Combinatorial Principles Research Articles

The Thin Set Theorem for Pairs Implies DNR

Answering a question in the reverse mathematics of combinatorial principles, we prove that the thin set theorem for pairs (TS(2)) implies the diagonally noncomputable set principle (DNR) over the base axiom system RCA0.

Strategies for network security

Security of networks has become an increasingly important issue in the highly connected world. Security depends on attacks. Typical attacks include both cascading failure of virus spreading and of information propagation and physical attacks of removal of nodes or edges. Numerous experiments have shown that none of the existing models construct secure networks, and that the universal properties of power law and small world phenomenon of networks seem unavoidable obstacles for security of networks against attacks. Here, we propose a new strategy of attacks, the attack of rules of evolution of networks. By using the strategy, we proposed a new model of networks which generates provably secure networks. It was shown both analytically and numerically that the best strategy is to attack on the rules of the evolution of networks, that the second best strategy is the attack by cascading failure models, that the third best strategy is the physical attack of removal of nodes or edges, and that the least desirable strategy is the physical attack of deleting structures of the networks. The results characterize and classify the strategies for network security, providing a foundation for a security theory of networks. Equally important, our results demonstrate that security can be achieved provably by structures of networks, that there is a tradeoff between the roles of structures and of thresholds in security engineering, and that power law and small world property are never obstacles of security of networks. Our model explores a number of new principles of networks, including some topological principles, probabilistic principles, and combinatorial principles. The new principles build the foundation for new strategies for enhancing security of networks, and for new protocols of communications and security of the Internet and computer networks etc. We anticipate that our theory can be used in analyzing security of real systems in economy, society and technology.

Reflection for lindelöf degree and inaccessible cardinals

We say that a cardinal function \({\phi}\) reflects an infinite cardinal \({\kappa}\), if given a topological space X with \({\phi(X) \geqq \kappa}\), there exists \({Y \in [X]^{\leqq \kappa}}\) with \({\phi(Y ) \geqq \kappa}\). We improve results of Hodel and Vaughan in [8], investigating under what conditions the cardinal function L (Lindelof degree) reflects an inaccessible cardinal. We show that L reflects all strongly inaccessible cardinals, and obtain several sufficient conditions under which L reflects a weakly inaccessible cardinal. We discuss some combinatorial principles relevant to the problem, like SSH (Shelah’s Strong Hypothesis) and SWH (Shelah’s Weak Hypothesis), and show that if \({0^\sharp}\) does not exist, then L reflects all weakly inaccessible cardinals.

Combinatorial principles between RRT 2 2 and RT 2 2

We study the strength of some combinatorial principles weaker than Ramsey theorem for pairs over RCA0. First, we prove that Rainbow Ramsey theorem for pairs does not imply Thin Set theorem for pairs. Furthermore, we get some other related results on reverse mathematics using the same method. For instance, Rainbow Ramsey theorem for pairs is strictly weaker than Erdős-Moser theorem under RCA0.

On idealized versions of Pr1(μ +, μ +, μ +, cf(μ))

We obtain an improvement of some coloring theorems from Eisworth (Fund Math 202:97---123, 2009; Ann Pure Appl Logic 161(10):1216---1243, 2010), Eisworth and Shelah (J Symb Logic 74(4):1287---1309, 2009) for the case where the singular cardinal in question has countable cofinality. As a corollary, we obtain an "idealized" version of the combinatorial principle Pr1(μ +, μ +, μ +, cf(μ)) that maximizes the indecomposability of the associated ideal.

NONSTANDARD MODELS IN RECURSION THEORY AND REVERSE MATHEMATICS

AbstractWe give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey’s Theorem for Pairs.

Split vector systems for ultra-targeted gene delivery: A contrivance to achieve ethical assurance of somatic gene therapy in vivo

Tightly controlled spatial localisation of therapeutic gene delivery is essential to maximize the benefits of somatic gene therapy in vivo and to reduce its undesired effects on the ‘bystander’ cell populations, most importantly germline cells. Indeed, complete ethical assurance of somatic gene therapy can only be achieved with ultra-targeted gene delivery, which excludes the risk of inadvertent germline gene transfer. Thus, it is desired to supplement existing strategies of physical focusing and biological (cell-specific) targeting of gene delivery with an additional principle for the rigid control over spread of gene transfer within the body.In this paper I advance the concept of ‘combinatorial’ targeting of therapeutic gene transfer in vivo. I hypothesize that it is possible to engineer complex gene delivery vector systems consisting of several components, each one of them capable of independent spread within the human body but incapable of independent facilitation of gene transfer. As the gene delivery augmented by such split vector systems would be reliant on the simultaneous availability of all the vector system components at a predetermined body site, it is envisaged that higher order reaction kinetics required for the assembly of the functional gene transfer configuration would sharpen spatial localisation of gene transfer via curtailing the blurring effect of the vector spread within the body. A particular implementation of such split vector system could be obtained through supplementing a viral therapeutic gene vector with a separate auxiliary vector carrying a non-integrative and non-replicative form of a gene (e.g., mRNA) coding for a cellular receptor of the therapeutic vector component. Gene-transfer-enabling components of the vector system, which would be delivered separately from the vector component loaded with the therapeutic gene cargo, could also be cell-membrane-insertion-proficient receptors, elements of artificial transmembrane channels capable of nucleic acid transfer or, perhaps, factors modifying existing cellular transmembrane channels (e.g., gap-junctional hemichannels) to serve as conduits for gene entry. In general, there are four possibilities for gene transfer in vivo using a split vector system: (1) simultaneous delivery of a mixture of the vector components to the same body site; (2) sequential delivery of the vector components to the same body site; (3) simultaneous delivery of the vector components to separate body sites; (4) sequential delivery of the vector components to separate body sites.It is hoped that, once experimentally confirmed, the combinatorial principle for tight control over localisation of gene transfer could be the critical element in attaining complete assurance of gene non-delivery to germline cells in somatic gene therapy in vivo.

Artificial receptors for artificial olfaction

Any attempt to mimic the behavior of natural olfaction is mainly based on the properties of the artificial sensor units. Since the beginning of artificial olfaction, the main requirement for sensors has been the partial crossselectivity. This trivial feature makes possible the application of the combinatorial selectivity principle to the arrays of gas sensors. The principle of cross-selectivity can actually be fulfilled by almost any kind of sensor technology. This resulted in a vast popularity of “electronic noses” and their application. However, it is important to consider the kind of the cross-selectivity offered by sensors technology. Current sensors combines together the capability of chemistry to synthesize molecules endowed with molecular recognition properties and a number of solid-state devices that actually measure the changes in some physical properties consequent to the chemical interaction event. From this point of view, it is important to consider that in practice a number of unwanted interaction can take place even if the receptor is designed to emphasize the selectivity towards a particular kind of molecule. These are mainly driven by Van der Waals forces. Most of the available transducers cannot discriminate interactions by their energy and since Van der Waals interactions are rather ubiquitous the sensors tend to be highly correlated. A scrutiny of literature on electronic noses shows that even in case of large arrays, PCA often is limited to two variables. In this presentation, these principles will be discussed considering the study case of metalloporphyrins. These are a rather versatile class of molecules whose gas sensing properties are actively exploited in Nature, for instance for the transport of molecules such as O2 and N2 in blood. The possibility to reduce the influence of common interactions will be illustrated. It has also to be considered that the reduction of correlation is necessary to increase the array size in artificial olfaction systems. Published: 16 April 2014

Infinite dimensional proper subspaces of computable vector spaces

This article examines and distinguishes different techniques for coding incomputable information into infinite dimensional proper subspaces of a computable vector space, and is divided into two main parts. In the first part we describe different methods for coding into infinite dimensional subspaces. More specifically, we construct several computable infinite dimensional vector spaces each of which satisfies one of the following:(1)Every infinite/coinfinite dimensional subspace computes Turing's Halting Set ∅′;(2)Every infinite/cofinite dimensional proper subspace computes Turing's Halting Set ∅′;(3)There exists x∈V such that every infinite dimensional proper subspace not containing x computes Turing's Halting Set ∅′;(4)Every infinite dimensional proper subspace computes Turing's Halting Set ∅′. Vector space (4) generalizes vector spaces (1) and (2), and its construction is more complicated. The same simple and natural technique is used to construct vector spaces (1)–(3). Finally, we examine the reverse mathematical implications of our constructions (1)–(4).In the second part we examine the limitations of our simple and natural method for coding into infinite dimensional subspaces described in the previous paragraph. In particular, we prove that our simple and natural coding technique cannot produce a vector space of type (4) above, and that any vector space of type (4) must have “densely many” (from a certain point of view) finite dimensional computable subspaces. In other words, the construction of a vector space of type (4) is necessarily more complicated than the construction of vector spaces of types (1)–(3). We also introduce a new statement (in second order arithmetic) about the existence of infinite dimensional proper subspaces in a restricted class of vector spaces related to (1)–(3) above and show that it is implied by weak König's lemma in the context of reverse mathematics. In the context of reverse mathematics this gives rise to two statements from effective algebra about the existence of infinite dimensional proper subspaces (for a certain class of vector spaces) of the form (∀V)[X(V)→A(V)] and (∀V)[X(V)→B(V)], that each imply ACA0 over RCA0, but such that the seemingly weaker statement (∀V)[X(V)→A(V)∨B(V)] is provable via WKL0 over RCA0. Furthermore, we highlight some general similarities between constructing of infinite dimensional proper subspaces of computable vector spaces and constructing solutions to computable instances of various combinatorial principles such as Ramsey's Theorem for pairs.

Some combinatorial principles for trees and applications to tree families in Banach spaces

Suppose that is a normalized family in a Banach space indexed by the dyadic tree S. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree‐families. More precisely, assuming that for any infinite chain β of S the sequence is weakly null, we prove that there exists a subtree T of S such that for any infinite chain β of T the sequence is nearly (resp., convexly) unconditional. In the case where is a family of continuous functions, under some additional assumptions, we prove the existence of a subtree T of S such that for any infinite chain β of T, the sequence is unconditional. Finally, in the more general setting where for any chain β, is a Schauder basic sequence, we obtain a dichotomy result concerning the semi‐boundedly completeness of the sequences .

Relativization makes contradictions harder for Resolution

We provide a number of simplified and improved separations between pairs of Resolution-with-bounded-conjunction refutation systems, Res(d), as well as their tree-like versions, Res⁎(d). The contradictions we use are natural combinatorial principles: the Least number principle, LNPn and an ordered variant thereof, the Induction principle, IPn.LNPn is known to be easy for Resolution. We prove that its relativization is hard for Resolution, and more generally, the relativization of LNPn iterated d times provides a separation between Res(d) and Res(d+1). We prove the same result for the iterated relativization of IPn, where the tree-like variant Res⁎(d) is considered instead of Res(d).We go on to provide separations between the parameterized versions of Res(1) and Res(2). Here we are able again to use the relativization of the LNPn, but the classical proof breaks down and we are forced to use an alternative. Finally, we separate the parameterized versions of Res⁎(1) and Res⁎(2). Here, the relativization of IPn will not work as it is, and so we make a vectorizing amendment to it in order to address this shortcoming.

Comparisons of Polychromatic and Monochromatic Ramsey Theory

AbstractWe compare the strength of polychromatic and monochromatic Ramsey theory in several set-theoretic domains. We show that the rainbow Ramsey theorem does not follow from ZF, nor does the rainbow Ramsey theorem imply Ramsey's theorem over ZF. Extending the classical result of Erdős and Rado we show that the axiom of choice precludes the natural infinite exponent partition relations for polychromatic Ramsey theory. We introduce rainbow Ramsey ultrafilters, a polychromatic analogue of the usual Ramsey ultrafilters. We investigate the relationship of rainbow Ramsey ultrafilters with various special classes of ultrafilters, showing for example that every rainbow Ramsey ultrafilter is nowhere dense but rainbow Ramsey ultrafilters need not be rapid. This entails comparison of the polychromatic and monochromatic Ramsey theorems as combinatorial principles on ω.

Large cardinals and basic sequences

The purpose of this paper is to present several applications of combinatorial principles, well-known in Set Theory, to the geometry of infinite dimensional Banach spaces, particularly to the existence of certain basic sequences. We mention also some open problems where set-theoretical techniques are relevant.

ℵn-FREE MODULES OVER COMPLETE DISCRETE VALUATION DOMAINS WITH ALMOST TRIVIAL DUAL

AbstractA module M over a commutative ring R has an almost trivial dual if there is no homomorphism from M onto a free R-module of countable infinite rank. Using a new combinatorial principle (the ℵn-Black Box), which is provable in ordinary set theory, we show that for every natural number n, there exist arbitrarily large ℵn-free R-modules with almost trivial duals, when R is a complete discrete valuation domain. A corresponding result for torsion modules is also obtained.

Formula omitted]-conservation of combinatorial principles weaker than Ramsey’s theorem for pairs

We study combinatorial principles weaker than Ramsey’s theorem for pairs over the RCA0 (recursive comprehension axiom) system with Σ20-bounding. It is shown that the cohesiveness (COH), ascending and descending sequence (ADS), and chain/antichain (CAC) principles are all Π11-conservative over Σ20-bounding. In particular, none of these principles proves Σ20-induction.

The combinatorial essence of supercompactness

We introduce combinatorial principles that characterize strong compactness and supercompactness for inaccessible cardinals but also make sense for successor cardinals. Their consistency is established from what is supposedly optimal. Utilizing the failure of a weak version of a square, we show that the best currently known lower bounds for the consistency strength of these principles can be applied.

On the consistency strength of the proper forcing axiom

In recent work, the second author extended combinatorial principles due to Jech and Magidor that characterize certain large cardinal properties so that they can also hold true for small cardinals. For inaccessible cardinals, these modifications have no effect, and the resulting principles still give the same characterization of large cardinals. We prove that the proper forcing axiom PFA implies these principles hold for ω 2 . Using this, we argue to show that any of the known methods for forcing models of PFA from a large cardinal assumption requires a strongly compact cardinal. If one forces PFA using a proper forcing, then we get the optimal result that a supercompact cardinal is necessary.

Optimization of Virtual Light-Harvesting Antennas Part I: Poly-Nuclear Complexes of Bis-(4′-substituted-2, 2′; 6′, 2′′-terpyridino) Group III and Group IV Elements

Semiempirical calculations were performed on complexes of 4-substituted-2,2′;6′,2′′-terpyridino group III and group IV elements. The energy values obtained for the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) were subjected to conditional sequence optimization (LUMODONOR < LUMOACCEPTOR and HOMODONOR < HOMOACCEPTOR). The virtual libraries of candidate antennas obtained were subjected to screening to find the structures with the highest drop potential and obtain the lead antennas. Hence, this study demonstrates the application of combinatorial principles in the field of optoelectronics.

Higher complexity search problems for bounded arithmetic and a formalized no-gap theorem

We give a new characterization of the strict $$\forall {\Sigma^b_j}$$ sentences provable using $${\Sigma^b_k}$$ induction, for 1 ? j ? k. As a small application we show that, in a certain sense, Buss's witnessing theorem for strict $${\Sigma^b_k}$$ formulas already holds over the relatively weak theory PV. We exhibit a combinatorial principle with the property that a lower bound for it in constant-depth Frege would imply that the narrow CNFs with short depth j Frege refutations form a strict hierarchy with j, and hence that the relativized bounded arithmetic hierarchy can be separated by a family of $$\forall {\Sigma^b_1}$$ sentences.

The provably total search problems of bounded arithmetic

We give combinatorial principles GIk, based on k-turn games, which are complete for the class of NP search problems provably total at the kth level Tk2 of the bounded arithmetic hierarchy and hence characterize the ∀ Σ ^ 1 b consequences of Tk2. Our argument uses a translation of first-order proofs into large, uniform propositional proofs in a system in which the soundness of the rules can be witnessed by polynomial time reductions between games. We show that ∀ Σ ^ 1 b ( α ) conservativity of T 2 i + 1 ( α ) over T 2 i ( α ) already implies ∀ Σ ^ 1 b ( α ) conservativity of T2(α) over T 2 i ( α ). We translate this into propositional form and give a polylogarithmic width CNF GI ¯ 3 such that if GI ¯ 3 has small R(log) refutations then so does any polylogarithmic width CNF which has small constant depth refutations. We prove a resolution lower bound for GI ¯ 3 . We use our characterization to give a sufficient condition for the totality of a relativized NP search problem to be unprovable in T 2 i ( α ) in terms of a non-logical question about multiparty communication protocols.