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Too small to represent and Chop in inaccessible algorithm


Overview

I want to apply a specified Chop function to every step in an function call in Mathematica (in my case LegendreP), especially when encountering a machine underflow error. The resulting function should be applicable to a dataframe.

The Setup

I am trying to calculate a function including an associated Legendre function with complex indices

function[t_] = 1/Sin[t] * LegendreP[-1/2 + V * I, l+1, Cos[t]]

where t is between 0 and Pi, V is of order of magnitude 10 and l is between 10 and 100. Beside the function I need its logarithmic derivative function'/function.

Ideally I want to do this by applying the function to a dataframe and appending the result in a separate column.

Append[#, "function" -> function[#timeframe]]

where timeframe is a column with all the t values.

The Problem

When I run this code for any l bigger than 12 and very small t~1e-5 values, the LegendreP Algorithm throws a machine underflow error General::munfl because it cannot execute a multiplication of extremly small complex numbers.

While for a single call or a plot, it seems to do some chopping or return Indeterminate, when I write it to a dataframe with

Append[#, "function" -> function[#timeframe]]

it just returns Failure and does not write anything to the dataframe.

What I have tried so far

I have tried to use Chop and Threshold, but this does not seem to apply to the single steps of the algorithm but only the final result.

The way I "solve" the problem at the moment is to catch the error and return 0 instead of my function. This is not ideal since the real or imaginary part of the step in question and the result might not be negligible while the other one is, or it might diverge instead of converge to 0.

Since the multiplication that raises the error lists numbers ~1e-300 or so, I doubt that the problem is solvable by increasing the precision.

My Goal

Ideally I’d like to call Chop, whenever Mathematica encounters a machine underflow. The behavior of Chop on complex numbers is exactly what I need. This way I should be able to preserve the real or imaginary part that does not vanish.

Is the error handling different, when applied to a dataframe as it relates to this question (for plots or even single evaluation points I don’t have the same issue) or can an indeterminate/NaN be written to a dataframe?

Is there a way to set a "global chop rule"?

Grateful for any hint 😀

Plotting a small gaussian | Small values and dealing with Machine Precision

I’ve defined the following:

k := 1.38*10^-16 kev := 6.242*10^8 q := 4.8*10^-10 g := 1.66*10^-24 h := 6.63*10^-27 

and

b = ((2^(3/2)) (\[Pi]^2)*1*6*(q^2)*(((1*g*12*g)/(1*g + 12*g))^(   1/2)) )/h  T6 := 20 T := T6*10^6 e0 := ((b k T6 *10^6)/2)^(2/3)  \[CapitalDelta] := 4/\[Sqrt]3 (e0 k T6 *10^6)^(1/2)  \[CapitalDelta]kev = \[CapitalDelta]*kev e0kev = e0*kev bkev = b*kev^(1/2) 

Then, I want to plot these functions:

fexp1[x_] = E^(-bkev *(x*kev)^(-1/2)) fexp2[x_] = E^(-x/(k*T)) fexp3[x_] = fexp1[x]*fexp2[x] 

and check that this Taylor expansion works:

fgauss[x_] =   Exp[(-3 (bkev^2/(4 k T*kev ))^(1/3))]*   Exp[(-((x*kev - e0kev)^2/(\[CapitalDelta]kev/2)^2))] 

which should, e.g., as expected:

Figure 10.1

This plot came from "Stellar Astrophysics notes" of Edward Brown (also it is a known approximation).

I used this line of command to Plot:

Plot[{fexp1[x],fexp2[x],fexp3[x],fgauss[x]}, {x, 0, 50},   PlotStyle -> {{Blue, Dashed}, {Dashed, Green}, {Thick, Red}, {Thick,      Black, Dashed}}, PlotRange -> Automatic, PlotTheme -> "Detailed",   GridLines -> {{{-1, Blue}, 0, 1}, {-1, 0, 1}},   AxesLabel -> {Automatic}, Frame -> True,   FrameLabel -> {Style["Energía E", FontSize -> 25, Black],     Style["f(E)", FontSize -> 25, Black]}, ImageSize -> Large,   PlotLegends ->    Placed[LineLegend[{"","","",""}, Background -> Directive[White, Opacity[.9]],      LabelStyle -> {15}, LegendLayout -> {"Column", 1}], {0.35, 0.75}]] 

but it seems that Mathematica doesn’t like huge negative exponentials. I know I can compute this using Python but it’s a surprise to think that Mathematica can’t deal with the problem somehow. Could you help me?

Small company owned by PC shadowrunners – which legal form it should be?


Intro:

We are playing SR3 in Seattle, but all players and GMs are from Europe & not skilled in law, not mentioning USA/Shadowrun law.

We are seeking for something somehow believable in USA/SR3 context (so we can find more about that later and build on that).

PCs got and took offer from friendly NPC (which "owes" them a lot and is trusted and good positioned) to buy some IDs and small company. They want run that as legal front for their new IDs to have "legal source of income" and a way for customers to be able pay by official money somehow. The company would own some buildings, cars and equipement too. PCs have large net of NPC contacts, which are willing and able "buy services" from that company (while paid by PCs for that) to keep the company alive, if needed. PCs should be able secretly run mentioned company while posing as "only ordinaly employees".

Antifraud and financial teams are not usually extra interested actively investigate such "small fishes".

Question:

GM does not know, what kind of legal form that should be called (like "Freds Services, inc." or so) and what to google for.


Background:

We play in really light "shadows", usually just some investigations, creating chaos here and there for distractration and such, PCs already killed few humans (mostly in "really excessive self defence"), but there are no visible traces to them and it is more exception, than a rule. The setting is soft and forgiving a lot, GM helps players to win many times. We just want to have fun and cinematic, not heavy drama and strugling. The "real action" takes less than half of gaming time, the rest is spend by buiding house, repairing and enhancing gear, cooking, gardening, chatting, dog shamman hiding bones on secret places … So the following plan should succeed and work for some time and offer a lot of small activities too.

The background is, that some well hidden NPC/company collected some IDs of peoples, who died, bud was not entered as dead in system, corpses dissapeared and those IDs was "kept alive", transfered to work for some existing company "Bettys laboratories, something" (which was build and kept just for this purpose), used for who know what for years (need a lend ID to spend a week with prostitues in resort? Here is ID for the week, for some price, so you real ID get no moral harm … style). And now the owner wants to close this "Bettys laboratories", sell all IDs to Shadows, divide the company to many idependent smaller independent sections/branches (via net of transactions) and sell them for lot of money. He have good law suport and knows what he is doing to hide the origins, but probably wants to start something different and bigger and so he want to capitalise this "Bettys laboratories" fast and discreetly. (GMs narative, why PC can buy legally clean company and new IDs, which are basically real.)

Our idea is, that that the "Fred’s Services" will own a house, some cars, properly hire a (innoncent, not knowing) secretary to take phones and transfer messages around, fill legal forms for taxes and make a chair in companys house warm. All income of this company will be "arranged" somehow and spend for paying the house, secretary and couple of "employees". The company would be able to officially send an invoice for some inoncent "services" (like consultations) to ABC.inc (or DEF.inc, or who is todays customer), which would pay for it and put it into official accounting, but those "services" would be simple fictional, intangible and just way to pay the real shadowruner. Basically money laundering of some kind. "Employees" would spend part of the wage to keep "conspiracy flat" paid, part for consumables (everybody needs food & groceries) and rest on "pleasures" (like prostitues, drinks, etc.) which they would then get back in "dirty money" to buy guns and other illegal equipement.

Interview q: Small possible length of stick from an array of stick lengths

I was asked this question in a phone interview recently and I bombed it completely. Zero clue how to approach it. I wasn’t able to find any similar patterns on google-ing. Thought maybe folks here might be able to help?


Statement: Given m sticks with different lengths. Combine these sticks to form longer sticks with the same length. What’s the smallest possible length of these newly unified sticks?

Conditions:

  • Must use all sticks
  • m < 50
  • max length of single stick less than 20

Example:

Input: 5 2 1 5 2 1 5 2 1 Output: 6 (Process: 1+5, 1+5, 1+5, 2+2+2) 

Input: 3 3 3 2 2 5 Output: 9 (Process: 3+3+3, 2+2+5) 

Input: 1 2 3 4 5 Output: 5 (Process: 2+3, 1+4, 5) 

Input: 1 3 4 5 Output: 13 (Process: 1+3+4+5) 

Does the ability score modifier go in the large box or the small oval?

When filling in the ability scores section of the standard character sheet as used in the D&D 5e Player’s Handbook, I’ve seen a lot of people put each ability score in the large box, and the corresponding ability score modifier in the small oval below it. This makes sense to some because that’s the order you’re writing it from top to bottom.

However, it seems more practical to put the ability score modifier in the large box, since that’s almost always the number you need to look at, and the ability score in the small oval.

Is there any official ruling on this, or other evidence to suggest which approach (if either) is official, standard, or more correct?

Attempt to merge 2 small fresh projects leads to freeze of GSA SER

Hello Sven,
I have tried several times, to make all projects Inactive, GSA SER is Stopped, no Threads is recognized on the status bar. I have also reset the Submitted records, and projects just keeps Options and Verified (105 and 130) to make projects for merge smaller. No matter what, anytime I try it, GSA SER got frozen. At the moment of trying is running only GSA SEO Indexer,  CapMonster, GSA Proxy Scraper and DropBox application, that feeds GSA SER by fresh lists.
Beyond mentioned apps is yet ran. I have set in all GSA apps count of threads to 20, however neither like that it have no possitive impact.

Is there anything else I can do, not to make GSA SER freezing all the time? That leads me to kill it in Task Manager and start it again.

My HW config is following: Intel i3-7130U @2,70GHz, 12GB RAM DDR4, 1TB M2 NvMe, 500Gbps WAN
System resources are following: 12%CPU, 35%RAM, 0%HDD, 0%LAN. 
My OS is MS Windows 10 PRO 64-bit. The machine is completely dedicated just for purpose of link-building.
Thanks for Your answer.

Regards,
Michal