Julia Knihs, Jeanette Patel, Joshua M. Sabloff
et al.
We define a new quantity, the Euler-normalized non-orientable genus, to connect a variety of ideas in the theory of non-orientable surfaces bounded by knots. This quantity is used to reframe non-orientable slice-torus bounds on the non-orientable $4$-genus, to bound below the Turaev genus as a measure of distance to an alternating knot, and to understand gaps between the $3$- and $4$-dimensional non-orientable genera of pretzel knots. Further, we make connections to essential surfaces in knot complements and the Slope Conjecture.
Douglas F. Copatti, Giuliano G. La Guardia, Waldir S. Soares
et al.
In this paper, we construct several new quantum Floquet codes on compact, orientable, as well as non-orientable surfaces. In order to obtain such codes, we identify these surfaces with hyperbolic polygons and examine hyperbolic semi-regular tessellations on such surfaces. The method of construction presented here generalizes similar constructions concerning hyperbolic Floquet codes on connected and compact surfaces with genus $g \geq 2$. A performance analysis and an investigation of the asymptotic behavior of these codes are also presented.
Murilo D. Forlevesi, Edson Denis Leonel, Emanuel F. de Lima
We explore the possibility of forming a oriented polar molecule directly from a pair of colliding atoms. The process comprises the photoassociation and vibrational stabilization along with the molecular orientation. These processes are driven by a single time-dependent, linearly polarized control field and proceeds entirely within the electronic ground state, leveraging the presence of a permanent dipole moment. The control field is found by means of an optimal quantum control algorithm with a single target observable given by the restriction of the orientation operator on a subset of bound levels. We consider a rovibrational model system for the collision of O + H atoms and solve directly the time-dependent Schrodinger equation. We show that the optimized field is capable of enhancing the molecular orientation already induced by the photoassociation and vibrational stabilization thus yielding oriented polar molecules that can be useful for many applications.
Spontaneous self-assembly of hard convex polyhedra are known to form orientationally disordered crystalline phases, where particle orientations do not follow the same pattern as the positional arrangement of the crystal. A distinct type of orientational phase with discrete rotational mobility has been reported in hard particle systems. In this paper, we present a new analysis method for characterizing orientational phase of a crystal, which is based on algorithmic detection of unique orientations. Using this method we collected complete statistics of discrete orientations along the Monte Carlo simulation trajectories and observed that particles were equally partitioned among them, with specific values of pairwise orientational differences. These features remained constant across the pressure range and did not depend on rotational mobility. The discrete mobility was characteristic of a distinct equilibrium thermodynamic phase, qualitatively different from the freely rotating plastic phase with continuous orientations. The high pressure behavior with frozen particle orientations was part of that the same description and not a non-equilibrium arrested state. We introduced a precise notion of orientational order and demonstrated that the system was maximally disordered at the level of unit cell, even though individual particles could only take few discrete orientations. We report the existence of this phase in five polyhedral shapes and in systematically curated shape families constructed around two of them. The symmetry mismatch between the particle and the crystallographic point groups was found to be a predictive indicator for the occurrence of this phase.
The construction of the Varchenko matrix for hyperplane arrangements, first introduced by Alexandre Varchenko, extends naturally to oriented matroids. In this paper, we generalize the theorem of Gao and Zhang by proving that the Varchenko matrix of an oriented matroid has a diagonal form if and only if the pseudohyperplane arrangement corresponding to the oriented matroid is in semigeneral position, i.e. it does not contain a degeneracy. Furthermore, we show that the Varchenko matrix of a pseudoline arrangement has a block diagonal form. This also provides an alternative combinatorial proof for the Varchenko matrix determinant formula in dimension two.
Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this paper, we calculate the non-orientable 4-genus of knots with crossing number 10.
Jose Ceniceros, Indu R. Churchill, Mohamed Elhamdadi
et al.
We extend the quandle cocycle invariant to oriented singular knots and links using algebraic structures called \emph{oriented singquandles} and assigning weight functions at both regular and singular crossings. This invariant coincides with the classical cocycle invariant for classical knots but provides extra information about singular knots and links. The new invariant distinguishes the singular granny knot from the singular square knot.
Varchenko introduced in 1993 a distance function on the chambers of a hyperplane arrangement that gave rise to a determinant whose entry in position $(C, D)$ is the distance between the chambers $C$ and $D$, and computed that determinant. In 2017, Aguiar and Mahajan provided a generalization of that distance function, and computed the corresponding determinant. This article extends their distance function to the topes of an oriented matroid, and computes the determinant thus defined. Oriented matroids have the nice property to be abstractions of some mathematical structures including hyperplane and sphere arrangements, polytopes, directed graphs, and even chirality in molecular chemistry. Independently and with another method, Hochstättler and Welker also computed in 2019 the same determinant.
Arrangement of interacting particles on a sphere is historically a well known problem, however, ordering of particles with anisotropic interaction, such as the dipole-dipole interaction, has remained unexplored. We solve the orientational ordering of point dipoles on a sphere with fixed positional order with numerical minimization of interaction energy and analyze stable configurations depending on their symmetry and degree of ordering. We find that a macrovortex is a generic ground state, with various discrete rotational symmetries for different system sizes, while higher energy metastable states are similar, but less ordered. We observe orientational phase transitions and hysteresis in response to changing external field both for the fixed sphere orientation with respect the field, as well as for a freely-rotating sphere. For the case of a freely rotating sphere, we also observe changes of the symmetry axis with increasing field strength.
Mireille Bousquet-Mélou, Andrew Elvey Price, Paul Zinn-Justin
We address the enumeration of planar 4-valent maps equipped with an Eulerian orientation by two different methods, and compare the solutions we thus obtain. With the first method we enumerate these orientations as well as a restricted class which we show to be in bijection with general Eulerian orientations. The second method, based on the work of Kostov, allows us to enumerate these 4-valent orientations with a weight on some vertices, corresponding to the six vertex model. We prove that this result generalises both results obtained using the first method, although the equivalence is not immediately clear.
Convolutional neural networks have many hyperparameters such as the filter size, number of filters, and pooling size, which require manual tuning. Though deep stacked structures are able to create multi-scale and hierarchical representations, manually fixed filter sizes limit the scale of representations that can be learned in a single convolutional layer. This paper introduces a new adaptive filter model that allows variable scale and orientation. The scale and orientation parameters of filters can be learned using back propagation. Therefore, in a single convolution layer, we can create filters of different scale and orientation that can adapt to small or large features and objects. The proposed model uses a relatively large base size (grid) for filters. In the grid, a differentiable function acts as an envelope for the filters. The envelope function guides effective filter scale and shape/orientation by masking the filter weights before the convolution. Therefore, only the weights in the envelope are updated during training. In this work, we employed a multivariate (2D) Gaussian as the envelope function and showed that it can grow, shrink, or rotate by updating its covariance matrix during back propagation training . We tested the new filter model on MNIST, MNIST-cluttered, and CIFAR-10 and compared the results with the networks that used conventional convolution layers. The results demonstrate that the new model can effectively learn and produce filters of different scales and orientations in a single layer. Moreover, the experiments show that the adaptive convolution layers perform equally; or better, especially when data includes objects of varying scale and noisy backgrounds.
The orientability problem in real Gromov-Witten theory is one of the fundamental hurdles to enumerating real curves. In this paper, we describe topological conditions on the target manifold which ensure that the uncompactified moduli spaces of real maps are orientable for all genera of and for all types of involutions on the domain. In contrast to the typical approaches to this problem, we do not compute the signs of any diffeomorphisms, but instead compare them. Many projective complete intersections, including the renowned quintic threefold, satisfy our topological conditions. Our main result yields real Gromov-Witten invariants of arbitrary genus for real symplectic manifolds that satisfy these conditions and have empty real locus and illustrates the significance of previously introduced moduli spaces of maps with crosscaps. We also apply it to study the orientability of the moduli spaces of real Hurwitz covers.
Hatim Hafiddi, Hicham Baidouri, Mahmoud Nassar
et al.
Today, service oriented systems need to be enhanced to sense and react to users context in order to provide a better user experience. To meet this requirement, Context-Aware Services (CAS) have emerged as an underling design and development paradigm for the development of context-aware systems. The fundamental challenges for such systems development are context-awareness management and service adaptation to the users context. To cope with such requirements, we propose a well designed architecture, named ACAS, to support the development of Context-Aware Service Oriented Systems (CASOS). This architecture relies on a set of context-awareness and CAS specifications and metamodels to enhance a core service, in service oriented systems, to be context-aware. This enhancement is fulfilled by the Aspect Adaptations Weaver (A2W) which, based on the Aspect Paradigm (AP) concepts, considers the services adaptations as aspects.
We report on optical orientation of Mn^2+ in bulk GaAs under application of weak longitudinal magnetic field (B <= 100 mT). The manganese spin polarization of 25% is directly evaluated using spin flip Raman scattering spectroscopy. The dynamical polarization of Mn^2+ occurs due to s-d exchange interaction with optically oriented conduction band electrons. Time-resolved photoluminescence uncovers nontrivial electron spin dynamics where the oriented Mn^2+ ions tend to stabilize the electron spin.