1
vote
1answer
14 views

Determinant of a tuple of vectors: is this a thing? If so, where can I learn more?

Let $k \leq n$ denote a pair of fixed but arbitrary natural numbers. Definition 0. Write $\varphi$ for the unique $\mathbb{R}$-linear function $$\Lambda^k\mathbb{R}^n \rightarrow \mathbb{R}$$ such ...
3
votes
2answers
37 views

A non continuous linear map $A:\Bbb{R}[X]\rightarrow \Bbb{R}$ such that $A(P)=P(1).$

I have a linear map $A:\Bbb{R}[X]\mapsto \Bbb{R}$ such that $A(P)=P(1)$, for the $p-$norm : $\Vert P\Vert=\bigl(\sum_{i=1}^n\vert a_i\vert^p\bigr)^{1/p}$ where $p\in[1,+\infty].$ For the cas ...
1
vote
1answer
47 views

Exercise from Kaplansky - Commutative Rings (1.2.3)

Exercise 3 in section 1-2: Let $R$ be an integral domain, $P$ a finitely generated non-zero prime ideal in $R$, and $I$ an ideal in $R$ properly containing $P$. Let $x$ be an element in the ...
0
votes
1answer
32 views

What do people mean by “finding the end of pi”

So I have been wondering. I have heard many times statements like "if we find the end of pi then we might be in a virtual reality" or "new computer can calculate X digits of pi". My question is pi ...
0
votes
1answer
14 views

Suppose that $A \in M_{m\times n}$ & $B, C \in M_{n\times m}$ are matrices that satisfy $BA= I_n$ and $AC=I_m$. Prove that $B=C$.

Suppose that $A \in M_{m\times n}$ & $B, C \in M_{n\times m}$ are matrices that satisfy $BA= I_n$ and $AC=I_m$. Prove that $B=C$. In my mind, a good way to go about this proof is proving that ...
1
vote
2answers
28 views

Prove $\left(1+\frac{x}{n}\right)^n < e^x$, where $x$ is any positive real number and $n$ is any positive integer.

I am having trouble with my homework problem, it says: Suppose that $n$ is a positive integer and that $x > 0$. Show that $$\left(1+\frac{x}{n}\right)^n < e^x.$$ I have proved the base ...
0
votes
1answer
8 views

If $M/N$ and $N$ are noetherian $R$-modules then so is $M$

Let $M$ be an $R$-module. I want to show only using the definition of noetherian that if $N$ is a noetherian submodule of $M$ such that $M/N$ is noetherian, then $M$ is noetherian. I know that if $M_1 ...
0
votes
0answers
10 views

Representation theorem for modular lattices?

Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices? Specifically, ...
0
votes
1answer
8 views

Maximal Solution to Differential Equation

For the differential equation $$\dot x = x(1-x), x(0)= \frac 12$$ Decide if the solution exists for all $t \ge 0$ or only on a finite time interval $0 \le t \lt T$. By the theorem, for the maximal ...
0
votes
0answers
6 views

Boundary value ODE with unknown functions?

Problem Let $f(x)$ be given such that $F'(x) = f(x)$. Also let $a(x) > 0$ in the interval $[0, L]$. $$ \left\{ \begin{array}{c l} -(a(x)u'(x))' = f(x),\ x \in (0,L) \\ u(0) = ...
0
votes
0answers
4 views

Brownian motion bounded variation

I am not sure right now, but does the process $X_t=c^{W_t}$ has bounded variation or not? $W_t$ has no bounded variation, therefore $c^{W_t}$ also has no bounded variation for $c>0$ and $W_t$ ...
0
votes
0answers
5 views

Jacobian of a bijective mapping?

Consider the smooth injective linear map $f:X \to Y$. To clarify my question, consider the two scenarios: If the Jacobian was nonvanishing, does it immediately follow that the map is also ...
1
vote
1answer
16 views

Comparing matrix norm with the norm of the inverse matrix

I need help understanding and solving this problem. Prove or give a counterexample: If $A$ is a nonsingular matrix, then $\|A^{-1}\| = \|A\|^{-1}$ Is this just asking me to get the magnitude of ...
0
votes
4answers
54 views

Without computing, is the integral of $\int_0^1 t(t-1)(t-2)\,dt$ positive or negative?

I have to graph the function, but I don't think I'm doing it right. Here is a picture of it Sorry, this is my first time using this site and I don't know how to use MathJax yet.
0
votes
0answers
2 views

Removing variables from convex linear program

I am solving linear program (possibly non-convex). Then we know that dual is always convex. Then I noticed that depending on objective functional I can sometimes remove particular variables from this ...
0
votes
0answers
12 views

Geometrical meaning of Tensor

i really need a geometrical interpretation or analogy for tensors. Bear in mind that i am not a mathematician, so complicated explanations will not suffice. Thank you!
2
votes
4answers
62 views

the product of an odd perfect number and some even perfect number is perfect

If $a$ were an odd perfect number ,does there exist an even perfect number $b$ such that $ab$ is a perfect number?
1
vote
1answer
22 views

Prerequisites for Spivak's Calculus on Manifolds

Hello I am a sophomore and I would like to know if Michael's other book named 'Calculus' is enough preparation for the Manifolds, if not please do tell what else should I be reading? I am taking an ...
1
vote
0answers
4 views

Extension of co-coercivity in strongly convex functions

I am studying strongly convex functions and they mention if $f(x)$ is strongly convex with Lipschitz gradients $L$, which means $\parallel \nabla f(y) - \nabla f(x)\parallel \leq L\parallel x - y ...
0
votes
0answers
10 views

Proof of independence of $\bar{X}$ and $S^2$: Trouble understanding $n$-dimensional Jacobian Result

I'm trying to work through the proof given here that the sample mean and sample variance of a random sample $X_1, X_2, ..., X_n \sim N(\mu,\sigma^2)$ are independent. The part I can't seem to follow ...
0
votes
1answer
13 views

Proving function is infinitely differentiable

On some interval, there is a function $h(x)$ that is differentiable and continuous. Then, there is another function $c(x)$ that is smooth(infinitely differentiable). Given that $h'(x) = c(h(x))$, show ...
2
votes
2answers
140 views

Assignment - Euler characteristic constant under barycentric subdivision

I am working on an assignment and I am stuck, mostly I have no clue how to quite attack it. I don't want the answer or anything just advice on angels at which I can go about this. For an n-dimensional ...
0
votes
1answer
25 views

Prove that $||a|-|b||\leq |a-b|$ for all real numbers

Prove that $||a|-|b||\leq |a-b|$ for all real numbers I was thinking divide it into $a\geq b$ and $a<b$, but then I realized I need to include situations when they are greater than zero and less ...
5
votes
2answers
166 views

-1 as the only negative prime

I was recently thinking about prime numbers, and at the time I didn't know that they had to be greater than 1. This got me thinking about negative prime numbers though, and I soon realized that, for ...
7
votes
1answer
390 views

What is the period of this sequence?

Consider the recurrence relation: $$x_{i+1}=p-1-((p \cdot i-1) \mod{x_i})$$ If $p$ is prime and $x_0=1$, what is the least period of the resulting (eventually periodic) sequence? My guess is the ...
0
votes
2answers
18 views

Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}

Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}$. I know that vectors that are orthogonal will have a dot product of 0. So here's what I was thinking: ...
0
votes
0answers
10 views

Externally tangent circle coordinates

I am trying to program following case as show in figure below. I have two circles at (x1,y1) and (x2,y2). Having coordinates, we can mention that the two circles are making an angle \theta_2. I need ...
0
votes
1answer
10 views

Addition vs. Substitution method for Linear Systems of Equations with Parameters

I thought that solving a 2x2 linear system of equations using either the substitution method or the addition method (adding the two equations to eliminate a variable, then using the substitution ...
0
votes
0answers
9 views

Hyperplane of an mn-dimensional space [on hold]

Can someone explain to me why the hyperplane of a $mn$-dimensional space would have dimension $(m-1)n$?
3
votes
0answers
101 views

How do find the numerical average of $x^x$ from $(-4,-2)$?

I wanted to find the approximate average of all real points in $(x)^{x}$ from $[-4,-2]$. This means I am ignoring all complex points and need average to be a real number. To first solve this I found ...
1
vote
1answer
34 views

Interpretation of set operations notation

I've been given a task that reads: Prove that given formulae is correct with the use of set theory axioms: $(\forall a)(\exists b)(\forall c)((c \in b) \iff (\exists d \in a)(c \subset d))$ ...
1
vote
7answers
53 views

Proving $\frac{n}{n+1} < \frac{n+1}{n+2}$ by induction?

I have the inequality $\frac{n}{n+1} < \frac{n+1}{n+2}$ I'm not sure how to go about proving it. I've started by testing with n = 1, which results in $\frac{1}{2} < \frac{2}{3}$ which is true ...
1
vote
0answers
9 views

Given 5 points on a sphere, divide the surface into 5 congruent connected regions containing one point

There are $5$ points on the surface of a sphere. Is it always possible to divide the surface into $5$ connected congruent regions such that each region contains one of the $5$ points?
1
vote
0answers
13 views

Zeros of this function?

Let $$f(z)=\gamma + z^{\beta_2-\beta_1}$$ where $\gamma\in \mathbb{R}$, $\beta_1\in \mathbb{Z}$, $\beta_2 \in \mathbb{Z}$ and $\beta_2 > \beta_1$. The variable $z$ takes complex values. Is there a ...
0
votes
1answer
25 views

Convergence of a series depending on a parameter

I have the following series $$\sum_{n=2}^{\infty} \frac{n}{(n-1)^2+\alpha 2^n}$$ I have to find for which $\alpha$ this series converges. I tried the ratio test but I get $\lim_{n \to \infty} ...
14
votes
1answer
265 views

Mathematical meaning of certain integrals in physics

While studying on texts of physics I notice that differentiation under the integral sign is usually introduced without any comment on the conditions permitting to do so. In that case, I take care of ...
0
votes
0answers
9 views

What does it mean that “$f = g$ in $k(h(t))$?”

Let $k$ be a field and consider the rational function field $k(t)$. I was just reading that if $f(t),g(t),h(t)\in k(t)$ are such that $f(h(t)) = g(h(t))$, then "$f = g$ in $k(h(t))$." What does that ...
1
vote
3answers
27 views

Integrate the following equation. (exponential function)

Integrate $$\frac{e^x -2}{e^{x/2}}$$ This is my calculation: but it is wrong....
0
votes
1answer
16 views

Prove that if $\mu (A) = \nu(A)$ for all $A \in s$, then this also holds for all $A \in M(s)$

Let $s$ be a collection of subsets of $X$. Assume that $\mu$ and $\nu$ are two measures on $M(s)$. Prove that if $\mu(A) = \nu(A)$ for all $A \in s$, then this also holds for all $A \in M(s)$, i.e., ...
0
votes
0answers
8 views

Cramer-Blackwell estimator for uniform distribution.

I've got two estimators of parameter $\alpha$ in the distribution $X=X_1,...,X_n$, where $X_i$s are i.i.d. uniform random variables on the interval of $(0,\alpha)$. These two estimators are: ...
4
votes
4answers
46 views

limit of form “$∞ \cdot 0$”

I am trying to formally prove that limit of $2^n\sin(π/2^n)$ as $n$ approaches infinity is $π$. Generally I can tell limit of each term of product of $∞$ and $0$ respectively, but am little confused ...
1
vote
2answers
16 views

Finding a linear transformation with a given null space

The problem statement is, Find a linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satisfying $4x_1-3x_2+x_3=0$ is the (i) null space of $T$ (ii) range of $T$. ...
1
vote
0answers
25 views

Prove that multiplication is well defined

Let $M = \mathbb{N} \ \mathbb{x} \ \mathbb{N}$. We define the following relation on $M$. Let $(a,b)R(a',b')$ iff $a + b'=a'+b$ We define the set of intergers $\mathbb{Z}$, to be the set of ...
0
votes
1answer
18 views

Convergence of product of random variables with distribution $U(0, e)$

Let $X_1, X_2, \ldots$ be a sequence of independent random variables with uniform distribution on $[0, e]$. Let $R_n:=\prod_{k=1}^n X_k$. By Kolmogorov's zero–one law $(R_n)$ converges with ...
0
votes
0answers
9 views

Finding different surface areas along a frustum pyramid

New to the site so excuse any clumsy writing: I am attempting to write an equation that finds the surface area along a pyramid as you move up and down the height of it. I know the initial areas of ...
3
votes
4answers
61 views

How to find unknowns $w_1,w_2,w_3$ that satisfy $t=w_1f_1 + w_2f_2 + w_3f_3$?

For any $i \in \{1,2,3\}$, let: $w_i \in [0,1]$ is an unknown number such that $\sum_{i \in \{1,2,3\}} w_i = 1$. $t$ is a known number in $[0,1]$. Suppose that $t = 0.8$. $f_i$ is also a known ...
0
votes
2answers
23 views

Prove or Disprove Θ

I want to prove or disprove that $3n^3 +n^2\log(n) = Θ(n^3)$. I'm aware that I will need to either prove or disprove both big-o and big-Ω to prove or disprove Θ. I am simply struggling to do so. Help ...
2
votes
1answer
23 views

When is a manifold also a vector space?

My question arises from this definition: Poincare group is the group of Minkowski space-time isometries. Which means that it leaves the space-time intervals unchanged. Now here is my understanding: ...
0
votes
1answer
26 views

Find all the numbers $a$ such that the number $an(n+2)(n+4)$ is an integer for all $n \in \mathbb{N}$

Find all the numbers $a$ such that the number $an(n+2)(n+4)$ is an integer for all $n \in \mathbb{N}$ It's trivial to see that if $a$ is irrational, we get no solution. Thus $a \in \mathbb{Q} ...
-1
votes
2answers
46 views

Evaluating $\lim_{x\to 0}{\frac{\sin^2x}{2x^2}}$ without L'Hospital

I have been trying to evaluate $$\lim_{x\to 0}{\frac{\sin^2x}{2x^2}}$$ Finally, I used the L'Hospital's Theorem and I got the answer $1/2$, but I wonder if there is a way to solve this without this. ...

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