In this paper we create and implement a method to construct a non-simple, symmetric 11-Venn diagram. By doing this we answer a question of Grunbaum. Daniel Kleitman's mathematics seems to be remote from this area, but in fact, his results inspired this solution, which holds promise of settling the general case for all prime numbers, and the promise to nd simple doilies with 11 and more curves as well.

Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of solid three-dimensional objects, (iii) in the theory of effective computability (Church-Turhrg thesis) related to the possible "solution of supertasks," and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for" physical applications are discussed: Cantorian "naive" (i.e., nonaxiomatic) set theory, contructivism, and operationalism, hr the arrthor's ophrion, an attitude of "suspended attention" (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same thne, physicists shouM be open to "bizarre" or "mindboggling" new formalisms, which treed not be operationalizable or testable at the thne of their " creation, but which may successfully lead to novel fields of phenomenology and technology.

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In this paper we close the gap between the results of [8] and [9] by showing that there are 11-doilies for vertex sets of size 275,286,...,462. In [8] it is shown that there are 11-doilies with vertex size 462,473,..., 1001; and in [9] it is shown that there are such diagrams with vertex size 231,242,..., 352. This is the third paper in a series showing that 11-doilies exist for any possible size of vertex set, answering a question of Griinbaum. We continue to extend the method that is developed in [7], [8], and [9]. The crucial step in the method of those paper is based on saturated chain decompositions of planar, spanning subgraphs of the p-hypercube, and on edge-disjoint path decompositions of planar, spanning subgraphs of the p-hypercube. We continue to study these type of decompositions. In constructing the doilies we use a basic structure, called the Venn graph of a doodle of the doily. Using the new one that we create here we show the existence of the above mentioned diagrams. In fact, without checking all the details we show that there are at least 2 7 (non-isomorphic) 11-doilies with these vertex sets.

Abstract Over the past three decades there has been interest in using Padé approximants K with n = deg(K) < deg(G) = N as “reduced-order models ” for the transfer function G of a linear system. The attractive feature of this approach is that by matching the moments of G we can reproduce the steady-state behavior of G by the steady-state behavior of K, for certain classes of inputs. Indeed, we illustrate this by finding a first-order model matching a fixed set of moments for G, the causal inverse of a heat equation. A key feature of this example is that the heat equation is a minimum phase system, so that its inverse system has a stable transfer function G and that K can also be chosen to be stable. On the other hand, elementary examples show that both stability and instability can occur in reduced order models of a stable system obtained by matching moments using Padé approximants and, in the absence of stability, it does not make much sense to talk about steady-state responses nor does it make sense to match moments. In this paper, we review Padé approximants, and their intimate relationship to continued fractions and Riccati equations, in a historical context that underscores why Padé approximation, as useful as it is, is not an approximation in any sense that reflects stability. Our main results on stability and instability states that if N ≥ 2 and ℓ,r ≥ 0 with 0 < ℓ + r = n < N there is a non-empty open set Uℓ,r of stable transfer functions G, having infinite Lebesque measure, such that each degree n proper rational function K matching the moments of G has ℓ poles lying in C − and r poles lying in C +. The proof is constructive.

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