text stringlengths 83 79.5k |
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H: Using a weighted table?
I'm writing a simple computer program that has to use the following weighted table to pick a random "payment" where the relative probabilities are given by "weight".
How does one interpret the table above and use it to determine how much to "pay"?
EDIT: After speaking with some folks it see... |
H: A puzzle with powers and tetration mod n
A friend recently asked me if I could solve these three problems:
(a) Prove that the sequence $ 1^1, 2^2, 3^3, \dots \pmod{3}$ in other words $\{n^n \pmod{3} \}$ is periodic, and find the length of the period.
(b) Prove that the sequence $1, 2^2, 3^{3^3},\dots \pmod{4}$ i.e... |
H: Fibonacci numbers and proof by induction
Consider the Fibonacci numbers $F(0) = 0; F(1)=1; F(n) = F(n-1) + F(n-2)$.
Prove by induction that for all $n>0$,
$$F(n-1)\cdot F(n+1)- F(n)^2 = (-1)^n$$
I assume $P(n)$ is true and try to show $P(n+1)$ is true, but I got stuck with the algebra. How do we reach $P(n+1)$ from... |
H: Proving pointwise convergence to a "Dirichlet-like" function
Question:
Let $\{r_1, r_2, \dots\}$ be the set of rationals in the interval $[0,1]$. For $x \in [0,1]$ and $n \in \Bbb N$, let $f_n(x)$ and $f(x)$ be given by the following:
$$ f_n(x) = \begin{cases}
1 & \text{ if } x= r_1, \dots, r_n \\
0 & \text{ other... |
H: Symbol for Euclidean norm (Euclidean distance)
Which symbol is more commonly used to denote the Euclidean norm: $ \left \| \textbf a \right \| $ or $ \left | \textbf b \right |$?
AI: As mentioned above, I don't know what is most common (statistically). However, ff you have a vector $V$ space over say the real numb... |
H: What is my mistake when evaluating this limit?
$\lim_{x\to 0^+} \frac{1}{x}=\infty$
But with l'Hospital's Rule
$\lim_{x\to 0^+} \frac{1}{x}=\lim_{x\to 0^+} \frac{0}{1}=0$
So where's my naive mistake?
AI: I believe l'Hopital's rule works only for indeterminate forms. |
H: What is the advantage and disadvantage of Hilbert System?
I have an exam in the course about higher order logic. I was looking for answer of the question "Explain the advantage and disadvantage of using Hilbert system". The disadvantage in the meaning of why it is hard to apply. Thanks.
AI: From a practical point o... |
H: Application of Hahn-Banach Theorem
Let $E$ be Banach space over $\mathbb{R}$. Let $u$ and $v$ be such that $||u||=||v||=1$ and $||2u+v||=||u-2v||=3$.
How do we show that there exists a linear functional $f$ defined on all of $E$ such that $||f||=1$, $f(u)=1$ and $f(v)=1$?
AI: Note first that $u$ and $v$ are linearl... |
H: Is there such a thing as discrete Riemannian geometry?
General relativity expresses gravity as a curvature in space time, created by the stress energy tensor.
$$T_{\mu\nu} \approx R_{\mu\nu} - \frac{R}{2} g_{\mu\nu}$$
Given I put the fact that energy is quantizied and can only change within discrete levels into the... |
H: Fast variance calculation
Suppose to have a sequence $X$ of $m$ samples and for each $i^{th}$ sample you want to calculate a local mean $\mu_{X}(i)$ and a local variance $\sigma^2_{X}(i)$ estimation over $n \ll m$ samples of $X$.
Excluding the "boundary" samples of $X$, one can incrementally calculate the local me... |
H: Evaluation of $\int_0^{2\pi} \frac{1}{1+3\cos^2(\theta)} \, d\theta$
For a test I had to evaluate $\int_0^{2\pi} \frac{1}{1+3\cos^2(\theta)}d\theta$. First I used substitution, with $z=e^{i\theta}$ and $d\theta=\frac{1}{iz}dz$, as shown:
$$\int_0^{2\pi} \frac{1}{1+3\cos^2(\theta)}d\theta=\int_{|z|=1}\frac{1}{1+\fra... |
H: An example of neither open nor closed set
I need a very simple example of a set of real numbers (if there is any) that is neither closed nor open, along with an explanation of why it is so.
AI: $[0,1)$
It is not open because there is no $\epsilon > 0$ such that $(0-\epsilon,0+\epsilon) \subseteq [0,1)$.
It is not c... |
H: another topology multiple choice
Let $S^1 = \{(x, y) \in \Bbb R^2 : x^2 + y^2 = 1 \}.$ Let
$D = \{(x, y) \in\Bbb R^2 : x^2 + y^2 \le 1 \}$ and $E = \{(x, y) \in\Bbb R^2 : 2x^2 + 3y^2 \le 1\}$ be also considered as subspaces of $\Bbb R^2.$ Which of the following statements are true?
a. If $f : D \to S^1$ is a contin... |
H: Real, imaginary parts of $G(w)=\frac{1}{2}(\sqrt{2}-\sqrt{2}e^{iw})$
For the function $G(w)=\frac{1}{2}(\sqrt{2}-\sqrt{2}e^{iw})$ , show that;
$\qquad\mathrm{Re}\,G(w)=\sqrt2\sin^2(w/2)\quad$ and $\quad\operatorname{Im}\,G(w)=-1/\sqrt2\sin w$.
I really need help with understanding this. Anything you could do to hel... |
H: is there unique name of inequality $ \leq $ and $ \geq $
I'm tutoring a 8th grader. Once that kid asked me if there is a unique name for $ \leq $ and $ \geq $.
Question goes like this: "Since it holds both equality $ = $ and inequality $<, >$ why is it still named inequality?"
How do I answer it?
AI: The detai... |
H: Proof of Bayesian Prediction Formula
I was wondering how to proof this formula, commonly used in Bayesian Prediction:
$$ \mathrm{P}(x|\alpha) = \int_\theta \mathrm{P}(x|\theta)\mathrm{P}(\theta|\alpha) \, \mathrm{d}\theta$$
The left hand side can be expressed as the following, through marginalizing:
$$ \mathrm{P}(x... |
H: Cauchy-Schwarz-type inequality
Let $f\in C^1[0,\infty)$ such that $f(x)\to 0$ as $x\to\infty$. Prove that $$\int_0^\infty f(x)^2\, dx\le \Biggl(\int_0^\infty x^2f(x)^2\, dx\Biggr)^{1/2}\Biggl(\int_0^\infty f'(x)^2\, dx\Biggr)^{1/2}\,.$$ Hint: write $f(x)^2=-\int_x^\infty \Bigl(f(t)^2\Bigr)'\, dt$.
So I tried doin... |
H: Is a right inverse of a surjective linear map linear?
On a finite dimensional vector space, the answer is yes (because surjective linear map must be an isomorphism). Does this extend to infinite dimensional vector space? In other words, for any linear surjection $T:V\rightarrow V$, AC guarantees the existence of ri... |
H: Formulate mathematical equation
I have a function:
$$f(n) = 1 - \left(\frac{1}{n}\right)$$
I want to add another variable $k$ to this equation such that:
if $k=n$ , then $f(n,k)=1$
if $n=1$, then $f(n,k)=0$
How can we do that ?
AI: Consider the function
$g(n,k) = 1 - \frac{1}{n} + \frac{1}{k}$
Then if $k = n$, you... |
H: Counting necklaces with a fixed number of each bead
I want to count the number of necklaces, with $n$ beads in total, where the alphabet of beads is $\{1,\ldots,k\}$, and where the number of beads with color $i$ is $n_i$. For example, if $n=4$, and $n_1 = n_2 = 2$, the following necklaces are the only possible
$$11... |
H: Infinite intersection and union, less than or equal to and less than
Can anyone show me why these sets are equal?
$$\{f \leq a \} = \bigcap_{k=1}^{\infty}\{f<a + \frac{1}{k}\}$$
$$\{f < a \} = \bigcup_{k=1}^{\infty}\{f \leq a - \frac{1}{k}\}$$
$f(x)$ is a function that can take on values over the extended real lin... |
H: Roots of unity in non-Abelian groups: when do they form subgroups?
I haven't studied group theory in earnest beyond first courses, so my notation may be nonstandard and my question may be a 'standard fact', so bear with me:
Consider a group $G$, and for each natural number $n \in \mathbb{N}$ define
$$G_n := \{g \in... |
H: Integral: $\int^\infty_{-\infty}\frac{\ln(t+1)}{t^2+1}\mathrm{d}t$
How to prove the following:
$$\int^\infty_{-\infty}\frac{\ln(t+1)}{t^2+1}\mathrm{d}t=\frac{\pi}{2}\left(\ln(2)+\frac{\pi}{2} i\right)$$
AI: Deform the contour to pick up the residue at $t=i$,
$$\begin{eqnarray*}
\int^\infty_{-\infty}\frac{\ln(t+1)}... |
H: Properties of ellipse x-y form
$$5x^2+8xy+5y^2=1$$
$$1\left(\frac{x-y}{\sqrt{2}}\right)^2+9\left(\frac{x+y}{\sqrt{2}}\right)^2=1$$
I know that these two forms are equal, showing that the equation is an ellipse.
I do know what happens when the ellipse is in the form of $ax^2 + by^2 = 1$ but not sure what happens in... |
H: Computing an integral.
How to compute $\displaystyle\iint\limits_{\Sigma} z\sqrt{1+z^2}dS$,
where $\Sigma=\Big\{(x,y,z)\Big| \cfrac{x^2}{2}+\cfrac{y^2}{2}+z^2=1,z\ge 0\Big\}$?
AI: Since in $\Sigma$, $z>0$ so our domain is $D:\sqrt{1-(1/2)x^2-(1/2)y^2}>0$ or $D: x^2+y^2<1$. Clearly $S: f=x^2+y^2+2z^2=2$ which is an ... |
H: question relating to analyticity and C-R equations and relationship between real differentiability, complex differentiability and C-R equations
So if function is analytic( ~ holomorphic) in Ω⊂C, then it satisfies C-R equations. And if f satisfies C-R equations and the functions u(x,y) and v(x,y) have first partial ... |
H: Getting wrong result calculating sums
I try to solve the following sum:
$$\sum_{k=0}^{n}\frac{2}{3^k} = \frac{2}{1} + \frac{2}{3} + \frac{2}{9} + \frac{2}{81} + ...$$
The recurring factor of all numbers is $\frac{1}{3}$, for example $\frac{2}{1} \times \frac{1}{3} = \frac{2}{3}$ and $\frac{2}{3} \times \frac{1}{3} ... |
H: Complete Lattice with unique negation
Suppose i have a complete lattice (meet and join exist for any two subsets of the lattice) and for every element of the lattice, there is a unique element that is its complement (hence a Boolean negation). If such a lattice is also modular, then it is Boolean. Negation is a str... |
H: If $\kappa$ is a cardinal, $\aleph$ is any $\aleph$-number, and if $\kappa\leq\aleph$ then $\kappa$ can be well ordered as well.
I'm having trouble understanding the statement:
If $\kappa$ is a cardinal, $\aleph$ is any $\aleph$-number, and if $\kappa\leq\aleph$ then $\kappa$ can be well ordered as well.
I unders... |
H: topology - analysis Book
I need some notion about topology(I'm very interested in boundary points, open sets) and few examples of solved exercises about limits of functions($f:\mathbb{R}^{n}\rightarrow \mathbb{R}^m$) using $\epsilon, \delta$ and also some theory for continous functions.
Please give me some links o... |
H: Radius of convergence (root, ratio)
Let $\sum c_n z^n$ be a power series. ($c_n,z_n \in \mathbb{C}$)
Let $\alpha = \lim \sup {|c_n|}^{1/n}$
Then this series is convergent if $|z|<1/{\alpha}$.
Let $\beta = \lim \sup |c_{n+1}/{c_n}|$
Here, series is convergent if $|z|<1/{\beta}$.
Since $\alpha≦\beta$, $|z|<1/{\alpha}... |
H: A criterion for convergence in the operator norm
Let $L:H\rightarrow H$ be a continuous linear operator and $R_n:H \rightarrow H$ a sequence of continuous linear operators, where $H$ is a Hilbert space. If the $\sum_{n=1}^{k} R_n$ converge pointwise to $L$, meaning for every $x\in H$ we have $$Lx=\sum_{k=1}^{\inft... |
H: Let both $a$ and $b$ belong to the set $\{ 1,2,3,4 \}$. What is the number of equations of the form $ax^2+bx+1=0$ which have real roots?
Let both $a$ and $b$ belong to the set $\{ 1,2,3,4 \}$. What is the number of equations of the form $ax^2+bx+1=0$ which have real roots?
for real roots, $a \gt 0$, $b^2-4{a}{c} ... |
H: Why are epsilon-delta proofs difficult?
What conceptual difficulties do students learning epsilon-delta proofs have, or, why are the proofs difficult?
Motivation
I have to teach people and never found the epsilonistics particularly difficult, so I have trouble relating to the students and understanding their diffic... |
H: Finitely but not countably additive set function
Let X be any countable! set and and let F be the cofinite set, i.e., $A \in F $ if A or $A^{c}$ is finite (this is an algebra).
Then show that the set function $\mu: F \rightarrow [0,\infty)$ defined as $\mu(A)=0$ if A is finite $\mu(A)=1$ if $A^{c}$ is finite is f... |
H: Existence of smooth function $f(x)$ satisfying partial summation
We know that $\sum_{i=1}^{n}i=\frac{n(n+1)}{2}$, $\sum_{i=1}^{n}\frac{1}{i}=\psi_{0} (n+1)-\psi_{0} (1)$, where $\psi_{0}(x)$ is the digamma function.
My problem is,
(1).Is there a transformation such that it maps
$x \to \frac{x(x+1)}{2}$ and $\frac{... |
H: Non-abelian group $G$ of order $p^3$
I just need some hints to prove this:
Let $|G|=p^3$ be a a non-abelian group. If every subgroup of $G$ is normal, then $p=2$ and $G=Q_8$.
I know the following facts about a non-abelian group $G$ of order $p^3$:
$Z(G)=G'=Z_p$
If $p$ is an odd number, then the function $\phi... |
H: Continuous solutions of $f(x) = \lambda \int_0^{\pi/2} \cos({x-y)}f(y)\,dy$
Possible Duplicate:
Eigenvalues of an operator
Find all the functions $f \in C([0,\frac{\pi}{2}])$ which are solutions of
$$
f(x) = \lambda \int_0^{\pi/2} \cos({x-y)}f(y)\,dy, \qquad \lambda \in \mathbb R.
$$
I just tried some algeb... |
H: Geometry of N-dimensional hypercubes and their N-M surfaces.
I'm working on a computer code to mesh a N-dimensional space with N-dimensional hypercubes and do some physics in it. I am wondering if I can produce a generic code (with C++ templates, but that is not the subject here) that is ok for N = 2, 3, 4...
My qu... |
H: Universal covers and homotopy
From Wikipedia (about homotopy of spheres):
$\pi_2 (S^1) = 0$: This is because $S^1$ has the real line as its universal cover which is contractible (it has the homotopy type of a point). In addition, because $S^2$ is simply connected, by the lifting criterion, any map from $S^2$ to $S... |
H: a problem about boundedness implying integrability
Let $X_k,k\in N$ be independent identically distributed random variables, and suppose that for some $p\in (0;2)$ the sequence $\{\frac{1}{n^{1/p}}\sum_{k\leq n} X_k\}_{n\ge 1}$ is a.s. bounded. Prove that $X_1$ is in $L^p$.
The question cannot be that hard, but I g... |
H: necessary conditions to be relatively compact
Let $X$ be a Banach space and $A\subset X$ be bounded. Suppose that, for any $\varepsilon>0$, $\exists\; F_\varepsilon$ a subspace of $X$ of finite dimension such that $$\text{dist}(x,F_\varepsilon)\leq\varepsilon,\quad\forall x\in A.$$
Then prove that $A$ is relatively... |
H: Calculation of residues
Let be $a\in\mathbb C\setminus\mathbb Z$ a fixed complex number, and define the following function:
$$f(z)=\frac{a\pi\cot\pi z}{z(a-z)}$$
It has simple poles in $z\in\mathbb Z\setminus\{0\}$ and in $z=a$, and a double pole in $z=0$. my textbook says that it is straightforward that:
$\operato... |
H: Prove that $J^T J+\lambda I$ is a singular matrix
If $J$ is a singular matrix, then $(J^T J)^{-1}$ is singular too. I'm trying to prove that $J^T J+\lambda I$ is a singular matrix, where $I$ denotes identity matrix. Any suggestions please?
Thanks
AI: Let $x$ be nonzero such that $Jx=0$. Then $J^\mathrm{T} Jx=0$, so... |
H: Axiom of Union?
I'm reading Comprehensive Mathematics for Computer Scientists 1. On the second chapter: Axiomatic Set Theory.
He first states the axiom of the empty set, the axiom of equality and then he proceeds to the axiom of union:
Axiom 3 (Axiom of Union) If $a$ is a set, then there is a set:
$\{$$x$ | there ... |
H: Cauchy's functional equation for $\mathbb R^n$
Suppose $f(x+y)=f(x)+f(y)$. If $f:\mathbb R\to \mathbb R$ and is measurable, then $f(x)=cx$. This is referred to as Cauchy's functional equation.
Suppose $f:\mathbb R^n\to \mathbb R^n$ instead. Does it still hold that $f$ is linear?
Wikipedia says that Hilbert's fifth ... |
H: Is it true that if function $f$ is analytic( ~ holomorphic) in $\Omega \subset\Bbb C$, then it satisfies C-R equations?
Is it true that if function $f$ is analytic ($\sim$ holomorphic) in $\Omega \subset \Bbb C$, then it satisfies the Cauchy-Riemann (C-R) equations? And is it t true that if $f$ satisfies C-R equati... |
H: How can I prove the properties of equality?
Say I have an arbitrary group, and a relation inside this group, e.g. $aba^{-1} = b^{-1}$. How can I prove that left-multiplying $a$ on both sides will also give a valid equation?
Thanks
AI: No justification is needed. In the formal predicate calculus with equality, the ... |
H: Finding the last digit of largest mersenne prime
I was wondering what the best way was to find the last digit of the largest known Mersenne prime $2^{6972593} - 1$? Is there any logical way to do this or will I have to just some how compute the answer and then find the last digit?
AI: The answer is yes, there is in... |
H: The different of a Galois extension of an algebraic number field
Since the ramification indices of a prime are the same in a Galois extension, the following proposition is likely to be true.
If it is true, how do we prove it?
Proposition
Let $K$ be an algebraic number field.
Let $L$ be a finite Galois extension of ... |
H: Writing 1/3 as a sum of other numbers
Is it possible to write $0.3333(3)=\frac{1}{3}$ as sum of $\frac{1}{4} + \cdots + \cdots\frac{1}{512} + \cdots$ so that denominator is a power of $2$ and always different?
As far as I can prove, it should be an infinite series, but I can be wrong.
In case if it can't be written... |
H: Solving an augumented matrix (A|B) with same matrix coeff
I am taking a linear algebra course at my university. We just have the first day of class but I am reading by myself, so I decided to solve some problems. I do not understand how to solve this problem:
"The two systems:
$a)$$2x + y = 3\tag{1}$
$4x + 3y= 5... |
H: Solving a Linear Equation over $2^X$
I was reading Vinberg's "A Course in Algebra" and in the opening line of chapter 2 he says, "Fix a field $\mathbb K$. We are going to abuse the language slightly and call elements of $\mathbb K$ numbers."
Anyway, linear algebra and solving systems of equations is present (it see... |
H: How to divide using addition or subtraction
We can multiply $a$ and $n$ by adding $a$ a total of $n$ times.
$$ n \times a = a + a + a + \cdots +a$$
Can we define division similarly using only addition or subtraction?
AI: To divide $60$ by $12$ using subtraction:
$$\begin{align*}
&60-12=48\qquad\text{count }1\\
&48... |
H: Ways to compare a set of values against a base value
I have a statistics doubt. Consider I have the following values.
Base Model Model 1 Model 2
5 7 4
10 12 8
15 17 12
20 23 21
25 30 27
30 ... |
H: 2D basis functions orthogonal under exponential kernel
In one dimension, the Laguerre polynomials are orthogonal under exponential weighting:
$$
\int_0^\infty L_n(x) L_m(x) e^{-x} \, dx = 0, n \ne m
$$
Does anyone know what the corresponding basis functions would be in 2 dimensions?
$$
\int_{-\infty}^\infty \int_{-... |
H: Pre-Calculus - Solving for $x^3$
In my Pre-Calculus class we were discussing $i$, and all of it's mysterious imaginary wonders, but I had a moment of confusion during the lecture.
This was the problem:
Solve for $x$.
$$
x^3 - 1 = 0
$$
Not too bad; what he said was to factor it:
$$
(x - 1)(x^2 + x + 1) = 0
$$
Then ... |
H: Optimization problem involving step function
I've got to optimize the following function with respect to $\phi$:
$q(\phi, x) = \frac{1}{n} \sum_{i=1}^{n}{H(y_i)}$
where
$y_i = k - \phi l - x_i$
and $H(.)$ denotes the Heaviside function. $k$ and $l$ are constants, and $x$ follows either (1) a continuous uniform dist... |
H: is the converse true: in a simply connected domain every harmonic function has its conjugate
The question is.
Is the converse true: In a simply connected domain every harmonic function has its conjugate?
I am not able to get an example to disprove the statement.
AI: No, to the question as stated by OP in the commen... |
H: Clarification on pairs of sets that make up a power set
From the Wikipedia article on power sets,
... one can select any element x of a set, S, and gather up the subsets of S itself into pairs (A-, A+) that are identical excepting that A- does not include x and A+ does.
So if S = {1,2,3}, and we choose x to be 3... |
H: How to find $k$'s quotient of $\frac{-18}{(-2)^k}$
I try to solve a geometric sum, but I'm stuck. The original terms are:
$$s = 9 - \frac{9}{2} + \frac{9}{4} - \frac{9}{8} + \cdots - \frac{9}{128}$$
After some work I found that the sum would be:
$$\sum_{k=1}^8 \frac{-18}{(-2)^k}$$
My next step towards finding the s... |
H: Lift of homeomorphism is homeomorphism
Suppose $p:Y \rightarrow X$ is the universal covering map of $X$.
Given a contiunuous $f: X \rightarrow X$ then a well known theorem for existence of lifts states that there exist a continuous lift
$$\tilde f : Y \rightarrow Y \text{ with } p \circ \tilde f = f \circ p.$$
If ... |
H: Dual of a vector space
Let $S$ be a subspace of a vector space $V$ so that $V/S$ is finite dimensional. How do we show that $S$ is the intersection of kernels of a finite number of elements in the dual of $V$? If $S$ is finite dimensional, what can we conclude about the dimension of $V$?
AI: Put $\,S=Span\{v_i\;\;|... |
H: Consistency of Peano axioms (Hilbert's second problem)?
(Putting aside for the moment that Wikipedia might not be the best source of knowledge.)
I just came across this Wikipedia paragraph on the Peano-Axioms:
The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying eith... |
H: How do you calculate "excess returns"?
(Although this is perhaps a very simple question, math-wise, it still has me puzzled so it's okay to ask for help here I guess. I really need to understand this, so any help is appreciated)
First, assume the S&P100 market index has performed like this on the first 3 days of th... |
H: definition of winding number, have doubt in definition.
could any one tell me why in the definition of index number or winding number of a curve $\gamma(t)$ around some point $a$ we take this integral : $$\frac{1}{2\pi i}\int_{\gamma}\frac{1}{z-a} $$ why not $$\frac{1}{2\pi i}\int_{\gamma}\frac{1}{z^2+2z+1+e^z-a}... |
H: natural solutions for $9m+9n=mn$ and $9m+9n=2m^2n^2$
Please help me find the natural solutions for $9m+9n=mn$ and $9m+9n=2m^2n^2$ where m and n are relatively prime.
I tried solving the first equation in the following way: $9m+9n=mn \rightarrow (9-n)m+9n=0 $
$\rightarrow m=-\frac{9n}{9-n}$
Thanks in advance.
AI: $$... |
H: Seemingly similar but different probability games
Burger King is currently running its "family food" game in which each piece can be modeled as a scratch off game where exactly one of three slots is a winner and you may only scratch one slot as your guess. As I was standing in line the other day I realized that the... |
H: Learning to differentiate, with respect to $f_{\overline{z}}$
Let $R,\phi$ be real valued $C^1$ functions over a complex variable $z$.
Consider the following function: $f=Re^{i\phi}$ .
Where we define $f_{\overline{z}}= \frac{1}{2}(f_x+if_y)$
I want to differentiate this function with "respect to $\overline{z}$".... |
H: A "Matrix Trigonometry"
$e^X$ for matrix $X$ is defined as an always-converging taylor series (provided that $X$ is a $n \times n $ complex matrix):
$$e^X:=\sum_{k=0}^{\infty}\frac{X^k}{k!} $$
A thought occurred to me that we might as well define $\cos(X):=\frac12(e^{iX}+e^{-iX})$ and $\sin(X):=\frac1{2i}(e^{iX}-e^... |
H: Can the dimension of the image of a linear map "increase"?
Suppose we have a linear transformation $f: V \to V$. How is it possible that $\dim(\operatorname{im}(f \circ f))$ is larger than $\dim(\operatorname{im}(f))+\dim(\operatorname{im}(f)) - \dim(V)$?
It's a question on a past exam, so there should be an exampl... |
H: Prove that $2^{pq} - 1 = (2^p - 1)(\sum^{q-1}_{i=0} 2^{pi})$ for two natural numbers $p, q$
How can I prove the below?
$$2^{pq} - 1 = (2^p - 1)\left(\sum^{q-1}_{i=0} 2^{pi}\right)$$ for two natural numbers $p, q$
It looks like I need proof by induction? But how? There's two variables?
AI: Setting $a=2^p$ then
$$... |
H: Can the completion of a domain be a non-domain?
Suppose $R$ is a domain, finitely generated over an algebraically closed field, and $\mathfrak{m}\subset R$ is a maximal ideal. Is $\underleftarrow{\lim} R/\mathfrak{m}^n$ necessarily a domain?
AI: This phenomenon of posing a question in the title and the posing its n... |
H: What preposition to use when fitting data?
Please could some real experts give an opinion on this question on English.SE: What preposition to use when fitting data?
Do we fit data with, by, or as a linear function?
Forgive me as I do not know how best to highlight this and I'm happy for this question be be delet... |
H: Test the convergence of $\sum_{n=0}^{\infty} \frac{n^{k+1}}{n^k + k}$
Problem: Test the convergence of $\sum_{n=0}^{\infty} \frac{n^{k+1}}{n^k + k}$, where $k$ is a positive constant.
I'm stumped. I've tried to apply several different convergence tests, but still can't figure this one out.
AI: Hint
$$ \frac{n^{k+1}... |
H: Find out the dimensions of the rectangular parallelepiped
The volume of a rectangular parallelepiped is given as 144 cc. Its surface area is given as 192 sq cm. And its corner to corner diagonal is given as 13 cm. How do I find out the three sides.
I have assumed the sides to be $a,b,c$. Now, $a^2+b^2+c^2=169, 2(ab... |
H: Can the coefficients of a given term in this family of power series have a common divisor?
Let $g_m(b)$ be the coefficient of $x^m$ the power series
$$\dfrac{1}{1-x-bx^2}$$
(When $b=1$, this is just the generating function of Fibonacci numbers.) Of course, $g_m(b)$ depends on both $m$ and $b$. Now fix $m$, and let ... |
H: What is the maximum number of consecutive composite numbers possible?
Can anyone tell me what is the maximum number of consecutive composite numbers possible? I mean can I get say 1000 consecutive natural numbers. Is there any general theorem that when I have a n-digit number there will always be p consecutive comp... |
H: On the number of possible solutions for a quadratic equation.
Solving a quadratic equation will yield two roots:
$$\frac{-\sqrt{b^2-4 a c}-b} {2 a}$$
and:
$$\frac{\sqrt{b^2-4a c}-b}{2 a}$$
And I've been taught to answer it like:
$$\frac{\pm\sqrt{b^2-4a c}- b}{2 a}$$
Why does it yields only two solutions? Aren't the... |
H: Proof of $(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2)>\frac{(ab+bc+ca)^3}{3}$
For positive real numbers $a$, $b$ and $c$, how do we prove that:
$$(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2)>\frac{(ab+bc+ca)^3}{3}$$
AI: By Cauchy-Schwarz:
$(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2) \ge (a^{1.5}b^{1.5}+b^{1.5}c^{1.5}+c^{1.5}a^{1.5})^2$
By the power... |
H: Distance Formula in Three Dimensions
The distance formula in 3-D space is defined as:
$$|P_1\, P_2| = \sqrt{(x_2- x_1)^2 + (y_2 -y_1)^2 + (z_2- z_1)^2}$$
My question is that if I have 2 points that have negative coordinates, do I have to use the absolute value on all the points?
For example my two points are $\,P(3... |
H: If $x$ and $y$ are prime , which of the following cannot be their sum
The question is
If $x$ and $y$ are prime numbers , which of the following can not be their sum? $5$,$9$,$13$,$16$ or $23$.
The answer is $23$.
How did they get this? As far as I can tell is that when prime numbers are added I am suppose to get... |
H: How to determine if a 3D triangle given by points is a right triangle?
How do I figure out if a triangle is a right triangle in 3-D space if you are given three points: $P = (3, -2, -3)$, $Q = (7, 0, 1)$, $R = (1, 2, 1)$?
I know that it is an isosceles triangle (two sides are the same $6$ units). But how would I ... |
H: Does an orientable subbundle of an orientable vector bundle always have a orientable complement?
If I have an orientable vector bundle $E$ and a subbundle $F$ on a manifold $M$, where both the bundles are orientable, does $F$ have a complement in $E$ which is also orientable? Does it have a complement bundle at all... |
H: Help understanding a step in the proof of Jordan's Lemma
I don't understand this step on the Wikipedia article on Jordan's Lemma:
$$ \begin{align}
I_R:=\biggl|\int_{C_R} f(z)\, dz\biggr|
&\le R\int_0^\pi\bigl|g(Re^{i\theta})\,e^{aR(i\cos\theta-\sin\theta)}\,ie^{i\theta} \bigr|\,d\theta\\
&=R\int_0^\pi \bigl|g(Re^{i... |
H: The intersection of two conics - matrix solution
I recently had to compute the intersection of two conics, and found it to be a long and complicated procedure. Looking at Wikipedia, the advice to find the points of intersection of two conics in general, which is echoed in the answer to this question, is to write th... |
H: Simplify the complex equation
For the function
$$G(w) = \frac{\sqrt2}{2}-\frac{\sqrt2}{2}e^{iw},$$
show that
$$G(w) = -\sqrt2ie^{iw/2} \sin(w/2).$$
Ive been told to use the equation below by use of the complex sine definition, but i would like to know the exact steps to get there and then how to simplify i... |
H: Finding a paper by John von Neumann written in 1951
There's a 1951 article by John von Neumann, Various techniques used in connection with random digits, which I would really like to read. It is widely cited, but I can't seem to find an actual copy of the paper, be it free or paying.
Is there a general strategy to ... |
H: need help understanding Cauchy-Riemann criteria
I don't understand why both limits have to be equal in both directions.
$$\lim_{ \Delta x \rightarrow 0} {\Delta u+i\Delta v \over \Delta x} |_{\Delta y = 0}
= \lim_{\Delta y \rightarrow 0}{\Delta u+i\Delta v \over i\Delta y}|_{\Delta x = 0} $$
Shouldn't the limit... |
H: Why is the last step of proof by induction necessary?
Why do we assume the condition holds up to certain number $n$ and prove it holds for $n+1$? Is there any example where something holds up to $n$ but fails for $n+1$?
AI: If I understand you correctly this is a confusion about how the variable n is quantified ove... |
H: inequality fraction
I saw this problem in an Australian maths olympiad:
$6/10 < a/b < 10/15$
The problem asked for the lowest possible value of $b\in \mathbb{Z}$.
I tried manipulating but couldnt derive one of their answers.
AI: That's how I did it. First of all, note that \begin{equation} \frac{11}{15} - \frac{7}{... |
H: how do we know that integral is non-elementary?
Possible Duplicate:
How can you prove that a function has no closed form integral?
Is there a condition that states that the indefinite integration is non-elementary?
AI: There is a decision procedure called the Risch algorithm that will either tell you that the in... |
H: expected value properties
I'm reading an algorithm proof in which there is a passage I can't understand.
Substantially we have 2 statistically independent random arrays $X$ and $Z$, and a third random array $Y$.
It's known that $Y=XZ$ and $E[Z] = 1$. What the author provides are textually the following passages:
\b... |
H: Can a transformation matrix be expressed in terms of the vector to be transformed?
I'm currently learning linear algebra with my friend via an online course, and we have a disagreement that we would like settled.
Upon learning that vectors can be projected onto lines by a simple function, and that this function is ... |
H: How many students turned up for renting the rooms?
Sara has a house which she wants to convert to a hostel and rent it out to students of a nearby women’s college. The house is a two story building and each floor has eight rooms. When one looks from the outside, three rooms are found facing North, three found facin... |
H: about the differentiability : the general case
Let $U$ be an open set in $\mathbb{R}^{n}$ and $f :U \subset \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a given function. We say that $f$ is differentiable at $x_{0}\in U$ if the partial derivatives of $f$ existi at $x_{0}$ and if
$$\displaystyle \lim_{x \rightarro... |
H: Continuity of function $\lim_{t \rightarrow x} f(t)$
Let a function $f:A \rightarrow \mathbb{R}$, defined on a set $A\subset \mathbb{R}$ without isolated points, has a finite limit $g(x)$ in each point $x\in A$.
I wish to show that $g: A \rightarrow \mathbb{R}$, $g(x)=\lim_{t \rightarrow x} f(t)$, is continuous.
AI... |
H: If $\sim$ is an equivalence relation on $X$, and there is a strict total order on $X/\sim$, what kind of ordering does $X$ have?
I would like to know if there's a special name for this kind of ordering. When I say there is a strict total order on $X/\sim$, what I mean is that two distinct elements in the same equiv... |
H: Convergence of a power series when radius of convergence is 1
Suppose radius of convergence of $\sum c_n z^n$ is 1. ($c_n, z \in \mathbb{C}$)
(i)Here, if $\{c_n\}$ is monotonically decreasing and $\lim c_n = 0$, then $\sum c_n z^n$ converges at every point on the circle $|z|$, except possibly at $z=1$.
(Of course, ... |
H: uniform sampling of the sphere w.r.t. inner products other than the usual one
I will have a small (usually 2 or 3, probably no bigger than 6) positive integer $n$
and symmetric positive-definite real $n$-by-$n$ matrices $A$,
and will want to sample an $\:\mathbf{R}^n$-valued$\:$ random variable $\mathbf{x}$ such ... |
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