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1

You can try checking out Problems in Real Analysis: Advanced Calculus on the Real Axis.


0

I think the best place to look is the book "Zeta Regularization Techniques with Applications" by Eliazade et. al. There are some other books by Eliazade, which I think you might find useful. The calculation you want is fairly straightforward. If it will help, I will sketch how to do it later.


0

Given the unexplanatory tone of many number theory expositions, it is, in fact, reasonable to ask this question, I think... insofar as there is a good explanation, apart from the artifactual one. That is, the Riemann Explicit Formula, as reformulated a bit by von Mangoldt, is an exact (not merely asymptotic) equality of a sum over primes and a sum over ...


0

In order to understand the applications of geometry to physics it is first of all necessary have a solid knowledge of geometry and physics! This means that you have to know quite well the subject and the tools you're planning to use. What does it mean in practice, it dependes on the level of formalism required. For example, Newtonian mechanics is ...


0

The following answer to my question was possible thanks to the links which Mike Miller gave. On page 630 of Dickson's "History of the Theory of Numbers", the following is written: "E. Lucas listed and treated the solvable equations $ax^4+by^4=cz^2$" in which $2$ and $3$ are the only primes dividing $a,b$, or $c$, viz." (he then lists out a bunch of values ...


1

This paper by Darmon and Merel, which proves the general case $x^n + y^n = z^3$ has no primitive solutions, cites Dickson, "History of the theory of numbers", page 630. For a good resource on the state of the art on the generalized Fermat problem (well, as of 2005), see Poonen, Schaefer, Stoll, "Twists of $X(7)$ and primitive solutions to $x^2+y^3=z^7$". ...


0

You can also read these books. I think you will be fascinated with these books. You can try: $1)$ Mathematical Thought From Ancient To Modern Times vol 3 $2)$Mathematical Thought From Ancient To Modern Times vol 1 $3)$Mathematical Thought From Ancient To Modern Times vol 2 $4)$Paul J. Nahin - An Imaginary Tale The Story of i the Square Root of Minus One ...


1

I assume you haven't been exposed to much trigonometry yet. There you will see the concept of 0 angles, and negative angles, and angles with sizes above $360^\circ$. From a strictly geometric viewpoint, these are not necessary. But from an analytic viewpoint, they are essential. For example, consider 2 planets moving in a perfectly circular orbits about ...


2

Roughly: calculus, multivariable calculus (including differential forms, at the level of, say, Spivak's Calculus on Manifolds, althought that's not the best book to learn from), a strong background in linear algebra, and some multilinear algebra (at least comparable to that in Spivak's Calculus on Manifolds perhaps a bit of abstract algebra, so that you ...


0

Of course, no deterministic algorithm can generate truly random numbers. The goal is to find an algorithm that gives output values that 'behave as if' they are independent and identically distributed according to some distribution--usually $Unif(0, 1).$ By 'behave as if' one means that the pseudo-random numbers pass a large battery of benchmark tests for ...


5

All of them. Define $b_1 = a_1$, $c_1 = 0$ and $$b_{n + 1} = \begin{cases}b_n & a_{n + 1} \le a_{n} \\ b_n + a_{n + 1} - a_n & a_{n + 1} > a_n\end{cases}$$ $$c_{n + 1} = \begin{cases}c_n + a_n - a_{n + 1} & a_{n + 1} \le a_{n} \\ c_n & a_{n + 1} > a_n\end{cases}$$ for $n > 0$. It is obvious that $b_n$ and $c_n$ are increasing and ...


0

Your $P_m(x)$ is exactly the $C_n(x)$ that appears in H.C. Williams' book "Edouard Lucas and Primality Testing" page 77 and 78. But $C_n(x)$ is defined in a much simpler way: $C_0(x)=2$, $C_1(x)=x$, $C_{i+1}(x)= xC_i(x) - C_{i-1}(x)$. HCW gives also a primality test theorem dealing with $N=A*3^n \pm 1$ , page 273, Theorem 11.3.1, and he makes use of: ...


0

Your algorithm finds the following first n: 4, 10, 17, 50, 170, 184, 194, 209, 641 . I do not have his book with me now, but I think that $x^3-3x$ does appear in Hugh C. Williams' book about the work of "Édouard Lucas", in a chapter dedicated to Chebitchef. It is also possible (but I should have to read again Lucas' books and papers) that Lucas did use such ...


0

Related question: You can see darij grinberg's answer for 1). The answer for 2) is negative. For example, you can see here while the K-dual of every coalgebra is an algebra, the K-dual of an infinite-dimensional algebra need not be a coalgebra. ON THE APPLICATIONS OF COALGEBRAS TO GROUP ALGEBRAS Unfortunately, the dual of an infinite dimensional ...


0

Visual Complex Analysis by Tristan Needham is genuinely one of the best books for visualizing algebra/complex functions that I've come across. http://www.amazon.com/Visual-Complex-Analysis-Tristan-Needham/dp/0198534469


0

I think this is his original proof. You can find it in this translated version of Gauss' book, starts at p. 82, article 125. I haven't read it. Just trying to help.


1

After asking this question on mathoverflow (the corresponding question is deleted now) it was pointed out in a comment that the proof is really simple. Proof. Let $p\in\text{rowspace}(A)$ and $s\in\ker(A)$ be two vectors realizing $\pi$ and $\sigma$. Since $$p\cdot s = \sum_i p_is_i = 0,$$ whenever there is a positive summand, there must also be a negative ...


1

According to this study by Jean-Philippe Villeneuve (in French): http://culturemath.ens.fr/histoire%20des%20maths/htm/villeneuve2009/theories-de-la-mesure.html the essential property of sigma-additivity that is used to define sigma-algebra was first used both by Émile Borel in 1898 for his work on the theory of functions in which there is a chapter on the ...


2

Module $2$ syllabus can be found in almost all books on Algebra but I recommend Dummit and Foote. For Module $1$, I would recommend "Linear Algebra done right" by Axler for abstract approach avoiding Matrices, and "Linear algebra done wrong " by Sergie Treil (Google it for e-copy). These two books covers all topics espically Sergie's book, but Axler is ...


1

Take a peek at some no-cost alternatives around the 'net, like Treil's "Linear Algebra done Wrong" (very nice, but might not be all the abstract you'd want). Often lecture notes are available, and add a different explanation that helps you over some rough spot.


2

http://www.amazon.com/gp/offer-listing/0936428066/?tag=wwwcampusboocom667-20&condition=used http://product.half.ebay.com/Between-Nilpotent-and-Solvable-by-Henry-G-Bray-John-F-Humphreys-David-Johnson-Paul-Venzke-and-W-E-Deskins-1982-Hardcover/717850&item=345068498941&tg=videtails These are some sites I found after searching, hope this helps.


-4

Basic Mathematics by Serge Lang for a start in Algebra, his Linear Algebra book was okay, look up other math books by Springer (publisher).


0

The easiest way to generate these formulas is to write everything in terms of eigenvalues. Then $$ \det(I+A) = \prod_{i=1}^n (1 + \lambda_i) = \sum_{i=0}^n e_i(\lambda_1,\dots,\lambda_n) ,$$ where the $e_i$ are the elementary symmetric polynomials described here: https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial Then you convert the elementary ...


1

You first prove that the right-shift operator $R$ is a linear operator on (doubly infinite) sequences. Then you define addition and scalar multiplication on operators so that you can given a sequence $f$ obeys a recurrence relation rewrite the recurrence relation as $(P(R))(f) = 0$ for some polynomial $P$, which is the characteristic polynomial. Now $P(R)$ ...


1

The term "logarithmic convex hull" is in use for this object, because it can be obtained by convexifying the image of a domain under the logarithm map, and then coming back. It comes up in complex analysis in several variables, due to the following fact: a domain $D\subset\mathbb{C}^n$ is a region of convergence of some power series (centered at $0$) if ...


1

Try books by Serge Lang, Spivak, and in that direction, I mean I'm throwing names out there because I have these books and the amount of sheer detail is staggering, this a start, branch out from here, locate common publishers and cross-check user reviews


0

http://www.purplemath.com/modules/index.htm http://www.sosmath.com/algebra/algebra.html Here you are as requested, hope it helps.


2

As pointed out in my earlier comment, we need to assume $s\in (0,1)$. Out of habit, I'm going using the convention $\mathbb{T}\cong[0,1)$. You can modified everything to your own convention. The expression $$[f]_{W^{2,s}(\mathbb{T})}=\left(\iint_{\mathbb{T}\times\mathbb{T}}\dfrac{\left|f(x)-f(y)\right|^{2}}{\left|x-y\right|^{1+2s}}dxdy\right)^{1/2}$$ ...


1

You need to tell us something more. Are you interested in very formal book with all the technicalities or rather something which mostly emphasises ideas and results. Is your background in pure mathematics or rather applications like physics etc. If you would choose second answer to both questions I strongly recommend Frankel, but it's actually a huge book. ...


1

Reading more mathematics is not going to help you understand more mathematics unless you read with pencil and paper next to you and try to fill the gaps in the proofs that you do not understand or find hard to follow. You don't get better at swimming by watching people swim, you must swim. To get more fluent in mathematics, you must do it as well.


2

Define $M(N)=\sum_{n\leq N}\mu(n)$. Then $M(N)\ll N^{1/2+\varepsilon}$ is equivalent to the Riemann Hypothesis (Aleksandar Ivic, The Riemann Zeta-Function, page 47). Define $S(N)=\sum_{n\leq N}\frac{\mu(n)}{n}$. We will prove $M(N)\ll N^{1/2+\varepsilon}$ if and only if $S(N)\ll N^{-1/2+\varepsilon}$. Proof: Suppose $M(N)\ll N^{1/2+\varepsilon}$. ...


0

http://tutorial.math.lamar.edu/Classes/Alg/Solving.aspx. Try this. It's a comprehensive website that explains a lot of maths. The link is the algebra page which deals with polynomials and such. You'll find the quadratic equations section in the list. Hope that helps I'm sorry I didn't read your question properly. Your proof is entirely correct, but it ...


2

From what I've been told, Brownian Motion and Stochastic Calculus by Karatzas and Shreve is the gold standard. Continuous Martingales and Brownian Motion by Revuz and Yor is also a great reference. What you have listed as background knowledge is sufficient.


0

I will say a good and extense explorer is http://metamath.org which in his start Metamath Proof Explorer Home Page also has a suggested reading in FAQ as a reference to A Primer for Logic and Proof by Holly P. Hirst and Jeffry L. Hirst.


0

A free online book A First Course in Linear Algebra by Robert A. Beezer


1

I am adding the answer I received by email from Doctor Tadashi Tokieda, he is the director of studies in mathematics at Trinity Hall, University of Cambridge, some bio here and here, I was following his Topology and Geometry open lectures at Youtube for the AIMS, so I dared to send him an email yesterday (I did not expect an answer, just tried) and received ...


6

If a function $m$ satisfies your condition, then $m(x) = 1$ for all $x \ne 0$. Just observe the following inequality: $$2^{m(x)}||x|| = ||2x|| = ||x + x|| \le ||x|| + ||x|| = 2||x||$$


0

Invariant Operators on Conformal Manifolds by Jan Slovák is available online: http://www.math.muni.cz/~slovak/ftp/papers/vienna.ps


1

UPDATE 2: I managed to prove rigorously that the conjecture I had formulated in the first update holds: The expressions for $\pi_j$ in both formulae are off by a factor of $t$. I also figured out the most plausible reason for this mistake. Consider the partition of the state space $\{S_0,\ldots, S_{t-1}\}$ mentioned above and let $q_{ij}\equiv p_{ij}^{(t)}$ ...


2

I would go with ACoPS because this book teaches you how to solve hard problems, and gives you ideas and techniques that are quite new that you can use them again or refine them so you can solve a broader class of questions.


0

I actually study analysis for my college using baby Rudin as a first reference book, but I've chosen another book as supplement e.g. now is watching S. Berberian' s book (the pdf-version) as a good support [http://www.amazon.com/Fundamentals-Analysis-Universitext-Sterling-Berberian/dp/0387984801]. I just can not find solutions...


0

The second and the third statement are basic facts in Sobolev Space theory. The first one is probably a corollary of the other two, but I think it can be proved directly. Anyway, Michel Willem's book on Functional Analysis, Birkhauser-Verlag, contains the proofs of these facts. You can also read Haim Brezis's book on functional analysis (Springer-Verlag) for ...


4

Gödel did epoch-making work in a number of fields: In pure logic, he was the first to prove the completeness of a system of the predicate calculus. In what we might call the proof theory of formal systems, he proved the incompleteness [different sense!] of any formal system strong enough to encode a certain amount of arithmetic. (This required developing ...


1

Try the book Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.


1

I found this exposition of the Smallest Eigenvalues of a Graph Laplacian by Shriphani Palakodety to be readable and informative. The article begins with a discussion of eigenvectors for the smallest eigenvalue, which in the case of the graph Laplacian happens to be zero. The number of eigenvectors for this eigenvalue gives the connected components of the ...


2

Unlike integration, differentiation is a very unstable operation. It is very hard to make assumptions on $\{f_n\}_n$ so that $\{f'_n\}_n$ converges. For instance, let $f_n(x)= \frac{\sin (nx)}{n}$: $\{f_n\}_n$ converges to zero uniformly, but the derivatives $f'_n$ are oscillating. The only "elementary" theorem about differentiation of sequences of ...


3

We have $$\frac{a_n}{\pi/2} = \prod_{k = n+1}^\infty \frac{(2k-1)(2k+1)}{(2k)^2} = \prod_{k = n+1}^\infty \biggl(1 - \frac{1}{4k^2}\biggr).$$ To estimate products, it is often convenient to take logarithms. Here we can get the easy upper bound $$\log \prod_{k = n+1}^\infty \biggl(1 - \frac{1}{4k^2}\biggr) = \sum_{k = n+1}^\infty \log \biggl( 1 - ...


1

Yes, "your" statement was very popular in classic Italian texbooks. Now I do not have them on my desk, but I believe that your theorem appears more or less explicitly in Prodi's book "Analisi matematica" and in the old treatise by Luigi Amerio. In Rudin's book it is not stated because Rudin approaches every topic from a rather abstract point of view. Limits ...


-1

When we define a limit point or to explain that some point is a limit point under some specific condition(e.g.: a limit of a function at a particular limit point) we always go back to the definition of the limit point, that is back to the existence of an arbitrary sequence $(a_n)$ in the domain such that $a_n \neq c$ for all $n\in \mathbb{N}$ and ...


1

You could try a Monte Carlo approach. Basically, you can simulate a large number of strong solutions and then evaluate the sample mean and variance of the specific instant of interest. Depending on the structure of the diffusion coefficient, it is possible to perform exact simulation. In this case, no approximation error will be propagated to your ...



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