Satisfiability in multi-valued circuits

Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time.Comment: 50 page

On the structure of positive maps II: low dimensional matrix algebras

We use a new idea that emerged in the examination of exposed positive maps between matrix algebras to investigate in more detail the difference between positive maps on $M_2(C)$ and $M_3(C)$. Our main tool stems from classical Grothendieck theorem on tensor product of Banach spaces and is an older and more general version of Choi-Jamiolkowski isomorphism between positive maps and block positive Choi matrices. It takes into account the correct topology on the latter set that is induced by the uniform topology on positive maps. In this setting we show that in $M_2(C)$ case a large class of nice positive maps can be generated from the small set of maps represented by self-adjoint unitaries, $2 P_x$ with $x$ maximally entangled vector and $p\otimes 1$ with $p$ rank 1 projector. We show why this construction fails in $M_3(C)$ case. There are also similarities. In both $M_2(C)$ and $M_3(C)$ cases any unital positive map represented by self-adjoint unitary is unitarily equivalent to the transposition map. Consequently we obtain a large family of exposed maps. We also investigate a convex structure of the Choi map, the first example of non-decomposable map. As a result the nature of the Choi map will be explained. This gives an information on the origin of appearance of non-decomposable maps on $M_3(C)$.Comment: Lemma 5 (in previous version, false) is removed. We would be very grateful for any remar

Equicontinuous Families of Markov Operators in View of Asymptotic Stability

Relation between equicontinuity, the so called e property and stability of Markov operators is studied. In particular, it is shown that any asymptotically stable Markov operator with an invariant measure such that the interior of its support is nonempty satisfies the e property

How far is it to a sudden future singularity of pressure?

We discuss the constraints coming from current observations of type Ia supernovae on cosmological models which allow sudden future singularities of pressure (with the scale factor and the energy density regular). We show that such a sudden singularity may happen in the very near future (e.g. within ten million years) and its prediction at the present moment of cosmic evolution cannot be distinguished, with current observational data, from the prediction given by the standard quintessence scenario of future evolution. Fortunately, sudden future singularities are characterized by a momentary peak of infinite tidal forces only; there is no geodesic incompletness which means that the evolution of the universe may eventually be continued throughout until another ``more serious'' singularity such as Big-Crunch or Big-Rip.Comment: REVTEX4, 4 pages, 2 figures, references change

Effective dynamics of the hybrid quantization of the Gowdy T^3 universe

The quantum dynamics of the linearly polarized Gowdy T^3 model (compact inhomogeneous universes admitting linearly polarized gravitational waves) is analyzed within Loop Quantum Cosmology by means of an effective dynamics. The analysis, performed via analytical and numerical methods, proves that the behavior found in the evolution of vacuum (homogeneous) Bianchi I universes is preserved qualitatively also in the presence of inhomogeneities. More precisely, the initial singularity is replaced by a big bounce which joins deterministically two large classical universes. In addition, we show that the size of the universe at the bounce is at least of the same order of magnitude (roughly speaking) as the size of the corresponding homogeneous universe obtained in the absence of gravitational waves. In particular, a precise lower bound for the ratio of these two sizes is found. Finally, the comparison of the amplitudes of the gravitational wave modes in the distant future and past shows that, statistically (i.e., for large samples of universes), the difference in amplitude is enhanced for nearly homogeneous universes, whereas this difference vanishes in inhomogeneity dominated cases. The presented analysis constitutes the first systematic effective study of an inhomogeneous system within Loop Quantum Cosmology, and it proves the robustness of the results obtained for homogeneous cosmologies in this context.Comment: 21 pages, 11 figures, RevTex4-1 + BibTe

Physical evolution in Loop Quantum Cosmology: The example of vacuum Bianchi I

We use the vacuum Bianchi I model as an example to investigate the concept of physical evolution in Loop Quantum Cosmology (LQC) in the absence of the massless scalar field which has been used so far in the literature as an internal time. In order to retrieve the system dynamics when no such a suitable clock field is present, we explore different constructions of families of unitarily related partial observables. These observables are parameterized, respectively, by: (i) one of the components of the densitized triad, and (ii) its conjugate momentum; each of them playing the role of an evolution parameter. Exploiting the properties of the considered example, we investigate in detail the domains of applicability of each construction. In both cases the observables possess a neat physical interpretation only in an approximate sense. However, whereas in case (i) such interpretation is reasonably accurate only for a portion of the evolution of the universe, in case (ii) it remains so during all the evolution (at least in the physically interesting cases). The constructed families of observables are next used to describe the evolution of the Bianchi I universe. The performed analysis confirms the robustness of the bounces, also in absence of matter fields, as well as the preservation of the semiclassicality through them. The concept of evolution studied here and the presented construction of observables are applicable to a wide class of models in LQC, including quantizations of the Bianchi I model obtained with other prescriptions for the improved dynamics.Comment: RevTex4, 22 pages, 4 figure

Improved bounds for Hadwiger's covering problem via thin-shell estimates

A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number $N\left(n\right)$ is such that every convex body in ${\mathbb R}^{n}$ can be covered by a union of the interiors of at most $N\left(n\right)$ of its translates. Despite continuous efforts, the best general upper bound known for this number remains as it was more than sixty years ago, of the order of ${2n \choose n}n\ln n$. In this note, we improve this bound by a sub-exponential factor. That is, we prove a bound of the order of ${2n \choose n}e^{-c\sqrt{n}}$ for some universal constant $c>0$. Our approach combines ideas from previous work by Artstein-Avidan and the second named author with tools from Asymptotic Geometric Analysis. One of the key steps is proving a new lower bound for the maximum volume of the intersection of a convex body $K$ with a translate of $-K$; in fact, we get the same lower bound for the volume of the intersection of $K$ and $-K$ when they both have barycenter at the origin. To do so, we make use of measure concentration, and in particular of thin-shell estimates for isotropic log-concave measures. Using the same ideas, we establish an exponentially better bound for $N\left(n\right)$ when restricting our attention to convex bodies that are $\psi_{2}$. By a slightly different approach, an exponential improvement is established also for classes of convex bodies with positive modulus of convexity