A Characterization of Locally Testable Affine-Invariant Properties via Decomposition Theorems

Let $\mathcal{P}$ be a property of function $\mathbb{F}_p^n \to \{0,1\}$ for a fixed prime $p$. An algorithm is called a tester for $\mathcal{P}$ if, given a query access to the input function $f$, with high probability, it accepts when $f$ satisfies $\mathcal{P}$ and rejects when $f$ is "far" from satisfying $\mathcal{P}$. In this paper, we give a characterization of affine-invariant properties that are (two-sided error) testable with a constant number of queries. The characterization is stated in terms of decomposition theorems, which roughly claim that any function can be decomposed into a structured part that is a function of a constant number of polynomials, and a pseudo-random part whose Gowers norm is small. We first give an algorithm that tests whether the structured part of the input function has a specific form. Then we show that an affine-invariant property is testable with a constant number of queries if and only if it can be reduced to the problem of testing whether the structured part of the input function is close to one of a constant number of candidates.Comment: 27 pages, appearing in STOC 2014. arXiv admin note: text overlap with arXiv:1306.0649, arXiv:1212.3849 by other author

Quantum kinetic theory of shift current electron pumping in semiconductors

We develop a theory of laser beam generation of shift currents in non-centrosymmetric semiconductors. The currents originate when the excited electrons transfer between different bands or scatter inside these bands, and asymmetrically shift their centers of mass in elementary cells. Quantum kinetic equations for hot-carrier distributions and expressions for the induced currents are derived by nonequilibrium Green functions. In applications, we simplify the approach to the Boltzmann limit and use it to model laser-excited GaAs in the presence of LO phonon scattering. The shift currents are calculated in a steady-state regime.Comment: 23 pages, 5 figures (Latex

L-Visibility Drawings of IC-planar Graphs

An IC-plane graph is a topological graph where every edge is crossed at most once and no two crossed edges share a vertex. We show that every IC-plane graph has a visibility drawing where every vertex is an L-shape, and every edge is either a horizontal or vertical segment. As a byproduct of our drawing technique, we prove that an IC-plane graph has a RAC drawing in quadratic area with at most two bends per edge

Generalized gradient expansions in quantum transport equations

Gradient expansions in quantum transport equations of a Kadanoff-Baym form have been reexamined. We have realized that in a consistent approach the expansion should be performed also inside of the self-energy in the scattering integrals of these equations. In the first perturbation order this internal expansion gives new correction terms to the generalized Boltzman equation. These correction terms are found here for several typical systems. Possible corrections to the theory of a linear response to weak electric fields are also discussed.Comment: 20 pages, latex, to appear in Journal of Statistical Physics, March (1997

The genus Bolbelasmus in the western and southern regions of the Mediterranean Basin (Coleoptera: Geotrupidae: Bolboceratinae)

The Bolbelasmus Boucomont, 1911 species of the western and southern regions of the Mediterranean Basin (Northern Africa, Iberian Peninsula and France) are revised. The following three new species are described: Bolbelasmus brancoi Hillert & Král sp. nov. and Bolbelasmus howdeni Hillert & Král sp. nov., both from Spain and Gibraltar, and Bolbelasmus nikolajevi Hillert, Arnone, Král & Massa sp. nov. from Egypt, Libya and Tunisia. Bolbelasmus vaulogeri (Abeille de Perrin, 1898) stat. restit. is removed from synonymy with B. bocchus (Erichson, 1841) and reinstated as a separate species. Bolbelasmus romanorum Arnone & Massa, 2010 is considered a junior subjective synonym of B. vaulogeri. Lectotypes for Bolboceras bocchus Erichson, 1841 and Bolboceras vaulogeri Abeille de Perrin, 1898 are designated. Relevant diagnostic characters (head, pronotum, elytron, external male genitalia) are illustrated. Identifi cation keys for both males and females, and an annotated list of the Western Palaearctic representatives of the genus Bolbelasmus are presented. Finally, fi rst records are given for B. gallicus (Mulsant, 1842) from Corsica and the Midi-Pyrénées region of France, B. keithi Miessen & Trichas, 2011 from the Greek island of Rhodes, and B. unicornis (Schrank von Paula, 1789) from the Tuscany province of Italy

Complexity of Coloring Graphs without Paths and Cycles

Let $P_t$ and $C_\ell$ denote a path on $t$ vertices and a cycle on $\ell$ vertices, respectively. In this paper we study the $k$-coloring problem for $(P_t,C_\ell)$-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved that 3-colorability of $P_5$-free graphs has a finite forbidden induced subgraphs characterization, while Hoang, Moore, Recoskie, Sawada, and Vatshelle have shown that $k$-colorability of $P_5$-free graphs for $k \geq 4$ does not. These authors have also shown, aided by a computer search, that 4-colorability of $(P_5,C_5)$-free graphs does have a finite forbidden induced subgraph characterization. We prove that for any $k$, the $k$-colorability of $(P_6,C_4)$-free graphs has a finite forbidden induced subgraph characterization. We provide the full lists of forbidden induced subgraphs for $k=3$ and $k=4$. As an application, we obtain certifying polynomial time algorithms for 3-coloring and 4-coloring $(P_6,C_4)$-free graphs. (Polynomial time algorithms have been previously obtained by Golovach, Paulusma, and Song, but those algorithms are not certifying); To complement these results we show that in most other cases the $k$-coloring problem for $(P_t,C_\ell)$-free graphs is NP-complete. Specifically, for $\ell=5$ we show that $k$-coloring is NP-complete for $(P_t,C_5)$-free graphs when $k \ge 4$ and $t \ge 7$; for $\ell \ge 6$ we show that $k$-coloring is NP-complete for $(P_t,C_\ell)$-free graphs when $k \ge 5$, $t \ge 6$; and additionally, for $\ell=7$, we show that $k$-coloring is also NP-complete for $(P_t,C_7)$-free graphs if $k = 4$ and $t\ge 9$. This is the first systematic study of the complexity of the $k$-coloring problem for $(P_t,C_\ell)$-free graphs. We almost completely classify the complexity for the cases when $k \geq 4, \ell \geq 4$, and identify the last three open cases

Optical excitations in hexagonal nanonetwork materials

Optical excitations in hexagonal nanonetwork materials, for example, Boron-Nitride (BN) sheets and nanotubes, are investigated theoretically. The bonding of BN systems is positively polarized at the B site, and is negatively polarized at the N site. There is a permanent electric dipole moment along the BN bond, whose direction is from the B site to the N site. When the exciton hopping integral is restricted to the nearest neighbors, the flat band of the exciton appears at the lowest energy. The higher optical excitations have excitation bands similar to the electronic bands of graphene planes and carbon nanotubes. The symmetry of the flat exciton band is optically forbidden, indicating that the excitons related to this band will show quite long lifetime which will cause strong luminescence properties.Comment: 4 pages; 3 figures; proceedings of "XVIth International Winterschool on Electronic Properties of Novel Materials (IWEPNM2002)

Exhaustive generation of kk-critical H\mathcal H-free graphs

We describe an algorithm for generating all $k$-critical $\mathcal H$-free graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove that there are only finitely many $4$-critical $(P_7,C_k)$-free graphs, for both $k=4$ and $k=5$. We also show that there are only finitely many $4$-critical graphs $(P_8,C_4)$-free graphs. For each case of these cases we also give the complete lists of critical graphs and vertex-critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the $3$-colorability problem in the respective classes. Moreover, we prove that for every $t$, the class of 4-critical planar $P_t$-free graphs is finite. We also determine all 27 4-critical planar $(P_7,C_6)$-free graphs. We also prove that every $P_{10}$-free graph of girth at least five is 3-colorable, and determine the smallest 4-chromatic $P_{12}$-free graph of girth five. Moreover, we show that every $P_{13}$-free graph of girth at least six and every $P_{16}$-free graph of girth at least seven is 3-colorable. This strengthens results of Golovach et al.Comment: 17 pages, improved girth results. arXiv admin note: text overlap with arXiv:1504.0697

Assigning channels via the meet-in-the-middle approach

We study the complexity of the Channel Assignment problem. By applying the meet-in-the-middle approach we get an algorithm for the $\ell$-bounded Channel Assignment (when the edge weights are bounded by $\ell$) running in time $O^*((2\sqrt{\ell+1})^n)$. This is the first algorithm which breaks the $(O(\ell))^n$ barrier. We extend this algorithm to the counting variant, at the cost of slightly higher polynomial factor. A major open problem asks whether Channel Assignment admits a $O(c^n)$-time algorithm, for a constant $c$ independent of $\ell$. We consider a similar question for Generalized T-Coloring, a CSP problem that generalizes \CA. We show that Generalized T-Coloring does not admit a $2^{2^{o\left(\sqrt{n}\right)}} {\rm poly}(r)$-time algorithm, where $r$ is the size of the instance.Comment: SWAT 2014: 282-29