Going higher in the First-order Quantifier Alternation Hierarchy on Words

We investigate the quantifier alternation hierarchy in first-order logic on finite words. Levels in this hierarchy are defined by counting the number of quantifier alternations in formulas. We prove that one can decide membership of a regular language to the levels $\mathcal{B}\Sigma_2$ (boolean combination of formulas having only 1 alternation) and $\Sigma_3$ (formulas having only 2 alternations beginning with an existential block). Our proof works by considering a deeper problem, called separation, which, once solved for lower levels, allows us to solve membership for higher levels

On Varieties of Ordered Automata

The Eilenberg correspondence relates varieties of regular languages to pseudovarieties of finite monoids. Various modifications of this correspondence have been found with more general classes of regular languages on one hand and classes of more complex algebraic structures on the other hand. It is also possible to consider classes of automata instead of algebraic structures as a natural counterpart of classes of languages. Here we deal with the correspondence relating positive $\mathcal C$-varieties of languages to positive $\mathcal C$-varieties of ordered automata and we present various specific instances of this correspondence. These bring certain well-known results from a new perspective and also some new observations. Moreover, complexity aspects of the membership problem are discussed both in the particular examples and in a general setting

From algebra to logic: there and back again -- the story of a hierarchy

This is an extended survey of the results concerning a hierarchy of languages that is tightly connected with the quantifier alternation hierarchy within the two-variable fragment of first order logic of the linear order.Comment: Developments in Language Theory 2014, Ekaterinburg : Russian Federation (2014

DFAs and PFAs with Long Shortest Synchronizing Word Length

It was conjectured by \v{C}ern\'y in 1964, that a synchronizing DFA on $n$ states always has a shortest synchronizing word of length at most $(n-1)^2$, and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for $n \leq 4$, and with bounds on the number of symbols for $n \leq 10$. Here we give the full analysis for $n \leq 6$, without bounds on the number of symbols. For PFAs the bound is much higher. For $n \leq 6$ we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding $(n-1)^2$ for $n =4,5,6$. For arbitrary n we give a construction of a PFA on three symbols with exponential shortest synchronizing word length, giving significantly better bounds than earlier exponential constructions. We give a transformation of this PFA to a PFA on two symbols keeping exponential shortest synchronizing word length, yielding a better bound than applying a similar known transformation.Comment: 16 pages, 2 figures source code adde

Spectroscopic applications and frequency locking of THz photomixing with distributed-Bragg-reflector diode lasers in low-temperature-grown GaAs

A compact, narrow-linewidth, tunable source of THz radiation has been developed for spectroscopy and other high-resolution applications. Distributed-Bragg-reflector (DBR) diode lasers at 850 nm are used to pump a low-temperature-grown GaAs photomixer. Resonant optical feedback is employed to stabilize the center frequencies and narrow the linewidths of the DBR lasers. The heterodyne linewidth full-width at half-maximum of two optically locked DBR lasers is 50 kHz on the 20 ms time scale and 2 MHz over 10 s; free-running DBR lasers have linewidths of 40 and 90 MHz on such time scales. This instrument has been used to obtain rotational spectra of acetonitrile (CH3CN) at 313 GHz. Detection limits of 1 × 10^–4 Hz^1/2 (noise/total power) have been achieved, with the noise floor dominated by the detector's noise equivalent power

Comparison of the clonality of urothelial carcinoma developing in the upper urinary tract and those developing in the bladder

PURPOSE: To identify the origin of synchronous and metachronous urothelial carcinoma (UC) of the bladder and upper urinary tract to get a better understanding of the basic mechanism behind the multifocality of UC, which may provide a sound bases for the future development of new strategies for detection, prevention and therapy. METHODS: Six patients with UC of the bladder and synchronous or metachronous UC of the upper urinary tract were studied. Genetic analysis involving the study of loss of heterozygosity (LOH) has been evaluated on their tumours using well characterised and new markers of UC (D9S171, D9S177, D9S303 and TP53). RESULTS: Five of the six patients demonstrated informative results. Four of five (80%) of patients had synchronous or metacharonous UC tumour and showed patterns of LOH consistent with tumorigenesis from monoclonal tumour origin. One of five (20%) patients exhibited a LOH consistent with oligoclonal tumorigenesis. CONCLUSION: These findings suggest that both the monoclonal and field cancerization theory of tumorigenesis may play a role in tumors of the urothelial tract. However, more data is needed

Future Foam

We study pocket universes which have zero cosmological constant and non-trivial boundary topology. These arise from bubble collisions in eternal inflation. Using a simplified dust model of collisions we find that boundaries of any genus can occur. Using a radiation shell model we perform analytic studies in the thin wall limit to show the existence of geometries with a single toroidal boundary. We give plausibility arguments that higher genus boundaries can also occur. In geometries with one boundary of any genus a timelike observer can see the entire boundary. Geometries with multiple disconnected boundaries can also occur. In the spherical case with two boundaries the boundaries are separated by a horizon. Our results suggest that the holographic dual description for eternal inflation, proposed by Freivogel, Sekino, Susskind and Yeh, should include summation over the genus of the base space of the dual conformal field theory. We point out peculiarities of this genus expansion compared to the string perturbation series.Comment: 23 pages, 6 figure