0
votes
1answer
9 views

Setting a function in function of another one

So I have H(E,X)=0.01EX and g(x)=0.02[x-0.001x^2) where G=H , so I want to redo the whole thing so everything is function of E, the result should be something like that Y(E)= 10E-0.4E^2 soo..how do ...
-1
votes
1answer
35 views

chosing between matrix theory and combinatroics

I have to take one more math course to finish my math minor , i am a computer science major and i want to know which course will benefit me more matrix theory or combinatorics and which takes more ...
1
vote
2answers
46 views

If $(1 + 2i)$ and $(3 - 2i)$ are two roots of $x^5 + ax^4 + bx^3 + cx^2 + dx + 4$, then $a$ =?

Consider the polynomial $x^5 + ax^4 + bx^3 + cx^2 + dx + 4$ where $a, b, c, d$ are real numbers. If $(1 + 2i)$ and $(3 - 2i)$ are two roots of this polynomial then what is the value of a? Well, I ...
2
votes
1answer
26 views

Formula for $\sum_{k=0}^n k^d {n \choose 2k}$

If $d \geq 1$ is an integer, is there a general formula for $$\sum_{k=0}^n k^d {n \choose 2k}\,?$$ We know that $\sum_{k=0}^n k {n \choose 2k} = \frac{n2^n}{8}$ and $\sum_{k=0}^n k^2 {n \choose 2k} = ...
0
votes
0answers
15 views

parallel transport_covariant derrivative

Let $u(s,t) $ be smooth and $X(t) $ vector field along $u(s_0,t) $. Denote with $P_s= P_s( u(s,t), u(s_0,t))$ parallel transport in $s$ direction along $u(s,t) $ from $u(s_0,t) $ to $u(s,t) $. Does ...
1
vote
0answers
15 views

Approximation of a quadratic form

Let $\mathbf{x}=(x_1,\cdots,x_n)^T\in\mathbb{R}^n$ and $A\in\mathbb{S}_{++}^n$ be a symmetric positive definite matrix. Also, let $Q\colon\mathbb{R}^n\to\mathbb{R}$ be the quadratic form given by $$ ...
-1
votes
1answer
22 views

Prove Every Function is a Relation

In my notes my professor has a question to prove that $\forall m,n \in \Bbb N^+$, $2^{mn}\ge n^m$. There is a suggestion that it can be proved by taking the logarithm of the inequality so that $mn ...
0
votes
1answer
13 views

What are numbers which are reverse digits of each other when divided by each other gives finite numbers after decimal point (non periodics)

Are there numbers which are reverse digits of each other when divided by each other gives a finite numbers after decimal point(non periodic)? For example (xyz are digits) xy/yx=abc.abcfinite and ...
0
votes
0answers
14 views

Laguerre's method and zero division

I'm trying to understand Laguerre's method for root finding and I have hit one road block. Suppose I have a polynomial $p(x) = x^4 + 1$ and an initial guess $x_0 = 0$. This results in division by ...
1
vote
0answers
13 views

Formula for calculation score based on distance

I try to write function for calculating scoring from distance in my game. I found something similar : link But I need the distance(x) to be between 0-31855000 meters and score between 0 to 1000. ...
0
votes
0answers
8 views

Get probability distribution from decision tree

I'm implementing decision tree based on CART algorithm and I have a question. Now I can classify data, but my task is not only classify data. I want have a probability of right classification in end ...
6
votes
3answers
70 views

Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$.

Question : Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$. Hint: consider decimal expansions ...
1
vote
1answer
25 views

Random conjecture

For any numbers $a$ and $b$ so that $a>b$ and $\text{gcd}(a, b)=1$, there exists a $c\in\mathbb{N^+}$ so that $a+bc$ is prime. I've only just tested this out with a few numbers, but I'm curious as ...
2
votes
1answer
44 views

How $\tan{\frac{A}{2}}\tan{\frac{B}{2}}=\frac{1}{2}$,then find $\angle C$

In $\Delta ABC$, if $$\tan{\dfrac{A}{2}}\tan{\dfrac{B}{2}}=\dfrac{1}{2}\\\sin{\dfrac{A}{2}}\sin{\dfrac{B}{2}}\sin{\dfrac{C}{2}}=\dfrac{1}{10}$$ Find the $\angle C$ My try: since ...
1
vote
0answers
30 views

solve this problem of trigonometry.

It is given : $$\sin(A-B)/\sin B = \sin(A + Y)/\sin (Y)$$ We have to prove $$\cot B - \cot Y = \cot(A + Y) + \cot(A - B).$$ Please help me solving this. I have tried to solve this by analyzing ...
0
votes
1answer
14 views

Cauchy sequence and uniform continuity

I read somewhere that because uniform continuous function maps cauchy sequence to cauchy sequence and cauchy sequence is bounded, so the function must be bounded. I am not sure if it is correct. My ...
2
votes
1answer
33 views

Help with integral from Apostol

I've been working on this all day and I'm still stumped. To state the problem (ref: Apostol Section 5.11, Question 29): Show $$ \int x^n\sqrt{ax+b}\,\,dx = \frac{2}{a(2n+3)}(x^n(ax+b)^{3/2} - nb\int ...
5
votes
1answer
24 views

Is a compact simplicial complex necessarily finite?

I'm aware that a finite simplicial complex is compact, and I am wondering whether the converse is true. If we have the topological realisation of a simplicial complex (not necessarily finite), $|K|$, ...
3
votes
2answers
40 views

Conjecture on twin primes

Let $p$ and $p+2$ be both prime. I conjectured (with my ignorance) that $$p^{\frac{p+1}{2}}\equiv -1\mod{(p+2)}$$ except for $p=17,41,71,137, 191, 239....$ I verified this on Mathematica. So for ...
1
vote
1answer
20 views

How to represent the notion that “at least one event occured”

Given two events, A and B, if it is the case that "at least one of the events occurred", I have seen it written that we can identify that probability as $$pr(A \cup B)$$ I'm struggling to understand ...
1
vote
1answer
18 views

Verify conditions of the Extension Theorem for a probability measure $\mathbb P$ of the Uniform[0,1] distribution

I am (self-)studying the book by Rosenthal called A first look at rigorous probability theory. My question is on verifying the conditions on a probability measure $\mathbb P$ of the Uniform[0,1] ...
0
votes
0answers
20 views

$\operatorname{Aut}(H_0)$ isomorphic to $\operatorname{GL}(3,\mathbb R)$?

Is $\operatorname{Aut}(H_0)$ (space of invertible endomorphisms over the space $H_0$) isomorphic to $\operatorname{GL}(3,\mathbb R)$ ? $H_0$ is the space of pure imaginary quaternions (which is ...
0
votes
1answer
25 views

finite and infinite, vector space, linear transformation

I have to answer these questions for homework and I don't know if I'm answering these correctly. I think most of them are correct, but a double check would be much appreciated. a) If $S$ is a set ...
0
votes
2answers
26 views

how do I figure out Which of the following is true?

I am studying for my exam and I am kind of stuck on this question, how is it that the answer is a)? can someone explain this please. Which one of the following is true? a) $$\sum_{k=0}^{n} ...
-1
votes
2answers
51 views

How to interpret $(2n)!$

It's all in question: how to interpret the factorial from $2n$? Is $(2n)!$ equal to $n!\times n!$ ? The problem is in Combinations if the combinations is $\binom{2n}3$. P.S. The main problem is ...
2
votes
2answers
90 views

A polynomial with integer coefficient

I'm struggling with this question: Suppose $P(x)$ is a polynomial with integer coefficients such that non of the values $P(1),...,P(2010)$ is divisible by $2010$. Prove that $P(n)\neq 0$ for all ...
0
votes
0answers
11 views

separable degree and the radical exponent

Let $\alpha$ be algebraic over $F$, where $charF=p\neq0$ and let $d$ be the radical exponent of $\alpha$. (which means $\alpha$ has multiplicity $p^d$) I am trying to show the following expression; ...
1
vote
2answers
16 views

Limit of Rational Expression with Repeating Integer

I'm wondering if someone could point me in the right direction for the following problem: I'm trying to express $(a + aa + \cdots + a\cdots a)/10^n$ as a summation that I can take the limit of as ...
1
vote
0answers
7 views

Symetric powers of $sl_2$ representations

I'd like to understand some special things about representations of $sl_2$ (which is considered as a Lie algebra over $\mathbb{C}$). First, it can be shown that for each $n\in \mathbb{N}$ there is ...
1
vote
2answers
36 views

finite sum of powers

Can you find what $\sum_{i=1}^{n}\left(n/i\right)^{i}$ is equal to? By simulation, I know that a loose upper bound is $2^n$. I am happy with a proof of such upper bound if an exact expression is not ...
0
votes
0answers
15 views

Approximation of a discontinuous function

I have a function from $R^n$ to $R$ such that $f_i(x_1,...,x_n)$ has a value of $1$ if $x_i$ is strictly above the $m$ th order statistics of $x_1,...,x_n$ $0$ if $x_i$ is strictly below the $m$ ...
0
votes
0answers
18 views

Some Cumulative Distribution and Density Function Question.

Suppose that x is normally distributed N(5.2,0.64), If Y=e^X; a.find the cumulative distribution function for Y. b.find the probability density function for Y. c.calculate the probability P(200 ...
0
votes
1answer
20 views

Inverse of a lower triangular Toeplitz matrix vs. the matrix size

I am recently trying to find the inverse of the lower triangular Toeplitz matrix ($\mathbf{A}$), with some special elements: $$ \mathbf{A}_{M\times M}=\left[\begin{array}{ccccc} 1\\ -a_{1} & 1\\ ...
0
votes
1answer
10 views

Divisibility by induction

I have learnt how to prove expressions by induction based on the use of three assumptions (a) n=1 (b) n=k (c) n=k+1 But can someone help me prove that 1+10^(2n-1) is divisible by 11
0
votes
2answers
27 views

Does singular values have anything to do with eigen-values of a square matrix?

We know from linear-algebra, how to calculate the singular values $\sigma_{n(A)}$ of a square-matrix, $A$ by square-rooting the eigen-values of $A^*A$ i-e $\sigma_{n(A)}=\sqrt{\lambda_{n(A^*A)}}$. ...
0
votes
0answers
12 views

Proving that the function set $\{ (2/l)^{1/2}\sin(n-\frac{1}{2})(\pi x/l) \}_1^{\infty}$ is an orthonormal set

I have the the following problem from my Fourier analysis book: Show that $\{ (2/l)^{1/2}\sin(n-\frac{1}{2})(\pi x/l) \}_1^{\infty}$ is an orthonormal set in $PC(0,l)$, i.e. class of piecewise ...
1
vote
1answer
25 views

Energy Transfer in a Mechanical System - Standard Pulley Scenario

I understand that this is quite a basic question, I am new to dynamics and have trouble starting off questions, I found it quite difficult to find an example question alike to the one below thus I am ...
0
votes
1answer
24 views

Eigen values and Eigen vectors

Let A be a 4x4 matrix with real entries such that $ \ -1,1,2,-2 \ $ are its eigen values.If $B=A^4-5A^2+5I$ ,where $I$ denotes the 4x4 identity matrix ,then which of the following statements are ...
1
vote
1answer
16 views

Conjecture about ordinal exponentiation: $n^ω=ω$ $\forall n\in\mathbb{N}$-{0,1}

Let $ω$ be the ordinal of the natural numbers. I think this is true: $n^ω=ω$ $\forall n\in\mathbb{N}-${0,1} Am i right? If I am wrong, is it true for any $n\in\mathbb{N}$?
0
votes
0answers
15 views

Is there a name for models whose every element is named by (one or more) variable-free terms?

Let $T$ denote a first-order theory. Is there a name for those models $M$ of $T$ such that for all $x \in M$, there is a variable-free term in the language of $T$ whose interpretation under $M$ is ...
1
vote
1answer
35 views

Asymptotics of sum of Binomial Coefficients (Binomial distribution) - Poisson approximation?

Let $$f(n):=\sum_{i=k}^n {n \choose i } p^i (1-p)^{n-i}$$ where $k\geq 2$ is a fixed Parameter and $p=p(n) \in (0,1]$ depends on $n$ where $np\leq 1$. We consider $n \rightarrow \infty$. I've found ...
0
votes
0answers
24 views

How to prove if XY is symmetric then Y is self adjoint? [on hold]

How to prove : If YX is symmetric then X is self adjoint. Given hint : Y is a symmetric matrix, $v^tYv > 0$ for all nonzero vectors v in R^n. v^t denotes the transpose of v. let X in ...
0
votes
2answers
14 views

line segment intersection

Do these two line segments intersect ? I'm confused because if you extend the below line then they will intersect otherwise not but we can't extend them as they are line segments. Is line segment ...
1
vote
0answers
15 views

Good introduction to number theory that develops and/or makes heavy use of commutative ring theory and lattice theory?

I'd like to learn some number theory, since it provides a lot of motivation for commutative ring theory and even some motivation for lattice theory (at least, that's the impression I'm under). ...
2
votes
1answer
40 views

If f and g are monotone functions, such that f is continuous and f(x)=g(x) for rationals x, then g is also continuous

Let f and g be monotone functions on R such that f is continuous and g(x) = f(x) for all rational numbers x, then g is also continuous on R. My idea is a such: Without loss of generality, we assume ...
0
votes
1answer
26 views

Contour Integrals, why give them a name?

I must ask because I can't find an answer, it's a bit of a soft question and for that I apologise. Why are they called contour integrals? In all the subheadings I've checked there is no special ...
1
vote
1answer
13 views

Solutions to a stochastic birth-death-immigration process

A population is undergoing a birth-death-immigration process. That is, the population size can increase by virtue of birth and immigration, and can decrease by virtue of death. The birth rate is ...
0
votes
0answers
24 views

proving regular expressions by induction [on hold]

i was given this question "Prove formally that L(R∗) = L((R∗)∗). [Hint: you may use proof by induction.]" ((R∗)∗) and (R∗) are basically regular expressions and L would represent it's language i ...
0
votes
0answers
17 views

how to find if a number is closer to a series of numbers?

Lets say you have a random series of real numbers. How to determine if a random test sample is closer to that random series within an acceptable threshold?
1
vote
6answers
126 views

Real life applications for logarithms [duplicate]

Can someone please tell me what purposes logarithms have in the everyday world? What non-theoretical applications are they in and when would one use them?

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