224,727 research outputs found
Sign-Rank Can Increase Under Intersection
The communication class UPP^{cc} is a communication analog of the Turing Machine complexity class PP. It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds.
For a communication problem f, let f wedge f denote the function that evaluates f on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem f with UPP^{cc}(f)= O(log n), and UPP^{cc}(f wedge f) = Theta(log^2 n). This is the first result showing that UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPP^{cc}, the class of problems with polylogarithmic-cost UPP communication protocols, is not closed under intersection.
Our result shows that the function class consisting of intersections of two majorities on n bits has dimension complexity n^{Omega(log n)}. This matches an upper bound of (Klivans, O\u27Donnell, and Servedio, FOCS 2002), who used it to give a quasipolynomial time algorithm for PAC learning intersections of polylogarithmically many majorities. Hence, fundamentally new techniques will be needed to learn this class of functions in polynomial time
Extensional and Intensional Strategies
This paper is a contribution to the theoretical foundations of strategies. We
first present a general definition of abstract strategies which is extensional
in the sense that a strategy is defined explicitly as a set of derivations of
an abstract reduction system. We then move to a more intensional definition
supporting the abstract view but more operational in the sense that it
describes a means for determining such a set. We characterize the class of
extensional strategies that can be defined intensionally. We also give some
hints towards a logical characterization of intensional strategies and propose
a few challenging perspectives
Yukawa couplings and masses of non-chiral states for the Standard Model on D6-branes on T6/Z6'
The perturbative leading order open string three-point couplings for the
Standard Model with hidden USp(6) on fractional D6-branes on T6/Z6' from
arXiv:0806.3039 [hep-th], arXiv:0910.0843 [hep-th] are computed. Physical
Yukawa couplings consisting of holomorphic Wilsonian superpotential terms times
a non-holomorphic prefactor involving the corresponding classical open string
Kaehler metrics are given, and mass terms for all non-chiral matter states are
derived. The lepton Yukawa interactions are at leading order flavour diagonal,
while the quark sector displays a more intricate pattern of mixings. While N=2
supersymmetric sectors acquire masses via only two D6-brane displacements -
which also provide the hierarchies between up- and down-type Yukawas within one
quark or lepton generation -, the remaining vector-like states receive masses
via perturbative three-point couplings to some Standard Model singlet fields
with vevs along flat directions. Couplings to the hidden sector and messengers
for supersymmetry breaking are briefly discussed.Comment: 52 pages (including 8p. appendix); 5 figures; 14 tables; v2:
discussion in section 4.1.3 extended, footnote 5 added, typos corrected,
accepted by JHE
Anabelian Intersection Theory I: The Conjecture of Bogomolov-Pop and Applications
We finish the proof of the conjecture of F. Bogomolov and F. Pop: Let
and be fields finitely-generated and of transcendence degree
over and , respectively, where is either
or , and is algebraically
closed. We denote by and their respective absolute Galois
groups. Then the canonical map \varphi_{F_{1}, F_{2}}: \Isom^i(F_1,
F_2)\rightarrow \Isom^{\Out}_{\cont}(G_{F_2}, G_{F_1}) from the isomorphisms,
up to Frobenius twists, of the inseparable closures of and to
continuous outer isomorphisms of their Galois groups is a bijection. Thus,
function fields of varieties of dimension over algebraic closures of
prime fields are anabelian. We apply this to give a necessary and sufficient
condition for an element of the Grothendieck-Teichm\"uller group to be an
element of the absolute Galois group of .Comment: 30 pages, comments welcome
Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems
We consider projection algorithms for solving (nonconvex) feasibility
problems in Euclidean spaces. Of special interest are the Method of Alternating
Projections (MAP) and the Douglas-Rachford or Averaged Alternating Reflection
Algorithm (AAR). In the case of convex feasibility, firm nonexpansiveness of
projection mappings is a global property that yields global convergence of MAP
and for consistent problems AAR. Based on (\epsilon, \delta)-regularity of sets
developed by Bauschke, Luke, Phan and Wang in 2012, a relaxed local version of
firm nonexpansiveness with respect to the intersection is introduced for
consistent feasibility problems. Together with a coercivity condition that
relates to the regularity of the intersection, this yields local linear
convergence of MAP for a wide class of nonconvex problems,Comment: 22 pages, no figures, 30 reference
Uniform Proofs of Normalisation and Approximation for Intersection Types
We present intersection type systems in the style of sequent calculus,
modifying the systems that Valentini introduced to prove normalisation
properties without using the reducibility method. Our systems are more natural
than Valentini's ones and equivalent to the usual natural deduction style
systems. We prove the characterisation theorems of strong and weak
normalisation through the proposed systems, and, moreover, the approximation
theorem by means of direct inductive arguments. This provides in a uniform way
proofs of the normalisation and approximation theorems via type systems in
sequent calculus style.Comment: In Proceedings ITRS 2014, arXiv:1503.0437
Tangential Extremal Principles for Finite and Infinite Systems of Sets, II: Applications to Semi-infinite and Multiobjective Optimization
This paper contains selected applications of the new tangential extremal
principles and related results developed in Part I to calculus rules for
infinite intersections of sets and optimality conditions for problems of
semi-infinite programming and multiobjective optimization with countable
constraint
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