83,762 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
Sign-rank can increase under intersection
https://arxiv.org/abs/1903.00544Accepted manuscrip
Minimal genera of open 4-manifolds
We study exotic smoothings of open 4-manifolds using the minimal genus
function and its analog for end homology. While traditional techniques in open
4-manifold smoothing theory give no control of minimal genera, we make progress
by using the adjunction inequality for Stein surfaces. Smoothings can be
constructed with much more control of these genus functions than the compact
setting seems to allow. As an application, we expand the range of 4-manifolds
known to have exotic smoothings (up to diffeomorphism). For example, every
2-handlebody interior (possibly infinite or nonorientable) has an exotic
smoothing, and "most" have infinitely, or sometimes uncountably many,
distinguished by the genus function and admitting Stein structures when
orientable. Manifolds with 3-homology are also accessible. We investigate
topological submanifolds of smooth 4-manifolds. Every domain of holomorphy
(Stein open subset) in the complex plane is topologically isotopic to
uncountably many other diffeomorphism types of domains of holomorphy with the
same genus functions, or with varying but controlled genus functions.Comment: 30 pages, 1 figure. v3 is essentially the version published in
Geometry and Topology, obtained from v2 by major streamlining for
readability. Several new examples added since v2; see last paragraph of
introduction for detail
On biunimodular vectors for unitary matrices
A biunimodular vector of a unitary matrix is a vector v \in
\mathbb{T}^n\subset\bc^n such that as well. Over the
last 30 years, the sets of biunimodular vectors for Fourier matrices have been
the object of extensive research in various areas of mathematics and applied
sciences. Here, we broaden this basic harmonic analysis perspective and extend
the search for biunimodular vectors to arbitrary unitary matrices. This search
can be motivated in various ways. The main motivation is provided by the fact,
that the existence of biunimodular vectors for an arbitrary unitary matrix
allows for a natural understanding of the structure of all unitary matrices
A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux
Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1
whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We
present a unifying perspective on ASMs and other combinatorial objects by
studying a certain tetrahedral poset and its subposets. We prove the order
ideals of these subposets are in bijection with a variety of interesting
combinatorial objects, including ASMs, totally symmetric self-complementary
plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux,
Catalan objects, tournaments, and totally symmetric plane partitions. We prove
product formulas counting these order ideals and give the rank generating
function of some of the corresponding lattices of order ideals. We also prove
an expansion of the tournament generating function as a sum over TSSCPPs. This
result is analogous to a result of Robbins and Rumsey expanding the tournament
generating function as a sum over alternating sign matrices.Comment: 24 pages, 23 figures, full published version of arXiv:0905.449
Convex Rank Tests and Semigraphoids
Convex rank tests are partitions of the symmetric group which have desirable
geometric properties. The statistical tests defined by such partitions involve
counting all permutations in the equivalence classes. Each class consists of
the linear extensions of a partially ordered set specified by data. Our methods
refine existing rank tests of non-parametric statistics, such as the sign test
and the runs test, and are useful for exploratory analysis of ordinal data. We
establish a bijection between convex rank tests and probabilistic conditional
independence structures known as semigraphoids. The subclass of submodular rank
tests is derived from faces of the cone of submodular functions, or from
Minkowski summands of the permutohedron. We enumerate all small instances of
such rank tests. Of particular interest are graphical tests, which correspond
to both graphical models and to graph associahedra
On sets of terms with a given intersection type
We are interested in how much of the structure of a strongly normalizable
lambda term is captured by its intersection types and how much all the terms of
a given type have in common. In this note we consider the theory BCD
(Barendregt, Coppo, and Dezani) of intersection types without the top element.
We show: for each strongly normalizable lambda term M, with beta-eta normal
form N, there exists an intersection type A such that, in BCD, N is the unique
beta-eta normal term of type A. A similar result holds for finite sets of
strongly normalizable terms for each intersection type A if the set of all
closed terms M such that, in BCD, M has type A, is infinite then, when closed
under beta-eta conversion, this set forms an adaquate numeral system for
untyped lambda calculus. A number of related results are also proved
Box Graphs and Singular Fibers
We determine the higher codimension fibers of elliptically fibered Calabi-Yau
fourfolds with section by studying the three-dimensional N=2 supersymmetric
gauge theory with matter which describes the low energy effective theory of
M-theory compactified on the associated Weierstrass model, a singular model of
the fourfold. Each phase of the Coulomb branch of this theory corresponds to a
particular resolution of the Weierstrass model, and we show that these have a
concise description in terms of decorated box graphs based on the
representation graph of the matter multiplets, or alternatively by a class of
convex paths on said graph. Transitions between phases have a simple
interpretation as `flopping' of the path, and in the geometry correspond to
actual flop transitions. This description of the phases enables us to enumerate
and determine the entire network between them, with various matter
representations for all reductive Lie groups. Furthermore, we observe that each
network of phases carries the structure of a (quasi-)minuscule representation
of a specific Lie algebra. Interpreted from a geometric point of view, this
analysis determines the generators of the cone of effective curves as well as
the network of flop transitions between crepant resolutions of singular
elliptic Calabi-Yau fourfolds. From the box graphs we determine all fiber types
in codimensions two and three, and we find new, non-Kodaira, fiber types for
E_6, E_7 and E_8.Comment: 107 pages, 44 figures, v2: added case of E7 monodromy-reduced fiber
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