764:(and I'm astonished by the incredible efficiency of his sharing) his entire system adds up to 22 logical operations- count my little plusses and EQs etc above and you get 24.. however I don't just arbitrarily invert without counting an invert, and (B * /D) can be shared between F and G, and (B XOR C) can be shared between D and G, so mine actually uses less logic =D. Unfortunately I think his point was just to demonstrate his logic-sharing process (unlike mine which was for sheer minimization) so it's not a fair contest. Nice find on the link by the way, do you actually read all of that? --
450:
514:
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No you can't just negate the final results..what about in the case of ( A OR B ) and both A and B are actually inverted? A' OR B' isn't the same as (A OR B)' it's the same as (A AND B)'. This is additional logic and isn't simply a boolean flip to be counted up and turned into either a single inverter
396:
That is precisely what I'm looking for as it accurately describes boundaries of the orbital field. Modification to M and N based on the Hill Radius results in an accurate description of the boundary of the
Gravitation field of Primary influence. in order to help you help me I should Provide a Truer
1518:
But more important, especially on test day, is the part about "this is just really throwing me". You would not have come this far without intellect and knowledge, and you have studied diligently. Go in knowing you are capable of passing, and don't need to be perfect to do so. A test is scribbles on
797:
Just read the whole thing. There is a solution for just 0-9 too later. He inverts terms because his model takes all 16 two-input one-output binary operations of the same cost, so negated inputs just means a different basic operation. (My opinion is that this is realistic, because if you have not
759:
This is obviously far larger than it needs to be, but his solution is so well-distributed into shared steps that if implemented his would actually have about the same number of gates as mine. Annoyingly he arbitrarily inverts his steps without counting the cost of an inverter, but it wouldn't add a
330:
solution meeting these requirements, or should it in some sense be "natural"? If the latter, one would expect the results for the cases I = 0Β° and I = 90Β° to be the limits of the general case 0Β° < I < 90Β° as I approaches these limiting values. However, from your description β which is not
616:
I meant indeed building the gate out of transistors; you need many more than for an OR gate. Another issue is that if the two inputs to an XOR gate flip synchronously, but there is a tiny time difference for the changes to arrive at the gate, it can briefly output a spurious value. For this
1795:
KSmrq - Thanks for the advice. It's often hard to keep these things in perspective sometimes. I do get to the gym every morning at around 7 to run around and get some exercise, which is helpful for the mind in the long run. I'll try to remember this and stay calm during the actual exam.
652:
Interesting results from my continuing work.. the equation can be compressed all the way down to C + (B EQ D), however then the 7 has a tail. I for one welcome our new digitΒ :) And astonishingly, directly connecting /D results in near-perfect results as well, but it's still flawed.
2240:
1596:
spaces are nested on spaces of measure 1 - and probably on all (totally) finite measure spaces - (apply HΓΆlder's inequality); with the counting measure, the inequality on the norms (and thus inclusion of the spaces) is reversed. To see this, use the fact that we have
798:
only AND, OR, XOR but also a fourth one like implication as basic operations, then you basically have all of those 16 operations only sometimes you get a negated result but you can keep track of that and only ever have to negate only the final results.) β
116:
Given an orbital path defined by the minimum constraints such that: M=Semi-Major Axis N=Semi-Minor Axis I=Angle of inclination from the
Elliptic plane of the system. L=Angle of Ascending Node (Herein Rendered of non importance to the final Equation)
1775:. I guess I'm not adding anything new here, really. These inclusions are cute, but their usefulness is quite limited since the are (of course!) not available on the line; which is a drag, really (imagine what Fourier analysis would look like!).
2247:
456:
As a pet project I'm trying to optimize the logic for a BCD-to-7-segment-display decoder. I've optimized it pretty far but I'm having trouble with the logic for segment D. So far what I have is (and ABCD represent the bits of the
345:
Sorry to be clear as mud, upon further reflection of the problem, the simpler description meet my needs. All the formulae I have already are
Parametric, and I am terrible with transforms to be honest. As for Limits on I :
2807:
473:
Is there some way to take advantage of this B-NOTB relationship and optimize it further? It feels like if I can rearrange it somehow then I can also take advantage of the Cs and Ds on both sides of the conjunction
683:
Doesn't look a lot more minimized than the first expression does it? Oh well, I'm satisfied since I can share B XOR C with the logic for segment G. Thanks for letting me use your math desk for my musingsΒ :)
592:, which gives the laws you can use in manipulating and simplifying Boolean expressions. I don't know a name for these specific transformations, but they are readily derived from the Boolean algebra laws. In
3334:
1328:
569:
transformation. It's an XOR because they could never both be true, but it doesn't really matter. Is there a name for this type of rule? And does anyone see any further optimization, or is this a dead end?
2097:
747:
Grr, postscript. OK I'm not pretending to understand how he reached his results, but they don't seem too impressive. For example, I worked out one chain of logic and here's his expression for segment A:
563:(the reasoning is that if B then SOMETHING2 needs to be true for the expression to be true, and if /B then SOMETHING1 needs to be true for the expression to be true. the or was a half-educated guess)
2638:
1519:
paper, nothing more. Try swimming, music, yoga, whatever helps you relax and calm your mind. Prepare as well as you can, get a good night's sleep, then let it all go and just do the best you can. --
236:
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66:
45:
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59:
1989:
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2809:. Note that there are no worries about taking q-th roots here. The point is that the q-th root function is increasing so it preserves inequalities. That it has the behavior we know on
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1184:
1130:
918:
874:
121:
Under the
Condition of I= 90 deg the corresponding surface would be comprised of 2 nested spheres with the outer Sphere having a Radius of M and the Inner Sphere having a Radius of N.
3143:
2081:
315:
Because of the cylindrical symmetry, that there is no dependence on ΞΈ. I do not understand your description of your second case well enough to determine an equation for this surface.
55:
51:
2028:
307:
1950:
162:, and only convert to Cartesian co-ordinates if and when necessary. In cylindruical co-ordinates the equations of the surfaces that you describe in your first and third cases are:
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3090:
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1773:
3000:
1867:
1834:
110:
I have been trying to unravel this one for a very long time, ~6 months, and as of yet only have appeared to have come close twice only to find it did not stand up to testing.
3234:
3043:
984:
951:
1710:
486:
For the digit 9, A = D = 1 and B = C = 0, so your formula evaluates then to (/0 + (0 XOR 1)) * (0 + 0 + /1) = 1 * 0 = 0. But shouldn't segment D be on for digit 9? Β --
2839:
2476:{\displaystyle \|x\|_{q}\leq \left(\|x\|_{\infty }^{q-p}\sum |x_{i}|^{p}\right)^{1/q}\leq \|x\|_{\infty }^{1-p/q}\|x\|_{p}^{p/q}\leq \|x\|_{\infty }^{1-p/q}\|x\|_{p}}
1594:
1075:
1033:
108:
I have also posted my query on some other math oriented boards and a couple science boards in hopes of getting an answer. With that said; Here lie my quandary.
25:
511:
All devices I could find in my neighbourhood using 7-segment digits (microwave, video recorder, several alarm clocks) display the digit 9 with a tail, thus:
499:
as a single disjunction for the tail segment on the 6 and you can use an XOR for the 9) . All the other digits work. But any ideas on how to optimize? : -->
361:
I wish I could describe it a lot better as I can see it in my minds eye but I'm having a devil of a time getting it from brain to hand(mouth) so to speak.
605:
What do you mean not pleasant to implement? As long as you're not actually building the gate, how is it any different than any other logical operation? --
125:
0 would be a Sphere
Negated in part by a Hyperboloid of one sheet, such that the Sphere has a Radius of M and the focii of Hyperboloid is dependant on I.
85:
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the
128:
Under the
Condition of I=0 the structure would degenerate in to a 2D torus such the the outer Circle has Radius M and the Inner Circle is of radius N.
532:
I forgot nines have tails when I was mapping it out. Also like I said it turns out to be better for logic minimization if it doesn't have a tail. --
397:
equation set for the Torus involved. Again I unfortunately do not know how to apply the inclination transform so I will make do without that part.
2643:
783:
The logic described in the fascicle implements the hex digits A-F, not just 0-9, which is why the logic implemented may seem somewhat redundant.
786:
Well it seems large because of the extensive term-sharing, but if it indeed is a full solution for 0-f then that beats the pants off mineΒ :( --
37:
1348:-norm raise each coordinate to a higher power, thus decreasing it, at the end it takes a higher root, decreasing the result further. --
2235:{\displaystyle \sum \left({\dfrac {|x_{i}|}{\|x\|_{\infty }}}\right)^{q}\leq \sum \left({\dfrac {|x_{i}|}{\|x\|_{\infty }}}\right)^{p}}
495:
There's no tail on the 9 (putting a tail on the 6 and not the 9 just makes things optimize beautifully- you can use a full half of the
1146:
The norms are clearly not equivalent, as there are sequences for which one is finite and the other is not (so even if you restrict to
371:
If you take a torus and rotate it around another axis than its own axis of rotational symmetry, you actually trace out a complicated
21:
3243:
1245:
714:
If anyone sees any glaring possibilities for further minimization please point them out but otherwise this is my final solution --
626:
Very interesting. This is purely a logic exercise so you're right it wouldn't matter- besides, that's why circuits are clocked --
356:
I know the parametric set for a torus of I=0 is: { x=x0+(M+N*cos(u))*cos(v), y=y0+(M+N*cos(u))*sin(v), z=z0+(N*sin(u)) }
1132:, but I don't know how to show this or how to prove or disprove the equivalency of the norms. Any aid would be appreciated! β
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I know the parametric set for a torus of I=0 is: { x=x0+(M+N*cos(u))*cos(v), y=y0+(M+N*cos(u))*sin(v), z=z0+(N*sin(u)) }
2553:
152:
3336:. It's really that simple. I suppose your calculation above failed because you have used inequalities which are too weak.
171:
3359:
Thanks for spelling it out. Gah. I thought it was going to be a little more technical. Please excuse my dullardness. β
1405:
1600:
1876:
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1955:
1547:
1468:
1365:
1353:
1335:
86:
17:
1238:, I'm pretty sure there's some well-known inequality which gives this instantly, but I don't remember which.
3148:
1189:
1149:
1083:
883:
841:
589:
3095:
2033:
1344:
Another solution: First normalize the vector so that its greatest coordinate is 1. Then, not only does the
1330:(for which Binomial expansions are handy), generalize to any number of summands, and pass to the limit. --
1994:
242:
151:
Since your surfaces all appear to have cylindrical rotational symmetry, it will be much easier to work in
2536:
is greater than 1 (which it very well may not be). Suggestions? Am I missing something so obvious here? β
1922:
135:
I do have a possibly simpler description. Take a Torus M-N of inclination I and
Revolve around z axis.
993:
Now, I know what everything here means, and the first part of the question is to do the same thing for
835:
Hi all, yet another real analysis question vexes me. Perhaps it shall vex thee, and ye can help me. =)
2930:
2858:
1239:
132:
I am hoping that someone will know the parametric equation set for {x,y,z} to create this construct.
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380:
1080:
Through various trials and errors, I have come to the conclusion that it's probably the case that
332:
566:
359:
I know neither how to incline the torus nor how to perform the second
Transform of revolution.
141:
I know neither how to incline the torus nor how to perform the second
Transform of revolution.
3005:
1504:
956:
923:
618:
597:
524:
487:
384:
336:
3240:. So multiplying a vector by a positive scalar multiplies the norm by that scalar. Therefore
1799:
Meni - Your "alternate solution" doesn't really work (as far as I can tell). After dividing
1678:
1508:
449:
2846:
2812:
1780:
1572:
1038:
996:
1503:= 1. We also know that a sequence space is a special case of a function space, using the
727:
375:
in space. Do I understand correctly that you looking for a parametric description of its
3145:(where the inequality is clearly strict if there is any other nonzero coordinate). Thus
735:
432:} This is unfortunately as Far As I have been able to get before my brain melts down.
372:
316:
734:
volume 4 has the 7-segment display as a worked-out example. Download it fast for the
326:
What is the origin of the conditions on the desired manifold? Would you be happy with
3362:
2539:
1135:
593:
433:
362:
142:
799:
739:
1077:, which I can (and have recently) done. However, this is just really throwing me.
1556:
Oops, sorry; yes, of course. The ones sequence is obviously unbounded for every
1489:
vector space are equivalent, but we also know that the sequence (1,1,1,β¦) has a
331:
perfectly clear; what is the meaning of "negated" here? β there appears to be a
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application that it is not an issue, but for some applications it may be. Β --
2802:{\displaystyle ||x||_{q}^{q}\leq ||x||_{q}^{q-p}||x||_{p}^{p}=||x||_{p}^{q}}
1242:
probably solves this easily, though. A different approach is to first prove
644:
74:
1569:
This is a classical exercise, and something that is neat to remember. The
596:, XOR is not a pleasant operator to implement, unlike (N)AND and (N)OR.Β --
1869:
floating around. I might be misunderstanding you, so here's what I have:
376:
113:
This combines both
Trigonometric and Astrometric sciences together.
3092:
has 1 as one of its summands and all others are non-negative. Thus
550:
OK through guesswork and a lot of truth tables, I figured out that
517:. I assume then you have a special reason for wanting the tailless
731:
643:
496:
448:
2855:
Yes, you have missed several very obvious things. I will denote
695:
Also in case anyone's curious here are my mostly final results:
458:
353:
Take a Torus M-N of inclination I and Revolve around z axis.
3329:{\displaystyle \|x\|_{q}=M\|y\|_{q}\leq M\|y\|_{p}=\|x\|_{p}}
2489:
Also, this "taking a higher root" deal only works if the sum
1323:{\displaystyle {\sqrt{a^{p}+b^{p}}}\geq {\sqrt{a^{q}+b^{q}}}}
79:
Welcome to the Knowledge Mathematics Reference Desk Archives
467:
The thing that's so tantalizing is that it's basically:
2640:. Now from what you have, and that inequality, we get
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2633:{\displaystyle ||x||_{\infty }^{p}\leq ||x||_{p}^{p}}
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Do we really know that the sequence (1,1,1,β¦) has a
231:{\displaystyle r^{2}+h^{2}=M^{2}{\mbox{ or }}N^{2}}
3328:
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738:tend to be taken down the net after some time. β
409:e of the system only defines the base for M and N.
301:
230:
1448:{\displaystyle \ell ^{p}\cap \ell ^{q}=\ell ^{p}}
1604:
1362:From the inequality it follows, of course, that
1459:-norm that can be arbitrarily greater than the
1668:{\displaystyle \sup |f|^{p}\leq ||f||_{p}^{p}}
1542:= 1? Perhaps it should be (1/1,1/2,1/3,β¦)? --
1186:, the ratio can be aribtrarily large). As for
1560:, the harmonic one is what I meant to say. --
8:
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1912:{\displaystyle x\in \ell ^{p}\cap \ell ^{q}}
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588:Maybe you should have a look at our article
559:(B AND SOMETHING2) OR (NOTB AND SOMETHING1)
920:. Is there an inequality between the norms
553:(B OR SOMETHING1) AND (NOTB OR SOMETHING2)
470:(B OR SOMETHING1) AND (NOTB OR SOMETHING2)
1984:{\displaystyle \|x\|_{\infty }<\infty }
1395:{\displaystyle \ell ^{p}\subset \ell ^{q}}
808:or no inverter at the end of the chain. --
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3190:{\displaystyle \|y\|_{q}\leq \|y\|_{p}}
2550:You basically got it, just notice that
1231:{\displaystyle \|x\|_{q}\leq \|x\|_{p}}
1179:{\displaystyle \ell ^{p}\cap \ell ^{q}}
1125:{\displaystyle \|x\|_{q}\leq \|x\|_{p}}
913:{\displaystyle \ell ^{p}\cap \ell ^{q}}
869:{\displaystyle 1\leq p<q<\infty }
65:
3138:{\displaystyle \sum |y_{i}|^{p}\geq 1}
2076:{\displaystyle \sum |x_{i}|^{p}\geq 1}
461:digit from high to low significance):
43:
2023:{\displaystyle \|x\|_{\infty }\geq 1}
831:Spaces of p- and q-summable sequences
405:=Hill Radius of the body in question.
349:I'll restate the simpler description.
302:{\displaystyle r^{2}+h^{2}=r(M+N)-MN}
7:
1945:{\displaystyle x\in \ell ^{\infty }}
728:Pre-Fascicle 0c (Boolean evaluation)
674:(/B * (C + /D)) XOR (B * (C XOR D))
573:(/B * (C + /D)) XOR (B * (C XOR D))
705:D: (D EQ (B XOR C)) + (/B * C * /D)
124:Under the Condition of I<90: -->
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2009:
1978:
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104:Advanced Parametric Equation in 3D
32:
2959:{\displaystyle \|y\|_{\infty }=1}
2887:{\displaystyle M=\|x\|_{\infty }}
709:F: (/C + (B * /D)) * (A + B + /D)
680:(D EQ (B XOR C)) + (/B * C * /D)
665:Aha! I do believe I have got it!
3085:{\displaystyle \sum |y_{i}|^{p}}
2920:{\displaystyle y={\frac {x}{M}}}
2529:{\displaystyle \sum |x_{i}|^{p}}
1768:{\displaystyle |f|^{p}|f|^{q-p}}
1515:. So that's two lines of attack.
990:? Are the two norms equivalent?
668:(/B + (C XOR D)) * (B + C + /D)
518:
512:
464:(/B + (C XOR D)) * (B + C + /D)
2995:{\displaystyle \|y\|_{\infty }}
1862:{\displaystyle \|x\|_{\infty }}
1829:{\displaystyle \|x\|_{\infty }}
876:and suppose we have a sequence
3229:{\displaystyle \|\cdot \|_{r}}
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275:
1:
1836:, I get this extra factor of
426:y=(M+N*cos(u))*(1+r)*sin(v)*,
33:
711:G: A + (B * /D) + (B XOR C)
556:is logically equivalent to:
423:x=(M+N*cos(u))*(1+r)*cos(v),
3386:
1481:Technically, we know that
3368:16:04, 24 July 2007 (UTC)
3351:15:53, 24 July 2007 (UTC)
3038:{\displaystyle |y_{i}|=1}
2851:15:43, 24 July 2007 (UTC)
2545:15:19, 24 July 2007 (UTC)
1785:15:27, 24 July 2007 (UTC)
1565:01:58, 24 July 2007 (UTC)
1552:22:18, 23 July 2007 (UTC)
1524:22:15, 23 July 2007 (UTC)
1473:21:18, 23 July 2007 (UTC)
1358:21:13, 23 July 2007 (UTC)
1340:21:07, 23 July 2007 (UTC)
1141:20:32, 23 July 2007 (UTC)
979:{\displaystyle \|x\|_{q}}
946:{\displaystyle \|x\|_{p}}
815:00:51, 29 July 2007 (UTC)
803:10:28, 25 July 2007 (UTC)
793:05:23, 25 July 2007 (UTC)
771:16:06, 24 July 2007 (UTC)
743:09:34, 24 July 2007 (UTC)
721:03:04, 24 July 2007 (UTC)
691:02:57, 24 July 2007 (UTC)
677:EQ (drum roll please...)
660:02:40, 24 July 2007 (UTC)
633:05:22, 24 July 2007 (UTC)
622:03:22, 24 July 2007 (UTC)
612:19:41, 23 July 2007 (UTC)
601:19:19, 23 July 2007 (UTC)
583:17:42, 23 July 2007 (UTC)
539:19:47, 23 July 2007 (UTC)
528:19:06, 23 July 2007 (UTC)
507:17:09, 23 July 2007 (UTC)
491:16:32, 23 July 2007 (UTC)
481:15:47, 23 July 2007 (UTC)
437:01:53, 24 July 2007 (UTC)
388:19:31, 23 July 2007 (UTC)
366:18:14, 23 July 2007 (UTC)
340:15:46, 23 July 2007 (UTC)
320:13:17, 23 July 2007 (UTC)
146:18:14, 23 July 2007 (UTC)
565:Like some kind of weird
153:cylindrical co-ordinates
18:Knowledge:Reference desk
1705:{\displaystyle |f|^{q}}
590:Boolean algebra (logic)
3330:
3230:
3191:
3139:
3086:
3039:
2996:
2960:
2921:
2888:
2835:
2803:
2634:
2530:
2477:
2236:
2077:
2024:
1985:
1946:
1913:
1863:
1830:
1769:
1706:
1669:
1590:
1449:
1396:
1324:
1232:
1180:
1126:
1071:
1029:
980:
947:
914:
870:
649:
453:
303:
232:
87:current reference desk
3331:
3231:
3192:
3140:
3087:
3040:
2997:
2961:
2922:
2889:
2836:
2834:{\displaystyle (0,1)}
2804:
2635:
2531:
2478:
2237:
2078:
2025:
1986:
1947:
1914:
1864:
1831:
1770:
1707:
1670:
1591:
1589:{\displaystyle L^{p}}
1450:
1397:
1325:
1233:
1181:
1127:
1072:
1070:{\displaystyle L^{q}}
1030:
1028:{\displaystyle L^{p}}
981:
948:
915:
871:
647:
452:
347:= 0Β° and I <= 90Β°.
304:
233:
3244:
3207:
3149:
3096:
3049:
3006:
2973:
2931:
2898:
2859:
2813:
2644:
2554:
2493:
2248:
2098:
2034:
1995:
1956:
1923:
1877:
1840:
1807:
1716:
1679:
1601:
1573:
1406:
1366:
1246:
1190:
1150:
1084:
1039:
997:
957:
924:
884:
842:
429:z=((N*(1+r))*sin(u))
381:topological boundary
243:
172:
3002:= 1, there is some
2798:
2760:
2725:
2681:
2629:
2591:
2456:
2418:
2389:
2302:
1664:
1455:and that it is the
1240:Jensen's inequality
986:valid for all such
699:A: A + C + (B EQ D)
3326:
3226:
3187:
3135:
3082:
3035:
2992:
2956:
2917:
2884:
2831:
2799:
2782:
2744:
2703:
2665:
2630:
2613:
2575:
2526:
2473:
2428:
2396:
2361:
2282:
2232:
2220:
2153:
2073:
2020:
1981:
1942:
1909:
1859:
1826:
1765:
1702:
1665:
1648:
1586:
1487:finite-dimensional
1445:
1392:
1320:
1228:
1176:
1122:
1067:
1025:
976:
943:
910:
866:
650:
648:Hail the tailed 7!
454:
333:jump discontinuity
299:
228:
216:
2915:
2219:
2152:
1505:Lebesgue integral
1318:
1281:
703:C: A + B + /C + D
564:
215:
93:
92:
73:
72:
3377:
3335:
3333:
3332:
3327:
3325:
3324:
3306:
3305:
3284:
3283:
3262:
3261:
3235:
3233:
3232:
3227:
3225:
3224:
3196:
3194:
3193:
3188:
3186:
3185:
3167:
3166:
3144:
3142:
3141:
3136:
3128:
3127:
3122:
3116:
3115:
3106:
3091:
3089:
3088:
3083:
3081:
3080:
3075:
3069:
3068:
3059:
3044:
3042:
3041:
3036:
3028:
3023:
3022:
3013:
3001:
2999:
2998:
2993:
2991:
2990:
2965:
2963:
2962:
2957:
2949:
2948:
2926:
2924:
2923:
2918:
2916:
2908:
2893:
2891:
2890:
2885:
2883:
2882:
2849:
2840:
2838:
2837:
2832:
2808:
2806:
2805:
2800:
2797:
2792:
2787:
2781:
2773:
2768:
2759:
2754:
2749:
2743:
2735:
2730:
2724:
2713:
2708:
2702:
2694:
2689:
2680:
2675:
2670:
2664:
2656:
2651:
2639:
2637:
2636:
2631:
2628:
2623:
2618:
2612:
2604:
2599:
2590:
2585:
2580:
2574:
2566:
2561:
2535:
2533:
2532:
2527:
2525:
2524:
2519:
2513:
2512:
2503:
2482:
2480:
2479:
2474:
2472:
2471:
2455:
2451:
2436:
2417:
2413:
2404:
2388:
2384:
2369:
2351:
2350:
2346:
2337:
2333:
2332:
2331:
2326:
2320:
2319:
2310:
2301:
2290:
2266:
2265:
2241:
2239:
2238:
2233:
2231:
2230:
2225:
2221:
2218:
2217:
2216:
2200:
2199:
2194:
2193:
2184:
2178:
2164:
2163:
2158:
2154:
2151:
2150:
2149:
2133:
2132:
2127:
2126:
2117:
2111:
2082:
2080:
2079:
2074:
2066:
2065:
2060:
2054:
2053:
2044:
2029:
2027:
2026:
2021:
2013:
2012:
1990:
1988:
1987:
1982:
1974:
1973:
1951:
1949:
1948:
1943:
1941:
1940:
1918:
1916:
1915:
1910:
1908:
1907:
1895:
1894:
1868:
1866:
1865:
1860:
1858:
1857:
1835:
1833:
1832:
1827:
1825:
1824:
1783:
1774:
1772:
1771:
1766:
1764:
1763:
1752:
1743:
1738:
1737:
1732:
1723:
1711:
1709:
1708:
1703:
1701:
1700:
1695:
1686:
1674:
1672:
1671:
1666:
1663:
1658:
1653:
1647:
1639:
1634:
1626:
1625:
1620:
1611:
1595:
1593:
1592:
1587:
1585:
1584:
1534:-norm for every
1509:counting measure
1493:-norm for every
1454:
1452:
1451:
1446:
1444:
1443:
1431:
1430:
1418:
1417:
1401:
1399:
1398:
1393:
1391:
1390:
1378:
1377:
1329:
1327:
1326:
1321:
1319:
1317:
1312:
1311:
1310:
1298:
1297:
1287:
1282:
1280:
1275:
1274:
1273:
1261:
1260:
1250:
1237:
1235:
1234:
1229:
1227:
1226:
1208:
1207:
1185:
1183:
1182:
1177:
1175:
1174:
1162:
1161:
1131:
1129:
1128:
1123:
1121:
1120:
1102:
1101:
1076:
1074:
1073:
1068:
1051:
1050:
1034:
1032:
1031:
1026:
1009:
1008:
985:
983:
982:
977:
975:
974:
952:
950:
949:
944:
942:
941:
919:
917:
916:
911:
909:
908:
896:
895:
875:
873:
872:
867:
760:lot if counted.
707:E: /D * (/B + C)
701:B: (C EQ D) + /B
562:
522:
516:
308:
306:
305:
300:
268:
267:
255:
254:
237:
235:
234:
229:
227:
226:
217:
213:
210:
209:
197:
196:
184:
183:
75:
38:Mathematics desk
34:
3385:
3384:
3380:
3379:
3378:
3376:
3375:
3374:
3316:
3297:
3275:
3253:
3242:
3241:
3216:
3205:
3204:
3177:
3158:
3147:
3146:
3117:
3107:
3094:
3093:
3070:
3060:
3047:
3046:
3014:
3004:
3003:
2982:
2971:
2970:
2940:
2929:
2928:
2896:
2895:
2874:
2857:
2856:
2845:
2841:is irrelevant.
2811:
2810:
2642:
2641:
2552:
2551:
2514:
2504:
2491:
2490:
2486:And I'm stuck.
2463:
2321:
2311:
2275:
2271:
2270:
2257:
2246:
2245:
2208:
2201:
2185:
2179:
2172:
2171:
2141:
2134:
2118:
2112:
2105:
2104:
2096:
2095:
2055:
2045:
2032:
2031:
2004:
1993:
1992:
1965:
1954:
1953:
1932:
1921:
1920:
1899:
1886:
1875:
1874:
1849:
1838:
1837:
1816:
1805:
1804:
1779:
1747:
1727:
1714:
1713:
1690:
1677:
1676:
1615:
1599:
1598:
1576:
1571:
1570:
1435:
1422:
1409:
1404:
1403:
1382:
1369:
1364:
1363:
1302:
1289:
1288:
1265:
1252:
1251:
1244:
1243:
1218:
1199:
1188:
1187:
1166:
1153:
1148:
1147:
1112:
1093:
1082:
1081:
1042:
1037:
1036:
1000:
995:
994:
966:
955:
954:
933:
922:
921:
900:
887:
882:
881:
840:
839:
833:
754:
712:
681:
675:
669:
574:
560:
554:
471:
465:
447:
445:Boolean Algebra
416:
404:
259:
246:
241:
240:
218:
201:
188:
175:
170:
169:
106:
101:
30:
29:
28:
12:
11:
5:
3383:
3381:
3373:
3372:
3371:
3370:
3354:
3353:
3343:Meni Rosenfeld
3339:
3338:
3337:
3323:
3319:
3315:
3312:
3309:
3304:
3300:
3296:
3293:
3290:
3287:
3282:
3278:
3274:
3271:
3268:
3265:
3260:
3256:
3252:
3249:
3223:
3219:
3215:
3212:
3202:
3201:has β-norm 1).
3184:
3180:
3176:
3173:
3170:
3165:
3161:
3157:
3154:
3134:
3131:
3126:
3121:
3114:
3110:
3105:
3101:
3079:
3074:
3067:
3063:
3058:
3054:
3034:
3031:
3027:
3021:
3017:
3012:
2989:
2985:
2981:
2978:
2955:
2952:
2947:
2943:
2939:
2936:
2914:
2911:
2906:
2903:
2881:
2877:
2873:
2870:
2867:
2864:
2853:
2830:
2827:
2824:
2821:
2818:
2796:
2791:
2786:
2780:
2776:
2772:
2767:
2763:
2758:
2753:
2748:
2742:
2738:
2734:
2729:
2723:
2720:
2717:
2712:
2707:
2701:
2697:
2693:
2688:
2684:
2679:
2674:
2669:
2663:
2659:
2655:
2650:
2627:
2622:
2617:
2611:
2607:
2603:
2598:
2594:
2589:
2584:
2579:
2573:
2569:
2565:
2560:
2523:
2518:
2511:
2507:
2502:
2498:
2484:
2483:
2470:
2466:
2462:
2459:
2454:
2450:
2446:
2443:
2440:
2435:
2431:
2427:
2424:
2421:
2416:
2412:
2408:
2403:
2399:
2395:
2392:
2387:
2383:
2379:
2376:
2373:
2368:
2364:
2360:
2357:
2354:
2349:
2345:
2341:
2336:
2330:
2325:
2318:
2314:
2309:
2305:
2300:
2297:
2294:
2289:
2285:
2281:
2278:
2274:
2269:
2264:
2260:
2256:
2253:
2243:
2229:
2224:
2215:
2211:
2207:
2204:
2198:
2192:
2188:
2183:
2175:
2170:
2167:
2162:
2157:
2148:
2144:
2140:
2137:
2131:
2125:
2121:
2116:
2108:
2103:
2072:
2069:
2064:
2059:
2052:
2048:
2043:
2039:
2019:
2016:
2011:
2007:
2003:
2000:
1980:
1977:
1972:
1968:
1964:
1961:
1939:
1935:
1931:
1928:
1906:
1902:
1898:
1893:
1889:
1885:
1882:
1871:
1870:
1856:
1852:
1848:
1845:
1823:
1819:
1815:
1812:
1797:
1792:
1791:
1790:
1789:
1788:
1787:
1762:
1759:
1756:
1751:
1746:
1742:
1736:
1731:
1726:
1722:
1699:
1694:
1689:
1685:
1662:
1657:
1652:
1646:
1642:
1638:
1633:
1629:
1624:
1619:
1614:
1610:
1606:
1583:
1579:
1544:Meni Rosenfeld
1528:
1527:
1526:
1516:
1476:
1475:
1465:Meni Rosenfeld
1442:
1438:
1434:
1429:
1425:
1421:
1416:
1412:
1389:
1385:
1381:
1376:
1372:
1360:
1350:Meni Rosenfeld
1342:
1332:Meni Rosenfeld
1316:
1309:
1305:
1301:
1296:
1292:
1285:
1279:
1272:
1268:
1264:
1259:
1255:
1225:
1221:
1217:
1214:
1211:
1206:
1202:
1198:
1195:
1173:
1169:
1165:
1160:
1156:
1119:
1115:
1111:
1108:
1105:
1100:
1096:
1092:
1089:
1066:
1063:
1060:
1057:
1054:
1049:
1045:
1024:
1021:
1018:
1015:
1012:
1007:
1003:
973:
969:
965:
962:
940:
936:
932:
929:
907:
903:
899:
894:
890:
865:
862:
859:
856:
853:
850:
847:
832:
829:
828:
827:
826:
825:
824:
823:
822:
821:
820:
819:
818:
817:
776:
775:
774:
773:
752:
751:
750:
749:
748:
710:
708:
706:
704:
702:
700:
698:
696:
679:
673:
667:
663:
642:
641:
640:
639:
638:
637:
636:
635:
572:
558:
552:
548:
547:
546:
545:
544:
543:
542:
541:
469:
463:
446:
443:
442:
441:
440:
439:
430:
427:
424:
421:
418:
414:
411:
410:
407:
406:
402:
399:
398:
391:
390:
360:
350:
348:
343:
342:
323:
322:
312:
311:
310:
309:
298:
295:
292:
289:
286:
283:
280:
277:
274:
271:
266:
262:
258:
253:
249:
238:
225:
221:
208:
204:
200:
195:
191:
187:
182:
178:
164:
163:
130:
129:
126:
122:
109:
105:
102:
100:
97:
95:
91:
90:
82:
81:
71:
70:
64:
48:
41:
40:
31:
15:
14:
13:
10:
9:
6:
4:
3:
2:
3382:
3369:
3366:
3365:
3364:
3358:
3357:
3356:
3355:
3352:
3348:
3344:
3340:
3321:
3313:
3307:
3302:
3294:
3288:
3285:
3280:
3272:
3266:
3263:
3258:
3250:
3239:
3221:
3213:
3203:
3200:
3182:
3174:
3168:
3163:
3155:
3132:
3129:
3124:
3112:
3108:
3099:
3077:
3065:
3061:
3052:
3032:
3029:
3019:
3015:
2979:
2968:
2967:
2953:
2950:
2937:
2912:
2909:
2904:
2901:
2871:
2865:
2862:
2854:
2852:
2848:
2844:
2825:
2822:
2819:
2794:
2789:
2774:
2761:
2756:
2751:
2736:
2721:
2718:
2715:
2710:
2695:
2682:
2677:
2672:
2657:
2625:
2620:
2605:
2592:
2587:
2567:
2549:
2548:
2547:
2546:
2543:
2542:
2541:
2521:
2509:
2505:
2496:
2487:
2468:
2460:
2452:
2448:
2444:
2441:
2438:
2425:
2419:
2414:
2410:
2406:
2401:
2393:
2385:
2381:
2377:
2374:
2371:
2358:
2352:
2347:
2343:
2339:
2334:
2328:
2316:
2312:
2303:
2298:
2295:
2292:
2279:
2272:
2267:
2262:
2254:
2244:
2242:, which gives
2227:
2222:
2205:
2190:
2186:
2173:
2168:
2165:
2160:
2155:
2138:
2123:
2119:
2106:
2101:
2094:
2093:
2092:
2090:
2086:
2070:
2067:
2062:
2050:
2046:
2037:
2017:
2014:
2001:
1975:
1962:
1933:
1929:
1926:
1904:
1900:
1896:
1891:
1887:
1883:
1880:
1846:
1813:
1802:
1798:
1794:
1793:
1786:
1782:
1778:
1760:
1757:
1754:
1744:
1734:
1724:
1697:
1687:
1660:
1655:
1640:
1627:
1622:
1612:
1581:
1577:
1568:
1567:
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1563:
1559:
1555:
1554:
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1549:
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1533:
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1525:
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1499:
1496:
1492:
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1436:
1432:
1427:
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1109:
1103:
1098:
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1061:
1058:
1055:
1047:
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1019:
1016:
1013:
1005:
1001:
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971:
963:
938:
930:
905:
901:
897:
892:
888:
879:
860:
857:
854:
851:
848:
845:
838:Suppose that
836:
830:
816:
813:
812:
806:
805:
804:
801:
796:
795:
794:
791:
790:
785:
784:
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769:
768:
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746:
745:
744:
741:
737:
736:pre-fascicles
733:
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725:
724:
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625:
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623:
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614:
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610:
609:
604:
603:
602:
599:
595:
594:digital logic
591:
587:
586:
585:
584:
581:
580:
571:
568:
557:
551:
540:
537:
536:
531:
530:
529:
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521:
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510:
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508:
505:
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468:
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318:
314:
313:
296:
293:
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281:
278:
272:
269:
264:
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1675:, and write
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344:
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134:
131:
115:
112:
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94:
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2091:, we have
1485:norms on a
567:de morgan's
26:Mathematics
2927:, so that
1463:-norm. --
762:Remarkably
335:there. Β --
1991:. Assume
317:Gandalf61
50:<<
3363:King Bee
2540:King Bee
1136:King Bee
726:Knuth's
500:_< --
434:Abyssoft
363:Abyssoft
143:Abyssoft
24: |
22:Archives
20: |
3197:(where
1919:. Then
1538:except
1402:, that
800:b_jonas
740:b_jonas
619:Lambiam
598:Lambiam
525:Lambiam
488:Lambiam
385:Lambiam
377:surface
337:Lambiam
99:July 23
89:pages.
67:July 24
46:July 22
2969:Since
1498:except
811:βͺfroth
789:βͺfroth
767:βͺfroth
717:βͺfroth
687:βͺfroth
656:βͺfroth
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608:βͺfroth
579:βͺfroth
535:βͺfroth
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477:βͺfroth
383:)? Β --
3236:is a
3045:, so
2087:<
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2030:, so
1952:, so
1873:Pick
1562:KSmrq
1521:KSmrq
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732:TAOCP
523:? Β --
497:K-map
158:, ΞΈ,
69:: -->
63:: -->
62:: -->
44:<
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3347:talk
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1336:talk
1035:and
953:and
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753:* '
730:for
415:Hill
403:Hill
56:July
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413:r=R
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60:Aug
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