A[x, y) by an algorithm of constant length
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From: Martin M. + I am right:if I am not tell me:or I am right on 26 May 2010 16:59 Brian Chandler wrote: Tim Golden BandTech.com wrote: On Jul 23, 12:36 pm, Hagen <k...(a)itwm.fhg.de wrote: On Jul 22, 10:24 am, Hagen <k...(a)itwm.fhg.de wrote: On Jul 21, 2:04 pm, Maarten Bergvelt <be...(a)math.uiuc.edu wrote: On 2009-07-21, Tim Golden BandTech.com snip Virgil: "In the polynomial ring A[x], x is not not ever a member of the base ring A." A[x, y) by an algorithm of constant length, P==NP q.e.d.. The definition of random sequences according to Martin-L6f uses the notion of recursive sequential test. ... randomness n 1: (thermodynamics) a thermodynamic quantity representing the amount of energy in a system that is no longer available for doing mechanical work; "entropy increases as matter and energy in the universe degrade to an ultimate state of inert uniformity" [syn: randomness, entropy, S] 2: the quality of lacking any predictable order or plan [syn: randomness, haphazardness, stochasticity, noise] randomness - Free On-line Dictionary of Computing (26 May 2007) : randomness 1. An inexplicable misfeature; gratuitous inelegance. 2. A hack or crock that depends on a complex combination of coincidences (or, possibly, the combination upon which the crock depends for its accidental failure to malfunction). "This hack can output characters 40--57 by putting the character in the four bit accumulator field of an XCT and then extracting six bits - the low 2 bits of the XCT opcode are the right thing." "What randomness!" 3. Of people, synonymous with "flakiness". The connotation is that the person so described is behaving weirdly, incompetently, or inappropriately for reasons which are (a) too tiresome to bother inquiring into, (b) are probably as inscrutable as quantum phenomena anyway, and (c) are likely to pass with time. "Maybe he has a real complaint, or maybe it's just randomness. See if he calls back." [Jargon File] randomness - Jargon File (4.4.4, 14 Aug 2003) : randomness n. 1. An inexplicable misfeature; gratuitous inelegance. 2. A hack or crock that depends on a complex combination of coincidences (or, possibly, the combination upon which the crock depends for its accidental failure to malfunction). "This hack can output characters 40--57 by putting the character in the four-bit accumulator field of an XCT and then extracting six bits -- the low 2 bits of the XCT opcode are the right thing." "What randomness!" 3. Of people, synonymous with flakiness. The connotation is that the person so described is behaving weirdly, incompetently, or inappropriately for reasons which are (a) too tiresome to bother inquiring into, (b) are probably as inscrutable as quantum phenomena anyway, and (c) are likely to pass with time. "Maybe he has a real complaint, or maybe it's just randomness. See if he calls back." Despite the negative connotations of most jargon uses of this term have, it is worth noting that randomness can actually be a valuable resource, very useful for applications in cryptography and elsewhere. Computers are so thoroughly deterministic that they have a hard time generating high-quality randomness, so hackers have sometimes felt the need to built special-purpose contraptions for this purpose alone. One well- known website offers random bits generated by radioactive decay. Another derives random bits from images of Lava Lite lamps. (Hackers invariably find the latter hilarious. If you have to ask why, you'll never get it.) randomness - Moby Thesaurus II by Grady Ward, 1.0 : 72 Moby Thesaurus words for "randomness": aimlessness, capriciousness, causelessness, chance, chanciness, changeableness, derangement, designlessness, disarrangement, disarray, disarticulation, discomfiture, discomposure, disconcertedness, disharmony, dishevelment, disintegration, disjunction, disorder, disorderliness, disorganization, disproportion, disruption, disturbance, dysteleology, entropy, erraticism, erraticness, fickleness, haphazardness, hesitancy, hesitation, incalculability, incertitude, incoherence, indecision, indecisiveness, indemonstrability, indeterminacy, indetermination, indeterminism, indiscriminateness, inharmonious harmony, irregularity, irresolution, luck, most admired disorder, nonsymmetry, nonuniformity, perturbation, promiscuity, promiscuousness, purposelessness, suspense, suspensefulness, turbulence, unaccountability, uncertainness, uncertainty, uncertainty principle, undecidedness, undeterminedness, unforeseeableness, unpredictability, unprovability, unsureness, unsymmetry, ununiformity, unverifiability, upset, vacillation, whimsicality Maarten: "You can substitute an element of A for for x" This is enough evidence to expose the polynomial construction as flawed. It's nothing of the sort, of course. In our village school we were taught by Mrs Brophy (I think) that "three into five won't go". You can't divide five by three. Later on, we got told to divide 5 by 3. This does not "expose elementary arithmetic as flawed". Virgil said: in *one ring* (the polynomials), x is not a member of *a different ring*. For if it were, all the polynomials would get evaluated in the base ring A, yielding values in A. Maarten said: "You *can* substitute an element of A for for x". Of course you can. You can take an expression like 3pi^2+7, and replace pi by e (of logarithm fame), getting a different expression 3e^2+7. No- one thinks that e=pi. *If* you replace x by an element of A, then all the polynomials get evaluated in the base ring A, yielding values in A. There is no contradiction between these statements. But anyway... Very good. Let's settle this one point for now. Is x a variable? Bill Dubuque answered this on June 6: Part of the problem here is that "variable" is an overloaded term. We're stuck with it for historical reasons. In this context X can denote any indeterminate, i.e. any element of some ring containing R that is transcendental (not algebraic) over R. It is better to avoid this confusion by defining polynomials by their coefficient sequences, i.e. they are functions N - R with finite support that are added pointwise and multiplied by Cauchy product (convolution). Then X = (0,1,0,0,0...), and X^n is the sequence having 1 in the n'th place and 0 elsewhere; r = (r,0,0,0...) for constants r in R. Now the question "what is X?" has a clear and rigorous answer. I mean nothing more by variable than the usual set theoretic usage. For instance a variable y can be declared as a member of a set S, which then means that while y has not taken on a specific value that it has taken on a specific set of possible values, namely the elements of S. The usual terminology is to state that y is in S, which is the same meaning as membership or element, but rather than a fixed element the variable y has freedom within S. The variable y also takes the interpretation of an unknown element, but this is for a particular usage or construction composed with y. (Longwinded as usual. "Declaring variables" is programming.) But in this case, you have already been told *in this post*, once by me and once by Bill Dubuque quoted above, that in A[x], x is *not* a variable ranging over values in A. I'll repeat that, just in case: x is not a variable ranging over values in A. By 'substitution' we mean merely to suppose that a variable (or constant for that matter) takes on a particular value, which may be another variable or constant, or an element of the containing set. This is done by assignment as y = v No it isn't, not in maths. We do not write y = v to mean "assign anything". Statements in mathematics are (static) statements, not the *instructions* intended by the misnomer "statement" in von Neumann programming languages. <snip ... Is x in the polynomial form an ordinary variable? I am answering yes here, and so the usual usage of 'variable' persists for x.... Yes, you have been insisting for quite some time that "x is a variable". Well here's a google search with 14 results for "not a variable" + tim (all in this thread of course): http://groups.google.com/group/sci.math/search?group=sci.math&q=%22not+a+variable%22+tim&qt_g=Search+this+group continue to discuss upon reaching agreement on this issue of x as a variable. Right, well here's how the discussion may well develop: Tim Golden: "I'm sure x must be a variable, and therefore word salad with ham and eggs..." Someone: No, x isn't a variable. Let me try to explain it this way [two pages] Tim Golden: "Thank you Someone for your contribution. As I see it, if x is a variable, the problem is that [two pages demolished] Someoneelse: But you have been told before: x is _not_ a variable... [Suppose a miracle] Me: Never mind about that. Consider the (ordinary, two-ended) bitstrings under the operations of XOR and non-carry multiplication (NCM) -- do you think they form a ring. TG: Oh yes [florid logorrhea deleted], they do seem to form a ring. Well-meaning poster: But the bitstrings are just another representation of the polynomial expressions with coefficients in GF2... TG: So really, this bitstring example just has the variable x inside it in disguise. WMP: Well, of course x is not a variable in the sense you misunderstand. TG: But since x is a variable, we can see that really the apparently infinite set of bitstrings contains only the values 0 and 1, the polynomial ring construction is flawed, and [...] Me: Odd, though, isn't it, that the bitstrings together with those two operations (XOR and NCM) _do_ comply with the axioms for a ring. TG: Well, I see that x is a variable, but let's try to get more opinions on this. Thank you for your contribution. Me: No trouble at all.... Brian Chandler |