Property Definition and 608 Threads

According to my notes, the absorption law states that p ∨ (p ∧ q) = p, p ∧ (p ∨ q) = p I have found a video where they were discussing a partial absorption such as ¬q ∧ (¬p∨q) = ¬q ∧ ¬p This is not in my notes, but is this correct? specifically, is the terminology used to decribe this property...

This is part of my note: Now, this is practice question: I want to ask why P(X = 2) is not zero, because from the note: P{X = a} = ##\int_{a}^{a} f(x) dx=0## ? If I differentiate F(x), I will get f(x) which is the pdf, then using the pdf to find P(X = 2), I think I will get zero as the...

First lets focus on ##|x|## which is defined as distance between ##x##and ##0##. But if we look into it closely $$13=|-11-2|$$ which is distance between -11 and 2 but $$13=|11-(-2)|$$ which means this is distance between 11 and -2. Which is it? In the same way $$x=|x-0|$$ is distance between 0...

I just learned that if two linear operators do not commute, this means when we use operators to characterize observables in quantum mechanics, the corresponding observables cannot both be definite at the same time. This seems hard to believe to me since I have a strong intuition, perhaps...

Let ##X## be a topological space, and let ##\mathscr{F}## be a sheaf on ##X##. Show that if ##\mathscr{U}## is an open cover of ##X## such that the restriction ##\mathscr{F}|_U## is flasque for every ##U\in \mathscr{U}##, then ##\mathscr{F}## is flasque. Note: A sheaf ##\mathscr{G}## on ##X##...

Could someone check whether my proof for this simple theorem is correct? I get to the result, but with the feeling of having done something very wrong :) $$\mathcal{L} \{f(ct)\}=\int_{0}^{\infty}e^{-st}f(ct)dt \ \rightarrow ct=u, \ dt=\frac{1}{c}du, \ \mathcal{L}...

I recall that there was an argument from Born expansion showing that exchange of odd spin between equal sign charges generates a repulsive potential, and if the charges are different or the spin is even the potential is attractive. I wonder, how does it work for non abelian gauge theory...

Hi PF! Everyone knows that: $${\varphi }^2 - \varphi - 1 = 0$$ But guess what? $${\varphi}^3-2{\varphi}^2+1=0$$ Generalizing this for all n-bonacci numbers: $$x^{n+1}+1 = 2x^n$$ where ##x## is the n-bonacci number and ##n## is the degree of the polynomial that the n-bonacci number is a root of...

So, I have to come up with some set which is lub A. Now, A is a subset of R, so each member of A is a Dedekind left set. So, A is a set of sets. Now, I propose that the following set would be lub of ##A##. $$ \alpha = \bigcup A = \{ \beta | \exists \delta \in A (\beta \in \delta) \} $$...

A Dedekind cut is a pair ##(A,B)##, where ##A## and ##B## are both subsets of rationals. This pair has to satisfy the following properties A is nonempty B is nonempty If ##a\in A## and ##c \lt a## then ##c \in A## If ##b \in B## and ## c\gt b## then ##c \in B## If ##b \not\in B## and ## a\lt...

Hello, First of all, I checked several other threads mentioning duality, but could not find a satisfying answer, and I don't want to revive years old posts on the subject; if this is bad practice, please notify me (my apologies if that is the case). For the past few days, I have had a lot of...

Hi, I have to verify the sifting property of ##\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} e^{-sa}e^{st} ds## which is the inverse Mellin transformation of the Dirac delta function ##f(t) = \delta(t-a) ##. let ##s = iw## and ##ds = idw## ##\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-iwa}e^{iwt}...

Hello! Reading Roger's book on supermanifolds one can find sketch of the proof for multiplicative property of super determinant. Which looks as follows All the words sounds reasonable however when it comes to the direct computation it turns out to be technical mess and I am about to give up. I...

Hello! I am reading "Differential Geometry and Mathematical Physics" by Rudolph and Schmidt. And they have and example of manifold (projective space). I believe that there is a typo in the book, but perhaps I miss something deep. Definitions are the following $$\mathbb{K}^n_\ast=\{\mathbf{x}\in...

Here is this week's POTW: ----- Let $X$ and $Y$ be topological spaces. If $Y$ is compact, show that the projection map $p_X : X \times Y \to X$ is closed. -----

Do todays physicists have a deeper understanding on mass and inertia on how inertia arises?

I started by taking a derivative: $$E = \sum_{i=1}^{\alpha} (E_i^{(p)} n_i) \ \ \ | \cdot \frac{\partial}{\partial n_i}$$ $$\frac{\partial E}{\partial n_i}=\sum_{i=1}^{\alpha} [\frac{\partial E_i^{(p)}}{\partial n_i}n_i + E_i^{(p)} \frac{\partial n_i}{\partial n_i}]$$ $$\frac{\partial...

In a 2012 article published in the Mathematical Gazette, in the game of golf hole score probability distributions were derived for a par three, four and five based on Hardy's ideas of how an hole score comes about. Hardy (1945) assumed that there are three types of strokes: a good (##G##)...

In some textbooks it is given that - Electric charge is the characteristic property of matter that causes it to experience a force when placed in an electromagnetic field. and In other textbooks it is given that - Electric Charge is the property of subatomic particles that causes it to...

As it is mentioned fluorescence is a singlet to singlet transition and this is the reason that fluorescence is a fast process. now consider the Eu doped phosphor material where 5D0--->7F2 and other transitions show the prominent intensity peaks in down-conversion process. those are not singlet...

I will quote this statement from another thread: In that thread number of other posters seemed to agree with this statement. So I tried to analyze it a bit. For the sake of my questions let's say we limit GR to Schwarzschild spacetime and if there are problems with gravitational potential...

Hi, I just have a quick question regarding the linear phase property of filters. It might be easier to provide some context before getting to the question, but feel free to skip to the bottom. Consider a system input as a discrete sequence obtained by sampling at t = 0, T, . . . , kT from an...

That may sound really silly, and that may be due to my lack of understanding of the operations itself, but: if ##|\vec{a}\times\vec{b}|=|\vec{a}|\cdot|\vec{b}|sin\theta##, being ##\theta## the angle between the two vectors, how could ##\vec{b}\times\vec{a}## be different? Wouldn't it be just the...

We have the function d from VxV to another set(not necessarily R) for which the following properties are to be satisfied: i) d(x,y)=0<=>x=y ii)d(x,y)=d(y,x) iii)d(x,z)≤(d2(x,y)+d2(y,z))1/2 ∀ x,y,z ∈ V. What do you say? Would this function have interesting properties on a set and theorems to be...

Commutative property of addition. If a & b are integers then, a+b = b+a 2+3 = 3+2 5. Does not work for subtraction. 2-3 = -1 3-2= 1 Having said that, what about the special case with negative numbers (when we also move their respective signs) -5 + 7 = 2 & 7 + (-5) = 2. 15 -7 = 8 & -7 + 15...

I came across this line in my java textbook-"Abstraction is the absolute property of a class".i want to know what does absolute property exactly mean and why it is considered an absolute property?Also how does it effect(or is useful) when we practically do programming?

This is why I don't like intellectual property laws: Hacker Mods Old Calculator to Access the Internet, CASIO Files DMCA Complaint

For a random variable Ti, SD (Ti) / E (Ti) ≤ 1 with SD (Ti) = (Var (Ti))1/2 and E (Ti) the expectation of Ti and Var (Ti) the variance of Ti. My question now is whether the following property then also applies. For any variable T, SD (T) / E (T) ≤ 1 where T = T1 + T2 + ... + TN and where...

I am currently writing my thesis and basically the conclusion is, in part, a statement about how this work can be built upon. I received a post-doc that is almost a natural extension of my thesis. One of the reasons I accepted it was so I could work on some ideas that I have had that I never...

Is there some property that I can look up which would tell me how much water a hygroscopic salt can absorb (per unit mass of salt; for example anhydrous lithium chloride) before it's saturated and won't absorb any more?

Summary:: If ##f(x)=-f(x+L/2)##, where L is the period of the periodic function ##f(x)##, then the coefficient of the even term of its Fourier series is zero. Hint: we can use the shifting property of the Fourier transform. So here's my attempt to this problem so far...

Spring has more potential energy when it is compressed or stretched from its initially balanced state. As external work is done, it stores energy in the form of potential energy. Here, we know energy is stored in spring but For the Earth-ball system, where the energy stored?

Definition: Let ##G## be a graph. ##G## is a functional graph if and only if ##(x_1,y_1) \in G## and ##(x_1,y_2) \in G## implies ##y_1=y_2##. Problem statement, as written: Let ##G## be a functional graph. Prove that ##G## is injective if and only if for arbitrary graphs ##J## and ##H##, ##G...

My only qualm is that the statement “Let G be a functional graph” never came into play in my proof, although I believe it to be otherwise consistent. Can someone take a look and let me know if I missed something, please? Or is there another reason to include that piece of information?

A recent https://mathhelpboards.com/potw-secondary-school-high-school-students-35/problem-week-411-apr-5th-2020-a-27196.html#post119308 asked about properties of a pair of positive integers $x$, $y$ such that $2x^2+x = 3y^2+y$. But it is not obvious that any such pairs exist. So the challenge...

Hey! :o When it is given that a signal $x(t)$ has a real-valued Fourier transformation $X(f)$ then is the signal necessarily real-valued? I have thought the following: $X_R(ω)=\frac{1}{2}[X(ω)+X^{\star}(ω)]⟺\frac{1}{2}[x(t)+x^{\star}(−t)]=x_e(t) \\ X_I(ω)=\frac{1}{2i} [X(ω)−X^{\star}(ω)]⟺...

In Weinberg's Cosmology, the comoving coordinate described as "A particle at rest in these coordinates will, therefore, stay at rest, so these are co-moving coordinates" Now when we write the proper distance ##s = a(t)\chi## where ##\chi## is the comoving coordinate. Taking the time...

In the brilliant.org website talking about quantum properties it is said that neutrons coming from a nuclear oven and passing through two permanent magnets of opposite polarity hit a surface only at the top and the bottom of it (there is no continuity) because the spin property is quantized and...

In chapter 3 of Sean Carroll's Introduction to General Relativity he 'makes the demand' of metric compatibility of a connection that ##\nabla_\mu g_{\lambda\nu}=0##. Metric compatibility becomes a phrase that is used frequently. However metric compatibility seems to arise naturally. One only...

Summary: Show that for this family of functions the following semigroup property with respect to convolution holds. Hi. My task is to prove that for the family of functions defined as: $$ f_{a}(x) = \frac{1}{a \pi} \cdot \frac{1}{1 + \frac{x^{2}}{a^{2}} } $$ The following semigroup property...

I was recently trying to explain to a grandchild the relative nature of velocity (the different paths of a coin dropped by a passenger on a train, as seen by the passenger on one hand and a trackside observer on the other), and the invalidity of the concept of absolute velocity. For some reason...

There is a property to geometric distribution, $$\text{Geometric distribution } Pr(x=n+k|x>n)=P(k)$$. I understand it in such a way: ##X## is independent, that's to say after there are ##(n+k-1)## successive failures, ##k## additional trials performed afterward won't be impacted, so these ##k##...

I am reading a lot of stuff on advanced algebra and running into these questions. Thank you

This is a bit of a vague question, but I was wondering if someone could explain. As far as I know, potential energy is formally a property of a system (for instance, the GPE of two gravitationally attracting particles). In many physics problems it happens to be the case that one of the bodies...

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...

Here is how I thought about it Consider a surface on which atoms bump into, and if I increase the number of atoms and at the same time allow the surface area to increase as well the pressure is still the same because these atoms occupy have size and thus occupy a certain area , and If they are...

Let a 3 × 3 matrix A be such that for any vector of a column v ∈ R3 the vectors Av and v are orthogonal. Prove that At + A = 0, where At is the transposed matrix.

What would you change about air to make it have the same density at sea level but the atmosphere would only be a few miles high instead of a several hundred miles high? I am a high school physics teacher. As I ponder this possibility, my first thought is I could increase the strength of the...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ... I am focused on Chapter 3: Convergent Sequences I need some help to fully understand some remarks by Garling made after the proof of Theorem 3.1.1...

Hi. I'm learning Quantum Calculation. There is a section about controlled operations on multiple qubits. The textbook doesn't express explicitly but I can infer the following statement: If ##U## is a unitary matrix, and ##V^2=U##, then ## V^ \dagger V=V V ^ \dagger=I##. I had hard time...