Definition Definition and 1000 Threads

I realize now that I took something for granted when I first learned it god knows when.. So I though of starting a discussion as to why were the order of operations defined the way they were? I mean, is there some kind of natural explanation as to why we should compute exponents first and...

Homework Statement A palindrome over an alphabet Ʃ is a string in Ʃ* that is spelled the same forward and backward. The set of palindromes over Ʃ can be defined recursively as follows: i) Basis: λ and a, for all a that are elements of Ʃ, are palindromes. ii) Recursive step: If w is a...

About the definition of "discrete random variable" Hogg and Craig stated that a discrete random variable takes on at most a finite number of values in every finite interval (“Introduction to Mathematical Statistics”, McMillan 3rd Ed, 1970, page 22). This is in contrast with the assumption that...

I have attached this definition that my book provides. My question is does that part "for each M > 0 there exists δ > 0 such that f(x) > M, mean that whenever you M close to the limit, you can find a δ that will give M1 that is closer to the limit?

So I was just working through Courant's calculus and am a bit confused as to where a few variables are pulled out of. Homework Statement Integration of f(x) = x We can see that a trapezoid is formed, so the relevant equation: 1/2(b-a)(b+a) is the value of this integral. To confirm that our...

My lecture notes say: Let f:[a,b]->R be bounded. F is said to rienmann integrable if: L=\int_{a}^{b} f(x)=U where : L=Sup(L(f,P)) and U=Inf(U,(f,P)) but everywhere else(internet) there's a definition with epsilon. I have the epsilon stuff later under "riemann...

I'm reading a book about Group Theory (by Mario Livio: The Equation that Couldn't be Solved ). On page 46 he explains that four rules and one operation define a group: The rules are Closure, Associativity, the existence of an Identity Element and finally the existence of an Inverse. He cites...

I am reading about the formal definition of a limit, and its corresponding proof, and there is one thing that I don't quite understand, yet. It says that delta depends on epsilon, but what I wonder is why is it not the other way around. Indeed, why does delta have dependency on epsilon?

Would I be correct in saying that: If Span(S)≠0 then S is linearly independant. If Span(S)=0 then S is linearly dependant. With S being a subset of V.

I know it may sound idiotic to ask questions about definition of something, but I'm going to do that now. I've seen the definition of categories in several different contexts, in all of them categories consisted of objects like groups, rings, R-modules (in particular, vector spaces), topological...

Homework Statement Hello! I need some help with a problem: Problem: Turbulent flow beteween parallel flat plates. It is defined: [ tex ] \tau = \mu \frac{d\bar{u}}{dy}-\rho\bar{u'v'} [ \tex ] The exercise gives that [ tex ] \tau = a y [ \tex ] and [ tex ] \rho\bar{u'v'} =...

We say a point x in X (which is a topological space) is an accumulation point of A if and only if any open set containing x has a non-empty intersection with A-{x}. Well, I'm creating examples for myself to understand the definition. Suppose X={a,b,c,d,e} and define...

While solving problems regarding Epsilon-delta definition of limit from my textbook i found that every answer was like ε= a×∂,where a was any constant.Is it necessary that ε should always be directly proportional to ∂ for limit to exist?? Cant they be inversely proportional? If they can please...

In Jackson's book he defines the capacitance of a conductor, "...the total charge on the conductor when it is maintained at unit potential, all other conductors being held at zero potential." I'm trying to get a more concrete definition in my head rather than the standard definition of...

Homework Statement Hi everyone, This is more of a definition clarification than a question. I'm just wondering if a branch is the same thing as a branch line/branch cut? I've come across a question set that is asking me to find branches, but I can only find stuff on branch lines/cuts and...

I'm having trouble conceptualizing exactly what a subspace is and how to identify subspaces from vector spaces. I know that the definition of a subspace is: A subset W of a vector space V over a field \textbf{F} is a subspace if W is also a vector space over \textbf{F} w/ the operations of...

I have a question that has stumped me a bit, i am not sure how to use the definition to calculate it, i can use the tables, but i don't think that's what is needed. Using the definition of the Laplace transform, determine the Laplace transform of I can do it with the table but i am not sure...

I mean the one saying that: (a,b) is defined to be the set: {{a},{a,b}} What exactly does this set definition of an ordered pair mean? Namely, how does it attribute the relevant "order" of terms to the concept of an ordered pair? Thanks!

This wasn't originally a homework problem as such, so sorry if its confusing, but I thought I would ask it here; Homework Statement Show that the the two methods of creating the n-torus are equivalent. 1) The n-torus as the quotient space obtained from ℝn by the relation x~y iff x-y \in...

I'm reading the book "Algorithms Design" and a recursion algorithm is defined as: T(n)\leqqT(n/q)+cn But in the Karatsuba’s Algorithm, the recurrence for this algorithm is T (n) = 3T (n/2) + O(n) = O(nlog2 3). The last equation is strange, since 3T(n/2) is bigger than the set. Why they define...

In text (Spivak) it says that a function is a collection of pairs of numbers with the following property: if (a,b) & (a,c) are both in the collection, then b=c; in other words, the collection must not contain two different pairs with the same first element. Now in an other text (Kolmogorov) I...

Hello, I’m designing a 2D wind tunnel model for my master thesis. It will be a profile equipped with a fixed hinged trailing edge flap. I’m going to measure at different angle of attacks and different flap settings at low speeds (about 70 to 100 m/s). The aim is to measure steady and unsteady...

I read in my metric spaces book that a separable space is that which has a countable, dense subset. This definition has no intuitive meaning to me. I'm able to show if a space is dense or not, and I think I can show a space is countable. But, I'm missing the "so what?!" I would like to...

"an electric field is the region of space surrounding electrically charged particles and time-varying magnetic fields" Why do the fields need to be time varying? Additionally, if light is a frequency of electromagnetic radiation, and EM is made of magnetic and electric fields occurring at...

Homework Statement f(x) = \left\{ {\begin{array}{*{20}{c}} {{x^2}\sin \frac{1}{x}}&{x \ne 0}\\ 0&{x = 0} \end{array}} \right. Is it differentiable at x=0? If it is, what's its value?Homework Equations The Attempt at a Solution I've calculated the derivative function for x not equal zero: f'(x)...

Generic "definition" of derivative? Hi. This is a theoric doubt I have since I went to class today. The professor "redifined" the derivative at point a. He draw a curve (the function) and the tangent at point a. Then he draw another two lines in the same point. Well, then he said that the error...

Hello, I am reading the paper of S. Ishihara, Quaternion Kaehlerian manifolds, I need it for understanding of totally complex submanifolds in quaternion Kaehlerian manifolds. I am afraid that I don't understand well the definition of quaternion Kaehler manifold, that is my question is the...

i am having trouble understanding some of the "basic" concepts of my linear algebra...any help would be greatly appreciated what is an orthogonal basis? and how to construct it? i keep stumbling upon questions asking about construction a orthogonal basis for {v1, v2} in W what i null A...

Hello all, This is very simple however I would like to understand why this is true. According to the definition of a limit, if we have limit of f(x) as x approaches infinity = a then for every ε>0 there exists a real number M such that if x>M then the absolute value of f(x)-a < ε. This...

I've been looking at the measure theoretic definition of a conditional expectation and it doesn't make too much sense to me. Consider the definition given here: https://en.wikipedia.org/wiki/Conditional_expectation#Formal_definition It says for a probability space (\Omega,\mathcal{A},P), and...

i see the definition of differential manifolds in some book for example, NAKAHARA but what is the definition of manifold in general! and what the definition of topological manifold.

Hi all, I am struggling to find any elementary material on the "gradient flow of a functional" concept. From introductions in advanced papers I seem to have understood that, assigned a functional F (u), the gradient flow is charactwerized by an equation of the type Du / Dt = P u , where P...

Homework Statement let the function f:ℝ→ℝ be differentiable at x=0. Prove that lim x→0 [f(x2)-f(0)] ______________ =0 x Homework Equations The Attempt at a Solution I am kind of lost on this one, I have tried manipulating the definition of a...

Hi all ! I am terribly sorry if this was answered before but i couldn't find the post. So that's the deal. We all know that while x→∞ (1+1/x)^x → e But I am deeply telling myself that 1/x goes to 0 while x goes to infinity. 1+0 = 1 and we have 1^∞ which is undefined. But...

Homework Statement I'm not sure if this is the appropriate board, but quantum mechanics people surely know about spherical harmonics. I need to implement the Wigner D matrix to do spherical harmonic rotations. I am looking at...

What is the general definition of energy? I already know that it means ability to perform work and that Work = ∫Force d(displacement) = Δ Kinetic Energy = -Δ Potential Energy ( in a conservative field "a closed path integral of the force = 0"), Σ Kinetic-Potential = constant, ∫Kinetic-Potential...

He starts using the term 'reducible', as it came out of nowhere, from the page 162 of the text. I know, roughly, what kind of thing he mean by this 'reducible' obejct. (That is that an element is factored into two elements that are not units.) And this should not be a problem if this term is...

In his Statistical Physics book, Landau introduces the specific heat as the quantity of heat which must be gained in order to raise the temperature of a body one by unit. I don't understand, how he directly jumps to the conclusion that that has to be (let's just say, for constant volume): C_V...

Question about Landau: Definition of "Number of states with energy" in an interval Hey! I am currently reading Landau's Statistical Physics Part 1, and in Paragraph 7 ("Entropy") I am struggling with a definition. Right before Equation (7.1) he gives the "required number of states with...

If A is a bounded operator on a Hilbert space H, isn't the following true of the residual spectrum \sigma_r(A): \lambda \in \sigma_r(A) iff (\forall \psi \in H, \psi \neq 0)((\lambda - A) \psi \neq 0) iff \ker (\lambda - A) = \{0\} iff \lambda - A is injective? So isn't the condition that...

Hi, All: Just curious: Rudin defines order in his "Baby Rudin" book ; an order relation < in a set S, as a relation* satisfying, for any x,y,z on S: 1) Either x<y , y<x , or y=x 2)If x<y and y<z , then x<z , i.e., transitivity. Just curious: why is Rudin only considering only...

Text book definition is "In the absence of forces, ("body") at rest will stay at rest, and a body moving at a constant velocity in a straight line continues doing so indefinitely". My thinking. Moon is orbiting Earth in a circular path, and not in a straight line. Still that motion follows...

Is this correct? A function ψ:A --> B is the set: ψ = { (x, y) | \forallx\inA\existsy\inB\ni(((x, y) \in ψ) \wedge ((x, z)\in ψ) \Rightarrow y = z)} Thanks.

Actually my sir asked me the definition of EMF so I just tell him that "Suppose a resistance(R) is connected across the terminals of a battery.A potential difference is developed across its ends.Current(or positive charge) flows from higher potential to lower potential across the resistance by...

The definition of 'Bounded above' states that: If E⊂S and S is an ordered set, there exists a β∈S such that x≤β for all x∈E. Then E is bounded above. The 'Least Upper Bound Property' states that: If E⊂S, S be an ordered set, E≠Φ (empty set) and E is bounded above, then supE (Least Upper...

Homework Statement In the Lectures, we are told that techniques like homogeneity and superposition work only for linear circuits, but in Chapter 3 of the Textbook (which is the only place I can find one) I see a definition of linearity as "A circuit is linear if and only if Homework...

I am doing an undergraduate project on bars and I am trying to derive the bar instability mode given by Mo et al. It says "whether or not a disk is globally stable depends on the global properties of the disk... it is not possible to write down a universal dispersion relation or stability...

I have been working in complex functions and this is a new animal I came across. Let γ be a piecewise smooth curve from -1 to 1, and let A=∫γa(x2-y2) + 2bxy dz B=∫γ2axy - b(x2-y2) dz Prove A + Bi = (2/3)(a-bi) In the past anything like this defined γ and I would find a parametric...

I will prove the following statement is true to show the flaw of 5.1 Definition in Baby Rudin. If in any case I'm wrong, please correct me. Thanks. Statment: Suppose f is a function defined on [a,a] with a \in \mathbb{R}. Then it is impossible to apply 5.1 Definition in Baby Rudin for this...

One of the definitions of the tensors says that they are multidimensional arrays of numbers which transform in a certain form under coordinate transformations.No restriction is considered on the coordinate systems involved.So I thought they should transform as such not only under rotations but...