What is Upper bound: Definition and 113 Discussions

In mathematics, particularly in order theory, an upper bound or majorant of a subset S of some preordered set (K, ≤) is an element of K which is greater than or equal to every element of S.Dually, a lower bound or minorant of S is defined to be an element of K which is less than or equal to every element of S.
A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound.
The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.

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  1. lua

    Upper bound for first excited state - variational principle

    I'm solving problem number 5 from https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/resources/mit8_05f13_ps2/. (a) Here I got: $$ \beta = \frac{\hbar^{\frac{1}{3}}}{(\alpha m)^\frac{1}{6}} $$ and: $$ E = \left ( \frac{\alpha \hbar^4}{m^2} \right )^\frac{1}{3}e $$ (b) Using Scilab I...
  2. docnet

    Derive an upper bound for |f(i)|

    ##\mathbb{D}## is open. Let ##\mathbb{A}:=\{z:|z-i/2|=\frac{1}{9}\}##. ##\mathbb{A}## is closed and contained in ##\mathbb{D}##. ##f## is analytic in ##\mathbb{D}##, so ##f## is analytic on the interior to and on ##\mathbb{A}##. By the Cauchy integral formula, ##f^{(4)}## exists at every point...
  3. C

    MHB Upper Bound of Sets and Sequences: Analyzing Logic

    Upper bound definition for sets: $ M \in \mathbb{R} $ is an upper bound of set $ A $ if $ \forall \alpha\in A. \alpha \leq M$ Upper bound definition for sequences: $ M \in \mathbb{R} $ is an upper bound of sequence $ (a_n)$ if $ \forall n \in \mathbb{N}. a_n \leq M$ Suppose we look at the...
  4. Kostik

    A Upper bound for wavelength of a photon inside an infinite square well

    Obviously a particle inside an ISW of width L cannot have arbitrarily precise momentum because ΔP ≥ ℏ/2ΔX ≥ ℏ/2L. Therefore you cannot have a particle with arbitrarily low momentum, since that would require ΔP be arbitrarily small. I need to show that a photon inside an ISW cannot have...
  5. P

    MHB Exploring Finite Group Theory: Finding the Upper Bound of Groups of Order

    In the context of group theory, there's a theorem that states that for a given positive integer \(n\) there exist finitely different types of groups of order \(n\). Notice that the theorem doesn´t say anything of how many groups there are, only states that such groups exist. In the proof of this...
  6. Adesh

    I Checking the integrability of a function using upper and lowers sums

    Hello and Good Afternoon! Today I need the help of respectable member of this forum on the topic of integrability. According to Mr. Michael Spivak: A function ##f## which is bounded on ##[a,b]## is integrable on ##[a,b]## if and only if $$ sup \{L (f,P) : \text{P belongs to the set of...
  7. M

    Engineering Plastic Analysis: Upper Bound Theorem

    Hi, I have a quick question about part 1 of this upper bound theorem question (in the attached image). Answer says that \lambda_c = 2.25 . First, we know that there is 1 redundancy and therefore there will be a maximum of 2 plastic hinges for failure. I have found that there needs to be...
  8. F

    Upper bound height and lower bound height of a 3-ary ordered tree

    how to find upper bound height and lower bound height of 3-ary ordered tree that have leaves of 101? ( the tree don't have to be complete tree, but have to be have 3 children) $$m^h \ge 101=3^h \ge 101$$ $$log \, m^h \ge 101=3^h \ge 101$$ $$h \ge 5$$ but how to know upper bound and lower...
  9. S

    I "Dumb" question : is there an upper bound on the energy of a photon?

    I was wondering if anybody knew if there was an upper bound on how much energy you can pack into a photon, if such a thing exists. I'm wanting to say no there isn't but it occurred to me that I did not know the answer. Sorry if this is an absurdly easy question but I don't remember reading...
  10. S

    Least Upper Bound Property ⇒ Archimedean Principle

    Hello! I was wondering if this proof was correct? Thanks in advance! Given: A totally ordered field, ##\mathbb{F}##. Claim: Least Upper Bound Property (l.u.b.) ⇒ Archimedean Principle (AP) --- Proof. I will show that the contrapositive is true; that is, if ##\mathbb{F}## does not have the AP...
  11. V

    Finding an upper bound that is not the supremum

    I just want to see if I did this correctly, the interval (0,1) has 2 as an upper bound but the supremum of S is 1. So M would be equal to 2? Thank you.
  12. G

    Upper bound for probability when Bayes risk is zero

    Homework Statement Bayes' risk is ##L^*=0## for a classification problem. ##g_n(x)## is a classification rule (plug-in) such that ##g_n=0## is ##\eta_n(x)\leq 1/2## and ##g_n=1##$ otherwise. The function ##\eta##is given by ##\eta(x)=\mathbb{E}(Y|X=x)##. Then ##\mathbb{P}(g_n(X)\neq Y)\leq...
  13. W

    Justification for upper bound in Taylor polynomial

    Homework Statement I've been reviewing some Taylor polynomial material, and looking over the results and examples here. https://math.dartmouth.edu/archive/m8w10/public_html/m8l02.pdf I'm referring to Example 3 on the page 12 (page numbering at top-left of each page). The question is asking...
  14. i_hate_math

    I Upper bound and supremum problem

    Claim: Let A be a non-empty subset of R+ = {x ∈ R : x > 0} which is bounded above, and let B = {x2 : x ∈ A}. Then sup(B) = sup(A)2. a. Prove the claim. b. Does the claim still hold if we replace R+ with R? Explain briefly. So I have spent the past hours trying to prove this claim using the...
  15. mooncrater

    I What is the Upper Bound of this Summation?

    There is this summation, that I've been trying to solve, but am not able to do so. It is : $$\sum\limits_{i=k}^{n} \frac {1}{(n-i)! m^{i-1}}$$ I would be happy to find it's upper bound too. So what I did was intensely naive. I made the denominator the minimum by making ##(n-i)! = 1## and...
  16. Zeeree

    Finding an upper bound for a contour integral (Complex)

    C1 1. Homework Statement : Using the ML inequality, I have to find an upper bound for the contour integral of \int e^2z - z^2 \, dz where the contour C = C1 + C2. C1 is the circular arc from point A(sqrt(3)/2, 1/2) to B(1/2, sqrt(3)/2) and C2 is the line segment from the origin to B...
  17. G

    MHB How to find the upper bound of an error by Taylor polynomial approximation

    I'm struggling about finding a way to find the upper bound of the error of Taylor polynomial approximation. I will explain better using a solved example I found... > $f: ]-3;+\infty[ \rightarrow \mathbb{R} $ $f(x)=ln(x+3) +1 $ >Find the upper bound of the error approximating the function in...
  18. C

    How to find upper bound for recurrence relation

    Homework Statement Find a tight upper bound for the recurrence relation using a recursion tree argument Homework Equations T(n)=T(n/2)+T(n/3)+c The Attempt at a Solution I don't know how to do this problem because the tree doesn't have symmetry. One side of the tree can keep going because of...
  19. A

    MHB Master algorithm design and upper bound proof

    Hello, I am currently preparing myself for exams and I have a past exam question which I can't solve. This question concerns online learning and the following picture illustrates it: Is anyone able to help me out and propose a solution to this question?
  20. davidbenari

    Finding an upper bound for the cosmological constant

    Homework Statement (Working with geometrised units) Consider the EFE ##G^{\alpha \beta }+\Lambda g^{\alpha \beta} = 8 \pi T^{\alpha \beta} ## work out (using weak-field considerations) an upper bound for the cosmological constant knowing that the radius of Pluto's orbit is 5.9 x 10^12 m...
  21. Alpharup

    B Doubt regarding least upper bound?

    I am using Spivak Calculus. I have a general doubt regarding the definition of least upper bound of sets. Let A be any set of real numbers and A is not a null set. Let S be the least upper bound of A. Then by definition "For every x belongs to A, x is lesser than or equal to S" Let M be an...
  22. P

    Supremum = least upper bound, anything > supremum?

    The supremum is defined as the "LEAST" upper bound. The word "least" makes me think, there is a "MOST" upper bound, or at least something bigger than a "least" upper bound. For a set of numbers, is there anything larger than a supremum? Supremum is analogous to a maximum, but I don't...
  23. W

    Cantelli's Inequality and Chebyshev's Inequality

    Homework Statement The number of customers visiting a store during a day is a random variable with mean EX=100and variance Var(X)=225. Using Chebyshev's inequality, find an upper bound for having more than 120 or less than 80customers in a day. That is, find an upper bound on P(X≤80 or X≥120)...
  24. evinda

    MHB How to get the desired upper bound

    Hello! (Wave) The backward Euler method We consider a uniform partition such that $[0,T_f]$ and $[a,b]$ $h=\frac{b-a}{N_x+1}, \tau=\frac{T_f}{N_t}$ $x_i=a+ih, i=0,1, \dots, N_x+1$ $t_n=n \tau, n=0,1, \dots, N_t$ $u_t-u_{xx}=0 \\ u(t=0,x)=u_0(x) \\ u(t,a)=0 \forall t \\ u(t,b)=0 \forall t$...
  25. T

    Upper bound on the Inflation's e-foldings

    It is not clear to me, why textbooks do not mention an upper bound for the e-foldings of the basic inflation theory. To my knowledge, in order to deal with the flatness problem, we require: \frac{Ω^{-1}(t_0)-1}{Ω^{-1}(t_i)-1} = \frac{Ω^{-1}(t_0)-1}{Ω^{-1}(t_e)-1}...
  26. S

    Proving least upper bound property implies greatest lower bound property

    Homework Statement Prove if an ordered set A has the least upper bound property, then it has the greatest lower bound property. Homework Equations Definition of the least upper bound property and greatest lower bound property, set theory. The Attempt at a Solution Ok, I think that my main...
  27. C

    Showing a function in R2 is unbounded (no least upper bound)

    Homework Statement Show that this function has no absolute max by showing that it is unbounded Homework Equations f(x,y) = (x-1)^2 + (y+2)^2 -4 The Attempt at a Solution my initial idea is to construct a sequence of points {(xk, yk)} so that the sequence {f(xk, yk)} becomes unbounded. to...
  28. evinda

    MHB Upper Bound for Recurrence Relation: $T(n) \leq c n^2 \log^2 n$

    Hello! (Wave) I want to find an asymptotic upper bound for the recurrence relation: $T(n)=9T \left (\frac{n}{3} \right ) + n^2 \log n $, $T(n)=c, \text{ when } n \leq 9$, using the following method: We choose a specific function $f(n)$ and we try to show that for an appropriate $c>0$ and an...
  29. R

    Understanding Least Upper Bound & Greatest Lower Bound in Q+

    Hey guys, I'm puzzling a bit over an example I read in Rudin's Principles of Mathematical Analysis. He has just defined least upper bound in the section I am reading, and now he wants to give an example of what he means. So the argument goes like this: Consider the set A, where A = {p}...
  30. T

    It has a horizontal asymptote at y = 7/8, and increasing for all x > -13/16.

    Homework Statement Find, with proof, the least upper bound of the set of real numbers E given by: E ={14n + 9/16n + 13: n \in N}  : Homework Equations The Attempt at a Solution So I said that 16n+13>14n+9 for all N From this I get n>-2 What do I do with this? I...
  31. S

    Find the torque of a rotating sheet and the upper bound of the torque

    Homework Statement Introduction to Classical Mechanics by David Morin - problem 9.43, page 424 A uniform flat rectangular sheet of mass m and side lengths a and b rotates with angular speed w around a diagonal. What torque is required? Given a fixed area A, what should the rectangle look...
  32. C

    Prove that an upper bound a is the least upper bound

    Homework Statement Let A be a non-empty subset of R (real numbers) and a an upper bound in R for A. Suppose that every open interval I containing a intersects A (so the intersection is non-empty). Show that a is a least upper bound for A. The Attempt at a Solution I've seen the prettier...
  33. evinda

    MHB Upper bound of the relative error

    Hello! :) I am looking at the following exercise: Let the linear system $Ax=b$ with $\begin{pmatrix} 2.001 & 2\\ 2& 2 \end{pmatrix}$ ,$b=\begin{bmatrix} 2.001 &2 \end{bmatrix}^T$ and y an approximate solution,so that $Ay-b=\begin{bmatrix} 0.001 &0 \end{bmatrix}^T$ .Find an upper bound of the...
  34. T

    Estimating upper bound from measurements with uncertainties

    Hello everyone, I have a large number of measurements with associated uncertainties, and I know that the real values are bounded above by some constant. How can I estimate the value of that constant, and the uncertainty on the estimate? Thanks
  35. P

    Exploring the Upper Bound of f^{n+1}(x) for x in [-1/2, 1/2]

    Let f = ln(\frac{1}{1-x}) show that if x \in [-1/2 , 1/2] then |f^{n+1}(x)| <= 2^{n + 1} * n! I am having a hard time seeing how 2^{n + 1} * n! comes into play. I have that the taylor series for f is \Sigma \frac{x^n}{n} If a take a derivative it becomes x^(n-1) and...
  36. D

    Find the upper bound on the relative speed of the Earth and the ether

    Homework Statement "The Michelson-Morley experiment was conducted using an interferometer with L1 = L2 = 40m, lambda = 632nm, and maximum fringe separation d = 0.0022 fringes. Find the upper bound on the relative speed of the Earth and the ether, and clearly state the significance of the...
  37. W

    What are upper and lower bounds and why are they important in mathematics?

    At: http://en.wikipedia.org/wiki/Upper_and_lower_bounds in example it says that "2 and 5 are both lower bounds for the set { 5, 10, 34, 13934 }, but 8 is not" Why "2"? as 2 is not in that set. Also, at: http://en.wikipedia.org/wiki/Supremum in example it says that "The...
  38. S

    Q* (the set of rational cuts) has least upper bound property or not?

    I am struggling to draw this point home: To prove that R has LUB property, we used the following reasoning: First we defined R to be set of cuts (having certain properties) where each cut corresponds to a real number and then we proved any subset A of R has LUB (least upper bound) property...
  39. M

    Operator norm upper bound

    Greetings everyone! I have a set of tasks I need to solve using using operator norms, inner product... and have some problems with the task in the attachment. I would really appreciate your help and advice. This is what I have been thinking about so far: I have to calculate a non trivial upper...
  40. M

    Lower and Upper bound proof in R

    I am getting lost in the proof in the 5th line when it says there are 10 numbers that have the same kth digit as x. Why 10? I don't understand where this number is coming from and it doesn't seem arbitrary. the rest of the proof...
  41. O

    MHB Least upper bound - greatest lower bound duality

    Hello everyone! There's a point I didn't get in Rudin's theorem 1.11 that says: Suppose S is an ordered set with the LUB property, and B $\subset$ S, B is not empty and B is bounded below. Let L be the set of lower bounds of B. Then a = sup L exists in S, and a - inf B. In particular inf B...
  42. M

    How to Prove an Upper Bound for a Set of Real Numbers?

    Homework Statement Let A be a set of real numbers. If b is the supremum (least upper bound) of the set A then whenever c<b there exist an a in A such that a>c. Homework Equations The Attempt at a Solution I considered two cases. The first one when the supremum b is attained by...
  43. B

    Least upper bound of open interval.

    I am having trouble understanding how there could be a least upper bound for an open interval. If I have (a,b) and i am looking for the least upper bound X which is the number that is less than or equal to the set of Y such that Y> all the numbers in the interval (a,b) when I think about it I...
  44. D

    Clarifications on the least upper bound property and the irrational numbers

    Hello everyone. I desperately need clarifications on the least upper bound property (as the title suggests). Here's the main question: Why doesn't the set of rational numbers ℚ satisfy the least upper bound property? Every textbook/website answer I have found uses this example: Let...
  45. B

    Least Upper Bound and Supremum

    Are the least upper bound and supremum of a ordered field same thing? If so, then why do we have two different terms and why do textbooks do not use them interchangeably. That also means that greatest lower bound and infimum are also the same thing.
  46. STEMucator

    Upper Bound Proof of Sup(SUT)=max{sup(S), sup(T)}

    Homework Statement Prove or disapprove, for non-empty, bounded sets S and T in ℝ : sup(SUT) = max{sup(S), sup(T)} Homework Equations The least upper bound axiom of course. The Attempt at a Solution Since we know S and T are non-empty and bounded in the reals, each of them...
  47. U

    Real Analysis Least Upper Bound Question

    Homework Statement If S1, S2 are nonempty subsets of ℝ that are bounded from above, prove that l.u.b. {x+y : x \in S1, y \in S2 } = l.u.b. S1 + l.u.b. S2 Homework Equations Least Upper Bound Property The Attempt at a Solution Using the least upper bound property, let us suppose...
  48. M

    Definite Integration with Upper bound as another integral

    i have a similar one. f(x) = \int\frac{dt}{\sqrt{1+t^3}} on (0, g(x)) g(x) = \int(1+sin(t^2))dt on (0, cos(x)) that is, these are definite integrals on the interval from zero up to the given function. the question is to solve f'(pi/2). the correct answer is -1 but i don't understand...
  49. R

    The least upper bound property and the irrationals.

    Hi Does anybody know if the irrational numbers have the least upper bound property?
  50. QuestForInsight

    MHB What is the definition of greatest/least upper bound in a partially ordered set?

    Let $a$, $b$ and $c$ be elements of a partially ordered set $P$. My book defines $c$ as the greatest upper bound of $a$ and $b$ if, for each $x \in L$, we have $x \le c$ if and only if $x \le a$ and $x \le b$. Similarly, it defines $c$ as the least upper bound of $a$ and $b$ if, for each $x \in...
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