Duals of Feynman integrals. Part I. Differential equations

We elucidate the vector space (twisted relative cohomology) that is Poincaré dual to the vector space of Feynman integrals (twisted cohomology) in general spacetime dimension. The pairing between these spaces — an algebraic invariant called the intersection number — extracts integral coefficients for a minimal basis, bypassing the generation of integration-by-parts identities. Dual forms turn out to be much simpler than their Feynman counterparts: they are supported on maximal cuts of various sub-topologies (boundaries). Thus, they provide a systematic approach to generalized unitarity, the reconstruction of amplitudes from on-shell data. In this paper, we introduce the idea of dual forms and study their mathematical structures. As an application, we derive compact differential equations satisfied by arbitrary one-loop integrals in non-integer spacetime dimension. A second paper of this series will detail intersection pairings and their use to extract integral coefficients.

Stability of symmetric powers of vector bundles of rank two with even degree on a curve

This paper treats the strict semi-stability of the symmetric powers [Formula: see text] of a stable vector bundle [Formula: see text] of rank [Formula: see text] with even degree on a smooth projective curve [Formula: see text] of genus [Formula: see text]. The strict semi-stability of [Formula: see text] is equivalent to the orthogonality of [Formula: see text] or the existence of a bisection on the ruled surface [Formula: see text] whose self-intersection number is zero. A relation between the two interpretations is investigated in this paper through elementary transformations. This paper also gives a classification of [Formula: see text] with strictly semi-stable [Formula: see text]. Moreover, it is shown that when [Formula: see text] is stable, every symmetric power [Formula: see text] is stable for all but a finite number of [Formula: see text] in the moduli of stable vector bundles of rank [Formula: see text] with fixed determinant of even degree on [Formula: see text].

The impacts of road traffic on urban air quality in Jinan based GWR and remote sensing

Traffic congestion and smog are hot topics in recent years. This study analyzes the impacts of road traffic characteristic parameters on urban air quality quantitatively based on aerosol optical thickness (AOD) and geographical weighted regression (GWR) models, including the road network density, road area occupancy, intersection number, and bus network density as main factors. There are some major research findings. Firstly, there exists a strong positive correlation between the peak congestion delay index (PCDI) and air quality, the correlation has R2 values of up to 0.4962 (R 0.70). Secondly, GWR refines the local spatial changes in the AOD and the road parameters, and the correlation R2 based GWR model all above 0.6. The correlation between AOD and the road area occupancy was the highest, and the correlations with the bus network density and the intersections number were higher than that with the road network density. Thus, bus route planning, bus emission reduction, road network planning, and signal timing (at intersections) have a greater impact on air quality than other policy, especially in areas with traffic jams. The results of this study could provide theoretical support for traffic planning and traffic control, and is promising in practice.

AN ALGORITHM OF COMPUTING COHOMOLOGY INTERSECTION NUMBER OF HYPERGEOMETRIC INTEGRALS

We show that the cohomology intersection number of a twisted Gauss–Manin connection with regularization condition is a rational function. As an application, we obtain a new quadratic relation associated to period integrals of a certain family of K3 surfaces.

Book Embedding of Infinite Family ((2h+3 2))-Crossing-Critical Graphs for h=1 with Rational Average Degree r∈(3.5,4)

A principal tool used in construction of crossing-critical graphs are tiles. In the tile concept, tiles can be arranged by gluing one tile to another in a linear or circular fashion. The series of tiles with circular fashion form an infinite graph family. In this way, the intersection number of this family of graphs can be determined. In this research, has been formed an infinite family graphs Q_((1,s,b) ) (n) with average degree r between 3.5 and 4. The graph formed by gluing together many copies of the tile P_((1,s,b) ) in circular fashion, where the tile P_((1,s,b) ) consist of two identical pieces of tile. And then, the graph embedded into the book to determine the pagenumber that can be formed. When embed graph into book, the vertices are put on a line called the spine and the edges are put on half-planes called the pages. The results obtained show that the graph Q_((1,s,b) ) (n) has 10-crossing-critical and book embedding of graph has 4-page book.

On the relation between action and linking

<p style='text-indent:20px;'>We introduce numerical invariants of contact forms in dimension three and use asymptotic cycles to estimate them. As a consequence, we prove a version for Anosov Reeb flows of results due to Hutchings and Weiler on mean actions of periodic points. The main tool is the Action-Linking Lemma, expressing the contact area of a surface bounded by periodic orbits as the Liouville average of the asymptotic intersection number of most trajectories with the surface.</p>