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A) Let $\tau$ be a Grothendieck pretopology on any category with fiber products. Define a new Grothendieck pretopology $\tau'$ where a cover $\{U_i \to X\}$ is a $\tau'$ cover if and only if there exists a refinement $\{V_{ij} \to X\}$ (i.e. there exists for each $ij$ an $X$-morphism $V_{ij} \to U_i$) such that $\{V_{i... |
I have been wondering about some of the different uses of Generalized Complex Geometry (GCG) in Physics. Without going into mathematical detail (see Gualtieri's thesis for reference), a Generalized Complex Geometry attempts to unify symplectic and complex geometry by considering the bundle $TM\oplus T^* M$ with its nat... |
Volume 4, Issue 1
March 1983, pages 1-63
pp 1-9 March 1983
The effective temperatures of the classical Cepheids RT Aur and T Vul have been determined by a comparison of their spectral scans with appropriate model atmospheres. The radii of the stars have been determined through the Wesselink method. Using these temperat... |
I would like to know what is the induced matrix norm of a matrix $A$, when the domain space is equipped with $l_2$ norm and the range space is equipped with $l_\infty$ norm, i.e., what is $\lvert| A \rvert|_{2 \infty}$? The expression for $\lvert| A \rvert|_{\infty 2}$ is given here.
In general, when $X$ is a normed sp... |
Does the sum $$\sum_{n=1}^{\infty}\frac{\tan n}{n^2}$$ converge?
note
$$ \cos(x) = \prod_{n=1}^{\infty}\left(1-\frac{4x^2}{\pi^2(2n-1)^2}\right) $$
then $$ \log(\cos(x)) = \sum_{n=1}^{\infty}\log\left(1-\frac{4x^2}{\pi^2(2n-1)^2}\right) $$ which gives $$ =-\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\frac{4^k x^{2k}}{\pi^{2k... |
Overview
The derivatives of functions are used to determine what changes to input parameters correspond to what desired change in output for any given point in the forward propagation and cost, loss, or error evaluation &mdash whatever it is conceptually the learning process is attempting to minimize. This is the conce... |
Research Open Access Published: Global well-posedness for nonlinear fourth-order Schrödinger equations Boundary Value Problems volume 2016, Article number: 25 (2016) Article metrics
981 Accesses
Abstract
This paper studies a class of nonlinear fourth-order Schrödinger equations. By constructing a variational problem an... |
Interested in the following function:$$ \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'm trying to understand the definition of tensor product of two vector spaces. So far, I've read the one using free vector spaces and a quotient space (here), and I think I understand it well. However, I want to understand the other definitions I can find, and it seems that a very common way to define it is through th... |
Yes there is a (slightly more) rigorous definition:
Given a model with a set of parameters, the model can be said to be overfitting the data if after a certain number of training steps, the training error continues to decrease while the out of sample (test) error starts increasing.
In this example out of sample (test/v... |
Steric hinderance is a major component in determining the feasibility and the rate of a chemical reaction. Wouldn't it be useful to measure it quantitatively then? This would make it easier to compare the property of two molecules. Are there currently ways to measure steric hindrance, or is it not possible for some rea... |
I found this interesting problem on calculating the limit of $\frac{\sin(xy^2)}{xy}$ on the positive coordinate axes $x$ and $y$. That is, compute the limit on the points $(x_0, 0)$ and $(0,y_0)$ when $x_0 > 0$ and $y_0 > 0$.
My approach was this:
If we first calculate the limit for $x$ axis, the the $x$ is a constant ... |
I'm going to find an example of
uniform algebra and show that satisfying the definition. Example: Show that The Gelfand transform $\widehat{f}$ is uniform algebra.
We know that:
A uniform algebra is a
closed subalgebra$\mathcal A$ of the complex algebra $C(X)$ that contains the constantsand separates points. Here $X$ i... |
So starting from the time dependent schrodinger equation I perform separation of variables and obtain a time and spatial part. The spatial part is in effect the time independent schrodinger equation.
Since we are dealing with a free particle I can take the time independent equation, set V = 0 and solve.
I can do this s... |
I want to show that a rigid body, with two components of its angular velocity vector and one component of its linear velocity vector,
in the absence of external forces and torques, has helical trajectories. This is usually taken for granted in various papers I have read (see for example this p.4 section 4, or this p.20... |
Business Insider's (long) article SpaceX's biggest rival has a 'genius' plan to cut its rocket launch costs more than 70% contains the statements sourced from ULA's CEO Tory Bruno:
Vulcan should lift 40 tons (nearly three school buses) into low-Earth orbit. That's less than SpaceX's Falcon Heavy, which can lift more th... |
This is a problem from Schlogl's book in the chapter on the HJM model:
Price option of the RAN instrument with 3 month coupons and maturity 3 years using Monte Carlo(Exercise 4 Range Accrual Note).
Is the code in the Quantlib library? If so can you tell me its location, thank you. If not can you give me some suggestion... |
I'm currently reading on self-balancing robots that use an IMU (gyroscopes + accelerometers) to estimate their current tilt angle.
Most documents that I have found say the same things:
You can't just take the arc-tangent of the accelerometers data to find the gravity direction because they are affected by "inertial noi... |
A local system of coefficients on a space $X$ is a functor $F\colon \Pi(X)\rightarrow Ab$ from the fundamental groupoid to the category of abelian groups. From this, one can define the homology groups of $X$ with local coefficients $H_*(X,F)$ as done e.g. in chapter VI of Whitehead's book "Elements of homotopy theory".... |
Extraordinary claims require extraordinary proofs which really is the reason why this sorts of discussion is important. Similarly, sometimes, you are so blinded to some sorts of a truth and are faced with something so different that you can misread entirely what is being said. If you read this morning's entry, you migh... |
Can you please help me and tell, how should I move on? Can this be proved by induction?
Every natural number $n\geq 8$ can be represented as $n=3k + 5\ell$.
Thank you in advance
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only ta... |
People are always fascinated by intelligent devices, and today they are software “chatbots”, which are becoming more and more human-like and automated. The combination of an immediate response and a permanent connection makes them an attractive way to expand or replace the web applications. The high-level diagram of an... |
I want to point out another interesting solution method, which also generalizes the result to the expectation of $X^{-m}$ for integer $m=1,2,3,\dots$. I will use moment generating functions (mgf) and the results from the paper by N Cressie et.al http://amstat.tandfonline.com/doi/abs/10.1080/00031305.1981.10479334?journ... |
Maass forms of levels 100 to 1000 with $0 \leq R\leq 1$
The horizontal axis is the spectral parameter $R$ with the Laplace eigenvaluesatisying $\lambda=1/4+R^2$. The vertical axis is the level $N$. Each pointcorresponds to a Maass form of weight 0 and trivial character on $\Gamma_0(N)$with the color showing whether the... |
Explanation of notation:
$H$ is the enthalpy of the system.
$\Delta$ means change of, so $\Delta H$ means change of the enthalpy
This symbol means standard condition, standard condition is defined as a pressure of $10\text{kPa}$ and reactants and products are in their standard state, or concentration of solutions are $... |
Amplitude of oscillator function
\(A(n)= \psi_n(0)\)
where \(\psi_n(z)=\frac{1}{\sqrt{N_n}}\)HermiteH\(_n(z)\, \exp(-z^2/2)\) is oscillator function, normalised solution of the stationary Schroedinger equation for the Harmonic oscillator. here, HermiteH\(_n\) denotes the \(n\)th Hermite polynomial and \(N_n\) is its no... |
There is a finite, circular area $A=\pi \times r^2$ with a given radius $r$ and a variable number of points $P_n$ ($n \in \mathbb{N}_{>0}$) that are to be placed inside of this area.
What is the distance $d$ to the closest dots when placing them in a way to maximise this closest distence while also placing them at mini... |
I'm reading through Halzen and Martin Chapter 6.3 and I have a few questions about what they're doing. We're calculating the invariant amplitude for $e^-\mu^- \rightarrow e^-\mu^-$ scattering. The text is (up to some rearranging to make the question clearer) as follows:
It is convienent to separate the sums over the el... |
Assume I have two neural networks, abstracted as two feature maps, parametrized by $\theta_x,\theta_y$ respectively. $\phi_x(x;\theta_x) \in \mathbb{R}^{h_1}$, $\phi_y(x;\theta_y) \in \mathbb{R}^{h_2}$ and we would like to perform canonical correlation analysis on the results of those feature maps, which means we want ... |
We develop all the necessary theory here without recourse to cycles/orbits/etc.
You are about to enter another dimension. A dimension not only of
sight and sound, but of mind. A journey into a wondrous land of
imagination.
Next stop, the Transparent-Transposition-Transformation-Tour!
Recall that the symmetric group $S_... |
Research Open Access Published: On a uniqueness theorem of Sturm–Liouville equations with boundary conditions polynomially dependent on the spectral parameter Boundary Value Problems volume 2018, Article number: 28 (2018) Article metrics
687 Accesses
3 Citations
Abstract
Inverse nodal problems for Sturm–Liouville equat... |
Research Open Access Published: The regularity criterion for weak solutions to the n-dimensional Boussinesq system Boundary Value Problems volume 2017, Article number: 44 (2017) Article metrics
642 Accesses
1 Citations
Abstract
We consider the Boussinesq system in the homogeneous spaces of degree −1. To narrow the gap ... |
Research Open Access Published: Existence and multiplicity of non-trivial solutions for the fractional Schrödinger–Poisson system with superlinear terms Boundary Value Problems volume 2019, Article number: 4 (2019) Article metrics
553 Accesses
Abstract
In this paper, we study the following fractional Schrödinger–Poisso... |
Rather than going to talk about algorithms and more about SAT, this answer tries to show that there is more than just one kind of logic and that there are also different kinds of approaches.
I won't go into detail much but rather provide a list with some topics and keywords that should form a good basis to do some rese... |
So Kirchhoff's second law states the sum of the emfs around a circuit is equal to the sum of potential drops. All explanations point to the conservation of energy. Fine. But why do electrons have to lose
all their gained energy from batteries to the components. Conservation of energy would still hold if they lost a bit... |
Neural networks (NNs) are used as approximators in reinforcement learning (RL). To update the policy in RL, the actor network's gradients w.r.t its weights are needed. Since NN doesn't have a mathematical expression to work with, how can its derivatives be calculated?
I think what you mean to ask is
how can differentia... |
So I'm working a problem that states:
A function $f$ is analytic in an open set $U$. Define $g$ by $g(z)=\overline{f(\overline{z})}$ (just because the notation can be hard to read, this is the the complex conjugate of the function $f$ defined at the complex conjugate of $z$). Show that $g$ is analytic in the oopen set ... |
What is the expected value of the determinant over the uniform distribution of all possible 1-0 NxN matrices? What does this expected value tend to as the matrix size N approaches infinity?
As everyone above has pointed out, the expected value is $0$.
I expect that the original poster might have wanted to know about ho... |
Exercise:Suppose that $a_k \geq 0$ for $k$ large and that $\sum_{k = 1}^{\infty} \frac{a_k}{k}$ converges.
Prove that $$\lim_{j \to \infty}\sum_{k = 1}^{\infty} \frac{a_k}{j+k} = 0$$
Attempt in proof:
Suppose $a_k \geq 0$ for any large $k$ and that $\sum_{k = 1}^{\infty} \frac{a_k}{k}$ converges. . Then give $\epsilon ... |
Published inAlgebras and Representation Theory
Let J(C) be the poset of order ideals of a cominuscule poset C where C comes from two of the three infinite families of cominuscule posets or the exceptional cases. We show that the Auslander-Reiten translation τ on the Grothendieck group of the bounded derived category fo... |
Angular velocity fixed at birth
Because of the Born rigidity of black holes, it follows directly from a consideration of the Ehrenfest paradox[5], that the angular velocity of a black hole can never be changed - it is fixed at birth. To explain this a little more fully, imagine a disk of radius \(R\) rotating with cons... |
Abstract
We prove that when subjected to periodic forcing of the form $$p-{\mu ,\rho ,\omega } (t) = \mu (\rho h(x,y) + \sin (\omega t)),$$ certain two-dimensional vector fields with dissipative homoclinic loops generate strange attractors with Sinai-Ruelle-Bowen measures for a set of forcing parameters (μ, ρ, ω) of po... |
Areas Related to Circles Class 10 Notes i.e. for chapter 12 provided here are extremely useful for the class 10 students to prepare and revise this chapter in a more efficient way. The most important topics from this chapter include the following topics which are explained in detail below.
Area of a circle Perimeter or... |
Quadratic Formula Complete the Square Factor
\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]
\[ x = \frac{-b}{2a} \pm \frac{\sqrt{b^2-4ac}}{2a} \]
Back to Top
Move c to the right side of the equation. \[ax^2+bx = -c \]
Factor a out of both terms on the left side. \[a(x^2+\frac{b}{a}x) = -c \] Divide both sides by a. \[x^2+\... |
Intuitive argument: by symmetry we should have $P(X_1 \ge 0 \mid W_1 = 0) = \frac{1}{2}$. However, $P(f(1, W_1) \ge 0 \mid W_1 = 0)$ is either $0$ or $1$ depending on whether $f(1,0) \ge 0$.
Of course this is not really a proof because I conditioned on an event of probability 0. So let's try to use the same idea in an ... |
Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:
1) $X$ is defined over $\overline{\mathbb{Q}};$
2) There exists a meromorphic function $\phi: X\to\mathbb{P}^1\mathbb{C} $ ramified at most at $0,1,$ and $\infty$;
3) $X$ is isomorphic to $\Gamma \backsl... |
What can we say about the quantum state from the number of zero and non-zero eigenvalues of the corresponding density matrix?
The number of zero eigenvalues has no significance, and is not really well defined anyway.
If the number of non-zero eigenvalues is not one, then there are many different ways to write the densi... |
Indeed, in one very specific sense we
don't have more freedom one way versus the other. So the basic rule of modeling a dynamical system says, "any Lagrangian or Hamiltonian or force law is allowed, as long as we recover the same equations of motion each way." And that freedom is the same for both the Lagrangian and Ha... |
Research Open Access Published: Blow-up criteria for Boussinesq system and MHD system and Landau-Lifshitz equations in a bounded domain Boundary Value Problems volume 2016, Article number: 90 (2016) Article metrics
856 Accesses
1 Citations
Abstract
In this paper, we prove some blow-up criteria for the 3D Boussinesq sys... |
GL(2)
Curves
group.alternating
The alternating group, denoted $A_n$ is the subgroup of the symmetric group of even permutations. It is of order $\frac{n!}{2}$, and is index $2$ in $S_n$. The group is solvable for $n\leq 4$ and simple for $n\geq 5$.
Not referenced anywhere at the moment. |
When there are more than one type of parentheses (say
[] and
()), the problem is NP-complete. The proof is the same as the proof for the palindrome problem except that in the reduction, 0s and 1s in $w$ are changed to
[s and
(s respectively, 0s in $v_i$ are changed to
]s and 1s in $r_k$ are changed to
)s.
When there ar... |
Edit: I'm leaving the old post below, but before I want to write the proof as suggested by Bruce from his book, which uses the ideas in a more efficient way.
Assume that $\|p-q\|<1$, with $p,q\in A$, a unital C$^*$-algebra. Let $x=pq+(1-p)(1-q)$. Then, as $2p-1$ is a unitary, $$\|1-x\|=\|(2p-1)(p-q)\|=\|p-q\|<1.$$So $x... |
To give you a bit of background on this, what you're trying to do is essentially to solve Richardsons first weather prediction problem (which he did by hand on a 3x3 grid laid over central Europe). For more on this you can see wikipedia.
The first weather models used the primitive equations (which are still used for te... |
This is from Nielsen and Chuang. If we have an ensemble of pure states that obey the following relationship for all $i, j$
$\vert \psi_i \rangle = \sum_{j} u_{ij}\vert \phi_j \rangle$
where $u_{ij}$ corresponds to the entries of a unitary matrix, we will obtain the same density matrix for both. That is, we will have
$\... |
Get your free trial content now! Video Transcript Transcript Combining Rational Number Like Terms
We're in Cambodia, located in Southeast Asia, to check in with our friend Laura. Laura’s an archaeologist and she's busy excavating a site that hasn't been touched in hundreds of years. While digging, Laura's discovered lo... |
February 15th, 2019, 01:32 PM
# 1
Banned Camp
Joined: Feb 2019
From: Casablanca
Posts: 23
Thanks: 2
weird series
Hello everyone
Here is a $Z_n$ Suite
$\displaystyle Z_3 = 1/2$
$\displaystyle Z_n = Z_ {n + 1} / \cos ({\pi} / {n})$
It's a bizarre suite decreasing to 0 without ever going.
I will define the n with which I ... |
Conveners Gamma-ray astrophysics: I David Paneque (Max Planck Institute for Physics, Munich) Gamma-ray astrophysics: II Francesca Calore (University of Amsterdam) Gamma-ray astrophysics: III Francesca Calore (University of Amsterdam) Gamma-ray astrophysics: IV David Paneque (Max Planck Institute for Physics, Munich)
Th... |
I'm not getting what acceleration concept is and how it relates to motion and how motion and acceleration can be in different direction? And what's behind the concept of negative and positive acceleration?
Let's say that we move along the straight line. Acceleration shows how fast velocity changes, it doesn't matter ho... |
In this example we use the package to infer the bias and coefficients in a logistic regression model using stochastic gradient Langevin Dynamics with control variates. We assume we have data \(\mathbf x_1, \dots, \mathbf x_N\) and response variables \(y_1, \dots, y_N\) with likelihood \[ p(\mathbf X, \mathbf y | \beta,... |
Get your free trial content now! Video Transcript Transcript Long Division of Polynomials “The World --- will end on ---th 30--! No more --- for ---- of the year! ---”Scientists have intercepted a cryptic radio message and rush to save mankind.Luckily, the scientists’ robot was able to detect a code embedded in the mes... |
Research Open Access Published: Schwarz boundary value problem for the Cauchy-Riemann equation in a rectangle Boundary Value Problems volume 2016, Article number: 7 (2016) Article metrics
912 Accesses
3 Citations
Abstract
In this paper, Schwarz problem for the inhomogeneous Cauchy-Riemann equation in a rectangle is inv... |
The symbol $B$ is explained in greater detail here: Letter codes in molecular term symbols. In this context, it indicates that it is the second excited state of the same multiplicity as the ground state. The ground state, labelled with $X$, is a doublet; the first doublet excited state is labelled with $A$, and the sec... |
Symmetries and Dualities in Name-Passing Process Calculi Abstract
We study symmetries and duality between input and output in the \(\pi \)-calculus. We show that in dualisable versions of \(\pi \), including \(\pi \) and fusions, duality breaks with the addition of ordinary input/output types. We illustrate two proposa... |
Sampling from Phase Space Distributions in 3D Charged Particle Beams
In the previous installment of this series, we explained two concepts needed to model the release and propagation of real-world charged particle beams. We first introduced probability distribution functions in a purely mathematical sense and then disc... |
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1. Measurement of the ratio of the production cross sections times branching fractions of B c ± → J/ψπ ± and B± → J/ψK ± and ℬ B c ± → J / ψ π ± π ± π ∓ / ℬ B c ± → J / ψ... |
Get your free trial content now! Video Transcript Transcript Compound Inequalities
Jerry and Larry want to ride the rollercoaster, but they have a "little big" problem. What does this have to do with
compound inequalities? Let me tell you.
Let’s look at the problem first: You are only allowed to ride the rollercoaster ... |
October 28th, 2018, 09:58 PM
# 1
Newbie
Joined: Jul 2018
From: morocco
Posts: 26
Thanks: 0
Math Focus: algebraic number theory
divisiblity of a binomial coefficient
Hello, how to show the following?? :
$C_{2^{m-1}}^{k}\times 2^k$ is divisible by $2^{m}$.
with for any $k$ such that $2^{m-1}\geq k \geq 1$.
October 29th, ... |
I have to show that, if $W_t$ is a 1-d Brownian motion then $\biggl(W_t, \int_0^t W_s ds\biggr)$ has normal distribution. Hint: apply Ito formula to this bivariate process. Any idea or suggestion on how to solve it?
I tried to show that with characteristic function approach, since the marginal distributions have both n... |
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Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems New general de... |
What is the exact shape of the universe? I know of the balloon analogy, and the bread with raisins in it. These clarify some points, like how the universe can have no centre, and how it can expand equally everywhere in all directions.
But they also raise some questions, like if you are on the surface of a balloon and t... |
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... |
Morias
\(\mathrm{mori}(z)=\displaystyle \frac{ J_0 (L_1 z)}{1-z^2}\)
where \(L_1\approx 2.4\) is first zero of the Bessel function, \(J_0(L_1)\!=\!0~\).
\(u\!+\!\mathrm i v=\mathrm{mori}(x\!+\!\mathrm i y)\)
For integer \(m>0\) and \(|z|\gg 1\), function mori\((z)\) can be approximated with
\(\displaystyle \mathrm{mori... |
You are correct on both questions. 1 you answered yourself. It is the correct rate to close out the trade. 2 you use a dollar discount rate because you are discounting dollars. The (1-K/X) term represents one dollar from the first trade and K/X dollars from the close out trade.
I performed spectral analysis on the stoc... |
In a paper by Joos and Zeh, Z Phys B 59 (1985) 223, they say:This 'coming into being of classical properties' appears related to what Heisenberg may have meant by his famous remark [7]: 'Die "Bahn" entsteht erst dadurch, dass wir sie beobachten.'Google Translate says this means something ...
@EmilioPisanty Tough call. ... |
In Hagan's paper on valuing CMS swaps (Convexity Conundrums: Pricing CMS Swaps, Caps, and Floors), there is:
So the swap rate must also be a Martingale, and
$$E \big[ R_s(\tau) \big| \mathcal{F}_0 \big]=R_s(0) = R_s^0$$
To complete the pricing, one now has to invoke a mathematical model (Black’s model, Heston’s model, ... |
Advanced Studies in Pure Mathematics Adv. Stud. Pure Math. Galois–Teichmüller Theory and Arithmetic Geometry, H. Nakamura, F. Pop, L. Schneps and A. Tamagawa, eds. (Tokyo: Mathematical Society of Japan, 2012), 579 - 600 $n$-nilpotent obstructions to $\pi_1$ sections of $\mathbb{P}^1 - \{0,1,\infty\}$ and Massey product... |
Linear quadratic mean-field-game of backward stochastic differential systems
1.
School of Mathematics, Shandong University, Jinan 250100, China
2.
Zhongtai Securities Institute for Financial Study, Shandong University, Jinan 250100, China
3.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong ... |
I had this question for math $$ \lim_{x\to 0}\frac{\sin{ax^2}}{\sin{bx^2}} $$
So I used squeeze theorem and got $-1<1/(\sin{bx^2})<1$ and I multiplied by $\sin{ax^2}$ and got $-\sin{ax^2}< \sin{ax^2}/\sin{bx^2}< \sin{ax^2}$
Which then equals $0 < \lim_{x\to 0}\sin{ax^2}/\sin{bx^2}<0$ and the answer I got is $0$, but th... |
Suppose $X_1,X_2,\ldots$ is a sequence of Cauchy random variables with density $$f(x)=\frac{1}{\pi(1+x^2)}, \hspace{3mm}x\in \mathbb{R}$$ and let $S_n=X_1+\ldots+X_n$.
It's easy to show that $\frac{S_n}{n}$ converges in distribution using characteristic functions (in fact, $S_n/n$ has the same distribution as $X$ for e... |
Let $A$ be a nilpotent matrix. Prove that $\det(I+A)=1$
Could someone at least give me a clue ?
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
Since $A$ is nilpotent, we ... |
The field you are describing is one possible solution to Maxwell's equations - a plane wave. It's a highly useful solution because all nonevanescent (propagating) fields in freespace / homogeneous mediums can be built from a superposition of plane waves.
But real solutions don't have to (and indeed never do) look like ... |
Let $K \subseteq \mathbb{R}^n$ be a "fat" convex body, i.e. one that contains a ball of radius 1. I'm interested in the following question about points $y \in K$: If you take a normally distributed $e \sim \mathcal{N}(0,\sigma^2)^n$ and add $e$ to $y$, what is the probability that $y + e \in K$? This question can be re... |
Research Open Access Published: The truncation regularization method for identifying the initial value of heat equation on a spherical symmetric domain Boundary Value Problems volume 2018, Article number: 13 (2018) Article metrics
819 Accesses
2 Citations
Abstract
In this paper, identifying the initial value for high d... |
This question already has an answer here:
How do you in general derive a formula for summation of n-squared, n-cubed, etc...? Clear explanation with reference would be great.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes... |
Find distance between intersections of circle and line?
02-07-2019, 03:48 PM (This post was last modified: 02-07-2019 04:56 PM by kevin3g.)
Post: #1
Find distance between intersections of circle and line?
The equations are y=2x+3 and (x+1)^2+y^2=55. I know you can put them in solve and find x and y twice, but is there ... |
Law of Sines Law of Cosines
Used when you know two angles and the included side (ASA), two angles and the non-included side (AAS) or two sides and the non-included angle (SSA).\[ \frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c} \]
Solve by cross multiplying.\[ b\sin{A} = a\sin{B} \]
Solve for a side:\[ b = \fr... |
Let $G=\pi(X,x)$ be the fundamental group of a compact orientable surface of genus $g\ge 2$. It is well known that a presentation of $G$ is $$G=\langle x_1,y_1,\dots,x_g,y_g \ | \ [x_1,y_1]\cdots [x_g,y_g]=1\rangle$$ (where $[x,y]=xyx^{-1}y^{-1}$ is the commutator).
Denote by $F$ be the free group with $2g$ generators ... |
The existing literature on generalized trigonometric functions is scarce and it seems there isn't a comprehensive account of generalized trigonometric functions anywhere. Also there isn't a unified accepted notation and different authors use different notations. The generalized sine and cosine functions $\sin_{pr}x$, $... |
In Hutchings and Taubes lecture note on Seiberg-Witten equation HERE, above equation (4.20) the authors claim that there is a version of Weitzenbock formula reads (where $\beta \in \Omega^{0,2}(M, E)$, M is a symplectic manifold with compatible $J$, $E$ is a line bundle with $U(1)$ connection $a$)
\begin{equation} \int... |
Research Open Access Published: Quasilinear elliptic equations with Hardy terms and Hardy-Sobolev critical exponents: nontrivial solutions Boundary Value Problems volume 2015, Article number: 171 (2015) Article metrics
1039 Accesses
2 Citations
Abstract
In this paper, we obtain one positive solution and two nontrivial ... |
Let $X$ be an affine, complex variety, $A$ be a $\mathbb{C}$-algebra (not necessarily noetherian) and $F_A$ is a coherent sheaf over $X \times \mbox{Spec}(A)$, flat over $\mbox{Spec}(A)$. Denote by $Y \subset X \times \mbox{Spec}(A)$ the scheme-theoretic support of $F_A$. Then, does there exist a non-empty open subset ... |
Abstract
Lingua originale English pagine (da-a) 397-410 Numero di pagine 14 Rivista Dynamic Systems and Applications Volume 22 Stato di pubblicazione Published - 2013 Fingerprint All Science Journal Classification (ASJC) codes Mathematics(all) Cita questo Dynamic Systems and Applications, 22, 397-410.
In: Dynamic Syste... |
Does this definite integral have a closed-form expression?
\begin{align*} I &= \int_0^\infty \sqrt{ \frac{1}{2} \frac{1}{x} \left( \frac{1}{(1+x)^2} + \frac{z}{(1+xz)^2} \right) } \, dx \\ &= \frac{1}{\sqrt{2}} \int_0^\infty \frac{1}{x} \sqrt{ \frac{x}{1+x} \left( 1-\frac{x}{1+x} \right) + \frac{xz}{1+xz} \left( 1-\fra... |
Cleve Moler is the author of the first MATLAB, one of the founders of MathWorks, and is currently Chief Mathematician at the company. He is the author of two books about MATLAB that are available online. He writes here about MATLAB, scientific computing and interesting mathematics.
Denormal floating point numbers and g... |
Suppose that $p$ is a prime with $p \equiv 7 \pmod 8$. If $t = \frac{p - 1}{2}$ , prove that $$2^t \equiv 1 \pmod p.$$
Any hints will be appreciated. Thanks so much.
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... |
$\newcommand{\HH}{\mathbb{H}}$Here is an expansion of what Anton is saying.
Suppose that $M$ is a closed hyperbolic three-manifold. It follows that the universal cover of $M$ is $\HH^3$: hyperbolic space. The covering map of $M$ comes with a deck group - namely there is an action of $\pi_1(M)$ on $\HH^3$ so that the qu... |
It's not just about integrating over wavelengths, it's also about integrating over space.
There are quite a few steps to get from the $L$ you have to the $R$ and I'm stuck on one integral's derivation but here goes:
The radiance shown here is the $\textit{spectral radiance}$ for a $\textit{perfect}$ blackbody ($\epsilo... |
If the natural density of $A = \{a_i\}$ exists, then we can show that it must be zero.
Let $\displaystyle S_{n} = \frac{|A \cup [1,n]|}{n}$
Now $\displaystyle \{\frac{n}{a_n}\}$ is a subsequence of $S_{n}$ and so if the limit is $\displaystyle 2\delta > 0 $ then we have that for all $\displaystyle n > N_0$, $\displayst... |
In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the ground field, which is a number. Such invariants are always multiplicative under disjoint union, this is a consequence of t... |
I have been considering a generalisation of the cosmological process involved in computing the temperature of the cosmic neutrino background.
It is well-known that (simplistically) this temperature $T_{\nu}$ differs from the temperature of the CMB $T_{\gamma}$ by a factor of $(4/11)^{1/3}$ due to the fact that shortly ... |
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