Natural Definition and 1000 Threads

Few days ago, I was thinking about why we need to define V*=Hom(V,K) for a K-vector space when the dimension of V is finite because then V* and V both will have the same dimension and will be isomorphic. So, I couldn't understand why such a thing would be even called a dual vector space if it's...

My question is: Show the limit of x_{n}=\frac{ln(1+\sqrt{n}+\sqrt[3]{n})}{ln(1+\sqrt[3]{n}+\sqrt[4]{n})} as n approaches infinity Solution: {x_n} = \frac{{\ln (1 + {n^{\frac{1}{2}}} + {n^{\frac{1}{3}}})}}{{\ln (1 + {n^{\frac{1}{3}}} + {n^{\frac{1}{4}}})}} = \frac{{\ln \left(...

Often there's these stories on TV about someone accidentally damaging a large diameter natural gas pipeline while digging a ditch etc. and then a huge raging fire. What I'm wondering about is how frequently are shutoff valves installed on these lines. My intuition says ~30 miles apart at most...

is there a straight formula for finding one of the natural (preferably fundamental) frequency of a steel flat with dimension say, L=1000 mm, B=100 mm and Thickness T=10 mm ? The flat is fixed at both ends by bolts and two weights of equal magnitude are suspended at 1/4 and 3/4 (i.e. 250 mm and...

is black hole a theoretical concept or a natural phenomenon?

Hi I am working on a problem that ends up having the natural log of a negative e which I'm confused on how to find the explicit solution. The Problem: Find an explicit solution with C. y'-e^{-y}cos(x)=0 My Conclusion: First of all, I'm confused how I should solve this explicitly if I'm...

Hello, everybody: I was wondering if you knew if there is a book, journal, webpage or a compendium of some sort in which different and diverse curious common and uncommon natural phenomena are explained with physics (preferably with math involved). For example, a place where to look if I'm...

Hi! I am trying to compute the natural frequency of a system and I also need to add the added mass term to my equation. The object I have is a torus and is basically in heaving motion. I found a good paper, which gives a formula for 3d added mass and 2d added mass. I would figure that I...

Which do you think would be more lucrative in the long run? I am in computer engineering, and I would like to do my own startup, but I am exploring other options as well.

$a,b,c,d,e,f,g \in N$ $a<b<c<d<e<f<g$ $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}+\dfrac{1}{e}+\dfrac{1}{f}+\dfrac{1}{g}=1$ please find one possible solution of a,b,c,d,e,f,g (you should find it using mathematical analysis,and show your logic,don't use any program)

I have seen a lot of examples of sets with same cardinality as the natural numbers. For instance the even numbers or the cartesian product. In any case the proof amounted to finding a way of labeling the elements uniquely. But I am curious - can anyone give me an example of a set, where this...

Greetings, This question makes reference to the stat mech book, “Fundamentals of Statistical and Thermal Physics”, by Reif, so people who have that book will probably understand where I am coming from most easily. However, the main points/questions of this post are independent of the book, so...

Is there a smallest possible amount of energy? Like, one where all energy quantities in the universe are a multiple of it and the only way to have less energy is to have none at all? If a photon had that energy, it would be the longest possible EMR wavelength. Or can energy be divided...

Homework Statement Determine whether Ʃ(n from 1 to infinity) ln(n)/n^3 converges or diverges using the limit comparison test. Homework Equations I must use the limit comparison test to solve this problem-not allowed to use other tests. The Attempt at a Solution I know that the...

I am reading a textbook on special relativity and the author says at one point that he switches to natural units where c=1. The way he quickly explains this is by arguing that if c=3 x 10^8 m/s then we are working in units where seconds = 3 x 10^8m . . . . This has always baffled me and I never...

Homework Statement Use x=-1/2 in the MacLaurin series for e^x to approximate 1/sqrt(e) to four decimal places.Homework Equations The Attempt at a Solution \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + x^2/2 + x^3/6 + ... For this particular power series, I have: \sum_{n=0}^\infty...

in HEP research and lit, is it common to think and write about the masses of the various leptons and quarks as relative to the invariant mass of some chosen standard particle? such as the electron? or something else?

Is there an exact reason for the non-uniformness or is it all just random?I mean in the early stages of Earth's development when It was still hot, shouldn't all types of substances have spread more or less evenly?

As science progresses and unlocks mystery after mystery is it natural to be scared of what they may discover next? I am already in awe at how insane our universe is how some of it just defies all common sense. The first time I felt scared was when I was about 16 and I first read about special...

Homework Statement I'm still not confident enough to answer this. Prove that if a set ##A## of natural numbers contains ##n_{0}## and contains ##k + 1## whenever it contains ##k##, then ##A## contains all natural numbers ##> n_{0}##. Homework Equations Proof by induction methods The Attempt...

Homework Statement Taken from Spivak's Calculus, Prologue Chapter, P.28 b) Notice that all numbers in Pascal's Triangle are natural numbers, use part (a) to prove by induction that ##\binom{n}{k}## is always a natural number. (Your proof by induction will be be summed up by Pascal's...

Homework Statement L(θ) = ∏(θ/(2√xi)*e^(-θ√xi)),i=1, n Homework Equations The Attempt at a Solution -> θ2∏(1/(2√xi)*e^(-θ√xi)) taking natural log of both sides lnL(θ) = nlnθ + ln∏(1/(2√xi)*e^(-θ√xi)) = nlnθ + Ʃln(1/(2√xi)*e^(-θ√xi)) Ok so from what I understand the...

Hi all, I need Explanation on the attached image from Van Dalen's Logic and Structure; specially on how the red part follows from the lines before it! Regards.

Find the smallest natural number with 6 as the last digit, such that if the final 6 is moved to the front of the number it is multiplied by 4.

According to most science jobs websites, like jobs.sciencecareers.org, the number of jobs in biology far exceeds the number of jobs in every other natural science, like physics or chemistry. What gives?

In Natural deduction in Predicate logic we have a rule which says [assume the set of hypotheses to be H) if H implies phi(x) then H implies [for all x phi(x)] such that x doesn't belong to FV(psi) for all psi in H [indeed such that x occurs free in no one of formulas in H] In other words, if...

Homework Statement Prove that if n is a natural number greater than 1, then n-1 is also a natural number. (Hint: Prove that the set {n | n = 1 or n in \mathbb{N} and n - 1 in \mathbb{N} } is inductive.)Homework Equations The Attempt at a Solution S(n) = \sum_{j = 2}^{n} j = 2 + 3 + \cdots...

In my very limited mathematical knowledge, at first the answer seems to be as N →∞ 1/N which we know equals zero. But, what if the number is 7, and that is what was picked?

Why is it that the magnetic component of light interacts more with metamaterials than with naturally occurring materials? It would be nice to explain this in terms of inductance, permeability, and permittivity within SRR metamaterials. I'm currently an undergraduate participating in...

Recently in the 'Challenge Forum' the following integral has been proposed... $$\int_{0}^{\infty} \frac{\ln x}{x^{2}+ a^{2}}\ d x\ (1)$$ Scope of this note is to illustrate a general procedure to engage integrals like (1) in elementary way, i.e. without use comnplex analysis tecniques. The...

Sorry about the intriguing title; this is just a continuation of the discussion in https://driven2services.com/staging/mh/index.php?threads/5216/ from the Discrete Math forum. The original question there was how to introduce mathematical induction in a clear and convincing way. Since the current...

ln(a * b) = ln(a) + ln(b) and ln(a/b) = ln(a) - ln(b) correct?

Hi there, First time posting in Chemistry, so go easy on me. I'm wondering if there's any fundamental difference between synthetic and naturally-occurring compounds. By "fundamental" I mean some physical/chemical difference besides the process in which they came to be. Related to this...

Realised I probably posted this in the wrong forum before, should've been here.. I often see a function's differential expressed in terms of convenient partial derivatives eg dU=(dU/dT) dT + (dU/dV) dV And I've seen it written that "any system is uniquely specified by two parameters, such...

is there any way to prove or disprove the statement: y=3x^2+3x+1 is prime for all x belongs to natural numbers...

Homework Statement The switch in the circuit in the figure has been open a long time before closing at t=0. Find iL(t) t>0 Express your answer in terms of t, where t is in milliseconds. https://www.flickr.com/photos/84781786@N03/8899251427/ The Attempt at a Solution We simplify...

I am trying to explain metric spaces and finding it hard to come up with simple to understand, interesting examples of metrics that are not translation invariant. The audience is people who are just now studying general metric spaces.

I want to verify this: 2ln(x)-ln(2x)=ln(x^2)-ln(2x)=ln\left(\frac{x^2}{2x}\right)=ln\left(\frac x 2\right) ln(2x)-ln(x)=\ln\left(\frac {2x}{x}\right)=ln(2) Thanks

So I have an exam tomorrow, and the teacher provided a review. f(x) = ln(x + y) I remember that d/dx ln[f(x)] = f'(x)/f(x) so would that not equal 2/(x + y) ? The answer she gave is 1/(x + y - 1) ... where that neg. one came from I have no idea. Come to think of it, there were no...

Hope this is in the right place... I'm trying to understand why the derivative of ln(x) is 1/x while the derivative of something like ln(4) is 0. My knee-jerk reaction is to view 4 as representative of x, thereby giving me F'(x) ln(4) = 1/4, not 0. That would be the case, except ln(4) is a...

"Quantum Optics," by Scully and Zubairy, attempts to explain the natural line width of an emitting atomic ensemble (in a resonant cavity with leakage) using random spontaneous emission events of the atoms (which in turn is due to interactions with the vacuum field). The basic idea and...

Given n numbers x1,x2...xn belong to N. x1+x2+x3...xn=m How many different combinations of x1,x2,x3...xn are there?Is there any formula useful here? Note:x1,x2,x3... need not be distinct and also can be 0. Thanks

Homework Statement \int^{1}_{0}\int^{e^x}_{e^-x}\frac{lny}{y}dydx The attempt at a solution So I am integrating ln(y)/y and I tried it by parts, first with u = ln(y), dv = 1/y, and therefore du = 1/y, and v = ln y but if I use that I get (ln(y))2-\int\frac{lny}{y} again. So I tried...

When deriving the natural level of output i am confused as to how the leap is made in the last two lines. P = P(1+μ)F(1-Y/L,z) => 1 = (1+μ)F(1-Y/L,z) => Yn = h(μ,L,z) I understand how the mark-up, Total Labour and variable factors are the factors of the natural level but how does...

Homework Statement The length L1, is not the wavelength of the fundamental frequency of the string. With the tension equal to F1, to which natural frequency does the wavelength equal to L1 correspond? Homework Equations I was reading online, and found that when a string vibrates at...

[FONT=Times]I figured I would just add this new problem over here, rather than starting a new thread. Im looking to solve integration leading to arctan or arcsin results. \int_{1}^{e}\frac{3dx}{x(1+\ln(x)^2}) Looking at this, it feels like this has an arctan in the result, but I would have to...

The natural logarithm can be expressed as lim n(x^(1/n)−1) n->ifinity Can someone please derive this for me? I've tried using the limit definition of e^x and then applying f^-1(x) but i can only get ((n*x)-1)^1/n any help is greatly appreciated thanks!

There are 3 long lived (over 10^9 years) isotopes which may capture electrons and have actually been observed to undergo single beta transmutation. Interestingly, all these 3 ALSO can undergo beta decay AND positron emission. Potassium 40 has a branching ratio of about 1000 electron...

please help i have to write a MATLAB code for DETECTION OF THE LOCATION AND SIZE OF A CRACK IN STEPPED CANTILEVER BEAMS BASED ON MEASUREMENTS OF NATURAL FREQUENCIES the reference paper is DETECTION OF THE LOCATION AND SIZE OF A CRACK IN STEPPED CANTILEVER BEAMS BASED ON MEASUREMENTS OF...

I have learned that beta-lactamase enzymes have very ancient origin. And they are just tinkered in the recent anthropogenic activity. So some original form must be out there. What are their natural target?