1,082 research outputs found
Connectivity and tree structure in finite graphs
Considering systems of separations in a graph that separate every pair of a
given set of vertex sets that are themselves not separated by these
separations, we determine conditions under which such a separation system
contains a nested subsystem that still separates those sets and is invariant
under the automorphisms of the graph.
As an application, we show that the -blocks -- the maximal vertex sets
that cannot be separated by at most vertices -- of a graph live in
distinct parts of a suitable tree-decomposition of of adhesion at most ,
whose decomposition tree is invariant under the automorphisms of . This
extends recent work of Dunwoody and Kr\"on and, like theirs, generalizes a
similar theorem of Tutte for .
Under mild additional assumptions, which are necessary, our decompositions
can be combined into one overall tree-decomposition that distinguishes, for all
simultaneously, all the -blocks of a finite graph.Comment: 31 page
Finding flows in the one-way measurement model
The one-way measurement model is a framework for universal quantum
computation, in which algorithms are partially described by a graph G of
entanglement relations on a collection of qubits. A sufficient condition for an
algorithm to perform a unitary embedding between two Hilbert spaces is for the
graph G, together with input/output vertices I, O \subset V(G), to have a flow
in the sense introduced by Danos and Kashefi [quant-ph/0506062]. For the
special case of |I| = |O|, using a graph-theoretic characterization, I show
that such flows are unique when they exist. This leads to an efficient
algorithm for finding flows, by a reduction to solved problems in graph theory.Comment: 8 pages, 3 figures: somewhat condensed and updated version, to appear
in PR
Partitioning 3-homogeneous latin bitrades
A latin bitrade is a pair of partial latin
squares which defines the difference between two arbitrary latin squares
and
of the same order. A 3-homogeneous bitrade has
three entries in each row, three entries in each column, and each symbol
appears three times in . Cavenagh (2006) showed that any
3-homogeneous bitrade may be partitioned into three transversals. In this paper
we provide an independent proof of Cavenagh's result using geometric methods.
In doing so we provide a framework for studying bitrades as tessellations of
spherical, euclidean or hyperbolic space.Comment: 13 pages, 11 figures, fixed the figures. Geometriae Dedicata,
Accepted: 13 February 2008, Published online: 5 March 200
Large Aperiodic Semigroups
The syntactic complexity of a regular language is the size of its syntactic
semigroup. This semigroup is isomorphic to the transition semigroup of the
minimal deterministic finite automaton accepting the language, that is, to the
semigroup generated by transformations induced by non-empty words on the set of
states of the automaton. In this paper we search for the largest syntactic
semigroup of a star-free language having left quotients; equivalently, we
look for the largest transition semigroup of an aperiodic finite automaton with
states.
We introduce two new aperiodic transition semigroups. The first is generated
by transformations that change only one state; we call such transformations and
resulting semigroups unitary. In particular, we study complete unitary
semigroups which have a special structure, and we show that each maximal
unitary semigroup is complete. For there exists a complete unitary
semigroup that is larger than any aperiodic semigroup known to date.
We then present even larger aperiodic semigroups, generated by
transformations that map a non-empty subset of states to a single state; we
call such transformations and semigroups semiconstant. In particular, we
examine semiconstant tree semigroups which have a structure based on full
binary trees. The semiconstant tree semigroups are at present the best
candidates for largest aperiodic semigroups.
We also prove that is an upper bound on the state complexity of
reversal of star-free languages, and resolve an open problem about a special
case of state complexity of concatenation of star-free languages.Comment: 22 pages, 1 figure, 2 table
Multi-party entanglement in graph states
Graph states are multi-particle entangled states that correspond to
mathematical graphs, where the vertices of the graph take the role of quantum
spin systems and edges represent Ising interactions. They are many-body spin
states of distributed quantum systems that play a significant role in quantum
error correction, multi-party quantum communication, and quantum computation
within the framework of the one-way quantum computer. We characterize and
quantify the genuine multi-particle entanglement of such graph states in terms
of the Schmidt measure, to which we provide upper and lower bounds in graph
theoretical terms. Several examples and classes of graphs will be discussed,
where these bounds coincide. These examples include trees, cluster states of
different dimension, graphs that occur in quantum error correction, such as the
concatenated [7,1,3]-CSS code, and a graph associated with the quantum Fourier
transform in the one-way computer. We also present general transformation rules
for graphs when local Pauli measurements are applied, and give criteria for the
equivalence of two graphs up to local unitary transformations, employing the
stabilizer formalism. For graphs of up to seven vertices we provide complete
characterization modulo local unitary transformations and graph isomorphies.Comment: 22 pages, 15 figures, 2 tables, typos corrected (e.g. in measurement
rules), references added/update
Design of Easily Synchronizable Oscillator Networks Using the Monte Carlo Optimization Method
Starting with an initial random network of oscillators with a heterogeneous
frequency distribution, its autonomous synchronization ability can be largely
improved by appropriately rewiring the links between the elements. Ensembles of
synchronization-optimized networks with different connectivities are generated
and their statistical properties are studied
The rigidity of periodic body-bar frameworks on the three-dimensional fixed torus
We present necessary and sufficient conditions for the generic rigidity of
body-bar frameworks on the three-dimensional fixed torus. These frameworks
correspond to infinite periodic body-bar frameworks in with a
fixed periodic lattice.Comment: 31 pages, 12 figure
The Hardness of Embedding Grids and Walls
The dichotomy conjecture for the parameterized embedding problem states that
the problem of deciding whether a given graph from some class of
"pattern graphs" can be embedded into a given graph (that is, is isomorphic
to a subgraph of ) is fixed-parameter tractable if is a class of graphs
of bounded tree width and -complete otherwise.
Towards this conjecture, we prove that the embedding problem is
-complete if is the class of all grids or the class of all walls
Factorization of Operators Through Orlicz Spaces
[EN] We study factorization of operators between quasi-Banach spaces. We prove the equivalence between certain vector norm inequalities and the factorization of operators through Orlicz spaces. As a consequence, we obtain the Maurey-Rosenthal factorization of operators into L-p-spaces. We give several applications. In particular, we prove a variant of Maurey's Extension Theorem.The research of the first author was supported by the National Science Centre (NCN), Poland, Grant No. 2011/01/B/ST1/06243. The research of the second author was supported by Ministerio de Economia y Competitividad, Spain, under project #MTM2012-36740-C02-02Mastylo, M.; Sánchez Pérez, EA. (2017). Factorization of Operators Through Orlicz Spaces. Bulletin of the Malaysian Mathematical Sciences Society. 40(4):1653-1675. https://doi.org/10.1007/s40840-015-0158-5S16531675404Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)Davis, W.J., Garling, D.J.H., Tomczak-Jaegermann, N.: The complex convexity of quasi-normed linear spaces. J. Funct. Anal. 55, 110–150 (1984)Defant, A.: Variants of the Maurey–Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Defant, A., Mastyło, M., Michels, C.: Orlicz norm estimates for eigenvalues of matrices. Isr. J. Math. 132, 45–59 (2002)Defant, A., Sánchez Pérez, E.A.: Maurey–Rosenthal factorization of positive operators and convexity. J. Math. Anal. Appl. 297, 771–790 (2004)Defant, A., Sánchez Pérez, E.A.: Domination of operators on function spaces. Math. Proc. Camb. Phil. Soc. 146, 57–66 (2009)Diestel, J.: Sequences and Series in Banach Spaces. Springer, Berlin (1984)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)Dilworth, S.J.: Special Banach lattices and their applications. In: Handbook of the Geometry of Banach Spaces, vol. 1. Elsevier, Amsterdam (2001)Figiel, T., Pisier, G.: Séries alétoires dans les espaces uniformément convexes ou uniformément lisses. Comptes Rendus de l’Académie des Sciences, Paris, Série A 279, 611–614 (1974)Kalton, N.J., Montgomery-Smith, S.J.: Set-functions and factorization. Arch. Math. (Basel) 61(2), 183–200 (1993)Kamińska, A., Mastyło, M.: Abstract duality Sawyer formula and its applications. Monatsh. Math. 151(3), 223–245 (2007)Kantorovich, L.V., Akilov, G.P.: Functional Analysis, 2nd edn. Pergamon Press, Oxford (1982)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Lozanovskii, G.Ya.: On some Banach lattices IV, Sibirsk. Mat. Z. 14, 140–155 (1973) (in Russian); English transl.: Siberian. Math. J. 14, 97–108 (1973)Lozanovskii, G.Ya.:Transformations of ideal Banach spaces by means of concave functions. In: Qualitative and Approximate Methods for the Investigation of Operator Equations, Yaroslavl, vol. 3, pp. 122–147 (1978) (Russian)Mastyło, M., Szwedek, R.: Interpolative constructions and factorization of operators. J. Math. Anal. Appl. 401, 198–208 (2013)Nikišin, E.M.: Resonance theorems and superlinear operators. Usp. Mat. Nauk 25, 129–191 (1970) (Russian)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domain and Integral Extension of Operators acting in Function Spaces. Operator Theory: Adv. Appl., vol. 180. Birkhäuser, Basel (2008)Pisier, G.: Factorization of linear operators and geometry of Banach spaces. CBMS Regional Conference Series in Mathematics, vol. 60. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1986)Reisner, S.: On two theorems of Lozanovskii concerning intermediate Banach lattices, geometric aspects of functional analysis (1986/87). Lecture Notes in Math., vol. 1317, pp. 67–83. Springer, Berlin (1988)Wojtaszczyk, P.: Banach Spaces for Analysts. Cambridge University Press, Cambridge (1991
Parameterized Compilation Lower Bounds for Restricted CNF-formulas
We show unconditional parameterized lower bounds in the area of knowledge
compilation, more specifically on the size of circuits in decomposable negation
normal form (DNNF) that encode CNF-formulas restricted by several graph width
measures. In particular, we show that
- there are CNF formulas of size and modular incidence treewidth
whose smallest DNNF-encoding has size , and
- there are CNF formulas of size and incidence neighborhood diversity
whose smallest DNNF-encoding has size .
These results complement recent upper bounds for compiling CNF into DNNF and
strengthen---quantitatively and qualitatively---known conditional low\-er
bounds for cliquewidth. Moreover, they show that, unlike for many graph
problems, the parameters considered here behave significantly differently from
treewidth
- …