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H: Get a Point on a Line which is Perpendicularly dropped from another Point
Let say I have this image,
(X1, Y1), (X2, Y2) and (X3, Y3) are known points. From point (X1, Y1), a line is dropped perpendicular to line (X2, Y2) and (X3, Y3). I need to calculate (Xn, Yn)
AI: you can easily find the equation of line joinin... |
H: choose n letters to form possible words from m letters, restraint no letter should be used more than twice
Given $m$ letters (distinct letters), choose $n$ ($n < m$) letters to form words (the word doesn't need to be correct), and no letter can be used more than twice.
How many distinct word can be formed?
AI: If ... |
H: for what value of $a$ has equation rational roots?
Suppose that we have following quadratic equation containing some constant $a$
$$ax^2-(1-2a)x+a-2=0.$$
We have to find all integers $a$,for which this equation has rational roots.
First I have tried to determine for which $a$ this equation has a real solution, s... |
H: What does it mean for a topological space $X$ to have a binary open cover?
Does someone know what's meant by a binary open cover of a topological space $X$? I can't find this definition of binary open cover. Could someone who knows it tell me? Thanks ahead for any help:)
AI: A binary open cover of a space $X$ is s... |
H: Why is this problem ill posed?
I would like to know why the following equation is ill posed .
$u_t=-ku_{xx}$ and $u(0,t)=u(L,t)=0$
Here , $u_t, u_{xx} $ denotes the partial derivatives of u with respect to time and space respectively .
Thank you for your help.
AI: The notion of wellposedness and illposedness depe... |
H: Finding sum of factors of a number using prime factorization
Given a number, there is an algorithm described here to find it's sum and number of factors.
For example, let us take the number $1225$ :
It's factors are $1, 5, 7, 25, 35, 49, 175, 245, 1225 $
and the sum of factors are $1767$.
A simple algorithm that is... |
H: antiderivative of $\frac{dy}{dx}$
Find the antiderivative of $\frac{dy}{dx}$=$e^{2x}-x$ and $y=5$ when $x=0$
the antiderivative formula is
$e^{kx}dx$=$\frac{1}{k}e^{kx}+c$
I did it ,
$\frac{dy}{dx}$=$e^{2x}-x$ and $y=5$ when $x=0$
$\frac{dy}{dx}=\frac{1}{2}e^{2x}+c$
but the right answer is $y=\frac{1}{2}(... |
H: Property of $2^n+1=xy$
I was wondering if the following were true. It makes sense but I'm having trouble concocting any formal reasoning.
Let $2^n+1=xy$ for some integers $x,y>1$ and $n>0$. For $a\in\mathbb{Z}^+$, does $2^a\mid (x-1)$ $\iff$ $2^a\mid (y-1)$?
Without loss of generality, one only needs to prove one... |
H: Visualizing Commutator of Two Vector Fields
I'm reading a book on calculus, the part about vector fields on
manifolds. It's a nice book, but with a severe drawback --- it has no pictures.
I like how vectors are treated algebraically, as derivatives over a local ring (ring of germs). But I still want to use "geomet... |
H: How to evaluate this definite integral: $\int_0^1 {(1-x^3)}/{(1-x^5)} \;\rm dx$
I need to compute:
$$\int_0^1 \frac{1-x^3}{1-x^5} dx$$
I tried integrating it by partial fractions but couldn't succeed. Is there any other way to integrate this?
AI: Use partial fraction:
$$
\frac{1-x^3}{1-x^5} = \frac{\sqrt{5}-1}{\sqr... |
H: whats the order of a distributional derivate?
I have to calculate the derivatives of order $\le 2$ of for example $f(x) = |x|$, is it the same as the second derivate, what does this "of order $\le 2$" mean? calculating distributionell derivatives is easy, for my example it would be
$$
\langle f', \phi \rangle = - ... |
H: An Integral Estimation
Let $g:\mathbb{R}^n \rightarrow \mathbb{R}$, let $t \in \mathbb{R}$, suppose $s \in [0, t]$ and let $e_i$ denote the standard $i^{th}$ basis vector. I have read the following claim:
$$
\frac{1}{t} \int^t_0 |g(x + se_i) - g(x)|ds \leq \max_{0 \leq s \leq t}|g(x + se_i) - g(x)|
$$
How can I se... |
H: What is $ \lim_{x\to 1}\frac{\sqrt[n]{x}+\sqrt[n]{x^2}-2}{e^x-e} $ without using L'Hospital?
I want to find the limit of this example using L'Hospital rule i get easily ans. but i want to find the limit without using L'Hospital
$$ \lim_{x\to 1}\frac{\sqrt[n]{x}+\sqrt[n]{x^2}-2}{e^x-e} $$
I tried to set the power o... |
H: Half-space membership in projective geometry?
Given a plane normal $n=(n_x,n_y,n_z)$ and a point on that plane $p_0\in\mathbb{R}^3$, testing whether another point $p\in\mathbb{R}^3$ is "above" the plane is easy:
$n\cdot(p-p_0) > 0$
Is there an equivalent in projective geometry?
AI: I am not sure if I understood you... |
H: Expressing $2002^{2002}$ as a sum of cubes
This is the problem: Determine the smallest positive integer $k$
such that there exist integers $x_1, x_2 , \ldots , x_k$ with
${x_1}^3+{x_2}^3+{x_3}^3+\cdots+{x_k}^3=2002^{2002} $. How to approach these kind of problems??
Thanks in advance!!
AI: $k=4$ is the smallest:
C... |
H: Is the Weak Law of Large Numbers only true for Bernoulli trials?
I wonder if the Weak Law of Large Numbers is only applicable if the random variable is binomially distributed.
(The random variable counts the relative frequency of an event A).
So, when you describe the Law, do you have to mention as a prerequisite ... |
H: What is the indefinite integral of the function $f(x) = \frac{\sin(x)}{x}$
I want to know if the function $f(x) = \frac{\sin(x)}{x}$ is integrable and if it is, then what's its integral?
My high school book says its a non-integrable function while WolframAlpha says its integral is Si$(x) +$constant.
Please shed som... |
H: Resources for kids education
Most of the questions on MSE about educational resources are for high school. At least I haven't found questions which deals with sites on very elementary mathematics.
I will be grateful if someone provides me with list of educational resources for kids - e.g. of the level of kindergar... |
H: Polynomials identity, factoring $x^{2^n}-1$
There is a proof that I can't solve.
Show that for any integer $k$, the following identity holds:
$$(1+x)(1+x^2)(1+x^4)\cdots(1+x^{2^{k-1}})=1+x+x^2+x^3+\cdots+x^{2^k-1}$$
Thanks for your help.
AI: This equation is a fancy way of stating existence and uniqueness of binary... |
H: Finding the number of homomorphisms between $C_6$ and $S_3$
Find the number of group homomorphisms between $C_6$ and $S_3$.
For all the group theory buffs this is probably a piece of cake, but how does one generally go about a question like this. Is there a general way to figure this out, or do you need to make u... |
H: Representation of smooth function
Is it true that any smooth function $f\colon \mathbb{R}^n \to \mathbb{R}^n$ can be represented as
$$
f(x) = \nabla U(x) + g(x)
$$
where $U(x)$ is a scalar function and $\langle g(x), f(x) \rangle \equiv 0$? Is this representation unique?
AI: Let me summarize my comments as an an... |
H: Single-digit even natural number solutions to the equation $a+b+c+d = 24$ such that $a+b > c+d$
Possible Duplicate:
Two algebra questions
How to approach the below question:
How many single-digit even natural number solutions are there for the equation $a+b+c+d = 24$ such that $a+b > c+d$?
AI: The single-digit ... |
H: a suggestion for several complex variable book
Could any one tell me name of some books on several complex variable for some one who will start reading the subject for the first time in his life. he has back ground on Differential geometry,complex analysis one variable, algebraic topology,commutative algebra.
AI: I... |
H: Limit of the form $\infty - \infty$
Consider:
$$\lim_{x \to \infty} \left(x - \ln(e^x + e^{-x})\right)$$
I wasn't sure how to treat the $\infty - \infty$ property. Can I exponentiate the function to get $$e^x - (e^x + e^{-x}) = \frac{1}{e^x}$$
$$\lim_{x \to \infty} \frac{1}{e^x} = 0$$
I feel like I have ignored th... |
H: Locally Path Connected space that is not Path Connected.
Can you give me an example of a space that is locally path connected but not path connected, if it exists ?
AI: Simple examples include $\mathbb{R} \setminus \{0\}$ with its natural topology as a subspace of $\mathbb{R}$, and any set with at least two points ... |
H: gradient in polar coordinate by changing gradient in Cartesian coordinate
I'm tried to do following and I can't see what went wrong.
$$\begin{bmatrix}
\hat r\\
\hat \theta
\end{bmatrix} = \begin{bmatrix}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{bmatrix}
\times
\begin{bmatrix}
\hat i\\
\hat j... |
H: Infinite limits of integration with discrete sequence
I'm reading a monograph where the following improper integral has infinite limits of integration (see pg. 125 of Seismic Inverse Q Filtering for $h(t)$ as Equation 7.19, and pg. 123 as the window function $w(t)$ as Equation 7.12):
$h(t) = {\left[ {\int\limits_{ ... |
H: Measurability of a map
Let $(X,\mathscr F)$ be some measurable space and $Y$ be a finite set with a $\sigma$-algebra $2^Y$. Let the map
$$
f:X\to Y
$$
be $\mathscr F|2^Y$-measurable. Consider sets $X^\mathbb N$ and $Y^\mathbb N$ endowed with product $\sigma$-algebras and extend
$$
f':X^\mathbb N\to Y^\mathbb N,... |
H: Example of an Hilbert space operator
There is a theorem in functional analysis, that says that for a selfadjoint compact operator $T:H\rightarrow H$, either $\lVert T\rVert $ or $-\lVert T\rVert$ is an eigenvalue. For finite dimensional $H$ it is easy to construct examples of operators such that $\lVert T \rVert$ a... |
H: Divergence Theorem on a curl
If $\vec{u}=4y\hat{i}+x\hat{j}+2z\hat{k}$, calculate the double integral
$$\iint(\nabla \times \vec{u})\cdot d\vec{s}$$ over the hemisphere given by,
$$x^{2}+y^{2}+z^{2}=a^{2}, \quad z\geq 0.$$
I approached it like this, $d\vec{s}$ can be resolved as $ds\vec{n}$ where $\vec{n}$ is the n... |
H: Given number of trailing zeros in n!, find out the possible values of n.
It's quite straightforward to find out number of trailing zeros in n!.
But what if the reverse question is asked?
n! has 13 trailing zeros, what are the possible values of n ?
How should we approach the above question ?
AI: Write $n$ in base... |
H: dotting gradient in spherical coordinates with displacement vector
The gradient in spherical coordinates is given by:
$\nabla f = \left(\frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}, \frac{1}{r \sin \theta} \frac{\partial f}{\partial \phi} \right)$
on the other hand the gradient is s... |
H: Pretty Basic: Determining the intersection.
I have to do the following problems, however my question is basicly: How would you go about interpreting these problems? At first I thought it might be something like this (for problem 1.A)
Which reads:
Determine the point of intersection of the graphs of each system
Th... |
H: Factoring multivariate polynomial
I'm trying to factor
$$x^3+x^2y-x^2+2xy+y^2-2x-2y \in \mathbb{Q}[x,y].$$
The hint for the exercise is to use the recursive multivariate polynomial form. So I'm using $\mathbb{Q}[x][y]$:
$$ x^3 + x^2(y-1) + x^1(y-2) + x^0(y^2-2y) $$
At this point I am stuck. Are their any general te... |
H: Identical balls arrangement in a circle
Six identical yellow balls and four identical red balls are to be arranged in the circumference of a circle. In how many ways it can be done?
AI: Out of 10 places we can choose 4 places for the red balls in $\binom {10}4$ =210 ways. If they are not symmetric, they come in g... |
H: determinant of matrix of transformation from Cartesian to orthogonal curvilinear
Let $(x_1, x_2)$ and $(y_1, y_1)$ be two orthogonal coordinate system with unit vectos $(\hat i_1, \hat i_2)$ and $(\hat e_1, \hat e_2)$ respectively defined by the
$x_1 = x_1(y_1,y_2)$ and $x_2 = x_2(y_1,y_2)$, $(x_1,x_2)$ be the Car... |
H: Multiplicative property of the GCD
I need to prove that
$$(ah,bk)=(a,b)(h,k)\left( \frac{a}{(a,b)},\frac{k}{(h,k)}\right)\left( \frac{b}{(a,b)},\frac{h}{(h,k)}\right)$$
I'm most certain I need to use $$\left(\frac{a}{(a,b)},\frac{b}{(a,b)}\right)=1 $$ and $(a ,b)=1 $ and $a \mid bc $ then $a \mid c$ but I'm not sur... |
H: What is EY if Y=a or Y=X with given probabilities p and 1-p?
Let $X$ be some random variable. Now define a new random variable $Y$ such that it has a probability $p$ of taking some fixed number $a$, and a probability $(1-p)$ of being determined by $X$.
What is the expected value of $Y$? Is it true that it is just $... |
H: The intersection of an abelian normal subgroup $A$ and subgroup $B$ is normal in $AB$
Let $G$ be a finite group with two subgroups $A$ and $B$. Show that if $A$ is Abelian and is normal in $G$, then $A\cap B$ is normal in $AB.$
It's been awhile since I did Abstract Algebra, (this is a prelim question) the above ... |
H: If the series $\sum_0^\infty a_n$ converges, then so does $\sum_1^\infty \frac{\sqrt{a_n}}{n} $
Problem:
Suppose that for every $n\in\mathbb{N}$, $a_n\in\mathbb{R}$ and $a_n\ge 0$. Given that
$$\sum_0^\infty a_n$$
converges, show that
$$\sum_1^\infty \frac{\sqrt{a_n}}{n} $$
converges.
Source: Rudin, Pri... |
H: Fractions in limits of a summation
What if on the sum there is a fraction in the limit?
$\sum_{m=k/12}^{k}$ or $\sum_{m=0}^{k/12+1}$
thank you very much!
what type of sequence is used for summing this type of interval?
AI: In the first case, the sum is over all integers $m$ that are in the closed interval whose end... |
H: Will the following expression be irrational, rational or integer?
Will the following expression be irrational, rational or integer?
$$\sqrt[3]{\sqrt a +b} - \sqrt[3]{\sqrt a -b}$$
where $a$ = $52$ and $b$ = $5$ .
By intuition, I think this will be an integer.
AI: Let's use the identity $(\alpha + \beta)^3 = \alpha^... |
H: Little Rudin series convergence exercise
Problem: If $\sum a_n$ converges, and $\{b_n\}$ is monotonic and bounded, prove that $\sum a_n b_n$ converges.
Source: Rudin, Principles of Mathematical Analysis, Chapter 3, Exercise 8.
AI: The sequence $\{b_n\}$ is monotonic and bounded, so it converges to some number $C$. ... |
H: Visualizing a Deformation Retraction
It's known that there exists a deformation retraction from the space $\mathbb{R}^3$\ $S^1$ to $S^2 \wedge S^1$, and I thought I had a visualization for it, but now it seems discontinuous. Can anyone help out with describing (or even better constructing explicitly) this map?
Tha... |
H: Integrating $\int_{\Gamma:|z|=1}\frac{e^z-1-z}{z^2}dz$, where $z\in\mathbb{C}$.
I recently came across the exercise of integrating
$$\int_{\Gamma:|z|=1}\frac{e^z-1-z}{z^2}dz,$$
where, naturally, $z\in\mathbb{C}$.
The first thing I thought of was using Cauchy's integral formula
$$f(z_0)=\frac{1}{2\pi i}\int_\Gamma\f... |
H: Generating correlated random numbers: Why does Cholesky decomposition work?
Let's say I want to generate correlated random variables. I understand that I can use Cholesky decomposition of the correlation matrix to obtain the correlated values. If $C$ is the correlation matrix, then we can do the cholesky decomposit... |
H: Find the number of rational roots of $f(x)$
There's a polynomial $f(x)$ such that its degree is 3.
All the coefficients of $f(x)$ are rational. If $f(x)$ is a tangent to $x$ axis, what can be the possible number of rational roots of $f(x) = 0$
options are : 0, 1, 2, 3, none
My approach :
Since y = 0 is a tangent to... |
H: Congruent Polynomials
If we have two Polynomials $f(x)$ and $g(x)$ with integer coefficients such that $$f(x) \equiv
g(x)\left( {\bmod n} \right),n \in {\mathbb{Z}^ + }$$
Does this mean that the coefficients of $f(x)$ are congruent coefficients of $g(x)$ mod n?
If it's True
Why ${x^{p}} \equiv x\left( {\bmod p} ... |
H: An application of Riesz' Lemma
How does one prove using Riesz' Lemma that an infinite dimensional subspace $Y$ of a Banach space $X$ contains a sequence $\{x_n:n\in \mathbb{N}\}$ in the unit ball of $Y$ such that $n \neq m$ implies that $\|x_n−x_m\|>1/2$?
AI: Pick any $x_1$ of norm 1. Let $F_1$ be the linear span o... |
H: Mean value theorem with $\ln(x)$
I understand how to do mean value theorum but I'm not sure how to apply it with $\ln(x)$.
$$f(x) = \ln(x), \ [1, 8]$$
How can I find a $c$ that satisfies the conclusion of the Mean Value theorem by using $\ln(x)$?
I know its $\dfrac{f(b)-f(a)}{b-a}$, then take derivative and fill ... |
H: difference between linear separable hyperplane and affine separable hyperplane
What is the differece between linear separable hyperplane and affine separable hyperplane?
How does one represent a hyperplane graphically in R^2?
AI: In general, affine = linear plus a translation. Any line in the plane is an affine hyp... |
H: How do you prove an equation has one root?
I have this equation:
$$9x + \cos x = 0$$
but I need to write out and prove why it has one real root. Could someone maybe give me a few pointers or what do I do exactly?
AI: Let $f(x)=9x+\cos x$ then $f$ is differentiable and $f'(x)=9-\sin(x)>0$.
So $f$ is strictly incre... |
H: What is a motivating way to introduce vectors?
What is a good way to introduce vectors on a linear algebra course so that students are motivated from the start? I need an opening which will have a real impact. Are there any motivating examples?
AI: You might like to take a look at an article by J. Kolecki from NASA... |
H: Max & Min Local Values without given interval
Now i know how to find the max and min local values using an interval if given, but in this question i am not given an interval. How do i go solving the min and max values without it?
$$f(x) = 4 + 6x^2 − 4x^3
$$
AI: $\lim_{-\infty}f=+\infty$ and $\lim_{+\infty}f=-\infty... |
H: Factoring $x^4z-2z^2-4x^6+x^2z$
We want to factor $8x^4y^4-2y^8-4x^6+x^2y^4 = -2y^8 + (8x^4+x^2)y^4 -4x^6$. We substitute $x^4$ with $z$:
Now we want to compute this $8x^4z-2z^2-4x^6+x^2z = -(x^2-2z)(4x^4-z)$ by hand.
Therefore we transform it into $-(2z^2-(8x^4+x^4)z+4x^6z^0)$ and use the quadratic formula on the ... |
H: A combinatorial problem of counting closed walks on grid
Consider an arbitrarily large $N \times N$ grid graph.
How can I express the number of closed walks starting from a reference vertex $v$ in terms of the length $L$ of the walk?
For example, for $L = 2$, there are 4 closed walks from $v$ to $v$.
For $L = 4$, t... |
H: Maximum value of the modulus of a holomorphic function
I'm looking for the maximum value of the modulus of a holomorphic function, and I am getting a bit stuck.
The function is $$(z-1)\left(z+\frac{1}{2}\right)$$ with domain $\,|z| \leq 1\,$
Now, I know by the maximum modulus principle the max value will occur on ... |
H: Prove that Pascals triangle contains only natural numbers, using induction.
I'm currently working my way through Spivak, and I'm stuck on the following.
Prove that Pascals triangle only contains natural numbers using induction and the following relation: $\left( {\begin{array}{*{20}c} n+1 \\ k \\ \end{array}} \righ... |
H: Unique Decomposition of Primes in Sums Of Higher Powers than $2$
Primes of the form $p=4k+1\;$ have a unique decomposition as sum of squares $p=a^2+b^2$ with $0<a<b\;$, due to Thue's Lemma.
What is known about sums of $n$ higher powers resulting in primes?
I tried $a^3+b^3+c^3$, asked Wolfram and found $3,17,29,... |
H: Diagonalizing a $2\times 2$ matrix
I am told that $A\in M_{2}(\mathbb{R})$ s.t $1-i$ is an eigenvalue
of $A$ with corresponding eigenvector $\begin{pmatrix}3+2i\\ 1+3i \end{pmatrix}$.
I wish to find a matrix $P$ s.t $P^{-1}AP=\begin{pmatrix}1-i & 0\\
0 & 1+i
\end{pmatrix}$. From what I understand since $\lambda_{1}... |
H: $p=4n+3$ never has a Decomposition into $2$ Squares, right?
Primes of the form $p=4k+1\;$ have a unique decomposition as sum of squares $p=a^2+b^2$ with $0<a<b\;$, due to Thue's Lemma.
Is it correct to say that, primes of the form $p=4n+3$, never have a decomposition into $2$ squares, because sum of the quadratic ... |
H: Boundary values of harmonic $u$ are $ u(e^{it}) = 5- 4 \cos t $; find $u(1/2)$ and $v(1/2)$.
My problem is the following:
Let $u$ be a continuous real-valued function in the closure of the unit disk $\mathbb{D}$ that is harmonic in $\mathbb{D}$. Assume that the boundary values of $u$ are given by
$$ u(e^{it}) = 5... |
H: If a principal divisor is defined over K, then is the function?
Let $X$ be an algebraic variety, $D$ a principal divisor of $X$ defined over $K$, i.e. the points of $D$ are in $X(K)$ and there is a function in $\overline{K}(X)$ whose divisor is $D$. Is $D$ necessarily the divisor of a function on $X$ defined over $... |
H: How many subsets are there?
I'm having trouble simplifying the expression for how many sets I can possibly have.
It's a very specific problem for which the specifics don't actually matter, but for $q$, some power of $2$ greater than $4$, I have a set of $q - 3$ elements. I am finding all subsets which contain at le... |
H: The limit of $\frac{n+\sqrt{n}+\cdots+\sqrt[n]{n}}{n}$
How do I compute the following limit or show it doesn't exists?
$$\lim_{n\rightarrow\infty}\frac{n+\sqrt{n}+\cdots+\sqrt[n]{n}}{n}$$
I've struggled with this problem for a while now so I would appreciate a complete solution.
AI: The limit is $2$.
Indeed by the ... |
H: The $p^{th}$ power of the elements of a basis of a finite separable field extension.
I came across the following claim.
Let $L/K$ be a finite, separable extension of characteristic $p$ fields. Suppose $a_1,\dots,a_d$ is a basis. Then, so is $a_1^p,\dots,a_d^p$.
To prove this, one just needs to consider $\sum_{i=1}... |
H: finding the distribution of two dimensional variable that is a function of two variables of a uniform distribution
I thought I understand the matter at hand but it seems I can't solve a basic exercise on the topic.
I've got a random variable $(X,Y)$ that has a uniform distribution over
$D = \left\{(x,y) : 0\leq x\... |
H: Interpretation of the number of non-negative integer solutions to an equation
So assume there are $N$ identical balls to be arranged into $r$ boxes. There could be empty boxes.
Howe many ways are there to arrange it?
So the text book give an example of solving $N$ identical balls into $r$ boxes without empty boxes... |
H: Integral with polygon
I've got another one weird integral
$$
\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\exp\left(-\inf\limits_{(a,b)\in P}((x-a)^2+(y-b)^2)^{1/2}\right)dxdy
$$
where $P$ is a convex polygon. Help me please, I don't know how to compute infimum.
AI: Expanding into an answer since it'... |
H: Integral of determinant
Good evening. I need help with this task
$$
\int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi{\det}^2\begin{Vmatrix}\sin \alpha x&\sin \alpha y&\sin \alpha z\\\sin \beta x&\sin \beta y&\sin \beta z\\\sin \gamma x&\sin \gamma y&\sin \gamma z\end{Vmatrix} \text{d}x\,\text{d}y\... |
H: Recursively enumerable languages are closed under the min(L) operation?
Define $\min(L)$, an operation over a language, as follows:
$$ min(L) = \{ w \mid \nexists x \in L, y \in \Sigma^+ , w=xy \} $$
In words: all strings in language L that don't have a proper prefix in $L$
Question: Recursively enumerable language... |
H: Simple permutation/combination question
In how many ways I can arrange six books on different subjects in a row such that the Math book is always to the left of history book (not necessarily adjacent) ?
AI: The total number of arrangements, with no restrictions, is just $6!$, since you are ordering six books.
Let u... |
H: point-free random variable: what does the functor preserve and reflect?
GC Rota, possibly in "Twelve problems in probability theory no one likes to bring up" (verify?), wrote that the point-free definition of a random variable goes in the opposite direction from that normally taught: it is a functor from a Borel al... |
H: Divisibility properties of Harmonic numbers
Consider the $n$th harmonic number $$H_n = \sum_{i=1}^n \frac{1}{i}$$ It is a well known result that the $n$th Harmonic number is not an integer. Are there more generalized results for the Harmonic numbers? More specifically, if we collect the terms under a common denomin... |
H: Number of specific partitions of a given set
Let U be the set $U=\{(1,2,3,\ldots,2^m)\}$. Let $A$ and $B$ partitions of $U$, such that $A \cup B$ is the set $U$, and their intersection is empty, and adding the elements of the first set is the same number of the addition of the elements of the second set $B$.
How ma... |
H: Parametric representation of surface of cylinder internally tangent to a sphere
In an exercise, we are given the cylinder, $x^2+y^2=ax$ and the sphere $x^2+y^2+z^2=a^2$, and are asked to calculate the surface area of the part of the cylinder that's inside the sphere. The recommendation in the exercise is to represe... |
H: Find the total number of digits in the following case.
Two numbers $2^k$ and $5^p$ are expanded first then written side by
side i.e. adjacent to each other. Find the total number of digits in
that case if $k = p = 2004$.
My approach :
$2004\times \log 5 = 1400 $
so number of digits would be $1401$.
$2004 \tim... |
H: Connectedness of $\mathbb{R}^2\setminus \lbrace 0\rbrace$
Is there a proof that $\mathbb{R}^2\setminus \lbrace 0\rbrace$ is connected without using the idea of path-connected sets ( i.e. using the definition of connected sets only ).
AI: $\Bbb R$ is connected, isn't it ? So $\Bbb R^2$ is as well, as a product of co... |
H: Find the remainder in the following case.
Find the remainder when $444^{444^{444}}$ is divided by $7$.
My approach :
$E(7) = 6 $
$444^{444} \pmod 6 = 0$
so , $444^0 \pmod 7 = 1$
AI: Yes your approach is indeed correct. I assume by $E(7)$ you mean the Euler totient function $\phi(7)$.
By Euler's theorem/ Fermat's l... |
H: Distance between $2$ random points in a segment.
On a straight line of length $10$ cm, two points A, B are selected at random uniformly and independently. What is the probability that the distance $AB > 4$ cm?
Edit: I edited the question to make it more clear. Note that apart from the clear answer to this question ... |
H: Question about integral curve on a manifold
In Warner's book on page 36 a curve $\gamma:(a,b)\rightarrow M$ is defined to be an integral curve iff
$$d\gamma(\frac{d}{dr}|_t)=X(\gamma(t))$$
Could anyone explain to me the left side in detail and break it into coordinates? Here $X$ is a vector field on $M$.
A curve ... |
H: Where is the most appropriate place to ask about the contribution of published papers?
I am trying to come to terms with a variety of new fields as I start doing mathematics research. Often I come across a paper and am lost as to what it's contribution (or perhaps significance is). I do not mean that I think it is ... |
H: Solving $\sqrt{3} \sin(\phi-\pi/6)=\sin(\phi)$
The question is to solve:
$$ \sqrt{3} \sin \left ( \phi - \frac \pi 6 \right )= \sin \phi $$
I tried to turn it into $a \sin \phi-b \cos\phi= \sin\phi$,
Then I got $ \frac 32 \sin\phi - \frac 3 4 \cos \phi= \sin\phi$,
Therefore, $ \frac 12 \sin\phi- \frac 3 4 \cos\phi=... |
H: Some doubt on series.
If i have a series of the form ( say fourier or anything ) , For example lets consider $$\sum_{k\in \mathbb Z} \exp\left( \frac {-ik\pi}{L}\right) a_k(t) ,$$ is it always possible to split it down to something like
$$\sum_{k=1}^\infty b_k(t) \sin\left( \frac{k\pi x}{L}\right)+c_k(t) \cos\left(... |
H: interpolation between 3 points
I have a 2D space that I can query that is between [0,0] and [1,1].
At each location in space is a data point X which is in R^2.
I have 3 known values at
{x1, y1} = D1
{x2, y2} = D2
{x3, y3} = D3
where {xn, yn) is between 0 and 1
My question is, if given a value {u,v} that is each in... |
H: Algebraic formula manipulation - solve for $y$ in the equation $0.25=1-(\sqrt[r]{|y-v|}×s)$
I need to find $y$ in the following formula. I used an online algebraic calculator, but the answer wasn't correct (it omitted the index):
$0.25=1-(\sqrt[r]{|y-v|}×s)$
EDIT
The online calculator gave the following:
$y=v+\dfra... |
H: Showing for $n=pq$, s.t $p,q$ primes, there is $a$ mod $(\bmod n)$ s.t. $a^{(n-1)/2} \neq (\frac{a}{n})$, where $(\frac{a}{n})$ is a Jacobi symbol
I'd really love your help with the following question:
For $n= pq$, where $p, q$ are both distinct odd prime numbers. I need to show that there is unit $a$ modulo $n$ ... |
H: What does "$(n,m)$-tensor" mean?
I know the meaning of tensor, but I forgot the meaning of "$(n,m)$-tensor". What do $n$ and $m$ refer to?
Thanks.
AI: An $(n,m)$-tensor on a finite-dimensional real vector space $V$ is (usually) defined to be a multilinear map $\Phi:\underbrace{V^{\ast}\times\cdots \times V^{\ast}}_... |
H: Find the derivative of the primitive of a discontinous function
I have problems solving the following task:
I have the function $f(x)=\sin\frac{1}{x}$ when x isn't $0$ and $f(0) = 1$
First, I must prove that $f(x)$ is integrable in every interval $[a, b]$.
Second, I have $F(x) = \int_0^x f(t)dt$. I must find the de... |
H: Solve for $x$: $2^x = x^3$
What category of equation is this?
What methods are available to solve it?
$2^x -x^3 = 0$ where $x\in\Bbb R$
AI: You can find a solution in terms of the Lambert W Function. Rewrite as:
$$
1 = \frac{x^3}{2^x} = x^3 \exp(-x\log 2)
$$
and take the real cube root:
$$
1 = x \exp \left(-\frac{x... |
H: Any manifold admits a morse function with one minimum and one maximum
I have heard the claim: "Any closed manifold admits a Morse function which has one local minimum and one local maximum" often used in talks without a reference.
This does not seem to be very easy to prove "hands on". Trying to perturb the minima/... |
H: Solve for $x$ in $v×b=1-(\sqrt[r]{b-(v×b)}×\dfrac{2^{a/2}}{\sqrt[r]{(1-x)×v}})$
This one is over my head (and causes an error in the best online calculator I could find and used previously).
$v×b=1-(\sqrt[r]{b-(v×b)}×\dfrac{2^{a/2}}{\sqrt[r]{(1-x)×v}})$
I need to know the formula for x. If needed, $a = 3$ and $b=0.... |
H: Prove that every number ending in a $3$ has a multiple which consists only of ones.
Prove that every number ending in a $3$ has a multiple which consists only of ones.
Eg. $3$ has $111$, $13$ has $111111$.
Also, is their any direct way (without repetitive multiplication and checking) of obtaining such multiple o... |
H: Evaluating $\int^1_0\frac{x^2}{4x+5}dx$
How can I integrate this function ?
$$\int^1_0\frac{x^2}{4x+5}dx$$
AI: Divide the $x^2$ by $4x+5$, using ordinary division of polynomials. We get
$$\frac{x^2}{4x+5}=\frac{1}{4}x-\frac{5}{16}+\frac{25}{16}\frac{1}{4x+5}.$$
Now the integration should be straightforward.
Alterna... |
H: Is this solution using Gaussian elimination or Jordan-Gauss?
I am currently completing a first year Linear Algebra course via correspondence. I am busy with the chapter on Gaussian Elimination. My textbook had the following exercise:
For which values of a will the following system have no solutions? Exactly one sol... |
H: Summation Identity for Stirling Numbers of the First Kind
For the Stirling numbers of the second kind, the following identity is well-known:
\begin{align}
S(n,l) = \sum_{k_1 + \cdots + k_l = n-l} 1^{k_1} 2^{k_2} \cdots l^{k_{l}},
\end{align}
where the sum is taken over non-negative integers satisfying $0 \leqslant ... |
H: Minimum possible value of the largest of $ab$, $1-a-b+ab$, $a+b-2ab$, provided $0\leq a\leq b\leq 1$
The minimum possible value of the largest of $ab$, $1-a-b+ab$, $a+b-2ab$ provided $0\leq a\leq b\leq 1$ is:
$1/3,\quad 4/9,\quad 5/9,\quad 1/9,\quad\text{ or }\quad \text{NONE}$
My approach :
I worked out that $\,... |
H: Searching for a secret, given a non-uniform distribution
Let $s$ be an unknown bit string of length $n$. Let $p(i, b)$ be the probability that $i$-th bit of $s$ is equal to $b \in \{0,1\}$. What's the fastest method to find $s$, given the distribution $p()$? "Fastest" meaning using the least (in the expected value ... |
H: Reference: semigroups theory for PDEs
I want to study semigroup theory that people use in PDEs. Can someone recommend a good book/lecture notes that I can use to teach me the basics (and also go into depths) in an accessible way?
Thanks.
Evans has a small section which I don't like to read and I am already aware of... |
H: Finding an irreducible quartic in $\mathbb{Z}_2[x]$
I'm trying to solve the following question, can you please help me?
Find an irreducible polynomial $p(x) \in \mathbb{Z}_2[x]$ with degree 4.
Thank you!
AI: An irreducible biquadratic cannot have a root, so the constant term must be $1$, and the number of nonzer... |
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