Infinitely repeated partitions of Liouville numbers

It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

It is a rather universal tacit and unquestioned belief—and even more so among physicists—that there is one and only one real line, namely, given by the coodinatisation of Descartes through the usual field R of real numbers. Such a dramatically limiting and thus harmful belief comes, unknown to equally many, from the similarly tacit acceptance of the ancient Archimedean Axiom in Euclid’s Geometry. The consequence of that belief is a similar belief in the uniqueness of the coordinatization of the plane by the usual field C of complex numbers, and therefore, of the various spaces, manifolds, etc., be they finite or infinite dimensional, constructed upon the real or complex numbers, including the Hilbert spaces used in Quantum Mechanics. A near total lack of awareness follows therefore about the rich self‐similar structure of other possible coordinatisations of the real line, possibilities given by various linearly ordered scalar fields obtained through the ultrapower construction. Such fields contain as a rather small subset the usual field R of real numbers. The concept of walkable world, which has highly intuitive and pragmatic algebraic and geometric meaning, illustrates the mentioned rich self‐similar structure.

A recent paper showed that given any positive integer k, the real line can be partitioned into k subsets, each of which is uncountable in every nonempty open interval in the real line. This article extends that result to show that the real line can be partitioned into an uncountable collection of such subsets, all but one of which have measure zero. This seems to be of interest in its own right since most of the well-known sets of measure zero are either countable or not dense. It is then shown that each of the sets in the partition can be partitioned further into an uncountable collection of subsets that are again uncountable in every nonempty open set of the real line. Indeed, this process can be repeated infinitely many times. Finally, all of the results for the real line are shown to extend to n-dimensional Euclidean spaces and to the classical lp spaces.

Field‐theoretic simulations of polyelectrolyte complexation

We study the propagation of monochromatic fields in a layered medium. The mathematical model is derived from Maxwell's equations. It consists of a nonlinear eigenvalue problem on the real axis with coefficients depending on the various layers.¶A systematic analysis is carried out to uncover the various mechanisms leading to the bifurcation of asymmetric solutions even in a completely symmetric setting. We derive two particular simple conditions for the occurence of asymmetric bifurcation from the symmetric branch. One of these conditions occurs at a matching of the refractive indices across the interface while the other corresponds to a switching of the peak from the core to the cladding.¶The rich bifurcation structure is illustrated by numerical calculations. Further stability considerations are included.

Recently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2n), starting from an elementary abelian regular subgroup. They have shown that the indices of consecutive groups in the chain depend on the number of partitions into distinct parts and have given a description, by means of rigid commutators, of the first n − 2 terms in the chain. Moreover, they proved that the (n − 1)-th term of the chain is described by means of rigid commutators corresponding to unrefinable partitions into distinct parts. Although the mentioned chain can be defined in a Sylow p-subgroup of Sym(pn), for p > 2 computing the chain of normalizers becomes a challenging task, in the absence of a suitable notion of rigid commutators. This problem is addressed here from an alternative point of view. We propose a more general framework for the normalizer chain, defining a chain of idealizers in a Lie ring over ℤm whose elements are represented by integer partitions. We show how the corresponding idealizers are generated by subsets of partitions into at most m − 1 parts and we conjecture that the idealizer chain grows as the normalizer chain in the symmetric group. As evidence of this, we establish a correspondence between the two constructions in the case m = 2.

The theory of vector measure has attracted much interest among researchers in the recent past. Available results show that measurability concepts of the Lebesgue measure have been used to partition subsets of the real line into disjoint sets of nite measure. In this paper we partition measurable sets in ℜn for n ≥ 3 into disjoint sets of nite dimension.

Summary We define an extension of parity from the integers to the rational numbers. Three parity classes are found—even, odd, and “none”. Using the 2-adic valuation, we partition the rationals into subgroups with a rich algebraic structure. The natural density provides a means of distinguishing the sizes of countably infinite sets. The Calkin-Wilf tree has a remarkably simple parity pattern, with the sequence “odd/none/even” repeating indefinitely. This pattern means that the three parity classes have equal natural density in the rationals. A similar result holds for the Stern-Brocot tree.

A nondistributive scator algebra in 1 + 2 dimensions is used to map the quadratic iteration. The hyperbolic numbers square bound set reveals a rich structure when taken into the three-dimensional (3D) hyperbolic scator space. Self-similar small copies of the larger set are obtained along the real axis. These self-similar sets are located at the same positions and have equivalent relative sizes as the small M-set copies found between the Myrberg-Feigenbaum (MF) point and -2 in the complex Mandelbrot set. Furthermore, these small copies are self similar 3D copies of the larger 3D bound set. The real roots of the respective polynomials exhibit basins of attraction in a 3D space. Slices of the 3D confined scator set, labeled [Formula: see text](s;x,y), are shown at different planes to give an approximate idea of the 3D objects highly complicated boundary.

Suppose thatV is a model of ZFC andU ∈ V is a topological space or a richer structure for which it makes sense to speak about the monadic theory. LetB be the Boolean algebra of regular open subsets ofU. If the monadic theory ofU allows one to speak in some sense about a family ofκ everywhere dense and almost disjoint sets, then the second-orderVB-theory of ϰ is interpretable in the monadicV-theory ofU; this is our Interpretation Theorem. Applying the Interpretation Theorem we strengthen some previous results on complexity of the monadic theories of the real line and some other topological spaces and linear orders. Here are our results about the real line. Letr be a Cohen real overV. The second-orderV[r]-theory of ℵ0 is interpretable in the monadicV-theory of the real line. If CH holds inV then the second-orderV[r]-theory of the real line is interpretable in the monadicV-theory of the real line.

The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts

We obtain an analytic approximation of the bound states solution of the Schrödinger equation on the semi-infinite real line for two potential models with a rich structure as shown by their spectral phase diagrams. These potentials do not belong to the class of exactly solvable problems. The solutions are finite series (with a small number of terms) of square integrable functions written in terms of Romanovski–Jacobi polynomials.

The space C(X) of all continuous complex- or real-valued functions on a topological space X plays an important role. This space, with pointwise multiplication, turns out to be an algebra. With a suitable topology, it is even a topological algebra. Thus the Stone-Weierstrass theorem can be formulated and proved. Furthermore, because of the lattice structure of the real line, there is a lattice structure on C(X). This is turn enables one to study some other, deeper properties of C(X). In particular, we prove the Banach-Stone theorem and other results which exploit the algebraic structure on C(X), which may not be available for C(X, 7), if X and Y are topological spaces and have no richer structure than that of the set of real numbers.

On some similarity of finite sets (and what we can say today about certain old problem)

ModMRF: A modularity-based Markov Random Field method for community detection