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I am thinking of the property in probability of inequality. In particular, we assume \begin{equation} P[\zeta>a]\leq b, \end{equation} where $a>0$, $b>0$ and $\zeta\in R$ is a random variable.
Now we would like to consider whether the inequality of $P[\zeta^2>a^2]\leq b$ holds.
In fact, for differential and monotonic t... |
In a previous post, I applied my rules-of-thumb for
response time (RT) percentiles (or more accurately, residence time in queueing theory parlance), viz., 80th percentile: $R_{80}$, 90th percentile: $R_{90}$ and 95th percentile: $R_{95}$ to a cellphone application and found that the performance measurements were not co... |
:Prefacing with mentioning that this method works exactly like Ross's above, however may appear a bit more tangible to use:
Assuming the coin is fair and $Pr(H)=Pr(T)=\frac{1}{2}$
This can be described with a Markov Chain with seven states (including a start and a placeholder state which can only be entered after a sin... |
In the Artin's book on Algebra, the author stated a theorem (Ch.9, Thm. 2.2):
"A finite subgroup $G$ of $GL(n,\mathbb{C})$ is conjugate to a subgroup of $U(n)$.
Here, $U(n)$ is the unitary group, i.e. if $\langle \,\,, \rangle$ is
on $\mathbb{C}^n$ given by $\langle (x_1,\cdots,x_n),(y_1,\cdots,y_n)\rangle=\sum_{}{x_i\... |
I’ve avoided doing any manifold (regretting it somewhat) courses, however do have some understanding. Let $p$ be a point on a surface $S:U\to \Bbb{R}^3$, we define:
The tangent space to $S$ at $p$, $T_p(S)=\{k\in\Bbb{R}^3\mid\exists\textrm{ a curve }\gamma:(-ε,ε)\to S\textrm{ with }\gamma(0)=p,\gamma'(0)=k\}$.
The tang... |
On the uniqueness of bound state solutions of a semilinear equation with weights
Departamento de Matemática, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile
$ \mbox{div}\big(\mathsf A\, \nabla v\big)+\mathsf B\, f(v) = 0\, , \quad\lim\limits_{|x|\to+\infty}v(x) = 0, \quad x\in\mathbb R... |
Attractors for first order lattice systems with almost periodic nonlinear part
Department of Mathematics, The University of Jordan, Amman 11942, Jordan
$ \begin{equation*} \overset{.}{u}+Au+\alpha u+f\left( u,t\right) = g\left( t\right) ,\,\,\left( g,f\right) \in \mathcal{H}\left( \left( g_{0},f_{0}\right) \right) ,t>\... |
The problem
Let us consider a row vector u of size $n\in\mathbb{N}$, containing only binary values (0,1): $$u=(u_1 \cdots u_n), n\in\mathbb{N}$$ $$\forall i \in \{1\ldots n\}, u_i \in\{0,1\}$$
I would like to define notions of
consistency and heterogeneity (for lack of better terminologies...), in order to quantify how... |
Difference between revisions of "Help:Editing"
m (→Tables)
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*[http://en.wikipedia.org/wiki/Wikipedia:Writing_better_articles Wikipedia:Writing better articles].
*[http://en.wikipedia.org/wiki/Wikipedia:Writing_better_articles Wikipedia:Writing better articles].
+ +
[[Category:Help:Editing]]
[[Categor... |
Trimester Seminar Venue: HIM, Poppelsdorfer Allee 45, Lecture Hall Thursday, December 13th, 2:30 p.m. Conjugacy classes and centralisers in classical groups
Speaker: Giovanni de Franceschi (Auckland)
Abstract
We discuss conjugacy classes and associated centralisers in classical groups, giving descriptions which underpi... |
1,764 69
Hi PF!
Given the ODE $$f'' = -\lambda f : f(0)=f(1)=0$$ we know ##f_n = \sin (n\pi x), \lambda_n = (n\pi)^2##. Estimating eigenvalues via Rayleigh quotient implies $$\lambda_n \leq R_n \equiv -\frac{(\phi''_n,\phi_n)}{(\phi_n,\phi_n)}$$ where ##\phi_n## are the trial functions. Does the quotient hold for all #... |
When choosing the public exponent $e$, if the value chosen is the first coprime after $\phi(n)/2$ then the resulting public and private exponents are equal.
Well, yeah, that'll always be true.
Why does this happen?
We have $e=d$ whenever we have both of the following true:
$$e^2 \equiv 1 \pmod{p-1}$$
$$e^2 \equiv 1 \pm... |
So one thing that I find really interesting is that if I have a vector $\vec V = V_x \hat i + V_y \hat j$ its length is just:
$$ |V|^2 = V_x^2 + V_y^2 $$
That is all well and good, but then if I transform the vector into a new basis, I can rewrite the vector in terms of a covariant basis as:
$$ \vec V = V^1 \vec b_1 + ... |
I am in the process of designing a reaction wheel to be used in a cubesat for my final year project. I have already chosen my motor ( Faulhaber 2610T006B ) , and am currently in the process of sizing my flywheel. But I have gotten stuck halfway through. The following is my working:
$$Desired \space slew \space rate: 3°... |
May 9th, 2017, 08:13 AM
# 1
Member
Joined: Dec 2016
From: United States
Posts: 53
Thanks: 3
Math Focus: Abstract Simulations
Building an intuition for infinite-dimensional points, lines, surface etc.
I come from a javascript background. My goal is to build simulations of "dream" stuff. Not in javascript
Yesterday my mi... |
I'm attempting to numerically solve the following in order to get a function of 2 variables, just looking at the real part of
$$\psi(x,t)=\frac{1}{\pi\sqrt{2}}\int_{-100}^{100}\frac{\sin{(k)}}{k}e^{i\left[kx-\frac{k^2}{2}t\right]}\,dk$$
where I have specific values for $t=0,2,4$ and I'd like to plot the function from $... |
So in Julie Bergner's work on $(\infty, 1)$-categories arXiv:0610239, she considers several model categories which model $(\infty, 1)$-categories, which are known to be equivalent. I'm guessing that there is another model which is equivalent to these. It is probably known to experts, and probably exists for easy reason... |
This is likely a simple question that I'm just missing, but nothing immediately came to mind.
When dealing with topological monoids, it is necessary to prove that the group operation is continuous. I.e. if we have a monoid $(G,\ast,e)$ with some topology $\tau$, then it is necessary that $-\ast-: G\times G\rightarrow G... |
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The final step to do with Siril is to stack the images. Go to the "stacking" tab, indicate if you want to stack all images, only selected images or the best images regarding the value of FWHM previously computed. Siril proposes several algorithms for stacking computation.
The final step to do with Sir... |
Genus 2 curves in isogeny class 448.a
Label Equation 448.a.448.2 \(y^2 + (x^3 + x)y = -2x^4 + 7\) 448.a.448.1 \(y^2 + (x^3 + x)y = x^4 - 7\) L-function data
Analytic rank: \(0\) Bad L-factors:
Good L-factors:
See L-function page for more information
\(\mathrm{ST} =\) $N(G_{1,3})$, \(\quad \mathrm{ST}^0 = \mathrm{U}(1)\... |
NEW CONJECTURE: There is no general upper bound.
Wadim Zudilin suggested that I make this a separate question. This follows representability of consecutive integers by a binary quadratic form where most of the people who gave answers are worn out after arguing over indefinite forms and inhomogeneous polynomials. Some r... |
If a given heap has $k$ inversions, what is the complexity of making it into a valid min heap? We could define an inversion as a tuple (node, descendant), where the node has a key value strictly higher than its descendant (all nodes have distinct key values).
This isn't an interview question, but is something that I th... |
Consider the scalar PDE for $u$ with Dirichlet boundary conditions:
$\mathrm{div}(\mathcal{K}\nabla u) = f\; \forall x\; \in \Omega \subset R^2$,
$u = 0 \; \forall \; x\;\in \partial\Omega$
where $\mathcal{K} \in R^{2\times 2}$ is positive definite symmetric.
For a start I assume
$\Omega$ is the unit square with a unif... |
If $\lambda_n,\mu_n \in \mathbb{R}$, $\lambda_n \sim \mu_n$ as $n \to +\infty$, and $\mu_n \to +\infty$ as $n \to +\infty$, is it true that $$ \sum_{n=1}^{\infty} \exp(-\lambda_n x) \sim \sum_{n=1}^{\infty} \exp(-\mu_n x) $$ as $x \to 0^{+}$?
In other words, is it true that $$ \lim_{x \to 0^+} \frac{\sum_{n=1}^{\infty}... |
The intersection of two groups $(U,\odot_U)$ and $(V,\odot_V)$ is first, and foremost, the
set $$ G = U \cap V \text{.}$$To turn this into a group, one would need to define a suitable operation $\odot_G$ on $G$. But where is that operation supposed to come from? Since $U$ and $V$ can be completely different groups, whi... |
There is no best general way to check if any two trigonometric expressions are equal. One can use
TrigReduce,
TrigExpand,
TrigFactor,
TrigToExp,
Together and
Apart (especially with the
Trig->True option),
Simplify,
FullSimplify, etc. All these functions have their advantages and we discuss some of them. For more compli... |
Axicons are conical prisms that are defined by the alpha and apex angle. As the distance from the Axicon to the image increases, the diameter of the ring increases, while the line thickness remains constant.
Given the input is a collimated beam, you can calculate the outer ring diameter and the line thickness an Axicon... |
This problem can be recast in terms of the famous problem of the number of ways to represent a positive integer as a sum of squares. With this perspective, we can see that the following more general statement is true for any $p > 1$ (so that each of the infinite series actually converges):$$\sum_{n=1}^{\infty} \frac{1}... |
I was asked to find the minimum and maximum values of the functions:
$y=\sin^2x/(1+\cos^2x)$; $y=\sin^2x-\cos^4x$.
What I did so far:
$y' = 2\sin(2x)/(1+\cos^2x)^2$
How do I check if they are suspicious extrema points? After this function is cyclical and therefore only section that is not $(-\infty,\infty)$ can there b... |
In the above question we asked for a strategy so you could guarantee you won as much money as possible. This week we explore this game a little further, introducing the concept of
Expected Value and showing how expected values relate to our card game!
Let's consider some simple examples first.
Example: Pretend you usua... |
The Annals of Applied Probability Ann. Appl. Probab. Volume 23, Number 6 (2013), 2161-2211. Limit theory for point processes in manifolds Abstract
Let $Y_{i}$, $i\geq1$, be i.i.d. random variables having values in an $m$-dimensional manifold $\mathcal{M}\subset\mathbb{R}^{d}$ and consider sums $\sum_{i=1}^{n}\xi(n^{1/m... |
14 0 Homework Statement A yo-yo is placed on a conveyor belt accelerating ##a_C = 1 m/s^2## to the left. The end of the rope of the yo-yo is fixed to a wall on the right. The moment of inertia is ##I = 200 kg \cdot m^2##. Its mass is ##m = 100kg##. The radius of the outer circle is ##R = 2m## and the radius of the inne... |
The inverse of an (invertible) upper triangular matrix is always upper triangular and the inverse of an (invertible) lower triangular matrix is always lower triangular. Why?
I don't have any work to show and this isn't homework.
Mathematics Stack Exchange is a question and answer site for people studying math at any le... |
Let $u$ be an element of $\mathbb{Z}[\sqrt 5]$ of norm 1, i.e. $u = r + s \sqrt 5$ with $r^2-5s^2 = 1$.
The multiplication by $u$ in $\mathbb{Z}[\sqrt 5]$ turns any element $y$ of norm $44$ into another element $uy$ of norm $44$.View this multiplication operation on $\mathbb{Z}[\sqrt 5]$ as the transformation of the pl... |
Buckling, When Structures Suddenly Collapse
Buckling instability is a treacherous phenomenon in structural engineering, where a small increase in the load can lead to a sudden catastrophic failure. In this blog post, we will investigate some classes of buckling problems and how they can be analyzed.
What Is Buckling?
H... |
LaTeX uses internal counters that provide numbering of pages, sections, tables, figures, etc. This article explains how to access and modify those counters and how to create new ones.
Contents
A counter can be easily set to any arbitrary value with
\setcounter. See the example below:
\section{Introduction} This documen... |
$\newcommand{\al}{\alpha}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\e}{\epsilon}$
Let $X,Y$ be Riemannian manifolds, such that $\dim(X) > \dim(Y)$.
I am trying to prove the following statement (mentioned by Gromov in his book on metric geometry):
There is no arcwise isometry (i.e length preserving map) from $X$ to $Y$.... |
(27 intermediate revisions by 2 users not shown) Line 8: Line 8:
* [[Siril:Tutorial_sequence|Work on a sequence of converted images]]
* [[Siril:Tutorial_sequence|Work on a sequence of converted images]]
* [[Siril:Tutorial_preprocessing|Pre-processing images]]
* [[Siril:Tutorial_preprocessing|Pre-processing images]]
−
*... |
A particle moves along the x-axis so that at time t its position is given by $x(t) = t^3-6t^2+9t+11$ during what time intervals is the particle moving to the left? so I know that we need the velocity for that and we can get that after taking the derivative but I don't know what to do after that the velocity would than ... |
This article provides answers to the following questions, among others:
What are the objectives of the “Charpy impact test”? Why does “notch impact energy” represent a measure of the toughness of a test specimen? What external influences effect the notch impact energy? What are “upper shelf”, “lower shelf” and “transit... |
Let’s understand the concept of per unit system by solving an example. In the one-line diagram below, the impedance of various components in a power system, typically derived from their nameplates, are presented. The task now is to normalize these values using a common base.
Now that you have carefully examined the sys... |
A partial attempt to address this issue is made by invoking the idea of quantum discord. The basic idea of quantum discord is the environment needn't be in a specific state prior to interacting with the system. All that is necessary is that it factorizes and there is no correlation.
Let's start with the simple example ... |
Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay
School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
We investigate the long-time behavior of solutions for the suspension bridge equation when the forcing term containing some heredita... |
A language $L'$ is $PSPACE$-hard if for every $L \in PSPACE$ we have $L \le_p L'$.
Here $L \le_p L'$ means that $L$ is polynomial-
time reducible to $L'$.
Why does we use time reductions instead of space reductions in this situation?
Computer Science Stack Exchange is a question and answer site for students, researcher... |
Is the congruence group $\Gamma(2)$ generated by the upper triangular matrix $(1, 2; 0, 1)$ and the lower triangular matrix $(1, 0; 2, 1)$ or does on need to also throw in the negation of the identity? To be specific, how do I check that the negation of the identity is not a word in the above matrices?
Yes, you need to... |
Article Gauge theories on compact toric surfaces, conformal field theories and equivariant Donaldson invariants
We show that equivariant Donaldson polynomials of compact toric surfaces can be calculated as residues of suitable combinations of Virasoro conformal blocks, by building on AGT correspondence between N=2 supe... |
V S Sunder
Articles written in Proceedings – Mathematical Sciences
Volume 113 Issue 1 February 2003 pp 15-51
We obtain (two equivalent) presentations — in terms of generators and relations — of the planar algebra associated with the subfactor corresponding to (an outer action on a factor by) a finite-dimensional Kac al... |
In short echo time spectroscopy sequences macromolecular signals may strongly influence the quantification of metabolite spectra they underlay. In this work we present a method for simulating a MM basis set that is tailored toward chosen sequences and sequence parameters with the aim to improve the accuracy of metaboli... |
Today’s post reviews a recent talk given at the Simons Institute for the Theory of Computing in their current workshop series Computational Challenges in Machine Learning.
tl;dr: Sufficiently regular functions (roughly: having Lipschitz,invertible derivatives) can be represented as compositions ofdecreasing, small pert... |
Determine spot size of our lasers and laser diode modules from user supplied working distances. Calculator provides circular or elliptical spot size approximations based on 1/e
2 beam diameter and beam divergence; for lasers, beam diameter is given for TEM 00 mode. Spot size visibility varies based on ambient light con... |
A convexified energy functional for the Fermi-Amaldi correction
1.
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States, United States
2.
Department of Mathematics and Computer Science, Benedict College, Columbia, SC 29204, United States
$ \mathcal{E}$,we prove that $ \mathcal{E}... |
Let $\Omega$ be a bounded smooth domain and define $\mathcal{C} = \Omega \times (0,\infty)$. Below, $x$ refers to the variable in $\Omega$ and $y$ to the variable in $(0,\infty)$. The map $\operatorname{tr}_\Omega:H^1(\mathcal C) \to L^2(\Omega)$ refers to the trace operator ($\operatorname{tr}_\Omega u = u(\cdot,0)$ f... |
Consider the following situation:
There are two boundaries -- one is denoted using grey lines, and the other is denoted using black lines. The boundaries are numerically represented using "vertices", with edges assumed to be connecting these vertices, but otherwise not necessary for the calculation.
At a given vertex (... |
Consider $GL_2$ as the affine group scheme with coordinate ring ${\mathbb Z}[x_1,x_2,x_3,x_4,y]/(\det\left(\begin{array}{cc}x_1& x_2\\ x_3& x_4\end{array}\right)y-1)$. The group scheme $PGL_2$ is then given by the subring $S$ of $GL_1$-invariants, which is the subring generated by all monomials of the form $x_ix_iy^2$.... |
A matrix whose number of rows does not equal to the number of columns, is called a rectangular matrix.
Rectangular matrix is a type of matrix and the elements are arranged in the matrix as number of rows and the number of columns. The arrangement of elements in matrix is in rectangle shape. Thus, it is called as a rect... |
I want to calculate / simplify:
$$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$
where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\xi}\,dx$) and where $f$ is an even function.
Looking here: wiki, we find that
$$\mathcal{F}[\log|x|](\xi)=-2\pi\gamma\delta... |
Little explorations with HP calculators (no Prime)
03-27-2017, 07:39 PM
Post: #41
RE: Little explorations with the HP calculators
Let's divide one of the right-triangles into four right-triangles (two with sides lengths equal to r and x and two with side lengths r and 1-x) and a square with side r. Then it's area is
S ... |
Information theory was born in 1948 with the publication of
A Mathematical Theory of Communication by Claude Shannon (1916 to 2001). Shannon was inspired in part by earlier work by Boltzmann and Gibbs in thermodynamics and by Hartley and Nyquist at Bell. Most of the theory and applications of information theory (compre... |
A new Chipotle Restaurant opened in my neighborhood, and while I was enjoying a delicious chicken bowl, I began wondering who was Chipotle’s primary consumer. I decided to answer this question using population statistics at a Zip code level. In addition to this, I decided to map out areas where a Chipotle would sit wel... |
The connection between symmetries and conservation laws can be viewed through the lens of both Lagrangian and Hamiltonian mechanics. In the Lagrangian picture we have Noether's theorem. In the Hamiltonian picture we have the so-called "moment map." When we consider the same "symmetry" in both viewpoints, we get the exa... |
Preprints (rote Reihe) des Fachbereich Mathematik Refine Year of publication 1996 (3) (remove)
285
On derived varieties (1996)
Derived varieties play an essential role in the theory of hyperidentities. In [11] we have shown that derivation diagrams are a useful tool in the analysis of derived algebras and varieties. In... |
I want to perform a simple linear interpolation between $A$ and $B$ (which are binary floating-point values) using floating-point math with IEEE-754 round-to-nearest-or-even rounding rules, as accurately as possible. Please note that speed is not a big concern here.
I know of two basic approaches. I'll use the symbols ... |
Update: The MathJax Plugin for TiddlyWiki has a new home: https://github.com/guyru/tiddlywiki-mathjax Some time ago I came across MathJax, a nifty, Javascript based engine for displaying TeX and LaTeX equations. It works by “translating” the equation to MathML or HTML+CSS, so it works on all modern browsers. The result... |
Assuming we have two sets of $n$ qubits. The first set of $n$ qubits is in state $|a\rangle$ and second set in $|b\rangle$. Is there a fixed procedure that generates a superposed state of the two $|a\rangle + |b\rangle$ ?
Depending on what precisely your assumptions are about a, b, I think this is essentially impossibl... |
Answer
In general, $\sin{(bx)} \ne b\cdot \sin{x}$.. This is because they have different periods and amplitudes. Refer to the image in the step by step part below for the graph.
Work Step by Step
RECALL: The function $a \cdot \sin{(bx)}$ has : amplitude = $|a|$ period = $\frac{2\pi}{b}$ Thus: The function $y=\sin{(2x)}... |
253 23 Homework Statement There is an infinite charged plate in yz plane with surface charge density ##\sigma = 8*10^8 C/m^2## and negatively charged particle at coordinate (4,0,0) Find magnitude of efield at coordinate (4,4,0) Homework Equations E= E1+E2
So I figured to get e-field at point (4,4,0), I need to find the... |
\[\]
\defl{Normal Distribution has the following properties:}
Symmetrical and a bell shaped appearance. The population mean and median are equal. An infinite range, $-\infty < x < \infty$ The approximate probability for certain ranges of $X$-values: $P(\mu - 1\sigma < X < \mu + 1\sigma) \approx 68\%$ $P(\mu - 2\sigma <... |
Dear Uncle Colin, I've come across a seemingly simple question I can't tackle: solve $x^2 + 2x \ge 2$. I tried factorising to get $x(x+2) \ge 2$, which has the roots 0 and -2, but the book says the answer is $x < -1-\sqrt{3}$ or $x > -1 + \sqrt{3}$.Read More →
A STEP question (1999 STEP II, Q4) asks: By considering the... |
$$\mathop {\lim }\limits_{n \to \infty } {{{a_n}} \over {{b_n}}} = 1$$Prove the statement implies $\sum {{a_n},\sum {{b_n}} } $ converge or diverge together.
My guess the statement is true.
if $\sum{{a_n}}$ diverges, then $\mathop {\lim }\limits_{n \to \infty } {a_n} \ne 0$
So, $$\eqalign{ & \mathop {\lim }\limits_{n \... |
Answer
11
Work Step by Step
We adjust equation 18.11b to obtain: $\frac{V_z{max}}{V_{min}} = (\frac{T_{max}}{T_{min}})^{1/(\gamma-1)}$ $\frac{V_{max}}{V_{min}} = (\frac{773}{293})^{1/(1.4-1)}\approx 11$
You can help us out by revising, improving and updating this answer.Update this answer
After you claim an answer you’... |
This page describes (some of) the differences of Welch (4th Ed) and its alternatives. For coverage comparisons, please look at bookcomparison, please.
Innovations in Approach
Every interested and modestly talented student can understand finance. I believe that our finance concepts are no more difficult than those in st... |
For a positive integer $d$, let $h(-d)$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. Is there a known asymptotic formula for the sum
$$\displaystyle \sum_{\substack{p \leq x \\ p \equiv 3 \pmod{4}}} h(-p)?$$
MathOverflow is a question and answer site for professional mathematicians.... |
Johannes Breitling
1,2,3, Anagha Deshmane 4, Steffen Goerke 1, Kai Herz 4, Mark E. Ladd 1,2,5, Klaus Scheffler 4,6, Peter Bachert 1,2, and Moritz Zaiss 4
1Division of Medical Physics in Radiology, German Cancer Research Center (DKFZ), Heidelberg, Germany, 2Faculty of Physics and Astronomy, University of Heidelberg, Hei... |
scipy.interpolate.UnivariateSpline.antiderivative¶
UnivariateSpline.
antiderivative(
n=1)¶
Construct a new spline representing the antiderivative of this spline.
Parameters: n: int, optional
Order of antiderivative to evaluate. Default: 1
Returns: spline: UnivariateSpline
Spline of order k2=k+n representing the antider... |
Is arithmetic with infinite numbers fictitious?
It depends on your definition of "arithmetic with infinite numbers" and "fictitious". The meaning of
fictitious that this question was written to oppose, was in reference to certain descriptions of how Robinson's nonstandard analysis is used for calculus. Those descriptio... |
For just answering the yes/no question, the easiest way is to use the Swiss knife of bijections, the Cantor-Schröder-Bernstein theorem, which just requires us to construct separate
injections in each direction $\mathbb R\to\mathbb N\times\mathbb R$ and $\mathbb N\times\mathbb R\to\mathbb R$ -- which is easy:
$$ f(x) = ... |
A first remark
This same phenomenon of 'control' qubits changing states in some circumstances also occurs with controlled-NOT gates; in fact, this is the entire basis of eigenvalue estimation. So not only is it possible, it is an important fact about quantum computation that it is possible. It even has a name: a "phase... |
I have multiple different log sums that I need to evaluate. How would I calculate the following without using a calculator or log tables?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join ... |
(See EDIT #3 below for an explicit formula, and EDIT #2 for a solution comparing Dirichlet coefficients)
Answer
19.03.15 20:20
We can get the first few terms in Mathematica in a straighforward manner as follows.
Let
z[s_]:=Zeta[3s]/Zeta[s]
t[s_,m_]:= Sum[f[n]/n^s,{n,1,m}]
Then we have
z[s] = f[1] + f[2]/2^s + f[3]/3^s ... |
Monomial symmetric polynomials on $n$ variables $x_1, \ldots x_n$ form a natural basis of the space $\mathcal{S}_n$ of symmetric polynomials on $n$ variables and are defined by additive symmetrization of the function $x^{\lambda} = x_1^{\lambda_1} x_2^{\lambda_2} \ldots x_n^{\lambda_n}$. Here $\lambda$ is a sequence of... |
$\newcommand{\Q}{\Bbb Q} \newcommand{\N}{\Bbb N} \newcommand{\R}{\Bbb R} \newcommand{\Z}{\Bbb Z} \newcommand{\C}{\Bbb C} \newcommand{\F}{\Bbb F} \newcommand{\p}{\mathfrak{p}} $ Let $A$ be an abelian variety over a number field $F$. It is expected that the $L$-function of $A$ has analytic continuation to $\Bbb C$ and sa... |
Given a group $G$, with its binary ("product") and unary ("inverse") operations:$$\begin{equation}\begin{split}\circ&\colon G^2\to G\\\operatorname{inv}&\colon G\to G\end{split}\end{equation}\tag{1}$$you can consider the restrictions of them on $H^2$ and $H$ respectively, where $H\subset G$:$$\begin{equation}\begin{spl... |
The extensional version of Intuitionistic Type Theory is usually formulated in a way that makes extensional concepts like functional extensionality derivable. In particular, equality reflection, together with $\xi$- and $\eta$-rules for $\Pi$ types are enough to get the standard formulation of $\textsf{funext}$
$\Pi_{x... |
Let $G$ be a split reductive algebraic group over an arbitrary field $k$ Suppose we have a split maximal torus $T$. There is a short exact sequence of groups$$1\to \mathrm{Inn}(G)\to \mathrm{Aut}(G)\to \mathrm{Out}(G)\to 1.$$where $\mathrm{Inn}(G) = G^{\mathrm{ad}}(k)$. This sequence splits in various ways, one choice ... |
Journal of Commutative Algebra J. Commut. Algebra Volume 8, Number 1 (2016), 89-111. A criterion for isomorphism of Artinian Gorenstein algebras Abstract
Let $A$ be an Artinian Gorenstein algebra over an infinite field~$k$ of characteristic either 0 or greater than the socle degree of $A$. To every such algebra and a l... |
I am having trouble understanding a mapping reduction and I would appreciate your help. Define
$\quad \begin{align} A_{TM} &= \{ \langle M, w \rangle \mid M \text{ Turing machine}, w \in \mathcal{L}(M)\} \\ S_{TM} &= \{ \langle M,w \rangle \mid M \text{ Turing machine}, w \in \mathcal{L}(M) \implies w^R \in \mathcal{L}... |
Normalized compression distance (NCD) is a way of measuring the similarity between two sequences.
A compression algorithm looks for patterns and repetitions in the input sequence to compress it. For example, instead of “abababab” we can say “(ab)x4”. This is how RLE works and this is the most simple and illustrative ex... |
I am trying to solve a set of DAEs.
\begin{equation} -4 \nu (\lambda(s))^{(-1 - 4 \nu)} \theta'(s) \lambda'(s) + (\lambda(s))^{(-4 \nu)} \theta''(s) = -\alpha_y \cos\theta(s) + \alpha_x \sin\theta(s) \end{equation}
\begin{equation} (\lambda(s))^{(-2 \nu)} \log(\lambda(s)) = f_s (\alpha_x \cos\theta(s) + \alpha_y \sin\t... |
The fractional Schrödinger equation with singular potential and measure data
1.
Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
2.
Departamento de Matemáticas, Universidad Autónoma de Madrid, Calle Francisco Tomás y Valiente 7, 28049 Madrid, Spain
We... |
Can I be a pedant and say that if the question states that $\langle \alpha \vert A \vert \alpha \rangle = 0$ for every vector $\lvert \alpha \rangle$, that means that $A$ is everywhere defined, so there are no domain issues?
Gravitational optics is very different from quantum optics, if by the latter you mean the quant... |
The most frequently used evaluation metric of survival models is the concordance index (c index, c statistic). It is a measure of rank correlation between predicted risk scores $\hat{f}$ and observed time points $y$ that is closely related to Kendall’s τ. It is defined as the ratio of correctly ordered (concordant) pai... |
Suppose that
$\mu_k$ is an increasing sequence of numbers such that $0 < \mu_1 \leq \mu_2 \leq ..$ with $\mu_k \to \infty$ as $k \to \infty$
$\sum_{k=1}^\infty |u_k|^2 < \infty$ and $\sum_{k=1}^\infty \sqrt{\mu_k}|u_k|^2 < \infty$ where $u_k$ is a given sequence of real numbers
I want to show that (if true) the sum $$\... |
Let $(R, \mathfrak{m})$ be a commutative Noetherian complete local rings ($R$ can be regular, if you need). Let $E(R/\mathfrak{p})$ be injective hull of $R/\mathfrak{p}$, if $\mathfrak{p}= \mathfrak{m}$ we simply write by $E$.
Question 1: What is $Hom (E(R/\mathfrak{p}), E)$?
By the isomorphism $Hom (Hom (M,N),E) \cong... |
What is the cut rule? I don't mean the rule itself but an explanation of what it means and why are proof theorists always trying to eliminate it? Why is a cut-free system more special than one with cut?
Suppose I have a proof of B starting from assumption A. And a proof of C starting from assumption B. Then the cut rul... |
The general algorithm is the Extended Euclidean algorithm.
Compute the gcd of $26$ and $5$ by Euclid's algorithm keeping track of coefficients:
$$\begin{align} 26 = & 1 \cdot 26 + 0 \cdot 21 \\ 21 = & 0 \cdot 26 + 1 \cdot 21 \end{align}$$
$21$ fits into $26$ $1$ time leaving a remainder of $5$, so we substract$1$ time ... |
This discussion is independent of my first answer, which I still think is the right form of the continuous version of the Borel-Cantelli lemma.
However, it seems that people are interested in the following question:
"What is the right condition making a family of random events $(A_t)_{t\geq 0}$ as surely stop happening... |
I tried construct DFA for this NFA
$\sum$ - alphabet set
$Q$ -states set
$\sigma(Q\times (\sum \cup {\epsilon} )) \to P(Q)$ state func
$q_0 = q_0$
$ F \subseteq Q, F = \{q_0\}$
Because every NFA has equal DFA lets constructDFA $M'$ for this given NFA.
alphabet - the same
$Q' = P(Q)$ - states
Current state is $R \in P(Q... |
So I've been working on the question below for a bit, and after arriving at what I thought was the answer, I found out that I was wrong, because the bounds I used for integration was incorrect.
Here is the problem statement.
So, the first thing I did was solve for the values of $\theta$ where an intersection occurs, an... |
I am so new on Mathematica. I try to find the first variation of this function according to $t$ on mathematica but I could not achieve. Here is the function ;
$$\int_{0}^{\infty}u\left(c\left(t\right)\right)\exp\left\{ -\int_{0}^{t}\theta\left(c\left(s\right)\text{d}s\right)\right\} \text{d}t$$
How can I find the first... |
Suppose that we want to make the best Bayesian inference about some value $\mu$ we have some normal prior about it. I.e. $\mu\sim N(\mu_0, \mu_0)$ with known parameters. To do so, we can choose parameters $(\mu_x, \sigma_x)$ that define a normal sampling method with mean $\mu_s=\mu \frac{\sigma_x^2}{\sigma_x^2+\sigma_0... |
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