Cayley Structures Research Articles

The Cayley Cubic and Differential Equations

We define Cayley structures as a field of Cayley's ruled cubic surfaces over a four dimensional manifold and motivate their study by showing their similarity to indefinite conformal structures and their link to differential equations. In particular, for Cayley structures an extension of certain notions defined for indefinite conformal structures in dimension four are introduced, e.g., half-flatness, existence of a null foliation, ultra-half-flatness, an associated pair of second order ODEs, and a dispersionless Lax pair. After solving the equivalence problem we obtain the fundamental invariants, find the local generality of several classes of Cayley structures and give examples.

On Almost Complex Structures on Six-dimensional Products of Spheres

In this paper, we discuss almost complex structures on the sphere S6 and on the products of spheres S3 × S3, S1 × S5, and S2 × S4. We prove that all almost complex Cayley structures that naturally appear from their embeddings into the Cayley octave algebra ℂa are nonintegrable. We obtain expressions for the Nijenhuis tensor and the fundamental form 𝜔 for each gauge of the space ℂa and prove the nondegeneracy of the form d𝜔. We show that through each point of a fiber of the twistor bundle over S6, a one-parameter family of Cayley structures passes. We describe the set of U(2) ×U(2)- invariant Hermitian metrics on S3 × S3 and find estimates of the sectional sectional curvature. We consider the space of left-invariant, almost complex structures on S3 × S3 = SU(2) × SU(2) and prove the properties of left-invariant structures that yield the maximal value of the norm of the Nijenhuis tensor on the set of left-invariant, orthogonal, almost complex structures.

Elevated effective dimension in tree-like nanomagnetic Cayley structures.

Using state-of-the-art electron-beam lithography, Ising-type nanomagnets may be defined onto nearly any two-dimensional pattern imaginable. The ability to directly observe magnetic configurations achieved in such artificial spin systems makes them a perfect playground for the realization of artificial spin glasses. However, no experimental realization of a finite-temperature artificial spin glass has been achieved so far. Here, we aim to get a significant step closer in achieving that goal by introducing an artificial spin system with random interactions and increased effective dimension: dipolar Cayley tree. Through synchrotron-based photoemission electron microscopy, we show that an improved balance of ferro- and antiferromagnetic ordering can be achieved in this type of system. This combined with an effective dimension as high as d = 2.72 suggests that future systems generated out of these building blocks can host finite temperature spin glass phases, allowing for real-time observation of glassy dynamics.

О ПОЧТИ (ПАРА) КОМПЛЕКСНЫХ СТРУКТУРАХ КЭЛИ НА СФЕРАХ S2,4 И S3,3

In this paper we study almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres in the space of purely imaginary octaves of the split Cayley algebra $\mathbf{Ca}'$. It is shown that the Cayley structures are non-integrable, their basic geometric characteristics are calculated. In contrast to the usual Riemann sphere $\mathbb{S}^6$, there exist (integrable) complex structures and para-complex structures on the pseudospheres under consideration.

Cayley Structures on S 6 as Sections if Twistor Bundle

We consider nearly Kahler structures on a 6-dimensional sphere as sections of the twistor bundle over the sphere and prove that for any point of the space of twistor bundle there is a one-parameter family of nearly Kahler structures passing through the point. We study some properties of such one-parameter families.

On Fano Schemes of Toric Varieties

Given a finite set of lattice points $\mathcal{A}$, we consider the associated homogeneous binomial ideal $I_\mathcal{A}$ and projective toric variety $X_\mathcal{A}$. We give a concise combinatorial description of all linear subspaces contained in the variety $X_\mathcal{A}$, or, equivalently, all solutions in linear forms to the system of binomial equations determined by $I_\mathcal{A}$. More precisely, we study the Fano scheme $\mathbf{F}_k(X_\mathcal{A})$ whose closed points correspond to $k$-dimensional linear spaces contained in $X_\mathcal{A}$. We show that the irreducible components of $\mathbf{F}_k(X_\mathcal{A})$ are in bijection to maximal Cayley structures for $\mathcal{A}$ of length at least $k$. We explicitly describe these irreducible components and their intersection behavior, characterize when $\mathbf{F}_k(X_\mathcal{A})$ is connected, and prove that if $X_\mathcal{A}$ is smooth in dimension $k$, then every component of $\mathbf{F}_k(X_\mathcal{A})$ is smooth in its reduced structure. Furthermore, in the special case $k=\dim X_\mathcal{A}-1$, we describe the nonreduced structure of $\mathbf{F}_k(X_\mathcal{A})$.

Virtual Resource Allocation Based on Link Interference in Cayley Wireless Data Centers

Cayley data centers are well known patterns of completely wireless data centers (WDCs). However, low link reliability and link interference will affect the construction of virtual networks. This paper proposes a virtual resource mapping algorithm on the basis of Cayley structures. First, we analyze the characteristics of Cayley WDCs and model networks in WDCs, where a virtual network is modeled as a traditional undirected graph, while a physical topology is modeled as a directed graph. Second, we propose a virtual resource mapping and coloring algorithm based on link interference called VRMCA-LI. We build a connection interference matrix for each node and use a coloring method to avoid interference. VRMCA-LI uses the same color for the nodes that are within the transmitting angle of a sending node and whose signal-to-noise ratio is less than a threshold. These nodes with the same color cannot be allocated to virtual nodes at the same time. The allocation of nodes and links are concurrent, which performs dynamic adjustment to save mapping time. Third, our experimental results show that VRMCA-LI outperforms WVNEA-LR and PG-VNE in terms of mapping time of virtual nodes, acceptance rate of virtual networks, and average node utilization rate.