I have a function for that I want to find the minimum. The function calculates the distance between two sets where a set is defined as matix of row vectors $D = [ d_1, d_2, ..., d_n]$, $d_n$ is a $m \times 1$ vector. The distance metric is:d(x_1, If a topological space X has a countable basis. Then if we have an open cover of X, can this cover be refined to a countable one February 10 1 If a topological space X has a countable basis. Then if we have an open cover of X, can this cover be refined to a countable one?Yes. Fix \cal B to be a countable basis. Let \cal U be our open cover. For U\in\cal U define V_U=\{V\in\mathca What does Equational Theorem Prover do February 10 1 http://www.cs.unm.edu/~mccune/eqp/What does EQP do? Is there any paper that explains what it does?README and other read files do not provide such information - it only talks of how to use it and not what theoretical background of it is.It seems to do Minimizing function of overlapping volumes February 10 I am implementing a method that performs alignment of slightly overlapping 3D volumes. To be more specific, I have a dataset of m x n volumes of size 1024 x 1024 x 100, and each volume overlaps for 100 pixels on the xy plane with the adjacent volumes Minimizing the following objective function with matrices February 10 2 Suppose A and B are known matrices, and we are to find matrix X that minimizes the following function,\frac{1}{2}||X||^2+\frac{1}{2}||X^TAX-B||^2$$Taking the relevant derivative w.r.t X gives us,$$ X +(X^TAX-B)(A+A^T)X$$I couldn't reduce t If I have an open cover, can I choose a countable sub-cover such that the union of the open sets strictly increasing February 10 2 If I have an open cover, can I choose a countable sub-cover such that the union of the open sets strictly increasing?Actually I want to use this to prove every noetherian space is compact. Thanks!In general you can't: Consider X = [0, 1] (which is Distance between theorems February 10 1 In automated proving one can define the best proof of a theorem as the one which minimizes the length of the proof. Given a set of known statements one could define the difficulty of a theorem as the minimum length among all its proofs deduced from k Can covering be done on two elements February 10 1 The covering rule is:$$B \bullet (B+C) = B$$and$$B+(B \bullet C)=B$$So does it follow from this rule that:$$B \bullet A \bullet \bar{C} + B \bullet D \bullet\bar{F} = B \bullet (A\bullet\bar{C}+D\bullet\bar{F}) = B? $$Or does the covering rule minimize this objective function February 10 I have a function to minimize and I don't understand how I should proceed. The function is coming from a publication.Background: In a 2D image, P_1 and P_2 represents 2 patches of colors (RGB) which overlap one on the other (these 2 patches have The distance covered by the Ant is: February 10 A set of concentric circles of integer radii1,2,...N is shown is the fig above. An ant starts at point A_1, goes round the first circle ,returns to A_1 ,moves to A_2, goes round the ... To test convergence of improper integral  \int_{0}^{\infty} \frac{x\log(x)}{(1+x^2)^2}\, \mathrm dx February 10 2 I have to test convergence of improper integral$$ \int_{0}^{\infty} \frac{x\log(x)}{(1+x^2)^2}\,\mathrm dx$$I write as \log(x) \leq x . So x\log(x) \leq x^2. So  \frac{x\log(x)}{(1+x^2)^2} \leq \frac{x^2}{(1+x^2)^2} . Now using comparison test Prove that for any integer n, if b^2 divides n, then b divides n. February 10 2 Prove that for any integer n, if b^2 divides n, then b divides n. Trying to figure out this proof. The proof I'm looking at is written as n = any integer, if 25|n \implies 5|n. I've been trying to figure this for days and have been runn y=e^x\sin x; find all points where slope of tangent line equals 0 February 10 1 I have the derivative already. Using the product rule, I got e^{2}\sin x+e^{2}\cos x. I can't figure out how to find all the points without graphing it.We need$$\dfrac{dy}{dx}=0\implies e^x(\sin x+\cos x)=0$$As for finite real x,e^x>0$$$\implies\ What does a variable superscript above a set mean February 10 I'm not entirely sure I've worded this correctly. An example of what I mean is...$$U = \{0,1\}^n$$What is the meaning of the superscript? Convergence of improper integral$\int_{2}^{\infty} \frac{1}{log(t)}dt$February 10 Convergence of improper integral$\int_{2}^{\infty} \frac{1}{log(t)}dt$How do i start? Is there any relation between a group being unimodular and having equivalent uniform structures February 10 1 Recall: A topological group is said to have equivalent uniform structures if its left and right uniform structures coincide. A locally compact group is said to be unimodular if left Haar measures and right Haar measures on it coincide.For locally com Why do characters on a subgroup extend to the whole group February 10 1 As background, I am trying to do exercise 3.10 in Deitmar's "Principles of Harmonic Analysis." I can do most of the problem but I'm stuck on the third part proving surjectivity. Given a locally compact abelian group$G,$a closed subgroup$H,$a Continuous functions on compact group and uniformity February 10 If$G$is a compact abelian group and$f\in C(G)$. Then$\forall \epsilon >0$,there exists an open neighbourhood$U$of$0\in G$, such that$\forall g\in G , \forall u_1,u_2\in U$, we have$|f(gu_1)-f(gu_2)|<\epsilon$.Is this statement right? how to For what values of$a, n$the number$2^a\cdot 3^n+1$is prime February 10 Respected All. Please forgive me as I am unable to understand if it is an off topic. I am stuck in a situation. Change the title if you find inappropriateWhat I am willing to know is that if there is any website where I can get complete information o Fraction walk through on hold February 10 1 $$0.824 = \frac{n/20\cdot 1}{n/20\cdot 1+(1-n/20)\cdot 0.5}$$ Please answer this question with step by step. 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