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Consider conditional expectation of
a real-valued r.v. $X$ given a sub
sigma algebra $\mathcal{B}$ of the
probability space $(\Omega,
\mathcal{F}, P)$.
Will independence between
$\sigma(X)$, i.e. the sigma algebra
of the r.v., and the given sub sigma
algebra $\mathcal{B}$ make $E(X \mid
\mathcal{B}) \equiv ... |
How to calculate the distance between a convex polytope and a point?
Polytope is specified as the solution to the system of linear inequalities.
I'm looking for the method that is computationally efficient.
|
I've got a few criteria and I need to find a simple (as simple as possible) function which fits those criteria:
$$\lim_{x \to -\infty }{f(x)} = C_1$$
$$f(0) = C_2$$
$$f'(0) = C_3$$
Where $C_1,C_2,C_3$ are constants... ($C_1$ will be $1$, and the other constants are determined from another function so the function is co... |
How would I write a summation for the average speed of an object, given $10$ speeds (which were calculated $r=\frac{d}{t}$)?
Would it be:
$$ \sum_{i = 1}^{10} \frac{d_i}{t_i}$$
or am I wrong? I'm not sure about the subscript i's after distance and time, and if they are actually number sequences that I could use. Than... |
This question explains that scale-invariance (or more accurately, similarity) is an important property of Euclidean geometry. Are there any other ways to define scale-invariant geometries in any sense? And how do they differ from Euclidean geometry?
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Let $A \subseteq P(\omega)$, where $\omega$ is the set of all natural numbers and $P(\omega)$ is the power set of $\omega$. If $\langle A,\subseteq\rangle$ is a well ordered set how can you prove that $A$ is a countable set.
Let $A$ be a set which elements are closed sets of real numbers. If $\langle A,\subseteq\rangl... |
We've just started this today in first year applied maths at university. Today we were given the problem of deriving the formula for the area of an ellipse. I've got as far as saying there's some relationship between the semi-major axis and the semi-minor axis on the one hand, and the area of the ellipse on the other, ... |
This is a task from an old programming contest the task is as follows, A list of system of linear equations is given in the inputs.If the system is solvable we have to output the solution in the form of 'A + Bk',else we have to output "no solutions".
Now lets consider the inputs:
x = 3 (mod 4)
x = 4 (mod 3)
x = 1 (mo... |
As we all know, the algebraic closure often has an infinite degree.
Also, this shows the necessary and sufficient condition for a Galois extension to be a finite extension of fields.
However, we may want to characterize the cases of the case where the extension is not Galois, which is actually my question.
And this que... |
Some definite integrals, such as $\int_0^\infty e^{-x^2}\,dx$, are known despite the fact that there is no closed-form antiderivative. However, the method I know of calculating this particular integral (square it, and integrate over the first quadrant in polar coordinates) is not dependent on the Riemann sum definition... |
Suppose I define a principal $G$-bundle as a map $\pi: P \to M$ with a smooth right action of $G$ on $P$ that acts freely and transitively on the fibers of $\pi$. Does it follow that $P$ is locally isomorphic to $M \times G$ with the obvious right action of $G$ on $M \times G$? Let's suppose $M$ is a manifold.
I know... |
I'll take a Calculus course next year, and my professor suggested reviewing high school algebra.
|
If $H$ is a Hilbert space and $x \in H$ then does it follow that $||x|| < \infty$?
|
This question is related to my bath towel, which I hang on a rope, so let's have fun (you can use your own towel to do this experiment in bath-o).
There is this rectangle with sides $a<b$. The rectangle is bent along a line that passes through the center of the rectangle. At which angle $\alpha$ (i. e., angle $\angle R... |
Does there exist a property which is known to be satisfied by only one integer, but such that this property does not provide a means by which to compute this number? I am asking because this number could be unfathomably large.
I was reading Conjectures that have been disproved with extremely large counterexamples? , d... |
I have been given the following definition of simplicial approximation in lectures:
Let $K, L$ be simplicial complexes and $f : |K| \to |L|$ be a continuous map of their polyhedra. A simplicial approximation of $f$ is a map $g$ of vertices of $K$ to vertices of $L$ such that $$f(\mathrm{st}_K(v)) \subseteq \mathrm{st}... |
[Context: I'm working through problem set one of Andrew Ng's Machine Learning course, question 2, trying to derive the Hessian myself. My trouble, though, is with the simple linear algebra. I'll ignore the regularization parameters in my question to keep things simple.]
I can find $H_{jk}$ by taking partial derivatives... |
I recently stumbled across the wikipedia page on equivalents to the Axiom of Choice. I noticed that every infinite Cartesian product of a non-empty family of non-empty sets being non-empty was equivalent to the axiom of choice. Which isn't hard to see, since any element from such a product is essentially a choice funct... |
Is it possible that a finite combinatorial problem may admit a closed form solution, and for it to be impossible in practice to prove the validity of this solution? I'm not sure if a rigorous definition can be given to the notion of a finite combinatorial problem, but I mean problems of the following nature:
Given a s... |
In my recent explorations I stumbled upon the following series
$$
\vartheta_{4}(0,e^{\alpha \cdot z})=1+2\sum_{k=1}^{\infty} (-1)^{k}\cdot e^{\alpha \cdot z\cdot k^{2}} ; \alpha \in \mathbb{R}, z \in \mathbb{C}
$$
This is one of the well known Jacobi theta functions/series with the peculiarity of having the variab... |
It's a well-known theorem (Corollary 8.10 in Lee Smooth) that given a smooth map of manifolds $\phi:M\rightarrow N$ and a regular value $p\in N$ of $\phi$, the level set $\phi^{-1}(p)\subset M$ is a closed embedded submanifold. Is the converse true? That is, given an embedded submanifold $S\subset M$, is there necessa... |
I am using Sipser terminology here.
Can anyone give examples of languages A and B such that we can prove B is undecidable using A in a proof by contradiction but we A is not $\leq_m$ B.
An example of this is $E_{TM}$ as defined on page 89 of Sipser's Introduction to the Theory of Computation 2nd ed. We use $A_{TM}$ in ... |
I know that we can assume that formulas are objects in $V_\omega$, and that notions such as formula and satisfiability for a standard model (when the universe is a set) are definable and absolute. Because of that it appears to me that we can define an absolute (metamathematical) formula that "decides" when a formula (a... |
The operators in this question are in Hilbert space, but I'm going to word it as a discrete spectrum and a finite dimensional (linear algebra) answer is fine by me.
Operators $A$ and $\Delta$ are bounded, self-adjoint and non-commutative, and $B\equiv e^{- A} \Delta e^{ A}$. Some of the eigenvalues of $A$ or $\Delta$ m... |
I'm having trouble showing the following:
If $f_n$ is a sequence of measurable functions such that $f_n$ converges to $f$ almost everywhere, then $f$ is measurable.
I was thinking of using $\limsup$ since I know that $\limsup f_n$ is measurable. But now I'm not sure how to continue my argument.
|
My question is how to interpret sigma algebra, especially in the context of probability theory (stochastic processes included). I would like to know if there is some clear and general way to interpret sigma algebra, which can unify various ways of saying it as history, future, collection of information, size/likelihood... |
Let $A$ be a 3x3 real symmetric matrix, and let $v_1, v_2$ be orthogonal eigenvectors of $A$. Suppose $v_3$ is orthogonal to both $v_1$ and $v_2$. Is $v_3$ necessarily an eigenvector of $A$?
To put it another way: by the spectral theorem we know that $\mathbb{R}^3$ has an orthonormal basis consisting of eigenvector... |
In the book, it said:
Let $n = a_{k}10^{k} + a_{k-1}10^{k-1} + a_{k-2}10^{k-2} + ... + a_110 + a_0$
Then, because $10 \equiv 0 \pmod{2}$ it follows that $10^j \equiv 0 \pmod{2^j}$
What congruence property did they use in this case? Is that:
If $a \equiv b \pmod{k_1}$ and $c \equiv d \pmod{k_2}$ then,
$ab \equiv cd \pm... |
In reading journal articles (in physics), I often come across recurring equations in the Introduction sections. Sometimes they don't mention its name. For example, I come across
$$\begin{eqnarray*}
E\Psi &=& {\skew{6}{\hat}{H}}\Psi\\
i\hbar\frac{\partial}{\partial t} &=& {\skew{6}{\hat}{H}}\Psi\\
\end{eqnarray... |
Possible Duplicate:
coordinate ring questions
Hello!
I need your help to solve this question
A subset X⊂k^n (k a field) is called algebraic if there exist polynomials f1,…,fm∈k[t1,…,tn] such that
X={x∈Kn|f1(x)=...=fm(x)=0}.
The coordinate ring k[X] of X is the ring of all functions f:X→k that can be represented by s... |
So the question states that the intersection of two prime ideals is always a prime ideal. Well this is false but I need an example to counter it. I looked online and found one "For example, inside $\mathbb Z, 2 \mathbb Z$ and $3\mathbb Z$ are prime, but there intersection, $6\mathbb Z$ is not prime"
so I just need some... |
I am having trouble with the following problem and would appreciate some help. It is a piece of a proof of Cauchy's integral formula for a circle that appears in Dieudonne's Treatise on Analysis, Vol. I. I am interested in this proof because it proceeds directly from the assumption that $f$ is complex differentiable ... |
Wikipedia describes the first-order vs. second-order logic as follows:
First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic has these variables as well as additional variables that range over sets of individuals.
It gives $\forall P\,\forall x (x ... |
I'm currently a Calculus III student. I enjoy math a lot, but only when I understand its beauty and meaning. However, so many times I have no idea what it is I am learning about, althought I am still able to solve problems pertaining to those topics. I'm memorizing the sounds this language makes, but I dont understand ... |
Two players A and B each has a fair coin and they start to toss simultaneously (counted as one round). They toss in $n$ ($\ge 1$) rounds and stop because they have accumulated the same number of heads (could be 0, the case that both of them did not get any head) for the first time. What is the distribution of $n$ and i... |
Can anyone share me the trick to solve this problem using congruence?
Thanks,
Chan
|
I recently showed my Algorithms class how to bound $\ln n! = \sum \ln n$ by integrals, thereby obtaining the simple factorial approximation
$$ e \left(\frac{n}{e}\right)^{n} \leq n! \leq en\left(\frac{n}{e}\right)^{n} $$
But one student, having seen Stirling's approximation, asked where the $\sqrt{2 \pi n}$
term comes... |
Let $W^{m,p}(\Omega) = \{ f \in L^p(\Omega): D^\alpha f \in L^p(\Omega) \text{ for multi-indices } |\alpha| \leq m\}$, where $D$ denotes the weak derivative. Let $W_0^{m,p}$ denote the closure of $C_c^\infty(\Omega)$ in $W^{m,p}(\Omega)$.
Why is it true that $W_0^{m,p}(\mathbb{R}^d) = W^{m,p}(\mathbb{R}^d)$, but in ge... |
What is the formula to get sum of series
$$1+(1+x)+(1+x)^2+.... +(1+x)^n,$$
where $n$ is Integer and $x$ is Rational
|
I'm really confused about congruence. I tried hard, but I kept failing :(
$$30! \equiv -1 \pmod{31} \text{ by Wilson's Theorem}$$
$$ \Longleftrightarrow 30.29.28.27.26.25! \equiv -1 \pmod{31}$$
$$ \Longleftrightarrow (-1).15.10.(-8).6.25! \equiv -1 \pmod{31}$$
$$ \Longleftrightarrow 15.4.5!.25! \equiv -1 \pmod{31}$$... |
Here is the original formula:
$$\frac{256}{2^x}=y$$
In order to solve for $x$, I've done this:
$$\log_{2}\left(\frac{256}{y}\right)=x$$
The problem is that $y$ can be zero. What should I do to solve for $x$?
|
In Cryptography, I find it commonly mentioned:
Let G be cyclic group of Prime order q and with a generator g.
Can you please exemplify this with a trivial example please!
Thanks.
|
let $a_n$ be a series and +5, -5 partial limits of it.
as well note that $\left|a_{n+1}-a_{n}\right| \leq \frac{1}{n}$ for all $n$.
prove that 0 is also a partial limit of $a_n$.
Thank you.
|
Let $\alpha = \frac{1+\sqrt{-19}}{2}$. Let $A = \mathbb Z[\alpha]$. Let's assume that we know that its invertibles are $\{1,-1\}$. During an exercise we proved that:
Lemma: If $(D,g)$ is a Euclidean domain such that its invertibles are $\{1,-1\}$, and $x$ is an element of minimal degree among the elements that are not... |
I'm stuck with this algebra question.
I try to prove that the exterior algebra $R$ over $k^d$, that is, the $k$-algebra that is generated by $x_1,\ldots,x_d$ and $x_ix_j=- x_jx_i$ for each $i,j$, has just one simple module which is not faithful.
I think the only simple module is $k$, but I am not really sure if my idea... |
Are there any known barriers to show the following invariant (perhaps by some sort of induction)?
Let $\Sigma$ be some finite alphabet with $|\Sigma| \geq 2$, let $M$ be some (deciding) deterministic Turing machine with input alphabet $\Sigma$, and let $L_0 \subseteq \Sigma^{\star}$ be some non-sparse, $\mbox{NP}$-comp... |
I've started learning about measure theory and I'm trying to get some intuitive grasp of the basic concepts. This is only succeeding partially so far.
There is an exercise which I don't quite understand. Here it is:
Let $\Omega$ be the set of all sequences $\omega = (\omega_1,\omega_2,\ldots)$ where $\omega_n \in \{0... |
I am trying to understand how definite integral works when the partitions of the function are of unequal length. I found this link and am stuck here:
$$I = \int_a^b f(x)dx = \lim_{mesh(P) \to 0} R(f,P,T)$$
I understand that $\int_a^b f(x)dx = \text{Area under the curve}$ but not how that is equal to $\lim\limits_{mesh... |
This is an attempt to refine my previous question into something precise enough to admit a resolution.
Let $P\;$ be a graph property, let $T_n\left(P\;\right)$ be the set of isomorphism classes of graphs on $n$ vertices which satisfy $P\;$, and let $f\left(n\right) = \left|T_n\left(P\; \right)\right|$, the number of ... |
Theorem:
If $x,y \in \mathbb{R}$ and $x > 0$, $\exists$ a positive integer $n$ such that $nx > y$
I read the proof by Rudin and understood it. I think it is very elegant and uses the LUB property of $\mathbb{R}$. But the way I tried to prove it is the following:
If $x > y$, then $n = 1$. If $x < y$, then $y > 0$ and it... |
The diagram shows the graph of $y=|mx+n|$
(i tried my best to do the same thing as my exercise book, actually 1 is propotional to 1 and 3 is propotional to 3, but 2 is not propotional to 2)
Find the value of $m$ and $n$.
Anyone can help me solve this problem? Thanks
|
How do I show that on a projective hypersurface of homogenous equation F = 0, the singular
points arise as common zeros of the partial derivatives dx_i(F)?
I have an assignment to calculate the singular points of the Steiner surface and it is not so hard using the above theorem. But I would like to understand why this ... |
I have been thinking about this for quite sometime.
Eisenstein Criterion for Irreducibility: Let $f$ be a primitive polynomial over a unique factorization domain $R$, say
$$f(x)=a_0 + a_1x + a_2x^2 + \cdots + a_nx^n \;.$$
If $R$ has an irreducible element $p$ such that
$$p\mid a_m\ \text{ for all }\ 0\le m\le n-1$$
$$... |
X is a projective variety, W is a quasi-projective variety over the algebraically closed field k.
I would like to construct a k-algebra isomorphism between O(XxW) and O(W) (the rings of regular functions).
On the level of varieties I know that the projection XxW->W maps closed sets to closed sets. I feel like this shou... |
Possible Duplicates:
Why is Euclidean geometry scale-invariant?
Proof that Pi is constant (the same for all circles), without using limits
The answer with the most up votes will be selected as the answers after 72-hours has passed.
Questions, feedback, requests -- just comment, thanks!!
|
In probability, events are considered to be closed under countable union and complement, so mathematically they are modeled by $\sigma$-algebra. I was wondering why events are considered to be closed under countably union and complement?
In Nate Eldredge's post, he has done an excellent job on explaining this, by us... |
There are $n$ symmetric polynomials in the eigenvalues of a square matrix. Two of these are the determinant and the trace, each of which have countless applications and interpretations in algebra and geometry.
What about the other symmetric polynomials? They are also similarity invariants, yet I've never seen them used... |
I'm trying to understand the proof of lemma 3.2 of this article by Keith Conrad on proofs of the simplicity of $A_n$. There is a step I don't follow.
When he says "Relabelling, we may write $\sigma=(1 2 \dots r) \pi$", I don't see why we could do that. Relabelling what?
As I tried to understand the statement, I thought... |
The characteristic polynomial of a matrix $A$ is defined as:
$$\chi(A) = \det(xI-A) = \sum_{i=0}^n (-1)^i\cdot \operatorname{tr}^{(i)}(A) \cdot x^{n-i}$$
The trace is the sum of the eigenvalues of a matrix, tr(A) = tr(1)(A). It is also the sum of the diagonal entries:
$$\operatorname{tr}(A) = \sum_{i=1}^n A_{ii}$$
The... |
I ask because, as a first-year calculus student, I am running into the fact that I didn't quite get this down when understanding the derivative:
So, a derivative is the rate of change of a function with respect to changes in its variable, this much I get.
Thing is, definitions of 'differential' tend to be in the form o... |
Is quantum logic producing interesting/different mathematics?
Is it different from the intuitionist approach to mathematics? How?
|
Fermat's little theorem says:
If $p$ is prime, and $a$ is an integer with $p \nmid a$, then $a^{p - 1 } \equiv 1 \pmod{p}$.
And this is a small part that I extracted from the book:
Let $n = 4k + r$ with $0 \leq r \leq 3$. Then by Fermat's little Theorem, we have
$$b^n \equiv b^{4k + r} \equiv (b^4)^kb^r \equiv ... |
While helping a friend out for an exam (last year of high school), I found an exercise that neither of us could solve. I've tried a couple of different approaches but nothing seemed to work. Could anyone tell me how to solve it, or at least some hints? This is the exercise:
Knowing that
$$\lim_{x \to a}\frac{x^2-\sqrt... |
The reduced residue system modulo $10$ is: $1, 3, 7, 9$
But how could we find these numbers?
The only thing I know is they're relatively prime to $10$.
What does it mean by "no two different elements of the set are congruent to modulo m"?
Thanks,
Chan
|
I'm thinking of modulo, but really don't know how to start? A hint would be sufficient.
Thanks,
Chan
|
I have a problem that is has the notation $lg^2 n$ and I just want to verify that it actually means / is the same as $(lg n)^2$
If it is not the same please tell me how to evaluate $lg^2 n$ (always assuming lg is base 2).
I do not have a source to quote from, I was just given a bunch of algorithms to sort out. Sorting... |
Show that if $c_1, c_2,\ldots, c_{\phi(m)}$ is a reduced residue system modulo $m$, $m \neq 2$, and $m$ is a positive integer, then $c_1 +\cdots+ c_{\phi(m)} \equiv 0 \pmod{m}$
From the problem statement, I only know that $\gcd(c_i, m ) = 1$.
Is there any related theorem that I missed?
A hint would be greatly appreci... |
Is there an explicit well-ordering of $\mathbb{N}^{\mathbb{N}}:=\{g:\mathbb{N}\rightarrow \mathbb{N}\}$?
I've been thinking about that for awhile but nothing is coming to my mind. My best idea is this:
Denote by $<$ the usual "less than" relation on $\mathbb{N}$. Since $\mathbb{N}^{\mathbb{N}}$ is the set of infinite
s... |
consider the simple function
$$f(x) = \left\{
\begin{array}{ll}
1 & \mbox{if $x \in [0,1]$}\\
10 & \mbox{if $x \in \mathbb{Q} \cap [1,2]$}\\
-10 & \mbox{if $x \in \mathbb{Q^c} \cap [1,2]$}\\
2 & \mbox{if $x \in [2,3]$}\
\end{array} \right.$$
is it correct to say that f is continuous ... |
I read a problem to determine the integral $\int_1^{100}x^xdx$ with error at most 5% from the book "Which way did the bicycle go". I was a bit disappointed to read the solution which used computer or calculator. I was wondering whether there is a solution to the problem which does not use computers or calculators. In p... |
I can't figure this problem out.
Let $b_1, b_2, \cdots, b_{\phi(m)}$ be the integers between $1$ and $m$ that are relatively prime to $m$. $B$ is the product of these integers. Prove that either $B$ is congruent to $1$ or $B$ is congruent to $-1 \pmod{m}$. Could you give me some hint where to start?
EDIT: Please avoid ... |
I realize that this is not a typical programing question but its still related. If anyone could help me out I would really appreciate it because I have a midterm coming up and this is the part that I don't understand. This is not a homework problem so don't worry about me trying to get out of my work. I just need someo... |
The goal is to maximize the following function:
\begin{align}
K_p(q) = q\log \frac{q}{p} + (1-q)\log \frac{1-q}{1-p}
\end{align}
where
\begin{align}
0 \leq q \leq 1
\end{align}
and $p \in (0,0.5)$ and is some constant. In other words, I'm maximizing the KL divergence in terms of one Bernoulli variable with success para... |
Reconstruction conjecture says that graphs (with at least three vertices) are determined uniquely by their vertex deleted subgraphs. This conjecture is five decades old.
Searching relevant literature, I found that the following classes of graphs are known to be reconstructible :
trees
disconnected graphs, graphs whose... |
I am trying to solve a problem which I have reduced down to find out one of integer solutions to a number of nonhomogeneous linear inequalities.
Can this be done efficiently ? If so: how?; if not: why?
|
Let $\phi(n) $ is the numbers of number that are relatively prime to n.
Then, how could we solve the equation $\phi(n) = k, k > 0?$
For example:
$\phi(n) = 8 $
I can use computer program to check all numbers that are relatively prime to $n$, but I think there must be an easier way to approach this problem.
Base... |
I understand the process in calculating a simple series like this one
$$\sum_{n=0}^4 2n$$
but I do not understand the steps to calculate a series like this one
$$\sum_{n=1}^x n^2$$
I have an awesome calculator and know how to use it, so I know the solution
$$\frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{6}$$
but I don't kno... |
The following is a quote from Lifting Markov Chains to Speed up Mixing, by Chen, Lovasz, and Pak:
...Thus we have described a (randomized) stopping rule that, for any starting node, stops in an expected number $O(\sqrt{n})$ steps, and the probability of stopping at each node is at least 1/2 its stationary probability.... |
I'm trying to do a piece-wise affine transform in Python.
I have one image with a set of points hand marked and another set of points where I wish to "move" my current points and the texture between.
This is my understanding of the transformation:
Find the delaunay triangulation of the moved points
For each pixel i... |
I've always used the method of Lagrange multipliers with blind confidence that it will give the correct results when optimizing problems with constraints. But I would like to know if anyone can provide or recommend a derivation of the method at physics undergraduate level that can highlight its limitations, if any.
|
I am a little muddled and am hoping I can get some clarification about forms in a complex manifold. Since I am only concerned with local issues, consider $M = \mathbb C^n$ as a complex manifold. So I have complex coordinates $z_1,\ldots,z_n$ and corresponding real coordinates $x_1, y_1,\ldots, x_n, y_n$ with $z_j = x... |
If $\displaystyle L = \frac{3+\sqrt{-3}}{2}$,
and if $x\equiv 1\pmod{L}$,
show that $x^3\equiv 1\pmod{L^4}$.
I have already shown that if $x\equiv 1\pmod{L}$,
then $x^3\equiv 1\pmod{L^3}$.
Thanks.
|
Show that if a set $A$ in a metric space is bounded, so is each subset
$B \subseteq A$.
|
Background:
Let $V$ be a vector space over a field $\Bbbk$. Let $V^*=Hom_\Bbbk(V,\Bbbk)$ be the vector space of linear functionals on $V$. Consider the map $V^*\times V\to End_\Bbbk(V)$ given by $(f,v_0)\to\phi$ where $\phi(v)=f(v)\cdot v_0$. This map is obviously bilinear and thus induces a linear map $V^*\otimes_\Bbb... |
[Disclaimer: I've studied a lot of logic but never been good at analysis, so that's the angle I'm coming from below]
In my attempt to find a precise version of the 'definitions' usually given when first introducing Leibniz notation in single or multivariable calculus or analysis, wherein there is no appeal to different... |
My textbook says that solutions for the equation $y'=-y^2$ must always be 0 or decreasing. I don't understand—if we're solving for y', then wouldn't it be more accurate to say it must always be 0 or negative. Decreasing seems to imply that we're looking at a full graph, even though the book is talking about individual ... |
What resources for "Calculus" do you recommend for beginners; they should describe the topics in a way easy to understand?
|
These are a series of questions about Turing machines.
First, are the number of a given Turing machine configurations (state + tape) countable?
Secondly, given that a computation history is a sequence of configurations, are the number of possible computation histories for a given Turing machine also countable?
My intui... |
Does a binary operation * exist such that (N;+) and (Z;*) are isomorphic?
N : set of natural numbers, Z : set of integers and + is the addition operation.
If yes, please give me an isomorphism function?
|
$\dot{\theta} \equiv \frac{d \theta(t)}{dt}$
I encountered this ODE
$\ddot{\theta} + \mu \dot{\theta}^2=0$
How do I find a solution for $\theta(t)$. I know $\dot{\theta}(t=0)=\omega_0$
|
For which values of m does the equation:
$$3 \ln x+m x^3 = 17$$
have $1$ solution? $2$ solutions? $0$ solution?
Thanks.
|
I have no idea about the underlying theory from which the mathematical induction was derived.
How to prove the mathematical induction is true?
|
The method of Lagrange multipliers is used to find the extrema of $f(x)$ subject to the constraints $\vec g(x)=0$, where $x=(x_1,\dots,x_n)$ and $\vec g=(g_1,\dots,g_m)$ for $m \leq n$.
Although many textbooks get the final equations by arguing that at an extrema, the variation of $f(x)$ must be orthogonal to the sur... |
Hi
I want to integrate this integral and ask if my work is correct or not.
$$\int^\infty_0 dx x^{\alpha-1} e^{-x} (a+bx)^{-\alpha}$$
I want to integrate it by parts, so I have
$$(a+bx)^{-\alpha} = v$$
$$-b\alpha(a+bx)^{-\alpha-1}dx = dv$$
$$x^{\alpha-1} e^{-x} dx = du$$
$$\Gamma(\alpha) = u$$
now the integral be... |
On the interval $(0, \infty)$,the function $f \geq 0$,$f' \leq 0$, and $f'' \geq 0$.Prove that $\lim\limits_{x \to \infty} xf'(x) = 0$.
|
I have a homework question where we have to prove the following definitions of a binomial coefficient are equal, algebraically.
This is what I got so far, and it's getting pretty complicated. And I could use some directions on how to continue. At the moment I think I'm just doing it wrong from the start and I'm overco... |
how can I please calculate an arc length of $\dfrac{e^x-e^{-x}}{2}$. I tried to substitute $\dfrac{e^x-e^{-x}}{2}=\sinh x$, which leads to $\int\sqrt{1+\cosh^2x}dx$, which unfortunately I can't solve.
Thank you very much.
C.
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We know the Stone-Weierstrass theorem for locally compact Hausdorff spaces (LCH) which states the following:
Theorem: Suppose $X$ is LCH. A subalgebra $\mathcal{A}$ of $C_0(X)$ is dense if and only if it separates points ($\forall x,y \in X : x\neq y \implies \exists f \in \mathcal{A}: f(x) \neq f(y)$) and vanishes no... |
Okay, so my text required me to actually prove both sides; The non-empty subset $W$ is a subspace of a $V$ if and only if $a\mathbf{x}+b\mathbf{y} \in W$ for all scalars $a,b$ and all vectors $\mathbf{x},\mathbf{y} \in W$. I figured out one direction already (that if $W$ is a subspace, then $a\mathbf{x}+b\mathbf{y}$ is... |
I recently found the primitive element theorem a little bit unnatural, and hence I was trying to replace the proofs of some theorems using the primitive element theorem, and here is one.
In the book by Jurgen Neukirch, the theorem that states (1) and (2) are equivalent, and uses the primitive element theorem:
$$\begin{... |
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