Related papers: On Differential equation of invariants of binary f…
In this paper, we deal with the problem of uniqueness of minimal system of binomial generators of a semigroup ideal. Concretely, we give different necessary and/or sufficient conditions for uniqueness of such minimal system of generators.…
This paper is concerned with a minimal resolution of the PROP for bialgebras. We prove a theorem about the form of this resolution (Theorem 15) and give, in Section 5, a lot of explicit formulas for the differential. Our minimal model…
We construct six unitary trace invariants for 2 by 2 quaternionic matrices which separate the unitary similarity classes of such matrices, and show that this set is minimal. We prove two quaternionic versions of a well known…
In this paper, we propose an approach to automatically compute invariant clusters for semialgebraic hybrid systems. An invariant cluster for an ordinary differential equation (ODE) is a multivariate polynomial invariant g(u,x)=0, parametric…
The ring of integers and the discriminant are determined for number fields which are simple radical extensions.
We consider a class of particular solutions to the (2+1)-dimensional nonlinear partial differential equation (PDE) $u_t +\partial_{x_2}^n u_{x_1} - u_{x_1} u =0$ (here $n$ is any integer) reducing it to the ordinary differential equation…
In this work we consider a given root of a family of n-degree polynomials as a one-variable function that depends only on the independent term. Then we prove that this function satisfies several ordinary differential equations (ODE). More…
We show that binary forms of degree n less than seven parameterize rings, generalizing a result of Levi on binary cubic forms parameterizing cubic rings, which can be related to results of Bhargava. The question of whether or not a binary…
Einstein equations for several matter sources in Robertson-Walker and Bianchi I type metrics, are shown to reduce to a kind of second order nonlinear ordinary differential equation $\ddot{y}+\alpha f(y)\dot{y}+\beta f(y)\int{f(y) dy}+\gamma…
We classify small binary bibraces, using the correspondence with alternating algebras over the field F2, up to dimension eight, also determining their isomorphism classes. These finite-dimensional algebras, defined by an alternating…
It has recently been shown that a minimal reversible nonsymmetric ring has order 256 answering a questioned original posed in a paper on a taxonomy of 2-primal rings. Answers to similar questions on minimal rings relating to this taxonomy…
We determine the structure of the graded ring of Siegel modular forms of degree 3. It is generated by 19 modular forms, among which we identify a homogeneous system of parameters with 7 forms of weights 4, 12, 12, 14, 18, 20 and 30. We also…
Entangling properties of a mixed 2-qubit system can be described by the local homogeneous unitary invariant polynomials in elements of the density matrix. The structure of the corresponding invariant polynomial ring for the special subclass…
Given an integer $d \ge 2$, what is the least $r$ so that there is a set of binary quadratic forms $\{f_1,\dots,f_r\}$ for which $\{f_j^d\}$ is non-trivially linearly dependent? We show that if $r \le 4$, then $d \le 5$, and for $d \ge 4$,…
Using the only admissible rank-two realisations of the Lie algebra of the affine group in one dimension in terms of the Lie algebra of Lie symmetries of the Ermakov-Pinney (EP) equation, some classes of second order nonlinear ordinary…
We produce minimal integrity bases for both isotropic and hemitropic invariant algebras (and more generally covariant algebras) of most common bidimensional constitutive tensors and -- possibly coupled -- laws, including piezoelectricity…
The goal and the main result of the paper is to provide a complete description of the field of rational differential invariants of one class of second order ordinary differential equations with scalar control parameter with respect to Lie…
The number $\nu_{n,d}(k)$ of linearly independed homogeneous invariants of degree $k$ for the $n$-ary form of degree $d$ is calculated.
In the present article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight $2,4$ and 6. We define Hecke operators on them, find some analytic relations between these…
Although degree bounds and algorithms for the generators of various invariant rings have been known for decades, little is known about the cardinality of minimal generating sets. Estimates of such would provide lower bounds for the runtime…