A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories, intensional definitions (which try to give the sense of a term) and extensional definitions (which try to list the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions.In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what a mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed.
I know the logic of proving/disproving mathematical statements, I learned it by reading books, texts regarding to the matter, lots of exercises ( in the subject of how to do mathematical proofs and in the subject of proving/disproving statements in my math courses [ e.g. Linear Algebra, Real...
Why, in lagrangian mechanics, do we calculate: ##\frac{d}{dt}\frac{\partial T}{\partial \dot{q}}## to get the (generalised) momentum change in time
instead of ##\frac{d T}{dq}##?
(T - kinetic energy; q - generalised coordinate; p - generalised momentum; for simplicity I assumed that no external...
I'm trying to wrap my head around the definition of a candela:
"The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540×10^12 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian."
I'm aware...
Hello there,
I have heard countless times about the word energy, but I still don't know what exactly an energy is. Like for example, we know the definition of a physical quantity called velocity which by definition is change in displacement (so we can easily describe what a velocity is). We...
I understand abstraction to be the process of representing a system, concept or process in simple terms, leaving out finer details which may obfuscate the bigger picture.
My understanding of layers of abstraction is that it refers to different depths of understanding of a concept where each...
Reduced graph states are characterized as follows from page 46 of this paper:
Proposition: Let ##A \subseteq V## be a subset of vertices for a graph ##G = (V,E)## and ##B = V\setminus A## the corresponding complement in ##V##. The reduced state ##\rho_{G}^{A}:= tr_{B}(|G\rangle\langle G|)## is...
I'm trying to learn about Abstract Wiener Spaces and Gaussian Measures in a general context. For that I'm reading the paper Abstract Wiener Spaces by Leonard Gross, which seems to be where these things were first presented.
Now, I'm having a hard time to grasp the idea/motivation behind the...
I was reading Fundamentals of Inket Printing and it said the following:
"The surface tension in a liquid causes a force to act in the plane of the free surface
perpendicularly to a free edge in that surface."
Can someone explain to me what this means? What's the direction of the force? I have...
In the paper "Steady-state spin synchronization through the collective motion of trapped ions" it states the following:
"Steady-state synchronization of atomic dipoles forms the foundation for ultra-stable optical lasers utilizing
narrow-linewidth atoms coupled to a lossy cavity mode. The...
Consider this Ito proces: $$dX_t = \mu(X_t,t)dt+\sigma(X_t,t)dW_t$$ with W_t being a wiener process. My question: What is the diffusion coefficient of X?
My motivation for asking: A lot if financial literature refer to "diffusion coefficient" and I haven't understood by googling it, because...
I do not want to get into a religious discussion. The question is an abstract one, not directed at any belief system. With that prologue: is it theoretically possible to give a rigorous and coherent definition of "supernatural" that would allow an observable event to be supernatural, independent...
I require your help to list all phenomena described as interferences in physics ( as teached nowadays ) with their citations in scholar documents if they are not well known by non-specialists. I am open to adjacent domains like information theory and mathematics.
There are already light...
Hi
I am a person who always have had a hard time picking up new definitions. Once I do, the rest kinda falls into place. In the case of abstract algebra, Stillwell's Elements of Algebra saved me. However, in the case of Spivak's Calculus on Manifolds, I get demotivated when I get to concepts...
In the book General Relativity for Mathematicians by Sachs and Wu, an observer is defined as a timelike future pointing worldline and a reference frame is defined as a timelike, future pointing vector field Z. In that sense a reference frame is a collection of observers, since its integral lines...
So I was reading this book, "Euclidean and non Euclidean geometries" by Greenberg
I solved the first problems of the first chapter, and I would like to verify my solutions
1. Homework Statement
Homework Equations
[/B]
Um, none that I can think of?
The Attempt at a Solution
(1)...
So I'm reading in the news and a group of scientists from the New Horizons missions
appear to be reopening "The Pluto Debate" with yet another new definition of planet
at the Lunar and Planetary Science Conference in March:rolleyes:. If accepted their definition
of planet would increase the...
I thought the definition was: a vectorial field that interacts with a moving charge
But wikipedia says it's a field generated by electric currents and magnetic materials...
https://en.m.wikipedia.org/wiki/Magnetic_field
Why is in the definition the way it is Generated?
When we had defined the...
Homework Statement
Hi, this is a question that has been bothering me for a while. (Im in calculus II at the moment)
Why do i need to derivate some functions by definition and other times i dont? for example if somebody asks me to calculate the partial derivatives of a branch function in a a...
In some elementary introductions to integration I have seen the Riemann integral defined in terms of the limit of the following sum $$\int_{a}^{b}f(x)dx:=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x^{\ast}_{i})\Delta x$$ where the interval ##[a,b]## has been partitioned such that...
Homework Statement
"For the given series, write formulas for the sequences an , Sn, Rn and find the limit as n->∞ (if it exists)
Homework Equations
∑∞1 ((1/n) - 1/(n+1)
The Attempt at a Solution
I know how to take the limit, that's no problem. I'm a bit confused about what an , Sn, Rn are...
I don't want to be an adult, this idea freaks me out and makes me feel old
Is 18 still an adolescent or is considered adult? When are you an adult and no more an adolescent?
20?
What's the difference between teen and adolescent?
And how did you dealed with the teen-adult transition? Was it...
I learned a lie group is a group which satisfied all the conditions of a diferentiable manifold. that is the real rigour definition or just a simplified one?
thanks
I was reading into traversable wormholes when I came across this definition from Matt Visser; 'If aMinkowski spacetimecontains a compact region Ω, and if the topology of Ω is of the form Ω ~ R × Σ, where Σ is a three-manifold of the nontrivial topology, whose boundary has topology of the form ∂Σ...
My professor says resistance depends on,
R\quad =\quad \frac { \rho L }{ A }
And is defined as,
R\quad =\quad \frac { V }{ I }
What does she mean? What is the difference?
A definition is denoted by \equiv, what is the difference between that and an ordinary equal sign?
why were quantities like momentum, force , potential energy, kinetic energy,work ,etc needed to be introduced in physics?
and why were they defined the way they are defined?.
would it not be possible to explain nature without defining these quantities or by using alternate physical quantities ?
A lot of web pages/books show how to use cosx=sin(Pi/2-x) and the chain rule to prove that the derivative of
cosx=-sinx. My question is how to use this identity and the defintion of the derivative to prove the same thing.
Or whether it is at all possible. Seeing that i get...
In his book, Euler gives the definition of a variable to be : "A variable quantity is an indeterminate or universal quantity, which includes within itself all completely determined values." What does he mean exactly in the last part of the sentence?
I have been trying for a while to read a precise definition of a Vector Multiplet (to whom ##N=2## Supergravity theories couple to in ##4D##) but was not lucky in finding a self-contained one. The best I got was that on https://en.wikipedia.org/wiki/Supermultiplet though it was on...
Hey guys, I need an explanation on the definition of the electric field. It was said in a post that " the definition of the electric field is defined in terms of how it is measured or tested". What do they mean by measured/tested?
If my understanding is correct the definition of a derivative is lim h->0 (f(x+h)-f(x))/h However, I've also seen this used: lim x->c (f(x)-f(x))/(x-c) are these both considered valid definition for the derivative or does the derivative have to tend towards zero? I am a bit confused because I...
I've been grappling with the idea in my head as to how I would explain to someone exactly what equality between two mathematical objects actually means. This maybe a very stupid question, so apologies in advance, but if I'm honest I struggle to come up with an answer that doesn't involve using...
When I learned the concept of specific heat capacity, I knew that 1J/(K*kg) means that it takes 1 Joule of energy to increase the temperature of a kilogram of matter by one Kelvin, but what does J/K, the unit of entropy, mean?
From my research I am showing that if an input signal becomes too slow (ie: a low slew rate) then the noise can cause multiple state changes.
But I am being told that if the slew rate is low then it will get rid of unwanted noise.
I read my results from this: (I lost the link but I did copy...
I would be grateful for some direction on this.
I wish to implement the following -
Given a deterministic signal (the feedback signal of a closed-loop stable system) I would like to plot the power spectral density.
The definition I am working with is this:
Implementation (MATLAB):
%...
Homework Statement
In writing the definition of ##e## i.e. ##e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n##, why do we denote the variable by 'n' despite the fact that the formula holds for n∈(-∞,∞)? Is there any specific reason behind this notation i.e. does the variable have...