Nonzero Determinant Research Articles

Constructing diffeomorphisms and homeomorphisms with prescribed derivative

We prove that for any measurable mapping T into the space of matrices with positive determinant, there is a diffeomorphism whose derivative equals T outside a set of measure less than ε. We use this fact to prove that for any measurable mapping T into the space of matrices with non-zero determinant (with no sign restriction), there is an almost everywhere approximately differentiable homeomorphism whose derivative equals T almost everywhere.

Involutions, links, and Floer cohomologies

AbstractWe develop a version of Seiberg–Witten Floer cohomology/homotopy type for a 4‐manifold with boundary and with an involution that reverses the structure, as well as a version of Floer cohomology/homotopy type for oriented links with nonzero determinant. This framework generalizes the previous work of the authors regarding Floer homotopy type for spin 3‐manifolds with involutions and for knots. Based on this Floer cohomological setting, we prove Frøyshov‐type inequalities that relate topological quantities of 4‐manifolds with certain equivariant homology cobordism invariants. The inequalities and homology cobordism invariants have applications to the topology of unoriented surfaces, the Nielsen realization problem for nonspin 4‐manifolds, and nonsmoothable unoriented surfaces in 4‐manifolds.

The determinant of semigroups of the pseudovariety ECOM

The purpose of this paper is to compute the nonzero semigroup determinant of the class of finite semigroups in which every two idempotents commute. This class strictly contains the class of finite semigroups that have central idempotents and the class of finite inverse semigroups. This computation holds significance in the context of the extension of the MacWilliams theorem for codes over semigroup algebras.

Cosmological constant Petrov type-N space–time in Ricci-inverse gravity

Our focus is on a specific type-N space–time that exhibits closed time-like curves in general relativity theory within the framework of Ricci-inverse gravity model. The matter-energy content is solely composed of a pure radiation field, and it adheres to the energy conditions while featuring a negative cosmological constant. One of the key findings in this investigation is the non-zero determinant of the Ricci tensor (Rμν), which implies the existence of an anti-curvature tensor (Aμν) and, as a consequence, an anti-curvature scalar (A≠R−1). Furthermore, we establish that this type-N space–time serves as a solution within modified gravity theories via the Ricci-inverse model, which involves adjustments to the cosmological constant (Λ) and the energy density (ρ) of the radiation field expressed in terms of a coupling constant. As a result, our findings suggest that causality violations remain possible within the framework of this Ricci-inverse gravity model, alongside the predictions of general relativity.

Enhancing Lie color algebras

Lie color algebras generalize Lie superalgebras. We adapt a construction of Bahturin and Pagon to create enhanced Lie color algebras. This construction offers a much-needed method for constructing simple Lie color algebras. To illustrate its applicability, we demonstrate how to enhance any simple Lie superalgebra and obtain a simple Lie color algebra. Additionally, we show that if a Lie color algebra has a nonzero determinant, this property extends to its enhancement. This ensures its universal enveloping algebra is semiprime. Extra conditions on the grading group provide a primeness criterion.

Some Properties of Chain and Threshold Graphs

Chain graphs and threshold graphs are special classes of graphs which have maximum spectral radius among bipartite graphs and connected graphs with given order and size, respectively. In this article, we focus on some of linear algebraic tools like rank, determinant, and permanent related to the adjacency matrix of these types of graphs. We derive results relating the rank and number of edges. We also characterize chain/threshold graphs with nonzero determinant and permanent.

A PACKAGE OF PROCEDURES AND FUNCTIONS FOR CONSTRUCTIONS AND INVERSION OF ANALYTIC MAPPINGS WITH UNIT JACOBIAN

The set of polynomial mappings of n-dimensional complex space into itself the Jacobian matrix of which has a constant nonzero determinant is known to be very vast in any dimension that exceeds one. The well-known Jacobian conjecture states that any such mapping is polynomially invertible. Even though the computation of the determinant of the Jacobian matrix is very well supported in modern computer algebra systems, the algorithmic inversion of a polynomial mapping is still a problem of considerable computational complexity. In this paper, we present a Mathematica package JC that can be used for construction and inversion of polynomial mappings and more general analytic mappings with the unit determinant of the Jacobian matrix. The package includes functions that allow one to algorithmically construct these mappings for a given dimension of the space of variables and a given degree of mapping components. The package, together with a library of datasets for testing it and results of computational experiments, is available for free public use at https://www.researchgate.net/publication/358409332_JC_Package_and_Datasets.

Higher-order symplectic integration techniques for molecular dynamics problems

The explicit symplectic difference schemes with a number of stages from 1 to 5 are considered for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. A general method for finding symplectic schemes of high order of accuracy using Gröbner bases is proposed. It is shown that it is possible to significantly increase the accuracy of symplectic schemes without increasing the number of stages by obtaining all possible real schemes with the aid of Gröbner bases. For the numbers of stages 2, 3, and 4, the parameters of Runge–Kutta–Nyström (RKN) schemes and partitioned Runge–Kutta (PRK) schemes are obtained using the Gröbner basis technique. It is shown that there exists only one explicit invertible (symmetric) two-stage RKN scheme. In the cases of the vanishing Vandermonde determinant, 20 new real four-stage RKN schemes and 18 new PRK schemes of Forest–Ruth and Yoshida were found in the analytic form with the aid of Gröbner bases. In the cases of the nonzero Vandermonde determinant, four symmetric four-stage RKN schemes were derived in the analytic form with the aid of the Gröbner basis technique. In addition, for the number of stages 4 and 5, new nonsymmetric schemes were found using the Nelder–Mead numerical optimization method. In particular, four new schemes were obtained for the number of stages 4. For the number of stages 5, three new schemes were obtained in addition to four schemes known earlier. For each specific number of stages, a scheme has been found that is the best in terms of the minimum of the leading term of the approximation error. The relations have been established between the polynomials entering the expansions of the errors of each scheme for the momentum and for the particle coordinate. Verification of the schemes was carried out on a problem that has an exact solution. It is shown that the symplectic five-stage RKN scheme provides a more accurate conservation of the total energy balance of the particle system than schemes of lower orders of accuracy. The stability studies of the schemes were performed using the Mathematica software package.

New integrable (2+1)- and (3+1)-dimensional shallow water wave equations: multiple soliton solutions and lump solutions

PurposeThis study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author derives multiple soliton solutions and a class of lump solutions which are rationally localized in all directions in space.Design/methodology/approachThe author uses the simplified Hirota’s method and lump technique for determining multiple soliton solutions and lump solutions as well. The author shows that the developed (2+1)- and (3+1)-dimensional models are completely integrable in in the Painlené sense.FindingsThe paper reports new Painlevé-integrable extended equations which belong to the shallow water wave medium.Research limitations/implicationsThe author addresses the integrability features of this model via using the Painlevé analysis. The author reports multiple soliton solutions for this equation by using the simplified Hirota’s method.Practical implicationsThe obtained lump solutions include free parameters; some parameters are related to the translation invariance and the other parameters satisfy a non-zero determinant condition.Social implicationsThe work presents useful algorithms for constructing new integrable equations and for the determination of lump solutions.Originality/valueThe paper presents an original work with newly developed integrable equations and shows useful findings of solitary waves and lump solutions.

Characteristics of the interaction behavior between solitons in (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation

Starting with Hirota bilinear form of (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation, a class of lump solutions, a strip soliton, a pair of resonance solitons as well as the rogue wave have been obtained through symbolic computation. Firstly, two lump solutions are derived, consisting of six free parameters, while the other three accord with non-zero determinant condition which guarantee locality and analyticity for the solutions. Secondly, the interaction phenomenon between a lump wave and a stripe soliton is reflected. Interaction figures show that the lump wave can be absorbed by the stripe wave or separate from it. Thirdly, by a novel ansatz of hyperbolic function and quadratic function, a fresh transformation for the rogue wave and a pair of resonance solitons have been discovered, that a peak profiled rogue wave emerges from one of the resonance wave and transfers to another, and finally fades away. At the same time, numerical simulations are demonstrated to visualize the mechanism of the above solutions.

English

Sanchez et al. (2005) have shown the structure of vector space genetic code over the Galois field GF(4) which relate to the physicochemical properties of the genetic code on proteins. From this vector space, an automorphism can be constructed to reflect the mutation process in the genetic code. This study investigates a type of transformation in the LK_ITB5a lipase gene mutation. The result of the study shows that there is a transformation matrix from the wild type gene lipase LK_ITB5a to mutant gene lipase LK_ITB5a H110F which is a diagonal matrix with non-zero determinant. This means that the transformation is an automorphism. Key words: Automorphism, vector space, mutational pathways &nbsp

Generating set for nonzero determinant links under skein relation

Traditionally introduced in terms of advanced topological constructions, many link invariants may also be defined in much simpler terms given their values on a few initial links and a recursive formula on a skein triangle. The crucial question to ask is how many initial values are necessary to completely determine such a link invariant. We focus on a specific class of invariants known as nonzero determinant link invariants and restate the objective by considering a set S of links subject to a closure condition under a given skein triangle. Then the aim is to determine a minimal set of initial generators so that S is the set of all links with nonzero determinant. We show that only the unknot is required as a generator if the skein triangle is unoriented. For oriented skein triangles, we show that the unknot and Hopf link orientations form a set of generators.

Lump solutions of a new generalized Kadomtsev–Petviashvili equation

A new generalized Kadomtsev–Petviashvili (GKP) equation is derived from a bilinear differential equation by taking the transformation [Formula: see text]. By symbolic computation with Maple, lump solutions, rationally localized in all directions in the space, to the GKP equation are presented. The obtained lump solutions contain a set of six free parameters, four of which should satisfy a nonzero determinant condition. As special examples, six particular lump solutions are constructed and depicted with [Formula: see text].

Khovanov homology and binary dihedral representations for marked links

We introduce a version of Khovanov homology for alternating links with marking data, ω, inspired by instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology introduced in Kronheimer and Mrowka (2011) for this marked Khovanov homology collapses on the E2 page for alternating links. We moreover show that for non-split links the Khovanov homology we introduce for alternating links does not depend on ω; thus, the instanton homology also does not depend on ω for non-split alternating links. Finally, we study a version of binary dihedral representations for links with markings, and show that for links of non-zero determinant, this also does not depend on ω.

On the Chebotarëv theorem over finite fields

Let p be a prime number and let Vp be the Vandermonde matrix with (i,j)-entry equal to ωij for 0≤i,j≤p−1, where ω is a primitive pth root of unity in the complex field. The classical Chebotarëv theorem says that all square submatrices of Vp have nonzero determinant. In this paper, we establish the Chebotarëv theorem over finite fields by imposing certain conditions.

Lump Solutions to a (2+1)-Dimensional Fifth-Order KdV-Like Equation

A (2+1)-dimensional fifth-order KdV-like equation is introduced through a generalized bilinear equation with the prime number p=5. The new equation possesses the same bilinear form as the standard (2+1)-dimensional fifth-order KdV equation. By Maple symbolic computation, classes of lump solutions are constructed from a search for quadratic function solutions to the corresponding generalized bilinear equation. We get a set of free parameters in the resulting lump solutions, of which we can get a nonzero determinant condition ensuring analyticity and rational localization of the solutions. Particular classes of lump solutions with special choices of the free parameters are generated and plotted as illustrative examples.

Rank Function and Outer Inverses

For the class of matrices over a field, the notion of `rank of a matrix' as defined by `the dimension of subspace generated by columns of that matrix' is folklore and cannot be generalized to the class of matrices over an arbitrary commutative ring. The `determinantal rank' defined by the size of largest submatrix having nonzero determinant, which is same as the column rank of given matrix when the commutative ring under consideration is a field, was considered to be the best alternative for the `rank' in the class of matrices over a commutative ring. Even this determinantal rank and the McCoy rank are not so efficient in describing several characteristics of matrices like in the case of discussing solvability of linear system. In the present article, the `rank--function' associated with the matrix as defined in [{\it Solvability of linear equations and rank--function}, K. Manjunatha Prasad, \url{http://dx.doi.org/10.1080/00927879708825854}] is discussed and the same is used to provide a necessary and sufficient condition for the existence of an outer inverse with specific column space and row space. Also, a rank condition is presented for the existence of Drazin inverse, as a special case of an outer inverse, and an iterative procedure to verify the same in terms of sum of principal minors of the given square matrix over a commutative ring is discussed.

A note on surjectivity of piecewise affine mappings

A standard theorem in nonsmooth analysis states that a piecewise affine function $$F:\mathbb {R}^n\rightarrow \mathbb {R}^n$$ is surjective if it is coherently oriented in that the linear parts of its selection functions all have the same nonzero determinant sign. In this note we prove that surjectivity already follows from coherent orientation of the selection functions which are active on the unbounded sets of a polyhedral subdivision of the domain corresponding to F. A side bonus of the argumentation is a short proof of the classical statement that an injective piecewise affine function is coherently oriented.

The snapback repellers for chaos in multi-dimensional maps

The key of Marotto's theorem on chaos for multi-dimensional maps is the existence of snapback repeller. For practical application of the theory, locating a computable repelling neighborhood of the repelling fixed point has thus become the key issue. For some multi-dimensional maps \begin{document}$F$\end{document} , basic information of \begin{document}$F$\end{document} is not sufficient to indicate the existence of snapback repeller for \begin{document}$F$\end{document} . In this investigation, for a repeller \begin{document}$\bar{\bf z}$\end{document} of \begin{document}$F$\end{document} , we start from estimating the repelling neighborhood of \begin{document}$\bar{\bf z}$\end{document} under \begin{document}$F^{k}$\end{document} for some \begin{document}$k ≥ 2$\end{document} , by a theory built on the first or second derivative of \begin{document}$F^k$\end{document} . By employing the Interval Arithmetic computation, we locate a snapback point \begin{document}${\bf z}_0$\end{document} in this repelling neighborhood and examine the nonzero determinant condition for the Jacobian of \begin{document}$F$\end{document} along the orbit through \begin{document}${\bf z}_0$\end{document} . With this new approach, we are able to conclude the existence of snapback repellers under the valid definition, hence chaotic behaviors, in a discrete-time predator-prey model, a population model, and the FitzHugh nerve model.

Lump Solutions to a (3+1)-Dimensional Potential-Yu–Toda–Sasa–Fukuyama (YTSF) Like Equation

The main motivation of our article is to implement generalized bilinear differential operators to create a (3+1)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) like equation linked with prime number $$p=3$$ . Specific features of lump solutions realistically localized in all directions of the (3+1)-dimensional potential-YTSF like equation is founded from bilinear closed form of it. Six free parameters are exist in the achieved lump solutions. Among them two parameters are owing to the translation invariance of the YTSF equation and the remaining parameters suit a non-zero determinant condition. Some 3D plots and several contour plots of the lump solutions with demanding choices of the existing parameters are made to show energy distribution of the lump waves and can be used to visualize the other properties of rogue wave phenomena.