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I am told that when transforming coordinates $\nabla (f(Ax)) = A^T(\nabla f)(Ax)$, however I read both of these as "the gradient of f, where f is a function of Ax, where A is a matrix and x is a vector", with the second multiplied with $A^T$.
This is obviously incorrect, so how am I misunderstanding the notation?
Thank... |
This exrcise is from the first section of Marden:
Exercise 12. Let the interior of a piecewise regular curve $C$ contain the origin $\cal O$ and be star-shaped with respect to $\cal O$. If the complex numbers $a_1, a_2, \ldots, a_m$ are given, then $n$ and $b_{m+1}, b_{m+2}, \ldots, b_n$ may be determined so that all ... |
I am looking at the following:
Show that a torsion-free divisible group $G$ is a vector space over $\mathbb{Q}$.
I have no problem verifying the axioms of vector spaces after noting that the term divisible group $G$ implies that a solution to $ny=x$ exist where $n$ is an integer and is unique by the fact that the gro... |
Assume $S_1$ and $S_2$ are two $n \times n$ (positive definite if that helps) matrices, $c_1$ and $c_2$ are two variables taking scalar form, and $u_1$ and $u_2$ are two $n \times 1$ vectors. In addition, $c_1+c_2=1$, but in the more general case of $m$ $S$'s, $u$'s, and $c$'s, the $c$'s also sum to 1.
What is the deri... |
Please help me with this.
Let $G$ be abelian group, and let $A_k$ be a family of subgroups of $G$. Prove that $G=\sum A_k$ (internal) if and only if every non-zero element $g\in G$ has a unique expression of the form $g=a_{k_1}+...+a_{k_n}$, where $a_{k_i} \in A_{k_i}$, the $k_i$ are distinct and each $a_{k_i}\neq 0$.... |
Can someone please let me know if my solution is correct:
Define:
1) Let $A = \{x \in \mathbb{R}^{2}: \text{all coordinates of x are rational} \}$. Show that
$\mathbb{R}^{2} \setminus A$ is connected.
My answer: just note that $A = \mathbb{Q} \times \mathbb{Q}$ so countable and there's a standard result that $\mathbb{R... |
I just read about the construction of the real numbers in Enderton's Elements of Set Theory, and am now trying to go through all the exercises. Enderton chooses to construct the reals with left sides of Dedekind cuts. His definition is $y$ is a Dedekind cut if $\emptyset\neq y\neq\mathbb{Q}$, $y$ has no largest member,... |
First, this was one of the four problems we had to solve in a project last year and I couldn’t find a suitable algorithm so we handle in a brute force solution.
Problem: The numbers are in a list that is not sorted and supports only one type of operation. The operation is defined as follows:
Given a position i and a po... |
I'm creating a computer program where I need to calculate the parametrized circumference of an ellipse, like this:
x = r1 * cos(t), y = r2 * sin(t)
Now, say I want this parametrized ellipse to be tilted at an arbitrary angle. How do I go about this? Any obvious simplifications for i.e 30, 45 or 60 degrees?
|
Slight Motivation: In Mac Lane and Freyd's books (the latter being a reprint of an older book called "Abelian Categories") they note that instead of defining any Objects in a category we may define an "arrows only" approach by considering the identity morphism associated to an object to be the object itself.
Question: ... |
Let $R$ be a commutative ring and $m\subseteq R$ be a maximal ideal.
Can you describe the set of prime ideals of the $R/m^2$. Are they all maximal ?
|
I was watching Beating Blackjack with Andy Bloch where he runs through the basic strategy charts that outline the best strategy with playing the game. Later he also talks about the methodologies to count, but that is not relevant to my question.
He states in that video that the basic strategy is an outcome of computer... |
Let $R$ be a commutative ring and $I,J$ ideals of $R$ such that $J$ is finitely generated and the rings $R/I$ and $R/J$ are Noetherian. Are the $R$-modules $R/J$, $J/IJ$, $R/IJ$ Noetherian? Is the ring $R/IJ$ is Noetherian?
|
the Helmholtz equation
$$\Delta \psi + k^2 \psi = 0$$
has a lot of fundamental applications in physics since it is a form of the wave equation $\Delta\phi - c^{-2}\partial_{tt}\phi = 0$ with an assumed harmonic time dependence $e^{\pm\mathrm{i}\omega t}$.
$k$ can be seen as some kind of potential - the equation is ana... |
If I can make two tools which would serve the same purpose and I need 10 times less time to make first tool but work with this tool is a half time slower than work with the second tool, after what period of time I will get advantage by making the tool with which I will work faster by 2 times (the second tool).
|
For a function $f(r, \vartheta, \varphi)$ given in spherical coordinates, how can the Fourier transform be calculated best? Possible ideas:
express $(r,\vartheta,\varphi)$ in cartesian coordinates, yielding a nonlinear argument of $f$
express $\vec k,\vec r$ in the $e^{i\vec k\vec r}$ term in spherical coordinates, yi... |
We all know that the complex integral calculus can be useful for computing real integrals. I was wondering if there are any similar example where we can use Green's theorem to compute one-variables integrals.
Now it is clear that if we have an integral $\int_a^b f dx$ on the real line we can view this as a curve integ... |
If $R$ is a semiprime ring, right principally injective and satisfies ACC on right annihilators of elements, is it self-injective?
I only know that it is right nonsingular.
|
Is it true that if a continuous-time stochastic process $X_t$ is weakly stationary then $|X_t|$ is also weakly stationary?
|
i've found this interesting page: superellipse and superellipsoid
and i used the formula for one of my computer graphics applications.
i used (the most usefull for computer graphics) the parametric formula:
but for correclty draw the superellipsod i had to calculate also some absolute values of sine and cosine and dea... |
Imagine 2 sets of 6 wires. How would I find how many possible connections there are?
Every wire must be used to be considered a connection.
//1st set
ABCDEF
GHIJKL
//2nd set
ABCDEF
HGIJKL
//visual layout of situation
0 1
0 1
0 1
0 1
0 1
0 1
|
I'd like a good way to solve an optimization problem I came across. It's a constrained knapsack problem: I want to find integers
$$1=a_1\le a_2\le\cdots\le a_t$$
$$a_1+a_2+\cdots+a_t=N$$
with $t$ as small as possible such that (with the $S_i\subseteq S={a_1,a_2,\ldots,a_t}$)
$$\sum_{e\in S_i}e=C_i$$
for a certain fixe... |
I'm looking for some help disproving an answer provided on a StackOverflow question I posted about computing the number of double square combinations for a given integer.
The original question is from the Facebook Hacker Cup
Source: Facebook Hacker Cup
Qualification Round 2011
A double-square number is an intege... |
Let's say I have $3$ points, $A, B,$ and $C$. Each point has a fixed distance to every other point.
If I want to find the right representation for those distances, I have to place $A$ and $B$ according to their distance, draw two circles respectively with center $A$ and $B$ and $d(A, C)$ radius and $d(B, C)$ radius.
... |
Problem:
$$ \Psi_{xx} - \Psi_{tt} - \Psi = \exp(3t) \cdot \delta(x)$$
No boundary conditions specified.
I solved the homogeneous portion, $\Psi_\mathrm{homogeneous}$, of this equation via separation of variables but my solution is just for some random case of $k^2$ where: $F''/F = G''/G + 4 = k^2$. I chose the case wh... |
In Allen Hatcher's book on page 117 (bottom) he says the following:
The boundary map $\partial : H_n (X, A)\rightarrow H_{n−1} (A)$ has a very simple description: If a class $[\alpha] \in H_n(X,A)$ is represented by a relative cycle $\alpha$, then $\partial[\alpha]$ is the class of the cycle $\partial \alpha$ in $H_... |
I have some maths homework due in on monday, therefore I can't ask my teacher for help and the three equations I have are a bit more advanced than what sir taught us. Here is the question I'm stuck on:
$$ y \leq 3y + 2 \leq 8 + 2y $$
I would appreciate it if someone could give me a solution and/or explanation of it ple... |
I can't seem to come up with a simple formula to head-tail adding two vectors in spherical coordinates. So I'd like to know:
Can anybody point out a way to do it in spherical coordinates (without converting back and forth from cartesian coordinates)?
For the sake of execution speed in a computer program, is it faster ... |
I hope this is the right place for help on this question! I expect this should be easy for this audience (and, no, this isn't homework).
I have a task that takes $X$ seconds to complete (say, moving a rock up a hill). However, sometimes the task is restarted from the beginning (say, an earthquake rolls the rock back to... |
I'm just beginning a course in discrete mathematics and I'm learning a few of the basic laws to prove propositions. I understand how to use propositions that use the logical connectives, AND , OR, NOT. However I'm not sure how to prove a proposition that has an implies operator. For example prove the following is a tau... |
Background
I am trying to do a reproducible scientific analysis. My conclusions are not dependent on the random number generator, but the RNG does change the results ~1% between runs. I would like to define starting points for the RNG algorithm to reproduce the same result each time.
I am using the "Mersenne-Twister" a... |
Is it legal to distribute the $\land$ or the $\lor$ operators over the $\implies$ operator? For example is it legal for me to to do the following?
$(p\land(p\implies q))\implies q $
$(p\land p) \implies (p\land q) \implies q$
Put more simply: Is the "AND" operator distributive over the "IMPLIES" operator.
|
I'm TA-ing an intro class on theoretical CS, and this week class only covered the simplest concepts, such as words and languages. I wanted to take this chance to present some combinatorics on words, and prove the cube-freeness of the Thue-Morse sequence (which I want to define as $w_0 = a, w_i = w_{i-1}w'_{i-1}$ where... |
The mean value of the fractional part $\{ x \}$ on $\mathbb{R}_+$ is clearly $\tfrac{1}{2}$. I'd like to know if a similar statement holds for $\{ f(x) \}$ where $f \colon X \to \mathbb{R}_{+}$ is a non-negative $C^{k}(X)$ function for some $k \geq 0$ and assumes all values of an infinite continuum subset of $\mathbb{R... |
I'm having an argument about what the notation of $\lim$ means.
Assume you have $f_n: X \rightarrow \mathbb{R}$. Are the following two sets equal:
$$\{ x \ |\ (f_n(x)) \ \text{converges} \} = \{ x \ |\ -\infty \lt \lim_{n \rightarrow \infty} f_n(x) \lt \infty\}$$
Edit:
You can assume the reals are not extended and th... |
So I got to the infamous "the proof is left to you as an exercise" of the book when I tried to look up how to get the Lagrange form of the remainder for a Taylor polynomial. Is this right?
Given
$R_{n}(x)=\frac{1}{n!}\int_{0}^{x}f^{n+1}(t)(x-t)^{n}dt$,
find out why
$R_{n}(x)=\frac{1}{(n+1)!}f^{n+1}(c)x^{n+1}$ for some ... |
I am currently converting a minimization problem from Matlab to C++, using the Numerical Recipes implementation of the Nelder and Mead Downhill simplex method. The function requires me to define a constant lambda for each variable, which represents the "characteristic length scale" of the variable.
I tried to find a mo... |
We know that $\text{exp}(-\alpha |x|^2)$ is a fixed point for the unitary Fourier transform if $\text{Re } \alpha > 0$.
I know many arguments to show this (contour-integration and differentiation).
Is there a not an elegant way where we can exploit that fact that the Gaussian is rotationally symmetric? A sketch would b... |
In Wikipedia, a cross product between two "vectors" is defined in terms of the angle between the vectors and their magnitudes.
As I learned cross product in linear
algebra, which I understand to be a
topic about vector space, I now
wonder if cross product is not
defined on vector space, but instead
can only be defined... |
Usually Jacobian matrix encapsulate partial derivatives of a vector f w.r.t. another vector v.
More generally how to define/get the Jacobian matrix of a matrix F w.r.t another matrix V?
Is it just taking the elements of F and V and arranging them as vectors and getting the usual Jacobian matrix of 2 vectors?
|
I'm looking for a proof that if f is $\mu$-integrable ($\mu(|f|) < \infty$, where $\mu(f)$=sup{$\mu(g):g \leq f,\ g \text{ simple}$}), and $\tau$ is measure preserving ($\tau^{-1}(A)$ measurable for every measurable A), then $f \circ \tau$ is integrable and $\int f d\mu = \int f \circ \tau \, d\mu$. I have tried provi... |
Given any triangle (vertices are known) and a distance $X$, how can I compute the triangle that is shrunk by $X$ from the original? By shrink, I mean edges of the shrunk triangle are exactly $X$ away from the original edges. So if $X$ is large enough, the shrunk triangle doesn't exist.
EDIT: the resulting triangle need... |
I am a computer science student and do not have any background about tensors. Recently, I started learning about tensor spaces and tensor products from the text "Tensor spaces and exterior algebra" by Takeo Yokonuma.
I had a question and I understand that its pretty basic - but I would like some help with that.
The aut... |
While reading about cardinality, I've seen a few examples of bijections from the open unit interval $(0,1)$ to $\mathbb{R}$, one example being the function defined by $f(x)=\tan\pi(2x-1)/2$. Another geometric example is found by bending the unit interval into a semicircle with center $P$, and mapping a point to its pro... |
Is there a monoidal category $\mathcal C$ whose unit object is $I$ (i.e. $I\otimes A\cong A\cong A\otimes I$ for all $A\in \text{Ob}_\mathcal C$), with an object "$-1$" such that
$$
(-1)\otimes(-1)\cong I ?
$$
(Edit: no one misunderstood, but I'm also asking that $-1\neq I$)
I'm struggling with that since I read this m... |
Let $ABC = D$ where $B$ and $D$ are symmetrical matrices. However their [rows x columns] values are not same. For example, $B$ is 2x2 and $D$ is 3x3 a matrix. Clearly, in this case, $A$ has to be a 3x2 matrix and $C$ must be a 2x3 matrix.
Prove or disprove that this holds iff $A^T = C$ i.e. if $A$ and $C$ are transpos... |
I have N sensors which are been sampled M times, so I have an N by M readout matrix. If I want to know the relation and dependencies of these sensors simplest thing is to do a Pearson's correlation which gives me an N by N correlation matrix.
Now let's say if I have the correlation matrix but I want to explore tho pos... |
When showing that a (natural family of) diagram of $R$-algebras for all rings $R$ commutes, why does it suffice to show that it commutes for all $R$ local with algebraically closed residue field?
My idea: Reduce to local rings (fpqc descent), then to strictly henselian rings (strict henselisation is faithfully flat).
... |
Let $x = i^{i^{i^{i^{.^{.^{.{^ \infty}}}}}}}$. This is the solution of the equation
$i^x - x = 0 $ . I used Euler's identity to find a solution. But I haven't yet found the real and imaginary parts of the solution. Are there more solutions? If so why did I miss them?
|
Prove that
$$\binom{2p}{p} \equiv 2\pmod{p^3},$$
where $p\ge 5$ is a prime number.
|
Given $BA = A$ and $AB = B$ and $A$ and $B$ are two square matrices, why does $BA +AB = A+B$ ?
Please explain.
ADDED:
The actual problem is it was given that,$BA = A$ and $AB = B$ hence find the value of $A^2 + B^2$.
I simply utilized the fact that if $BA = A$ and $AB = B$ then $A$ and $B$ are idempotent matrices, hen... |
In a normal problem solving I run into a sum:
$\sum_{k=1}^n \frac{10}{10+k}$
Let $n=20$ or something like that (not huge). Browsing a list of sums and another, I'm not finding a formula for this one.
I'm wondering is it possible to find a formula for this type of sum, or if this type of sum has to be computed by a com... |
I've got the following grammar I'm attempting to convert to CNF.
S -> T01 | USV | epsilon
U -> X
V -> S1S | X | 1
X -> 0XS | 0
T -> TV | XT | UTU
I know that when I eliminate epsilon-productions I get...
S -> T01 | USV | UV
U -> X
V -> S1S | S1 | 1S | X | 1
X -> 0XS | 0X | 0
T -> TV | XT | UTU
I understand that we're... |
how can i predict a chaos with feigenbaum constant in the logistic map?
|
What is the difference between a vector $pq$ and an equation of vector $pq$?
|
Suppose $p$ is a prime, $\chi$ and $\lambda$ are characters on $\mathbb{F}_p$.
How can I show that $\sum_{t\in\mathbb{F}_p}\chi(1-t^m)=\sum_{\lambda}J(\chi,\lambda)$ where $\lambda$ varies over all characters such that $\lambda^m=id$?
($J$ is the Jacobi sum, defined as $J(\chi,\lambda)=\sum_{a+b=1}\chi(a)\lambda(b)$ )
... |
I am taking a Computer Science class soon that requires a solid knowledge of the basics of probability. I've only had minimal exposure to probability in classes I've taken in the past, so I need to get up to speed quickly. Can anyone recommend some good self-study resources (e.g. books, online classes, web sites) tha... |
Here's my problem: I have this function $\psi(\lambda)=\frac{e}{\pi\lambda}Im(e^{-\omega(\lambda-1)^{\frac{1}{4}}})$ (where $\omega=e^{-i\frac{\pi}{4}}$) defined on the interval $[1,\infty]$ and want to compute its integral (it should be 1).
I've been suggested to consider the complex function $e^{-\omega(z-1)^{\frac{1... |
What would be the formula to estimate the rate of failure of some test as a percentage chance of failure from the number of runs of the test until the first failure was seen?
For example, considering 0 to mean failure and 1 to mean success, the following are possible samples from which each should have an estimated fai... |
I have got discrete time model of the point x movement to point y stated as:
$$z = x + s\left( \frac{y - x}{||y-x||} \right)$$
where
z - is new position
s- is some offset for example 0,04
|| y - x || - euclidean norm operator
z,x,y are points in some m-dimensional R space
How to use this model (how to calculate) in
... |
I just wanted to clear this up, because i think a question I'm attempting may be somewhat badly written. It talks about a set of functions being 'bounded pointwise by 1' - but isn't that just 'bounded by 1'? I don't understand in what sense it is bounded pointwise by a single value, because surely as 1 is constant, it ... |
I have to prove the triangle inequality
$(|x_1 - z_1|^p + |x_2 - z_2|^p)^{1/p} \leq (|x_1 - y_1|^p + |x_2 - y_2|^p)^{1/p} + (|y_1 - z_1|^p + |y_2 - z_2|^p)^{1/p}$ for $p \geq 1$ on $\mathbb{R}^2$.
Two things are confusing me:
1) How we manipulate the expression, seeing as we can't use the standard triangle inequality $... |
Essentially what the title says - where to me a Hilbert space is a complete (Hermitian) inner product space, am I safe to assume every such real Hilbert space is of uncountable dimension over $\mathbb{R}$, or is there a countable-dimension example?
Thanks a lot :)
|
I am attempting to integrate a Bessel function of the first kind multiplied by a linear term: $\int xJ_n(x)\mathrm dx$ The textbooks I have open in front of me are not useful (Boas, Arfken, various Schaum's) for this problem. I would like to do this by hand. Is it possible? I have had no luck with expanding out $J_... |
An object with known initial orientation and angular velocity $\omega_0$ is subjected to constant angular acceleration $\alpha$. Its final orientation at time $t$ is some rotation of its initial orientation. How can I find this rotation?
Angular velocity and acceleration are 3D vectors. Note, $\alpha$ is not necessaril... |
Here is what our professor showed us in our linear algebra class to introduce the idea of determinants:
Suppose we have an $n$-dimensional vector space $V$. Then we can create a function from $V^n$ to $\mathbb{R}$ called $vol$ (for "volume") satisfying these properties:
$vol$ is multilinear
$vol$ is alternating (i.e. i... |
The notion that the union of two finite sets is again finite is something I took as intuitively true for quite a while. A proof using arithmetic is relatively straight forward.
Suppose $A$ and $B$ are finite sets, so let $m=|A|$ and $n=|B|$. Now $A\cup B=A\cup(B-A)$, and since $B-A$ is a subset of a finite set, it is a... |
Today matematics are moving away from understanding basics & core methodology. Level of the difficulties are reducing. Questions are becoming more simpler. Hence I would like to see the difficulty level in years Ranging 1940-1970. These Question papers will help improve students to improve in understanding rather than ... |
I think singleton sets $\{x\}$ where $x$ is a member of $\mathbb{R}$ are both open and closed.
Singleton sets are open because $\{x\}$ is a subset of itself. There are no points in the neighborhood of $x$.
I want to know singleton sets are closed or not.
|
My professor and I have came to a disagreement. My problem is with question number 8. I am pretty sure that you have to use the Bayes'thm for the question. I have tried ONCE to convince him, but regrettably, I have failed. Here is the question(s). I have included the previous questions because some of the information (... |
p is a non-zero real number. The forms of generalized mean with exponent p and $l_p$ norm seem similar except whether to take mean before taking p-root. I am trying to compare them together.
Can generalized mean with exponent p become a norm on the real sequence space?
Given a sequence, generalized mean with exponent p... |
I'm taking a first course in discreet mathematics and there are a lot of new rules/laws to remember.
My question is what is the best way to remember the rules of logic?
How would you remember the:
Absorption Laws
The Domination Laws
Identity Laws
Are there any well known mnemonics, like SOHCAHTOA in trigonometry?
|
I'm trying to help a friend of mine solve this problem. It's been like 10 years since I took calculus please help:
Find min/max values of $y = \sin x + \cos x$ on $[0, 2\pi)$
Thanks.
I've got this much:
$f'(x) = \cos x - \sin x$.
$0 = \cos x - \sin x$.
$\sin x = \cos x$.
$\tan x = 1$.
Thanks
Evaulate at end point... |
One sphere with radius $R$, named big sphere, two points on it: $a (\text{(longitude)}_a, \text{(latitue)}_a), b(\text{(longitude)}_b, \text{(latitude)}_b)$, $\text{distance}\space (a,b)=r$, $a$ is center & $r$ is radius. There is another sphere, named little sphere. Now what is the equation of circle of the intersect... |
How does one integrate
$$\int \dfrac{du}{(a^2 + u^2)^{3/2}}\ ?$$
The table of integrals here: http://teachers.sduhsd.k12.ca.us/abrown/classes/CalculusC/IntegralTablesStewart.pdf
Gives it as: $$\frac{u}{a^2 ( a^2 + u^2)^{1/2}}\ .$$
I'm getting back into calculus and very rusty. I'd like to be comfortable with some of ... |
I understand that Diophantine Analysis is an enormous field! Without first determining the solution set, suppose I'd like to calculate the number of non-negative integer solutions $(x,y,z)$ of
\begin{eqnarray}
ax^{p} + by^{q} \leq cz^{r}
\end{eqnarray}
for positive integers $a,b,c,p,q$ and $r$ (as a function of $z$). ... |
Suppose we have invariant of graph $G$ that tells us number of subgraphs with $i$ vertices and $j$ edges for every setting of $i$ and $j$. Is there a name for it?
I searched for "subgraph polynomial", but that comes up as something else.
|
Is there a way to produce parametric equations for a curve?(If we do know cartesian coordinates of course)
|
What is the geometrical "meaning" of gaussian curvature?
|
Let $A$ be a commutative noetherian ring, and suppose that $A$ is $I$-adically complete with respect to some ideal $I\subseteq A$. Is it true that for any ideal $J\subseteq I$, the ring $A$ is also $J$-adically complete?
Edit. Recall that a ring $A$ is $I$-adically complete if the canonical morphism $A\to \varprojlim... |
I'm looking for a concrete example of a complete (in the sense that all Cauchy sequences converge) but non-archimedean ordered field, to see that these two properties are independent (an example of archimedean non-complete ordered field is obviously the rationals).
Thank you in advance.
|
This is a puzzle that was sent to me a while back, I am told it is really hard, but supposedly solvable, I cant solve it, but I am interested in the solution, or any tips on how to proceed.
In front of you is an entity named Adam. Adam is a
solid block with a single speaker, through which
he hears and communicates.... |
I was wondering if you would point me to a book where the theory of second order homogeneous linear difference equation with variable coefficients is discussed. I am having difficulties in getting rigorous methods to solve some equations, see an example below.
In particular, given the recurrence relation
$X_{n+2} = \fr... |
Let $n \geq 1$. $V=Z_2^n \setminus \{0\}$. $A \cup B = V$. $A \cap B = \Phi$.
Is it true that either A or B contains a sub-space of dimension $n-1$ (without the zero element)?
(I reduced a homework question to this one).
|
I did a homework question, which was choosing the shortest path to the other points from a point using Dijkstra's Algorithm.
I ended up with the following, whilst an online applet resulted in something else (upper graph is my try, lower one is the applet's one):
(The vertices in the second image have different names, ... |
I have the feeling that an engineer should be more comfortable with math than I'm feeling I am right now. I also have a tendency toward learning math not required by the math classes I took, and was always leaning toward more fundamental way of doing engineering (math & physics.) Though I'm "fundamentalist" I feel much... |
I am studying for an exam and I don't know how to determine a general solution for a differential equation. For example,
$$
x = t x^\prime + (x^\prime)^2
$$
Can anyone help?
|
Suppose I have a value $a_1$ such that $-L < a_1 < L$. I need to compute $a_2$ from $a_1$ and $b$, while satisfying the following requirements:
$-L < a_2 < L$
if $b = 0$ then $a_2 = a_1$
if $b < 0$ then $a_2 < a_1$
if $b > 0$ then $a_2 > a_1$
a chaining requirement, explained below.
The operation can be written as $a... |
In this book I'm using the author seems to feel a need to prove
$e^{u+v} = e^ue^v$
By
$\ln(e^{u+v}) = u + v = \ln(e^u) + \ln(e^v) = \ln(e^u e^v)$
Hence $e^{u+v} = e^u e^v$
But we know from basic algebra that $x^{a+b} = x^ax^b$.
Earlier in the chapter the author says that you should not assume $e^x$ "is an ordinary powe... |
I am a bit perplexed in trying to find values $a,b,c$ so that the approximation is as precise as possible:
$$\sum_{k=n}^{\infty}\frac{(\ln(k))^{2}}{k^{3}} \approx \frac{1}{n^{2}}[a(\ln (n))^{2}+b \ln(n) + c]$$
I can see from Wolfram that $\ln(x)$ could be written in many different ways and that the sum starting from $1... |
I'm trying to understand how you derive a context free grammar (CFG) from a Pushdown Automata (PDA)?
I have the following PDA.
I believe the following context-free language can be derived..
{0^n1^n|n≥0}
My problem is I don't understand how you can build the context free grammar from a PDA and vice versa. How would I b... |
Below are the steps the teacher took to solve:
$y = \sqrt{3}\sin x + \cos x$ find min and max on $[0, 2\pi)$
Step 1: = $2\sin(x + \pi/6))$
How does the teacher get this first step?
Note: No calculus please.
|
I'm in the process of reading a paper and I believe there's a mistake, but it could also be me not noticing something.
Here's the deal:
Let $\bar\Pi_k$ be the set of all polynomials of degree at most $k$ that satisfy $p(1) = 1$.
Next, let $\mu_i, i = 1...n$ be a finite set of numbers in the open interval $(-1,1)$.
In ... |
I have a general question about the applications of mathematics.
What are some applications of mathematics that are not scientific, perhaps maybe literary or philosophical, or political.
I am basically asking for the scope of mathematics, and looking for concrete examples.
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Hi
I'm trying to prove some properties of $L^1$, the space of $\mu$-integrable functions.
I have $$ f_n \in L^1$$ and $$ \sum_{n \geq 1} || f_n ||_{1} \lt \infty$$ and I used it to prove $$ \sum_{n\geq 1} f_n(x) \lt \infty $$ $\mu$-almost-everywhere.
The next thing I would like to show is that $$ \sum_{n\geq 1} f_n(x) ... |
What makes a context free grammar ambiguous?
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My question is as to which is the best method to find extremal eigenvalues of a real symmetric matrix? Currently I am using Lanczos Iterations followed by Bisection Method. Does anyone have a better suggestion?
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Is everything that you can write in math (that makes mathematical sense) an expression? If not, what would be examples of non-expressions? And would all expressions be composed of expressions themselves?
Also, are operators (like the differential operator) by themselves expressions?
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If $f$ is a $\mu$-measurable function and I change its values on a $\mu$-negligible set, i.e. on $Y \subset Z$ with $\mu (Z) = 0$, why is $f$ still measurable?
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Suppose $K$ is a $p$-adic field (finite extension of the $p$-adics), and let $n$ be any integer (independent of what $p$ is). Define $U$ to be the set of all $x$ in $K$ such that $|x| = 1$ and such that $x = y^n$ for some $y$ in $K$. I would like to show that $U$ is an open set and that as a multiplicative group $U$ ha... |
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