# All Questions

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This is a lighter version of the questions that I asked yesterday. If you have the sums $f(n) = 1^{29} + 2^{29} + 3^{29} + \cdots + (10^n)^{29}$ and $g(n) = 1^{2} + 2^{2} + 3^{2} + \cdots + ... 0answers 42 views ### Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay the same I would like to compute, in MATLAB, the basis Z for the nullspace of an m by n matrix A, such that if one row of A is added (resulting in A_a), the basis for A_a is n-m-1 of the n-m columns of Z, ... 0answers 38 views ### Conditional probability of dependent random variables Let$ X \sim f_X(x), Y \sim f_Y(y) $are two dependent random variables and their corresponding PDFs. I want to find a probability $$P(Y\ge 0 | X+Y\ge 0) .$$ If these variables were independent I'd ... 0answers 29 views ### Number of real singular points Let$f\in\mathbb{R}[X,Y]$be a real polynomial in two variables. Are there bounds on the number of singular points of$f$which take into account that$f$might have high degree but be rather sparse? ... 1answer 27 views ### Information on special matrices similar to Jacobi matrices Jacobi matrices are well known and deeply investigated mathematical objects from various point of view. One can arrive at these operators while studying discrete systems of particles interacting with ... 0answers 96 views ### field of constants of a curve [on hold] I'm trying to gain some intuition about the field of constants of a curve. If$C$is over a field$k$, then it is defined as the set of elements of$k(C)$algebraic over$k$. If I understood ... 2answers 139 views ### What is the easiest way to compute Ozsváth-Szabó tau invariant of a knot? Suppose that we have a knot$K$with 40 crossings which is not a cable knot or an alternating knot. Then, what is the easiest way to compute Ozsváth-Szabó's invariant$\tau(K)$? Are there any ... 0answers 12 views ### Bifurcations in flows on two dimensional torus I want to have a research about bifurcations which are appeared in flows on two dimensional torus. Especially bifurcations that can not be seen in flows of$\mathbb{R}^2$. Can anyone introduce me ... 1answer 176 views ### continuity of the Boltzmann entropy in the Wasserstein metric For Lebesgue-absolutely continuous probability measures$\rho\ll \mathcal{L}^d$in the whole space$\mathbb{R}^d$with finite second moments (i-e$\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let $$... 0answers 33 views ### Is this infinite family of non-trivial snarks resulting from the first Celmins-Swart? Non-trivial snark is cubic graph with chromatic index 4, girth at least 5 and doesn't to contain three edges whose deletion results in a disconnected graph, each of whose components is nontrivial. ... 0answers 42 views ### Characterisation of vector fields solution to a simple equation This question is complementary to another question I asked on math.stackexchange. I believe it is more subtle than it seems - it will become clearer when I provide more context - and probably hides ... 0answers 69 views ### Existence of non-trivial characters on compact abelian group [closed] Does for every compact (compact metric) abelian group (G, \odot ) there exist a non-trivial homomorphism \varphi : (G, \odot ) \to (\mathbb{C} , \cdot ) such that |\varphi (g) |=1 for all ... 1answer 150 views ### An elementary functional inequality Let g be a C^1 function with g(0)=0 and g(t)>0 for all t>0. I am surprised that for all such g the following seems to hold \frac{\int_0^t(g'(s))^2ds}{g^2(t)}\geq \frac{1}{t} for ... 1answer 203 views ### Travelling Salesman Problem: Can the nearest neighbor algorithm be n times longer than the optimal solution? This is inspired by a recent question. Given a positive integer n\in\mathbb{N}, is there a setting of finitely many points and a designated "starting point" s in \mathbb{R}^2 such that the ... 1answer 270 views ### Computing an eigencuspform in S_2(\Gamma_0(1776)) Consider$$\bar{\rho}:G_{\mathbb Q}\longrightarrow\operatorname{GL}_2(\mathbb F_7)$$the residual 7-adic Galois representation attached to the elliptic curve y^2=x^3+x^2-4x-4 of conductor 48. Then ... 0answers 129 views ### Is anything known about a ternary equivalent of groups? Group theory studies the properties of algebraic structures that combine a set of elements with a binary operation. Different structures such as Monoids, Semigroups, Groups, Rings, Fields etc demand ... 2answers 1k views ### Arctangents of odd powers of the golden ratio While trying to answer this MSE question, I found that arctangents of many odd powers of the golden ratio \varphi=\frac{1+\sqrt5}2 are expressible as rational linear combinations of arctangents of ... 0answers 77 views ### Optimal strategy for game of 'online sorting' into a poset Consider a single-player game played with an arbitrary finite poset, and a random number generator with a known distribution: Each turn, the RNG produces a number, and the player must assign that ... 1answer 80 views ### Elliptic pde with bilaplacian; boundary conditions. I am interested in the solvability of$$ \Delta^2 u + u = f(x) \mbox{ in } \Omega $$with \partial_\nu u = \Delta u=0 on \partial \Omega where f(x) is some smooth bounded function on ... 0answers 21 views ### property of orthonormal systems and sequences in Hilbert space [closed] Problem: Let H be a separable Hilbert space and {e_n} a complete orthonormal system of H. Prove that, if {y_k} is a bounded sequence in H, the condition \lim_{k→∞} (e_n , y_k ) = 0 for ... 0answers 124 views ### Conditions for splitting of short exact sequence? Are there conditions under which the short exact sequence$$0\rightarrow E (K)/mE (K)\rightarrow H^1_{Sel}(K,E_m)\rightarrow \Sha(E|K)_m\rightarrow 0$$splits? I assume K to be a number field and ... 0answers 54 views ### Tensor product of bounded analytic functions I asked this question on math.SE, but couldn't get an answer. Let H^\infty(\mathbb{D}) denote the set of functions holomorphic and bounded on \mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}. ... 0answers 68 views ### sum of the series and infinity [closed] If you have the sums f(n) = 1^{59} + 2^{59} + 3^{59} + \cdots + (10^n)^{59} and g(n) = 1^{5} + 2^{5} + 3^{5} + \cdots + (10^n)^{5}, for large enough n, f(n) is approximately \frac{1}{60} ... 0answers 54 views ### Choice of framing in Gravitational Chern Simons I was trying to understand formula(2.21) in Witten's paper "Quantum Field Theory and Jones Polynomial"(link: https://projecteuclid.org/euclid.cmp/1104178138) (Page 360). There, it was mentioned, the ... 0answers 14 views ### Bayes' Rule where the probabilities are taken as conditional [migrated] I'm encountering some difficulty beginning statistics work with a basic Bayes' Rule problem. You can see the problem and answer on page 16 here, but I've explained it below. ... 2answers 65 views ### Discrete optimization problem Suppose you had N many fixed points X_1, X_2, ..., X_N in some Euclidean space R^d and from these coordinates you had to choose n many of them (n \leq N also being fixed) to form a subset ... 1answer 46 views ### Can every hyperelliptic genus 3 surface be minimally immersed in flat T^3 Every minimally immersed genus 3 surface in flat T^3 must be hyperelliptic, as the Gauss map gives the degree 2 covering map. How about the converse of this problem? The only thing I can find is ... 0answers 50 views ### Let p be prime, is there a divisor d of p-1 or p+1 with gcd(d!,p+d)=1 such that p+d is prime ? [closed] My previous question was incomplete. Please accept my apologies. 2answers 265 views ### Logarithmic integral, π(x) and x/(\ln x) The function \text{Li} (logarithmic integral) is defined for x>0 by$$ \text{Li}(x)=\int_2^{x}\frac{dt}{\ln t}. $$The prime number theorem, proven by Hadamard and de la Vallée-Poussin in 1896 ... 4answers 682 views ### Number of \mathbb F_p points constant mod p? I have some affine varieties X defined over \mathbb Z, and associated integers c(X), with the property that \# X_{\mathbb Z/p} \equiv c(X) \bmod p for all p. (In particular c(X) is usually ... 0answers 12 views ### heavy subgraph searching result in pseudopatterns in tensor [closed] I encounter problem while trying to find heavy subgraph in tensor. I'm trying to maximize H(x,y)=1/2 summation a(ijk)x(i)x(j)y(k) Why do I only find pseudopatterns in heavy subgraph searching in ... 0answers 88 views ### References for modular curves over finite fields [closed] I'm looking for a detailed reference for modular curves over finite fields, such as X(N), X_1(N), and X_0(N). There seems to be a lot of literature dealing with them over \mathbb{C}, but I'm ... 0answers 99 views ### Number of primes one larger than divisors of a fixed number, which is LCM of 1,2,3,…,L I am hoping someone can estimate the number of primes that come up this way: take a number L, then let$$ C = \operatorname{lcm} (1,2,3,\ldots,L). $$We know that C has quite a lot of divisors; ... 0answers 16 views ### How should strongly correlated covariates for logistic regression be treated? [closed] I have to build a logistic regression for multiple covariates (predictor variables), two of which are strongly correlated. How should they be treated? Am I to exclude one of them from the regression? ... 2answers 155 views ### Permutation covering of a G-lattice Let G be a finite group. By a G-lattice we mean a finitely generated free abelian group L with an action of G. We say that L is a permutation G-lattice if L has a {{\mathbf{Z}}}-basis ... 1answer 103 views ### Stationary distribution of last passage percolation Consider last passage percolation model on \mathbb{Z}^2. I am interested to know if there is any known result for the stationary distribution of passage times, given some distribution for the ... 0answers 32 views ### Finding the equation of a curve from two given points [closed] " A curve is traced by a point P(x,y) which moves such that its distance from the point A(-1,1) is three times its distance from the point B(2,-1). Determine the equation of the curve. " 0answers 83 views ### Can I find the gap between the two least eigenvalues of this special matrix A(t)?‎ I am interested in finding the gap between the two least eigenvalues of A(t), a Hermitian N\times N sparse ‎matrix whose diagonal elements are a_it+b_i\,(1\leq i\leq N), and all off-diagonal ... 0answers 201 views ### Whitehead group [closed] Whitehead group (WG) is known for some groups (e.g. free abelian group, cyclic group, Braid group etc). For example: The Whitehead group of the trivial group is trivial. The Whitehead group of a ... 0answers 58 views ### Messages on rotating wheels [closed] The question is: Would it be possible display a message (image, logo, text, ...) on a rotating wheel so that it would become readable once rotating at a certain speed, knowing that our brain will ... 1answer 39 views ### Find inverse and determinant of a symmetric matrix - for a maximum-likelihood estimation Evaluate the determinant \det \Omega and find the inverse matrix \Omega^{-1} of:$$\Omega = \begin{bmatrix} \beta_1^2(1+\theta_1^2) & \beta_1 \beta_2 & ... & \beta_1 \beta_{k-1} ... 1answer 100 views ### rationality of residues of differentials Let$C$be a smooth curve over a field$k$,$\overline{C}$the smooth compactification and$S=\overline{C} \setminus C$. We think of$S$as a reduced divisor defined over$k$. Take the sheaf of ... 0answers 35 views ### Independence of inverse system to define continuous cohomology for profinite groups I have a problem concerning cohomology of profinite groups as it is defined e.g. in Gille's and Szamuely's "Central Simple Algebras and Galois Cohomology" on page 86. For a profinite group ... 0answers 29 views ### Boundary conditions of PDE from SV model with stochastic interest rate The PDE for the American put option price$P(S,\sigma ,r,t)is \begin{align*} 0 =& P_t+P_SS(r-\delta)+P_\sigma a(\sigma)+P_r\alpha (r,t) \\ +& \frac{1}{2}P_{SS}S^2\sigma ^2 + ... 0answers 98 views ### Ricci flow in complex analysis [closed] Occasionally, I find a paper http://arxiv.org/abs/math/0505163 written by Chen, Lu and Tian. In this paper, the uniformalization theorem was proved by Ricci flow. I think it is a very interesting ... 0answers 224 views ### Mathematical theories of changes - except from calculus? [closed] Unfortunately motion is regarded as displacement in geometry: By a motion or displacement in the general sense is not meant a change of position of a single point or any bounded figure, but a ... 0answers 78 views ### Prove bijection beetween sets [closed] Prove that if the set X $is infinite, and a subset$ Y $is finit, there is a bijection$ X \setminus Y \to Y $. It seems a simple task, but no ideas yet. At first I thought that between these ... 2answers 850 views ### How to prove that this equation has only one solution? I can't find a way to prove that the following equation has only one solution : $$X = \frac{2^Q - 1}{2^{P+Q} - 3^P}$$ with$X,P,Q$integers$> 0$. One trivial solution is$X = 1, P = 1, Q = ...
I need help with this excercise Let $k[X_1,\ldots,X_d]$ be the polynomial ring in $X_1,\ldots,X_d$ over a field $k$, and let $F_1,\ldots,F_m$ be forms of degree $n$. Assume that ...
The spherical harmonics are given by $$Y^m_l(\phi,\theta):=N^m_l e^{im\theta}P^m_l(\cos \phi)$$ where $P^m_l$ are the associated Legendre Polynomials and $N^m_l$ is the normalisation. From ...