domains Definition and 114 Threads

Hi PF, $$\int \cos ax\,dx,\quad a\in{\mathbb R-\{0\}}\quad x\in{\mathbb{R}}$$ Let's make $$u=ax,\quad du=adx$$ and apply $$\int \cos u\,du=\sin u+C$$ $$\frac{1}{a}\int \cos ax\,adx=\frac{1}{a}\sin u+C$$ Substituting the definition of u $$=\frac{1}{a}\sin ax+C$$ Doubts: (i) Have I written well...

In another thread This has me curious about "ordering other than our normal ordering." What does this mean? I take it that "normal ordering (of integers)" is ... 0, 1, 2, 3... Do mathematicians consider alternate orderings like ...0, 2, 1, 3... That doesnt seem to make sense to me, that's...

Hi! This thread might well be similar to: https://www.physicsforums.com/threads/thread-about-jacksons-classical-electrodynamics-3rd-edition.910410/ I'm self-studying Vanderlinde and having a great time. However, I think that I am conflating and confusing many different things. Let me just ask...

This is a surface level question and I don't want to go into detail. Imagine an algorithm which when used with a sensor output gives the statistical moments of a variable in nature (for example mean and standard deviation of a variable). The sensor measures this once in a while (like once in a...

Hi PF! Suppose I have three pieces of data: x1 = 0:3:12 and x2 = 0:4:16 and x3 = 0:5:20 with corresponding functions y1 = x1.^2 and y2 = x2.^3 and y3 = x2.^4. How would you average these the "functions" with data (x1,y1) and (x2,y2) and (x3,y3)? My thoughts are: 1) linearly interpolate y1, y2...

Hello, I have attached my question and the work. I believe the answer is correct. Looking for verification. Thank you!

Of the various notions of convergence for sequences of functions (e.g. pointwise, uniform, convergence in distribution, etc.) which of them can describe convergence of a sequence of functions that have different domains? For example, let ##F_n(x)## be defined by ##F_n(x) = 1 + h## where ##h...

Hello, I'm a teacher and will be doing a lesson on "Graphing the inverse cosine function." In the lesson, I show the students a cosine function graphed from 0 to 360 degrees ( I use degrees to really drive home the point that this is a mapping between two different sets, namely angles and...

I've tried to solve the following circuit using mesh equation, but the solution seems to differ from my attempted answer. Mesh circuit as follows: My mesh equation is: -10+3(i1)+2s(i1-i2)=0 (for the mesh on the left) and -10+12(i2)+6s(i2)+2s(i2-i1)=0 (right mesh) However the answer seems to...

Hi, Which technology specializations within the mechanical engineering domain will be most demanded in the future? (I am looking for different technologies rather than for applications enabled by the technology. For example not 'robotics', but rather the technologies used for robotics) Thanks...

##z^2\leq x^2+y^2, z\geq x^2+y^2## I know the shapes of those inequalities, but the question is: How do i find if the point are external the shape or internal?

Please refer to the simplified circuit in the attached figure. The overall goal is to control the current through sense resistor R1 by adjusting a reference voltage V2, thus creating an electronic load where the power is dissipated in Q1. All load current is returned locally in the isolated loop...

Modern physics is comprised of a quartet of theory specializations. I. Classical Physics + Special Relativity S.R. II. Quantum Mechanics. QM III. Quantum Field Theory QFT IV. General Relativity GR Under what practical conditions do the predictions of each of these theories become...

Homework Statement For any harmonic load: $$F(t)=F_0\cdot \sin(\omega t)$$ What is the corresponding Frequency domain equivalent? My lecture notes is suggesting: $$ F(t)=F_0 \cdot e^{i \omega t} $$ But I am failing to see how they are equal? The lesson is about Stochastic Response of...

Hi PF! I looked through the documentation on their website, but under the tab "Solve partial differential equations over arbitrarily shaped regions" I am redirected to a page that does not specify how to create a region. Any help is greatly appreciated. Also, if it helps, the domain is a...

Hi PF! I have a list of numbers, and I'm trying to plot each constant value on a 1-unit long interval. What I'm trying is this a = Table[i, {i, 2, 8}]; For[i = 1, i < 5, i++, Plot[a[[i]], {x, i - 1, i}] // Print] but that generates plots that are not on the same axes. Any help?

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help in order to fully understand the proof of Proposition 4.3.12 ... ... Proposition 4.3.12 reads as follows:In the above proof by Bland we read...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help in order to formulate a proof of Proposition 4.3.5 Part (iii)... ... Proposition 4.3.5 reads as...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help to fully understand the proof of Lemma 4.3.10 ... ... Lemma 4.3.10 and its proof read as...

I am reading The Basics of Abstract Algebra by Paul E. Bland ... I am focused on Section 7.2 Euclidean, Principal Ideal, Unique Factorization Domains ... ... I need help with the proof of Theorem 7.2.20 ... ... Theorem 7.2.20 and its proof reads as...

I am reading The Basics of Abstract Algebra by Paul E. Bland ... I am focused on Section 7.2 Euclidean, Principal Ideal, Unique Factorization Domains ... ... I need help with the proof of Theorem 7.2.14 ... ... Theorem 7.2.14 and its proof reads as follows: In the above proof by Bland we...

I am reading The Basics of Abstract Algebra by Paul E. Bland ... I am focused on Section 7.2 Euclidean Domains, Principal Ideal Domains, Unique Factorization Domains ... ... I need help with the proof of Lemma 7.2.13 ... ... Lemma 7.2.13 reads as...

I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with an aspect of the proof of Proposition 1.5 ... ... Proposition 1.5 and its proof read as...

I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with an aspect of the proof of Proposition 1.5 ... ... Proposition 1.5 and its proof read as follows: At the end of the above proof...

(There is a better formatted word version of all of this in the attachments) Hi I was trying to solve some problems in the website AoSP when I came across this problem https://artofproblemsolving.com/wiki/index.php?title=1990_AHSME_Problems/Problem_8 which made me question few things in my...

So, here's my question. I read somewhere that all universal truths on empty domains are vacuously true, whereas all existential are false. However, if all statements of the form (∀x∈A)(P(x)) , where A is an empty set, are vacuously true, then the statement (∃x∈A)(P(x)) should also be true...

I'm doing a research project over the summer, and need some help understanding how to construct an inverse Fourier transform (I have v. little prior experience with them). 1. Homework Statement I know the explicit form of ##q(x)##, where $$ q(x) = \frac{M}{2 \pi} \int _{- \infty}^{\infty} dz...

Homework Statement Okay, I have two examples that are confusing me. I am not sure where all the numbers that must be excluded from the denominators so that we're not dividing by zero are coming from. a) x2 + 6x +5 / x2 - 25 b) x-7 / x-1 multiplied by x2-1 / 3x-21 Homework Equations None...

Dear all, I have a question concerning calculating the following limit: \lim_{x \rightarrow 0} f(x) = \lim_{x \rightarrow 0} \frac{\sin{(x)}}{x} = 1 Obviously, x=0 is not part of the domain of the function. One way to calculate the limit is using l'Hospital. Another way for these kinds of...

Nearly every analysis reference I come across defines the derivative for functions on an open interval ##f:(a, b) \rightarrow \mathbb{R}##. I understand that, in constructing the definition of ##f## being differentiable on a point ##c##, we of course want it to first be a point it's domain, so...

Homework Statement Homework Equations log_2 x = y 2^y = x 3^2^y The Attempt at a Solution log_2 x = y 2^y = x log_2 {log _3 {log _2 { log_3 {2^y} } } } what am I suppose to do?

I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ... I need some help with understanding Example 1.4.1 ... Example 1.4.1 reads as follows: In the above text by Alaca and Williams we read the...

I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ... I need some help with understanding Example 1.4.1 ... Example 1.4.1 reads as follows: In the above text by Alaca and Williams we read the...

Homework Statement Let ##R## be a principal ideal domain and suppose ##I_1,I_2,...## are ideals of ##R## with ## I_1 \subseteq I_2 \subseteq I_3 \subseteq ...## The Question has two parts: 1. to show that ##\cup _{i=0}^{\infty}I_i## is an ideal. 2. to show that any ascending as above must...

I'm wondering if the issue of bacteria and archaea having a single last universal common ancestor (LUCA) with eukarya diverging later, or all three having distinct common ancestors, has been clarified. I've seen a number of texts indicating the eukarya diverged much later, but also some...

Hello.If one’s goal was to contact a technologically superior extraterrestrial being (or a group of beings, or a civilization), which domain(s) of astronomy should one specialize in precisely?1) Which type of astronomy should one specialize in? Computational astronomy Experimental astronomy...

Hello! I have been studying some pertubation theory lately which i found very useful. I then started thinking about how to approximate solutions to a 2d boundary value problem if the difficulty lies in the geometry of the boundary(I.e. not rectangular), and not in the diff. equation itself(i.e...

Homework Statement question: State two different functions that have same range but different domain. Then tell me what is the range of those two functions. The attempt at a solution Y = x/2 Y= x/2 +1 I don't know if that is correct or not. Any suggestion will help.

I just finished working through compositions of functions, and what properties the inner and outer functions need to have in order for the whole composition to be injective or surjective. I checked Wikipedia just to make sure I'm right in thinking that for a composition to be injective or...

Can you guess them? OK, .com as #1 is a no-brainer. How about the other nine? OK, here they are, in order from #1 to #10: (from Verisign's Domain Name Industry Brief for March 2015) By the way, the first .com was symbolics.com in 1985. Happy 30th birthday!

Homework Statement Can someone explain me what are high field domain or can give me a source from where I can understand it.It will be very helpful. Homework EquationsThe Attempt at a Solution

I understand that only three elements are magnetic: Iron, Cobalt and Nickel, iron being the strongest. This element in it's purest state is un-magnetized, right? Composed of a bunch of crystal magnets or domains. So if you have three ores of each element at their purest form they won't...

Let $R$ be an integral domain. Suppose that $R_1$ and $R_2$ are proper subrings of $R$ and that both $R_1$ and $R_2$ are unique factorization domains (UFDs). Let $R_3$ be the subring of $R$ that is generated by $R_1$ and $R_2$. Is $R_3$ necessarily a UFD? (The subring generated by two subrings...

OK, so I posted this a few days ago: https://www.physicsforums.com/threads/subtracting-the-overlap-of-functions.784184/#post-4925108 What I've come to discover is that I want to understand how I can subtract f(x) on domain [b,c] from g(x) on domain [a,d]. I want to be able to disregard both...

This isn't a homework problem, but rather a bit of confusion regarding something in the textbook we're using; if this isn't the right place, feel free to move it. From Artin's Algebra pages 422/423 (slightly paraphrased): Let ##Q=\begin{bmatrix}1&\\3&1\end{bmatrix}##...

Hey guys, I have a couple more questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: For the first one, I got a dotted graph (with 1,4 as the initial value) and then precipitous jumps in the graph. So I said that...

I am reading Beachy and Blair's book: Abstract Algebra (3rd Edition) and am currently studying Proposition 5.211. I need help with the proof of the proposition. Proposition 5.211 and its proof read as follows: In the proof we read the following: " ... ... ... Since \phi ( \mathbb{Z} )...

If the critical points corresponding to the global min/max of a function ##f:\mathbb{R}^2\rightarrow\mathbb{R}## lie in a subset ##A## of ##\mathbb{R}^2##, then the global min/max of ##f## in ##A## correspond to the global min/max of ##f##. If the global min/max of ##f## lie outside of ##A##...

Hallo, I'm trying to compare the distance between two distributions that I got from a Kernel smoothing density estimate (ksdensity in matlab). I was thinking of using the kullback leibler divergence, but I realized that the domains of my distributions are different (see attached). Can I...

Homework Statement a) f(x,y)=√ [1−(x2+y2)] b) f(x,y)=2cos(4x+y2) The Attempt at a Solution a) The domain is such that x2 + y2 must not be greater than 1 In other words, this is expressed (as stated as an option on the answer sheet) as "xy-plane without the line x=y" Why is this...