# Tag Info

## Hot answers tagged reference-request

6

This is based on the following formula for the harmonic numbers: $$H_n = \int_0^1 \frac{1-t^{n}}{1-t} \, dt$$ Why is this true? In the integer case, we can just remember the identity $$1-t^{n} = (1-t)(1+t+t^2+\dotsb+t^{n-1});$$ then the integral is $$\int_0^1 \sum_{k=1}^{n} t^{k-1} \, dt = \sum_{k=1}^n \int_0^1 t^{k-1} \, dt = \sum_{k=1}^n \frac{1}{k} = ... 6 Because S^\infty is contractible, and we have a fiber bundles S^1 \to S^\infty \to \Bbb{CP}^\infty, the long exact sequence of homotopy groups of a pair shows that its only nonvanishing homotopy group is \pi_2(\Bbb{CP}^\infty) = \Bbb Z. See Hatcher, 4.50. Example 4.44 is a construction of this bundle. \Bbb{RP}^\infty is more elementary: if \tilde ... 4 This is Theorem 6.12 of Paul Mitchener's article C^\ast-categories. 4 See : Stewart Shapiro, Foundations without Foundationalism : A Case for Second-Order Logic (1991). 3 Yes, \rightharpoonup is usually used to indicate weak convergence. 3 A very famous textbook using the notation [n]=\{1,2,\dots,n\} is Enumerative Combinatorics (Volume 1) by Richard Stanley. It's in the list of notation at the beginning of the book, and is first used in Example 1.1.16. 3 My attempt at (2):$$\int_0^1 \frac{f(x_0)-f(t)}{1-t}dt = \int_0^1 \frac{\sum_{n=0}^\infty a_n x_0^n - \sum_{n=0}^\infty a_n (x_0t)^n}{1-t}dt = \int_0^1 \frac{\sum_{n=0}^\infty a_n x_0^n( 1 - t^n) }{1-t}dt = \int_0^1 \sum_{n=0}^\infty a_n x_0^n\sum_{j=0}^{n-1}t^j dt = \sum_{n=0}^\infty a_n x_0^n\sum_{j=0}^{n-1} \int_0^1t^j dt = \sum_{n=0}^\infty a_n ...

3

I like Lorenz Halbeisen's "Combinatorial Set Theory" book. It also gives some basic introduction to logic, and how it is used in set theory. The book itself is very thorough and the parts I have read were mostly well-written. Let me also add, that if you felt a bit shaky on the way logic was used in the set theory proofs, then perhaps it's best not to skip ...

2

I don't think it is true, Let $X= \mathbb{Q}$ which is clearly meager & $Y$ is singleton. and let $f:X\to Y$ be constant map, so has satisfied all the desired properties, but $Y$ is not meager, because any set is is open in $Y$ since it is singleton.

2

A standard reference is do Carmo. However, general tensors are not thorougly approached as far as I remember: he just mentions it en passant. Another good reference, but that may not be appropriate since you are not acquainted with Riemannian Geometry, is Helgason. Differently from do Carmo, he treats tensors extensively. Now, tangent bundle and tangent ...

2

MIT has an open course-ware (OCW) website. http://ocw.mit.edu/courses/mathematics/ There is multiple single variable calculus courses. University of Wisconsin has lecture notes available. It is made into a .pdf form. http://www.math.wisc.edu/~angenent/Free-Lecture-Notes/free221.pdf

2

There is nothing wrong with using valley in this context. It is a visual metaphor which will be readily understood. An alternative word might be trough, which is a bit more common in this abstract sense; but is still a visual metaphor.

2

I seem to be in a good position to answer this, so will give a try. I had the good fortune to work on the natural extension of Rosenthal's random walk on $SO(2n+1)$, namely the case where the rotation angle $\theta \neq \pi$. Even more luckily I got help from fellow graduate student at the time, Bob Hough, who contributed the critical insight that allowed us ...

2

I'm studying random geometric graphs (RGG's) in the context of ad-hoc wireless networks. I am not sure that I can help you but I will tell you what I know. Erdos-Renyi (or Bernoulli) random graphs are one example of a random graph but there are many others. Indeed, since the probability that a distinct pair of vertices share an edge is the same for all such ...

2

There are two nice introductions. First, as I mentioned in comments, it is Introduction to Random Graphs, a recent book on the classical theory of random graphs, which presupposes much milder prerequisites than, e.g., Bollobas' classic. (The book is not sold yet, but you can find a draft on one of the authors' webpages) I also very much like the monumental ...

2

Density of a sum is convolution of densities $$f_{X_1+X_2}(z)=\int f_{X_1}(y)f_{X_2}(z-y)dy$$ To see why support of $Y$ doesn't have to be $G$, consider $X_i=Binomial(1,1/2)+Uniform(-\epsilon,\epsilon)$ for $\epsilon<1/3$. Then $Y$ has non-zero density on the neighborhood of $1/2$ for even $n$ while density of $X$ is zero on that neighborhood. In general, ...

2

I seem to have found an answer regarding this. According to the conference paper The Bhattacharyya Metric as an Absolute Similarity Measure for Frequency Coded Data. (Thacker et al. 1997), the significance of the Bhattarcharyya distance: As the Bhattacharyya measure is equivalent to the Matusita measure we see that the Bhattacharyya metric is a ...

1

Maybe authors avoid putting the words "higher-order logic" in their titles to avoid frightening potential readers $\ddot{\smile}$. I'd recommend Peter Andrews' An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof

1

Non-standard topology normally starts with a topological space $\langle X,\tau\rangle$ and forms a non-standard extension ${^*X}$, so you have the topology on $X$ to begin with. However, you can then use the extension to prove things about $X$ just as you can use the hyperreals to prove things about $\Bbb R$. This PDF of an M.Sc. thesis at the Islamic ...

1

This result is a corollary of the following applied to the case where $\lambda$ is real: Theorem: Let $L : \mathcal{D}(L)\subseteq \mathcal{H}\rightarrow\mathcal{H}$ be a densely-defined symmetric linear operator on a complex Hilbert space $\mathcal{H}$ for which $(L-\lambda I)$ and $(L-\overline{\lambda}I)$ are surjective for some ...

1

The case $p=2$ is special, because the mean minimizes $\int |u-c|^2$ among all $c\in\mathbb{R}$. It does not have such a property for $p\ne 2$, and there is no reason for the stated inequality to hold. As PhoemueX noted, the Sobolev space is not really relevant: any $L^p$ function can be approximated by smooth functions in $L^p$ norm, and the quantities in ...

1

The equation is a single thing. The plural of maximum is maxima. There are multiple places of maxima. So the proper sentence is: "$t=\frac{(2n+1)\pi}{2}$ represents the places of maxima". "represets" is singular, "places" and "maxima" plural. Better explanation of period of 5/2? The period isn't 5/2. There were 5/2 periods. I would say "the function ...

1

A standard example is the derivative operator $\frac{d}{dx}$ on the f.d. vector space $P_k$ of polynomials of degree $\leq k$, with respect to the standard basis $\{1, x, \ldots, x^k\}$. In fact, since differentiation decreases the degree of a polynomial, $\frac{d}{dx}$ is actually strictly upper triangular (its diagonal entries are all zero) and nilpotent ...

1

To answer your second question: Start from $\frac{f(x_0)-f(x_0t)}{1-t}$ and replace $f$ with its series representation. A little algebra yields $$\frac{\sum a_n x_0^n-\sum a_n x_0^nt^n}{1-t} =\frac{\sum a_n x_0^n(1-t^n)}{1-t} =\sum a_n x_0^n(1+t+\dots+t^{n-1}).$$ You can integrate the last expression term-by-term in $[0,1]$ to get $$\sum a_n ... 1 I noticed, that the diagrams are wrong (I'm sorry!). They do not show P(\tilde B_n < x) -P(N \le x) but P(\tilde B_n = x) -\phi(x) (the difference in the density functions). So far I have the idea to use the Edgeworth series which states:$$P(\tilde B_n\le x) = \Phi(x) + \frac 1{\sqrt n} k_3 (1-x^2) \phi(x) + ...

1

As I see it, you have $\mathbb{R}^{n}$ along with a group action of $S_{n}$ acting by permuting coefficients of your vectors (wrt lets say the standard basis). For each permutation $\pi$, you have a map $\varphi_{\pi}:\mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R}$ by $\varphi_{\pi}:(x,y) \mapsto x \cdot y^{\pi}$ (each of these is actually a ...

1

I guess the easiest option would be to apply Lyapunov theory: https://en.wikipedia.org/wiki/Lyapunov_function https://en.wikipedia.org/wiki/Lyapunov_stability As a short summary: take $f(x)$ as a candidate Ljapunov function, which puts some reasonable restrictions on the shape of your optimization landscape (i.e. locally positive-definite function). All ...

1

Logic textbook "less verbose" and quite general : Dirk van Dalen, Logic and Structure (5th ed - 2013). As an alternative (with less topics covered) : Derek Goldrei, Propositional and Predicate Calculus : A Model of Argument (2005) Set theory : Derek Goldrei, Classic set theory (1996).

1

You could have a go with H-D Ebbinghaus, J Flum: Finite Model Theory and T Jech: Set Theory.

1

Rewrite the function into a form more amenable to differentiation \eqalign{ f &= \log(\det(V)) + {\rm tr}(V^{-1}S) \cr &= {\rm tr}(\log(V)) + S:V^{-1} \cr } where the colon (:) denotes the Frobenius product. Letting $d$ denote the derivative operator with respect to $\sigma_i\,\,$ differentiate the above to obtain \eqalign{ df &= ...

Only top voted, non community-wiki answers of a minimum length are eligible