0
votes
0answers
5 views

Should I pick the higher dice?

Assuming I start with $n$ dice that have been rolled once, is it beneficial to choose the higher dice when I roll less than $n$ dice again (assuming I want a high roll)? In some board games, dice ...
-1
votes
0answers
5 views

Equivalence of a integer domain

If R is a commutative ring with 1. Suppose that for all polynomial $P(X)\in{R[X]\setminus{R}}$ has at most $n$ roots, with $n=grad(f)$ then $R$ is a integer domain. Any sugestion, please.
0
votes
1answer
11 views

Why do you need to understand distribution, dual space and functionals for fourier transform

I'm currently learning about treatment of dirac delta as a "tempered distribution", but the notation in this book is no short of unbelievable (Walter & Shen, Wavelets), explanations are very ...
0
votes
1answer
24 views

How to simplify recursive eq?

I know how to programatically calculate this, but im not sure how it can be simplified for documentation. Can someone help? $R = (X\cdot 1) + (X\cdot 2) + (X\cdot 3) + (X\cdot 4) + (X\cdot 5) + ...
1
vote
1answer
10 views

Conditional Expectation and Poisson

Let $X$ be the number of requests you receive at your web site per minute, where $X ∼$ POI$(10)$. Each request, independently of all other requests, is equally likely to be routed to one of $N$ web ...
0
votes
0answers
6 views

Show that $X_n \stackrel{d}{\longrightarrow} 0$ iff $\{\varphi_n(t)\}$ converges to 1 in some neighbourhood of $t=0$.

$X_n$ is a sequence of random variables, and $\{\varphi_n(t)\}$ is the corresponding sequence of characteristic functions. Show that $X_n \stackrel{d}{\longrightarrow} 0$ iff $\{\varphi_n(t)\}$ ...
1
vote
2answers
38 views

Solving the power equation $A^X = \frac{(1-X)}{(1+X)}$

I want to solve the following power equation (get $X$ value): $A^X = \frac{(1-X)}{(1+X)}$, where $X\neq 0$, $A\in {\mathbb R}$ (a real number) I think $X$ should be a complex number.
0
votes
0answers
11 views

sorting young's tableau better than n^3

Young's tableau takes $O(n^3)$ to sort. However, if we build heap and then sort it we can do better than $n$-cube ? i.e. $$n^2 + n^2(\log(n^2)) = n^2 + 2n^2\log n = n^2\log n$$ ? or am I missing some ...
0
votes
0answers
8 views

Multilinear quaternion interpolation

I'm looking for literature to study more on multilinear quaternion interpolation. Looking for 'trilinear interpolation' and 'tricubic interpolation' on Google Scholar or arxiv doesn't yield much ...
1
vote
3answers
22 views

Finding kernel of a particular homomorphism

Let $f:\mathbb Z \to S_8$ be a homomorphism such that $f(1)=(1426)(257)$ , then how to compute $\ker(f)$ and $f(20)$ ? I know that $f(n)=f^n(1)$ but this seems too tedious ; please help
0
votes
0answers
17 views

Question on radius of convergence

Can anyone help me with the following problem: I have a solid geometric picture of what is going on in my head, but I can't seem to turn that into a proof.
0
votes
1answer
38 views

not following two steps in proof that $\int_{0}^{\infty} cos(x^2) = \frac{\sqrt{2 \pi}}{4}$

Hi: I'm reading some notes I found on complex analysis on the internet. In the example, they prove that $$\int_{0}^{\infty} \cos(x^2) = \int_{0}^{\infty} \sin(x^2) = \frac{\sqrt{2\pi}}{4}.$$ I ...
0
votes
0answers
21 views

Differentiability and $L^1, L^2$ spaces

If $f\in L^1(\mathbb{R})$ then $\frac{d}{dx}\{f(x)\}\in L^1(\mathbb{R})$ where we have given that $f$ is of compact support.
0
votes
0answers
17 views

Proof that $S^\perp$ is a subspace of a vector space $V$

Just doing some review for a final exam and would like some feed back on the following proof if anyone would like to help me out. First the premise. Let $V$ be a finite dimensional inner product ...
0
votes
1answer
19 views

Can you uniquely define a tangent line at a point for a 3D csurve?

Let f be a function of the form: $x=f_x(t); y=f_y(t);\text{ and }z=f_z(t)$. Does the derivative set of the 3 functions mean the tangent at a point on the curve of f? Thank you in advance.
0
votes
0answers
7 views

Does consistent estimators have in-variance property?

If $(T_n)$ is a sequence of consistent estimators of a parameter $\theta$ ( i.e. for every $ \epsilon >0$ , $\lim_{n \to \infty} P [ \space |T_n -\theta|< \epsilon ]=1$ ) , then is it true that ...
0
votes
0answers
31 views

Is it true that $\int_{-C} f(x, y)ds = -\int_C f(x, y) ds$

I think it is more of a convention question, right ? $$\int_{-C} f(x,y)ds = -\int_C f(x,y) ds$$
0
votes
0answers
14 views

Adding matrices with transposition

Let's say I have two arrays as follows: A = [1,2,3] B= [1,2,3] Can I add $A + B^t$ ?
-1
votes
0answers
16 views

f(x)= {$ \uparrow$ when $x \in K$, 2 when otherwise } is partial recursive?

Recently i learned some material on partially recursive topic. Partially computable functions are also called partial recursive, and computable functions, i.e., functions that are both total and ...
0
votes
1answer
26 views

Find a subgroup of $S_4$ that is isomorphic to V, the Klein group.

So I know that the Klein group is the group with 4 elements that is not cyclic but I'm stuck from there onwards?
-1
votes
1answer
31 views

What is the value of a limit of sums with $\sin k$

What is the value of the following limit? $$\lim_{t\to \infty} \sum_{k=1}^{\lfloor 10^t π \rfloor} \sin k $$ I don't know what to do. I need your help. Thank you.
0
votes
0answers
40 views

Example of Something That's Not A Manifold

Two examples of non-manifolds that I know are the cross and the cone. Also the sphere with a hair isn't a topological manifold. But what's an example of a topological space $X$ such that $X$ is not a ...
0
votes
1answer
16 views

Condition for Orientability of Manifold

Let $M^n$, $n>2$ be a manifold and let $f:D\rightarrow M$ be an embedding of the closed $n-$disk in $M$. Prove or Disprove: $M$ orientable iff $M-f(D)$ is orientable. $M$ is orientable iff all ...
1
vote
2answers
26 views

Base of Subspace with vectors

Let E be the vector subspace of $R^3$ generated by it vectors $v1 = (1,2,0)$ and $v2 = (-1,0,2)$ How can find a basis of E between the following vectors? $$w1=(-2,-12,8), w2=(-12,-2,-8), ...
0
votes
0answers
33 views

Alternatives to the notation $\|x\|$ for the norm of $x$?

For aesthetic reasons, I don't like the notation $\|x\|$ for the norm of $x$. Have any alternatives been proposed?
0
votes
1answer
24 views

The number of subgroups conjugate to a given subgroup of a finite group

Let $H$ and $K$ are subgroups of $G$ conjugate to each other. $A$ is defined as $$A = \{a \in G \mid aHa^{-1} = H \}$$ for all $a\in G$. Prove that $A$ is a subgroup of $G$ and prove that if $G$ ...
3
votes
4answers
54 views

Propositional logic problem about a conversation of four people who lie or tell the truth

This is obviously elementary but can't figure it out. I am taking a course named Logic and Introduction to Analysis next semester and wanted to do some reading beforehand but to figure out how deep ...
1
vote
0answers
19 views

A point minimizing total great circle distance to a given set of points on a hemisphere

If you have a set of points on a hemisphere, how do you find a point on that hemisphere that has the minimum total great circle distance to the points in the set.
2
votes
0answers
33 views

Connection between integral expression and the factorial of infinity

Does the fact that $$\int_{-\infty}^{\infty}\exp\left(-\frac{1}{2}x^2\right)\mathrm{d}x=\sqrt{2\pi}$$ Have something to do with the fact that the regularized factorial of infinity is also ...
0
votes
1answer
35 views

Why is $x\mapsto x$-th prime number a partial recursive function?

I think that partial recursive functions correspond to all computable functions. Thus, if we can write a computer program to represent a function, the function is partial recursive. In computability ...
0
votes
0answers
8 views

Reference for p-capacitary functions

Consider $\Omega_1 $ and $\Omega_2$ two open bounded and convex sets in $R^n$with $\Omega_1 \supset \overline{\Omega}_2$. The unique weak solution of the problem $$ \begin{cases} \Delta_p u = 0 ...
0
votes
2answers
19 views

Can zero rows in matrices be ignored in calculations of matrix products?

I understand that when calculating the product of 2 matrices you need to account for the dimensions. But when there is an empty row in one of the matrices, why does it need to be accounted for? What ...
0
votes
1answer
66 views

Why didn't Fermat provide proofs of his theorems?

Apparently Fermat stated but didn't provide proofs of various theorems named after him, including Fermat's little theorem, Fermat's theorem on sums of two squares, Fertmat's polygonal number theorem, ...
0
votes
0answers
10 views

Presheaf of real valued functions

Seen as how a Presheaf of real valued functions on a topological space X associates a function f:U→ℝ to each open set U, what function maps the empty set to ℝ since the empty set is by definition an ...
0
votes
0answers
24 views

Prove that $\exists$ U: $P$ is self adjoint if and only if $P=P_U$

Suppose $P \in L(V)$ is such that $P^2 = P$. Prove that there is a subspace U of V such that $P= P_U$ if and only if P is self adjoint. First suppose that $P = P_U$ Show this implies that P is self ...
0
votes
3answers
59 views

If pi (or tau) is a constant for a circle, what is the equivalent of a sphere called?

I never learned this in math class, but was just thinking that a sphere should have some equivalent to pi, and that it too should be a constant... what would that be? Thinking more about it, if the ...
2
votes
2answers
21 views

Proof of multiplicative inverse for polar complex numbers [duplicate]

Use the polar form of complex numbers to show that every complex number $z\neq0$ has multiplicative inverse $z^{-1}$. If $z=a+bi$, then the polar form is $z=r(\cos(\alpha)+i\sin(\alpha))$. I can do ...
-1
votes
2answers
57 views

Prove that $T-\sqrt{2}I$ is invertible.

Suppose $T \in L(v)$ is such that $\|Tv\| \leq \|v\|, \forall v \in V$. Prove that $T-\sqrt{2}I$ is invertible. I know that I need to show there exists a $R^{-1}(T-\sqrt{2}I) = I$ where $R = ...
0
votes
1answer
15 views

Probability inequality exchanging sum with cardinality

Let $P_{XY}$ be the joint probability distribution of discrete random variables $X$, $Y$. Then I would like to prove the following inequality: $$ \sum_{y}\max_xP_{XY}(x,y)\leq |Y|\max_xP_X(x) $$ ...
1
vote
2answers
20 views

Line parallel to a plane and have 45 degrees between another

I need to find a direction vector for a line parallel to a plane $x+y+z = 0$ and that have $45$ degrees with the plane $x-y = 0$ So, i've assumed the vector $\vec V_r = (a,b,c)$ and since it is ...
1
vote
2answers
16 views

Determining the maximum % below average

Is there a way to determine the maximum percentage of values that fall below the average in a given sample? How would someone go about this? How does this relate to what Markov's inequality and ...
2
votes
0answers
18 views

Finding points where a smooth map between differential manifolds is or is not an immersion.

I am having trouble answering questions pertaining to immersions on smooth manifolds. For example: Given the unit sphere $S^2$ around the origin in $R^3$ and the map $f: S^2 \rightarrow R^3$ given ...
0
votes
0answers
12 views

Statistics: How would I correlate many variables to a few coefficients?

I have around 550 asymmetrical sigmoid curves fitted to a function with 4 varying coefficients. Each of these curves represent strength as a function of time and temperature for a different compound. ...
2
votes
1answer
68 views

A certain “harmonic” sum

Is there a simple, elementary proof of the fact that: $$\sum_{n=0}^\infty\left(\frac{1}{6n+1}+\frac{-1}{6n+2}+\frac{-2}{6n+3}+\frac{-1}{6n+4}+\frac{1}{6n+5}+\frac{2}{6n+6}\right)=0$$ I have thought of ...
1
vote
3answers
23 views

Interest Theory- Annuity Withdrawals/Deposits

"Consider an investment of $5,000 at 6% convertible semiannually. How much can be withdrawn each half−year to use up the fund exactly at the end of 20 years?" To solve this problem, an equation of ...
4
votes
0answers
22 views

Proper Forcing and Sequence of Names for Reals.

I read something that seems to suggest the following is true: If $\mathbb{P}$ is a proper forcing, $|\mathbb{P}| = \aleph_1$, and $\mathsf{CH}$ holds, then there exists a sequence $\{(p_\xi, ...
1
vote
0answers
20 views

How can we show a data set satisfies the manifold assumption?

In machine learning, we often assume that a data set lies on a low-dimensional manifold (the manifold assumption), but is there any formal proof saying that assuming the data set satisfies certain ...
0
votes
1answer
26 views

The ring of fractions $K(x)$ is the field generated by $K$ and $x$.

I would like to show that the ring of fractions $K(x)$ of $K[x]$ in an extension $L$, where $K\subset L$ fields, is the field generated by $K$ and $x$ (let's call it by $\tilde{K(x)}$). I know just ...
1
vote
0answers
29 views

are all md5 hashes reachable by hashing hashes [on hold]

If you hashed every possible md5 hash would the output be every known md5 hash or are there deadzones? ...
1
vote
0answers
20 views

Gradient descent via polynomial approximation

It seems that most proofs of convergence for gradient descent algorithms rely on strong conditions on the first and second derivatives of the function, for instance that $$|f''(x)| \leq K$$ over the ...

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