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In this chapter we learned about left and right adjoints, and about joins and meets. At first they seemed like two rather different pairs of concepts. But then we learned some deep relationships between them. Briefly:
Left adjoints ... |
We have a sequence $a_1,a_2,...,a_n$ and $\forall i\in N;~~~ a_i\in\{1,2,3,...,n\}$. A sequence is $GOOD$ if we have that: For Every $k \in N$, we have $a_k≥a_i$ for every $i<k$, or $a_k≤a_i$ for every $i<k$. it means that every element is larger than all previous elements or smaller than all previous elements.
We want... |
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Tue, 23 Jul 2019
About ten years ago I started an article, addressed to my younger self, reviewing various books in category theory. I doubt I will ever publish this. But it contained a long, plaintive digression about Categories, Allegories by Peter Freyd and Andre Scedrov:
In lig... |
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Isomorphisms are very important in mathematics, and we can no longer put off talking about them. Intuitively, two objects are 'isomorphic' if they look the same. Category theory makes this precise and shifts the emphasis to the 'iso... |
tNIRS-1 非侵襲脳酸素モニタ
従来法（MBL法、SRS法）は、透過光の光強度を測定し、その情報から酸素飽和度などを算出しますが、測定部位の形状やプローブの装着状態の違いに影響されやすく、定量性の点で応 用に限界がありました。
TRS法は、生体に照射する光に短パルス光 を用い、透過光の時間応答波形を測定します。測定部位の形状やプローブの装着状態に影響されにくく、酸素飽和度などを精度良く安定的に測定できます。特に定量性・再現性に優れているため、日をまたいだデータの比較も可能になります。
$$R(t,\mu_a,\mu'_s) =\Biggl(\frac{4\pi c}{3\mu'_s}\Biggr)^{-\frac{3}{2}}\... |
The cross section method for determining triple integral bounds
One tricky part of calculating define triple integrals is determining the bounds of integration. To help you in this endeavor, we outline a couple methods for reducing the task of finding these bounds to the simpler task of finding bounds on a double integ... |
Principle of Mathematical Induction Contents 1 Theorem 2 Proof 3 Contexts 4 Terminology 5 Informal Analogy 6 Warning 7 Also defined as 8 Also known as 9 Also see 10 Historical Note 11 Sources Theorem
Let $\map P n$ be a propositional function depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that: $(1): \quad... |
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In this chapter we learned about left and right adjoints, and about joins and meets. At first they seemed like two rather different pairs of concepts. But then we learned some deep relationships between them. Briefly:
Left adjoints ... |
I am interested in modeling the following experiment:
A binomial trial with $n$ Bernoulli experiments is run.For the k positive outcomes a second (independent) binomial trial with $k$ runs with a
different success probability is run.
For example: I throw $n$ darts with probability $p_1$ of hitting the bull's eye. The n... |
What you wrote is an expectation value, which means an average on some state $|\psi\rangle$ over all possible eigenvalues of the operator under analysis, weighted with the probability of that eigenvalue occurring on that state $\psi$.
So, yes, $\hat H$ is an observable (we reserve this term for operators and the quanti... |
Necessary Condition for Twice Differentiable Functional to have Minimum Theorem
Let $J\sqbrk y$ be a twice differentiable functional.
Let $\delta J \sqbrk {\hat y; h} = 0$.
Suppose, for $y=\hat y$ and all admissible $h$
$\delta^2 J\sqbrk{y;h}\ge 0$
Then $J$ has a minimum for $y=\hat y$ if
Proof
By definition, $\Delta J... |
The terms
heavy and light are commonly used in two different ways. We refer to weight when we say that an adult is heavier than a child. On the other hand, something else is alluded to when we say that oak is heavier than balsa wood. A small shaving of oak would obviously weigh less than a roomful of balsa wood, but oa... |
I'm looking for the right argument why the function $ \cos\sqrt z$ is analytic on the whole complex plane. As far as I understand, a holomorphic branch of $\sqrt z$ can only be found on the cut plane (without negative numbers) since the Argument function isn't continuous everywhere. Hence $ \cos\sqrt z$ is at least hol... |
The rubber duck explains coherence spaces
I've spent a chunk of the past week, at least, trying to understandthe idea of a
coherence space (or coherent space). This appearsin Jean-Yves Girard's Proofs andTypes, and it's amodel of a data type. For example, the type of integers and the typeof booleans can be modeled as c... |
For discussion of specific patterns or specific families of patterns, both newly-discovered and well-known.
gmc_nxtman Posts: 1147 Joined: May 26th, 2015, 7:20 pm Kazyan wrote:
Component found in a CatForce result:
Code: Select all
x = 42, y = 67, rule = LifeHistory
A$.2A$2A5$7.A$8.2A$7.2A19$24.2A$24.2A2$39.A$37.A3.A$3... |
I am preparing for my exam in Formal languages and Automata theory and I'm looking at some old exam questions right now. I need help with the following question:
For each of the following languages answer whether it is regular, context-free but not regular, or not context-free. A brief, informal explanation is sufficie... |
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Tue, 31 Oct 2017
[ The Atom and RSS feeds have done an unusually poor job of preserving the mathematical symbols in this article. It will be much more legible if you read it on my blog. ]
Lately I've been enjoying
He continues a little later:
As you can see, it is not written in th... |
MathRevolution wrote:
Attachment:
The attachment
GEOMETRY.jpg is no longer available
If a smaller circle is inscribed in an equilateral triangle and a lager circle circumscribed about the triangle shown as above figure, what is the ratio of the smaller circle’s area to the larger circle’s area?
A. 1:2
B. 1:√3
C. 1:3
D.... |
I have searched but have not found a solution to the following problem. I am trying to shift the location of a dot accent over a greek letter using the
textgreek package. However, the dot is shifted left of the letter. I've seen solutions to solve this problem in math mode, however I would like to use the greek letter ... |
Given a distribution $\mu$ on a finite set, let us denote by $T(\mu)$ the average depth of a leaf in a Huffman tree of $\mu$ (depth is measured by the number of edges from root to leaf); we assume that no element has zero probability. Then$$H(\mu) \leq T(\mu) < H(\mu)+1,$$where $H(\mu) = \sum_i \mu_i \log_2 (1/\mu_i)$ ... |
I'm trying to get consistent normals along a 3D Bezier curve $B(t)$, where for any point I compute the normal as:
$$ \begin{align} \vec{a} &= B'(t) \\ \vec{b} &= B''(t) \\ \vec{c} &= \vec{a} + \vec{b} \\ \vec{r} &= \vec{c} × \vec{a} \\ \vec{n} &= \vec{r} × \vec{a} \\ \end{align} $$
So, get the derivative at a point for... |
As you are confused let me start by stating the problem and taking your questions one by one. You have a sample size of 10,000 and each sample is described by a feature vector $x\in\mathbb{R}^{31}$. If you want to perform regression using Gaussian radial basis functions then are looking for a function of the form $$f(x... |
Let's review the basic concept of a confidence interval.
Suppose we want to estimate an actual population mean \(\mu\). As you know, we can only obtain \(\bar{x}\), the mean of a sample randomly selected from the population of interest. We can use \(\bar{x}\) to find a range of values:
\[\text{Lower value} < \text{popu... |
In 1998 C. Cachin proposed an information-theoretic approach to steganography. In particular, in the framework of this approach, so-called perfectly secure stegosystems were defined, where messages that carry and do not carry hidden information are statistically indistinguishable. There was also described a universal s... |
I have the following proof so far:
In step 9 I'm not sure how to prove P from the steps I have before. I thought that I could use ∨ Elim but I don't think I can now.
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... |
Solutions to elementary discrete dynamical systems biology problems
The following is a set of solutions to the elementary discrete dynamical systems biology problems. Let us know if you have a better solution.
Let $t$ be time in years. Let $m_t =$ the mass of the fish in grams in year $t$. The dynamical system where th... |
Assume we have the following setup:
A client with trusted storage and computing capabilities (e.g. a smartcard) A server with trusted computing and short-term storage capabilities (e.g. RAM + CPU, possibly with something like Intel SGX). The server has no trusted large-scale long-term storage capabilities and may only ... |
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Isomorphisms are very important in mathematics, and we can no longer put off talking about them. Intuitively, two objects are 'isomorphic' if they look the same. Category theory makes this precise and shifts the emphasis to the 'iso... |
The easiest way which is widely known to calculate a modular inverse is finding the smallest $k\in\mathbb N$ such that the following expression is an integral integer:
$$\frac{1+k\cdot \varphi}{e}$$
If this is an integral integer, it is the inverse of $e$ modulo $\varphi$ and thus $d$ and you'll find it with at most $e... |
This is a side question which is more motivated by teaching than research.
First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" quantities; on the other hand, decimal expansions -- especially infinite -- are more likely to be series).
Secondly, is it na... |
Main Page The Problem
Let [math][3]^n[/math] be the set of all length [math]n[/math] strings over the alphabet [math]1, 2, 3[/math]. A
combinatorial line is a set of three points in [math][3]^n[/math], formed by taking a string with one or more wildcards [math]x[/math] in it, e.g., [math]112x1xx3\ldots[/math], and repl... |
How can I prove that non-regular languages are closed under concatenation using only the non-regularity of $L=\{a^nb^n|n\ge1\}$ ?
You can't prove it because it isn't true: the class of non-regular languages isn't closed under concatenation.
Let $X\subseteq \mathbb{N}$ be any undecidable set containing $1$ and every eve... |
Let $\pi$ is a $\Bbb{P}$-name for a partial order, i.e. there is a name $\pi'$ and $\pi''$ such that $$1\Vdash_\Bbb{P} \pi '' \in \pi\land (\text{$\pi'$ is a partial order of $\pi$ with largest element $\pi''$}).$$ We call $\pi$ is full for $\searrow$ $\omega$-sequences if whenever $p\in \Bbb{P}$, $\rho_n\in\operatorna... |
Special cases of the multivariable chain rule
The general statement of the multivariable chain rule is the following.
Chain Rule: For differentiable functions $\vc{g}: \R^m \rightarrow \R^k$ and$\vc{f}: \R^k \rightarrow \R^n$ (confused?), the derivative matrix of the composition$\vc{h}=\vc{f} \circ \vc{g}$ (i.e,. $\vc{... |
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Sun, 15 Oct 2017
[ I started this article in March and then forgot about it. Ooops! ]
Back in February I posted an article about how there are exactly 715 nondecreasing sequences of 4 digits. I said that !!S(10, 4)!! was the set of such sequences and !!C(10, 4)!! was the number of ... |
[Top][All Lists] [Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index] RE: [Axiom-developer] about Expression Integer
From: Page, Bill Subject: RE: [Axiom-developer] about Expression Integer Date: Fri, 24 Feb 2006 08:12:16 -0500 Ralf,
On Friday, February 24, 2006 5:12 AM you wrote:
> ...
> Let us t... |
I had a problem when considering symmetry breaking in an SO(4) gauge theory:
$\mathcal{L} = \left| D_\mu\phi \right|^2$
where $D_\mu$ is the SO(4) covariant derivative. Then assuming there is some potential that has a minimum such that we can choose the ground state to be:
$\langle \phi \rangle = \begin{pmatrix} 0 & 0 ... |
Difference between revisions of "Haar system"
m (some TeX)
m (some TeX)
Line 3: Line 3:
\chi_1(t) \equiv 1\quad \text{ on } [0,1];
\chi_1(t) \equiv 1\quad \text{ on } [0,1];
$$
$$
− −
if $n=2^m+k$, $k=1,\dots, 2^m$, $m=0,1,\dots$, then
if $n=2^m+k$, $k=1,\dots, 2^m$, $m=0,1,\dots$, then
Line 16: Line 14:
At interior po... |
I analyzed a Pratt & Whitney F100 turbofan last semester in my aerothermodynamics course, so allow me to answer this question. The short answer:
the un-compressed air provides the majority of an engine's total thrust since the compressed air powers the engine.
Correction: I forgot to mention that the fans also compress... |
Suppose we are given that a sequence of functions $f_n(z)$ convergences pointwise to $f(z)$ on the interval $[0,1]$. Suppose further that all of these functions are given by power series centered at 0 with radius of convergence $R > 1$.
To fix notation, say $f_n(z) = a_{n, 0} + a_{n, 1}z + a_{n, 2} z^2 + \dots$
and $f(... |
There are
many properties that are equivalent to uniqueness of factorization in $\,\Bbb Z.\:$ Below is a sample off the top of my head (by no means complete). Each provides a slightly different perspective on why uniqueness holds - perspectives that becomes clearer when one sees how these equivalent properties bifurcat... |
How to determine if a vector field is conservative
A conservative vector field (also called a path-independent vector field) is a vector field $\dlvf$ whose line integral $\dlint$ over any curve $\dlc$ depends only on the endpoints of $\dlc$. The integral is independent of the path that $\dlc$ takes going from its star... |
Elementary number theory
The branch of number theory that investigates properties of the integers by elementary methods. These methods include the use of divisibility properties, various forms of the axiom of induction and combinatorial arguments. Sometimes the notion of elementary methods is extended by bringing in th... |
I am trying to derive a relation between the angle from the center of ellipse $C$ and its true anomaly (angle from the focal point $F$) (
alpha vs.
beta in the picture), for a general ellipse. For some reason, it seems I am doing some mistake somewhere.
What I tried is this. When we take the polar description with rega... |
I have been trying to solve the following problem for quite some time now: Let X denote the Fermat curve of degree d in $\mathbb{P}^2$, defined by the homogenous polynomial $$x^d+y^d+z^d=0$$. Let $F:X \rightarrow \mathbb{P}^1$ be defined by $F([x:y:z]) = [x:y]$, show that F has d branch points, and find the d correspon... |
Investigations concerning random Morse functions led me to the following problem. Consider the classical GOE of $m\times m$ real symmetric matrices $A$ with independent Gaussian entries with zero means and variances
$$ \boldsymbol{E}(a_{ii}^2)=2 \boldsymbol{E}(a_{ij}^2)= 2 $$
for all $i \neq j$. Consider the function
$... |
I'm learning about the quantum computer basics and got confused about qubits and the hadamard-gate. What I understood:
A qubit can (naturally) be in the states $\lvert 0 \rangle$, $\lvert 1 \rangle$ or any superposition $\alpha \lvert 0 \rangle + \beta \lvert 1 \rangle$
The hadamard-gate transforms a qubit from state $... |
Chaochen Wang$^1$, Yingsong Lin$^1$, Masumi Okuda$^2$, Shogo Kikuchi$^1$
1.Aichi Medical University School of Medicine 2.Hyogo College of Medicine Metronidazole (MNZ) has been broadly prescribed as therapy for \(H. pylori\) eradication worldwide.
Second line regimen using MNZ is covered under national health insurance ... |
How to interpret a formula with free variables ?
In mathematics, we usually have two kind of "equations" :
$(x+1)(x-1)=x^2-1$
$x^2-2x+1=0$
The first one is an
identity and it is clearly implicitly universally quantified; i.e. it must be read as :
$\forall x [(x+1)(x-1)=x^2-1]$.
If we consider for simplicity the interpr... |
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The Lockman Hole project : LOFAR observations and spectral index properties of low-frequency radio sources
(2016-12-11)
The Lockman Hole is a well-studied extragalactic field with extensive multi-band ancillary data covering a wide range in frequency, essential for characterising the p... |
Background
For a system consisting of two molecules (monomers or fragments are also used) X and Y, the binding energy is
$$\Delta E_{\text{bind}} = E^{\ce{XY}}(\ce{XY}) - [E^{\ce{X}}(\ce{X}) + E^{\ce{Y}}(\ce{Y})]\label{eq:sherrill-1} \tag{Sherrill 1}$$
where the letters in the parentheses refer to the atoms present in ... |
Mapping is Injection and Surjection iff Inverse is Mapping/Proof 2 Theorem
Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Then:
$f: S \to T$ can be defined as a bijection in the sense that: the inverse $f^{-1}$ of $f$ is such that: That is, such that $f^{-1} \subseteq T \times S$ is itself a mapping. Proof
Let... |
Mathematics - Functional Analysis and Mathematics - Metric Geometry
Abstract
The following strengthening of the Elton-Odell theorem on the existence of a $(1+\epsilon)-$separated sequences in the unit sphere $S_X$ of an infinite dimensional Banach space $X$ is proved: There exists an infinite subset $S\subseteq S_X$ an... |
We call a compact complex manifold Moisezon manifold, if its dimension coincides with the algebraic dimension, i.e. it has as many algebraically independent meromorphical functions as its complex dimension. Boris Moisezon himself gave a proof to the following theorem: Let $X$ be a Moisezon mainfold, then for $X$ to be ... |
Computing Ranges in Constant Time
Suppose have some sequence of elements. We want to be able to answer questions about any of its ranges in time \(O(1)\). For example, we have the following sequence:
\[ A = \{ 5, 2, 4, 7, 6, 3, 1, 2 \} \]
What is the minimum/maximum element in the range from index 0 to 3? What is the s... |
The question was a simple one: given a string (e.g.) "examplesgnome" consisting of two substrings (in this case, "examples" and "gnome"), how can we swap the two substrings in-place? In the interview, we covered the third method in more detail, but touched upon the other two. Afterwards, I calculated their computationa... |
The Erdos-Rado sunflower lemma The problem
A
sunflower (a.k.a. Delta-system) of size [math]r[/math] is a family of sets [math]A_1, A_2, \dots, A_r[/math] such that every element that belongs to more than one of the sets belongs to all of them. A basic and simple result of Erdos and Rado asserts that Erdos-Rado Delta-sy... |
(a) The idea of potentials is familiar from mechanical and electrical systems. In an electric field the work required to move a charge $q$ from one location with potential $\theta_1$ to one with $\theta_2$ is $q(\theta_2-\theta_1)$. This expression has the form of a constant factor ($q$) times the change in potential. ... |
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In this chapter we learned about left and right adjoints, and about joins and meets. At first they seemed like two rather different pairs of concepts. But then we learned some deep relationships between them. Briefly:
Left adjoints ... |
Let us consider the parameter p of population proportion. For instance, we might want to know the proportion of males within a total population of adults when we conduct a survey. A test of proportion will assess whether or not a sample from a population represents the true proportion from the entire population.
Critic... |
Defining parameters
Level: \( N \) = \( 8 = 2^{3} \) Weight: \( k \) = \( 21 \) Nonzero newspaces: \( 1 \) Newforms: \( 2 \) Sturm bound: \(84\) Trace bound: \(0\) Dimensions
The following table gives the dimensions of various subspaces of \(M_{21}(\Gamma_1(8))\).
Total New Old Modular forms 43 21 22 Cusp forms 37 19 1... |
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Isomorphisms are very important in mathematics, and we can no longer put off talking about them. Intuitively, two objects are 'isomorphic' if they look the same. Category theory makes this precise and shifts the emphasis to the 'iso... |
Prove that $$\int_0^\infty \frac{\sqrt{x}}{x^2+1}\log\left(\frac{x+1}{2\sqrt{x}}\right)\;dx=\frac{\pi\sqrt{2}}{2}\log\left(1+\frac{\sqrt{2}}{2}\right).$$ I managed to prove this result with some rather roundabout complex analysis (writing the log term as an infinite sum involving nested logs), but I am hoping for a mor... |
Suppose $f : [a, b] \rightarrow R$ is continuous on $[a, b]$ and convex on the open interval $(a, b).$ Show that $f$ is convex on the closed interval $[a, b].$
closed as off-topic by uniquesolution, Claude Leibovici, mrp, Namaste, Shailesh Mar 13 '17 at 14:19
This question appears to be off-topic. The users who voted t... |
I have this problem where a transmitting antenna radiated uniformly in all directions at a radius of r. The station broadcasts at 10 kilowatts. How much power is obtained by a reciever antenna 20km away? The antenna of the reciever is 50cm^2. I am unsure of how to set up an equation for this problem, I know the distanc... |
The question is as follows, apologies in advance, I don't know how to do the LaTex thing in posts.
Let $X_1,\ldots,X_n, Y_1,\ldots,Y_n$ be independent random variables such that $X_i \sim N(\mu_1,\sigma^2)$ and $Y_j \sim N(\mu_2,\sigma^2)$. Both $\mu_1$ and $\mu_2$ are known but $\sigma^2$ is not.
Find the maximum like... |
I've been reading this article where the IR radiance of an IR-window ($MgF_2$, 1.75mm thick, passband 3-5$\mu m$) is calculated. At some point during the calculation they mention in a footnote:
The spectral emittance has been approximated by means of a step function: $\epsilon_\lambda$ = 0.08 and 0.06 from 3$\mu$ to 4$... |
In electron liquids, the compressibility $K$ is defined as $\frac{1}{K}=-V\left(\frac{\partial P}{\partial V}\right)_N=n^2\frac{\partial \mu}{\partial n}$, where $P$, $V$, $n$ and $\mu$ are pressure, volume, density and chemical potential. However, in thermodynamics we learned, I can only find the definition of isother... |
Inflection points, concavity upward and downward
A
point of inflection of the graph of a function $f$ is a point
where the second derivative $f''$ is $0$. We have to wait a
minute to clarify the geometric meaning of this.
A piece of the graph of $f$ is
concave upward if the curve
‘bends’ upward. For example, the popula... |
Let G be a directed graph with a countable number of vertices, and suppose G is strongly connected (given any two vertices v and w, there exists a path from v to w). Fix a base vertex v
0∈G, and let L n denote the number of loops of length n based at v 0; that is, the number of sequences of vertices v 0, v 1, ..., v n ... |
Algebra is a branch of mathematics deals with symbols and rules for manipulating those symbols. Algebra involves algebraic expressions or manipulating equations. Studying algebra helps you to think logically and critically to solve many problems both in studies and in real-life situations. It opens up the other subject... |
The emf is not the work done per unit charge integrated along a closed loop by a source that is not electrostatic, it's just:
$$ \mathscr E =\oint \vec{f}_s \cdot d\vec{\ell},$$
where the integral is around the circuit, and $\vec{f}_s$ is the net force per unit charge on the conduction charges that move about within th... |
Let
$H$ be a separable $\mathbb R$-Hilbert space $L\in\mathfrak L(H,\mathfrak L(H,\mathbb R))$ $T\in\mathfrak L(H)$ be nonnegative, self-adjoint and nuclear (trace-class)
Note that$^1$ $$\operatorname{tr}\left(\left(L\otimes_\pi\operatorname{id}_H\right)T\right)=LT,\tag1$$ where on the left-hand side $L$ is considered ... |
@Julio's excellent answer describes a flight path angle, and explains that it is the angle between the tangential direction (perpendicular to the radial vector to the central body) and the current velocity vector.
I've first tried to get the angle from this expression, but it's obviously wrong, since $\arccos$ is an ev... |
Ratio of Area of Triangle Inscribed in a Circle to Triangle Enclosing the Circle
\[2x\]. Draw a line from the centre of the circle - which is also the centre of the triangles - to a vertex of the triangle as shown.
This line will bisect the angle at the vertex, producing an angle of 30 degrees. From the centre of the c... |
Taylor polynomials: formulas
Before attempting to illustrate what these funny formulas can be used for, we just write them out. First, some reminders:
The notation $f^{(k)}$ means the $k$th derivative of$f$. The notation $k!$ means $k$-
factorial, which by definition is$$k!=1\cdot 2\cdot 3\cdot 4\cdot \ldots\cdot (k-1)... |
The curve is not hyperbolic. A hyperbolic curve is a result of an equation $f(x)=\dfrac{b}{a}\sqrt{x^2-a^2}$. Which is not the case here.
The enzyme catalysis (Michaelis-Menten model) can be described by the two step reaction.
$$\ce{E + S<=>[k_f][k_r] ES ->[k_{cat}] E + P}$$
In Michaelis-Menten kinetics you make either... |
Axiom:Axiom of Empty Set Axiom $\exists x: \forall y: \paren {\neg \paren {y \in x} }$ Also defined as
It can equivalently be specified:
$\exists x: \forall y \in x: y \ne y$
The equivalence is proved by Equivalence of Definitions of Empty Set.
Also known as
This axiom is also known as the
Axiom of Existence, but there... |
What's a meaningful "correlation" measure to study the relation between the such two types of variables?
In R, how to do it?
Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up.Sign up ... |
With $s \in \mathbb{C}, a \in \mathbb{R}$,
numerical evidence strongly suggests that the complex zeros in the critical strip of:
$$\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$$
all reside on the line $\Re(s)=\frac12$ for $a \lt 0$ or $a\ge 1$. There also exist a finite few complex zeros for each $... |
If you want to understand the classical limit of a harmonic oscillator, it is more meaningful to consider coherent states $$|\alpha \rangle = e^{-|\alpha|^2/2}\sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n \rangle$$ for $\alpha$ an arbitrary complex number. Such states satisfy $a |\alpha \rangle = \alpha |\alpha \rang... |
This argument involving $\Gamma(\omega)$ is made since in a naive approach, in order to obtain the Hawking radiation, one generally drops the effective potential. Let me make this statement clearer:
In a curved spacetime, the Lagrangian of a free massless scalar field is:\begin{equation}\mathcal{L}=\frac{1}{2}g^{\mu\nu... |
The formula in complex analysis is
$$\int f(\gamma(t))\cdot(\gamma'(t)dt$$
and the formula in the real variable setting, for a gradient field, is:
$$\int F\cdot dr$$ $$=\int f_x\,dx + f_y\,dy + f_z\,dz,$$
where the integrand is said to be an "exact differential" (or total differential.)
Are the formulas essentially the... |
Using Kjetil's answer answer, with process91's comment, we arrive at the following procedure.
Derivation
We are given two unit column vectors, $A$ and $B$ ($\|A\|=1$ and $\|B\|=1$). The $\|\circ\|$ denotes the L-2 norm of $\circ$.
First, note that the rotation from $A$ to $B$ is just a 2D rotation on a plane with the n... |
Dear community,
In light of the recent work of DeBacker/Reeder on the depth zero local Langlands correspondence, I was wondering if there is an attempt to "geometrize" the depth zero local Langlands correspondence.
In particular, in Teruyoshi Yoshida's thesis, one can see a glimpse of this for $GL(n,F)$, where $F$ is a... |
How do I integrate this?
$$\int_0^{2\pi}\frac{dx}{2+\cos{x}}, x\in\mathbb{R}$$
I know the substitution method from real analysis, $t=\tan{\frac{x}{2}}$, but since this problem is in a set of problems about complex integration, I thought there must be another (easier?) way.
I tried computing the poles in the complex pla... |
Use cylindrical coordinates to evaluate the triple integral
$$\iiint_E \sqrt{x^2+y^2}dV, $$ where $E$ is the solid bounded by the circular paraboloid $z=16−4(x^2+y^2)$ and the $xy$-plane. Please Help I am confused.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and profes... |
How these processes are different from simple IIR 1 order filtering, FIR filters in terms of amplitude and phase characteristics?
Yes, integration and differentiation can be linear filters.You can start from
laplace properties that say:
$ \int_{0}^{t} {x(t)dt} \longrightarrow \frac{X(s)}{s} \\ \frac{d}{dt}x(t) \longrig... |
Visit the forum if you have a language query!
formula
Definition from Dictionary, a free dictionary
Now and then it's good to pause in our pursuit of happiness and just be happy.Guillaume Apollinaire
Wikipedia
See also fórmula Contents English Etymology Pronunciation NounWikipedia (mathematics) Any mathematical rule ex... |
Are there any differences between the study of Calculus done by Newton and by Leibniz. If so please mention point by point.
Newton's notation, Leibniz's notation and Lagrange's notation are all in use today to some extent they are respectively:
$$\dot{f} = \frac{df}{dt}=f'(t)$$ $$\ddot{f} = \frac{d^2f}{dt^2}=f''(t)$$
Y... |
Consider a smooth convex/compact domain $D\subset \mathbb{R}^n$ and a smooth, concave function $F:D\to \mathbb{R}$. Then we can define the function that simply takes the volume of the upper contour sets determined by the argument:
$$G(t) = \int_{\{x\in D \; : \; F(x) \ge t\}} d\lambda$$
where $\lambda$ denotes the Lebe... |
Related rates
In this section, most functions will be functions of a parameter $t$which we will think of as
time. There is a convention comingfrom physics to write the derivative of any function $y$of $t$ as $\dot{y}=dy/dt$, that is, with just a dot over thefunctions, rather than a prime.
The issues here are variants a... |
Assume you have a fixed ($d=O(1)$ for that matter) degree matrix polynomial $$P(X)=A_0+A_1\cdot X+A_2\cdot X^2+\ldots+A_dX^d$$
Where $A_0,A_1,\ldots A_d\in\mathbb N^{n\times n}$ are given as input. Also given is some constant $\epsilon$.
Can we find a matrix $X_0\in \mathbb R^{n\times n}$ such that $||P(X)||<\epsilon$,... |
I would like to prove that the number of simple jump discontinuities of any function is countable.
Can someone point me some material where the proof is or explain the proof here?
Thanks.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. ... |
Written by
Carlo Luschi & Dominic Masters
Posted
Apr 24, 2018
The team at Graphcore Research has recently been considering mini-batch stochastic gradient optimization of modern deep network architectures, comparing the test performance for different batch sizes. Our experiments show that small batch sizes produce the b... |
I have earlier posted the same question here on
math stackexchange but without any answer. As the question concerns tensors, I guess that I have come to the right place i.e. to physicists.
Say I have the following equation of motion in the
Cartesian coordinate system for a typical mass spring damper system:
$$M \; \ddo... |
Double integrals where one integration order is easier
Suppose you need to calculate the double integral $\iint_\dlr f(x,y)\,dA$ for some function $f(x,y)$ and the region $\dlr$ shown below.
To calculate the double integral, you can write it as an iterated integral. For example, let's say that in the region $\dlr$, the... |
Let $G$ be a semisimple group over $\mathbb C$, and $X=G/H$ be a spherical homogeneous space of $G$. Let $T\subset B\subset G$ be a maximal torus and a Borel subgroup. Let $S=S(G,T,B)$ denote the corresponding set of simple roots.
Let ${\mathcal{P}}(S)$ denote the set of subsets of $S$. Let $M$ denote the weight lattic... |
SPPU Electronics and Telecom Engineering (Semester 4)
Control Systems December 2015
Control Systems
December 2015
Total marks: --
Total time: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons (2) Marks are given to the right of every question (3) Draw neat diagrams wherever necessary
(1)... |
Review Questions Review Questions Creating Expressions Equations
Q10.01 Create the symbolic math variables a, b, c and x. Use these variables to define the symbolic math expressions:
Q10.02 Create the symbolic math variables a, b, c and x. Use these variables to define the symbolic math equations:
Q10.03 Create the sym... |
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