0
votes
0answers
2 views

Sum of skew symmetric and symmetric parts of tensors

Denoting the skew-symmetrisation and symmetrisation of a $(0,p)$-tensor $X_{a_1 \ldots a_p}$ by $X_{[a_1 \ldots a_p]}$ and $X_{(a_1 \ldots a_p)}$ respectively, is it always true that $X_{a_1 \ldots ...
0
votes
0answers
3 views

Is it true that solving a triangular system using forward or backward substitution numerically stable?

The system is $TX = B$, where $T$ is a triangular matrix, $X$ is a unknown matrix, and $B$ is the RHS matrix. I know the system $Tx = b$ is backward stable where $b$ is a RHS vector. Detail check ...
0
votes
0answers
6 views

Probability formula for 6 die throw

Suppose you roll a fair 6-sided dice four times. Let C be the event that the value of each roll is at least as large as the value of the previous roll. What is the probability of C? I know that ...
0
votes
0answers
8 views

Function from $\mathbb{R}^9$ to $\mathbb{R}^6$ with zero set the orthogonal $3\times 3$ matrices

I am trying to construct a $C^\infty$ function from $\mathbb{R}^9$ to $\mathbb{R}^6$ with zero set the orthogonal $3\times 3$ matrices. I am thinking about mapping $M$ to $MM^T-I$, but am not sure ...
0
votes
0answers
5 views

Formula on the basis of discrete data

I have a table of numbers for fixed values of 4 parameters $x, y, z, t$, at this $x$ belongs to finite set of natural numbers, $y\in\{1;2\}$, $z\in\{5;10;15;20;25\}$ and $t\in\{1,2,3\}$. Is there a ...
0
votes
0answers
10 views

alternating binomial sums

So we know that $\sum_{i=0}^t\binom{m}{i}\binom{n-m}{t-i}=\binom{n+m}{t}$ by a simple counting argument. Now is there any bound on the quantity $\sum_{i=0}^t(-1)^i\binom{m}{i}\binom{n-m}{t-i}$? Can ...
0
votes
0answers
8 views

Find the equation of the line which is

Find the equation of the line perpendicular to the line joining the points $A(3,6)$ and $B(-6,9)$, which divides the line $AB$ in the ratio of $2:1$. My attempt: Equation of $AB$ is ...
0
votes
0answers
11 views

Maximum hyperrectangle

Is there a way to determine the coordinates of the maximum hyper-rectangle in n-D space subject to linear constraints and $x_i\ge0$ ? Example: Argument Maximum of $x_1 x_2 x_3$ Given ...
-1
votes
0answers
7 views

Given the normal versor $n$ to a curve determine its curvature and torsion.

Given a curve $\alpha: I \rightarrow \mathbb{R^3} $ and its normal versor $n(s)$ who is known $ \forall s \in I$ and given the Frenet relations $$t'=kn$$ $$n'=-kt-\tau b$$ $$b'=\tau n$$ where $t$ is ...
0
votes
0answers
9 views

Sets: Would it be true if one could test for the absence of rights as well as for the presence of rights?

This is a computer security question I have, however, it applies more to sets in math (which I don't understand). Suppose two subjects s1 and s2 are created and the rights in A[s1, o1] and A[s2, o2] ...
0
votes
0answers
17 views

Determine supremum and infimum of $\{x \in \Bbb R\,:\, x < 3/x\}$

Set $S := \{x \in \Bbb R\,:\, x < 3/x\}$. (a) Determine whether $\sup S$ exists, and determine its value if it exists. Justify your answer. (b) Determine whether $\inf S$ exists, and determine ...
0
votes
1answer
20 views

Is the condition $\;P(X^2>1|X>0)\;$ the same as $\;P(X>1)\;$?

I saw two examples for this question the condition $\;X>0\;$ means that $\;X^2>1\;$ is true only when $\;X>1\;$, and the probability is $\;P(X^2>1|X>0)=P(X>1)\;$ But I also saw the ...
0
votes
1answer
6 views

example of a graph such that K(G)=δ(G)=Δ(G)E(G)

example of a graph such that K(G)=δ(G)=Δ(G), where K(G) is the number of components,δ(G) is the minimum degree of G and Δ(G) is the maximum degree in G.
1
vote
0answers
6 views

Is composition of regular epimorphisms always regular?

In a finitely complete and cocomplete category. Does it always hold that the composition of two regular epimorphisms is regular? And if it's not the case, what kind of additional constraints can make ...
-4
votes
1answer
12 views

$F$ need not be normal over $K$ [on hold]

Prove that if $F$ is normal over an intermediate field $E$ and $E$ is normal over $K$, then $F$ need not be normal over $K$. No clue.
0
votes
0answers
26 views

Calculate $\int_0^1 \ \int_0^1 \ x \sin \lvert x^2-y^2 \lvert \; dx \; dy$

$$\int_0^1 \ \int_0^1 \ x \ \sin \lvert x^2-y^2 \lvert dx \ dy $$ $$\int_0^1 \frac{1}{2} \Big[ \sin \lvert x^2-y^2 \lvert \Big]_0^1 \ dy= \int_0^1 \frac{1}{2} \Big( \sin \lvert 1-y^2 \lvert - ...
2
votes
1answer
39 views

What is the derivative of $z^{-1}$ with respect to $\bar{z}$?

I asked a question here a few days ago but it wasn't answered and, as often happens with me, in trying to answer it myself I just confused myself out of understanding what I thought I knew. What is ...
0
votes
0answers
10 views

Calculating 5 different ranges for people resource management

I am working on a project for my company. My team is building a project charter template. In this template needs to be a drop down that estimates how many full-time employee days(FTE) will be ...
2
votes
3answers
25 views

A Compact Hausdorff Space with no Manifold Structure?

What is an example of a compact Hausdorff space that cannot be given the structure of a (i) differential manifold (ii) topological manifold?
0
votes
0answers
12 views

When do two subbases generate the same topology

Let $X$ be a set. If $\mathcal B_1$ and $\mathcal B_2$ are bases of subsets of $X$, it is well-known that $\mathcal B_1$ and $\mathcal B_2$ generate the same topology if and only if for any pair of ...
0
votes
1answer
28 views

To prove that $G$ is the group the condition is not necessary $\forall a,b, c \in G(ba=ca\to b=c)$.?

$1.$ Let $G$ be a finite semigroup such that $\forall a,b, c \in G(ab=ac\to b=c)$. Then $G$ is Group. ? I know the following result : If $G$ be a finite semigroup such that $\forall a,b, c \in ...
0
votes
0answers
8 views

Basics of complete ring

I would like to learn basics of complete rings. Let $A$ be a commutative ring with unit which has ideals $I_i$. Can $A$ be complete with respect to the $I_1$-adic topology but not with respect to ...
1
vote
0answers
20 views

Product of Factory A and Factory B in a bag

Factory $A$ produces $10\%$ defective items and $B$ produces $20\%$ defective items . A bag contains $4$ items from $A$ and $5$ from $B$. If $2$ items selected at random from bag find probability that ...
0
votes
2answers
17 views

how to find out the following statements are true or false?

Let $p(x)$ be an odd degree polynomial and let $q(x)=(p(x))^2+ 2p(x)-2$ a) The equation $q(x)=p(x)$ admits atleast two distinct real solutions. b) The equation $p(x)=0$ admits atleast two distinct ...
0
votes
0answers
19 views

how to scale a value from 0-1 to 0-5

I have these facts: the user (X) rates the item (I) by (4/5) the item (II) is 0.5 similar to item (I), 0.5 means 50% (the scale is from 0 to 1) then I can say (according to my business model) that ...
0
votes
0answers
5 views

Faces of a Bipyramid over a a Simplicial Polytope

Is there a simple way of expressing the number of faces of a bipyramid built over a polytope that is known to be simplicial, using the number of faces of the original polytope? This seems an easy ...
-3
votes
0answers
44 views

Solve out $a,b$ from system of equations [on hold]

I encounter some question I don't know how to solve out $a,b$ from $$\begin{cases}(a^2-1)a=(b^2-1)b\\ (a^2-1)(3a^2+1)= (b^2-1)(3b^2+1)\end{cases}$$ Someone suggests me to edit the question to get ...
-2
votes
0answers
15 views

conditional expectation and equality (question)

Let X,Y random variable and f is the density of Y. $P(X<u)=E(P(X<u)|Y)=\int P(X<u|Y=x)f(x)dx$ Is it true? Thank you
0
votes
0answers
8 views

time series smoothing techniques

I am currently analyzing time series data relating to financial observations (revenue and number of deposits in particular). I would like to smoothen the data because a lot of outliers are present. ...
0
votes
0answers
27 views

Collective name for algebraic structures

I am doing a thesis about various algebraic structures, primarely about groups, rings and modules (with maybe hint of algebras). However always having type out ALL of them constantly gets very tedious ...
-2
votes
0answers
22 views

how to get length of side of right triangle

Please help in determining the length of the sides of the triangle marked [?].
0
votes
3answers
12 views

How can I find $y$ coordinate of a straight line at a specific $x$ value

Lets say I have a straight line between $p_1=(-2, -0.5)$ and $ p_2=(0.25, 0.5)$. How can I find the value of $y$ when $x=-1$? I have tried to solve this the whole day without finding an answer, I ...
4
votes
3answers
42 views

Evaluating the limit $\lim_{x \to 0}\left(x+e^{\frac{x}{3}}\right){}^{\!\frac{3}{x}}$

$$y=\left(x+e^{\frac{x}{3}}\right)^{\frac{3}{x}}$$ I'm looking at this equation, and need to solve for the limit as $ \to 0$. I've graphed it, and know the solution is $e^4$, but am clueless as ...
0
votes
0answers
17 views

Probability of balls of colors in two urns

In one urn A there is $2$ red balls and $3$ white balls. In another urn B there is $3$ red balls and $1$ white ball. $4$ balls are taken out and returned from urn A and $5$ balls from urn B. I could ...
0
votes
0answers
6 views

How can I find the stability of the equilibria of this vector field?

Consider the vector field given by $y' = y - y^{3}$. This clearly has equilibria at the points $y = 0, \; y = 1, \; y = -1$. How would I find the stability of these points though? I understand that I ...
1
vote
0answers
11 views

Combination solution

Given $N_{tot}=82$ where $N = [N_9 \: N_8 \:N_7 \:N_6 \:N_5 \:N_4 \:N_3 \:N_2 \:N_1 \:N_0]$ how many possible combinations are there if $N$ must be odd and $N_i < N_{i-1}$ i.e. one solution is $[1 ...
0
votes
1answer
25 views

P.I.D. and a nontrivial ideal, Quotient ring has finitely many ideals [on hold]

A ring $R$ is a P.I.D. Let $I$ be a nontrivial ideal in $R$. Prove that $R/I$ has finitely many ideals.
3
votes
5answers
39 views

About inner products, norms and metrics

Do these three kinds of vector spaces, those with an inner-product, those with a norm and those with a metric, are the same sets of vector spaces? At least for finite dimensional vector spaces all of ...
2
votes
2answers
21 views

Find the new variance

In a sample of size $21$ the sample mean is $58$ and the sample variance is $10.7$. If an observation of value $52$ is added to the sample, what now is the sample variance of the observations? I ...
0
votes
0answers
25 views

XOR binary matrix multiplication $AX=B$?

Let $A$, $B$, and $X$ be binary matrices, where $A$ and $B$ are of size $n \times m$ with $n > m $. $X$ is an $m \times m$ matrix. Compute $X$ such that $AX=B$. ps: $A$ is not a quadratic matrix. ...
1
vote
0answers
20 views

Matrix of rotation

I am haunted by a question. Consider a vector $v=\begin{bmatrix} a\\ b \\ c \end{bmatrix}$ is firstly multiplied by $R_1= \begin{bmatrix} \cos(\theta_1) &-\sin(\theta_1)&0\\ \sin(\theta_1) ...
0
votes
1answer
20 views

Why is $m(n)\approx\log_2(n)$?

Why is $m(n)\approx\log_2(n)$ ? If $m(n)=\inf\{m:2^{-m}m^{-3/2}\le\frac1n\}$, taking log of $m(n)$ I get $-m(n)-\frac32\log_2(m(n))\le-\log_2(n)$ (This appears in the solution of an exercise in ...
1
vote
0answers
15 views

What's stopping me from gluing cosheaves of spaces? (reality check)

Let $X$ be a locally ringed space and let $\mathcal{Z}: Open(X) \to \mathsf{Lrs}_{/X}$ be a cosheaf of spaces over $X$. Cosheaf means the dual to sheaf, i.e. the colimit of the value of $\mathcal{Z}$ ...
-4
votes
0answers
15 views

Probibility of an event occuring over multiple mutually exclusive draws

Consider a super bowl box. If you have 8 boxes out of 100 and there is a winner each of the 4 quarters. What are the odds of winning atleast once?
2
votes
1answer
22 views

$\sigma$-algebra produced by a subclass of a class.

im studying the book 'probability & measure' by Patrick Billingsley. in chapter 2 there's an exercise 2.9 say's: show that: If $B\in\sigma(A)$, then there exists a countable subclass $A_B$ of $A$ ...
4
votes
5answers
90 views

integrate $\int \frac{dx}{x\sqrt{1-x}}$

$$\int \frac{dx}{x\sqrt{1-x}}$$ $$\int \frac{dx}{x\sqrt{1-x}}$$ $u=1-x$ $du=-dx$ $$-\int \frac{du}{(1-u)\sqrt{u}}$$ $a(1-u)+b\sqrt{u}=1\Rightarrow a-au+b\sqrt{u}=1$ $a=1\Rightarrow ...
2
votes
0answers
8 views

Is multiplication of a correlated random variable and a independent random variable, an independent random variable

I have a random variable that is a multiplication of two random variables as bellow: $$A_n=B_n\times C_n$$ $B_n$s are identically distributed with zero mean and are correlated for different $n$s and ...
4
votes
2answers
58 views

Let $S = \{n\in\mathbb{N}\mid 133 \text{ divides } 3^n + 1\}$. Find three elements of S.

Question: Let $S = \{n\in\mathbb{N}\mid 133 \;\text{divides} \; 3^n + 1\}$ $a)$ Find three different elements of $S$. $b)$ Prove that $S$ is an infinite set. My intuition is find the prime factors of ...
0
votes
1answer
16 views

Canonical form and fundamental solution of pdf

Can someone help with these two PDE problems? Thank you. Reduce to Canonical form and find the fundamental solution if possible. $$y^2u_{xx} + x^2u_{yy} = 0.$$ What type of transformation should I ...
1
vote
0answers
8 views

Intersections of connected components of real curves

Let $C_1,C_2\subset \mathbb{P}^2(\mathbb{R})$ be real algebraic curves each of degree $d$. By Bezout's Theorem these curves have at most $d^2$ points of intercestion. Since we are in the real case, ...

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