Odd Polynomial Research Articles

Interlacing of zeros of odd period polynomials

The work of Conrey, Farmer and Imamoglu shows that all nontrivial zeros of the odd period polynomial of a Hecke eigenform of level one lie on the unit circle. We further show that nontrivial zeros of odd period polynomials of Hecke eigenforms of different weights share certain interlacing property.

On the construction of certain odd degree irreducible polynomials over finite fields

On the construction of certain odd degree irreducible polynomials over finite fields

Reaction-diffusion equations on metric graphs with edge noise

We investigate stochastic reaction-diffusion equations on finite metric graphs. On each edge in the graph a multiplicative cylindrical Gaussian noise driven reaction-diffusion equation is given. The vertex conditions are the standard continuity and generalized, non-local Neumann-Kirchhoff-type law in each vertex. The reaction term on each edge is assumed to be an odd degree polynomial, not necessarily of the same degree on each edge, with possibly stochastic coefficients and negative leading term. The model is a generalization of the problem in [14] where polynomials with much more restrictive assumptions are considered and no first order differential operator is involved. We utilize the semigroup approach from [15] to obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph.

Hierarchical stability conditions for linear systems with interval time-varying delay

This paper proposes the hierarchical stability conditions for linear systems with an interval time-varying delay. In the matter of the delay, the upper and lower bounds are restricted, but there is no constraint on its derivative. Firstly, based on the state vectors and multiple integral state vectors involved in the delay-related generalized free-matrix-based integral inequalities (GFIIs), the hierarchical Lyapunov-Krasovskii functional (LKF) candidates are constructed. Then, the GFIIs are applied for the approximation of the integral quadratic terms, which will introduce the delay related nonlinear terms in the LKF differential. Next, the novel adjusted matrix-valued high and odd degree polynomial negative definite conditions (NDCs) are provided to acquire the hierarchical linear matrix inequality (LMI) conditions and cope with the nonlinear terms introduced by the GFIIs. Eventually, the advantages of the obtained stability conditions are checked through some numeric examples.

Bi-Unitary Superperfect Polynomials over 𝔽2 with at Most Two Irreducible Factors

A divisor B of a nonzero polynomial A, defined over the prime field of two elements, is unitary (resp. bi-unitary) if gcd(B,A/B)=1 (resp. gcdu(B,A/B)=1), where gcdu(B,A/B) denotes the greatest common unitary divisor of B and A/B. We denote by σ**(A) the sum of all bi-unitary monic divisors of A. A polynomial A is called a bi-unitary superperfect polynomial over F2 if the sum of all bi-unitary monic divisors of σ**(A) equals A. In this paper, we give all bi-unitary superperfect polynomials divisible by one or two irreducible polynomials over F2. We prove the nonexistence of odd bi-unitary superperfect polynomials over F2.

An Extremal Problem for Odd Univalent Polynomials

An Extremal Problem for Odd Univalent Polynomials

A polynomial constitutive model of shape memory alloys based on kinematic hardening

This paper describes the derivation of a phenomenological model for shape memory alloys under the framework of classical plasticity theory. The proposed model combines the Souza constitutive approach with kinematic hardening; the model requires solving only one nonlinear equation rather than several nonlinear ones, therefore increasing the computational efficiency and convergence. Moreover, the original Souza model is improved by adding an odd polynomial function to describe the phase transformation of the shape memory alloys, making it possible to use a lower number of parameters for the inverse identification of the constitutive properties of SMAs from simple tensile tests. A tangent stiffness formulation is also derived to simulate the variation of the elastic modulus during the phase transformation. The tangent stiffness formulation proposed here extends the one used in classical plasticity and improves the convergence of the proposed model. The reliability and fidelity of the model described in this work are benchmarked against experimental data and other models. The numerical results show that the proposed phenomenological approach can describe well the pseudoelasticity and shape memory effect of shape memory alloys. The formulation described in this paper can be readily generalized to finite strains and other formulations based on existing formulations related to classical plasticity theory.

Virtual Knots with Properties of Kishino's Knot

Satoh and Taniguchi defined an invariant of virtual knots $J_n$ for a non-zero integer $n$. It is called the $n$-writhe. The $n$-writhes give the coefficients of some polynomial invariants for virtual knots including the index polynomial, the odd writhe polynomial and the affine index polynomial. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The unknotting number by the virtualization is called the virtual unknotting number. Kishino's knot is a virtual unknotting number one knot which has the trivial $n$-writhe and the trivial Jones polynomial. In this paper, we construct infinitely many virtual knots which have the same properties as Kishino's knot.

The zeros of odd period polynomials for newforms on $$\Gamma _0(2)$$

The zeros of odd period polynomials for newforms on $$\Gamma _0(2)$$

Asymptotic behavior of Ext for pairs of modules of large complexity over graded complete intersections

Let M and N be finitely generated graded modules over a graded complete intersection R such that \({\text {Ext}}_R^i(M,N)\) has finite length for all \(i\gg 0\). We show that the even and odd Hilbert polynomials, which give the lengths of \({\text {Ext}}^i_R(M,N)\) for all large even i and all large odd i, have the same degree and leading coefficient whenever the highest degree of these polynomials is at least the dimension of M or N. Refinements of this result are given when R is regular in small codimensions.

Detection of subcritical Hopf and fold bifurcations in an aeroelastic system via the Describing Function method

The paper deals with Limit Cycle Oscillations (LCOs) in a Pitch & Plunge model of the wing dynamics, where the stiffness on the plunge displacement is assumed to be an odd fifth-order polynomial. First, it is shown that the model dynamics can be equivalently described via a Lur’e system, i.e., the feedback interconnection between a linear time-invariant subsystem, whose transfer function depends on the wind speed, and a nonlinear memoryless one. Then, the Describing Function (DF) method is used to establish the existence of Predicted Limit Cycles (PLCs) whose amplitude can be analytically expressed in terms of the wind speed, once the intersections of the Nyquist diagram of the linear subsystem with the real axis are obtained. The related bifurcation diagram permits to predict that a stable equilibrium point and two LCOs (one unstable and one stable) coexist for wind speeds smaller than the flutter velocity, at which a subcritical Hopf bifurcation occurs, and larger than the value at which a cyclic fold bifurcation is detected. The closeness of each PLC to the related LCO is measured via the so-called distortion index, which also admits a direct expression in terms of the wind speed. The comparison of the predicted bifurcation diagram with that obtained via numerical simulations confirms the quite good accuracy of the PLCs obtained via the DF method, especially when the distortion index is small.

Eigenvalue embedding problem for quadratic regular matrix polynomials with symmetry structures

In this paper, we consider structure-preserving eigenvalue embedding problem (SEEP) for quadratic regular matrix polynomials with symmetry structures. First, we determine perturbations of a quadratic matrix polynomial, unstructured or structured, such that the perturbed polynomials reproduce a desired invariant pair while maintaining the invariance of another invariant pair of the unperturbed polynomial. If the latter is unknown, it is referred to as no spillover perturbation. Then we use these results for solving the SEEP for structured quadratic matrix polynomials that include: symmetric, Hermitian, ⋆-even and ⋆-odd quadratic matrix polynomials. Finally, we show that the obtained analytical expressions of perturbations can realize existing results for structured polynomials that arise in real-world applications, as special cases.

A New Approach for Estimating Parameters in PV Cell Models based on Odd Polynomial Regression

A New Approach for Estimating Parameters in PV Cell Models based on Odd Polynomial Regression

Critical functions and inf-sup stability of Crouzeix-Raviart elements

In this paper, we prove that Crouzeix-Raviart finite elements of polynomial order p≥5, p odd, are inf-sup stable for the Stokes problem on triangulations. For p≥4, p even, the stability was proved by Á. Baran and G. Stoyan in 2007 by using the macroelement technique, a dimension formula, the concept of critical points in a triangulation and a representation of the corresponding critical functions. Baran and Stoyan proved that these critical functions belong to the range of the divergence operator applied to Crouzeix-Raviart velocity functions and the macroelement technique implies the inf-sup stability.The generalization of this theory to cover odd polynomial orders p≥5 is involved; one reason is that the macroelement classes, which have been used for even p, are unsuitable for odd p. In this paper, we introduce a new and simple representation of non-conforming Crouzeix-Raviart basis functions of odd degree. We employ only one type of macroelement and derive representations of all possible critical functions. Finally, we show that they are in the range of the divergence operator applied to Crouzeix-Raviart velocities from which the stability of the discretization follows.

Weak universality of dynamical Φ34: polynomial potential and general smoothing mechanism

We consider a class of stochastic reaction-diffusion equations on the three dimensional torus. The non-linearities are odd polynomials in the weakly non-linear regime, and the smoothing mechanisms are very general higher order perturbations of the Laplacian. The randomness is the space-time white noise without regularisation. We show that these processes converge to the dynamical Φ34(λ) model, where the coupling constant λ has an explicit expression involving non-trivial interactions between all details of the smoothing mechanism and the non-linearity, even though they all formally vanish in the limit.

Failure of Approximation of Odd Functions by Odd Polynomials

We construct a Hilbert holomorphic function space H on the unit disk such that the polynomials are dense in H, but the odd polynomials are not dense in the odd functions in H. As a consequence, there exists a function f in H that lies outside the closed linear span of its Taylor partial sums \(s_n(f)\), so it cannot be approximated by any triangular summability method applied to the \(s_n(f)\). We also show that there exists a function f in H that lies outside the closed linear span of its radial dilates \(f_r, ~r<1\).

Iterated differences sets, Diophantine approximations and applications

Let v be an odd real polynomial (i.e. a polynomial of the form ∑j=1ℓajx2j−1). We utilize sets of iterated differences to establish new results about sets of the form R(v,ϵ)={n∈N|‖v(n)‖<ϵ} where ‖⋅‖ denotes the distance to the closest integer. We then apply the new Diophantine results to obtain applications to ergodic theory and combinatorics. In particular, we obtain a new characterization of weakly mixing systems as well as a new variant of Furstenberg-Sárközy theorem.

Stochastic reaction\u2013diffusion equations on networks

We consider stochastic reaction–diffusion equations on a finite network represented by a finite graph. On each edge in the graph, a multiplicative cylindrical Gaussian noise-driven reaction–diffusion equation is given supplemented by a dynamic Kirchhoff-type law perturbed by multiplicative scalar Gaussian noise in the vertices. The reaction term on each edge is assumed to be an odd degree polynomial, not necessarily of the same degree on each edge, with possibly stochastic coefficients and negative leading term. We utilize the semigroup approach for stochastic evolution equations in Banach spaces to obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph. In order to do so, we generalize existing results on abstract stochastic reaction–diffusion equations in Banach spaces.

On Estimating Nonlinear Frequency Modulated Radar Signals in Low SNR Environments

Estimating the signal transmitted by a noncooperating radar is of great use in radar countermeasures. In this article, we propose to improve the noise resilience of the radar signal estimation by constraining the estimate to be in a low-dimensional signal space. In particular, we focus on estimating nonlinear frequency modulated radar signals that have good target resolution characteristics. A flexible 3-D odd polynomial frequency signal model is introduced as the low-dimensional signal space to compute accurate estimates at low signal-to-noise ratios (SNRs) with small number of intercepted signals. The quality of the low-dimensional signal estimates is compared with that of the estimate from the unconstrained high-dimensional space computed using principal component analysis estimator, which is the method in predominant use in the multisensor scenario. It is shown that signal estimation through low-dimensional signal spaces yields more accurate estimates at low SNRs with a smaller number of sensors compared to searching in the unconstrained high-dimensional signal space.

On the Direct Limit from Pseudo Jacobi Polynomials to Hermite Polynomials

In this short communication, we present a new limit relation that reduces pseudo-Jacobi polynomials directly to Hermite polynomials. The proof of this limit relation is based upon 2F1-type hypergeometric transformation formulas, which are applicable to even and odd polynomials separately. This limit opens the way to studying new exactly solvable harmonic oscillator models in quantum mechanics in terms of pseudo-Jacobi polynomials.