March 2018 - mathfun@mailman.xmission.com

Re: [math-fun] SQRT occurring in inventory management optimization ?
by Henry Baker 06 Mar '18

06 Mar '18

Thanks! There appear to be many different SQRT examples in optimal inventory management, but the SQRT's seem to come from two different sources: fixed cost amortizations and random walks from statistical deviations in Gaussian/normal distributions. Here's the simplest example I can think of: withdrawing cash from an ATM machine. s is constant spend rate per year; s = total $ spent in one year. r is simple interest rate per year. A is ATM fee per withdrawal; A is independent of amount of cash withdrawn. C is the amount of cash withdrawn on each trip to the ATM. We want to find the optimal amount of cash C* to withdraw on each trip to the ATM. C* minimizes the total ATM fees in one year + total interest foregone in one year. Trips to the ATM in one year = s/C Total ATM fees in one year = A*(s/C) Mean amount of cash on hand = C/2 (spend rate is constant) Simple interest foregone in one year = r*(C/2) Total costs in one year: T = A*(s/C) + r*(C/2) dT/dC = -s*A/C^2 + r/2 = 0 Solving for C, we get C* = C = sqrt(2*s*A/r) Suppose A=$5, r=3%, s=$30/year, C*=$100. i.e., even though I spend only $30 cash per year, I'm still best off withdrawing $100 every trip to the ATM. #trips to the ATM in one year = 30/100 = .3 trips/year ATM costs per year = $5*0.3 = $1.50 Mean cash on hand = $50 Simple interest foregone per year = $50*3/100 = $1.50 So, with optimum cash withdrawal amounts, the ATM fees equal the interest foregone. https://en.wikipedia.org/wiki/Economic_order_quantity At 06:37 AM 3/6/2018, Seb Perez-D wrote: >Square roots do appear in optimal inventory >https://www.wiley.com/legacy/wileychi/waters/supp/Equations.pdf > >In particular (S,s) models >https://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.24.1.183 > >Cheers, > >Seb > >On 5 March 2018 at 21:33, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> I seem to recall a classical inventory management problem whose optimum >> solution involves a sqrt function. >> >> Since I never took a Operations Research course, I think that this problem >> showed up in a calculus class & used Lagrange multipliers. >> >> But after Googling a while, I couldn't find it. >> >> Anyone here recall a similar problem?

1 0

Re: [math-fun] Math-Fun topics
by Bill Gosper 06 Mar '18

06 Mar '18

Should be safe. Most of the antinukes have switched to you-know-what. Won't the next breakthrough be quantum computing? If it works. --rwg On 2018-03-02 09:44, Eugene Salamin via math-fun wrote: > Surely discussion of nuclear reactors should be an allowed math-fun topic. > > -- Gene > > > > On Friday, March 2, 2018, 9:28:38 AM PST, Richard Howard > <rich(a)richardehoward.com> wrote: > > Sorry about that--will do. > > There is the slight twist in that I was trying to respond to the issue of *new > technologies* that are down the road. > Leaving out geoengineering or pocket nuclear reactors and including Bitcoin > seems to be leaving the elephant in the living room from the "technology" > point of view. > > Adversarial AI is ok, though--right? > > --R > > On Fri, Mar 2, 2018 at 11:35 AM, <rcs(a)xmission.com> wrote: > >> We've had some new folks join us recently, >> who might be unaware of some of my arbitrary rules. >> >> The topics of politics and global warming are banned. >> >> Rich

6 5

[math-fun] SQRT occurring in inventory management optimization ?
by Henry Baker 06 Mar '18

06 Mar '18

I seem to recall a classical inventory management problem whose optimum solution involves a sqrt function. Since I never took a Operations Research course, I think that this problem showed up in a calculus class & used Lagrange multipliers. But after Googling a while, I couldn't find it. Anyone here recall a similar problem?

2 1

Re: [math-fun] SQRT occurring in inventory management optimization ?
by Dan Asimov 05 Mar '18

05 Mar '18

I don't know that one, but I once had a pure math problem that used a square root, too. Does anyone know which one I'm thinking of? —Dan ----- I seem to recall a classical inventory management problem whose optimum solution involves a sqrt function. Since I never took a Operations Research course, I think that this problem showed up in a calculus class & used Lagrange multipliers. But after Googling a while, I couldn't find it. Anyone here recall a similar problem? -----

1 0

[math-fun] everywhere
by françois mendzina essomba2 05 Mar '18

05 Mar '18

Hello... %e^z=(sqrt(1+z^2/n^2)+z/n)^n; sin(z)=(1/(2*%i))*((sqrt(1-z^2/n^2)+%i*z/n)^n-(sqrt(1-z^2/n^2)-%i*z/n)^n); cos(z)=(1/(2))*((sqrt(1-z^2/n^2)+%i*z/n)^n+(sqrt(1-z^2/n^2)-%i*z/n)^n); n-----infinity FME...

2 1

[math-fun] Averaging functions on trees
by Dan Asimov 04 Mar '18

04 Mar '18

Let G be an infinite connected graph G = (V, E) where V is a countably infinite set and E is a subset of V x V - diagonal, satisfying 1) and 2): ----- 1) Every edge e in E has 2 endpoints, and any 2 vertices v,w in V bound at most one edge. 2) If for any v in V, deg(v) denotes the number of edges v belongs to, then for all v we have 2 <= deg(v) < oo. ----- Let C denote all functions V(G) —> reals: C = {f : V(G) —> R} For any v in V let the star star(v) of v be the set of vertices w at the other end of all edges vx containing v: star(v) = {x in V | vx is in E} Define A : C —> C by letting A(f) at a vertex v be the average of f(x) on the star of v: A(f)(v) = (Sum_{x in star(v)} f(x)) / |star(v)|. Question: Which f in C approach a limit under iteration of A: lim A^n(f) = f^ n—>oo for some f^ in C ??? —Dan

2 1

Re: [math-fun] Infinite games
by Dan Asimov 03 Mar '18

03 Mar '18

Here's another (potentially) infinite game: Starting from zero, players write down successive ordinals in turn. The loser is the first person whose turn is illegible. —Dan >From: "Adam P. Goucher" <apgoucher(a)gmx.com> ----- Here's a fun game which works with any finite number k of players: Players take turns* to name positive integers that haven't already been named. When you name an integer, you subsequently 'own' it. Play continues (possibly beyond time omega) until every integer is owned. Clearly, it must stop before time omega_1. * For a positive integer n, Turn(n) is the sum (reduced modulo k) of the base-k digits of n. For an ordinal alpha, Turn(alpha) is the sum (again reduced modulo k) of the Turns of the coefficients of alpha in Cantor normal form. This is really fair. An integer n is described as 'dominated' by a player if there exists an infinite set S containing n such that all finite sums of elements from S are owned by that player. The winner is the person who dominates the smallest integer. (A winner exists by Hindman's theorem, and is unique by the fact that you can only dominate an integer you own.) > From: "Dan Asimov" <dasimov(a)earthlink.net> > One kind of infinite game discussed by logicians is determined by the choice of a > subset X of the half-open unit interval [0, 1). > > Then each player in turn selects a bit from {0,1}, and if the countable bitstring > thus created lies in X then the first player (W) wins; if not, the second player (B) wins. > > The Axiom of Determinacy states that every such game has a strategy for one player > or the other. Meanwhile, the Axiom of Choice can be used to show there exists a > subset X of [0,1) whose corresponding game has no strategy for either player. > > I'm hoping for a totally explicit game perhaps played by placing counters on > unoccupied cells of a board, as in Hex (or Conway's Angels & Devils game). > ----- > On Saturday, March 3, 2018, 4:36:13 AM PST, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: > If your game has length \omega , how do you know who wins? WFL > On 3/3/18, Dan Asimov <dasimov(a)earthlink.net> wrote: > > I am curious: Do people know of interesting infinite two-player games, > > preferably without chance playing a role? > > > > I'm mainly interested in discrete 2-player games with one player going > > first and with play alternating thereafter. Such that in some sense, > > except for that first player / second player thing, the challenge > > facing each player is as symmetrical as possible. > > > > Though I will not rule out continuous games. -----

1 0

Re: [math-fun] Infinite games
by Dan Asimov 03 Mar '18

03 Mar '18

One kind of infinite game discussed by logicians is determined by the choice of a subset X of the half-open unit interval [0, 1). Then each player in turn selects a bit from {0,1}, and if the countable bitstring thus created lies in X then the first player (W) wins; if not, the second player (B) wins. The Axiom of Determinacy states that every such game has a strategy for one player or the other. Meanwhile, the Axiom of Choice can be used to show there exists a subset X of [0,1) whose corresponding game has no strategy for either player. I'm hoping for a totally explicit game perhaps played by placing counters on unoccupied cells of a board, as in Hex (or Conway's Angels & Devils game). —Dan ----- On Saturday, March 3, 2018, 4:36:13 AM PST, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: If your game has length \omega , how do you know who wins? WFL On 3/3/18, Dan Asimov <dasimov(a)earthlink.net> wrote: > I am curious: Do people know of interesting infinite two-player games, > preferably without chance playing a role? > > I'm mainly interested in discrete 2-player games with one player going > first and with play alternating thereafter. Such that in some sense, > except for that first player / second player thing, the challenge > facing each player is as symmetrical as possible. > > Though I will not rule out continuous games. -----

2 1

[math-fun] Infinite games
by Dan Asimov 03 Mar '18

03 Mar '18

I am curious: Do people know of interesting infinite two-player games, preferably without chance playing a role? I'm mainly interested in discrete 2-player games with one player going first and with play alternating thereafter. Such that in some sense, except for that first player / second player thing, the challenge facing each player is as symmetrical as possible. Though I will not rule out continuous games. —Dan

3 2

[math-fun] Cuba audio: ultrasonic+ultrasonic=audible
by Henry Baker 02 Mar '18

02 Mar '18

It's possible that multiple *different* ultrasonic spying devices may have interfered with one another in the recent Cuba incident! (Spy v Spy)*nonlinear => Intermodulation Distortion + Oops! This obviously violated the spooks' Hypocritic Oath: First, Do No Harmonics! https://spqr.eecs.umich.edu/papers/YanFuXu-Cuba-CSE-TR-001-18.pdf On Cuba, Diplomats, Ultrasound, and Intermodulation Distortion University of Michigan Tech Report CSE-TR-001-18 Chen Yan 1 , Kevin Fu 2 , and Wenyuan Xu 1 1 Department of Systems Science and Engineering, Zhejiang University 2 Computer Science & Engineering, University of Michigan March 1, 2018 Abstract This technical report analyzes how ultrasound could have led to the AP news recordings of metallic sounds reportedly heard by diplomats in Cuba. Beginning with screen shots of the acoustic spectral plots from the AP news, we reverse engineered ultrasonic signals that could lead to those outcomes as a result of intermodulation distortion and non-linearities of the acoustic transmission medium. We created a proof of concept eavesdropping device to exfiltrate information by AM modulation over an inaudible ultrasonic carrier. When a second inaudible ultrasonic source interfered with the primary inaudible ultrasonic source, intermodu- lation distortion created audible byproducts that share spectral characteristics with audio from the AP news. Our conclusion is that if ultrasound played a role in harming diplomats in Cuba, then a plausible cause is intermodulation distortion between ultrasonic signals that unintention- ally synthesize audible tones. In other words, acoustic interference without malicious intent to cause harm could have led to the audible sensations in Cuba.

1 0