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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4
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1answer
47 views

Unique factorization in fields

Suppose $A$ is a commutative $R$-algebra and that is also a field. Define: For $x,y \in A$, say that $x$ divides $y$ iff $xr = y$ for some $r \in R$. Call $x,y \in A$ associates iff each divides the ...
0
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0answers
20 views

Linear Algebra and Its Applications Gilbert Strang-Solutions-Unable to find

I am trying to find Linear Algebra and Its Applications Solution Handbook by Gilbert Strang but I am unable to find it any where. I am more focused on this particular book and not anyone else since it ...
0
votes
1answer
18 views

Reciprocity for Lagrange multipliers

Does anyone know of a textbook with explicit examples of Lagrange multiplier problems of the following type? Compare the results of : (a) optimizing $f(x,y)$ [max or min] subject to the ...
0
votes
0answers
18 views

Integral kernels of self-adjoint operators

If the integral kernel $k(x, y)$ of an operator $T : C^\infty_c(M) \to \mathcal{D}'(M)$ is symmetric ($M$ is a compact manifold), then the operator $T$ is symmetric. Is the converse true? That is, ...
1
vote
0answers
25 views

Inclusionwise maximal linear subvarieties of a projective variety

Let $X\subseteq\mathbb P^n$ be a complex, projective variety. A linear subspace $L\subseteq\mathbb P^n$ will be called a maximal linear subspace of $X$ if $L\subseteq X$ and for any linear subspace ...
1
vote
1answer
14 views

Picture of the first homology group

I have to draw a picture of the base of the first homology group over $\mathbb{Z}$ of $C$, where $C$ is a compact Riemann surface of positive genus. How can i draw it? Is there some free programm wich ...
0
votes
1answer
53 views

Reverse Order Laws of M-P pseudoinverse

When I was writing a literature survey on Moore-Penrose pseudoinverse (literatures like this one, and this one), I encountered with the following equality which was named as reverse order law: ...
4
votes
1answer
65 views

Suggest a follow up book to Axler's Linear Algebra Done Right?

So I know that a similar question has probably been asked about alternatives or compliments to this book, but I think my situation is different enough to warrant slightly different advice. So I've ...
0
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0answers
29 views

Is this type of Markov chain known?

I am looking at a situation where we have $N$ urns and $K\le N$ balls. Consider some allocation of the balls to the urns. When any urn contains two or more balls, we call it a colliding urn. The ...
1
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0answers
56 views

Is this ordered square mathematically interesting?

Build a $5\times5$ (say) square using $\frac{1-\left(\frac{3}{2}\right)^n}{(2 m-1) 2^n-1}+1$, $$ \left( \begin{array}{ccccc} 0.5\hfill & 0.9\hfill & 0.944444 & 0.961538 & 0.970588 \\ ...
0
votes
0answers
19 views

Are there prerequisites to “Geometry: Euclid and Beyond” By Robin Hartshorne?

I saw that the "Geometry: Euclid and Beyond" By Robin Hartshorne seemed like a good book to learn geometry. Is this text suitable for someone who doesn't have a lot of geometry knowledge ? Thank you
2
votes
1answer
71 views

Learning Combinatorics Further

I have completed most of the basic parts in Combinatorics like Generalised Permutation & Combination, Recurrence relations, Pigeonhole Principle, Formal power series, Stirling no, Catalan no, ...
1
vote
1answer
30 views

Holomorphic maps between smooth algebraic curves

I am looking for a reference for the following statement: Let $X$ be a smooth projective curve over $\mathbb{C}$. Every holomorphic function $f: X \to \mathbb{P}^1_{\mathbb{C}}$ is in fact a morphism ...
0
votes
1answer
22 views

Best trigonometry book for complete beginner

What are some best trigonometry books for complete beginner I can't decide between S.L loney and I.M Gelfand which would be better for understanding concepts from scratch
0
votes
2answers
49 views

Prerequisite knowledge required to efficiently understand 'The Art & Craft of Problem Solving'

I would like to improve as a Mathematician and I am currently doing A-Level Mathematics in England. I have come to learn that this book is a fantastic resource for anybody serious about a career in ...
1
vote
1answer
49 views

book on real analysis.

could anyone suggest me a problem book that comprise interesting problems on topics like series , sequence , functional identities. a good problem book on real analysis is what I'm looking for.
1
vote
0answers
24 views

How many shuffles are really needed for bridge?

According to the Gilbert-Shannon-Reeds model (which apparently models reality well), one should riffle shuffle seven times to achieve a suitably randomized $52$ card deck. However, it occurs to me ...
6
votes
1answer
56 views

Distance between theorems

In automated proving one can define the best proof of a theorem as the one which minimizes the length of the proof. Given a set of known statements one could define the difficulty of a theorem as the ...
0
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0answers
17 views

Theory around the Cellular Sheaf

I have lately stumbled upon cellular (co)sheaves, which look very interesting. To understand them better, I would like references that systematically develop the theory behind them (preferably in ...
11
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0answers
155 views

Textbooks on higher category theory

What textbooks on higher category theory are there? What books do you recommend? I am looking for self-contained introductions, no research reports. There are lots of informal summaries and arXiv ...
0
votes
1answer
31 views

How to prove this result with sequences and series?

Let $\left(v_n\right)_{n\geqslant1}$ and $\left(w_n\right)_{n\geqslant1}$ be two real sequences with strictly positive values, i.e., $v_n> 0$ and $w_n >0$. Assume that $v_n\sim w_n$ and the ...
3
votes
3answers
79 views

Is there such a thing as pre- abstract-algebra books?

I am taking Abstract Algebra now and it seems most of its examples come from elsewhere. For example, I know what permutations are in counting, but the kind Algebra studies seem to be somewhat ...
3
votes
2answers
43 views

Scheme whose points over $x\colon\mathrm{spec}(R)\to X$ are the isomorphisms $x^*(F)$ and $x^*(E)$?

If one has two vector bundles $E\to X$ and $F\to X$ over a scheme $X$, why is there a scheme $S$ over $X$ with points of $S$ over a point $x\colon\mathrm{spec}(R)\to X$ is precisely the set of ...
3
votes
1answer
59 views

Categorical Foundations text

I've heard that someone's thought up a way of using category theory, involving something called topoi, as a foundation for mathematics. If this is true then are there any texts which explain such a ...
2
votes
2answers
57 views

Recommendation of multivariable calculus books

I am looking for some suggestions on a good calculus book I shall keep on hand all the time. I am a graduate student who will be commencing research in the area of theoretical PDE (nonlinear). ...
2
votes
2answers
81 views

Algebra Text Recommendations [closed]

I am looking for any recommendations or suggestions for a good book covering an introduction to the following; Relation , sets and functions, divisibility theory and modular arithmetic , groups, ...
1
vote
1answer
63 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 ...
1
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0answers
10 views

All riemannian isometries between open subsets of $\mathbb{R}^n$ are affine

I heard that there is a theorem of Liouville (Something like "Liouville's rigidity theorem") which states the following: Every Riemannian isometry between open subset of $\mathbb{R}^n$ is affine. ...
0
votes
0answers
32 views

Can someone please suggest an easy book for complex analysis?

Can someone please suggest an easy book for complex analysis? By an easy book I mean to say that the book should give the proofs of the important theorems like Cauchy ,Morera etc in an elegant ...
3
votes
1answer
35 views

References for the operator $(I-\Delta)^{\alpha /2}$

I am studying PDEs involving fractional differential operators, and I have found a few properties for the operator $(I-\Delta)^{\alpha /2}$ scattered through scientific papers. I wonder if there is a ...
4
votes
2answers
495 views

Is there an opposite to the Squeeze theorem?

I'm familiar with the squeeze theorem (AKA Two Policemen and a Drunk---no matter how wobbly, the drunk will reach the same destination as the policemen). Is there is an opposite theorem that tugs or ...
3
votes
3answers
90 views

For a prime $p\ge 17$ is $\dfrac{p^2-1}{24}$ ever a prime?

It was indicated in the comments of this MO question that if $p\ge5$ is a prime then $24|p^2-1$. Indeed $p=6k\pm1$ and $p^2-1=36k^2\pm12k+1-1=12k(3k\pm1)$ and exactly one of $k$ and $3k\pm1$ is even. ...
0
votes
0answers
64 views

Comments on Eilenberg and Steenrod's “Foundations of algebraic topology” and other similar books for recomendation

The biggest obstacle for me to learn geometry and topology is the haziness of textbooks. I took algebraic topology last semester and the textbook we used in class was Rotman's "An introduction to ...
2
votes
0answers
33 views

Is there a measure theoretic version of Stokes's theorem?

Is there a way to generalize Stokes's theorem on manifolds to general measure spaces? This idea came from trying to generalize the fundamental theorem of calculus to general function/infinite ...
3
votes
1answer
45 views

Could all iterates of $s(n)=2n+1$ be composite for some starting $n$?

Let $s(n)=2n+1$ and $\sigma(n)=\{n,s(n),s^2(n),s^3(n),\ldots\}$, where $s^3$ denotes functions composition, $s^3(n)=s(s(s(n)))$. For example $\sigma(11)=\{11,23,47,95,\ldots\}$. As another example ...
2
votes
0answers
52 views

When can/can't one switch limits? [closed]

Is there some nice article where I can find isolated some concrete theorems and counterexamples on when can one switch the order of two integrals, differentiate under the integral sign, differentiate ...
3
votes
2answers
103 views

Functional equation $f'(x)=cf(x+1)$ has a solution if and only if $c\leq 1/e$

In Contests in Higher Mathematics: Miklos Schweitzer Competitions, 1962-1991 by Gabor J Szekely, problem F.57 there is the study of $f~:~[0,\infty)\to (0,\infty)$ such that: $\exists c>0, \forall ...
0
votes
2answers
109 views

Convergence test for a special type of series

I have two null sequences $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ with $a_n \ge 0$ and $b_n \le 0$ for all $n\in \mathbb N$. Let $(c_n)_{n\in\mathbb N}$ be a sequence with either $c_n=a_n$ ...
5
votes
0answers
29 views

When a given family of curves are geodesics of some affine connection?

Let $M$ be a two-dimensional manifold and let $\mathcal C$ be a family of smooth paths on $M$. How to understand whether this family is actually a family of (possibly reparametrized) geodesics of some ...
7
votes
2answers
73 views

How do theorems arise?

I am reading complex analysis where I have come across Maximum Modulus Principle which states that if an analytic function $f$ assumes its maximum in a point of the domain $S$ then $f$ is constant ...
2
votes
1answer
21 views

Reference request: infinite-dimensional manifolds

The following books develop various aspects of the theory of infinite-dimensional manifolds: Lang, Fundamentals of Differential Geometry. Kriegl & Michor, The Convenient Setting of Global ...
1
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0answers
46 views

Mathematical structures with name reffering to a country

I am looking for a list of mathematical structures (not theorems) that refer to a country or nationality. I only know of Polish spaces and Polish groups. Does anyone have other examples? Note: many ...
2
votes
1answer
39 views

When is shear useful?

I'd never heard of the shear of a vector field until reading this article. Shear is the symmetric, tracefree part of the gradient of a vector field. If you were to decompose the gradient of a vector ...
1
vote
0answers
26 views

Best book to understand representation theory.

I have tried to read representation and character theory from a few books but none of them was working for me, like Lieback and Serre. I want to understand representation and character to use them and ...
1
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0answers
9 views

Good resources for learning to recognize word problems in statistics?

I've got a number of books and resources for statistics theory, but I've always had problems with the approaches needed in answering questions, specifically for probability theory where counting ...
0
votes
0answers
17 views

Derivatives and integrals of polynomials of two variables

Suppose I have a real-valued two dimensional polynomial $p(x,y)$ of order d. The partial derivatives inherit a nice structure, in particular knowing $\partial p/\partial x$ tells you $p$ up to the ...
1
vote
1answer
27 views

Applications of differential equations.

I'm trying to explain to a friend the power of differential equations in modelling math stuff. Does anyone have some really engaging/"juicy" examples? I looked at this question and found some good ...
0
votes
0answers
23 views

Book Recommendation - Exercise book for Calculus / Analysis I

I am looking for exercise books that provides problems and exercises that together cover those topics: convergence of sequences, series etc. limit points etc. continuous functions derivatives, ...
0
votes
0answers
29 views

Reference request: history of analytic geometry

I am searching a book in the domain of the history of math, that describes the historical origins of analytic geometry, starting from Descartes (?), and that describes also its development (e.g. the ...
2
votes
1answer
51 views

Solvability of the equation $2a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + a_{4}^{n}$ when $n \geq 5$ is prime?

As a natural extension of the question titled Solvability of $a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + a_{4}^{n}$ when $n \geq 5$ is prime?, I wonder if the equation $$2a_{1}^{2} = a_{2}^{n} + a_{3}^{n} + ...