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Optimality conditions for $ E $-differentiable vector optimization problems with the multiple interval-valued objective function
1.
Faculty of Mathematics and Computer Science University of Łódź, Banacha 22, 90-238 Łódź, Poland
2.
Department of Mathematics, Hadhramout University, P.O. BOX : (50511-50512), Al-Mahrah, Ye... |
Table of Contents: Electric Charge Gauss Law Coulombs law Electric Field Intensity Equipotential Surface Electric Potential Energy What is Electrostatics?
Study of stationary electric charges at rest is known as
electrostatics. An electroscope is used to detect the charge on a body. Pith ball electroscope is used to de... |
this is a mystery to me, despite having changed computers several times, despite the website rejecting the application, the very first sequence of numbers I entered into it's search window which returned the same prompt to submit them for publication appear every time, I mean ive got hundreds of them now, and it's stil... |
Fix an integer $k >0 $ and would like to know the maximum number of different ways that a number $n$ can be expressed as a sum of $k$ squares, i.e. the number of integer solutions to $$ n = x_1^2 + x_2^2 + \dots + x_k^2$$ with $x_1 \ge x_2 \ge \dots \ge x_k$ and $x_i \ge 0$ for every $i$.
What I'd really like to know a... |
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... |
In our final section we place stronger restrictions on the choice of potential, we assume that they are Gaussian. Although this appears to be restrictive, we may approximate general potentials by Gaussians. When the potentials are chosen to have this nice form the problem can be rephrased in terms of linear operators, ... |
Difference between revisions of "Main Page"
(→The Problem)
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== Bibliography ==
== Bibliography ==
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# H. Furstenberg, Y. Katznelson, “[http://math.stanford.edu/~katznel/hj43.pdf A density version of the Hales-Jewett theorem for k=3]“, Graph Theory and Combinatorics (Cambridge, 1988). Di... |
also, if you are in the US, the next time anything important publishing-related comes up, you can let your representatives know that you care about this and that you think the existing situation is appalling
@heather well, there's a spectrum
so, there's things like New Journal of Physics and Physical Review X
which are... |
I came up with this proof for my number theory class. Is it valid?
Proposition: $u\in U_m \Rightarrow u^{\varphi(m)}=1$ (Where $U_m$ is the multiplicative group of integers modulo $m$) Attempted Proof: Lemma:$\forall a,b\in U_m$, $(a\cdot b)\in U_m$
Consider two numbers $a,b\in U_{m}$. Being in $U_m$, we know that they... |
A Matrix Commuting With a Diagonal Matrix with Distinct Entries is Diagonal
Problem 492
Let\[D=\begin{bmatrix}d_1 & 0 & \dots & 0 \\0 &d_2 & \dots & 0 \\\vdots & & \ddots & \vdots \\0 & 0 & \dots & d_n\end{bmatrix}\]be a diagonal matrix with distinct diagonal entries: $d_i\neq d_j$ if $i\neq j$.Let $A=(a_{ij})$ be an $... |
Hey guys! I built the voltage multiplier with alternating square wave from a 555 timer as a source (which is measured 4.5V by my multimeter) but the voltage multiplier doesn't seem to work. I tried first making a voltage doubler and it showed 9V (which is correct I suppose) but when I try a quadrupler for example and t... |
I am studying about the moduli space of a 2 monopole system from Harvey's notes, and Manton's paper. In both of these, (Harvey section 6.2), after constructing the Lagrangian for a two dyons system, the author replaces the electric charge $e$ with $\dot{\chi}$. $$e \to \dot{\chi}$$ to obtain the the equation of motion ... |
After delivering a short introductory paragraph, Rubini delved right into formulas for special determinants and relations between them. Intriguingly, he did this without providing the reader with any of the basic properties of determinants or any explanation of his notation. In order for his readers to understand this ... |
Seminar Number Theory 摘要 报告人简介 2019-10-14
Title: Cycles on Shimura
varieties via Geometric Satake
Speaker: Liang Xiao (Peking University)
Abstract: I will explain a
joint work with Xinwen Zhu on constructing algebraic cycles on special fibers
of Shimura varieties using geometric Satake theory. The talk will focus on
ex... |
An Example of a Linear Operator with a Closed Graph that is Unbounded
Recall from The Closed Graph Theorem that if $X$ and $Y$ are Banach spaces and if $T : X \to Y$ is a linear operator then $T$ is bounded if and only if $\mathrm{Gr}(T)$ is closed, that is, if $(x_n)$ is a sequence in $X$ that converges to $x \in X$ a... |
I'm confused about the definitions of black-box zero knowledge and non-black-box zero knowledge. I have searched and found explanations but still a bit confused about it. From the context, their definition (with auxiliary input) of black-box zero knowledge proof system between the prover $P$ and (malicious) verifier $V... |
My main problem is that I'm unsure whether I can add
cosfi1 + cosfi2=cosfi. I calculated them both, one for RC and the second for RL.
Here is my overall (exhaustive) attempt on this problem (please correct if you think something is wrong). Data given: $$R_1=20\Omega;R_2=3\Omega;L=12.73*10^{-3}H;C=212.2*10^{-6} F; f=50H... |
Compact Sets in Hausdorff Topological Spaces
Recall from the Compactness of Sets in a Topological Space page that if $X$ is a topological space and $A \subseteq X$ then $A$ is said to be compact in $X$ if every open cover of $A$ has a finite subcover.
On the Closed Sets in Compact Topological Spaces page we proved a ve... |
Assume $M$ to be a compact $n$-dimensional manifold, endowed with a complete metric.
Let us consider the space $C^\infty(M)$ endowed with the standard $C^\infty$ topology, i.e. generated by the seminorms $$\sup_K\left|\frac{\partial}{\partial_x^\alpha}f\right|,$$ over the compact subsets of $M$. Now consider the follow... |
I was reading the Chapter 14 of the textbook by Philip Phillips on Advanced Solid State Physics, when he introduced the mean-field treatment of the quantum rotor model, he used the method first used in this paper: Quantum fluctuations in two-dimensional superconductors.
Here my question is related to the Hubbard-Strato... |
I actually don't think that this view of light being in a quantum superposition is anything new: what Discover magazine is describing (I believe) is the
stock standard picture of how one would describe a system of cells, molecules, chloroplasts, fluorophores, whatever interacting with the quantised electomagnetic field... |
Glioblastoma, or glioblastoma multiforme, is a particularly aggressive and almost invariably fatal type of brain cancer. It is infamous for causing the deaths of U.S. Senators John McCain and Ted Kennedy, as well as former U.S. Vice President Joe Biden’s son Beau. Though glioblastoma is the second-most common type of b... |
I'm building regression models. As a preprocessing step, I scale my feature values to have mean 0 and standard deviation 1. Is it necessary to normalize the target values also?
Let's first analyse why feature scaling is performed. Feature scaling improves the convergence of steepest descent algorithms, which do not pos... |
We claim that the vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r, \mathbf{v}_{r+1}$ are linearly dependent.Since the vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r$ are linearly dependent, there exist scalars (real numbers) $a_1, a_2, \dots, a_r$ such that\[a_1 \mathbf{v}_1+a_2\mathbf{v}_2+\cdots +a... |
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J/Ψ production and nuclear effects in p-Pb collisions at √sNN=5.02 TeV
(Springer, 2014-02)
Inclusive J/ψ production has been studied with the ALICE detector in p-Pb collisions at the nucleon–nucleon center of mass energy √sNN = 5.02TeV at the CERN LHC. The measurement is performed in... |
In this article I will try to introduce the complete derivation behind the
Kalman Filter, one of the most popular filtering algorithm in noisy environments. In addition, the following article will be about the Extended Kalman Filter, how it’s used in localisation algorithms, when we have known and unknown correspondenc... |
Bernoulli actions of type III
Speaker：Stefaan Vaes
Time： Fri 16:00-17:00, 2019-9-13
Venue：Lecture Hall, Jin Chun Yuan West Bldg.
In this lecture, I focus on ergodic theory of group actions. We consider the translation action of a discrete group G on the product space {0,1}^G equipped with the product of probability mea... |
I heard a couple of times that there is no dynamics in 3D (2+1) GR, that it's something like a topological theory. I got the argument in the 2D case (the metric is conformally flat, Einstein equations trivially satisfied and the action is just a topological number) but I don't get how it is still true or partially simi... |
Under the auspices of the Computational Complexity Foundation (CCF)
We revisit the problem of hardness amplification in $\NP$, as
recently studied by O'Donnell (STOC `02). We prove that if $\NP$ has a balanced function $f$ such that any circuit of size $s(n)$ fails to compute $f$ on a $1/\poly(n)$ fraction of inputs, t... |
Following the ideas in the suggested reading by Trurl, here is an outline of how to go about it in your case. The main complication in comparison with the linked random walk question is that the backward step is not $-1$. I'm going to go ahead and divide your amounts by 6 to simplify them, so if you win, you win $1$, a... |
The Real Number Line
One way to represent the real numbers $\mathbb{R}$ is on the real number line as depicted below.
We will now state the important geometric representation of the absolute value with respect to the real number line.
Definition: If $a$ and $b$ are real numbers, then we say that the distance from $a$ t... |
NTS ABSTRACTSpring2019
Return to [1]
Contents Jan 23
Yunqing Tang Reductions of abelian surfaces over global function fields For a non-isotrivial ordinary abelian surface $A$ over a global function field, under mild assumptions, we prove that there are infinitely many places modulo which $A$ is geometrically isogenous ... |
LPC Hands-on Advanced Tutorials Session (HATS) on MET Instructions for HATS on MET at FNAL LPC in March 2014 Contact:
Tai Sakuma
Introduction
This topic includes instructions for LPC HATS on MET, which takes place in March 2014.
Hands-on 1: Check out CMSSW and MET recipe Computing Environment at FNAL LPC
This exercise ... |
The Fundamental Theorem of Algebra
Recall from the Properties of Polynomials page that a polynomial is a function in the form $p(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$, and that if $a_n \neq 0$ then $\mathrm{deg} (p) = n$. We will now look at some more theorems regarding polynomials, the first of which is extremely i... |
Contraction and Dilation Transformation Operators
We will now begin to look at some more interesting aspects of matrices and vectors. One such use arises in linear transformations or linear maps. We will look more into these later on, but for now, we will outline a few common linear transformations before moving onto a... |
Algebraic Geometry Seminar Spring 2017
The seminar meets on Fridays at 2:25 pm in Van Vleck B113.
Contents Algebraic Geometry Mailing List Please join the AGS Mailing List to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department com... |
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... |
Having had enough of the different forms of Riemann Curvature Tensor, I finally decided to recommend the “most correct” one here.
In mathematics books, we say \(\text{Operator }R:\Gamma(TM)\times\Gamma(TM)\times\Gamma(TM)\rightarrow\Gamma(TM)\) is \(\text{the curvature of connection } D\) (Here connection is a general ... |
Background
When labels are shown with feynmp, they extend beyond the box of the graph. One can use
\fmfframe to allocate an invisible frame to accommodate these, but it seems like you need to manually adjust all of the parameters.
Question Is there an easy way of automatically placing ones Feynman graphs in an appropri... |
Given that for a known mean $\mu$ and unknown variance $\sigma^2$ the normal distribution is
$$X_i|\sigma^2 \sim \mathcal{N}(\mu, \sigma^2) = \frac{1}{\displaystyle\sigma\sqrt{2\pi}}\exp\left[-\displaystyle\frac{1}{2}\left(\frac{x_i - \mu}{\sigma}\right)^2\right]$$
$$\sigma^2 \sim \text{Inv-}\chi^2(\nu_p, \sigma_p^2).$... |
Evaluating Limits of Sequences
We will now look at some examples of evaluating limits of sequences using various approaches and theorems that we have already looked at. More examples can be found on the Evaluating Limits of Sequences Examples 1 and Evaluating Limits of Sequences Examples 2 pages.
Example 1 Evaluate the... |
Why negative interest rates might not work Matthew Martin9/04/2014 01:58:00 PM
Tweetable
1so that we can eliminate liquidity traps by implementing negative interest rates. Here's why I'm uncertain whether negative interest rate policy is expansionary.
To start, let's go through the standard logic: the fisher equation r... |
The only trick here is getting used to how discrete sums are turned into integrals.
Suppose you let energy be a function of momentum $p$ and position $q$. Then you can rewrite the discrete quantum partition function as
$Z_{quantum}=\sum_{p,q}e^{- \beta E(p,q)},$
where the sum is over each of the $N$ positions and $N$ m... |
First, notice that, if you use some properties of the trace operator,
\begin{align*}p(\mu, \Sigma) &\propto \lvert \Sigma\rvert^{-((\nu_0+d)/2+1)}\exp\Big(-\frac{1}{2}\text{tr}(\Lambda_0\Sigma^{-1})-\frac{\kappa_0}{2}(\mu-\mu_0)'\Sigma^{-1}(\mu-\mu_0)\Big) \\&= \lvert \Sigma\rvert^{-((\nu_0+d)/2+1)}\exp\Big(-\frac{1}{2... |
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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 ... |
The term functional is used in at least two different meanings.
One meaning is in the mathematical topic of functional analysis, where one in particular studies linear functionals.This meaning is not relevant for the discussion at page 299 in Ref. 1.
Another meaning is in the topics of calculus of variations and (class... |
If a body is acted on by a force which obeys the inverse square law, will it mean the body follows an ellipse, if so, equating Newton's law of universal gravitation $(f=(GMm/(r^2)))$ to centripetal force $(mv^2)/r$ is erroneous.
Yes, if you are considering uniform circular motion, then you can use $$ mv^2\,r^{-1}=GMm\,... |
What is an assumption of a statistical procedure?
I am not a statistician and so this might be wrong, but I think the word "assumption" is often used quite informally and can refer to various things. To me, an "assumption" is, strictly speaking, something that only a theoretical result (theorem) can have.
When people t... |
Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms
1.
School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China
2.
Department of Mathematics and Computer Science, John Jay College of Criminal Justice, CUNY, New York, NY ... |
Tagged: determinant of a matrix Problem 718
Let
\[ A= \begin{bmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{bmatrix} . \] Notice that $A$ contains every integer from $1$ to $9$ and that the sums of each row, column, and diagonal of $A$ are equal. Such a grid is sometimes called a magic square.
Compute the determinan... |
+Corollaries to the Equivalence of Norms in a Finite-Dimensional Linear Space]
Recall from the Equivalence of Norms in a Finite-Dimensional Linear Space page that if $X$ is a finite-dimensional linear space then any two norms $\| \cdot \|_1$ and $\| \cdot \|_2$ on $X$ are equivalent, that is, there exists positive numb... |
I'm looking for strategies for evaluating the following sums for given $z$ and $m$: $$ \mathcal{S}_m(z):=\sum_{n=1}^\infty \frac{H_n^{(m)}z^n}{n}, $$ where $H_n^{(m)}$ is the generalized harmonic number, and $|z|<1$, $m \in \mathbb{R}$.
Using the generating function of the generalized harmonic numbers, an equivalent pr... |
I have asked about this before and have really been struggling with identifying what makes a model parameter and what makes it a latent variable. So looking at various threads on this topic on this site, the main distinction seems to be:
Latent variables are not observed but have an associated probability distribution ... |
I came across the following in Kevin Murphy's "a probabilistic perspective on machine learning". I am struggling to understand the derivation of the conditional probability for $z_i$. I tried different forms of Bayes' formula try to derive this line (24.10), but failed so far. any hints much appreciated
$z_i$ is the in... |
There are trivial examples that arise just by taking products of irreducible Einstein metrics with different Einstein constants. Whether an irreducible example exists is a much harder question. I do not know the answer to that, but I can think about it. (
However, see below, where I answer this question.)
Of course, no... |
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Now showing items 1-10 of 26
Production of light nuclei and anti-nuclei in $pp$ and Pb-Pb collisions at energies available at the CERN Large Hadron Collider
(American Physical Society, 2016-02)
The production of (anti-)deuteron and (anti-)$^{3}$He nuclei in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV has ... |
GolfScript (23 chars)
{:^((1${\.**2^?%}+*}:f;
The sentinel result for a non-existent inverse is
0.
This is a simple application of Euler's theorem. \$x^{\varphi(2^n)} \equiv 1 \pmod {2^n}\$, so \$x^{-1} \equiv x^{2^{n-1}-1} \pmod {2^n}\$
Unfortunately that's rather too big an exponential to compute directly, so we have... |
While computer simulations have a wide range of uses, their goals are generally similar: find the simplest model that recreates the properties of the system under investigation. For scientific systems, this involves matching observed or experimental phenomena as precisely as necessary.
But what about movie simulations?... |
The Radius of Convergence of a Power Series
Recall from the Power Series page that we saw that a power series will converge at it's center of convergence $c$, and that it is possible that a power series can converge for all $x \in \mathbb{R}$ or on some interval centered at the center of convergence. If a power series ... |
I've been struggling with a detail in Second Quantization which I really need to clear out of my head. If I expand the S-matrix of a theory with an interaction Hamiltonian $ H_I(x) $ then I have
$$ S - 1= \int^{+\infty}_{-\infty} d^4x\, H_I(x) + \int^{+\infty}_{-\infty} \int^{+\infty}_{-\infty} d^4 x\, d^4 y\, T[ H_I(x... |
The constrained CSBM definition is as follows. There is a distribution $D = D_x \times D_{\omega|x} \times D_{r|\omega,x}$, where $r: A \to [0, 1] \cup \{ -\infty \}$ takes values in the unit interval augmented with $-\infty$, and the components of $r$ which are $-\infty$-valued for a particular instance are revealed a... |
I am a first year student and a learner of hyperbolic geometry. I was wondering if you could suggest some exciting topics to research about in this field (some people suggested fundamental polygons and areas of hyperbolic triangles).
Any other exciting topics to be suggested? I am a first year student, but I don't mind... |
The setting is basically earth. Our planet is unfortunately on a collision course with a large asteroid. However, humans have discovered and decoded a message from an ancient, advanced alien race (our parents). These aliens have left behind technology and knowledge that could save the planet, but this technology is enc... |
Group Homomorphisms
Definition: Let $(G, \cdot)$ and $(H, *)$ be two groups. A Homomorphism between these groups is a function $f : G \to H$ such that for all $x, y \in G$ we have that $f(x \cdot y) = f(x) * f(y)$.
Definition: Let $(G, \cdot)$ and $(H, *)$ be two groups. Then: 1) A Monomorphism from $G$ to $H$ is a hom... |
Linear Independence and Dependence Examples 1
Recall from the Linear Independence and Dependence page that a set of vectors $\{ v_1, v_2, ..., v_n \}$ is said to be
Linearly Independent in $V$ if the vector equation $a_1v_1 + a_2v_2 + ... + a_nv_n = 0$ implies that $a_1 = a_2 = ... = a_n = 0$, that is, the zero vector ... |
Moscow-Beijing topology seminar Speaker Introduction 2019-10-30
Title: Stringc structures
and modular invariants
Speaker: 黄瑞芝 Huang Ruizhi （中科院）
Abstract: Spin structure and
its higher analogies play important roles in index theory and mathematical
physics. In particular, Witten genera for String manifolds have nice ge... |
The degeneracy or non-degeneracy of these states depends on the problem's hamiltonian as well as on the system's Hilbert space, so this question doesn't really have an answer. However, the angular momentum states will typically be degenerate in the sense that multiple (linearly independent) states will have the same an... |
This page on Wikipedia -- Quasars mentions that the "The largest known [quasar] is estimated to consume matter equivalent to 600 Earths per minute". However, there is no citation for this comment. How can I find out where this information came from? I've commented in the Talk section for the page.
Tricky to say for sur... |
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... |
Let $X$ be a Banach space and $B(X)$ be its space of all (bounded) operators. A
nuclear functional on $B(X)$ is a linear functional $u:B(X)\to{\mathbb C}$ that can be represented in the form$$u(A)=\sum_{n=1}^\infty \lambda_n\cdot f_n(Ax_n),\qquad A\in B(X),$$where $\lambda_n\in{\mathbb C}$, $x_n\in X$, $f_n\in X^*$ are... |
I'm training a neural network (or any ML model with non-convex gradient-based optimization) to predict a continuous outcome variable. Currently, I use the mean squared error loss function, i.e., if $y$ is the true outcome and $\hat{y}$ is the model prediction, I minimize the expected loss $$\text{E}[(y-\hat{y})^2]$$
Ho... |
Current browse context:
cs.IT
Change to browse by: References & Citations Bookmark(what is this?) Computer Science > Information Theory Title: Constructions of transitive latin hypercubes
(Submitted on 28 Feb 2013 (v1), last revised 28 Aug 2019 (this version, v3))
Abstract: A function $f:\{0,...,q-1\}^n\to\{0,...,q-1\}... |
Another Comparison Theorem for Integrals of Step Functions on General Intervals
Recall from the The Limit of the Integral of a Decreasing Sequence of Nonnegative Step Functions Approaching 0 a.e. on General Intervals page that if $(s_n(x))_{n=1}^{\infty}$ is a decreasing sequence of nonnegative step functions that conv... |
Introduction to Sound waves
Sound is a form of energy arising due to mechanical vibrations. Hence sound waves require a medium for its propagation sound cannot travel a medium for its propagation. Sound cannot travel in vacuum. The sound waves are propagated as longitudinal mechanical waves through solids liquids and g... |
Segment of a Circle: A region bounded by a chord and a corresponding arc lying between the chord’s endpoints is known as segment of a circle. It is to be noted that the segments do not contain the center point. Definition:
The segment of a circle divides it into two region namely major segment and minor segment. The se... |
In this MathStackExchange post the question in the title was asked without much outcome, I feel.
Edit: As Douglas Zare kindly observes, there is one more answer in MathStackExchange now.
I am not used to basic Probability, and I am trying to prepare a class that I need to teach this year. I feel I am unable to motivate... |
Table of Contents
The Closure of a Set and the Distance from Points to a Set
Recall from the The Distance Between Points and Subsets in a Metric Space page that if $(S, d)$ is a metric space, $A \subseteq S$, and $x \in S$ then we can define a function $f_A : S \to \mathbb{R}$ for all $x \in S$ by:(1)
We said that the ... |
Given an abelian variety $A$ defined over $\mathbb{Q}$, for a positive integer (we can suppose prime) $\ell$, let $A[\ell]$ denote the groupof points of $A$ that are annihilated by $\ell$, the division field $\mathbb{Q}(A[\ell])$ is obtained by adjoining to $\mathbb{Q}$ the
coordinates of the points of $A[k]$.
Why the ... |
Considering that the top/truth quark is the only quark with higher mass than the massive bosons, is the W boson in its decay different than the off-shell bosons that mediate the weak decay of other particles?
My usual disclaimer: I use these questions as a way of learning for myself about different aspects of physics, ... |
So i'm asked to calculate the generating function $Z(J)$ for a Lagrangian density $$L = -\left( \partial \phi \right) ^2 + m^2\phi^2 + f\left(x\right) \phi$$ for a fixed function $ f(x) $ , and express it in terms of the usual propagator and $f$ in the case $f=0$. My problem is in the role of the function $f(x)$.
We ar... |
Mathematics is rapidly growing as old subfields are deepened and new subfields are created. It is simultaneously integrating, as direct connections are found between subfields previously regarded as distant. It is dramatically increasing its role in other disciplines, both in science and beyond. These are exciting time... |
Ancillary-file links:
Ancillary files (details): python3_src/KW_scaldims.py python3_src/LICENSE python3_src/README python3_src/TNR.py python3_src/cdf_ed_scaldimer.py python3_src/cdf_scaldimer.py python3_src/custom_parser.py python3_src/ed_scaldimer.py python3_src/initialtensors.py python3_src/modeldata.py python3_src/p... |
The Mean Value Theorem
The Mean Value Theorem
We are now going to look at a very important theorem in Calculus known as The Mean Value Theorem.
Theorem 1 (The Mean Value Theorem): If $f$ is a function that satisfies the following conditions: a) $f$ is a continuous function on the closed interval $[a, b]$. b) $f$ is dif... |
I know asking for proof-verification on MO is a tricky thing. On one hand interesting research level proofs are usually subject of articles and can not be discussed here in detail. On the other hand most simple proof which can be written on a forum are not "high-level" enough for MO and other places are more appropriat... |
On the monotonicity of the period function of a quadratic system
1.
Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275
$ \dot x=- y + x y,\quad \dot y=x + 2 y^2-c x^2, \quad -\infty < c < +\infty.$
We show that this system has two isochronous centers for $c=1/2$, and its period function has only one c... |
HESS upper limits for Kepler's supernova remnant Date2008 Author
Büsching, I.
De Jager, O.C.
Holleran, M.
Raubenheimer, B.C.
Venter, C.
H.E.S.S. Collaboration
MetadataShow full item record Abstract
Aims. Observations of Kepler's supernova remnant (G4.5+6.8) with the HESS telescope array in 2004 and 2005 with a total li... |
Performing Topology Optimization with the Density Method
Engineers are given significant freedom in their pursuit of lightweight structural components in airplanes and space applications, so it makes sense to use methods that can exploit this freedom, making topology optimization a popular choice in the early design ph... |
We claim that the statement is false.As a counterexample, consider the matrices\[A=\begin{bmatrix}1 & 0\\0& 0\end{bmatrix} \text{ and } B=\begin{bmatrix}0 & 0\\0& 1\end{bmatrix}.\]Then\[A+B=\begin{bmatrix}1 & 0\\0& 1\end{bmatrix}\]and we have\[\det(A+B)=\begin{vmatrix}1 & 0\\0& 1\end{vmatrix}=1.\]
On the other hand, th... |
Lower central quotients can be extracted from group homology via spectral sequence built up from free simplicial resolution of a group. So, if your complex variety is aspherical, you probably know those Hodge structures because everything is as natural and functorial as it can be.
By a classical result of Magnus, for f... |
I have a basic confusion concerning the mean-field theory of quantum phase transitions in Fermi systems. Consider as an example the BCS theory of superconductivity in a Dirac fermion system, considered by Sachdev in Sec. 17.1 of his QPT book. The variational ground state energy is given by
\begin{align} E_{BCS}=\frac{J... |
On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms
1.
Department of Mathematics and Statistics, Federal University of Campina Grande, 58109-970, Campina Grande, PB, Brazil
2.
Department of Mathematics - State University of Maringá, 87020-900 ... |
Step Functions
We will soon look at Riemann-Stieltjes integrals where the integrator $\alpha$ is a step function, however, we will first need to formally define what exactly a step function is.
Definition: A Step Function $\alpha$ on the interval $[a, b]$ is a piecewise constant function containing finitely many pieces... |
Chamley-Judd revisited Matthew Martin9/17/2014 04:52:00 PM
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If 1+1=3, then 2=1.This is a valid theorem. We can prove it: we know that 2=3-1, and it is postulated in the theorem that 3=1+1, therefore 2=1+1-1, which implies 2=1. Neither the postulate nor the conclusion are factually correct, but the theorem is n... |
A
Porous medium can be defined to specify porousity characteristics of the computational domain.
To model a porous mediuam, navigate to
Advanced Concepts and under Porous Media add a new porous medium. The following models are supported:
This porousity model takes non-linear effects into account by adding inertial term... |
Compact Sets in Metric Spaces are Complete
Recall from the Complete Metric Spaces page that a metric space $(M, d)$ is said to be complete if every Cauchy sequence in $M$ converges in $M$. Furthermore, if $S \subseteq M$ then we said that $S$ is complete if every Cauchy sequence in $S$ converges in $S$.
We will now loo... |
Before looking at the perimeter and area of a circle, a basic understanding of perimeter and area is needed.
Perimeter is associated with any closed figure like triangle, quadrilateral, polygons or circles. It is the measure of distance covered while going around the closed figure on its boundary.
For example, the peri... |
Let $X$ and $Y$ be topological spaces. A function $f: X \rightarrow Y$ is defined as continuous if for each open set $U \subset Y$, $f^{-1}(U)$ is open in $X$. This definition makes sense to me when $X$ and $Y$ are metric spaces- it is equivalent to the usual $\epsilon-\delta$ definition. But why is this a good definit... |
I will try to answer as many questions as I can. I won't presume to give you complete exhaustive answers, but maybe they will be nonetheless useful to you.
What variables determine the range of temperatures over which matter is liquid? My understanding of thermodynamics is that matter changes from a solid to a gas when... |
A quantum time crystal does not from my reading of Wilczek and others appear to require entanglement, but the idea is interesting. It does seems plausible that a quantum time crystal model could be developed with entanglement.
A quantum time crystal is just a periodicity in a system in time that has a lattice structure... |
The force you experience is of the form $\vec{F} = - Gmr\vec{u_r}$, and we also know that in the surface, $r=R$, it is $\vec{F}=- gm\vec{u_r}$, so
$$\vec{F} = -gm\frac{r}{R}\vec{u_r}$$
This is a conservative force that can be derived from a potential
$$U = \frac{1}{2}gm\frac{r^2}{R}$$
Because this is a central force, a... |
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