This tag is for questions seeking external references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

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9
votes
4answers
6k views

A good book on Statistical Inference?

Anyone can suggest me one or more good books on Statistical Inference (estimators, UMVU estimators, hypotesis testing, UMP test, interval estimators, ANOVA one-way and two-way...) based on rigorous ...
3
votes
1answer
138 views

Looking for an article on general principles of discrete mathematics

In his article 2 cultures Timothy Gowers states that the structure in combinatorics is there in the form of somewhat vague general statements that allow proofs to be condensed in the mind, and ...
0
votes
1answer
98 views

abide law — how to say and generalization

Suppose that some algebraic operations $+$ and $\oplus$ satisfy the abide law, i.e. $(a_0+a_1)\oplus(b_0+b_1)=(a_0\oplus b_0)+(a_1\oplus b_1)$. How should I say this, “$+$ abides by $\oplus$” or ...
3
votes
1answer
197 views

Anyone knows the name of the Hungarian undergraduate Math Seminars book?

I had this book bookmarked somewhere and now I have lost it somewhere. The only faint description I can remember was something about a culture of Seminars in Hungary and it was decided that a series ...
8
votes
2answers
878 views

Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function

To everyone: don't bother writing up another answer, i'm giving this bounty Antonio's answer. It just doesn't let me yet (24 hours delay). If you map the nth roots of unity $z$ with the function ...
8
votes
6answers
6k views

A Math function that draws water droplet shape?

I just need a quick reference. What is the function for this kind of shape? Thanks.
6
votes
1answer
2k views

Basic facts about ultrafilters and convergence of a sequence along an ultrafilter

Could you help, please. I need the information about the ultrafilters, namely, any ideas how one can see that they exist and a proof of the fact that for any ultrafilter every sequence on a compact ...
16
votes
4answers
940 views

Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$

I was reviewing some matrices and found this interesting if $r = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$ then $rr=-I$, also $$\exp{(\theta r)} = \cos\theta I + \sin\theta r$$ No wonder, the ...
1
vote
1answer
399 views

Number of operations in grade school algorithm for multiplying 577 and 423

Given 577 x 423 with grade school algorithm you calculate 577 x 3 = 1731 577 x 2 = 1154 577 x 4 = 2308 These are 3 multiplications of a number by a single digit. Then, you go on and add 1731 + ...
1
vote
1answer
163 views

Appropriate book for propositional logic

I am not looking for a good book but an appropriate book that is suitable for my logic course. Currently the professor only offers lectures. (Not sure why, perhaps there is no universal approach to ...
1
vote
2answers
632 views

Induction from normal subgroups

Let $G$ be a (finite) group and $N$ a normal subgroup. Given an irreducible representation $\pi$, how can I decompose $Ind_N^G \pi$? I'd be happy also about a good reference for this.
6
votes
1answer
237 views

Triangle from lengths of angle bisectors

According to http://www.cut-the-knot.org/triangle/TriangleFromBisectors.shtml it is impossible to construct a triangle from the lengths of its angle bisectors. Is there a more comprehensive account of ...
4
votes
2answers
347 views

Models of hyperbolic geometry

Wikipedia states the following: [The Poincaré half-plane model of hyperbolic geometry] is named after Henri Poincaré, but originated with Eugenio Beltrami, who used it, along with the Klein model ...
12
votes
2answers
707 views

Looking for an André Weil excerpt

I just wasted the last hour on google looking in vain for an excerpt of Weil's writings describing the process of discovering mathematics. I believe he once beautifully described the feeling of loss ...
1
vote
1answer
168 views

Reference to a list of affine Cartan matrices

I can find a list of affine Dynkin diagrams in some books but cannot find a list of affine Cartan matrices. We can write down affine Cartan matrices using affine Dynkin diagrams. But are there a list ...
0
votes
2answers
174 views

Ask for references on the comparison of $|A\circ B|$ and $|A|\circ| B|$

Let $A,B$ be complex matrices of the same size. I am looking for some references on the comparison of $|A\circ B|$ and $|A|\circ| B|$, where $|A|=(A^*A)^{1/2}$, "$\circ$" stands for Hadamard product. ...
2
votes
4answers
490 views

Self contained reference for norm and trace

I have learnt about the concepts of norm and trace of an element with respect to a finite extension, say $L/K$, of fields in terms of the determinant and trace (resp.) of the corresponding scalar ...
20
votes
7answers
2k views

Definition of definition

I was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topic. ...
4
votes
0answers
381 views

Understanding orientability of vector bundles

I'm having trouble understanding how orientability of vector bundles work. The book I'm reading, Spivak's A comprehensive introduction to differential geometry, is not very clear on this. Edit: ...
2
votes
1answer
106 views

Arrangements of congruent rectangles

I have stumbled on an interesting problem. How many congruent rectangles on a plane can be arranged in such a way that they all touch each other but never overlap? I pondered this problem for a few ...
6
votes
1answer
649 views

Understanding of exterior algebra

Consider the following definition from Loring W. Tu's An Introduction to Manifolds: For a finite-dimensional vector space $V$, say of dimension $n$, define ...
6
votes
0answers
403 views

Reference for the range of possible values in Hahn-Banach Theorem

This is the usual formulation of Hahn-Banach theorem (some books use sublinear function instead, but it probably does not make much difference): Let $X$ be a vector space and let $p:X\to{\mathbb R}$ ...
2
votes
2answers
203 views

Direct sum of an algebra and its opposite

I hate to do this, but I cannot seem to remember/find a particular result that I thought was true. Forgive me if I have some points wrong, since this is the point of my asking. I thought I remembered ...
0
votes
1answer
183 views

Some book to learn ODE, theory and computations

I'm not so good with computations, does someone know some book that has good computational tutorials and the basic theory of any introductory course?
4
votes
1answer
56 views

Stricter permutation patterns

A lot of work has been done on patterns in permutations, where a permutation is said to match a given pattern if it contains a subsequence of elements ordered according to the pattern (e.g., $\pi=(2\ ...
10
votes
1answer
1k views

Reference book for Artin-Schreier Theory

The aim of the question is very simple, I would like to study Artin-Schreier Theory, but I have had embarassing difficulties in finding a book which could help me in doing that. In specific I'm ...
27
votes
1answer
661 views

A Universal Property Defining Connected Sums

I once read (I believe in Ravi Vakil's notes on Algebraic Geometry) that the connected sum of a pair of surfaces can be defined in terms of a universal property. This gives a slick proof that the ...
4
votes
3answers
784 views

Article or book about the history of spherical geometry?

I teach a course on non-Euclidean geometry to high schoolers. I'm looking for an article or book that gives a thorough and interesting history of spherical geometry and trigonometry. I'm looking for ...
8
votes
1answer
439 views

PDE - Feynman-Kac vs. finite difference methods

I've heard that in greater than three dimensions, it's more efficient to solve a second-order parabolic PDE using a Monte-Carlo method based on the Feynman-Kac formula that it is to use finite ...
3
votes
4answers
418 views

Generalization $\zeta_\varphi(s)=\sum_{k=0}^\infty {\exp(I\varphi*k) \over (1+k)^s} $

This is more a reference-request for some fiddling/exploration with the $\zeta$-function. In expressing the $\zeta$ and the alternating $\zeta$ (="$\eta$") in terms of matrixoperations I asked myself, ...
4
votes
2answers
206 views

“Change-of-base” between enriched categories

I would like to prove that a monoidal functor $$\Phi\colon \mathbf{V}\to \mathbf{V'}$$ induces a functor $$\Phi^\#\colon \mathbf{V}\text{-Cat}\to \mathbf{V'}\text{-Cat}$$ and in particular I ...
5
votes
1answer
760 views

The only two rational values for cosine and their connection to the Kummer Rings

I am trying to learn about Kummer Rings, and in particular what makes $n=3,4,6$ so special. (That is the Gaussian and Eisenstein integers) The only $\theta\in [0,\frac{\pi}{2}]$ which are rational ...
18
votes
14answers
11k views

Requesting abstract algebra book recommendations [closed]

I've taken up self-study of math. (How smart can that be?) I've just about finished a course in real analysis which spent a lot of time on metric spaces and some time revisiting calculus. I was ...
5
votes
1answer
327 views

Fiber of jet bundle of a fiber bundle

Given a fiber bundle $p:E\to B$ with fiber $V$ and structure group $G$, one can define the corresponding $k$-jet bundle $E^k\subset J^k(B,E)$ of jets of local sections of $E$. On Wikipedia there is a ...
3
votes
1answer
208 views

$ \cos(\hat{A})BC+ A\cos(\hat{B})C+ AB\cos(\hat{C})=\frac {A^2 + B^2 + C^2}{2} $

What more can be said about the identity derived from law of cosines (motivation below)$$ \cos(\widehat{A})BC+ A\cos(\widehat{B})C+ AB\cos(\widehat{C})+=\frac {A^2 + B^2 + C^2}{2} \tag{IV}$$ RHS ...
8
votes
1answer
539 views

Summation formula name

What is the name of the following summation formula? $$\sum_{k = 1}^n f(k)) = \int_1^{n + 1} f - \frac{f(n + 1) + f(0)}2 + \int_1^{n + 1} f'w,$$ where $w$ is the “sawtooth” function, defined by ...
16
votes
2answers
913 views

$\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$

Let $A$ be an integral domain of finite Krull dimension. Let $\mathfrak{p}$ be a prime ideal. Is it true that $$\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$$ where $\dim$ ...
45
votes
6answers
10k views

The Langlands program for beginners

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its ...
17
votes
3answers
1k views

Are the fractional parts of $\log \log n!$ equidistributed or dense in $[0,1]$?

Are there any results relevant to the distribution of the sequence $\{\log \log n!\}$ for integers $n$, where $\{x\}$ denotes the fractional part of $x$? For instance, it is known that for irrational ...
6
votes
1answer
352 views

Capacity theory beginner resources

I'm currently studying a book on shape optimization: Variation et optimisation de formes: Une analyse géométrique By Antoine Henrot, Michel Pierre. The book introduces at some point capacity, and uses ...
4
votes
2answers
148 views

Article or book explaining rigorously facts about the mapping class group

I would like to know more about relationships between the mapping class group of an orientable surface with negative Euler's characteristic and moduli spaces. In particular, I would like to have a ...
6
votes
2answers
1k views

Homology of the loop space

Let $X$ be a nice space (manifold, CW-complex, what you prefer). I was wondering if there is a computable relation between the homology of $\Omega X$, the loop space of $X$, and the homology of $X$. I ...
6
votes
1answer
2k views

Any great *Introductory* books for Finite (Element/Difference) Methods

I sort of have two questions on the same subject so I just thought to ask both in one thread. I am looking to do some research on Finite Element Methods (FEM) and I am recently started looking into ...
0
votes
1answer
107 views

(pre)sheaves on path connected neigh

Presheaves are defined by using neighborhoods of a point. Is there a way to restrict this construction to path connected neighborhoods of points? What is the name of the object which assigns other ...
4
votes
3answers
1k views

What is “entire finite complex plane"?

The question is from the following problem: If $f(z)$ is an analytic function that maps the entire finite complex plane into the real axis, then the imaginary axis must be mapped onto A. the ...
2
votes
0answers
51 views

Reduction for the Maximum Subforest Problem

I'm trying to find (without luck) the reduction for the Maximum Subforest Problem. INSTANCE: Tree $G=\left(V,E\right)$ and a set of trees H. SOLUTION: A subset $E'\subseteq E$ such that ...
2
votes
3answers
179 views

Are magic squares inevitable?

Consider: If we are given a reasonably well-behaved statistical population of real numbers, then samples that are not small will have mean approximately equal to that of the population, right? Then: ...
12
votes
2answers
722 views

Closed-form Expression of the Partition Function $p(n)$

I feel like I have seen news that a paper was recently published, at most a few months ago, that solved the well-known problem of finding a closed-form expression for the partition function $p(n)$ ...
3
votes
1answer
152 views

What is the reference for the General Chernoff inequality in Wikipedia?

Where is the reference mentioned in the Wikipedia article for the General Chernoff inequality? I cannot find it.
16
votes
4answers
589 views

Banach spaces over fields other than $\mathbb{C}$?

Sorry, this is a rather vague question. I was just wondering if there is any kind of theory about normed (if possible Banach) spaces over fields other than the real or complex numbers. I'm guessing ...