.

Most contradictions are not contradicional anymore,

as the truth values belong to different layers.

The good news (in my theory):

The liar´s paradox, Cantor´s diagonal argument, Russell´s set and Goedel´s incompleteness theorem

are valid no more.

The bad news: There is no more absolute truth

and we have to get used to a new mathematics

where numbers might have multiple prime factorisations.

Over all, infinity and paradoxes will be much easier to handle in layer theory,

finite sets and natural numbers more complicated, but possible

(but it will be a new kind of natural numbers...).

The theory was in the beginning just a ´Gedankenexperiment´,

and my formal description and axioms may still be incorrect an incomplete.

Perhaps someone will help me?

Here my axioms of layer logic:

Axiom 0: There is a inductive set T of layers: t=0,1,2,3,…

(We can think of the classical natural numbers, but we need no multiplication)

Axiom 1: Statements A are entities independent of layers, but get a truth value only in connection with a layer t,

referred to as W(A,t).

Axiom 2: All statements are undefined (=u) in layer 0.

VA: W(A,0)=u

(We need u to have a symmetric start.)

Axiom 3: All statements in positive layers have either the truth value ´w´ (true)

or ´-w´ (false).

Vt>0:VA: W(A,t)= either w or –w.

(We could have u in all layers, but things would be more complicated).

Axiom 4: Two statements A an B are equal in layer logic,

if they have the same truth values in all layers t=0,1,2,3,...

VA:VB: ( A=B := Vt: W(A,t)=W(B,t) )

Axiom 5: (Meta-)statements M about a layer t are constant = w or = -w for all layers d >= 1.

For example M := ´W(-w,3)= -w´, then w=W(M,1)=W(M,2)=W(M,3)=...

(Meta statements are similar to classic statements)

Axiom 6: (Meta-)statements about ´W(A,t)=...´ are constant = w or = -w for all layers d >= 1.

Axiom 7: A statement A can be defined by defining a truth value for every layer t.

This may also be done recursively in defining W(A,t+1) with W(A,t).

It is also possible to use already defined values W(B,d) and values of meta statements (if t>=1).

For example: W(H,t+1) := W( W(H,t)=-w v W(H,t)=w,1)

A0-A7 are meta statements, i.e. W(An,1)=w.

Although inspired by Russell´s theory of types, layer theory is different.

For example there are more valid statements (and sets) than in classical logic

and set theory (or ZFC), not less.

And (as we will see in layer set theory) we will have the set of all sets as a valid set.

Last not least a look onto the liar in layer theory:

Classic: LC:= This statement LC is not true (LC is paradox)

Layer logic: We look at: ´The truth value of statement L in layer t is not true´

And define L by (1): Vt: W(L, t+1) := W ( W(L,t) -= w , 1 )

Axiom 2 gives us: W(L,0)=u

(1) with t=0 gives us: W(L,1) = W ( u-=w , 1 ) = -w

(2) with t=1 : W(L,2) = W ( -w-=w , 1 ) = w

(3) with t=2 : W(L,3) = W ( -w-=w , 1 ) = -w

L is a statement with different truth values in different layers,

but L is not paradox.

Set theory is very nice in layer theory,

but that at another time.

What do you think about it,

is it worth further investigation - or too far-fetched?

Yours

Trestone

For example : There are 230 non isomorphic groups of order 96.....and only 1non isomorphic group of order 97.

Once you understand what PvsNP problem is actually all about, you might as well try and solve it.

In loose terms, the P vs. NP problem actually seeks an answer to this simply stated question:

"Is finding a solution to a math problem equally hard in comparison to verifying that it IS a solution ?"

Math guys usually "search" for a solution to their problem (e.g. solving some equation), but this can apply to "searching" any data set.

Imagine a program that searches for a solution to some equation. That program will most certainly consist of two major parts: a searching part (the solver) and a verifying part (the verifier). The solver tries to construct a solution by some rules and a verifier checks that it actually *is* a solution.

This solution constructing part is like when you do all sorts of manipulations (factoring, cancelling common terms, ...) to solve an equation, and this verifying part is more like when you plug in some values for your solution back to the original equation to check if both sides turn out equal.

The first part will usually take up much time, as finding a solution to some equations is sometimes hard, but once the right solution is constructed, the verifier will take only a fraction of that time to check if that actually IS a solution. The PvsNP asks if those two parts are actually the same thing, because it would be nice of course, that solving an equation is as easy as checking the result.

Another way to look at it, it's basically a question about searching trough (potentially large) sets of data. In that context the PvsNP asks this:

"Is there a systematic way of searching trough a large data set ?"

(a large data set means for example, a data set not completely searchable in the course of one persons lifetime, for example the whole Internet)

Of course, people have been trying to answer this for decades ever since the computer era started, but with no luck, in my opinion because of the way the final solution needs to be presented.

It is widely believed that P is not equal to NP, because otherwise it would have baffling implications for say cryptography and code breaking. As there is a huge number of potential passwords that one can make up, a positive answer to PvsNP means that a brute force search is not necessary when trying to guess someone's password and there is also a systematic way how to obtain it. On the other hand, if P is not equal to NP than it means that there is no such thing.

Also, in this digital age, when almost everything is stored on a computer (music, pictures, texts, ...) if P = NP is true then we could generate any piece of music, any picture, anything ... by means of a computer program that would solve P vs NP, we just "search" for it, provided we have a computer program that recognizes that something is "a piece of music".

Finally, the PvsNP can be restated in terms of creativity as: "Can creativity be effectively automated ?"

The hardest thing about solving the problem is actually proving that either case is true. There are of course up till now many false starts and dead ends, and people today that are still trying are trying to prove that in fact P does not equal NP. Richard Karp, one of the most renowned computer scientists once said that this problem will someday be solved (either way) by someone under thirty using a completely new method. So, until then, you might try and solve it for yourself.

I need help to understand the computer science application of Algebra (rings, fields, groups, etc.)

The symbols we use in mathematics to form equations are just an aid in clearly forming an argument and communicationg it to others. We are clearly restricted when we use this formal language. If we could only cast out any mention of this language and symbols when doing mathematics, then we would be on the right track in truly understanding reality's ways.

The notion of quantity, form, change, space, shape, order, etc. are all independent of their symbolic representation. The language can easily change trough time, but these notions will not.

Computation as we know it, is merely a formal manipulation or transformation of symbols. It can be done by hand or by a computer. Either way, there is always a notion of a conciever and an executor present, when talking about computation. These two are usually one and the same, but I like to think about them as separate entities. The executor, follows a fixed set of rules to transform given string of symbols, that a conciever has conceived having some end goal in mind. The executor blindly follows these rules and eventually, (if he's in luck and didn't get stuck somewhere blindly following the rules),he will get a transformed string of symbols representing the final result.

And the conciever is the one that anticipates this result, again as a string of symbols.

So, when doing computation, the main assumption is that, when we manipulate symbols, we manipulate the notions that they represent. Just like in the primitive times, when people practiced magic, they believed that the symbols they use in their spells represent objects from the real world.

They believed that drawing these symbols in some special sequence will result in a spell being cast, and as a result something in the real world will change according to the spell's intention. So, in an amusing way, doing mathematics can be regarded as "doing magic", not in the real world, but in the world of ideas.

Computers process strings of symbols by following a fixed set of rules that we call a program. The conciever is the programmer, and the executor is of course the computer. The processing by a computer is usually done in a one-by-one

fashion, but is much faster that doing it by hand. Computers can be seen as manipulators of symbols, or executors of programs, but the acctual thing we are after is the "manipulated" idea after the computer has done millions and millions of manipulations on it (that would be too tedious to do by hand).

So "ideas" are the ones that we are after when doing computation, because we hope that this mechanical grinding away of symbols will tell us something new and interesting about reality and nature, although this point of view was refuted a hundred years ago by Godel's famous incompleteness theorems. These theorems show that there is definately something more to mathematics and computation than just "symbol grinding". Remarkably, Godel showed this using only using some basic facts from NUMBER THEORY, nothing fancy.

And what about nature and reality ?

What are nature's rules, and what "language" is used to set these rules ? Nature is the executor, but who is the conciever ? And what is the final result ? Is it LIFE maybe ?

The answers to these questions are certainly beyond human comprehension, but there is, as always a lot if speculation about it! But, when we finally find this out, only then we can make a significant progress in truly understanding this "manipulation of ideas" notion and and "reality's ways" in general that mathematicians are still desperately and vaguely trying to capture by the notion of "computation".

Hi,

We are working on the theory of GCR-Lightlike Submanifolds of indefinite Kaehler manifolds and Sasakian Manifolds. Till now we have studied Totally Umbilical, Totally geodesic, Mixed geodesic GCR-lightlike submanifolds, GCR-lightlike Product, sectional curvature and Holomorphic sectional curvatures of GCR-Lightlike submanifolds and found expressions for Ricci tensor also. Now I am looking for new topic for GCR-Lightlike submanifolds. So please suggest some topics on which we can continue our research.

Thanks.

Set theory

What is set theory, and where is it applicable?

Domination graph theory is the most popular topic for research.