By Ricciardi T.

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The significance of this example resides in our imminent demonstration that the highlighted condition, “like-length,” forces M4 to have infinitely many states. 2 2. , a nonnegative integer k of the form k = n2 . 1 Online Automata and Their “Languages” M 4: a A0 A1 a A2 b ... a a Ak b b B1 b 39 a,b a ... b b B2 ... a b Bk b ... a ... C a,b M 5: A0 a A1 a A2 a A3 a a A7 A6 a a A5 A4 a A8 a A9 a A10 a ... Fig. 3 Graph-theoretic representations of two simple infinite OAs. The significance of this example resides in our imminent demonstration that the structure of the set of integers that are perfect squares forces M5 to have infinitely many states.

Reinforcing the preliminaries. Before continuing with our development, we illustrate our definitions with three simple finite OAs and one infinite one. We hope that these examples will hone the reader’s intuition and serve as concrete hooks to stabilize our rather quick journey into the land of abstraction. 1. 2. 1. , composed of instances of the letter a) whose lengths are divisible by 3. 2. , strings of 0’s and 1’s) that contain four or more consecutive 1’s in at least one place. 4 We intentionally use diverse terminology in describing these languages, so that the reader will get familiar with alternative modes of describing the same concept.

We argue next that no finite OA recognizes the language L2 = L(M5 ). The simplest way of seeing this begins by assuming, for contradiction, that there is a finite OA M = (Q, {a, b}, δ , q0 , F) such that L2 = L(M). , ai ≡M a j . On the one hand, we note that 2 δ (q, a2i+1 ) = δ (q0 , ai a2i+1 ) = δ (q0 , ai 2 +2i+1 ) = δ (q0 , a(i+1) ), 2 which M should accept, because a(i+1) ∈ L2 . On the other hand, we note that 2 2 δ (q, a2i+1 ) = δ (q0 , a j a2i+1 ) = δ (q0 , a j 2 +2i+1 ), which M should not accept, because j2 < j2 + 2i + 1 < j2 + 2 j + 1 = ( j + 1)2 , so that a j +2i+1 ∈ L2 (the exponent of a falls strictly between two adjacent perfect squares).

### A sharp Hölder estimate for elliptic equations in two variables by Ricciardi T.

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