現代数学の系譜工学物理雑談古典ガロア理論も読む46

[過去ﾛｸﾞ] 現代数学の系譜工学物理雑談古典ガロア理論も読む46 (692ﾚｽ)
上下前次1-新
通常表示 512ﾊﾞｲﾄ分割ﾚｽ栞
抽出解除ﾚｽ栞

このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索歴削→次ｽﾚ栞削→次ｽﾚ過去ﾛｸﾞﾒﾆｭｰ

ﾘﾛｰﾄﾞ規制です｡10分ほどで解除するので､他のﾌﾞﾗｳｻﾞへ避難してください｡

480: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2017/11/22(水) 19:17:55.55 ID:mEHYOxL2

>>479 つづき

１）
さて、まず、[HT08b] より
(抜粋)
P91
1. INTRODUCTION.
We often model systems that change over time as functions from the real numbers R (or a subinterval of R) into some set S of states, and it is often our goal to predict the behavior of these systems.
Generally, this requires rules governing their behavior, such as a set of differential equations or the assumption that the system (as a function) is analytic.
With no such assumptions, the system could be an arbitrary function, and the values of arbitrary functions are notoriously hard to predict.
After all, if someone proposed a strategy for predicting the values of an arbitrary function based on its past values, a reasonable response might be,
“That is impossible. Given any strategy for predicting the values of an arbitrary function, one could just define a function that diagonalizes against it: whatever the strategy predicts, define the function to be something else.”
This argument, however, makes an appeal to induction:
to diagonalize against the proposed strategy at a point t, we must have already defined our function for all s < t in order to determine what the strategy would predict at t.

In fact, the lack of well-orderedness in the reals can be exploited to produce a very counterintuitive result: there is a strategy for predicting the values of an arbitrary function, based on its previous values, that is almost always correct.
Specifically, given the values of a function on an interval (?∞, t), the strategy produces a guess for the values of the function on [t,∞), and at all but countably many t, there is an ε > 0 such that the prediction is valid on [t, t + ε).
Noting that any countable set of reals has measure 0, we can restate this informally: at almost every instant t, the strateg predicts some “ε-glimpse” of the future.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1510442940/480

481: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2017/11/22(水) 19:18:43.01 ID:mEHYOxL2

>>480 つづき

Nevertheless, we choose this presentation because we find it the most interesting, as well as pedagogically useful.
For instance, “predicting the present” is a very natural way to think of the problem of guessing the value of f (t) based on f |(?∞, t).

2. THE μ-STRATEGY.
(詳細は略すので、原文ご参照。ここに引用するには、数学記号が複雑過ぎるので。)

P92
3. PREDICTING THE PRESENT.
(詳細は略すので、原文ご参照。ここに引用するには、数学記号が複雑過ぎるので。)

Corollary 3.4. If T = R and ∇ is <, then W0 is countable, has measure 0, and is nowhere dense.

What Corollary 3.4 tells us is that, if we model the universe as a function from the real numbers into some set of states, then the μ-strategy will correctly predict the present from the past on a set of full measure.
(In the following section, we show that, on a set of full measure, it correctly predicts some of the future as well.)
Note that these results concerning T = R are also valid when T is any interval of reals.
One needs to be cautious about interpreting this as meaning that the μ-strategy is correct with probability 1.
For a fixed true scenario, if one randomly selects an instant t in the interval [0,1] (or in R, under a suitable probability distribution), then
Corollary 3.4 does tell us that the μ-strategy will be correct at t with probability 1.
However, if one fixes the instant t, and randomly selects a true scenario, then the probability that the μ-strategy is correct at t under that scenario might be 0 or might not even exist, depending on how one defines the notion of a random scenario.

P95
W = {t ∈ R | the μ-strategy does not guess well at t }.
Theorem 5.1. The set W is countable, has measure 0, and is nowhere dense.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1510442940/481

上下前次1-新書関写板覧索設栞歴

ｽﾚ情報赤ﾚｽ抽出画像ﾚｽ抽出歴の未読ｽﾚ

ぬこの手ぬこTOP 0.033s