New answers tagged

2

It seems to be The Art of Combinatorics, Volume IV: Arrangements and Methods by Douglas B. West; see here: http://www.math.illinois.edu/~dwest/. “Four advanced graduate textbooks and research references on classical and modern combinatorics. Preliminary versions available by special arrangement for use in specialized graduate courses; not available for ...


1

Following @JJacquelin, by variation of the constant, the equation becomes $$\sin(x)f''(x)+2\cos(x)f'(x)-\sin(x)f(x)+\sin(x)f(x)=\tan(x).$$ Then with $g(x)=f'(x)$, $$\sin(x)g'(x)+2\cos(x)g(x)=\tan(x).$$ We multiply by $\sin(x)$ to get an exact differential on the left $$\sin(x)^2g'(x)+2\sin(x)\cos(x)g(x)=(\sin^2(x)g(x))'=\sin(x)\tan(x).$$ Now we ...


1

$$\frac{d^2y}{dx^2} +y=\tan(x)$$ The first task is to solve the related homogeneous equation : $$\frac{d^2Y}{dx^2} +Y=0$$ I suppose that you can solve it. The second task is to find one particular solution of $\frac{d^2y_p}{dx^2} +y_p=\tan(x)$. The index $p$ means that we look for only one function (any one) among the infinity of functions $y$ which are ...


0

The online notes on differential geometry by Balazs Csikos are really quite good for self-study: http://www.cs.elte.hu/geometry/csikos/ under Lecture Notes, then under BSM Lecture Notes. Assumed prerequisites are multivariate calculus, linear algebra, and just a little topology.


0

It can be computed by using the complex error function (aka the Faddeeva function): $$1-{\rm{erf}}(z)=e^{-z^2}w(iz)$$ Matlab and C packages for the Faddeeva function are available in the Matlab Central.


0

The topics that you want to study use mostly the very essential ideas from Linear Algebra. Yes, over $\mathbb{R}$ and $\mathbb{C}$ is all you need. Since you seem to be a theoretically minded person with interest in geometry related subjects, I would recommend Gelfand's "Lectures on Linear Algebra". Strang's textbook is excellent, but probably not the style ...


1

The paper is: Bronisław Knaster (1893-1980) and Kazimierz [Casimir] Kuratowski (1896-1980), Sur quelques propriétés topologiques des fonctions dérivées [On some topological properties of derivative functions], Rendiconti del Circolo Matematico di Palermo (1) 49 (1925), 382-386. JFM 51.0208.01 I mentioned it in my answer to Examples of dense sets in the ...


1

It may fall into the more "undergraduate mathematics" but the Lotka–Volterra equations are a really good topic for something like this.


1

As David C. Ullrich and Jason pointed out, the answer is not always affirmative. Let me quote a Theorem that gives a partial affirmative answer to your question, with somewhat different assumptions. Theorem: Assume that $f$ satisfies the Fourier inversion formula, i.e. $f(x) = \int_{\mathbb{R}}\hat{f}(\xi)e^{2\pi x\xi}\,d\xi$ and that $$|\hat{f}(\xi)| ...


2

What you are asking for is the theory of hyperbolic 2-dimensional orbifolds. It's a very big theory. If you do not add additional hypotheses, it becomes somewhat unreasonable to expect a good description of the theory. Even if you add enough hypotheses to tame the question, it still requires a lot of mathematics to even describe the classification. Let me ...


7

We indeed have a local extension, given by the Taylor expansion. That is, given $x_0\in\mathbb R$ there exists $\epsilon>0$ such that $$\sum_{n=0}^\infty\frac{f^{(n)}(x_0)}{n!}(z-x_0)^n$$ converges for $|z-x_0|<\epsilon$, and we call the limit $g(z)$. This defines an analytic function $g$ on a neighborhood of $\mathbb R$ which clearly extends $f$. (We ...


12

Yes, a real analytic function on $\Bbb R$ extends locally to a complex analytic function, except that (in my opinion) "locally" doesn't/shouldn't mean what you say it does. If $f$ is real analytic on $\Bbb R$ then there exists an open set $\Omega\subset\Bbb C$ with $\Bbb R\subset\Omega$, such that $f$ extends to a function complex-analytic in $\Omega$. This ...


0

Read Frenkel's Langlands Coorespondence for Loop Groups, https://math.berkeley.edu/~frenkel/loop.pdf Standard college algebra is the only prerequisite.


2

Many possibilities.. Malthusian population model, Prey predator models, pursuit models, dynamic systems, or Black-Scholes model etc.


0

https://www.researchgate.net/publication/251217821_General_Theory_of_Embeddings But you have to have an account to see it


2

usually graduate students study PDE (Partial Differential Equations) rather than ODE. You may want to try doing a project on Sturm–Liouville theory. https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory It is a slightly advanced part of ODE that may be suitable for graduate students. If that is too hard, you can try Bernoulli differential equation: ...


1

I think one reason Complex Analysis is so nice is because being holomorphic/analytic is an extremely strong condition. As opposed to real analysis, differentiability is a rather weak condition, so we have functions that are differentiable once but not twice etc. Real analysis is full of nasty counterexamples like the Weierstrass function which is continuous ...


1

Well, any triangle-free regular graph has constant link. Or take the line graph of such a graph.


1

For each pair of disjoint open intervals $U,V\subset\mathbb{R}$ with rational endpoints, $f^{-1}(U)$ and $V$ are both $F_\sigma$, so $f^{-1}(U)\times V$ is an $F_\sigma$ subset of $\mathbb{R}^2$. Furthermore, since $U$ and $V$ are disjoint, $f^{-1}(U)\times V$ is disjoint from the graph of $f$. I claim that $\bigcup_{(U,V)}f^{-1}(U)\times V$ is equal to ...


1

For the model defined in the following way: Fix some graph $G$. For any $p \in [0,1]$, each vertex $v$ in $G$ is in the random induced subgraph with probability $p$ independent of other vertices. This model is an instance of site percolation. Two popular introductory textbooks to Percolation Theory are Percolation by Bela Bollobás and Oliver Riordan and ...


2

This is a general fact about operads. Let $\mathtt{P}$ be any operad (e.g. the pre-Lie operad) in a symmetric monoidal category $\mathsf{C}$. If I understand your question correctly, you have two definitions of "free algebra" you want to compare: The forgetful functor $U : \mathtt{P}\text{-alg} \to \mathsf{C}$ has a left adjoint $F_\mathtt{P}$, and a "free ...


-2

I would recommend one of two books for you, Proofs and Fundamentals by Ethan D. Bloch Topology by James B. Munkres The first book, Proofs and Fundamentals, is an extremely quick read, and it is very informative. I read the core chapters (1-6) over a winter break. The second book is a reference book for the fundamentals of point set topology. It contains ...


2

I would not limit yourself just to those universities if you want good analysis lectures. I would highly recommend looking into Francis Su's lectures from Harvey Mudd College. His is a one-semester course and he ends roughly when they get into single-variable differentiation. A first glance at MIT's Open Courseware doesn't show any video lectures in ...


2

Chapter 1 of Stein and Shakarchi's Real Analysis textbook has a similar result, and the proof includes a helpful picture. But I think it's worth taking the time to understand Tao's proof. Dyadic cubes are quite useful.


1

This is a one-liner in Sage (try the online calculator): if you type Gamma(5).generators() then you get the response [ [1 5] [-24 5] [-109 40] [11 -5] [-39 25] [ 6 -5] [-64 105] [0 1], [ -5 1], [ -30 11], [20 -9], [-25 16], [ 5 -4], [-25 41], [ 11 -20] [-89 235] [ 16 -45] [ 21 -80] [ 5 -9], [-25 66], [ 5 -14], [ 5 -19] ] (This ...


1

Try the book Galois Theory for Beginners: A Historical Perspective by Bewersdorff. Read an MAA review here.


2

Copyright is very rarely an issue in mathematics, as long as proper attribution is given. It would be absurd to try to copyright a theorem, insofar as someone else could rediscover it for themselves, and often someone else does. If you're duplicating large portions of a published textbook to the open Internet, that could in principle be a copyright ...


0

I hope I understand the exact question. The version you have provided is "after October 1, 2001". On the very same site, you can find a version with copyright in 1990, 1991, 1993 a version with copyright in 1986


0

This seems to wrong without further assumptions, indeed: Fix any $L > 0$, aned let $X_1,\dots, X_n$ be (almost surely) constant r.v.'s taking value $L$ with probability $1$. Then, for this very value of $L$, and any integer $k > 1$, $$ \mathbb{E}[\lvert X_i\rvert^k] = L^k \leq \frac{k!}{2}L^k= \frac{1}{2} \mathbb{E}[ X_i^2] L^{k-2} k!. $$ Yet, we ...


0

In the second sequence above, just take any $\mathcal{O}_{\mathbb{P}^n}(-1)$ in the middle and its image in $\mathcal{O}_{\mathbb{P}^n}$ and take the quotient to get the first exact sequence.


0

I'm not sure how helpful it will be for your specific purposes, but I found Brualdi to be a good introductory text


1

OK, I now have most of an answer, which I will post here. That said... any references would really be appreciated. I am still looking for references on the matter; I don't want to have to rederive this theory myself. Anyway, yes, it seems to be an alternate characterization of a $\lambda$-ring, via the usual way of converting between $e_n$ (elementary ...


0

I do not think there is a commonly agreed on definition. Here is one, based on the discussion in William Thurston, Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. A domain $\Omega$ in $R^n$ has piecewise smooth boundary if for each $x\in cl(\Omega)$ ...


-2

I got stuck on the definition of piecewise smooth boundary while reviewing material for a calculus course I am taking. I can't intuitively see the what the definition means but it happens to solve your question. The definition is: if the boundary of a set is a finite union of piecewise, simple closed curves, then the set has a piecewise smooth boundary and ...


0

This book will do: Stochastic Geometry and Its Applications, 3rd Edition Sung Nok Chiu, Dietrich Stoyan, Wilfrid S. Kendall, Joseph Mecke


1

Yes. We can recover the category $\mathcal{T}$ from the concrete category $(C,U)$, where $C = \text{Alg}(\mathcal{T})$ and $U\colon C\to \text{Set}$ is the forgetful functor. The point is that the $n$-ary operations (arrows $X^n\to X$ in $\mathcal{T}$) correspond exactly to the natural transformations $U^n\to U$, i.e. $\mathcal{T}$ is the full subcategory of ...


0

I start the thread with an observation: The Probabilistic Method book by Noga Alon has a treament of random graphs on page 155 where they calculate the expected value and the threshold function for graph-theoretic property such as connectedness


1

I think you can try to follow the OCW course: http://ocw.mit.edu/courses/mathematics/18-466-mathematical-statistics-spring-2003/syllabus/ You are certainly right that most introductory statistics books do not use measure theory rigorously, and it is a bit annoying if you knew it already. But if you go through the proofs in the above notes, you will find ...


0

The book, An Introduction to Inverse Problems with Applications, mentioned in Francisco Moura Neto's answer certainly appears both applied and gentle as an introduction. The main prerequisite seems to be linear algebra, but some exposure to multivariable calculus, numerical methods and differential equations would be valuable too. Another option is the ...


4

By the Cauchy-Schwarz inequality, \begin{align*} 1 &= \left(\sum_{k = 1}^n x_k\right)^2 \\ &= \left(\sum_{k = 1}^n \frac {x_k} {\sqrt{y_k}} \sqrt{y_k}\right)^2\\ &\le \left(\sum_{k = 1}^n \frac{x_k^2}{y_k}\right) \cdot \left(\sum_{k = 1}^n y_k\right) \\ \end{align*} from which the result follows. The fact that a) the terms can naturally be ...


0

http://www.maa.org/press/maa-reviews/mathematics-its-content-methods-and-meaning This one should be ok, I've read it and I didn't find any.


0

Two sources come to mind: first, Knopp's Theory and application of infinite series like user247608 suggested (very cheap), and Pete Clark's free notes which can be found here: http://math.uga.edu/~pete/3100supp.pdf


1

Here is a neat proof from Qiaochu Yuan's answer to this question: If $L$ is diagonalizable with eigenvalues $\lambda_1, \dots \lambda_n$, then it's clear that $(L - \lambda_1) \dots (L - \lambda_n) = 0$, which is the Cayley-Hamilton theorem for $L$. But the Cayley-Hamilton theorem is a "continuous" fact: for an $n \times n$ matrix it asserts that $n^2$ ...


3

My favorite : let $k$ be your ground field, and let $A = k[X_{ij}]_{1\leqslant i,j\leqslant n}$ be the ring of polynomials in $n^2$ indeterminates over $k$, and $K = Frac(A)$. Then put $M = (X_{ij})_{ij}\in M_n(A)$ the "generic matrix". For any $N=(a_{ij})_{ij}\in M_n(k)$, there is a unique $k$-algebra morphism $\varphi_N:A\to k$ defined by ...


-4

Your question is how to define $$\lim_{x\to x_0} |f-g|=0$$ for two continuous functions $f,g: X\to Y$, where $X,Y$ are general topological spaces. If $X,Y$ are metric spaces and we define a metric $d_y$ on $Y$ and a metric $d_x$ on $X$. Then we can formalize the notion $$\lim_{x\to x_0} |f-g|=0$$ as below: For any prescribed $\epsilon>0$ there is ...


3

Here is the construction of a (co)homology theory for lattices which yields the Chech (co)homology in the case of topological spaces when applied to the lattice of open subsets. Recall that the Chech theory works with open coverings. An analogue of an open covering in a lattice $L$ is a subset $C$ of $L$ satisfying the property that the join of the elements ...


0

I haven't read much of it yet, but here's the table of contents: Preliminaries Sets and set operations The real numbers as a field The order axioms Absolute values Quantifiers Logical connectives Negation of quantified statements The principle of finite induction A deeper look at induction Analytic Geometry of Straight Lines and Curves A synopsis of ...


1

For free Lie Lie algebras see here; a survey on free pre-Lie algebras (also called free left-symmetric algebras) is given in the article monomial bases of free pre-Lie algebras, which has many references, e.g. the book of Reutenauer on free Lie algebras.


1

Not a book exactly but free if you have access: Y.T. Cheng’s undergraduate thesis on continued fractions.



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