# All Questions

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### How does the work of a pure mathematician impact society? [closed]

First, I will explain my situation. In my University most of the careers are doing videos to explain what we do and try to attract more people to our careers. I am in a really bad position, because ...
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### Periods and commas in mathematical writing

I just realized that I am a barbarian when it comes to writing. But I am not entirely sure, so this might be the right place to ask. When typing display-mode formulae do you guys add a period after ...
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### Elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...
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Numerical evidence suggests: $$\int_0^{\frac12}\int_0^{\frac12}\frac{1}{1-x^2-y^2} dy \, dx= \frac{G}{3}\qquad (1)$$ Couldn't find the indefinite integral, though maple simplifies (1) to $$\int ... 2answers 135 views ### Image of poset with Hausdorff interval topology Given a poset (P,\leq) the interval topology \tau_{\text{int}}(P) on P is generated by$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$where \downarrow x = ... 1answer 116 views ### Product of posets with Hausdorff interval topology Given a poset (P,\leq) the interval topology on P is generated by$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$where \downarrow x = \{y\in P: y\leq x\} and ... 1answer 80 views ### Order-preserving images of (\mathcal{P}(\kappa),\subseteq) Is there a cardinal \kappa \neq \emptyset and a connected poset P of cardinality \leq \kappa such that there is no surjective order-preserving map from (\mathcal{P}(\kappa),\subseteq) onto ... 12answers 10k views ### Is pi a good random number generator? Part of what I do is study typical behavior of large combinatorial structures by looking at pseudorandom instances. But many commercially available pseudorandom number generators have known defects, ... 11answers 3k views ### Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points? Is there an infinite field k together with a polynomial f \in k[x] such that the associated map f \colon k \to k is not surjective but misses only finitely many elements in k (i.e. only ... 3answers 527 views ### Generalization's of Greene's Theorem for the Robinson-Schensted correspondence One important property of the Robinson-Schensted correspondence (RS) is that the longest increasing subsequence of the permutation \sigma is \lambda_1, the first entry of the shape ... 12answers 9k views ### why is it so cool to square numbers? (in terms of finding the standard deviation) When we want to find the standard deviation of \{1,2,2,3,5\} we do$$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$. Why ... 3answers 527 views ### Looking for “large knot” examples This question is about knots and links in the 3-sphere. I want to find an example of a "large" knot or link with some special properties. I'm looking for some fairly specific examples, but I'm also ... 5answers 6k views ### Computing the Galois group of a polynomial Does there exist an algorithm which computes the Galois group of a polynomial p(x) \in \mathbb{Z}[x]? Feel free to interpret this question in any reasonable manner. For example, if the degree of ... 3answers 562 views ### A fibrant-objects structure on Top (Sorry for the crossposting, but I'm really interested in this question). One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ... 2answers 512 views ### Ergodic theory and dynamical systems books references I am arranging a weekly meeting of 2 hours with postgraduate students in ergodic theory (for a period of 3 weeks). I am asking here for an advice of a book (or maybe a set of papers) to look at ... 2answers 688 views ### An integrality question about expressing an integer as a product of numbers below n Let n\ge 2 be a natural number. Suppose that N is a natural number, composed only of primes below n, and that can be expressed as$$ N= \prod_{j=1}^{n} j^{x_j} $$where x_1, \ldots, ... 6answers 5k views ### Book on mathematical “rigorous” String Theory? I've been looking high and low for a mathematical Book on String Theory. The only Book I could find was "A Mathematical Introduction to String Theory" by Albeverio, Jost, Paycha and Scarlatti. I only ... 8answers 1k views ### Natural examples of sequences of adjoint functors I am looking for examples of sequences of adjoint functors. That are (possibly bounded) sequences$$(...,F_{-1}, F_{0}, F_1, F_2,...)$$such that each F_n is left adjoint to F_{n+1}. We call such ... 1answer 3k views ### Geometric interpretation of characteristic polynomial The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ... 5answers 2k views ### Generating finite simple groups with 2 elements Here is a very natural question: Q: Is it always possible to generate a finite simple group with only 2 elements? In all the examples that I can think of the answer is yes. If the answer is ... 1answer 2k views ### An etale version of the van Kampen theorem Let V be a smooth connected algebraic variety over an algebraically closed field k. Let W_1, W_2 be closed subvarieties of V of positive codimension whose intersection W_1 \cap W_2 has ... 3answers 2k views ### Number of Normal subgroups In a p-Group Dear all, Does someone know of any paper/method that enables us counting/estimating the number of normal subgroups of some p-group of order p ^n  ( n is some natural number ? ) . Is there anyway ... 4answers 2k views ### Is there a universal countable group? (a countable group containing every countable group as a subgroup) This recent MO question, answered now several times over, inquired whether an infinite group can contain every finite group as a subgroup. The answer is yes by a variety of means. So let us raise the ... 2answers 1k views ### The difference between a handle decomposition and a CW decomposition Let M be a compact finite-dimensional smooth manifold. I have a question about the relationship between the statements that a Morse function induces a handle decomposition for M, and that it ... 3answers 704 views ### Computing the Euler characteristic of the complex projective plane using differential topology I am trying to compute \chi(\mathbb{C}\mathrm{P}^2) using only elementary techniques from differential topology and this is proving to be trickier than I thought. I am aware of the usual proof for ... 2answers 209 views ### Left determined model structure on delta-generated topological spaces Consider the class of cofibrations of the Quillen model structure, restricted to delta-generated topological spaces (the full subcategory of topological spaces generated by the colimits of simplices). ... 6answers 8k views ### Reading materials for mathematical logic Hi everyone, the summer break is coming and I am thinking of reading something about mathematical logic. Could anyone please give me some reading materials on this subject? 1answer 489 views ### A chain homotopy that does not arise from a homotopy of spaces? Algebraic topologists like to cook up algebraic invariants on topological spaces in order to answer questions, so they are often concerned with how strong those invariants are. Currently, I am ... 0answers 176 views ### Is this (funny) combinatorial optimization problem NP-hard ? (cutting numbers and placing them in urns) The parameters of the problem are m numbers which are integers (these numbers are denoted b_i), n urns and in each urn, we can place C numbers. We assume nC \geq m so that the problem is ... 2answers 1k views ### (co)homology of symmetric groups Let S_n=\{\text{bijections }[n]\to[n]\} be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the \mathbb{Z}-modules H_k(S_n;\mathbb{Z})? Using GAP, we ... 0answers 11 views ### Distance minimization and submodular functions I have a question about square distance minimization and submodular functions. Suppose I have a random variable X = (X_1,...,X_n) over \mathbb{R}^n. Let x be a realization of this random ... 0answers 160 views ### A “slice-map” type problem for symmetric tensors in the square of a nuclear C*-algebra Throughout: let \otimes denote the minimal (i.e. spatial) \newcommand{\Cst}{{\rm C}^*}\Cst-tensor product of two \Cst-algebras. Let B be a unital, nuclear \Cst-algebra and let A\subset B ... 1answer 108 views ### Stinespring's dilation without C^{\ast}-algebras Does Stinespring's dilation theorem hold if the algebra of interest is a topological \ast-algebra instead of the usual C^{\ast}-algebra? I will now state the version of Stinespring's dilation ... 0answers 22 views ### exponential growth of an ordered structure (like a dcpo) [on hold] Here is a paper that relates hyperbolic spacetimes to a special type of Domain (dcpo) called an interval domain. Inflation is a well understood aspect of the history of our spacetime and can be ... 0answers 94 views ### When do partial derivatives p_x, p_y of a polynomial over \mathbb{C} not have any common factor? When do partial derivatives p_x, p_y of a polynomial over \mathbb{C} not have any common factor? Is there a general approach for any number of variables, aka when is the variety defined by the ... 2answers 186 views ### Gauss--Lucas type theorem for tracts and higher derivatives of a polynomial The Gauss--Lucas Theorem states that all zeros of a degree n complex polynomial p(z) are contained in the convex hull of the zeros of p. By iteration, this implies that the zeros of ... 19answers 9k views ### Math paper authors' order It seems in writing math papers collaborators put their names in the alphabetical order of their last name. Is this a universal accepted norm? I could not find a place putting this down formally. 0answers 230 views ### Bounding the degrees in a Bézout relation for integer polynomials Let A and B be two polynomials in \mathbf Z[X] which generate \mathbf Z[X], that is assume that there exist polynomials U and V in \mathbf Z[X] such that$$ A \cdot U + B \cdot V=1. $$... 6answers 908 views ### Sum of n vectors in (\mathbb Z/n)^k Let n,k be positive integers. What is the smallest value of N such that for any N vectors (may be repeated) in (\mathbb Z/(n))^k, one can pick n vectors whose sum is 0? My guess is ... 1answer 108 views ### Concerning Jump process (Lévy process) Consider X= \left( X_t \right)_{t\geq 0} is a Lévy process whose characteristic triplet is \left( \gamma, \sigma ^2, \nu \right) and where its Lévy measure is$$ \nu \left( dx\right) = A ...
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I am trying to understand the notion of Radon measure, but I am a little bit lost with the different conventions used in the litterature. More precisely, I have a doubt about the very definition of ...
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### A slight generalization of Mehta's integral.

I am trying to find the value of following integral \int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty}\prod_{i=1}^ne^{-\frac{t_i^2}{2}+\alpha_i t_i}\prod_{1\le i<j\le ...
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### Conditions for underlying space of an orbifold $\Bbb T^n/\Gamma$ to be a sphere?

Given a $n$-dimensional torus, is it always possible to find a discrete action to produce an orbifold such that its underlying space is the $n$-dimensional sphere? Or does it only happens for specific ...
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### Exceptional Lie algebras

I have some questions regarding the exceptional Lie algebras e(n), n=6,7,8. Can anybody explain to me what prevents us from constructing e(9) from e(8)? One can use the e(8) lattice vectors and try ...
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### Complement and fibers

Let $\mathcal M \rightarrow S$ be a projective irreducible scheme over the spectrum of a DVR and $U\subset \mathcal M$ an open subscheme surjective on $S$. Is it true for both points (generic and ...
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### is there a general statement about structures on spheres relating to division algebras?

It is classical to take a division algebra over $\mathbb{R}$ and defining an H-space structure on the unit spheres by restricting and normalizing. There are commutative division algebras of dimension ...
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### Cohomology of complexes

I sometimes see cohomologies of complexes of sheaves. What is the definition of these? Say if $\mathcal F^*$ is a complex of sheaves on $C$, what is $H^i(C,\mathcal F^*)$?