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    I Discrete Optimization Problem?

    Consider the expression: $$A = \frac{ M! }{ r_1!\ r_2! }$$ where M = r_1 + r_2 , where r_1 = (M - 2r_2) $$A = \frac{ (r_1 + r_2)! }{ r_1!\ r_2! } \\ \ \\ \ = \frac{ ((M-2r_2) + r_2)! }{ (M-2r_2)!\ (r_2)! } \\ \ \\ \ = \frac{ (M-r_2)! }{ (M-2r_2)!\ r_2! } $$ Then, for a...
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    I Measuring Pressure For Compressible Fluid Systems?

    Is measuring pressure for a compressible fluid system angle dependent? For a compressible fluid, Bernoulli's Law gives us a relation between two points along a closed system. More specifically it gives us the relation between two cross sections belonging to two distinct points in the closed...
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    B What Does It Mean For A Force To Act On A Particle?

    I know this is kind of a dumb question but please forgive me it's been awhile. Is it enough for a particle to merely "feel" an external force F to state that "F is acting on the particle"? ie if the particle was confined in a potential well and experiences F but does not move. or does...
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    I Why does a metal ring, slid onto a solenoid, fly off?

    Our professor did a demonstration today for our physics II class. She was demonstrating electromagnetic induction with a solenoid and a metal ring. When she slid the ring down the solenoid and passed alternating current through the solenoid, the metal ring was flung upward off the solenoid. My...
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    I Why are accelerating cars less fuel efficient?

    What is it about acceleration that makes it less fuel efficient?
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    B Best fit curve associated with the combination formula

    Sorry if this is more of a HW question (if so then moderator please move my question. Thanks!) Hi, I'm trying to get an expression for a best fit curve of the combination formula (##_nC_r##). As far as I can tell, the curve is a simple parabolic curve, and its shape doesn't change. It's just...
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    I How do you move floors and ceilings in discrete math?

    oops, sorry, here's the other side ##\frac{log(stuff)}{8t} < 1 ## ##\frac{stuff}{10^{8t}} < 10 ##
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    I How do you move floors and ceilings in discrete math?

    The title more accurately should have been "How do you cancel floors and ceilings in discrete functions" For instance, ##\frac{log{\frac{3x}{-6(z)}}}{8t} < 1## If I wanted to get rid of the log, I'd just raise the expression by base 10. ##\frac{(\frac{3x}{-6(z)})}{10^{8t}} < 10^1## But what...
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    A Finding the bounds of a ratio

    I know I stated that simplification was my goal. But I suppose that's not the only way to achieve my "true goal", which I'm still getting closer to.. (sorry) I postulate that there is a relationship between the numbers in the given set, that dictates its closeness to the value 1. I just want to...
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    A Finding the bounds of a ratio

    ##\{100\ ,\ 1\}## yields... ##numerator: log_2{ \frac{ 101! }{ 100! \cdot 1! } } = log_2{ \frac{ 101! }{ 100! } } = log_2{101} =7## ##ratio: \frac{7}{102} = 0.0686## (remember the numerator is logged by base 2)
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    A Finding the bounds of a ratio

    For any set of values I've tested so far, the ratio has not exceeded the value 1. In fact i haven't found it to ever reach 1. I wanted to know what properties of a given set determined its closeness to the value 1. I hope this makes sense.
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    A Finding the bounds of a ratio

    The fraction within the log is always greater than or equal to one, and only gets larger and larger with increasing ##R## and ##\bar{r}##. But I don't know how I would go about expressing the rate of change with respect to the denominator of the ratio. Hmm, I guess... what I'm looking for is...
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    A Finding the bounds of a ratio

    Sorry in advance if I've posted in the wrong section. given the set ##\{r_i, r_{ii}, r_{iii}, ... , r_R\}## where ##r \ \epsilon \ \mathbb{Z}_+ \ , \ r_i \geq r_{i+1}## How would you go about finding the bounds of something like this, or determining if it even has any? ##( \...
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    Simplifying a Product

    Homework Statement This is a child thread I'm creating from a previous topic: https://www.physicsforums.com/threads/combinatorics-problem.871661/#post-5473920 In that thread, I was helped to come up with the expression for the number of arrangements of R distinct types of objects given the...
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    Combinatorics Problem

    Hmm? I mean the bounds are just 1 to however many colors I have right?
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    Combinatorics Problem

    Ah okay, from the point you make then I can see that I am over-counting somewhere, but having a number of different colors greater than two.... Hmm... ooh is it a multiplicitive system?? Since, for each case of pen_type arrangement, there are .. sub-states for each arrangement of the other...
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    Combinatorics Problem

    The purpose of mentioning the colors is so that one would treat them the same. I don't quite follow. Can you elaborate?
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    Combinatorics Problem

    Homework Statement I have: 4 Blue pens 16 Green pens 7 Red pens 11 Yellow pens If I lay out all the pens in a single row, how many different arrangements does this system have? Homework Equations $$_nC_r = \frac{n!}{r!(n-r)!}$$ The Attempt at a Solution Procedure: Basically the number of...
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    I Solving This Polynomial

    Is there a numerical method to solve something like this?
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    I Solving This Polynomial

    I want to keep this question conceptual and qualitative (for now). I have the following polynomial $$\frac{(ar-1)(ar-2)(ar-3)(ar-4)(ar-5)}{(r-1)(r-2)(r-3)(r-4)(r-5)} = P$$ where r is the variable I'd like to solve for and P, a are just real constants. I was wondering whether or not I could use...
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    I What's The Discrete Math Derivative Equivalent?

    $$ƒ = b^n$$ $$ b,n,I ∈ ℤ $$ Condition: Upon choosing a base value b.. $$ n | b^n ≤ I $$ (n is determined based off the value of b to yield the highest ƒ without going over I) $$1<b<L , L<<I$$ where I is some large number, and L is also sufficiently large such that we want to avoid going...
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    How Signals Are Sampled and Stored As A Fourier Transform?

    yyup! I mean the only thing I'm tripped up on is turning that table into a function. I can take a given function and express it with sines and cosines via the Fourier Transform, but I need to know how to turn those amplitude values into a function for me to use the Fourier Transform.
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    How Signals Are Sampled and Stored As A Fourier Transform?

    Of course, here you go: In my PDE class we were always given a base function to put in terms of sin and cos. But when taking a bunch of samples, all I end upwith is a table/array over some time T. How might I use this to store it in a fourier Transform function (dft)? If you could specify what...
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    How Signals Are Sampled and Stored As A Fourier Transform?

    This question is a little basic but.. how are signals stored in a Fourier Transform function f(t)? In my PDE class we were always given a base function to put in terms of sin and cos. But when taking a bunch of samples, all I end up with is a table/array over some time T. How might I use this...
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    I Alternative Ways To Find The Remainder of A/B?

    Are you just multiplying it by ten because it's quick and easy to do? I meant 371 / 19. But I think I get the flow. But just for my understanding what would you do if it was 155 / 19?
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    I Alternative Ways To Find The Remainder of A/B?

    Where were you getting the double tens, and the single three from? Were the double tens systematically determined or just randomly? what if the dividend was 371, what would the double tens then be?
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    I Alternative Ways To Find The Remainder of A/B?

    I guess this would be a Number Theory question. Short of actually going through the division process, is there another way to find the decimal remainder of an arbitrary set of integers { A , B } $$\frac{A}{B} , A > B$$
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    Which is more processor-intensive?

    What's more process intensive: 5465411784154564 / 3 or 5465411784154564 / 5746845218 ie a big number / small number or a big number / big number? does it matter if I want to take the modulus instead? ------------------ EDIT: Accidentally messed up my title, if a moderator could change it...
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    Optimization Problem -- Can a number always be expressed as the ratio of two other numbers?

    Take a number r that is n-digits long where n is finite. so if r =2385813... $$r_1r_2r_3...r_n$$ $$r_1 = 2$$ $$r_2 = 3$$ $$r_3 = 8$$ etc.. I postulate (since I don't know this is true): Every such number can be expressed as a division between two other numbers, say a and b. $$r = \frac{a}{b}$$...
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