0
votes
0answers
7 views

Are multi-valued functions a rigorous concept or simply a conversational shorthand?

In Brown and Churchill's book, the concept of multivalued functions is not discussed in a very rigorous way (if at all). But I can see that branch cuts have importance in complex analysis, so I want ...
0
votes
0answers
12 views

Function whose power series coefficients contain logarithms

Is there a function that can be expressed as a power series $$f(x)=\sum_{n=0}^\infty a_n x^n$$ whose coefficients $a_n$ are expressions containing $\log n$ or something similar?
0
votes
0answers
20 views

Why can we make this integral change of limits? Is it obvious?

When deriving the equation for the impulse-momentum theorem, the following occurs: $...=\int\limits_{t_1}^{t_2}\frac{d\vec p}{dt}dt = \int\limits_{\vec p_1}^{\vec p_2}d\vec p=...$ I know the $dt$s ...
0
votes
0answers
20 views

Represent a prime number $p$ congruent to $1$ $\pmod{3}$ by a sum of a square and $3$ times a square

I want to have a proof of the fact that each prime number is the sum of a square and three times a square (Euler).
1
vote
1answer
21 views

Finding the perimeter of the room

If the length and breadth of a room are increased by $1$ $m$, the area is increased by $21$ $m^2$. If the length is increased by $1$ $m$ and breadth is decreased by $1$ $m$ the area is decreased by ...
2
votes
1answer
15 views

Proving independence of random variables

If $X$ and $Y$ are independent exponential random variables with parameter $\lambda$ and $\mu$. Let $Z=\min(X,Y)$, prove that $Z$ and $\mathbf 1_{\{X<Y\}}$ are independent. I don't know, how ...
3
votes
1answer
28 views

An algebraic topology proof of a result from analysis

A colleague of mine recently brought up the following result from real analysis: Theorem: If $f:\mathbb{R}^2\to\mathbb{R}^2$ is continuous and $|f(x)-f(y)|\geq |x-y|$ for all $x,y$, then $f$ is onto. ...
0
votes
0answers
5 views

Solutions to functional equation $f(a_1t+x_1,a_2t+x_2,x_1,x_2)=g(b_1t+x_1,b_2t+x_2,x_1,x_2)+h(t)$.

Find all functions $f$, $g$, and $h$ such that $$ f(a_1t+x_1,a_2t+x_2,x_1,x_2)=g(b_1t+x_1,b_2t+x_2,x_1,x_2)+h(t) $$ Is it possible (and what I was hoping for) to express the solution as ...
0
votes
1answer
12 views

Vectors with given angle and magnitude

Give an example of vectors $\mathbf{v}$ and $\mathbf{w}$ such that the angle between $\mathbf{v}$ and $\mathbf{w}$ is $\frac{2\pi}{3}$ and $||\mathbf{v} \text{ x } \mathbf{w}||=\sqrt{3}$. Should I ...
0
votes
0answers
18 views

Weak convergence plus strong convergence

Let $H$ be a Hilbert space and let $(x_n), (y_n)$ be sequences in $H$ such that $(x_n)$ converges strongly to $x$ and $(y_n-x_n)$ converges weakly to 0. I can show that $(y_n)$ converges weakly to ...
0
votes
0answers
12 views

Paraboloid Curvature calculation methods

If we have a paraboloid generated as a surface of revolution of the 2d function $f(x)=ax^2+b$, the equation of the 3d graph is $f(x,y)=ax^2 + ay^2+b$. The gaussian curvature of a 3d graph $f(x,y)$ is ...
1
vote
0answers
14 views

A certain two-step subgroup of a nilpotent group

Let $\Gamma$ be a finitely-generated, torsion-free, nilpotent group, of nilpotency class $n\ge 2$. Is there an $N \lhd \Gamma$, such that (i) $N$ is two-step nilpotent, (ii) $\Gamma / N$ is torsion ...
9
votes
0answers
37 views

How to Self-Study Higher Math Without Solutions

I've been lurking on this site for several months, and as someone studying higher mathematics independently (i.e., outside of a college/institutional setting), this forum has probably been the best ...
0
votes
0answers
25 views

Exponential Growth Differential Equation

A population of buffalo grows exponentially (the rate of growth is determined by the population itself) but has a carrying capacity. Its population (in tens of thousands) at a time t ( in years ) is ...
2
votes
1answer
37 views

Improper integral of a cosine

I'm trying to follow some equations in an electrical engineering paper that I'm reading. I'll spare you the details, but at one point I come across: $$\lim_{ T \rightarrow \infty }\int_{-T/2}^{T/2} ...
0
votes
0answers
18 views

Are there axiomatizations of first order logic or set theory defined in first order logic or set theory?

There are several axiomatizations for number theory, group theory, and other theories represented in first order logic. Further, these theories are also representable in set theory such as $\sf ZFC$ ...
0
votes
3answers
17 views

Finding the angles of a parallelogram.

In a parallelogram, one angle is $2/5th$ of the adjacent angles. Determine the angles of the parallelogram. I tried the following, Let the adjacent angles be $2x$ Let the other angle be $y$ ...
0
votes
0answers
13 views

Why abstract index notation should not be confused with the Ricci calculus?

Considering this answer, it is mentioned that the range of indices $a, b, c,\dots$ are seen as abstract and coordinate-free and linear operations can be represented with them; and the range of indices ...
0
votes
3answers
24 views

Converting exponents to scientific notation

I have to solve or estimate the answer to an equation that is as follows: $$P_\text{blocks} = \frac{398 \cdot 19^{65}}{\prod^{66}_{i=0} 78804 - i}$$ It doesn't take long to realize that this is an ...
1
vote
1answer
29 views

Existence of solution of ordinary differential equation

I am reading a proof of the existence of solutions for ordinary differential equations and I have some basic doubt. I'll copy the statement, the part of the proof I don't understand and my question: ...
4
votes
2answers
28 views

Relation between continuous maps and convergence of sequences

I am studying metric spaces and I know that in a normed space $E$ a map $T:E \to E$ is contínuous if and only if $T(x_n) \to T(x)$ for every convergent sequence $x_n \to x$ in $E$. In my notes there ...
0
votes
0answers
4 views

Computer software for solving mixed strategy Nash equilibrium

Is there any computer software available for solving for mixed strategy Nash equilibria for two players given each player's payoff matrix?
0
votes
0answers
29 views

Is there a discrete initial topology on the set of real numbers?

Consider the real numbers R first as just a set with no structure. Then consider it as a topological space R* with the usual topology. The question is: is there a function f from R to R* whose ...
0
votes
1answer
14 views

Find the smallest positive number $p$ for which the equation $\cos(p\sin x)=\sin(p \cos x)$ has a solution $x\in[0,2\pi].$

Find the smallest positive number $p$ for which the equation $\cos(p\sin{x})=\sin(p\cos{x})$ has a solution $x$ belonging $[0,2\pi]$. I am not able to solve this problem. Please help me.
0
votes
0answers
14 views

$X$ ~ $uniform(0,1)$, $f_Y (y | X=x) = I( x<y<x+1 )$ ( for $0<x<1$ )

$X$ ~ $uniform(0,1)$, $f_Y (y | X=x) = I( x<y<x+1 )$ ( for $0<x<1$ ). Find.... a) What is the distribution of $Y$, given $X = x$? b) What is $f(x,y)$? Distribution of $(X,Y)$? c) $f_Y ...
-2
votes
2answers
20 views

Grade 12 Calculus Lines Intersections Question [on hold]

Give the equations of two lines that meet at the point (-1, 5, 2) and which meet at right angles, but do not use that point in either of the equations. Explain your reasoning
0
votes
2answers
19 views

The perimeter of a rectangle is 48 meters and its area is 135 m^2. Determine the sides of the rectangle.

The perimeter of a rectangle is 48 $m$ and its area is $135$ $m^2$. Determine the sides of the rectangle. I tried the following, Perimeter$=$$48$ $m$ Let the length be $x$ m and the breadth be $y$ m ...
0
votes
0answers
35 views

Calculus book by Abel prize winner?

There's a new calculus book titled calculus with applications by Peter Lax.I really liked his linear algebra and functional analysis book,and I was curious if this book was as good as them .Is the ...
0
votes
1answer
29 views

How to prove that : $ \mathrm{Hom} ( A(G), H) \simeq \mathrm{Hom} (G , I(H)) $?

How do we show that the functor $ A : \mathrm {Gr} \to \mathrm {Gr} $ defined by $ A (G) = G / [G, G] $ is a left adjoint functor of the inclusion functor : $ I : \mathrm {Ab} \to \mathrm {Gr} $ ?. ...
0
votes
1answer
9 views

Hamiltonian cycle adjacency sum Proof

Let $C$ be a Hamiltonian cycle on a graph with vertices labeled {$1,...,9$}. Prove that there are $3$ vertices adjacent in $C$ whose labels sum to at least $12$. I understand why this fact is true by ...
0
votes
1answer
30 views

L'Hopital's Rule with $\lim \limits_{x \to \infty}\frac{2^x}{e^\left(x^2\right)}$

(a) Show that $$\lim \limits_{x \to \infty}\frac{2^x}{e^\left(x^2\right)}$$ is a standard indeterminate form, but that L'Hopital's Rule does not give you any information about the limit. (b) Show ...
-3
votes
2answers
21 views

I need a formula to compute the 3rd ratio, given 2 other ratios (non-linear or non-proportional) [on hold]

Basically I need a formula to compute, for example... $\$50,000 = 1\%$ $\$3,000 = 5\%$ What does $\$20,000 = (\text{ ? }\%)$ I don't specifically need the answer to $\$20,000$, but a formula to ...
1
vote
1answer
25 views

Solving a particular system of differential equations

The problem I'm trying to solve is this: $X'(t) \in \mathbb{R}^3 \,, \, \omega = (\omega_1,\omega_2,\omega_3) $ Find the general solution for $$X'(t) = \omega \times X(t)$$ After doing the cross ...
0
votes
0answers
10 views

Monotonic perimeters?

Let C and D be two compact convex sets in the plane with respective perimeters Per(C) and Per(D) . If C is properly contained in D does it follow that ...
0
votes
0answers
16 views

Mysterious membership-relation question in Enderton's `Elements of Set-Theory'

The question I am currently working on states (mysteriously): ``Simplify $\in^{-1}_{\omega} [\{7,8\}]$". For those who have the text, this is exercise #18 in Enderton's `Elements of Set Theory' on ...
0
votes
2answers
20 views

Uniform convergence on compact sets allows switching the limit and the integral.

Why does uniform convergence on compact sets allows switching the limit and the integral?
1
vote
1answer
13 views

Cobb Douglas Difficulty

Show that the Cobb-Douglas production function, for Labour costs L and Capital costs K, $P(L, K) = AL^{\alpha}K^{1-\alpha}$ satisfies the equation: $$L\frac{\partial P}{\partial L} + ...
1
vote
0answers
9 views

Given a process what is the stochastic differential equation it fulfils?

Given the process $X_t = (2+t+\exp(W_t))_t$ where $W_t$ is Brownian motion. How can I find the SDE that it fulfils. I am actually looking for two functions $\sigma, \tau$ such that $X_t = X_0 + ...
1
vote
3answers
64 views

Prove that the limit of $2^{\frac{-1}{\sqrt{n}}}=1$

Prove that the limit of $2^{\frac{-1}{\sqrt{n}}}=1$. I need to show that for each $\epsilon$ there exists an $n_0 \in \mathbb{N}$ such that $ \forall n \geq n_0: |2^{\frac{-1}{\sqrt{n}}}-1|\lt ...
0
votes
1answer
13 views

A field being a sigma field if and only if it's a monotone class

The exercise is as follows: "The limit of an increasing (or decreasing) sequence An of sets is defined as its union ∪nAn (or the intersection ∩nAn). A monotone class is defined as a class ...
1
vote
0answers
12 views

Brianchon's theorem proof

Can I prove Brianchon's theorem using Ceva's? I am also wondering if parabola and hyperbola can be inscribed in a hexagon?
0
votes
0answers
14 views

Proof for Homologous cycles

Prove that two cycles that surround the same holes differ by a boundary i.e. the relation for calling two cycles homologous as mentioned here. ...
0
votes
1answer
29 views

Symbolic Integration involving hypergeometric functions

What's the best way to symbolically evaluate this integral? $$\frac{1}{\hbar}\int_{-\infty}^\infty e^{iux/\hbar}\Psi^{*}_n(p-u/2)\Psi_n(p+u/2)\,du$$ where: $$\Psi_n(p)=\frac{1}{(1+\alpha ...
3
votes
0answers
33 views

How to show this expression is always a perfect square?

The number of tilings of an $m \times n$ board with $2 \times 1$ dominoes (each placed either horizontally or vertically on two squares of the board) has been shown to be $$\sqrt{\prod_{j=1}^m ...
2
votes
2answers
32 views

Mr. and Mrs. Ahuja weigh x and y kg. Find their present weights.

Mr. and Mrs. Ahuja weigh $x$ kg and $y$ kg respectively. They both take a dieting course at the end of which Mr. Ahuja loses $5$ kg and weighs as much as the wife weighed before the course. Mrs. Ahuja ...
0
votes
1answer
16 views

Calculating the number of isomorphic classes of complete bipartite graph

How many isomorphism classes of complete bipartite graphs have exactly 10 vertices? I don't understand what the question is asking or how to go about solving it.
6
votes
5answers
163 views

What does conditional probability $P(A|B)$ mean when $P(B)=0$?

Does anyone ever ascribe a value to $P(A|B)$ when $P(B)=0$? I realize that it being undefined makes sense, but I also feel like there should be a sensible definition.
0
votes
0answers
27 views

range of one increasing computation function? [on hold]

We know that that the range of any recursive partial function is recursively enumerable. I see in page 154 in books "Problems in Set Theory, Mathematical Logic and the Theory of Algorithms" that the ...
3
votes
1answer
55 views

How would Johann Bernoulli have tutored Euler?

Early in Euler's life (when he was still a child/teenager), the Euler family friend Johann Bernoulli would tutor Euler in mathematics. Do we know how Johann Bernoulli would have tutored the young ...
0
votes
0answers
19 views

convergence interval of an infinite series without the general term

I am trying to find the convergence of an infinite series of which I do not have the nth term. Instead of applying the ratio test for the nth term, I divided the first two terms, then the next two and ...

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