0
votes
1answer
28 views

What are prime and primitive polynomials?

Please, I am not a mathematician so highly mathematical textbook language will not make sense, that is why I am forced to post this question here. I am reading about Checksum and CRC data integrity ...
-2
votes
0answers
19 views

Improper integral complex analysis

Show that $$\int_{-\infty}^{\infty}\frac{x\sin\left(qx\right)}{x^2+1}dx =\pi sgn(q)e^{-|q|},q \in \mathbb{R}$$ using a semi-circular contour in the upper half-plane.
2
votes
1answer
16 views

Complex numbers inequality $|a_nb_n+…+a_mb_m|≤a_n * \max_{n≤k≤m} |b_n+b_{n+1}+…b_k|$

$a_n,...,a_m \in R $, such that $a_n≥a_{n+1}≥...≥a_m≥0$ and any $b_n,...,b_m \in C$ How to conclude $$|a_nb_n+...+a_mb_m|≤a_n * \max_{n≤k≤m} |b_n+b_{n+1}+...b_k|$$ It is said in the book, that if the ...
1
vote
1answer
23 views

What is wrong with this line integral? (Line integral change of variables)

I have this line integral $$\oint_{\partial D}(f\nabla f)\cdot\hat{n}\,ds$$ which I would like to "change variables" so that the final result is in terms of a line integral of $g$ on $[0,2\pi]$ where ...
3
votes
0answers
33 views

Do $L^p$ spaces measure something natural?

Obviously much of analysis on $\mathbb R^n$ considers $L^p$ spaces or other Banach spaces derived from them. The definition of $L^p(\mathbb R^n)$ looks very natural, but I've been bothered for some ...
1
vote
0answers
15 views

Do two rational parametric curves intersect only finitely many times?

Suppose there are two rational parametric curves $f = (f_1, \ldots, f_n)$ and $g = (g_1, \ldots, g_n)$ in $\mathbb{R}^n$. I read somewhere that such a parametric expression can always be transformed ...
0
votes
0answers
16 views

What is the Code in mathematica for getting a graph with following condition? [on hold]

Let X=(V(X),E(X)) is a graph with following properties. It's an undirected graph. Contains isolated vertex,{1} in E(X). Graph need to be labeled with V(X). Vertex Style is circular with labeling ...
0
votes
0answers
13 views

A good version of truncated real radical ideal?

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1,...x_n$. $\mathbb{R}[X]_t$ is the truncated set such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], \deg(f) ...
0
votes
0answers
9 views

Using integer arithmetic, how do I determine which product pair is larger without calculating the product?

How do I to determine, given pairs $\{m_1, M_1\}$ and $\{m_2, M_2\}$, where $m_n \le 0$, $M_n \gt 0$ and $\{m_n, M_n \in \mathbb{Z}\}$, which pair $\{m_1, M_2\}$ or $\{m_2, M_1\}$ will have the ...
0
votes
0answers
40 views

Solve $x = \sqrt[x]2$

How does one solve $x = \sqrt[x]2$ for $x$? This can be otherwise stated as $x = 2^{1/x}$ Raising both sides to the power of $x$: $x^x = (2^{1/x})^x$ $x^x = 2$ But I don't know where I can go from ...
0
votes
1answer
25 views

Number of solutions to $\sin^{-1}x + |x| = 1$ and friends

The problem: If $$\begin{align} \sin^{-1}x\;+\;|x| &\;=\;1 \quad \text{has} \quad n_1 \quad \text{solutions}\\ \cos^{-1}x\;+\;|x| &\;=\;1 \quad \text{ ... } \quad n_2 \quad \text{...} ...
1
vote
2answers
31 views

Divergence change of variables (to polar)

I would wish to simplify this integral $$\oint_{\partial D}(f\nabla f)\cdot\hat{n}\,ds$$ in terms of a line integral of $g$ on $[0,2\pi]$ where $g(\theta)=f(e^{i\theta})$. Background info: ...
0
votes
1answer
437 views

Simplex - Help with a proof

I am trying to prove the following (basic) claim about simplex. If you check my proof and help me with the part where I stuck, I would appreciate it very much. Let $X$ be a finite set and define ...
0
votes
1answer
12 views

Modulus of numbers for negative divisors

How does the Google calculator or the Wolfram calculator calculate the modulus of the numbers with negative divisors? For example: -4 % -3 = -1 and 4 % -3 = -2 The same topic has been mentioned ...
3
votes
3answers
675 views

Positive symmetric matrices and positive-definiteness

Is a symmetric real matrix with diagonal entries strictly greater than $1$ and off-diagonal entries positive but strictly less than $1$ necessarily positive-semidefinite?
5
votes
2answers
2k views

Stochastic matrices?

Let $A$ be a symmetric stochastic matrix, such that the sum over the columns, for each row, is 1, and all elements are positive. $A$ dimensions are $n \times n$ Let's say that $B$ is a matrix which ...
0
votes
0answers
22 views

Finite module over Noetherian ring faithfully flat?

If I have Noetherian rings $B=A^G\subset A$ (for some action of finite group $G$, maybe not relevant) and $A$ is finite as $B$-module. Is it always true that $A$ is faithfully flat over $B$? EDIT: ...
3
votes
2answers
2k views

$A^TA$ is always a symmetric matrix?

Through experience I've seen that the following statement holds true: "$A^TA$ is always a symmetric matrix?", where $A$ is any matrix. However can this statement be proven/falsefied?
0
votes
1answer
21 views

Let $X$ have a Poisson distribution with parameter $\lambda$.

Let $X$ have a Poisson distribution with parameter $\lambda$. (a) Show that the moment-generating function of $$Y = \dfrac{(X − \lambda)}{\sqrt{\lambda}}$$ is given by $$M_Y(t)=exp(\lambda ...
1
vote
0answers
25 views

Why say “exists smooth structure” instead of “is a smooth manifold”?

As some of you know I started to learn differential geometry some weeks ago. Now I came across this theorem (you can find it in Lee's Intro to Smooth Manifolds but it is in many other books also: ...
0
votes
1answer
51 views

Let $X$ be a random variable with mean $0$ and finite variance $\sigma^2$. By applying Markov’s inequality show that

I am looking for confirmation that I am working in the correct direction as well as pointers for points where I have gone astray. Here is the problem. (a) Let $X$ be a random variable with mean $0$ ...
2
votes
1answer
3k views

Help with proving that the transpose of the product of any number of matrices is equal to the product of their transposes in reverse

Specifically I am trying to show that (An)T = (AT)n where A is an mxm square matrix and n is a positive integer. This is where I'm stuck: To prove the theorem I would like to show that ((An)T)ij = ...
0
votes
0answers
4 views

Transcritical bifurcation for map function

Question: Determine the transcritical bifurcation for $x_{n+1}=\alpha x_{n}\left ( 1-x_{n} \right )^{2}$ I have determined the fixed point to be $x^{\ast}=0$ and $x^{\ast}=1$ Also, for ...
0
votes
3answers
39 views

Given that the sum of two of its roots is zero,solve the equation: $6x^4-3x^3+8x^2-x+2=0$

Solution:-let $\alpha,\beta,\gamma,\delta$ be the roots of equation. It is given that,$\alpha+\beta=0$ and,$\alpha+\beta+\gamma+\delta=3/6=1/2$,this implies $\gamma+\delta=1/2$ and ...
38
votes
11answers
3k views

Besides mathematical induction how else can you prove these two functions are equal?

Using mathematical induction, I have proved that $$1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$$ for every integer $n > 0$. I would like to know if there is another way of proving this result ...
0
votes
0answers
13 views

How to find the eigen values

How to find the eigen values of the graph having vertex set as $\{1,2,.......n\}$ and edge set as $\{(I,I+1)\}$ $ \cup (1,n)$ ? where $1\le l \le n$. Here I am considering the Laplacian matrix of ...
4
votes
2answers
2k views

Folding a rectangular paper sheet

You are given a rectangular paper sheet. The diagonal vertices of the sheet are brought together and folded so that a line (mark) is formed on the sheet. If this mark length is same as the length of ...
0
votes
1answer
21 views

Is there a relation between the branches of the Lambert function?

Is it possible to express $W_{-1}(z)$ exactly by a closed-form expression, allowing the principal branch function $W_0$ ?
0
votes
5answers
77 views

Solve for $a,b,c$ given $1/a + 1/b = 1/c$ and a couple other constraints

Suppose $a, b, c \in \{4, 5, \ldots, 21\}$ and satisfy $1/a + 1/b = 1/c$. Suppose also that $\gcd(b,c)=1$. Find $a, b, c$. I tried this problem and I got the answer, but I can't to understand how ...
2
votes
1answer
53 views

What type of singularity is $z=0$ for $f(z)=1/{\cos\frac{1}{z}}$

Consider the function $\displaystyle f(z)=\frac{1}{\cos\frac{1}{z}}$. Test what kind of singularity at $z=0$ ? For singularities of $f(z)$ , $\cos \frac{1}{z}=0\implies z=\frac{2}{\pi (2n+1)}$. ...
-2
votes
1answer
23 views

Explain Pigeon holes principle in your own words.

my own words explanation: If there is four pigeonholes in which six pigeons uses to lay their eggs, then there is atleast one pigeonhole housing two or more pigeons. Does my wordings correct and ...
-1
votes
0answers
9 views

do all sub functions have to be monotonic increasing in order for the function to be monotonic increasing? [on hold]

As stated in the question, Do all sub functions have to be monotonic increasing in order for the function to be monotonic increasing? I came across this exercise in class where I was asked to prove ...
2
votes
0answers
17 views

Convolution of a function and its inverse

I want to calculate the convolution of a function and its inverse, $$f(t) * f^{-1}(t)$$ e.g. $f(t)=1/(t-2i)$ I've heard that the answer can be a delta function. What requirements are necessary for ...
5
votes
2answers
49 views

Numerical Calculation of Eigenvalues of a large real Symmetric tridiagonal matrix

If I have an $N \times N$ matrix where every entry is zero except for along the super-diagonal and sub-diagonal, where the each entry is the conjugate of the last, like the following $5 \times 5$ ...
0
votes
1answer
46 views

How to evaluate or approximate this kind of recursion: $a(n+1) = m \cdot \exp(-K \cdot a(n) / m)$?

I came up on a recursive definition of a function, given by $$a(n+1) = m \cdot \exp(-K \cdot a(n) / m),\ n \geq 2$$ with $m$ and $K$ being fixed integers ($m$ large). The first terms of the recursion ...
-3
votes
1answer
23 views

$P(T ≤ 5 | T ≥ 2)$ from CDF [on hold]

If for discrete random variable T the CDF is defined as $$F(t) = \begin{cases} 0, & \text{t<1}\\ 1/4, & \text{1≤t<3}\\ 1/2, & \text{3≤t<5}\\ 3/4, & \text{5≤t<7}\\ 1, ...
2
votes
0answers
9 views

Quotient of a graph and digraph? What are the equivalence classes of a graph?

I want to understand quotient of a graph (also called quotient graph), my teacher says that the terms quotient of a graph and a modulo of a graph should be synonyms (even though modulo of a graph ...
3
votes
1answer
29 views

Class of the derivative of a bilinear map

This question is more conceptual than other thing. We know that if $f:U\subset \mathbb{R}^n \to \mathbb{R}^m$, where $U$ is a open subset of $\mathbb{R}^n$, is differentiable, then the derivative of ...
1
vote
2answers
53 views

If $f$ is continuous with $ \int_0^{\infty}f(t)\,dt<\infty$ then which are correct?

Let $f:[0,\infty)\to [0,\infty)$ be a continuous function such that $\displaystyle \int_0^{\infty}f(t)\,dt<\infty$. Which of the following statements are true ? (A) The sequence $\{f(n)\}$ is ...
0
votes
1answer
37 views

Uniformly bound exists for a continuous function sequence in a neighbhorhood of a convergent point?

Assume that $$\lim_{n\to\infty}f_n(x_0) < \infty$$ and also that $\forall n\ge1$, $f_n(x)$ is continuous in a neighborhood of $x_0$, say $(x_0-\delta_1, x_0+\delta_1)$. Besides, for any fixed $x$ ...
0
votes
0answers
12 views

Boundary of $\sum_{j}x_j(x_j-x_i)$ for $x_i \in[0,1]$

Does $\sum_{j}x_j(x_j-x_i)$ for $x_i\in[0,1]$ and $0\le i,j\le N-1$ have a upper and lower boundary? And how to calculate them? Thanks!
0
votes
0answers
10 views

How can the supremum of a set A of Dedekind cuts ever not be a member of set A?

I'm reading about Dedekind cuts in Rudin's Principles of Mathematical Analysis and it has led me to the following question: Let's say I have some set $A$ of Dedekind cuts. Specifically, $A = ...
6
votes
3answers
192 views

Eigenvalues of Matrix vs Eigenvalues of Operator

I'm having some trouble reconciling the concept of eigenvalues of operators with eigenvalues of matrices: Say you have an $n\times n$ matrix $A$. It represents a linear operator $T:V\to V$ with ...
3
votes
1answer
2k views

Find an inverse of $a$ modulo $m$ for each of these pairs of relatively prime integers

How would I find the inverse of a given number $a$ modulo $m$, given that $\gcd(a,m)=1$? a) $a = 2$, $m = 17$ $17 = 2 \cdot 8 + 1$ $2 = 1 \cdot 2 + 0$ $1 = 17 - 8 \cdot 2$ <-How do I know ...
0
votes
2answers
31 views

How to determine which pair will have the smallest and largest product WITHOUT actually multiplying?

Is it possible to determine, given pairs $\{m_1, M_1\}$ and $\{m_2, M_2\}$, where $m_n \le 0$, $M_n \gt 0$ and $\{m_n, M_n \in \mathbb{Z}\}$, which pair $\{m_1, M_2\}$ or $\{m_2, M_1\}$ will have the ...
1
vote
2answers
21 views

find $c$ and $b$ in terms of $x$ and $a$

I have this geometry problem. Supose any $\triangle{ABC}$, where $\overline{CE} \perp \overline{AB}$; $\overline{CM}$ is median; $n$ is $proy_{\overline{CM}}\overline{AB}$; $\angle{CMA}$ is ...
1
vote
0answers
13 views

Determine whether the set of points is achiral

I have a set of points in 3D space with given coordinates. Points have labels, thus not all of them are identical in general. How to check whether this set of points achiral or not?
-3
votes
0answers
18 views

How do I interpret my Anova results? [on hold]

Anova1 Here is the SPSS Anova I did. I am unsure how to interpret and write up the results. Thanks for any help in advance. This is the written section I wrote to go along with my data. I am not ...
0
votes
0answers
19 views
+150

Weight Modification for Computationally-Efficient Nonlinear Least Squares Optimization

There was a time where I could figure this out for myself, but my math skills are rustier than I thought, so I have to humbly beg for help. Thank you in advance. I am solving a weighted nonlinear ...
2
votes
0answers
20 views

Center of mass of convex polyhedron.

Given a set of 2d points we can find it's convex hull. Now I'm going to explain an algorithm I thought of and you guys let me know where I've gone wrong (because it probably is wrong) So our goal ...

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