0
votes
0answers
5 views

Is Spivak wrong here, or am I just missing something?

Chapter 1 Problem 18 has the reader doing various proofs with second-degree polynomial functions of the form $x^2 + bx + c$. My issue lies with problem 18d, but it uses knowledge from 18b and 18c, so ...
0
votes
0answers
2 views

Proving that a polilinear operator is differentiable

A Polilinear map operator is $P:X^1 \times ... \times X^n \to Y$ such that the foolowing applies: $\lambda, \mu \in R$ $$ P( \lambda x_1^1 + \mu x_2 ^1, x^2,...,x^n)= \lambda P(x_1^1,x^2,...x^n)+ \mu ...
0
votes
0answers
7 views

Trying to make sense of this proof in Hatcher

So I'm trying to understand this proof in Hatcher's Algebraic Topology. Lemma: The composition $\Delta_n(X)\xrightarrow{\partial_n} \Delta_{n-1}(X)\xrightarrow{\partial_{n-1}}\Delta_{n-2}(X)$ is zero ...
0
votes
1answer
18 views

Meaning of such that.

The use of this term confuses me, I've seen this like $A=\{(x,y) :x,y\in\Bbb R\ \text{and } P(x,y) \}$ and $B=\{(x,y)\in \Bbb R^2:P(x,y)\}$ for some predicate $P$. Is there any difference between ...
0
votes
0answers
10 views

Proper name and access of the constituents of an equation.

In any math problems for example: ... B = (1 + B) A = (A + B) ... How can I definitely define a variable that points to the ...
0
votes
1answer
28 views

algebra question..

If $f : \mathbb{R}\rightarrow \mathbb{R}$, and $f(x)=\frac{2}{4^{x}+2}$ Find the value of $$f\left [ \frac{1}{11} \right ]+f\left [ \frac{2}{11} \right ]+ \cdots +f\left [ \frac{10}{11} \right ]$$ I ...
0
votes
0answers
8 views

Does the concept of Lie derivative by bivector fields exist?

A cursory glance at the internet shows that perhaps the closest (if not exactly) to what I'm seeking is Albert Nijenhuis' generalization of the ordinary Lie derivative. He constructed a way to take ...
0
votes
0answers
24 views

Possible values of a $f(x)=(ab-b^2-2)x+\int_{0}^{x} x^2(cos^{4}t+sin^{4}t)dt$

If $f(x)=(ab-b^2-2)x+\int_{0}^{x} x^2(\cos^{4}t+\sin^{4}t)\, dt$ is a decreasing function of $x$, $x$ is a real number. What are the possible values of $a$? $b$ is independent of $x$. I tried Newton ...
-1
votes
2answers
12 views

continuously differentiable multivariable functions

What does it really mean to say a function $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is continuously differentiable? A function $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuously differentiable if ...
2
votes
1answer
21 views

Permutation count of AABBC

Given a string: $AABBC=A^2B^2C^1$ I am trying to find the Total Permutations (this may be incorrect): $\dfrac{5!}{2!\cdot2!}=30$ My question is how would I find the partial sums (perhaps the ...
0
votes
2answers
33 views

Does anyone know when I would use this symbol ($\supseteqq$) and meaning?

Does anyone know what this symbol means? Where would one use it? Someone recently asked me but I do not know what it means. I have seen it with just one line underneath to denote subset. With an ...
3
votes
2answers
11 views

Does the radius of the quadrant pass from the center of the inscribed circle?

In the following picture: The smaller circle is inscribed inside the quadrant, whose radius (OB) is 8. The original question (but not the question of this post) is that "find the radius of the ...
1
vote
1answer
15 views

Computing eigenvalue of the adjacency matrix of a path

Let $A\in \{0,1\}^{n \times n}$ be the adjacency matrix of a path of length $n$, i.e. having ones on the two off-diagonals, and zeros elsewhere. How does one compute the eigenvalues of this? I know ...
1
vote
0answers
20 views

Number of different keyboard layouts?

Pretty really stupid question probably, but if I would have a keyboard with the 30 main keys (A-Z,. shift) in how many different keyboard arrangements could I put them provided their possible ...
-2
votes
0answers
33 views

Why $ (\cos(\theta) \frac{\partial}{\partial x} + \sin(\theta) \frac{\partial}{\partial y} ) \frac{\partial}{\partial \theta} =0$?

Why $ (\cos(\theta) \frac{\partial}{\partial x} + \sin(\theta) \frac{\partial}{\partial y} ) \frac{\partial}{\partial \theta} =0$ ?
0
votes
1answer
22 views

Finding the series for $(\ln(1+z))^2$

So, I'm supposed to use product of infinite series methods to find the series for $(\ln(1+z))^2$. I'm given that the answer has the form $$z^2 \sum_{l=0}^{\infty} c_l z^l$$ and I'm given that the ...
0
votes
1answer
23 views

Evaluations of a Definite Integral with cosine function

How do you evaluate this integral? Does it involve an elliptical integral? What technique do I use to evaluate this integral? $$\int _{ 0 }^{ 2\pi }{ \sqrt { 5-4\cos { \theta } } d\theta } $$
0
votes
0answers
14 views

Create solid torus with geometric algebra

I have created an algorithm for tracking a vessel's centerline in 3 dimensions, and traveling through that centerline. My supervisor asked me whether I could add a small theoretical section on ...
1
vote
0answers
13 views

How Kriging, Bochner theorem and Positive definite (PD) function are related?

This question referes to the link: https://en.wikipedia.org/wiki/Kriging I can understand the relation between Bochner's theorem and PD function. But could not properly understand and connect all ...
1
vote
1answer
39 views

Computing $\operatorname{Tr} \bigl( \bigl( (A+I )^{-1} \bigr)^2\bigr)$

Suppose that $A \in \mathbb{R}^{n \times n}$ is a symmetric positive semi-definite matrix such that $\operatorname{Tr}(A)\le n$. I want a lower bound on the following quantity $$\operatorname{Tr} ...
0
votes
1answer
6 views

Optimizing space for many shapes within an irregular shape

So let's say in a state, there are 50 schools dispersed throughout, given by Latitude Longitude points. How would we create distinct zones that optimize the space around each school? The goal is to ...
4
votes
0answers
30 views

Basic question about $\sigma$-fields

Billingsley's text "Probability and Measure" has the following exercise problem: Problem 2.5(b): For a collection of sets $\mathcal{A},$ let $\mathcal{F}(\mathcal{A})$ be the intersection of all ...
1
vote
0answers
11 views

Representation of conjugate directions

Is there a way to represent conjugate directions on a Mohr circle of curvature? ( Surface Theory, Second fundamental form, M = 0 ) Directions given by double angles AOB, AOC. Is this attempt ...
3
votes
1answer
25 views

Convergence of Integrals of Exponential Functions

Let $f$ be a non-negative real valued function on $[a,b]$, and let $p:[a,b]\to(1,\infty)$ such that $f^p\in L^1([a,b])$. Let $p_n:[a,b]\to(1,\infty)$ be a (uniformly bounded) sequence of ...
0
votes
1answer
33 views

If $\displaystyle\lim_{n\to\infty}|x_n|^{1/n} = L < 1$ then $\displaystyle\lim_{n\to\infty}|x_n|=0$.

Let $(x_n)$ be a sequence of real numbers such that $\displaystyle\lim_{n\to\infty}|x_n|^{1/n} = L < 1$. Show that $\displaystyle\lim_{n\to\infty}|x_n|=0$. Remark: Not use that exist $0<r<1$ ...
3
votes
1answer
20 views

Convergence of fixed-point iteration for $p$ times continuously differentiable function

I am stuck at this problem: Let $\alpha\in\Bbb{R}$ be some number that satisfies $g(\alpha)=\alpha$ for some function $g$ that is $p$ times continuously differentiable on some neighborhood of ...
2
votes
0answers
12 views

Inter-event time between 1st and 2nd events for Poisson Process

Say we have two independent Poisson process $N_1$ and $N_2$ with parameter $\tau_1$ and $\tau_2$. Now I want to determine the inter-arrival time between the 1st and 2nd events. Given that processes ...
1
vote
0answers
4 views

2HC - Edge Disjoint Hamilton Paths

G is a directed graph and s and t are 2 vertices in G. $2HC = \{(G, s, t): \; G $ has at least 2 edge-disjoint Hamilton paths $\}$ Prove that $2HC$ is NP-Complete. I'm trying to reduce UHAMPATH to ...
-3
votes
5answers
60 views

Evaluate $ \lim_{ x \to 0}{ \frac{e^{e/x}-e^{-e/x}}{e^{1/x}+e^{-1/x}} }$ [on hold]

How to evaluate limit of the following function at x=0 ? $$ \lim_{ x \to 0}{ \frac{e^{e/x}-e^{-e/x}}{e^{1/x}+e^{-1/x}} } $$
6
votes
1answer
91 views

Have I just discovered an easy way to square numbers?

Choose any number, $x$: say, $x = 876$ (you can pick any $n$ digit number) Now, square the number -> $876 * 876 = 767376$ But now, If I ask you the square of $ x + 1$ --> $876 + 1 = 877$. You can't ...
1
vote
0answers
22 views

Does Bezout's Identity hold for Zero cases?

In some places I see Bezout's Identity stated for any two non-zero numbers $a$ and $b$. In other places it is stated that $a$ and $b$ are not both zero (so one of them can be). But doesn't Bezout's ...
1
vote
6answers
57 views

Does the function $|x^2-4|/x$ have critical points?

Does the function $|x^2-4|/x$ have critical points? I tried differentiating and putting the derivative equal to 0.But I'm still a bit confused (as I got no solution).
1
vote
0answers
8 views

Representing numbers by quasilexicographic ordered strings, formula for size of conversion between different alphabets

Let $X_r = \{ 0, 1, \ldots, r-1 \}$ and $X_b = \{ 0, 1, \ldots, b-1 \}$ be two finite alphabets with order's given by their numerical value. Consider the quasilexicographic (or shortlex) order on ...
1
vote
1answer
20 views

Terms of a certain recurrence

Let $a_1, a_2\dots $ be a sequence of reals such that $a_1 = a_2 = 1$, and $$a_{n + 2} = \frac{a_{n + 1}^3 + k}{a_n}$$ for $n \ge 1$, for some fixed $k$. It appears to be the case that all of these ...
5
votes
2answers
45 views

Proving that $(A\setminus C)\cap(B\setminus C)\cap(A\setminus B)=\emptyset$

For each $A,B,C$ how would I prove that $(A\setminus C)\cap(B\setminus C)\cap(A\setminus B)=\emptyset$ ? My thoughts are if $x\in (A\setminus C)\cap(B\setminus C)\cap(A\setminus B)$, then $x\in ...
1
vote
0answers
19 views

What is the expectation of minimum of two correlated gaussian random variables?

I am doing research on supply chain management and got stuck on the following problem. Let, $D_{1}$ and $D_{2}$ are gaussian random variables. So, what is the expected value of the following ...
1
vote
1answer
19 views

Sorting triangles by hypotenuse length

I have some points in $xy$ space and I need to sort distances between these points. If I calculate real distance, then I need to perform $\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$ and this is very time ...
2
votes
0answers
30 views

Bound on the difference of two determinants

Let $A$ and $B$ be two real, $n\times n$ matrices. Using Hadamard's inequality, it is not hard to show that $$ \left|\det A - \det B \right| \leq \|A-B\|_{2} \frac{\|A\|_{2}^n -\|B\|_{2}^n}{\|A\|_2 ...
2
votes
1answer
17 views

Using addition and subtraction in algebraic proving in set theory

I am trying to prove (using algebraic way) the following statement: A∆B=A iff B=∅ So it goes like this in one direction: A∆B=A A∆B∆A=A∆A (I added ∆A to both sides) B∆A∆A=A∆A (commutativity) B∆∅=∅ ...
1
vote
0answers
18 views

Literature concearning characteristic classes?

It sounds the literature about characteristic classes is not very abundant (am I wrong?). Whenever I look for books dealing with this matter I'm always lead to the same material like the classical ...
3
votes
3answers
83 views

How can I intuitively interpret this vector operation?

In reading through some very old source code that I inherited and came across a three-dimensional Euclidean vector operation that I can't seem to gain an intuition for. Transcribing the program code ...
1
vote
2answers
20 views

Max Mod Principle

I'm stuck with the following exercise: Let $f$ be holomorphic on an open set containing $\bar{D}$, the closed unit disk. Prove that there exists a $z_0 \in \partial D$ such that ...
3
votes
0answers
32 views

Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
3
votes
2answers
53 views

Help in finding the integral function.

Can somebody provide a hint in finding the following integral? $$\displaystyle \int \dfrac{1}{(x^3+1)^3} \text{ d}x$$ I thought of using partial fractions but that isn't making any sense.
2
votes
1answer
13 views

Determinant of a Certain 3 by 3 Block Matrix with Scaled Identity Blocks

What is the determinant or/and eigenvalues of the following 3 by 3 block matrix: $$\left[\begin{array}{ccc} \frac{3}{4}I & \frac{1}{4}I & \frac{1}{4}I \\ \frac{1}{4}I & \frac{3}{4}I & ...
1
vote
0answers
19 views

Eigenvalue of Block matrix: Adjacency of complete bipartite Graph

Let $A\in \{0,1\}^{mn \times mn}$ be the adjacency matrix of a complete bipartite graph with $m$ and $n$ vertices each, i.e. let $A$ be the matrix consisting of two blocks $A_1\in \{0,1\}^{m \times ...
3
votes
1answer
34 views

If ${a_i} \to 0$ and $\{ {X_i}\} _{i = 1}^\infty $ is a sequence of iid random variables with zero mean, does ${a_i}{X_i} \to 0$ almost surely?

My problem is slightly more specific than the title of this question: Let $0 < \beta < 1$ and let $\{ {X_i}\} _{i = 1}^\infty $ be a sequence of i.i.d. random variables with $E({X_i}) = 0$. In ...
1
vote
0answers
11 views

Best hinged solids to enclose unit volume, if hinges are expensive

In three-dimnesional space, you are asked to construct a polytope (a geometric object with flat sides) enclosing a volume of $1$. The cost of the model is the sum of the areas of the faces, plus some ...
1
vote
0answers
28 views

Logarithm Propr

I'm having a bit of trouble proving the following property: Theorem If $Re(z)>0 $ and $ Re(w)≥0$, then $\log(zw)=\log(z)+\log(w)$, where log is the principal branch. I know that $\log (zw) = ...
4
votes
2answers
74 views

Why is it that the product of first N prime numbers + 1 another prime? [duplicate]

Recently I came across this proof for fact that primes are infinite. It's a proof by contradiction. The proof assumes that primes are finite and there is a prime M which is larger than any prime out ...

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