2,394 research outputs found
On the Positivity Problem for Simple Linear Recurrence Sequences
Given a linear recurrence sequence (LRS) over the integers, the Positivity
Problem} asks whether all terms of the sequence are positive. We show that, for
simple LRS (those whose characteristic polynomial has no repeated roots) of
order 9 or less, Positivity is decidable, with complexity in the Counting
Hierarchy.Comment: arXiv admin note: substantial text overlap with arXiv:1307.277
A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces
Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating
4-dimensional super-symmetric gauge theory for a gauge group G with certain
2-dimensional conformal field theory. This conjecture implies the existence of
certain structures on the (equivariant) intersection cohomology of the
Uhlenbeck partial compactification of the moduli space of framed G-bundles on
P^2. More precisely, it predicts the existence of an action of the
corresponding W-algebra on the above cohomology, satisfying certain properties.
We propose a "finite analog" of the (above corollary of the) AGT conjecture.
Namely, we replace the Uhlenbeck space with the space of based quasi-maps from
P^1 to any partial flag variety G/P of G and conjecture that its equivariant
intersection cohomology carries an action of the finite W-algebra U(g,e)
associated with the principal nilpotent element in the Lie algebra of the Levi
subgroup of P; this action is expected to satisfy some list of natural
properties. This conjecture generalizes the main result of arXiv:math/0401409
when P is the Borel subgroup. We prove our conjecture for G=GL(N), using the
works of Brundan and Kleshchev interpreting the algebra U(g,e) in terms of
certain shifted Yangians.Comment: minor change
Proposal to demonstrate the non-locality of Bohmian mechanics with entangled photons
Bohmian mechanics reproduces all statistical predictions of quantum
mechanics, which ensures that entanglement cannot be used for superluminal
signaling. However, individual Bohmian particles can experience superluminal
influences. We propose to illustrate this point using a double double-slit
setup with path-entangled photons. The Bohmian velocity field for one of the
photons can be measured using a recently demonstrated weak-measurement
technique. The found velocities strongly depend on the value of a phase shift
that is applied to the other photon, potentially at spacelike separation.Comment: 6 pages, 4 figure
Torus fibrations and localization of index II
We give a framework of localization for the index of a Dirac-type operator on
an open manifold. Suppose the open manifold has a compact subset whose
complement is covered by a family of finitely many open subsets, each of which
has a structure of the total space of a torus bundle. Under an acyclic
condition we define the index of the Dirac-type operator by using the
Witten-type deformation, and show that the index has several properties, such
as excision property and a product formula. In particular, we show that the
index is localized on the compact set.Comment: 47 pages, 2 figures. To appear in Communications in Mathematical
Physic
Surface Operators in N=2 Abelian Gauge Theory
We generalise the analysis in [arXiv:0904.1744] to superspace, and explicitly
prove that for any embedding of surface operators in a general, twisted N=2
pure abelian theory on an arbitrary four-manifold, the parameters transform
naturally under the SL(2,Z) duality of the theory. However, for
nontrivially-embedded surface operators, exact S-duality holds if and only if
the "quantum" parameter effectively vanishes, while the overall SL(2,Z) duality
holds up to a c-number at most, regardless. Nevertheless, this observation sets
the stage for a physical proof of a remarkable mathematical result by
Kronheimer and Mrowka--that expresses a "ramified" analog of the Donaldson
invariants solely in terms of the ordinary Donaldson invariants--which, will
appear, among other things, in forthcoming work. As a prelude to that, the
effective interaction on the corresponding u-plane will be computed. In
addition, the dependence on second Stiefel-Whitney classes and the appearance
of a Spin^c structure in the associated low-energy Seiberg-Witten theory with
surface operators, will also be demonstrated. In the process, we will stumble
upon an interesting phase factor that is otherwise absent in the "unramified"
case.Comment: 46 pages. Minor refinemen
Termination of Triangular Integer Loops is Decidable
We consider the problem whether termination of affine integer loops is
decidable. Since Tiwari conjectured decidability in 2004, only special cases
have been solved. We complement this work by proving decidability for the case
that the update matrix is triangular.Comment: Full version (with proofs) of a paper published in the Proceedings of
the 31st International Conference on Computer Aided Verification (CAV '19),
New York, NY, USA, Lecture Notes in Computer Science, Springer-Verlag, 201
Spatial Statistical Data Fusion (SSDF)
As remote sensing for scientific purposes has transitioned from an experimental technology to an operational one, the selection of instruments has become more coordinated, so that the scientific community can exploit complementary measurements. However, tech nological and scientific heterogeneity across devices means that the statistical characteristics of the data they collect are different. The challenge addressed here is how to combine heterogeneous remote sensing data sets in a way that yields optimal statistical estimates of the underlying geophysical field, and provides rigorous uncertainty measures for those estimates. Different remote sensing data sets may have different spatial resolutions, different measurement error biases and variances, and other disparate characteristics. A state-of-the-art spatial statistical model was used to relate the true, but not directly observed, geophysical field to noisy, spatial aggregates observed by remote sensing instruments. The spatial covariances of the true field and the covariances of the true field with the observations were modeled. The observations are spatial averages of the true field values, over pixels, with different measurement noise superimposed. A kriging framework is used to infer optimal (minimum mean squared error and unbiased) estimates of the true field at point locations from pixel-level, noisy observations. A key feature of the spatial statistical model is the spatial mixed effects model that underlies it. The approach models the spatial covariance function of the underlying field using linear combinations of basis functions of fixed size. Approaches based on kriging require the inversion of very large spatial covariance matrices, and this is usually done by making simplifying assumptions about spatial covariance structure that simply do not hold for geophysical variables. In contrast, this method does not require these assumptions, and is also computationally much faster. This method is fundamentally different than other approaches to data fusion for remote sensing data because it is inferential rather than merely descriptive. All approaches combine data in a way that minimizes some specified loss function. Most of these are more or less ad hoc criteria based on what looks good to the eye, or some criteria that relate only to the data at hand
Larger Corner-Free Sets from Combinatorial Degenerations
There is a large and important collection of Ramsey-type combinatorial
problems, closely related to central problems in complexity theory, that can be
formulated in terms of the asymptotic growth of the size of the maximum
independent sets in powers of a fixed small (directed or undirected)
hypergraph, also called the Shannon capacity. An important instance of this is
the corner problem studied in the context of multiparty communication
complexity in the Number On the Forehead (NOF) model. Versions of this problem
and the NOF connection have seen much interest (and progress) in recent works
of Linial, Pitassi and Shraibman (ITCS 2019) and Linial and Shraibman (CCC
2021).
We introduce and study a general algebraic method for lower bounding the
Shannon capacity of directed hypergraphs via combinatorial degenerations, a
combinatorial kind of "approximation" of subgraphs that originates from the
study of matrix multiplication in algebraic complexity theory (and which play
an important role there) but which we use in a novel way.
Using the combinatorial degeneration method, we make progress on the corner
problem by explicitly constructing a corner-free subset in
of size , which improves the previous lower bound
of Linial, Pitassi and Shraibman (ITCS 2019) and which gets us
closer to the best upper bound . Our new construction of
corner-free sets implies an improved NOF protocol for the Eval problem. In the
Eval problem over a group , three players need to determine whether their
inputs sum to zero. We find that the NOF communication
complexity of the Eval problem over is at most ,
which improves the previous upper bound .Comment: A short version of this paper will appear in the proceedings of ITCS
2022. This paper improves results that appeared in arxiv:2104.01130v
L^2 torsion without the determinant class condition and extended L^2 cohomology
We associate determinant lines to objects of the extended abelian category
built out of a von Neumann category with a trace. Using this we suggest
constructions of the combinatorial and the analytic L^2 torsions which, unlike
the work of the previous authors, requires no additional assumptions; in
particular we do not impose the determinant class condition. The resulting
torsions are elements of the determinant line of the extended L^2 cohomology.
Under the determinant class assumption the L^2 torsions of this paper
specialize to the invariants studied in our previous work. Applying a recent
theorem of D. Burghelea, L. Friedlander and T. Kappeler we obtain a Cheeger -
Muller type theorem stating the equality between the combinatorial and the
analytic L^2 torsions.Comment: 39 page
Essential self-adjointness of magnetic Schr\"odinger operators on locally finite graphs
We give sufficient conditions for essential self-adjointness of magnetic
Schr\"odinger operators on locally finite graphs. Two of the main theorems of
the present paper generalize recent results of Torki-Hamza.Comment: 14 pages; The present version differs from the original version as
follows: the ordering of presentation has been modified in several places,
more details have been provided in several places, some notations have been
changed, two examples have been added, and several new references have been
inserted. The final version of this preprint will appear in Integral
Equations and Operator Theor
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