0
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0answers
2 views

Which method do i use ? Interpolation

I have table of 5 values i.e abscissa and ordinates are given. I have been asked to find derivative at particular point and also second derivative at that value. That value is between my given equi ...
0
votes
0answers
2 views

Eigenvalues of matrix with all $1$'s.

Let $A$ be the $n \times n$ matrix over a field of characteristic 0, all of whose entries are 1. What are the eigenvalues of $A$, counted with their multiplicities?
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0answers
6 views

Two straight lines one being a tangent to $y^2=4ax$ and the other to $x^2=4by$ are at right angles.Find the locus of their point of intersection.

Two straight lines one being a tangent to $y^2=4ax$ and the other to $x^2=4by$ are at right angles.Find the locus of their point of intersection. I tried but could not reach final answer.The tangent ...
0
votes
0answers
4 views

Product of measures on rectangular cylinders

Let $(\Omega_i,\mathcal A_i,\mu_i)$ be a $\sigma$-finite measure space. Why is \begin{equation} \begin{split} ...
0
votes
0answers
4 views

Looking for set of combinatorics problems

I'm preparing to Mathematics for Computer Science exam. What I learned from past edition of exams is fact of very often occurence of old problems. I mean more or less known problems, but possible to ...
0
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0answers
12 views

Drunk Passenger Probability question

I don't know how to solve this question. A line of 100 airline passengers are waiting to board a plane. They each hold a ticket to one of the 100 seats on the flight. For convenience, lets say that ...
1
vote
0answers
3 views

constant coefficient difference equations LTI, why do I need the initial conditions?

Consider the following difference equation $$y_{n}=-\sum_{k=1}^{q}a_{k}y_{n-k}+\sum_{m=0}^{p}b_{m}x_{n-m}$$ I know that this is supposed to be LTI iff $y_{-q}=y_{-q+1}=...=y_{-1}=0$. How does one go ...
1
vote
1answer
13 views

Need help proving this geometry problem.

My friend asked me one question yesterday.It is as follows. Let there be two triangles ABD and ACD.D is a point on base BC such that BD=CD(given).Also,clearly side AD is common.Now we know median ...
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votes
2answers
22 views

How to prove that $\{0\} \cup \{1, \frac{1}{2}, \frac{1}{3}, …\}$ is not isolated

(As a subset of $\mathbb{R}$.) I am having trouble proving that $0$ is not an isolated point. If there exists an open ball with radius $\epsilon$ about $0$, I have to find a point of the form ...
0
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0answers
15 views

Epsilon-Delta Limit for Trigonometric Function

I'm studying an Epsilon-Delta proof for a trigonometric function: $$\lim_{x \to 1/9} \sin(x) = \sin(1/9)$$ This is the procedure from my (Italian) book: $$−\epsilon < \sin(x) − \sin(1/9) < ...
1
vote
0answers
7 views

Finding ring isomorphisms

Let $A$ be a ring with $0\neq 1$ such that $x^4=1, \forall x\in A$, with $x\neq 0$. My question is: to which ring is $A$ isomorphic? $A$ can be, for example, isomorphic to $\mathbb{Z_2}$. The question ...
0
votes
0answers
9 views

Embedding of a topological manifold

We know the celebrated 'Whitney embedding theorem' for smooth manifold that says any n-dimensional manifold can be smoothly embedded in $\ \mathbb R^{2n} \ $. Now my question is: Is there similar ...
0
votes
0answers
11 views

Diophantine equation by matrice?

I want to learn how solve simple ax+by=c with matrices (assuming that's the fasted method?), but it's difficult to find correct learning material. I've been through this process: 4386x + 89744y ...
1
vote
3answers
27 views

Does the => truth table break mathematical induction?

Since $F \Rightarrow F$ and $F \Rightarrow T$ both evaluate to $T$ with the truth table for $\Rightarrow$, does this not break mathematical induction? For example, once you show the base case holds ...
0
votes
1answer
8 views

Derivative of Poisson that approximates Binomial

Instead of a standard urn ball problem, I have many urns and balls. Many. One might say, a continuum of balls $B$ and urns $U$. The likelihood of a single urn having $x$ matches is, under the ...
2
votes
0answers
13 views

Supremum of a functional on $C^1([0,1])$

I've tried to solve the following problem (1.1.30 from Berkeley Problems in Mathematics, no solution is provided therein), but I'm not sure if my solution is correct. I wanted to ask you for your ...
0
votes
0answers
5 views

Eigen-decomposition of augmented block rectangular matrix

I have a rectangular matrix $\mathbf{X}_{n\times p}$ where the eigenvector decomposition of its inner product with itself is $$ \mathbf{X}^T\mathbf{X} = \mathbf{P}^T\mathbf{\Lambda P} $$ where ...
0
votes
2answers
14 views

Matrix exponential question

Wiki https://en.wikipedia.org/wiki/Matrix_exponential said: if a matrix A is diagonal $$A=\begin{bmatrix} a_1 & 0 & \ldots & 0 \\ 0 & a_2 & \ldots & 0 \\ \vdots & \vdots ...
0
votes
1answer
19 views

Why is the integral of a square always larger than the square of an integral?

I learned in physics that $\langle x^2 \rangle - \langle x \rangle ^2 = \sigma_x^2 \ge 0$ and thus $\langle x^2 \rangle \ge \langle x \rangle ^2$. In the case of continuous distribution, it becomes ...
0
votes
0answers
3 views

Proof for the fact that the method of characteristic equations is a valid method to solve recurrence relations

Could someone kindly point me to a proof of the fact that the method is characteristic equations is a valid way of solving recurrence relations? It seems fairly arbitrary to me. I would be grateful ...
0
votes
1answer
12 views

null empty set has 2 subsets?

The question in the book was: How many subsets does $\{\emptyset\}$ have? a) 0, b) 1, c) 2, d) 3. The answer was c. How can an empty set have 2 subsets?
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votes
3answers
15 views

The dice is rolled 10 times and the results are added with given conditions.

Q: A dice has one of the first 6 prime number on each its six sides ,with no two sides having the same number .the dice is rolled 10 times and the results added.the addition is most likely to be ...
2
votes
4answers
19 views

How would one work this out? (Fractions)

One fifth of all value lamps are already defected at the time of purchase. How many do you have to buy to ensure that you have 16 functioning lamps.? Anyone have any advice on how to layout this ...
0
votes
0answers
5 views

Project point on plane - Unique identfier?

I have a number of planes (in $\mathbb{R}^3$), each represented by a point $\vec{P_i}$ which lies within each plane and the normal vector $\vec{n_i}$. If I project a point $\vec{Q}$ (which does not ...
1
vote
0answers
8 views

Coupled partial diffential equation, with boundaries specification

Please, help me to find a books or samples to learn how to solve such coupled equations $$\begin{eqnarray} \frac{\partial T_1(x,t)}{\partial t}&=& \alpha_1 \frac{\partial^2 T_1(x,t)}{ ...
0
votes
0answers
14 views

Find a formula for $f(x, y)$ given the following assumptions…?

I've been going through some examples in my textbook ready for a uni exam in a few days, and I am having difficulty with a few of the questions, in particular this one: A gene is a sequence of ...
1
vote
1answer
9 views

Law of a random variable (characterization)

If $X$ is a real random variable defined on $(\Omega,\mathcal{F},\mathbf{P})$ then there exist several characterizations of the law of $X$ being $\mu$ : $X \sim \mu$ if and only if for every ...
0
votes
1answer
13 views

$(w^2+x^2).(y^2+z^2)$ is always divisible by which of the max no. Where w;x;y;z are positive odd integers?

Q $(w^2+x^2).(y^2+z^2)$ is always divisible by which of the max no where w,x,y,z are positive odd integers? Options given: 20;8;4;2 My Approach: I Choose ($9^2$+$5^2$).($7^2$+$3^2$) to get ...
0
votes
0answers
3 views

Projective dimension on factor ring

If $pdim_A M$ is the projective dimension of $M$ as an $A$-module how can i prove that if $A/I=A'$ then $$pdim_A M\leq pdim_A A' + pdim_{A'} M$$ If the left summand are finite? On a side note: if $M$ ...
0
votes
0answers
18 views

Taking limit inside integration

What the conditions, other than DCT and MCT, under which $$lim \int f_n(x) \, dx = \int lim f_n(x) \, dx $$ ? f_n are measurable functions DCT- Dominated Convergence Theorem MCT- Monotone ...
0
votes
1answer
23 views

Root test for convergence: $\displaystyle{\lim_{n\to\infty} (a+bi)^n}$

$$\lim_{n\to\infty} (a+bi)^n$$ where $i$ is the imaginary unit. I'm having trouble with this question. I get to $a+bi$ but I have no clue how to finish it in order to determine if it converges ...
1
vote
0answers
10 views

Trapezoidal Rule yielding the exact value of the integral

It is clear that if a function $f(x)$ is linear over the domain $a \leq x \leq b$, then one application of the trapezoidal rule, over the same domain, will yield the exact value of ...
0
votes
2answers
29 views

Is it true that $|f(x)|\leq |f^2(x)|$?

Is the following true for all $x\in\mathbb{R}$ and for all real functions f? $$\left| f(x)\right| \leq \left| f^2(x)\right|$$ Also, is it true that $|f(x)|\leq |f^3(x)|$?
0
votes
0answers
4 views

Divergence based robust inference

I have learnt that the inference based on minimizing the following divergence is robust to outlying observations for some specific range of $\alpha\in\mathbb{R}$. $$D_{\alpha}(g,f) = ...
1
vote
4answers
38 views

Integral $\int{ \frac{1}{\sqrt {1 - e^{2x}} } dx}$

I need a hint how to start solving this integral: $$\int{ \frac{1}{\sqrt {1 - e^{2x}} } dx}$$
0
votes
1answer
16 views

For the line $y = mx$, let $m = tan(\theta)$. Write $f(x, mx)$ as a function of $\theta$..?

I have a problem, and I am not sure how to solve it. This is the problem from my book: let $f(x, y)$ be given by the function: $$ f(x, y) = \begin{cases} \frac{2xy}{x^2 + y^2}, & (x, ...
0
votes
2answers
10 views

Is a matrix similar to its RREF?

Let a matrix be denoted by A and its RREF be denoted by R. Then, is it true that R is similar to A? I am trying to find out Jordan canonical form of a large matrix. If I can somehow prove that a ...
0
votes
1answer
19 views

General ODE question

Find the general solution $y(t)$ of the ordinary differential equation $$y''+\omega^2 y=\cos \omega t,$$ where $\omega>0$. I'm relatively new to ODEs and PDEs but can someone show me the ...
-1
votes
1answer
26 views

3x + 1 problem other repititions

The Collatz problem: Pick an integer x > 0 if x even: $x = x / 2$, if x odd: $x = 3*x + 1$ repeat 2.) as long as you want This algorithm seems to always end up with the loop 4-2-1 My actual ...
-1
votes
1answer
11 views

Find the population by the end of the same year…

The population of a type of insect is known to be 200,000 on 1st January in a particular year. Each month, the population increases by 75,000. Find a.) the total population by the end of the same ...
0
votes
1answer
11 views

Terminology - “Sample space” vs “sample set”?

Given that a "sample space" is defined as the set of possible outcomes of a given random experiment, is there a fundamental reason to use the term "sample space" instead of "sample set" in probability ...
0
votes
0answers
4 views

Galloway (extension of Myers theorem)

Let $M^n$ be a complete riemannian manifold, $n\geq2$. Let $\gamma:[0,L]\longrightarrow M$ be a stable arc length parametrized geodesic. Suppose there exists $c>0$ such that: ...
0
votes
0answers
6 views

Correct approach for Laplace of periodic function

I have a piecewise function $g(x)$ $g(x)=0$ for $x<0$ $g(x)=1$ for $x\geq0$ Now, i want to apply the Laplace transform for the following function $$f(t)=g(sin(\pi.t)).sin(\pi.t)$$ I now tend ...
0
votes
0answers
13 views

Love's equation $f(x)+\frac{1}{\pi} \int_{-1}^{1} \frac{f(t)}{1+(x-t)^2}dt=1, \ \ (|x|\geq 1)$

Let us consider Love's equation: $$f(x)+\frac{1}{\pi} \int_{-1}^{1} \frac{f(t)}{1+(x-t)^2}dt=1, \ \ (|x|\geq 1)$$ Is $f(x)$ a two times differentiable function?
0
votes
0answers
12 views

Hooke's Law - Modelling a 'Bungee Jump'

A question in my textbook asked me to find a general equation for depth fallen by an object of mass $75kg$ thrown from a bridge whilst tied to an elastic rope. Below the bridge there is a stream of ...
0
votes
0answers
8 views

Monoid filtration

I lately been introduced to monoid filtrations and I have a couple of questions. Let $(\mathfrak{M},\star,1_\mathfrak{M})$ be a monoid with total order, $(A,+)$ the additive subgroup and ...
3
votes
1answer
17 views

If $N$ is nilpotent then there exists $A$ such that $A^2=I+N$

Suppose $N\in M_{3\times 3}^{\mathbb{C}}$ is a nilpotent matrix. Prove that there exists $A\in M_{3\times 3}^{\mathbb{C}}$ such that $A^2=I+N$. Hint: find $A$ in the form $A=P(N)$ where $P$ is a ...
0
votes
0answers
11 views

Endomorphisms of $\mathbb C^{\times}$

What are all continuous multiplicative endomorphisms of $\mathbb C$? As $\mathbb C^{\times} \simeq \mathbb R_{>0}^{\times}\times S^1$ this question can be reduced to the description of continuous ...
0
votes
1answer
13 views

Is Bs(1,-1) linear?

I would like to prove that the Baumslag-Solitar group $BS(1,-1)=\langle a,b| bab^{-1}=a^{-1}\rangle$ is embeddable in $GL_n(\mathbb{Z})$ for some nonnegative integer $n.$ So i tried to find two ...
0
votes
1answer
15 views

Explain Kraft McMillan inequality and how it is applied.

I am going through some questions and answers regarding Information Theory and I found this question and its solution. Can some one explain this solution to me. We would like to encode a sequence of ...

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