Large Cardinals Research Articles

QUOTIENTS OF STRONGLY PROPER FORCINGS AND GUESSING MODELS

AbstractWe prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the ω1-approximation property. We prove that the existence of stationarily many ω1-guessing models in Pω2(H(θ)), for sufficiently large cardinals θ, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss [13].

Big fundamental groups: Generalizing homotopy and big homotopy

The concept of big homotopy theory was introduced by J. Cannon and G. Conner in [1] using big intervals of arbitrarily large cardinality to detect big loops. We find, for each space, a canonical cardinal that is sufficient to detect all big loops and all big homotopies in the space.

Exploiting clustering algorithms in a multiple-level fashion: A comparative study in the medical care scenario

Clustering real-world data is a challenging task, since many real-data collections are characterized by an inherent sparseness and variable distribution. An appealing domain that generates such data collections is the medical care scenario where collected data include a large cardinality of patient records and a variety of medical treatments usually adopted for a given disease pathology.This paper proposes a two-phase data mining methodology (MLC) to iteratively analyze different dataset portions and locally identify groups of objects with common properties. Discovered cohesive clusters are then analyzed using sequential patterns to characterize temporal relationships among data features. To support an automatic classification of new data objects within one of the discovered groups, a classification model is created starting from the computed cluster set. A mobile application has been also designed and developed to visualize and update data under analysis as well as categorizing new unlabeled data objects.The experimental evaluation conducted on real datasets in the medical care scenario showed the effectiveness of MLC to discover interesting knowledge items and to easily exploit them through a mobile application. Results have been also discussed from a medical perspective.

On strong forms of reflection in set theory

In this paper we review the most common forms of reflection and introduce a new form which we call sharp‐generated reflection. We argue that sharp‐generated reflection is the strongest form of reflection which can be regarded as a natural generalization of the Lévy reflection theorem. As an application we formulate the principle sharp‐maximality with the corresponding hypothesis . The statement is an analogue of the (Inner Model Hypothesis, introduced in ) which is compatible with the existence of large cardinals.

Left division in the free left distributive algebra on many generators

Left distributive algebras arise in the study of classical structures such as groups, knots, and braids, as well as more exotic objects like large cardinals. A long-standing open question is whether the set of left divisors of every term in the free left distributive algebra on any number of generators is well-ordered. A conjecture of J. Moody describes a halting condition for descending sequences of left divisors in the free left distributive algebra on an arbitrary number of generators. In this paper we present progress toward an affirmative answer to the open question mentioned above, namely we prove that the many generator form of Moody's conjecture holds if each member of the free left distributive algebra has a "division form" representation in an expanded algebra.

Superstrong and other large cardinals are never Laver indestructible

Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σn-reflecting cardinals, Σn-correct cardinals and Σn-extendible cardinals (all for n ? 3) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if ? exhibits any of them, with corresponding target ?, then in any forcing extension arising from nontrivial strategically <?-closed forcing $${\mathbb{Q} \in V_\theta}$$Q?V?, the cardinal ? will exhibit none of the large cardinal properties with target ? or larger.

Uncountable groups with restrictions on subgroups of large cardinality

The aim of this paper is to investigate the behaviour of uncountable groups of regular cardinality ℵ in which all proper subgroups of cardinality ℵ belong to a given group class X. It is proved that if every proper subgroup of G of cardinality ℵ has finite conjugacy classes, then also the conjugacy classes of G are finite, provided that G has no simple homomorphic images of cardinality ℵ. Moreover, it turns out that if G is a locally graded group of cardinality ℵ in which every proper subgroup of cardinality ℵ contains a nilpotent subgroup of finite index, then G is nilpotent-by-finite, again under the assumption that G has no simple homomorphic images of cardinality ℵ. A similar result holds also for uncountable locally graded groups whose large proper subgroups are abelian-by-finite.

New Tree-Structured Optical Codes for Incoherent Optical-CDMA Applications

Optical codes with large cardinality and tree structure of multiple levels of subsets of codewords for adjustable performance and cardinality have recently been studied. These codes support contemporary applications, such as a scalable, multihop, multilevel network architecture with rapid code-switching/dropping and multicode-keying encryption for improving physical-layer confidentiality in optical code-division multiple-access systems and networks. In this paper, to further enlarge such code collection, two new families of optical codes are constructed from the rich families of cyclic maximal-distance-separable (MDS) codes. The code performances are algebraically analyzed using both the distance and weight distributions of the MDS codes and validated with computer simulation.

Large cardinals need not be large in HOD

We prove that large cardinals need not generally exhibit their large cardinal nature in HOD. For example, a supercompact cardinal κ need not be weakly compact in HOD, and there can be a proper class of supercompact cardinals in V, none of them weakly compact in HOD, with no supercompact cardinals in HOD. Similar results hold for many other types of large cardinals, such as measurable and strong cardinals.

Bounded stationary reflection

We prove that, assuming large cardinals, it is consistent that there are many singular cardinals μ such that every stationary subset of μ+ reflects but there are stationary subsets of μ+ that do not reflect at ordinals of arbitrarily high cofinality. This answers a question raised by Todd Eisworth.

Guessing more sets

Let κ be a regular uncountable cardinal, and λ a cardinal greater than κ with cofinality less than κ. We consider a strengthening of the diamond principle ⋄κ,λ that asserts that any subset of some fixed collection of λ+ elements of Pκ(λ) can be guessed on a stationary set. This new principle, denoted by ⋄κ,λ[λ+], implies that the nonstationary ideal on Pκ(λ) is not 2(λ+)-saturated. We establish that if λ is large enough and there are no inner models with fairly large cardinals, then ⋄κ,λ[λ+] holds. More precisely, it is shown that if 2(κℵ0)≤λ+ and both Shelah's Strong Hypothesis SSH and the Almost Disjoint Sets principle ADSλ hold, then ⋄κ,λ[λ+] holds. The paper also contains ZFC results. Suppose for example that 2κ≤λ+, there is a strong limit cardinal τ with cf(λ)<τ≤κ, and either κ is a successor cardinal greater than ρ+3, where ρ is the largest limit cardinal less than κ, or κ is a limit cardinal and σκ<λ<(σκ)+κ for some cardinal σ≥2. Then ⋄κ,λ[λ+] holds.

Inverse limit reflection and the structure of L(Vλ+1)

We extend the results of Laver on using inverse limits to reflect large cardinals of the form, there exists an elementary embedding Lα(Vλ+1) → Lα(Vλ+1). Using these inverse limit reflection embeddings directly and by broadening the collection of U(j)-representable sets, we prove structural results of L(Vλ+1) under the assumption that there exists an elementary embedding j : L(Vλ+1) → L(Vλ+1). As a consequence we show the impossibility of a generalized inverse limit X-reflection result for X ⊆ Vλ+1, thus focusing the study of L(ℝ) generalizations on L(Vλ+1).

Alternating plane graphs

A plane graph is called alternating if all adjacent vertices have different degrees, and all neighboring faces as well. Alternating plane graphs were introduced in 2008. This paper presents the previous research on alternating plane graphs. There are two smallest alternating plane graphs, having 17 vertices and 17 faces each. There is no alternating plane graph with 18 vertices, but alternating plane graphs exist for all cardinalities from 19 on. From a small set of initial building blocks, alternating plane graphs can be constructed for all large cardinalities. Many of the small alternating plane graphs have been found with extensive computer help. Theoretical results on alternating plane graphs are included where all degrees have to be from the set {3,4,5}. In addition, several classes of “weak alternating plane graphs” (with vertices of degree 2) are presented.

NUMA-Aware Scalable and Efficient In-Memory Aggregation on Large Domains

Business Intelligence (BI) is recognized as one of the most important IT applications in the coming big data era. In recent years, non-uniform memory access (NUMA) has become the de-facto architecture of multiprocessors on the new generation of enterprise servers. Such new architecture brings new challenges to optimization techniques on traditional operators in BI. Aggregation, for example, is one of the basic building blocks of BI, while its processing performance with existing hash-based algorithms scales poorly in terms of the number of cores under NUMA architecture. In this paper, we provide new solutions to tackle the problem of parallel hash-based aggregation, especially targeting at domains of extremely large cardinality. We propose a NUMA-aware radix partitioning (NaRP) method which divides the original huge relation table into subsets, without invoking expensive remote memory access between nodes of the cores. We also present a new efficient aggregation algorithm (EAA), to aggregate the partitioned data in parallel with low cache coherence miss and locking costs. Theoretical analysis as well as empirical study on an IBM X5 server prove that our proposals are at least two times faster than existing methods.

LARGE CARDINALS AND LIGHTFACE DEFINABLE WELL-ORDERS, WITHOUT THE GCH

AbstractThis paper deals with the question whether the assumption that for every inaccessible cardinal κ there is a well-order of H(κ+) definable over the structure $\langle {\rm{H}}({\kappa ^ + }), \in \rangle$ by a formula without parameters is consistent with the existence of (large) large cardinals and failures of the GCH. We work under the assumption that the SCH holds at every singular fixed point of the ℶ-function and construct a class forcing that adds such a well-order at every inaccessible cardinal and preserves ZFC, all cofinalities, the continuum function, and all supercompact cardinals. Even in the absence of a proper class of inaccessible cardinals, this forcing produces a model of “V = HOD” and can therefore be used to force this axiom while preserving large cardinals and failures of the GCH. As another application, we show that we can start with a model containing an ω-superstrong cardinal κ and use this forcing to build a model in which κ is still ω-superstrong, the GCH fails at κ and there is a well-order of H(κ+) that is definable over H(κ+) without parameters. Finally, we can apply the forcing to answer a question about the definable failure of the GCH at a measurable cardinal.

The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $${\theta}$$ θ -supercompact

We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $${\theta}$$?-supercompact, for any desired $${\theta}$$?. In addition, we prove several global results showing how the entire class of weakly compactcardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or with the class of nearly $${\theta_\kappa}$$??-supercompact cardinals $${\kappa}$$?, for nearly any desired function $${\kappa\mapsto\theta_\kappa}$$????. These results answer several questions that had been open in the literature and extend to these large cardinals the identity-crises phenomenon, first identified by Magidor with the strongly compact cardinals.

The large cardinals between supercompact and almost-huge

I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a high-jump cardinal. A high-jump cardinal is defined as the critical point of an elementary embedding $${j: V \to M}$$ such that M is closed under sequences of length $${\sup\{{j(f)(\kappa)\,|\,f: \kappa \to \kappa}\}}$$ . Some of the other cardinals analyzed include the super-high-jump cardinals, almost-high-jump cardinals, Shelah-for-supercompactness cardinals, Woodin-for-supercompactness cardinals, Vopěnka cardinals, hypercompact cardinals, and enhanced supercompact cardinals. I organize these cardinals in terms of consistency strength and implicational strength. I also analyze the superstrong cardinals, which are weaker than supercompact cardinals but are related to high-jump cardinals. Among the results, I highlight the following. Furthermore, I prove some results relating high-jump cardinals to forcing, as well as analyzing Laver functions for super-high-jump cardinals.

Results on the Erdős–Falconer distance problem in [formula omitted] for odd q

The Erdős–Falconer distance problem in Zqd asks one to show that if E⊂Zqd is of sufficiently large cardinality, then the set of distances determined by E satisfies Δ(E)=Zq. Previous results were known only in the case q=pℓ, where p is an odd prime, and as such only showed that all units were obtained in the distance set. We give the first such result over rings Zq where q is no longer confined to be a prime power, and despite this, we show that the distance set of E contains all of Zq whenever E is of sufficiently large cardinality.

Enhancing Optical-CDMA Confidentiality With Multicode-Keying Encryption

Optical codes with large cardinality and tree structures of multiple subsets of codewords for adjustable code performance and cardinality have recently been proposed. As studied in this paper, these characteristics support multicode-keying encryption for enhancing physical-layer confidentiality in optical code-division multiple-access systems and networks. The concept of the multicode-keying encryption technique is introduced. The associated all-optical hardware is designed and validated with OptiSystemTM simulation. The theoretical analyses of confidentiality improvement by means of rapid codeword switching and multicode-keying encryption are formulated.

The tree property at both ℵω+1and ℵω+2

We force from large cardinals a model of ${\rm ZFC }$ in which $\aleph _{\omega +1}$ and $\aleph _{\omega +2}$ both have the tree property. We also prove that if we strengthen the large cardinal assumptions, then in the final model $\aleph _{\omega +2}$ e