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A proof of divisor of Polynomial by Polynomial's GCD with its derivative have the same roots as itself

Problem If $Q(x)$ is the greatest common divisor of the polynomials $P(x)$ and $P'(x)$, where $P'(x)$ is the derivative of $P(x)$, then the polynomial $\frac{P(x)}{Q(x)}$ has the roots of $P(x)$ as ...
Yinuo An's user avatar
  • 370
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0 answers
11 views

How to compute Contour Integral Numerically

So I am trying to compute the integral from $0$ to $a$ on the real axis, shown in the picture, for a function that is completely analytic on the upper half plane except at $E + i\epsilon$ where $\...
Aziz's user avatar
  • 11
1 vote
0 answers
9 views

Projection onto a real subspace within a complex vector space

Given a complex vector space $V$, choose a finite set of vectors from $V$: $$S = \{v_1, v_2, ..., v_n\}.$$ Now consider the following "real subspace" of V: $$R = \{\sum_{i=1}^n a_i v_i: a_i \...
pillow47's user avatar
  • 161
1 vote
0 answers
19 views

A method of solving systems of linear equations in C.

It is known that a complex number $a + bi$ may be represented as a matrix in the form below:$$ \begin{bmatrix} a & -b \\ b & a \end{bmatrix} $$ Suppose that we have a system of linear ...
iceyspinglass's user avatar
1 vote
0 answers
7 views

Continuity of $L_p$ norms

I am solving Exercise 3.21 of section 4 in Erhan Cinlar's book - Probability and Stochastics. The question is: Fix a random variable $X$. Define $f(p)=\|X\|_p$ for $p\in[1,\infty]$. Show that the ...
Mshirur's user avatar
  • 23
2 votes
1 answer
21 views

Sum of Repetends of Prime Reciprocals

Just a recreational mathematician here with a random question. All reciprocals of primes are periodic and there is some rational number that approximates them exactly up to their period. For example, $...
Reuben Danyali's user avatar
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0 answers
12 views

Solutions to this odd Diff Eq with an integral in it?

I’ve come to this unusual Diff Eq in a statistical physics problem. A real positive density function $\rho\left(r,v\right)$ is a function of two real positive variables $0\leq r<\infty$ and $0\leq ...
Jerry Guern's user avatar
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2 votes
0 answers
20 views

Probability of Coffee being Hotter than some Temperature?

My cousin and I have been long debating this question: Will a cafe with fresh but inferior quality coffee beans have "better coffee?" compared to a cafe with superior quality coffee beans ...
stats_noob's user avatar
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1 vote
0 answers
13 views

How to prove the continuity of the path lifting function of covering spaces.

Let $p : E \to B$ be a covering map, the path lifting function $\varphi : E \times_p B^I \to E^I$, where $E \times_p B^I := \{(e, \gamma) \in E \times B^I : \gamma(0) = p(e)\}$ is a pullback of $E$ ...
HWC's user avatar
  • 47
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0 answers
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holomorphic on every $2$-dimensional complex plane $\implies$ holomorphic on the entire $m$-dimensional complex plane

Denote by $D_{a, b}^{m}$ the spherical shell $$D_{a, b}^{m}=\{z\in\mathbb C^m~|~a<|z|^2<b\}.$$ A complex manifold $M^{n}$ is said to obey the Hartogs phenomenon, if for any $1>a \geq 0$, any ...
HeroZhang001's user avatar
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-1 votes
2 answers
18 views

Confusion in using applying variance formula

Consider the question An elevator’s weight capacity is 1000 pounds. Three men and three women are riding the elevator. Adult male weight is normally distributed with mean 172 pounds and standard ...
Richard Gene's user avatar
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0 answers
4 views

Nonsingularity about the control gramian

I am recently learning Linear system theory and design by Chen (1999 edition), and on page 146 he said that every entry of $W_c(t)$ is analytic of $t$, where $W_c(t)$ is the control gramian $W_c(t)=\...
xinyang li's user avatar
1 vote
2 answers
68 views

Is addition always closed?

I am currently studying group theory on my own and had two questions: Are there any mathematical structures for which addition is not closed? For example, the sum of two integers is always an integer,...
AM_'s user avatar
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12 views

Upper bound for a operator norm in terms of any open ball.

Is the following inequality true for any bounded operator in a normed space? $$ \sup_{y \in B(x,r)} \|Ty\| \geq \|T\| r $$
Javier Bejarano's user avatar
1 vote
0 answers
15 views

Minimal number of binary operators to mock an arbitrary truth table

Considering: $$G(X,A,B)=(X∧A)∨(¬X∧B)$$ For all boolean unary operator $F(X)$, there exists a unique pair of $A,B$ so that $G(X,A,B)=F(X)$ for all $X$. That is, by assigning correct $A,B$, function $G$ ...
Xiaoyou Zhai's user avatar
1 vote
0 answers
12 views

Spaces of Probability Measures with Weak Convergence

I am not fluent in measure theory but have to address a question that I have come across. Let $X_i$, $i=1,2,\dots, K$ be compact metric spaces. So, for each $i$, by Prokhorov's theorem, the space of ...
GA-Student's user avatar
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0 answers
23 views

Question on exercise 1.1.12 of Terence Tao's Random Matrix Theory notes

I understood Example 1.1.3. by explicitly writing down every element in hyperplane V. However, in Exercise 1.1.12 below, there seems to be no such primitive solution. Let $\mathbb{F}$ be a finite ...
Nao Tomita's user avatar
1 vote
0 answers
31 views

Is my proof rigorous enough? Spivak, Prove:$\lim_{x \to a}f(x)=\lim_{h \to 0}f(a+h)$ (Chapter 5, Problem 10)

I'm aware that the answers to this question already exist on this site I would just like to know if my proof is rigorous enough (or incorrect). Prove that $\lim_{x \to a}f(x)=\lim_{h \to 0}f(a+h)$. (...
Edward Falls's user avatar
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0 answers
19 views

A surjective continuous open map has a continuous right inverse

My question arises from this post and similar questions. It's clear that for an open, onto and continuous function $f:X\rightarrow Y$, not every right inverse is continuous, but my question is if ...
H4z3's user avatar
  • 800
1 vote
0 answers
7 views

Finding $\min_{Q^TQ = I}(n Tr((Q^T \Lambda Q)^2) - Tr(Q^T \Lambda Q)^2)$

Let $n\in \mathbb{N}$, $c > 1$ and $T = T (n) = \lceil cn \rceil$. Let $p \in (0, 2]$. Given diagonal $T \times T$ matrix $\Lambda$ with diagonal entries $\lambda_j = j^p$, $j = 1, \ldots, T$, let $...
Kakashi's user avatar
  • 2,254
2 votes
0 answers
30 views

Is it really necessary to go to all this trouble to split division algebras?

Let $D$ be a finite-dimensional central simple algebra over a field $K$. $D$ is said to split over a field extension $L/K$ if we have $D_L \cong D \otimes_K L \cong M_n(L)$ for some $n$. In a few ...
Qiaochu Yuan's user avatar
2 votes
1 answer
31 views

Connection Between Derivations of Finite and Infinite Binomial Expansion

At first when learning the binomial expansion you learn it in the case of working as a shortcut to multiplying out brackets - anti-factorising if you will. In these cases what you are expanding takes ...
Ardavan Hamisi's user avatar
2 votes
1 answer
29 views

Is there a continuous bijective mapping of $R$ into a compactum

I wonder if there are generally no bijective, continuous mappings of the form $$f: \mathbb{R} \rightarrow K$$ if $K$ is a topological compact space.$$$$ My considerations I realize that the inverse ...
Noctis's user avatar
  • 344
0 votes
1 answer
16 views

Rational quaternions are division sub algebra of any cyclic division algebra over a number field

If $L$ is a finite cyclic Galois extension of degree $n$ over a number field $K$, $\theta\in\mathrm{Aut}(L/K)$ is a generator, and $a\in K$ nonzero is such that $a^n$ is the least power of $a$ that is ...
Ama's user avatar
  • 83
1 vote
0 answers
15 views

Signed Measures and Radon-Nikodym Theorem with Total Variation Measure

This is a question from Richard Bass, Real Analysis for Graduate Students. If $\mu$ is a signed measure on $(X,A)$ and $|\mu|$ is the total variation measure, prove that there exists a real-valued ...
Tyler88's user avatar
  • 11
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0 answers
29 views

Can mathematics explain the reason behind my finding regarding the binary matrix arising from BCH code?

I am a PhD student and I work in the area of applied mathematics. I am studying the different kind of uniform column weight binary matrices with prescribed inner products between its columns. I am ...
Dark Forest's user avatar
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0 answers
5 views

What term do you use to describe to the rate of change of higher order tensor functions with respect to their inputs?

If the rate of change of a scalar valued function is the gradient, and the rate of change of a vector valued function is the Jacobian, what is the rate of a change of a matrix valued function? To add ...
David's user avatar
  • 101
1 vote
0 answers
15 views

evaluation and coevaluation, FDVS case

Let $V$ be a finite dimensional vector space over a field $\mathbf F$. We may obtain the evaluation map $\text{ev}:V\otimes V^*\to\mathbf F$ via universal property of the tensor product; the ...
node196884's user avatar
0 votes
0 answers
10 views

Looking for an example of a real (hyper)cubic surface with special restriction

Need help looking for an example of a cubic surface over $\mathbf{R}$: it has two connected components, one of the components is convex. Here's my thoughts so far: The model in my head is the ...
Degenerate D's user avatar
0 votes
1 answer
18 views

Clustering for a real problem - location matters!

I am working on a clustering problem and need some help to develop an appropriate mathematical model. Here are the details of my problem: Locations: I have a set of 141 locations, each defined by ...
juasmilla's user avatar
  • 101
0 votes
1 answer
46 views

When will "Retract $\iff$ Deformation Retract" hold true?

Problem. Given topological space $X$ and subspace $A\subset X,$ under what conditions will we be able to make the following claim: $A$ is a retract of $X$ if and only if $A$ is a deformation retract ...
JAG131's user avatar
  • 917
-1 votes
0 answers
31 views

Use Parseval's identity to show a series converges [closed]

$$\sum_{n=1}^\infty \sqrt{a_n^2+b_n^2}$$ Let f be a 2L-periodic function and such that its derivative f' is continuous. If $a_n$ and $b_n$ are the Fourier coefficients for f, show that the series ...
Laura Freitas's user avatar
-5 votes
0 answers
31 views

Help me with this problem please [closed]

Let ABC be a triangle with AB = 18, BC = 24, and CA = 20. D is placed on AB such that AD = 15. E is placed on BC such that EC = 20. Call the intersection of lines AE and DC point F. Compute: $(\text{...
Sebas Domenech's user avatar
-1 votes
1 answer
22 views

Lebesgue integral of L^1 function is differentiable

Let $f\in L^1(\mathbb{R})$, and define the function $$ F(x)=\int_a^xf(t)dt. $$ I want to prove that $F$ is almost everywhere differentiable and that $F'(x)=f(x)$ where $F$ is differentiable. I am ...
Laurent Claessens's user avatar
2 votes
2 answers
50 views

Can a group have a subgroup whose complement is closed under the group operation?

Does there exist a group $G$ and a subgroup $H$ of $G$ which is not equal to $G$, such that the set-theoretic complement $G - H$ is closed under the group operation? I have tried to come up with some ...
user107952's user avatar
  • 21.4k
0 votes
0 answers
38 views

Is every geodesic constant-speed?

I am learning differential geometry with "Pressley, Elementary Differential Geometry". He establishes that (1) geodesics are curves of constant speed, and that (2) every meridian on a ...
user358572's user avatar
2 votes
1 answer
28 views

If a sequence $a_n$ satisfies the following two properties, does $\sum_{k=1}^{\infty} (\sum_{n=1}^{\infty} \frac{1}{a_n^k} - L)$ converge?

Let $a_n$ be a positive, increasing sequence satisfying the following two properties: $S_k :=\displaystyle\sum_{n=1}^{\infty} \frac{1}{a_n^k}$ converges for all $k \in \mathbb{N}$. And $\displaystyle\...
Cristof012's user avatar
0 votes
1 answer
34 views

Binomial identity involving square binomial coefficient [closed]

I want to prove this identity, but I have no idea... Could someone please post a solution? Thank you. $$\sum_{k=0}^{n} \binom{-1/2}{n+k}\binom{n+k}{k}\binom{n}{k}= \binom{-1/2}{n}^2$$ (Maybe -1/2 can ...
anonymoususer's user avatar
-4 votes
0 answers
41 views

Definition of smooth functions: one widespread missunderstanding [closed]

Definition of smooth functions: one widespread missunderstanding During my time participating in this website , I have saw a misunderstanding when trying to explain that a function is smooth: For some ...
Joako's user avatar
  • 1,548
0 votes
1 answer
24 views

Proving existence of a linear combination by using partial fraction decomposition

For $n$ integer and $t>1$, we define $A_n(t)$ as follows: \begin{align*} A_n(t) = \frac1{(1+t)^n} + \frac1{(1-t)^n} \end{align*} Now, we want to compute $A_n(t)\cdot A_p(t)$: \begin{align*} A_n(t) \...
edrezen's user avatar
  • 243
0 votes
1 answer
31 views

General question: Is there theory/literature for the solution of symmetric PDEs?

I'm currently struggling with $x^2 \frac{\partial^2f}{ \partial y^2} = y^2 \frac{\partial^2f}{\partial x^2}.$ I managed to get some of the solutions to the first order PDE $x \frac{\partial f}{\...
Emanuel Landeholm's user avatar
0 votes
1 answer
29 views

Jacobian of the row de-meaning of a matrix X, with respect to matrix X

Let $\mathbf{X}$ $\in \mathbb{R}^{M \times N}$ be a long matrix, with $M<N$. Let $\mathbf{Y}$ $\in \mathbb{R}^{M \times N}$ represent $\mathbf{X}$ after each row of $\mathbf{X}$ has been "de-...
Cal's user avatar
  • 3
3 votes
1 answer
51 views

Show that $\sum_{k=1}^n{2^{2k-1}\binom{2n+1}{2k}B_{2k}(0)}=n$

Lately, I've been working on a proof (whose context is not necessary to discuss) and I only need one last thing in order to finish it. To be more specific, for completeness it would suffice to show ...
Vaskara_GRek_O's user avatar
1 vote
0 answers
15 views

Conservative idempotent magma - proof attempt

I need help with checking proof about idempotent and conservative magmas. Let magma be any ordered pair $(M, \odot)$, where $M$ is nonempty set and $\odot$ binary operation on $M$. Now I need to ...
Oliver Bukovianský's user avatar
1 vote
0 answers
26 views

Asymptotic growth of the Fibonacci primes

Define $\kappa(n)$ to be the number of prime Fibonacci numbers less than $F_n$. Is it the case that $\kappa(n) \sim a\log(n)$ for some constant $a$? I know that $F_n$ prime $\implies n$ prime (except ...
Cristof012's user avatar
1 vote
2 answers
32 views

Trouble Understanding Difference in Epsilon-Delta Arguments: Why One Works But The Other Fails (Spivak Calculus Problem 5-10c)

In the problems for the limits chapter (5) of Spivak's Calculus, we are asked to prove: $\lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} f(x^3)$. The relevant to the question proof alternative is: ...
Stephen Premel's user avatar
3 votes
1 answer
78 views

Spivak's proof of the parallelogram law of forces

In Physics for Mathematicians, Spivak offers a "proof" of the parallelogram law for forces, i.e. the rule that forces combine according to the rule of standard vector addition. I say "...
Jack M's user avatar
  • 28k
1 vote
2 answers
47 views

Orthocenter: The "Bad Boy" of Distinguished Points in a Triangle

It is a well-known fact that the altitudes of a triangle $ABC$ (with vertices $A,B,C)$ intersect at exactly one point, the orthocenter. The proof known to me (see eg here) involves the construction of ...
user267839's user avatar
  • 7,541
0 votes
1 answer
22 views

Simplifying Sentences that Precede a Conditional

Suppose we have the following proposition: Suppose that $x \in \mathbb Z$, that $a$ is even, and that $b$ is odd. If $x^2-ax+b$ is even, then $x$ is odd. I am not interested in proving the proposition;...
LateGameLank's user avatar
0 votes
0 answers
26 views

Different approach to calculating longest run of heads probability

In need of implementing a solution for this, I've read through this prior stack exchange discussion that describes an approach to the problem. I've also read this paper The Longest Run of Heads Author(...
koalacombatsystems's user avatar

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