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There is also the (old) book of Seneta: Non-negative Matrices and Markov Chains. First edition: 1973. See chapters 3-4.

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This is a site for mathematical questions. Let me try to state your question in mathematical terms, and you tell us whether I translated it correctly or not. Let $\mathbf{x}=(x_1,x_2,x_3,y)^T$ be a time dependent vector in $R^4$, satisfying the differential equation $$\mathbf{x}'=A(\mathbf{x})\mathbf{x},$$ where $A$ is the matrix $$A=\left(\begin{array}{cccc}... 2 Posted from the comments (1 2 3), by request. Of course, the spinor formula itself arguably is an explicit formula, for some values of explicit, so I'll take explicit to mean polynomial in the entries, as you suggest—in which case I imagine one can prove rigorously that the answer is ‘no’. If some special formulæ are instead of interest, Jessica Fintzen, ... 2 This is not an answer that points to a more recent and more accessible account of the triangulability of surfaces, but rather a way to make the account in the first chapter of Ahlfors' & Sario's book more accessible, if sufficient time is available. It should be noted that the proof given by Ahlfors & Sario works for all (connected, 2nd countable) ... 3 See Lemma 3.2 in the following paper: R. W. Thomason, Les K-groupes d'un schéma éclaté et une formule d'intersection excédentaire, Invent. Math. 112, 195--215 (1993), DOI. 0 Let$$g^m_n := f_n \psi_m(f_n).$$The assumptions mean that (f_n)_n is a bounded sequence in L^p(a,b) and that g_n^m is relatively compact in L^p(a,b) for each m. We use the Frechet-Kolmogorov theorem characterizing compactness in L^p spaces to show that this transfers to (f_n)_n. (Then not only (f_n)_n has a convergent subsequence, but also ... 2 Perhaps, Proposition 1.28 in https://arxiv.org/abs/1411.7994 may help. 4 Erdmann's article Schur algebras of finite type shows that S(n,r) has finite representation type in prime characteristic p if and only if n=2 and r < p^2 or n \ge 3 and r \le 2p or p=2, n=2 and r=5 or 7. In these cases the quiver and relations for (the basic algebra Morita equivalent to) each block are found explicitly. Quivers and ... 2 This list is certainly far from being complete, but it contains some important results obtained in the last 20 years. The following thesis discusses some recent results obtained by Bell (see Section 5): Michelle Roshan Marie Ashburner (2008). A Survey of the Classification of Division Algebras over Fields. Master Thesis, University of Waterloo This is a ... 6 I can recommend Topics in Chromatic Graph Theory (Encyclopedia of Mathematics and its Applications) with editors Lowell W. Beineke and Robin J. Wilson. It is from 2015, and if you are interested in chromatic topological graph theory topics, there are three relevant chapters for you: Chapter 1: Colouring graphs on surfaces, chapter 4: Hadwiger's conjecture, ... 7 The problem is discussed in a more general setting (operator ideals in Banach spaces) for so-called analytic semigroups (parabolic problems) in Blunck, S.; Weis, L., Operator theoretic properties of differences of semigroups in terms of their generators, Arch. Math. 79, No. 2, 109-118 (2002). ZBL1006.47036. The paper semms to be freely accessible. The idea ... 6 A more precise form of this result is originally due to Miyaoka and Mori, see Theorem 3.6 page 67 in Olivier Debarre: Higher-dimensional algebraic geometry, Universitext. New York, NY: Springer. xiii, 233 p. (2001). ZBL0978.14001. 1 As mentioned in the answer by user6976, there is the idea of development of algebraic geometry to (essentialy) any general algebraic system. This is carried out(following Plotkin's work) by E. Daniyarova, A. Miasnikov, V. Remeslennikov and co-authors in a series of papers. A more recent survey (2016) of this area, called Universal Algebraic Geometry, of ... 5 Example 2 of arXiv:0704.2826 considers the analogous problem for the continuous-time random walk, in the more general case that the curve has the form g(t)=a+b\sqrt{T-t} with a+b\sqrt T\geq 0. The random walk starting at the origin stays below that curve for all t<T with probability$$P(T,a,b)=1-\frac{\int_{-\infty}^{-a/\sqrt T} e^{-y^2/2}dy}{\int_{-...

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A great and current reference is "Algorithms for embedded graphs" from Éric C. de Verdière, it is a 66 page synthesis of his course notes from 2017 (find here: http://monge.univ-mlv.fr/~colinde/cours/all-algo-embedded-graphs.pdf). Covers topological graph theory plus related algorithms e.g. to minimize edge length of embedded graphs. See this quote ...

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I would like to add one important aspect: it was known that $\mu(G)$ and $\sigma(G)$ can deviate by a large amount for larger values $k$. Now we have the proof of the improved (sharper) bound $\mu(G)\leq\sigma(G)$, but even though this is an improvement, Kaluza and Tancer also showed that a large gap exists already for small values of $k$: They showed there ...

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Stuck - On some mathematical aspects of deterministic classical electrodynamics The article found through this link provides an exposition of all of electromagnetism using differential forms and exterior calculus. All of Maxwell's equations are developed using differential forms from exterior calculus; undoubtedly all of this was known to Grassmann in his ...

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Replying to this somewhat late but I think there is important context to this. The Lagrangian Grassmannian and the quadric are both examples of R-spaces (symmetric R-spaces in particular) as such they are given by $G/P$ for $G$ a semisimple Lie Group and $P$ some parabolic. So this question boils down to $\mathfrak{sp}(4,\mathbb{R}) \cong \mathfrak{so}(2,... 8 Kaluza and Tancer have actually proved$\mu(G)\leq\sigma(G)$in 2019: See their proof in the preprint "Even maps, the Colin de Verdière number, and representations of graphs" on arxiv. Here is the link https://arxiv.org/pdf/1907.05055.pdf You are right, the invariant$\sigma(G)$of Holst and Pendavingh does not seem to have an established name yet. 3 I can recommend two books to you which I think give a rather good coverage of the foundations for determinacy and infinite games Hugh Woodin: The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. Walter de Gruyter, Berlin 1999. Gerald Sacks (ed.): Mathematical Logic in the 20th Century, SUP and World Scientific Publishing Singapore and ... 4 Coincidentally, a day after you asked this question, Alex Simpson gave a nice talk (video, slides) where he gave a synthetic formulation of probability theory. In this formulation, random variables are a primitive notion, not maps from a sample space to a measurable space. Hence there's no need to keep track (or even mention) sample spaces at all. That's ... 2 Since the garden variety Fréchet nuclear spaces (the Schwartz space, the space of smooth functions on the torus, the space of smooth fuctions with support in a given compact set, etc., Edit: see also Jochen's comment below) are isomorphic as topological vector spaces to$\mathscr{s}$, the space of rapidly decaying sequences, and since the proof of the Montel ... 2 Additional references: [Sch99, §III.7.2, Corollary 2]: every bounded subset of a nuclear space is precompact. [Pie, §4.4.7]: In each nuclear or dual nuclear locally convex space$E$all bounded subsets are precompact. References. [Pie72]: Albrecht Pietsch, Nuclear Locally Convex Spaces (1972), Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, ... 2 So directed graphs are not well-quasi-ordered by butterfly minors; see the intro of [BPP]. Furthermore, there are reasons to think that many of the FPT results for graph minors may not hold in the directed setting (ie [PW]). Yet, perhaps surprisingly, it may still be possible to get a structure theorem for butterfly minors! There is an ongoing project to do ... 3 To see what you might expect for a relation, consider the case of a$2\times 2$matrix$M=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$, with eigenvalues$\lambda_\pm=\tfrac{1}{2}(a+d)\pm\sqrt{4bc+(a-d)^2}$. Knowledge of$a$and$d$is a constraint on the sum of the eigenvalues (the trace of$M$), but there is no other constraint. I'm pretty sure this ... 4 Q1: The state of the art as reported in 2014, see arXiv:1406.2097, is that the only numbers which are known to be Tarski numbers of some groups are 4,5,6. Tarski numbers$<4$are forbidden, which suggests that 7 is the answer to the question "What is the smallest non negative integer that we do not know yet is the Tarski number of a group?". Q2: ... 4 In Addition to Geck-Pfeiffer: Small values of the a-function are also contained in Geck, Jacon - Representations of Hecke algebras at roots of unity. In particular, for$G_2$it's in Table 1.3.; for$F_4$it's table 1.2.; other values are available through combinatorially formulas (for example type$A$is completely covered in section 2.8) and Remark 1.3.11 ... 15 Thurston approaches 3-manifolds by cutting them up along various surfaces (one first cuts along spheres [Kneser-Milnor] and then along tori [Jaco-Shalen-Johannson]) into pieces which each admit a locally homogeneous geometric structure, modelled on a homogeneous space with an invariant Riemannian metric. A compact, simply connected manifold with such a ... 2 This is the monograph of J. C. Cha which discusses the rational version of filtrations of knot concordance groups comparing with the original paper. Cha, Jae Choon. The structure of the rational concordance group of knots. Vol. 182. American Mathematical Soc., 2007. The following article includes a very nice application of Cochran-Orr-Teichner's work. Here,... 5 There are summarys of parts of Cochran, Teichner and Orr's paper in: These lecture notes of Peter Teichner, typed up by Julia Collins and Mark Powell; Mark Powell's 2011 Edinburgh PhD thesis; Julia Collins' 2012 Edinburgh PhD thesis. You'll find all of these references (and I'm sure more besides) at Andrew Ranicki's webpage. Disclaimer: I'm not a geometric ... 7 The Electronic Journal of Combinatorics has many Dynamic Surveys one of which is The Graph Crossing Number and its Variants: A Survey by Schaefer which first appeared in 2013 and has been updated as recently as Feb 14, 2020. From the bottom of page 40 onto page 41 you will find this conjecture for complete bipartite graphs discussed (with many references). ... 4 It is a fascinating conjecture. The following might be a good reference for you: In 1997, Richter & Thomassen showed that $$\lim_{n\to\infty}cr(K_{n,n})\left(\begin{array}{c} n \\ 2 \end{array}\right)^{-2}$$ exists and is at most$1/4$. If the conjecture is true, the value of this limit is exactly$1/4$. (R.B. Richter, C. Thomassen, "Relations ... 6 Schur Weyl duality holds in the super case, as well. There is the double centralizer property, thus a positive answer to Q1, and also a characterization of the kernel as those ideals of$\mathbb C[S_d]$which correspond to partitions that don't fit inside the (m,n)-hook. See the paper "Hook Young diagrams with applications to combinatorics and to ... 4 This is a result of Sergeev, but I can only find the article in Russian at the moment. If I remember right, the map is surjective, and the kernel can be worked out from the fact that a Schur functor applied to the standar representation of$\mathfrak{gl}(m|n)$vanishes iff the Young diagram contains the box at position (m+1,n+1), i.e. if they do not fit in ... 1 We studied this in a formal computability/complexity theoretic setting here: A Pauly, D Seon & M Ziegler: Computing Haar Measures CSL 2020 Our work only deals with the compact case, not the locally compact one. To summarize: While the usual proofs of the existence of the Haar measure are not actually constructive, there are constructive proofs that ... 0 The smallest interesting case of$k=2$reduces to a family of Pell equations paramaterized by$b$: $$(2c-1)^2 - b^3(2a)^2 = 1.$$ This gives infinitely many solutions. For example, for$b=2$, we have a series of solutions indexed by$n$: $$c_n + a_n\sqrt{8} = \frac{(17+6\sqrt{8})^n+1}2.$$ Numerical values of$c_n$are listed in OEIS A055792. 2 The discussion from Section 13.1 in the book of Gilbarg and Trudinger shows that$u \in C^{1,\,\alpha}\left(B_{3/4}^+\right)$. From here one can apply Schauder estimates for linear equations. For example, one can pass the divergence on the left hand side and view$u$as a solution to a non-divergence form linear equation with Hölder continuous coefficients (... 1 You might be interested in extensions to the Sylvester Schur theorem, which by your constraints shows that c is bigger than k^2 as the set of consecutive integers in the product must have a single multiple of q^2 for some prime q bigger than k. A paper of Saradha and Shorey from 2003, Almost Squares and Factorizations in Consecutive Integers, shows the ... 3 You may already know this, but numbers of the form$a^2b^3$are called powerful numbers. A closely related question that might provide information on your question is to ask for binomial coefficients that are powerful. A Google search of "powerful number" and "binomial coefficient" brought up the following paper of Granville: On the ... 3 Consider the quotient map$\pi:\mathrm{X}\times\mathrm{X}\rightarrow\mathrm{S}^2\mathrm{X}=(\mathrm{X}\times\mathrm{X})/\mathrm{S}_2$, and a point$p$of$\mathrm{X}\times\mathrm{X}$. Then the tangent space of$\mathrm{S}^2\mathrm{X}$at$\pi(p)$equals the tangent space of$\mathrm{T}_p(\mathrm{X}\times\mathrm{X})/(\mathrm{S}_2)_p$at the origin (see for ... 6 A Markov chain on the symmetric group with this transition graph (but with directed edges and weights) was investigated by Lam and Williams. This has since received considerable attention, and has been connected to "TASEP on a ring" if you are looking for search words (it doesn't appear that the graph itself has a name in this context). I should ... 2 I figured out the details, and wrote it up here. I did not manage to find a good reference. There are a few nice surveys on RSK and on crystals, but a survey covering how different tableau operators interact would be nice to see someone type up. For the interested, these properties above were needed for this project, where we look at a type of skew q-... 2 Brownian motion, i.e. Wiener measure, is a good source of ideas and examples here. For instance if$W_t$is 1-dimensional standard Brownian motion at time$t$and $$P(\forall x\,F(x)=x^2)=1$$ and$Y=W_1$then$F$and$Y$are independent but $$E(F(Y))=1\ne E(F(E(Y)))=E(F(0))=0.$$ (Actually here we just need that$W_1$is a standard normal random variable.) 0 This one looks far more interesting than google translate. I tested it at least for the German-English-German and the French-English-French and I found it more reliable for natural language processing and far more accurate: https://www.deepl.com/translator 7 I like a lot the book from Beineke & Wilson (editors) "Topics in Topological Graph Theory" from 2009 for that purpose. Take a look at the article "Open Problems" from Archdeacon in this book. It is just like 5 pages or so, but inspired me a lot. I think you could find it very useful. 2 Call the two series$S_1, S_2$. Start out by letting$g_1=1$. Whatever we do afterwards, this makes sure that$S_1\ge 1$. Next, fix an$M$such that$u_M e^{-1\cdot u_1}\ge 2$, and then give$g_2, \ldots, g_M$a common small value that will give us $$e^{-\sum_{j=2}^M g_j u_j}\ge \frac{1}{2} .$$ This guarantees that$S_2\ge 1\$. Now just continue in this way....

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If you are interested in tangles in the sense of Robertson and Seymour, this is just to provide some perspective on it. I am working on this for my Ph.D. project and I thought maybe it is considered helpful if I share this high-level, intuitive perspective here (it is not a detailed definition): The best and shortest description, I think, is given in the ...

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Pohlers's 1989 book 'Proof Theory, An Introduction' gives a very clean, streamlined approach (based on work by Tait.) Takeuti's presentation in his 'Proof Theory' is closer to Gentzen's original proof, but is much less readable than Pohlers.

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$$B=\left[\matrix{1&1&1&1\\1&1&-1&-1\\-1&-1&-1&1\\-1&1&-1&1}\right]$$ and probably many other solutions. I'm also voting to close because you didn't pose a research-level problem. If you have an interesting general case, pose that.

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