2
votes
1answer
478 views

Proof that every natural number is the sum of 9 cubes of natural numbers

What types of proof are there of this result and where can I read about it? I think that the Hardy-Littlewood circle method can prove that every number is the sum of something like $100000$ cubes, ...
3
votes
1answer
203 views

Morrey space and Campanato space.

I'd like to know a lot about Morrey space and Campanato spaces. For example, I'd like to know how can I see the details presents here. I'd like some reference about this. I thank you very much.
0
votes
2answers
80 views

Explain this statement $\bar 0 \in \partial f(x^*)$ where $\partial f(x^*)$ is subgradient

I haven't understood this theorem "$x^*$ is global minimum iff $\bar 0\in \partial f(x^*)$". What does it mean? Visually? P.s. Studying Nonlinear-optimization -course, 2.3139.
1
vote
1answer
127 views

Mertens' asymptotic formula for $\prod \left(1-p^{-1}\right)$ without constant

I've heard that there is an easy way to derive the asymptotic $$\prod_{p\le x} \left(1-\frac{1}{p}\right) \sim \frac{c}{\log(x)}$$ if one isn't interested in deriving $c=e^{-\gamma}$. I don't see how ...
0
votes
1answer
58 views

How to find the derivative with respect to the transformed co-ordinates.

I am stuck with something very simple , would be glad to get help . Suppose if i have a transformation matrix J , how do i find the derivative with respect to new co-ordinates , and derivative of ...
1
vote
0answers
64 views

Optimized Algorithm for Distance Matrix Solution

I've been looking for an optimized algorithm for solving a distance matrix (a hollow, skew symmetric matrix), but I haven't been able to find anything but papers discussing repopulating sparse ...
12
votes
1answer
349 views

$\tilde{u}=0,\ a.e.\ x\in\Gamma$?

Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain. Let $u\in H_0^1(\Omega)$ and $\tilde{u}\in L^2(\Omega)$ satisfies $$\int_\Omega\nabla u\nabla v=\int_\Omega \tilde{u}v,\ \forall\ v\in ...
0
votes
2answers
3k views

Find the directions in which the directional derivative has the value 1

Can anyone show me how to adjust my work below so that it is a correct answer? This is question number 14.6.28 in the 7th edition of Stewart Calculus. Find the directions in which the directional ...
136
votes
2answers
4k views

Can we ascertain that there exists an epimorphism $G\rightarrow H$?

Let $G,H$ be finite groups. Suppose we have an epimorphism $$G\times G\rightarrow H\times H$$ Can we find an epimorphism $G\rightarrow H$?
2
votes
1answer
530 views

Integrate reciprocal of a quadratic when the quadratic cannot be factorised

How do you go about integrating the reciprocal of a quadratic when it cannot be factorised (ruling out the use of partial fractions)? E.g. $$\int \frac{1}{x^2-x+1} dx$$
1
vote
3answers
2k views

How do we find the inverse of a function with $2$ variables?

$$f(m,n) = (2m+n, m+2n)$$ What do we have to do to find the inverse of this function? I don't even know where to begin.
2
votes
1answer
127 views

How can I learn more about projective geometry

I am here for a simple question for which I have done some search on this site and google. Considering I was successful on my search, there is not a coherent explanation about these. I would like to ...
0
votes
3answers
73 views

Seeking a proof of $ \sum\limits_{0\le i\le k} \binom{n}{i}.\binom{m}{k-i} = \binom{m+n}{k}$

I'm trying to prove that $$\sum_{0\le i\le k} \binom{n}{i}.\binom{m}{k-i} = \binom{m+n}{k}.$$ I got the constants $m!$ and $n!$ out of the sum but I couldn't proceed.
3
votes
1answer
88 views

What is the range of the function $f:[-1,1] \rightarrow \mathbb{R}$ defined by $f(x) = 12x^2 + 5x\sqrt{1-x^2} - 10$?

I was navigating through math exercises about functions and I got with this question If $f:[-1,1] \rightarrow \mathbb{R}$ defined by $f(x) = 12x^2 + 5x\sqrt{1-x^2} - 10$. Give the range of $f(x)$ ...
13
votes
4answers
18k views

Difference between Fourier series and Fourier transformation

Whats the difference between Fourier transformations and Fourier Series? As I've been working with Fourier Series in my maths lectures yet a friend of mine also doing engineering has been working with ...
1
vote
1answer
100 views

Real Positive Zeros of Equation

During my research on physical problem, I faced the following simple equation: $r^{2k+1}+ab\,r-a=0$ With: $-1\leq k\leq1\:,\:0<r\:,\: a,b\in\mathbb{R}$ I need to put bounders on $a,b,k$ such ...
0
votes
1answer
187 views

Union, intersection and complement of a set

Is this equation true? $$\mathcal C \bigcap_{M\in A}M=\bigcup_{M\in A}\mathcal CM$$ where C is the complement, M is a set, A is a set of sets. I don't know how to start proving or disproving it ...
1
vote
3answers
200 views

Is the inverse of a function the reflection of the function about the line $y=x$?

So if we have $f(x)$: $y=x$ when $x \ne 1$ and $y = 0$ when $x = 1$. The inverse would be: $y=x$ when $x \ne 0$ and $y=1$ when $x = 0$ ?
0
votes
1answer
42 views

What is $\left(\delta_{ab}\right)^{-1}$?

I have an expression that involves the Wigner 3j coefficient: $$\left(\matrix{a&b&0\\0&0&0}\right)^{-1}$$ This simplifies to: ...
1
vote
1answer
747 views

Linear least squares with non-negativity constraint

I am interested in the linear least squares problem: $$\min_x \|Ax-b\|^2$$ Without constraint, the problem can be directly solved. With an additional linear equality constraint, the problem can be ...
0
votes
1answer
86 views

If $X_n$ converges to $X$ almost surely how can we proof that $1/X_n$ will converge to $1/X$ almost surely?

If $X_n$ converges to $X$ almost surely how can we prove that $\cfrac 1{X_n}$ will converge to $\cfrac 1{X}$ almost surely?
0
votes
1answer
320 views

Every non-empty subset of $\mathbb{R}$ bounded above has a largest element

I restarted my analysis book from page 1 trying to relearn everything because I feel like my knowledge is too fragmented. This true false question asks exactly what the title says. I don't know 100% ...
1
vote
1answer
159 views

The material derivative

Let $u:S(t) \to \mathbb{R}$ be a scalar field on a surface $S(t)$ parametrised by time. The material derivative is $$Du = u_t + v \cdot \nabla u$$ where $v$ is the velocity. I fail to understand the ...
1
vote
1answer
36 views

If $G$ is a tree and $\forall v_1, v_2 : dist(v_1, v_2)$ is even, where $v_1, v_2-$ are leaves $\Rightarrow \exists!$ a maximal independent set.

If $G$ is a tree and $\forall v_1, v_2 : dist(v_1, v_2)$ is even, where $v_1, v_2-$ are leaves of the tree $\Rightarrow \exists!$ a maximal independent set. Give some clue please! Thanks anyway!
5
votes
4answers
196 views

Find $\lim_{n\to\infty} (1+\frac{1}{2}+…+\frac{1}{n})\frac{1}{n}$

Find the following limit: $$\lim_{n\to\infty} \left(1+\frac{1}{2}+...+\frac{1}{n}\right)\frac{1}{n}$$ My intuition says that this goes to zero, because $1/n$ goes much faster to zero than the ...
3
votes
1answer
470 views

How to prove that the image of a continuous curve in $\mathbb{R}^2$ has measure $0$?

How to prove that the image of a continuous curve in $R^2$ has measure $0$? This is an exercise given in Real Analysis-Stein & Shakarchi. A hint is given as follow: Cover the curve by rectangles, ...
7
votes
4answers
427 views

Spectrum of $\mathbb{Z}[x]$

Can someone point me towards a resource that proves that the spectrum of $\mathbb{Z}[x]$ consists of ideals $(p,f)$ where $p$ prime or zero and $f$ irred mod $p$? In particular I remember this can be ...
2
votes
1answer
66 views

How to show that $(x+y)^{a} \geq x^{a} + y^{a}$ given that $a \geq 1$ and $x,y \geq 0$?

How to show that $(x+y)^{a} \geq x^{a} + y^{a}$ given that $a \geq 1$ and $x,y \geq 0$? Also, how to prove that the reverse inequality holds when $0 \leq a \leq 1$?
0
votes
1answer
120 views

Gradually rising or falling numbers

I'm looking for a number series I can use for gradually rising or falling numbers. The number series should not be linear and should converge to a number at some point. (Sorry I'm really scared of ...
2
votes
1answer
660 views

How to simultaneously solving trig equations?

I was doing some math work, and have to solve two trig equations simultaneously, but have no idea how to approach this, can anyone help, just need to be pointed in the right direction. I have to ...
0
votes
2answers
60 views

Limit in metric space.

I have , In $X$ $\lim_{n\to \infty} f_n=f_0 $ and similarly $g_n$ converges to $g_0$, $d$ is a metric defined in $X$ . I have to show that $\lim_{n\to \infty} d(f_n, g_n)=(f_0, g_0)$ This is what i ...
6
votes
1answer
316 views

Prove that $\int_0^1[f''(x)]^2dx\ge4.$

Let $f$ be a $C^2$ function on $[0,1]$ such that $f(0)=f(1)=f'(0)=0,f'(1)=1.$ Prove that $\int_0^1[f''(x)]^2dx\ge4.$ Find all $f$ for equality to occur.
1
vote
0answers
78 views

Is there an action functional whose critical points are the geodesics of a arbitrary connection on TM?

Geodesics of the Levi-Civita connection may be defined as the critical points of the action functional $S[\gamma]=\int \lvert\dot{\gamma}\rvert\,dt$ (or square it, if you like). The Euler-Lagrange ...
0
votes
1answer
107 views

Approximate rational number of radical combination

Suppose that there is a radical combination $a\sqrt{b}+c\sqrt{d}$ where a,b,c,d are natural. Each term part $\sqrt{b}$ cannot be transformed into the form of $s\sqrt{q}$. The question is, 1) Suppose ...
2
votes
4answers
107 views

How to prove the convergence?

I have problem with proving that $$a_{n} = - 2\sqrt {n+1} + \left({\frac{1}{\sqrt {1}}+{\frac{1}{\sqrt{2}}}+\cdots+{\frac{1}{\sqrt{n}}}}\right)$$ converges. I have tried to transform it to the ...
1
vote
2answers
178 views

Geometry problem - squares and triangle

In $\triangle ABC$ $BE$ and $AD$ are heights. Create rectangle $AEPQ$ that $AC = AQ$ and create rectangle $DBNM$ that $BC = BN$. Show that Area of square $LKBA$ is equal to sum of areas of rectangles. ...
6
votes
4answers
8k views

Poisson Distribution of sum of two random independent variables $X$, $Y$

$X \sim \mathcal{P}( \lambda) $ and $Y \sim \mathcal{P}( \mu)$ meaning that $X$ and $Y$ are Poisson distributions. What is the probability distribution law of $X + Y$. I know it is $X+Y \sim ...
0
votes
1answer
112 views

Monte Carlo sampling a binomial expansion

I want to figure out the following question $$ 1 = (10 - 9)^{100} = 10^{100}-100 \times 10^{99} \ 9 + \frac{100 \times 99}{2} 10^{98} \ 9^{2} - \frac{100 \times 99 \times 98 }{3}10^{97} 9^{3} ...
0
votes
1answer
136 views

Solving the initial value problem of a differential equation

Let $x′′- q(t) x = 0$, $0\le t \lt\infty$ , $x(0)=1$, $x'(0)=1$, where $q(x)$ is monotonically increasing continuous function, then what will be the solution?
0
votes
0answers
417 views

Proof of closure axiom necessary or is it implied by other axioms?

Sometimes I see books saying that you need to prove the closure axiom to decide if a set is a group. Other times I see books saying you only need to prove the identity, inverse and associativity ...
2
votes
1answer
124 views

Lie Groups question from Brian Hall's Lie Groups, Lie Algebras and their representations.

In page 60 of Hall's textbook, ex. 8 assignment (c), he asks me to prove that if $A$ is a unipotent matrix then $\exp(\log A))=A$. In the hint he gives to show that for $A(t)=I+t(A-I)$ we get ...
7
votes
1answer
1k views

Why are continuous functions not dense in $L^\infty$?

Why are the continuous functions not dense in $L^\infty$? I mean both concretely (i.e. a counter example) and intuitively why is this the case.
0
votes
1answer
709 views

General method for determining stability of equilibrium points

Given a system of ODEs, $\mathbf{x}' = A\mathbf{x}$, one way to determine the stability of an equilibrium point is to look at the eigenvalues of the Jacobian matrix. However, there are cases in ...
4
votes
4answers
304 views

Book on matrix computation

I'm taking a machine learning course and it involves a lot of matrix computation like compute the derivatives of a matrix with respect to a vector term. In my linear algebra course these material is ...
2
votes
1answer
81 views

What is the $P( |X-10| > 2)$ of a normal distribution when mean is 10, and standard deviation is 6?

I couldn't figure out this question: What is the $P( |X-10| > 2)$ of a normal distribution when mean is 10, and standard deviation is 6?
2
votes
2answers
192 views

Basic introduction to algebraic topology using simplicial complexes

What is a basic, graduate-level introduction to algebraic topology? I think Hatcher is a great book, but I want to learn the subject from the point of view of simplicial complexes. Primarily, I want ...
4
votes
2answers
277 views

For two uncorrelated random variables $X,Y$, why does $\rho(X+Y,2X+2Y)=4?$

Given two uncorrelated random variables $X,Y$ with the same variance $\sigma^2 $ I need to compute $\rho= \frac{COV(X,Y)}{\sigma(X)\sigma(Y)}$ between $X+Y$ and $2X+2Y$. I know it should be a ...
0
votes
0answers
58 views

Write expectation

How can I write the folowing expectation $E[f(X_t,X_s)]$ by means of a Lebesgue integral and the density of $X_t$? where $f$ is a "nice" function and $X_t$ is a process without undependent increments! ...
2
votes
1answer
153 views

Finding the Probability that a randomly chosen integer is a square or psuedo-square

Definition: $n$ is a psuedo-square if the legendre symbol $(\frac{n}{p}) = 0$ or $= 1$ for $p = 3, 5, 7, 11$. I want to find the probability of $n$ being a square or psuedo-square. I know that any ...
1
vote
0answers
44 views

geometric charaterization of complex interpolation spaces $(H,Y)_\theta$ where $H$ is a Hilbert space?

Let $C$ be the class of Banach spaces $X$ such that there exists $0<\theta<1$, a Hilbert space $H$ and a Banach space $Y$ such that $$ X=(H,Y)_\theta $$ (complex interpolation of Calderon). ...

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