text stringlengths 256 16.4k |
|---|
If $f(x) = \Omega(n)$ and $g(x)= O(n)$, what would be the order of growth of $f(x) \cdot g(x)$ ?
First I figured it should $\Theta(n)$ , as two extremes would cancel each other and the order of growth will be same as $n$
But, where I came across this question, the answer given was $\Omega(n)$, and no proof was mentione... |
Polarized Light
Earlier, we discussed
polarization as the direction of a wave's displacement with respect to its motion. We mentioned that an electromagnetic wave is transversely polarized, which is fairly unambiguous because both the electric and magnetic fields oscillate perpendicular to the direction the wave travel... |
We shall show that, if at least one of $|a_1|,\ldots,|a_k|$ is larger than one, then the sequence $\displaystyle b_n=\frac{1}{n}(a_1^n+\cdots+a_k^n)$ is not bounded, and hence $\displaystyle\sum_{n=1}^\infty\frac{1}{n}(a_1^n+\cdots+a_k^n)$ diverges.
Assume, without loss of generality that$$|a_1|=\cdots=|a_m|=r>s=|a_{m+... |
My understanding is that a KDF is a function that takes a master secret and generates multiple keys. It is secure as long as the keys are "independent". If this is true, the following definition would generate a secure KDF, right?
Assuming we have access to a completely random function, $R: \mathcal{K} \rightarrow \mat... |
Bravais Lattices¶
ATK has built-in support for all 14 three-dimensional Bravais lattices along with an additional possibility to specify the unit cell directly. To allow QuantumATK to take advantage of the symmetries of the lattice and define the relevant high-symmetry points in the Brillouin zone, e.g. for band struct... |
Journé's theorem for $C^{n,\omega}$ regularity
1.
Department of Mathematics, West Chester University, West Chester, PA 19383
Mathematics Subject Classification:Primary: 37D20, 58A05; Secondary: 41A0. Citation:V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete & Continuous Dynamical Systems - A, 2008, 2... |
There is a theorem which says that if $V$ is a vector space over a field $\mathbb{K}$ and $V$ admits an infinite linearly independent set, then any two bases $B$ and $B'$ have the same cardinality. So I wanted to prove this theorem. The question is followed by a hint which says that the following lemma can be applied t... |
Assessment | Biopsychology | Comparative |Cognitive | Developmental | Language | Individual differences |Personality | Philosophy | Social |
Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology |
In statistics and probability theory, the
covariance matrix is a matrix of cov... |
I am interested in data analysis. While my working data (actually it's shopping mall's daily sale) is accumlating, I wish to find some statistical laws underlying business phenomena. I left school for more than 10 years, and I am working in a business enviroment, I have no choice but to study by myself from level 0.
I ... |
I am trying to solve Exercise 3 a) given here. The problem states:
Let $\mathcal{M}$ be an infinite $\sigma$-algebra. Prove that $\mathcal{M}$ contains an infinite sequence of nonempty, disjoint sets. (Hint: if $\mathcal{M}$ contains an infinite sequence of strictly nested sets, then we’re done, so assume that no such ... |
Force in the Direction of Displacement
The work done by a constant force is proportional to the force applied times the displacement of the object.
learning objectives
Contrast displacement and distance in constant force situations
Work Done by a Constant Force
When a force acts on an object over a distance, it is said... |
To summarise the predictive power of a classifier for end users, I'm using some metrics. However, as the users input data themselves, the amount of data and class distribution varies a lot. So to inform users about the strength of the metrics, I'd like to include confidence intervals.
Background on the metrics
Suppose ... |
Mathematics can be used to describe processes we see everywhere! We can use maths to model how cancerous tumours form, how diseases spread in populations and how the patterns on animals coats form.
My favourite application of such mathematical models is in Mathematical Ecology. Here we use mathematics to describe proce... |
Let $K$ be a maximal subfield of $\mathbb C$ which doesn't contain $\sqrt{2}$ (one exists by Zorn's Lemma). Then $\mathbb C$ is algebraic over $K$, since if a complex number $\alpha$ is a transcendental over $K$, then adjoining $\alpha$ cannot result in $\sqrt{2}$ appearing, since squaring any expression of $\sqrt{2}$ ... |
Analysis of a Plane Blown Off Course oto a town B. The wind is blowing at 20km/h on a bearing of 300 o.
If the pilot aims the plane on to fly to B, he will be blow off course by the wind. What will be the ground speed and the bearing actually flown by the plane?
We can draw the vector diagram shown.
If we construct a p... |
The bias-variance decomposition can be expressed as:
\begin{align} \newcommand{\Var}{{\rm Var}} E[(y_0 - \hat{f}(x_0)) ^ 2] &= (E[\hat{f}(x_0)] - f(x_0)) ^ 2 + E[(\hat{f}(x_0) - E[\hat{f}(x_0)]) ^ 2] + \sigma ^ 2\\ &= [{\rm Bias}(\hat{f}(x_0))] ^ 2 + \Var(\hat{f}(x_0)) + \Var(\varepsilon) \end{align}
I tried to verify ... |
Alright, I have this group $\langle x_i, i\in\mathbb{Z}\mid x_i^2=x_{i-1}x_{i+1}\rangle$ and I'm trying to determine whether $x_ix_j=x_jx_i$ or not. I'm unsure there is enough information to decide this, to be honest.
Nah, I have a pretty garbage question. Let me spell it out.
I have a fiber bundle $p : E \to M$ where ... |
First, the permanent of an $n\times n$ integer matrix with $O(n)$-bit coefficients is an integer with $O(n^2)$ bits, hence if we know it modulo an integer with $\Omega(n^2)$ bits (with the implied constant depending on the constant in the input bit size), we know it outright.
Your problem with $p$ allowed to have $O(\l... |
The integral test applied to the
harmonic series
. Since the area under the curve
y = 1/ x
for
x ∈ [1, ∞)
is infinite, the total area of the rectangles must be infinite as well.
In
mathematics, the integral test for convergence is a method used to test infinite series of non-negative terms for convergence. It was devel... |
Category:Derivatives Let $I\subset\R$ be an open interval.
Let $f : I \to \R$ be a real function.
Let $f$ be differentiable on the interval $I$.
$\displaystyle \forall x \in I: f' \left({x}\right) := \lim_{h \mathop \to 0} \frac {f \left({x + h}\right) - f \left({x}\right)} h$ Subcategories
This category has the follow... |
Author Message TAGS:
Manager
Joined: 27 May 2010
Posts: 200 Re: For every positive integer x, f(x) represents the greatest prime fact [#permalink]
Show Tags 18 Jul 2019, 22:56
For every positive integer x, f(x) represents the greatest prime factor of x!, and g(x) represents the smallest prime factor of 2x+1. What is (g... |
I'm thinking to the famous problem of cancellation property in Grp, i.e: $$G_1 \times G_2 \cong G_1 \times G_3 \Rightarrow G_2 \cong G_3. $$ Clearly there are many counterexamples like $\prod_{i \in \omega}\mathbb{Z}_i$ or $ \oplus_{i \in \omega}\mathbb{Z}_i$ but these counterexamples can be bypassed by giving a defini... |
Limit on num_vars?
Post Reply
2 posts • Page
1of 1
I have a strange issue, I have added four new parameters to cosmomc putting them in parameter positions 14, 15 ,16 and 17. This has moved \Omega_\Lambda, Age/Gyr, \Omega_m, \sigma_8, z_{re} and H_0 to parameter positions 18, 19, 20, 21, 22, and 24.
The problem is when ... |
calculateEffectiveMassTensor¶
calculateEffectiveMassTensor(
configuration, kpoint=None, spin=None, band_indices=None)¶
Calculates the effective mass tensor \(\frac{d^2 E}{dk_i dk_j}\) at the given kpoint, spin, and bands.
Parameters: configuration(
BulkConfiguration) – The configuration for which to calculate the effec... |
I saw another question on here concerning light speeds and specifically concerning different types of light waves etc. The typical answer dealt with the speed of light in a vacuum but how do we know that space and especially deep space is truly a vacuum? I mean Einstein implies space is able to be contorted or bent whi... |
I am trying to implement a project about the BGM model, suggested in the book "The Concepts and Practice of mathematical finance" by Mark Joshi.
My question is related to the forward volatility structure, particularly about the covariance matrix. First of all, the book assumes that forward $f_j$ has volatility $$ K_j \... |
Both your teacher and your answer are wrong!
There are two steps to this problem. First, we need to convert between mass fractions and mole fractions. Second, we need to convert from a
per-amount basis to a per-volume basis. "Per-amount" is my made up word for all units such as mass fraction, mole fraction, molality, e... |
Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term
1.
Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
2.
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China
In this pape... |
The mathese in your question makes it difficult to understand, it is best to be more concrete rather than abstract. The answer is yes, this is how higher dimensional Dirac operators are standardly constructed.
If you have the Dirac algebra (clifford algebra) on a 2n-dimensional space
$$ \{ \gamma_\mu \gamma_\nu \} = 2 ... |
I have a data set where many of the actual values are zero, so I can't use MAPE. It's not a time series, so I can't use MASE ala our very own Rob Hyndman.
Is there another alternative to MAPE that I could use?
Cross Validated is a question and answer site for people interested in statistics, machine learning, data anal... |
The Latin Modern Math and TeX Gyre Math support projects are complete. Together, these fonts provide a total of 5 fonts for typesetting mathematics, all produced by the GUST e-foundry. Details can be found here.
That page lists an additional six fonts, produced by other foundries, which support the mathematics opentype... |
I am looking at the Harvard notes for the complex analysis, and I do not follow how they arrive at the circled:
EDIT: Can also someone show me how to get to the last line? I am a bit confused about how the $\text{sgn}(b)$ emerges there.
Mathematics Stack Exchange is a question and answer site for people studying math a... |
Here are a couple quick and dirty ways to count these operators:
Compute the conformal block expansion of the four-point function $\langle \phi\phi\phi\phi\rangle$. This will only contain blocks with $\Delta-\ell=d-2$. This is done in http://arxiv.org/abs/1009.5985, equation 64. Compute the character of the conformal g... |
$L$ isn't recognizable. We'll first establish a couple of preliminary results
I. $\overline{L}$ is recognizable
The complement of $L$,$$\overline{L}=\{\langle M\rangle\mid M \text{ halts on at least one input}\} $$is recognizable. Define a recognizer TM as follows:
R(<M>) =
for n = 1, 2, 3, ...
for each x in {x_1, x_2,... |
Your score is simply the sum of difficulties of your solved problems. Solving the same problem twice does not give any extra points. Note that Kattis' difficulty estimates vary over time, and that this can cause your score to go up or down without you doing anything.
Scores are only updated every few minutes – your sco... |
So I'm trying to take the derivative of $\frac {\sin(x^2)}{3x}$.
Here are my steps:
$$\frac {d}{dx}\left[\frac {\sin(x^2)}{3x}\right]$$ Use the Quotient Rule: $$\frac {3x\frac {d}{dx}[\sin(x^2)]-\sin(x^2)\frac {d}{dx}[3x]}{3x^2}$$
Simplify second part:
$$\frac {3x\frac {d}{dx} [\sin(x^2)]-3\sin(x^2)}{3x^2}$$
Use Chain ... |
Abstracts
We consider space-bounded computations on a random-access machine (RAM) where the input is given on a read-only random-access medium, the output is to be produced to a write-only sequential-access medium, and the available workspace allows random reads and writes but is of limited capacity. The length of the ... |
This is more of an extended comment, rather than an answer completely separate from what Urs and Moshe have already said. The axioms of AQFT are designed to capture a mathematical model of the
physical observables of a theory, while OTOH gauge invariance is a feature of a formulation of a theory, though perhaps an espe... |
[Questions about Machine Learning] Chapter I Mathematics Fundamentals
In this chapter, we will discuss some basic mathematics knowledge that you need to know for further study.
Q. What are the relations between scalar, vector, matrix, and tensor?
A. A vector is an ordered finite list of numbers. Vectors are usually wri... |
Suppose one has the limit of a completely non-continuous when $x<0$ such as $$\lim _{x\to-\infty}\left|{x}^{\frac{1}{2x}}\right|$$ Which is undefined at any negative fraction with an odd denominator (such as $-2$, $-2/3$, $-3/5$).
If you treat this function as a sequence at any integer it is undefined. However, by usin... |
Interested in the following function:$$ \Psi(s)=\sum_{n=2}^\infty \frac{1}{\pi(n)^s}, $$where $\pi(n)$ is the prime counting function.When $s=2$ the sum becomes the following:$$ \Psi(2)=\sum_{n=2}^\infty \frac{1}{\pi(n)^2}=1+\frac{1}{2^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{3^2}+\frac{1...
Consider a random binary str... |
I've been given a proof for convergence of a monotone increasing sequence bounded from above, and am attempting to replicate it for a decreasing sequence bounded from below (we're working in the real space). I'm hoping this is a valid proof, any improvements/criticisms would be helpful:
Let $\left\{x_n|n \in \mathbb{N}... |
The ISO/IEC 9899:1990 edition of the C standard contains:EXAMPLE The following functions define a portable implementation of rand and srand.static unsigned long int next = 1;int rand(void) // RAND_MAX assumed to be 32767{next = next * 1103515245 + 12345;return (unsigned int)(next/65536) % 32768;}void srand(unsigned int... |
In your original question you didn't have the assumption of nilpotence. I want to answer the question both with and without this assumption, because it is easier without the nilpotency assumption an both proofs use the same idea. The answer is "no" in both cases.
A group $G$ is
residually finite if for every $g\in G$ t... |
For what follows, let us restrict ourselves to discussing
discrete probability distributions.
When learning about things like relative entropy (aka Kullback-Leibler divergence) $D(\cdot || \cdot)$, one typically learns that it does not technically define a metric, since $D(P||Q) \ne D(Q||P)$, for some distributions $P,... |
In a typical step-growth polymerization, the Anderson–Schulz–Flory distribution is the probability mass function that describes the number fraction or weight fraction of polymers of length $x$ at a given extent of reaction $p$. The extent of reaction, $p$, is a value between 0 and 1, and $x$ is the degree of polymeriza... |
Completeness is a property of proof systems. Roughly, we say that a proof system is complete iff truth implies derivability. Yet, the proofs for FOL we usually find in logic textbooks, i.e. "Henkin-proofs", do not seem to make any reference to any deductive system at all.
Consider the following proof sketch:
Lemma 1 (M... |
The Genetic and Evolutionary Computation Conference (GECCO) the leading conference in the area of evolutionary computation. In 2016 it will take place in Denver, Colorado, USA, from July 20-24. This year 381 papers were submitted to 15 different tracks, and 1740 reviews assigned. Approximately 36% of papers have been a... |
The practical thing that you are missing here is that the air compressor is cooled constantly so that it doesn't melt during operation (can't be adiabatic). So the upper limit on the temperature is driven by the operating temperature of the compressor. Then the other posters comments about heat transfer out of the tank... |
Learning Objectives
By the end of this section, you will be able to:
Explain the difference between the solar day and the sidereal day Explain mean solar time and the reason for time zones
The measurement of time is based on the rotation of Earth. Throughout most of human history, time has been reckoned by positions of... |
Are there any differences between the study of Calculus done by Newton and by Leibniz. If so please mention point by point.
Newton's notation, Leibniz's notation and Lagrange's notation are all in use today to some extent they are respectively:
$$\dot{f} = \frac{df}{dt}=f'(t)$$ $$\ddot{f} = \frac{d^2f}{dt^2}=f''(t)$$
Y... |
I want to compute the following integral
$$- \frac{1}{M(\lambda_1-\lambda_2)}\int\limits_{-\infty}^t(e^{\lambda_1(t-t')}-e^{\lambda_2(t-t')})(\beta\omega A\sin\omega t' +g)\;dt'$$
Here the integral that's not trivial is
$$\int_{-\infty}^t (e^{\lambda_1(t-t')}-e^{\lambda_2(t-t')}) \sin\omega t'\; dt'.$$
If I use $\sin \... |
Find the values of $x$ such that $$2\tan^{-1}x+\sin^{-1}\left(\frac{2x}{1+x^2}\right)$$ is independent of $x$.
Checking for $x\in [-1,1]$ In the taken domain $\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ comes out to be $2\tan^{-1}x$ hence the taken function comes out to be equal to $4\tan^{-1}x$ hence the function is clear... |
Definition:Mellin Transform
Jump to navigation Jump to search
The
Definition
The
Mellin Transform of $f$, denoted $\mathcal M \left({f}\right)$ or $\phi$, is defined as: $\displaystyle \mathcal M \left\{{f \left({t}\right)} \right\} \left({s}\right) = \phi \left({s}\right) = \int_0^{\to +\infty} t^{s-1} f \left({t}\rig... |
To send content items to your account,please confirm that you agree to abide by our usage policies.If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.Find out more about sending content to .
To send content items to your Kindle, first ensure no-rep... |
A elementary study of Quantum Mechanics, following $[1]$, yields in the realization that the basic algebraic structure are the complex vector spaces $\mathbb{C}^{n}$. Then a contravariant vector (using the terminology of Differential Geometry/General Relativity) is simply defined as an element of $\mathbb{C}^{n}$ ;
$$\... |
Let $a_n = n^{1/n}$ and $b_n = \dfrac{a_{n+1}}{a_n}$. I am trying to prove that $\{b_n\}_{n=5}^{\infty}$ is increasing. Here is what I know and what I've tried:
The first few elements of $b_n$ aren't increasing (because $a_n$ is decreasing only after $n \geq 3$, for instance.) and so it wouldn't be correct to prove tha... |
Let $f\colon X \to Y$ and $g\colon Y \to X$ be functions. Assume $g \circ f$ is bijective. Prove $f$ is injective and $g$ is surjective.
Approach:
if $g \circ f$ is bijective then $g \circ f$ is one to one
if $g \circ f$ is bijective then $g \circ f$ is onto
so we know $$g ◦ f (a_1)=g ◦ f (a_2) \text{ this implies $a_1... |
The number of variables is the number of elements you need to change to change the value of your function. In your case, knowing that $x,y,z$ are functions of $t$, you are implicitly saying that $\bf F$ can be viewed as a function of $t$ only too.
In symbols: you have a function of four variables$$\mathbf{F}:\bf{R}^4\l... |
Markov Chain Monte Carlo Background
What if we know the relative likelihood, but want the probability distribution?
But what if \( \int f(x)dx \) is hard, or you can’t sample from \( f \) directly?
This is the problem we will be trying to solve.
First approach
If space if bounded (integral is between \( a,b \)) we can ... |
Let $X$, respectively $Y$, be a space endowed with the $\sigma$-additive complete measure $\mu_x$, respectively $\mu_y$.
If $f: X\times Y\to\mathbb{R}$ is $\mu_x\otimes\mu_y$-measurable$^1$ and Lebesgue summable in one variable, is its integral measurable in the other variable? I mean if, for example, $$\int_{Y}f(x,y)d... |
Hello guys! I was wondering if you knew some books/articles that have a good introduction to convexity in the context of variational calculus (functional analysis). I was reading Young's "calculus of variations and optimal control theory" but I'm not that far into the book and I don't know if skipping chapters is a goo... |
Please help with a question that I am working on just now...:)
If $z=2e^{i\theta}$ where $0<\theta<\pi$, how can I find the real and imaginary parts of $w=(z-2)/(z+2)$? Hence, how can I calculate w and deduce that this complex number always lies parallel to the real axis in the Argand plane?
The following information c... |
This question already has an answer here:
The two circles are:
1) $$(x-2)^2 + (y+1)^2 = 25$$ 2) $$(y-2)^2 + (x+1)^2 = 25$$
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 this community
... |
So far, I have encountered two main definitions for the Lebesgue integral of a non-negative measurable function $f$.
1) $\int_A f d \mu = \sup _{h \leq f, \space h \space simple} \{ \int_A h d \mu \}$
2) $\int_A f d \mu = \sup \{ \inf _{x_i \in E_i} \sum_{i=1}^N f(x_i) \mu(E_i) \} = \sup \{ \sum_{i=1}^N (\inf _{x_i \in... |
Let $u_\epsilon \in W^{1,p}(\Omega)$ be a sequence with $\Omega \subset \mathbb{R}^n$ bounded. Let $u_\epsilon \rightharpoonup^* u$ weakly* in $L^\infty (\Omega)$ (for a subsequence) with $u \in W^{1,p}(\Omega)$ and assume $\sup_\epsilon \vert \nabla u_\epsilon\vert_{L^p (\Omega; \mathbb{R}^n)} < C$.
Proof that $\nabla... |
I have been stuck on a chemistry problem for a long time now and if anyone here can help me I would be eternally grateful.
40.Du tillsätter $\pu{100 ml}$ $\pu{0,01 M}$ $\ce{Na2SO4}$-lösning i $\pu{100 ml}$ $\pu{0,02 M}$ $\ce{CaCl2}$-lösning. Det bildas en $\ce{CaSO4}$-fällning. Hur mycket av $\pu{0,01 M}$ $\ce{Na2SO4}$... |
(11C) Tree Heights
11-02-2018, 01:24 PM (This post was last modified: 11-02-2018 01:26 PM by Gamo.)
Post: #1
(11C) Tree Heights
This program was adapted from the Hand-Held-Calculator Programs
for the Field Forester.
More detail information attached here.
Procedure:
1. Enter slope distance to base of tree [A] -->Display... |
Learning Objectives
Express momentum as a two-dimensional vector Write equations for momentum conservation in component form Calculate momentum in two dimensions, as a vector quantity
It is far more common for collisions to occur in two dimensions; that is, the angle between the initial velocity vectors is neither zero... |
2019-09-04 12:06
Soft QCD and Central Exclusive Production at LHCb / Kucharczyk, Marcin (Polish Academy of Sciences (PL)) The LHCb detector, owing to its unique acceptance coverage $(2 < \eta < 5)$ and a precise track and vertex reconstruction, is a universal tool allowing the study of various aspects of electroweak an... |
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... |
Assume $f$ : E -> $\mathbb{R}$ is a continuous function defined on some set E $\subset$ $\mathbb{R}$ and if {$a_j$}$_{j\in\mathbb{N}}$ $\subset$ E is a Cauchy sequence is it true that {$f(a_j)$}$_{j\in\mathbb{N}}$ is also a Cauchy sequence?
Not necessarily.
A sufficient condition such that $\{f(x_j)\}$ is again Cauchy ... |
Let $(E_i,T_i)_{i\in I}$ be a family of topological spaces, and let $(E,T)$ be the product of these spaces. Let $f_i:E\to E_i\ (i\in I)$ be a family of mappings. Suppose that each $f_i$ is $(T,T_i)-$ continuous. Denote the product of these mappings by $\displaystyle\prod_{i\in I}f_i,$ or shortly $f.$
Denote the image o... |
Garabedian,Typically, the "swap curve" refers to an x-y chart of par swap rates plotted against their time to maturity. This is typically called the "par swap curve."Your second question, "how it relates to the zero curve," is very complex in the post-crisis world.I think it's helpful to start the discussion with a gov... |
I came across John Duffield Quantum Computing SE via this hot question. I was curious to see an account with 1 reputation and a question with hundreds of upvotes.It turned out that the reason why he has so little reputation despite a massively popular question is that he was suspended.May I ...
@Nelimee Do we need to m... |
The bounty period lasts 7 days. Bounties must have a minimum duration of at least 1 day. After the bounty ends, there is a grace period of 24 hours to manually award the bounty. Simply click the bounty award icon next to each answer to permanently award your bounty to the answerer. (You cannot award a bounty to your ow... |
Category:Definitions/Geometric Rotations
A
rotation is defined usually for either: $n = 2$, representing the plane
or:
$n = 3$, representing ordinary space. $\map {r_\alpha} O = O$
That is, $O$ maps to itself.
Let $P \in \Gamma$ such that $P \ne O$.
Let $OP$ be joined by a straight line.
Let a straight line $OP'$ be co... |
The
marginal pdf will be calculated over the area defined by a triangle as mentioned in the comments. The reason for it lies in the boundary constraints $0 < x < y < 2$, where the bivariate joint pdf is defined. To see this, we can mentally slide along the $x$ axis from $0$ to $2$, and see how at any given point, the $... |
First, you are confusing Lie groups with Lie algebras. Casimir elements are objects that can be attached to certain Lie algebras.
Second, Casimir elements do not always exist. For any Lie algebra $\mathfrak{g}$, there is a canonical bilinear form, the Killing form
$$B(x, y) = \text{tr}(\text{ad}_x \text{ad}_y)$$
where ... |
I also do not understand the textbook solution.
In an equilibrium situation the ball is effectively rolling down a plane inclined at angle $\theta$ while the plane is itself accelerating upwards along the plane. In the ground frame of reference the centre of mass of the ball is stationary, so the resultant force on it ... |
Let's assume we restrict consideration to symmetric distributions where the mean and variance are finite (so the Cauchy, for example, is excluded from consideration).
Further, I'm going to limit myself initially to continuous unimodal cases, and indeed mostly to 'nice' situations (though I might come back later and dis... |
Introduction
Built at the Jet Propulsion Laboratory by an Investigation Definition Team (IDT) headed by John Trauger, WFPC2 was the replacement for the first Wide Field and Planetary Camera (WF/PC-1) and includes built-in corrections for the spherical aberration of the HST Optical Telescope Assembly (OTA). The WFPC2 wa... |
This MSE question made me wonder where the Leibnitz notation $\frac{d^2y}{dx^2}$ for the
second derivative comes from. It does not arise immediately as the obvious generalization of $\frac{dy}{dx}$. Did Leibnitz use it himself? Or was it introduced later?
This MSE question made me wonder where the Leibnitz notation $\f... |
Overview
So far in the course we have discussed how a source could create a wave pulse, a repeating wave or a harmonic wave. By knowing the motion of the source, we have seen that the disturbance keeps its shape and propagates with a speed \(v_{wave}\). These discussions all assumed that the medium was unperturbed befo... |
Problem for Secondary Group from Divisional Mathematical Olympiad will be solved here.
Forum rules
Please
don't post problems (by starting a topic)in the "Secondary: Solved" forum. This forum is only for showcasing the problems for the convenience of the users. You can post the problems in the main Divisional Math Olym... |
CentralityBin () CentralityBin (const char *name, Float_t low, Float_t high) CentralityBin (const CentralityBin &other) virtual ~CentralityBin () CentralityBin & operator= (const CentralityBin &other) Bool_t IsAllBin () const const char * GetListName () const virtual void CreateOutputObjects (TList *dir, Int_t mask) vi... |
If you had taken the $\operatorname{lcm}$ operation more seriously, and used $\operatorname{lcm}(1..n)=\operatorname{lcm}(1,2,3,4, \dots, n)$ as the place values (for $\operatorname{lcm}(1..(n-1)) \neq \operatorname{lcm}(1..n)$, i.e. for prime powers $n=p^r$) in your mixed radix number representation, then you would ha... |
I have this piecewise function:
$$f(x)=\begin{cases} \dfrac{x^2-x-2}{|x-2|}, & x \neq 2 \\ 0, & x = 2\text{.} \end{cases}$$
I can't figure out how to graph it. I punched these numbers into my calculator, and it created a parabola, but I haven't been able to get there on my own without the calculator.
So far I have plot... |
Let $U \subset \mathbb{C}$ be open. I want to show that we can write $U$ as a countable union of open connected sets.
If $U$ is empty there is nothing to prove. Else it contains some ball, and hence a point in $\mathbb{Q} + i\mathbb{Q}$.
Denote $U_{\mathbb{Q}} = U \cap \mathbb{Q} + i\mathbb{Q}$.
For each $q \in U_{\mat... |
I've been given this as a homework assignment, and have no idea how to proceed. Can anyone help? The question is:
(1) Let $\phi: M_{n}(\mathbb{C}) \rightarrow M_{n}(\mathbb{C})$ be an endomorphism such that $$M \in \operatorname{GL}_{n}(\mathbb{C}) \implies \phi (M) \in \operatorname{GL}_{n}(\mathbb{C}).$$ Show that, f... |
There is a classic problem:
Suppose that $X_1,\ldots,X_n$ form an i.i.d. sample from a uniform distribution on the interval $(0,\theta)$, where $\theta>0$ is unknown. I would like to find the MLE of $\theta$.
The pdf of each observation will have the form: $$ f(x\mid\theta) = \begin{cases} 1/\theta\quad&\text{for }\, 0... |
It can be shown starting from Maxwell's equations that the electromagnetic field satisfies the wave equation: $$\square^2 \mathbf{E} = \frac{1}{\epsilon_0}\nabla\rho+\mu_0\frac{\partial\mathbf{J}}{\partial t}$$ $$\square^2 \mathbf{B}=\mu_0\nabla\times\mathbf{J}$$ In particular, when the RHS of the above equations are z... |
(50g) Simpson's rule for f(x,y) - Printable Version
+- HP Forums (
https://www.hpmuseum.org/forum)
+-- Forum: HP Software Libraries (
/forum-10.html)
+--- Forum: General Software Library (
/forum-13.html)
+--- Thread: (50g) Simpson's rule for f(x,y) (
/thread-11818.html) (50g) Simpson's rule for f(x,y) - peacecalc - 11... |
A. Enayat, J. D. Hamkins, and B. Wcisło, “Topological models of arithmetic,” ArXiv e-prints, 2018. (under review)
@ARTICLE{EnayatHamkinsWcislo2018:Topological-models-of-arithmetic, author = {Ali Enayat and Joel David Hamkins and Bartosz Wcisło}, title = {Topological models of arithmetic}, journal = {ArXiv e-prints}, ye... |
A. Enayat, J. D. Hamkins, and B. Wcisło, “Topological models of arithmetic,” ArXiv e-prints, 2018. (under review)
@ARTICLE{EnayatHamkinsWcislo2018:Topological-models-of-arithmetic, author = {Ali Enayat and Joel David Hamkins and Bartosz Wcisło}, title = {Topological models of arithmetic}, journal = {ArXiv e-prints}, ye... |
Divided by Positive Element of Quotient Field Theorem Then: $\forall z \in K: \exists x, y \in D: z = \dfrac x y, y \in D_{>0}$ Proof
By definition:
$\forall z \in K: \exists x, y \in D: z = \dfrac x y, y \in D_{\ne 0}$ Suppose $z = x' / y'$ such that $y' \notin D_{>0}$.
Then $y' < 0$ as $D$ is totally ordered.
Then:
\... |
A. Enayat, J. D. Hamkins, and B. Wcisło, “Topological models of arithmetic,” ArXiv e-prints, 2018. (under review)
@ARTICLE{EnayatHamkinsWcislo2018:Topological-models-of-arithmetic, author = {Ali Enayat and Joel David Hamkins and Bartosz Wcisło}, title = {Topological models of arithmetic}, journal = {ArXiv e-prints}, ye... |
Let $K$ be a compact metric space, and let $A \subset K$. Prove that $A$ is compact if and only if, for every continuous function $f:K \to \mathbb{R}$, there exists a $q \in A$ so that $$f(q) = \max_{a \in A} \{f(a)\}.$$ i.e. $f$ attains its maximum on the restriction to $A$.
Here's an attempt:
The forward direction is... |
Prove:
Let $a_n, b_n$ be positive sequences so that $\limsup \frac {a_n}{b_n} \lt\infty$. If $\sum b_n$ converges then $\sum a_n$ converges.
So far I was thinking that from $\limsup \frac {a_n}{b_n} \lt\infty$, we know that $a_n$ doesn't have a subsequential limit of positive infinity and $b_n$ doesn't have a subsequen... |
I am trying to show that
$$\int_{S(t)} (\textbf{n} \cdot \sigma ) \cdot\textbf{u} \ dS = \int_{V(t)}(\nabla\cdot\sigma) \cdot \textbf{u} \space + \sigma : \nabla \textbf{u} \space dV$$
where $\textbf{u,n}$ are vectors and $\sigma$ is a symmetric second order tensor.
Using the divergence theorem
$$\int_{S(t)} (\textbf{n... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.