The reference-request tag has no wiki summary.
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Reference: Qualifiers for defining trees
What is a good authoritative (i.e. has standard usage) reference accessible on the Net for checking the definitions of the multitude of terms used to describe trees in combinatorics?
Examples: ...
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2answers
63 views
Proof that the $d$-th powers generate the $d$-th symmetric power of a vector space
Let $V$ be a $\mathbb{C}$-vector space of finite dimension. Denote its $d$-th symmetric power by $V^{\odot d}$. I am looking for a proof that $V^{\odot d}$ is generated by the elements $v^{\odot d}$ ...
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1answer
23 views
About $P_{{L},{M}}$, projection transformation onto subspace $L$ along subspace $M$ .
I need help to study following theorem:
For every idempotent matrix $E\in\mathbb{C}^{n\times n}$, $R(E)$ and $N(E)$ are complementary subspaces
with $E = P_{{R(E)},{N(E)}}$. Conversely, if $L$ and ...
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32 views
Can anybody suggest me references for functional analysis? My main concern is to do examples as much as i can.
Can anybody suggest me references for functional analysis? My main concern is to workout examples as much as i can do. Also if there is any web resources to help mein this regard ?
Thanks for giving ...
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0answers
51 views
Why topological stratification is useful?
My main focus is on the applications of stratification in complex/abstract Algebraic geometry especially, from the scheme-theoretic viewpoint and (Added) Moduli spaces.
I have a vague feeling that ...
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3answers
216 views
Can the Bourbaki series be used profitably by undergraduates?
Can the Bourbaki series be used profitably by undergraduates and high school students?Are we the target audience? I came across the N.Bourbaki texts while surfing the internet(I have not had the ...
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0answers
27 views
Sufficient condition for surjectivity of a morphism of group schemes
Let $G$ be a group scheme over a field $F$, and let $f:G\to G$ be a homomorphism. Written in my notes, I have the following statement:
To check surjectivity (on $F$-rational points), it suffices ...
3
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1answer
28 views
Gliders, static structures in various (dynamic) systems
Structures, i.e. symmetries over time, appear in various systems:
gliders in cellular automata, like Game of Life or Rule 110,
unmatched string's parts in rewrite systems – unchanged in multiple ...
4
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2answers
59 views
are fibres of a flat bijective map reduced?
Let $f: X \to Y$ be a flat map of algebraic varieties or of complex analytic spaces which is bijective on closed points (or just bijective in the secnond case). Suppose both $X$ and $Y$ are reduced. ...
3
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1answer
48 views
$K$-theory of smooth manifolds: continuous vs. smooth vector bundles
Suppose I have a smooth manifold $M$, and want to consider the $K$-theory $K^0(M)$. I could define this in the usual way (by taking the Grothendieck group of the monoid of equivalence classes of ...
3
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1answer
100 views
Free Graph Theory Resources
What freely available graph theory resources are there on the web? In particular, I am interested in books and lecture notes containing topics such as trees, connectivity, planar graphs, the ...
3
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6answers
106 views
Where do I start with self learning linear algebra
I'm a physics major and I'd probably go with mathematical/theoretical physics path.
Where do I start with self learning linear algebra? I'm good with proofs but I'm not comfortable with learning ...
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1answer
82 views
+100
Why can't you simulate isotropic fluid flow on a square lattice?
There are easy methods for discrete simulations of gas dispersion in two dimensions. If you take a large square lattice, each cell of which is assumed to contain at most one gas molecule, and you ...
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0answers
22 views
How to characterize the image of $PGL(V)$ in $\mathbb{P}(W)$ for an irreducible $GL(V)$-representation $W$
I'd like to ask two versions of my question, one a very specific case that I suspect may have a fairly easy and classical answer, the other the general case which I would like to find references on.
...
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1answer
103 views
Linear Algebra Text
I intend to study linear algebra during my summer vacation.I will be thankful if anyone could please point me to a good text.I do not care about the difficulty as long as the book is self contained.I ...
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0answers
17 views
Reference Request - Spaces of Smooth Vectors
I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a ...
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1answer
75 views
List of important and influential publications in logic
Wikipedia surprisingly does not contain a good list. Under philosophy of mathematics it is rather vacant too. A graduate level list is here, but I was curious more about the most influential papers. ...
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2answers
78 views
$\mathcal{C}^1$ implies locally Lipschitz in $\mathbb{R}^n$
According to wikipedia a function $f\colon \mathbb{R}^n\to\mathbb{R}^n$ that is continuously, is also locally Lipschitz.
I there someone who knows a good reference which contains a proof of this ...
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2answers
31 views
Freudenthal suspension theorem- Weak excision lemma
http://www.math.uchicago.edu/~amwright/HomotopyGroupsOfSoheres.pdf
I'm trying to understand this theorem on page 6. Apparantly you can use that to prove the Freudenthal suspension theorem for ...
4
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1answer
115 views
Projecting onto (lightface) Borel sets
Suppose $A \subseteq \omega^{\omega} \times \omega^{\omega}$ is Borel. If we project $A$ onto $\omega^\omega$, we get a $\mathbf{\Sigma^{1}_{1}}$ set $\{y: \exists x (y,x) \in A\}$. What if we project ...
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0answers
23 views
Bivariate recurrence relation
Consider the following recurrence relation:
$$A(h,0)=1\\
A(h,h)=c^h\\
A(h,r)=A(h-1,r)+(c-1)\cdot A(h-1,r-1).$$
Obviously, this is just a generalization of A008949, where $c=2$. Since I'm pretty sure ...
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1answer
49 views
a good book on inverse problems for engineers
I'm looking for a book on inverse problems which is suitable for engineers, both introduction and practical applications are required. Currently I'm looking to Parameter Estimation and Inverse ...
3
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1answer
19 views
On one quantity in a finite graph
Let $(G,E)$ be a finite undirected graph, and $d$ be the usual shortest path distance on $G$. The graph is not necessary connected, so $d(v',v'') = \infty$ if there are no paths from $v$ to $v'$. For ...
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2answers
51 views
Problem regarding infinite sum of remainders.
Before here @math.SE there was a question regarding a problem on a maths magazine. I decided to look at the link provided, and one problem proposed was (if I'm not recalling this wrongly):
Find
...
3
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4answers
120 views
Reference request: Vector bundles and line bundles etc.
I am interested in learning algebraic geometry and I talked to one of my professors today (who is, in part, an algebraic geometer) and he recommended I understand the analytic analogue of the ideas in ...
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2answers
120 views
Popular science book on rigorous axiomatic approach
Can you recommend a popular science book that deals with axiomatic foundation of mathematical areas and consequences? The areas could be geometry, algebra/numbers, set theory, etc.
It should be exact ...
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0answers
63 views
Eternity Puzzle Decoded
The story is about 13 years ago, when i was 17 and i remember that it was about solving the Eternity puzzle that I've seen in a newspaper.. in those days did not have a computer and I remember passing ...
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1answer
53 views
Reference for classification of small groups
There are various online resources for the classification of groups of small order, such as this one or that one.
Is there any nice reference in the literature which contains such a classification ...
2
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1answer
36 views
Probability as allocation of resources?
If we have probabilities for disjoint events:
$A, B, ..., \text{i.e.:}\space P(A), P(B), ..., \text{and}\space P(A) + P(B) + \ldots = 1$
then does this in fact mean, that there is a system, that has ...
3
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2answers
76 views
Natural & important probability measures on $\mathcal{C}[0,1]$, in particular the Wiener measure
Which probability measures on $\mathbf{\mathcal{C}[0,1]}$ are known? (Here $\mathcal{C}[0,1]$ is the space of continuous real-valued functions defined on the unit interval.)
I'm pretty sure the ...
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0answers
53 views
explicit formula for norm map of Kummer extensions
Since it is particularly easy to write down a basis of a Kummer extension $K=k(\mu)/k$ (where $\mu^n=a \in k$) as a $k$-vector space, I suspect that it is should not be terribly hard to write an ...
5
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2answers
126 views
$SU(2)$ Lie group
I have been studying Lie groups for a bit of fun for a while now and think they are fascinating. I have recently been told that $SU(2)$ can be used in some way to keep track of navigational systems in ...
1
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1answer
81 views
determinant of a sum
I need a formula for the determinant of the sum of two matrices:
$\det(\mathbb{I}+M)$. On the internet I found it for the first order but i need it at second or even third order. Where can I find the ...
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2answers
67 views
Divergent series and $p$-adics
If we naïvely apply the formula $$\sum_0^\infty a^i = {1\over 1-a}$$ when $a=2$, we get the silly-seeming claim that $1+2+4+\ldots = -1$. But in the 2-adic integers, this formula is correct.
Surely ...
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3answers
196 views
Why does the Hilbert curve fill the whole square?
I have never seen a formal definition of the Hilbert curve, much less a careful analysis of why it fills the whole square. The Wikipedia and Mathworld articles are typically handwavy.
I suppose the ...
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1answer
107 views
Harvard math 55 materials
I would like to know what Harvard math 55 go through.So can anyone please point me to their course and their problem sets ?Your help is appreciated.
4
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3answers
129 views
Please suggest a functional analysis book to refresh my knowledge
I would like to ask you to recommend me a good modern textbook on functional analysis to refresh what I already know. I am a computer science student and for the last two semesters we've been having a ...
7
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1answer
75 views
Ties between Lie algebras and ring theory
I would like to get a general understanding of the relationship between (noncommutative) ring theory and Lie algebra theory. All Lie algebras are finite dimensional and over a field $k$ of ...
6
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4answers
211 views
Rudin or Apostol
I have an option to choose between the two books Mathematical Analysis by Tom Apostol and Principles of Mathematical Analysis by Walter Rudin as I was gifted Rudin by a friend and ended up buying the ...
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0answers
61 views
About the Platonic Solids in all dimensions
I am asking about the Platonic solids in all dimension, some reference about the proofs of many of the statement made in here.
I would like to here about how to think about higher dimensions mainly ...
3
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1answer
61 views
Alexander Multivariate Polynomial of a General Torus Link
In his book Singular Points of Complex Hypersurfaces, Milnor quotes the Alexander (multivariate) polynomial of the torus link $T_{p,pq}$
\begin{align}
\Delta_{T_{p,pq}}(t_1, \dots, t_p) = \frac{((t_1 ...
1
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1answer
46 views
combinatorial optimization? What is the name of this problem and where can I find material for studying it?
I'm looking for material to study the following problem,
Suppose I have $N$ numbers, and I know that the sum of $M$ of those number equals $k$. The goal is to find all combinations of cardinality (is ...
5
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1answer
59 views
Group of finite ideles
Let $K$ be a number field. Let $\mathbb{I}_f$ denote the group of finite ideles, and let $\phi: K^{\times} \rightarrow \mathbb{I}_f$ be the diagonal embedding. On page 167 of his notes on Class Field ...
7
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8answers
322 views
Where to begin with foundations of mathematics
I would like to know more about the foundations of mathematics, but I can't really figure out where it all starts. If I look in a book on axiomatic set theory, then it seems to be assumed that one ...
3
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1answer
86 views
Proof of a Theorem in Gao's 'Invariant Descriptive Set Theory'
Theorem 1.7.5 on p.35 of Gao's Invariant Descriptive Set Theory reads
Theorem 1.7.5 (Kleene)
If $A\subseteq X \times \omega^{\omega}$ is $\Pi^{1}_{1}$ and $$x \in B \Longleftrightarrow \exists y ...
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1answer
33 views
Exponential stability of inhomogeneous linear ODE's
Can anybody give me a good reference which under suitable assumptions
discusses exponential stability of $0$ for the equation
$\dot{u}_t = A(t)u(t) + b(t)$
Here $u_t\in\mathbb R^n$ is the unknown, ...
5
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3answers
74 views
Mathematics applied to biology
Can anyone suggest reference material on mathematics applied to biology, in particular the study of the behavior of say simple unicellular organisms or cells? Ideally the level of complexity should be ...
5
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0answers
46 views
A scalar product in the space of oriented volumes?
Let $L\colon \mathbb{R}^n \to \mathbb{R}^N$ be an injective linear map. By the Cauchy-Binet formula, $\det(L^TL)$ equals the sum of the squares of all minors of $L$ of order $n$: this looks just like ...
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33 views
About regularity of the maximal operator of Hardy-Littlewood
What difficulties arise when you consider the centered maximal operator, such that you can't prove that it maps BV into BV?
Does someone have some reference which that maps BV into BV?
PS: The ...
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0answers
19 views
paper about linear independence in altered Vandermonde and Cauchy Matrices
Both Vandermonde and Cauchy matrices with $n$ rows and $k$ ($n \geq k$) columns have the property that any $k$ rows are linearly independent (assuming the coefficient are independent). It seems to me ...
