Incidence Geometry Research Articles

Incidence geometry in a Weyl chamber I: GLn

We study the central hyperplane arrangement whose hyperplanes are the vanishing loci of the weights of the first and the second fundamental representations of gln restricted to the dual fundamental Weyl chamber. We obtain generating functions that count flats and faces of a given dimension. This counting is interpreted in physics as the enumeration of the phases of the Coulomb and mixed Coulomb-Higgs branches of a five-dimensional gauge theory with 8 supercharges in presence of hypermultiplets transforming in the fundamental and antisymmetric representation of a U(n) gauge group as described by the Intriligator-Morrison-Seiberg superpotential.

Incidence geometry in a Weyl chamber II: SLn

We study the polyhedral geometry of the hyperplanes orthogonal to the weights of the first and the second fundamental representations of sln inside the dual fundamental Weyl chamber. We obtain generating functions that enumerate the flats and the faces of a fixed dimension. In addition, we describe the extreme rays of the incidence geometry and classify simplicial faces.From the perspective of supersymmetric gauge theories with 8 supercharges in five-dimensional spacetime, the poset of flats is isomorphic to the network of mixed Coulomb-Higgs branches. On the other hand, the poset of faces is conjectured to be isomorphic to the network of crepant partial resolutions of an elliptic fibration with gauge algebra sln and “matter representation” given by the sum of the first two fundamental representations.

An infinite family of incidence geometries whose incidence graphs are locally X

We construct a new infinite family of incidence geometries of arbitrarily large rank. These geometries are thick and residually connected and their type-preserving automorphism groups are symmetric groups. We also compute their Buekenhout diagram. The incidence graphs of these geometries are locally X graphs, but more interestingly, the automorphism groups act transitively, not only on the vertices, but more strongly on the maximal cliques of these graphs.

Dextran-Thyme Magnesium-Doped Hydroxyapatite Composite Antimicrobial Coatings

The dextran-thyme magnesium-doped hydroxyapatite (10MgHAp-Dex-thyme) composite layers were prepared by a dip-coating procedure from stable suspensions and further analyzed for the first time. Different characterization techniques were employed to explore the physical-chemical features of the 10MgHAp-Dex-thyme suspensions and derived coatings. Information regarding the 10MgHAp-Dex-thyme suspensions was extracted on the basis of dynamic light scattering, zeta potential, and ultrasound measurements. The crystalline quality of the biocomposite powders—resulting after the centrifugation of suspensions—and the layers deposited on glass was assessed by X-ray diffraction in symmetric and grazing incidence geometries, respectively. The chemical structure and presence of functional groups were evaluated for both powder and coating by Fourier transform infrared spectroscopy in attenuated total reflectance mode. The extent of the antimicrobial effect range of the biocomposite suspensions and coatings was tested against different Gram-positive and Gram-negative bacteria (Staphylococcus aureus, Enterococcus faecalis, Escherichia coli, and Pseudomonas aeruginosa) and fungus (Candida albicans) strains with promising results.

Property Testing and Expansion in Cubical Complexes

We consider expansion and property testing in the language of incidence geometry, covering both simplicial and cubical complexes in any dimension. We develop a general method for the transition from an explicit description of the cohomology group, which need not be trivial, to a testability proof with linear ratio between errors. The method is demonstrated by testing functions on 2-cells in cubical complexes to be induced from the edges.

δ-A15 and bcc phases coexist in sputtered chromium coatings with moderate oxygen contents

Thin films of nanocrystalline Cr with grain sizes of around 20 nm were grown on glass substrates by D.C. magnetron sputtering in an argon atmosphere containing oxygen. The thickness of the films ranged between 30 and 135 nm, and the oxygen content between 11% and 13%. We studied the chemical composition and crystalline structure by means of X-ray spectroscopy, X-ray reflectivity and X-ray diffraction in both Bragg-Brentano and grazing incidence geometries. Our results show the coexistence of metastable δ-A15 and bcc phases in all the Cr films. The combination of the two X-ray diffraction modes allowed us to observe that growth of the δ-A15 phase in the [200] direction is favored when a certain critical film thickness is exceeded for equivalent values of oxygen content. Our results show the importance of the coexistence of both crystalline Cr phases and being aware of their textures, in order to avoid possible misinterpretations of phase transformations associated with critical stress, thickness or oxygen content.

Cover-up: a probabilistic privacy-preserving graph database model

A new graph database model is introduced that allows for an efficient and straightforward privacy-preserving mechanism. A probabilistic graph database model is also proposed, perhaps less suitable for lossless storage, but adapted for the use of statistical analysis that preserves the privacy of the individuals behind the data. Parallels are drawn to concepts in combinatorics such as clique complexes and incidence geometries.

Probabilities of incidence between lines and a plane curve over finite fields

We study the probability for a random line to intersect a given plane curve, over a finite field, in a given number of points over the same field. In particular, we focus on the limits of these probabilities under successive finite field extensions. Supposing absolute irreducibility for the curve, we show how a variant of the Chebotarev density theorem for function fields can be used to prove the existence of these limits, and to compute them under a mildly stronger condition, known as simple tangency. Partial results have already appeared in the literature, and we propose this work as an introduction to the use of the Chebotarev theorem in the context of incidence geometry. Finally, Veronese maps allow us to compute similar probabilities of intersection between a given curve and random curves of given degree.

Real-time Observation of Interface Atomic Structures by an Energy-Dispersive Surface X-ray Diffraction

Surface X-ray diffraction is a powerful tool for studying the atomic structure of buried interfaces nondestructively. The analysis is often limited to the static structures, since the acquisition of crystal truncation rod (CTR) profile dataset is lengthy. Recently, high-speed methods have been developed by several groups, aiming for the in operando study of interface phenomena. Our method uses energy-dispersive convergent X-rays and area detector, and allows the quantitative structure analysis during irreversible phenomena in a typical time frame of 1 s. In this review, the energy-dispersive method is compared with the other high-speed methods which use high-energy X-rays with a grazing incidence geometry and transmission geometry, and then two examples of the real-time monitoring are presented, the photo-induced wettability transition of the rutile-TiO2(110) surface and an electrochemical reaction on the Pt(111) electrode surface, to show the capability of the energy-dispersive method.

Relationship between Basal Plane Dislocation and Local Basal Plane Bending in PVT-Grown 4H-SiC Crystals

Efforts to improve the quality and size of 4H-SiC wafers have resulted in size of 6” and dislocation densities of the order of 103 cm-2. The presence of significant quantities of crystallographic defects in bulk 4H-SiC crystals continues to be a major obstacle for achieving high-quality 4H-SiC substrates. Of particular note is the nucleation and distribution of basal plane dislocations (BPDs). BPDs are deformation-induced and are mostly generated during post-crystal growth cooling or high temperature annealing processes. BPDs have been associated with the formation of Shockley stacking fault leading to an increase in forward voltage degradation and are undesirable. Depending on the PVT growth conditions, internal stresses due to non-uniformity of temperature field cause lattice plane bending. During cooldown, this bending is accommodated by the inhomogeneous distribution of BPDs in a PVT grown 4H-SiC boule and persists in wafers sliced from the boule. Crystal bending affects subsequent wafer processing especially polishing and device fabrication. Therefore, analysis of the BPD distribution (sign and density) enables us to better understand basal plane bending, which provides insight into further improvement of the crystal growth process. Synchrotron Monochromatic Beam X-ray Topography (SMBXT) studies in grazing incidence geometry reveal the appearance of black and white contrast of BPDs which are associated with opposite sign of Burgers vectors. Based on the principle of ray tracing, black contrast BPDs are generated by focusing of diffracted X-rays when the extra half plane is extending away from the imaged surface, while defocusing of X-rays contribute to the generation of white contrast BPDs due to the extra half plane extending towards the surface [1]. The inhomogeneous distribution of the two types of BPD contrast will thus result in concave/convex shape of the basal planes. In our study, we have evaluated the ratio of the black/white contrast of BPDs along 11-20 direction across multiple 6-inch diameter 4H-SiC substrates sliced from the same and different boules and predicted how the lattice is tilted in different substrate wafers. High resolution X-ray diffraction measurements are currently underway to investigate the variation of rocking curve peak shift along 11-20 direction and thus to further verify our result.

Controlling acoustic waves using magneto-elastic Fano resonances

We propose and analyze theoretically a class of energy-efficient magnetoelastic devices for analog signal processing. The signals are carried by transverse acoustic waves while the bias magnetic field controls their scattering from a magnetoelastic slab. By tuning the bias field, one can alter the resonant frequency at which the propagating acoustic waves hybridize with the magnetic modes, and thereby control transmission and reflection coefficients of the acoustic waves. The scattering coefficients exhibit Breit-Wigner/Fano resonant behavior akin to inelastic scattering in atomic and nuclear physics. Employing oblique incidence geometry, one can effectively enhance the strength of magnetoelastic coupling, and thus countermand the magnetic losses due to the Gilbert damping. We apply our theory to discuss potential benefits and issues in realistic systems and suggest routes to enhance the performance of the proposed devices.

Highly Sensitive Refractive Index Sensor Using Dual Resonance in Subwavelength Grating/Waveguide With Normally Incident Optical Geometry

A highly sensitive refractive index sensor with normally incident optical geometry was experimentally demonstrated using dual resonance in a subwavelength grating (SWG) on a waveguide. Two eigenmodes within the SWG and waveguide were utilized for refractive index sensing with the reduction of noise such as background noise. The finite-difference time-domain numerical method was used to estimate the refractive index sensing performance of our sensor. The calculated electric field clarified that the incident light could excite modes in the SWG and waveguide with normally incident optical geometry, and the modes formed two resonance reflection peaks. The calculation showed that these dual resonance peaks depended differently on the refractive index of the ambient around the sensor, and the intensity difference between both resonance peaks varied with the slight change in refractive index of the ambient. The sensor was fabricated using traditional electron-beam lithography techniques, and two reflection resonant peaks were experimentally obtained. The difference between the dual resonance peak intensities varied significantly with the minuscule change in refractive index of the ambient around the sensor, and the resolution of the refractive index reached $7.65\times 10^{\mathbf {-4}}$ , assuming a spectrometer intensity sensitivity of 1%.

Coherent Magnetization Rotation of a Layered System Observed by Polarized Neutron Scattering under Grazing Incidence Geometry

The in-plane magnetic structure of a layered system composed of polycrystalline grains smaller than the ferromagnetic exchange length was studied to elucidate the mechanism controlling the magnetic properties considerably different from the bulk using polarized neutron scattering under grazing incidence geometry. The measured result, together with quantitative analysis based on the distorted wave Born approximation, showed that the in-plane length of the area with a uniform orientation of moments ranging from 0.5–1.1 μ m was not significantly varied during the process of demagnetization followed by remagnetization. The obtained behavior of moments is in good agreement with the two-dimensional random anisotropy model where coherent magnetization rotation is dominant.

Generation and Transformation of Light Beams and Pulses, Containing Polarization Singularities, in Media with Nonlocality of Nonlinear Optical Response (Scientific Summary)

We discuss analytically found expressions, which relate the values of two parameters characterizing the topological type of linear and circular polarization singularities in nonparaxial light fields to the values of the complex amplitude components of the electric field and their first spatial derivatives. The necessary conditions are described for the formation of light pulse at the doubled frequency in the bulk of an isotropic gyrotropic medium with a frequency dispersion of quadratic nonlinearity and they impose constraints on the spatial transverse structure of the non-uniformly polarized incident radiation pulse and its time structure. The electric field of a light beam reflected from the surface of an isotropic gyrotropic medium with spatial dispersion of quadratic nonlinearity is investigated in the case of an arbitrary structure and geometry of incidence of non-uniformly polarized fundamental light beams. The ranges of the parameters of an elliptically polarized Gaussian beam and a medium with local and nonlocal cubic nonlinearity are determined, at which the lines of circular polarization singularity appear in cross sections of self-focused beam.

Covariants of SL4(ℂ) and incidence geometry of ℙ3(ℂ)

We determine a minimal generating set for the algebra of covariants of an arbitrary number of points, lines and planes in three-dimensional projective space. Moreover, we discuss the corresponding minimal set of geometric incidence relations.

On generalized quadrangles with a point regular group of automorphisms

A generalized quadrangle is a point-line incidence geometry such that any two points lie on at most one line and, given a line ℓ and a point P not incident with ℓ, there is a unique point of ℓ collinear with P. We study the structure of groups acting regularly on the point set of a generalized quadrangle. In particular, we provide a characterization of the generalized quadrangles with a group of automorphisms acting regularly on both the point set and the line set and show that such a thick generalized quadrangle does not admit a polarity. Moreover, we prove that a group G acting regularly on the point set of a generalized quadrangle of order (u2,u3) or (s,s), where s is odd and s+1 is coprime to 3, cannot have any nonabelian minimal normal subgroups.

Outlaw distributions and locally decodable codes

Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. Despite decades of study, the optimal trade-off between query complexity and codeword length is far from understood. In this work, we give a new characterization of LDCs using distributions over Boolean functions whose expectation is hard to approximate (in L∞ norm) with a small number of samples. We coin the term “outlaw distributions” for such distributions since they “defy” the Law of Large Numbers. We show that the existence of outlaw distributions over sufficiently “smooth” functions implies the existence of constant query LDCs and vice versa. We give several candidates for outlaw distributions over smooth functions coming from finite field incidence geometry, additive combinatorics and hypergraph (non)expanders. We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters.

Chemiplasmonics for high-throughput biosensors.

The sensitivity of ELISA for biomarker detection can be significantly increased by integrating fluorescence with plasmonics. In surface-plasmon-coupled emission, the fluorophore emission is generally enhanced through the so-called physical mechanism due to an increase in the local electric field. Despite its fairly high enhancement factors, the use of surface-plasmon-coupled emission for high-throughput and point-of-care applications is still hampered due to the need for expensive focusing optics and spectrometers. Here, we describe a new chemiplasmonic-sensing paradigm for enhanced emission through the molecular interactions between aromatic dyes and C60 films on Ag substrates. A 20-fold enhancement in the emission from rhodamine B-labeled biomolecules can be readily elicited without quenching its red color emission. As a proof of concept, we demonstrate two model bioassays using: 1) the RhB-streptavidin and biotin complexes in which the dye was excited using an inexpensive laser pointer and the ensuing enhanced emission was recorded by a smartphone camera without the need for focusing optics and 2) high-throughput 96-well plate assay for a model antigen (rabbit immunoglobulin) that showed detection sensitivity as low as 6.6 pM. Our results show clear evidence that chemiplasmonic sensors can be extended to detect biomarkers in a point-of-care setting through a smartphone in simple normal incidence geometry without the need for focusing optics. Furthermore, chemiplasmonic sensors also facilitate high-throughput screening of biomarkers in the conventional 96-well plate format with 10-20 times higher sensitivity.

Two cryptomorphic formalizations of projective incidence geometry

Incidence geometry is a well-established theory which captures the very basic properties of all geometries in terms of points belonging to lines, planes, etc. Moreover, projective incidence geometry leads to a simple framework where many properties can be studied. In this article, we consider two very different but complementary mathematical approaches formalizing this theory within the Coq proof assistant. The first one consists of the usual and synthetic geometric axiom system often encountered in the literature. The second one is more original and relies on combinatorial aspects through the notion of rank which is based on the matroid structure of incidence geometry. This paper mainly contributes to the field by proving the equivalence between these two approaches in both 2D and 3D. This result allows us to study the further automation of many proofs of projective geometry theorems. We give an overview of techniques that will be heavily used in the equivalence proof and are generic enough to be reused later in yet-to-be-written proofs. Finally, we discuss the possibilities of future automation that can be envisaged using the rank notion.

Spatial integration and differentiation of optical beams in a slab waveguide by a dielectric ridge supporting high-Q resonances.

We show that a very simple structure consisting of a single subwavelength dielectric ridge on the surface of a slab waveguide enables spatial integration and differentiation of the profile of optical beams propagating in the waveguide. The integration and differentiation operations are performed in reflection and in transmission, respectively, at oblique incidence of the beam impinging on the ridge. The implementation of these operations is associated with the resonant excitation of a cross-polarized eigenmode of the ridge. We demonstrate that the quality factor of the resonances strongly varies along the dispersion curves and allows one to achieve the required tradeoff between the integration (or differentiation) quality and the amplitude of the resulting beam. The presented rigorous numerical simulation results confirm high-quality integration and differentiation. The proposed integrated structure may find application in ultrafast all-optical analog computing and signal processing systems.