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I would like to gain some intuition regarding the modules of Kähler differentials $\Omega^j_{A/k}$ of an affine algebra $A$ over a (say - algebraically closed) field $k$.
Let us recall the definition: let $A^e = A\mathrel{\otimes_k} A$, let $f:A^e\to A$ be the map defined by $f(a\otimes b) = ab$, and let $I = \ker f$. ... |
Taken a break from my personal project and getting some work done. I would like to find some examples, tutorials on relation algebra.
Anything with good examples will be very useful
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I'm trying to prove that a graph $G$ contains $k$ independent edges iff $q(G-S) \le |S|+|G|-2k$
for all sets $S \subset V(G)$ (where $q(G-S)$ is the odd components). I'm having trouble with verifying whether the following direction is correct though, so, does this look valid for showing that if a graph contains $k$ ind... |
I'm wondering, for no particular reason: are there differentiable vector-valued functions $\vec{f}(\vec{x})$ in three dimensions, other than the constant function $\vec{f}(\vec{x}) = \vec{C}$, that have zero divergence and zero curl? If not, how would I prove that one doesn't exist?
I had a vague memory of learning som... |
Suppose $A$ is a normed vector space and $a_i \in A$ and we have the following sequence
$$ \Vert a_1\Vert \leq \Vert a_2 \Vert \leq \cdots \leq \Vert a_n \Vert $$
for any $1\leq i<n-1$. How can I show the following is not necessarily true
$$\Vert a_{i+1} -a_i \Vert \leq \Vert a_{i+2} - a_i \Vert ?$$
It's... |
Is there any relation between the eigenvalues of possibly non-Hermitian matrix A and those of exp(A)?
For hermitian matrices, they are just exponentials of the corresponding values. But in general, any relation?
Thanks.
|
I had to do some homework problems involving the polynomial ring $R=\mathbb{Z}+x\,\mathbb{Q}[x]\subset\mathbb{Q}[x]$. This is an integral domain but not a UFD. Further, $x$ is not prime in $R$.
One of the problems was to describe to $R/(x)$.
Since $x$ is not a prime element, we know $(x)$ is not a prime ideal. So at ... |
In particular, I'm told if $k$ is commutative (ring), $R$ and $S$ are commutative $k$-algebras such that $R$ is noetherian, and $S$ is a finitely generated $k$-algebra, then the tensor product $R\otimes_k S$ of $R$ and $S$ over $k$ is a noetherian ring.
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What are the major and minor contributions of Galois Theory to Mathematics? I mean direct contributions (like being aplied as it appears in Algebra) or simply by serving as a model to other theories.
I have my own answers and point of view to this question, but I think it would be nice to know your opinion too.
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I apologize for my total ignorance in the sphere of mathematics and the possibly very silly question I'm about to ask. My mathematical knowledge level is quite limited (pretty much finished with some slightly more advanced stuff then grade 12) so please if possible limit too much terminology to about that level of math... |
Given:
$$F(F(n)) = n$$
$$F(F(n + 2) + 2) = n$$
$$F(0) = 1$$
where n is a non-negative integer. $$F(129) = ?$$
How can we solve such kind of functional equations? Is there any simpler approach other than doing iteration over the values manually?
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If $G$ is a group acting on a set $S$, then the "orbit" of a point $x$ in $S$ is defined as the set of all elements of the form $gx$ where $g \in G$. My question: why was the word "orbit" chosen for this concept? It is not obviously related to previous uses of this word, such as the path of a planet around the Sun.
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The area/arc-length is given by an integral or the integral defines the area/arc-length is one the first things we learn in Calculus, but that is done in the Cartesian coordinates, next one moves to polar coordinates and the area is transformed/redefined by a new integral using the Jacobian. My question is : Since area... |
Does there exist for all subsets of the natural numbers, with infinite elements, a function that maps each element in that subset to each of the natural numbers, and the other way around?
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Does anyone know any available reference to learn how works CORDIC algorithm to implement a division?
Thank you!
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Suppose we have the following situation: $I = \oplus_{i=1}^{p-1}I_1$ is a decomposition of an $R$-ideal in invertible modules of rank $1$, $I_iI_j \subset I_{i+j \pmod{p-1}}$ and $I_1^i = I_{i \pmod{p-1}}$.
Then it is clear that $\text{Sym}_RI_1$ surjects onto $R$, but why is the kernel of the natural map $\text{Sym}_R... |
My question is ; How can I solve the following integral question?
$\displaystyle \int_{\sqrt2/2}^{1}\int_{\sqrt{1-x^2}}^{x}\frac{1}{\sqrt{x^2+y^2}}dydx$
Thanks in advance,
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I am a computational physics postgrad student, working with libraries like ATLAS and MAGMA. I have a matrix which is upper-triangular, and is the result of a Cholesky decomposition. I need to convert the upper-triangular matrix in to a symmetric, square matrix where the elements of the lower triangle are
mat(j,i) = m... |
Why are spectral sequences called "spectral"? Is that use of "spectral" related to other uses in math, such as spectra in homotopy theory, the spectrum of a ring in algebraic geometry or the spectrum of an operator? Why are those things called spectral, for that matter?
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Suppose $A$ and $B$ are two commuting square matrices over an algebraically closed field. It is true that the spectrum of $A+B$ is contained in the set $\{\lambda_1+\lambda_2:\lambda_1 \in \sigma(A), \lambda_2 \in \sigma(B)\}$ where $\sigma(A)$ denotes the spectrum of $A$?
If so, can you provide a reference for this? c... |
Is there any general methods available for transforming a 2nd order Boundary value problem such as
$F\left(x,y,\frac{\text{dy}}{\text{dx}},\frac{d^2y}{\text{dx}^2}\right)=0$
$y(a)=y_0$ and $y(b)=y_1$
into an initial value problem?
Regards
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I would like to know if every sufficiently differentiable function is convex near a local minimum. The background to my question is that I became curious if one could motivate the usefulness of convex optimization techniques by saying that at least locally they work for all continuous functions.
Unfortunately I discove... |
Does every unfolding of an icosahedron has the same number of edges to be glued to construct it back to the solid?
If yes, what are those numbers for Platonic solids?
If no, which unfoldings have the least number of edges to be glued for Platonic solids?
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I'm looking at this second order quasilinear PDE:
$\alpha_{xx} - \alpha_{yy} - m\alpha^2 = 0$
Attempted Strategies:
Fourier Transform resulted in convolution due to the $\alpha^2$ term, I don't think that would be pretty to work on.
Separation of Variables resulted in not being able to separate the variables.
I tried ... |
I was reading the Wikipedia article of Dynamic programming, however, I'm having a hard time understanding the explanation given in the example for a type of balanced 0-1 matrix.
The problem is stated as:
Consider the problem of assigning values, either zero or one, to the positions of an n × n matrix, n even, so that... |
At one step of an algorithm (i=1: something big) I have to draw a vector from the normal distribution $X_p^{(i)}\sim N((A^{(i)} + D^{(i)}_p)^{-1}m, (A^{(i)} + D^{(i)}_p)^{-1})$ for $p=1:P$, with a large $P$ (say 5,000).
$A^{(i)}$ is symmetric positive definite and rank $K$, $A^{(i)}$ is $K\times K$ symmetric positive ... |
I am taking a beginner linear algebra class and would like a hint on a homework question. I don't need the answer, just a little guidance.
Consider the differential equation:
$$\frac{d^2x}{dt^2} - \frac{dx}{dt} - x = \cos(t).$$
We are told that the differential equation has a solution of the form
$$x(t) = a\sin(t) +... |
I've never really understood why math induction is supposed to work.
You have these 3 steps:
Prove true for base case (n=0 or 1 or whatever)
Assume true for n=k. Call this the induction hypothesis.
Prove true for n=k+1, somewhere using the induction hypothesis in your proof.
In my experience the proof is usually alg... |
Here's the question:
Solve the recurrence by obtaining a $\theta$ bound for $T(n)$ given that $T(1) = \theta(1)$.
$T(n) = n + T(n-3).$
Attempted Solution:
\begin{align*}
T(n) &= T(n-6) + (n-3) + n \\
&= T(n-9) + (n-6) + (n-3) + n \\
&= T(n-(n-1)) + [(n-n) + (n-(n-3)) + (n-(n-6)) + \cdots + n]\\
&= T(1) + [0 + 3 + 6 + ... |
Possible Duplicate:
$S^n \backslash S^m $ homotopy equivalent to $ S^{n-m-1} $
I'm trying to show that, if we embed $S^m$ in $S^n$ as the subspace $\{ (x_1,x_2, \ldots, x_{m+1}, 0, 0, \ldots, 0) \}$, $S^n-S^m$ is homotopy equivalent to $S^{n-m-1}$.
In order to show the homotopy equivalence, I tried to use the inclu... |
The ODE: $y' = (1-2x)y^2$
Initial Value: $y(0) = -1/6$
I've solved the particular solution, which is $1/(x^2-x-6)$. I don't understand what they mean about the solution is defined, because when I graph $1/(x^2-x-6)$, it's only discontinuous at $x = -2, 3$.
What does "Determine the interval in which the solution is defi... |
I was reading my daily reddit and came accross this link to a new double major at Oxford, Computer Science and Philosophy.
https://www.ox.ac.uk/admissions/undergraduate/courses-listing/computer-science-and-philosophy
I just wish my university had a double major like this, i would have taken it as an undergrad for sure ... |
Let $f(x)$ be a real-valued function on $\mathbb{R}$ such that $x^nf(x), n=0,1,2,\ldots$ are
Lebesgue integrable.
Suppose $$\int_{-\infty}^\infty x^n f(x) dx=0$$ for all $n=0,1,2,\ldots. $
Does it follow that $f(x)=0$ almost everywhere?
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Since I often draw wrong conclusions, I need a confirmation.
My argument is that for nonzero prime $\mathfrak{p}\subset A$, $A/\mathfrak{p}$ is injective being a field (hence divisible), and $A/p^{k}$ is its quotient hence again is injective, and the f.g. torsion module is a finite direct sum of such quotients hence is... |
Let $R=\mathbb{Q}[x,y,z]$, then every simple $R$-module $M$ is finite dimensional over $\mathbb{Q}$.
Had this been over $\mathbb{C}$ (complex field), it would have been rather easy. I have tried to use a theorem which says simple modules over $R$ is isomorphic to $R/I$, where I is a maximal regular ideal. But I don'... |
Application field can vary from biology\biochemistry, to computer science\coding theory, the more unexpected a connection to a field, the better. And paper preferably should be not very large one.
Thank you.
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If $f$ is continuous at $(0,0)$, is $g(x) := f(x, \sin x)$ continuous at $0$?
Intuitively this seems true because as you approach 0 the $\sin$ function also approaches 0 from both sides. Any push on how to prove this would be appreciated!
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I'm looking for a formula or quick trick for computing the branch locus of a hypersurface under projection. My specific case:
Let $V\subset\mathbb{P}^3$ be a surface of degree $d$, $p\in\mathbb{P}^3$ (it's actually a node of $V$, if that's relevant), and I'm attempting to compute the branch locus of the projection $V\... |
I saw on wikipedia that the determinant of a rotation matrix is always one (possibly by definition?), but it doesn't say anything about the determinant of the Jacobian of such a matrix.
Since applying a rotation shouldn't change the integral of a rotationally invariant function over $\mathbb{R}^d$, if $(v_1,...,v_d) =... |
If $g(x):=f(x, kx^m)$ is continuous at $0\;\;\;$ $\forall k\in R$, $\;\;\;\forall m\in N$, then $f(x,y)$ is continuous at $(0,0)$.
I'm not quite sure what is meant here by at 0. This means $f(x, kx^m) = 0$? Seems there could be an issue in this statement when k = 0?
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Let $f,g \in L^2(\mathbb{R^n})$, $\{ f_n \}, \{ g_m \} \subset C^\infty_0(\mathbb{R}^n)$ (infinitely differentiable functions with compact support) where $f_n \to f$ in $L^2$, and $g_n \to g$ in $L^2$.
Then $f_n \star g_m(x) \to f \star g(x)$ pointwise as $n,m \to \infty$ (where $\star$ denotes convolution).
Is it tru... |
I need to prove the following:
If $a\gt 0$, $n\in \mathbb{N}-\{0\}$, and if $f$ analytic on $(0, 2a)$ with $$|f^{(n)}(0)| = \frac{n!}{a^n}\cdot \sup\left\{|f(z)|\mid : |z|=a\right\},$$ then $f(z) = rz^n$ for some $r\in \mathbb{C}$.
I have tried to borrow ideas from the proof of Liouville's Theorem and go backwards. H... |
If all entries of an invertible matrix $A$ are rational, then all the entries of $A^{-1}$ are also rational. Now suppose that all entries of an invertible matrix $A$ are integers. Then it's not necessary that all the entries of $A^{-1}$ are integers. My question is:
What are all the invertible integer matrices such th... |
Possible Duplicate:
When do the multiples of two primes span all large enough natural numbers?
If $m,n$ are positive integers and $(m,n)=1$.
What's the largest integer $N$ that cannot be expressed as $N=am+bn$, where $a$ and $b$ are nonnegative integers?
The answer is $N=mn-m-n$. How to prove it?
|
from my understanding,every set has at least two subsets; the null set and the original set itself.
My question is, what is the power set of the null set? Shouldn't it be just itself?
|
If $q\colon\mathbb{C}\to\mathbb{C}$ is a polynomial, $f\colon\mathbb{C}\to\mathbb{C}$ is analytic on all of $\mathbb{C}$, and if there exists $a\gt 0$ such that $|f(z)| \lt a|q(z)|$ for every $z\in \mathbb{C}$, then $f = bq$ for some $b\in \mathbb{C}$.
Can an arbitrary analytic function (on all of $\mathbb{C}$) rep... |
If $A$ is an $n \times n$ matrix such that $A^2=0$, is $A+I_{n}$ invertible?
This question yielded two different proofs from my professors, which managed to get conflicting results (true and false). Could you please weigh in and explain what's happening, and offer a working proof?
Proof that it is invertible: Consider ... |
I needed to find the derivative of $\cos^3(x)$ and find it in a list of given derivatives, but I could not find $-3\sin(x)\cos^2(x)$, so I waited until I did more problems and the only one left was $-3\cos^3(x)\tan(x)$.
If anyone could show me the steps to convert $-3\sin(x)\cos^2(x)$ into $-3\cos^3(x)\tan(x)$, that wo... |
So I used the split step method on the Schrodinger equation and have produced the following equation:
$\Psi(x,t+dt)=F^{-1} \left\{ e^{-i\frac{\hbar^2k^2}{2m}\frac{dt}{\hbar}}F\left\{e^{-iV(x)\frac{dt}{\hbar}}\Psi(x,t) \right\}\right\}$
Which when scaled to dimensionless the time evolution step can be written as:
$\Psi(... |
If we have
An equation $E$ for which the conditions for the existence of a solution are satisfied but we can't prove the uniqueness of the solutions.
A perturbed equation $E_p$ of $E$ which the existence and the uniqueness of solution are satisfied.
The solutions $S_p$ of $E_p$ converge strongly to the solutions $S$... |
As per the title, I'm looking for the name and for a way to construct a ${\rm GF}(2)$ square matrix of size $N$ with the following properties:
All rows/columns should be linearly independent
On each row and each column there should be $N/2$ ones and zeros
Can anybody provide some pointers? It doesn't have to be a gen... |
I've decided to learn the basics of proofs and here is my first attempt. Could I improve or simplify my proof in any way? Is my formal language correct? Thanks!
Let $n$ be any integer. $$n(n+2) = (n+1)^2 - 1$$ I will prove this identify by induction.
First, check with $n=1$;
$$1 \times 3 = 2^2 + 1 \equiv
3 = 3$$
Induc... |
I know $(\{0, 1\}, \oplus)$ (where $\oplus$ is the XOR operator) is a well-known abelian group.
But what if i "extend" the underlying set?
For example, i can get $(\{0, 1, 2, 3\}, \oplus)$, or in general $(\{0, 1, ... , 2^n-1\}, \oplus)$ and they are groups as well.
Have these groups been studied? Have they any applica... |
I am a student of Computer Science and currently learning about optimization. We've been introduced to the Fritz-John and Karush-Kuhn-Tucker conditions for convex optimizations.
I think I understand what they mean, in the sense that these conditions have to be fulfilled in order for a point to be an optimum.
However, I... |
As in Wikipedia:
In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminates) and constants.
So, it's only considered a polynomial if it has both variables and constants?
$x^2 + x$ or $x + x$ are not polynomials (as they only have variables)?
|
I have been doing a basic math course on Real analysis...I encountered with a problem which follows as " Prove that $na \pmod1$ is dense in $(0,1)$..where $a$ is an Irrational number , $n\ge1$...
I tried to prove it using only basic principles...first of all I proved that above defined sequence is infinite..and also it... |
What is the difference between a "time series" and a (discrete-time) stochastic process?
|
Let $T_1$ and $T_2$ be two first-order logical theories (over the same signature) such that $T_1 \subseteq T_2$ and both are recursively axiomatized.
My question is the following: is it possible that $T_1$ is finitely axiomatizable and decidable, while its extension $T_2$ is undecidable?
I would like to know whether th... |
For an assignment, I've been asked to evaluate $7^{10507} \bmod 13$.
I know it's possible to do this using binary fast exponentiation - in fact the question refers to a previous question where I calculated $7^{12} \bmod 13$ to help prove $13$ is prime using Fermat's Little Theorem.
I could simply perform the exponentia... |
I'm having a little difficulty wrapping my head around Lie theory (I'm a computer scientist, so perhaps that's to be expected).
Specifically, considering the following definition from Wikipedia for the exponential map. What is the significance of the identity element? I think that this could be to do with the derivativ... |
Let $A$ and $B$ be two operators on a Banach space X.
I am interested in the relationship between the spectra
of $A$, $B$ and $AB$.
In particular, are there any set theoretic inclusions
or everything can happen in general like:
$\sigma(A) \subset \sigma(AB)$, and conversely,
$\sigma(B) \subset \sigma(AB)$, and converse... |
I am looking for a book in projective geometry, using the apparatus of linear algebra, complex analysis, and, perhaps, modern algebra, in full. The counterexample is the Hartshorne's book on projective geometry that starts out with a list of axioms, as in high school geometry book. I'm looking for something in more Nee... |
I am curious as to what changes do we need to make to the hypotheses of the
inverse function theorem in order to be able to find the global differentiable inverse to a differentiable function. We obviously need $f$ to be a bijection, and $f'$ to be non-zero. Is this sufficient for the existence of a global differentiab... |
Given a differential matrix equation, ie $X'=A(z)X+B(z)$ where both $A$ and $B$ are matrix of size $n\times n$ with coefficients that are holomorfic functions in a convex open set $\Omega$ and continuous on the closure $\bar \Omega$, and an initial data: $X(z_0)=u$, I know there exists a solution.
However, I haven't b... |
In a number of real analysis texts (I am thinking of Folland in particular), three different kinds of measures are defined.
Positive measures: Take values in $[0, +\infty]$
Signed measures: Take values in either $(-\infty, \infty]$ or $[-\infty, \infty)$, but cannot assume both $+\infty$ and $-\infty$.
Complex measure... |
As a newbie to vector bundles, it seems like all vector bundles I have run into ( not that many, I admit) need only two charts to be trivialized; one of these
charts will contain the "trouble point" (the point that prevents the top space from
being a global product), and one chart that will not contain the trouble... |
an arbitrary point P is chosen on side BC of triangle ABC and perpendiculars PU and PV are drawn from P to other two sides of the triangle. (It may be that U or V lies on an extension of AB or AC and not on the actual side of the triangle. This can happen, for instance, if angle A is obtuse and point P is very near B... |
I have just started out proofs so I am not fully grasping this concept yet.
Question:
Is the set $\mathbb{R}^2$, with addition and multiplication defined below a field? Explain.
$(a, b) + (c, d) = (a + c, b + d)$
$(a, b) (c, d) = (ac, bd)$
Attempted solution:
We know: $f\colon F \times F \rightarrow F$
$$f(x, y) = ... |
I am stuck on this question from a problem set. The question itself has two parts so I post them under the same topic (they're pretty much related to each other too - at least in the book). I hope it's ok.
Q1: Let $f:[a,\infty] \rightarrow \mathbb R$ derivative function. $\lim_{x\rightarrow\infty} f'(x) = \infty$. Show... |
Need some help with the steps in converting the derivatives of the following functions.
derivative of $\cos(\tan(x))$ to $\frac{-\sin(\tan (x))}{\cos^2(x)}$
I can get $-\sec^2(x) \cdot (\sin(\tan(x))$ using chain rule, but then I am stuck. I guess I just need help on understanding how $\sec^2(x) = \frac{1}{\cos^2(x... |
Need some clarification on this one.
Question:
Is the set [0, infinity) with the usual addition and multiplication axioms, a field?
Attempted solution:
Yes it is a field.
let 0 <= r, s < infinity
then by commutative addition:
r + s = s + r
s + r = r + s
and by commutative multiplication:
rs = sr
sr = rs
This is prett... |
I need some help proving the following:
Let $\mathbb{F} = \{0, 1, a, b\}$ be a field with four elements. Prove that $a^2 = b$.
You can use $a \cdot 0 = 0$ without proving it.
Attempted solution:
$$
a^2 = b \\
aa = b \\
aa + 0 = b + 0
$$
We know $a \cdot 0 = 0$ so we can substitute for zero
$$
aa + a \cdot 0 = b
$$... |
I'm taking classes in homological algebra now, and the book (together with the lecturer) seem to assume more category theory than I already know.
A "fact" that is used freely in the book ("Homological algebra" by Weibel, by the way), is that every abelian group is (isomorphic to) the direct limit of its finitely genera... |
Given this recurrence equation:
$u_1 = 0, u_2 = 1$
$u_n = 4u_{n−1} + 4u_{n−2}$
Is the correct characteristic equation:
$4x+4 = 4$
EDIT:
Complete solve:
The characteristic equation is
$x^2-4x-4=0$
We solve the quadratic equation...
$\alpha = 5$
$\beta=-1$
So:
$u_n = c_1 \alpha^n + c_2 \beta^n$
We solve the equation...
... |
Most students come to calculus with an intuitive sense of what a tangent line should be for a curve. It is easy enough to give a definition of a tangent to a circle that is both elementary and rigorous. (A line that intersects a circle exactly once.) Yet, when we talk about a curve, such as a polynomial, I think one mu... |
Here's a problem I have. Consider having 4 vectors in $R^5$ such that $3\vec{v_{1}}+\vec{v_{2}}=4\vec{v_{4}}$. Let $T$ be a linear transformation from $R^4$ to $R^5$ such that $T(e_{j})=v_{j}$ where $e$ is a normal vector.
(a) Find a nonzero vector in kernel of $T$ -- this I solved as $\begin{bmatrix}
3\\
1\\
0\\ ... |
I'm trying to teach math to my 7 year old daughter. I'm teaching following type of equations.
$$\cdots - x = y$$
I'm able to explain her the rule that:
when $\cdots- x = y$, we can always take $x$ (value on the left of equation) to the other side of $=$ sign and flip( $-$ to $+$ and vice versa), $-$ to $+$ and get... |
I knew this at one point, and in fact it is embarassing that I have forgotten it. I am wondering what filtartion of the total space of a fibration we use to get the Serre SS. I feel very comfortable with the Serre SS, I am just essentially looking for a one line answer. I checked have Hatcher, Mosher and Tangora and S... |
I have a solution to a logic problem involving propositions that I don't undersand how a particular step was carried out.The professor called the step I'm having trouble with reverse distribution.
Prove: $(p \lor q ) \land\lnot (p \lor \lnot q)\leftrightarrow p$
$(p \lor q ) \land (p \lor \lnot q) $
$p\lor(q \land \ln... |
Assume $\alpha + \beta = 1$, and one is trying to find a lower bound on the $k$th term in the binomial expansion of $(\alpha + \beta)^m$. The terms are of the form $\binom{m}{k} \alpha^k \beta ^{m-k}$.
Assume that $k \ll m \ll 1/\alpha$. More formally, and in big-O notation, $k = o(m)$ and $m = o(1/\alpha)$. Can we der... |
I started off substituting $u = e^x$ so $du = e^x\, dx$, giving me
$$\int \frac{u^8}{u^{16}+36}\,du.$$
So then I substituted $v = u^8$ and $dv = 8u^7du$, which gives me
$$\int \frac{1}{v^2 + 36}\,dv.$$
At this point I am unsure what to do.
|
I've been studying some linear algebra for a test and I've stumbled across this particular issue.
I'm given the transformation matrix $M_{BB'}$, corresponding to some $f \colon \mathbb{R}^n \to \mathbb{R}^m$ with respect to the (ordered) bases $B=(B_1, \dotsc, B_n)$ and $B'=(B'_1, \dotsc, B'_n)$.
As I understand, if I ... |
I needed some help integrating this:
$$\int\frac{2\,dx}{x\ln(6x)}.$$
I have never seen the dx within the problem like that, I am assuming I can't just move it to the outside can I?
Can I start by factoring out the 2 and then setting $u = \ln(6x)$ and $du = (1/x)\; dx$.
|
The Mathematical Contest in Modeling (MCM) is coming up in a couple of weeks. With the idea that the competition is ultimately about teaching students to become better at mathematical modeling, I was wondering if anyone else on the site who has participated in the MCM or who coaches would be willing to share strategie... |
I have several problems like this to do, but I am having a hard time understanding how they get from one step to the next in the example (see note in picture). If I can understand this example I think I'll be good. Also want to point out that this is just an example, not the actual homework problem.
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I am wondering if there are some conclusions as to when a series solution using the separate variable method to a PDE exists; i.e. for what requirements on the PDE, what requirements on the initial and boundary conditions so that one can assume the solution as products of terms involving only a single dependent variabl... |
Given that $f(x) = \dfrac{x}{x+ (2/x)}$
The derivative function is given by $f'(x) = \dfrac{Ax^2 + Bx + C}{(x^2+D)^2}$
where
$A=$
$B=$
$C=$
$D=$
From rearranging the original equation, I can get $D=2$, $C=0$ and $A=0$, but I cant seem to find $B$. Maybe someone could help?
EDIT: I found the answer to be $4$, but don't... |
A search on wikipedia shows:
$$\mu(n) = \sum_{k=1,gcd(k,n)=1}^{n} e^{2\pi i \frac{k}{n}}$$
But that uses complex numbers... and requires finding out the gcd...
How useful will be a method, if that could find the exact value for $\mu(n)$ function, by just knowing all the values from $\mu(1)$ till $\mu(n-1)$, which does... |
If we have a function, say, $y=x^2$, then $dy/dx=2x$.
Now $y=1,4,9,16...$ for $x=1,2,3,4...$
and $dy/dx=2,4,6,8...$ for $x=1,2,3,4...$
Now as $dy/dx$ represents rate of change of y w.r.t x but I can't understand how these particluar
values of $dy/dx$ represent change ?
I mean for x=3, the rate of change is 6. What does... |
My book, professor, and friends make this theorem look very significant:
If A is an m by n matrix, then the following are either all true or all false:
Each vector b in R^m is a linear combination of the columns of A.
For each b in R^m, the equation Ax = b has a solution.
The columns of A span R^m.
A has a pivot posit... |
Is $(\sin \phi)^2$ is equal to $\sin^2\phi$?
Can any one tell what is the ans for the below expression
$\sin^260$ + $\cos^260$ + $\tan^245$ + $\sec^260$ - $\csc^260$
|
I'm interested in sufficient conditions for a notion of sequential convergence to be induced by a topology. More precisely: Let $V$ be a vector space over $\mathbb{C}$ endowed with a notion $\tau$ of sequential convergence. When is there a topology $\mathcal{O}$ on $V$ that makes $V$ a topological vector space such tha... |
So I'm revising my counting, and I looked at this problem and after a little bit of brainstorming I came up with the following solution.
Number of 8 digit numbers containing both a 5 and a 6
= Number of 8 digit numbers - Number of 8 digit numbers lacking both a 5 and a 6
= 90000000 - $7 * 8^7$
= 75319936
Does this seem... |
I thought you have to say a mapping is onto something... like, you don't say, "the book is on the top of"...
Our book starts out by saying "a mapping is said to be onto R^m", but thereafter, it just says "the mapping is onto", without saying onto what. Is that simply the author's version of being too lazy to write the ... |
To solve $y^2 + 2 = x^3$ you can factor $(y - \sqrt{-2})(y + \sqrt{-2}) = x^3$ and then check that they are relatively prime and by unique factorization both must be cubes then you can solve it.
What about $y^2 + 5 = x^3$ which does not have unique factoring?
|
Can a polynomial size Context free grammar describe the finite language {$w \pi(w)$ : $\pi(w)$ is fixed string permutation, $|w|=n$ is fixed} over alphabet of {0,1}?
One case this is possible is when $\pi(w)$ is the string reverse permutation (CFG is describing palindromes).
Are there other choices for $\pi(w)$ with po... |
I'm going through Hartshorne's Geometry, and one of the exercises has stumped me for a good few hours. The problem is a version of one of Pappus's theorems:
Let $A$, $B$, $C$, be points on a line $l$, and let $A'$, $B'$, $C'$ be points on a line $m$. Assume $AC'\parallel A'C$ and $B'C\parallel BC'$. Show that $AB'\par... |
Consider a large set of points with coordinates that are uniformly distributed within a unit-length segment. Consider a Voronoi diagram built on these points. If we consider only non-infinite cells, what would be (if any) the typical (most frequent) number of edges (that is, neighbors) for those cells? Is there a limit... |
Does anybody know a simple and differentiable function that converts a 3D vector u = (x, y, z) to another vector that is orthogonal to u.
To be more precise, I am looking for three differentiable functions {f, g, h} such that the vector u = (x, y, z) is orthogonal to v = (f(x,y,z), g(x,y,z), h(x,y,z)) and v is zero on... |
Asymmetric graph is a graph that has only trivial automorphism. Asymptotically, almost all finite graphs are asymmetric. I'm looking for upper bounds and lower bounds on the growth rate of the number of asymmetric graphs on n nodes.
Also, I'm looking for a function $f(n)$ that exactly counts the number of asymmetric ... |
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