578 research outputs found
On log concavity for order-preserving and order-non-reversing maps of partial orders
Stanley used the Aleksandrov-Fenchel inequalities from the theory of nixed volumes to prove the following result. Let P be a partially ordered set with n elements, and let x β P. If Ni* is the number of linear extensions , β : P + (1 , 2,...,n) satisfying β (x) = i, then the sequence N*1,β¦,N*n is log concave (and therefore unimodal). Here the analogous results for both order-preserving and order-non-reversing maps are proved using an explicit injection. Further, if vc is the number of order-preserving maps of P into a chain of length c, then vc is shown to be 1-og concave, and the corresponding result is established for order-non-reversing maps
Monotonic functions of finite posets
Classes of monotone functions from finite posets to chains are studied. These include order-preserving and strict order-preserving maps. When the maps are required to be bijective they are called linear extensions. Techniques for handling the first two types are closely related; whereas for linear extensions quite distinct methods are often necessary, which may yield results for order-preserving injections.
First, many new fundamental properties and inequalities of a combinatorial nature are established for these maps. Quantities considered here include the range, height, depth and cardinalities of subposets. In particular we study convexity in posets and similarly pre-images of intervals in chains. The problem of extending a map defined on a subposet to the whole poset is discussed.
We investigate positive correlation inequalities, having implications for the complexity of sorting algorithms. These express monotonicity properties for probabilities concerning sets of relations in posets. New proofs are given for existing inequalities and we obtain corresponding negative correlations, along with a generalization of the xyz inequality. The proofs involve inequalities in distributive lattices, some of which arose in physics. A characterization is given for posets satisfying necessary conditions for correlation properties under linear extensions.
A log concavity type inequality is proved for the number of strict or non-strict order-preserving maps of an element. We define an explicit injection whereas the bijective case is proved in the literature using inequalities from the theory of mixed volumes.
These results motivate a further group of such inequalities. But now we count numbers of strict or non-strict order-preserving maps of subposets to varying heights in the chain.
Lastly we consider the computational cost of producing certain posets which can be associated with classical sorting and selection problems. A lower bound technique is derived for this complexity, incorporating either a new distributive lattice inequality, or the log concavity inequalities
Random structures for partially ordered sets
This thesis is presented in two parts. In the first part, we study a family of models
of random partial orders, called classical sequential growth models, introduced by
Rideout and Sorkin as possible models of discrete space-time. We analyse a particular
model, called a random binary growth model, and show that the random partial
order produced by this model almost surely has infinite dimension. We also give
estimates on the size of the largest vertex incomparable to a particular element of
the partial order. We show that there is some positive probability that the random
partial order does not contain a particular subposet. This contrasts with other existing
models of partial orders. We also study "continuum limits" of sequences of
classical sequential growth models. We prove results on the structure of these limits
when they exist, highlighting a deficiency of these models as models of space-time.
In the second part of the thesis, we prove some correlation inequalities for mappings
of rooted trees into complete trees. For T a rooted tree we can define the proportion
of the total number of embeddings of T into a complete binary tree that map the
root of T to the root of the complete binary tree. A theorem of Kubicki, Lehel and
Morayne states that, for two binary trees with one a subposet of the other, this
proportion is larger for the larger tree. They conjecture that the same is true for
two arbitrary trees with one a subposet of the other. We disprove this conjecture
by analysing the asymptotics of this proportion for large complete binary trees.
We show that the theorem of Kubicki, Lehel and Morayne can be thought of as a
correlation inequality which enables us to generalise their result in other directions
Properties of Noncommutative Renyi and Augustin Information
The scaled R\'enyi information plays a significant role in evaluating the
performance of information processing tasks by virtue of its connection to the
error exponent analysis. In quantum information theory, there are three
generalizations of the classical R\'enyi divergence---the Petz's, sandwiched,
and log-Euclidean versions, that possess meaningful operational interpretation.
However, these scaled noncommutative R\'enyi informations are much less
explored compared with their classical counterpart, and lacking crucial
properties hinders applications of these quantities to refined performance
analysis. The goal of this paper is thus to analyze fundamental properties of
scaled R\'enyi information from a noncommutative measure-theoretic perspective.
Firstly, we prove the uniform equicontinuity for all three quantum versions of
R\'enyi information, hence it yields the joint continuity of these quantities
in the orders and priors. Secondly, we establish the concavity in the region of
for both Petz's and the sandwiched versions. This completes the
open questions raised by Holevo
[\href{https://ieeexplore.ieee.org/document/868501/}{\textit{IEEE
Trans.~Inf.~Theory}, \textbf{46}(6):2256--2261, 2000}], Mosonyi and Ogawa
[\href{https://doi.org/10.1007/s00220-017-2928-4/}{\textit{Commun.~Math.~Phys},
\textbf{355}(1):373--426, 2017}]. For the applications, we show that the strong
converse exponent in classical-quantum channel coding satisfies a minimax
identity. The established concavity is further employed to prove an entropic
duality between classical data compression with quantum side information and
classical-quantum channel coding, and a Fenchel duality in joint source-channel
coding with quantum side information in the forthcoming papers
Convexity and Liberation at Large Spin
We consider several aspects of unitary higher-dimensional conformal field
theories (CFTs). We first study massive deformations that trigger a flow to a
gapped phase. Deep inelastic scattering in the gapped phase leads to a
convexity property of dimensions of spinning operators of the original CFT. We
further investigate the dimensions of spinning operators via the crossing
equations in the light-cone limit. We find that, in a sense, CFTs become free
at large spin and 1/s is a weak coupling parameter. The spectrum of CFTs enjoys
additivity: if two twists tau_1, tau_2 appear in the spectrum, there are
operators whose twists are arbitrarily close to tau_1+tau_2. We characterize
how tau_1+tau_2 is approached at large spin by solving the crossing equations
analytically. We find the precise form of the leading correction, including the
prefactor. We compare with examples where these observables were computed in
perturbation theory, or via gauge-gravity duality, and find complete agreement.
The crossing equations show that certain operators have a convex spectrum in
twist space. We also observe a connection between convexity and the ratio of
dimension to charge. Applications include the 3d Ising model, theories with a
gravity dual, SCFTs, and patterns of higher spin symmetry breaking.Comment: 61 pages, 13 figures. v2: added reference and minor correctio
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