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Vector analysis text book.

It is "Calculus: Early Transcendental Functions" by Robert T Smith and Roland Minton. I found it here on page 876. It seems to be a matter of luck whether Google shows this page on a preview ...
Stefan's user avatar
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1 vote

Where do i start learning recursion for mathematical olympiads?

Not sure if this is useful, but there is a textbook called "Further Pure Mathematics 2" published by Pearson, which is intended for high-school students studying Edexcel AS and A Level ...
Tom's user avatar
  • 333
0 votes

Realtionship between automorphism number of a graph and subgraph count

I think that a way to see $S'_{H,N}$ is just the set of all bijections (i.e., functions $f$ from an arbitrary subset of nodes of size $m$ of the graph $G$), that ''map'' on $H$. Therefore for each ...
DivergentSeries's user avatar
0 votes

Is there a closed form for the integral $\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\mathrm{d}z$?

If we start by taking $ z \rightarrow e^u$ to get $$ \int^{\infty}_{-\infty} u e^{\lambda u - w\cosh(u)} du = \Omega(w,\lambda) $$ We can then find $$ \int \Omega(w,\lambda) d\lambda = \int^{\infty}_{-...
Aidan R.S.'s user avatar
0 votes

Problem with an example in Werner Amrein book

Hint for the first: You presumably know that $\operatorname{supp} (E) \subset \mathbb{R}$. Hint for the second: The resolvent map $\rho(A) \ni z \mapsto (A-z)^{-1}$ and the operator norm are ...
jd27's user avatar
  • 2,295
0 votes

Periods of certain lagged Fibonacci sequences

Defining the RNG First off, as the comments in random.cs state, Microsoft's implementation is based on an algorithm from Numerical Recipes (specifically the 2nd edition: newer editions recommend a ...
trillian's user avatar
  • 101
0 votes

Is there a closed form for the integral $\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\mathrm{d}z$?

Yes, it has a closed form in terms of trascendental functions. What you are looking for is the derivative of this expression: $$ \int_{0}^{\infty}x^{s-1}\exp\left(-\alpha x^h-\beta x^{-h}\right)dx = \...
Bertrand87's user avatar
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1 vote
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Finding References for known results about mixing time of a Markov chain arising from Gaussian elimination

There are at least $N=c2^{n^2}$ invertible matrices mod 2, where $c=1/2(1-1/4)(1-1/8)\cdots (1-2^{-k})\cdots>0$. Since there are less than $n^2$ possible moves in each step, after $t$ steps the ...
Yuval Peres's user avatar
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0 votes

Topology textbook with a solution manual

Four other than Munkres (who benefits from fair-sized readership) come to mind (as a fellow self-student who appreciates solutions after over-extended attempts). Two introductory: Introduction to ...
isaac's user avatar
  • 325
1 vote

Law of large numbers result for largest component in Erdos-Renyi

See Lemma 2.12 here: https://www.math.cmu.edu/~af1p/BOOK.pdf Essentially the limit you ask for is like $1/(p-1-\log p)$ (But you should use c, not p, since p is usually the notation for the edge ...
dbal's user avatar
  • 511
2 votes
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Law of large numbers result for largest component in Erdos-Renyi

With apologies, I will switch the notation, to avoid writing $\frac pn$ for the edge probability (which is almost always $p$, and $p = \frac pn$ would be silly). Let's say we are looking at $G(n,p)$ ...
Misha Lavrov's user avatar
1 vote
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References to study Clifford algebras

Only two of the six books I checked give an explicit proof that you can generate a basis for the entire Clifford algebra from an orthogonal vector basis: Ian Porteous in Clifford Algebras and the ...
Nicholas Todoroff's user avatar
0 votes

What is the optimal definition of the convolution of two measurable functions?

Your extension is valid, but the usual generalisation is to the convolution of distributions or tempered distributions. In particular, if $f\in L^p$ and $g\in L^q$ with $1\leq p,q\leq\infty,$ $p^{-1}+...
Lieven's user avatar
  • 1,158
1 vote

Stone's Representation Theorem equivalences

That the Stone representation theorem implies the ultrafilter lemma is almost immediate: Every set algebra has ultrafilters on it (for any point in the underlying set, take the collection of sets in ...
4 votes
Accepted

The rank of Sylow subgroup of special linear groups over finite fields

The Sylow $\ell$-subgroup of $SL_n(q)$ is fairly easy to write down, when $\ell\neq 2$ and $\ell\nmid (q-1)$. (The reason for this latter distinction is that the Sylow $\ell$-subgroups of $GL_n(q)$ ...
David A. Craven's user avatar
2 votes
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Reference on asymptotic expansion of Dawson integral

Here is a simple derivation using Watson's lemma, assuming $x>0$. To begin, note that $$ {\rm e}^{ - x^2 } \int_0^x {{\rm e}^{y^2 } {\rm d}y}\; \mathop = \limits^{y = xt} \;x\int_0^1 {{\rm e}^{ - ...
Gary's user avatar
  • 32.6k
2 votes

Reference on asymptotic expansion of Dawson integral

I do not remember where and when I saw an interesting probabilitic approach for the generation of this unusual series expansion. It was based on $X\sim\mathcal N(\sqrt 2,\frac 1 {x^2})$ and the ...
Claude Leibovici's user avatar
0 votes

What can I say about $P(B|A_1 \cap A_2)$ if I know that $P(B|A_1), P(B|A_2) \approx 1$?

The event $A_1\cap A_2$ may be much smaller than either of the events $A_1$ or $A_2$. So even if most outcomes of the events are also outcomes of event $B$, that does not require that any outcomes of ...
0 votes

Uniform random triangles in the square.

Probability density and cumulative distribution functions for uniform distribution: $$ f_X(x)=\begin{cases}1 & x\in[0,1] \\ 0 & \text{otherwise}\end{cases}\\ F_X(x)=\begin{cases}0 & x<0\...
evn's user avatar
  • 21
1 vote
Accepted

Cauchy Product of Summable Sequences is Cesàro Summable

Here we prove: Lemma. Let $A_n = \sum_{k=0}^{n} a_k$ and $B_n = \sum_{k=0}^{n} b_k$. Then $$ \sum_{n=0}^{N} \sum_{k=0}^{n} \sum_{j=0}^{k} a_j b_{k-j} = \sum_{n=0}^{N} A_n B_{N-n}. $$ Proof of Lemma. ...
Sangchul Lee's user avatar
0 votes

reference for linear algebra books that teach reverse Hermite method for symmetric matrices

question today, May 19, 2024. All zero on the diagonal $$ H = \left( \begin{array}{rrr} 0 & 7 & 8 \\ 7 & 0 & 4 \\ 8 & 4 & 0 \\ \end{array} \right) $$ The steps obey: ...
Will Jagy's user avatar
  • 140k
2 votes

Definition of a local coordinate system as a mapping of region of $\mathbf{R}^n$ to the manifold (and not the inverse)

The maps are bijective and smooth so locally you do have unique coordinates. It doesn't matter which direction the arrows go. If two charts overlap, then by definition of a smooth atlas you have ...
Sam Kirkiles's user avatar
  • 2,833
1 vote

Reference for result and proof that $R(g_*\circ f_*)\cong Rg_*\circ Rf_*$ for morphisms of ringed spaces $X\xrightarrow{f}Y\xrightarrow{g}Z$

I suppose you mean $R(g\circ f)_* \cong Rg_* \circ Rf_*$. This is Corollary $11.12$ in Scholze's Algebraic Geometry II.
kobe's user avatar
  • 42.2k
2 votes

Geometric constructions problems problem book recommendation.

A : You might want to know about "Possibility theorems" involving Euclidean Constructions & "Impossibility theorems" & Constructing various interesting Patterns & the ...
Prem's user avatar
  • 11k
0 votes

Does a terminal Gorenstein cDV singularity imply a DuVal singularity on the general elephant?

Both the Gorenstein and the non-Gorenstein case are in Main Theorem (I) in Reid's "Minimal Models of Canonical 3-Folds", (0.6). If you prefer, this is also in Theorem (a) in Reid's "...
SeparatedScheme's user avatar
1 vote

Books on numerical integration

The book Quadrature Theory by Brass and Petras seems quite modern, but I have no experience with it.
lhf's user avatar
  • 217k
1 vote

Books on numerical integration

Somewhat new "Tea Time Numerical Analysis" , last updated in 2021. Available at https://open.umn.edu/opentextbooks/textbooks/741 Endre Suli and David Mayers have written "An ...
Prem's user avatar
  • 11k
1 vote

What can I say about $P(B|A_1 \cap A_2)$ if I know that $P(B|A_1), P(B|A_2) \approx 1$?

The general principle that you are suggesting is not true. Here is an example. Suppose I am the greatest chess player that ever lived. I have two students. They will be attending a conference with 998 ...
Jason Swanson's user avatar
0 votes

Almost sure convergence of means of arbitrary sample sets?

Here is a sufficient condition for the above to hold. Proposition. Suppose $X$ has finite variance $\sigma^2$ and suppose that $$\sum_n \frac 1{|I_n|} < +\infty$$ Then $(X_{I_n})_n$ converges ...
Olivier Bégassat's user avatar
4 votes
Accepted

Analysing a Lebesgue integral inequality for $|t^{-n} \phi(x/t)|$, where $\phi \in C_c^\infty \cap L^1$ with $\| \phi \|_1 = 1$.

The essential ingredients are that $\phi\in L^1(\mathbb{R}^n)$ and that $\phi$ has compact support (no need to assume any regularity or boundedness). The compact support condition is telling us that ...
Severin Schraven's user avatar
4 votes

Analysing a Lebesgue integral inequality for $|t^{-n} \phi(x/t)|$, where $\phi \in C_c^\infty \cap L^1$ with $\| \phi \|_1 = 1$.

Since $\phi \in C_{c}(\mathbb{R}^{n})$, there is $R > 0$ such that $\phi(x) = 0$ if $|x| \geq R$, so $\phi_t(z) = 0$ if $|z| \geq tR$, now, using the change of coordinates $z = ty$ you have $$ \...
Raul Fernandes Horta's user avatar
2 votes

Comprehensive Linear Algebra Text

caffeinemachine's comment recommends Linear Algebra by Kenneth Hoffman and Ray Kunze (first published by Prentice-Hall in 1961), but it seems that only Pearson Education India still reprints this book....
Tsundoku's user avatar
  • 217
2 votes

Exercise books in linear algebra

There are several options in addition to those already suggested in other answers: Luis Barreira, Claudia Valls: Exercises In Linear Algebra. World Scientific, 2016. Each chapter contains a ...
Tsundoku's user avatar
  • 217
1 vote

Seeking Reference for Identity Involving Binomial Coefficients

You won't find any reference, the claim is not true. $$ 1\leq 2\\ 1+3\leq 2+2\\ $$ but $$ 3=\binom{1}{2}+\binom{3}{2}\not\leq \binom{2}{2}+\binom{2}{2}=2. $$
Sil's user avatar
  • 16.8k
2 votes
Accepted

Projective Representation Theory of Order 16 Nonabelian Groups

According to a computer calculation (in Magma) that I just did, the two groups $$\mathtt{SmallGroup}(16,3) = \langle x,y,z \mid x^4=y^2=z^2=1,yz=zy,y^x=yz,z^x=z \rangle$$ and $$\mathtt{SmallGroup}(16,...
Derek Holt's user avatar
3 votes
Accepted

Distributing elements of a multi-set to triplets with certain properties possible?

Yes, such a partition exists! Let's first build some intuition for why these constraints are so tight. Assuming that triples are supposed to have three distinct elements: Observe that for any distinct ...
Haran's user avatar
  • 10.3k
0 votes

Projective Representation Theory of Order 16 Nonabelian Groups

I think I may have solved this. Every other nonabelian metacyclic group of order $16$ with the exception of $Q_{16}$ has a dihedral presentation: $$G = \langle a,b \: | \: a^m=b^s=e, \: ba=a^rb \...
Isochron's user avatar
  • 1,355
3 votes
Accepted

Given $f \in L^p_{\text{loc}}$ and $\phi \in S$ (Schwartz class), do we have that $f \ast \phi \in C^\infty$? What if $f,\phi$ have compact support?

Claim 1 is false. If neither function has to have compact support, then the convolution may not be well-defined, e.g., take $\phi\in S$ nowhere zero, and $f=1/\phi$ which is continuous and therefore ...
Lieven's user avatar
  • 1,158
2 votes
Accepted

Linear independence of PDFs

Here is an example where MGFs make the proof simple. Suppose $f(x) = a N(\mu_1, \sigma_1)(x) + bN(\mu_2, \sigma_2)(x) = 0$ for all $x \in \mathbb{R}$, where $a, b \in \mathbb{R}$. Taking MGFs of both ...
Kakashi's user avatar
  • 1,831
2 votes

How useful is algebraic topology to computer science (Dieck vs Lee)?

On the face of it, I would guess that the part of dynamical systems that requires algebraic topology is somewhat removed from the part that is relevant to automata. More specifically, the most obvious ...
Lieven's user avatar
  • 1,158
0 votes

Six points in a full circular quadrilateral belong to one circle

COMMENT.-The problem can be stated in another equivalent way. From a point outside the given circle $R$ we draw two lines that determine the points $A,B,C,D$ of the circle which in turn determines the ...
Piquito's user avatar
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1 vote
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Algorithm to find if intersection of convex sets is empty

For a wide variety of convex sets that can be described by linear systems of equations and inequalities, second order cone constraints, and semidefiniteness constraints, you can apply a primal-dual ...
Brian Borchers's user avatar
0 votes

Find the Harmonic Mean Using Parabola

I found another case where the harmonic mean is the horizontal line $FH$, this situation occurs when the points $A,B,M$ are on the same line
زكريا حسناوي's user avatar
0 votes
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The midpoint of distance between the centers of similarity of two circles is the focal point of their tangent parabola

Our answer addresses OP’s request for a synthetic proof. $\underline{\text{Lemma:}}$ Radius of a circle that is internally tangential at two points to the parabola $\space x^2=4ay\space$ is equal to $...
YNK's user avatar
  • 4,302
0 votes

Find the Harmonic Mean Using Parabola

It seems that this method I'm talking about is just another version of the Cross ladder theorem.
زكريا حسناوي's user avatar
2 votes

Thurston's metric on $\widetilde{SL(2,\mathbb{R})}$ is twisted. So is this paper "wrong"?

After taking a carefull read in P. Scott's The Geometries of 3-manifolds, I'm posting a (partial) answer myself. Basically, I just misunderstood concepts. Many sources call Thurston's model geometries ...
Derso's user avatar
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0 votes
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Calculating the n-th power of any 2×2 matrix

I found a reference, see McLaughlin, where the applications listed in the post have already been done. However, I did not find a reference of the above proposition, which was concluded independently ...
Vaskara_GRek_O's user avatar
0 votes

Is there any online source for proofs than $(1+\frac{1}{n})^n$ and $(1+\frac{1}{x})^x$ is $e$?

This limit definition comes from the fact that $\frac{d}{dx}(e^x)=e^x$, so we can start by stating the formula for a slope with a really small $\Delta$, $\frac{f(x+\Delta)-f(x)}{\Delta }$ so this ...
assaf peretz's user avatar
3 votes
Accepted

How is the discriminant defined for $x^3+y^3+z^3+u^3+(ax+by+cz+du)^3+exyzu$?

I have good news & bad news. The good news is that there is a method to get the Discriminant. The bad news is that the eventual expression is going to be impractical & very hard to write out. ...
Prem's user avatar
  • 11k
3 votes
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Character tables of Coxeter Groups

For Weyl groups of type $A$, $B$ and $D$ GAP has built-in generic parameterized tables via ...
ahulpke's user avatar
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