1
vote
2answers
97 views

Sum of a cosine series

I have a short question: It turns out that the following holds: $\sum_{i=1}^{k-1} \cos(\frac{2\pi i}{k}) = - 1$. Why is that? Thank you!
1
vote
1answer
124 views

Distributional derivative of a characteristic function

I need some help with this exercise about distributional derivatives: If we have $N=2$, and a function $g=\chi_{C}$, where $\chi$ is the characteristic function, and $C$ is the unitary cube ...
1
vote
1answer
93 views

how do you rewrite a recursive formula to find its roots

Let $(x_n)$ be the sequence defined by $x_1=2$ and the recursive formula $x_{n+1} = \frac12 + \sqrt{x_n}$. Rewrite the recursive formula in the form $$ x_n - x_{n+1} = ax_{n+1}^2 + bx_{n+1} + c$$ ...
1
vote
2answers
302 views

Using Euler's method and Taylor polynomial to solve differential equation

Consider the initial value problem $dy/dx=x+y^2$ with $y(0)=1$ a) Use Euler's Method with step-length $h=0.1$ to find an approximation to $y(0.3)$. HINT 1: :Numerical methods. HINT 2: ...
0
votes
1answer
72 views

The question about strong law of large number

In my lecture notes about proof of strong law of large number. The last step to show $\sum_n^\infty \frac{var(Y_n)}{n^2}$ converges is $$E(X^2\sum_{n>max\{|X|,1\}}^\infty\frac{2}{n(n+1)} )\leq ...
1
vote
1answer
35 views

Difference of a likelihood function for a vector and a single value

$p(x\mid C)$ is defined as the probability density of a point $x$ given that it belongs to a class $C.$ But what of $p(\mathbf{x}\mid C)$ where $\mathbf{x}$ is a vector? I'm finding hard to ...
2
votes
2answers
291 views

Ideas about an Ordinary Differential Equations research work (University level)

Good afternoon to everyone, I need some ideas about a Ordinary Differential Equations research work. It is for the ODE subject that I am doing at my Mathematics degree in my University. They asked me ...
9
votes
1answer
184 views

Is there anything to be learned from the spectrum of a cohomology ring?

Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"? As $H^*$ is a graded ring, does this question ...
-1
votes
2answers
92 views

How to find $\lim_{x \rightarrow 0} x^2\ln x$?

I can't find the limit of $x^2 \ln x$ as $x$ approaches $0$. I can use only notable special limits and I'm not allowed to use l'Hopital's method.
3
votes
1answer
54 views

$L^2$ function on finite interval implies $L^1$?

Let $a,b\in\mathbb{R}$. Suppse $f:\mathbb{R}\rightarrow\mathbb{C}$ is an $L^2$ function on the finite interval $(a,b)$. That is, $$\int_{a}^b|f(x)|^2dx<\infty$$ Is it always true that $f$ is an ...
2
votes
1answer
115 views

Why do we need primitive roots?

What is the most motivating way to introduce the order of a modulo n? Apart from simplifying powers of residues is there any other use of the order? Are there any examples which have a real impact on ...
1
vote
1answer
116 views

How to find the normal plane from a tangent plane?

$$f(x,y,z)=\frac{x^2}{4} +\frac{y^2}{9} +\frac{z^2}{25}=3 $$ I found the tangent plane from this surface at $P(2,3,5)$ by using the gradient vector, $\nabla F=\langle f_x, f_y, f_z\rangle$. I was ...
4
votes
1answer
73 views

Is Hausdorffness necessary for the classical ascoli theorem? [duplicate]

Munkres - topology p.278 I exactly followed the argument in the text, and I cannot find where I used hausdorffness. Where in the argument used Hausdorffness? The reason why I am asking is that the ...
0
votes
1answer
52 views

Problem with a loop in Mathematica

I am trying to create a loop in Mathamtica, where i get all $i$'s (with a certain bound, of course) for which the following expression holds: $7^{41\ast i}\equiv 3^{41}\, mod\, p$, where $p$ is just ...
1
vote
1answer
75 views

Stokes' theorem and partial derivatives

Applying Stokes' theorem to a surface, I obtained the following equations, $$R_y - Q_z = xe^{y}-e^{x}\cos(z)$$ $$P_z - R_x = -2y\sin(z)-e^{y}$$ $$ Q_x-P_y = e^{x} \sin(z) - 2\cos(z)$$ where the field ...
5
votes
1answer
170 views

Physical Meaning of Symplectic Vector Fields

The mathematics of symplectic (as well as Hamiltonian) vector fields is something that has been quite clear to me for some time, but recently I have been thinking much more about what certain ...
1
vote
2answers
38 views
1
vote
1answer
26 views

Linear Equation- Equation of a line passing through points

What is the slope-intercept form of the equation of the line passing through the points (-1,-4) with an undefined slope?
2
votes
1answer
111 views

Markov chain and hitting times

I have a Problem about hitting times. That's the following: Let $A\subset E$ and the first passage time $T_A$ and the hitting time $H_A$. Define: $T_A =\inf\{n\geq 0;X_n \in A\}$ and $H_A ...
0
votes
1answer
137 views

What is the Krull dimension of this local ring

I want to know what is the dimension of this ring $\mathbb C[x,y]_{(0,0)}/(y^2-x^7,y^5-x^3)$. I don't know how to do that. If I suppose $y^2=x^7$ I will get a higher degree of $x$.
0
votes
1answer
23 views

prove a limit about convergence of norma

Given $f_n:\Bbb R \to \Bbb R $ converge to 0 on norma 2. Show that: $$\lim_{n\to\infty}{1\over n}\int_{-n}^n|f_n|dx=0$$ I think it has something to do with C-S inequality but i'm having troubles with ...
1
vote
1answer
71 views

Almost uniform convergence of $f_n(x) = x^n$ on the interval $[0, 1]$?

Consider $f_n(x) = x^n$ on the interval $[0, 1]$. This converges pointwise to $$f = \begin{cases} 0, & \mbox{if } 0 \le x < 1\ \\ 1, & \mbox{if } x = 1 \end{cases}$$ Now I know $f_n(x)$ ...
1
vote
1answer
67 views

Is the submanifold compact?

Let $M$ be the following subsets of $\mathbb R^4$:$$M= \{(x,y,z,w), 2x^2+2=z^2+w^2, 3x^2+y^2=z^2+w^2 \}$$ we know $M$ is a submanifold of $\mathbb R^4$, is $M$ compact?
0
votes
1answer
125 views

Closures of Relations

How to prove that the transitive closure of a symmetric closure of a relation is greater than the symmetric closure of a transitive closure of a relation?
0
votes
2answers
53 views

Problem with integrating by parts

\begin{eqnarray*} \int x^3\cos 4x \ dx &=& \int x^3(8\cos^4 x - 8\cos^2x+1) \ dx\\ &=&\int 8x^3\cos^4x \ dx - \int 8x^3\cos^2x + \int x^3 \ dx\\ &=&8\int x^3\cos^4x \ dx - ...
0
votes
1answer
45 views

symmetry in the homotopy relation

Suppose $\alpha, \beta : I \to X$ are paths and suppose $\alpha $ is homotopic to $\beta$, $\alpha \cong \beta$. So, can find a continuous function $F(s,t) = f_t(s)$ such that $$ f_t(0) = \alpha(0) ...
1
vote
0answers
86 views

What is it that determines the degree of a tile, and to what extent does the shape of a tile determine its degree?

I'd be interested to learn if/what the geometric or algebraic approach to acquiring a tile of degree 2, 3,..., n (i.e. a tile of an arbitrary degree) would be? Asked another way- what is it that ...
4
votes
3answers
100 views

$(p^m-1) \mid (p^n-1) \Leftrightarrow m \mid n$

Prove that $(p^m-1) \mid (p^n-1) \Leftrightarrow m \mid n$. The $\Leftarrow$ part is ok, the $\Rightarrow$ part should be easy but I'm stuck with it! Thanks in advance.
2
votes
2answers
141 views

Is $\lim_{n \rightarrow \infty} a_{n+1}/a_n=L \implies \lim_{n \rightarrow \infty} \sqrt [n] {a_n}=L$ true? If not, is there a counter example? [duplicate]

We were told, in recitation class, about a test for sequences convergence (not series) Which goes as follows: if $\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n}=L$ then $\lim_{n \rightarrow \infty} ...
0
votes
1answer
126 views

Prove that subsequences of a sequence with infinite range contained in a compact set X converges to a point in X.

I have a question about a step regarding the following proof from Rudin theorem 3.6a), Proof: Let E be the range of the sequence ${p_n}$ since E is a infinite subset of a compact set it has a limit ...
2
votes
2answers
47 views

Maximize expected return

making a practice exam I had to make the following problem which I couldn't solve unfortunately... Problem In the springtime, a student has $N$ days to find a summer job for one month. Each day, ...
0
votes
1answer
47 views

Possible number of names from a certain alphabet

I am trying to solve the following problem, but I am a bit stuck. The question is as follows. The language of a certain island has only the letters A, B, C, D, E. Every place name must start and ...
3
votes
1answer
86 views

convergence on $L^p$ space

Let $ \displaystyle{ f \in L^p (\mathbb R^n), 1\leq p <\infty }$ and let $ \upsilon \in \mathbb R^n$. For $h>0$ define $\displaystyle{ f_h(x) = \frac{1}{h} \int_0^h f(x+s \upsilon) ds }$. ...
2
votes
0answers
123 views

Local trivializations for orthonormal frame bundle

Let $(E,\pi, M)$ be a real vector bundle of Rank $N$. Then one can define its frame bundle $GL(E)$ as follows: $GL(E)_x:=\{\text{ordered bases of }E_x\}$ (for $x\in M$). $GL(E):=\bigcup_{x\in M} ...
2
votes
2answers
57 views

Question about measure theory.

Suppose $( \Omega_1, \mathcal{F}_1), ( \Omega_2, \mathcal{F}_2) $ are measurable spaces then My question is with regard to the definition of the collection $\mathcal{G}$. I don't really ...
1
vote
2answers
71 views

Kronecker product of $A$ and $B$ is similar to the Kronecker product of $B$ and $A$

Given two matrices $A$ and $B$. How would one prove that the Kronecker product of $A$ and $B$ is similar to the Kronecker product of $B$ and $A$?
0
votes
1answer
114 views

Least Squares & Normal Equations

I encountered the problem below, and I know how to do the least squares for a system of equations with no solutions (inconsistent system) where the number of equations (rows) is greater than the ...
1
vote
2answers
99 views

Optimal strategy in a match picking game

I faced this exercise making a practice exam and couldn't find the answer: Question: 2 players play the following game with matches. Each player can take in turn some matches from a bunch of ...
2
votes
3answers
611 views

Find the inverse of a matrix with a variable

$$X= \begin{pmatrix} 2-n & 1 & 1 & 1 & \ldots & 1 & 1 \\ 1 & 2-n & 1 & 1 & \ldots & 1 & 1 \\ \vdots & \vdots & \vdots & \vdots & \ddots ...
2
votes
2answers
48 views

Order of an element as a product

I apologise if this has been asked before, but I wasn't able to find it.. I am trying to prove that if $g\in G$, where $G$ is an arbitrary group,$|g| = n$, where $n = ab$, with $\gcd(a, b) = 1$, then ...
0
votes
2answers
710 views

How can I determine if three 3d vectors are creating a triangle

I have a basic homework vectors question which I can't figure. let's say I have three 3d vectors: $$\vec{a}=2\mathbf{i}-2\mathbf{j}-\mathbf{k}\\ \vec{b}=6\mathbf{i}-3\mathbf{j}+2\mathbf{k}\\ ...
3
votes
1answer
314 views

Evaluate $\int_0^1\int_p^1 \frac {x^3}{\sqrt{1-y^6}} dydx$

I have been working on this sum for a while. The question asks to evaluate the double integral. $$\int_0^1\int_p^1 \frac {x^3}{\sqrt{1-y^6}} dydx$$ where $p$ is equal to $x^2$. I know that I have to ...
2
votes
1answer
78 views

$\nabla \cdot \hat n$ where $\hat n$ is a unit vector normal to a cylinder of radius $R$ and with a length $L=\infty$

I'd like to calculate $\nabla \cdot \hat n$ where $\hat n$ is a unit vector normal to a cylinder of radius $R$ and with a length $L=\infty$. What I've thought of is: $\hat n= \hat R $ and using: ...
10
votes
2answers
423 views

Minimal difference between classical and intuitionistic sequent calculus

Consider propositional logic with primitive connectives $\{{\to},{\land},{\lor},{\bot}\}$. We view $\neg \varphi$ as an abbreviation of $\varphi\to\bot$ and $\varphi\leftrightarrow\psi$ as an ...
0
votes
0answers
30 views

Prove that $|G| = |Z(G)| + \sum_{i' \in I'}|G:C_G(x_{i'})|$

Let $G$ be a finite group. Define $\rho: G \to Sym(G)$ by $\rho(g) = c_g,$ where $c_g:G \to G (y \mapsto gyg^{-1}). $ Denote $(\rho(g))(x) =g(x)$ For $x\in G,$ define $G_x: = \{g\in G| g(x) = x\}$ ...
0
votes
1answer
55 views

Sequence in span with disjoints supports has (?) block subsequence.

Assume $b_k \in <e_i, \text{with coefficients} \ a_i^k \geq 0>$ is a sequence and $a_i^k$ have disjoint supports(support is the set where $a_i^k \neq 0$). Is there a way to prove or disprove ...
1
vote
0answers
118 views

Solving construction problems?

I recently encountered 'construction problems' in geometry. These were quite new to me and I didn't know the requirements they expected and prerequisites to solve them. I'll explain with an example. ...
1
vote
4answers
109 views

Complex numbers problem

I have to solve where n is equal to n=80996.
1
vote
3answers
499 views

Prove that $\operatorname{trace}(ABC) = \operatorname{trace}(BCA) = \operatorname{trace}(CAB)$

Prove that $\operatorname{trace}(ABC) = \operatorname{trace}(BCA) = \operatorname{trace}(CAB)$ if $A,B,C$ matrices have the same size.
1
vote
0answers
41 views

Let $V$ be a normed vector space over $\mathbb{C}$, is there an inner product structure on $V$ such that the two spaces have the same topology.

Let $V$ be a complete normed vector space over $\mathbb{C}$. Let $\tau_1$ be the topology induced by its norm. Is there an inner product structure on $V$ such that the topology induced by the norm of ...

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