0
votes
0answers
2 views

Method of successive approximations to solve y'=y^2

(a) Show that all the successive approximations for the problem y'=y^2, y(0) = l, exist for all real x. (b) Find a solution of the initial value problem in (a). On what interval does it exist? ...
0
votes
0answers
3 views

Is okay to have different solution to differential equation?

Suppose I have the following differential equation: $ydx - xdy - dx = 0$ Now, I could divide it by Integrating factor $x^2$ to get: $(xdy - ydx)/(x^2) - dx/x^2 = 0$ Use the inspection rule to get: ...
1
vote
0answers
10 views

Probability that a natural number is a sum of two squares?

Some natural numbers can be expressed as a sum of two squares: $$2=1^2+1^2$$ $$25=3^2+4^2$$ $$50=7^2+1^2$$ If one chooses a random natural number, what would be the probability that that number is a ...
0
votes
0answers
5 views

The supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ when $x+y=2n$ for some fixed $n\in \mathbb N $

Let $S$ be the set of all tuples $(x,y)$ such that $x+y=2n$ for a fixed $n\in \mathbb N$. Then what is the supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ $?$ I substituted $y=2n-x$ ...
0
votes
0answers
4 views

Cohomology space

Let M be a compact Riemannian manifold without boundary. a) If M is a sphere, prove that the cohomology space of order 1 is trivial $H ^{ 1} (M, R) = 0$ b) If $\omega = \delta\theta$ is the ...
0
votes
1answer
8 views

The Sum of the first five terms of an arithmetic sequence is 65/2…

The sum of the first five terms of an arithmetic sequence is 65/2. Also, five times the seventh terms is the same as six times the second term. Find the first term and the common difference of the ...
-3
votes
0answers
7 views

numerical analize ,regula falsi

A nice method to find an approximate solution is to successively cut intervals in half, as follows: let's first rewrite this as $$f(x) = 3x + \sin x - e^x = 0$$ Now pick two values, $a$ and $b$, such ...
0
votes
0answers
4 views

Approximaing Gamma function

For $c>1$ and $0<\theta<1$, we wish to approximate (upper bound) following Gamma function: $$\int_c^{c\theta}x^{-3}e^{-x}dx $$
0
votes
0answers
7 views

Altering a model to account for another variable

A question in my textbook asked me to find a general equation for depth fallen by an object of mass $75kg$ thrown from a bridge whilst tied to an elastic rope. Below the bridge there is a stream of ...
0
votes
1answer
10 views

Security of such cryptosystem design?

Is one able to reveal $m$ when $$С = (m + r)^e \bmod N$$ $C$ is known $r$ is known $e$ is known $N$ is known and not prime
0
votes
0answers
8 views

Prove that $\sin\theta_1.\sin\theta_2.\sin\theta_3=\frac{r^2_1}{16R^2}$

If $2\theta_1,2\theta_2,2\theta_3$ are the angles subtended by the circle escribed to the side $a$(opposite to vertex $A$) of a triangle at the centers of the inscribed triangle and the other two ...
1
vote
0answers
12 views

A question about finitely generated projective modules

Let $A$ be a commutative ring with unity and let $P$ be a finitely generated projective $A$-module. For $any$ $A$-module $M$, how does one show that $\operatorname{Hom}_A(P,A) \otimes_A M \simeq ...
1
vote
1answer
18 views

How do we call a pair of sets $A,B$ such that there is some injection $f: A \to B$?

Let $A,B$ be sets and let $f: A \to B$. If $f$ is a surjection, then we may simply write $f(A) = B$ or say in a more laborious way that $f$ maps $A$ onto $B$, to mean the same thing. However, if $f$ ...
0
votes
1answer
10 views

Generalizing results about limits of sums, products, and quotients of sequences to abstract topological spaces?

Introductory real analysis books usually state some properties about limits of sequences of numbers. Suppose $\lim x_n=x$ and $\lim y_n=y$. Then: $\lim (x_n+y_n) = x+y$ $\lim c x_n = cx$, for $c \in ...
0
votes
1answer
10 views

Statics Find the range value of P

Answer is $29.3 N ≤ P ≤ 109.3 N$ I tried solving it for quite some time already which I don't understand why I don't get the values. Can someone help me? $Fv$ Vertical force $Fh$ Horizontal force ...
0
votes
1answer
11 views

Give a good reason to define a function from A to B as a triple (F, A, B) rather than a functional set of pairs with domain A and image included in B.

The operative part of this question is "good reason": either an example or an argument, without preconceptions or fallacies. The object is comparing two definitions for "a function f from A to B", ...
2
votes
1answer
29 views

$A^k = I$ implies diagonalizable? [duplicate]

If $A$ is a square complex matrix with $A^k = I$ (where $I$ is the identity matrix of the same size as $A$) for some positive integer $k$, does it follow that $A$ is diagonalizable?
0
votes
0answers
6 views

Kiselev's Book I Plainimetry Question 242 - Question in the Description

Two lines passing through a point Μ are tangent to a circle at the points A and B. Through a point С taken on the smaller of the arcs AB, a third tangent is drawn up to its intersection points ...
3
votes
1answer
14 views

$\mathbb{C}$-algebra automorphism of $M_n(\mathbb{C})$ has form $X \mapsto AXA^{-1}$.

As the title suggests, what is the easiest way to see that any $\mathbb{C}$-algebra automorphism of $M_N(\mathbb{C})$ has the form $X \mapsto AXA^{-1}$ for some fixed $A \in GL_n(\mathbb{C})$?
0
votes
0answers
8 views

Definition of prime model extension over a set

Standard definition of prime model over a set is that: $M\vDash T$ is said to be a prime model extension of a set $A$ if $A\subset M$ and any partial elementary map $A\rightarrow N$ ($N\vDash$) extend ...
0
votes
0answers
5 views

$ G $ is $ p $-supersolvable group . $ Q \in Syl_{2}(G^{\prime}) $. Show $ Q \unlhd G $.

Let $ G $ is a finite $ p $-supersolvable group for odd prime numbers . Suppose $ Q \in Syl_{2}(G^{\prime}) $. Now i'll show $ Q \unlhd G $. Since $ G $ is $ p $-supersolvable, then $ G^{\prime} $ ...
1
vote
0answers
14 views

Finding cubed roots of complex number

Is this correct? $a^3 =r^3e^{i3\theta}= 5\sqrt{5}e^{i\arctan(11/2)}$ $$\implies r=\sqrt{5}, 3\theta = \arctan(11/2)+2\pi n,n\in\Bbb Z$$ $$\theta = \frac{\arctan(11/2)+2\pi n}{3}$$ $$\theta = ...
0
votes
2answers
13 views

$4\sin^2\frac{\theta}{2}.S=(n+1)\sin n\theta-n \sin (n+1)\theta$, and $4\sin^2\frac{\theta}{2}.C=-1+(n+1)\cos n\theta-n \cos (n+1)\theta$

If $S\equiv \sin\theta+2\sin2\theta+3\sin3\theta+......+n\sin n\theta$ and $C\equiv \cos\theta+2\cos2\theta+3\cos3\theta+......+n\cos n\theta$,prove that $4\sin^2\frac{\theta}{2}.S=(n+1)\sin ...
-1
votes
0answers
9 views

Fulton Algebraic Curves, Exercise 2.32(a).

Let $R$ be a DVR satisfying the conditions of Problem 2.30. Any $z \in R$ then determines a power series $\lambda_i X^i$. If $\lambda_0, \lambda_1, \dots$ are determined as in Problem 2.30(b). Show ...
0
votes
1answer
10 views

Number of partitions containing $k$ occurrences of a given number

Consider the ordered partitions of $N$ with size $m$ ($m \leq N$), that is, the set $\mathcal{P}_m^N$ of all vectors $\vec{n} \in \mathbb{N}^m$ such that $\sum_{i=1}^m n_i = N$. In how many of these ...
0
votes
1answer
18 views

Prove $2a^Tb \leq \|a \|^2 + \|b\|_*^2$ with dual norm

We know we can easily (**) prove $$ 2a^Tb \leq \|a \|_2^2 + \|b\|_2^2, \forall a,b $$ Is there a way to prove the following: $$ 2a^Tb \leq \|a \|^2 + \|b\|_*^2, \forall a,b $$ where ...
0
votes
2answers
25 views

Dirichlet's Test in convergence

Can Dirichlet's test be applied to establish the convergence of 1-1/2-1/3+1/4+1/5+1/6-..... where the number of signs increase by one in each block? Can somebody give me some ideas or hints?
0
votes
0answers
6 views

Three party simple economic system, how to maximize production

Say we're given three economic entities, $A,B,C$, capable of producing two goods, $\mathbb{1}, \mathbb{2}$. Say, given a unit time $\mathscr{U}$, $A$ can produce $a_1$ of $\mathbb{1}$, $B$ can produce ...
0
votes
2answers
46 views

Is there a bijective function $f: \mathbb{R}-\mathbb{Q} \to \mathbb{Q}$?

Is there a bijective function $f: \mathbb{R}-\mathbb{Q} \to \mathbb{Q}$? If it exists, then example. If not, then proof?
0
votes
1answer
14 views

Expected number of customers sitting on correct places

In a shop customers are given a seat number before entering the shop in the order 1,2,3,...,n but after entering the shop they sit in a random order not related to their seat number. what is the ...
0
votes
0answers
7 views

Does this method show that the projections $K$ and $L$ of an enumeration are primitive recursive?

In the fifth edition of Boolos et al's Computability and Logic, Exercise 6.5 asks the following (modified to provide background definitions): Define $K(n)$ and $L(n)$ to be the first and second ...
0
votes
0answers
10 views

norm on $\mathcal{B}(H)$

Given a Hilbert space $H$ and $a$ be a real numbers $\geq‎‎‎ 1$ , let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is $$ \Vert T ...
0
votes
0answers
17 views

Strongly Equivalent metrics

How to show any two metrics to be strongly equivalent? Please suggest me the proper way to show this. Also i want to know how to find the constants in the respective definition.
0
votes
0answers
12 views

Find a smooth path along which a given function on the plane is not differentiable at the origin

From Bamberg & Sternberg’s A Course In Mathematics For Students of Physics, Exercise 6.1d: Let $F(x,y) = \frac{x^3y}{x^2+y^2}$ for $(x,y) \neq (0,0)$ and $F(0,0)=0$. Invent a smooth curve ...
0
votes
2answers
31 views

Finding eigenvalues and eigenvectors of $2 \times 2$ matix

I having a few issues finding the eigenvectors for the following matrix: $$ \begin{bmatrix} -1 & -1\\ 0 & -2 \\ \end{bmatrix}$$ I calculated the eigenvalues to be ...
0
votes
1answer
22 views

Geometric Progression sums and sums of squares

Sum of the first $4$ terms in GP is $30$ and the sum of their squares is $340$. Find the numbers. How do I solve this?
1
vote
1answer
25 views

Trying to Understand a Remark about Zariski Topology

I'm reading some notes in which following remark is given: The Zariski topology is quite different from the usual ones. For example, on affine space $ \mathbb A^n$ a closed subset that is not equal ...
4
votes
1answer
20 views

Showing a subset of $S_n$ is a subgroup

Let $P$ be the set of all the elements of $S_n$ which can be written as $\sigma\mu\sigma^{-1}\mu^{-1}$ for $\sigma, \mu \in S_n$. Show this is a subgroup. This doesnt seem to be as simple as ...
0
votes
2answers
7 views

how Find the probability that the committee will consist of the following all dentists

A committee of four people is to be formed from six doctors and eight dentists. Find the probability that the committee will consist of only dentists
0
votes
2answers
11 views

Ring homomorphism from field

If we have homomorphism from field K to ring R, does that mean that we have ring homomorphism but K is a field? I have trouble understanding this. Thank You very much for your help.
0
votes
0answers
9 views

If G is finite p-group then $d(G)=d(\frac{G}{\Omega_1(Z(G))})$

Let $G$ be a finite p-group such that $G$ has no non-inner automorphism of order p leaving Φ(G) elementwise fixed If $\Omega_1(Z(G))\le G'\le \Phi(G)$ how we can get $d(G)=d(\frac{G}{\Omega_1(Z(G))})$ ...
1
vote
1answer
32 views

What is the geometric meaning of the inner product of two functions?

When it comes to inner product I have thus far only dealt with vectors, and so the concept is very intuitive because one can easily visualize two vectors and how they get multiplied, and it is clear ...
1
vote
0answers
14 views

Quotient Space $X^*$ is homeomorphic to the Subspace $S^2$ of $\mathbb R^3$

Let $X$ be the closed unit ball $\{ x^2 + y^2 \leq 1 \}$ in $\mathbb R^2$ and let $X^*$ be the partition of $X$ consisitingof all the one point set $\{ x \times y \}$ for which $x^2 + y^2 < ...
1
vote
0answers
9 views

Tangent to dual curve.

Let $C$ be a smooth projective plane curve, let $P \in C(k)$, and let $\ell$ denote the tangent line to $C$ at $P$. Let $C^*$ denote the dual curve to $C$, in the dual plane $(\mathbb{P}^2)^*$ (the ...
0
votes
0answers
8 views

Inverse of the sum of a symmetric positive definite matrix and a diagonal (but with different entries) matrix

Suppose we have symmetric positive definite $A$ with the size of $d \times d$, giving the SVD $A=V\Sigma U^T$ , if $D$ is an identity matrix, ie $D=I$, then $(A^T A + \gamma I)^{-1}=U (\Sigma^2 + ...
0
votes
0answers
12 views

A billinear transformation

Let $$w(z)=(az+b)/(cz+d)$$.Then $w(z)$ maps a straight line of $z-$plane to the circle $|w|=1$ in $w-$plane if $1.|b|=|d|$ $2. |a|=|c|$ $3. |a|=|d|$ $4. |b|=|c|$ My work: I started by considering ...
0
votes
0answers
19 views

Chance of wining Uno 7 Times in a row

I am Intrigued to determine the odds of winning such a game with 4 players.. and in 7 times consecutively Would be great to have a clear answer on this Thks Phil
3
votes
1answer
21 views

Minimum value function

It's just a very simple question, is there a function defined and that tells you the minumum and maximum value of a list of variables, like: min(4, 3) = 3 min(2, 19) = 2 max(1, 10, 3) = 10 Is that the ...
1
vote
0answers
2 views

Lebesgue-integrability of piecewise function with random variable

This function is Lebesgue-integrable:$$\chi(x)= \left\{ \begin{array}{ll} 1 & \text{if}~x~\text{is rational}\\ 0 & \text{if}~x~\text{is irrational}. \end{array} ...
0
votes
0answers
5 views

Prove of negative transitive relation

Prove that a binary relation R on X is negatively transitive if and only if for each x, z∈X, xRz implies that ∀y∈X, xRy or yRz.

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