Science topics: Mathematics
Science topic

Mathematics - Science topic

Mathematics, Pure and Applied Math
Questions related to Mathematics
Question
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 :
As you may recall, the title of the journal was: "Electronic Journal of Cross Sections in Mathematics" acronymed by EJCSM
Sincerely,
Dear Dr. Dejenie A. Lakew,
Your idea is excellent. I may support the idea by contributing mathematics in industrial applications.
Question
(a+b) (a+b) = a2+2ab+b2
(a+b)square
you calculate dela
Question
I have prepared a paper on Graph theory but I don't know how or where to publish it.
Whoever it may concern I am N V Nagendram working as Assistant Professor in Mathematics and i got published 14 papers in the academic year 2010 - 2011 after continuous and constant work done in Near Rings under Algebra of Mathematics. please visit lbrce.ac.in CSS Dept. Faculty / publications you can have my profile.
For your answer after writing an article to how it is to be sent ?
Ans: please go through international journal for graph theory topic you can find many journal titles go through each and every journal inside till you get the publication on your specialization of topic if you find then you goto submission online from there only. it will ask you title, author, co-author if any, Abstract , Key words and Subject specification/classification code mention thereof. upload your paper in the form of either MSWORD / LATEX or required form there mentioned by publisher.
So for this you can not ask every time to where my paper / article to be sent what you need to do is on regular basis you must have to had habit of reading journal by opening site international journal of topic name then you will find and read increase your reading capacity about journals. it is a good habit for a researcher on any topic. ok!
Now i am going to give you one journal name here you please send this article to that journal.
"International Journal of Mathematics Archive (IJMA)." or "International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN(Print):2249-6955 ISSN(Online): 2249-8060 " like this you have to thorough with net web site on your selected topic. bye yours .............N V Nagendram
Question
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 don't think there is any other way ,in the definition itself they say only if f'(xo) not equal to zero.
Question
I know parallel lines can't touch but what if they share all of the same points? Just wondering if there are any exceptions.
@Deepak Anand: The problem is the use of the word "Euclidean". This implies many things, including Euclid's 5th Postulate wherein parallel lines do not intersect. If you want to investigate other geometries (there are many) in which there are no parallel lines or there are many parallel lines, all passing through the same point. These are important and just as valid as Euclidean Plane Geometry. They just have different assumptions and contexts. You might want to do a Google search on Vanishing Point or on Projective Geometry which are probably relevant to the point you would like to make.
Question
One is euler's method.
Broadly, there are three classes of methods for solving (systems of ordinary) differential equations: special methods, numerical and graphical methods, and qualitative methods.
1. Special methods. They are rare, are only available for standard problems, and often require an initial transformation to a standard form before applying a solution method. However, it can be difficult to recognize the kind of differential equation you have so you can choose the right transformation. Possibly the best guide to these methods is "Ordinary Differential Equations" by Tenenbaum and Pollard, is available in a low-cost (US$12-25) Dover edition, and can often be found in college libraries. 2. Numerical and graphical methods. This includes Euler's method. The methods are broadly applicable, and there are excellent numerical solvers available for computers that are based upon the same idea as Euler's method, but are faster and more accurate. They are used to compute particular solutions or solution curves, and usually include methods for creating graphs when they can be useful. However, this approach may not give you insight into your problem. I recommend that you use such methods once you understand what are differential equations and what it means to find their solutions. 3. Qualitative methods. They are broadly applicable, are intended to give you insight into the kinds of solutions that exist for your differential equation, and the nature of the long-run behavior of the solutions to the equation (e.g. stabilize, grow or decrease without limit, oscillate, etc.). However, they may not give you numbers. They are based on patching together solution curves of (linearised versions of) the differential equation near coordinates where the kinds of solutions change. A modern first course in ordinary differential equations will usually introduce you to all three classes of methods, while a traditional course will emphasize special methods. As I said above, Tenenbaum and Pollard offer a textbook for a traditional first course. There are many texts available for a modern first course, such as the one by Blanchard, Devaney, and Hall, but they are usually expensive (about US$125).
If you are eventually headed towards graduate studies in mathematics, the above approach will be OK for a first course, but there are much better texts available. Authors of excellent introductory texts for mathematics majors include V. I. Arnold or Birkhoff and Rota.
Question
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 ?
I'm not sure about your question, but an important factor if you are considering using moment descriptors is to make sure that moments exist in your domain. (For example Lorentzian curves, which are common in my work, have no defined moments).
Question
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.
S. K's definition is for a bijection, not an injection. An injection is one-one into, not necessarily onto. Cardinality arguments show that there are many undefinable functions. If by definable one means definitions by an expression in some language where the expression contains a finite number of symbols and the number of symbols is finite, then there are a countable number of definable functions. But the number of injective functions is uncountable.
Question
Hi i send you one problem.
If for (4) you don't need that it is 'strictly' increasing then you can just use Q(t)=t/3
Question
May i know a book which gives a basic results or informations about Matrix theory?
- Matrix Theory, Joel N. Franklin‏
- Elementary matrix theory, Howard Whitley Eves
- Introduction to matrix analysis, Richard Ernest Bellman
- Matrix theory, James M. Ortega
- Matrix Theory, David W. Lewis‏
How do I improve my English and my mathematical representation
Question
First of all, your English for a German is better than my German for an American. I mostly understand what you are saying, which is the basis for communication. If you want to personally improve your English, I recommend that you practice, practice, practice. If you want to publish in English I recommend finding a person who is very fluent in English and German and writing in German. Second, just because English is the language of science does not mean that you cannot write to German journals and have a friend write the English translation for you. In fact, some people would like the opportunity to do translation work for their professional resume and if you get published that would help them out. Finally you can try to get a quick translation by using technology like translate.google.com, which by no means is a perfect piece of software, but it does well enough. Better yet that technology (by both Google and others) is getting better. For example: ------------------------ Guten Tag Herr Guettinger. Ich empfehle auch Englisch Lernsoftware von einer Firma Berlitze. Ich bin mit ihrer Software, um ein wenig Spanisch zu lernen. Es ist erschwinglich, im Gegensatz zu Rosetta Stone. Mit freundlichen Grüßen, Jimmy ------------------------- So as you can see, you have a lot of options.
Question
as I say vogels method is usually better but not always, have you seen a contradiction example?
thanks a lot
Golabi is the best method for it
Question
What are the various fields
Hi John,
I admire your bravery of challenging the "ridiculous" theories and I have browsed your website and sharing notes. But I think you go to extremes to some extent.
1. You said you had corrected some mistakes of Newton, Leibniz & Cauchy, and give your own definition of derivative. However, even though your New Calculus is well defined as you said, it may be classified as another type of calculus. Only when the "Old" Calculus' foundations have been PROVED wrong could I admit that is a fake.
Infinity does not exist in reality, but it doesn't mean we couldn't use this definition to help us cope with some practical problems.
2. Could you tell me some other advantages of studying your New Calculus? I mean other than the easy-understanding, for the epsilon-delta is quiet difficult for students as you said, is there any possible applications? Maybe we could test your derivative and make a comparison of the speed and accuracy with the "old", that may make sense in computer applications.
Frankly in China, undergraduates like me have few concepts of challenging the theories in our textbooks. So it's a great honor for me to have approached your ideas, regardless of the right or wrong. Maybe I have misunderstand your ideas, and wish we all have a fine discussion in the Research Gate.
Sincerely
Question
My question is : can we use adomian decomposition method for solving non linear equations in complex space ?
Thanks a lot
Dear Hamed,
I think that the Adomian decomposition method has been used for solving non linear equations in complex space.
Question
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!
You are posing a problem with two unknowns. First, I don't know what kind of works you are going to publish. Second, this PubCo is not well-known, but looks similar to Lambert Academic Publishing. Seems, it publishes only monographs. I have experience with LAP.
There are mathematical objects or situations for which one needs an infinite number of definitions, statements or descriptions?
Question
Mathematical objects or situations are special numbers, for example, Quantities in general, functions .....
My God, First of all thank you for the great number of contributions. Since I've stirred up a hornet's nest. I did not receive notification of your contributions. Who knows what to do? Can someone help me? Now back to my question. The intuition for my question arose from the post: "Is there a finite definition for every real number?" I probably still not understood. Now something very important to my question: I think of the primes, I think of the twin primes. I mean this is still an infinite process. Are there any other mathematical things? Yes, there is only the Fields Medal. See you soon
Question
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.
Dear Steven,
Every mathematical paper contains always some definitions and notations introduced by the corresponding author. In general, these definitions must be only considered in the paper context. Unfortunately, there are a limited symbol set to be used, and this fact oblige us to term different objects by the same symbols. This inconvenient does not matter whenever the author takes care of defining them.
The great french mathematician Henri Poincaré says: "Mathematics is the art of denoting different things by the same name". Of course, he was thinking in equivalence classes and analogies. Analogies are also particular cases of equivalences.
The father of normed spaces, Banach, wrote the following:
A mathematician is a person who can find analogies between theorems;
a better mathematician is one who can see analogies between proofs
and the best mathematician can notice analogies between theories.
One can imagine that the ultimate mathematician is one who can
see analogies between analogies.
(Stefan Banach 1892 - 1945)
Best regards.
Juan Esteban
P.S. I have sent you my paper about Cantor's theorem via e-mail.
Question
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?
Normal linear systems involve derivatives on the input and output signals. The transfer function is the quotient of the Laplace transforms of the output and input. They are rational functions of a complex variable, normally noted by "s". The poles of the transfer function are very important in describing the properties of the system. If we introduce delays, at any of the involved derivatives, terms with exponentials will appear. In this situation the discovery of the poles is a difficult task. Frequently, the Lambert function is used. The situation becomes more involved if the system is fractional. In this case, fractional powers of s will appear.
Question
If f(x) is an increasing real valued function, what about its inverse?
As many have pointed out, f(x) has an inverse if f(x) is strictly increasing. In that case, its inverse is also strictly increasing. Let´s see:
Let´s call g(x) to the inverse of f(x).
f(x) is strictly increasing; that means that the following sentence is true:
"x1 < x2 if and only if f(x1) < f(x2)" for any values x1 and x2 in the domain of f(x) .
g(x) is also strictly increasing only if it satisfies that same sentence,
so we need to prove that the following sentence is true:
"y1 < y2 if and only if g(y1) < g(y2)"
but proving it is easy if one takes into account the following:
For some values x1 and x2 it´s true that y1 = f(x1) and y2 = f(x2) (we used the fact that f and g are inverses).
and replacing those values y1 and y2 in g(y1) and g(y2)
we obtain g(y1) = g( f(x1) ) = x1 and g(y2) = g( f(x2) ) = x2 (we used the fact that f and g are inverses).
Now, that means that
y1 < y2 is the same as f(x1) < f(x2)
and
g(y1) < g(y2) is the same as x1 < x2
and thus, we can say that the pair of sentences
"x1 < x2 if and only if f(x1) < f(x2)" and "y1 < y2 if and only if g(y1) < g(y2)"
are the same sentence!!
and finally, since the former is true, so is the later.
Question
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?
I'm sorry……I couldn't catch your point. Maybe my English is poor, so you didn't know the question clearly. Let me say it again. I mean that g(x) is already given. We need to make sure if there exists a f(x) which allows f(f(x))=g(x). What's more,if we also have h(h(x))=g(x),can we make sure f(x)=h(x) for any x?
Question
Commuting vector fields means that their Lie bracket equals zero.
Use the well known formul for commuting vector fields, $X,Y$:
$\exp(t(X+Y))=\exp(tX)\exp(tY)$
Question
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.
@Filipe. I've read your notes. The idea is sugestive but I can't find out how to accept some context-free languages as {(a^n)(b^m)(a^p)(b^q):n+m=p+q}
On the other hand in the definition of transition function \delta on your example 1, you set \delta(q_1,M)={q_2}, but M is not in the alphabet ... and you cannot collect all possible M's in the alphabet because the M's are an infinite set whereas alphabets must be finite. Perhaps I've misunderstood some point.
Question
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| ?
My conjecture is that the only possible isometries of the Hilbert Cube (in this metric) are
f(x_i) = (y_i) such that y_i = x_i or y_i = 1 - x_i
where (x_i) and (y_i) denote sequences in I^\infty and x_i, y_i denote their i-th terms, respectively.
Question
Write what you want
Dear Hanspeter,
It is the metamorphosis of another that degenerated.
Question
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)
What you see on top, is
11.1 Partielle Ableitungen
in English
3 Differentials in several variables
That is a partial differentiation.
Greetings
Question
Few examples are FFT,wavelet ect
There are many examples, to mention some: Laplace transform, Fourier transform, wavelets, z-transform, Hankel Transform, Hilbert transform, and the list follows a reference that can be of some help is Transform Methods in Applied Mathematics by PETER LANCASTER K STUTIS SALKA USKAS , 1996 by John Wiley & Sons, Inc.
Question
y = 1=(x²+c) is a one-parameter family of solutions of the fi rst-order Di erential Equation; y+ 2xy² = 0. Find a solution of the fi rst-order Initial Value Problem(IVP) consisting of this di fferential equation and the given initial condition. Give the largest interval I over which the solution is de fined. y(2) =1/3
Good work
Question
We are working on coupled fibonacci sequences of higher orders.
applications?, for what?, math doesn't need applications. Applications need Math.
Question
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.
Why not using LabView from NI?
But also Matlab supports this (Data Acquisiton Toolbox, Real time workshop, ...) or together with a DSpace solution?
You have to check what is the most economic solution for your problem.
Question
Can anybody suggest me the problems where research could be done in the field of Linear Integral equation.
There are many institutes working on mathematical area of research. the following are the few:
Chennai mathematics institute
Institute of mathematics and applications, bhuwaneshwar
Institute of mathematics, Gurgaon
The institute of mathematical sciences, Chennai
CNR Rao Advanced Institute of mathematics, statistics and computer sciences
I suggest u to do research on application point of view rather than core mathematics
Bessel Functions
Question
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.
Check this out Ara: F. Bowman, Introduction to Bessel Functions, Dover Publications Inc, 1958 N.Y.
Question
All types, especially scalene
See ACM Algorithm 736 in (1994)
Question
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}.
Ask Heng Huat Chan in Singapore..he is an expert...
Question
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.
Your question is a bit provoking I guess...
From a physics point of view, I think you mean an absolute zero property.
No zero temperature is possible (as absolute temperature in Kelvin), no zero energy is possible (as absolute energy since the vibration in the lattice still has some energy involved), maybe no zero mass (debate on the mass of a photon still open)...
But I do not think of zero as a problem. Further zero is extremely useful to compare equalities, A=B means A-B=0.
Question
How the no of generator relate with the order of a cyclic group?
An ifinite cyclic group has only two generaters and a finite group of order n has phi(n) generators
Can anybody suggest me some open problems in mathematics involving matrix theory
Question
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
Hello Sir, I advice you to refer X. Zhan's paper "Open problems in Matrix Theory."
Question
I feel weird without good looking math expressions
For me personally, LaTeX is fine, but as I can see it is next to unknown to many researchers. But there is another possibility: use pictures (jpg, png) attached to your posts. I'm sure you know how to create them.
Question
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:
Question
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 assume you have already used "fsolve" built-in solver in Matlab. If the roots (lamda1, 2 and 3) are not plausible/ difficult to optimize, you can use contour plot technique to visualize the real roots of the coupled simultaneous nonlinear equations. Please visit MathWorks and Matlab central for examples. Hope this help.
Question
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?
Linear Algebra by Gilbert Stang ,,,,,,,,,Read online MIT LECT. NOTES
Question
I need a definition for a search.
For more details.It is better to refer:
Question
I have these equations with parameters. I have to plot graph for numerical solutions
Thnx.... I have the trial version. I m working on models , so i need the graphs to see the stability directly with given parameter values in the system of 3 equations stated earlier.
Question
I have three data sets,A,B and C.A is dependent on B and C therefore has (C*B number of data points).
For quantiative data:
Well, you want to find some relationship between B,C and A
this
means:
you can use a model of this type
A=f(B,C)=a_0 + a_1 B + a_2 C + a_3 B*C + a_4 B^2+ a_5 C^2 + a_6 B*C^2 + a_7 *C*B^2+....
and with MRA you can make statistical tests which terms are relevant and determine the coefficients a_i for this terms.
Question
Pls mention the book names
Thank u
Question
C= inv( Trans(A) * inv(B) * A )
A is a rectangular matrix, and B is a large square matrix.
You can indeed avoid the calculation of inv(B) if some orthonormality conditions hold among the columns of A. See "Moore–Penrose pseudoinverse".
Question
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!
Before giving bonus, first you have to decide if h is a function of 2 or 3 variables.
Question
Number theory and computer science question
Here is my interpretation of your question:
"Continued fractions are very useful and as a novice I'm very impressed playing around with them. Hey, we can find patterns even on pi, wikipedia says! And periodic expansion of sqrt(2) is a miracle! Why nobody shares my enthusiasm?"
Correct?
Loop as only one vertices or more than one vertex?
Question
Graph theory
Hello, This is not my area. Perhaps the two websites are helpful. http://en.wikipedia.org/wiki/Graph_theory http://en.wikipedia.org/wiki/Glossary_of_graph_theory#Walks Regards
Question
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.
Thank you, Henk Smid.
Question
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.
Anybody got solution
Question
My interest is to solve nonlinear problems using HAM(homotopy analysis method) and prefect it in theory.
And what's this "homotopy analysis" please?
Question
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?
Nothing else
Question
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.
no comment
Question
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.
give me
Question
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,
I'm interested in Oscillation of functional differential equations (dynamic equations, neutral, neutral delay, delay differential equations, inequalities). I'm ready to cooperate in this area and related topics.
Question
Yang-Fourier transforms and Yang-Laplace transforms are new tools to deal with fractal differential equations and dynamical systems in fractal space.
I want to share and present new ideas to you to extend this transformation. You can contact through e-mail
Question
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.
Consider the case of a number of nonlinear algebraic equations in several unknowns:
F_j(x_1,...,x_U)=0 for j=1,...,E
where U < E (more equations than unknowns).
If there is a solution (X_1,...,X_U) for the unknowns, then it suffices perhaps to solve only U of those equations.
Here, I write perhaps for the following reason:
For given x_2,...,X_U there may be several solutions of F_j(x_1,...,x_U)=0 for x_1 due to the nonlinear nature of the equations. For instance, there are normally two solutions for x_1 if F_j is quadratic in x_1. But it could be that only one of the values for x_1 satisfies also the rest of the equations F_k=0 for k different from j for the given x_2,...,X_U.
Also, there may be the problem of which of the E equations to choose for obtaining as many equations as there are unknowns.
Furthermore, there may be several solutions whence uniqueness of the solution is not guaranteed.
Additionally, it may be that the system does not possess a unique solution or any solution at all.
One possible solution is to replace the system of equations by a minimization problem in the least-square sense:
L_E(x_1,...,x_U) = sum_{j=1,..,E} F_j(x_1,...,x_U)^2 = min
Then, any of the local minima of the sum is nonnegative because the sum contains only nonnegative terms. The global minimum is the minimum of all the local minima and may be zero in case of the existence of the solution of the original system, or positive otherwise.
As in the linear case, one may minimize other norms of the vector function F, i.e. ||F||=min.
Of course, the minimization problem may be quite involved numerically.
An interesting question is whether it is preferable to consider an alternative minimization problem instead:
Choose U of the F_j renumbered in such a way that they correspond to F_1,...,F_U and minimize
L_U(x_1,...,x_U) = sum_{j=1,..,U} F_j(x_1,...,x_U)^2 = min
under the E-U contraints F_j=0 for j=U+1,...,E.
The constraints may be added via Lagrange parameter lambda_j and then, one has to minimize
M(x_1,...,x_U,lambda_{U+1},...,lambda_E) =
L_U(x_1,...,x_U) + sum_{j=U+1,...,E} lambda_j F_j(x_1,...,x_U) = min
Making M stationary with respect to all its E arguments then leads to E equations for the U unknowns x_k and the E-U unknowns lambda_j.
There is still a further possibility that may be worth investigating that I would like to introduce via an example:
Consider simultaneously three quadratic equations in a single variable:
F_1 = a x^2 + b x + c = 0
F_2 = d x^2 + e x + f = 0
F_3 = g x^2 + h x + i = 0
This nonlinear system may be converted to a linear one
a y + b x + c = 0
d y + e x + f = 0
g y + h x + i = 0
by introducing the new variable y=x^2. This linear system of 3 equations for 2 unknowns x,y may be solved by any of the standard methods.
The original system has only a solution if a solution (x,y) of the linear system satisfies y=x^2.
Again, one may tackle the linear system by minimizing L_3(x,y), but now with the constraint y=x^2, that also may be added via a Lagrange multiplier mu, say.
Thus, one may try to linearize the original system by introducing further variables for the nonlinear terms and add additional
constraints for the defining equations of the new variables.
Question
Hallo,
there is a new proof of the 3n+1-Problem !
The paper ist available

Perhaps there is a flaw in the proof

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
no
Question
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!
In this moment i have not the proof complete, this week i finish the semester an in vacations i think in solve this problem and other about sucesions. When i finished i load in a pdf the solution.
Thanks for the interest
Question
Is there any skew symmetric matrix of odd order which is non singular over a finite field?
try to find it
Question
whether there is a definition of the point?
certainly a well-defined?
Point: its is a ball in the n-dimensions with center (0,0,...,0) "n-times" and radius 0
Question
Hi I am an M.Sc graduate in mathematics and computer sciences. Suggest me some good universities in Asia to do P.hD
It is good to hear that from you. Of course, I don't know any university of India with a good reputation but as i am also in a need of that i will inform you when i come to realize there is any. Hope U do the same for me.
Stay cool........
Question
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?
Best Regards
Gholamreza Soleiman
It is difficult to help since the letter "f" seems to have two different meanings in your question. Is "f" a function of two variables (x, y)
or a function of one variable t.
Maybe is your question the following. Let f be a real function
of two real variables (x, y) and g be another real function of
one real variable t. Now consider the set S of real numbers (x, y, t) such
that g(t)=f(x,y). Is it possible to find two real functions X and Y of t such that
(x, y, t) belongs to S exactly when x=X(t) and y=Y(t) ?
To this question, the answer is most of the time no. Take for instance
g(t)=t and f(x, y)=x+y. If the answer to the question was yes, then,
for each t, you would have exactly one couple (x, y), namely (X(t), Y(t)), such that x+y=t. But this is not true for instance for t=1 since (x, y)=(1, 0)
and (x, y)=(0, 1) are two different couples such that x+y=1.
By the way, "the chain rule" means something else in mathematics.
graceful labeling
Question
iam doingresearch in graph theory. will you help me in this. iam doing in graceful labeling.
Please refer the book  " Dynamic Survey on Graph Labeling" by J. A. Gallian- Electronic Journal of combinatorics.
Question
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....
Sheba: The answer supplied earlier is not correct! It should be
(2/7) + (5/7). (4/6) . (2/5) + (5/7). (4/6). (3/5). (2/4). (2/3).
In the earlier answer, instead of (3/7), the multiplier should be (5/7) in the second as well as the third terms of the expression. It was perhaps a typing error. .
Question
lyapunove type of the difference equation
What is difference equation. Please throw light on it.
Dr. Balkishan Sharma
Question
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.
---------------------------
---------------------------------
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
A very useful and sincere initiative indeed,Thank you
Question
i am doing m.tech in computer science,want to do phD in maths.is there any possibility for this
If you can take GRE subject test in math and can get acceptable marks,there are plenty of universities that really consider your applications for Ph.D(also,know about your research interests before jumping into this journey)
Question
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.
Mathematics is the branch of science that deals with logic, decision-making, assumptions, deductions, the clarity of thought and ability to solve the problems in a calculative manner. It is the branch of Mathematics that deals with the finding and properties of derivatives and anti-derivatives of functions by methods originally based on the summation of infinitesimal differences. The two main types are differential calculus and integral calculus.
Question
prove that the equatin of a circle is y=mx + c
the gradient multiplied by x added to the y intercept must be equal to y.
Question
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.
Dear Dr. Dejenie A. Lakew,
Communication in Mathematics teaching has its great roles while teaching courses where mathematics communicates about the practical engineering applications. For example: Laplace transform, State space, Graph Theory, Boolean Difference, Mod Theory, and many such which leads to realize the real concepts of the systems.
Question
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.
Hello, We might have a common interest, please find it out.
Question
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.
Dear Dr. Dejenie A. Lakew,
Communication in Mathematics teaching has its great roles while teaching courses where mathematics communicates about the practical engineering applications. For example: Laplace transform, State space, Graph Theory, Boolean Difference, Mod Theory, and many such which leads to realize the real concepts of the systems.
analytic real-valued functions
Question
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
Full details are here : http://en.wikipedia.org/wiki/Analytic_function I have verified the content of this source, it is ok..
Question
y= -x/(a^2-x^2)
what is dy/dx?
where a is a constant
Given how unlikely it is to get a useful explanation here, I suggest just using www.wolframalpha.com for cheating instead, you would've gotten Pathak's solution with for instance "dy/dx of y= -x/(a^2-x^2)" or "derivative of y = -x/(a^2-x^2)". An added plus is that it won't judge you for being too lazy to execute the straightforward derivation algorithm yourself. And also that it's very likely to be correct, so you don't have to rely on majority voting.
Question
Like hyperbolic and circular trigonometric functions can we able to generalize trigonometric ratios with respect to a general curve?
I think this is a problem of Differential Geometry. By setting new cordinates, (a new curvelinear system not nececary orthogonal, as the plane-orthgonal cordinates) to a curve ''under conditions'' you have the same curve from another point of view. Some properties of the curve remain the same such as its lenght or its curvature some others not (I think).
I like to think as follows: Concider y=e^x then this curve is the same as log(x) because these two are catoptric to each other with respect to the line y=x. Hence we have the same curve. The only thing we did is to transfer -all curve- say e^x in the plane to his catoptric. The curve x^2-y^2=1 is the same curve as y=1/x. The reason is that, we take x^2-y^2=1 and we rotate it by an angle a=pi/4...etc
However this can see one in a better way with surfaces. A change of the parameters live some values unchanged and others not. (Some of them which remain the same, is the Gauss curvature-K and its mean curvature-H).
So, I think, your answer will be given by a Differential Geometry book, that contains plane curves, space curves, surfaces and its generalizations which are Riemann spaces.
Question
Hai
for M/M/1 queue the limiting distribution is obtained by recursive arguments.
For a complete solution of the difference-differential equations refer to Gross and Harris (1998),Fundamentals of Queueing Theory, 3rd ed., Wiley, New York.
Question
Solve this Indices problem friends....
The answer to you question is infinity, as you are just summing up 1/6 to infinity ? Perhaps you meant 6+1/(6+1/(6+1/(6+...))) ? Well, for this the n-th iteration is defined by recursion a(1) = 6; a(n+1) = 6 + 1/a(n). Since this sequence is increasing and bounded it converges to a (positive) limit x, you can replace a(n) (for each large enough n) by x to get x = 6+1/x which is equivalent to x^2 - 6x - 1 = 0. Solving this you get x = 3-Sqrt(10) or x = 3+Sqrt(10). And since x must be positive, the solution is x = 3+Sqrt(10) which is about 6.16228.
Question
Solve this homogeneous problem
I tried a lot but still can't get it .please tell me the ans
Question
Is Zeta[2+n^2]-1 a Normal[mu,sigma] ?? (Zeta is Zeta Riemann funtion)
Question
the following link introduce the Vieta jumping method and some of its application
Question
Need to know.
D^3+d+1=0 solve it
Question
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?
I shall have a look at these in detail!
Question
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.
@Samuel Paraview seems cool but i have never used it before and it looks complex.Any suggestions on how to go about it?
Question
Can we relate Grobner Bases for ideals to Computational Mathematics or Applied Mathematics?........thanks
Groebner bases can also used to prove the some geometric theorem by mathematical mechnazation.
Question
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
Predator-Prey Lotka-Volterra Model:
x'=ax-bxy
y'=-cy+dxy
where a stands for the reproductive power of the preys; -bxy and dxy model the encounters of preys and predators (which are negative for the preys and positive for predators); and -cy models the competition between predators, which is more powerful than their reproductive capacity.
Then if one try to introduce a new actor, a new equation is needed. Let z denote the scavengers What are the relations of scavengers and remaining actors? If you assume that there is no influence, then first two equations will remain the same. If you assume that scavengers do competition between them (-ez) and are fed by death bodies (+fx+gy)
and the third eqution is thus:
z'=-ez+fx+gy
Another considerations of the relations of species led to other equations.
Question
What are the main differences between finsler spaces and riemann spaces
In Riemannian geometry the metric tensor depends only on the points x of tha manifold M (g=g(x)), whereas in Finsler geometry the metric tensor depends on both a point x of M and a tangent vector y to M at x (g=g(x,y)).
Question
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.
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,
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
Hello,
here some more details about the new set theory,
that can be defined using layer logic:
This "layer set theory" is different in many points to ZFC:
It has only one kind of infinity and the set of all sets is an ordinary set.
The central idea is to treat “x is element of set M” (x e M) as a layer statement:
It is true in layer t+1 that set x is element of the set M, if the statement A(x) is true in layer t.
(There may be still some gaps in my formalization of layer logic and set theory,
but I hope, that this is owing to my limited capabilties and not to gaps in layer logic:
help welcome!)
Equality of layer sets:
W (M1=M2, d+1) = W ( For all t: W(xeM1,t) = W(xeM2,t) , 1 )
Especially: W (M=M, d+1)=w for d>=0.
The empty set 0:
W(x e 0, t+1) := W( W( x e 0, t ) = w , 1 ) = -w for t>=0.
The full set All:
W(x e All, t+1) := W( W( x e All, t ) = w v W( x e All, t ) = u v W( x e All, t ) = -w , 1 ) = w for t>0 and =u for t=0.
So other than in most set theories in layer theory the full set is a normal set.
Axiom M1 (assignment of statements to sets):
W(x e M, t+1) := W ( W ( A(x), t ) =w1 v W ( A(x), t ) =w2 v W ( A(x), t ) =w3 , 1 )
with w1,w2,w3 = w,u,-w
For every layer set M there exists a layer logic statement A(x) witch fulfils for all t=0,1,2, …:
W(x e M, t+1) = W ( W ( A(x), t ) = w v W ( A(x), t ) = -w , 1 )
W(x e M, 0+1) = W ( W ( A(x), 0) = w v W ( A(x), 0 ) = -w , 1 )
= W (u=w v u=-w, 1 ) = -w
Axiom M2 (sets defined by statements):
For every layer logic statement A(x) about a layer set x there exits a layer set M so that for all t=0,1,2,3,… holds:
W(x e M, t+1) := W ( A(x), t ) (or the expressions of axiom M1).
Definition M3 (definition of meta sets):
If F is a logical function (like identity, negation or f.e. FoW(xeM1,t) = W(xeM1,t)=w )
then the following equation defines a meta set M: (M1=M is allowed):
W(x e M, t+1) := W ( F o W(x e M1, t), 1 )
Consequences of the axioms and definitions:
In layer 0 all sets are u:
W( x e M, 0 ) = u (as all statements in layer 0).
In layers > 0:
W(x e M, t+1) := w if W ( A(x), t ) = w else W(x e M, t+1) := -w
For all x and (normal layer) sets M holds: W(x e M, 1) = u (as W(A(x),0)=u).
For all x and meta sets M holds: W(x e M, 1) = w or –w
Last not least let´s look upon the Russell set:
Classic definition: RC is the set of all sets, that do not have themselves as elements
RC:= set of all sets x, with x –e x
In layer theory: W(x e R, t+1) := W ( W ( x e x, t ) = -w v W ( x e x, t ) = u , 1 )
W(x e R, 0+1) = W ( W ( x e x, 0 ) = -w v W ( x e x, 0 ) = u , 1 ) = W (u=-w v u=u , 1 ) = w
Therefore W(R e R,1) = w
W(R e R,2) = W ( W ( R e R, 1 ) = -w v W ( R e R, 1 ) = u , 1 ) = W (w=-w v -w=u , 1 ) = -w
And so W(R e R,3) = w, W(R e R,4) = -w , …
R is a set with different elements in different layers, but that is no problem in layer set theory.
As All, the set of all sets, is a set in layer theory, it is no surprise,
that the diagonalisation of cantor is a problem no more (I just give the main idea):
Be M a set and P(M) its power set and F: M -> P(M) a bijection between them (in layer d)
Then the set A with W(x e A, t+1) = w := if ( W(x e M,t)=w and W(x e F(x),t)=-w )
A is a subset of M and therefore in P(M).
So it exists x0 e M with A=F(x0).
First case: W(x0 e F(x0),t)=w , then W(x0 e A=F(x0), t+1) = -w (no contradiction, as in another layer)
Second case: W(x0 e F(x0),t)= -w then W(x0 e A=F(x0), t+1) = w (no contradiction, as in another layer)
If we have All as M and identity as Bijektion F we get for the set A:
W(x e A, t+1) = w := if ( W(x e All,t)=w and W(x e x),t)=-w ) = if ( W(x e x),t)=-w )
This is the layer Russell set R (I omitted the ´u´-value for simplification)-
and no problem.
So in layer theory we have just one kind of infinity – and no more Cantor´s paradise …
Yours,
Trestone
Question
Integration
no
Question
For example : There are 230 non isomorphic groups of order 96.....and only 1non isomorphic group of order 97.
In general, no formula is known for the number of groups of order n. The number is sometimes incredibly huge. For example, there exist about 50 billion groups of order 1024. If |G|=pq where p,q are prime numbers and p < q, then there exists one or two groups of order pq, according to whether or not p divides q-1.
Question
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.
What if it's only solvable as long as the solution is known to exist? Can't run a successful search for music that doesn't exist. I think the only factual solution would be (P=NP) as long as (P-NP>0).
Question
Definition and aplication
Sobolev spaces are both distribution spaces and Banach spaces.
You can solve PDE's by:
1- Distribution calculus (convolutions, Fourier transform...) to find weak solutions.
2- Sobolev imbeddings of Sobolev spaces in $C^k$ spaces of smooth functions.
Some nice books, in addition to Sudev's list:
1. F. G. Friedlander, M. Joshi, Introduction to the Theory of Distributions.
2. Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations.
Question
I need help to understand the computer science application of Algebra (rings, fields, groups, etc.)
Group theory , fields and rings are very much useful in cryptoanalysis. For so many cryptosystems(cryptoanalysis) , we use groups and fields. (i.e., for encrypting a message and decrypting a message). Main aim is to determine how efficient an algorithm in terms of complexity.
Question
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".
Majority of people working in the area can do nothing but manipulating symbols. They are always formally right because they never break the formal grammar. So many people are doing this that if you wish to do something in this manner, you can be sure beforehand, that this is already done. However, they never dare to try to do something non-trivial. All non-trivial ideas as well as the reality lie beyond this grammar. Those able to see it, are not confined in formal grammar. Symbols an all their combinations constitute a discrete set which has dimension zero, whereas the world is continuous and has greater dimension (I believe, 4), therefore it cannot be embedded into 0-dimensional grammar or symbolic logic. Let computer do what can be done in 0 dimensions and do what it cannot. Human brain is presently the only instrument to work in higher (than 0) dimensions. Therefore it does not obey formal rules blindly. I believe that the time of pure formalism is over. Further progress depends on those who can go beyond blind formalities.
Question
Abstract algebra
Total no. Of homomorphisms will be gcd(m,n)... I wl find the no. Of onto homomorphisms...
Question
The butterfly effect
What about the brownian mouvement ?
Question
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.
Dear Cenap
Thank you very much for your kind suggestion. We will definitely try to work on this space.
Regards
Set theory
Question
What is set theory, and where is it applicable?