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
2answers
351 views

Good economics textbooks.

I would like a suggestion for the most mathematically fun/interesting mathematical economics textbook, preferably using abstract math. I want to prove theorems to complete my economics minor. I have ...
0
votes
1answer
22 views

Text Suggestion for the given topics.

Can anyone suggest me a suitable text(s) for the given topics ? Sample Surveys and Design of Experiments: Sampling and non-sampling errors. Conventional sampling techniques (SRSWR/SRSWOR, ...
1
vote
3answers
139 views

Good introduction to cardinals?

is there a good text book to cardinals? I am more interested in how cardinal works than cardinality. Because it seems in undergrad they cut off at proof of $\mathbb{R}$ is uncountable and does not go ...
0
votes
1answer
65 views

If $|t| = |W(-\ln z)| = 1$ and $t^n =1$ then $z^{z^{z^{…}}}$ is convergent

Let $z \in \mathbb{C}$ and $W$ be the Lambert W function. In this post I was told if $|t| = |W(-\ln z)| = 1$ and $t^n =1$ for some $n \in \mathbb{N}$ than the iterated exponential $z^{z^{z^{...}}}$ ...
3
votes
4answers
344 views

Measure theory for self study. [duplicate]

I have good knowledge of Elementary Real analysis. Now I'd like to study measure theory by myself (self-study). So please give me direction for where to start? Which book is good for starting? I have ...
4
votes
1answer
65 views

Study materials to help understand the generalized Stokes' theorem both intuitively and rigorously?

Dear MSE: My goal is to understand the generalized Stokes' theorem both intuitively and rigorously. Could someone give advice or recommend study materials to help understand the generalized Stokes' ...
1
vote
1answer
30 views

Which functions arise from a probability measure in this way?

Given a probability measure $\mathbf{P}$ on the interval $I=[0,1]$, we get a corresponding function $f:(\mathbb{R}_{>0})^2 \rightarrow \mathbb{R}_{>0}$ as follows: $$f(x,y) = \int_\mathbf{P} x^q ...
0
votes
0answers
9 views

Source proof of equivalence between relation algebra and three variable first order logic

I am reading up on relation algebra and a lot about first order logic. Being a bit of a heavy torsion on the latter one as I am not used to it. Either way I read on wikipedia that up to 3 variable ...
0
votes
0answers
39 views

How can I study probability?

I want to have a deep understanding of probability. I've tried William Feller's first book on Probability, and E.T Jaynes' Probability theory - the logic of science (which is very different from most ...
6
votes
0answers
74 views

Rigorous justification for “complex” change of variable in integration

Suppose that I have $X_1,\ldots,X_n$ i.i.d. $\sim $ $X$ and $Y_1,\ldots,Y_n$ i.i.d. $\sim$ $Y$ for some continuous $X$ and $Y$. Consider the r.v.'s $\bar{X}=\frac{1}{n}\sum_jX_j$ and $\bar{Y}=\frac{1}{...
8
votes
3answers
85 views

Continued fraction for $c= \sum_{k=0}^\infty \frac 1{2^{2^k}} $ - is there a systematic expression?

I want to use the convergents of the continued fraction for $$c= \sum_{k=0}^\infty \frac 1{2^{2^k}} $$ - but of course a numeric software is very limited here, so I hope there exists a systematic ...
1
vote
1answer
28 views

Reference request: product Borel $\sigma$-algebra of non-separable metric spaces

The following is a proposition in Folland's Real Analysis about product sigma algebra: Here $\mathcal{B}_X$ denotes the Borel $\sigma$-algebra on $X$. Could anyone come up with an example that ...
0
votes
1answer
28 views

Passing from classical formulation to weak formulation for a general PDE

I am reading a paper dealing with a general elliptic PDE that I need to transform from classical formulation to weak formulation: $$\left\{\begin{matrix} - \sum_{i=1}^n \sum_{j=1}^n (a_{ij} u_{x_i})_{...
2
votes
1answer
47 views

Five exponentials theorem

The six exponentials theorem is proved in most textbooks on transcendental number theory, and the four exponent conjecture is an open problem. Is there any good/accessible exposition of the five ...
2
votes
0answers
27 views

GRE Math Resources

I am planning to give the MATH GRE. I intend to go to physics graduate schools but I may have to given the MATH GRE according to my college's requirement. I very much like maths, but I'm overloaded ...
0
votes
1answer
31 views

Good, simple reference for Riesz-Fischer Theorem.

I am looking for a good, simple reference for the proof of Riesz-Fischer Theorem ($L^p$ spaces are complete). An example of a not so good reference in my opinion is Royden, where he uses "rapidly ...
5
votes
2answers
83 views

Formal systems in which $\forall x \in \mathbb{R}(x \neq 0 \rightarrow x^{-1} \neq 0)$ is true, but the contrapositive is disallowed.

Question. Are there any formal systems out there for which $$\forall x \in \mathbb{R}(x \neq 0 \rightarrow x^{-1} \neq 0)$$ is true, but the contrapositive $$\forall x \in \mathbb{R}(x^{-1} =...
2
votes
1answer
130 views

The following is a necessary condition for a number to be prime, from its digit expansion. Has it been referred somewhere?

Concerning a numbers’ digits we know some necessary conditions on them for the number to be prime, besides the last digit having to be odd (except for prime 2). For example in decimal representation ...
6
votes
1answer
1k views

Correspondence theorem for rings.

Could someone provide a reference that includes a full and honest proof of the Correspondence Theorem for rings? Let $A$ be a multiplicative ring with identity and $I$ an ideal of $A$. There is a ...
1
vote
0answers
24 views

What are some methods of proving undefinability results? (Reference)

I'm trying to prove some results regarding undefinability of functions from the natural numbers in certain structures, but besides texts on elemental logic and number theory, i haven't found anything ...
9
votes
2answers
631 views

An extension of the birthday problem

Th birthday problem (or paradox) has been done in many way, with around a dozen thread only on math.stackexchange. The way it is expressed is usually the following: "Let us take $n$ people "...
10
votes
1answer
2k views

Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.

This is solely a reference request. I have heard a few versions of the following theorem: If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ ...
0
votes
0answers
18 views

Ask for an example of the following type of parabolic PDE with analytic solution

I aim to find an example of the following type semi-linear PDE with analytic solution to test my numerical method: $$\begin{cases} u_t=Lu+g(t,x,u,u_x), &(t,x)\in [0,T]\times \mathbb{R}\\ u(0,x)=f(...
0
votes
0answers
9 views

Bounds for transition density and its derivative

Suppose the process $X_t$ has a transition density $p(t,x,y)$, which is continuously differentiable w.r.t $y$. In my proof, I use the following properties of $p$ and $p_y$: There exist functions $\...
12
votes
2answers
243 views

modern calculus or analysis text that emphasizes Landau notation?

Is there a comprehensive calculus or analysis textbook or problem book, written in the last twenty years, that emphasizes the use of Landau notation (big and little oh), especially for making ...
6
votes
0answers
159 views

A new type of Arithmetic-Harmonic mean for $n$ numbers

Let's introduce the following iterative procedure. Take two numbers $x_0$ and $y_0$. $$a_0=\frac{x_0+y_0}{2}~~~~~~~~~~~b_0=\frac{2x_0y_0}{x_0+y_0}$$ $$x_1=\frac{x_0+a_0+b_0}{3}~~~~~~~~~~~y_1=\frac{...
0
votes
0answers
48 views

What's about the derivative of the Riemann zeta function?

The derivative of the Riemann Zeta function is $$\zeta'(s)=-\sum_{n=2}^\infty\frac{\log n}{n^s}$$ for $\Re s>1$. Question. Can you refers us in a short post, from a divulgative viewpoint (but ...
2
votes
1answer
57 views

Literature on fields $\mathbb{Q}(\sqrt[4]{D})$

Can anyone direct me to literature discussing extension fields of the form $K = \mathbb{Q}(\sqrt[4]{D})$ where $D$ is squarefree? I'm particularly interested in results regarding the class and unit ...
2
votes
0answers
31 views

Historically accurate alternatives to men of mathematics? [migrated]

I have heard that the book "Men of Mathematics" by E. Bell is a very entertaining book composed of biographies of several influential mathematicians, and is in fact one of the most popular popular ...
-1
votes
0answers
108 views

Implicit arguments in informal math, how to explain?

Let we have three categories $Z$, $C$, and $D$. And let $Z$ have partially ordered Hom-sets each Hom-set having a least element. Let also every object of $D$ be an ordered set and has least element. I ...
4
votes
1answer
89 views

Applications of PDEs in many variables

One reason that solving systems of partial differential equations is so important is the many applications of PDEs in science and engineering (eg. the heat equation, the wave equation, etc.). Often ...
0
votes
1answer
31 views

Book recommendation for engineer turning towards mathematics (Abstract Algebra)

I will be taking abstract algebra course in a month from now. I am first time taking and algebra course and will be sitting with math majors. Can someone suggest me a book suitable for me i.e for ...
3
votes
2answers
95 views

The graph of the function $f(x)= \left\{ \frac{1}{2 x} \right\}- \frac{1}{2}\left\{ \frac{1}{x} \right\} $ for $0<x<1$

Let for reals $$\{x\}=\text{Frac}(x)$$ the fractional part function, take for example the more common definition, the first (there is a different definition as you see in this MathWorld's Page, ...
-3
votes
0answers
25 views

Sources of good problems for measure theoretic probability [duplicate]

Please let me know sources of good problems for measure theoretic probability, like the ones in the book by Rick Durret. Thanks
0
votes
1answer
34 views

Find functions $F(\mathbf{x})$ invariant under a map $\mathbf{x} \to \mathbf{x'}$

We introduce a map $\mathbf{x} \to \mathbf{x'}$, defined as (for example on $\mathbb{R}^3$): $$x'=f(x,y,z) \\ y'=g(x,y,z) \\ z'=h(x,y,z)$$ Note that $f,g,h$ are not all linear (or at least, I'm not ...
1
vote
0answers
48 views

$A = \sum_{n=0}^\infty a_n$ and $b_n \to B$ implies $\sum_{k=0}^n a_k b_{n-k} \to AB$

This question is motivated by the answer With $y_n$ a sequence of real numbers, prove that if $y_n=x_{n-1}+2x_{n}$ converges then $x_n$ also converges, where essentially the following fact is used: ...
0
votes
1answer
38 views

Books on mathematical biology

Can someone recommend good books on mathematical biology for self study. Especially to understand SIR models and stochastic models such as branching process. For SIR models I want more of beginner ...
7
votes
1answer
51 views

Reference Request: Complete Proof of Braikenridge–Maclaurin Theorem

Where can I find a reference to a complete proof of the Braikenridge–Maclaurin theorem, which is stated as: If the three pairs of opposite sides of (an irregular) hexagon meet at three collinear ...
0
votes
0answers
18 views

higher derivatives of $R^m \to R^n$ [duplicate]

What's a good source (paper, book, website,...) where I can learn more about higher derivatives of functions $R^m \to R^n$ as multilinear functions or tensors? Thanks.
2
votes
2answers
138 views

formal definition of “fractal” or standardized categories?

fractals are many decades old and come up in a wide variety of contexts and can be generated in so many different ways. however, a formal definition of fractal seems really slippery/ difficult. are ...
3
votes
1answer
54 views

What is the difference between high dimensional and low dimensional chaos?

Often I read of high and low dimensional chaos. But, I don't know what is their difference. I have thought the following answer. Let us consider a time series $\{x_i\}_{i\in\mathbb N}$. According to ...
1
vote
0answers
32 views

Existence of solutions of $\Delta u = f$ such that $|u|_{L^\infty} < \infty$ if $|f|_{L^\infty} < \infty$.

I am struggling with figuring out the details of proposition 7.1. in the paper Curvature and Uniformization - R. Mazzeo and M. Taylor. Setting is as follows. Let $\Omega$ be a noncompact Riemann ...
3
votes
1answer
48 views

What is the difference between operator theory and functional analysis?

In my undergrad mind they are the same subject because functional analysis studies functional spaces like Banach and Hilbert spaces. Operators are function, so shouldn't they be the same subject? ...
0
votes
0answers
26 views

Legendre's Conjecture Theme (Part II)

This is a continuation of this question. My main question is that, in the previous question we were mainly concerned about the sign of, $$f_{2}(n)=\pi\left((n+1)^2\right)+\pi\left(n^2\right)-2\pi\left(...
2
votes
0answers
65 views

Legendre's Conjecture Theme (Part I)

Main Question Recently I have been thinking about the Legendre's Conjecture. I noticed that a proof of the conjecture can be obtained if we can prove any one of the following, Conjecture 1. For ...
0
votes
2answers
72 views

Need Guidance for engineer taking rigorous analysis course for first time

I will be taking analysis course in a month from now. Topics are given below. I am doing engineering and had been through calculus courses but nothing like sort of analysis before. Many of my friends ...
2
votes
0answers
57 views

All but a finite number of finite simple groups are groups of matrices over $\mathbb{F}_q$

In the introduction to this honors thesis, http://people.math.gatech.edu/~jrabinoff6/papers/building.pdf I found this statement: Matrix groups defined over the finite fields $\mathbb{F}_q$ ...
2
votes
2answers
69 views

Evaluate $\int_0^{\pi/2}(\sin x)^n e^{-(2+\cos x)\log k}dx$ for fixed integers $n,k\geq 1$

My question is the following Question. Can you compute some of the following $$c_{n,k}=\int_0^{\pi/2}(\sin x)^n e^{-(2+\cos x)\log k}dx$$ where $n\geq 1$ is a fixed integer and $k\geq 1$ is ...
0
votes
1answer
21 views

When does a stationary point process on group $G$ have $0$ or $\infty$ many points a.s.?

For $G=\mathbb{R}^d$ I know that a stationary point process $X$ either has 0 or infinitely many points, a.s. Daley and Vere-Jones refer to this as the 0-Infinity dichotomy. They hint that this fact is ...