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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2
votes
1answer
38 views

Reference request for a theorem on maps to normal varieties with equidimensional fibers being open

I am requesting a reference for a proof.. I believe that it is due to Chevalley. A theorem by Chevalley says that if $f: X \rightarrow Y$ is a dominant morphism of irreducible varieties, then there is ...
-4
votes
1answer
39 views

Looking for a paper of Arhangel'Skii A V, Bella A. [closed]

I'm Looking for a paper of Arhangel'Skii A V, Bella A. Its title is this: The diagonal of a first countable paratopological group, submetrizability, and related results[J]. Applied General ...
1
vote
0answers
42 views

special kinds of homotopies

Let $X$ and $Y$ be two homotopy-equivalent topological spaces. That is, there exists maps $f:X \to Y$ and $g:Y \to X$ such that $g \circ f \simeq 1_X$ and $f \circ g \simeq 1_Y$. $1_X$ and $1_Y$ are ...
2
votes
0answers
38 views

Expected maximum degree Erdős–Rényi graph

Consider an Erdős–Rényi random graph $\mathrm{ER}(N,p)$, where $N$ is the number of nodes and $p$ the probability of placing an edge between each distinct pair of nodes. I'm interested in finding ...
2
votes
0answers
24 views

List of simple roots in the H-basis for various Lie algebras?

There are four usual bases one can use to express the roots and weights of a given algebra. The $\alpha$-basis, where we write the roots and weights in terms of the simple roots $\alpha_i$. The ...
1
vote
0answers
29 views

Where to find “non-standard” characteristic functions?

Well, the title says it all. I need the characteristic function of the (generalized) arcsine distribution. I desperately searched the internet for it but haven't found anything. Is there some standard ...
2
votes
3answers
64 views

Help finding specific book

I'm studying Engineering and I'm in my second year, studying Multivariable Calculus, but my University is kind of hard teaching me fresh calculus with topology and analysis, and is kind of hard, so I ...
0
votes
0answers
8 views

Logical systems and formal proof

Is there any good book dealing with various formal systems and a book for formal proofs. Or atleast some good notes. This page on wikipedia also says: 'This article needs attention from an expert in ...
1
vote
0answers
14 views

Would a class in Linear Optimization/Programming be useful for a CS degree?

Would a class in Linear Optimization/Programming be useful for a CS degree? if it is useful, how useful it is or what is it used for. Can someone please help me decide. Thanks in advance
1
vote
0answers
41 views

Expected distance of biased 1d random walk with 0 drift

There were many question about biased 1d random walks earlier, but as far as I can tell none of these are directly related. Let $p,q>0$ such that $p+q=1$. Let $X_0=0$ and $X_{i+1} = X_i+1$ with ...
1
vote
0answers
11 views

What's a good reference for the geometry of G/B and G/P?

Is there a reference that develops the geometry of G/B and G/P where G is a Lie group, B a Borel, P a parabolic? More specifically, I would like a reference that does the Geometry section of Fulton's ...
0
votes
0answers
11 views

Is there a good reference for the Borel-Weil Theorem?

I haven't been able to find a reference that proves the Borel-Weil Theorem carefully and completely, preferably in the language of sections of line bundles on G/B. Does anyone know of a good source?
1
vote
0answers
19 views

Ref. Request — Non-Transitive Lie Group Actions, Applications to Orbifolds/Groupoids

I'm working on a problem where I have a (highly) non-transitive Lie group action on a manifold, and I am trying to deduce the geometric structure of the quotient space. I've been looking at some ...
0
votes
2answers
61 views

Books on number theory

I had a very short introduction to number theory in one of the classes I took and I learned a bit about divisibility and congruence, not much further than Fermat's little theorem, and I would like to ...
2
votes
1answer
40 views

In dual basis, why the functions are of the form $\sum_{i=1}^{n}a_ix_i$?

My book says (Linear Algebra - Lipschutz): Let V vector space where $dim(V) = n$. Any functional $\phi$ of $V*$ has the representation $\phi(x_1, x_2, ..., x_n) = a_1x_1 + a_2x_2 + ... + a_nx_n$. Why? ...
0
votes
2answers
43 views

Good reference material for graph theory [duplicate]

Could anyone please suggest either a book or some reference material either online or as printed material for graph theory?
2
votes
0answers
22 views

Linear vs smooth actions of finite groups on spheres, Euclidean spaces and closed disks

I would like to know examples (with references, if possible) of the following: (1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ but admitting no effective linear action on ...
-1
votes
0answers
25 views

Prereq's for Stillwell Elements of Number Theory?

I just finished high school and am planning to introduce myself to abstract algebra using a historically motivated approach before starting it formally at university. I am particularly attracted to ...
0
votes
2answers
47 views

How to learn cryptography [duplicate]

I have just started learning Cryptography.I looked on the Wikipedia and found topics like "Public key Cryptosystem","Symmetric Cryptography" ,"Cryptanalysis" etc. Below this in the reference section ...
0
votes
0answers
18 views

Prerequisites for Stillwell's Elements of Algebra

I recently posted a thread asking about Cox's Galois Theory text but apparently it is too advancd for me at the moment so I did some searching and found this book by Stillwell. It seems to take a ...
2
votes
1answer
42 views

Is $B(H)$ the weak-$*$ closure of $K(H)$?

I am getting the following result: If $H$ is a Hilbert space, then the weak-$*$ closure of $K(H)$, the space of compact operators on $H$, is $B(H)$, the space of bounded operators on $H$. Is this ...
2
votes
1answer
38 views

Fibonacci and Lucas numbers congruence relation?

The wikipedia page for Lucas Numbers seems to suggest that if $F_n ≥ 5$ is a Fibonacci number then no Lucas number is divisible by $F_n$. Here is the link. However, the page does not give any ...
0
votes
1answer
33 views

If $L$ splits $D=I\otimes N$ does $L$ split $I$ and $N$?

Let $D$ be a central simple division algebra over the field $F$. Let $D\sim I\otimes N$ where $I$ and $N$ are division algebras in Br$(F)$ and $\sim$ is equivalence in Br$(F)$. I am interested in this ...
0
votes
1answer
54 views

Random permutation and isolated points on the line

Let $[n]=\{1,\dots,n\}$ be the (ordered) set of the $n$ first integers, and $\mathcal{S}_n$ denote the set of permutations of $[n]$. Let $1\leq k \leq \frac{n}{4}$ be an integer. If I draw uniformly ...
0
votes
1answer
29 views

Seeking the Recommendation on Complexity Theory books

S.E advisers, I am a rising college junior in US with a major in mathematics and an aspiring applied mathematician in the fields of theoretical computing. I just recently got a research project on ...
0
votes
1answer
36 views

Prerequisites for Cox's Galois Theory

I am planning to read Cox's book 'Galois theory' but I don't know any abstract algebra. In the preface which you can read here there is no indication of the assumed knowledge. Could someone please ...
1
vote
3answers
126 views

a good modern topology book

I want to study an advanced modern book on topology, but I couldn't find any. I've already studied the first chapters of Munkres' book, but it is not as advanced as books such as Engelking's topology, ...
0
votes
0answers
70 views

Reference request for well known theorem in combinatorics

From where, I can find the proof of the following theorem. I have to to cite it, in my research article. Theorem: The combination $ {n} C {r}$ is the number of possibilities for ...
0
votes
0answers
24 views

Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f(u) = 0, $$ $$ u|_\Gamma = u_0 $$ by Newton’s method when its convergence is global and monotonic. ...
3
votes
1answer
26 views

Statistics books with motivation and historical tidbits about the development of the concepts?

I've seen some questions asking books on statistics. I'm looking something a little different, I'm specifically looking for a book in statistics that teaches the important concepts well and also ...
0
votes
2answers
29 views

If $u$ is a Sobolev function then $\nabla u = 0$ on $\{ u = c\}$.

There is a result of the form: If $u$ is a Sobolev function on some domain then $\nabla u = 0$ on $\{ x \mid u(x) = c\}$ where $c$ is constant. Can someone point me to a specific reference? I ...
0
votes
1answer
42 views

Examples of how abstract algebra is used to find concrete solutions to a mathematical model?

All references I've seen to abstract algebra show how it helps in the representation of mathematical models...are there any examples of using abstract algebra to calculate actual solutions to a ...
0
votes
0answers
22 views

Looking for resources on these topics from Linear Algebra

I am looking for Characteristic roots and characteristic vectors of a linear transformation or of a matrix, Algebraic and Geometric multiplicity of a characteristic value, Cayley-Hamilton theorem, ...
0
votes
1answer
77 views

Questions on CW-complexes

I am trying to proof the following two statements. If $X_1 \subset \dots \subset X_i \subset \dots$ is a infinite sequence of CW-complexes, then $X = \bigcup X_i$ is a CW-complex and each $X_i$ is a ...
0
votes
1answer
11 views

Differentiability of Regulated Functions

In general, what can be said about differentiability of (real-valued) regulated functions, i.e. such for which the left and right limit exist at every point? Such functions are necessarily continuous ...
0
votes
0answers
28 views

Need recommendation for following topics in combinatorics

I have to do following topics for my exam .I have 2 months time .However i have never done any combinatorics except that of high school (Permutations ,Combinations etc ) .I want a book which covers ...
9
votes
1answer
162 views

Ramanujan's transformation formula connected with $r_{2}(n)$

Let $r_{2}(n)$ denote the number of ways in which a positive integer $n$ can be expressed as the sum of squares of two integers. Here the sign as well as order of summands matters. Also by convention ...
2
votes
2answers
59 views

Trying to understand the limit of regular polygons: circle vs apeirogon (vs infinigon?)

In the definition of regular polygon at the Wikipedia, there is this statement about the limit of a n-gon: "In the limit, a sequence of regular polygons with an increasing number of sides becomes ...
0
votes
0answers
20 views

References for microlocal versions of some theorems

I am trying to introduce myself into Microlocal Analysis. In particular, motivated by some results in Inverse Problems, I would like to find good references for the microlocal versions of Helgason and ...
4
votes
2answers
59 views

Order Properties on Open Sets

Considering the subset order on the open sets of a topological space, it seems natural to ask what kind of total orders exist as suborders of the subset order. One possibility is that each total order ...
0
votes
0answers
12 views

Sub gaussian concentration for Lipschitz functions

It is well know that: if $f:\mathbb{R}^m\to\mathbb{R}$ is a Lipschitz function with Lipschitz constant $L$, and $X_1,\dots X_m$ are i.i.d random variables s.t. $X_i\sim N(0,1)$, then for any $t>0$ ...
3
votes
2answers
124 views

Functional Analysis Question?

I have a question about functional analysis. I know in finite dimensional space $\mathbb{C}^n$, all bases have the same cardinality. However let us consider $L^{2}[-\pi,\pi]$ which has TWO bases ...
5
votes
0answers
67 views

Infinite Sums which turn out to be Riemann Integrals

I'm looking for examples of infinite series which look hard to evaluate at first, but become very simple when viewed as a Riemann integral. An example would be $$\frac{1}{n+1}+\frac{1}{n+2}+ \ldots ...
1
vote
0answers
54 views

Recommendations for a thorough logic textbook

I'm looking for a (possibly introductory) textbook on logic that covers the motivation behind conventions in logic, like the definition of the implication. Prof. J. Lau has an excellent webpage, ...
0
votes
0answers
9 views

A reference to study Boundary conditions of diffusion processes

I am trying to learn about Wentzell Boundary condition and (A,L) diffusion in the sense of Watanabe's paper (On the existence and uniqueness of diffusion processes with Wentzell's boundary condition ...
0
votes
1answer
19 views

An inductive limit of amenable groups is amenable

It is a Theorem that an inductive Limit of amenable Groups is amenable. Could someone sketch me a proof of this, or give me a reference? I couldn't find one. Thanks in advance. Edit: I wanted it for ...
6
votes
2answers
144 views

Cancellation problem: $R\not\cong S$ but $R[t]\cong S[t]$ (Danielewski surfaces)

I would like to understand why the two rings $$ R={\mathbb{C}[x,y,z]}/{(xy - (1 - z^2))} \\ S=\mathbb{C}[x,y,z]/{(x^2y - (1 - z^2))} $$ are not isomorphic, but $R[t]\cong S[t]$. This example is ...
5
votes
0answers
53 views

a new(?) operation using products of multiplicities

Does the operation $$n \odot m := \prod_{p \text{ prime}} p^{v_p(n) \cdot v_p(m)}$$ on positive integers have a common name? Has this operation been studied somewhere? Notice that $\odot$ is ...
2
votes
0answers
15 views

Understanding $SL_3(D)$ where D is a central division algebra

Suppose that $K$ is a non-archimedean local field of positive characteristic and $D$ is a four-dimensional central division algebra over $K$. The group $SL_{3}(D)$ can be embedded as a $K$-form of ...
-1
votes
0answers
20 views

Topological entropy and degree of smooth mappings

Where can I find the literature "Topological entropy and degree of smooth mappings" by Misiurewicz. Thanks for any help.