All Questions

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0answers
2 views

Is converse of Lewy theorem true?

In complex analysis, there is a result named Lewy's theorem, which states that: If $u=(u_1,u_2):\subseteq \mathbb{R^2}\to \mathbb{R}^2$ is one-one and harmonic in a neighborhood $U$ of origin ...
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0answers
3 views

Conceptual Question on Cramer rao lower bound for performance measure

In system identification, parameter estimation I have found in several papers that an analytical bound is derived which is the CRB of the error variance of the estimates. For, optimal performance of ...
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0answers
4 views

Barycentric Coordinates of the circumcenter of an arbitrary triangle

Given points $A(1, 0, 0), B(0, 1, 0), C(0, 0, 1)$ in barycentric coordinates, and points $P(x_P, y_P, z_P), Q(x_Q, y_Q, z_Q), R(x_R, y_R, z_R)$, what would the barycentric coordinates of the ...
2
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1answer
9 views

I must be misunderstanding bijection on uncountable sets. (As it relates to stereographic projection)

At least one of these things must be false: The existence of a bijection between two sets implies that those sets have an equal number of elements. A bijection exists between {all points in a ...
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0answers
8 views

eigen values and eigen vectors of the projection

Assume that $W$ is n-dimensional subspace of an m-dimensional vector space $V$. Find all eigenvalues and all eigenvectors of the projection operators $P_W$. Here is my ideas: Since $W$ is ...
1
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0answers
5 views

Show the following is a functor.

Suppose $p: Y \to X$ and $p': Y' \to X$ are covering maps, and let $\phi: Y \to Y'$ be a homeomorphism such that $p'\phi=p$. Show that the functors $p^{-1}$ and $(p')^{-1}$, from $\Pi_1(X)$ to the ...
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0answers
5 views

Primality Test for Safe Primes

Is this proof acceptable ? Theorem Let $N$ be of the form $N=2p +1$ with $p$ prime , then $N$ is prime iff $N \mid 2^{2p}-1$ Proof In one direction , if $2p+1$ is a prime then by Fermat ...
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0answers
3 views

Doubt in a step of the proof of Rado-Kneser-Choquet theorem

I am trying to prove Rado-Kneser-Choquet theorem, which states that if $f$ is sense preserving self homeomorphism of the unit circle $\partial D$. Then harmonic extension $F$ of $f$ is self ...
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1answer
6 views

Probability from multiple trials

This questions is from a practice mid-term that I don't have a solution to. A monkey in a research lab is given 6 tiles with the letters AAABNN. On each trial the monkey randomly arranges the ...
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0answers
14 views

How can I prove that $P \neq EXP$

It seems like $P\neq EXP$ is much easier than $P \neq NP$. How can I prove $P \neq EXP$? (Well, after all I want to know any proof technique of proving there does not exist any algorithm of certain ...
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0answers
8 views

Laplace Transform Homework

Evaluate the following integrals (you may use the table of the Laplace transform) (a)integral of t^4 e^(−(s+2)t)dt, s > −2 (b)integral of sin3t e^(−(s^2+16)t)dt I solve the first one by using t^n ...
0
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1answer
13 views

Meaning of Exponential map

I've been studying differential geometry using Do Carmo's book. There's the notion of exponential map, but I don't understand why it is called "exponential" map. How does it has something to do with ...
2
votes
2answers
22 views

the inequality $\frac{a^4}{a^3+b^3}+\frac{b^4}{b^3+c^3}+\frac{c^4}{c^3+a^3}\ge \frac{a+b+c}2$

How to show that $$\frac{a^4}{a^3+b^3}+\frac{b^4}{b^3+c^3}+\frac{c^4}{c^3+a^3}\ge \frac{a+b+c}2$$ for $a,b,c>0$? I tried to prove $$\frac{a^4}{a^3+b^3}\ge \frac {5a}4+\frac{-3b}4$$ but could not ...
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0answers
13 views

Intuitive Understanding of Second Derivative/Concavity.

In Calculus, I understand that derivatives (simply explained) are a rate of change; a slope of a function at a certain point. However, I am struggling to understand the explanations behind the second ...
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0answers
15 views

Help me with this question

Length of AB, BC and CD are equal. length of AD=9,AE=6. Find the length of $CE^2$
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0answers
2 views

Why the doubly non-central F distribution does not have a mean or variance if the denominator degree of freedom is less than or equal 2 ??

Normally the doubly non-central F distribution is generated by the division of two non-central chi squared Random Variables,, so what is the the problem of using any famous formula to get the mean of ...
2
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2answers
27 views

How to solve differential equation $dy/dx = y^2/(1+y^2)$ by inegration

My first question is, how does one solve the following differential equation: $$y' = y^2/(1+y^2)$$ My second question is, would it be possible to solve this using ordinary integration method, ...
0
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1answer
17 views

Are there properties of vector space equipped with two norms?

I am interested in a vector space equipped with two norms$ \lvert \lvert \cdot \rvert \rvert$ and $ \lvert \lvert \cdot \rvert \rvert ^*$ satisfies that there is $M>0$ such that $ \lvert \lvert x ...
0
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3answers
16 views

Nature of roots of a quadratic.

Question: If the equation $x^2+2x+1+\lambda=0$ has real and unequal roots, determine the nature of the roots of the equation $(\lambda+2)(x^2+2x+1+\lambda)=2\lambda(x^2+1)$. My attempt: Taking ...
0
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1answer
7 views

Bounded Linear Maps on Normed Vector Spaces

Let $A$ be an $m\times n$ matrix $(\alpha_{jk};\;j=1,...m,k=1,...,n).$ As we know, $$[Bx]_j = \sum_{k=1}^n\alpha_{jk}x_k,\;\;\;\;\;j=1,...,m,\;\;\;x=(x_1,...,x_n),$$ defines a bounded linear operator ...
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0answers
15 views

Continuity of translation on $L^1$ on the reals ($\int |f(x+h)-f(x)|\,dx\to 0$)

Let $f$ be a real valued, Lebesgue integrable function on $\mathbb{R}$. prove $$\lim_{t \to 0} \int_{\mathbb R} |f(x+t)-f(x)|\, dx=0.$$ I solved it in this way. is it correct? Since $f(x)$ is ...
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0answers
7 views

$\ell^p$ with pointwise multiplication — example of $C^\ast$ algebra?

I was trying to think of some examples of $C^\ast$-algebras and I think $\ell^p$ with pointwise multiplication would be a good example. My reasoning is that if $a_n, b_n$ are in $\ell^p$ then ...
0
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0answers
15 views

Solving for a differential equation

I need assistance in solving this differential equation related to an electronics course: $$V= R\frac{dQ}{dt} + \frac{Q}{C}$$ The solution is supposed to be $$Q = CV(1-e^\frac{-t}{RC})$$ This is my ...
1
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2answers
16 views

Two questions concerning divisibility

I was looking at some proof questions and had difficulty answering a few of them How do I prove these statements below: 1) $3 \mid (10^{n+1} + 10^n + 1)$ 2) $(a-b) \mid (a^n - b^n)$
4
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1answer
23 views

Find $\bigcap_{n = 1}^{\infty} (-\frac{1}{n}, \frac{2}{n})$

Find the $\bigcap_{n = 1}^{\infty} (-\frac{1}{n}, \frac{2}{n})$ I know that $\{0\}$ is in the intersection because it lies between negative and positive numbers. I want to show that for $x > 0$, ...
1
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1answer
12 views

Lang problem about automorphism of polynomial ring

STATEMENT: Let $A$ be a commutative entire(integral domain) ring and $X$ a variable over $A$. Let $a,b\in A$ and assume that $a$ is a unit in $A$. Show that the map $X\mapsto aX+b$ induces a unique ...
1
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0answers
8 views

necessary conditions for having a equatoins

suppose that we have a normal extension $K|F$ and we have $\alpha_1,...\alpha_m \in F'and \notin K$ when $F'$ is the algebraic closure of $F$.is it true that for every $\alpha_i$ , $F'|K(\alpha_i)$'s ...
1
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2answers
15 views

How do I prove converse of these two claims?

Prove or disprove the claim, and prove or disprove the converse: Claim 1: ∀n ∈ ℕ, (Ǝk ∈ ℕ, n = 5k + 2) ⇒ (Ǝj ∈ ℕ, n^2 = 5j + 4) Claim 2: ∀m,n ∈ ℕ, (Ǝk ∈ ℕ, m = 7k + 3) ∧ (Ǝj ∈ ℕ, n = 7j + 4) ⇒ ...
1
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1answer
25 views

A formula for second derivative

I want to prove that if $f''$ is continuous at $x_0$, then $$f''(x_0)=\displaystyle\lim_{h\to 0}\dfrac{f(x_0+2h)+f(x_0)-2f(x_0+h)}{h^2}$$ Any hint to prove it? I can't use l'Hopital or Taylor. ...
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0answers
7 views

Calculating dihedral 6 factor group

Let H={$\rho_{0}, \rho_{2}, \rho_{4}$}, a subgroup of D6, the group of symmetries. Where $\rho_{0}$=identity permutation, $\rho_{2}$=(1,3,5)(2,4,6) and $\rho_{4}$=(1,5,3)(2,6,4). I am trying to ...
1
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1answer
9 views

How to show this Legendre Symbol Problem

Let $n \in \mathbb{N}$ and $p$ an odd prime number such as $p \nmid n$, Prove that: $\exists x, y \in \mathbb{Z};\,\, \gcd(x,y) = 1$ such as $x^{2} +ny^{2} \equiv 0\, (\mod p) \iff ...
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1answer
7 views

Number of preimages of a group element under a homomorphism.

I was reading the first chapter of Robert Gilmore's "Lie groups, physics and geometry" and I came across a brief statement regarding the number of preimages of an element under an homomorphism which I ...
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0answers
6 views

every $f \in F^k_p$ has a Taylor expansion

$F_p$ is the set of germs of functions on a manifold M which vanish at $p \in M$. Let $F^k_p$ be the ideal of $C^\infty(p)$ generated by $f_1,... \,f_k$, where $f_i \in F_p$. (i.e $F^k_p$ is $\sum ...
0
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4answers
56 views

Factoring $x^4 + 4x^2 + 16$

I was putting together some factoring exercises for my students, and came across one that I am unsure of how to factor. I factored $x^6 - 64$ as a difference of squares, and then tried it as a ...
1
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1answer
9 views

Is there a nice characterization of those categories in which $a \subseteq b$ implies $a \leq b$?

Given arrows $a:A \rightarrow Y$ and $b : B \rightarrow Y$, define that: $a \subseteq b$ iff for all $Z$ and all $g,g' : Y \rightarrow Z$, we have: $$gb=g'b \rightarrow ga=g'a$$ $a \leq b$ iff ...
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0answers
10 views

Prove that a finite group is polycyclic if and only if solvable

I'm learning group theory but I'm not sure how to solve this problem.
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0answers
10 views

Can O(ndlog(d)) + O(n^2d) + O(n^3) be simplified?

Can $O(nd \log(d)) + O(n^2 d) + O(n^3)$ be simplified without increasing the time complexity? Can $O(nd \log(d)) + O(n^2 d)$ become $O(n^2 d \log(d))$? Can it all become $O(n^3 d \log(d))$? Please ...
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0answers
12 views

Sets, Topology and Applying Cantor's Intersection Theorem

I am trying to solve the problem related to the Sierspinski triangle. The triangle is shown as follow. Let $S$ be the intersection of all the finite stages a). Show that $S$ is a nonempty compact ...
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0answers
20 views

extension of a continuous function

Please is it true that if $f:K\to\mathbb{R}$ is a continuous function of a comact set $K\subset\mathbb{R}^m$ then $f$ can be extended to a continuous function of some open neighbourhood of K ? ...
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2answers
15 views

Merge Sort the sequence.

I have no idea how to do this problem. Can someone show me how to do this? The problem is I don't know how to do backward substitution. I can't seem to find resources on it, that pertain to discrete ...
1
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2answers
31 views

Help showing $\sum_{n=2}^{\infty}(\log n)^{-\log n}$ converges.

I need to show that $\sum_{n=2}^{\infty}(\log n)^{-\log n}$ converges without using the integral test. Any help?
2
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1answer
18 views

For $f(x) = ax^2 + bx +c$, why is it written $a(x-h)^2 + k$

I'm going to have to teach how to graph quadratic equations. Since we've already done a lot of work with the Quadratic Formula, the students are more or less familiar with the standard notation of a ...
0
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1answer
13 views

underdetermined homogeneous system of linear equations has always infinitely many solutions

I want to prove that an underdetermined homogeneous system of linear equations always has infinitely many solutions. I know that an homogenous system of linear equations always has the trivial ...
1
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1answer
10 views

A hand of six cards is dealt from a standard poker deck. Find formula for p_(XYZ) (x,y,z).

A hand of six cards is dealt from a standard poker deck. Let X denote the number of aces, Y the number of kings, and Z the number of queens. a) write a formula for p_(XYZ) (x,y,z). b) Find ...
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0answers
17 views

homology and cohomology with coefficients of ring and field

(1). Let $R$ be a ring. Let $X$ be a topological space. Then $H^n(X;R)$ is a module over $R$. Also $H_n(X;R)$ is a module over $R$.Is this statement correct? (2). Let $F$ be a field. Let $X$ be a ...
1
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1answer
10 views

Algorithm Running Time

Given an algorithm with a running time $$T(n)=5T(n/2)+n^2$$ So the number of nodes at a depth $i$ would be: $5^i$ The input size for each node at $i$ would be: $n/2^i$ Agreed. Then it states that ...
0
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1answer
13 views

Deflating (factoring) a 6th degree polynomial

What is the procedure to factor a 6th degree polynomial of a complex variable? $$P(z)=1+x^2+x^3+x^4+x^5+x^6$$ I do have the correct answer but no idea how to get to it. The answer is: ...
0
votes
1answer
14 views

no. of distinct real roots of the equation $x^2=x\sin x+\cos x$

$(1)$ The no. of Distinct real solution of the equation $x^4-4x^3+12x^2+x-1=0$ $(2)$ The no. of distinct real roots of the equation $x^2=x\sin x+\cos x$. $\bf{My\; Try}$ For $(1)$ one:: Let ...
0
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0answers
11 views

The Answer to the problem Prove that there is a 1-1 correspondence between the set of subgroups of Z/NZ and the set of the positive divisors of N [duplicate]

I need to Prove that there is a 1-1 correspondence between the set of subgroups of Z/NZ and the set of the positive divisors of N My attempt: We first define $B=\{d>0: divisor of N\}$, ...
-5
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1answer
28 views

Calculating the mass of the earth [on hold]

The formula I am using to calculate the mass of the earth is: M = ar2/G = 5.98 × 1024 kg. a being the acceleration of gravity (9.8 m/s squared), r being the radius of the earth ((6.4) *(10^6)), and ...

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