0
votes
1answer
26 views

How can I prove that it is an Entire Function

Prove that if $$ f(z)=\left\{ \begin{array}{ll} \frac{\cos z}{z^2-(\pi /2)^2} & \hbox{when} \; z\neq \mp \pi/2\\ -\frac{1}{\pi}, & \hbox{when} \;z= \pi/2. \end{array} \right. $$ ...
1
vote
3answers
19 views

Proof about composed functions (elementary number theory)

Let f : X → Y and g : Y → X be functions and assume $g ◦ f = I_X$. Prove of g is injective then $f ◦ g = I_Y$. Approach if g is the left inverse of f then there exists $x\in X$ and $y \in Y$ such ...
1
vote
0answers
11 views

Find Unique Index for a Subset S

I'm looking for a way to assign a unique number to a particular subset of S. S is a set of n distinct integers from 1 through n. Now, take the set of all subsets of length k where order doesn't ...
-2
votes
1answer
31 views

Challanging problems on [Grade-12]Complex Number [on hold]

recently we are introduced to interesting world of complex number but except for 3-5 problems in the my books,all the problems are just plug-and chug,expression manipulation,etc.. which bores me out ...
0
votes
1answer
20 views

There are 40 available time slots for examinations. You need to schedule the A and B exams according to the following rules:

NOTE: This is homework so would appreciate if I could get some explanations instead of just straight answers. Really struggling with this question and to be honest, don't really know where to even ...
0
votes
1answer
19 views

counting the forecasts of 20 chess games

I have a Question... The results of 20 chess games (win, lose, draw) have to be predicted. How many different forecasts can contain exactly 15 correct results? I don't really understand this ...
0
votes
0answers
16 views

How to interpret multi-conditional piecewise functions.

I'm trying to simulate hysteresis and the its inverse for a control problem. This is a model found in [Tao & Kokotovic, Adaptive Control Systems with Actuator and Sensor Nonlinearities] to ...
0
votes
0answers
7 views

Why we can not get a Plucker mapping under the Supergrassmannian?

Given a grassmannian, we can obtain a Plucker mapping. Reading the book "Gauge Field Theory and Complex Geometry", I know that we can not get a Plucker mapping under the supergrassmannian generally. ...
0
votes
1answer
16 views

Taylor series integration

I am having trouble with the following question: Integrate the Taylor series $$e^{(-t^2)} = \sum^\infty_{n=0} \frac{(-t^2)^n}{n!}$$ term-by-term to obtain the Taylor series for erf (error function) ...
0
votes
1answer
17 views

Prove $-\frac{\ln(1-x^2)}{1-x^2}=H_1x^2+H_2x^4+H_3x^6+H_4x^8+\cdots$

$H_n$ is nth the harmonic numbers $x<1$ (1) $$-\frac{\ln(1-x^2)}{1-x^2}=H_1x^2+H_2x^4+H_3x^6+H_4x^8+\cdots$$ A different approach of representing $\ln(x)$ let expand out the series ...
3
votes
0answers
32 views

An analytic method to prove a specific curve is closed

In my study of Hamiltonian dynamics I have come across a Hamiltonian dynamic system with a solution curve I know to be closed via computer and via intuition but I require a rigorous way to prove this, ...
-3
votes
1answer
38 views

What is the solution for the Logarithmic CompSci question? [on hold]

Facts: My alphabet contains m digits. Each digit is represented in memory by 1 byte. A word is any string of digits. Words are separated by a single non-alphabetic character. It is represented in ...
0
votes
0answers
20 views

General Solution by complexification $\,y''-3y' =2e^{t}\sin 2t $

General Solution by complexification $\,y''-3y' =2e^{t}\sin 2t $ I could use some help here as I don't know what complexification is and I tried googling it but the material was sort of confusing. ...
-1
votes
0answers
41 views

There are $101$ positive integers that sum to $300$. Can we find a subset of these integers that sums to $100$? [duplicate]

We are given a set of $101$ positive integers that sum to $300$. Since summation of $101$ distinct numbers cannot be $300$, repetition among the $101$ positive integers exists. Can we choose a group ...
0
votes
2answers
32 views

How to find area of isosceles triangle when given two heights? [on hold]

So I know the sine and cosine theorem and I tried using them but I got nowhere. (I got to an equation which I can't solve and I know there must be an easier method since we have not studied how to ...
0
votes
0answers
6 views

Proving that largest root (obtained via P.C.A.) is a symmetric function

Suppose, we are given $\textbf{X} = (X_1, X_2, \ldots,X_m)$ and $\textbf{Y} = (Y_1, Y_2, \ldots, Y_n)$. Also, we are given, S = pooled variance. If we implement Principal Component Analysis (P.C.A.) ...
0
votes
1answer
19 views

Goldberg compound probability problem- guess the correct colour ball from an urn

I am working my way through an example problem from Goldberg's "Probability: An Introduction". There are x red balls and x green balls in an urn. Total number of balls in the urn is 5. You must guess ...
0
votes
1answer
18 views

Find Surface Area Via a Line Integral (Stokes' Theorem)

I am trying to use Stokes' Theorem to calculate the surface area of a parametrized surface via a line integral. The surface is the part of $z= x^2+y^2$ below the plane $z=5$. I know this can be done ...
1
vote
2answers
23 views

build absolute value equations know solution

We have absolute value equations with unknown coefficients: $$|x + a| = b$$ and we know the solutions: $$x = 11 \text{ and } x = 5$$ We need to find $a$ and $b$. From $$11 + a = b \\ 5 + a = -b$$ we ...
1
vote
2answers
21 views

Can anyone help me with this finite sum?

I have to calculate the sum $\displaystyle\sum_{k=1}^n \displaystyle\frac{3^k}{3^{2k+1}-3^k-3^{k+1}+1}$ We can re-write the sum as follows $\displaystyle\sum_{k=1}^n ...
1
vote
0answers
10 views

Logic - logical connective for (~ABC) + (A~BC) + (AB~C)?

Is there a logical connective that says 'True, if and only if 1 proposition is true'. Or perhaps even better, is there one that describes 'True, if and only if n propositions is true'? Where n is an ...
1
vote
0answers
15 views

Formula relating dimension of fiber of morphism between varieties

Let $f: X \to Y$ be a morphism of (irreducible) varieties, where the dimension of every fiber dim$f^{-1}(y)=n$ is the same. Must it follow that dim$X=$ dim$Y+n$? The reason I am asking this is that ...
0
votes
1answer
15 views

Problem with Molien's formula for covariants

If $E$ and $H$ are finite-dimensional faithful representations (over $\mathbb{C}$) of a finite group $G$, with $H$ irreducible. The Molien formula describer the Poincaré series of the covariants as $$ ...
0
votes
2answers
16 views

Undetermined coefficients, particular solution $y'' +8y' +15y = 4$ vs $y'' -\, 3y' = 8$

I have a question about finding the particular solution using method of undetermined coefficients. For $y'' +8y' +15y = 4$, the guess for the particular solution is $y_p(t) = A$; but for $y'' -\, 3y' ...
0
votes
0answers
26 views

Computing average of a distribution.

Suppose on a day we toss an unfair coin $60$ times (each trial are independent) and we assume getting a head "H" is favorable event and the probability of getting a head is $0.04$. Then the case is ...
2
votes
2answers
72 views

Complicated series converges to $\pi$.

How do I get this result? $$\frac {426880 \sqrt {10005}}{\large \sum_{k = 0}^{\infty}\frac {(6k)!(545140134k + 13591409)}{(k!)^3 (3k)! (-640320)^{3k}}} = \pi$$ It seems formidable. Context: I came ...
1
vote
1answer
20 views

Divisibility - what is A+B?

Is there an easy to solve this problem? I can find the answer by using a complicated rule that I don't understand. Even if I try to remember this rule, I probably will forget about it a year later. ...
0
votes
0answers
5 views

Are there any cubic bezier curve that cannot imitate by multiple quadratic bezier curve?

I want to make a line curve system with bezier curve. And I want to use only quadratic bezier curve so it can be extend and control easily, it can add control point anywhere and more intuitive But ...
0
votes
2answers
30 views

Complex logarithms when computing real-valued integral

My question arise when I try to calculate real-valued integral, specifically, I want to evaluate the integral \begin{equation} \int_0^1 \frac{\ln \left(\frac{x^2}{2}-x+1\right)}{x} dx \end{equation} ...
8
votes
2answers
47 views

How do I prove that $f_n\to f$ in $L^p$?

Let $\{f_n\}$ be a sequence in $L^p([0,1])$ for $p\geq 1$. Suppose there exists $f\in L^p([0,1])$ satisfying $\lim_{n\to\infty} \int_0^1 f_n(x)g(x)dx = \int_0^1 f(x)g(x)dx$ for any $g\in L^2([0,1])$. ...
0
votes
0answers
10 views

Increasing or decreasing theta based on direction of vector

I am a programmer, and I have some holes in my math knowledge that I am working on filling in. Right now I'm working with a simple process involving drawing curves and straight lines. The line that is ...
1
vote
1answer
22 views

Polar equations (further question)

Consider the polar equations of the form $r=a\cos(b\theta)$ and $r=a\sin(b\theta)$. What is the nature of the graphs of these two polar equations and then summarize some generalizations with respect ...
-1
votes
0answers
22 views

If $U$ and $V$ are disjoint, then why is $\mathcal{F}(U\cup V)=\mathcal{F}(U)\times \mathcal{F}(V)$?

Let $X$ be a topological space such that $U,V$ are disjoint open sets on it. Let $\mathcal{F}$ be a sheaf on $X$. Then Vakil's notes say that $\mathcal{F}(U\cup V)=\mathcal{F}(U)\times \mathcal{F}(V)$ ...
1
vote
1answer
11 views

simplification question on Diff. Eq. Solution

Can anyone explain this simplification from y^-2 to y? If you distribute the x^4 through, you obtain y^-2 = (2/x) + Cx^4. This leads to y^2 will equal the reciprocal of what I just wrote. Where in ...
-1
votes
0answers
17 views

use the definition of limits to find f(x), f(y) in a function with multiple variables [on hold]

I have the next function $f(x,y)$ = $(x^2+6xy+7y^2)/(6x^2+7y^2)^{(1/2)}$ and I need find the derivative of this by using the definition of limits and I have no Idea of how to start or develop this ...
0
votes
2answers
25 views

Show that there are at most two rational points on $(x - a)^2 + (y - b)^2 = r^2$ for $a, b$ irrational.

For any given irrational numbers $a, b$ and real number $r \gt 0$, show that there are at most two rational points (points whose coordinates are both rational numbers) on the circle $(x - a)^2 + (y - ...
0
votes
1answer
37 views

Collatz Conjecture, sufficient to show odd numbers reach $1$?

The famous conjecture: Let $$ f(n) = \begin{cases} n/2 & \quad \text{if } n \text{ is even}\\ 3n+1 & \quad \text{if } n \text{ is odd}\\ \end{cases} $$ The Collatz Conjecture ...
1
vote
7answers
1k views

What is “8 log 2”?

When someone says "8 Log 2" what does this equate to in writing? Does it mean the following? $$ \log _{2} 8 $$ And if so, what is the value of this? When I plug those numbers into this log ...
0
votes
1answer
13 views

Proof about inverse functions in elementary number theory

Let f : X → Y and g : Y → X be functions. Assume g ◦ f is invertible. Prove that f has a left inverse and g has a right inverse. Approach I already proved in a previous exercise that if g ◦ f is ...
0
votes
1answer
11 views

What effect does a negation have on a proposition in a bracket.

Say for example ¬ (p ∧ ¬q}, what does the negation outside the bracket do to the proposition inside the bracket?
0
votes
0answers
3 views

Elastic Body Simple Deformation

In continuum mechanics we can consider a reference frame $B = [0,1]$ along with a homogeneous deformation $F$ where $x = Fp$ for $x \in \mathbb{R}$ and $p \in [0,1]$ and $F = 2$ so $F[B] = [0,2]$. ...
0
votes
1answer
6 views

build absolute value equations know solution

We have absolute value equations with unknown coefficients: |x + a| = b and we know solutions: x = 11 and x = 5 We need ...
0
votes
1answer
20 views

Clarification about a Convergence in Probability example proof

Let $U \sim \mathrm{Uniform}[0,1]$. $$ X_n= \begin{cases} 3 & U \leq \frac{2}{3}-\frac{1}{n} \\ 8 & \text{otherwise} \end{cases} $$ $$ Y = \begin{cases} 3 & U ...
0
votes
2answers
9 views

Using scale transformation on the exponential distribution

In my textbook, they show that: $$Y=\frac{X}{\lambda}\sim\text{Expo}(\lambda)$$ Where $X \sim\text{Expo}(1)$. I am confused about why they divide by $\lambda$ instead of multiply to transform from ...
2
votes
2answers
57 views

I have to find $I=\int_{0}^{\pi}\ln(1-2a \cos x+a^2)\, dx$ [duplicate]

I have to find $$I=\int_{0}^{\pi}\ln(1-2a \cos x+a^2)\,dx$$ Can someone help me to solve it?
1
vote
2answers
32 views

About converging sequence of norm-one elements of $\ell^2$

I got stuck in this question for days, so any help/hint is appreciated. Assume that $T$ is the unit ball in $\ell^2$ and $(x_n)_{n=1}^{\infty} \subset T$ and that $(x_n(i))_{n=1}^{\infty}$ converges ...
12
votes
0answers
50 views

Moving half of the nuts

An even number of nuts is divided into three nonempty piles. In each step, we are allowed to take half the nuts from a pile with an even number of nuts, and put them on another pile. Can we always ...
0
votes
0answers
8 views

Interpolating a vector about an arc (Slerp)

In the following image, how can I solve for $k_0$? I know that $\mathbf v_1$ is a unit vector and $k_1 = \sin tω/\sin ω$.
1
vote
0answers
17 views

Determine the equilibrium temprature [on hold]

By solving the heat equation determine the equilibrium temperature distribution for the circular ring $\theta\in[0,2\pi]$ by both (a) directly setting $u_t=0$, and finding the equilibrium solution, ...
1
vote
1answer
11 views

In queueing-theory, M/M/1/K, if in steady state, is the mean arrival rate equal to mean departure rate(not service rate)?

I am trying to calculate the Waiting time of a box that contains two queueing systems(qs) in serial. The arrivals are on left, they enter qs1, exit , then enter qs2, then exit qs2 i.e exits the whole ...

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