Polygonal Space Research Articles

Degenerate band edges in periodic quantum graphs

Edges of bands of continuous spectrum of periodic structures arise as maxima and minima of the dispersion relation of their Floquet--Bloch transform. It is often assumed that the extrema generating the band edges are non-degenerate. This paper constructs a family of examples of $\mathbb{Z}^3$-periodic quantum graphs where the non-degeneracy assumption fails: the maximum of the first band is achieved along an algebraic curve of co-dimension 2. The example is robust with respect to perturbations of edge lengths, vertex conditions and edge potentials. The simple idea behind the construction allows generalizations to more complicated graphs and lattice dimensions. The curves along which extrema are achieved have a natural interpretation as moduli spaces of planar polygons.

Integrable Evolutions of Twisted Polygons in Centro-Affine ℝm

Abstract We show that discrete $W_m$ lattices are bi-Hamiltonian, using geometric realizations of discretizations of the Adler–Gel’fand–Dikii flows as local evolutions of arc length-parametrized polygons in centro-affine space. We prove the compatibility of two known Hamiltonian structures defined on the space of geometric invariants by lifting them to a pair of pre-symplectic forms on the space of arc length parametrized polygons. The simplicity of the expressions of the pre-symplectic forms makes the proof of compatibility straightforward. We also study their kernels and possible integrable systems associated to the pair.

Extreme contractions on finite-dimensional polygonal Banach spaces. II

We introduce the concept of weak L-P property for a pair of Banach spaces, in the study of extreme contractions. We give examples of pairs of Banach spaces (not) satisfying weak L-P property and apply the concept to compute the exact number of extreme contractions between a particular pair of polygonal Banach spaces. We also study the optimality of our results on the newly introduced weak L-P property for a pair of Banach spaces, by considering appropriate examples.

Confocal and Histological Features After Poly(Ethylene Glycol) Diacrylate Corneal Inlay Implantation.

PurposeTo evaluate the in vivo biocompatibility of photopolymerized poly(ethylene glycol) diacrylate (PEGDA) intrastromal inlays in rabbit corneas.MethodsSixty-three eyes of 42 New Zealand rabbits were included. Manual intrastromal pockets were dissected in 42 eyes. PEGDA inlays were obtained using a specifically designed photomask and were inserted in the intrastromal pocket of 21 eyes (inlay group); the remaining 21 right eyes did not receive any implant (pocket-only group). Twenty-one eyes with no intervention were used as controls. In vivo confocal microscopy (IVCM) was performed at every visit. After 2 months, rabbits were sacrificed and corneas removed for histological analysis.ResultsCorneas remained clear in all but two animals, and five cases of corneal neovascularization were seen (P = 0.2). Inlays remained stable without evidence of lateral or anterior migration, and no other complications were observed. No changes in anterior and posterior keratocyte density (P = 0.3 and P = 0.1, respectively) or endothelial cell density (P = 0.23) were observed between groups during the study time by IVCM. On pathology samples, thinning of the epithelium over the inlay area and epithelial hyperplasia over the edges were observed. A polygonal empty space with no evidence of PEGDA hydrogel within the midstroma was seen in the inlay group. Keratocytes were normal in shape and number in the vicinity of the PEGDA implant area.ConclusionsPhotopolymerized PEGDA intrastromal inlays have shown relatively good safety and stability in rabbit corneas. Inlays were biostable in the corneal environment and remained transparent during follow up.Translational RelevanceThe investigated PEGDA is promising for the development of biocompatible intrastromal implants.

Geometric presentation for the cohomology ring of polygon spaces

We describe the cohomology ring of the moduli space of a flexible polygon in geometrically meaningful terms. We propose two presentations, both are computation friendly: there are simple rules for cup product.

Designing Spatial Kramers–Kronig Media Using Transformation Optics

Spatial Kramers–Kronig (KK) relations have been recently proposed to achieve perfect absorption without using any gain elements or highly anisotropic materials. However, such perfect absorption is limited to the surface of an infinitely long planar slab in the Cartesian coordinate, and the cases of surfaces with bends or sharp corners are still challenging. In this article, we apply conformal transformation optics methods to design spatial KK media, which can achieve smooth perfect absorption both in cylindrical spaces and in polygonal spaces with sharp corners. Moreover, it is demonstrated that we can also control the scattering direction of the designed polygonal spatial KK media to realize invisibility. Our work opens new avenues for perfect absorption devices based on transformation optics and spatial KK relations.

A Twelve Concentric Multilevel Twenty-Four Sided Polygonal Voltage Space Vector Structure for Variable Speed Drives

Generating dense multilevel voltage space structure using polygons of higher sides is one of the novel and elegant method to suppress low-order harmonics and to obtain refined sinusoidal voltages using voltage source inverters for variable frequency drive application without passive filters. Apart from the advantages of conventional multilevel inverter topologies, schemes generating polygonal voltage space vector structure can equip full dc-bus utilization, low switching frequency strategies to eliminate low-order harmonics, and increased linear modulation range. In this paper, an induction motor drive scheme generating a highly dense multilevel 24-sided polygonal voltage space vector structure using a single dc source, which can eliminate harmonics till 23rd order from phase voltages, is presented. Cascaded power circuit topology with a flying capacitor inverter fed with a dc source and two low voltage floating capacitor fed H-bridge inverters is used. Detailed experimentally validated results, under scalar as well as indirect rotor field oriented control of induction motor are provided. Studies on voltage ripple and reactive energy in various floating capacitors, harmonic performance of output voltage for wide range of speed operation, are also included.

A Switched Capacitive Filter-Based Harmonic Elimination Technique by Generating a 30-Sided Voltage Space Vector Structure for IM Drive

This paper proposes a novel polygonal voltage space vector structure (SVS) having 30 sides, for a star-connected induction motor drive. The SVS eliminates the presence of harmonics up to 25th order from motor phase voltage throughout the entire modulation range, providing a torque profile devoid of lower order pulsations. Linear modulation is extended till 99.63% of base speed without exceeding the motor phase voltage rating. Topology consists of a dc-link fed primary inverter and two equal low voltage modular capacitor fed secondary inverters. Here the harmonics generated by the primary inverter is canceled by the secondary inverter which acts as a switched capacitive filter. Detailed description of the SVS generation and timing calculations are provided in this paper. Effectiveness of the proposed scheme is validated using experimental results, inverter loss calculations, and harmonic analysis.

𝐾-theory and immersions of spatial polygon spaces

Abstract For ℓ {\ell} a generic n-tuple of positive numbers, N ⁢ ( ℓ ) {N(\ell)} denotes the space of isometry classes of oriented n-gons in ℝ 3 {\mathbb{R}^{3}} with side lengths specified by ℓ {\ell} . We determine the algebra K ⁢ ( N ⁢ ( ℓ ) ) {K(N(\ell))} and use this to obtain nonimmersions of the 2 ⁢ ( n - 3 ) {2(n-3)} -manifold N ⁢ ( ℓ ) {N(\ell)} in Euclidean space for several families of ℓ {\ell} . We also use obstruction theory to tell exactly when N ⁢ ( ℓ ) {N(\ell)} immerses in ℝ 4 ⁢ n - 14 {\mathbb{R}^{4n-14}} for two families of ℓ {\ell} ’s.

DELAYED SCLEROSING GRANULOMATOUS REACTION TO HYALURONIC ACID REINFORCED WITH POLY-HYDROXY-ETHYL-METHACRYLATE INJECTION

Objective On the basis of manufacturers’ and some authors’ claims, all commonly used injection materials for aesthetic correction and different formulations of hyaluronic acid (HA), with or without adjunctive substances, result in no immunogenic reactions or other complications; nevertheless, unexpected, late or early adverse reactions have been reported. Overall, HA reinforced with hydroxyethyl-methacrylate (HEMA) can promote the formation of late foreign body granulomas (FBGs). The authors report on the histological (conventional and confocal laser scanning microscopy) features of a case occurred 10 years after the injection of HA+HEMA in the lower lip of a female patient. Findings The nodular lesion was mainly composed by several almost empty and polygonal spaces, surrounded by fibrous collagen and sparse multinucleated giant cells, pointing at long-standing FBG. The polygonal spaces were 20–120m in size and partly filled with translucent particles, with a broken-glass appearance. Conclusions HA is a constituent of several normal tissues and, as such, does not lead to adverse reactions. When FBG is present, one should argue that additional components were bound to HA. HEMA has been used as a stabilizer of HA-based fillers but it is known to induce transient macrophagic reaction, fibroblast proliferation with scarce collagen deposition and multinucleated giant cells. The morphological features of the present case are consistent with previous injection of HA+HEMA and the prolonged time interval from injection to clinical manifestations indicates the adverse reaction is slowly progressive. Also, it was postulated that macrophages would incorporate foreign particles, thus keeping the foreign particles in a latent stage. Subsequently, additional priming events (e.g., supervening infections) would be needed to re-activate macrophages, lead to multinucleated giant cell accumulation and finally to wide granulomatous reaction. Such pathogenetic mechanism may explain the prolonged course of the disease, with only late development of clinically detectable nodular lesions.

Compactifications of $${{\cal M}_{0,n}}$$ Associated with Alexander Self-Dual Complexes: Chow Rings, ψ-Classes, and Intersection Numbers

An Alexander self-dual complex gives rise to a compactification of $${{\cal M}_{0,n}}$$ , called an ASD compactification, which is a smooth algebraic variety. ASD compactifications include (but are not exhausted by) the polygon spaces, or the configuration spaces of flexible polygons. We present an explicit description of the Chow rings of ASD compactifications. We study the analogs of Kontsevich’s tautological bundles, compute their Chern classes, compute top intersections of the Chern classes, and derive a recursion for the intersection numbers.

An n-dimensional Klein bottle

AbstractAn n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.

Legal problems of environmental safety of Kazakhstan

The purpose of the state policy in the field of environmental safety is to ensure the protection of natural systems, the vital interests of society and the rights of the individual from threats arising from human and natural impact on the environment. The President’s Address to the people of Kazakhstan noted the health, education and well-being of the citizens of the Republic of Kazakhstan as the fourth long-term priority. Among this priority are the issues of disease prevention and promotion of healthy lifestyles, combating drug addiction and drug trafficking, reducing tobacco and alcohol consumption, improvingthe health of women and children, improving nutrition, clean environment and ecology. In accordancewith this, it is very appropriate to signal scientists and the whole society in connection with the contradictionsbetween the importance of the use of natural resources for humans and the need to protect thenatural environment. The task arises to mitigate the negative consequences and damage caused to theenvironment by particularly important natural resources, to minimize their consequences and to createmeasures for the reproduction of natural resources. To do this, there are needs to address the followingchallenges: reducing the anthropogenic impact that leads to climate change and ozone depletion; preservationof biodiversity and prevention of desertification and land degradation; rehabilitation of ecologicaldisaster zones, military space polygons and test complexes; prevention of pollution of the Caspian seashelf; prevention of degradation and pollution of water resources; elimination and prevention of naturalpollution, air pollution, radioactive, bacteriological and chemical, including transboundary pollution;industrial and household waste; prevention of natural and man-made emergencies.Key words: environment, national security, natural resources, national interests, threat to watersecurity.

Archimedean Tiling Patterns Self-Assembled from X-Shaped Rod–Coil Copolymers with Hydrogen Bonds

On the basis of self-consistent field theory (SCFT), we demonstrate that X-shaped rod–coil molecules with hydrogen-bonding groups self-assemble into Archimedean tiling patterns including [36], [44], [63], [3.4.6.4], and [(3.6)2], and dual tiling patterns including dual-[32.4.3.4]. The rod blocks form the edges of polygons, coil blocks fill in the inner spaces of polygons, and the existence of hydrogen bonds further decreases the domain size. A new mechanism is proposed to guide the formation of Archimedean tiling patterns: the fabrication of tiling patterns is controlled by the relationship between the length–diameter ratio β and volume fraction fR of rigid cores in X-shaped supra-macromolecules. This study provides a concept of macromolecular tiling for producing Archimedean tilings and suggests X-shaped rod–coil supra-macromolecules with hydrogen-bonding groups as an ideal platform for the fabrication of two-dimensional nanoscale patterns.

Relative topological complexity of a pair

For a pair of spaces X and Y such that Y⊆X, we define the relative topological complexity of the pair (X,Y) as a new variant of relative topological complexity. Intuitively, this corresponds to counting the smallest number of motion planning rules needed for a continuous motion planner from X to Y. We give basic estimates on the invariant, and we connect it to both Lusternik–Schnirelmann category and topological complexity. In the process, we compute this invariant for several example spaces including wedges of spheres, topological groups, and spatial polygon spaces. In addition, we connect the invariant to the existence of certain types of axial maps.

Polygons with prescribed edge slopes: configuration space and extremal points of perimeter

We describe the configuration space $$\mathbf {S}$$ of polygons with prescribed edge slopes, and study the perimeter $${\mathcal {P}}$$ as a Morse function on $$\mathbf {S}$$ . We characterize critical points of $${\mathcal {P}}$$ (these are tangential polygons) and compute their Morse indices. This setup is motivated by a number of results about critical points and Morse indices of the oriented area function defined on the configuration space of polygons with prescribed edge lengths (flexible polygons). As a by-product, we present an independent computation of the Morse index of the area function (obtained earlier by Panina and Zhukova).

The Limit Point of the Pentagram Map

Abstract The pentagram map is a discrete dynamical system defined on the space of polygons in the plane. In the 1st paper on the subject, Schwartz proved that the pentagram map produces from each convex polygon a sequence of successively smaller polygons that converges exponentially to a point. We investigate the limit point itself, giving an explicit description of its Cartesian coordinates as roots of certain degree three polynomials.

On right-angled polygons in hyperbolic space

We study oriented right-angled polygons in hyperbolic spaces of arbitrary dimensions, that is, finite sequences $$( S_0,S_1,\ldots ,S_{p-1})$$ of oriented geodesics in the hyperbolic space $$\varvec{H}^{n+2}$$ such that consecutive sides are orthogonal. It was previously shown by Delgove and Retailleau (Ann Fac Sci Toulouse Math 23(5):1049–1061, 2014. https://doi.org/10.5802/afst.1435 ) that three quaternionic parameters define a right-angled hexagon in the 5-dimensional hyperbolic space. We generalise this method to right-angled polygons with an arbitrary number of sides $$p\ge 5$$ in a hyperbolic space of arbitrary dimension.

Spherical geometry and the least symmetric triangle

We study the problem of determining the least symmetric triangle, which arises both from pure geometry and from the study of molecular chirality in chemistry. Using the correspondence between planar $n$-gons and points in the Grassmannian of 2-planes in real $n$-space introduced by Hausmann and Knutson, this corresponds to finding the point in the fundamental domain of the hyperoctahedral group action on the Grassmannian which is furthest from the boundary, which we compute exactly. We also determine the least symmetric obtuse and acute triangles. These calculations provide prototypes for computations on polygon and shape spaces.

Kähler structures on spaces of framed curves

We consider the space $${\mathcal {M}}$$ of Euclidean similarity classes of framed loops in $${\mathbb {R}}^3$$ . Framed loop space is shown to be an infinite-dimensional Kahler manifold by identifying it with a complex Grassmannian. We show that the space of isometrically immersed loops studied by Millson and Zombro is realized as the symplectic reduction of $${\mathcal {M}}$$ by the action of the based loop group of the circle, giving a smooth version of a result of Hausmann and Knutson on polygon space. The identification with a Grassmannian allows us to describe the geodesics of $$\mathcal {M}$$ explicitly. Using this description, we show that $${\mathcal {M}}$$ and its quotient by the reparameterization group are nonnegatively curved. We also show that the planar loop space studied by Younes, Michor, Shah and Mumford in the context of computer vision embeds in $${\mathcal {M}}$$ as a totally geodesic, Lagrangian submanifold. The action of the reparameterization group on $${\mathcal {M}}$$ is shown to be Hamiltonian, and this is used to characterize the critical points of the weighted total twist functional.