Set Definition and 1000 Threads

I cannot understand a corollary from this paper on surreal numbers: Does it say that there are no countable surreal numbers greater than the germs of Hardy fields?

Example 1.2.24: Let C be a set in n-dimensional space, and let Q(C) = ∫...∫Cdx1dx2...dxn. If C= {(x1,x2,...,xn : 0 ≤ x1 ≤ x2 ≤ ... ≤ xn ≤ 1}, then Q(C) = ∫01 ∫0xn∫0xn-1...∫0x3∫0x2dx1dx2...dxn = 1/n!. (Apologies for the poor formatting of the integration bounds.) I haven't the foggiest idea how...

Want to understand how set C contains ##N## x H. H is only defined to be a set with element e and as the domain/range of function k. Is this enough information to conclude that the second set in the cartesian product W is H and not a subset of H? My thinking is to show that ##N## and H satisfy...

Hello all, The question I am tackling is as follows: I was wondering if any of you could look over my solution and tell me if my logic is correct. Any feedback would be great!

For this, I am trying to understand why the set ##(0,2)## has no maximum. Is it because if we say for example claim that ##a_0 = 1.9999999999## is the max of the set, then we could come along and say that ##a_0 = 1.9999999999999999999999999999## is the max, can we continue doing that a...

In the context of the mappings of a set S into itself, when S is not number system with a zero, what is the customary definition for "zero mapping"? ( ChatGPT says that its a mapping that maps each element of S to some single element of S , i.e. maps all elements to some constant. )

I am thinking why the following holds: Let f be a smooth function with f: Ω⊂R^m→R. Why is the set {(x,y)∈Ω×R|y=f(x)} a manifold? Would be helpful if you are providing me some guidance or tips:)

We are told that ##(x-4)(z-2)=0## The ##0## refers to what variable? The ##z=2## i think the sketch on diagram may be misleading (not to scale). Why have the ##x## crossing the axis on negative side and same applies to ##z##... To put this into context, we may have ##(x,y,z)=(4,y,z)## where...

Let E be a finite nonempty set and let ## \Omega := E^{\mathbb{N}}##be the set of all E-valued sequences ##\omega = (\omega_n)_{n\in \mathbb{N}}F##or any ## \omega_1, \dots,\omega_n \in E ## Let ##[\omega_1, \dots,\omega_n]= \{\omega^, \in \Omega : \omega^,_i = \omega_i \forall i =1,\dots,n...

I want to fine-tune the Pixel2Style2Pixel model with my custom data set, but I keep getting an error when I'm trying to load in the pre-train weights. Here is my code : # Load the pre-trained model os.chdir("/content/pixel2style2pixel") from models.psp import pSp config = { "lr": 0.0001...

This is basically a physics problem but I will try my best to highlight the mathematics behind it. Suppose I have two functions: $$T(z,B)=\frac{\text{z}^3 e^{-3 A(\text{z})-B^2 \text{z}^2}}{4 \pi \int_0^{\text{z}} \xi ^3 e^{-3 A(\xi )-B^2 \xi ^2} \, d\xi },$$ $$\phi(z,B)=\int_0^z...

my question is how can I approch the problem ? And what is explicitly the set f(Eα)? {f(x) ∈ [a, b] such that what ??}

Question: Suppose I have a data file for the acceleration of an object after every ## \Delta t_i##, how do I obtain the displacement of it? Context: Integral in a PID loop, although not exactly what I am asking as one is sum of error: $$\int_0^T \int_0^T \ddot {\vec \theta(t)}dtdt$$ the other...

How to set the parameters - photon index and metal abundance in the CIAO-sherpa tool for an x-ray source?

In physics there is a notation ##\nabla_i U## to refer to the gradient of the scalar function ##U## with respect to the coordinates of the ##i##-th particle, or whatever the case may be. A question asks me to prove that $$\nabla_1U(\mathbf{r}_1- \mathbf{r}_2 )=-\nabla_2U(\mathbf{r}_1-...

TL;DR Summary: Eight Exact To Get 49 Please advise if I am wording this problem correctly and what are the solution (is there some equation for combinations )/ Answers:SET of numbers 1,2,3,4,5,6,7,8,9,10,11,12 months of the year. Within the exact time frame of 10 years I MUST choose a number...

Hi everyone Can someone show me how to add MPAndroid Chart to Android Studio Dolphin? I've tried following the instructions given at https://www.geeksforgeeks.org/how-to-create-a-barchart-in-android/ Navigate to the Gradle Scripts > build.gradle(Module:app) and add the below dependency in the...

Let ##Q_{n}(x)## be the inverse of an nth-degree polynomial. Precisely, $$Q_{n}(x)=\displaystyle\frac{1}{P_{n}(x)}$$, It is of my interest to use the set notation to formally define a number, ##J_{n}## that provides the maximum number of solutions of ##Q_{n}(x)^{-1}=0##. Despite not knowing...

I am confused by a question. I thought "right handed set" only applied to sets of three vectors. However I have been given 2 vectors and asked "check whether they are perpendicular to each other and if they form a right handed set. If they don't form a right handed set, the second vector must be...

Let ## p ## be an odd prime. Then ## 23 ## is a quadratic residue modulo ## p ## if ## (23|p)=1 ##. Applying the Quadratic Reciprocity Law produces: ## (23|p)=(p|23) ## if ## p\equiv 1\pmod {4} ## ## (23|p)=-(p|23) ## if ## p\equiv 3\pmod {4} ##. Now we consider two cases. Case #1: Suppose ##...

I have a shape based on the idea of having a string, a closed loop, that is sitting outside a rectangle. The string is d longer than the circumference of the rectangle. I noted the sides as a and b. The shapes is formed by drawing a line around the rectangle with the string as a constraint of...

Square matrices are closed under addition and their own form of multiplication, but in general do not commute. What algebraic structure then describes this, along with polynomials of matrices and allows us to amend with other operations, such as differentiation or integration defined on these...

I have been considering the properties of a Diffractive Optical Element (DOE) consisting of a very large number of concentric rings of equal (small) width, where the thicknessses of the rings are such as to produce random phase shifts in the range 0 to 2pi. I think I understand the behaviour of...

My interest is only on part (a). Wah! been going round circles to try understand why the radius = ##2##. I know that the given sequence is both bounded and monotonic. I can state that its bounded above by ##1## and bounded below by ##0##. Now when it comes to the radius=##2##, i can also say...

The problem goes as follows: Let ##M, N## be sets and ##f : M \rightarrow N##. Further let ##L \subseteq M## and ##P \subseteq N##. Then show that ##L \subseteq f^{-1}(f(L))## and ##f^{-1}(f(P)) \subseteq P##. Obviously, I would simply use the definition of a functions inverse to obtain...

[FONT=times new roman]Problem Statement : I copy and paste the problem as it appears in the text (Lang, Basic Mathematics, 1971). Attempt : There are several questions in both a) and b) above. I type out the question and my answer each time. a) (i) Show that addition for ##E## and ##I## is...

help me in my problem set qs

Proof: Let ## p ## be the prime divisor of two successive integers ## n^{2}+3 ## and ## (n+1)^{2}+3 ##. Then ## p\mid [(n+1)^{2}+3-(n^{2}+3)]\implies p\mid (2n+1) ##. Observe that ## p\mid (n^{2}+3) ## and ## p\mid (2n+1) ##. Now we see that ## p\mid [(n^{2}+3)-3(2n+1)]\implies p\mid...

There is a massive and continuous fireball as fire crackers are exploding, generating thousands of shockwaves. Perhaps they are bouncing off each other and cancelling each other out like noise cancelling headphones,

Hi PF! The autocorrelation coefficient ##\rho## is defined as $$\rho_k \equiv \frac{\sum_{t=k+1}^T (x_t - \bar x)(x_{t-k} - \bar x)}{\sum_{t=1}^T(x_t-\bar x)^2}$$ Now suppose we calculate ##\rho## through ##T##, but are then given a new data at time ##T + \Delta t##. Is there a way to...

Dear Everybody, I am having some trouble with proving this set ##S=\{(x,y)\in \mathbb{R}^2: 3x^2-4xy+5y^2 \leq 5\}## is bounded. Find a real number ##R>0## such that ##\sqrt{x^2+y^2}\leq ## for all ##(x,y)\in S.## My attempt: ##3x^2-4xy+5y^2 =3x^2+(x-y)^2-(x+y)^2+5y^2 \\ \leq...

In general relativity, rotation of mass gives rise to framedraging effects, just like linear motion does, because of the off-diagonal components in the mass-energy-momentum tensor. So around Bonnor beams there is framedragging, as well around a rotating mass. Now imagine a spherical rotating...

Okay so I know, that if the radius is 0, the z coordinate will run from -1 to +1. If the radius tends to one, the z coordinate will tend to 0. But I still cannot imagine how this set looks like, help would be appreciated. Thank you.

Summary: Determine the absorbing states & communication classes of the given matrix. Hello everyone, If we have a state space of S = {1,2,3,4} and the following matrix: \begin{bmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 1/3 & 2/3\\ 1 & 0 & 0 & 0\\ 0 & 1/2 & 1/2 & 0\\ \end{bmatrix} Now, given the...

## y-x \gt 1 \implies y \gt 1+x## Consider the set ##S## which is bounded by an integer ##m##, ## S= \{x+n : n\in N and x+n \lt m\}##. Let's say ##Max {S} = x+n_0##, then we have $$ x+n_0 \leq m \leq x+(n_0 +1)$$ We have, $$ x +n_0 \leq m \leq (x+1) +n_0 \lt y+ n_0 $$ Thus, ##x+n_0 \leq m \lt...

Proof: Suppose for the sake of contradiction that ## aa_{i}\equiv aa_{j}\pmod {n} ## for some ## i,j\in\mathbb{Z} ## such that ## 1\leq i<j\leq n ##. Then ## aa_{i}\equiv aa_{j}\pmod {n}\implies n\mid (aa_{i}-aa_{j})\implies n\mid [a(a_{i}-a_{j})] ##. Note that ## n\mid b ## if ## n\mid (ab) ##...

Find solution here; Ok i just want clarity for part (a), My approach is as follows, since we want positive integer values that satisfy the problem then, ##\dfrac {n^2-1}{2}≥1## I had earlier thought of ##\dfrac {n^2-1}{2}≥0## but realized that ##0## is an integer yes but its not a positive...

I would like to use VFDs for motor sizes ranging from 0.4kw to 15kw. My idea here is to reduce the generator foot print to the lowest possible value. My thinking is that, theoretically, I could use a 25kva generator set to drive a 20hp motor (80% operating capacity) and since the VFD eliminates...

My interest is on question 9. b(i) Find the question and solution here; I understand that ##a## should be less than ##2## because when ##a=2##, the two equations shall have same gradients which implies that the two lines are parrallel to each other. Now to my question, this solution does not...

Is the statement that I have circle is true ? Because I feel like the solution in my textbook is wrong, I only learn that empty set is a subset of every set but it is not an element of a set.

Hello there! Reading the textbook on differential geometry I didn't get the commentary. In Chapter about vector bundles authors provide the following example Let ##M=S^1## be realized as the unit circle in ##\mathbb{R}^2##. For every ##x\in S^1##, the tangent space ##T_x S^1## can be identified...

Can someone show me how to include the set minus operation in latex code ... obviously there is a difficulty that the symbol \ within latex delimiters just leaves a space ... ... so B \ B' doesn't result in B \ B' Peter (SOLVED) B \text{ \ } B' works ... ... ... ... ... B \text{ \ } B'

ok we shall have ##(A-B)∩(A-C)= [1,y]∩[1,2,x,y]=[1,y]## correct?

Dear Everybody, I am confused by ##1/n C##, where C is a cantor set in base 3 and ##n\geq2##. I can understand the construction of the normal Cantor set. How do I comprehend this set with this extra condition. Do I multiply the set with ##1/n## or not? Thanks, Cbarker1 mentor note...

I know that for a set to be bounded it is bounded above and below, for the bound below is it 0 and n cannot equal 1 and u paper bound is inf but how do I prove that it is bounded?

a) a subset ##L\subset \mathbb{Q}## is a Dedekind cut if ##L## is proper, ##L## has no maximal element, and $$\forall a,b\in \mathbb{Q}, [(a<b)\land( b\in L)\Longrightarrow a\in L]$$ b) Is ##P=\{x^4|x\in L\}## a Dedekind cut? P is proper: $$(a\in L)\Longrightarrow (a^4\in P)\Longrightarrow...

Hi Everybody, I am having some difficulties on the prove this problem. I picked a nice example when I was trying to think about the proof. Let ##s=3## and ##t=2##. Then ##u1=c1v1+c2v2, u2=c3v1+c4v2, u3=c5v1+c6v2##. Then a linear combination of u: ##K1u1+K2u2+K3u3=0##. I grouped both linear...

I was asked to give an estimate for a three dimensional permeate problem and I need an assist in how to setup a model equation. Picture a 6" underground cylindrical bore. The depth of the cylindrical bore is 150 feet. The bore is sealed at the bottom and the permeability below the bore bottom...

In Gravitation by Misner, Thorne and Wheeler (p.139), stress-energy tensor for a single type of particles with uniform mass m and uniform momentum p (and E = p2 +m2) ½ ) can be written as a product of two 4-vectors,T(E,p) = (E,p)×(E,p)/[V(E2 – p2 )½ ] Since Einstein equation is G = 8πGT, I am...