Large Cardinals Research Articles

Local analysis, cardinality, and split trick of quasi-biorthogonal frame wavelets

The notion of quasi-biorthogonal frame wavelets is a generalization of the notion of orthogonal wavelets. A quasi-biorthogonal frame wavelet with the cardinality r consists of r pairs of functions. In this paper we first analyze the local property of the quasi-biorthogonal frame wavelet and show that its each pair of functions generates reconstruction formulas of the corresponding subspaces. Next we show that the lower bound of its cardinalities depends on a pair of dual frame multiresolution analyses deriving it. Finally, we present a split trick and show that any quasi-biorthogonal frame wavelet can be split into a new quasi-biorthogonal frame wavelet with an arbitrarily large cardinality. For generality, we work in the setting of matrix dilations.

A Local Relaxation Method for Cardinality Constrained Portfolio Optimization Problem

The NP-hard nature of cardinality constrained mean-variance portfolio optimization problems has led to a variety of different algorithms with varying degrees of success in reaching optimality given limited computational resources and under the presence of strict time constraints in practice. The proposed local relaxation algorithm exploits the inherent structure of the objective function. It solves a sequence of small, local, quadratic-programs by first projecting asset returns onto a reduced metric space, followed by clustering in this space to identify sub-groups of assets that best accentuate a suitable measure of similarity amongst different assets. The algorithm can either be cold started using the centroids of initial clusters or be warm started based on the output of a previous result. Empirical result, using baskets of up to 3,000 stocks and with different cardinality constraints, indicates that the proposed algorithm is able to achieve significant performance gain over a sophisticated branch-and-cut method. One key application of this algorithm is in dealing with large scale cardinality constrained portfolio optimization under tight time constraint, such as for the purpose of index tracking or index arbitrage at high frequency

Improving the performance of Naive Bayes multinomial in e-mail foldering by introducing distribution-based balance of datasets

E-mail foldering or e-mail classification into user predefined folders can be viewed as a text classification/categorization problem. However, it has some intrinsic properties that make it more difficult to deal with, mainly the large cardinality of the class variable (i.e. the number of folders), the different number of e-mails per class state and the fact that this is a dynamic problem, in the sense that e-mails arrive in our mail-folders following a time-line. Perhaps because of these problems, standard text-oriented classifiers such as Naive Bayes Multinomial do no obtain a good accuracy when applied to e-mail corpora. In this paper, we identify the imbalance among classes/folders as the main problem, and propose a new method based on learning and sampling probability distributions. Our experiments over a standard corpus (ENRON) with seven datasets (e-mail users) show that the results obtained by Naive Bayes Multinomial significantly improve when applying the balancing algorithm first. For the sake of completeness in our experimental study we also compare this with another standard balancing method (SMOTE) and classifiers.

Some consequences of reflection on the approachability ideal

We study the approachability ideal I [ κ + ] \mathcal {I}[\kappa ^+] in the context of large cardinals and properties of the regular cardinals below a singular κ \kappa . As a guiding example consider the approachability ideal I [ ℵ ω + 1 ] \mathcal {I}[\aleph _{\omega +1}] assuming that ℵ ω \aleph _\omega is a strong limit. In this case we obtain that club many points in ℵ ω + 1 \aleph _{\omega +1} of cofinality ℵ n \aleph _n for some n > 1 n>1 are approachable assuming the joint reflection of countable families of stationary subsets of ℵ n \aleph _n . This reflection principle holds under M M \mathsf {MM} for all n > 1 n>1 and for each n > 1 n>1 is equiconsistent with ℵ n \aleph _n being weakly compact in L L . This characterizes the structure of the approachability ideal I [ ℵ ω + 1 ] \mathcal {I}[\aleph _{\omega +1}] in models of M M \mathsf {MM} . We also apply our result to show that the Chang conjecture ( κ + , κ ) ↠ ( ℵ 2 , ℵ 1 ) (\kappa ^+,\kappa )\twoheadrightarrow (\aleph _2,\aleph _1) fails in models of M M \mathsf {MM} for all singular cardinals κ \kappa .

The Axiom of Infinity and Transformations j: V → V

AbstractWe suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnková–Blass, showing that if the preservation properties of Lawvere's tranformation are strengthened to the point of requiring it to be an exact functor, such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V → V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V → V known to be equivalent to the Axiom of Infinity.

TRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY

This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.

Tallness and level by level equivalence and inequivalence

AbstractWe construct two models containing exactly one supercompact cardinal in which all non‐supercompact measurable cardinals are strictly taller than they are either strongly compact or supercompact. In the first of these models, level by level equivalence between strong compactness and supercompactness holds. In the other, level by level inequivalence between strong compactness and supercompactness holds. Each universe has only one strongly compact cardinal and contains relatively few large cardinals (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Forbidden intervals

Many classical statements of set theory are settled by the existence of generic elementary embeddings that are analogous the elementary embeddings posited by large cardinals. [2] The embeddings analogous to measurable cardinals are determined by uniform, κ-complete precipitous ideals on cardinals κ. Stronger embeddings, analogous to those originating from supercompact or huge cardinals are encoded by normal fine ideals on sets such as [κ]<λ or [κ]λ.The embeddings generated from these ideals are limited in ways analogous to conventional large cardinals. Explicitly, if j: V → M is a generic elementary embedding with critical point κ and λ supnЄωjn(κ) and the forcing yielding j is λ-saturated then j“λ+ ∉ M. (See [2].)Ideals that yield embeddings that are analogous to strongly compact cardinals have more puzzling behavior and the analogy is not as straightforward. Some natural ideal properties of this kind have been shown to be inconsistent:Theorem 1 (Kunen). There is no ω2-saturated, countably complete uniform ideal on any cardinal in the interval [ℵω, ℵω).Generic embeddings that arise from countably complete, ω2-saturated ideals have the property that sup . So the Kunen result is striking in that it apparently allows strong ideals to exist above the conventional large cardinal limitations. The main result of this paper is that it is consistent (relative to a huge cardinal) that such ideals exist.

A new family of 2D wavelength/time codes with large cardinality for incoherent spectral amplitude coding OCDMA networks and analysis of its performance

A new family of two-dimensional wavelength/time codes (2D-W/T-MQC/MQCs) and its system structure of encoder/decoder are proposed, which is based on tunable optical fiber delay lines (TOFDLs) and fiber Bragg gratings (FBGs). Multiple-access interference (MAI) can fully be eliminated by using a wavelength/time balanced detector structure at the receivers. Furthermore, the performance of the system is also analyzed by taking into account the phase-induced intensity noise (PIIN), shot noise, and thermal noise. The simulation results reveal that the new code family possesses higher signal-to-noise ratio (SNR) and lower bit-error rate (BER) than the family of one-dimensional spectral amplitude coding modified quadratic congruence codes (1D-SAC-MQCs) and that of the two-dimensional wavelength/spatial M-matrices codes (2D-W/S-M-Matrices) so that a larger number of subscribers can be supported simultaneously. Additionally, the 2D-W/T-MQC/MQC system requires less signal power for each light source under the same error-free condition. As a result, the network based on the new code family will support more active users and utilize the frequency bandwidth more efficiently.

Combined Maximality Principles up to large cardinals

AbstractThe motivation for this paper is the following: In [4] I showed that it is inconsistent with ZFC that the Maximality Principle for directed closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the Maximality Principle for <κ-closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal κ (instead of ∞), and if so, how strong it is. It turns out that it is consistent in many cases, but the consistency strength is quite high.

Antidiamond principles and topological applications

We investigate some combinatorial statements that are strong enough to imply that $\diamondsuit$ fails (hence the name antidiamonds); yet most of them are also compatible with CH. We prove that these axioms have many consequences in set-theoretic topology, including the consistency, modulo large cardinals, of a Yes answer to a problem on linearly Lindelöf spaces posed by Arhangel’skiĭ and Buzyakova (1998).

Smoke and mirrors: Combinatorial properties of small cardinals equiconsistent with huge cardinals

Since the work of Godel and Cohen many questions in infinite combinatorics have been shown to be independent of the usual axioms for mathematics, Zermelo Frankel Set Theory with the Axiom of Choice (ZFC). Attempts to strengthen the axioms to settle these problems have converged on a system of principles collectively known as Large Cardinal Axioms. These principles are linearly ordered in terms of consistency strength. As far as is currently known, all natural independent combinatorial statements are equiconsistent with some large cardinal axiom. The standard techniques for showing this use forcing in one direction and inner model theory in the other direction. The conspicuous open problems that remain are suspected to involve combinatorial principles much stronger than the large cardinals for which there is a current fine-structural inner model theory for. The main results in this paper show that many standard constructions give objects with combinatorial properties that are, in turn, strong enough to show the existence of models with large cardinals are larger than any cardinal for which there is a standard inner model theory.

Piecewise training for structured prediction

A drawback of structured prediction methods is that parameter estimation requires repeated inference, which is intractable for general structures. In this paper, we present an approximate training algorithm called piecewise training (PW) that divides the factors into tractable subgraphs, which we call pieces, that are trained independently. Piecewise training can be interpreted as approximating the exact likelihood using belief propagation, and different ways of making this interpretation yield different insights into the method. We also present an extension to piecewise training, called piecewise pseudolikelihood (PWPL), designed for when variables have large cardinality. On several real-world natural language processing tasks, piecewise training performs superior to Besag's pseudolikelihood and sometimes comparably to exact maximum likelihood. In addition, PWPL performs similarly to PW and superior to standard pseudolikelihood, but is five to ten times more computationally efficient than batch maximum likelihood training.

The Way toward Wisdom: an Interdisciplinary and Intercultural Introduction to Metaphysics. By Benedict M. Ashley, O.P.

The Way toward Wisdom: an Interdisciplinary and Intercultural Introduction to Metaphysics. By Benedict M. Ashley, O.P.

Large cardinals and definable well-orders on the universe

AbstractWe use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle , at a proper class of cardinals κ. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.

ARONSZAJN TREES AND FAILURE OF THE SINGULAR CARDINAL HYPOTHESIS

The tree property at κ+states that there are no Aronszajn trees on κ+, or, equivalently, that every κ+tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ+and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper, we reconcile the two. We prove from large cardinals that the tree property at κ+is consistent with failure of the singular cardinal hypothesis at κ.

Indestructibility and stationary reflection

AbstractIf κ < λ are such that κ is a strong cardinal whose strongness is indestructible under κ ‐strategically closed forcing and λ is weakly compact, then we show that A = {δ < κ | δ is a non‐weakly compact Mahlo cardinal which reflects stationary sets}must be unbounded in κ. This phenomenon, however, need not occur in a universe with relatively few large cardinals. In particular, we show how to construct a model where no cardinal is supercompact up to a Mahlo cardinal in which the least supercompact cardinal κ is also the least strongly compact cardinal, κ 's strongness is indestructible under κ ‐strategically closed forcing, κ 's supercompactness is indestructible under κ ‐directed closed forcing not adding any new subsets of κ, and δ is Mahlo and reflects stationary sets iff δ is weakly compact. In this model, no strong cardinal δ < κ is indestructible under δ ‐strategically closed forcing. It therefore follows that it is relatively consistent for the least strong cardinal κ whose strongness is indestructible under κ ‐strategically closed forcing to be the same as the least supercompact cardinal, which also has its supercompactness indestructible under κ ‐directed closed forcing not adding any new subsets of κ (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Operational set theory and small large cardinals

A new axiomatic system OST of operational set theory is introduced in which the usual language of set theory is expanded to allow us to talk about (possibly partial) operations applicable both to sets and to operations. OST is equivalent in strength to admissible set theory, and a natural extension of OST is equivalent in strength to ZFC. The language of OST provides a framework in which to express “small” large cardinal notions—such as those of being an inaccessible cardinal, a Mahlo cardinal, and a weakly compact cardinal—in terms of operational closure conditions that specialize to the analogue notions on admissible sets. This illustrates a wider program whose aim is to provide a common framework for analogues of large cardinal notions that have appeared in admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics, and systems of recursive ordinal notations that have been used in proof theory.

MINIMUM CONNECTED r-HOP k-DOMINATING SET IN WIRELESS NETWORKS

In this paper, we study the connected r-hop k-dominating set problem in wireless networks. We propose two algorithms for the problem. We prove that algorithm I for UDG has (2r + 1)3 approximate ratio for k ≤ (2r + 1)2 and (2r + 1)((2r + 1)2 + 1)-approximate ratio for k > (2r + 1)2. And algorithm II for any undirected graph has (2r + 1) ln (Δr) approximation ratio, where Δr is the largest cardinality among all r-hop neighborhoods in the network. The simulation results show that our algorithms are efficient.

Epireflections and supercompact cardinals

We prove that the existence of arbitrarily large supercompact cardinals implies that every absolute epireflective class of objects in a balanced accessible category is a small-orthogonality class. In other words, if L is a localization functor on a balanced accessible category such that the unit morphism X → L X is an epimorphism for all X and the class of L -local objects is defined by an absolute formula, then the existence of a sufficiently large supercompact cardinal implies that L is a localization with respect to some set of morphisms.