# All Questions

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### Integral calculation - where the $i$ came from?

$$\frac{e^{it(c-n)}}{i(c-n)} |_{-\pi}^\pi = \frac{e^{i\pi(c-n)} - e^{-i\pi(c-n)}}{i(c-n)} =\frac{2\sin(\pi(c-n))}{i(c-n)}$$ Correct answer is:$$\frac{2\color{red}{i}\sin(\pi(c-n))}{i(c-n)}$$ Why? :/...
3answers
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### How would you prove this theory of computation problem?

I have trouble proving the following statement, I'm supposed to do it for our theory of computation course but since I've been trying for days I'm looking for a hint : What is the smallest value ...
2answers
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1answer
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### Marbles in a bag (Combinatorics)

There are 10 numbered from 1 to 10 marbles. The marbles are placed in an opaque bag and shuffled. One random marble is taken out, its number is written on a piece of paper, marble is then returned to ...
0answers
19 views

### Simple odds calculation

I'm stuck with a simple expression creation problem. I'd like to express odds by removing values from $100$ and arriving at a number. Every variable I use has value, that can either be $+10$ or $-10$, ...
3answers
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1answer
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### A question about the definition of a stalk

In the definition of the stalk of a presheaf of abelian groups on a topological space, at a point, one uses the fact that the open sets containing that point form a poset, which is directed via ...
0answers
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Let a Complex no. $\alpha$ and $\displaystyle \frac{1}{\bar{\alpha}}$ lie on the Circles $\left|z-z_{0}\right|=r$ and $\left|z-z_{0}\right|=4r^2$ respectively. If $\displaystyle z_{0}=x_{0}+iy_{0}... 2answers 151 views ### A is a proper subset of B implies NOT(B subset of A) Proof This is not homework. I'm just studying for my Discrete Mathematics course. I'd like to know how to prove the following using element wise proofs: Given a proper subset: $$A \subset B \iff (A \... 1answer 58 views ### Totally geodesic submanifold I'm reading "Introduction to symplectic topology", D.McDuff, D.Salamonand and I have a problem with the exercise 1.26. According to the definition, a submanifold L of a Riemannian manifold (\mathbb ... 4answers 97 views ### How to prove this limit is 0? Let f:[0,\infty )\rightarrow \mathbb{R} be a continuous function such that: \forall x\ge 0\:,\:f\left(x\right)\ne 0. \lim _{x\to \infty }f\left(x\right)=L\:\in \mathbb{R}. \forall \... 0answers 80 views ### Is it correct to say that if \lim\limits_{x \to a}f(x) = 0 it is an Infinitesimal? I think I'm misuderstanding something here, because to my understanding the definition of infinitesimal given in my textbook does not convey the same thing as in other sources. I've read the ... 1answer 34 views ### Do we have proj_u(a) + proj_u(b) = proj_u(a+b)? Let a, b, u be vectors in \mathbb{R}^3. For two vectors r, u in \mathbb{R}^3, let proj_u(r) be the projection of r on the line of u in \mathbb{R}^3. Do we have proj_u(a) + proj_u(b) = ... 2answers 3k views ### Difference between sum and direct sum What is the difference between sum of two vectors and direct sum of two vectors? My textbook is confusing about it. Any help would be appreciated. Thanks in advance! 1answer 47 views ### Is my proof for an invertible matrix correct? I'm a bit confused in class over some rules for proofs, so I was wondering if this was the correct proof for the following question: Suppose that A and B are similar n\times n matrices and ... 0answers 33 views ### Liapunov 's Direct Method (S.L.Ross ) Consider the Autonomous Non-Linear system$$ \frac{dx}{dt} = P(x,y), \frac{dy}{dt} = Q(x,y).$$Assume that the given system has an isolated critical point (0,0). Let P \text{ and }Q ... 2answers 62 views ### Probability of a disintegration The half-life of Uranium-238 is 5×10^9 years. What is the probability than a uranium atom disintegrates in any one year? I think I have to use Poisson's law but I don't know how to apply it in ... 1answer 51 views ### Closure function of a matroid I need some help to understand: If M is matroid and e is an element in that matroid, what is the closure function of M\setminus e? And what is the closure function of M/e? Any help would be ... 3answers 110 views ### y''+y'^{2}+y=0 equation solution How would you solve this differential equation y''+y'^{2}+y=0? I can't apply the ansatz method (or more formally apply the characteristic polynomial method). Thanks 2answers 33 views ### Let f: \Bbb R \to \Bbb R be such that f^{-1} (a, \infty) and f^{-1} (- \infty, b) are open for any a,b \in \Bbb R. Let f: \Bbb R \to \Bbb R be such that f^{-1} (a, \infty) and f^{-1} (- \infty, b) are open for any a,b \in \Bbb R. Show that f is continuous. My Try: We first take an arbitary open subset (... 1answer 112 views ### Problem in Gradient operator and Kronecker delta function I have this expression$$\nabla_{i}\nabla_{j}\Big(\frac{1}{r}\Big)$$Where r is a distance. I tried this, but encountering manipulations of \delta_{ij} with \hat{r_i},\hat{r_{j}} and still ... 1answer 51 views ### Characteristic polynomial of recurrence relation \lambda^4 + \lambda^3 - 9\lambda^2 + 11\lambda -4 = 0 The characteristic polynomial of this recurrence relation is$$λ^4 + λ^3 - 9λ^2 + 11λ- 4 = 0$$or$$(λ − 1)^3 \cdot (λ + 4) = 0.$$So the solution is of the form a_n = α({-4})^n + β~n^2 + γ~n + δ.... 3answers 43 views ### Prove that f is continuous [closed] Let F a set non-empty and closed. Give x∈\mathbb{R} and let f(x)=inf{|x-y|, y∈F }.Prove that f is continuous and {x∈\mathbb{R}, f(x)=0} = F. 1answer 35 views ### For which real x is this (monster) series convergent? I'm practicing for an exam, and got to this example:$$\sum_{n=1}^{+\infty} \left (\frac{x^2n^2-2|x|^3n}{1+2xn^2} \right)^{7n}$$I rearranged the expression to try to check for which x the ... 0answers 79 views ### Hilbert style proof for \Box A \vee \Box B \rightarrow \Box(A\vee B) in K. I have to find a formal Hilbert style proof for \Box A \vee \Box B \rightarrow \Box(A\vee B) on modal logic, K. I can use all classical propositional tautologies, Modus Ponens and Distribution axiom.... 2answers 435 views ### Doing Michael Spivak's Exercises I am doing Spivak's Calculus, and I find it EXTREMELY difficult. I usually ask questions here because I cannot do the problems on my own. How long should it take to do a Spivak problem? Is it ... 3answers 54 views ### Meaning of derivatives I need to know the meaning of the higher order derivatives of a polynomial. Let make an example. Assume we have a polynomial of degree n. Then$$ f(x)=a_0+a_1x+a_2x^2+\ldots+a_nx^n $$We know that ... 2answers 798 views ### Integration by parts using u-substitution and square root I have to use the technique of integration by parts to evaluate the integrals. I'm having trouble with a particular problem:$$ \int x (\sqrt{x+2}) dx$$I'm using u-substitution, but since I'm also ... 1answer 22 views ### Volume of Revolution$f(x) = x^2$Suppose you are given$y = f(x)$I want to use double integrals, instead of the traditional washers. Suppose even better,$f(x) = x^2$Find the volume of$f(x) = x^2$,$x = 0$,$x = 4$,$y = 0\$ &...

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