# All Questions

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### Size issue in localization of a category.

I have read on nlab and other sources that I can't backtrack that the localization process of a category can lead to size issue. More especially, starting from a locally small category $\mathsf C$ and ...
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### Integral of a differential 1-form along a curve (clarification on the definition)

Let's denote with $(e_1,\dots,e_d)$ the usual basis of $\Bbb R^d$, and with $({e_1}^*,\dots,{e_d}^*)$ the dual basis of its dual space $\Bbb {(R^d)}^*$. Let $U$ be an open subset of $\Bbb R^d$ and ...
Aaagain, I fail to understand the trivial: Using compactness argument it is straightforward to show: $$\lambda^{*}([a,b]) \geq b-a$$ And everything is OK. But, regarding $\lambda^{*}([a,b]) \leq ... 0answers 8 views ### Proof:$A,B \subset \Bbb{R} \wedge A,B \mbox { are bounded above by } \leq \to A \cup B \mbox{ is bounded above by }\leq$Let$\Bbb{R}$is a complete ordered field by$\leq$, I must proof the following: Prop.: let$A,B \subset \Bbb{R}$then: $$A,B \mbox { are bounded above by }\leq \mbox{}\to A \cup B \mbox{ is ... 1answer 23 views ### Is dxdy really a multiplication of dx and dy? In this link Is dy/dx not a ratio? it was told that \frac{dy}{dx} cannot be seen as a quotient, even though it looks like a fraction. My question is: does dxdy in the double integral represent a ... 0answers 17 views ### Which answer is true? Suppose \{E_n\} is a sequence of measurable sets in measure space (X,M,\mu) such that for every n, \mu(E_n)>\frac{1}{100}. For the set F = \{x \in X: x belongs to an infinite number ... 0answers 55 views ### On the evaluation of a beautiful series! Find value of \sum_{n=1}^\infty \left\{e-(1+\frac1n)^n\right\} exactly!? I found this question from here: 'Titu Andrescu-Problems in Real Analysis-p.114' 0answers 14 views ### Finding the elements of this subgroup I'm trying to find all of the elements of the subgroup of C* generated by (1+i)/Sqrt[2]. I know how to find elements in a subgroup in Z and U, but I don't know where to start here. 0answers 13 views ### An exercise in operator theory Let H be a Hilbert space and P be a projection to a finite dimensional subspace K of H, for a T\in B(H), if ||PTP||=1, then, for arbitrary \epsilon>0, there exists a vector \alpha ... 0answers 23 views ### Exact arctan value. Can't be solved ?? I am kind of stuck here. I need to know the exact value of$$ \arctan\left(\sqrt{2} \over 4\right). $$I am looking into double angle formula's, half angle formulas but honestly, I can't find it? Is ... 0answers 13 views ### Dummy variable in the probability generating function I'm struggling to understand what the purpose of the dummy variable t in the probability generating function is? I know it takes a value between 0 and 1, and have heard it described as a 'relative ... 0answers 10 views ### product of cumulative distribution function For all integer N>1, I am trying to show that for a gaussian (or even better any type) cumulative distribution function F(\theta;\mu,\sigma) (\mu and \sigma are the mean and standard ... 0answers 13 views ### Difference between sum of alternate terms of a decreasing series. I'm given a series say 5,7,8,4,2,10 First arrange terms is decreasing order i.e 10,8,7,5,4,2. Then calculate sum of numbers at alternate positions ie s_1=10+7+4 s_2=8+5+2 Required answer ... 0answers 19 views ### IEEE 754 as a mathematical space Integer operations in computers (i.e. 32-bit integers) probably can be represented best by modular arithmetic (because of integer overflows/underflows). What about IEEE 754 floating point arithmetic? ... 0answers 9 views ### completion of the canonical module For a local Noetherian Cohen-Macaulay ring (R,m,k) the canonical module is defined to be any maximal Cohen-Macaulay module of finite injective dimension and of type 1. The canonical module is ... 0answers 18 views ### Existence of unique periodic solution of ODE How to prove or disprove that the ODE$$ y'(x) = y( x )^9 + ( 1+\sin( x)) y(x) +\cos(x) =0 $$has the unique periodic solution? PS. Its fieldplot done with Maple 3answers 50 views ### Proof: a^2 - b^2 = (a-b)(a+b) holds \forall a,b \in R iff R is commutative We want to show that for some ring R, the equality a^2 - b^2 = (a-b)(a+b) holds \forall a,b \in R if and only if R is commutative. Here's my proof --- I'm not sure if the first part stands ... 0answers 7 views ### gr_I (R) finite? Is …⊕R(-1) ⨁m⨁R_1 ⨁… necessarily an ideal When is gr_I (R) (I mean "associated graded ring of I") finite? When is gr_I (M) (M is R-module) finite? Is Exercise 2.8 from Marley's note on "GRADED RINGS AND MODULES" true? Exercise 2.8: Let R ... 0answers 10 views ### Canonical “orientification” of a manifold? Canonical complexification of a manifold? Maybe this is a silly question(i'm pretty new to both geometry and category theory) but i was wondering: 1)Consider the category of orientable smooth manifold on \mathbb{R}, if you forget the ... 0answers 14 views ### existence of spanning trees in complete graphs implies choice? it is known that the existence of spanning trees in arbitrary (connected) graphs implies the Axiom of Choice. I was wondering if this result still holds if we restrict ourselves to spanning trees of ... 0answers 6 views ### Notations in Oka family definition Definition. An ideal family F in a ring R with R \in F is said to be an Oka family (strongly Oka family) if, for a \in R and I, A \lhd R, (I, a), (I:a) \in F \Rightarrow I \in F ... 0answers 3 views ### Inducing highest weight modules I have a question regarding highest-weight modules: Let be \mathfrak{g} a Lie algebra, \mathfrak{b} a Borel subalgebra, \mathfrak{h} a Cartan subalgebra and U(\mathfrak{g}) its universal ... 1answer 11 views ### Geometry of k-forms and k-vectors In this question I was trying to see why k-forms are selected as the way to generalize vector calculus rather than k-vectors and a comment providing links to other questions made me end up with ... 0answers 12 views ### Looking for hints on how to prove the convergence of this iterative estimator! [duplicate] Let X_n be a Poisson process with mean \lambda^*. The following sequence estimates the parameter of the Poisson process: X_{n+1} = \hat{\lambda}_{n+1} + ... 3answers 44 views ### Modular Arithmetic - Are we allowed to distribute the Modularity? Assume I have a problem such as "Prove that \displaystyle103^{53} + 53^{103} is divisible by 39." This would mean I wanted to prove that \displaystyle103^{53} + 53^{103}\equiv0\pmod{39}. My ... 2answers 26 views ### Solution of a simple linear diophantine equation I'm having a slight problem with a simple equation of the sort a_1+a_2+a_3...=n. Where n,a_1, a_2, a_3... \in N. I do know how to solve these equations when they are of the regular form. But if ... 0answers 6 views ### Is there a single well-ordered set whose discrete space equals to its order topology? Let (X,\leq_X ) and (Y,\leq_Y) be two well-ordered sets, such that the discrete space of X is the same as the order topology on (X,\leq_X) and the same goes for Y. I'd like to prove that ... 5answers 90 views ### Can be solved without L'Hopital? Can this limit be solved without l'hopital?$$\lim_{x\to0}\frac{\sqrt[3]{8+h}-2}{h}$$thanks anyway ;) 0answers 14 views ### imagine of set in complex plane I) I must find the image of the set 0 < Re(z) < \pi under the function \cos z = \frac{e^{iz}+e^{-iz}}{2}. So I find image of simple: a) z=0+iy, where y is a real number: \cos(0+iy) = ... 0answers 9 views ### Examples of C^* algebras Let X be locally compact and look to C_0(X). Suppose thar this C^* algebra ia generated by the sequence (p_n) of projections. My task is now to prove that the element ... 1answer 12 views ### surface that is created by the intersection of paraboloid and plane Find the surface that is created by the intersection of the paraboloid x^2+y^2-z=0 and the plane z=2.$$x^2+y^2-z=0 \Rightarrow x^2+y^2=zz=2$$Is the intesection: x^2+y^2=2? So is the asked ... 0answers 27 views ### Expressing the price of gasoline algebraically [on hold] I am stuck on this problem , the question is of the following [x represents the number of gallons of gas sold at 2.65 per gallon, and y represents the total cost of the gasoline(in dollars) answers ... 1answer 20 views ### Cubic diophantine equation How can I solve the equation x^3+x-1=y^2 in positive integers? I know this equation defines an elliptic curve but this seems to be a non-elementary way to solve the question. Is there a more ... 0answers 7 views ### How can i Taylor expand to get a difference approximation formula using How can i Taylor expand to get a difference approximation formula using y'''(0)=ay(-h)+by(0)+cy(h)+dy(2h)+O(h^p) where O(h^p) needs to be as high as possible? i.e how can i Taylor expand the ... 0answers 15 views ### How to find multiplicative orders of all elements in field \Bbb F (say \Bbb F_{13})? I am working on some finite fields and I was referring to some online class material. Is there any way to find the multiplicative orders of all elements in a field \Bbb F? 1answer 22 views ### Maximise the happiness among children Given N different candies and M childrens. Each ith children demands for K[i] different candies and will ONLY BE HAPPY IFF HE GET ALL THOSE DIFFERENT CANDIES WHICH HE DEMANDED. Now i want that ... 1answer 11 views ### Isogeny of an elliptic curve Let E be an elliptic curve over \mathbb{Q} and p be a prime. Then what does it mean by "E has a \mathbb{Q}-isogeny of degree p"? 1answer 13 views ### Convolution of two sums (fourier transform) This question is from the book "Advanced Engineering Mathematics" by Stroud. I can't seem to get the required answer for this. I've derived the two Fourier transform equations for them. . U and ... 2answers 12 views ### Let \{A_α\} be a collection of connected subspaces of X; let A be con. sub. of X. Show that if A∩A_α≠ ∅ ∀α, then A∪(∪ A_α) is connected. Let \{A_\alpha\} be a collection of connected subspaces of X; let A be connectted subspace of X. Show that if A\cap A_\alpha \neq \emptyset for all \alpha, then A\cup(\cup ... 0answers 6 views ### How to evaluate the derivate of a hypergeometric function w.r.t. one of its parameters? I have to numerically evaluate the derivative of the hypergeometric function w.r.t. its first and second parameters \large\frac{\partial}{\partial a}{_2F_1}\left(a , b ,c;z\right) and ... 0answers 13 views ### Sequences, sets and element position in the set. I have a sequence Q with the length of N. This is the fragment of this sequence: 68 70 72 74 76 78 80 The sequence has been divided into the sets of 4 elements ... 2answers 42 views ### Prove that if a set A \subset \mathbb R^n is connected, then it has the Intermediate Value Property. Prove that if a set A \subset \mathbb R^n is connected, then it has the Intermediate Value Property. Intermediate Value Property: let f be a real-valued continuous function on a domain A if ... 2answers 28 views ### \lim_{n \to \infty}x^{1/n}=1 for every x \in \mathbb{R} and x>0? Two questions: 1) Let x \in \mathbb{R} and y \in \mathbb{R} and x>0 and y>0. Let n \in \mathbb{N}. We know that For every real x>0 and every integer n>0 there is one and ... 3answers 19 views ### Limit Question for (\sqrt{x^2+a^2}-a)/(\sqrt{x^2+b^2}-b) Can someone help me solve this limit?$$\lim_{x\to0}\frac{\sqrt{x^2+a^2}-a}{\sqrt{x^2+b^2}-b}$$with$a>0$and$b>0$. 1answer 25 views ### Show that f is Lebesgue integrable on R? [on hold] Show that$f(x)=1/x^{1/2}, x \in (0,1]$is Lebesgue integrable? 1answer 40 views ### Integral of$1/(x^2+a^2)^{3/2}$? What should I substitute to calculate the integral of$1/(x^2+a^2)^{3/2}$? With a being constant. Or is there a better way than substituting for this? I tried$u=x^2+a^2$but then I'm left with a ... 1answer 23 views ### Why generalize vector calculus with$k$-forms instead of$k$-vectors? The motivation usually given to differential forms is that they generalize vector calculus nicely. That's true, but there are also$k$-vectors, i.e., objects from$\Lambda^k(V)$instead of ... 1answer 25 views ### Please can you check my proof of$f$is Lipschitz I tried to prove: If$f$is differentiable on$[a,b]$and if$f'$is continuous on$[a,b]$then$f$is Lipschitz continuous on$[a,b]$. Please can you tell me if my proof is correct: Proof: Since ... 0answers 9 views ### rigidified line bundles and the picard functor Let$S$be a scheme, and$f : X \to S$be an$S$-scheme such that 1)$f$is quasi-compact and separated 2) for every$S$-scheme$T$we have an isomorphism$\mathscr{O}_T \to f_{T*} ...
I am a little confused about the following notation: $L' = \{xy|x\in L \ , y\in L^R\}$. I think this expression is not equivalent with palindrome but I am not entirely sure. For example, I think the ...