2
votes
0answers
75 views

Projective modules are stably isomorphic if localization at a monic polynomial is isomorphic

Let $R$ be a ring and let $P,Q$ be finitely generated projective $R[x]$-modules. Let further $f\in R[x]$ be a monic polynomial. If $P_f\cong Q_f$ (localization on $f$) then $P$ and $Q$ are stably ...
3
votes
3answers
350 views

sets with no asymptotical density over $\mathbb N$

Let's consider the Natural density on $\mathbb N$ defined by: Take $ A\subset \mathbb N$; define the sequence $x_n= \dfrac{|A\cap[1,n]|}{n}$, and then if $\lim\limits_{n\to\infty} x_n$ exists, ...
1
vote
1answer
134 views

Outer automorphism group of a free product

Suppose $G=\mathbb{Z}\ast C_n=\langle a, b; b^n\rangle$ is the free product of the infinite cyclic group with a finite cyclic group. Then $G$ has finite outer automorphism group. However, my proofs ...
4
votes
1answer
196 views

Surface infinitesimals and its intuitive manipulation?

The excess pressure in the concave side of any liquid bubble or drop with surface tension of the liquid being $T$ is $\frac {4T}r$ and $\frac {2T}r$ respectively. I wanted to derive it using a ...
1
vote
2answers
113 views

How to compute the rational group of this elliptic curve?

How to compute the rational group of this elliptic curve: $$E:\quad y^2=(x+3)x(x-1).$$ Ps: I am not familar with elliptic curves. (1,0), (0,0), (-3,0), (-1, 2), (-1, -2), (3, 6), (3, -6) are ...
10
votes
2answers
692 views

Why $f(x) = \sqrt{x}$ is a function?

Why $f(x) = \sqrt{x}$ is a function (as I found in my textbook) since for example the square root of $25$ has two different outputs ($-5,5$) and a function is defined as "A function from A to B is a ...
1
vote
1answer
67 views

Help me go from English to Logic

The positive-definiteness axiom used for just about all the definitions of inner-product spaces that I've seen goes like this: $$\langle \mathbf{x},\mathbf{x}\rangle \ge 0 \text{ with equality only ...
1
vote
1answer
152 views

fundamental decomposition theorem an cyclic modules

In Robert Ash's Abstract Algebra book, chapter 4 section 4.6.3 (Fundamental Decomposition Theorem), it states that for any finitely generated module $M$ over PID $R$, there exist ideals such that $M ...
0
votes
1answer
24 views

Key technique to find the nature of root(s)

What is the key technique to be used to solve these types of problems?
2
votes
1answer
71 views

was flatness really used in this argument? (Matsumura, Theorem 7.2)

Let $A$ be a ring and $M$ an $A$-module. Then $M$ is faithfully flat over $A$ $\Leftrightarrow$ $M$ is flat over $A$ and $M \otimes N=0 \Rightarrow N=0$. This is part of theorem 7.2, p. 47 in ...
2
votes
1answer
346 views

a finitely additive measure that is continuous at $\phi$ is $\sigma$-additive

Let $ (\Omega,M)$ a $\sigma$-algebra of events, and let $P$ be a finitely additive measure. We say that $P$ is continuous at an event $ A\in M$ if $A_n,B_n\in M$ are sequences of events such that ...
0
votes
1answer
41 views

Scaling a scale

I have a list of input values from 0 to 100. Each value must be scaled to a short scale, that start from 20 to 80. So: 0=>must be 20 100=> must be 80 and ...
1
vote
1answer
54 views

Given $D$, find $K$ such that $(D+1) + DK$ and $K+1$ are perfect squares

That's the problem I am facing: Given $D>0$, find the minimum (or all) $K$ ($K>0$) such that both $(D+1) + DK$ and $K+1$ are perfect squares How can I attack this problem?
15
votes
1answer
380 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
0
votes
1answer
42 views

Integration proeblem

$$\int_0^{\infty} dr \, \cos{k r} \,= \frac12 \text{Re} \int_{-\infty}^{\infty} dr \, e^{i k r} \,$$ How can we write the above integral? $Re$ represents here the real part? Didi we do in the ...
6
votes
2answers
2k views

Recognizing that a function has no elementary antiderivative [duplicate]

Is there a method to check whether a function is integrable? Of-course trying to solve it is one but some questions in integration may be so tricky that I don't get the correct method to start off ...
1
vote
2answers
128 views

Open set of $\Bbb R$ which is not bounded below can be written as atmost countable collection of disjoint segments

Suppose I have a open set of $\Bbb R$ which is not bounded below but bounded above. Now, I want to show that that open set can be written as atmost countable collection of open intervals. I have ...
1
vote
1answer
234 views

How to solve non-homogeneous recurrence relation?

The relation is $$T(n) = T(n-1)+T(n-2)-T(n-3)+1 \quad \quad (1)$$ I tried in this way but stuck at a point . Please Help $$T(n+1) = T(n)+T(n-1)-T(n-2)+1 \quad \quad (2)$$ Subtracting $(2)$ from ...
2
votes
1answer
101 views

Question on Lagrange's theorem from homework

In one of my homework problems: In $Z^∗_{13}$ let H = {1, 5, 12, 8}. List the right cosets Ha. (The elements of $Z^∗_{n}$ are those i, 1 ≤ i ≤ n − 1 which are relatively prime to n. Multiplication is ...
1
vote
2answers
190 views

How do you call $\succsim$?

actually, I study economics, not math. As some of you may know, there is a sign for comparing goods: $\succsim$. My professor read $x\succsim y$ like "x is at least as good as y". I asked her if ...
1
vote
5answers
96 views

Locating possible complex solutions to an equation involving a square root

Note after reviewing answers. This question illustrates in a non-trivial way how the choice of how to compute the square root in the complex plane can make real difference to the answer you get. In ...
4
votes
0answers
84 views

Miranda Pag. 66 Plugging Holes

Let $X$ be a Riemann surface. A hole chart on $X$ is a complex chart $\phi: U \mapsto V$ on $X$ such that $V$ contains an open punctured disc $D_0=\{z: 0 < ||z-z_0 || < \epsilon \}$ with the ...
1
vote
0answers
29 views

How I can define an equivalence relation on the graph of a function?

Let $a∈ℝ$ being fixed. Let $g:ℝ→ℝ$ be a real analytic function with infinitely many zeros. I want to classify the points $c,d$ such that $g(a)=(a-u)g′(c)≠0$ and $g(a)=(a-w)g′(d)≠0$ where $c∈(u,a)$ and ...
1
vote
1answer
195 views

Prove that $M/Tor(M) $ is torsion-free.

Suppose $M$ is an $R$-module where $R$ is an integral domain.Define $Tor(M)$ be the set containing torsion elements of $M$. Prove that $M/Tor(M) $ is torsion-free. I have manage to prove that ...
3
votes
1answer
219 views

Defining $\{a_i\}$ as $(1+x+⋯+x^k)^n =\sum_{i=0}^{kn}a_ix^i$, then is the 'special' difference-sequence $\{d^Na_i\}$ a unimodal sequence?

Question : Letting $k,n$ be positive integers, let's define a sequence $\{a_i\}\ (i=0,1,\cdots, kn)$ as $$(1+x+\cdots+x^k)^n=\sum_{i=0}^{kn}a_ix^i.$$ Then, is the 'special' difference-sequence ...
2
votes
4answers
501 views

How to evaluate the limit $\lim\limits_{x \to 1} \left(\frac{2}{1-x^2} - \frac{3}{1-x^3}\right) $, and others?

$$ \lim_{x \to 1} \left( \frac{2}{1-x^2} - \frac{3}{1-x^3} \right)$$ In my opinion the function is not defined at $ x = 1 $ but somehow when I look at the graph, it's continuous and there is no ...
6
votes
2answers
151 views

Differential — Mathematically conform?

In calculus, I know that one defined the differential quotient $$\frac{dy}{dx} := \lim\limits_{h \rightarrow 0}{\frac{y(x+h)-y(x)}{h}}$$ I learned that it is not a quotient, but can be treated as one ...
1
vote
3answers
62 views

non-existence limit proof

Find the limit and prove your answer is correct $$\lim_{n\to\infty}\frac{n^3+2}{n^2+3}$$ By divide everything by $n^3$ I got $$\lim_{n\to\infty}\frac{n^3+2}{n^2+3}=\frac10 $$ which is undefined. ...
0
votes
1answer
48 views

Find the sign of $\int_{k}^{k+2}\frac{(\sin {\pi x})^c}{x} dx$

Let $$I(k)=\int_{k}^{k+2}\frac{(\sin {\pi x})^c}{x} dx,$$ where $c>0$ is an odd integer, prove that $I(2n)>0,I(2n+1)<0,\forall n\in\mathbb N.$ I can prove this for $c=1,$ ...
0
votes
1answer
45 views

highest power of Prime

How to find the highest power of prime $p$ in $N!$, when .$p^r-1<N<p^r+p$ $p^r-p<N<p^r$ I know that the highest power of prime contained in $N!$ is given by: $$s_p(N!) = \left \lfloor ...
0
votes
0answers
36 views

Find conditions in which $f$ is invertible over its entire image

Let us consider the following bijective applications: $$f_{k}:A_{k}→B_{k}, k=1,2,\ldots,m$$ We construct the following application: $$f=(f_1,f_2,\ldots,f_m):A_1×A_2\times\cdots ...
1
vote
2answers
417 views

what to do next recurrence relation when solving exponential function?

find gernal solution of :$a_n = 5a_{n– 1} – 6a_{n –2} + 7^n$ Homogeneous solution: $$a_n -5a_{n– 1} + 6a_{n –2} = 7^n$$ put $a_n=b^n$: $$b^n -5b^{n– 1} + 6b^{n –2} =0 \\b^{n-2} (b^2-5b^{} + 6b) =0 ...
1
vote
2answers
250 views

How to find a formula for this bijection?

We know that there is a bijection of the set $S:= \{ 2^m 3^n \mid m, n \in\mathbb Z, m,n\geq 0\}$ onto the set $\mathbb N$ of natural numbers. How to find a simple formula for such a map?
1
vote
1answer
115 views

Relations that are: reflexive but not transitive; transitive but not symmetric; symmetric but not reflexive

I have an incomplete answer to my question. Can anyone help me answer the last two parts. My question is: Find example of a set $S$ and three relations $R_1$, $R_2$, $R_3$ on it such that ...
4
votes
3answers
476 views

For a general plane, what is the parametric equation for a circle laying in the plane

Given a general equation for a plane through the origin $$\vec{n}\cdot\vec{r}=0$$ With no assumptions made on $\vec{n}$ except having unit modulus, real $3\times1$ vector. How can you describe a unit ...
2
votes
3answers
101 views

If $\,x-\frac 1 x=k, \, k$ being any integer,then $\,\,x^5-\frac {1}{x^5}=?$

I am stuck with the following problem which one of friends gave me : If $\,x-\frac 1 x=k, \, k$ being any integer,then $\,\,x^5-\frac {1}{x^5}=?$ The options are $\,\,k^5+4k^3+4k, ...
1
vote
2answers
49 views

Two possible answers for x

I was trying to solve a question on maxima-minima and I finally ended up getting this equation: $$\ln(1/x)=1;$$ If I take anti-log on both sides I get 1/x=e and therefore x=1/e. But if ...
0
votes
1answer
52 views

Conservative Field

I need help with the following question . I have this vector field: $$F(x,y):=y\frac{x^2\cos(xy)+y^2\cos(xy)-1}{x^2+y^2}\vec{i}+x\frac{x^2\cos(xy)+y^2\cos(xy)+1}{x^2+y^2}\vec{j}$$ and 2 domains: ...
1
vote
4answers
483 views

$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$

Question is to prove : $f : S^1 \to \mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$ I guess it would be helpful to use intermediate value theorem Assuming $f(x)\neq f(-x)$ then given ...
7
votes
1answer
225 views

What's the intuition behind the tangent bundle?

Well, when we work with a smooth manifold $M$ we can associate with each point $p\in M$ a vector space $T_p M$ of all vectors at $p$ tangent to $M$: this is the space of linear functionals obeying ...
0
votes
0answers
52 views

using entropy to calculate the relatedness of two columns in a database

There are two columns(x, y) in a database, I want to define the "relatedness" of the two columns. First i try to use I(x, y) (mutual information) to define the relatedness, then: date, ...
1
vote
2answers
107 views

Find a chief series for dihedral group $D_{2n}$

The question is : Find a chief series for dihedral group $D_{2n}$. Is each normal subgroup of $C_n$ normal in $D_{2n}$?
0
votes
1answer
100 views

Hermiticity of a Matrix representing a Clifford alg. element

I'm interested in representations of the elements of a Clifford algebra, $\gamma^a$ with $a\in \{1,...,n\}$, such that $(\gamma^a)^2 =\pm 1$. If these were numbers, for those $\gamma^a=1$ one have ...
3
votes
2answers
154 views

Find roots of polynomial with degree $\ge 5$

During our research we came up with the problem of computing the root of a polynomial of degree $\ge 5$ exactly. The coefficients are, except for the linear and constant term, all non-negative and ...
1
vote
2answers
116 views

Solve $x(x+1)=y(y+1)(y^2+2)$ for $x,y$ over the integers

Solve $$x(x+1)=y(y+1)(y^2+2)$$ , for $x,y$ over the integers
3
votes
2answers
525 views

Characteristic Function of Inverse Gaussian Distribution

The pdf of Inverse Gaussian distribution, IG$(\mu,\lambda)$, is : $$p_X(x)=\sqrt\frac{\lambda}{2\pi x^3}\exp\left[\frac{-\lambda}{2\mu^2x}(x-\mu)^2\right];\quad x>0,\lambda,\mu>0$$ I have to ...
1
vote
1answer
351 views

Number of possible triangles

Given six line segments of length 2, 3, 4, 5, 6, 7 units, the number of triangles that can be formed by these segments is $(A)^6C_3 – 7$ $(B)^6C_3 – 6$ $(C)^6C_3– 5$ $(D)^6C_3 – 4$ I know that ...
5
votes
2answers
225 views

Group theory with analysis

I've studied group theory upto isomorphism. Topics include : Lagrange's theorem, Normal subgroups, Quotient groups, Isomorphism theorems. I too have done metric spaces and real analysis properly. ...
0
votes
1answer
98 views

How to minimize this non-linear function?

minimize the following function: $$\sum^n_{v=1}\left(S_{1v} - t \frac{(1-p_v)\sin r_v }{1-p_v\cos r_v }\right)^2 + \left(S_{2v} - t \frac{(1-p_v)\sin (6r_v) }{1-p_v\cos (6r_v) }\right)^2$$ subject ...
1
vote
2answers
43 views

Learning about geometric bezier splines

I want to understand geometric Bezier splines. I have basically no advanced maths but am willing to learn. Can anybody suggest a good starting point for a complete beginner to start learning what I ...

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