0
votes
0answers
3 views

Application Farkas Lemma

Let $A$ be a $m \times n$ matrix and $C$ a $k \times n$ matrix. Let $b \in \mathbb{R}^m$ and $d \in \mathbb{R}^k$. Show that exactly one of the following holds: a) There exists an $x \in ...
0
votes
0answers
8 views

Markov Chain with transition matrix Q=(I+P)/2

I have finite irreducible Markov Chain with transition matrix P. 1)Prove that Markov Chain with transition matrix Q=(I+P)/2 is irreducible and aperiodic (I is identity matrix); 2)Prove that P and Q ...
0
votes
0answers
4 views

wave equation on a circular domain

Consider the wave equation for the displacement $$\text{u(r,$\theta $,t)}$$ in a circular domain $$\text{0 $<$ r $<$ a, -$\pi $ $<$ $\theta $ $<$ $\pi $}$$ How do I use the separation ...
0
votes
0answers
3 views

Algebra with element having empty spectrum?

The definition of the spectrum makes sense for any algebra. I guess we can go to the unitization to make sense of it even non-unital algebras. Recalling the well-known fact that for normed algebras, ...
0
votes
3answers
11 views

How to denote an average of data satisfying a given condition

I have to write an expression for the following. I have values, one fpr each of the last 5 months: ...
1
vote
1answer
7 views

What is the interpretation of the following optimization problem?

Suppose we have $N$ variables $x_1,\ldots,x_N$. Let $\mathbf{A}$ a $M \times N$ matrix, and $\mathbf{b}$ a $M \times 1$ vector. I have the following minimization problem: \begin{array}{rl} \min ...
4
votes
1answer
13 views

Matrix ring $M_2(\mathbb{C})$, $\mathbb{C}^2$ with $M_2(\mathbb{C})$-module structure.

Let $R$ be the $2 \times 2$ matrix ring $M_2(\mathbb{C})$. let $M = \mathbb{C}^2$ with its natural $R$-module structure (just given by the usual action of $2 \times 2$ matrices on $2$-dimensional ...
1
vote
0answers
14 views

Open and Closed covering

Let $X$ be a compact Hausdorff and totally disconnected space and $A$ be a closed subset of $X$ contained in an open set $U$. Then we can find a finite set $\{V_1,\cdots,V_n\}$, where each $V_i$ is ...
0
votes
0answers
7 views

Expectation Maximization Algorithm and low number of observations

I am observing a strange behaviour with EM algorithm and I would like to see whether anyone else has had a similar experience or not or if I could be guided through a reference or anything like that ...
0
votes
0answers
9 views

Grand canonical derivative.

I've been trying to work out how to find the density in the thermodynamic limit of a nearest neighbour magnetic lattice gas in the grand canonical ensemble. I'll with hold the Hamiltonian for the ...
3
votes
0answers
26 views

Does equation $f(g(x))=a f(x)$ have a solution?

Suppose $g: [0,1] \to [0,1]$ is a strictly increasing, continuously differentiable function with $g(0)=0$ and $g(1) < 1$, and $a \in (0,1)$. Is there a function $f:[0,1] \to \mathbb{R}$ such that ...
0
votes
1answer
7 views

Convex Homotopy

Suppose $f , g : X \to U \subset \mathbb R^2$ are two mappings from a topological space $X$ to a convex set $U$. Prove that $f$ and $g$ are homotopic, using only the definition of the product ...
2
votes
0answers
11 views

Connection and reduction of the structure group

I am writing a memoir about gauge theory. I have trouble with a small proof which should be simple and have the feeling that I am missing something obvious. I want to show that the set of connections ...
0
votes
0answers
7 views

How to make use of symmetric of sparse matrix to solve this kind of problem?

I have the following matrix to be solved: $$\left\{ \matrix{ {a_{11}}{x_1} + {a_{12}}{x_2} + \cdots + {a_{1n}}{x_n} = {y_1} \hfill \cr {a_{21}}{x_1} + {a_{22}}{x_2} + \cdots + {a_{2n}}{x_n} = ...
1
vote
0answers
4 views

When can we move a Fréchet derivative under a Lebesgue integral?

Under what conditions can we move a Fréchet derivative under a Lebesgue integral? Specifically, when does $$ G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt) $$ where $$ ...
0
votes
0answers
5 views

The coproduct of a family of objects of a Preorder (seen as a category)

If the coproduct of a family of objects of a Poset (seen as a category) is the least upper bound, who is the coproduct of a family of objects of a Preorder (seen as a category)? My intuition ...
1
vote
1answer
11 views

Projectile to clear a semispherical mound

I have only just stumbled across this site a few days ago while searching for other things. This is a great resource. I have a question for nearly 30 years now when me and my friend were trying to ...
0
votes
1answer
10 views

Dependence of product of matrix and a vector, on the rank of a Matrix

What is the significance of the rank of a matrix, say $A$, when I am multiplying a vector, say $x$, by $A$? In other words, let $x$ be a column vector of suitable dimension and let $rank (A)=m$. What ...
0
votes
3answers
26 views

Math question quick and easy

Let $x=17$ $n=130$. Find $y; (1\leq y \leq n-1)$ that satisfies :$$xy=1(modn)$$ Now I'm nlt sure if I should use one of Euler's theorem's for prime numbers? Can anyone help? Or try something with ...
-1
votes
2answers
12 views

Region of Convergence of power series

The power series $\sum_{n=0}^\infty 2^{-n} z^{2n} $ converges if a)$|z|\le 2$ b)$|z|\lt 2$ c)$|z|\le\sqrt2$ d)$|z|\lt\sqrt 2$ I tried this problem,my answer is d).I am not sure whether it is correct ...
1
vote
0answers
11 views

How do I show $f_0 \sim f_1 \Rightarrow f_0 + g \sim f_1 + g$

Consider two smoothly homotopic maps $f_1,f_2:M \to S^1$ from a compact smooth $n$-manifold $M$ to the unit circle. How do I show $$f_0 \sim f_1 \Rightarrow f_0 + g \sim f_1 + g$$ for all $g:M \to ...
1
vote
1answer
20 views

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 ...
1
vote
0answers
8 views

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 ...
0
votes
1answer
9 views

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 ...
0
votes
0answers
8 views

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 ...
-2
votes
0answers
17 views

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 ...
2
votes
1answer
12 views

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 ...
-2
votes
2answers
36 views

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?
2
votes
2answers
34 views

$\frac{1}{{1 + \left| {\left\| A \right\|} \right|}} \le \left| {\left\| {{{(I - A)}^{ - 1}}} \right\|} \right|$

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 ...
0
votes
0answers
26 views

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. $$A := \begin{bmatrix}-6 & 3 & 7 \\ 0 & -1 & 0 \\ -14 & 12 & 15\end{bmatrix}$$
2
votes
2answers
16 views

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 $M \cong R$?
0
votes
1answer
8 views

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 ...
3
votes
0answers
26 views

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 ...
0
votes
1answer
12 views

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 ...
-7
votes
0answers
34 views

How to prove that one guy in all groups [on hold]

I dont know how even think about it. Anyone? thanks
4
votes
2answers
48 views

$3^x + 4^y = 5^z$

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?
2
votes
0answers
24 views

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." ...
0
votes
1answer
11 views

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 ...
0
votes
0answers
4 views

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 ...
0
votes
0answers
6 views

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 ...
0
votes
0answers
12 views

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.
0
votes
0answers
3 views

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 ...
3
votes
1answer
45 views

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 ...
6
votes
0answers
29 views

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 ...
-3
votes
0answers
24 views

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
1
vote
2answers
30 views

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 $$p\{N>n\}=p\{X_1+...+X_n\leq x\} \ \ ?$$
2
votes
2answers
35 views

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 ...
-5
votes
1answer
32 views

tetrahedron problem with center and reflections [on hold]

I can i solve it? help me please i dont know how to do it. thanks
-4
votes
1answer
42 views

Prove that for all prime numbers [duplicate]

Prove that for all prime numbers $p>2000$ the sum $1 + 2\cdot2000 + 3\cdot2000 + … + (p-1)\cdot2000$ is divisible by $p$.
0
votes
1answer
15 views

Let X have density 2t on 0 < t < 1 and Y be uniform on the interval (0, 10) and independent of X. Find the density of Y/X.

Let X have density 2t on 0 < t < 1 and Y be uniform on the interval (0, 10) and independent of X. Find the density of Y/X. I have no ideas how to solve it

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