We revisit the quantization of U(1) holonomy algebras using the abelian C algebra based techniques which form the mathematical underpinnings of current efforts to construct loop quantum gravity. In particular, we clarify the role of “smeared loops” and of Poincare invariance in the construction of Fock representations of these algebras. This enables us to critically re-examine early pioneering ...

A loop whose inner mappings are automorphisms is an automorphic loop (or A-loop). We characterize commutative (A-)loops with middle nucleus of index 2 and solve the isomorphism problem. Using this characterization and certain central extensions based on trilinear forms, we construct several classes of commutative A-loops of order a power of 2. We initiate the classification of commutative A-loo...

We construct and study the conformal loop ensembles CLE(κ), defined for 8/3 ≤ κ ≤ 8, using branching variants of SLE(κ) called exploration trees. The CLE(κ) are random collections of countably many loops in a planar domain that are characterized by certain conformal invariance and Markov properties. We conjecture that they are the scaling limits of various random loop models from statistical ph...

We construct the Bismut-Chern character form associated to a complex vector bundle with connection over a smooth manifold in the framework of supersymmetric quantum field theories developed by Stolz and Teichner [ST07]. We show that this differential form comes up via a loop-deloop process when one goes from 1|1D theory over a manifold to 0|1D theory over its loop space by crossing with the sta...

The main function of parallelizing compilers is to analyze sequential programs, in particular the loop structure, to detect hidden parallelism and automatically restructure sequential programs into parallel subtasks that are executed on a multiprocessor. This article describes the design and implementation of an efficient parallelizing compiler to parallelize loops and achieve high speedup rate...

The FALC-Loop web server provides an online interface for protein loop modeling by employing an ab initio loop modeling method called FALC (fragment assembly and analytical loop closure). The server may be used to construct loop regions in homology modeling, to refine unreliable loop regions in experimental structures or to model segments of designed sequences. The FALC method is computationall...

Nonlinear, adaptive, or otherwise complex control techniques are increasingly relied upon to ensure the safety of systems operating in uncertain environments. However, the nonlinearity of the resulting closed-loop system complicates verification that the system does in fact satisfy those requirements at all possible operating conditions. While analytical proof-based techniques and finite abstra...

A left Bol loop is a loop satisfying x(y(xz)) = (x(yx))z. The commutant of a loop is the set of all elements which commute with all elements of the loop. In a finite Bol loop of odd order or of order 2k, k odd, the commutant is a subloop. We investigate conditions under which the commutant of a Bol loop is not a subloop. In a finite Bol loop of order relatively prime to 3, the commutant generat...

A bstract We present a loop-by-loop method for computing the differential equations of Feynman integrals using recently developed dual form formalism. give explicit prescriptions fibration multi-loop forms. Then, we test our formalism on simple, but non-trivial, example: two-loop three-mass elliptic sunrise family integrals. obtain an ? -form equation within correct function space in sequence r...

A (1,-1)-matrix will be called a bent type matrix if each row and each column are bent sequences. A similar description can be found in Carlisle M. in which the authors use the properties of bent type matrices to construct a class of bent functions. In this paper we give a general method to construct bent type matrices and show that the bent sequence obtained from a bent type matrix is a genera...