212,276 research outputs found
Renormalization Invariants and Quark Flavor Mixings
A set of renormalization invariants is constructed using approximate,
two-flavor, analytic solutions for RGEs. These invariants exhibit explicitly
the correlation between quark flavor mixings and mass ratios in the context of
the SM, DHM and MSSM of electroweak interaction. The well known empirical
relations , can
thus be understood as the result of renormalization evolution toward the
infrared point. The validity of this approximation is evaluated by comparing
the numerical solutions with the analytical approach. It is found that the
scale dependence of these quantities for general three flavoring mixing follows
closely these invariants up to the GUT scale.Comment: 23 pages, 7 figure
Laplacian spectral characterization of some double starlike trees
A tree is called double starlike if it has exactly two vertices of degree
greater than two. Let denote the double starlike tree obtained by
attaching pendant vertices to one pendant vertex of the path and
pendant vertices to the other pendant vertex of . In this paper, we prove
that is determined by its Laplacian spectrum
FPTAS for Counting Monotone CNF
A monotone CNF formula is a Boolean formula in conjunctive normal form where
each variable appears positively. We design a deterministic fully
polynomial-time approximation scheme (FPTAS) for counting the number of
satisfying assignments for a given monotone CNF formula when each variable
appears in at most clauses. Equivalently, this is also an FPTAS for
counting set covers where each set contains at most elements. If we allow
variables to appear in a maximum of clauses (or sets to contain
elements), it is NP-hard to approximate it. Thus, this gives a complete
understanding of the approximability of counting for monotone CNF formulas. It
is also an important step towards a complete characterization of the
approximability for all bounded degree Boolean #CSP problems. In addition, we
study the hypergraph matching problem, which arises naturally towards a
complete classification of bounded degree Boolean #CSP problems, and show an
FPTAS for counting 3D matchings of hypergraphs with maximum degree .
Our main technique is correlation decay, a powerful tool to design
deterministic FPTAS for counting problems defined by local constraints among a
number of variables. All previous uses of this design technique fall into two
categories: each constraint involves at most two variables, such as independent
set, coloring, and spin systems in general; or each variable appears in at most
two constraints, such as matching, edge cover, and holant problem in general.
The CNF problems studied here have more complicated structures than these
problems and require new design and proof techniques. As it turns out, the
technique we developed for the CNF problem also works for the hypergraph
matching problem. We believe that it may also find applications in other CSP or
more general counting problems.Comment: 24 pages, 2 figures. version 1=>2: minor edits, highlighted the
picture of set cover/packing, and an implication of our previous result in 3D
matchin
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