362 research outputs found
A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics
Motivated by questions in mass-action kinetics, we introduce the notion of
vertexical family of differential inclusions. Defined on open hypercubes, these
families are characterized by particular good behavior under projection maps.
The motivating examples are certain families of reaction networks -- including
reversible, weakly reversible, endotactic, and strongly endotactic reaction
networks -- that give rise to vertexical families of mass-action differential
inclusions. We prove that vertexical families are amenable to structural
induction. Consequently, a trajectory of a vertexical family approaches the
boundary if and only if either the trajectory approaches a vertex of the
hypercube, or a trajectory in a lower-dimensional member of the family
approaches the boundary. With this technology, we make progress on the global
attractor conjecture, a central open problem concerning mass-action kinetics
systems. Additionally, we phrase mass-action kinetics as a functor on reaction
networks with variable rates.Comment: v5: published version; v3 and v4: minor additional edits; v2:
contains more general version of main theorem on vertexical families,
including its accompanying corollaries -- some of them new; final section
contains new results relating to prior and future research on persistence of
mass-action systems; improved exposition throughou
Phase Diagram for Anderson Disorder: beyond Single-Parameter Scaling
The Anderson model for independent electrons in a disordered potential is
transformed analytically and exactly to a basis of random extended states
leading to a variant of augmented space. In addition to the widely-accepted
phase diagrams in all physical dimensions, a plethora of additional, weaker
Anderson transitions are found, characterized by the long-distance behavior of
states. Critical disorders are found for Anderson transitions at which the
asymptotically dominant sector of augmented space changes for all states at the
same disorder. At fixed disorder, critical energies are also found at which the
localization properties of states are singular. Under the approximation of
single-parameter scaling, this phase diagram reduces to the widely-accepted one
in 1, 2 and 3 dimensions. In two dimensions, in addition to the Anderson
transition at infinitesimal disorder, there is a transition between two
localized states, characterized by a change in the nature of wave function
decay.Comment: 51 pages including 4 figures, revised 30 November 200
Heegaard Floer homology and integer surgeries on links
Let L be a link in an integral homology three-sphere. We give a description
of the Heegaard Floer homology of integral surgeries on L in terms of some data
associated to L, which we call a complete system of hyperboxes for L. Roughly,
a complete systems of hyperboxes consists of chain complexes for (some versions
of) the link Floer homology of L and all its sublinks, together with several
chain maps between these complexes. Further, we introduce a way of presenting
closed four-manifolds with b_2^+ > 1 by four-colored framed links in the
three-sphere. Given a link presentation of this kind for a four-manifold X, we
then describe the Ozsvath-Szabo mixed invariants of X in terms of a complete
system of hyperboxes for the link. Finally, we explain how a grid diagram
produces a particular complete system of hyperboxes for the corresponding link.Comment: 231 pages, 54 figures; major revision: we now work with one U
variable for each w basepoint, rather than one per link component; we also
added Section 4, with an overview of the main resul
Non-Perturbative Gravity and the Spin of the Lattice Graviton
The lattice formulation of quantum gravity provides a natural framework in
which non-perturbative properties of the ground state can be studied in detail.
In this paper we investigate how the lattice results relate to the continuum
semiclassical expansion about smooth manifolds. As an example we give an
explicit form for the lattice ground state wave functional for semiclassical
geometries. We then do a detailed comparison between the more recent
predictions from the lattice regularized theory, and results obtained in the
continuum for the non-trivial ultraviolet fixed point of quantum gravity found
using weak field and non-perturbative methods. In particular we focus on the
derivative of the beta function at the fixed point and the related universal
critical exponent for gravitation. Based on recently available lattice
and continuum results we assess the evidence for the presence of a massless
spin two particle in the continuum limit of the strongly coupled lattice
theory. Finally we compare the lattice prediction for the vacuum-polarization
induced weak scale dependence of the gravitational coupling with recent
calculations in the continuum, finding similar effects.Comment: 46 pages, one figur
Subquadratic-Time Algorithm for the Diameter and All Eccentricities on Median Graphs
On sparse graphs, Roditty and Williams [2013] proved that no O(n^{2-?})-time algorithm achieves an approximation factor smaller than 3/2 for the diameter problem unless SETH fails. We answer here an open question formulated in the literature: can we use the structural properties of median graphs to break this global quadratic barrier?
We propose the first combinatorial algorithm computing exactly all eccentricities of a median graph in truly subquadratic time. Median graphs constitute the family of graphs which is the most studied in metric graph theory because their structure represents many other discrete and geometric concepts, such as CAT(0) cube complexes. Our result generalizes a recent one, stating that there is a linear-time algorithm for computing all eccentricities in median graphs with bounded dimension d, i.e. the dimension of the largest induced hypercube (note that 1-dimensional median graphs are exactly the forests). This prerequisite on d is not necessarily anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is O(n^{1.6456}log^{O(1)} n)
Subquadratic-time algorithm for the diameter and all eccentricities on median graphs
On sparse graphs, Roditty and Williams [2013] proved that no
-time algorithm achieves an approximation factor smaller
than for the diameter problem unless SETH fails. In this article,
we solve an open question formulated in the literature: can we use the
structural properties of median graphs to break this global quadratic barrier?
We propose the first combinatiorial algorithm computing exactly all
eccentricities of a median graph in truly subquadratic time. Median graphs
constitute the family of graphs which is the most studied in metric graph
theory because their structure represents many other discrete and geometric
concepts, such as CAT(0) cube complexes. Our result generalizes a recent one,
stating that there is a linear-time algorithm for all eccentricities in median
graphs with bounded dimension , i.e. the dimension of the largest induced
hypercube. This prerequisite on is not necessarily anymore to determine all
eccentricities in subquadratic time. The execution time of our algorithm is
.
We provide also some satellite outcomes related to this general result. In
particular, restricted to simplex graphs, this algorithm enumerates all
eccentricities with a quasilinear running time. Moreover, an algorithm is
proposed to compute exactly all reach centralities in time
.Comment: 43 pages, extended abstract in STACS 202
Rational Orthogonal versus Real Orthogonal
The main question we raise here is the following one: given a real orthogonal
n by n matrix X, is it true that there exists a rational orthogonal matrix Y
having the same zero-pattern? We conjecture that this is the case and prove it
for n<=5. We also consider the related problem for symmetric orthogonal
matrices.Comment: 23 page
Rational Design of DNA Sequences with Non-Orthogonal Binding Interactions
Molecular computation involving promiscuous, or non-orthogonal, binding interactions between system components is found commonly in natural biological systems, as well as some proposed human-made molecular computers. Such systems are characterized by the fact that each computational unit, such as a domain within a DNA strand, may bind to several different partners with distinct, prescribed binding strengths. Unfortunately, implementing systems of molecular computation that incorporate non-orthogonal binding is difficult, because researchers lack a robust, general-purpose method for designing molecules with this type of behavior. In this work, we describe and demonstrate a process for the rational design of DNA sequences with prescribed non-orthogonal binding behavior. This process makes use of a model that represents large sets of non-orthogonal DNA sequences using fixed-length binary strings, and estimates the differential binding affinity between pairs of sequences through the Hamming distance between their corresponding binary strings. The real-world applicability of this model is supported by simulations and some experimental data. We then select two previously described systems of molecular computation involving non-orthogonal interactions, and apply our sequence design process to implement them using DNA strand displacement. Our simulated results on these two systems demonstrate both digital and analog computation. We hope that this work motivates the development and implementation of new computational paradigms based on non-orthogonal binding
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