16 research outputs found
Spectra of infinite graphs via freeness with amalgamation
We use tools from free probability to study the spectra of Hermitian
operators on infinite graphs. Special attention is devoted to universal
covering trees of finite graphs. For operators on these graphs we derive a new
variational formula for the spectral radius and provide new proofs of results
due to Sunada and Aomoto using free probability.
With the goal of extending the applicability of free probability techniques
beyond universal covering trees, we introduce a new combinatorial product
operation on graphs and show that, in the non-commutative probability context,
it corresponds to the notion of freeness with amalgamation. We show that Cayley
graphs of amalgamated free products of groups, as well as universal covering
trees, can be constructed using our graph product.Comment: 45 pages, 4 figures, comments welcome; v2:Key references and new
results have been included. In particular, we realized that a result on the
band structure of the spectrum, which we believed to be new, had actually
appeared in previous literature (see the bibliographic note just before Sec
1.1); v3:Minor edits, added discussions, generalized Thm 1.6 from adjacency
matrices to graph Jacobi matrice
Achieving Starvation-Freedom with Greater Concurrency in Multi-Version Object-based Transactional Memory Systems
To utilize the multi-core processors properly concurrent programming is
needed. Concurrency control is the main challenge while designing a correct and
efficient concurrent program. Software Transactional Memory Systems (STMs)
provides ease of multithreading to the programmer without worrying about
concurrency issues such as deadlock, livelock, priority inversion, etc. Most of
the STMs works on read-write operations known as RWSTMs. Some STMs work at
high-level operations and ensure greater concurrency than RWSTMs. Such STMs are
known as Object-Based STMs (OSTMs). The transactions of OSTMs can return commit
or abort. Aborted OSTMs transactions retry. But in the current setting of
OSTMs, transactions may starve. So, we proposed a Starvation-Free OSTM
(SF-OSTM) which ensures starvation-freedom in object based STM systems while
satisfying the correctness criteria as co-opacity. Databases, RWSTMs and OSTMs
say that maintaining multiple versions corresponding to each key of transaction
reduces the number of aborts and improves the throughput. So, to achieve
greater concurrency, we proposed Starvation-Free Multi-Version OSTM (SF-MVOSTM)
which ensures starvation-freedom while storing multiple versions corresponding
to each key and satisfies the correctness criteria such as local opacity. To
show the performance benefits, We implemented three variants of SF-MVOSTM
(SF-MVOSTM, SF-MVOSTM-GC and SF-KOSTM) and compared it with state-of-the-art
STMs.Comment: 68 pages, 24 figures. arXiv admin note: text overlap with
arXiv:1709.0103
Sums and differences of correlated random sets
Many fundamental questions in additive number theory (such as Goldbach's
conjecture, Fermat's last theorem, and the Twin Primes conjecture) can be
expressed in the language of sum and difference sets. As a typical pair of
elements contributes one sum and two differences, we expect that for a finite set . However, in 2006 Martin and O'Bryant showed that a
positive proportion of subsets of are sum-dominant, and Zhao
later showed that this proportion converges to a positive limit as . Related problems, such as constructing explicit families of
sum-dominant sets, computing the value of the limiting proportion, and
investigating the behavior as the probability of including a given element in
to go to zero, have been analyzed extensively.
We consider many of these problems in a more general setting. Instead of just
one set , we study sums and differences of pairs of \emph{correlated} sets
. Specifically, we place each element in with
probability , while goes in with probability if
and probability if . If , we
call the pair a \emph{sum-dominant -pair}. We prove
that for any fixed in , is a
sum-dominant -pair with positive probability, and show that
this probability approaches a limit . Furthermore, we show that
the limit function is continuous. We also investigate what
happens as decays with , generalizing results of Hegarty-Miller on phase
transitions. Finally, we find the smallest sizes of MSTD pairs.Comment: Version 1.0, 19 pages. Keywords: More Sum Than Difference sets,
correlated random variables, phase transitio
Sets Characterized by Missing Sums and Differences in Dilating Polytopes
A sum-dominant set is a finite set of integers such that .
As a typical pair of elements contributes one sum and two differences, we
expect sum-dominant sets to be rare in some sense. In 2006, however, Martin and
O'Bryant showed that the proportion of sum-dominant subsets of
is bounded below by a positive constant as . Hegarty then extended
their work and showed that for any prescribed , the
proportion of subsets of that are missing
exactly sums in and exactly differences in
also remains positive in the limit.
We consider the following question: are such sets, characterized by their
sums and differences, similarly ubiquitous in higher dimensional spaces? We
generalize the integers in a growing interval to the lattice points in a
dilating polytope. Specifically, let be a polytope in with
vertices in , and let now denote the proportion of
subsets of that are missing exactly sums in and
exactly differences in . As it turns out, the geometry of
has a significant effect on the limiting behavior of . We define
a geometric characteristic of polytopes called local point symmetry, and show
that is bounded below by a positive constant as if
and only if is locally point symmetric. We further show that the proportion
of subsets in that are missing exactly sums and at least
differences remains positive in the limit, independent of the geometry of .
A direct corollary of these results is that if is additionally point
symmetric, the proportion of sum-dominant subsets of also remains
positive in the limit.Comment: Version 1.1, 23 pages, 7 pages, fixed some typo
Pseudospectral Shattering, the Sign Function, and Diagonalization in Nearly Matrix Multiplication Time
We exhibit a randomized algorithm which given a square complex
matrix with and , computes with high probability
invertible and diagonal such that and
in
arithmetic operations on a floating point machine with bits of precision. Here is the number of arithmetic
operations required to multiply two complex matrices numerically
stably, with for every , where
is the exponent of matrix multiplication. The algorithm is a variant
of the spectral bisection algorithm in numerical linear algebra (Beavers and
Denman, 1974). This running time is optimal up to polylogarithmic factors, in
the sense that verifying that a given similarity diagonalizes a matrix requires
at least matrix multiplication time. It significantly improves best previously
provable running times of arithmetic operations for
diagonalization of general matrices (Armentano et al., 2018), and (w.r.t.
dependence on ) arithmetic operations for Hermitian matrices
(Parlett, 1998).
The proof rests on two new ingredients. (1) We show that adding a small
complex Gaussian perturbation to any matrix splits its pseudospectrum into
small well-separated components. This implies that the eigenvalues of the
perturbation have a large minimum gap, a property of independent interest in
random matrix theory. (2) We rigorously analyze Roberts' Newton iteration
method for computing the matrix sign function in finite arithmetic, itself an
open problem in numerical analysis since at least 1986. This is achieved by
controlling the evolution the iterates' pseudospectra using a carefully chosen
sequence of shrinking contour integrals in the complex plane.Comment: 78 pages, 3 figures, comments welcome. Slightly edited intro from
previous version + explicit statement of forward error Theorem (Corolary
1.7). Minor corrections mad