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### Doubt regarding the radius of convergence of Cauchy Product

Let $\sum \limits _{n=0}^\infty a_nz^n$ and $\sum \limits _{n=0}^\infty b_nz^n$ be two power series with complex coefficients. If the radius of convergence of the first series is $R_1$ and the radius ...
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### An interesting recurrent equality, possibly easier to solve in its differential form?

I encountered an interesting inequality that I'm not sure how to approach. Here $c$ is a positive constant. $$f(n+1) - f(n) = c f(n)\sum_{m=0}^n f(m)$$ I am not familiar with techniques to solve ...
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### How many integer values of $m\in[-20;22]$ are there such that the equation $\log_2(x^2+m+x\sqrt{x^2+4})=(2m-9)x-1+(1-2m)\sqrt{x^2+4}$ has a solution?

How many integer values of parameter $m$ on $[-20; 22]$ are there such that the equation $$\log_2(x^2 + m + x\sqrt{x^2 + 4}) = (2m - 9)x - 1 + (1 - 2m)\sqrt{x^2 + 4}$$ has a solution? [For context, ...
1 vote
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### Why does Galois theory most naturally take place in the context of fields?

At least as far as I can tell, historically Galois theory was a more computational tool than it appears now, and https://hsm.stackexchange.com/questions/8099/how-did-the-modern-understanding-of-galois-...
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### Maximize (z-x) such that x^2 + y^2+ z^2 =1.

Here I tried the coordinate geometry, as in the equation represents a sphere. From there (z-x) would imply the distance between the z coordinate and x coordinate so that the difference is maximum. ...
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### Given, $a,b,c\in\mathbb{R}$. Let $p(x)=ax^2+bx+c$. Find equivalent condition that always has an integer value ($p(x)$'s value) for $x\in\mathbb{Z}$.

Problem : Given, $a,b,c\in\mathbb{R}$. Let $p(x)=ax^2+bx+c$. Then, find an equivalent condition that always has an integer value ($p(x)$'s value) for an arbitrary integer $x$. [Problem is represented ...
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### Integral of Complex Numbers

I was wondering about doing integration the unorthodox way. For example I took up $$\int\sqrt{1-x^2}\,\mathrm dx$$ and instead of substituting $x$ for $\sin t$ I tried doing it for $\sec t$ which ...
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### Exemple where tower property of conditional expectation is NOT verify

Question: Let Ω={a,b,c}. Give an example for $X, F_1, F_2$ in which $E(E(X|F_1)|F_2) \neq E(E(X|F_2)|F_1)$ My ansmer: I am not at all sure of my answer. If you have any shorter and niver answer i will ...
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### Goldbach Conjecture and Triangular Number

Goldbach conjecture states that every even number greater than 2 is sum of two prime numbers. We know that every positive integer can be represented as a sum of three triangular numbers. Is it ...
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### Two fair dice tossed simultaneously till combination $(4,3)$ is obtained. Obtain the probability that number of tosses required is atmost $2$.

Original Question: Two fair dice are tossed simultaneously till a combination $(4,3)$ is obtained. Obtain the probability that the number of tosses required is atmost $2$. My Reasoning : The dice are ...
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### Does Lipschitz-kind "non-scalar ODEs" and PDEs could stand having finite-duration solutions?

Does Lipschitz-kind "non-scalar ODEs" and PDEs could stand having finite-duration solutions? Intro Recently I have found on these papers by Vardia T. Haimo (1985) Finite Time Controllers ...
1 vote
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### Hyperplane contains an inversible matrix

Let $H$ be an hyperplane of $M_n(\Bbb K)$ $(n\ge 2)$ Show that there exists $A \in M_n(\Bbb K)$ such that $H=\{M\mid \text{tr}(AM)=0\}$ Deduce that $H$ contains an invertible matrix
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### Discussion, a sieve of co-primes that contains all primes.

Discussion Claim: "All prime numbers are contained in the set [P]n ∪ [G]": [P]n is composed of the set ...
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### Solution of $\theta$ when $\tan(\theta)-\sin(\theta)=\frac{\sqrt3}{2}$

I came across this trigonometry problem. If, $$\tan(\theta)-\sin(\theta)=\frac{\sqrt3}{2}$$ What is the value of $\theta$ I got the solution that $\theta$ will be $\frac{\pi}{3}$ by expanding the ...
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I'm studying this article and and I'm having doubts about understanding two statements at the end of the proof of Lema 3.3 Why does $G = UP$ implies $C_G(U) = U \mathrm{core}_G(P)$? And why does $\... -1 votes 1 answer 18 views ### Tangent plane to the surface and the Taylor polynomial Let$f(x,y) = a \sin(x+y^2) + e^{bx+y^2}$, what are the values of$a$and$b$such that the tangent plane to the surface$z = f(x,y)$at$(0,0,f(0,0))$be horizontal and the second order Taylor ... 2 votes 0 answers 10 views ### Can we calculate the opponent's hidden values in this statistical battle? First-order statistical battle Imagine there is a game in which the user should guess what values the opponent is hiding from the user. In the first battle, the opponent has two hidden values ... 5 votes 3 answers 140 views ### Inequality involving sums with binomial coefficient I am trying to show upper- and lower-bounds on $$\frac{1}{2^n}\sum_{i=0}^n\binom{n}{i}\min(i, n-i)$$ (where$n\geq 1$) in order to show that it basically grows as$O(n)$. The upper-bound is easy to ... 0 votes 0 answers 20 views ### What conditions are required to guarantee that my matrix is skew-Hermitian? Consider the equation $$\frac{\partial \boldsymbol{T}}{\partial t}=\kappa \space \frac{\partial^2 \boldsymbol{T}}{\partial x^2}$$ with $$\boldsymbol{T}=(T_1,T_2,...T_N)^T$$ Let$\$ T_i=\sum_{j=1}^...

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