On the uniform one-dimensional fragment

The uniform one-dimensional fragment of first-order logic, U1, is a recently introduced formalism that extends two-variable logic in a natural way to contexts with relations of all arities. We survey properties of U1 and investigate its relationship to description logics designed to accommodate higher arity relations, with particular attention given to DLR_reg. We also define a description logic version of a variant of U1 and prove a range of new results concerning the expressivity of U1 and related logics

One-dimensional fragment of first-order logic

We introduce a novel decidable fragment of first-order logic. The fragment is one-dimensional in the sense that quantification is limited to applications of blocks of existential (universal) quantifiers such that at most one variable remains free in the quantified formula. The fragment is closed under Boolean operations, but additional restrictions (called uniformity conditions) apply to combinations of atomic formulae with two or more variables. We argue that the notions of one-dimensionality and uniformity together offer a novel perspective on the robust decidability of modal logics. We also establish that minor modifications to the restrictions of the syntax of the one-dimensional fragment lead to undecidable formalisms. Namely, the two-dimensional and non-uniform one-dimensional fragments are shown undecidable. Finally, we prove that with regard to expressivity, the one-dimensional fragment is incomparable with both the guarded negation fragment and two-variable logic with counting. Our proof of the decidability of the one-dimensional fragment is based on a technique involving a direct reduction to the monadic class of first-order logic. The novel technique is itself of an independent mathematical interest

On the uniform one-dimensional fragment over ordered models

The uniform one-dimensional fragment U1 is a recently introduced extension of the two-variable fragment FO2. The logic U1 enables the use of relation symbols of all arities and thereby extends the scope of applications of FO2. In this thesis we show that the satisfiability and finite satisfiability problems of U1 over linearly ordered models are NExpTime-complete. The corresponding problems for FO2 are likewise NExpTime-complete, so the transition from FO2 to U1 in the ordered realm causes no increase in complexity. To contrast our results, we also establish that U1 with an unrestricted use of two built-in linear orders is undecidable

Critical scaling in standard biased random walks

The spatial coverage produced by a single discrete-time random walk, with asymmetric jump probability $p\neq 1/2$ and non-uniform steps, moving on an infinite one-dimensional lattice is investigated. Analytical calculations are complemented with Monte Carlo simulations. We show that, for appropriate step sizes, the model displays a critical phenomenon, at $p=p_c$. Its scaling properties as well as the main features of the fragmented coverage occurring in the vicinity of the critical point are shown. In particular, in the limit $p\to p_c$, the distribution of fragment lengths is scale-free, with nontrivial exponents. Moreover, the spatial distribution of cracks (unvisited sites) defines a fractal set over the spanned interval. Thus, from the perspective of the covered territory, a very rich critical phenomenology is revealed in a simple one-dimensional standard model.Comment: 4 pages, 4 figure

Collapse of Primordial Filamentary Clouds under Far-Ultraviolet Radiation

Collapse and fragmentation of primordial filamentary clouds under isotropic dissociation radiation is investigated with one-dimensional hydrodynamical calculations. We investigate the effect of dissociation photon on the filamentary clouds with calculating non-equilibrium chemical reactions. With the external radiation assumed to turn on when the filamentary cloud forms, the filamentary cloud with low initial density ($n_0 \le 10^2 \mathrm{cm^{-3}}$) suffers photodissociation of hydrogen molecules. In such a case, since main coolant is lost, temperature increases adiabatically enough to suppress collapse. As a result, the filamentary cloud fragments into very massive clouds ($\sim 10^5 M_\odot$). On the other hand, the evolution of the filamentary clouds with high initial density ($n_0>10^2 \mathrm{cm^{-3}}$) is hardly affected by the external radiation. This is because the filamentary cloud with high initial density shields itself from the external radiation. It is found that the external radiation increases fragment mass. This result is consistent with previous results with one-zone models. It is also found that fragment mass decreases owing to the external dissociation radiation in the case with sufficiently large line mass.Comment: 26 pages, 15 figures, accepted by PAS

The fragmentation of expanding shells II: Thickness matters

We study analytically the development of gravitational instability in an expanding shell having finite thickness. We consider three models for the radial density profile of the shell: (i) an analytic uniform-density model, (ii) a semi-analytic model obtained by numerical solution of the hydrostatic equilibrium equation, and (iii) a 3D hydrodynamic simulation. We show that all three profiles are in close agreement, and this allows us to use the first model to describe fragments in the radial direction of the shell. We then use non-linear equations describing the time-evolution of a uniform oblate spheroid to derive the growth rates of shell fragments having different sizes. This yields a dispersion relation which depends on the shell thickness, and hence on the pressure confining the shell. We compare this dispersion relation with the dispersion relation obtained using the standard thin-shell analysis, and show that, if the confining pressure is low, only large fragments are unstable. On the other hand, if the confining pressure is high, fragments smaller than predicted by the thin-shell analysis become unstable. Finally, we compare the new dispersion relation with the results of 3D hydrodynamic simulations, and show that the two are in good agreement.Comment: 9 pages, 9 figures, accepted by MNRA

Analysis of simple 2-D and 3-D metal structures subjected to fragment impact

Theoretical methods were developed for predicting the large-deflection elastic-plastic transient structural responses of metal containment or deflector (C/D) structures to cope with rotor burst fragment impact attack. For two-dimensional C/D structures both, finite element and finite difference analysis methods were employed to analyze structural response produced by either prescribed transient loads or fragment impact. For the latter category, two time-wise step-by-step analysis procedures were devised to predict the structural responses resulting from a succession of fragment impacts: the collision force method (CFM) which utilizes an approximate prediction of the force applied to the attacked structure during fragment impact, and the collision imparted velocity method (CIVM) in which the impact-induced velocity increment acquired by a region of the impacted structure near the impact point is computed. The merits and limitations of these approaches are discussed. For the analysis of 3-d responses of C/D structures, only the CIVM approach was investigated

Recursive tilings and space-filling curves with little fragmentation

This paper defines the Arrwwid number of a recursive tiling (or space-filling curve) as the smallest number w such that any ball Q can be covered by w tiles (or curve sections) with total volume O(vol(Q)). Recursive tilings and space-filling curves with low Arrwwid numbers can be applied to optimise disk, memory or server access patterns when processing sets of points in d-dimensional space. This paper presents recursive tilings and space-filling curves with optimal Arrwwid numbers. For d >= 3, we see that regular cube tilings and space-filling curves cannot have optimal Arrwwid number, and we see how to construct alternatives with better Arrwwid numbers.Comment: Manuscript accompanying abstract in EuroCG 2010, including full proofs, 20 figures, references, discussion et

Collapse and Fragmentation of Magnetized Cylindrical Clouds

Gravitational collapse of the cylindrical elongated cloud is studied by numerical magnetohydrodynamical simulations. In the infinitely long cloud in hydrostatic configuration, small perturbations grow by the gravitational instability. The most unstable mode indicated by a linear perturbation theory grows selectively even from a white noise. The growth rate agrees with that calculated by the linear theory. First, the density-enhanced region has an elongated shape, i.e., prolate spheroidal shape. As the collapse proceeds, the high-density fragment begins to contract mainly along the symmetry axis. Finally, a spherical core is formed in the non-magnetized cloud. In contrast, an oblate spheroidal dense disk is formed in a cloud in which the magnetic pressure is nearly equal to the thermal one. The radial size of the disk becomes proportional to the initial characteristic density scale-height in the r-direction. As the collapse proceeds, a slowly contracting dense part is formed (approximately < 10% in mass) inside of the fast contracting disk. And this is separated from other part of the disk whose inflow velocity is accelerated as reaching the center of the core. From arguments on the Jeans mass and the magnetic critical mass, it is concluded that the fragments formed in a cylindrical elongated cloud can not be supported against the self- gravity and it will eventually collapse.Comment: 20 pages, figures available upon request, LaTeX, NIGAST040

Krausz dimension and its generalizations in special graph classes

A {\it krausz $(k,m)$-partition} of a graph $G$ is the partition of $G$ into cliques, such that any vertex belongs to at most $k$ cliques and any two cliques have at most $m$ vertices in common. The {\it $m$-krausz} dimension $kdim_m(G)$ of the graph $G$ is the minimum number $k$ such that $G$ has a krausz $(k,m)$-partition. 1-krausz dimension is known and studied krausz dimension of graph $kdim(G)$. In this paper we prove, that the problem $"kdim(G)\leq 3"$ is polynomially solvable for chordal graphs, thus partially solving the problem of P. Hlineny and J. Kratochvil. We show, that the problem of finding $m$-krausz dimension is NP-hard for every $m\geq 1$, even if restricted to (1,2)-colorable graphs, but the problem $"kdim_m(G)\leq k"$ is polynomially solvable for $(\infty,1)$-polar graphs for every fixed $k,m\geq 1$