Narrow Results

We consider paths in the square lattice and use a valuation called the winding number in order to exhibit some combinatorial properties on these paths. As a corollary, we obtain a characteristic property of non-crossing closed paths, generalizing in this way a result of Daurat and Nivat (2003) on the boundary properties of polyominoes concerning salient and reentrant points. Moreover we obtain a similar result for hexagonal lattices and show that there is no other regular lattice having that property.

In this paper, we study several closely related invariants associated to Dirac operators on odd-dimensional manifolds with boundary with an action of the compact group $H$ of isometries. In particular, the equality between equivariant winding numbers, equivariant spectral flow and equivariant Maslov indices is established. We also study equivariant $\eta$-invariants which play a fundamental role in the equivariant analog of Getzler’s spectral flow formula. As a consequence, we establish a relation between equivariant $\eta$-invariants and equivariant Maslov triple indices in the splitting of manifolds.

We have studied the topological expression of the reduced quantum trajectory using ϕ-mapping topological current theory and reduced density matrix theory. Moreover, we have proposed a new topological expression called the reduced quantum topological trajectory. Ignoring the coefficients of the winding number, we have found that the expression for the reduced quantum topological trajectory is the same for coherence as it is for decoherence. Under certain conditions, the vorticity of the reduced quantum topological trajectory is a topological invariant.

Recently, the measurement of the polar Kerr effect (PKE) in the quasi two-dimensional superconductor Sr₂RuO₄, which is motivated to observe the chirality of p_x + ip_y-wave pairing, has been reported. We clarify that the PKE has intrinsic and extrinsic (disorder-induced) origins. The extrinsic contribution would be dominant in the PKE experiment.

We derive a rigorous quantum formula called topological trajectory to describe orbital angular momentum (OAM) index based on linear momentum density of Laguerre–Gauss (L–G) light beam. By considering the correspondence between optics and quantum theory, we construct a coherent state from two L–G modes light beam. The light beams with fractional OAM are described by the coherent state. By making use of topological trajectory, we present the conditions that OAM index is fractional.

In this paper, we analyze the shapes of forward curves and yield curves that can be attained in the two-factor Vasicek model. We show how to partition the state space of the model, such that each partition is associated to a particular shape (normal, inverse, humped, etc.). The partitions and the corresponding shapes are determined by the winding number of a single curve with possible singularities and self-intersections, which can be constructed as the envelope of a family of lines. Building on these results, we classify possible transitions between term structure shapes, give results on attainability of shapes conditional on the level of the short rate, and propose a simple method to determine the relative frequency of different shapes of the forward curve and the yield curve.

In general, topological quantum field theory (TQFT) is studied in detail from the n-dimensional cobordism (nCob) to the Hilbert vector space. However, we study the TQFT in the different way in this paper, that is, the TQFT is studied from the Hilbert vector space to nCob. To do this, the theory called the ϕ-mapping topological current theory is used. The relation between the objects and zero points of the Hilbert states in the Hilbert vector space is studied in this frame. The relation between the morphism and topological current is revealed too.

The creations and manipulations of vortexes in ferroelectric materials with external stimuli are expected to be used in the design and fabrication of sensing materials and multifunctional electronic devices. In this work, we investigated the surface charge-induced multi-vortex evolution using the phase-field simulations in BiFeO₃. A combination of domain morphology, polarization distribution and winding number calculation was considered. The results show that vortex and anti-vortex exist simultaneously in pairs, and the total value of winding numbers is always 0. In addition, the minimum distance $\mathrm{\Delta }l$ between the surface charge regions is 9nm when the vortex domains are independent of each other. This work provides a reference for the manipulation of ferroelectric vortex induced by surface charges, which lays a theoretical foundation for the design and fabrication of high-density vortex memories.

The numerical range of a matrix is a compact convex set in the complex plane that contains the convex hull of the spectrum of the matrix. A key question is whether the origin belongs to the convex hull of the spectrum or the numerical range. This question arises in many computational and engineering applications such as the stability analysis of dynamic systems and the study of scattering matrices. Yet, there lacks an analytical criterion for this problem. Many relevant applications involve parameter-dependent matrices. In this paper, we provide a simple sufficient analytical criterion for the zero-inclusion problem, for a loop of parameter-dependent matrices. We prove the following theorem: Let $A:[0,1]\to GL(n,ℂ)$ (the set of $n\times n$ invertible complex matrices) be continuous with $A(0)=A(1)$, thus the winding number of $detA$ is well defined. If the winding number is not divisible by n, then the origin belongs to the convex hull of the spectrum, thus the numerical range of $A(\varphi )$ for some $\varphi \in [0,1]$. Our criterion may find potential applications in physics, engineering, and computation.

For generic maps from compact surfaces with boundary into the plane we develop an explicit algorithm for minimizing both the number of cusps and the number of components of the singular locus. More precisely, we minimize among maps with fixed boundary conditions and prescribed singular pattern, by which we mean the combinatorial information of how the 1-dimensional singular locus meets the boundary. Each step of our algorithm modifies the given map only locally by either creating or eliminating a pair of cusps. We show that the number of cusps is an invariant modulo 2 and can be reduced to at most one, and we compute the minimal number of components of the singular locus in terms of the prescribed data. Applications include a discussion of pseudo-immersions as well as the computation of state sums in Banagl’s positive topological field theory.