# All Questions

32 views

Consider that $f: \mathbb{R}^N \to \mathbb{R}$ is infinite times differentiable and is smooth over the entire domain. Take $N$ to be large. Also at point $\mathbf{x_0}$: $$... 0answers 49 views ### numerical and functional mixed optimization problem \max f - \min f +\int_{-1}^1 (f'(x)-x)^2dx Given a function g(x) and its domain, we want to get another function f(x) whose derivative is approximately g(x), but so that f(x) itself has small variation. For example, for ... 0answers 141 views ### The topology of the classifying space of U(n) In \textit{Atiyah and Bott: The Yang-Mills equations over riemann surfaces}, there is a sentence about the topology of BU(n) (i.e. the classifying space of U(n)): Here K(\mathbb{Z},n) means the ... 0answers 47 views ### Method used in fmincon() of Matlab? [on hold] We are using the Matlab optimization toolbox function fmincon() to solve a constrained minimization with only equality constraints. We wish to find out which particular constrained optimization method ... 0answers 268 views ### Who know about Rumek proof [on hold] Rumek has actually shown that the usual ZFC axioms for mathematics are in fact inconsistent. For example, ​Rumek has been able to prove from the ZFC axioms that both 2+2 = 4 and 2+2 = 5. ... 2answers 276 views ### Aspheric functors and Grothendieck fibrations Following Grothendieck, let us say that a category is aspheric if its nerve is weakly contractible and a functor u : \mathcal{A} \to \mathcal{B} is aspheric if for every object b in \mathcal{B}, ... 2answers 145 views ### Are Anderson T-motives motives for the function field analogy? this question is related to this one Geometry for Anderson's motives?, though the previous one doesn't answer exactly my question. Let \mathbb{C}_{\infty} be the function field analog of ... 0answers 41 views ### Calculus question on limts [on hold] Can't solve this algebraically. Answer would be greatly appreciated. Thanks. lim x->0 ( ( (sin^2 x)(1-cos x) ) / 2x^4 ) 0answers 33 views ### Ferenczi: minimal, uniquely ergodic, sublinear complexity systems are not strongly mixing The following result is on page 26 of this paper by Ferenczi [PDF]. Corollary 3. A minimal and uniquely ergodic system of sub-affine complexity cannot be strongly mixing (i.e., \mu(T^nA \cap B) ... 0answers 79 views ### Free groups and varietal product [on hold] I will be so thankful if some one help me. My knowledge in free group is not deep. Suppose S is the variety of p-groups of class at most 2 and exponent p. Question one) For any n, is there a ... 0answers 62 views ### Internal categories in an endofunctor category Here we see the definition of an internal category in a monoidal category. We also know that endofunctor categories support a monoidal product which is actually functor composition. It is the case ... 0answers 12 views ### 3D Vector projection on a Plane [migrated] I want to Project a Vector on to a Plane. Assume, you have a Central Point (1,1,1) and you want to move (0,0,3) in z-direction. How can I project the end of this movement (point) on a plane with ... 0answers 119 views ### What books I have to study to get into Riemann hypothesis (from almost zero) [on hold] Could anyone help me with my own self-learning process to get in Riemann hypothesis from the level of 1st year of technical Bsc college? What minimal number of books (and which one) I have to study to ... 0answers 57 views ### Are all complex zeros of Li_s(i)\, + \, Li_{1-\overline{s}}\,(-i) equal to the \rho's? Take the well known square relationship for polylogarithms:$$Li_s(z)\, + \, Li_{s}(-z)=2^{1-s}Li_s(z^2)$$Assume z=i:$$Li_s(i)\, + \, Li_{s}(-i)=2^{1-s}Li_s(-1)=-2^{1-s}\,\eta(s)$$with ... 0answers 112 views ### excess intersection theory Can the excess intersection theory be applied to the following problem: I have a non-singular irreducible variety X of dimension k and degree d and k+1 hyperplane sections of X, ... 2answers 344 views ### Is fixed point property for posets preserved by products? Recall that a Partially Ordered Set (poset) P has the fixed point property (FPP) if any order preserving function f:P\longrightarrow P has a fixed point. Theorm : Suppose P and Q are posets ... 0answers 73 views ### What are the first non-maximal non-group-subgroup simple subfactors? Definition: For an irreducible (finite index) subfactor (\mathcal{N} \subset \mathcal{M}), an intermediate (\mathcal{N} \subset \mathcal{P} \subset \mathcal{M}) is normal if the biprojections ... 1answer 51 views ### Schur norm for self-adjoint operators If A is a n \times n complex matrix then the Schur norm of A is given by$$ || A||_S := \max_{||B||=1} ||A*B||,$$where ||. || is the operator norm and * is the Hadamard (entry-wise) ... 0answers 44 views ### The relation on the set of functions Let \varphi: \mathbb{R}^{2} \to \mathbb{R} be a symmetric (not necessarily continuous) function (so, \varphi(x,y)=\varphi(y,x) \forall (x,y)\in \mathbb{R}^{2}), let \mathcal{F} be the set of ... 1answer 153 views ### Finite subgroups of mapping class groups Given a closed, oriented surface \Sigma of genus greater than 1, let Mod(\Sigma) denote the mapping class group of orientation preserving diffeomorphisms of \Sigma up to isotopy. Given any ... 1answer 70 views ### RUCar^{V}-semiproperness implies properness This is a claim in Shelah's Proper and Improper Forcing, more specifically Claim 2.3(1) of Chapter X (p. 484). The proof of the claim is "Easy" but I cannot quite figure it out. There must be some ... 1answer 121 views ### Optimal lower bounds for the sum of digits in base b Let b \geq 2 be an integer and let s_b(n) be the sum of the digits of the base-b representation of the nonnegative integer n (e.g., s_{10}(726)=7+2+6). From the weak law of large numbers, it ... 0answers 116 views ### A compactification of the non-negative rationals with the discrete topology Let S be the set of non-negative rational numbers. (If it makes any difference, feel free to take the non-negative dyadic rationals instead.) Let B=\ell_\infty(S); as a {\rm C}^*-algebra this is ... 1answer 271 views ### Paper by Mumford In the paper of "The spectrum of difference operators and algebraic curves", by P. van Moerbeke and D. Mumford, Acta Mathematics, vol.143, 1979, (link: ... 0answers 67 views ### True lies logic [on hold] Santa says lie on all days except one day... Once he made 3 statemnts on consequtive days : 1) I lie on monday and tuesday. 2) today is thursday, saturday or sunday. 3) I lie on wednesday and friday. ... 0answers 39 views ### Direct sum of simple modules [on hold] I would like to ask the simplest example of a simple module M in which M \oplus M has infinite sub-modules. Thanks a lot. 0answers 100 views ### Understanding a program for computing Khovanov homology I would like to understand how a computer program for computing Khovanov homology works. The particular program I have in mind is by John Baldwin: https://web.math.princeton.edu/~baldwinj/Kh.cpp The ... 1answer 330 views ### A question on BSD conjecture If E is an elliptic curve over \mathbb{Q}, and K is an imaginary quadratic field. If rank E(K)\leq 1 and both E and the quadratic twist of E by K satisfy the full BSD conjecture, does ... 0answers 24 views ### reducing an n-order differential equation to a first order system of equations using either sagemath or sympy I want to reduce a n-order ordinary differential equation into a first order system of equations. This is in preparation for numerical analysis. I use both Sympy and Sagemath for Computer Algebra, but ... 1answer 274 views ### Examples of continuous differential equations with no solution Several theorems regarding differential equations are true not only in finite dimension but also in infinite dimension Banach spaces. This is in particular the case for the Cauchy–Lipschitz theorem. ... 1answer 25 views ### Making a system of second-order ODEs chaotic Consider a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions X = (x_1, x_2, ..., x_N), we have X'' ... 2answers 150 views ### A question on certain elliptic PDE Consider the elliptic PDE "CR"$$\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$And its consequence "LAP"$$U_{xxxx}+U_{yyyy}=0$$. Somehow, these equations are similar to the Cauchi ... 1answer 137 views ### Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q) I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form  \left( ... 0answers 35 views ### Machine learning approach [on hold] I will illustrate my question my example let us say, I want to build a classifier to label each word in a huge set or words into either "GOOD" or "BAD" word. I have a set of GOOD words and I have ... 0answers 32 views ### Lagrange error bound for the approximation of sinx≈x [on hold] The problem is to bound the error of the approximation of sinx≈x on the interval [-1,1]. Here is what I tried/understand: -I know this is a n=1, x is a 1st degree Taylor polynomial of sin x -On the ... 0answers 46 views ### Different definitions of linkless graphs Robertson, Seymour and Thomas defined linkless embeddings of graphs as follows: An embedding of G is linkless if every pair of disjoint circuits of G have zero linking number (see here). However ... 1answer 62 views ### Difference between straight and piecewise linear and continuous embeddings of graphs / complexes in d-dimensional space? Here is my main question: what is the difference between "straight" and "piecewise linear" and "continuous" embeddings of graphs/complexes in d-dimensional space? Moreover I would like to know if any ... 1answer 178 views ### A random variation on Polya's orchard problem Polya's orchard problem is as follows: "How thick must the trunks of the trees in a regularly spaced circular orchard grow if they are to block completely the view from the center?" See, ... 1answer 80 views ### Can the graph Laplacian be well approximated by a Laplace-Beltrami operator? It seems rather well known that given a Laplace-Beltrami operator \mathcal{L}_{M} on a manifold M we can approximate its spectrum by that of a graph Laplacian L_{G} for some G (where G is ... 1answer 101 views ### Classification of indecomposable inclusions (H \subset G) with G decomposable Definition: A group G is indecomposable if: G = G_1 \times G_2 \Rightarrow \exists i \ G_i = 1. We can generalize the notion of indecomposable from groups to inclusion of groups as ... 1answer 149 views ### Asymptotic formula for the number of ways to write a number as the sum of k triangular numbers How would one derive an asymptotic formula for the number of representations of a number n as the sum of k numbers of the form \frac{m(m + 1)}{2} I think that one could use the circle method, ... 0answers 45 views ### Uniqueness of the tensor product decomposition of subfactors A subfactor (N \subset M) is indecomposable if (for N_i \subset M_i):$$(N \subset M) = (N_1 \otimes N_2 \subset M_1 \otimes M_2) \Rightarrow \exists i \ N_i = M_i$$Then, a subfactor (N ... 1answer 218 views ### On the positivity of matrices For given n, the following n\times n real matrix M=M^{T} is called positive, if x^{T}M x\geq 0 holds for all non-negative real x_1,x_2,\cdots,x_n, where x=(x_1,x_2,\cdots,x_n)^T. ... 0answers 25 views ### Family of (Cumulative Distribution) Functions I'm looking for a 2 (or more)-parameter family of functions F with the following properties: For each f \in F, f(0)=0, f(1)=1, and f is (weakly) increasing. F is closed under products. ... 0answers 37 views ### Integrating a differential form over a box [on hold] I've only ever seen examples of integrating a differential form over a curve C involving defining a parameterization. I have seen people integrate 1 forms over a box without defining a ... 0answers 61 views ### Smoothness in Ecalle's method for fractional iterates Some four years ago I answered my own question on fractional iteration, concluding that there is a half iterate of sine, that is f(f(x)) = \sin x, which is real analytic for 0 < x < \pi but ... 1answer 225 views ### Is there an analog of the Barratt-Eccles construction for E_∞-groups and E_∞-rings? The Barratt-Eccles operad is an operad in simplicial sets that provides a particularly nice model of an E∞-operad; algebras in spaces over the Barratt-Eccles operad model E∞-spaces, i.e., homotopy ... 1answer 423 views ### Reference request: Book of topology from “Topos” point of view Question: Is there any book of topology in the modern language of topos theory? I refer to the (systematic, formalist) study of the category of sheaves on a site or the study of topology in a ... 0answers 44 views ### The regularity of Dirichlet form in Besov space Let C_{0}(\mathbb{R}^n) denote the set of continuous functions in \mathbb{R}^n with compact support, C_0^\infty(\mathbb{R}^n) denote the set of infinite differentiable functions in ... 0answers 64 views ### Are (odd) perfect numbers divisible by a repdigit (in another base)? How about by a repunit? [This question was cross-posted from MSE.] A positive integer N is said to be a perfect number if$$\sigma(N) = 2N, where $\sigma(x)$ is the sum of the divisors of $x$. For example, $6$ is ...

15 30 50 per page