An instanton-invariant for 3-manifolds

To an oriented closed 3-dimensional manifoldM withH ₁(M, )=0, we assign a ₈-graded homology groupI _*(M) whose Euler characteristic is twice Casson's invariant. The definition uses a construction on the space of instantons onM×.     

Ricci flow for homogeneous compact models of the universe

Using quaternions, we give a concise derivation of the Ricci tensor for homogeneous spaces with topology of the 3-dimensional sphere. We derive explicit and numerical solutions for the Ricci flow PDE and discuss their properties. In the collapse (or expansion) of these models, the interplay of the various components of the Ricci tensor are studied.     

A general transformation for compact waveguide coupler by using homogeneous media

A general transformation is proposed to design the compact waveguide coupler with homogeneous media. By dividing the coupling region into several triangle blocks and engaging the transformation, material inhomogeneity of the coupler can be eliminated, and thus the device only requires homogeneous and anisotropic media. Also, it is found that the electromagnetic field in the coupling region can be controlled artificially and the field in the two waveguides is little influenced. Thus, much freedom and flexibility is provided in the design of waveguide coupler. Besides, the general transformation can also be extended to design waveguide bender with homogeneous media. Furthermore, Full wave simulation based on finite element method is performed to verify the performance of the waveguide coupler.     

The local geometry of compact homogeneous Lorentz spaces

In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, Zeghib also described the geometric structure of the manifolds. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation and curvatures. In particular, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.     

A new model for spherically symmetric charged compact stars of embedding class 1

In the present study we search for a new stellar model with spherically symmetric matter and a charged distribution in a general relativistic framework. The model represents a compact star of embedding class 1. The solutions obtained here are general in nature, having the following two features: first of all, the metric becomes flat and also the expressions for the pressure, energy density, and electric charge become zero in all the cases if we consider the constant \(A=0\), which shows that our solutions represent the so-called ‘electromagnetic mass model’ [17], and, secondly, the metric function \(\nu (r)\), for the limit n tending to infinity, converts to \(\nu (r)=C{r}^{2}+ ln~B\), which is the same as considered by Maurya et al. [11]. We have investigated several physical aspects of the model and find that all the features are acceptable within the requirements of contemporary theoretical studies and observational evidence.     

Random tilings of spherical 3-manifolds

A set of random tilings for the compact Euclidean 3-manifolds have been considered recently. In this paper, non-deterministic triangulations of spherical 3-manifolds based on recursive procedures are introduced. Simplicial structures for lens, prism and Poincaré spherical dodecahedron spaces are described. The configurational entropies for the random structures depend only on the information in 2D and are consistent with the Bekenstein–Hawking bound.     

Flat 3-webs via semi-simple Frobenius 3-manifolds

We construct flat 3-webs via semi-simple geometric Frobenius manifolds of dimension three and give geometric interpretation of the Chern connection of the web. These webs turned out to be biholomorphic to the characteristic webs on the solutions of the corresponding associativity equation. We show that such webs are hexagonal and admit at least one infinitesimal symmetry at each singular point. Singularities of the web are also discussed.     

Remarks on the entropy of 3-manifolds

We give a simple combinatoric proof of an exponential upper bound on the number of distinct 3-manifolds that can be constructed by successively identifying nearest neighbour pairs of triangles in the boundary of a simplicial 3-ball and show that all closed simplicial manifolds that can be constructed in this manner are homeomorphic to S3. We discuss the problem of proving that all 3-dimensional simplicial spheres can be obtained by this construction and give an example of a simplicial 3-ball whose boundary triangles can be identified pairwise such that no triangle is identified with any of its neighbours and the resulting 3-dimensional simplicial complex is a simply connected 3-manifold.     

A compact permanent magnet array with a remote homogeneous field

We present the design and construction of a single sided magnet array generating a homogeneous field in a remote volume. The compact array measures 11.5 cm by 10 cm by 6 cm and weights approximately 5 kg. It produces a B(0) field with a 'sweet spot' at a point 1cm above its surface, where its first and second spatial derivatives are approximately zero. Unlike other sweet spot magnets of this general type, our array has B(0) oriented parallel to its surface. This allows an ordinary surface coil to be used for unilateral measurements, giving the potential for dramatic SNR improvement.     

Dolbeault cohomology groups of compact pseudo-Kähler homogeneous manifolds

It is well known that a pseudo-Kähler structure is one of the natural generalizations of a Kähler structure. In this paper, we consider the Dolbeault cohomology groups of compact pseudo-Kähler homogeneous manifolds.     

Collapsing and monopole classes of 3-manifolds

By applying the L²$L^2$-estimate of the scalar curvature of a Riemannian 3-manifold with a Seiberg–Witten monopole class to a collapsing sequence of metrics, we obtain conditions to be a monopole class on certain 3-manifolds. This also gives a relation between a maximizing sequence of the Yamabe constants and the collapsing on a 3-manifold with a non-torsion monopole class.     

A criterion for a topological group to admit a continuous embedding in a locally compact group

It is proved that a topological group admits a continuous embedding in a locally compact group if and only if the group in question admits a separating family of unitary representations defining a symmetric Hopf-von Neumann algebra whose intrinsic group coincides with the set of nonzero characters of the predual algebra. To the blessed memory of Leonid Romanovich Volevich The author is indebted to Victor Kac for a pdf-file of the paper [13]. The research was partially supported by the Russian Foundation for Basic Research under grant no. 08-01-00034.     

The Ricci tensor of SU(3)-manifolds

Following the approach of Bryant [R. Bryant, Some remarks on G₂-structures. e-print: math.DG/0305124] we study the intrinsic torsion of a SU(3)-manifold deriving a number of formulae for the Ricci and the scalar curvature in terms of torsion forms. As a consequence we prove that in some special cases the Einstein condition forces the vanishing of the intrinsic torsion.     

Applying TQFT to count regular coverings of Seifert 3-manifolds

I give a formula for computing the number of regular Γ$\Gamma$-coverings of closed orientable Seifert 3-manifolds, for a given finite group Γ$\Gamma$. The number is computed using a 3d TQFT with finite gauge group, through a cut-and-glue process.     

EVA-enhanced embedding medium for histological analysis of 3D porous scaffold material

When sectioning a 3D porous scaffold made of a soft elastomeric material embedded in paraffin medium, it is not easy to obtain a section because of the different mechanical properties of the paraffin and tissue/scaffold. We describe a new embedding material for histological analysis of various biomaterials that is composed of paraffin and ethylene vinyl acetate (EVA) resin (0, 3, 7, and 13 wt.%). 3D porous poly(l-lactide--caprolactone) (PLCL) and chitosan scaffolds were fabricated to test the sectioning efficiency of the paraffin/EVA embedding material. The new embedding material was characterized by rheological analysis and solvent solubility testing in xylene and n-hexane. The hydrophilicity of the new material was assessed by contact angle measurement and its surface roughness was measured using AFM analysis. The staining efficiency of sections embedded in a paraffin/EVA mixture was determined by eosin staining of the chitosan scaffold and chitosan/collagen hybrid scaffold using a fluorescently labeled collagen. Section roughness decreased with increasing EVA content. The softening temperature of the paraffin/EVA mixture was similar to that of paraffin (50–60 °C by rheometer). The paraffin/EVA mixture dissolved completely in xylene after 30 min at 50 °C, and after 30 min in n-hexane at 60 °C. Therefore, the new embedding medium can be used for histological analysis of various biomaterials and natural tissues.     

嵌埋Ni颗粒的BaTiO_3/SrTiO_3薄膜的相变特性

采用高温X射线原位衍射和变温介电谱对SrTiO3基底上外延生长的BaTiO3(嵌埋Ni颗粒)薄膜进行了相变特性分析。从X射线衍射和介电谱的分析结果得出,BaTiO3的相变温度点转变为弥散的温度区间。在这个弥散的相变温度区间内,由于基底和薄膜之间的失配,以及嵌埋Ni颗粒的应力作用,薄膜的介电响应弥散剧烈,并偏离德拜弛豫。分析Cole-Cole图获知,BaTiO3薄膜在四方相转变为立方相的相变过程中同时存在几种极化机制,在高温状态下介电损耗随温度增大而增大。降温过程中,薄膜没有立即恢复四方相,可能是基底和Ni颗粒的共同作用影响了相变弛豫。     

Quantum groups at odd roots of unity and topological invariants of 3-manifolds

Topological invariants of three-manifolds are constructed using quantum groups associated with theA, B, C andD series of Lie algebras at odd roots of unity. These invariants are also explicitly computed for the lens spaces.     

Quantum algorithm for neighborhood preserving embedding

Neighborhood preserving embedding (NPE) is an important linear dimensionality reduction technique that aims at preserving the local manifold structure. NPE contains three steps, i.e., finding the nearest neighbors of each data point, constructing the weight matrix, and obtaining the transformation matrix. Liang et al. proposed a variational quantum algorithm (VQA) for NPE [Phys. Rev. A 101 032323 (2020)]. The algorithm consists of three quantum sub-algorithms, corresponding to the three steps of NPE, and was expected to have an exponential speedup on the dimensionality n. However, the algorithm has two disadvantages: (i) It is not known how to efficiently obtain the input of the third sub-algorithm from the output of the second one. (ii) Its complexity cannot be rigorously analyzed because the third sub-algorithm in it is a VQA. In this paper, we propose a complete quantum algorithm for NPE, in which we redesign the three sub-algorithms and give a rigorous complexity analysis. It is shown that our algorithm can achieve a polynomial speedup on the number of data points m and an exponential speedup on the dimensionality n under certain conditions over the classical NPE algorithm, and achieve a significant speedup compared to Liang et al.'s algorithm even without considering the complexity of the VQA.