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Upper semicontinuity of global attractors for the perturbed viscous Cahn-Hilliard equations
DOI: http://dx.doi.org/10.12775/TMNA.2008.051
Abstract
It is known that the semigroup generated by the initial-boundary
value problem for the perturbed viscous Cahn-Hilliard equation with $\varepsilon> 0$ as a parameter admits a... |
$$ \sum_{n=1}^\infty \frac{1}{n^2 3^n} $$ I tried to use the regular way to calculate the sum of a power series $(x=1/3)$ to solve it but in the end I get to an integral I can't calculate.
Thanks
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related ... |
Consider a large number $N$ of distinguishable particles distributed among $M$ boxes.
We know that the total number of possible microstates is $$\Omega=M^N$$ and that the number of microstates with a distribution among the boxes given by the configuration $[n_1, n_2, ..., n_M]$ is given by $$\frac{N!}{\prod_{j=1}^M (n_... |
I am interested to write down a derivation of Lagrange equations from Newton's second law for a non-holonomic system of particles. Here, I mention my derivation where I am stuck right at the last step.
Consider a system of $N$ particles where their position vectors are written as
$$\mathbf{r}_i=\mathscr{R}_i(q_1(t),\do... |
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.
Yeah it does seem unreasonable to expect a finite presentation
Let (V, b) be an n-dimensional, non-degenerate s... |
Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resist... |
While there is no confirmation that quark stars exist, is there any theoretical limit analogous to (but different from) the Tolman–Oppenheimer–Volkoff limit for neutron stars?
In other words, what is the maximum pressure for quark matter?
Physics Stack Exchange is a question and answer site for active researchers, acad... |
Gravitational force exerted by ring
Let's say that we wanted to find the gravitational
force exerted by a disk of mass \(M\) on a particle of mass \(m\) a vertical height \(h\) above the center \(O\) of the disk as illustrated in Figure 2. To find this force we'll use Newton's law of gravity and the concept of a defini... |
Inclusion Mapping is Surjection iff Identity Theorem
Let $T$ be a set.
Let $S\subseteq T$ be a subset.
Let $i_S: S \to T$ be the inclusion mapping.
Then:
where $I_S: S \to S$ denotes the identity mapping on $S$.
Alternatively, this theorem can be worded as: $i_S: S \to S = I_S: S \to S$ Proof $(1): \quad \Dom {i_S} = S... |
A comparison sort cannot require fewer than $\Theta (n\log n)$ comparisons on average. However, consider this sorting algorithm:
sort(array): if length(array) < 2: return array unsorted ← empty_array i ← 0 while i < length(array) - 1: if array[i] > array[i + 1]: push(unsorted, pop(array, i + 1)) else: i ← i + 1 return ... |
№ 8
All Issues Volume 65, № 12, 2013
Ukr. Mat. Zh. - 2013. - 65, № 12. - pp. 1587–1603
We introduce the notion of Kravchuk derivations of the polynomial algebra. It is proved that any element of the kernel of a derivation of this kind gives a polynomial identity satisfied by the Kravchuk polynomials. In addition, we de... |
First we make some computations:$$\lim_{x\to 0, y\to y_0}x^3\log\Big(1+\frac{|y|^\alpha}{x^4}\Big)=\lim\left(x^3\sqrt{1+\frac{|y|^\alpha}{x^4}}\right)\frac{2\log(u)}{u}=\lim x\sqrt{x^4+|y|^\alpha}\cdot \frac{2\log(u)}{u},$$where $u=\sqrt{1+\frac{|y|^\alpha}{x^4}}$. If $(x,y)\to(0,a)$ and $u$ remains bounded, the last p... |
Happy near year, and best wishes to those close and \(\varepsilon\)-far! December concluded the year with 4 new preprints, spanning quite a lot of the property testing landscape:
Testing Stability Properties in Graphical Hedonic Games, by Hendrik Fichtenberger and Anja Rey (arXiv). The authors of this paper consider th... |
The brachistochrone problem is a very famous problem in the history of physics which was first solved by an excellent mathematician named Jean Bernoulli. He posed this problem as a challenge to the greatest mathematicians of Europe during the period of the Renaissance. He stated the problem as such:
We are given two fi... |
It's hard to say just from the sheet music; not having an actual keyboard here. The first line seems difficult, I would guess that second and third are playable. But you would have to ask somebody more experienced.
Having a few experienced users here, do you think that limsup could be an useful tag? I think there are a... |
Conjugate Heat Transfer
In this blog post we will explain the concept of
conjugate heat transfer and show you some of its applications. Conjugate heat transfer corresponds with the combination of heat transfer in solids and heat transfer in fluids. In solids, conduction often dominates whereas in fluids, convection usu... |
In a
balanced chemical equation, the total number of atoms of each element present is the same on both sides of the equation. Stoichiometric coefficients are the coefficients required to balance a chemical equation. These are important because they relate the amounts of reactants used and products formed. The coefficie... |
Nano Express Open Access Published: Improvement of Bipolar Switching Properties of Gd:SiO x RRAM Devices on Indium Tin Oxide Electrode by Low-Temperature Supercritical CO 2 Treatment Nanoscale Research Letters volume 11, Article number: 52 (2016) Article metrics
1362 Accesses
5 Citations
1 Altmetric
Abstract
Bipolar sw... |
This post describes some generative machine learning algorithms.
Gaussian Discriminant Analysis
Gaussian Discriminant Analysis is a
supervised classification algorithm. In this algorithm we suppose the p(x|y) is multivariate Gaussian with parameter mean \(\overrightarrow{u}\) and covariance \(\Sigma : P(x|y) \sim N(\ov... |
2019-05-20 15:18 Detailed record - Similar records 2019-01-23 09:13
nuSTORM at CERN: Feasibility Study / Long, Kenneth Richard (Imperial College (GB)) The Neutrinos from Stored Muons, nuSTORM, facility has been designed to deliver a definitive neutrino-nucleus scattering programme using beams of $\bar{\nu}_e$ and $\bar... |
This is a very simple question; I apologize if it has already been asked here. Define the following function (superficially similar to a theta function):
$$\varsigma(x)=\sum_{n=1}^\infty e^{-xn^3}$$
I am interested in knowing the Laurent series about $x=0$ of this series if it exists, i.e. I would like to know if there... |
Suppose chiral fermions $\psi$ interacting with gauge fields $A_{\mu,L/R}$. With $P_{L/R} \equiv \frac{1\mp\gamma_{5}}{2}$ and $t_{a,L/R}$ denoting the generators, the corresponding action reads
$$ S = \int d^{4}x\bar{\psi}i\gamma_{\mu}D^{\mu}\psi, \quad D_{\mu} = \partial_{\mu} - iA_{\mu,L}^{a}t_{a,L}P_{L} - i\gamma_{... |
Question:
The pendulum is released from {eq}60^0 {/eq} position and then strikes the initially stationary cylinder of mass {eq}m_2 {/eq} When OA is vertical. Determine the maximum spring compression {eq}\delta {/eq} when {eq}m_1=3kg,m_2=2kg,OA=0.8m,k=6kN/m {/eq}
Assume the bar of the pendulum is light so that the mass ... |
№ 8
All Issues Volume 61, № 11, 2009 Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities
Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1443-1454
We determine the exact values of the best (α, β)-approximations and the best one-sided approximations of classes of dif... |
Advanced topics in information theory From CYPHYNETS
(→July 14: Rate distortion theory - I)
(→July 14: Rate distortion theory - I)
Line 86: Line 86:
* There is also an information theoretic definition,
* There is also an information theoretic definition,
-
<math>R^I(D) = \min_{p(x, \hat{x}), \sum
+
<math>R^I(D) = \min_... |
Fractions, a topic that we had learned in our basic schooling and would still be present in some corner of our head. But with time and no practice on this topic could have rusted the concept. This article is to renew those concepts for the GMAT exam and make you ready to solve any problem related to fractions in minima... |
Skills to Develop
Apply the Binomial Theorem.
A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find... |
Background: So, just for fun, I was trying to analyze the types of soultion one may recieve from a quadratic equation, the solutions from $\mathbb{Z},\mathbb{R},\mathbb{C}$ was all rather easy, but when it comes to solutions in $\mathbb{Q}$, I tried applying the rational root theorem but that one has a few criteria tha... |
All:
Say $f$ is a measurable (integrable, actually) function over the Lebesgue-measurable set $S$, with $m(S)>0$.
Now, since $m(S)>0$, there exists a non-measurable subset $S'$ of $S$, and we can then write:
$$S=S'\cup (S\setminus S').$$
How would we then go about dealing with this (sorry, I don't know how to Tex an in... |
I take it that we call $TAUT$ the problem of given a DNF formula, decide if it is a tautology (if you do not want to restrict to DNF, this will still work as this only makes the problem more general).
The answer of your questions easily follows from the definition of $coNP$. Remember that a language $L \subseteq \{0,1\... |
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Now showing items 1-10 of 24
Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ... |
Linear Algebra
A matrix is a 2-D array of numbers.
A tensor is a n-D array of numbers.
Matrices are associative but not commutative.
Not all square matrices have inverse. A matrix that is not invertible is called Singular or Degenerative.
A matrix is “singular” if any of the following are true:
-Any row or column conta... |
A few basic calculations:
The current through Rled, assuming the transistor is fully saturated:$$I_R = \frac{3.7V - 2.4V - 0.3V}{39 \Omega} = 25.6 mA$$
Looking at a 2N3904 Datasheet, they define saturation as the point hfe=10.
Thus:$$I_b = 2.56 mA$$
This means your micro controller needs a control signal of:$$V_m = 0.6... |
Verify Simulations with the Method of Manufactured Solutions
How do we check if a simulation tool works correctly? One approach is the Method of Manufactured Solutions. The process involves assuming a solution, obtaining source terms and other auxiliary conditions consistent with the assumption, solving the problem wit... |
First the definitions:
The point $p$ is an
$\omega$-cluster pointfor a subset $A$ of a topological space $X$ if every neighbourhood of $p$ contains infinitely many points of $A$.
A space is
countably compactif every countable open cover has a finite subcover.
Now the question:
How does one prove that every countable in... |
I am new to option pricing and following problem came up that I don't understand how to handle.
A derivative will pay out dollar amount equal to $$\frac1T\ln \frac{S_T}{S_0}$$ at maturity, where $S_T$ is distributed log-normally, and the expected return is $\mu$ and volatility is $\sigma$ and $T$ is the time.
So what i... |
Seeing that in the Chomsky Hierarchy Type 3 languages can be recognised by a DFA (which has no stacks), Type 2 by a DFA with one stack (i.e. a push-down automaton) and Type 0 by a DFA with two stacks (i.e. with one queue, i.e. with a tape, i.e. by a Turing Machine), how do Type 1 languages fit in...
Considering this ps... |
PhD Thesis Research
My thesis research is in the field of underwater acoustics and signal processing. In particular, this research explores the possibility using a measured acoustic field with some bandwidth to create new fields that have much lower or much higher frequency content. This is made possible by mathematica... |
The crucial observation is that if $A \vDash B$ then also $\sigma(A) \vDash \sigma(B)$. This follows since all $\sigma$ does is rename variables and flip some variables. For example, if $\sigma(x) = \lnot y$, $\sigma(y) = z$, and $\sigma(z) = x$, then $A(x,y,z) \vDash B(x,y,z)$ implies also $A(\lnot y, z, x) \vDash B(\... |
Skills to Develop
Evaluate 2 × 2 determinants. Use Cramer’s Rule to solve a system of equations in two variables. Evaluate 3 × 3 determinants. Use Cramer’s Rule to solve a system of three equations in three variables. Know the properties of determinants.
We have learned how to solve systems of equations in two variable... |
Click edit button to change this text. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Ut elit tellus, luctus nec ullamcorper mattis, pulvinar dapibus leo.
Lily Chambers
Co-Editor in Chief
Lily Chambers transferred to d.tech during her Sophomore year from Woodside High School. She enjoys yoga and hiking on the... |
I have received important information from Michael Filaseta, with which we can answer:
1,2) Pick a prime $p$ big enough, put $c=p-f(0)$ and consider the polynomial $F(X)=f(X)+c$, which has $F(0)=p$. Specifically, if $f(X)=a_nX^n+\ldots+a_0$, by picking $p>|a_n|+\ldots+|a_1|$ we can guarantee that $F(X)$ has all its roo... |
I've been playing around with some finite fields to test how rapid brute-force is when solving discrete logarithm problems occurring in DH methods.
Working in $\mathbb{F}_{101}$, pick a private key $\alpha=88$, $\beta$ arbitrary and let $g=41$ such that $A=g^{\alpha}\equiv 87 \pmod{101}$. I believe $g$ is not a primiti... |
We know $\frac{1}{81}$ gives us $0.\overline{0123456790}$
How do we create a recurrent decimal with the property of repeating:
$0.\overline{0123456789}$
a) Is there a method to construct such a number?
b) Is there a solution?
c) Is the solution in $\mathbb{Q}$?
According with this Wikipedia page: http://en.wikipedia.or... |
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1. Observation of a peaking structure in the J/psi phi mass spectrum from B-+/- -> J/psi phi K-+/- decays
PHYSICS LETTERS B, ISSN 0370-2693, 06/2014, Volume 734, Issue 37... |
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract In ordinary quantum mechanics for finite systems, the time evolution induced by Hamiltonians of the form $$H = \frac{{P^2 }}{{2m}} + V(Q)$$ is studied from the point of view of *-automorphisms of the CCRC*-a... |
In the theory of chemical reactions, it is often possible to isolate a small number or even a single degree of freedom in the system that can be used to characterize the reaction. This degree of freedom is coupled to other degrees of freedom (for example, reactions often take place in solution). Isomerization or dissoc... |
OpenCV #004 Common Types of Noise Digital Image Processing using OpenCV (Python & C++) Highlights: We will give an overview of the most common types of noise that is present in images. We will show how we can generate these types of noise and add them to clean images. Then, we will show how we can filter these images u... |
Main Page See also Wikipedia:Introduction, Wikipedia:Manual of Style, Wikipedia:Tutorial, Help:Editing, and Help:Starting a new page Contents 1 Introduction 2 Minor edits 3 Major edits 4 Wiki markup 5 More information on editing wiki pages Introduction
Editing most Wikipedia pages is not difficult. Simply click on the ... |
Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resist... |
Strophoid
A third-order plane algebraic curve whose equation takes the form
$$y^2=x^2\frac{d+x}{d-x}$$
in Cartesian coordinates, and
$$\rho=-d\frac{\cos2\phi}{\cos\phi}$$
in polar coordinates. The coordinate origin is a node with tangents $y=\pm x$ (see Fig.). The asymptote is $x=d$. The area of the loop is
$$S=2d^2-\f... |
I'm working on the following problem for my PDE's class as study for a test.
Sketch the Fourier series of $$ f(x) = 2x^2$$ On the interval of $[-1,1]$.
My professors answer key states that the Fourier series for this function is $2x^2$ repeated over the interval $[-1,1]$ (I'd post the graph, but I'm not sure how.)
My u... |
I'm trying to understand BRST complex in its Lagrangian incarnation i.e. in the form mostly closed to original Faddeev-Popov formulation. It looks like the most important part of that construction (proof of vanishing of higher cohomology groups) is very hard to find in the literature, at least I was not able to do so. ... |
Alice is a soccer coach who occasionally bring her soccer team to explore Caveland (that can be modeled as an undirected unweighted connected graph) for special event, e.g. for initiation ceremony, to celebrate birthdays, etc. Caveland has $N$ junctions and $M$ tunnels.
Caveland is quite prone to flooding, but that doe... |
In
scattering theory, a part of mathematical physics, the Dyson series, formulated by Freeman Dyson, is a perturbative series where each term is represented by Feynman diagrams. This series diverges asymptotically, but in quantum electrodynamics (QED) at the second order the difference from experimental data is in the ... |
Recall that an operator \(T \in \mathcal{L}(V)\) is diagonalizable if there exists a basis \(B\) for \(V\) such that \(B\) consists entirely of eigenvectors for \(T\). The nicest operators on \(V\) are those that are diagonalizable with respect to some orthonormal basis for \(V\). In other words, these are the operator... |
I'm having trouble evaluating the following integral: $$ \int^\pi_0 \frac{\cos^9(x)}{\sin^3(x)+\cos^3(x)}dx $$
I tried to convert it into an algebraic function by multiplying the numerator and denominator by $\sec^{11}(x)$ and substituting $\tan(x)=t$ as
$$ \int^\pi_0 \frac{\cos^9(x)\cdot \sec^{11}(x)}{(\sin^3(x)+\cos^... |
The expectation values$$ \langle p | \vec E(\vec x) | p\rangle $$and similarly for $\vec B(\vec x)$ vanish for a simple reason: the state $|p\rangle$ is by definition translational symmetric (translation only changes the phase of the state, the overall normalization) so the expectation values of any field in this state... |
Matrix factorization is a simple embedding model. Given the feedback matrix A \(\in R^{m \times n}\), where \(m\) is the number of users (or queries) and \(n\) is the number of items, the model learns:
A user embedding matrix \(U \in \mathbb R^{m \times d}\), where row i is the embedding for user i. An item embedding m... |
This question already has an answer here:
I have been trying to follow this derivation from Sakurai and Shankar, pulling from both. I would like to see how the following derivation can be extended to orbital angular momentum, and thus find that $\ell$ is an integer. The details are omitted but the core of the proof is ... |
Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resist... |
This is cool: In today’s
Nature, Toby Cubitt, David Perez-Garcia, and Michael Wolf published a paper, “Undecidability of the spectral gap.” A short writeup is in Nature News, and an extended paper is on arXiv. It shows a problem in quantum physics–the spectral gap problem–to be undecidable by reducing the halting probl... |
On the profile of solutions for an elliptic problem arising in nonlinear optics
1.
Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing, 100080, China
2.
School of Mathematics, The University of New South Wales, Sydney 2052, Australia
3.
School of Mathematics and Statistics, University of Sydney, NSW 20... |
According to the textbook by Silberschatz,
A relation schema R is in third normal form with respect to a set $F$ of functional dependencies if, for all functional dependencies in $F^+$ of the form $\alpha$ → $\beta$, where $\alpha$ ⊆ R and $\beta$ ⊆ R, at least one of the following holds:
$\bullet$ $\alpha$ → $\beta$ i... |
I'm taking an undergrad GR course, and our text (Lambourne) mentions covariant and contravariant vectors and tensors ad-nauseum, but never really gives a formal definition for what they are, and how they are unique from each other in any physical sense (other than their difference in transformations). Is there any phys... |
We can express any ket vector \(|\psi⟩\) (representing all the different possible states) in its “component form” as \(|\psi⟩=\sum_{i=1}^{N}\psi_i|i⟩\) where \(|i⟩\) are all the different possible basis vectors one could decompose \(|\psi⟩\) with respect to, \(\psi_i\) are the different components (which, in general, c... |
So far, we have a fairly small collection of examples of groups: the dihedral groups, the symmetric group, and \(\mathbb{Z}_n\). In this section, we'll look at products of groups and find a way to make new groups from the groups we already know.
A very famous group - though not a very complicated one - is the
Klein Fou... |
Given $x \sim N(0, I)$ and $f : \mathbb{R}^{n} \longrightarrow \mathbb{R}$ which is $L$-Lipschitz, we have that $f - \mathbb{E}f$ is a sub-Gaussian random variable, specifically that $$ \| f(x) - \mathbb{E} f(x) \|_{\psi_2} \leq C L \:. $$
I want to show that if in addition $f \geq 0$, then for all $p > 0$, the more ge... |
Damping in Structural Dynamics: Theory and Sources
If you strike a bowl made of glass or metal, you hear a tone with an intensity that decays with time. In a world without damping, the tone would linger forever. In reality, there are several physical processes through which the kinetic and elastic energy in the bowl di... |
I came across a version of a proof that One Time Pads have perfect secrecy and have a few questions with this version of the proof. The proof is attributed to Dan Boneh (the proof starts on slide 10),
Definition: A cipher (E,D) over (K,M,C) has perfect secrecy if $\forall m_{0}, m_{1} \in$ M, $(\left\vert{m_{0}}\right\... |
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Now showing items 1-1 of 1
Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
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Now showing items 1-10 of 20
Measurement of electrons from beauty hadron decays in pp collisions at root √s=7 TeV
(Elsevier, 2013-04-10)
The production cross section of electrons from semileptonic decays of beauty hadrons was measured at mid-rapidity (|y| < 0.8) in the transverse momentum range 1 < pT <8 GeV/c w... |
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Now showing items 1-6 of 6
Forward-backward multiplicity correlations in pp collisions at √s = 0.9, 2.76 and 7 TeV
(Springer, 2015-05-20)
The strength of forward-backward (FB) multiplicity correlations is measured by the ALICE detector in proton-proton (pp) collisions at s√ = 0.9, 2.76 and 7 TeV. The measurement... |
Prove that this sequence converges. I can't do it.
Let $\{a_n\}$ be a sequence of positive real numbers that converges to a number $A$. Prove that $\{(a_1\cdots a_n)^{1/n}\}$ converges to $A$.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fie... |
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In probability theory, a log-normal (or lognormal) distribution i... |
Seeing that in the Chomsky Hierarchy Type 3 languages can be recognised by a DFA (which has no stacks), Type 2 by a DFA with one stack (i.e. a push-down automaton) and Type 0 by a DFA with two stacks (i.e. with one queue, i.e. with a tape, i.e. by a Turing Machine), how do Type 1 languages fit in...
Considering this ps... |
4:28 AM
@MartinSleziak Here I am! Thank you for opening this chat room and all your comments on my post, Martin. They are really good feedback to this project.
@MartinSleziak Yeah, using a chat room to exchange ideas and feedback makes a lot of sense compared to leaving comments in my post. BTW. Anyone finds a
\oint\fr... |
Lesson overview
In this lesson, we'll prove when the value of the p-series \(\sum_{n=1}^∞\frac{1}{n^p}\) converges to a finite value and when its diverges to infinity. We'll show that when \(0<p<1\), the p-series diverges; and when \(p>1\), the p-series converges.
A graph of the function \(y=\frac{1}{x^p}\) is shown in... |
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial\Omega$. Consider the following initial-boundary value problem for the heat equation:
\begin{equation} \begin{cases} u_t=\Delta u\quad\quad\quad\;\; \text{in}\;\Omega\times(0,\infty) , \\ u=0 \quad\quad\quad\;\;\;\;\; \text{on}\;\partial\O... |
Flatland Fidget Spinner
Freddy the Flatland Photographer wants to report on fun new things in Flatland for the Flatland Financial Times. He saw a really nice picture of a Fidget Spinner in Flatland Weekly, and he would like to publish a similar picture. Actually, he likes the picture so much he would like to use the ex... |
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In this chapter we learned about left and right adjoints, and about joins and meets. At first they seemed like two rather different pairs of concepts. But then we learned some deep relationships between them. Briefly:
Left adjoints ... |
I want to solve the one-touch American call at $t = 0$ with level $B,$ maturity $T$ under the following assumption: $$d S= rSd t + \sigma SdW,\quad S_0<B.$$ We have following formula: $$V(S_0,0) = \left(\dfrac{B}{S_0}\right)^{2r/\sigma^2}N(d_2)+\dfrac{S}{B}N(d_1),$$ where we ormit $d_1,d_2.$
Is there any easy way or re... |
Prediction for Fitted Multiple Point Process Model
Given a fitted multiple point process model obtained by
mppm, evaluate the spatial trend and/or the conditional intensity of the model. By default, predictions are evaluated over a grid of locations, yielding pixel images of the trend and conditional intensity. Alterna... |
Say I have a family of hash functions that are weakly universal, i.e. the probability of two non-identical keys $x\neq y$ are mapped to the same hash-value is bounded by $k/m$ when I have $m$ bins and $k$ is some constant (I am mainly interested in $k=1$ and $k=2$):
$$\Pr_h\left[h(x)=h(y)\right] \leq \frac{k}{m} $$
I h... |
Matthew Needham Dr M D Needham Research interests
I am a Chancellors Fellow in the School of Physics and Astronomy at the University of Edinburgh. My main research interest is in indirect searches for physics beyond the Standard Model of Particle Physics in the decays of beauty mesons and neutrinos.
One of the puzzling... |
The summer gets off to a flying start, with three property testing papers, spanning differential privacy, distribution testing, and juntas in Gaussian space!
On closeness to \(k\)-wise uniformity, by Ryan O’Donnell and Yu Zhao (arXiv) In this paper, the authors consider the following structural question about probabili... |
The geometry of multiplying complex numbers is usually a depiction, not a definition. I don't understand why you claim it's "not defined well," though. It's perfectly defined. If you need any clarification on how it works, you can always ask a question. Also, please do be aware that the geometry of multiplying complex ... |
Sorry if this seems like a dumb question, but what what type of logarithm is $\log$ in Wikipedia articles? Cheers.
I think it depends on the situation. For a mathematician, $\log n$ probably means the natural logarithm. For a computer scientist, $\log n$ probably denotes the base $2$ logarithm, etc.
I think sometimes p... |
Radio astronomy
When we view the Earth or the night sky, we
can see plants, people, cars and cities, your coffee mug, the Moon, the stars. This is all we have ever seen with our two eyes sine we've lived our whole lives seeing things with visible light. But a bee can see something that we cannot see. They see light—inv... |
Consider a $C^k$, $k\ge 2$, Lorentzian manifold $(M,g)$ and let $\Box$ be the usual wave operator $\nabla^a\nabla_a$. Given $p\in M$, $s\in\Bbb R,$ and $v\in T_pM$, can we find a neighborhood $U$ of $p$ and $u\in C^k(U)$ such that $\Box u=0$, $u(p)=s$ and $\mathrm{grad}\, u(p)=v$?
The tog is a measure of thermal resist... |
Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] along with all its transformations: shifts, stretches, compressions,... |
Electrochemistry, which is the study of the interaction of electricity and chemical reactions, is a central theme in the story of the discovery of cisplatin. In the redox chemistry module, we saw how electrons are transferred from one species to another in a chemical reaction, and we also learned some of the formalisms... |
When I showed to my brother how I proved \begin{equation} \int_{0}^{\!\Large \frac{\pi}{2}} \ln \left(x^{2} + \ln^2\cos x\right) \, \mathrm{d}x=\pi\ln\ln2 \end{equation} using the following theorem by Mr. Olivier Oloa \begin{equation}{\large\int_{0}^{\!\Large \frac{\pi}{2}}} \frac{\cos \left( s \arctan \left(-\frac{x}{... |
I will not give you the numerical solution, but I will below explain some analytical simplifications that I believe are required to solve the numerical problem. The strategy is simple: try to express all the parameters of the integral in term of dimensionless variables. To achieve a discussion in term of $\delta = \Del... |
Suppose we wish to find the zeros of \(f(x) = x^3 + 4x^2-5x-14\). Setting \(f(x)=0\) results in the polynomial equation \(x^3 + 4x^2-5x-14=0\). Despite all of the factoring techniques we learned in Intermediate Algebra, this equation foils us at every turn. If we graph \(f\) using the graphing calculator, we get
The gr... |
Inverse Cauchy problem for fractional telegraph equations with distributions Abstract
$$u^{(\alpha)}_t-r(t)u^{(\beta)}_t+a^2(-\Delta)^{\gamma/2} u=F_0(x)g(t), \;\;\; (x,t) \in {\rm R}^n\times
(0,T],$$ with given distributions in the right-hand sides of the equation and initial conditions is studied. Our task is to dete... |
Seeing that in the Chomsky Hierarchy Type 3 languages can be recognised by a DFA (which has no stacks), Type 2 by a DFA with one stack (i.e. a push-down automaton) and Type 0 by a DFA with two stacks (i.e. with one queue, i.e. with a tape, i.e. by a Turing Machine), how do Type 1 languages fit in...
Considering this ps... |
We can derive Lagrange equations supposing that the virtual work of a system is zero.
$$\delta W=\sum_i (\mathbf{F}_i-\dot {\mathbf{p}_i})\delta \mathbf{r}_i=\sum_i (\mathbf{F}^{(a)}_i+\mathbf{f}_i-\dot {\mathbf{p}_i})\delta \mathbf{r}_i=0$$
Where $\mathbf{f}_i$ are the constrainded forces and are supposed to do no wor... |
The Annals of Statistics Ann. Statist. Volume 9, Number 3 (1981), 544-554. A Nonparametric Control Chart for Detecting Small Disorders Abstract
We consider sequential observation of independent random variables $X_1,\cdots, X_N$ whose distribution changes from $F$ to $G$ after the first $\lbrack N\theta \rbrack$ variab... |
$\textbf{The Problem:}$ Let $m(X)<\infty$ and $f$ bounded and measurable. For $1\leq q<p<\infty$ prove that $\|f\|_{p}^{p}\leq\|f\|_{q}^{q}\|f\|_{\infty}^{p-q}.$
$\textbf{My Thoughts:}$ At first sight I was trying to somehow bring in Holder's inequality here, but I got nowhere. So I observed that the result would follo... |
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