0
votes
0answers
5 views

SVD and base changes matrices

I'm not hugely comfortable with linear algebra, so wanted to double check that the following reasoning was correct. Does it hold that, given two matrices R and B U R B Ut=U R Ut U B Ut= U R Ut D I ...
0
votes
0answers
16 views

I have calculate an integral. Is it correct?

Could you please help me to check this integral? Is it correct? \begin{align} I&=\int \limits_{\gamma_0}^{\infty}\frac{exp(-x D_6)}{(x A_3 +1)(x A_4+1)^{m}}dx \\ &=\int ...
1
vote
0answers
11 views

Divisibility of group exponents when the subogroup has finite index.

Let $G$ be a group (not necessary finite) and $H$ a subgroup of $G$ of index $n$. Show that $exp(G)|exp(H)\cdot n$. Remarks. -The case when $H$ is normal is a consequence of the group structure on ...
-1
votes
1answer
10 views

Double-check my answer in calculating gravitational potential energy

"A satellite with a mass of 200kg maintains its orbit at an altitude of 300km above the Earth's surface. The Earth has a radius of $6.38 * 10^6m$ and a mass of $5.97 * 10^24$" I was told this was the ...
0
votes
0answers
11 views

Prove the Basis of Column Vectors

Let $V$ be a vector space over field $\mathbb{F}$. Let $B=\{u_1,...,u_n\}$ be an ordered basis of $V$. Show that $\{[u_1]_c,...[u_n]_c\}$ is a basis of $M_{n,1}(\mathbb{F})$ for every ordered basis ...
0
votes
1answer
16 views

The cardinal of the set of all measures on $\mathbb{R}$

It is a very simple question that I don't know how to do: Let $M = \{\mu \colon \mathcal{B}(\mathbb{R})\to \mathbb{R} \colon \mu \text{ is a measure}\}$ $$|M| = \ ?$$ Any help will be appreciated.
1
vote
0answers
9 views

Existence of nonstandard elementary extensions of $PA$?

My question follows from the 1958 result of MacDowell–Specker (located originally in Modelle der Arithmetik, J. Symbolic Logic Volume 38, Issue 4 (1973), 651-652) of the proof of the following ...
1
vote
0answers
15 views

How to compute the volume of such a triangular prism shaped object

like this : where $z_i$ are not equal.
0
votes
0answers
20 views

Where did I go wrong with this definite integration?

I'm trying to solve the definite integral $\int_0^n\pi^{ex}dx$ Wolfram says that the answer is $\frac{\pi^{en}-1}{e \ln(\pi)}$, but I got $\frac{\pi^{en}-1}{\ln(\pi)}$. Can anyone help me figure out ...
0
votes
0answers
7 views

Error analysis of kernel density estimation

Let $X$ be a random variable with true density $f$, $Y = \{y_i\}_{i=1}^n$ be a realization of a random variable in $d$-dimensional space $R^d$, and $\hat{f}$ is the density estimator of $Y$ using ...
2
votes
0answers
11 views

Find optimal matrix to maximize the expected expression

I am interested of the following function with $Q$. $$f(Q)=h^TQh-\frac{1}{(h^TQh+1)^2}-g^TQg+\frac{1}{(1+g^TQg)^2}$$ where $h$ and $g$ are both given $N\times 1$ vectors. And $Q$ is $N \times N$ ...
0
votes
1answer
10 views

Why is $\hat{I}$ contained in the Jacobson radical $J(\hat{R})$?

Suppose $I$ is an ideal of a commutative ring $R$, and $\hat{R}$ is the $I$-adic completion. I don't follow why $\hat{I}$ is in $J(\hat{R})$. I know $\hat{R}$ is complete wrt the $\hat{I}$-adic ...
0
votes
0answers
27 views

Nice approximations of sums by integrals.

Let $f(x):\Bbb Z^+\rightarrow \Bbb R^+$ be a non-monotone function. If for every $m\in\Bbb N$, $$S(m) =\sum_{n=1}^N\frac{1}{(1+f(n))^m}$$ be sum of interest, then is there a way to study this ...
1
vote
1answer
9 views

Height and coheight of an ideal

Given an ideal $\mathfrak{a}$, Matsumura defined the height of $\mathfrak{a}$ as: $$\text{ht}(\mathfrak{a})=\inf_{\mathfrak{p}\in V(\mathfrak{a})}\text{ht}(\mathfrak{p})$$ He states that: ...
2
votes
1answer
22 views

tesnsor product of two copies of $\Bbb{R}$ over $\Bbb{R}$

I would like to know what would be tensor product of set of reals over reals would be? That is, $\Bbb{R} \otimes_\Bbb{R} \Bbb{R}$ I think it should be $\Bbb{R}^{2}$ as tensor product combines two ...
0
votes
0answers
17 views

Branch of the cube root function in $\mathbb{C} \backslash [0,\infty)$

Let $D=$$\mathbb{C} \backslash[0,\infty)$ and define $f:D\to \mathbb{C}$ by $$f(z) = \begin{cases} z^{1/3} & \text{if }\operatorname{Im} z \geq 0, \\[6pt] \tfrac{1}{2}(-1+i\sqrt 3)z^{1/3} & ...
0
votes
0answers
4 views

Comparing marginal on product space with other measure

My previous post Unifying the treatment of discrete and continuous random variable, got successfully answered and allowed me to get further in my results. However I am facing a question that I can't ...
0
votes
4answers
25 views

Absolute convergence in a metric space

Let $(X,d)$ be a metric space, $(a_n)$ and $(b_n)$ are sequences in $(X,d)$. If $\sum_{n=1}^\infty d(a_n,b_n)$ is absolutely convergent, what do I say about the convergence of $(a_n)$ and $(b_n)$?
0
votes
0answers
12 views

sum of two curvatures

I have a question about sum of two relative curvature and torsion. The problem is as follow. Suppose we have three points, Ps, ...
0
votes
1answer
24 views

Prove an upper bound for the determinant of a matrix A

a) Let A be a 3x3 real matrix with all $0\le a_{ij} \le 1$. Show that det(A) $\le$ 2 and find such matrices with det(A) = 2. b) Let A be a nxn matrix with all $0\le a_{ij} \le 1$. Estimate ...
1
vote
1answer
37 views

Does $\lim_{x\to \infty}f(x)$ exist?

For any $x\in \Bbb R$, $h(x)\le f(x)\le g(x)$ and $\displaystyle\lim_{x\to \infty}(g(x)-h(x))=0$. Then does $\displaystyle\lim_{x\to \infty}f(x)$ exist? Thanks for your help.
0
votes
2answers
25 views

Calculating interaction beween 100 objects with each other.

The other day I was thinking about how many interactions 100 objects would have with each other. By that I mean if we are using a computer to draw the scene with 100 point lights, the total result ...
0
votes
0answers
20 views

Let X be a space with topology T, let U be the collection that U={(X\A)|A in T}

For X as a finite set, when does U form a topology? For X=S3 with its usual( Euclidean) topology, when does U form a topology?
0
votes
1answer
11 views

How to calculate number of elements in HomSet

Im giving category theory a chance but have very limited math background, I'm learning from the book "Category theory for the sciences" but got lost on page 16 :) ...
1
vote
1answer
10 views

How to apply non-linear regression to Logistic (sigmoid) curve

I've been looking at a useful way to represent Doppler shift from a satellite passing over a ground station. I've calculated the Doppler shift frequency values at 1-second interval for the duration of ...
1
vote
2answers
23 views

Function with zero description

Is there a nice expression (possibly differentiable outside $0$) for a function $f(x)$ that satisfies the following property other than the delta? $$f(x)\neq0\iff x=0$$ $$f(x)=0\iff x\neq0$$ Is it ...
1
vote
2answers
16 views

Can we ignore predicates in a statement if they aren't used?

Prove/disprove: $$\forall a>0:a\in\mathbb R: \exists N\in\mathbb R:\forall x\in \mathbb R:\exists z\in\mathbb R:\forall n\in \mathbb N:|n-99|<N\Rightarrow n>10 \vee \frac {n^2} 4 \le 25$$ ...
1
vote
1answer
29 views

$L^1$ and $L^2$ norm inequaliy

Consider real valued function $f$ defined on $[0, T]$. L1 norm and L2 norm of function $f$ are given by $$ \|f\|_1=\int_0^T |f(t)| \, dt $$ and $$ \|f\|_2=\sqrt{\int_0^T |f(t)|^2 \, dt } $$ Then we ...
-4
votes
0answers
13 views

Calculation of rate of interest on monthly installments. [on hold]

In a Bankers scheme I have to pay Rs 5222 per month for a period of 10 years (120 months), Banker will return Rs 10,00,000 after 10 years. what is the rate of interest.
-2
votes
0answers
20 views

circles word problems! geometry section [on hold]

In the given figure, triangle $ABC$ is inscribed in a circle with center $O$. if angle $\angle ACB$ equals 65 degree then what is angle $\angle ABC$?
2
votes
2answers
20 views

Find position on line given start and end points

I'm trying to solve this problem for myself which is for latitudes and longitudes. For the illustration below - I know ALL the variables EXCEPT $(a_1,b_1)$ which the latitude and longitude I need to ...
0
votes
0answers
16 views

Evaluate $\displaystyle\sum_{i=1}^nF^k_{p_i}$

Let $F_{p_n}$ denote the $p_n$-th Fibonacci Number where $p_n$ is $n$-th prime. Now define, $$\mathfrak{S}_k=\displaystyle\sum_{i=1}^nF^k_{p_i}$$ Is there any formula for $\mathfrak{S}_k$? I ...
-2
votes
0answers
22 views

Questions about the map on $\pi_1$ induced by the inclusion $i : S^1 \to S^2$ [on hold]

Let $A = S^1$, $X = S^2$ and let $i : A \to X$ be the inclusion. What homomorphism does $i$ induce on $\pi_1$? Is this homomorphism injective/surjective? Does inclusion of spaces $i$ have a ...
0
votes
1answer
14 views

About rings with a right multiplicative identity $1$ such that every element in $R$ \ $\{0\}$ has a left-inverse

$R$ be a ring with a right multiplicative identity $1$ such that for every $a \in R$ \ $\{0\}$ , $\exists x \in R$ such that $xa=1$ i.e. every element in $R$ \ $\{0\}$ has a left-inverse ; then is it ...
2
votes
0answers
55 views

Is this a valid proof for $1+1=2$? [duplicate]

I am extremely new to proofs, and quite bad at them. In studying and practicing the different types of proofs, I developed this very rough proof that $1+1=2$, one of the simplest mathematical truths I ...
0
votes
3answers
22 views

Metrics (Distances) on $\mathbb{F}$ Theorem Proof

I had a question regarding a Theorem I had come across that described metrics (distances) on ordered field $\mathbb{F}.$ Here it is: Theorem: If $\mathbb{F}$ is an ordered field, then $d(x,y)=|x-y|$ ...
0
votes
0answers
9 views

Find the largest ball where some conditions are true

Given a set $ \Omega\subseteq\mathbb{R}^n $ and $ f:\Omega\to\mathbb{R}^n $, with $ f\in C^0(\Omega) $, i need to find the largest ball, centered in $ x_0\in \Omega $, where some condition on $ f $ ...
2
votes
1answer
55 views

A combinatorial proof of Wilson's Theorem

I am looking for a combinatorial proof of Wilson's Theorem. Something along the lines of this kind of proof. $\textbf{Combinatorial proof of Fermat's Little Theorem}$ First consider a $p$ -tuple and ...
0
votes
1answer
18 views

Detail in the derivation of the Miller-Rabin test

In the derivation of the Miller-Rabin Primality (or actually, "probably composite") test, Wikipedia http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test as well as other sites (including ...
1
vote
2answers
40 views

Prove that U and W are contained in U+W

Let $V$ be a vector space and $U\leq V,W\leq V$ Prove that $U$ and $W$ are contained in $U+W$. I do not understand the requirement of the question. Is it prove that $U,W$ is the subset of $U+W$ or ...
0
votes
1answer
22 views

Finding the equation of a 3D surface bounded by 3 known curves

I am trying to find the equation of a 3D surface as illustrated below. The boundaries of this surface is comprised of two planar elliptical arcs $AB$ and $AC$ as well as a 3D arc $BC$ which is a 3D ...
0
votes
0answers
18 views

Merely conditionally convergent series [on hold]

What is the definition of 'Merely conditionally convergent series'? Is it exactly same as 'conditionally convergent series'? Or different??
0
votes
2answers
37 views

Find the area of a region using Green's Theorem.

Let $R$ be the region enclosed by the curve parameterized by $g(t) = (t^4 - t^2, t^6 - t^2)$, where $0 \leq t \leq 1.$ Find the area of $R$ using Green's Theorem. In order to use the green's theorem, ...
-1
votes
0answers
19 views

Movement of Horse Position during a race

I am trying to determine how to trace a horses position in running during a race and sort them in order of the horses have the fastest foot speed. Here is a sample of the data: ...
0
votes
0answers
39 views

A basic O.D.E doubt

Consider the non-autonomous O.D.E $\dot{x}(t) = \int h(x(t),y)\mu(t,dy)=F(x(t),t)$ where $\mu(t,dy) = \delta_{y_n}(dy)$ when $t \in [t_n,t_{n+1})$ where $y_n$ and $t_n$ are given sequences s.t. ...
1
vote
1answer
17 views

Question around the following relation: $T(n,1) = n$, for a positive integer $n$, and for all $k\geq 1,\ T(n,k+1)=n^{T(n,k)}$.

I'm beginning the studies on number theory and then i'm facing the following problem that i couldn't solved yet: given a positive integer $n$ and being $T(n,1)=n$ and, for all $k\ge1$, ...
-4
votes
0answers
19 views

Sigma equation statistics problem [on hold]

I've been solving this for days and can't prove the equation.
0
votes
1answer
28 views

Space of continuous functions linear operator eigenvalues

Let $V$ be the vector space of continuous functions from $\mathbb R$ to $\mathbb R$. Let $T$ be the linear operator on $V$ defined as $$(Tf)(x)=\int_0^x f(t)\,dt$$ Prove that $T$ doesn't have ...
1
vote
0answers
12 views

Not Univalent on the open disk, but univalent on any annulus

This is a problem from old qualifying exam. Let $f$ be an analytic function on the open unit disk $D$. Suppose that $f$ is not univalent on $D$. Show that $f$ cannot be univalent on any ...
2
votes
2answers
36 views

Does there exist a subfield $S$ of $\mathbb C$ such that $\mathbb R \subset S \subset \mathbb C$?

Does there exist a subfield $S$ of $\mathbb C$ such that $\mathbb R \subset S \subset \mathbb C$ ? ; I kind of have a feeling that there does not exist any such $S$ but cannot prove . Thanks in ...

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