Unary FA-presentable semigroups

Automatic presentations, also called FA-presentations, were introduced to extend nite model theory to innite structures whilst retaining the solubility of interesting decision problems. A particular focus of research has been the classication of those structures of some species that admit automatic presentations. Whilst some successes have been obtained, this appears to be a dicult problem in general. A restricted problem, also of signicant interest, is to ask this question for unary automatic presentations: auto-matic presentations over a one-letter alphabet. This paper studies unary FA-presentable semigroups. We prove the following: Every unary FA-presentable structure admits an injective unary automatic presentation where the language of representatives consists of every word over a one-letter alphabet. Unary FA-presentable semigroups are locally nite, but non-nitely generated unary FA-presentable semigroups may be innite. Every unary FA-presentable semigroup satises some Burnside identity.We describe the Green's relations in unary FA-presentable semigroups. We investigate the relationship between the class of unary FA-presentable semigroups and various semigroup constructions. A classication is given of the unary FA-presentable completely simple semigroups.PostprintPeer reviewe

The finiteness dimension of local cohomology modules and its dual notion

Let \fa be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_{\fa}(M), the finiteness dimension of M with respect to \fa, and, its dual notion q_{\fa}(M), the Artinianess dimension of M with respect to \fa. When (R,\fm) is local and r:=f_{\fa}(M) is less than f_{\fa}^{\fm}(M), the \fm-finiteness dimension of M relative to \fa, we prove that H^r_{\fa}(M) is not Artinian, and so the filter depth of \fa on M doesn't exceeds f_{\fa}(M). Also, we show that if M has finite dimension and H^i_{\fa}(M) is Artinian for all i>t, where t is a given positive integer, then H^t_{\fa}(M)/\fa H^t_{\fa}(M) is Artinian. It immediately implies that if q:=q_{\fa}(M)>0, then H^q_{\fa}(M) is not finitely generated, and so f_{\fa}(M)\leq q_{\fa}(M).Comment: 14 pages, to appear in Journal of Pure and Applied Algebr

Homotopy invariance through small stabilizations

We associate an algebra \Gami(\fA) to each bornological algebra \fA. The algebra \Gami(\fA) contains a two-sided ideal I_{S(\fA)} for each symmetric ideal S\triqui\elli of bounded sequences of complex numbers. In the case of \Gami=\Gami(\C), these are all the two-sided ideals, and I_S\mapsto J_S=\cB I_S\cB gives a bijection between the two-sided ideals of \Gami and those of \cB=\cB(\ell^2). We prove that Weibel's $K$-theory groups KH_*(I_{S(\fA)}) are homotopy invariant for certain ideals $S$ including $c_0$ and $\ell^p$. Moreover, if either $S=c_0$ and \fA is a local $C^*$-algebra or $S=\ell^p,\ell^{p\pm}$ and \fA is a local Banach algebra, then KH_*(I_{S(\fA)}) contains K_*^{\top}(\fA) as a direct summand. Furthermore, we prove that for $S\in\{c_0,\ell^p,\ell^{p\pm}\}$ the map K_*(\Gamma^\infty(\fA):I_{S(\fA)})\to KH_*(I_{S(\fA)}) fits into a long exact sequence with the relative cyclic homology groups HC_*(\Gamma^\infty(\fA):I_{S(\fA)}). Thus the latter groups measure the failure of the former map to be an isomorphism.Comment: 32 pages. The original paper has been split into two parts, of which this is the first part. The second part is now arXiv:1304.350

Remarks on (1−q)(1-q) expansion and factorization approximation in the Tsallis nonextensive statistical mechanics

The validity of (1-q) expansion and factorization approximations are analysed in the framework of Tsallis statistics. We employ exact expressions for classical independent systems (harmonic oscillators) by considering the unnormalized and normalized constrainsts. We show that these approxiamtions can not be accurate in the analysis of systems with many degrees of freedom.Comment: Latex, 6 pages, 2 figure

Nonlinear dynamics of Bose-condensed gases by means of a low- to high-density variational approach

We propose a versatile variational method to investigate the spatio-temporal dynamics of one-dimensional magnetically-trapped Bose-condensed gases. To this end we employ a \emph{q}-Gaussian trial wave-function that interpolates between the low- and the high-density limit of the ground state of a Bose-condensed gas. Our main result consists of reducing the Gross-Pitaevskii equation, a nonlinear partial differential equation describing the T=0 dynamics of the condensate, to a set of only three equations: \emph{two coupled nonlinear ordinary differential equations} describing the phase and the curvature of the wave-function and \emph{a separate algebraic equation} yielding the generalized width. Our equations recover those of the usual Gaussian variational approach (in the low-density regime), and the hydrodynamic equations that describe the high-density regime. Finally, we show a detailed comparison between the numerical results of our equations and those of the original Gross-Pitaevskii equation.Comment: 11 pages, 12 figures, submitted to Phys. Rev. A, January 200