0
votes
0answers
7 views

Eigenvalues of $A=\begin{bmatrix}2&1\\\alpha&0\end{bmatrix}$

Determine (with respect to $\alpha $) all Eigenvalues (in $\space \Bbb C$)of the Matrix: $A=\begin{bmatrix}2&1\\\alpha&0\end{bmatrix}$ $det(A-\lambda I)=\begin{vmatrix}2-\lambda&1&\\ ...
0
votes
1answer
11 views

What does it mean to say “f(x) ~ g(x), i.e. f(x) behaves like g(x) when x --> infinity”?

The limit as x approaches infinity of f(x)/g(x) is infinity, then f(x) grows faster than g(x). Same if the limit as x approaches infinity of g(x)/f(x) = 0. Would f(x) behave like g(x) if the limit as ...
3
votes
0answers
8 views

minimum eigenvalue for difference of two matrices

Let $A$ a positive definite matrix, and $B$ a matrix constructed from $A$ by setting all its off-diagonal elements to zero. Then is there a way to see for which values of positive scalars $a$ and $b$ ...
0
votes
0answers
3 views

Understanding Symmetric tensor field

I am reading an article in which author calls some basic tensor analysis result. He states in general we define on $\mathbb R^N$ that $$ \mathcal T^k(\mathbb R^N):=\{\xi:\,\mathbb ...
0
votes
0answers
10 views

Understanding Cohn's Radon-Nikodym proof from his book on measure theory

The part of the proof which i dont get is $\nu(A)=\int_{A} g \ d\mu$ where g is Radon-Nikodym derivative. He has a set of functions for which $\int_{A} f \ dx \le \nu(A) $, he has just shown the $g$ ...
0
votes
0answers
7 views

Asymptotical stability of a discrete dynamical system

There is a linear time invariant discrete system, \begin{align} x_{k+1}&=\tilde{A}x_k, \end{align} where $\tilde{A}$ is a block matrix represented by \begin{align} \tilde{A}= ...
0
votes
0answers
6 views

Gaussian Process with explicit basis functions

I am considering the Gaussian process with explicit basis functions as discussed in the book (section 2.7): http://www.gaussianprocess.org/gpml/chapters/RW2.pdf Has anyone tried to derive formulas ...
0
votes
0answers
7 views

Difference bewteen tilt angles and change of attitude (euler angles)

I would like to know if there is a difference, when estimating the change of atitude of a body, between tilt angles and attitude changes/variations from one epoch to an other. Is it the same thing? If ...
0
votes
0answers
11 views

Analytic version of Hahn-Banach using geometric version

When studying the Hahn-Banach theorem, one can demonstrate the geometric version from scratch and use it to prove the analytic version, as is outlined in Hahn-Banach theorem: 2 versions. To do so, it ...
2
votes
0answers
17 views

how to solve an affine differential equation

Is there a general way to solve $y'=Ay+b$, with $y, b \in \mathbb{R}^n$, $A$ a matrix, and where $A$ and $b$ are constant? I'm tempted to make the substitution $z = y+A^{-1}b$, and then use the matrix ...
2
votes
2answers
33 views

What's special about the cauchy product?

I am reading chapter 3 in Rudin's Principles of Mathematical Analysis. In it, Rudin defines the Cauchy product of two series. That is, given $\sum a_n$ and $\sum b_n$, $c_n=\sum_{k=0}^n a_k b_{n-k}$ ...
0
votes
0answers
6 views

Inducing representations from a subgroup of finite index.

Let $G$ be a group and $H$ a subgroup of finite index. Let $(\sigma , W) $ be a irreducible representation of $H$ (which need not be finite dimensional). 1) When can we extend this representation of ...
0
votes
1answer
12 views

Boolean equation - bitwise AND operator

I have equation: (x AND B) XOR x = C where x - is unknown variable, B and C are constant. I need just one solution x that will satisfy this equation. How I can do this?
0
votes
0answers
6 views

Why is $ab=ba=a^\ast b=ab^\ast=0$ ( orthogonal elements in a $C^\ast$-algebra)?

Let $a,b$ be elements in a $C^\ast$-algebra $A$, such that $a^\ast ab^\ast b=b^\ast ba^\ast a=0$, $a^\ast abb^\ast=bb^\ast a^\ast a=0$, $aa^\ast b^\ast b=b^\ast b aa^\ast =0$ and $aa^\ast ...
1
vote
2answers
30 views

Why is the topology on $\mathbb{R}$ formed by the basis $[a,b)$ normal?

Why is the topology on $\mathbb{R}$ formed by the basis $[a,b)$ normal? I need to prove that two disjoint closed sets are contained wtihin two open disjoint sets. First, I tried to understand how a ...
0
votes
0answers
13 views

KKT system with rank-deficient constraints

I have an optimization problem of the following form: $$ \begin{aligned} \operatorname*{minimize}_x & \quad \frac{1}{2}||x - a||^2 \\ \operatorname{subject~to} & \quad ...
1
vote
1answer
41 views

Trigonometric equation

I was solving this trigonometric equation $$\tan^2x+\cot^2x=2-\cos^{2014}(2x) $$ I solve it by inequality $|a|+\frac{1}{|a| }\geq 2$. I provided an answer below. Can someone give an alternative ...
0
votes
1answer
7 views

Rate of change for no stretch/compression

I am reading about cloth simulation from here. This is what one of the part says - Shouldn't the condition for no compression/stretching be Wu = 0 If there is no stretch/compression along ...
0
votes
3answers
30 views

Points on 3d line

Say we have $2$ points on a 3d line, point $A(x,y,z)$ and point $B(x,y,z)$. If we want to get the coordinates of a third point, beyond point $B$ but a certain distance from point $A$, how would we do ...
0
votes
1answer
19 views

Iterative solutions of linear systems

I do not understand that why $M$ must be invertible for $x^{(k+1)}$ to be uniquely specified in equation below: $$ Mx^{(k+1)} = Nx^{(k)} + b \quad (k=0,1,\ldots).$$ Why $M$ must be invertible? And ...
0
votes
1answer
42 views

Function with increasing property.

Prove that $\frac{1}{2}(x+2)^{-3/2}-(\frac{1}{2}x+3)(x+3)^{-3/2}$ is increasing function for $x\ge4$. I tried it by taking its first derivative but by first derivative for me its difficult to say it ...
3
votes
1answer
45 views

Application of the Jensen's inequality

Claim: If $x, y, z >0$ and $x+y+z = 3\pi, $ then $x^x + y^y + z^z > 81.$ My attempt: Let $f(w) = w^w$, so $f$ is convex on $(0, \infty).$ By Jensen's inequality, $f(x\frac{x}{3\pi}+ ...
0
votes
0answers
11 views

Generalization of Cantor Pairing function to triples and n-tuples

Is there a generalization for the Cantor Pairing function to (ordered) triples and ultimately to (ordered) n-tuples? It's however important that the there exists an inverse function: computing z from ...
0
votes
2answers
29 views

Sum of Converging sequence

I'm given this sequence where it goes: $$ 1,\; \frac1a,\; \frac1{a(a+b)},\; \frac1{a^2(a+b)},\; \frac1{a^2(a+b)^2}, \frac1{a^3(a+b)^2}, \dotsc $$ where $a$ and $b$ are any positive integers How ...
2
votes
4answers
72 views

Prove using mathematical induction that $x^{2n} - y^{2n}$ is divisible by $x+y$

Prove using mathematical induction that $x^{2n} - y^{2n}$ is divisible by $x+y$. Step 1: Proving that the equation is true for $n=1 $ $x^{2\cdot 1} - y^{2\cdot 1}$ is divisible by $x+y$ Step 2: ...
0
votes
1answer
14 views

Curve of intersection between surfaces

I want to calculate the curve of intersection between the following surfaces $$\Sigma: z=10e^{-x^2-\frac{1}{4}.y^2}$$ $$\alpha: z=2x-6$$ I can substitute $x=t$ $z=2t-6$ I equate the surfaces to ...
2
votes
0answers
17 views

Some analogs of the pentagonal number theorem

There are the following analogs of the famous identity $$ \prod_{n\geqslant1}(1-q^n)=\sum_{n\in\mathbb Z}(-1)^nq^{\frac{3n^2-n}2}. $$ Let $v_2(n)$ denote the 2-adic valuation of $n$, that is, the ...
0
votes
1answer
10 views

Find point on circle's tangent based on point on circle, radius and angle

The circle is centered at (0,0)"P" with a radius of 5. I have a point on the circle at (4,-3)"A". How would I find the points "B1" and "B2" on the tangent through point "A" given an arbitrary angle ...
5
votes
2answers
234 views

Fooled around with integrals and found something nice

One time I was bored and played around a bit with integrals and wolfram alpha and tested the following integral: http://www.wolframalpha.com/input/?i=integral_0%5E1+ceil%28x*sin%281%2Fx%29%29 Note: ...
2
votes
0answers
16 views

Generalization of the Vitali-Hahn-Saks Theorem

Is there a generalization of the Vitali-Hahn-Saks Theorem for nets of measures? I do not find any related literature.
1
vote
1answer
31 views

Stadium Seating - Geometric Sequences

A circular stadium consists of sections as illustrated, with aisles in between. The diagram show the tiers of concrete steps for the final section, Section K. Seats are to be place along every step, ...
5
votes
2answers
40 views

geometrical interpretation of a line integral issue

I was wondering : if the geometrical interpretation of a line integral is that the line integral gives the area under the function along a path, then why the line integral is equal to zero when the ...
2
votes
0answers
16 views

Probability of histogram bars

Say I collect data that follows a Normal distribution $f(z)$ in a histogram with bins of width $w$. I want to calculate the probability that the number of hits $N_i > N_j$. My naive approach would ...
2
votes
0answers
23 views

Is there a programmatic way to calculate cascaded sigma functions?

Let my format be sigma(function,from,to) = f(n) for example sigma(sigma(1 , j = 1 , j = i) , i = 1 , i = n) = (n^2)/2 ...
3
votes
1answer
27 views

Does the centroid of a triangle ever fall outside of its Morley's triangle?

Let $T$ be a triangle, and $M$ its (first) Morley triangle:                     (Image from Bruce Shawyer web page.) Q1. Does the ...
2
votes
0answers
10 views

How do I demonstrate Jordan measurability of a compact convex polytope?

Ex 1.1.9 in Tao's An introduction to measure theory asks us to show that any compact convex polytope in $\mathbb{R}^d$ is Jordan measurable. Is the following an efficient (or even valid) approach to ...
5
votes
1answer
58 views

Prime Numbers and a Two-Player Game

In this question, $\mathbb{N}_0$ is the set of all nonnegative integers. The notation $\mathbb{N}$ is reserved for the set of all positive integers. Alex and Beth is playing the following game. ...
1
vote
0answers
12 views

Properties of a Semi-modulo! operation

Let $A$ be an integer with its representation in base $p^n$ ($p$ may be prime number but not necessarily) described as: $$A=(a_ma_{m-1}\ldots a_1a_0)_{p^n}$$ We know $A\equiv (a_n\ldots ...
1
vote
1answer
14 views

Amount of match combinations of creating a 5 v 5 team from a pool

This question is inspired by the popular games: Dota 2, Heroes of the Storm and League of Legends; where players have to create two teams of 5 from a pool of "Heroes" in each match. How many ...
1
vote
0answers
10 views

Determine mutual location of two coordinate systems, given two sets of points

My problem is: we've got tracking device and a robot. Tracking device provides set of $n$ points in cartesian coordinates(taken from marker on robot arm) and robot driver returns position of TCP(tool ...
1
vote
0answers
10 views

composition and strong limits of completely positive maps is completely positive

I have two claims about completely positive maps. Let $X$, $Y$, $Z$ be $C^\ast$-algebras. 1)Let $f:X\to Y$ and $g:Y\to Z$ be completely positive maps. I want to know, why $g\circ f$ is completely ...
1
vote
1answer
44 views

AMT - Three whole numbers add up to 149 and multiply to give 987. What is the largest of the three number

So about this question I'm not too sure... Can't find out what I should start off with. If anyone can help me I'll be very greatly appreciated. The question is: Three whole numbers add up to 149 ...
1
vote
0answers
16 views

how to vectorize this kind of loop for array. [on hold]

I'm newbie to Matlab. I have some questions. How i vectorise this loop ...
3
votes
0answers
24 views

k! perfect matches

Let $G=($ $ A \cup B $ , $E$ $)$ be a bipartite graph with perfect matching. Denote $|A| = n$. Prove that if every vertex in A has degree $\geq$ $k$ then G has at least $k!$ perfect matches. ...
1
vote
2answers
18 views

a symmetric bilinear form has a basis such that it's matrix with respect to it is diagonal

I'm reviewing a proof regarding $f$, a symmetric bilinear form having a basis such that it's matrix with respect to this basis is diagonal. Here's a summarization: For $n=1$ there's nothing to ...
3
votes
0answers
18 views

Choice of order in the Leibniz rule is arbitrary?

One of the rules which characterizes the exterior derivative is that, for $\varphi$ a real-valued function and $\omega$ a $k$-form, we have $$d(\varphi \cdot \omega) = d\varphi \wedge \omega + ...
3
votes
3answers
72 views

Evaluate $\iint_{R}(x^2+y^2)dxdy$

$$\iint_{R}(x^2+y^2)dxdy$$ $$0\leq r\leq 2 \,\, ,\frac{\pi}{4}\leq \theta\leq\frac{3\pi}{4}$$ My attempt : Jacobian=r $$=\iint_{R}(x^2+y^2)dxdy$$ $$x:=r\cos \theta \,\,\,,y:=r\cos \theta$$ ...
2
votes
0answers
8 views

How are $M_n(A)$ and $M_n(\mathbb{C})\otimes A$ identified as Banach algebras?

When $A$ is a Banach algebra, one can make $M_n(A)$ into a Banach algebra by equipping it with various norms. One can also make the algebraic tensor product $M_n(\mathbb{C})\otimes A$ into a Banach ...
0
votes
0answers
18 views

How is Lipschitz continuity for Fréchet derivatives defined?

Let $(X,||\cdot||_X)$ be a Banach space and $X^*$ it's dual of linear functionals $X\to\mathbb{R}$. The Fréchet derivative $\nabla f(x)$ of a function $f:X\to\mathbb{R}$ at $x$ is an element of $X^*$. ...
-2
votes
0answers
36 views

Help me understand probability..I don't get it!! Exam tomorrow

Postcodes can be made from 4 digits --> 1234 How many different postcodes beginning with 2 are possible? How do I truly understand probability questions? Nothing beyond the level of ordered / ...

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