1
vote
0answers
34 views

What can we say about the integral curve of a vector field on the warped product manifold?

Let $Z=(X,Y)$ be a vector field defined on the warped product $M×_{f}N$ where $f$ is defined on $M$. The integral curve of $X$ on $M$ is $\alpha$ and the integral curve of $Y$ on $N$ is $\beta$. I ...
3
votes
1answer
178 views

Find a differentiable $f$ such that $f'$ is not continuous. [duplicate]

I'm trying to solve this problem: Find a differentiable function $f:\mathbb{R} \longrightarrow \mathbb{R}$ such that $f':\mathbb{R} \longrightarrow \mathbb{R}$ is not continuous at any point of ...
0
votes
1answer
72 views

Help needed for mathematical formulation

I'm trying to write a simple mathematical formulated which expresses the following: Let F={f1,f2,...,fn} be an ordered set of flights, each f being associated with a begin time begin(f) and an end ...
2
votes
0answers
72 views

$ \lim_{n\to\infty} e^{-n}\sum_{k=1}^n \frac{n^k}{k!} $ [duplicate]

How can be evaluated this limit: $$ \lim_{n\to\infty} e^{-n}\sum_{k=1}^n \frac{n^k}{k!} .$$ Thank you.
0
votes
1answer
63 views

Newton Raphson Method: approximating root

How do we start from approximating a root using this technique? I know of two, viz - a table of x vs f(x), and see where f(x) changes sign - plot a graph, and see where the graph cuts the x axis But ...
0
votes
1answer
26 views

Two points of a vector

I have a source point of a vector (x, y), the vector's size, and the angle of it. What's the formula to calculate the X and Y values of the point the vector will get to from the source point? I tried: ...
5
votes
5answers
480 views

Trigonometry equation $\sin(x)+\cos(x)-\tan(x)=0.4$

There's some way to find $x$ here ? $$\sin(x)+\cos(x)-\tan(x)=0.4$$
3
votes
1answer
64 views

How to sketch $y = \sqrt{x-1} + \sqrt{6-x}$

How to sketch $y = \sqrt{x-1} + \sqrt{6-x}$ My solution: It does not have any roots Domain = [1,6] Increasing till 3.5 and then decreasing How to go on further? Please help.
3
votes
1answer
98 views

Preservation of separatedness of a scheme of finite type over a field by shrinking the base field

This is a generalization of this question. Let $k$ be a field. Let $k'$ be an extension field of $k$. Let $X$ be a $k$-scheme of finite type. Suppose $X\times_k k'$ is separated over $k'$. Is $X$ ...
3
votes
1answer
107 views

A generalization of Cayley-Bacharach Theorem

This is exercise 19.4.B on Ravi Vakil's notes. Let $C$ be a regular plane curve of degree $e>2$, and $D_1,D_2$ be two plane curves of same degree $d$ not containing $C$. By Bezout's theorem $D_i$ ...
2
votes
1answer
45 views

The dimension of space of morhisms as the number of orbits

All groups are finite, all representations are over $\mathbb{C}$ (just in this context, of course), $G$ is a group, $K,H\subset G$ - its subgroups. By $\mathbb{C}$ we denote the trivial representation ...
3
votes
0answers
118 views

The sum of Gaussian functions

Suppose there is a normal distribution and the Gaussian function is $F(x)=\exp(-c\|x-b\|^2)$ where $c$ is a constant and $x,b\in \mathbb{R}^N$, b means the mean value. ...
1
vote
2answers
134 views

Example of rings with idempotent and non-zero Jacobson radical

I am looking for a simple (as simple as possible) example of a commutative ring (with identity) $R$ such that its Jacobson radical is non-zero but idempotent. (The simplest example that I know is ...
1
vote
1answer
63 views

Generating function for lattice points in a sphere

This is a note in Sedgewick's Analytic Combinatorics: The number of lattice points with integer coordinates that belong to the closed ball of radius n in d-dimensional Euclidean space is ...
0
votes
2answers
148 views

Existence of d-regular graphs

It is well known that if $0<d<n$ and $d\cdot n$ is even then there exist $d$-regular graphs on $n$ vertices. My question is: What is the easiest way to prove this?
9
votes
2answers
190 views

Prove $1^{2007}+2^{2007}+\cdots+n^{2007}$ is not divisible by $n+2$

Prove that for any odd natural number $n$, the number $1^{2007}+2^{2007}+\cdots+n^{2007}$ is not divisible by $n+2$.
2
votes
0answers
36 views

What's the pair of numbers made using 1-9 which has maximum product? [duplicate]

The task is to find a pair of numbers whose digits are 1-9, no digit repeated such that their product is maximum possible. eg. 123 and 456789 is one such pair. Question: Please suggest a way to do ...
4
votes
4answers
319 views

Trouble solving $\int\sqrt{1-x^2} \, dx$

I am trying to learn how to solve integrals and I've got the hang out of a lot of examples, but I haven't got the slightest idea how to solve this example, this is how far I've got: $$ \int\sqrt{1 - ...
7
votes
5answers
193 views

Solving equations of type $x^{1/n}=\log_{n} x$

First, I'm a new person on this site, so please correct me if I'm asking the question in a wrong way. I thought I'm not a big fan of maths, but recently I've stumbled upon one interesting fact, which ...
3
votes
1answer
93 views

Show that $p$ is prime if the following limit property holds

Let $n$ be a positive integer. Show that $n$ is prime if and only if $$\lim_{r\to \infty}\lim_{s\to\infty} \lim_{t\to\infty} ...
1
vote
1answer
191 views

Basic question about my understanding of the Lyapunov equation

Consider the system $\dot{x}(t) =Ax(t)$ where $A \in \Bbb R^{n\times n}$. Now let $P$ be a symmetric matrix and define $V(x) = x^T Px$. Then $V(x)$ satisfies $$\frac{d}{dt}V(x) = -x^TQ x,$$ where $Q = ...
2
votes
1answer
69 views

Compute $E[X_1|Y]$

Let $X_1, \ldots ,X_n$ be i.i.d uniform random variables on $[0,1]$ and define $Y=max\{X_1, \ldots ,X_n\}$. Compute $E[X_1|Y]$. I can find the distribution of $Y$ but I'm stuck trying to find the ...
0
votes
1answer
35 views

Finding polynomials sattisfying $P\bigr(-c + K/(u+c)\bigl) (u+c)^2/K =P(u)$

Is there any simple way to find the polynomials satisfying the functional relation \begin{align*} P\left(-c + \frac{K}{u+c}\right) \frac{(u+c)^2}{K} = P(u) \tag{*} \end{align*} Where $K = ...
14
votes
5answers
526 views

Evaluate the limit $\lim\limits_{n \to \infty} \frac{1}{1+n^2} +\frac{2}{2+n^2}+ \ldots +\frac{n}{n+n^2}$

Evaluate the limit $$\lim_{n \to \infty} \dfrac{1}{1+n^2} +\dfrac{2}{2+n^2}+ \ldots+\dfrac{n}{n+n^2}$$ My approach : If I divide numerator and denominator by $n^2$ I get : $$\lim_{ n \to ...
2
votes
1answer
90 views

What does the logarithm of a hyperbolic line look like?

At a fixed point $p$ of hyperbolic $n$-space $H$, there is the exponential map from flat $n$-space to $H$ taking straight lines through the origin of the flat space to hyperbolic lines through $p$ in ...
0
votes
1answer
492 views

How many ways to find the center of an inscribed circle?

I want to find the coordinates of center of the inscribed circle triangle $ABC$, where $A(-274,-253)$, $B(-1,7)$, $C(14,7)$. I tried First way. We have $c = AB=377$, $a = BC=15$, $b = AC=388$. Let ...
13
votes
4answers
295 views

Coefficients of a polynomial also are the roots of the polynomial?

How many real solutions $(r_1, r_2, \cdots, r_n)$ are there such that $(r_1, r_2, \cdots, r_n)$ are the roots of the polynomials $x^{n} + r_1 x^{n-1} + r_2 x^{n-2} + \cdots + r_n$ For $n = 2, 3, 4$ I ...
1
vote
2answers
77 views

$f=u+iv$ is entire. for some $K>0$, $ u > 0$ for all $z \in \Omega = \{ z : |z| \geq K \}$. Prove that $f(z)$ is constant.

Let $f=u+iv$ be an entire function. Suppose for some $K>0$, we have that $ u > 0$ for all $z$ belonging to $\Omega = \{ z : |z| \geq K \}$. Prove that $f(z)$ is constant. I guess that I have to ...
0
votes
1answer
30 views

Are all substitution-closed sets schematic?

I assume everyone is familiar with propositional logic and its language. Let $S$ be a set of wffs of propositional logic. The set $S$ is said to be substitution-closed iff whenever $x$ is in $S$, ...
0
votes
1answer
90 views

quick guide to understand theory of computation

Can someone tell me some quick guides in understanding theory of computation. I know this is not the place to ask such question
1
vote
2answers
68 views

How to sketch $y = \frac1{\sqrt{x-1}}$

How to sketch $y = \frac1{\sqrt{x-1}}$ My way:(which does not work here) I normally solve these problems by squaring and converting them to equations of 2 degree curves(such as parabola, hyperbola, ...
0
votes
1answer
287 views

Find the volume $y=x^2 [0,2]$

Find the volume of the solid whose base is region bounded between the curve y=x^2 And x-axis [0,2] And whose cross section taken perpendicular the x-axis are squares My solution V= the integral ...
4
votes
2answers
274 views

Two series involving the Gamma function

The last piece I am left with in my proof is to compute the following two series: $$\sum_{i=1}^{n-1}\dfrac{\Gamma(i-d)\Gamma(n-i+d)}{\Gamma(i+1)\Gamma(n-i)(n-i-d)}, \sum_{i=1}^{n-1} ...
1
vote
2answers
193 views

Maclaurin Series confusion

Using the Maclaurin expansion formula: to find the Maclaurin series for $sin(3x)$, I can get the correct answer by using $x^n$ in the formula above (in the tail-end of the formula). Similarly, to ...
2
votes
1answer
44 views

Dependence on parameters in propositional logic

Warning: my background is mostly in probability and analysis, and not in logic. When reading or writing a complex proposition, with long chains of "for all... there exists... for all...", I tend to ...
1
vote
2answers
69 views

Figuring out whether a ring is a field

Given a ring, how do you test whether it is a field? What properties would you look at?
1
vote
2answers
72 views

Constructing equivalent matrices with rows and columns exchanged

I am trying to construct all inequivalent $8\times 8$ matrices (or $n\times n$ if you wish) with elements 0 or 1. The operation that gives equivalent matrices is the simultaneous exchange of the i and ...
0
votes
1answer
889 views

integral of Dirac delta function with sine

It i well known that the Dirac Delta Function has the following property $\int_{-\infty}^{\infty}\delta(t-a)f(t)dt=f(a)$ if $g(t)=\int_{0}^{t}\sin(t-\tau)\delta(\tau-\pi)d\tau$ then $g(t) = ...
0
votes
3answers
191 views

checking the diffeomorphism between 2 surfaces

I tried to show that a surface $x^4+y^2+z^2=1$ and the unit sphere are diffeomorphic. Since a diffeomorphism between them is not chosen easily, I would apply a theorem using invertibility of ...
2
votes
4answers
180 views

Divisibility by seven

Given number n, whose decimal representation contains digits only $1, 6, 8, 9$. Rearrange the digits in its decimal representation so that the resulting number will be divisible by 7. If number is m ...
0
votes
2answers
250 views

110 people divided into 3 groups of different sizes

I need to distribute 110 people to 3 groups of different sizes as follows: 50 people to group 1 40 people to group 2 20 people to group 3 Instead of simply filling each group up to the desired size ...
2
votes
2answers
361 views

Predicting what the graph of a function looks like?

How can you predict what the function $$f(x) = \frac{(x - 5)(x + 4)(x - 3)^2(x)}{(x-5)(x)}$$ looks like before you plot it?
2
votes
0answers
81 views

Clarification of the Jacobian

Well, that was cool if not tedious but I understand the Jacobian and its application to changing coordinate systems. $${J_{POLAR}= \rho}$$ $$ {J_{cyl}= \rho}$$ and $${J_{sphere}=\rho^2\sin\phi}$$ ...
1
vote
1answer
127 views

Denoise using wavelet transform

My mathematical class task is to de-noise a function using wavelet transform. I am to select a function $f(x)$ and noise function with zero-mean $n(x)$. I am to add noise like this: $$f_{noise}(x) = ...
2
votes
2answers
98 views

How many $T_1$ topologies can be constructed on a finite set with n elements?

The problem is about the total number of topologies on a finite set. How many $T_1$ topologies can be constructed on a finite set with n elements?
0
votes
1answer
145 views

Mathematics Debt problem

Imagine that there is a group of three friends: $A$, $B$ and $С$. $A$ owes $B\,$ $20$ rubles and $B$ owes $C\,$ $20$ rubles. The total sum of the debts is $40$ rubles. We can see that the debts are ...
0
votes
1answer
95 views

Differences between methods for solving linear equation system

I have a huge linear equation system in this form: F=K.Δ as usual form of problems in the finite element method, where the F vector and K are known and Δ vector is unknown. There are several ...
0
votes
2answers
127 views

continuous map from indiscrete space

Show that a continuous map from indiscrete space X to a $T_0$ space must be constant function. I was trying to prove it by contradiction. If the map f is not constant then there is two points x, y ...
1
vote
1answer
48 views

convergence, but not almost uniform convergence.

Can anyone help me with finding an example of function convergence, which is not an almost uniform convergence?
2
votes
2answers
98 views

About the construction of semidirect products

I need help with the following question: We have $G_0=H_0[N]_\rho$ a semidirect product, and $\pi:H\to H_0$ an epimorphism such that $K=\ker(\pi)\unlhd H$, and $H/K\cong H_0$. We have to construct a ...

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