Independent Polynomial Research Articles

On the Non-Commuting Graph of the Group U6n

Let G be a finite group. The non-commuting graph of G is a simple graph Γ(G) whose vertices are elements of G∖Z(G), where Z(G) is the center of G, and two distinct vertices a and b are joint by an edge if ab≠ba. In this paper, we study the non-commuting graph of the group U6n. The independent number, clique and chromatic numbers of the non-commuting graph of the group U6n, Γ(U6n), are determined. Additionally, the resolving polynomial, total eccentricity and independent polynomials of Γ(U6n) are computed. Finally, the detour and eccentric connectivity indices of Γ(U6n) are found.

On the Independent Uphill Domination Polynomial of a Graph

Objectives: The concept of independent uphill domination polynomial is newly defined to deal with the present graph theory in this paper. It is characterized and analyzed within various contexts and types of graphs, its roots are discovered, its relation to different graph transforms is discussed and the study of this graph can help reveal new structures of graphs. Methods: The independent uphill domination polynomial is defined as is an independent uphill dominating set of size in . The uphill paths are used in identifying the criterion of independent uphill dominating sets. The computation of specific values of the polynomial is performed for basic graphs, and the polynomial’s response to various flip operations is observed. Findings: Checking the standard graphs for some properties of the independent uphill dominance polynomial shows the features of the independent uphill dominance polynomial as an uphill dominance polynomial of a certain type of digraph. Families of polynomials can be best understood when it comes to stability and characteristics by analyzing their roots on different graphical planes. They demonstrate how the structure of the graph and the polynomial change as a result of the different graph operations. Novelty: Incorporating uphill paths into independent dominating sets extends common domination polynomials and provides a distinct perspective distinct from previous investigations on graph dominance. Hence, this work enriches the subject and suggests new avenues for studying related polynomials and their uses by encompassing more structural properties of graphs. Keywords: Domination, Polynomial, Uphill domination, Independent domination, Root, Book graph, Independent uphill domination polynomial

Beyond conventional models: integer and fractional order analysis of nonlinear Michaelis-Menten kinetics in immobilised enzyme reactors

PurposeThis study explores the immobilisation of enzymes within porous catalysts of various geometries, including spheres, cylinders and flat pellets. The objective is to understand the irreversible Michaelis-Menten kinetic process within immobilised enzymes through advanced mathematical modelling.Design/methodology/approachMathematical models were developed based on reaction-diffusion equations incorporating nonlinear variables associated with Michaelis-Menten kinetics. This research introduces fractional derivatives to investigate enzyme reaction kinetics, addressing a significant gap in the existing literature. A novel approximation method, based on the independent polynomials of the complete bipartite graph, is employed to explore solutions for substrate concentration and effectiveness factor across a spectrum of parameter values. The analytical solutions generated through the bipartite polynomial approximation method (BPAM) are rigorously tested against established methods, including the Bernoulli wavelet method (BWM), Taylor series method (TSM), Adomian decomposition method (ADM) and fourth-order Runge-Kutta method (RKM).FindingsThe study identifies two main findings. Firstly, the behaviour of dimensionless substrate concentration with distance is analysed for planar, cylindrical and spherical catalysts using both integer and fractional order Michaelis-Menten modelling. Secondly, the research investigates the variability of the dimensionless effectiveness factor with the Thiele modulus.Research limitations/implicationsThe study primarily focuses on mathematical modelling and theoretical analysis, with limited experimental validation. Future research should involve more extensive experimental verification to corroborate the findings. Additionally, the study assumes ideal conditions and uniform catalyst properties, which may not fully reflect real-world complexities. Incorporating factors such as mass transfer limitations, non-uniform catalyst structures and enzyme deactivation kinetics could enhance the model’s accuracy and broaden its applicability. Furthermore, extending the analysis to include multi-enzyme systems and complex reaction networks would provide a more comprehensive understanding of biocatalytic processes.Practical implicationsThe validated bipartite polynomial approximation method presents a practical tool for optimizing enzyme reactor design and operation in industrial settings. By accurately predicting substrate concentration and effectiveness factor, this approach enables efficient utilization of immobilised enzymes within porous catalysts. Implementation of these findings can lead to enhanced process efficiency, reduced operating costs and improved product yields in various biocatalytic applications such as pharmaceuticals, food processing and biofuel production. Additionally, this research fosters innovation in enzyme immobilisation techniques, offering practical insights for engineers and researchers striving to develop sustainable and economically viable bioprocesses.Social implicationsThe advancement of enzyme immobilisation techniques holds promise for addressing societal challenges such as sustainable production, environmental protection and healthcare. By enabling more efficient biocatalytic processes, this research contributes to reducing industrial waste, minimizing energy consumption and enhancing access to pharmaceuticals and bio-based products. Moreover, the development of eco-friendly manufacturing practices through biocatalysis aligns with global efforts towards sustainability and mitigating climate change. The widespread adoption of these technologies can foster a more environmentally conscious society while stimulating economic growth and innovation in biotechnology and related industries.Originality/valueThis study offers a pioneering approximation method using the independent polynomials of the complete bipartite graph to investigate enzyme reaction kinetics. The comprehensive validation of this method through comparison with established solution techniques ensures its reliability and accuracy. The findings hold promise for advancing the field of biocatalysts and provide valuable insights for designing efficient enzyme reactors.

Strictly positive polynomials in the boundary of the SOS cone

We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which corresponds to polynomials that vanish at at least one point. We focus on the part of the boundary which is not shared, corresponding to strictly positive polynomials.For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4 variables, this boundary has been completely characterized by G. Blekherman. For the cases of a polynomial f in more variables or of higher degree, results by G. Blekherman, R. Sinn and M. Velasco and other authors based on a conjecture by Ottaviani and Paoletti give bounds for the maximum number of linearly independent polynomials that can appear in an SOS decomposition of f, or equivalently the maximum rank of the matrices in the Gram spectrahedron of f. We show that the same bounds can be obtained from the Eisenbud-Green-Harris conjecture. Combining theoretical results and computational techniques, we compute examples that allow us to prove the optimality of the bounds for all degrees and number of variables. Additionally, we give examples for the following problems: examples in the boundary of the cone that are the sum of less than n squares and have common complex roots, and examples of polynomials in the boundary with SOS length larger than the expected one from the dimension.

Multidimensional polynomial patterns over finite fields: Bounds, counting estimates and Gowers norm control

We examine multidimensional polynomial progressions involving linearly independent polynomials over finite fields, proving power saving bounds for sets lacking such configurations. This jointly generalises earlier results of Peluse (for the single dimensional case) and the author (for distinct degree polynomials). In contrast to the cases studied in the aforementioned two papers, a usual PET induction argument does not give Gowers norm control over multidimensional progressions that involve polynomials of the same degrees. The main challenge is therefore to obtain Gowers norm control, and we accomplish this for all multidimensional polynomial progressions with pairwise independent polynomials. The key inputs are: (1) a quantitative version of a PET induction scheme developed in ergodic theory by Donoso, Koutsogiannis, Ferré-Moragues and Sun, (2) a quantitative concatenation result for Gowers box norms in arbitrary finite abelian groups, motivated by (but different from) earlier results of Tao, Ziegler, Peluse and Prendiville; (3) an adaptation to combinatorics of the box norm smoothing technique, recently developed in the ergodic setting by the author and Frantzikinakis; and (4) a new version of the multidimensional degree lowering argument.

Independent domination polynomial for the cozero divisor graph of the ring of integers modulo n

Independent domination polynomial for the cozero divisor graph of the ring of integers modulo n

Effective rationality for local unitary invariants of mixed states of two qubits

We calculate the field of rational local unitary invariants for mixed states of two qubits, by employing methods from algebraic geometry. We prove that this field is rational (i.e. purely transcendental), and that it is generated by nine algebraically independent polynomial invariants. We do so by constructing a relative section, in the sense of invariant theory, whose Weyl group is a finite abelian group. From this construction, we are able to give explicit expressions for the generating invariants in terms of the Bloch matrix representation of mixed states of two qubits. We also prove similar rationality results for the local unitary invariants of symmetrically mixed states of two qubits. We also provide a sketch of how to generalize our results to the case of an arbitrary number of qubits. Our results apply to both complex-valued and real-valued invariants.

Sums of random polynomials with differing degrees

Let μ \mu and ν \nu be probability measures in the complex plane, and let p p and q q be independent random polynomials of degree n n , whose roots are chosen independently from μ \mu and ν \nu , respectively. Under assumptions on the measures μ \mu and ν \nu , the limiting distribution for the zeros of the sum p + q p+q was computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021), p. 124719] as n → ∞ n \to \infty . In this paper, we generalize and extend this result to the case where p p and q q have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of μ \mu and ν \nu , scaled by the limiting ratio of the degrees of p p and q q . Additionally, our approach provides a complete description of the limiting distribution for the zeros of p + q p + q for any pair of measures μ \mu and ν \nu , with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment.

Galois groups of random additive polynomials

We study the distribution of the Galois group of a random q q -additive polynomial over a rational function field: For q q a power of a prime p p , let f = X q n + a n − 1 X q n − 1 + … + a 1 X q + a 0 X f=X^{q^n}+a_{n-1}X^{q^{n-1}}+\ldots +a_1X^q+a_0X be a random polynomial chosen uniformly from the set of q q -additive polynomials of degree n n and height d d , that is, the coefficients are independent uniform polynomials of degree deg ⁡ a i ≤ d \deg a_i\leq d . The Galois group G f G_f is a random subgroup of GL n ⁡ ( q ) \operatorname {GL}_n(q) . Our main result shows that G f G_f is almost surely large as d , q d,q are fixed and n → ∞ n\to \infty . For example, we give necessary and sufficient conditions so that SL n ⁡ ( q ) ≤ G f \operatorname {SL}_n(q)\leq G_f asymptotically almost surely. Our proof uses the classification of maximal subgroups of GL n ⁡ ( q ) \operatorname {GL}_n(q) . We also consider the limits: q , n q,n fixed, d → ∞ d\to \infty and d , n d,n fixed, q → ∞ q\to \infty , which are more elementary.

The Perfect Codes of Non-coprime and Coprime Graphs

In this paper, we focus on the perfect and total perfect codes of the non-coprime and coprime graphs associated to the dihedral groups and finite Abelian groups. We used the advantage of independent sets and tried to present the independent polynomial for them.

Developing stem taper of Shorea robusta in the far-western Terai of Nepal

Tree taper functions expressed in terms of height and diameter at breast height (BDH) can provide accurate and timely information on current growing stock. Taper equations are very important for calculation of growing stock in forest inventories, however, in the context of Nepal, these are still unavailable at the required level. The study aimed to develop taper equations, so that those can be used to predict the diameter anywhere along the stem, and estimate tree volumes at desired sections. The input data was a destructive sampling method of 81 sample trees of Shorea robusta distributed in ten locations of the far western Terai of Nepal (Kailali and Kanchanpur districts).
 Using two independent B-Spline and 5th-degree polynomial taper models, the upper stem diameters were imputed. Both models were applied on the whole dataset, irrespective of DBH size, to get common fitted taper models. Later, the same models were tested for three different DBH classes to compare and evaluate the best-fit model. 
 Taper models, developed under the B-spline polynomial model of 3rd degree, were found to be highly dependent on the tree (DBH) sizes. Thus, better models could be developed by classifying the whole dataset based on different DBH classes. On the other hand, developed models under the 5th degree polynomial taper model did not exhibit their dependency on the tree (DBH) sizes and a better model was found for the unclassified datasets, i.e. for all DBH sizes.

Hierarchical Divergence-Conforming Vector Bases for Pyramid Cells

Divergence-conforming hierarchical vector bases for the pyramid consist of face- and volume-based functions obtained by a simple procedure that uses a new paradigm recently introduced by this author to produce pyramid bases. In order to define the bases’ order, the procedure starts by mapping the pyramids into a cube of a new Cartesian space, which we call the grandparent space, where the basis functions and their divergences take on polynomial form. Then we get the face-based functions of zero polynomial order and the volume-based functions of the first order. Functions of arbitrarily high order are obtained by multiplying the vector functions of the lowest order by independent scalar polynomials of higher order. Our face-based functions conform to those of other differently shaped elements to allow the use of hybrid meshes, while the multiplicative construction technique generates right away the volume-based basis functions. The completeness of the bases is demonstrated and all the basis functions we obtain are suitably normalized; their expression involves orthogonal polynomials which are easy to implement and alleviate the loss of linear independence.

Modeling the flexoelectric effect in semiconductors via a second-order collocation MFEM

In this paper, the flexoelectric effect in flexoelectric semiconductors (FSs) is studied via a second-order collocation mixed finite element method (MFEM) with the consideration of the strain gradient, electric field gradient, and drift-diffusion motion of carriers. In the present second-order collocation MFEM, the mechanical strain is assumed to be an independent quadratic polynomial function of local coordinates, and the kinematic constraint between the mechanical strain and displacement is approximately satisfied through a collocation method at 9 Gaussian quadrature points in each element. The setup for the electric field and electric potential also follows similar collocation strategy. Thus, except for the fundamental field variables, no additional DOFs are introduced. The accuracy of the proposed second-order collocation MFEM is verified by comparing numerical results with analytical solutions for a cantilever beam example. By performing the numerical experiments using the second-order collocation MFEM, the flexoelectric effect in FSs is simulated. Besides, we propose a novel mechanism for the converse flexoelectric effect in FSs which is induced by the diffusion motion of electrons. Numerical results indicate that by intensifying the diffusion motion of electrons, the converse flexoelectric effect, associated with the electric field gradient, could be enhanced.

Low degree Lorentz invariant polynomials as potential entanglement invariants for multiple Dirac spinors

A system of multiple spacelike separated Dirac particles is considered and a method for constructing polynomial invariants under the spinor representations of the local proper orthochronous Lorentz groups is described. The method is a generalization of the method used in Johansson (2022) for the case of two Dirac particles. All polynomials constructed by this method are identically zero for product states. The behaviour of the polynomials under local unitary evolution that acts unitarily on any subspace defined by fixed particle momenta is described. By design all of the polynomials have invariant absolute values on this kind of subspaces if the evolution is locally generated by zero-mass Dirac Hamiltonians. Depending on construction some polynomials have invariant absolute values also for the case of nonzero-mass or additional couplings. Because of these properties the polynomials are considered potential candidates for describing the spinor entanglement of multiple Dirac particles, with either zero or arbitrary mass or additional couplings. Polynomials of degree 2 and 4 are derived for the cases of three and four Dirac spinors. For three spinors no non-zero degree 2 polynomials are found but 67 linearly independent polynomials of degree 4 are identified. For four spinors 16 linearly independent polynomials of degree 2 are constructed as well as 26 polynomials of degree 4 selected from a much larger number. The relations of these polynomials to the polynomial spin entanglement invariants of three and four non-relativistic spin-12 particles are described. Moreover, it is described how degree 4 polynomials for five spinors can be constructed and how degree 2 polynomials can be constructed for any even number of spinors.

Spectra of algebras of analytic functions, generated by sequences of polynomials on Banach spaces, and operations on spectra

We consider the subalgebra of the Fréchet algebra of entire functions of bounded type, generated by a countable set of algebraically independent homogeneous polynomials on the complex Banach space $X.$ We investigate the spectrum of this subalgebra in the case $X = \ell_1.$ We also consider some shift type operations that can be performed on the spectrum of this subalgebra in the case $X = \ell_p$ with $p \geq 1$.

Unisolvence of Symmetric Node Patterns for Polynomial Spaces on the Simplex

Finite elements with polynomial basis functions on the simplex with a symmetric distribution of nodes should have a unique polynomial representation. Unisolvence not only requires that the number of nodes equals the number of independent polynomials spanning a polynomial space of a given degree, but also that the Vandermonde matrix controlling their mapping to the Lagrange interpolating polynomials can be inverted. Here, a necessary condition for unisolvence is presented for polynomial spaces that have non-decreasing degrees when going from the edges and the various faces to the interior of the simplex. It leads to a proof of a conjecture on a necessary condition for unisolvence, requiring the node pattern to be the same as that of the regular simplex.

Effects of circular plates on forced vibration of orthogonally stiffened cylindrical shell in wavenumber-frequency domain

A modified variational method for orthogonally stiffened cylindrical shell coupled with multiple circular plates is proposed in this paper. A unified variational modelling procedure is developed to accommodate three dimensional deformation restrains between the shell and substructures, including circular plates, ring and axial stiffeners. The vibration displacements are analytically expanded with Fourier series in the circumferential direction. The input energies due to external loads and the responses of the shell in circumferential wavenumber domain can be naturally obtained, which significantly facilitate the effect analysis of substructures. Close orthogonal independent polynomials are employed for the displacements in the axial direction to achieve satisfactory solutions. The convergence and accuracy of the present method are validated by comparing the analytical and numerical results. The effects of circular plates on vibration responses under different external forces are investigated. The contributions of different circumferential modes to the forced responses are analyzed, and reasons for the effects of the circular plates are then revealed in the circumferential wavenumber-frequency domain. The results indicate that the circular plates not only lead to frequency shift, but also coupling of the circumferential modes of the shell, which will respectively account for frequency shifts and increase of the resonance peaks.

A generalization of Gerzon’s bound on spherical s-distance sets

Using the method of linearly independent polynomials, we derive an upper bound for the cardinality of a spherical s-distance set $${\mathcal F}$$ where the sum of distinct inner products of any two elements from $${\mathcal F}$$ is zero. Our result generalizes the well-known Gerzon’s bound for the cardinality of an equiangular spherical set to a significantly broader class of spherical s-distance sets.

Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets: Collinearity Testing and Related Problems

We present subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets A, B, and C of points in the plane, that satisfy a certain polynomial equation or two equations. The best known instance of such a problem is testing for the existence of a collinear triple of points in \(A\times B\times C\), a classical 3SUM-hard problem that has so far defied any attempt to obtain a subquadratic solution, whether in the (uniform) real RAM model, or in the algebraic decision-tree model. While we are still unable to solve this problem, in full generality, in subquadratic time, we obtain such a solution, in the algebraic decision-tree model, that uses roughly \(O(n^{28/15})\) constant-degree polynomial sign tests, for the special case where two of the sets lie on two respective one-dimensional curves and the third is placed arbitrarily in the plane. Our technique is fairly general, and applies to many other problems where we seek a triple that satisfies a single polynomial equation, e.g., determining whether \(A\times B\times C\) contains a triple spanning a unit-area triangle. This result extends recent work by Barba et al. (2017) and by Chan (2018), where all three sets A, B, and C are assumed to be one-dimensional. As a second application of our technique, we again have three n-point sets A, B, and C in the plane, and we want to determine whether there exists a triple \((a,b,c) \in A\times B\times C\) that simultaneously satisfies two independent real polynomial equations. For example, this is the setup when testing for collinearity in the complex plane, when each of the sets A, B, C lies on some constant-degree algebraic curve. We show that problems of this kind can be solved with roughly \(O(n^{24/13})\) constant-degree polynomial sign tests.

Implementations and the independent set polynomial below the Shearer threshold

The independent set polynomial is important in many areas of combinatorics, computer science, and statistical physics. For every integer Δ≥2, the Shearer threshold is the value λ⁎(Δ)=(Δ−1)Δ−1/ΔΔ. It is known that for λ<−λ⁎(Δ), there are graphs G with maximum degree Δ whose independent set polynomial, evaluated at λ, is at most 0. Also, there are no such graphs for any λ>−λ⁎(Δ). This paper is motivated by the computational problem of approximating the independent set polynomial when λ<−λ⁎(Δ). The key issue in complexity bounds for this problem is “implementation”. Informally, an implementation of a real number λ′ is a graph whose hard-core partition function, evaluated at λ, simulates a vertex-weight of λ′ in the sense that λ′ is the ratio between the contribution to the partition function from independent sets containing a certain vertex and the contribution from independent sets that do not contain that vertex. Implementations are the cornerstone of intractability results for the problem of approximately evaluating the independent set polynomial. Our main result is that, for any λ<−λ⁎(Δ), it is possible to implement a set of values that is dense over the reals. The result is tight in the sense that it is not possible to implement a set of values that is dense over the reals for any λ>λ⁎(Δ). Our result has already been used in a paper with Bezáková (STOC 2018) to show that it is #P-hard to approximate the evaluation of the independent set polynomial on graphs of degree at most Δ at any value λ<−λ⁎(Δ). In the appendix, we give an additional incomparable inapproximability result (strengthening the inapproximability bound to an exponential factor, but weakening the hardness to NP-hardness).