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0

The trick is to apply the usual trace estimate $$ \|f\|_{L^p(\Gamma)} \le C \|f\|_{W^{1,p}(\Omega)} $$ to all the derivatives consecutively. Let $\alpha$ be a multi-index with $|\alpha|<m$ then $$ \|D^\alpha f\|_{L^p(\Gamma)} \le C \|D^\alpha f\|_{W^{1,p}(\Omega)} . $$ By carefully inspecting, which derivatives appear in the right-hand side, i.e., which ...


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My first suggestion would have been Schaum's outline. However since you have gone through that already, another book I am quite fond of (which I think covers a good portion of the topics you mentioned) is "Linear Algebra Problem Book" by Paul Halmos: http://www.amazon.co.uk/Algebra-Problem-Dolciani-Mathematical-Expositions/dp/0883853221 It possesses a ...


3

Let $x = \{x_k\}$ denote an arbitrary sequence in $\ell^p$. For $j \in \Bbb N$, let $x^{(j)} = \{x^{(j)}_k\}$ denote the sequence given by $$ x^{(j)}_k = \begin{cases} x_k & k \leq j\\ 0 & k > j \end{cases} $$ Note in particular that $x^{(j)} \in V$ for all $j$. Claim: In the space $\ell^p(\Bbb N)$, $x^{(j)} \to x$ as $j \to \infty$. Proof: We ...


3

Consider $(\mu_n)_{n\geqslant 1}$ a Cauchy sequence for the metric $\rho$. Then for each $f$ (measurable) and bounded by $1$, the sequence $\left( \int_X f(x)\mathrm d\mu_n(x)\right)_{n\geqslant 1} $ is Cauchy. In particular, for each measurable subset $A$ of $X$, the sequence $(\mu_n(A))_{n\geqslant 1}$ is convergent. By the Vitali–Hahn–Saks theorem, we ...


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Clarity seems to be a rare commodity in the literature on linear logic. However, Girard's Proofs and Types is the place to start for coherence spaces.


3

Partial answer: For $f$ one has: $$\begin{align} f(y)-f(x) &= \int_x^y f'(s) ds \\ &= (y-x) \int_0^1 f'(x+(y-x)t)dt \\ &= (y-x) E\left[f'\left(x+(y-x)\Theta\right)\right] \end{align}$$ with $\Theta$ uniformly distributed in $[0,1]$. Thus $$\begin{align} E[f(X+Y)]-E[f(X)] &= E[f(X+Y) - f(X)] \\[1em] &\left\downarrow\ f(y)-f(x) = (y-x) ...


1

There was just a typographical error in the textbook. Instead of $E[|X|\cdot \mathbf{1}_{[a,b]}(Y)]$ it has to be $E[|X| \cdot E[\mathbf{1}_{[a,b]}(Y)]]$. Then it is obvious, that $$E[|X| \cdot E[\mathbf{1}_{[a,b]}(Y)]] = E[|X| \cdot P(a \le Y \le b)]$$


1

I strongly recommend Scorpan's The Wild World of 4-manifolds. As the title suggests, it's mainly centered on dimension 4, but in its first part, it does a superb job at explaining what is special about low dimensions.


2

You need to show how an element $g \in L^1([0,1])$ corresponds to a bounded linear funtional $\varphi$ on $C([0,1])$. The usual way to do this is: $$ \varphi(f) = \int_0^1 f(t)\;g(t)\;dt,\qquad f \in C([0,1]) $$ So to complete this you have to show that the map $g \mapsto \varphi$ is what I claimed.


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The classical textbooks on set-point topology are John Kelley & Sam Sloan, General Topology. James Dugundji, Topology. An slightly easier textbook is John B. Conway, A Course in Point Set Topology.


2

While I second Noah Schweber's recommendation to read 'Computability and Logic', I also recommend this free textbook recently put together by a group of logicians aimed at providing a free and rigorous introduction to logic which goes into computability theory and meta-logic as well. http://people.ucalgary.ca/~rzach/static/open-logic/open-logic-complete.pdf ...


2

By far my favorite book on mathematical logic is "Computability and Logic" by Boolos, Burgess, and Jeffries, now in its fifth edition (I learned logic from the fourth ed., which is available used for cheap). I cannot recommend it highly enough. A brief outline of the book: the first 8 chapters cover basic computability theory. This nicely complements the ...


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You could start by reading the article A brief history of loop rings by E. G. Goodaire and trace back the papers citing this one. See also Advances in loop rings and their loops from the same author.


1

I'm basing myself on McCleary's book A user's guide to spectral sequences, sections 1.3 and 2.4. He actually calls them "spectral sequences of algebras", not "multiplicative spectral sequences". But it's probable that other sources give similar definitions (in the end, it all depends on what your applications are, I guess). Yes, the product is almost* ...


4

I suggest Peter May's Concise course on algebraic topology. You will find e.g. categorical formulations (and proofs) of the van Kampen theorem and the classification of covering spaces.


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The closest thing I've found is Strom's Modern Classical Homotopy Theory, although I haven't read much of it. Chapter 1 is called Categories and Functors, so that's a good start. This is the only introductory algebraic topology textbook I know of that explicitly uses the language of homotopy limits and colimits.


2

Rotman's An Introduction To Algebraic Topology is a great book that treats the subject from a categorical point of view. Even just browsing the table of contents makes this clear: Chapter 0 begins with a brief review of categories and functors. Natural transformations appear in Chapter 9, followed by group and cogroup objects in Chapter 11. The aspect I ...


4

Spanier's book is relatively old (so I know it does not quite answer your question), but excellent. It uses category theory from the get-go. Riehl's "Categorical homotopy theory" is very well-written, though it may be a bit too advanced if you hadn't seen a bit of algebraic topology already. Riehl's book is focused on the categorical aspect via Quillen model ...


0

Ronald Brown's text Topology and Groupoids is probably what you want from a topology text. He gives an introduction to general topology and the fundamental groupoid using the language of category theory throughout. It's an excellent textbook.


2

Let $f : A \to X$ be a based map of based spaces. The homotopy pushout $X \coprod_A \text{pt}$ is called the homotopy cofiber, cofiber, or mapping cone of $f$; I'll denote it by $X/A$. Iterating this construction produces the cofiber sequence or Puppe sequence $$A \to X \to X/A \to \Sigma A \to \Sigma X \to \Sigma X/A \to \dots$$ which is in some sense the ...


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Well, I think this is a solution to the problem: 1 - If there was convergence, then, by the Uniform Boundedness Principle, as $\sup_R \|S_R f \|_1 < \infty$, then the operators $S_R$ should be bounded in $L^1$. It can be shown that, conversely, it is enough that the latter condition holds for the convergence to hold. 2 - As the multiplier for $S_R$ is ...


2

It seems the following. A space $X$ is almost $\omega_1$-Lindelöf [Par][Mat, p. 92], if every open cover of cardinality at most $\omega_1$ has a countable subfamily whose union is dense in $X$. It is easy to check that a space $X$ has Property A iff $X$ is almost $\omega_1$-Lindelöf. See the diagrams at [Mat, p.92] and at [DRRT, p.94] about the relations of ...


2

I am not sure if this exactly what you want, but a homotopical perspective on homology is given in the book partially titled Nonabelian Algebraic Topology, EMS Tracts, vol 15 (2011) (pdf available there). The main results do not assume singular homology, but nevertheless give results such as the Relative Hurewicz Theorem, (!), and results on second ...


0

I thought you might enjoy the fact that the value of the infinite product $$\prod_p\frac{p^2+1}{p^2-1}$$ is $\frac{5}{2}$.


1

Suppose that $x=\langle x^{(j)}:j\in J\rangle\in (E^I)^J$, where each $x^{(j)}=\langle x_i^{(j)}:i\in I\rangle\in E^I$. Similarly, let $y=\langle y^{(j)}:j\in J\rangle\in (E^I)^J$, where each $y^{(j)}=\langle y_i^{(j)}:i\in I\rangle\in E^I$. Then we want to define $\mathfrak{W}$ so that $$\begin{align*} ...


0

The book Fabian, Habala, Hájek, Montesinos, Zizler: Banach Space Theory, The Basis for Linear and Nonlinear Analysis (CMS Books in Mathematics) has many exercises at the end of each chapter. It does not have solutions, but most of the exercises come with some hint (in some cases rather detailed). Google Books link, DOI: 10.1007/978-1-4419-7515-7 This book ...


1

One can also prove the theorem using Nevanlinna theory. See here, for example.


1

The biggest number that is not too big to be the limit is one way to describe the limit superior, $\limsup_{x\to a}f(x)$, and the smallest number that is not too small to be the limit is one way to describe the limit inferior, $\liminf_{x\to a}f(x)$. Any number $l\in[\liminf,\limsup]$ is a number that is neither too big nor too small to be the limit. And if ...


0

As a fellow undergraduate, I'm working my way through the book Introduction to Abstract Algebra (4th Edition) by W. Keith Nicholson. The book contains all of the areas of Algebra you mentioned and more, and there are a plethora of problems for each of the sections on Group Theory, Ring Theory, Field Theory, Galois Theory, and so on. Each section has ...


7

It's called the prime constant. When you enter that number into WolframAlpha and you see the $\mathcal{P} = 0.41468250985111166$, notice that in the bottom right-hand corner of that cell it says "$\mathcal{P}$ is the prime constant", which links to the Wolfram Mathworld page explaining what it is.


0

Not an answer, but too long for a comment: Something you may be interested in is the modal logic approach to forcing, which is probably the most important and ubiquitous technique in modern set theory, as developed in the book by Fitting and Smullyan (see the review http://www.jstor.org/stable/2586777?seq=2#page_scan_tab_contents, which includes a critique ...


2

user, Interesting choice for a topic of self-study! Good luck! Steen and Seebach's Counterexamples in Topology is phenomenal, a true must-have, and rigorously defines a lot of worthwhile spaces (which will also help come exam time if you have a professor who loves counterexamples!) It's relatively inexpensive and makes for great bathroom reading ...


0

The simplest way to show that for a non-compact manifold the resolvent does not have to be compact is to give a counterexample. Following your (very good) choice of the manifold $M=\mathbb{R}^n$, lets show that $(-\Delta - \lambda I)^{-1}$ is not compact. For simplicity, assume $n=2$. Assume $\vec{\mathbf{x}}, \vec{\mathbf{y}} \in \mathbb{R}^2$. Then the ...


2

For $M=\mathbb R^n$ take a smooth function with compact support $\phi\in C_0^\infty(M)$. Then $$ \phi = (-\Delta+\lambda I)^{-1}((-\Delta+\lambda I)\phi) $$ obviously. Now consider translations $\phi_k(x):=\phi(x+kv)$ for $v\in \mathbb R^n$, $k\in \mathbb N$. Take the norm of $v$ large enough such that the supports of different $\phi_k$'s are empty. The ...


3

This would be the corona of $K_n$ and $K_1$, usually denoted $K_n \circ K_1$. The original definition was by Harary and Frucht in 1970 in their paper "On the Corona of Two Graphs" See this question for an application of it to more general graphs: Eccentricity in corona product


5

Mertens' Theorem says: $$\lim_{n \rightarrow \infty} \ \frac{1}{\log p_n} \prod_{k=1}^{n} \frac{1}{1 - \displaystyle{\frac{1}{p_k}}} = e^{\gamma}.$$ Euler's product formula for the $\zeta$ function and his evaluation of $\zeta(2) = \pi^2/6$ says that $$\zeta(2) = \lim_{n \rightarrow \infty} \ \prod_{k=1}^{n} \frac{1}{1 - \displaystyle{\frac{1}{p^2_k}}} ...


3

This can be found in the proof of Theorem 5.2 of the following paper: Handel, David. "On products in the cohomology of the dihedral groups." Tohoku Mathematical Journal, Second Series 45, no. 1 (1993): 13-42. In particular, for $m$ even and $n>0$, we have: $$ H^n(D_m;\mathbb{Z}) \;=\; \begin{cases} (\mathbb{Z}/2)^{(n-1)/2} & \text{if }n\equiv 1\pmod ...


0

Surprisingly, the following (in my opionion) quite elegant proof is still missing: Look at the $F$-vector space $F[x]/(f)$. The map $$\phi : F[x]/(f)\to F[x]/(f),\quad g + (f)\mapsto x\cdot g + (f)$$ is well-defined and $F$-linear. Let $m_\phi = \sum_{i=0}^d a_i x^i\in F[x]$ be the minimal polynomial and $\chi_\phi\in F[x]$ the characteristic polynomial of ...


0

Proof: Since $K$ is compact, we may impose the following assumptions on $K$: $K$ consists of a finite union of closed balls. The poles of $K$ are contained in the interior $K^\circ$. Let $z_1,z_2,\ldots,z_k$ be the poles of $f$ in $K$ with multiplicities $m_1,m_2,\ldots,m_k$. Define $q(z)=(z-z_1)^{m_1}\cdots(z-z_k)^{m_k}$, and define $g(z)=f(z)/q(z)$, ...


0

I presume by "ratio" you mean something like $\frac{|N(A) \cap N(C)|}{|N(A)| + |N(C)|}$ where $N(X)$ is the neighbor set of $X$. This is an example of "collaborative filtering," and it's a technique applied in machine learning. You can read about it here or here, for example. I presume you are describing a situation of the following form: if $A$ and $C$ ...


1

Donald L. Cohn, Measure Theory (Birkhäuser 1980) First Edition Theorem 8.3.6, page 275 (2nd Edition, p. 259) alternate proof: Exercise 5, page 277. Outline: (a) Every Borel subset of a Polish space is Borel isomorphic to a Borel subset of $\{0,1\}^{\mathbf N}$. (b) Each uncountable Borel subset of a Polish space has a Borel subset that is Borel ...


0

the book Hardy wright theory of numbers has the equation you stated.author is probably Mordell and Hammond.


0

An answer can also be found in the section 6.3 of the book "L. Comtet, Advanced Combinatorics, Springer 1974" and asymptotic formula $f(n)\sim e^{-2}36^{-n}(3n)!$ can be derived from a general asymptotic formula in the paper "C. J. Everett, P. R. Stein, The asymptotic number of integer stochastic matrices, Disc. Math. Vol. 1, No. 1 (1972) 55-72".


2

Here are three (somewhat standard) references: Jurgen Jost, "Riemannian Geometry and Geometric Analysis." Peter Li, "Lecture Notes on Geometric Analysis." Thierry Aubin, "Nonlinear Analysis on Manifolds. Monge-Ampere Equations." Jost's book is on its sixth edition. Aubin's book has a first and second edition, although my understanding is that the first ...


1

The metric you described is the standard metric on the projective space: in the real case it can be visualized as the angle between lines (thinking of the elements as lines). It arises as the quotient of the spherical metric on $S^n$ by the group of isometries $\{x\mapsto \alpha x, \ |\alpha|=1\}$ where $\alpha$ belongs to the ground field, $\mathbb{R}$ or ...


0

The largest distributive lattice of rank $n$ has $2^n$ elements, so the largest distributive sublattice of any lattice of rank $n$ has size at most $2^n$. For any geometric lattice of rank $n$ this bound can be achieved by taking the sublattice generated by $n$ atoms forming a basis for the underlying matroid.


2

This is not true. In fact, Kolmogorov constructed (1923) an example of a $L^1$ function whose Fourier series diverges almost everywhere (later improved to everywhere divergent). Om the other hand, if $f\in L^p$ for some $p>1$, it's a deep theorem by Carleson (the $p=2$ case) and Hunt ($p>1$) that the Fourier series of $f$ converges pointwise almost ...


1

Much longer comment turned answer: Howard Eves' An Introduction to the History of Mathematics seems like a perfect fit. The chapters that would be of particular use are chapters 11 through 14 (relevant sections included on the side): Chapter 11: The Calculus and Related Concepts [11.9 Newton; 11.10 Leibniz] Chapter 12: The Eighteenth Century and the ...


0

My reference is this post. Note that it is not necessary for $A$ to have an identity. Let $\mathscr{C}$ and $\mathscr{D}$ denote, respectively, the collection of ideals of $A$ containing $I$, and the collection of ideals of $A/I$. Define: $f:\mathscr{C}\to\mathscr{D}$ by $f(J) = \{a + I \mid a\in J\}\subset A/I$. Define $g:\mathscr{D}\to\mathscr{C}$ by ...


1

This is maybe not a full answer, but too long for a comment. Any two Morse functions are homotopic through the obvious homotopy $t\mapsto t\,f+(1-t)g$. This homotopy will not be through Morse functions. This is easy to see for a closed manifold as the number of critical points cannot change through such a homotopy. The homotopies can be done through ...



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