An analogue of Euler''s identity and new combinatorial properties of n-colour compositions

An analogue of Euler's partition identity: “The number of partitions of a positive integer ν into odd parts equals the number of its partitions into distinct parts” is obtained for ordered partitions. The ideas developed are then used in obtaining several new combinatorial properties of the n-colour compositions introduced recently by the author.     

Forcing closed unbounded subsets of ω2

It is shown that there is no satisfactory first-order characterization of those subsets of ω₂ that have closed unbounded subsets in ω₁,ω₂ and GCH preserving outer models. These “anticharacterization” results generalize to subsets of successors of uncountable regular cardinals. Similar results are proved for trees of height and cardinality κ⁺ and for partitions of [κ⁺]², when κ is an infinite cardinal.     

Eigenvalue interlacing and weight parameters of graphs

Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. In particular, it has been successfully used for proving a number of results about the relation between the (adjacency matrix or Laplacian) spectrum of a graph and some of its properties. For instance, some characterizations of regular partitions, and bounds for some parameters, such as the independence and chromatic numbers, the diameter, the bandwidth, etc., have been obtained. For each parameter of a graph involving the cardinality of some vertex sets, we can define its corresponding weight parameter by giving some “weights” (that is, the entries of the positive eigenvector) to the vertices and replacing cardinalities by square norms. The key point is that such weights “regularize” the graph, and hence allow us to define a kind of regular partition, called “pseudo-regular,” intended for general graphs. Here we show how to use interlacing for proving results about some weight parameters and pseudo-regular partitions of a graph. For instance, generalizing a well-known result of Lovász, it is shown that the weight Shannon capacity Θ^* of a connected graph Γ, with n vertices and (adjacency matrix) eigenvalues λ₁ > λ₂ … λ_n, satisfies

where Θ is the (standard) Shannon capacity and v is the positive eigenvector normalized to have smallest entry 1. In the special case of regular graphs, the results obtained have some interesting corollaries, such as an upper bound for some of the multiplicities of the eigenvalues of a distance-regular graph. Finally, some results involving the Laplacian spectrum are derived.     

The Dempster–Shafer calculus for statisticians

The Dempster–Shafer (DS) theory of probabilistic reasoning is presented in terms of a semantics whereby every meaningful formal assertion is associated with a triple (p, q, r) where p is the probability “for” the assertion, q is the probability “against” the assertion, and r is the probability of “don’t know”. Arguments are presented for the necessity of “don’t know”. Elements of the calculus are sketched, including the extension of a DS model from a margin to a full state space, and DS combination of independent DS uncertainty assessments on the full space. The methodology is applied to inference and prediction from Poisson counts, including an introduction to the use of join-tree model structure to simplify and shorten computation. The relation of DS theory to statistical significance testing is elaborated, introducing along the way the new concept of “dull” null hypothesis.     

Partition numbers

We continue [21] and study partition numbers of partial orderings which are related to (ω)/fin. In particular, we investigate P_f, be the suborder of ((ω)/fin)^ω containing only filtered elements, the Mathias partial order M, and (ω), (ω)^ω the lattice of (infinite) partitions of ω, respectively. We show that Solomon's inequality holds for M and that it consistently fails for P_f. We show that the partition number of (ω) is C. We also show that consistently the distributivity number of (ω)^ω is smaller than the distributivity number of (ω)/fin. We also investigate partitions of a Polish space into closed sets. We show that such a partition either is countable or has size at least D, where D is the dominating number. We also show that the existence of a dominating family of size ₁ does not imply that a Polish space can be partitioned into ₁ many closed sets.     

On an observer-related unequivalence between spatial dimensions of a generic Cremonian universe

Given a generic Cremonian space-time, its three spatial dimensions are shown to exhibit an intriguing, “two-plus-one” partition with respect to standard observers. Such observers are found to form three distinct, disjoint groups based on which one out of the three dimensions stands away from the other two. These two subject-related properties have, to our knowledge, no analogue in any of the existing physical theories of space-time; yet, in one of them, the so-called Cantorian model, a closer inspection may reveal some traits of such a “space split-up.”     

Sample Size for Testing Homogeneity of Two a Priori Dependent Binomial Populations Using the Bayesian Approach

This paper establishes new methodology for calculating the optimal sample size when a hypothesis test between two binomial populations is performed. The problem is addressed from the Bayesian point of view, with prior information expressed through a Dirichlet distribution. The approach of this paper sets an upper bound for the posterior risk and then chooses as “optimum ”the combined sample size for which the likelihood of the data does not satisfy this bound. The combined sample size is divided equally between the two binomials. Numerical examples are discussed for which the two proportions are equal to either a fixed or to a random value.     

On small homotopies of loops

Two natural questions are answered in the negative:

• “If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic?”

• “Can adding arcs to a space cause an essential curve to become nulhomotopic?”

The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and π₁-shape injective.

Gravitational instanton in Hilbert space and the mass of high energy elementary particles

While the theory of relativity was formulated in real spacetime geometry, the exact formulation of quantum mechanics is in a mathematical construction called Hilbert space. For this reason transferring a solution of Einstein’s field equation to a quantum gravity Hilbert space is far of being a trivial problem.

On the other hand ^(∞) spacetime which is assumed to be real is applicable to both, relativity theory and quantum mechanics. Consequently, one may expect that a solution of Einstein’s equation could be interpreted more smoothly at the quantum resolution using the Cantorian ^(∞) theory.

In the present paper we will attempt to implement the above strategy to study the Eguchi–Hanson gravitational instanton solution and its interpretation by ‘t Hooft in the context of quantum gravity Hilbert space as an event and a possible solitonic “extended” particle. Subsequently we do not only reproduce the result of ‘t Hooft but also find the mass of a fundamental “exotic” symplictic-transfinite particle m1.8 MeV as well as the mass M_x and M (Planck) which are believed to determine the GUT and the total unification of all fundamental interactions respectively. This may be seen as a further confirmation to an argument which we put forward in various previous publications in favour of an alternative mass acquisition mechanism based on unification and duality considerations. Thus even in case that we never find the Higgs particle experimentally, the standard model would remain substantially intact as we can appeal to tunnelling and unification arguments to explain the mass. In fact a minority opinion at present is that finding the Higgs particle is not a final conclusive argument since one could ask further how the Higgs particle came to its mass which necessitates a second Higgs field. By contrast the present argument could be viewed as an ultimate theory based on the existence of a “super” force, beyond which nothing else exists.     

Euler's partition theorem and the combinatorics of ℓ-sequences

Euler's partition theorem states that the number of partitions of an integer N into odd parts is equal to the number of partitions of N in which the ratio of successive parts is greater than 1. It was shown by Bousquet-Mélou and Eriksson in [M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions II, Ramanujan J. 1 (2) (1997) 165–185] that a similar result holds when “odd parts” is replaced by “parts that are sums of successive terms of an ℓ-sequence” and the ratio “1” is replaced by a root of the characteristic polynomial of the ℓ-sequence. This generalization of Euler's theorem is intrinsically different from the many others that have appeared, as it involves a family of partitions constrained by the ratio of successive parts.In this paper, we provide a surprisingly simple bijection for this result, a question suggested by Richard Stanley. In fact, we give a parametrized family of bijections, that include, as special cases, Sylvester's bijection and a bijection for the lecture hall theorem. We introduce Sylvester diagrams as a way to visualize these bijections and deduce their properties.In proving the bijections, we uncover the intrinsic role played by the combinatorics of ℓ-sequences and use this structure to give a combinatorial characterization of the partitions defined by the ratio constraint. Several open questions suggested by this work are described.     

A theory of tensor products for module categories for a vertex operator algebra, III

This is the third part in a series of papers developing a tensor product theory for modules for a vertex operator algebra. The goal of this theory is to construct a “vertex tensor category” structure on the category of modules for a suitable vertex operator algebra. The notion of vertex tensor category is essentially a “complex analogue” of the notion of symmetric tensor category, and in fact a vertex tensor category produces a braided tensor category in a natural way. In this paper, we focus on a particular element P(z) of a certain moduli space of three-punctured Riemann spheres; in general, every element of this moduli space will give rise to a notion of tensor product, and one must consider all these notions in order to construct a vertex tensor category. Here we present the fundamental properties of the P(z)-tensor product of two modules for a vertex operator algebra. We give two constructions of a P(z)-tensor product, using the results, established in Parts I and II of this series, for a certain other element of the moduli space. The definitions and results in Parts I and II are recalled.     

Landmarks in graphs

Navigation can be studied in a graph-structured framework in which the navigating agent (which we shall assume to be a point robot) moves from node to node of a “graph space”. The robot can locate itself by the presence of distinctively labeled “landmark” nodes in the graph space. For a robot navigating in Euclidean space, visual detection of a distinctive landmark provides information about the direction to the landmark, and allows the robot to determine its position by triangulation. On a graph, however, there is neither the concept of direction nor that of visibility. Instead, we shall assume that a robot navigating on a graph can sense the distances to a set of landmarks.

Evidently, if the robot knows its distances to a sufficiently large set of landmarks, its position on the graph is uniquely determined. This suggests the following problem: given a graph, what are the fewest number of landmarks needed, and where should they be located, so that the distances to the landmarks uniquely determine the robot's position on the graph? This is actually a classical problem about metric spaces. A minimum set of landmarks which uniquely determine the robot's position is called a “metric basis”, and the minimum number of landmarks is called the “metric dimension” of the graph. In this paper we present some results about this problem. Our main new results are that the metric dimension of a graph with n nodes can be approximated in polynomial time within a factor of O(log n), and some properties of graphs with metric dimension two.     

Planar and axially symmetric configurations which are circumvented with the maximum critical mach number

The structure of planar and axially symmetric configurations which, by satisfying a number of geometrical constraints, are circumvented in a boundless space or in a cylindrical channel by an ideal (non-viscous and non-thermally conducting) gas with a maximal critical Mach number M* is found. The analysis is carried out using the “rectilinearity property” of a sonic line in “subsonic” flows (SF), the “principle of a maximum” for an SF and “comparison theorems” which are either taken from /1/ or serve as a generalization of the corresponding assertions from /1/. Following /1/, configurations are considered which have a plane or axis of symmetry parallel to the velocity V_∞ of the approach stream, while flows in which (including the boundary) the Mach number M 1 are said to be “subsonic”. As usual, by M* we mean a value of M_∞ such that the inequality M1, which is satisfied in the whole stream when M_∞ M*, is violated when M_∞>M*.

The configurations investigated include closed bodies and the leading (trailing) parts of a semi-infinite plate or a circular cylinder in an unbounded flow and in a channel as well as lattices of symmetric profiles. Both in /1/, where the structure of closed planar and axially symmetric bodies was found, as well as in /2/, where such bodies were constructed numerically, the generatrices of all the configurations investigated contain the end planes or the segments replacing them of the maximum permissible slope (in modulus) and the “free” streamlines with M 1. Now, however, unlike in /1, 2/, segments of the horizontals are added to it in the general case. Furthermore, in the case of flows in channels and lattices, the configurations which have been found can be circumvented with the development of finite domains of advancing sonic flow.     

Fuzzy logic in measurements

Our main interest in this paper is to translate from “natural language” into “system theoretical language”. This is of course important since a statement in system theory can be analyzed mathematically or computationally. We assume that, in order to obtain a good translation, “system theoretical language” should have great power of expression. Thus we first propose a new frame of system theory, which includes the concepts of “measurement” as well as “state equation”. And we show that a certain statement in usual conversation, i.e., fuzzy modus ponens with the word “very”, can be translated into a statement in the new frame of system theory. Though our result is merely one example of the translation from “natural language” into “system theoretical language”, we believe that our method is fairly general.     

The motion of a material point in friedmann-lobachevsky space

Some simple instances of the motion of a material point in Friedmann-Lobachevsky space /1/ are constructed and investigated, on the assumption that the space, like Galilean space, is empty (i.e., contains no matter) and that the forces acting on the point, including the gravitational forces, constitute a factor extraneous to the space. Thus, the problem is being considered in the context of a rather unusual “relativistic” mechanics, distinct from relativisitc mechanics proper. As will be seen later, the difference is quantitatively small and can be regulated by slowly varying cosmological factors in the pseudo-Euclidean metric of the space of special relativity theory.     

Quasilinear conflict-controlled processes with additional restrictions

A class of conflict-controlled processes [1–3] with additional (“phase” type) restrictions on the state of the evader is considered. A similar unrestricted problem was considered in [4]. Unlike [5, 6] the boundary of the “phase” restrictions is not a “death line” for the evader. Sufficient conditions for the solvability of the pursuit and evasion problems are obtained, which complement a range of well-known results [5–10].^‡     

Statistical inference in non-nested econometric models

The purpose of this paper is to discuss some procedures that are available for testing non-nested (or separate) hypotheses in the statistics and econometrics literature. Since many of these techniques may also be exploited in other disciplines, it is hoped that an elaboration of the principal theoretical findings may make them more readily accessible to researchers in other disciplines. Several simple examples are used to illustrate the concepts of nested and non-nested hypotheses and, within the latter category, “global” and “partial” non-nested hypotheses. Two alternative methods of testing non-nested hypotheses are discussed and contrasted: the first of these is Cox's modification of the likelihood-ratio statistic, and the second is Atkinson's comprehensive model approach. A major emphasis is placed on the role of the Cox principle of hypothesis testing, which enables a broad range of hypotheses to be tested within the same framework. The problem associated with the application of the comprehensive model approach to composite non-nested hypotheses is also highlighted; Roy's union-intersection principle is presented as a viable method of dealing with this problem. Simulation results concerning the finite-sample properties of various tests are discussed, together with an analysis of some attempts to correct the poor size of the Cox and related tests.     

Enumeration of Lozenge Tilings of Hexagons with Cut-Off Corners

Motivated by the enumeration of a class of plane partitions studied by Proctor and by considerations about symmetry classes of plane partitions, we consider the problem of enumerating lozenge tilings of a hexagon with “maximal staircases” removed from some of its vertices. The case of one vertex corresponds to Proctor's problem. For two vertices there are several cases to consider, and most of them lead to nice enumeration formulas. For three or more vertices there do not seem to exist nice product formulas in general, but in one special situation a lot of factorization occurs, and we pose the problem of finding a formula for the number of tilings in this case.     

Multi-step quasi-Newton methods for optimization

Quasi-Newton methods update, at each iteration, the existing Hessian approximation (or its inverse) by means of data deriving from the step just completed. We show how “multi-step” methods (employing, in addition, data from previous iterations) may be constructed by means of interpolating polynomials, leading to a generalization of the “secant” (or “quasi-Newton”) equation. The issue of positive-definiteness in the Hessian approximation is addressed and shown to depend on a generalized version of the condition which is required to hold in the original “single-step” methods. The results of extensive numerical experimentation indicate strongly that computational advantages can accrue from such an approach (by comparison with “single-step” methods), particularly as the dimension of the problem increases.     

A quantum mechanical approach to a fuzzy theory

Recently, we proposed a general measurement theory for classical and quantum systems (i.e., “objective fuzzy measurement theory”). In this paper, we propose “subjective fuzzy measurement theory”, which is characterized as the statistical method of the objective fuzzy measurement theory. Our proposal of course has a lot of advantages. For example, we can directly see “membership functions” (= “fuzzy sets”) in this theory. Therefore, we can propose the objective and the subjective methods of membership functions. As one of the consequences, we assert the objective (i.e., individualistic) aspect of Zadeh's theory. Also, as a quantum application, we clarify Heisenberg's uncertainty relation.