On the Lazy Set object

The aim of this article is to employ the Lazy Set algorithm as an example for a mathematical framework for proving the linearizability of distributed systems. The proof in this approach is divided into two stages of lower and higher abstraction level. At the higher level a list of "axioms" is formulated and a proof is given that any model theoretic structure that satisfies these axioms is linearizable. At this level the algorithm is not mentioned. At the lower level, a Simpler Lazy Set algorithm is described, and it is shown that any execution of this simpler algorithm generates a model of these axioms (and is therefore linearizable). Finally the linearization of the Lazy Set algorithm is obtained by proving that any of its executions has a {\em reduct} that is an execution of the Simpler algorithm. So the reduct executions are linearizable and this entails immediately linearizability of the Lazy Set algorithm itself

Kishon's Poker Game

We present an approach for proving the correctness of distributed algorithms that obviate interleaving of processes' actions. The main part of the correctness proof is conducted at a higher abstract level and uses Tarskian system executions that combine two separate issues: the specification of the serial process that executes its protocol alone (no concurrency here), and the specification of the communication objects (no code here). In order to explain this approach a short algorithm for two concurrent processes that we call "Kishon's Poker" is introduced and is used as a platform where this approach is compared to the standard one which is based on the notions of global state, step, and history

Ladder gaps over stationary sets

For a stationary set S subseteq omega_1 and a ladder system C over S, a new type of gaps called C-Hausdorff is introduced and investigated. We describe a forcing model of ZFC in which, for some stationary set S, for every ladder C over S, every gap contains a subgap that is C-Hausdorff. But for every ladder E over omega_1 setminus S there exists a gap with no subgap that is E-Hausdorff. A new type of chain condition, called polarized chain condition, is introduced. We prove that the iteration with finite support of polarized c.c.c posets is again a polarized c.c.c poset

A Delta^2_2 well-order of the reals and incompactness of L(Q^{MM})

A forcing poset of size 2^{2^{aleph_1}} which adds no new reals is described and shown to provide a Delta^2_2 definable well-order of the reals (in fact, any given relation of the reals may be so encoded in some generic extension). The encoding of this well-order is obtained by playing with products of Aronszajn trees: Some products are special while other are Suslin trees. The paper also deals with the Magidor-Malitz logic: it is consistent that this logic is highly non compact

Lusin sequences under CH and under Martin's Axiom

Assuming the continuum hypothesis there is an inseparable sequence of length omega_1 that contains no Lusin subsequence, while if Martin's Axiom and the negation of CH is assumed then every inseparable sequence (of length omega_1) is a union of countably many Lusin subsequences

Coding with ladders a well-ordering of the reals

Any model of ZFC + GCH has a generic extension (made with a poset of size aleph_2) in which the following hold: MA + 2^{aleph_0}= aleph_2+ there exists a Delta^2_1-well ordering of the reals. The proof consists in iterating posets designed to change at will the guessing properties of ladder systems on omega_1. Therefore, the study of such ladders is a main concern of this article

Poset algebras over well quasi-ordered posets

A new class of partial order-types, class \gbqo^+ is defined and investigated here. A poset $P$ is in the class $W^+$ iff the free poset algebra $F(P)$ is generated by a better quasi-order $G$ that is included in the free lattice $L(P)$. We prove that if $P$ is any well quasi-ordering, then $L(P)$ is well founded, and is a countable union of well quasi-orderings. We prove that the class $W^+$ is contained in the class of well quasi-ordered sets. We prove that $W^+$ is preserved under homomorphic image, finite products, and lexicographic sum over better quasi-ordered index sets. We prove also that every countable well quasi-ordered set is in $W^+$. We do not know, however if the class of well quasi-ordered sets is contained in $W^+$. Additional results concern homomorphic images of posets algebras.Comment: 28 page

Traversal times for resonant tunneling

The tunneling time of particle through given barrier is commonly defined in terms of "internal clocks" which effectively measure the interaction time with internal degrees of freedom of the barrier. It is known that this definition of the time scale for tunneling is not unique in the sense that it depends on the clock used to define it. For the case of resonance tunneling, a particular choice that in the limit of a high/broad square barrier yields the original result of Buttiker and Landauer (Phys. Rev. Lett. 1982, 49, 1739) is correlated to the lifetime of the resonance state. This is illustrated for analytically solvable one-dimensional double barrier models and for a realistic model of electron tunneling through a static water barrier. The latter calculation constitutes a novel application of this concept to a 3-dimensional model, and the observed structure in the energy dependence of the computed traversal time reflects the existence of transient tunneling resonances associated with instantaneous water structures. These models, characterized by the existence of shape resonances in the barrier, make it possible to examine different internal clocks that were proposed for measuring tunneling times in situations where a "clock independent" intrinsic time scale (the resonance life time) for the tunneling time exists. It is argued that this time may be used in order to estimate the relative importance of dynamical barrier processes that affect the tunneling probability.Comment: 20 pages, 4 figures. J. Phys. Chem., in pres

Preliminary evidence for a virial shock around the Coma galaxy cluster

Galaxy clusters, the largest gravitationally bound objects in the Universe, are thought to grow by accreting mass from their surroundings through large-scale virial shocks. Due to electron acceleration in such a shock, it should appear as a $\gamma$-ray, hard X-ray, and radio ring, elongated towards the large-scale filaments feeding the cluster, coincident with a cutoff in the thermal Sunyaev-Zel'dovich (SZ) signal. However, no such signature was found until now, and the very existence of cluster virial shocks has remained a theory. We find preliminary evidence for a large, $\sim 5$ Mpc minor axis $\gamma$-ray ring around the Coma cluster, elongated towards the large scale filament connecting Coma and Abell 1367, detected at the nominal $2.7\sigma$ confidence level ($5.1\sigma$ using control signal simulations). The $\gamma$-ray ring correlates both with a synchrotron signal and with the SZ cutoff, but not with Galactic tracers. The $\gamma$-ray and radio signatures agree with analytic and numerical predictions, if the shock deposits $\sim 1\%$ of the thermal energy in relativistic electrons over a Hubble time, and $\sim 1\%$ in magnetic fields. The implied inverse-Compton and synchrotron cumulative emission from similar shocks can significantly contribute to the diffuse extragalactic $\gamma$-ray and low frequency radio backgrounds. Our results, if confirmed, reveal the prolate structure of the hot gas in Coma, the feeding pattern of the cluster, and properties of the surrounding large scale voids and filaments. The anticipated detection of such shocks around other clusters would provide a powerful new cosmological probe.Comment: Replaced with published versio

Martin's Axiom and Delta^2_1 well-ordering of the reals

Assuming an inaccessible cardinal kappa, there is a generic extension in which MA + 2^{aleph_0} = kappa holds and the reals have a Delta^2_1 well-ordering