Remarks on The Logistic Lattice in Random Number Generation. Neal R. Wagner
|
|
- Hester Day
- 6 years ago
- Views:
Transcription
1 Remarks on The Logistic Lattice in Random Number Generation Nea R. Wagner 1. Introduction Pease refer to the quoted artice before reading these remarks. I have aways been fond of this particuar random number generator (RNG) because of its unusua character, as expained in the main artice. These remarks just present a few additiona experiments that I ve carried out. 2. The Logistic Equation The ogistic equation is given by. I experimented on cyce engths with various hardware, but not with the IEEE standard for doube cacuations. Tabe 1 shows cyce engths obtained from random starting vaues. Notice that of the time it headed into the cyce of ength. (The software checked that this was not the other cyce of ength :!" #.) This cyce structure is simiar in form to that given in the earier artice for other hardware Cyce Structure Number Hard- Preci- Cyce Percent Average of starting ware sion ength occurrence initia run vaues IEEE doube % standard % (Pent II) % % % Tabe 1. Cyces of the ogistic equation. $( I aso carried out experiments on the interva of numbers around %$'& that is mapped into and then into the cyce ) by. The foowing is a sma piece of C code that checks for this interva. Beow getdoube() fetches the next doube vaue, and putdoube(x) prints the vaue of x in exponent and hex (interna) form. x = getdoube(); y = 4.*x*(1.0 - x); putdoube(y); z = 4.*y*(1.0 - y); putdoube(z); Here are resuts of a run. Notice that 3ff is an exact one, and a zero bits is an exact zero. The number 3fefffff ffffffff is as cose to as one can get from beow. User input is in bod red, whie the bue comments inside square brackets are not part of the output.
2 y: Dec: e-01, Hex: 3fefffff ffffffff [amost one] z: Dec: e-16, Hex: 3cbfffff ffffffff y: Dec: e+00, Hex: 3ff [exacty one] z: Dec: e+00, Hex: y: Dec: e-01, Hex: 3fefffff ffffffff [amost one] z: Dec: e-16, Hex: 3cbfffff ffffffff y: Dec: e+00, Hex: 3ff [exacty one] z: Dec: e+00, Hex: %$(, The above shows that the entire interva %$* ++++,+++ -,. /0%$'&,,,!.-, of ength. #& - is mapped to $(, and then on into the cyce. The hex vaues show that no arger interva wi map this way. This might not sound ike so much, but two smaer intervas map into the above interva, and so on with doube the number of intervas at each stage, each smaer than those at the previous stage. To get the two intervas that map into the interva around %$'&, one can sove for 2, 3 in the formua 1, to get :9 ; <" - 1. Now pug in the endpoints of the first interva above to get the endpoints of the two intervas that map into it. The foowing resuts, using the same C code as before, gives vaues for the two intervas that map into the interva around %$'&. Again the hex vaues show that no arger interva wi map this way. (Beow, amost one means as cose to as you can get with a doube vaue.) y: Dec: e-01, Hex: 3fe e y: Dec: e-01, Hex: 3fe y: Dec: e-01, Hex: 3fdfffff fbff812c y: Dec: e-01, Hex: 3fdfffff fc y: Dec: e-01, Hex: 3fe e y: Dec: e-01, Hex: 3fe ffaede y: Dec: e-01, Hex: 3fdfffff fbff812c y: Dec: e-01, Hex: 3fdfffff fc Tabe 2 beow shows the first interva and the two intervas mapping into it.
3 E Interva = Length What maps where >? ( , ) , for < >? = ( , ) , for < >? = ( , ) Tabe 2. Intervas mapped to 1., for in = 3. The Re-mapped Logistic Equation Whie I was studying the ogistic equation, I had an intuitive feeing that the equation coud be re-structured in a way that woud improve its performance. I didn t know exacty what I was ooking for. I spent an entire afternoon one day scratching around, trying a sorts of ideas, drawing endess pictures. The resut was the equation in this section. It very much out-performs the ordinary ogistic equation of the previous section in terms of the ength of its cyces and the average initia run up to a cyce. The re-structured or re-mapped ogistic equation is defined by: BDC F G -IH H - H H KJ LM H H ON - 6; H H 6PKJ LMKNRQ H H?, (1) where NS T;U V" 9 -. The earier artice gives a graph of the origina (expanded by a factor of two) and re-mapped versions (Figure 1 of the earier artice). Tabe 3 beow gives the cyces obtained from & random starting vaues of the remapped equation using the IEEE standard. (In the earier artice, I ony used. & random starting vaues.) Cyce Structure Number Hard- Preci- Cyce Percent Average of starting ware sion ength occurrence initia run vaues IEEE doube % standard % (Pent II) % % % Tabe 3. Cyces of the re-mapped ogistic equation. In the earier artice, I stated: In infinite precision, this re-mapped equation behaves exacty ike the origina, but with foating point numbers there is no onger any convergence to the cyce ) of ength. This was my intuition a aong, but I sti don t have a proof of it. However, numerica experiments simiar to those above show that the conjecture must be true. So here again I carried out experiments in the re-mapped setting. The numbers must be )X$Y&Z\[]%$Y&_^[0 near but ess than and near but greater than W. So what corresponds a]ax^b[ to an interva and 6_\[]]c. is repaced in this re-mapped setting by the intervas ` I used the foowing C code to evauate the behavior under using different vaues of [. #define BETA x = getdoube(); putdoube(x); x = fabs(x);
4 ` if ( x <= BETA ) y = (2.0*x*(2.0-x)); ese y = (-2.0*(1.0-x)*(1.0-x)); putdoube(y); y = fabs(y); if ( y <= BETA ) z = (2.0*y*(2.0-y)); ese z = (-2.0*(1.0-y)*(1.0-y)); putdoube(z); Here are resuts of the second run. Notice that 3fefffff ffffffff is the argest number ess than, whie bfefffff ffffffff is the smaest number greater than a. The number is usuay referred to as minus zero, that is, 8, but in fact the hardware treats it just ike the usua vaue of zero. (Again, amost one means as cose to as you can get with a doube vaue.) x: Dec: e-01, Hex: 3fefffff ffffffff [amost one] y: Dec: e-32, Hex: b z: Dec: e-32, Hex: x: Dec: e+00, Hex: 3ff [exacty one] y: Dec: e+00, Hex: [exacty -zero] z: Dec: e+00, Hex: [exacty zero] x: Dec: e-01, Hex: bfefffff ffffffff [amost -one] y: Dec: e-32, Hex: b z: Dec: e-32, Hex: x: Dec: e+00, Hex: bff [exacty one] y: Dec: e+00, Hex: [exacty -zero] z: Dec: e+00, Hex: [exacty zero] The above resuts show that no proper intervas with eft end at a or with right end at wi map into, but ony the singe vaues W and. 4. Runs Into the Cyces (0.75) and (-0.5) The ordinary ogistic equation has the obvious cyce ) of ength and one other cyce of ) ength, namey %$. &. In the re-mapped ogistic equation, the corresponding cyces are and 8%$Y&,. With the ordinary ogistic equation in singe precision (foat type), the origina artice showed that a arge number of initia vaues ( +! ) end up in, whie there are no runs at a into %$. &. Using the re-mapped ogistic equation, there were very few runs into ), and again none at a into 6d%$'&. In the artice, I stated without proof: The cyce search in singe precision is exhaustive because every cyce%$ wi eventuay enter the range from!" to. I meant to say incusive. &Xec at the end, that is, `. I aso meant to incude, except for the cyce. Here is a proof, with )X$ a tighter statement of the assertion. (Beow, f means a cyce containing just, and. &g]c means those for which X$. &hqi.) %]c Here are cases based on the vaue of kj If m, then it is in the cyce ), the specia case. If m %$. & then it is in the cyce )%$. & %$. &Xec, so it is certainy in the range `. If jn )X$. &g]c then oqx$. &, so no cyce can be entirey inside %$. &X]c. :
5 ` If %$ Oj? )X$ - &g0%$. & then \j? )%$. &X]]c so no cyce can be entirey inside - &X0%$. &, and in fact in this case, f jn )%$. &X]]c. If pj %0%$ - &qc then rf s T hit, so Irf vu -. Using the Mean Vaue Theorem, A <"x ku there is an r Qw such 8 <"/ f R r r, so - or ou -. Ifz'}{ we f ~ take iterates %$ A f < y Dz'P6{, etc., then eventuay either - & z'} ){ f ~ X$. & z(}v{ T %$ - & z'} ){< f ƒ %$ so that, or so that. &, so that iterates go into )X$. &g]c. Thus no cyce can be entirey inside )%0X$ - &xc. This %$ competes the proof. This shows that every cyce except must eventuay be inside. &X]c (incuding the cyce %$. &.) It aso shows that no cyce can be entirey inside %0%$ - &qc, or entirey inside )X$ - &g0%$. &, or entirey inside %$. &X]c. However, much to my surprise, there is a run of ength &g -, ending in the cyce )%$. &, that is entirey inside )%0X$. &xc. The same situation occurs with the re-mapped ogistic equation. So now I ook into possibe runs into %$. & (ordinary ogistic equation) and 6dX$Y& (remapped ogistic equation) in doube precision. First, focus on the ordinary ogistic equation, and work backwards from %$. &. In infinite precision, there are in genera two vaues that map by into any singe vaue. We can use the equation ƒ o7 9 " - 1 to get approximate vaues, athough to quaify, the vaue must be exact. Cacuating, both X$ - & and %$. & map into %$( %$. &. There are two vaues that map approximatey into X$ - & : %$(+!!,% -. XVt+ -- V+! and +,t.- +txv.. t,. -. However, as the cacuation beow shows, ony the second of these vaues actuay works. (Beow, the input consists of exact hex vaues, rather than approximate decima vaues.) 3feddb3d 742c2655 x : Dec: e-01, Hex: 3feddb3d 742c f1(x): Dec: e-01, Hex: 3fd succes f2(x): Dec: e-01, Hex: 3fe sive vaues 3feddb3d 742c2656 x : Dec: e-01, Hex: 3feddb3d 742c f1(x): Dec: e-01, Hex: 3fcfffff fffffff5 f2(x): Dec: e-01, Hex: 3fe7ffff fffffffa 3fb e9ecd56 x : Dec: e-02, Hex: 3fb e9ecd56 f1(x): Dec: e-01, Hex: 3fd f2(x): Dec: e-01, Hex: 3fe For the number that starts with %$'+,!!% -,., you can see two successive hex vaues, one too %$( sma and one too arge, so that no vaue can work exacty. The number that starts with +,t.- +t %$ has a cose hex vaue that works exacty. So ony one precursor vaue from - & works. Continuing to move backwards, at each stage the vaue near works, whie the vaue near does not. One gets a singe run into %$. & of ength &X - : 00118bc4 418cafe1 x : Dec: e-308, Hex: 00118bc4 418cafe1 f 1(x): Dec: e-308, Hex: 00318bc4 418cafe1 f 2(x): Dec: e-307, Hex: 00518bc4 418cafe1 f 3(x): Dec: e-306, Hex: 00718bc4 418cafe1... [480 ines omitted, a ending with 18bc4 418cafe1 ] f484(x): Dec: e-17, Hex: 3c918bc4 418cafe1 f485(x): Dec: e-16, Hex: 3cb18bc4 418cafe1
6 f486(x): Dec: e-16, Hex: 3cd18bc4 418cafe0 f487(x): Dec: e-15, Hex: 3cf18bc4 418cafdb f488(x): Dec: e-14, Hex: 3d118bc4 418cafc8 f489(x): Dec: e-14, Hex: 3d318bc4 418caf7b f490(x): Dec: e-13, Hex: 3d518bc4 418cae47 f491(x): Dec: e-13, Hex: 3d718bc4 418ca978 f492(x): Dec: e-12, Hex: 3d918bc4 418c963a f493(x): Dec: e-11, Hex: 3db18bc4 418c4943 f494(x): Dec: e-11, Hex: 3dd18bc4 418b1567 f495(x): Dec: e-10, Hex: 3df18bc f6 f496(x): Dec: e-09, Hex: 3e118bc f497(x): Dec: e-09, Hex: 3e318bc f498(x): Dec: e-08, Hex: 3e518bc4 3ff234b9 f499(x): Dec: e-08, Hex: 3e718bc4 3b22c33c f500(x): Dec: e-07, Hex: 3e918bc4 27e4fd53 f501(x): Dec: e-06, Hex: 3eb18bc3 daede659 f502(x): Dec: e-06, Hex: 3ed18bc2 a f503(x): Dec: e-05, Hex: 3ef18bbd d7a0f869 f504(x): Dec: e-05, Hex: 3f118baa 99e912e4 f505(x): Dec: e-04, Hex: 3f318b5d a3b24972 f506(x): Dec: e-03, Hex: 3f518a29 d563b9c0 f507(x): Dec: e-03, Hex: 3f71855b 44e5d92f f508(x): Dec: e-02, Hex: 3f91722b 8b740eb3 f509(x): Dec: e-02, Hex: 3fb e9ecd56 f510(x): Dec: e-01, Hex: 3fd f511(x): Dec: e-01, Hex: 3fe The notation f 3(x) stands for f(f(f(x))), and so on for vaues other than!. The argest possibe IEEE doube ess than is 3fefffff ffffffff. Appy to this vaue to get /$(, t+ - +t&,, - &.. This is the smaest vaue, other than, that can occur in a run using the ogistic equation, athough one coud choose a smaer initia vaue. Thus everything in the ist above preceding f486(x) coud ony occur as an initia vaue. A numbers before f485(x) end with 18bc4 418cafe1, and in this range each successive hex vaue is obtained by adding to the previous vaue, that is, by mutipying by, since for numbers very cose to, f is exacty mutipication by. The smaest possibe IEEE doube is - $ -- &,.!t&,t&.- %]!t, so that the run cannot be made one onger. Of course a subsequent vaues in the run wi be the cyce vaue X$. &. 8%$'& With the re-mapped there is an exact corresponding run of ength &X - into the cyce : 00218bc4 418cafe1 x : Dec: e-308, Hex: 00218bc4 418cafe1 f 1(x): Dec: e-307, Hex: 00418bc4 418cafe1 f 2(x): Dec: e-307, Hex: 00618bc4 418cafe1 f 3(x): Dec: e-306, Hex: 00818bc4 418cafe1... [504 ines omitted] f508(x): Dec: e-02, Hex: 3fa1722b 8b740eb3 f509(x): Dec: e-01, Hex: 3fc e9ecd56 f510(x): Dec: e-01, Hex: 3fe f511(x): Dec: e-01, Hex: bfe There is one type of addition to this run. In the case of the re-mapped ogistic equation, at any stage of the run, you can start with the negative of an entry. There is no precursor to this vaue, but the$'! subsequent vaues are the same positive ones as above. Thus, for exampe, the. /.!. V,+!%.,+ numbers, X$ˆV!,!+. #&+ - x&,& V!&, %$'&, dx$y& form an initia run up to 8%$'&. Otherwise the same remarks appy to these cacuations as to the ones for the ordinary ogistic equation.
7 - %$. & The same behavior occurs with IEEE foat cacuations, except that the run into or 8%$'& is ony 64 ong. Here the smaest possibe IEEE foat is $. & +,!&!t. The ist of vaues beow shows both the ordinary and re-mapped ogistic equations side-by-side. Here before the f52(x) terms, each term is obtained from the previous one by adding , that is, mutipying by. Any run for the re-mapped ogistic equation can start with a negated entry, as above. In the origina artice, the exhaustive search of the ordinary equation for foat X$ precision. &g]c, so it found the cyce )X$. &,, but missed the run ony covered the range in the interva ` beow atogether, and simiary for the re-mapped equation. Ordinary Logistic Equation Re-mapped Logistic Equation 008c5e22 010c5e22 x : e-38, Hex: 008c5e e-38, Hex: 010c5e22 f 1(x): e-38, Hex: 018c5e e-37, Hex: 020c5e22 f 2(x): e-37, Hex: 028c5e e-37, Hex: 030c5e22 f 3(x): e-37, Hex: 038c5e e-36, Hex: 040c5e22... [46 ines omitted, a ending with c5e22 ] f50(x): e-08, Hex: 328c5e e-08, Hex: 330c5e22 f51(x): e-08, Hex: 338c5e e-07, Hex: 340c5e22 f52(x): e-07, Hex: 348c5e e-07, Hex: 350c5e21 f53(x): e-06, Hex: 358c5e1f e-06, Hex: 360c5e1f f54(x): e-06, Hex: 368c5e e-06, Hex: 370c5e15 f55(x): e-05, Hex: 378c5def e-05, Hex: 380c5def f56(x): e-05, Hex: 388c5d e-04, Hex: 390c5d55 f57(x): e-04, Hex: 398c5aed e-04, Hex: 3a0c5aed f58(x): e-03, Hex: 3a8c514f e-03, Hex: 3b0c514f f59(x): e-03, Hex: 3b8c2ada e-03, Hex: 3c0c2ada f60(x): e-02, Hex: 3c8b915c e-02, Hex: 3d0b915c f61(x): e-02, Hex: 3d8930a e-01, Hex: 3e0930a3 f62(x): e-01, Hex: 3e e-01, Hex: 3f f63(x): e-01, Hex: 3f e-01, Hex: bf Discussion Suppose one tried to use the ordinary ogistic equation directy as a RNG. Start with a seed (some initia doube vaue). Then ++X$'+! of the time, you get some miion vaues before cycing ( ) or ending in ( ). This is compared to a singe cyce of about V biion numbers with an od-fashioned! - -bit generator. (Of course you have to iterate maybe times between samping for a random number.) Anyway, this isn t so terribe, perhaps, except for the remaining X$'. of the time. The main artice provides three ways to improve the basic ogistic equation based RNG: 1. Iterate the generator to fi with random noise. 2. Use the re-mapped ogistic equation that has much onger cyces and initia runs, and has no convergence to. 3. Use a couped attice of ogistic equation generators that wi perturb one another. Of the above, item 1 is essentia, but fairy easy to think of. Item 2 is the one I m most proud of, because it s unexpected and origina. But item 2 is not essentia as ong as one uses item 3, which reay is essentia. The basic idea of item 3 was suggested to me by Kay Robbins. (Couped attices of ogistic equations are a standard simpe mode of chaos, but I didn t know that initiay.)
NCH Software VideoPad Video Editor
NCH Software VideoPad Video Editor This user guide has been created for use with VideoPad Video Editor Version 4.xx NCH Software Technica Support If you have difficuties using VideoPad Video Editor pease
More informationEDT/Collect for DigitalMicrograph
May 2016 (Provisiona) EDT/Coect for DigitaMicrograph Data Coection for Eectron Diffraction Tomography EDT/Coect Manua 1.0 HREM Research Inc. Introduction The EDT/Coect software has been deveoped by HREM
More informationTopology of Musical Data
Topoogy of Musica Data Wiiam A. Sethares Department of Eectrica and Computer Engineering, University of Wisconsin, Madison, USA, sethares@ece.wisc.edu November 27, 2010 Abstract Techniques for discovering
More informationPrior Subspace Analysis for Drum Transcription
Audio Engineering Society Convention Paper Presented at the 4th Convention 23 March 22 25 Amsterdam, he Netherands his convention paper has been reproduced from the author's advance manuscript, without
More informationBackground Talent. Chapter 13 BACKGROUND CASTING AGENCIES. Finding Specific Types THE PROCESS
Chapter 13 Background Taent Note that whie The Screen Actors Guid has changed the designation of extra to that of background actor, for the purpose of this chapter, the terms extra, extra taent, background
More informationOperation Guide 5200
MO1103-EA Getting Acquainted ongratuations upon your seection of this ASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Be sure to keep a user documentation handy for
More informationOperation Guide
MO0503-EA Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy and keep it on hand for ater reference when
More informationRunning a shared reading project. A scheme of activities to help older children share picture books with younger ones
CFE Leves Eary Senior phase (Ages 3-16) Running a shared reading project A scheme of activities to hep oder chidren share picture books with younger ones Resource created by Scottish Book Trust Contents
More informationOperation Guide 3271
MO1106-EA Operation Guide 3271 Getting Acquainted ongratuations upon your seection of this ASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Be sure to keep a user documentation
More informationUsing wordless picture books in schools and libraries. Ideas for using wordless picture books in reading, writing and speaking activities
CfE eves Eary to Fourth (Ages 3-16) Using wordess picture books in schoos and ibraries Ideas for using wordess picture books in reading, writing and speaking activities Resource created by Scottish Book
More informationOperation Guide 3197
MO1004-EA Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Keep the watch exposed to bright ight The eectricity
More informationDrum Transcription in the presence of pitched instruments using Prior Subspace Analysis
ISSC 2003, Limerick. Juy -2 Drum Transcription in the presence of pitched instruments using Prior Subspace Anaysis Derry FitzGerad φ, Bob Lawor*, and Eugene Coye φ φ Music Technoogy Centre, Dubin Institute
More informationTRANSCENSION DMX OPERATOR 2 USER MANUAL
TRANSCENSION DMX OPERATOR 2 USER MANUAL I. PRODUCT DESCRIPTIONS Thank you for using our company the 192 CH DMX OPERATOR. To optimize the performance of this product, pease read these operating instructions
More informationDown - (DW Sampler Hold Buffer * Digital Filter * Fig. 1 Conceptual bunch-by-bunch, downsampled feedback system.
Bunch-by-Bunch Feedback for PEP II* G. Oxoby, R. Caus, N. Eisen, J. Fox, H. Hindi, J.Hoefich, J. Osen, and L. Sapozhnikov. Stanford Linear Acceerator Center, Stanford University, Stanford, CA 94309 I.
More informationSection 2 : Exploring sounds and music
Section 2 : Exporing sounds and music Copyright 2014 The Open University Contents Section 2 : Exporing sounds and music 3 1. Using stories and games to introduce sound 3 2. Working in groups to investigate
More informationConcerto in B-flat Major Opus 4 Number 6. G.F. Handel ( )
Concerto in B-fat Major Opus Number 6 G.F. Hande (685-759) Transcribed for fu-sized ever harp tuned to Eb or Bb by Darhon Rees-Rohrbacher DFM A Dragonfower Music Pubication i This pubication is printed
More informationReal-Time Audio-to-Score Alignment of Music Performances Containing Errors and Arbitrary Repeats and Skips
IEEE/ACM TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. XX, NO. YY, 2015 1 Rea-Time Audio-to-Score Aignment of Music Performances Containing Errors and Arbitrary Repeats and Skips Tomohiko
More informationAmerican English in Mind
An integrated, four-skis course for beginner to advanced teenage earners of American Engish American Engish in Mind engages teenage students of Engish through: American Engish in Mind features: M O H DV
More informationOperation Guide 2531
MO0404-EC Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to carefuy read this manua and keep it on hand for ater reference when
More information25th DOE/NRC NUCLEAR AIR CLEANING AND TREATMENT CONFERENCE
DEEP BED CHARCOAL FILTER RETENTION SCREEN IN-PLACE REPLACEMENT AND REPAIR Wiiam Burns and Rajendra Paude Commonweath Edison Company LaSae County Station Raymond Rosten and Wiiam Knous Duke Engineering
More informationImportant Information... 3 Cleaning the TV... 3
Contents Important Information... 3 Ceaning the TV... 3 Using the Remote Contro... 4 How to Use the Remote Contro... 4 Cautions... 4 Instaing the Remote Contro Batteries... 4 The Front and Rear Pane...
More informationLONG term evolution (LTE) has now been operated in
IEEE/ACM TRANSACTIONS ON NETWORKING 1 A Pricing-Aware Resource Scheduing Framework for LTE Networks You-Chiun Wang and Tzung-Yu Tsai Abstract Long term evoution (LTE) is a standard widey used in ceuar
More informationDiploma Syllabus. Music Performance from 2005
Dipoma Syabus Music Performance from 2005 SPECIAL NOTICES This Music Performance Dipoma Syabus from 2005 is a revised version of the Performing sections of the Dipoma Syabus from 2000. It is vaid wordwide
More informationLEGEND SERIES. DIMENSIONS In inches (mm)
LEGEND SERIES MODEL LGS -Singe Preset Counter/Rate Indicator MODEL LGD - Dua Preset Counter/Rate Indicator MODEL LG - Four Preset atch/counter/rate Indicator MODEL LGM - Six Preset Counter/Rate Indicator
More informationOperation Guide 3172
MO1007-EC Congratuations upon your seection of this CASIO watch. Appications The buit-in sensors of this watch measure direction, barometric pressure, temperature and atitude. Measured vaues are then shown
More informationOperation Guide 4717
MO0812-EB Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. This watch does not have a Time Zone that corresponds
More informationOperation Guide 3270/3293
MO1109-EA Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Be sure to keep a user documentation handy
More informationGeometric Path Planning for Automatic Parallel Parking in Tiny Spots
13th IC Smposium on Contro in Transportation Sstems The Internationa ederation of utomatic Contro September 1-14, 01. Sofia, ugaria eometric Path Panning for utomatic Parae Parking in Tin Spots Héène Vorobieva
More informationHigh. Achievers. Teacher s Resource Book
B High Achievers Teacher s Resource Book Contents Vocabuary Worksheets page Grammar Worksheets page Speaking Worksheets page Festivas page Tests page Speaking Tests page Introduction This Teacher s Resource
More informationUse with VIP2K Monitor Program Version 1.4 (vip2k14.hex) at U2, and the matching state machine file (vip-2716.hex) at U7 on the VIP2K CPU board.
VIP2K Video Display Memory Map - by Chuck Yakym 1/3/2019 Use with VIP2K Monitor Program Version 1.4 (vip2k14.hex) at U2, and the matching state machine file (vip-2716.hex) at U7 on the VIP2K CPU board.
More informationOperation Guide
MO1603-EA 2016 ASIO OMPUTER O., LT. Operation Guide 5484 5485 Getting Acquainted ongratuations upon your seection of this ASIO watch. To get the most out of your purchase, be sure to read this manua carefuy.
More informationOperation Guide 2804
MO007-EA Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to carefuy read this manua and keep it on hand for ater reference when necessary.
More informationThe Basics of Monitor Technology (1)
The Basics of Monitor Technoogy 2-187-799-12(1) Preface In recent years, the editing systems and equipment used by broadcasters, production houses and independent studios have improved dramaticay, resuting
More informationGetting in touch with teachers
Getting in touch with teachers Advertising with INTO INTO Media pack 2017 2018 The INTO The Irish Nationa Teachers Organisation (INTO), was founded in 1868. It is the argest teachers trade union in Ireand.
More informationB. Please perform all warm- ups/exercises and Open Up Wide as close to tempo markings as provided.
Greetings Percussionists: The 201 DUMINE audition music for the Texas Tech University Marching Band- Goin Band from aiderand consists of warm- upsexercises ( GB15 Warm- up for A and Tripet- Grid and Dupe
More informationVocal Technique. A Physiologic Approach. Second Edition
Voca Technique A Physioogic Approach Second Edition Voca Technique A Physioogic Approach Second Edition Jan E. Bicke, D.M.A. 5521 Ruffin Road San Diego, CA 92123 e-mai: info@purapubishing.com Website:
More informationUSER S GUIDE About This Manual. (Light) 12/24-Hour Format. described below. Setting GMT differential. Longitude
MO0302-A USER S GUIDE 2611 Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to carefuy read this manua and keep it on hand for ater
More informationOperation Guide
MO1302-EB Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Warning! The measurement functions buit into
More informationMMS-Übungen. Einführung in die Signalanalyse mit Python. Wintersemester 2016/17. Benjamin Seppke
MIN-Fakutät Fachbereich Informatik Arbeitsbereich SAV/BV (KOGS) MMS-Übungen Einführung in die Signaanayse mit Python Wintersemester 2016/17 Benjamin Seppke MMS-Übungen: Einführung in die Signaanayse mit
More informationOperation Guide 3143
MO0804-EA Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. This watch does not have a time zone that corresponds
More information3,81 mm Wide Magnetic Tape Cartridge for Information Interchange - Helical Scan Recording - DDS-2 Format using 120 m Length Tapes
Standard ECMA-198 2nd Edition - June 1995 Standardizing Information and Communication Systems 3,81 mm Wide Magnetic Tape Cartridge for Information Interchange - Heica Scan Recording - DDS-2 Format using
More informationD-ILA PROJECTORS DLA-X95R DLA-X75R DLA-X55R DLA-X35
D-ILA PROJECTORS DLA-X95R DLA-X75R DLA-X55R DLA-X35 D L A-X S e r i e s DLA-X95R 4K-resoution D-ILA Projector JVC D-ILA projector premium mode that adopts high-grade parts reaises 4K-resoution* 1 and industry
More informationDocuCom PDF Trial. PDF Create! 6 Trial
In The Name of Aah, the most Mercifu, the most Compassionate We, the Engish teachers in the 3 rd area, Azzoun Sannyria, are reay peased to produce this modest magazine which contains some enjoyabe topics
More informationAuburn University Marching Band
Aubun Univesity Maching Band Dea Pospective Dum Line Membe, Thank you fo you inteest in the 2018 Aubun Dum Line! The Aubun Univesity Maching Band has a poud tadition of exceent pefomances and geat schoo
More informationFalcons team update. Presentation Portugal Workshop 2015
Facons team update Presentation Portuga Workshop 2015 Contents ASML s mission for Robocup Software updates New vision system Mission Vision for 2015-2016 Inside ASML there is a growing awareness for the
More informationOperation Guide 5008
MO080-EA Operation Guide 008 Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. This watch does not have
More informationv z :,& 9.-b OF WlS D O C U M M 0s ~L~~~ BS
.2 MW KLYSTRON FOR ASYMMETRC STORAGE RNG B FACTORY W. R. Fowkes, G. Caryotakis, E. Doye, E. Jongewaard, C. Pearson, R. Phiips, J. Sackett, E. Wright Stanford Linear Acceerator Center, Stanford University,
More informationUNIQUE LIGHTING SOLUTIONS. LED PRODUCTS for the SIGN INDUSTRY
UNIQUE LIGHTING SOLUTIONS LED PRODUCTS for the SIGN INDUSTRY RIGHT LIGHT SERIES MORE LUMENS per DOLLAR Modue Lumens LED L70 Light Feet / W Feet / 0W Modue Mode Number Coor Lifetime (hrs) Ange SV---W. 90
More informationThe optimal multi-stage contest
MPRA Munich Persona RePEc Archive The optima muti-stage contest Fu, Qiang and Lu, Jingfeng UNSPECIFIED November 2006 Onine at http://mpra.ub.uni-muenchen.de/946/ MPRA Paper No. 946, posted 07. November
More informationOperation Guide 5135
MO1006-EA Operation Guide 5135 Getting Acquainted ongratuations upon your seection of this ASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. This watch does not have
More informationLibrary and Information Sciences Research Literature in Sri Lanka: A Bibliometric Study
Journa of the University Librarians Association of Sri Lanka. Vo. 12, 2008 Library and Information Sciences Research Literature in Sri Lanka: A Bibiometric Study Author Gunasekera, Chamani MLIS (Coombo),
More informationSpecifications. Lens. Lens Shift. Light Source Lamp. Connectors. Digital. Video Input Signal Format. PC Input Signal Format.
Projection Distance Chart Dispay size (16:9) Projection distance Screen diagona (inch) W (mm) H (mm) Wide (m) Tee (m) 60 1,328 747 1.78 3.66 70 1,549 872 2.09 4.28 80 1,771 996 2.40 4.89 90 1,992 1,121
More informationModbus Register Tables for SITRANS RD300 & WI100
AG021414 Modbus Register Tables for SITRANS RD300 & WI100 WARNING: As is typical with most instruments, the addition of serial communications carries an inherent risk; it allows a remote operator to change
More informationOperation Guide
MO0908-EA Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Warning! The measurement functions buit into
More informationOperation Guide 4719
MO0801-EA Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Keep the watch exposed to bright ight The eectricity
More informationOperation Guide 3147
MO1203-D Operation Guide 3147 Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Appications The buit-in
More informationEnergy meter MRE-44S. MRE-44S/DC24V energy meter
MRE-44S MRE-44S/DC24V energy meter Comprehensive consumption data anaysis in rea time High resoution and accuracy (cass 0.) even in harmonicay distorted grids Aso anayses harmonics (optiona, up to 50 Hz)
More informationTexas Music Educators Association 2017 Clinic/Convention San Antonio, Texas 9-12 February 2017
Texas Music Educators Association 2017 Cinic/Convention San Antonio, Texas 9-12 February 2017 Create and Pay the Day Away. Singing Games for the Upper Eementary Music Cassroom Mícheá Houahan and Phiip
More informationModal Bass Line Modules
Moda Bass Line Modues We wi now take a ook at a stye of jazz known as moda tunes. Moda tunes are songs buit on one or two chord changes that ast at east eight bars each. The exampe we use in our study
More informationRX-V890. Natural Sound Stereo Receiver. Contents OWNER S MANUAL
RX-V890 Natura Sound Stereo Receiver CAUTION OWNER S MANUAL Contents PROFILE OF THIS UNIT... 6 SPEAKER SETUP FOR THIS UNIT... 7 CONNECTIONS... 9 ADJUSTMENT BEFORE OPERATION... 15 BASIC OPERATIONS... 18
More informationv 75 THE COMMUNICATIONS CIRCUIT REVISITED'
THE COMMUNICATIONS CIRCUIT REVISITED' A s we enter the twenty-first century the spread of internet has aready concusivey shown that digita transmission of texts is here to stay. Indeed its significance
More informationMulti-TS Streaming Software
Appication Note Thomas Lechner 1.2017 0e Muti-TS Streaming Software Appication Note Products: R&S CLG R&S CLGD R&S SLG The R&S TSStream muti-ts streaming software streams a number of MPEG transport stream
More informationOperation Guide 3150
MO0805-EA Getting Acquainted ongratuations upon your seection of this ASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Keep the watch exposed to bright ight The eectricity
More informationNAIVE - Network Aware Internet Video Encoding
NAIVE - Network Aware Internet Video Encoding Hector M. Bricefio MIT hbriceno@cs. mit. edu Steven Gorter Harvard University sjg @ cs. harvard. edu Leonard McMian MIT mcmian@cs.mit. edu Abstract The distribution
More information(12) (10) Patent N0.: US 7,043,320 B1 Roumeliotis et a]. (45) Date of Patent: May 9, 2006
United States Patent US007043320B1 (12) (10) Patent N0.: Roumeiotis et a]. (45) Date of Patent: May 9, 2006 (54) METHOD AND APPARATUS FOR PLANNING 5,369,570 A * 11/1994 Parad..... 705/8 A MANUFACTURING
More informationHorizontal Circuit Analyzing
THE HA2500 Horizonta Circuit Anayzing Rea Answers - Rea Profits - Rea Fast! HA2500 Universa Horizonta Anayzer Why A Universa Horizonta Anayzer For Your Business? Today s CRT video dispay monitors support
More information1/ 19 2/17 3/23 4/23 5/18 Total/100. Please do not write in the spaces above.
1/ 19 2/17 3/23 4/23 5/18 Total/100 Please do not write in the spaces above. Directions: You have 50 minutes in which to complete this exam. Please make sure that you read through this entire exam before
More informationITU BS.1771 Loudness Meter BLITS Channel Identification for 5.1 Surround Sound
RW-6seiter_IBC_009_GB_RZ_V5.qxp 0: Seite Functions of the various modes: ypica dispay patterns and their interpretation a few exampes: 900 900S 900D 900SD Mode 960 960S 960D 960SD 900 900S 900D 900SD F
More informationOperation Guide 3220
MO1007-EA Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Keep the watch exposed to bright ight The eectricity
More informationOperation Guide 2628
MO0608-EA Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. This watch does not have a time zone that corresponds
More informationOperation Guide 3195
MO0911-EA Operation Guide 3195 Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Keep the watch exposed
More informationHome & Garden Shows. Oak Brook v N. Shore v Naperville v Arlington Lake Co. v Tinley Park v Crystal Lake
2019 Home & Garden Shows Oak Brook v N. Shore v Napervie v Arington Lake Co. v Tiney Park v Crysta Lake Thank you very much for your interest in our 2019 Home & Garden Shows. We ve incuded the foorpans
More informationMeasuring Product Semantics with a Computer
San Jose State University SJSU SchoarWorks Facuty Pubications Art and Art History & Design Departments October 1988 Measuring Product Semantics with a Computer De Coates San Jose State University, dcoates@decoates.com
More informationSpectrum Management. Digital Audio Broadcasting. Content Protection. Video Streaming. Quality of Service
Wecome 3 Spectrum Management Impementation of the Digita Dividend technica restraints to be taken into account Jan Doeven, KNP 4 Digita Audio Broadcasting The evoution of DAB Frank Herrmann, Larissa Anna
More informationIntercom & Talkback. DanteTM Network Intercom BEATRICE R8. Glensound. Network Intercom. Eight Channel Rackmount Intercom.
G ensound Dante Intercom & Takback Eight Channe Rackmount Intercom Highights Dante and AES67 Compiant Simpe To Use Inteigeabe Loudspeaker 48kHz Crysta Cear Digita Audio Mains/ PoE Powered Low Noise Microphone
More informationOperation Guide 3017
MO0602-EA Operation Guide 3017 Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Keep the watch exposed
More informationNATURAL SOUND AV RECEIVER AMPLI-TUNER AUDIO-VIDEO
MN L/UTO FM UTO/MN L MONO U C NTURL SOUND V RECEIVER MPLI- UDIO-VIDEO NTURL SOUND V RECEIVER RX V9S CEM DSP 7ch PUT SELECTOR VOLUME 6 8 8 STNDBY/ON PUT MODE 6 db SPEKERS B PROGRM EFFECT EXT. DECODER /B/C/D/E
More informationMuslim perceptions of beauty in Indonesia and Malaysia Neil Gains Warc Exclusive Institute on Asian Consumer Insight, February 2016
Musim perceptions of beauty in Indonesia and Maaysia Nei Gains Warc Excusive Institute on Asian Consumer Insight, February 2016 Tite: Musim perceptions of beauty in Indonesia and Maaysia Author(s): Nei
More informationINSTRUCTIONS FOR AUTHORS
INSTRUCTIONS FOR AUTHORS In this docuent authors wi find instructions for the preparation of papers according to the standard forat required for their pubication. 1. LANGUAGES The officia conference anguages
More informationHeritage Series. Heritage Heritage Heritage Heritage Extender. Heritage 1000
Heritage Series Heritage 4 Heritage 3 Heritage Heritage Extender Heritage Heritage 4 The Midas Heritage 4 is an evoution of the award winning Heritage 3 with an additiona 6 more busses, which has resuted
More informationOperation Guide 3063
MO0801-EA Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Keep the watch exposed to bright ight The eectricity
More informationKeller Central Percussion
Kee Centa Pecussion Font Ensembe Execise Packet The fooing pages incude basic to intemediate technique and coodination execises fo the maching pecussion idiom. A stong gasp of these fundamentas is essentia
More informationThe Lively Bard. Twenty Up-Tempo Arrangements of Welsh Airs Based on Collected Tunes of Edward Jones Harper to King George IV
The Livey Bard Twenty Up-Tempo Arrangements of Wesh Airs Based on oected Tunes of Edward Jones Harper to King George IV or ute or Vioin With Lever Harp Accompaniment (Lever Harp Tuned in Eb or Bb) With
More informationProfessional HD Integrated Receiver Decoder GEOSATpro DSR160
Professiona HD Integrated Receiver Decoder GEOSATpro DSR160 User Manua V1.00-C Preface About This Manua This manua provides introductions to users about how to operate the device correcty. The content
More informationDownloaded from 1. English Communicative Code No. 101 CLASS - IX EXAMINATION SPECIFICATIONS. Summative Assessment II 30%
Downoaded from 1. Engish Communicative Code No. 101 CLASS - IX EXAMINATION SPECIFICATIONS Division of Syabus for Term II (October-March) Tota Weightage Assigned Summative Assessment II 30% Section Marks
More informationOperating Instructions VEGAPULS 63 Foundation Fieldbus
Operating Instructions VEGAPULS 63 Foundation Fiedbus Radar Contents Contents 1 About this document 1.1 Function... 5 1.2 Target group... 5 1.3 Symboism used... 5 2 For your safety 2.1 Authorised personne...
More informationOperation Guide
MO0312-EA Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to carefuy read this manua and keep it on hand for ater reference when
More informationORF 307 Network Flows: Algorithms
ORF 307 Network Flows: Algorithms Robert J. Vanderbei April 5, 2009 Operations Research and Financial Engineering, Princeton University http://www.princeton.edu/ rvdb Agenda Primal Network Simplex Method
More informationOperation Guide 3090
MO0706-EA Operation Guide 3090 Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Keep the watch exposed
More informationPlease note that not all pages are included. This is purposely done in order to protect our property and the work of our esteemed composers.
Pease note that not a pages are incuded This is purposey done in order to protect our property and the work of our esteemed composers f you woud ike to see this work in its entirety, pease order onine
More informationOperation Guide
MO0605-EA Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Keep the watch exposed to bright ight The eectricity
More informationDESIGN REVIEW BOARD Staff Report
DESIGN REVIEW BOARD Staff Report Agenda Item F.2 Meeting Date: Apri 9, 2013 TO: Goeta Design Review Board FROM: Joe Pearson II, Assistant Panner; Phone Number: 961-7573 SUBJECT: 13-029 DRB; Union Bank
More informationconsul 32L04SE SUNDAY "LAKE"
consu 32L04SE9483 2.5836 SUNDAY "LAKE" 010 Magnetic Survey for DOME EXPLORATION (CANADA) LIMITED on PROJECT 191 Sunday Creek Area, Ontario (To Accompany Maps 82-120 A to H) RECEIVED AUG 3 01982 MINING
More informationImage Generation in Microprocessor-based System with Simultaneous Video Memory Read/Write Access
Image Generation in Microprocessor-based System with Simutaneous Video Memory Read/rite Access Mountassar Maamoun 1, Bouaem Laichi 2, Abdehaim Benbekacem 3, Daoud Berkani 4 1 Department o Eectronic, Bida
More informationOperation Guide
MO0605-EA Getting Acquainted Congratuations upon your seection of this CASIO watch. To get the most out of your purchase, be sure to read this manua carefuy. Warning! The measurement functions buit into
More informationConducteur d'émotions...
2010 Conducteur d'émotions... www.rea-cabe.com Dear CUSTOMERS, In ess than 10 years, REAL CABLE has become a major payer in cabe manufacturing market thanks to our strong experience in this industry and
More informationA motor behavioral evaluation method for children with developmental disorders during music therapy sessions: A pilot study.
Curr Pediatr Res 6; (&): 3-7 ISSN 97-93 www.currentpediatrics.com A motor behaviora evauation method for chidren with deveopmenta disorders during music therapy sessions: A piot study. Zu Soh, Ryo Migita,
More informationU C A RX-V995 AV RECEIVER AMPLI-TUNER AUDIO-VIDEO OWNER S MANUAL MODE D EMPLOI
U C RX-V995 V RECEIVER MPLI-TUNER UDIO-VIDEO OWNER S MNUL MODE D EMPLOI CUTION RISK OF ELECTRIC SHOCK DO NOT OPEN CUTION: TO REDUCE THE RISK OF ELECTRIC SHOCK, DO NOT REMOVE COVER (OR CK). NO USER-SERVICELE
More informationOperating Instructions VEGAPULS ma/hart
Operating Instructions VEGAPULS 61 4 20 ma/hart Radar Contents Contents 1 About this document 1.1 Function... 5 1.2 Target group... 5 1.3 Symboism used... 5 2 For your safety 2.1 Authorised personne...
More informationRX-V793. Natural Sound AV Receiver Récepteur audiovisuel Son Naturel OWNER S MANUAL MODE D EMPLOI
U C RX-V79 Natura Sound V Receiver Récepteur audiovisue Son Nature Thank you for seecting this YMH V receiver. Nous vous remercions d avoir porté votre choix sur ce récepteur audiovisue YMH. OWNER S MNUL
More information