0
votes
0answers
2 views

Is it possible to represent {$0, ±m, ±2m, ±3m, \ldots$} in an augmented matrix?

An augmented matrix of a system consists of the coefficient matrix with an added column containing the constants from the right sides of the equations. Source: Linear Algebra and Its Applications, ...
1
vote
0answers
11 views

What *is* the coordinate ring K[V] of an algebraic variety V?

I've been trying to understand this for a while. If I understand it, we let V be an algebraic variety (set?) then define I(V) to be the ideal generated by V. The coordinate ring is K[V] = K[X]/I(V), ...
0
votes
0answers
3 views

Is there a more relaxed bound for this inequality?

I have the following inequality: $$4p^2-||sa+qb||^2>0$$ where $p,s,q$ are real scalars and $a,b$ are real vectors. I know that $||a||\le F_1$ and $||b||\le F_2$. I want to express the inequality ...
1
vote
0answers
6 views

The largest product of two n-digit numbers which is palindrome

Project Euler: 4 is stated as follows: Largest palindrome product A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 ...
0
votes
1answer
18 views

number of function $f$ from $f:\mathbb{A}\rightarrow \mathbb{A}$ and satisfying $f(f(x))=x$

Let $A=\{1,2,3,4\}\;,$ Then total number of function $f$ from $f:\mathbb{A}\rightarrow \mathbb{A}$ and satisfying $f(f(x))=x$ $\bf{My\; Try::}$ If $f(x)=x\;,$ Then $f(f(x))=x.$ So there are ...
0
votes
0answers
6 views

Guessing mathematical probabilities by tests

I'm stuck with a (maybe simple) problem. I have 4 values possible for a test, and I can do as much test as I want. What is the minimum number of tests required to be at least at 95% sure it is the ...
0
votes
0answers
10 views

Find a recurrence relation and the Fourier-Legendre Series

Rodrique's Formula for the $n$th Legendre Polynomial is $$P_n\left(x\right)=\dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}\left(\left(x^2-1\right)^n\right)$$ The Fourier-Legendre series of a function f is ...
0
votes
0answers
4 views

Evaluation sum indexed by non decreasing sequences

During solving a problem from probability theory, I've met the following sum to evaluate: $$p_n(N) = \frac{1}{N!}\sum_{0\leqslant k_1\leqslant\ldots\leqslant k_n\leqslant N}\frac{k_1\cdot\ldots\cdot ...
1
vote
1answer
7 views

Infimum inequality comparing restrictions.

Suppose $f$ is a continuous function on the real line. Say we have two collections of sets $\{A_k\}_{k=1}^{n}$ and $\{B_k\}_{k=1}^{m}$, where $n>m$ and \begin{align} \bigcap_{k=1}^{n} A_k &= ...
0
votes
0answers
2 views

About properties of duality set

Let $X$ be a Banach space. For every $x\in X,$ the non-empty dual duality set $\mathcal{J}(x)$ is defined as:$$\mathcal{J}(x):= \left\{j(x) \in X': \langle x, j(x)\rangle = \|x\|^{2} = \|j(x)\|^{2} ...
2
votes
0answers
11 views

Count integer squares coordinates

Let $n$ be given an natural number. We want to find the number of squares which have corners with integer coordinates between $0$ and $n$. For example $n=1$, there is only one square; $n=2$ there are ...
0
votes
0answers
6 views

Intersection of normed speces and desity

Let $(X_n, \|\cdot\|_n)$ be a sequence of normed spaces. My first question is, whether it is possible to norm $X= \cap_n X_n$. My idea would be to take $\|\cdot\|_X = \sup \|\cdot\|_n$ if it is ...
0
votes
0answers
10 views

solving equation invloving both algebraic and trigonometric terms

x*sin(3)+3*sin(x)-x*log(3)+3*log(x)=10 I need to know a method that finds all the possible values of x that satisfy the above equation.
1
vote
0answers
6 views

How should I calculate the MLE based on a random sample from $PAR(\theta,2)$

Consider a random sample of size $n$ from a Pareto distribution, $X_i \sim PAR(\theta, \kappa =2)$. I have to compute the MLE, $\hat \theta$, to three decimale places. So I started doing the ...
0
votes
0answers
7 views

Is there an invariant that completely classify all knots?

Is there a set of invariants which completely classify all knots? And such that every object of this set represents a knot?
1
vote
2answers
11 views

sum of all positive integral values of $a\;,$ for which equation $\lfloor x \rfloor ^3+x-a=0$ has solution

The sum of all positive integral values of $a\;,$ Where $a\in \left[1,1500\right]$ for which the equation $\lfloor x \rfloor ^3+x-a=0$ has solution, Where $\lfloor x \rfloor $ Represent floor ...
0
votes
0answers
4 views

How do I determine value of a single attribute from a cluster of attributes?

If I have price and demand for a cluster of attributes, how can I dissemble the cluster to logically determine the driving factor to in turn say this $Attribute_1$ has a value of 1, $Attribute_2$ has ...
0
votes
2answers
12 views

Find intersection point of two straight lines

I want to find the intersection point of two lines where, one of the lines is parallel to y axis. I know we can find the intersection point of two line by solving the equation y=m(x-Px)+Py where m is ...
0
votes
0answers
6 views

Consistency of ZFC + “for every function there exists a class inaccessible to it”

Is ZFC + the following statement consistent (and if so, is it equiconsistent to some known large cardinal): For every function $f:ORD \rightarrow ORD$ such that: $f(\alpha)\geq \alpha$, $\alpha ...
0
votes
0answers
4 views

Computing $PAQ = LU$ using Gaussian elimination with complete pivoting

Suppose $PAQ = LU$ is computed via Gaussian elimination with complete pivoting. Show that there is no element in $e_i^{T}U$ i.e., row $i$ of $U$, whose magnitude is larger than $|\mu_{ii}| = ...
0
votes
0answers
4 views

Does marginalizing on a Bayesian network preserve its original independence assumptions?

I know that marginalizing over a Bayesian network causes changes to the graph (e.g. marginalizing node $c$ in the V-structure given by $a \rightarrow c \leftarrow b$ results in $a$ and $b$ being ...
0
votes
0answers
7 views

Computing homology group using Mayer-Vietoris sequence

Suppose I am given an exact sequence: $$0\to G\xrightarrow{f} \mathbb{Z} \xrightarrow{g} \mathbb{Z} \xrightarrow{h} H\to 0 $$ where the first $\mathbb{Z}=H_3(A\cup B)$ and the second ...
0
votes
2answers
19 views

Finding the points on a curve, closest to a specific point

Find the point(s) on the curve $y^3=x^2$ closest to the point $P=(0,4).$ I understand that there is a way to solve this, using the distance formula, however this turns out to seem rather complicated. ...
0
votes
0answers
7 views

ODE shooting method

I have the following exercise problem: (Gewöhnliche Differentialgleichungen und dynamische Systeme by Mathias Wilke,Jan W. Prüss Page 82,8) Look at the following boundary value problem: $$x''(t) + ...
1
vote
0answers
6 views

Probability - Finding the support of a joint transformation

$$ f(x,y) = \left\{ \begin{array}{ll} 12xy(1-y) & \quad 0< x < 1, 0<y<1 \\ 0 & \quad elsewhere \end{array} \right. $$ Let $Z = XY^2$ ...
1
vote
2answers
22 views

How many ways are there for W women and M men to sit on N chairs, if no man can sit next to woman?

So, we have: W - count of women M - count of men N - count of chairs standing in a row (N > M + W) Each person sits on her chair, and only two men or two ...
0
votes
2answers
16 views

A group of order pq with a single subgroup of order p

Given a group $G$ of order $pq$ (such that $p,q$ are primes and $p < q$) that have a single subgroup of order p (named $H$) prove that $G$ is cyclic
0
votes
2answers
13 views

Prove that $aRb$ if $a = 2^kb$ is an equivalence relation.

Let $R$ be a relation on the set of integers given by $aRb$ if $a = 2^kb$, for some integer $k$. show that $R$ is an equivalence relation. I don't understand how it will be equivalence. Is it the ...
0
votes
1answer
11 views

Can you multiply a matrix out of another one?

This is actually from a computer graphics problem. I calculate a transformation matrix by multiplying a few other ones. ...
0
votes
0answers
6 views

Lagrange interpolation, constant term of polynomial

I have a question about how to use the lagrange interpolation, $2x^3+8x^2+4$ The question is: What is the constant term of the $f \in Z_{13}[x]$ polynomial with degree at most 3, if f(1)=2, f(2)=3, ...
0
votes
0answers
9 views

Concerning the notation $\chi (U)$ in one of the hipotesis for some propierties of curl and divergence

I have the following excercise: Let $U \subset \mathbb{R}^3$ be open, $X \in \chi (U)$ and $f \in C^{\infty}(U)$, prove the following: $$curl(\nabla f)=0 \\ div(curl(X))=0 \\ curl(f(X))= ...
1
vote
0answers
8 views

Find the conditional expectation $E[X=k|Y=n]$

Suppose X and Y are stochastic variables with a simultaneous distribution $$P(X=k, Y=n) = \frac{e^{-1}2^{n-k}}{3^nk!(n-k)!},\ \ \ \ \text{for 0 $\leq$ k $\leq$ n and n $\geq$ 0 }$$ Determine the ...
0
votes
1answer
7 views

Low torsion in orientable manifolds?

The final sentence on page 170 of Stillwell's Classical Topology an Combinatorial Group Theory is: Poincaré justified the term "torsion" by showing that $(m-1)$-dimensional torsion is present only ...
0
votes
0answers
13 views

Why is the fundamental period $T_0$ of the complex exponential $e^{i\omega_0t}$, $T_0 = \frac{2 \pi}{|\omega_0|}$?

Assuming that $\omega_0 \in \mathbb{C}$. I realize that in order for $e^{i\omega_0t}$ to be perioric, it must be true that $e^{i\omega_0(t + T)} = e^{i\omega_0t}$ for some $T$ and for all $t$, from ...
0
votes
1answer
8 views

Proof of a theorem regarding group homomorphisms and kernels

I am looking for a proof of the following theorem: "Let $H<G$ then $H\unlhd G$ $\iff$ there exists a group $K$ and a group homomorphism: $\phi : G \rightarrow K$ such that $ker(\phi) = H$ There ...
0
votes
0answers
8 views

Probability of moving counters into bags, using factorials.

Bag P and bag Q each contain n counters, where n > 2. The counters are identical in shape and size, but coloured either black or white. First, k counters (0 < k < n) are drawn at random from bag ...
0
votes
0answers
6 views

How can I find the probability of choosing a club AND then a four from a standard deck, without replacement.

Please help me find a way to determine the probability of choosing first a club, then a four, from a standard deck of cards, without replacement.
0
votes
0answers
5 views

Gradient and critic points of a open set

Let $B = B(0;\delta)\in \mathbb{R}^{n} $ and $f: B \rightarrow \mathbb{R}$ $f$ is diferentiable. Let $a \in B$ Prove that if $f(x) \leq f(a)$ $\forall x \in B$ then $\triangledown f(a) = 0$. Well, i ...
0
votes
1answer
25 views

Express $\sin^3x$ in terms of cosines of multiples of $x$

I am studying complex numbers, and I have been trying to figure that out. Just not getting it. I keep getting $\frac{1}{-i8 (2\cos(3x) - 2\cos(x) - i4\sin(x))}$.
0
votes
0answers
3 views

Any duality theorem for digraphs between cut space and cycle space?

Acyclic directed graphs (DAGs) have no cycles and DAGs are always weakly connected digraphs. There are no directed cycles in DAGs by their definition Directed acyclic graph (DAG) is a digraph ...
2
votes
0answers
17 views

$P(x)=a_nx^n+…+a_1x+a_0 -$ polynomial with integer coefficients. $P(1)=P(2)=0 \Rightarrow \exists a_i<-1$

Let $P (x) -$ polynomial with integer coefficients. It is known that the numbers $1$ and $2$ are its roots. Prove that there exists a factor that is less than $-1$. My wor so far: Let ...
0
votes
0answers
11 views

Integral computation with Mathematica and Sympy differ

To compute the integral: $I = \int_{0}^{+oo} ue^{Au^{2}+Bu}du$ where $A<0$ and $B>0$ I have tried both Mathematica and Sympy but they yield different results: Mathematica yields: $ I = ...
0
votes
0answers
7 views

String Isomorphism : Detect Bijection between two sets of Permutations

Summary: We label/color a string in two different ways. Coloring/labeling preserves the structure of the string. For each, coloring we obtain a set of permutations from oracle . Since ...
1
vote
1answer
18 views

Let $T: V \rightarrow V$ be a linear map, where $nullity(T) = dim(V) - 1$. Prove there is a $\lambda$ such that $T^{2}(v) = \lambda T(v)$.

Let $T: V \rightarrow V$ be a linear map, where $nullity(T) = dim(V) - 1$. Let $w$ be a vector from the image of $T$. If $T(w) \neq 0$, prove there is a non-zero number $\lambda$ such that $T^{2}(w) ...
0
votes
0answers
4 views

How is the following integral related to confluent hypergeometric functions?

I am solving an integral that appears in a physics paper. $$ -\int_0^{\infty}dt\,\frac{e^{-t}}{t}\bigg[\bigg(1+\frac{3}{N}t\bigg)^N-1\bigg] $$ The paper does not give the full solution, it only gives ...
2
votes
2answers
20 views

Difference of subsets of integers with $A-A=2 \mathbb{Z}\setminus \{-2k,2k\}$

Is there any subset $A$ of integers such that $A-A= 2\mathbb{Z}\setminus \{-2k,2k\}$, for some integer $k$? ($A-A=\{a_1-a_2: a_1,a_2\in A\}$, and $2\mathbb{Z}$ is the set of even integers.)
0
votes
0answers
14 views

How do I solve the Principle component Analysis

So i am given the Principal Component Analysis: Y = $$(X − 1x^T )G$$ where X (n × p) is the data matrix, 1 is a vector of length n consisting of ones, and G is the orthogonal matrix containing the ...
0
votes
2answers
30 views

For $f:M\to N$ to be continuous its sufficient that $x_n\to a\implies f(x_n)_n$ is convergent in N

In order to prove: For $f:M\to N$ to be continuous its sufficient that $x_n\to a\implies f(x_n)_n$ is convergent in N I'm supposing that $x_n$ is convergent, that is: $$\forall \epsilon>0, ...
0
votes
0answers
15 views

Is this proof regarding product of connected spaces correct?

Let $X,Y$ be connected spaces, and consider their product $X\times Y$. I want to show that their product is connected. The posts I've read here regarding this question often include creating "slices" ...
2
votes
0answers
10 views

$X_t$ measurable wrt $\sigma$-algebra and “revaled information”

Studying stochastic processes, it is mentioned that if $(X)_t$ is a process and $(\mathcal{X})_t$ a filtration, then if the process is adapted to the filtration, the informal way to think about it is ...

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