1
vote
0answers
7 views

What is common notation for “disjoint union of copies of $\mathbb{R}$”?

I'm looking at a question out of Lee's Smooth Manifolds: Show that a disjoint union of uncountably many copies of $\Bbb{R}$ is locally Euclidian and Hausdorff but not second countable. My ...
0
votes
0answers
10 views

Watches on a table

From Peter Winkler's 'Mathematical puzzles', taken from an All USSR Mathematical Competition, 1976: 50 accurate watches lie on a table. Prove that there exists a moment in time when the sum of the ...
1
vote
0answers
13 views

Is this what universal covering spaces are used for?

From the perspective of real analysis, we have: $$\int_{-1}^{1}\frac{1}{1+x^2} = \mathrm{tan}^{-1}(1)-\mathrm{tan}^{-1}(-1) = 2\mathrm{tan}^{-1}(1) = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}$$ Something ...
0
votes
0answers
13 views

Volume of solid lies under $z=x^2+y^2$

Find the volume of solid lies under $z=x^2+y^2$ above $x$-$y$ plane and inside the cylinder $x^2+y^2=2x$. I know, for volume we have to us $V=\iiint { \mathrm dx\mathrm dy\mathrm dz}$ but i was not ...
1
vote
2answers
8 views

order of multiple quantifiers

Problem: For a, b, c, d restricted to the universe of positive integers, explain why ∀a ∃b ∀c ∃d a/b = c/d is true, but ∀a ∃d ∀c ∃b a/b = c/d is false. I understand that the order of ...
1
vote
0answers
14 views

How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}} $?

How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}} $ ? I tried substituting $x=1/t$ but that's making it more complicated.Any suggestions?
0
votes
3answers
29 views

Reduce Number of digit

I have 24 digit number like 281564785148270616103860.I want to reduce this number with any optimization technique to 12 digit number . So i can use this to decode with the same technique.Any answer ...
0
votes
1answer
8 views

Is there any approach to computer vision that doesn't make use of geometry?

I've long been interested in applying my background in functional analysis (especially wavelets) and other related areas to actually create something with "real world" value (not that I don't enjoy ...
1
vote
1answer
23 views

Where to learn the algebra behind the use of differential operators in calculus

Algebra with differential operators is often used as a shortcut in calculus problems. I have previously asked about manipulations such as these: $$\int x^5e^xdx =\frac1Dx^5e^x=e^x\frac{1}{1+D}x^5=e^x(...
-4
votes
2answers
32 views

How to solve $x^3+y^3=z^4$?

Any three positive integers that are greater than 2 satisfying $x^3+y^3=z^4$ . You can not have two or more of the same number.
0
votes
1answer
15 views

Give a geometric comparison of the solutions to

Give a geometric comparison of the solutions to $x_1 + 3x_2 - 5x_3 = 4$ $x_1 +4x_2 -8x_3 = 7$ $-3x_1 -7x_2 + 9x_3 = -6$ and $x_1 + 3x_2 -5x_3 = 0$ $x_1 +4x_2 -8x_3 = 0$...
1
vote
0answers
18 views

What is the difference between derevative w.r.t a vector and directional derivative?

Say we have a scalar-valued function $f: \mathbb R^3 \rightarrow \mathbb R$, such that: $$f(\mathbf x) = \mathbf x^T\mathbf a$$ $\mathbf x$ and $\mathbf a$ are two vectors. The derivative of $f$ ...
1
vote
0answers
22 views

Explanation about proof of euler's function

Explanation about reduced residue system theorem Theorem 3-8 $$\varphi(m)=m\prod_{p|m}(1-1/p)$$ Proof: By reduced residue system theorem 3-7, if $$m=\prod_{i=1}^{r}p_i^{a_i}$$ then $$\varphi(m)...
0
votes
1answer
14 views

Find a vector $w$ such that $Aw = v_1 + 3v_2$

if $A$ is a $3 \times 3$ matrix and $\vec{v_1},\vec{v_2},\vec{y_1},\vec{y_2}$ are vectors so that $A\vec{y_1} = \vec{v_1}$ and $A\vec{y_2} = \vec{v_2}$ find a vector $\vec{w}$ so that $A\vec{w} = \vec{...
0
votes
0answers
13 views

The ideal of the image of homogeneous polynomials

Let $k$ be an algebraically closed field, and $f_0,\dots,f_m \in k[x_0,\dots,x_n]$ be homogeneous polynomials of the same degree. Denote by $I\subset k[x_0,\dots,x_m]$ the kernel of the homomorphism ...
0
votes
0answers
18 views

Why can the equation Ax = b not be solved for every b

Let $A$ be a $3 \times 2$ matrix . Explain why the equation $A\vec{x} = \vec{b}$ cannot be solved for every $\vec{b}$ in $\mathbb{R}^3$. What about a $4 \times 3$ matrix? I'm not sure how to answer ...
0
votes
1answer
13 views

Using Taylor Polynomial to Show How An Expression Of Only Real Numbers Can Be Approximated

I am studying for my graduate level GQE and looking at problems from old exams. The following question (from an unknown original source) reads: Suppose a,b,c and d are positive real numbers with a $&...
-3
votes
0answers
23 views

trigonometry to calculate height

Can not solve this problem I can help Edit : From the top of a hill a person finds losangulos Depression three consecutive stones and indicator of kilometers of a straight road level are α, β and γ....
0
votes
0answers
17 views

What vectors can be generated by permuting and halving?

$x$ is a vector in the unit simplex in $\mathbb{R}^n$, i.e: $$x = (x_1,\dots,x_n)\,\,\,\,\,\,\,\,;\,\,\,\,\forall i: x_i\geq 0\,\,\,\,;\,\,\,\,\,\,\,\,\sum_{i=1}^n x_i = 1$$ Initially, $x=(0,0,\dots,0,...
1
vote
3answers
42 views

Is it correct to interpret the “dx” in the standard notation for integrals as the Lebesgue measure?

Ok so I am in my Calc I class for the summer and we are just beginning to talk about integrals. I know a little bit about measure theory and the Lebesgue integral and why is it more general than the ...
1
vote
1answer
10 views

Find union and intersection of family or index

For each $n∈ℕ$, let $βn = \{\ldots, -3n, -2n, -n, 0, n, 2n, 3n,\ldots\}$, and let $β=\{βn:n∈ℕ\}$. My attempt: For union, it would be all integers. As for intersection, $βn1=\{\ldots, -3, -2, -1, ...
0
votes
1answer
9 views

Conditions for Non-negativity

Let's consider $A$ to be a square symmetric matrix whose entries are non-negative real numbers that sums to one. Even more, we shall consider its diagonal elements to be equal to zero. The question is:...
1
vote
1answer
11 views

Show that Borsuk lemma need not hold if $f$ is not injective

The following lemma is called Borsuk lemma which can be found in Munkres' topology (Lemma 62.2). (Borsuk lemma) Let $a$ and $b$ be points of $S^2$. Let $A$ be a compact space, and let $f:A\to S^2\...
0
votes
1answer
11 views

How to quickly find the basis of Null space in GF2

I have a matrix $A$ contains only 1 and 0. The operations is defined as in GF2, e.g., 1+0=1, 1+1=0, 0+0=0. I know how to make $A$ into row echelon form. For example, my $A$ now becomes ...
1
vote
1answer
27 views

What does neighborhood/ball/closure mean in a non-metric and/or finite space?

I'm trying to understand these fundamental concepts of topology. I understand what open closed and boundary mean in a continuous space with a metric, but I'm having issues understanding the meanings ...
2
votes
0answers
28 views

Elementary proof for non-existence of a pointwise convergent subsequence of $\{\sin (nx)\}$

My teacher showed this proof using the dominated convergence theorem or Fourier analysis, but I wonder if there is an elementary proof of this problem. My teacher said it is difficult to solve this in ...
2
votes
2answers
22 views

Differential Equation Initial Value Problem

Here is a pretty standard initial value problem that I'm having a little trouble with. $$(\ln(y))^2\frac{\mathrm{d}y}{\mathrm{d}x}=x^2y$$ Given $y(1)=e^2$, find the constant $C$. So I separated and ...
0
votes
4answers
36 views

Values of x that satisfy this inequality

$||x-2|-3|>1$. I have made some cases but still the complete values don't come,plus I don't have any idea of how to sketch the graph for the lefy hand side of the inequality.
1
vote
3answers
39 views

Sides of the triangles are in G.P.

Question:- The sides of a triangle are in G.P. and it's largest angle is twice the smallest one. Prove that the common ratio of the G.P. lies in the interval $(1,\sqrt{2})$ Attempt at a solution:- ...
0
votes
1answer
12 views

The validity of normalization in homogeneous inequalities?

I'm going through a book on inequalities right now, and the author describes normalization with the following example. Prove that $a^2 + b^2 + c^2 \ge ab + bc + ca$ Of course the fundamental ...
0
votes
0answers
22 views

Model for headphone quality as a function of price

I've looked for any mathematical model that shows headphone quality as a function of the price of the headphones, but have not found anything yet. Of course, "quality" is quite subjective, which could ...
0
votes
2answers
29 views

If C/A is abelian, then C/B is abelian.

I am trying to prove that if groups A $\lhd$ B $\lhd$ C, A $\lhd$ C, and C/A is abelian, then C/B and B/A are abelian. Clearly, B/A is abelian since it is a subgroup of the abelian group C/A. ...
7
votes
1answer
27 views

Is this true for functions with certain conditions?

Let $f$ and $g$ be real-valued functions defined for all real values of $x$ and $y$, and satisfying the equation $f(x + y) + f(x − y) = 2f(x)g(y)$ for all $x$, $y$. Is it true that if $f(x)$ is not ...
1
vote
1answer
11 views

Showing Residual Sum of Squares for Multiple Linear Regression is 0

Problem: I have the linear regression model: $y_i=\beta_0+\sum_{k=1}^p \beta_kx_{ik}+\epsilon_i$ where $\epsilon_i\sim N(0,\sigma^2)$, for $i = 1,2,\ldots ,n$. I want to prove that the residual sum ...
0
votes
1answer
8 views

Inverse of a quasipositive matrix with negative spectral bound

A square matrix is quasipositive if all off-diagonal elements are nonnegative. The spectral bound of a square matrix is defined as $$s(A) = \max\{\Re (\lambda) : \lambda \mbox{ is an eigenvalue of } A\...
2
votes
3answers
88 views

Getting 2016.20162016…to infinity after dividing

Dividing 2240000/1111 seems to give 2016.20162016... to infinity. That is, 2016 keeps repeating. Can someone please tell me what the chances of this surprising effect is? Thank you!
0
votes
0answers
22 views

Probabilistic parameters determination

Before posing the question itself, it is indispensable to give the definition from which it arises. First of all, let us restrict our attention to the vectors $\overrightarrow{x} = (x_{1},x_{2},\ldots,...
0
votes
0answers
18 views

Decomposition of a monomial ideal

I have to find a primary decomposition of the following ideal and I proceeded in this way: $$(x^2z,x^2y^3,xt^2)=(x)\cap(t^2,x^2z,x^2y^3)=(x)\cap(t^2,x^2)\cap(t^2,z,z^2y^3)=(x)\cap(t^2,x^2)\cap(t^2,z,...
0
votes
0answers
16 views

Symmetric brace algebras - unshuffle sequences

I'm studying brace algebras in this article: Symmetric Brace Algebras. In the following definition, what do the authors mean by "unshuffle sequences"? Definition 2. A symmetric brace algebra is a ...
1
vote
1answer
22 views

For which integers $q \ge p\ge 1$ with $q^2-2p^2=2$ is $2p^2+1 \pm pq$ an integer square?

The title says it all… I’m looking to prove (in an elementary way, if possible) the following question: Conjecture: If $q$ and $p$ are positive integers such that $q^2-2p^2=2$ and $2p^2+1 \pm pq$ is ...
5
votes
1answer
17 views

Parallel planes and existence of a regular tetrahedron

Could somebody please guide with an appropriate approach for the following problem? Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.
1
vote
2answers
26 views

If the radioactive isotope strontium $240$ has a half life of $120$ years, how long until it decays to only $60\%$ of its original radioactivity?

I been trying to solve this problem for hours and the only thing i came up with, was a formula for their relationship. $1/2A = A_0 e^{120r}$ $\ln(1/2 = e^{120}r)$ $\ln(1/2) = 120r$ $r = \ln(0.5)/...
2
votes
4answers
23 views

Existence of a basis $B$ such that $M(\phi,B)=E$

Considering the matrix $$A=\begin{bmatrix}{2}&{1}&{0}\\{1}&{0}&{-1}\\{0}&{-1}&{-2}\end{bmatrix}\in M_3(R).$$ And $\phi:R^3 \times R^3\longrightarrow R$ the bilinear ...
0
votes
0answers
16 views

How to represent $GF(q)$ element in Matlab

Everyone, I am trying to simulate in Matlab the Galois field $GF(q)$ where $q$, for example, is $8$. I know the elements of $GF(8)$ are $0,1,\alpha,\alpha^2,\alpha^3,\ldots,\alpha^6$. Also, we can ...
1
vote
2answers
34 views

Proof/Reasoning why the sgn function which counts inversions has the following property?

$\mathrm{sgn}(\pi\circ\sigma)=\mathrm{sgn}(\pi)\cdot \mathrm{sgn}(\sigma)$ I am familiar with how to count inversions and any insight for why this formula holds true would be very helpful.

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