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H: $\frac{dy}{dt}-y=(7e^t + 25e^{6t})$ in terms of y, when $y(0)=7$
Here's the problem:
$$\frac{dy}{dt}-y=7e^t + 25e^{6t}$$ in terms of $\,y\,$, when $\,y(0)=7\,$
AI: To solve:
$$y'-y=7e^t+25e^{6t}$$
Multiply by $e^{-t}$:
$$e^{-t}y'-e^{-t}y=7+25e^{5t}$$
So that the right side is the result of the product rule, as fol... |
H: Evaluating $\sum_{k=1}^n k^x$
Possible Duplicate:
Finite Sum of Power?
Is there a general expression for $\sum_{k=1}^n k^x$ for any integer value of $x$? The table for $x=1,2,\dots 10$ is given here. Is there formula for any value of $x$?
AI: For arbitrary natural $x\in\mathbb{N}$, the general formula is given b... |
H: Where is the square in the Least square regression method?
I'm having a serious doubt in the least square regression problem. I guess its got to do with the notation of norm. Is the least square formulation $||b - \mathbf{A}x||^2$ or is it $||b - \mathbf{A}x||_2$. Or are they the same ?
AI: Usually it is $||b - \m... |
H: finding a basis for U
My question is a finding basis.I used the definition but I am not sure if it did work or not.
Let $ { v_1 ,..., v_n} $ denote a set of linearly independent vectors in V. Let
U={u $ \in V $ : u= $ \sum_{i=1}^n\left (a_iv_i\right )$ where $a_i \in F $ for all i and $ \sum_{i=1}^n\left (a_i\r... |
H: Prove that if $n$ is not the square of a natural number, then $\sqrt{n}$ is irrational.
Possible Duplicate:
$\sqrt a$ is either an integer or an irrational number.
I have this homework problem that I can't seem to be able to figure out:Prove: If $n\in\mathbb{N}$ is not the square of some other $m\in\mathbb{N}$, ... |
H: How to find Profit percent in this case?
A trader allows a discount of 20% on his goods and still makes a profit of 25%.Find the profit percent made, if he sells his goods at:
(1) the Marker price
(2) at 10% discount
What i've tried:
let MP = 100
D=20%
=> SP = 80
SP = 80
P = 25%
SP CP
125 100
80 ?
... |
H: factor theorem with two variables
is it possible to use the factor theorem when there is more than one variable? I believe so; however, don't know how to check every case.
Example:
$x^2-y^2$
AI: If you have a quadratic form with integer coefficients $a,b,c,$ as in
$$ f(x,y) = a x^2 + b x y + c y^2, $$ then $f... |
H: Small step in proving Hilbert's (Weak) Nullstellensatz
I'm reading up on a proof of Hilbert's Nullstellensatz which uses the Artin-Tate lemma. I followed all of it except for one step, which is probably quite elementary, but my brain may be too fried from the rest of the proof to find the logic.
Let $k$ be an alge... |
H: Recursive number of divisors function
Does there exist a recursive function, or a recurrence relation, for the number-of-divisors function?
For example, something like this:
$\sigma_0(n) = \sigma_0(n-1) + \sigma_0(n-2)$
AI: It's multiplicative, so if $m,n$ are coprime, that is have gcd 1, then $\sigma(mn) = \sigma(... |
H: How to perform logical inclusive OR operation on hexadecimal numbers?
In logic there is so called OR operation that is quite clear to me as long as it is in the binary system. For example, if I want to OR such binary values as "101" (which corresponds to decimal "5") and "110" (which corresponds to decimal "6"), yo... |
H: Place of functions in order of operations?
So I came upon something that I thought was interesting, and my google search didn't come up with a good answer.
Say you have some function $f(x)$. Where does evaluating $f(x)$ come in the order of operations? PEMDAS says that things in Parenthesis come first, so in $(3+2... |
H: What is Baye's Theorem in simplest way
I am currently planning to take a course on artificial intelligence, there Bayes theorem is the basic. Now I tried to understand bayes theorem so many times. But to understand that I have to understand conditional probability, joint probability and total probability. So can an... |
H: Inequality. $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+\frac{3n}{a^2+b^2+c^2} \geq 3+n$
I can't find any solution for this inequality which can be found here
Exercise 1.3.5 Let $a,b,c$ be positive real numbers such that $a+b+c=ab+bc+ca$ and $n \leq 3$. Prove that
$$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+\frac{... |
H: Verifying an error estimate on a multidimensional function using its jacobian
This is taken from Nocedal & Wright's Numerical Optimization 2nd edition, pg 279 Theorem 11.3:
Suppose $r:R^n\rightarrow R^n$ is continuously differentiable in a convex set $D\subset R^n$. Let $x^*\in D$ be a nondegenerate solution of th... |
H: A question dealing with the compactification of the real line
I've been reviewing my old exams, and I came across a question that I was unable to answer. Can anyone help me out on this one?
Let $X = \mathbb{R}$, and let $cX$ be the compactification corresponding to the algebra $A \subset C_{b}(X)$ of all uniformly ... |
H: How to construct a one-to one correspondence between$\left [ 0,1 \right ]\bigcup \left [ 2,3 \right ]\bigcup ..$ and $\left [ 0,1 \right ]$
How can I construct a one-to one correspondence between the Set $\left [ 0,1 \right ]\bigcup \left [ 2,3 \right ]\bigcup\left [ 4,5 \right ] ... $ and the set $\left [ 0,1 \ri... |
H: Modulo of (Power of 2 divided by a number)
I wanted to calculate the power of $2$ raised to a number $a$, divided by another number $b$ and then take the modulo $K$ of this quantity.
Meaning, I basically wanted $(2^a/b) \mod K$. Take an example where $a = 4$, $b = 3$ and $K = 98765431$. For this I calculated $3^{-... |
H: Are the Lie algebras $\mathfrak{sl}_2(\Bbb{R})$ and $\mathfrak{su}(2)$ simple?
I believe I have proven that the real Lie algebras $\mathfrak{sl}_2(\Bbb{R})$ and $\mathfrak{su}(2)$ are simple. However I have tried searching for results that mention this but they only talk of these Lie algebras as being semi-simple. ... |
H: Algorithm to generate a prime number which is n-digits long
Is there an algorithm which, given the number of digits n, generates a prime number which is n-digits long, in polynomial time complexity?
AI: It is believed (though it is a much stronger result than anyone has been able to prove) that for $N$ large e... |
H: Number of possible rod cuts of a long rod .
Basically I am trying to understand the concept of dynamic programming via Rod Cutting example.
How the number of ways in which a rod of length $n$ units can be cut is ${2}^{n-1}$ and not $2^n$?
Consider the smallest cut be of one unit and there can also be a case w... |
H: Asymmetric Normal Probability Distribution
I'm looking for a continuous probability distribution a little bit like the normal distribution but asymmetric.
In my opinion this distribution applies to phenomenons related to response time in environments marked by resource contention.
Examples I have in mind are:
In... |
H: Closed form representation of an irrational number
Can an arbitrary non-terminating and non-repeating decimal be represented in any other way? For example if I construct such a number like 0.1 01 001 0001 ... (which is irrational by definition), can it be represented in a closed form using algebraic operators? Can ... |
H: Generalization of $\frac{x^n - y^n}{x - y} = x^{n - 1} + yx^{n - 2} + \ldots + y^{n - 1}$
I thought about a generalization for the formula
$$\frac{x^n - y^n}{x - y} = x^{n - 1} + yx^{n - 2} + \ldots + y^{n - 1}$$
It can be written as
$$\frac{x^n - y^n}{x - y} = x^{n - 1} + yx^{n - 2} + \ldots + y^{n - 1} = \sum_{i ... |
H: Follow up on Fréchet derivative
I don't know whether my doubt warrants a separate question thread as it shares similarities with my last question. So in that case I apologise.
Fréchet derivative
The thing is as Siminore said a Fréchet derivative in general for infinite dimensional spaces is defined as a "Continuous... |
H: Is there a distance generalizing the Hamming distance to $[0, 1]^n$?
I'd like to compute a distance between two vectors of $[0, 1]^n$ ($n$ numbers between 0 and 1).
I'm looking for a distance generalizing the Hamming distance which I would have used if the numbers were bits.
Actually, I don't need a real distance, ... |
H: Which vectors in $\Bbb{C}^3$ are linear combinations of $(1, 0, -1), (0, 1, 1)$, and $(1, 1,1)$?
If $\Bbb{C}$ is the field of complex numbers, which vectors in $\Bbb{C}^3$ are linear combinations of $(1, 0, -1)$, $(0, 1, 1)$, and $(1, 1,1)$?
I didn't understand the question actually. It's an exercise in Hoffman's... |
H: Solve $x^2+x^2t^2+2xt^2+t^2=1$ for $x$.
Solving
$$x^2+x^2t^2+2xt^2+t^2=1$$
for $x$ yields
$$x=\frac{1-t^2}{t^2+1}.$$
I had to use software because I could not manipulate it algebraically, and sadly the program did not tell me how it reached that solution.
What is the method?
AI: This is a quadratic equation in $x$.... |
H: Prove that a set is Bounded
In the process of constructing a proof, I obtain the following set. Let C be the set of continuous and nondecreasing functions defined on $[0,1]$. Let the set $B$ be given as follows:
$$
B=\{b_i\in[0,1]: b_i=\sup_{t\in[0,1]}|h_i(t)|,\,h_i \in C\}
$$
My question is whether the set $B$ is ... |
H: Maximize function of two variables
Let $f: X \times Y \rightarrow \mathbb{R}$ be a continuous function, where $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ are compact sets.
Say under which conditions we have that
$$ \max_{x \in X} \max_{y \in Y} f(x,y) = \max_{y \in Y} \max_{x \in X} f(x,y) $$
AI: Always :... |
H: Is the integral closure of $k[[t]]$ in a finite extension of $k((x))$ necessarily a free module?
In Milne Prop 2.29, it is said that the integral closure $B$ of a PID $A$ in a separable finite extension of its fraction field is a free $A$-module.
On the other hand, I have read here that if the base ring is a compl... |
H: Is a variable transformed from a random variable a random variable?
Let a random variable $X$ be uniform distribution in $(0,1)$ and another variable $Y=X^2$. I calculate the sample space of $Y$ is also $(0,1)$, and the cdf and pdf of $Y$ are $F_Y(y)=\sqrt{y}$ and $f_Y(y) = \frac{1}{2\sqrt{y}}$. I think $Y$ should ... |
H: Nonexistence of an injective $C^1$ map between $\mathbb R^2$ and $\mathbb R$
I am getting bored waiting for the train so I'm thinking whether there can exist a $C^1$ injective map between $\mathbb{R}^2$ and $\mathbb{R}$. It seems to me that the answer is no but I can't find a proof or a counterexample... Can you he... |
H: consider the group $G=\mathbb Q/\mathbb Z$. For $n>0$, is there a cyclic subgroup of order n
consider the group $G=\mathbb Q/\mathbb Z$. Let $n$ be a positive integer. Then is there a cyclic subgroup of order $n$?
not necessarily.
yes, a unique one.
yes, but not necessarily a unique one.
never
AI: Another hint... |
H: $n+1$ is a divisor of $\binom{2n} {n}$
I want to show that $n+1$ is a divisor of $\displaystyle\binom{2n} {n}$, for all $n\in\mathbb{N}.$
I have tried to show it by induction and Pascal's rule but it did not worked out.
I would appreciate some help.
AI: From the Wikipedia page for Catalan numbers: http://en.wikiped... |
H: When $\min \max = \max \min$?
Let $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ be compact sets.
Consider a continuous function $f : X \times Y \rightarrow \mathbb{R}$.
Say under which condition we have
$$ \min_{x \in X} \max_{y \in Y} f(x,y) = \max_{y \in Y} \min_{x \in X} f(x,y). $$
From this we have tha... |
H: Is Complex Analysis equivalent Real Analysis with $f:\mathbb R^2 \to \mathbb R^2$?
Am I correct in noticing that Complex Analysis seems to be a synonym for analysis of functions $\mathbb R^2 \to \mathbb R^2$?
If this is the case, surely all the results from complex analysis carry over to the study of these $\mathb... |
H: Testing for irreducibility over $R=\mathbb Q[x]/(1+x^2)$
Let $R=\mathbb Q[x]/I$ where $I$ is the ideal generated by $1+x^2$. Then is
$y^2 +1$ is irreducible over $R$ ?
$y^2+y+1$ is irreducible over $R$ ?
$y^2-y+1$ is irreducible over $R$ ?
$y^3+y^2+y+1$ is irreducible over $R$ ?
I am completely stuck on it. Ple... |
H: How does one draw a triangle $ABC$ by using straightedge and compass knowing two sides and its median?
I want to construct one triangle $ABC$ by using edge and compass. I know that $\overline{AB}=c, \overline{AC}=b$ and the measure of the median relative to side $BC$ is $m_{a}$.
Here is what I have thought. First I... |
H: Clarification on normal subgroups and quotient groups
The following corollary is quite confusing to me.
Let $N$ be a normal subgroup of a group, $G$ and let $\bar{G}$ denote the set of cosets of $N$ in $G$. Let $\pi: G \rightarrow \bar{G}$ be the canonical homomorphism. Let $a_{1},...a_{k}$ be elements of $G$ such ... |
H: Problem understanding a proof about powers in ordered fields
I am reading through a textbook on Analysis and have come across a question that I can't seem to make any headway with. A proof is outlined, but I can't make any sense out of it.
The problem is as follows: Let $n$ be a natural in $E^{1}$, and $p,a>0$ be e... |
H: The determinant of a block Matrix of Order $2n$
Let us consider the $2n \times 2n$ block matrix $$\mathbf{X} = \pmatrix{\mathbf{P} & \mathbf{Q}\\ \mathbf{R} & \mathbf{S} }$$ where $\mathbf{P} , \mathbf{Q} , \mathbf{R} , \mathbf{S}$ are square matrices of order $n$ which commute pairwise. Show that $$\det \mathbf{X... |
H: Prove that the equation is autonomous
Let $\dot{x} = f(t,x)$, such that $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is continuous. Prove that if for every solution to the equation, $x(t)$ and for every $c \in \mathbb{R}$, $x(t + c)$ is also a solution, then $f$ is autonomous.
AI: Let's assume that the existence and un... |
H: Why is this quotient space not Hausdorff?
I am trying to show that the following space is not Hausdorff. Consider the topological space $S^1$, and let $r$ be an irrational number. Consider the action of $\mathbb{Z}$ on $S^1$ given by
$$
S^1\times\mathbb{Z}\to S^1; (e^{ix}, n)\mapsto e^{i(x+2\pi n r)}.
$$
Let $S^1/... |
H: Proving that $a,n$ and $b, n$ relatively prime implies $ab,n$ relatively prime
Question:
Suppose $a,b \in \Bbb N$, $\gcd (a,n) = \gcd(b,n) = 1$. The question is to prove or give a counterexample: $\gcd(ab,n) = 1$.
My Work:
This is what I have so far (for $\alpha, \beta, \gamma, \delta \in \Bbb Z$):
\begin{align*... |
H: Prime elements in $\mathbb{Z}/n\mathbb{Z}$
I'm tring to determine the prime elements in the ring $\mathbb{Z}/n\mathbb{Z}$.
AI: Every ideal of $\mathbb{Z}/n\mathbb{Z}$ is of the form $m\mathbb{Z}/n\mathbb{Z}$, where $m$ is a divisor of $n$. And its residue ring is isomorphic to $\mathbb{Z}/m\mathbb{Z}$.
Hence a prim... |
H: Localization at a point - an exercise
I'm trying to practice with localization of rings. In particular, I want to determine the generator t of the principal maximal ideal of the localization of the coordinate ring of a curve at a point. Let's say I have a curve C given by $y^2=(x-1)(x-2)(x-3)$.
The coordinate ring... |
H: Rigorous way to find the limit of this difference?
This is a question from an old released exam.
By the triangle inequality, $s-r<1$, so I eliminate answers D and E. Intuitively, since the lower angle between $1$ and $r$ is fixed at $110^\circ$, $s$ will always be a little longer than $r$, so I eliminate A and C t... |
H: Irreducible polynomial in $\mathbb{F}_{p}[x]$
I'm studing for an exam and I am stuck on the following practice problem.
Consider the the ring $R=\mathbb{F}_{p}[x]$. How many irreducible polynomials of degree 4 exist in $R$?
AI: Irreducible quartic polynomials (assumed monic) in $R$ are exactly the minimal polyno... |
H: Fourier transform of an exponentially attenuated function
Suppose I have a function $f(t) \in L^2(\mathcal{R})$ and it is specified by:
$$f(t) = \int_{-\infty}^{\infty} H(\omega) \exp(-\beta t \omega) \exp(i \omega t)\,d\omega$$
Suppose $H(\omega)\in L^2(\mathcal{R})$, and we know it is among the subset of $L^2(\ma... |
H: How many weak/strong limit cardinals exist under different assumptions?
I am trying to hastily answer the question in the title because I need it for another problem. I have been consulting multiple sources and am confused about the standard meanings of the following terms and the relations among them: successor ca... |
H: simple inclusion exclusion problem
Here is the question.
Each of the 90 students participated in at least one of the three track events A, B and C. If 20 students participated in A, 40 students participated in B, and 60 students participated in C, and if 5 students participated in all three events, how many stud... |
H: Is "$K$ convex + absorbing $\not\Rightarrow$ $0\in \mathrm{Int }\, K$" dependent on AC?
I have encountered the following problem in Dirk Werner's "Funktionalanalysis" (English translation by me):
Definition: A convex set $K\subset X$ is called absorbing, if given $x\in X$ there exists $\lambda>0$ such that $\lambda... |
H: From which areas of mathematics does consumer theory in microeconomics spawn?
In my intermediate microeconomics class last year, I was rather surprised by the math involved in building consumer theory. In consumer theory you do things like define a binary relation $\succsim_a$ over a set of outcomes $X$ where $x\su... |
H: Difference between E( expression ) and E[ expression ] for expected value
I don't understand the difference in notation between E( expression ) and E[ expression ] for [expected value][1]. The Wikipedia article seems to indicate E( ) is used for an arbitrary function of X, but I'm not sure what the difference in me... |
H: How to exhibit models of set theory
Even though $\mathbb{N}$ cannot be defined by first order means, it can be defined by second order means. Anyway: it can be defined, and there is no doubt, which abstract structure $\mathbb{N}$ represents.
Accordingly $\mathbb{Z}$, $\mathbb{Q}$, and $\mathbb{R}$ can be characteri... |
H: orthonormal vector fields
In a Riemannian manifold $(M^n,g)$, is it true that given a point $p$ we can find $n$ vector fields $X_i$ on a neighborhood $U$ of $p$ such that $g(X_i,X_j)(x)=\delta_{ij}$ with $x \in U$?
It seems that if I do Gram-Schmidt to the coordinate basis would work, but then the sectional cuvatur... |
H: Why is the Continuum Hypothesis (not) true?
I'm making my way through Thomas W Hungerfords's seminal text "Abstract Algebra 2nd Edition w/ Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis (There does not exist a set with a cardinality less than the reals and no set strictly gr... |
H: What's the difference between algebra and $\sigma$-algebra?
The title is quite misleading, I don't have a better one though. It's clear by definition that $\sigma$-algebra is also an algebra. Here is my question, for those algebras which are not $\sigma$-algebras, say, $\mathcal{A}_0$ on $\Omega$, (where $\mathcal{... |
H: About the solution of the infinite recurrence $f(x,f(x,f(x,f(x,f(...))))=a$
On internet I found some recreational problems as $$3=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$$ $$\frac{1}{2}=\frac{1}{x+\frac{1}{x+...}}$$ $$2=x^{x^{x^{...}}}$$ And the trick to solve them was just to reuse the equation itself inside the e... |
H: Realizing polar function via Newtonian gravitation
Let $r=f(\theta)$ be a smooth, $2\pi$-periodic function, representing
a curve star-shaped with respect to the origin.
Maybe something like this:
$r=f(2^\theta)-1.7$ (idea from here)
Which functions $f$ can be re... |
H: Prerequisite reading for Weil's Basic Number Theory
I'm interested as to what would constitute prerequisite reading for André Weil's book Basic Number Theory. For simplicity (and generality), you can assume that the reader can read anything that requires only the knowledge of undergraduate algebra, analysis, and el... |
H: Let X be a metric space and let $f : X\rightarrow R$ be a continuous function. Pick out the true statements
Let $X$ be a metric space and let $f : X\rightarrow R$ be a continuous function.
Pick out the true statements.
(a) $f$ always maps Cauchy sequences into Cauchy sequences.
(b) If $X$ is compact, then $f$ alway... |
H: isometry $f:X\to X$ is onto if $X$ is compact
Possible Duplicate:
Isometries of $\mathbb{R}^n$
A question about isometry
A isometric map in metric space is surjective?
How to prove that isometry $f:X\to X$ is onto if $X$ is compact??
What about the case when $X$ is non-compact?
Any help would be appreciated.
AI:... |
H: How to prove projection of a point onto a line(perpendicular to the line) is the most minimum distance?
I've a differential geometry math problem that I can't solve.
The problem is:
Prove that the projection of a 2D point on a 2D line which is perpendicular to the line, is the most minimum distance between the poin... |
H: Let P denote the set of all polynomials in the real variable x which varies over the interval [0, 1]. What is the closure of P in C[0, 1]
Let P denote the set of all polynomials in the real variable x which varies
over the interval [0, 1]. What is the closure of P in C[0, 1] (with its usual supnorm
topology)
please... |
H: Let B be the closed ball in R2 with centre at the origin and radius unity. Pick out the true statements
Let B be the closed ball in $\Bbb R^2$ with centre at the origin and radius unity.
Pick out the true statements.
(a) There exists a continuous function $f : B \to\Bbb R$ which is one-one.
(b) There exists a conti... |
H: If $S$ is integral over integrally closed $R$, can $\mathfrak{a}S$ be principal without $\mathfrak{a}\triangleleft R$ being principal?
Let $R\subset S$ be integral domains, with $R$ integrally closed in its field of fractions, and $S$ integral over $R$. Suppose that the fraction field of $S$ is a finite Galois exte... |
H: Fourier Coefficients in arbitrary Hilbert Spaces
Say we have an orthonormal basis $\{e_n\}$ for a infinite Hilbert Space $H$. I want to prove that any vector $x=\sum_{n=1}^\infty\langle x, e_n\rangle e_n$. I don't know where to start. Could I have any help?
AI: What is the meaning of
$$
x=\sum_{n=1}^\infty \la... |
H: Average price per person
I am having trouble judging where I am going wrong with the following question
The price of lunch for 15 people was $207.00 including a 15% gratuity for service.What was the average price per person EXCLUDING the gratuity ?
a)11.73 b)12.00 c)13.80 d)14.00 (Ans=12.00)
Here is how I a... |
H: Right adjoint to the product in an over category
Let $\mathbb{C}$ be a small category and $X\in \mathbb{C}$ an object. The category $\widehat{\mathbb{C}}$ of presheaves on $\mathbb{C}$ is cartesian closed, i.e. each product
\begin{equation}
-\times A:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}
\end{equati... |
H: To Mock A Mockingbird: Two barbers (logic puzzle)
I suspect an error in the solution given to a logic problem in the book To Mock a Mockingbird (Chapter 3 Problem 2).
Problem
Suppose I told you that the town contains two barbers, Arturo and
Roberto, and that Arturo shaves all and only those inhabitants who
sha... |
H: Which spheres are fiber bundles?
The Hopf fibration is a fiber bundle with total space $S^3$, and there are similar constructions for $S^7$ and $S^{15}$. Are there any other ways to regard a sphere as a nontrivial fiber bundle?
My guess is no. Perhaps it would help to look at the long exact sequence of homotopy g... |
H: Higher order Derivatives.
Now we know that given two Banach spaces $E$ and $F$ and a function $\ f:E \to F $ , the derivative $ Df(x) $ is a linear map from $E$ to $F$ at some point $ x $ in $E$. Briefly $ \ Df: E \to L(E,F) $ where $ L(E,F) $ is space of all linear mappings from $E$ to $F$. Now my question is how... |
H: Bizarre formula for arc length
I'm reading a book on computer science/math and I found this formula for arc lengths that I've not been able to decipher: $$\left|\int_p^q\left\| {df(x)\over dx} \right\| dx\right|$$ where $\lVert \cdot\rVert$ is the Euclidean norm.
AI: Yes that's the usual definition. It depends on y... |
H: Homomorphism induced by a transitive action is surjective?
Must an homomorphism induced by a transitive action be surjective?
Thanks.
AI: No. If $X=\{1,\dots,n\}$ for some positive integer $n$, then the cyclic group $C_n$ of order $n$ with generator $g\in C_n$ acts on $X$ by the rule $g\cdot i=i+1$ for all $1\leq i... |
H: What branch of the Math can help me with this?
I would love to focus on the branches of the Math that can help me with:
generation of entropy, i suppose that most of the works are based on statistic since even a big part of the cryptographic world starts from this and is tested with the help of the statistic. But ... |
H: two functions $ f(x) $ and $ g(x) $
let be two functions $ f(x) $ and $ g(x) $ with an infinite set of roots $ a_{n} $ and $b_{n} $ so $ f(a_{n}) =0= g(b_{n}) $
also they satisfy the same functional equation $ f(1-s)=f(x) $ and $ g(s)=g(1-s) $
then we use some numerical method to evaluate the roots and we check tha... |
H: finding $P(Y_i=X_n)$ in exponential distribution
Suppose $X_1,X_2,\cdots,X_n$ are independent random variables and we show order statistics of this random variables with $Y_1,Y_2,\cdots,Y_n$.
$X_1,X_2,\cdots,X_{n-1}$ have exponential distribution with mean $1$ and $X_n$ has exponential distribution with mean $\the... |
H: Easy polynomials question?
Please do this without using the quadratic formula.
If $\alpha$ and $\beta$ are zeroes of the polynomial $x^2 -6x + a$ then find the value of "$a$" if $3\times \alpha + 2\times \beta = 20$
Thank you for the help
There is also a second question of this sort, i dont get that either. Would h... |
H: Small math help with polynomials
If one solution of the equation $3x^2 = 8x + 2k + 1$ is $7$ times the other. Find the solutions and the value of $K$.
Note: This isn't a homework question. I'm skipping ahead in my textbook.
Thank you for the help.
Also please help me with this question no.2 also.
Find the other zer... |
H: projection of a function on an orthogonal set
I'm currently trying to solve an equation of the form
$$f(x) = \sum_m\,a_m\,\varphi_m(x)$$
and it's required me to project this equation on a different set of functions
$$\{\phi_m(x)\}$$
that is orthonormal on the interval (a,b).
How do I execute such kind of projecti... |
H: How long will it take to fill this cistern?
Pipes $A$ and $B$ call fill a cistern in $20$ and $30$ minutes and C can empty it in $15$ minutes. If the three are opened and closed after the other successively for $1$ min each in that order, how soon will the cistern be filled?
So from this we can say that part filled... |
H: A quotient field is finitely generated as a module?
When is the following true and why?
If a commutative ring $A$ is a finitely generated module over a commutative ring $B$, then its quotient field $K(A)$ is a finitely generated module over $K(B)$.
AI: It is always true that $K(A)$ is finitely generated as a modul... |
H: Is $\tan(\pi/2)$ undefined or infinity?
The way I have understood, $0/0$ is undefined or indeterminate because, if $c=0/0$ then $c\cdot 0=0$, where $c$ can be any finite number including $0$ itself.
If we also observe a fraction $F=a/b$ where $a,b$ are positive real numbers, the value of $F$ increases with the dec... |
H: If $\sum\frac{a_n}{b_n}$ and $\sum\frac{a_n^2}{b_n^2}$ both converge then $\sum\frac{a_n}{a_n + b_n}$ converges
If $\sum\limits_{n=0}^\infty\frac{a_n}{b_n}$ converges and $\sum\limits_{n=0}^\infty\frac{a_n^2}{b_n^2}$ converges, where $(a_n + b_n)b_n \ne 0$ for every $n \in \mathbb{N}$ , then show that $\sum\limits... |
H: Profinite completion of Special linear group
Is the profinite completion of $SL_2(\mathbb{Z})$ equal to $SL_2(\hat{\mathbb{Z}})$?
AI: No. The congruence kernel of the natural epimorphism $\widehat{\mathrm{SL}_2(\mathbb{Z})} \to \mathrm{SL}_2(\widehat{\mathbb{Z}})$ is a free profinite group of countably infinite ran... |
H: Intuition behind gradient VS curvature
In Newton's method, one computes the gradient of a cost function, (the 'slope') as well as its hessian matrix, (ie, second derivative of the cost function, or 'curvature'). I understand the intuition, that the less 'curved' the cost landscape is at some specific weight, the bi... |
H: Can all primes be written as a Mersenne prime?
Do all primes can be written in form of Mersenne prime?
If not, why Mersenne form is important?
AI: Trivially, the answer to the first question is no, since for example $13$ is a prime which cannot be written in the form $2^n-1$.
For the second question, one of the re... |
H: Uncountable minus countable set is uncountable
Problem statement
Prove that if $A$ is an uncountable set and $B$ is a countable set, then $A\setminus B$ must be uncountable.
What I think
The statement does not mention $A$ and $B$ relationship. I think there are two possibilities:
If $A \cap B = \emptyset $, then $... |
H: Partial linear dependence
let $S=\{s_1, s_2, s_3 \}$, if $s_1$ can be represented as a linear combination of $s_2$ and $s_3$, $s_2$ can be represented as a linear combination of $s_1$ and $s_3$ but $s_3$ can not be represented as a linear combination of $s_1$ or $s_2$ or $s_1$ and $s_2$, can we call $S$ a linearly ... |
H: Showing that $ 1<\sin\frac{\alpha}{2}+\sin\frac{\beta}{2}+\sin\frac{\gamma}{2}$
I would like to show that:
$$ 1<\sin\frac{\alpha}{2}+\sin\frac{\beta}{2}+\sin\frac{\gamma}{2}$$
where $\alpha, \beta, \gamma$ are the angles of a triangle.
I know that the inequality $$ 1<\cos \alpha+\cos \beta+\cos \alpha $$
is a direc... |
H: counter-examples in measure theory and set topology
The boundary of a subset of Euclidean space has empty interior, and furthermore has Lebesgue measure zero.Well,this is generally not true,but I can't find an explicit counter-example right now.
Similarly,in set topology.Please show that there exists a metric space... |
H: Find $y=\sqrt{x}$ where $x$ and $y$ positive integers in polynomial time?
Let $x$ be a positive integer and let $y$ be a real number such that
$$y=\sqrt{x}$$
Objectives:
If $y$ is an integer, find it in polynomial time.
If $y$ is not an integer, prove that there is no integer solution in polynomial time.
Is there... |
H: Compute derivative involving mod
Consider two angles $\alpha_1$ and $\alpha_2$. Let
$$\theta=\alpha_1-\alpha_2.$$
If $\alpha_1, \alpha_2\in [0,2\pi)$, it is obvious that $\theta\in(-2\pi,2\pi)$. Because I need to compute an error angle $\theta-\theta_0$ where $\theta_0\in[0,2\pi)$ is a constant angle, I need to res... |
H: Proving that one quantity *eventually* surpasses the other
I want to prove the following; $$\forall t>0\ \ \forall m\in \mathbb{N} \ \ \exists N \in \mathbb{N} \ \ \forall n\geq N: \ (1+t)^n > n^m.$$
For readers who hate quantifiers, here's the version in words: "$(1+t)^n$ eventually surpasses $n^m$ for some $t>0$ ... |
H: Does $i^i$ and $i^{1\over e}$ have more than one root in $[0, 2 \pi]$
How to find all roots if power contains imaginary or irrational power of complex number? How do I find all roots of the following complex numbers?
$$(1 + i)^i, (1 + i)^e, (1 + i)^{ i\over e}$$
EDIT:: Find the value of $i^i$ and $i^{1\over e}$ in ... |
H: confused about law of exponent (contradiction)
Is the following statement true?
$$
(-1.2)^{1.6} = \sqrt[10]{(-1.2)^{16}}.
$$
Is it because $a^{0.5} = a^{1/2} = \sqrt{a}$?
AI: I'll take $\sqrt{}$ as example because for odd numbers, you get a bijection from $\mathbb{R}$ into itself (with positive number mapped to p... |
H: Normal Subgroup Problem
Possible Duplicate:
$gHg^{-1}\subset H$ whenever $Ha\not = Hb$ implies $aH\not =bH$
Suppose that $H$ is a subgroup of $G$ such that whenever $H\circ a\neq H\circ b $ then $ a\circ H\neq b\circ H$.
Prove that $ g\circ H\circ g^{-1} \subset H$, $\forall g \in G$.
AI: Let $g \in G$.
Suppose ... |
H: Proving that these functions are proportional
This is a past exam question, that I don't have a solution to:
Consider the equation:
$$
\ddot{x} + p(t) \dot{x} + q(t)x = 0
$$
when $q$ and $p$ are continuous on $\mathbb{R}$, and there is $a \in \mathbb{R}$ such that
$$
\forall t \in \mathbb{R}, \space p(t) \leq -a <... |
H: Which of the following subsets of $\mathbb{R}^2$ are compact
Which of the following subsets of $\mathbb{R}^2$ are compact?
(a) $\left\{ (x, y) : xy = 1 \right\} $
(b) $\left\{ (x, y) : x^{2/3} + y^{2/3} = 1 \right\}$
(c) $\left\{(x, y) : x^2 + y^2 < 1\right\}$
clearly a and c are not compact. not sure about b
A... |
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