# All Questions

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### the average speed of the car

A car started its journey from town A to town B with a constant acceleration. From the time the speed of 80 km/h was reached, for half of the total travel time the car continued at this speed. The ...
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### Is a map from a smooth curve, bijective away from the singularities, already a normalization?

Over $\mathbb{C}$, I am considering a map between projective curves $f: C \rightarrow C'$, where $C$ is smooth. Suppose that $f$ is surjective as well as bijective away from the singularities of ...
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### Does “pseudo-independent implies independent” imply that $R$ is a field?

(All my rings are unital.) Suppose $R$ is a commutative ring and that $M$ is an $R$-module. Definition. Call a subset $X \subseteq M$ pseudo-independent iff for all proper subsets $Y$ of $X,$ the ...
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### Proving max-mod principle by contradiction

This is a homework exercise I have to make which I am kind of stuck on. First let $U$ be open and connected, $\overline{D}$ be the closure of the disk $D$ contained in $U$ and ...
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### What is the maximal number of residents of the city

The population of the city Roachrach consists of cockroaches and cucarachas. The number of residents of the city is no greater than 2,000,000. Each cockroach is acquainted with 1,000 cucarachas, and ...
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### Perfect pairing induces isomorphism of tensor products

Let $M, N$ be $R$-modules and $(\cdot, \cdot): M \times N \to R$ be a perfect pairing. Wikipedia sais that this means that the map $\varphi: M \to \text{Hom}_R(N, R), m \mapsto (n \mapsto (m, n))$ is ...
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### Proving the equation has no root.

How to show that for $a\in \mathbb R$, the equation $x^2+12a^2+4ax-8a+8=0$ has no root?
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### $\frac{1}{{1 + \left| {\left\| A \right\|} \right|}} \le \left| {\left\| {{{(I - A)}^{ - 1}}} \right\|} \right|$

Let a matrix norm $\left| {\left\| . \right\|} \right|$ have the property that $\left| {\left\| I \right\|} \right| = 1$ and $\left| {\left\| A \right\|} \right| < 1$. Why does the following ...
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### How to find one matrix, which is subject to $B^3 = A$. How much is such matrices?

Here I have a problem with row echelon form. $$A := \begin{bmatrix}-6 & 3 & 7 \\ 0 & -1 & 0 \\ -14 & 12 & 15\end{bmatrix}$$
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### Cyclic projective module

Let $R$ be an integral domain. If $M$ is a cyclic $R$-module which is also projective, then must there necessarily be an isomorphism of $R$-modules $M \cong R$?
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### Complexity of Newton iteration problem for a d-dimensional problem

If we assume that we have $f:\mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ and we want to use the Newton iteration method to solve $f(x)=0_{\mathbb{R}^{d} }$. Is there any theorem regarding the ...
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### Is there an established notation for this “replacement” operation?

If $S$ is a set, define $$(x \to y) \cdot S := \begin{cases} (S \setminus \{x\}) \cup \{y\} & \text{ if } x \in S \text{ and } y \not \in S; \\ S & \text{ otherwise.} \end{cases}$$ In other ...
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### Combinations for pairing groups

I have a little bit of a complex question and I don't know anything about combinatorics, but I'm working on software problem and I'm trying to figure out how my algorithm will scale. I'm having to ask ...
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### How to prove that one guy in all groups [on hold]

I dont know how even think about it. Anyone? thanks
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### $3^x + 4^y = 5^z$

This is an advanced high-school problem. Find all natural $x,y$, and $z$ such that $3^x + 4^y = 5^z$. The only obvious solution I can see is $x=y=z=2$. Are there any other solutions?
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### What exactly is wrong with this argument (Lucas-Penrose fallacy)

Argument "For every computer system, there is a sentence which is undecidable for the computer, but the human sees that it is true, therefore proving the sentence via some non-algorithmic method." ...
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### Prove that $\sum_{n=0}^\infty e^{-nz}$ is analytic in the right half plane $\text{Re}(z)>0$

Consider$$\sum_{n=0}^\infty e^{-nz}$$ Using Weierstrass theorem, prove that the series is analytic in $\text{Re}(z)>0$. I know that $f$ is analytic if it satisfies Cauchy–Riemann equations. Could ...
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### How to deal with a barrier function when constrained variables reach their bounds?

I am implementing an algorithm of Dang and Xu's, Non-convex Quadratic Programming Problem with Box Constraints'' and I'm hoping that somebody could verify what I'm doing. Their algorithm minimizes ...
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### anlyalytic paths through convergent cauchy sequence II

Assume we have a Cauchy sequence $\{\vec{a_i}:i\in\mathbb{N}\}$ converging to $\vec{0}$ in $\mathbb{C}^n$ such that $|\vec{a_i}|<|\vec{a_j}|$ whenever $i>j$. Can we find an analytic path ...
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### Invertibility for a matrix that I don't know

I would like to know why $(e^{-At}-I)^{-1}$ is invertible when matrix A is Hurwitz.
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### Direct Product of Chernikov Groups is Chernigov group?

A group $G$ is said to be Chernikov if it contains a normal subgroup N such that $G/N$ is finite and $N$ is direct product of finitely many Prufer groups. The problem is the following: If $G$ is a ...
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### A logic problem about set theory

In a group of n people, subgroups with common interest are formed (football,tennis,snooker). The number of subgroups equals $2^{n-1}$. Any 3 subgroups have a common member. Prove that there is a ...
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### Groups with finite automorphism groups.

An easy argument shows that for any finite group $G$ the cardinal of $Aut(G)$ is less than $(|G|-1)!$. In particular the automorphisms group of a finite group is finite. Basically my question is about ...
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### minimal volume of pyramid x0,y0,z0

I dont know how should i even start. I tried to think about something but get nothing. can someone help me please? Thanks
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### Why $p\{N>n\}=p\{X_1+…+X_n\leq x\}$.

Let $(X_k)$ a sequence iid of random variable uniform on $[0,1]$. Let $x\in]0,1[$ and $N=\min\{n\geq 1\mid X_1+...+X_n>x\}$. Why $$p\{N>n\}=p\{X_1+...+X_n\leq x\} \ \ ?$$
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### a linear differential equation with periodic coefficients

Let $$y' = a(x) y + b(x)$$ be a linear differential equation with continuous, periodic coefficients $a, b: \mathbb{R} \to \mathbb{R}$ that both have a period of $T > 0$. Also, we assume that ...
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### tetrahedron problem with center and reflections [on hold]

I can i solve it? help me please i dont know how to do it. thanks
Prove that for all prime numbers $p>2000$ the sum $1 + 2\cdot2000 + 3\cdot2000 + … + (p-1)\cdot2000$ is divisible by $p$.