Finite Plane Research Articles

Three new classes of generalized almost perfect nonlinear power functions

Generalized almost perfect nonlinear (GAPN) functions are algebraic generalization of almost perfect nonlinear functions and have important applications in cryptography and finite geometry. Based on the well known binomial theorem and some combinatory formulas, the conjecture proposed by Kuroda is partially disproved and three new classes of GAPN power functions for odd characteristic are presented.

Design and analysis of an optical full-adder based on nonlinear photonic crystal ring resonators

In this paper, first we have used a photonic crystal ring resonator in a two-dimensional structure composed of GaAs rods for designing a channel drop filter (CDF). Then the CDF is used to design a 1-bit full-adder. The structure has two nonlinear ring resonators which are coupled to eight waveguides. The nonlinear ring resonators have Kerr-type nonlinearity which causes the resonant mode of photonic crystal resonators to depend on intensity of input power launched into the structure. The proposed structure has three input and two output ports which are dubbed “A”, “B” and “C” as input and “Sum” and “Carry” as output. The maximum delay time of the structure is about 3 ps. The structure has been simulated and analyzed using finite difference time domain (FDTD) and plane wave expansion (PWE) methods. The main advantages of the proposed structure are compactness, reduction of the delay time, reduction of the optical power intensity required at input ports and reduction of the backward reflection toward the input ports.

Gradient elasticity theory for fiber composites with fibers resistant to extension and flexure

A model for the mechanics of an elastic solid, reinforced with bidirectional fibers is presented in finite plane elastostatics. The fiber’s resistance to stretch and flexure are accounted for with variational computations of first and second gradient of deformations, respectively. Within the framework of strain-gradient elasticity, the Euler equation and necessary boundary conditions are formulated. A rigorous derivation of the corresponding linear theory is also developed from which, a complete analytical solution is obtained for small deformations superposed on large. In particular, we assimilate plane bias extension test results indicating that the proposed model successfully predicts smooth transitions of shear strain fields unlike those depicted by the first gradient theory where significant discontinuities are present. The proposed model can serve as an alternative 2D Cosserat theory of non-linear elasticity.

Elliptic semiplanes and regular graphs with girth 5

A (k, g)-graph is a k-regular graph with girth g and a (k, g)-cage is a (k, g)-graph with the fewest possible number of vertices. The cage problem consists of constructing (k, g)-graphs of minimum order n(k, g). We focus on girth g=5, where cages are known only for degrees k≤7. Considering the relationship between finite geometries and graphs we establish upper constructive bounds on n(k, 5), for k∈{13,14,17,18,…} that improve the best so far known.

Development and application of CATIA-based interference simulation system for tube bending

Aiming at the urgent demands for precise and intelligent forming of complex bent tubular components, this paper reviews the developments of tube bending related simulation technologies. Combined with the advantages of finite element (FE) simulation analysis and geometry motion simulation analysis, the present work is proposed to establish a knowledge base of tube bending simulation based on the knowledge engineering technology, and to develop an interference simulation system of tube bending based on CATIA software. Through the application of interference simulation system in continuous tube multi-bending in civil aircrafts, the extraction of model data, selection of process parameters, FE simulation of single bending segment, geometry simulation of continuous multi-bending, generation of simulation reports and numerical control codes can be automatically realized, and the bending quality and interference situation can be simultaneously obtained. Using the interference simulation system, the product efficiency of aircraft duct components can be greatly improved, and the labour cost and working intensity of technicians can be significantly reduced.

Realization of tunable optical channel drop filter based on photonic crystal octagonal shaped structure

Photonic crystals (PhCs) are artificial dielectric nanostructure materials. In these structures, a periodic modulation of the material dielectric constant results in a photonic band gap (PBG). By employing defects in the photonic crystals, light can steer in specific direction. Consequently, defects are used in most of the PhC structures. In this work, a novel channel drop filter (CDF) based on two dimensional (2-D) photonic crystal with octagonal shaped structure is designed and numerically analyzed by using the finite difference time domain (FDTD) and plane wave expansion (PWE) techniques. The proposed filter is an octagonal shaped structure of Si rods between two waveguides; consisting of 19*18 silicon rods with the refractive index of 3.5 and rod radius of 0.16a; where a is a lattice constant and equals 0.6 μm. By analyzing the structure, wide ranges of TE photonic band gap (PBG) would be achieved. It will be stated that the proposed CDF has appropriate characteristics and could be used in the future wavelength division multiplexing (WDM) communication systems.

Design of a label-free photonic crystal refractive index sensor for biomedical applications

Two photonic crystal optical refractive index sensors have been proposed in this paper. These sensors can be used for detection of basal cell cancer. A rod-type square lattice of GaAs is used for this purpose. The sensor topologies are based on a photonic crystal ring-shaped resonator coupled to two and waveguides respectively. Instead of using ultra-high quality factor cavities for improving the sensitivity, the resonance profile is localized and concentrated on the analyte to achieve good distinction. A 13 nm cavity resonance frequency shift for normal and cancer cells is obtained for detection of basal cell carcinoma. Finite difference time domain method and plane wave expansion methods are used to simulate and analyze the structures. The proposed biosensors have a sensitivity equal to 720 and 638 nm/RIU. The simplicity of the design and its high sensitivity make it a suitable choice for bio-sensing applications.

Asymptotically approaching the Moore bound for diameter three by Cayley graphs

The largest order n(d,k) of a graph of maximum degree d and diameter k cannot exceed the Moore bound, of the form M(d,k)=dk−O(dk−1) for d→∞ and any fixed k. Known results in finite geometries on generalised (k+1)-gons imply, for k=2,3,5, existence of an infinite sequence of values of d such that n(d,k)=dk−o(dk). Thus, for k=2,3,5 the Moore bound can be asymptotically approached in the sense that lim supd→∞n(d,k)/M(d,k)=1; moreover, no such result is known for any other value of k≥2. The corresponding graphs are, however, far from vertex-transitive, and there appears to be no obvious way to extend them to vertex-transitive graphs giving the same type of asymptotic result.The second and the third author (2012) proved by a direct construction that the Moore bound for diameter k=2 can, in a similar sense, be asymptotically approached by Cayley graphs. Subsequently, the first and the third author (2015) showed that the same construction can be derived from generalised triangles with polarity.By a detailed analysis of regular orbits of suitable groups of automorphisms of graphs arising from polarity quotients of incidence graphs of generalised quadrangles with polarity, we prove that for an infinite set of values of d there exist Cayley graphs of degree d, diameter 3, and order d3−O(d2.5). The Moore bound for diameter 3 can thus as well be asymptotically approached by Cayley graphs. We also show that this method does not extend to constructing Cayley graphs of diameter 5 from generalised hexagons with polarity.

Numerical simulation of damage fields and residual stresses after hole punching

The article studies the process of reinforcing the technological hole in the aluminum aviation alloy D16CHT for various levels of plastic deformation corresponding to the level of plastic deformation of 1 %, 2 %, and 3 % by technological hole processing. In the finite elemental package of Abaqus Student Edition, the dynamic task of simulating the technological hole processing is solved by creating the corresponding finite element of the model and the calculation of the contact problem with an explicit integration scheme. To improve the quality of the calculation level, the division of the models into zones with different parameters of the finite element geometry is used. Based on the results of numerical simulation, the distribution of damageability in the zone of the technological hole for various levels of plastic deformation, studied in the present work, was obtained. The parameter of damageability, which depends on the level of plastic deformation, was considered as a scalar quantity, and it was estimated by changing the modulus of elasticity of the first kind. The fields of distribution of residual stresses and equivalent plastic deformation were obtained. The dependence of the change in the principal residual stresses on the inner surface of the hole is gained depending on the depth of the hole. On the basis of numerical calculations, we obtained the effect of the residual strain on the initial level of material damage in the area of the technological hole. The redistribution of the equivalent plastic deformation and residual stresses during the subsequent loading of the technological hole with tensile loads is analyzed.

One-factorisations of complete graphs arising from ovals in finite planes

In this paper we use the geometry of finite planes to set up a procedure for the construction of one-factorisations of the complete graph. Let π be a projective plane of order n−1 with n even containing an oval Ω, and regard Ω as the vertex set of the complete graph Kn. Then any one-factorisation of Kn has a representation by a partition of the external points to Ω whose components are of size n2 and meet every tangent to Ω in a unique point. Our goal is to construct such partitions from nice geometric configurations in the Desarguesian plane of order q with q=ph and p>2 prime.

On elation Laguerre planes with a two-transitive orbit on the set of generators

We study finite elation Laguerre planes with a group of automorphisms fixing a generator and acting two-transitively on the set of remaining generators. For odd order, this assumption characterizes the Miquelian Laguerre planes, but there are non-Miquelian examples if the order is even.

Nonlocal vertices and analyticity: Landau equations and general Cutkosky rule

We study the analyticity properties of amplitudes in theories with nonlocal vertices of the type occurring in string field theory and a wide class of nonlocal field theory models. Such vertices are given in momentum space by entire functions of rapid decay in certain (including Euclidean) directions ensuring UV finiteness but are necessarily of rapid increase in others. A parametric representation is obtained by integrating out the loop (Euclidean) momenta after the introduction of generalized Schwinger parameters. Either in the original or parametric representation, the well-defined resulting amplitudesare then continued in the complex space of the external momenta invariants. We obtain the alternative forms of the Landau equations determining the singularity surfaces showing that the nonlocal vertices serve as UV regulators but do not affect the local singularity structure. As a result the full set of singularities known to occur in local field theory also occurs here: normal and anomalous thresholds as well as acnodes, crunodes, and cusps that may under certain circumstances appear even on the physical sheet. Singularities of the second type also appear as shown from the parametric representation. We obtain the general Cutkosky discontinuity rule for encircling a singularity by employing contour deformations only in the finite plane. The unitarity condition (optical theorem) is thendiscussed as a special application of the rule across normal thresholds and the hermitian analyticity property of amplitudes.

3-Dimensional Optical Orthogonal Codes With Ideal Autocorrelation-Bounds and Optimal Constructions

Several new constructions of 3-dimensional optical orthogonal codes are presented here. In each case the codes have ideal autocorrelation $\mathbf{ \lambda_a=0} $, and in all but one case a cross correlation of $ \mathbf{\lambda_c=1} $. All codes produced are optimal with respect to the applicable Johnson bound either presented or developed here. Thus, on one hand the codes are as large as possible, and on the other, the bound(s) are shown to be tight. All codes are constructed by using a particular automorphism (a Singer cycle) of $ \mathbf{ PG(k,q)} $, the finite projective geometry of dimension $ k $ over the field of order $ \mathbf{q} $, or by using an affine analogue in $ AG(k,q) $.

Event plane dependence of the flow modulated background in dihadron and jet-hadron correlations in heavy ion collisions

Di-hadron and jet-hadron correlations are commonly used in relativistic heavy ion collisions to study the soft component of jets in a quark gluon plasma. There is a large correlated background which is described by the Fourier decomposition of the azimuthal anisotropy where $v_n$ is the $n$th order coefficient. The path length dependence of partonic energy loss can be studied by varying the angle of the high momentum trigger particle or jet relative to a reconstructed event plane. This modifies the shape of the background correlated with that event plane. The original derivation of the shape of this background only considered correlations relative to the second order event plane, which is correlated to the initial participant plane. We derive the shape of this background for an event plane at an arbitrary order. There is a phase shift in the case of jets restricted to asymmetric regions relative to the event plane. For realistic correlations between event planes, the correlation between the second and fourth order event planes leads to a much smaller effect than the finite event plane resolution at each order. Finally, we assess the status of the rapidity even $v_1$ term due to flow, which has been measured to be comparable to $v_2$ and $v_3$ terms.

Thermal Stress Analysis in a Finite Perforated Plane

In this paper, hoop stress around a triangular or square hole in the finite isotropic plane under thermal loading is studied. The employed method is based on the analytical solution of finite plane with a circular hole. The thermal and stress distribution in this perforated plate is calculated by means of series and airy stress functions, respectively. The unknown coefficients in these functions are obtained by using the least square boundary collocation method and applying the boundary conditions. By applying the Muskhelishvili’s method and conformal mapping, the analytical thermal stress around the square and the triangular hole in finite plate is obtained. The results show the effect of plane’s size, hole’s size of square hole on thermal stress around the hole edges.

Design and analysis of 1xN symmetrical optical splitters for photonic integrated circuits

Optical power splitters play a momentous role because of its capability to completely eliminate active devices in the Passive Optical Network (PON) link which progressively improves the broadband optical communication. Communication link between the service provider and the user premises of PON networks depends on the splitter. Even though various types of splitters based on optical fibre are available, we report the design and simulation results of 1 × 2, 1 × 4 and 1 × 8 symmetrical splitters based on photonic crystal ring resonators. Photonic crystal optical splitters will provide technological compatibility for integration in Photonic Integrated Circuits (PIC). The designed layout operates at 1490 nm wavelength, which is the recommended ITU-T standard for down streaming in PON architecture. Finite difference time domain (FDTD) method and Plane wave expansion (PWE) method is used for simulation. The upshot of this simulation exhibits good splitting of optical power in terms of its overall efficiency, splitting ratio and propagation delay. Its miniaturised size and capacity to integrate as PIC, the reported layout makes itself as a best and suitable candidate for PON architecture.

An undergraduate laboratory study of the polarisation of annihilation photons using Compton scattering

An experiment for the advanced undergraduate laboratory which allows students to study the effect of photon polarisation in Compton scattering and to explore quantum entanglement is described. The quantum entangled photons are produced through electron–positron annihilation in the S-state, and their polarisations are analysed using the Compton scattering cross-section dependence on the photon polarisation. The experiment was equipped with off-the-shelf detectors and electronic units. Finite geometry effects are discussed and investigated with the use of a Geant4-based simulation.

Surface Green’s functions in finite plane elastostatics of harmonic materials

The closed-form representations of surface Green’s functions corresponding to the action of a concentrated force applied at the boundary of a region occupied by a particular class of compressible hyperelastic materials of harmonic type, has been derived. In our analysis, we consider both a bounded region in the form of a circular disk and an unbounded region with either an elliptical hole or a parabolic boundary.

Theoretical Modeling of an Ion-Beam-Driven Kelvin–Helmholtz Instability in a Plasma Cylinder Having Negatively Charged Dust Grains

An ion beam propagating through a magnetized plasma cylinder having negatively charged dust grains drives the Kelvin–Helmholtz instability (KHI) via Cerenkov interaction. The frequency and the growth rate of the KHI increase with the relative density of negatively charged dust grains $\delta $ . However, for large relative density of dust grains $\delta $ , the growth rate of KHI saturates. Moreover, beam density plays an important role in enhancing the growth rate of the KHI. It is also observed that finite boundary effects modify the dispersion properties of the KHI. The growth rate of KHI in a plasma cylinder is destabilized under the effect of dust charge fluctuations for relatively large velocity shears and low density of negatively charged dust grains $\delta $ . A comparative study of results is done for finite and infinite geometries of plasma waveguide.

A Characterization of Hermitian Varieties as Codewords

It is known that the Hermitian varieties are codewords in the code defined by the points and hyperplanes of the projective spaces $\mathrm{PG}(r,q^2)$. In finite geometry, also quasi-Hermitian varieties are defined. These are sets of points of $\mathrm{PG}(r,q^2)$ of the same size as a non-singular Hermitian variety of $\mathrm{PG}(r,q^2)$, having the same intersection sizes with the hyperplanes of $\mathrm{PG}(r,q^2)$. In the planar case, this reduces to the definition of a unital. A famous result of Blokhuis, Brouwer, and Wilbrink states that every unital in the code of the points and lines of $\mathrm{PG}(2,q^2)$ is a Hermitian curve. We prove a similar result for the quasi-Hermitian varieties in $\mathrm{PG}(3,q^2)$, $q=p^{h}$, as well as in $\mathrm{PG}(r,q^2)$, $q=p$ prime, or $q=p^2$, $p$ prime, and $r\geq 4$.