3
votes
1answer
32 views

At which points the tangent lines of the function $y=\cos x$ are parallel to $-\frac{1}{2}x+1$?

I did the following: $$\cos' x=-1/2 \\ -\sin x=-1/2 \\ \sin x = 1/2$$ So, I guess that in this case I have to find values such that $\sin x = 1/2$, these values are $\pi/5$ and $5\pi /6$ (I know ...
1
vote
1answer
22 views

Show that all solutions remain in the interval for all time

I really have no idea on how to get started with these, there's no similar example in my book. Do I need to compute $\frac{dy}{dx}$? Any help would be greatly appreciated. Maybe there's just some ...
0
votes
1answer
18 views

Confusion about probability question

Some friends are sitting together playing a game that involves rolling dice. On one turn, a player rolls six 6-sided dice and gets one of each number showing. Another player sees this and asks "what ...
1
vote
3answers
85 views

Does $\tan x\cdot \cos x$ equal $\sin x$?

Is it true that $\tan x\cdot \cos x = \sin x$? If I put $x=30$ in my calculator then I don't get the same answer as $\sin 30$, why is this? Don't the two cosines cancel out? I'm probably missing ...
0
votes
1answer
21 views

How to show eigenvalues and singular values are the same?

Let $A$ be Hermitian and positive semi-definite i.e. $x^TAx \ge 0$ Let $K=A^TA=A^2$ $Ax=\lambda x$. Then $Kx=A^2x=\lambda^2 x$ Then, every eigenvector $x$ of $A$ is also eigenvector of $K$ with ...
0
votes
3answers
52 views

Prove that $1 \cdot 1!+2 \cdot 2!+\cdots+n \cdot n!=(n+1)!-1$

Prove that $1 \cdot 1!+2 \cdot 2!+\cdots+n \cdot n!=(n+1)!-1$ whenever $n$ is a positive integer. Basis step: $P(1)$ is true because $1 \cdot 1!=(1+1)!-1$ evaluate to $1$ on both sides. Inductive ...
0
votes
1answer
23 views

Solve in positive integers $a^2-b^2+4a=0$

Solve in positive integers $a^2-b^2+4a=0$ I tried considering the residues in mod4 but not so helpful. Any help/hint on how to approach this problem ? Thanks !
0
votes
0answers
8 views

On non-degenerate bilinear forms on infinite dimensional vector spaces

For any non-degenerate bilinear form $(\cdot,\cdot)$ on a vector space $V$ and a linear functional $f$, there exists $v \in V$ such that $f(v)=(v,w)$ for all $w \in V$. It's easy in ...
1
vote
0answers
19 views

Can a connected regular simple graph have two maximal cliques of different order?

Can a connected regular simple graph have two maximal cliques of different order? I know of examples of regular simple graphs having two maximal cliques of different order. But will connectedness ...
0
votes
1answer
17 views

Partially ordered permutations

If I have a set $X$ of length five such that $X=\{1,2,3,4,5\}.$ I want to find the number of permutations where the relative order of the $2$ sets $\{2,4\}$ and $\{2,5\}$ is maintained. In other ...
-1
votes
0answers
14 views

Markov Chain States

Given a MC with five states $\{1,2,3,4,5\}$ and transition matrix \begin{bmatrix} 0.5& 0.5 & 0 & 0 & 0 \\ 0.75 & 0.25 & 0 & 0 & 0 \\ 0 & 0.25 & ...
0
votes
0answers
11 views

“Transference” Argument

In the proof of the Iwaniec-Martin theorem (giving a bound in $L^p$ for the Riesz transform, $\|R_j\|_p=\cot(\frac{\pi}{2p^*})$ the proof of this equality is given by proving the inequalities $\leq$ ...
0
votes
1answer
26 views

How many ways to assign 10 digits to 6 containers.

So if we had 6 containers eg (a,b,c,d,e,f) how many ways could we assign the digits 0-9 to these containers. For example one way might be: a = 4 b = 5 c = 0 d = 3 e = 8 f = 7 Is there a ...
-4
votes
1answer
27 views

Mathematical Induction problem involving binomial coefficients [on hold]

I cannot solve it by using mathematical induction. Please help me with it. $$\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\ldots+\binom{n}{n}=2^n$$ ...
1
vote
2answers
20 views

$\sigma$-algebra generated by the set $A:=\{n,n+1,n+2\}$ where $n\in \Bbb{N}-\{1\}$.

I am taking a course on measure theory and I have some issues regarding $\sigma$-algebra. Here is the exercise, we are taking place on $\Bbb{N}$ and I need to characterize the $\sigma$-algebra ...
1
vote
1answer
41 views

Calculating the number of terms in arithmetic sequence

I know that if I have a set of numbers, let's say+ $1,2,3,4,5$ I can find the number of terms by subtraction the last term $5$ from the first terms $1$ and then add $1$: $(5-1)+1 = 5$, then the ...
1
vote
2answers
25 views

Limit problems in two variable function

How would one find the $\alpha$ and $\beta$ for which $$\frac{x^{\alpha}y^{\beta}} {\sqrt{x^2 + y^2}} \to 0$$ as $(x,y) \to (0,0)$ ? I understand the $\epsilon$-$\delta$ definition of a limit but ...
0
votes
1answer
19 views

Arithmetic functions proof

Theorem: Let $f$ be an arithmetic function such that $f(1)=1$. Then there exists a unique arithmetic function $g$ such that $f\ast g =\epsilon$. The arithmetic function $g$ is called the Dirichlet ...
3
votes
2answers
50 views

Proving that the delta function is the derivative of the step function.

I want to prove $\frac{\mathrm{d} }{\mathrm{d} x}\Theta =\delta (x)$ using this representation of the delta function: $\delta(x)= \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ikx}dk $ This should be ...
0
votes
1answer
12 views

How to get the value of 'scaled' binomial distribution?

People kindly told me that there is not a equivalent popular distribution for $aX$ when $X$ is distributed as Binomial, but it is just a 'scaled' distribution. Here, $a$ is a positive constant. ...
1
vote
1answer
35 views

Prove by Induction - Sequence [on hold]

The sequence $x_1, x_2, x_3, \ldots$ is such that $x_1 = 1 $ and $$x_{n+1} \space = \frac{1+4x_n}{5 + 2x_n}$$ Prove by induction that $x_n > 1$ for all $n \ge 1$. I have absolutely no clue how ...
6
votes
1answer
37 views

Is there a standard notation for the product from right to left?

I am considering a product of the matrices $(A_i)_{1\leq i\leq n}$ in reverse order $$P=A_nA_{n-1}\dots A_1,$$ and I was wondering if there was a standard notation for it, like ...
1
vote
0answers
9 views

dependent “time change” of a.s. convergent random variables

Let $(X_n)$ be a sequence of random variables, s.t. $\frac{X_n}{n^p}\to X$ a.s. for some $p>0$. Now let $(Y_t)$ be a discrete stochastic process, s.t. $\frac{(Y_t)^p}{t}\to Y>0$ a.s. We only ...
0
votes
1answer
22 views

directional derivative problem

for a point M(4,1) and a function $z = x y^2 - (x^2/y^3)$ I was tasked with finding a directional derivative in the direction which creates a 30 degree angle with the $x$ axis....I find it a little ...
2
votes
1answer
22 views

Example 5, Sec. 23 in Munkres' TOPOLOGY, 2nd edition: What is the closure of this set?

What is the closure in $\mathbb{R}^2$ of the set $$ \left\{ \ x \times y \ \in \mathbb{R} \times \mathbb{R} \ \colon \ x > 0, \ y = \frac{1}{x} \ \right\}? $$ I know that each point of the set is ...
6
votes
3answers
628 views

I want to learn mathematics to extend myself.

I am a middle school student that is highly fascinated by mathematics and its elegance. However, I have found that to appreciate a lot of mathematics a lot of knowledge is required. Currently, I can ...
0
votes
0answers
17 views

Trace in Einstein notation

I know quite well what the trace of a matrix is; however, I am not quite sure I understand the meaning of the 'trace' concept when applied to tensors. I would be very grateful to you if: 1) You could ...
0
votes
2answers
19 views

Mistake in proof of sum of divisors function $\sigma(n)$

The proof derives the correct result, but I cannot see how the first equality is correct. To begin we use the formula $\sigma(n)=\sum_{d\mid n}d$ This is the first step in the proof: $$\sum_{1\leq ...
0
votes
0answers
10 views

Commutation of Convolution, Restriction and Differentiation

Let $B$ be the open unit ball in $\mathbb R^n$ centered at zero and let $K=\bar{B}\cap (\mathbb R^{n-1}\times\{0\})$. Suppose you are given $u\in C^{1,\alpha}(B)$ such that $u|_K=f\in C^2(K)$. For a ...
0
votes
1answer
28 views

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions :

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions : $ f(x+t,y) = f(x,y) + ty~~;~~f(x,t+y) = f(x,y) + tx~~;~~f(0,0) = K$. Then $\forall ~~x,y \in \mathbb R, ...
1
vote
1answer
56 views

How to define the isomorphism?

Let $R$ be a ring, then For $R[x]/\langle x-1\rangle \cong R$, we define the map, $\varphi$ : $R[x]\rightarrow R$, defined by $\varphi(f) =f(1)$ For $R[x]/\langle x\rangle \cong R$, we define the ...
1
vote
1answer
17 views

“p-adic absolute value” in polynomial ring

I'm working with Gouvea's book on p-adic numbers. In problem 34 I'm asked to give a "p-adic" (I put it in quotes as in my understanding its just a p-adic-like) valuation and absolute value for an ...
1
vote
1answer
36 views

The moduli space of stable maps

I am reading Katz' book Enumerative Geometry and String Theory. I have a few questions regarding the moduli space of degree $d$ genus $0$ stable maps into $\mathbb P^n$, denoted ...
0
votes
1answer
26 views

Changing the basis of a transformation

Given $T: \mathbb{R}^2 \to \mathbb{R}^2 : T\begin{bmatrix} x \\ y \end{bmatrix} \to \begin{bmatrix} 2x+y \\ x-3y \end{bmatrix}$ with standard basis $\mathcal{B}$ and basis ...
0
votes
1answer
16 views

How to formulate the product of two generating functions without their final terms?

I know that if we have two generating functions like so: $A(z) = \sum_{n=0}^\infty a_nz^n$ and $B(z) = \sum_{n=0}^\infty b_nz^n$ Then we can write $A(z)B(z) = \sum_{n=0}^\infty(a_0b_n + a_1b_{n-1} ...
0
votes
2answers
45 views

Please help solve this equation [on hold]

$$3.8r - 0.057r^2 + 0.00038r^3 + 0.00000095r^4 = 95$$ Please help.
2
votes
2answers
29 views

Laurent series of $e^{e^{\frac{1}{z}}}$ around $z=0$

Actually I need only the $res(f;0)$ where $f = e^{e^{\frac{1}{z}}}$ I thought of finding the Laurent series of $e^{e^{\frac{1}{z}}}$ around $z=0$ Any other Ideas if you have ?
0
votes
0answers
8 views

Suspension flow and topological equivalence(s)

Let $M$ be a compact smooth manifold, $\tau:M\to\mathbb{R}_{\geq 0}$. Let $f:M\to M$ is a surjective piecewise-smooth map. There is a standard construction of suspension allowing to extend $f$ to a ...
-2
votes
1answer
36 views

Characteristic of a field [on hold]

If $K$ is an infinite field, then $Char K = 0$ but the reverse is not sure. Examples of $Char K = 0$ but not infinite field $K$?
1
vote
1answer
16 views

Composition of equivalence relations

As part of a HW assignment in the course elementary set theory, I was given the following question: Let $A$ be a group and let $T$ and $S$ be two equivalence relations on $A$. Prove: $S\circ ...
-1
votes
4answers
21 views

Why is $\lim_{x\to e^+} (\ln x)^{1/(x-e)} =e^{1/e}$

$$\lim_{x\to e^+} (\ln x)^{1/(x-e)} =e^{1/e}$$ I started by taking ln on both side, which brings the power down, by I tried using L'Hopital, but it doesn't seem to work.
0
votes
0answers
14 views

Find the minimizer of the functional

Find the minimizer of the functional $ l= \int u(t) $ with $u(1)=u(1)=0 $ subject to $g=\int $$\sqrt{1+u'(t)} dt $ I want to solve it using E-L equation first $l^*=l- \lambda g$ then i used e-l ...
0
votes
1answer
21 views

relative homotopy groups

I study relative homotopy groups and I have a question: Let $A\subseteq X$ (not necessarily CW complex) and $\pi_{n}(X,A)$. Is it always possible to find a pointed space Y for which ...
1
vote
0answers
26 views

PDE question: heat equation (third order??)

I am familiar with the usual heat equation, however, my lecturer gave me this problem and it does not look like anything I have ever seen (in my whole entire life and I am not just being dramatic). ...
0
votes
3answers
59 views

What is $n+1$ factorial or $(n+1)!$? [on hold]

I have to prove by induction but first I want to know what $(n+1)!$ is? I know that $n!=n \cdot (n-1) \cdot (n-2)...$
0
votes
0answers
11 views

Finding #Groupoid like subsets

Given $S=\{x \in \mathbb{R}: 1 \leq |x| \leq 100\}$, find all subsets $M$ of $S$ such that for all $x$, $y$ in $M$, their product $xy$ is also in $M$. My attempt: If any number with magnitude ...
8
votes
1answer
231 views

Bourbaki and set inclusion

Which notation ($\subset$ or $\subseteq$) was preferred by Bourbaki for set inclusion (not proper)? A side question: Was the notation for subset one of the many notations invented by Bourbaki?
0
votes
1answer
16 views

Poisson Probability (Shopkeeper Sales)

SOLUTIONS: (A) 0.1804 (B) 0.0166 (C) 0.3233 Mean = 2/7*5 (a) x = 3 (b) x > 5 I'm still unsure how to approach each question, because I still get the wrong answers.
1
vote
0answers
12 views

Find the equation of intersection of a torus and a circle on a plane without using iterative methods.

I have the equation of a circle on the plane (where $p_0$ is the centre, $\theta$ is the angle of the circle, and $w$ and $v$ are pair of orthogonal vectors from $p_0$ to the circle (having equal ...
0
votes
1answer
28 views

Bitwise ops - The relationship between $a$, $b$, $a \wedge b$, $a \vee b$ and $a \oplus b$

In computer programming, the term bitwise operation is used to denote the use of boolean operators (and $\wedge$, or $\vee$, exclusive or $\oplus$) on corresponding bits of two numbers. Bits, in this ...

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