143,009 research outputs found

    Perfect Sets and ff-Ideals

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    A square-free monomial ideal II is called an {\it ff-ideal}, if both δF(I)\delta_{\mathcal{F}}(I) and δN(I)\delta_{\mathcal{N}}(I) have the same ff-vector, where δF(I)\delta_{\mathcal{F}}(I) (δN(I)\delta_{\mathcal{N}}(I), respectively) is the facet (Stanley-Reisner, respectively) complex related to II. In this paper, we introduce and study perfect subsets of 2[n]2^{[n]} and use them to characterize the ff-ideals of degree dd. We give a decomposition of V(n,2)V(n, 2) by taking advantage of a correspondence between graphs and sets of square-free monomials of degree 22, and then give a formula for counting the number of ff-ideals of degree 22, where V(n,2)V(n, 2) is the set of ff-ideals of degree 2 in K[x1,…,xn]K[x_1,\ldots,x_n]. We also consider the relation between an ff-ideal and an unmixed monomial ideal.Comment: 15 page

    A Comparison of Quantum Oracles

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    A standard quantum oracle SfS_f for a general function f:ZN→ZNf: Z_N \to Z_N is defined to act on two input states and return two outputs, with inputs ∣i⟩\ket{i} and ∣j⟩\ket{j} (i,j∈ZNi,j \in Z_N ) returning outputs ∣i⟩\ket{i} and ∣j⊕f(i)⟩\ket{j \oplus f(i)}. However, if ff is known to be a one-to-one function, a simpler oracle, MfM_f, which returns ∣f(i)⟩\ket{f(i)} given ∣i⟩\ket{i}, can also be defined. We consider the relative strengths of these oracles. We define a simple promise problem which minimal quantum oracles can solve exponentially faster than classical oracles, via an algorithm which cannot be naively adapted to standard quantum oracles. We show that SfS_f can be constructed by invoking MfM_f and (Mf)−1(M_f)^{-1} once each, while Θ(N)\Theta(\sqrt{N}) invocations of SfS_f and/or (Sf)−1(S_f)^{-1} are required to construct MfM_f.Comment: 4 pages, 1 figure; Final version, with an extended discussion of oracle inverses. To appear in Phys Rev

    Tree decompositions with small cost

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    The f-cost of a tree decomposition ({Xi | i e I}, T = (I;F)) for a function f : N -> R+ is defined as EieI f(|Xi|). This measure associates with the running time or memory use of some algorithms that use the tree decomposition. In this paper we investigate the problem to find tree decompositions of minimum f-cost. A function f : N -> R+ is fast, if for every i e N: f(i+1) => 2*f(i). We show that for fast functions f, every graph G has a tree decomposition of minimum f-cost that corresponds to a minimal triangulation of G; if f is not fast, this does not hold. We give polynomial time algorithms for the problem, assuming f is a fast function, for graphs that has a polynomial number of minimal separators, for graphs of treewidth at most two, and for cographs, and show that the problem is NP-hard for bipartite graphs and for cobipartite graphs. We also discuss results for a weighted variant of the problem derived of an application from probabilistic networks

    A limit theorem for the six-length of random functional graphs with a fixed degree sequence

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    We obtain results on the limiting distribution of the six-length of a random functional graph, also called a functional digraph or random mapping, with given in-degree sequence. The six-length of a vertex v∈Vv\in V is defined from the associated mapping, f:V→Vf:V\to V, to be the maximum i∈Vi\in V such that the elements v,f(v),…,fi−1(v)v, f(v), \ldots, f^{i-1}(v) are all distinct. This has relevance to the study of algorithms for integer factorisation

    Renormalization of the Fayet-Iliopoulos Term in Softly Broken SUSY Gauge Theories

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    It is shown that renormalization of the Fayet-Iliopoulos term in a softly broken SUSY gauge theory, in full analogy with all the other soft terms renormalizations, is completely defined in a rigid or an unbroken theory. However, contrary to the other soft renormalizations, there is no simple differential operator that acts on the renormalization functions of a rigid theory and allows one to get the renormalization of the F-I term. One needs an analysis of the superfield diagrams and some additional diagram calculations in components. The method is illustrated by the four loop calculation of some part of renormalization proportional to the soft scalar masses and the soft triple couplings.Comment: Latex2e, 14 pages, uses axodraw.sty. References adde
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