0
votes
0answers
2 views

Is there a name for the closed form of $\sum_{n=0}^{\infty} \frac{1}{1+ a^n}$?

I hope this is not a duplicate question. If we modify the well known geometric series, with $a>1$, to $$ \sum_{n=0}^{\infty} \frac{1}{1+a^n} $$ is there a closed form with a name? I suspect ...
1
vote
0answers
4 views

Maximal $n$ such the the additive partition with a given product is unique.

Given $n$, there are many tuples with $a + b + c = n,a < b < c$. For large $n$, different tuples may give the same products. E.g. $2+8+9=19=3+4+12,2\times8\times9=144=3\times4\times12$. What is ...
0
votes
0answers
3 views

How to solve a bi-quadratic equation with symbolic coefficients?

I have the following equation with the symbolic coefficients specified using 'syms' which i have been trying to solve in MATLAB:- ...
0
votes
0answers
4 views

How to estimate magnitude of expontent?

When given an exponent, such as 6^12, is there a simple way to approximate how large(magnitude) the result is, without performing the calculation? Is this method accurate for large exponents?
2
votes
0answers
6 views

cross-sections of a sphere bundle

Let $B$ be a CW-complex (or a manifold) and $B_0$ a CW-subcomplex (or a submanifold) of $B$. Let $\xi=(E,p,B)$ be a fibre bundle with fibre $S^n$. Choose a basepoint $*\in S^n$. Let $\Gamma(\xi)$ be ...
0
votes
0answers
8 views

Map to $RP^2 \vee S^1$ nullhomotopic

Let $R$ be $S^{1}\vee S^{1}$. Call the first circle by $a$ and second one by $b$. Let $X$ be space by attaching two $2$-cells to $R$ one via the boundry map $a^{3}$ and the other via the boundry ma ...
1
vote
1answer
7 views

Function determining temperature of points along a curve

Let $T=x^2+y^2+z^2$ be the function determining the temperature at the point $(x,y,z)$. Find a function that determines the temperature at the points along the curve $\vec\alpha(t)=(4\cos t, 4 \sin t, ...
2
votes
2answers
26 views

0's Exponents are impossible?

I've had something that's been bugging me, and I tried research and asked my math teacher. None had sufficient answers. The concept of $0$ is that when $0$ goes to any exponent except for $0$, it ...
0
votes
0answers
10 views

Generalization of a Result on Modular Inverses

Yesterday, I attempted to solve the general system of linear congruences (I'm not sure why I've never tried this before.) \begin{align*} x &\equiv a \pmod{A} \\ x &\equiv b ...
3
votes
0answers
36 views

Fruitful advice to get back to study Mathematics again? [on hold]

I have completed masters in Pure Mathematics a year back.I was preparing for an exam for pursuing a PhD program in the same .The results came out in this year in the month of April and found that I ...
-2
votes
1answer
23 views

How would you divide a polynomial by another polynomial whose power is greater than its nominator?

I have a polynomial which is: $$\frac{(x^3-4x)}{(4x^2-4x+1)} = -10$$ Is there a way to do this? I have thought about doing long division which was not helpful...
2
votes
0answers
19 views

Approximating nice functions with wild ones

Let $X$ and $Y$ be toplogical spaces, and call a function $f:X\to Y$ wild if the preimage $f^{-1}(\{y\})$ is dense in $X$ for every $y\in Y$ -- or, equivalently, if the image of every nonempty open ...
0
votes
1answer
19 views

Congruence using extended GCD

$$\eqalign{ x &\equiv 5 \mod 15\cr x &\equiv 8 \mod 21\cr}$$ The extended Euclidean algorithm gives $x≡50 \bmod 105$. I understand now that if we combine the two it implies ...
1
vote
0answers
14 views

Weierstrass-$\wp$ Function Asymptotics

Given the Weierstrass-$\wp$ function, $$\wp(2x+1+\tau \mid 1, \tau),$$ with half-periods $1$ and $\tau=\omega_2/ \omega_1$, I want to look at the case where $\rm{Re}(\tau) \in \mathbb{Z}$ and I want ...
1
vote
1answer
22 views

Definite integral with limits from zero to infinity

Let $ I=\int\limits_{0}^{\infty}e^{-(x^2+\frac{1}{x^2})}dx$ and $J=\int\limits_{0}^{\infty}x^2e^{-(x^2+\frac{1}{x^2})}dx$.If $J=\frac{p}{q}I$,where p and q are natural numbers and are coprime to each ...
0
votes
0answers
7 views

Two contests, an extension of the Coupon Collector's problem

Coupon Collector's Problem Let $X$ be the number of coupons drawn with replacement from an urn containing $N$ distinct coupons until each coupon has been drawn at least once, winning the coupon ...
1
vote
0answers
33 views

Find positive integer x,y,z such that $2x^{2x}-1=y^{z+1}$

Find all positive integer x,y,z such that $2x^{2x}-1=y^{z+1}$ I have tried to use LTE lemma but it didn't work. I think my problem is $z+1$. I can not control it. When I use LTE lemma, the purpose ...
0
votes
2answers
23 views

Finding the correct slope.

To determine the slope of the graph of this relation do I take the two points as (4,20), (0,0) and then proceed to take 20-0=20 and 4-0=4, to divide 20 by 4 to get the slope of 5m? For the ...
0
votes
0answers
7 views

Functions that are “balanced” on the support of a permutation

Let $F = GF(2^n)$. Let $P(x), Q(x) \in F[x]$ be such that $P(x)$ is a permutation, while $Q(x)$ is not a permutation. For $\lambda \in F^*$ define the function $g_\lambda(x) = Tr(\lambda Q(x))$. Let ...
0
votes
1answer
10 views

Draw figures for the 5 different lattices with 5 elements.

I can only think of 4 lattices. Those are: a 1-1-1-1-1 (a chain), a 1-3-1, a 1-1-2-1 and a 1-2-1-1 (if this notation isn't clear, I'll provide images). I really can't figure out what the 5th lattice ...
-3
votes
0answers
24 views

Vector spaces and nontrivial subspace.

Give an example of a subset of $\mathbb{R}^2$ that is a nontrivial subspace of $\mathbb{R}^2$? $\mathbb{R}^2$ as $\{(a, b) \mid a, b \in \mathbb{R}\}$
2
votes
0answers
26 views

Why we define the adjoint operator

Suppose in vector space $A: X\rightarrow Y$ is a linear map, the adjoint operator $A^{'}: Y^{'}\rightarrow X^{'}$ is defined as: $f(Ax)=(A^{'}f)(x)$. As I can understand, the adjoint operator just ...
1
vote
0answers
30 views

Help me to prove the determinant formula

Actually it is about the question of n-linear function, but it is so relevant to the determinant formula. Here is the notation of the theorem. If $n>1$ and $A$ is an $n \times n$ matrix over $K$, ...
1
vote
0answers
12 views

Difference between torsion and zero divisor

I'm not understanding what the difference is between a zero divisor and a torsion element of a module. My best guess is that the torsion elements are "vectors" and zero divisors are scalars. This ...
0
votes
1answer
46 views

Harder-Than-Seems Inverse of $f(x)=x^3-x-12$?

This may seem simple but I have had long days of frustration with finding the inverse of this: $$f(x)=x^3-x-12.$$ I got this on some homework and it did not ask for the inverse. However i wanted to ...
0
votes
0answers
16 views

Probability Ques.

From previous experience, Bob’s Programming teacher takes down the attendance 40% of the time. Bob’s classmate, Marty, comes late to class (i.e. after the attendance is taken down) 20% of the time. ...
-1
votes
0answers
6 views

Qualitative Ordinary differential equations [on hold]

Reduce the following systems of equation to a systems of first order ODE’s: 〖( d^2 y)/〖dt〗^2 〗^+3 dz/dt+2y=0 〖( d^2 z)/〖dt〗^2 〗^+3 dy/dt+2z=0
0
votes
2answers
16 views

In an inner product space, if the matrix is symmetric, is an eigenspace necessarily orthogonal to the range space?

Say I have 3 distinct eigenvalues for a symmetric matrix. By the Spectral Theorem, the three eigenspaces are mutually orthogonal. But, if I just wanted to compute the first eigenspace, ...
1
vote
1answer
16 views

harmonic functions on the disk which agree on the real are identical?

The question is whether it is possible to find two distinct harmonic functions on the unit disk $\mathbb{D}=\{z: \ |z|<1 \}$ such that they agree on $\mathbb{D} \cap \mathbb{R}$. If yes, please ...
1
vote
2answers
37 views

Why is this congruence true?

$$\eqalign{ x &\equiv 5 \mod 15\cr x &\equiv 8 \mod 21\cr}$$ The extended Euclidean algorithm gives $x≡50 \bmod 105$. How/why? I am trying to understand how this is true when ...
0
votes
0answers
12 views

Need help with a design calculations equal spacing of circles

I need the spacing between the circles to match. Design need 6 circles, their diameter is not fixed, but the spacing between circles need to be identical. As the circles are moved up and down their ...
0
votes
0answers
6 views

Solving a boundary-value problem where the function is not differentiable at the boundary?

Let us say we have a initial-boundary value problem $$ \frac{\partial u}{\partial t} = Lu $$ on $(0, T]\times [0, \infty)$ with initial condition $u(0, x)=h(x)$. I don't specify $L$ here in the hope ...
3
votes
1answer
20 views

the sequence of derivative cannot satisfy $|f^{(n)}(z_0)| > n!n^n$

Let $f: \Omega \to \mathbb{C}$. Prove that for any $z_0 \in \Omega$, the sequence of derivatives cannot satisfy $|f^{(n)}(z_0)| > n!n^n$ In this problem, I intend to prove by contradiction, and I ...
1
vote
3answers
32 views

How do I interpret this question: Do I multiply, divide, subtract first?

Which of the following expresses $6p+2py-4p$ in its simplest form? (A) $2p+2py$ (B) $4py$ (C) $4p^3y$ (D) $10p+2py$ Im not really sure how to go about it...
1
vote
0answers
18 views

Separating vectors from linear combination

Suppose I have a linear combination of vectors as follows $ \mathbf{s} = \alpha_1\mathbf{x}_1 + \dots + \alpha_m\mathbf{x}_m + \beta_1\mathbf{y}_1 + \dots + \beta_n\mathbf{y}_n $ where $\alpha_i, ...
4
votes
1answer
30 views

$2$-adic sequence converging to $\sqrt{-7}$.

I am trying to construct a sequence in $\mathbb Q_2$ that is formed of rational numbers and converges to $\sqrt{-7}$, to prove that $(\mathbb Q, |\cdot|_2)$ is not complete. My lecturer stated that ...
0
votes
0answers
18 views

How to solve this mechanics problem when friction depends on velocity?

Consider a mass $m$ at position $x(t)$ on a rough horizointal table attached to the origin by a spring with constant $k$ (restoring force $-kx$) and with a dry friction force $f$ $$\begin{cases} ...
2
votes
0answers
23 views

What's wrong with this calculation involving pullbacks of divisors on surfaces?

Beauville, Complex Analytic Surfaces, Proposition I.8(b), reads: Let [S and] $S'$ be a surface, $g : S \to S'$ a generically finite morphism of degree $d$, and $D$ and $D'$ divisors on $S$. Then ...
0
votes
1answer
17 views

How to get the derivative of a matrix function?

I want to get the derivative of a matrix function as follow: $$\frac{\partial f(\boldsymbol{AX})}{\partial \boldsymbol{X}}$$ which $f(\cdot)$ is a scalar function, and the result as I think should be ...
0
votes
1answer
22 views

combination of 5 digit numbers

looking at 5-digit number when digits can be $1,2,...,9$ and with repetition, $|\Omega|=9^5$ the event of $5$ distinct digits is $9\times 8\times 7\times 6\times 5$? and the event 2 digits the same ...
0
votes
0answers
10 views

Convexity of $ t \mapsto \log \left[ \int_0^1 f^{pt} g^{q(1-t)} dx \right] $

Let $f,g \geq 0$ be bounded, measurable functions on $[0,1].$ For real $p,q>0$ with $p^{-1}+q^{-1} = 1$, I want to show the convexity of $$ h(t) = \log \left[ \int_0^1 f^{pt} g^{q(1-t)} dx ...
0
votes
0answers
17 views

Determine clockwise or anticlockwise

I have a central point define by an x and y and I have an object which is moving around it with a location defined by an x and a y. I'm trying to determine if the object is moving clockwise or ...
0
votes
1answer
13 views

Non-finitely generated, non-divisible, non-projective, flat module, over a polynomial ring

(1) Let $R=k[x_1,\ldots,x_n]$. I wish to find an example of a non-finitely generated, non-divisible, non-projective, flat $R$-module. Notice that $k(x_1,\ldots,x_n)$ is NOT an example of what I am ...
1
vote
0answers
6 views

Will numerical routines for the Exponential Integral function E_n work when n is continuous?

So I am a mathematical biologist of sorts. I rely heavily on Mathematica which often provides analytic results couched in terms of special functions which I then try to go and learn about. Right now ...
0
votes
1answer
14 views

Solving Equations system question

We get this equation and need to solve Solve in $\mathbb{Z} $ the given equation $ y(y -x )(x+1) = 12\ $
4
votes
1answer
38 views

Computing $\sum_{n=-\infty}^\infty \int_{-\pi}^\pi e^{-x^2} \cos(nx) dx $

I'm trying to find $$ \sum_{n=-\infty}^\infty \int_{-\pi}^\pi e^{-x^2} \cos(nx) dx. $$ About the only thing I can think of is the well-known identity $$ \sum_{n=-k}^k \cos(nx) = \sum_{n=-k}^k ...
5
votes
4answers
35 views

calculate the limit $\lim_{x\to0} (\frac 1{x^2}-\cot^2(x))$

The answer of the given limit is 2/3,but i cannot reach it.I have tried to use the L'Hospital rule, but i couldnt drive it to the end.Please give a detailed solution! $$\lim_{x\to0} \left(\frac ...
0
votes
1answer
30 views

If $gcd(a,b)=1$, then there exists integers x and y such that $xa + yb = 1$

Did not find this from this website... If $$ gcd(a,b)=1,$$ then there exists integers x and y such that $$xa+yb=1.$$ Now, the tip is to use particular corollary, that states: The class $[m]_{n}$ ...
1
vote
1answer
8 views

Finding the horizontal and vertical tangents of a parametric equation.

Find the points at which the polar curve $r=2+2\sin{(\theta)}$ has a horizontal or vertical tangent line. Translate the parametric equation to Cartesian coordinates: $$ r^2=2r+2r\sin{(\theta)} ...
0
votes
1answer
13 views

GNS construction and representations

I am currently reading about C* from the following notes ( http://www.math.uvic.ca/faculty/putnam/ln/C%2A-algebras.pdf ). In the proof of GNS construction theorem 1.12.4 page 50 there is something I ...

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