Determinantal Equations Research Articles

A Finite Integral Transform-Based Generalized Eigenvalue Formulation for Buckling of Orthotropic Rectangular Plates with Rotational Restraints

Although the finite integral transform (FIT) method has been developed for buckling analysis of rectangular thin plates, the existing formulation involves solving complex nonlinear determinantal equations, leading to potentially questionable numerical results. To address this challenge, a new FIT-based formulation is established that transforms the solution of a complex nonlinear equation into that of a straightforward generalized eigenvalue problem. New analytical solutions for buckling of orthotropic rectangular thin plates with rotational restraints are obtained. The solution procedure is implemented via the following four steps: imposing the two-dimensional FIT on the governing equation; inputting deflection conditions of four edges, by which the relations between the transformed deflections and specific unknowns are provided; imposing the one-dimensional FIT on the elastic conditions to replace the unknowns with the transformed deflections, a generalized eigenvalue problem is constructed; determining the final analytical solutions by solving the generalized eigenvalue problem. Comprehensive highly accurate buckling load/mode solutions for typical rotationally restrained orthotropic plates are presented as benchmarks. The effects of boundary conditions, rotational fixity factors, and loading ratios on the buckling behaviors of orthotropic plates are expediently investigated adopting the present solutions.

Threshold conditions for transversal modes of tunable plasmonic nanolasers shaped as single and twin graphene-covered circular quantum wires

We implement the lasing eigenvalue problem (LEP) approach to study the electromagnetic field in the presence of a circular quantum wire (QW) made of a gain material and wrapped in graphene cover and a dimer of two identical graphene-covered QWs, at the threshold of stationary emission. LEP delivers the mode-specific eigenvalue pairs, namely the frequencies and the threshold values of the QW gain index for the plasmon and the wire modes of such nanolasers. In our analysis, we use quantum Kubo formalism for the graphene conductivity and classical Maxwell boundary-value problem for the field functions. The technique involves the resistive boundary conditions, the separation of variables in the local coordinates, and, for the dimer, the addition theorem for the cylindrical functions. For single-wire plasmonic laser, we derive approximate engineering expressions for the lasing frequencies and threshold values of the gain index that complement the full-wave computations. For the dimer, we derive separate determinantal equations for four different classes of symmetry of the lasing supermodes and solve them numerically. Our investigation of the mode frequencies and thresholds versus the graphene and QW parameters shows that plasmon modes or, for the dimer, plasmon supermodes have lower frequencies and thresholds than the wire modes provided that the QW radius is smaller than 10 μm, however in thicker wires they are comparable. Only the plasmon-mode characteristics are well-tunable using the graphene chemical potential. In the dimer, all lasing supermodes form closely located quartets, however, they quickly approach the single-wire case if the inter-wire separation becomes comparable to the radius. These results open a way for building essentially single-mode plasmonic nanolasers and their arrays and suggest certain engineering rules for their design.

On Finding Determinantal Equations Involving the Coordinates of the Object Point, Point of Incidence, Point of Observation and the Image Point in Cases of Reflection and Refraction

This paper is concerned with the procedure of finding determinantal equations, each of which involves the coordinates of the object point, point of incidence, point of observation, and the image point in cases of reflection and refraction of light. An algorithm has been presented for the said purpose and it has been subsequently employed for the generation of relevant determinantal equations by considering some particular cases of reflection and refraction, based on which the readers will be able to reach the desired goal for all other remaining cases of reflection and refraction. Overall, two sets of such determinantal equations have been found to prevail in the present study. One resulted from handling the following cases: (i) Reflection at a plane reflecting surface, (ii) Refraction at a plane surface of discontinuity regardless of whether light moves from a rarer medium to a denser medium or from a denser medium to a rarer medium. The second one has been found while going through the following cases: (i) Reflection at a concave spherical reflecting surface, (ii) Reflection at a convex spherical reflecting surface, (iii) Refraction at a concave spherical surface of discontinuity regardless of whether light passes from a rarer medium to a denser medium or from a denser medium to a rarer medium, (iv) Refraction at a convex spherical surface of discontinuity regardless of whether light moves from a rarer medium to a denser medium or from a denser medium to a rarer medium

Addressing a novel problem on the basis of the generalized vectorial laws of reflection and refraction along with offering novel treatments of derivation of some results in geometrical optics

The generalized vectorial laws of reflection and refraction have been applied in this paper to develop novel determinantal equations involving the coordinates of the object point, image point, and point of incidence in cases of reflection and refraction of light when reflection (refraction) takes place from a plane reflecting surface (plane surface of discontinuity) or from a concave (convex) spherical reflecting surface (spherical surface of discontinuity). Establishment of the theoretical proof of the validity of the principle of reversibility of light has also been accomplished by making use of the novel determinantal equations developed in an earlier work by the author. Furthermore, looking at the interest of the readership, novel treatments for the derivation of the relationship between object distance and image distance in case of reflection of light at a plane reflecting surface as well as for the derivation of the relationships between real depth (height) and apparent depth (height) when refraction of light takes place at a plane surface of discontinuity are also offered in this paper.

Rank 3 quadratic generators of Veronese embeddings

Let $L$ be a very ample line bundle on a projective scheme $X$ defined over an algebraically closed field $\Bbbk$ with ${\rm char}\,\Bbbk \neq 2$. We say that $(X,L)$ satisfies property $\mathsf {QR}(k)$ if the homogeneous ideal of the linearly normal embedding $X \subset {\mathbb {P}} H^{0} (X,L)$ can be generated by quadrics of rank less than or equal to $k$. Many classical varieties, such as Segre–Veronese embeddings, rational normal scrolls and curves of high degree, satisfy property $\mathsf {QR}(4)$. In this paper, we first prove that if ${\rm char}\,\Bbbk \neq 3$ then $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (d))$ satisfies property $\mathsf {QR}(3)$ for all $n \geqslant 1$ and $d \geqslant 2$. We also investigate the asymptotic behavior of property $\mathsf {QR}(3)$ for any projective scheme. Specifically, we prove that (i) if $X \subset {\mathbb {P}} H^{0} (X,L)$ is $m$-regular then $(X,L^{d} )$ satisfies property $\mathsf {QR}(3)$ for all $d \geqslant m$, and (ii) if $A$ is an ample line bundle on $X$ then $(X,A^{d} )$ satisfies property $\mathsf {QR}(3)$ for all sufficiently large even numbers $d$. These results provide affirmative evidence for the expectation that property $\mathsf {QR}(3)$ holds for all sufficiently ample line bundles on $X$, as in the cases of Green and Lazarsfeld's condition $\mathrm {N}_p$ and the Eisenbud–Koh–Stillman determininantal presentation in Eisenbud et al. [Determinantal equations for curves of high degree, Amer. J. Math. 110 (1988), 513–539]. Finally, when ${\rm char}\,\Bbbk = 3$ we prove that $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (2))$ fails to satisfy property $\mathsf {QR}(3)$ for all $n \geqslant 3$.

Novel determinantal equations leading to the generalized vectorial laws of reflection and refraction

This paper makes use of two novel determinantal equations, each involving the coordinates of the point of incidence as well as those of a pair of points lying respectively on the incident ray and reflected (refracted) ray for the establishment of the theoretical proofs of the generalized vectorial laws of reflection and refraction discovered by the author in 2005. The treatment of derivation offered is novel and useful from the view point of academic interest. The present work will enhance the theoretical foundation of the generalized vectorial laws of reflection and refraction, there by enriching the traditional optical physics literature as well.

Del Pezzo surfaces, rigid line configurations and Hirzebruch–Kummer coverings

We prove the equisingular rigidity of the singular Hirzebruch–Kummer coverings X(n, $$\mathcal {L}$$ ) of the projective plane branched on line configurations $$\mathcal {L}$$ , satisfying some technical condition. In the case, $$\mathcal {L}=$$ the complete quadrangle, we give explicit equations of the Hirzebruch–Kummer covering $$S_n(=$$ the minimal desingularisation of $$X (n, \mathcal {L}))$$ in a product of four Fermat curves of degree n. Since $$S_n$$ is the $$(\mathbb {Z}/n)^5$$ covering of the Del Pezzo surface $$Y_5$$ of degree 5 branched on the 10 lines, these equations are derived from explicit equations of the image of $$Y_5$$ in $$(\mathbb {P}^1)^4$$ .We describe more generally determinantal equations for all Del Pezzo surfaces of degree $$9-k \le 6$$ as subvarieties of the k-fold product of the projective line.

CCOMP: An efficient algorithm for complex roots computation of determinantal equations

In this paper a free Python algorithm, entitled CCOMP (Complex roots COMPutation), is developed for the efficient computation of complex roots of determinantal equations inside a prescribed complex domain. The key to the method presented is the efficient determination of the candidate points inside the domain which, in their close neighborhood, a complex root may lie. Once these points are detected, the algorithm proceeds to a two-dimensional minimization problem with respect to the minimum modulus eigenvalue of the system matrix. In the core of CCOMP exist three sub-algorithms whose tasks are the efficient estimation of the minimum modulus eigenvalues of the system matrix inside the prescribed domain, the efficient computation of candidate points which guarantee the existence of minima, and finally, the computation of minima via bound constrained minimization algorithms. Theoretical results and heuristics support the development and the performance of the algorithm, which is discussed in detail. CCOMP supports general complex matrices, and its efficiency, applicability and validity is demonstrated to a variety of microwave applications. Program summaryProgram Title: CCOMPProgram Files doi:http://dx.doi.org/10.17632/x6fx4zssft.1Licensing provisions: GPLv3Programming language: PythonNature of problem: The determination of the resonances of a physical system, arising from determinantal type equations. These resonances arise from the non trivial solution of a homogeneous system of equations, whose system matrix depends on a parameter z, which can be either real or complex depending on the application. The values of z for which the determinant of the aforementioned system matrix is zero, are the resonances of the physical system.Solution method: An open-source software is developed for the efficient detection of all complex roots inside a prescribed complex domain D. The program is written in Python programming language [1] in conjunction with NumPy [2], SciPy [3], and Matplotlib [4]. The key to the method presented is the efficient determination of the candidate points inside D which in their close neighborhood the existence of a minimum is guaranteed. Once these points are detected, the algorithm proceeds to a two-dimensional minimization problem with respect to the minimum modulus eigenvalue of the system matrix, which is a positive function inside D. If the minimization yields global minima (near zero), the roots are found. For local minima (values of minimum modulus eigenvalue function away from zero), no roots exist. For all other points which have not been flagged as candidates, the algorithm does not proceed to the minimization problem.Additional comments including restrictions and unusual features: Python library psutil is used to compute memory consumption. [1]The Python programming language, https://www.python.org/.[2]NumPy, http://numpy.org/.[3]SciPy, http://scipy.org/.[4]Matplotlib, http://matplotlib.org/.

Darboux integrability of determinant and equations for principal minors

We consider equations that represent a constancy condition for a 2D Wronskian, mixed Wronskian–Casoratian and 2D Casoratian. These determinantal equations are shown to have the number of independent integrals equal to their order—this implies Darboux integrability. On the other hand, the recurrent formulae for the leading principal minors are equivalent to the 2D Toda equation and its semi-discrete and lattice analogues with particular boundary conditions (cut-off constraints). This connection is used to obtain recurrent formulae and closed-form expressions for integrals of the Toda-type equations from the integrals of the determinantal equations. General solutions of the equations corresponding to vanishing determinants are given explicitly while, in the non-vanishing case, they are given in terms of solutions of ordinary linear equations.

The distribution of solutions of a determinantal equation

In 1964, Linnik and Skubenko established the equidistribution of the integral points on the determinantal surface , where is a matrix with independent entries and is an increasing parameter. Their method involved reducing the problem by one dimension (that is, to the determinantal equations with a matrix). In this paper a more precise version of the Linnik-Skubenko reduction is proposed. It can be applied to a wider range of problems arising in the geometry of numbers and in the theory of three-dimensional Voronoi-Minkowski continued fractions. Bibliography: 24 titles.

Alternating trilinear forms on a nine-dimensional space and degenerations of (3, 3)-polarized Abelian surfaces

We give a detailed analysis of the semisimple elements, in the sense of Vinberg, of the third exterior power of a nine-dimensional vector space over an algebraically closed field of characteristic different from 2 and 3. To a general such element, one can naturally associate an Abelian surface X, which is embedded in eight-dimensional projective space. We study the combinatorial structure of this embedding and explicitly recover the genus 2 curve whose Jacobian variety is X. We also classify the types of degenerations of X that can occur. Taking the union over all Abelian surfaces in Heisenberg normal form, we get a five-dimensional variety which is a birational model for a genus 2 analog of Shioda's modular surfaces. We find determinantal set-theoretic equations for this variety and present some additional equations which conjecturally generate the radical ideal.

Exact Eigenfrequencies in Concentric Prolate Spheroidal-Spherical Metallic Cavities

In this letter the eigenfrequencies in prolate spheroidal-spherical and spherical-spheroidal cavities are calculated. The metallic boundaries of the cavities are perfect conductors. The matrix equations are constructed using expansions connecting the spherical and the spheroidal eigenvectors. Then, by truncating the resulting matrices, the exact eigenfrequencies are obtained as the converged roots of determinantal equations. This method allows to study elongated spheroidal cavities that existing methods fail to handle. The method is validated by comparison with published results from the literature. Original results for spheroids having large eccentricities are given for various values of the parameters.

Generic Identication of Binary-Valued Hidden Markov Processes

The generic identication problem is to decide whether a stochastic process (Xt) is ahidden Markov process and if yes to infer its parameters for all but a subset of parametrizationsthat form a lower-dimensional subvariety in parameter space. Partial answers so far availabledepend on extra assumptions on the processes, which are usually centered around stationarity.Here we present a general solution for binary-valued hidden Markov processes. Our approach isrooted in algebraic statistics hence it is geometric in nature. We nd that the algebraic varietiesassociated with the probability distributions of binary-valued hidden Markov processes are zerosets of determinantal equations which draws a connection to well-studied objects from algebra. Asa consequence, our solution allows for algorithmic implementation based on elementary (linear)algebraic routines.

Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes

We study the secant varieties of the Veronese varieties and of Veronese reembeddings of a smooth projective variety. We give some conditions, under which these secant varieties are set-theoretically cut out by determinantal equations. More precisely, they are given by minors of a catalecticant matrix. These conditions include the case when the dimension of the projective variety is at most 3 and the degree of reembedding is sufficiently high. This gives a positive answer to a set-theoretic version of a question of Eisenbud in dimension at most 3. For dimension four and higher we produce plenty of examples when the catalecticant minors are not enough to set-theoretically define the secant varieties to high degree Veronese varieties. This is done by relating the problem to smoothability of certain zero-dimensional Gorenstein schemes.

High rank linear syzygies on low rank quadrics

We study the linear syzygies of a homogeneous ideal $I \subseteq S = {\rm Sym}_{{\bf k}}(V)$, focussing on the graded betti numbers $$ b_{i,i+1} = {\rm dim}_{{\bf k}} {\rm Tor}_i (S/I,{\bf k})_{i+1}. $$ For a variety $X$ and divisor $D$ with $V = H^0(D)$, what conditions on $D$ ensure that $b_{i,i+1} \ne 0$? Eisenbud has shown that a decomposition $D \sim A +B$ such that $A$ and $B$ have at least two sections gives rise to determinantal equations (and corresponding syzygies) in $I_X$; and conjectured that if $I_2$ is generated by quadrics of rank $\le 4$, then the last nonvanishing $b_{i,i+1}$ is a consequence of such equations. We describe obstructions to this conjecture and prove a variant. The obstructions arise from toric specializations of the Rees algebra of Koszul cycles, and we give an explicit construction of toric varieties with minimal linear syzygies of arbitrarily high rank. This gives one answer to a question posed by Eisenbud and Koh.

Linear determinantal equations for all projective schemes

We prove that every projective embedding of a connected scheme determined by the complete linear series of a sufficiently ample line bundle is defined by the 2-minors of a 1-generic matrix of linear forms. Extending the work of Eisenbud-Koh-Stillman for integral curves, we also provide effective descriptions for such determinantally presented ample line bundles on products of projective spaces, Gorenstein toric varieties, and smooth n-folds.

Exact 3D elasticity solution for free vibrations of an eccentric hollow sphere

An exact three-dimensional elastodynamic analysis for describing the natural oscillations of a freely suspended, isotropic, and homogeneous elastic sphere with an eccentrically located inner spherical cavity is developed. The translational addition theorem for spherical vector wave functions is employed to impose the zero traction boundary conditions, leading to frequency equations in the form of exact determinantal equations involving spherical Bessel functions and Wigner 3j symbols. Extensive numerical calculations have been carried out for the first five clusters of eigenfrequencies associated with both the axisymmetric and non-axisymmetric spheroidal as well as toroidal oscillation modes for selected inner-outer radii ratios in a wide range of cavity eccentricities. Also, the corresponding three-dimensional deformed mode shapes are illustrated in vivid graphical forms for selected eccentricities. The numerical results describe the imperative influence of cavity eccentricity, mode type, and radii ratio on the vibrational characteristics of the hollow sphere. The existence of “multiple degeneracies” and the trigger of “frequency splitting” are demonstrated and discussed. The accuracy of solution is checked through appropriate convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data in the existing literature.

Gauss-Bruhat decomposition as an example of Thomas decomposition

Both the Gauss-Bruhat decomposition and the LU-decomposition of the general linear group over a field are examples of a Thomas decomposition of systems of polynomial equations and inequations into disjoint triangular systems, a recently rediscovered method of the nineteen-thirties, applied to the inequation det (A) ≠ 0 for an n × n-matrix of indeterminants. More specifically it is shown that the cells of the two decompositions can be described by determinantal equations and inequations yielding simple systems in the sense of Thomas of a rather special type, which are called split and allow counting solutions over any finite field.

The multivariate Selberg beta distribution and applications

The classical beta distribution defined on (0,1) is based on Euler's integral of the second type and its generalization to several variables, defined on a simplex, is based on Dirichlet's generalized form of Euler's integral. We define the multivariate Selberg beta random vector X=(X 1, …, X p ) ∼ MSBeta1 (α, β, γ, p), p≥1, defined on (0, 1) p . This distribution, based on the Selberg integral, a generalized form of the Dirichlet integral, has close relationships with the distributions of roots of determinantal equations and also has several interesting applications, such as the multivariate Gini mean difference.

A Home-Made Hartshorne-Serre Correspondence

We provide an elementary proof of the Hartshorne-Serre correspondence for constructing vector bundles from local complete intersection subschemes of codimension two. This will be done, as in the correspondence of hypersurfaces and line bundles, by patching together local determinantal equations in order to produce sections of a vector bundle