Nilpotency Property Research Articles

Lie Properties of Symmetric Elements Under Oriented Involutions

We study the Lie nilpotency and Lie n-Engel properties of symmetric elements under oriented involutions. In some cases, the results obtained are similar to the ones in the case of the classical involution, but interesting new situations arise.

Decidability and undecidability in cellular automata

We survey results on decidability questions concerning cellular automata. Properties discussed include reversibility and surjectivity and their variants, time-symmetry and conservation laws, nilpotency and other properties of the limit set and the trace, properties chaoticity related such as sensitivity to initial conditions and mixing of the space, and dynamics from finite initial configurations. We also discuss briefly the tiling problem and its variants, and consider the influence of the dimension of the space on the decidability status of the questions.

Nilpotent ideals in polynomial and power series rings

Given a ring R and polynomials f(x), g(x) ∈ R[x] satisfying f(x)Rg(x) = 0, we prove that the ideal generated by products of the coefficients of f(x) and g(x) is nilpotent. This result is generalized, and many well known facts, along with new ones, concerning nilpotent polynomials and power series are obtained. We also classify which of the standard nilpotence properties on ideals pass to polynomial rings or from ideals in polynomial rings to ideals of coefficients in base rings. In particular, we prove that if I ≤ R[x] is a left T-nilpotent ideal, then the ideal formed by the coefficients of polynomials in I is also left T-nilpotent.

Globally nilpotent differential operators and the square Ising model

We recall various multiple integrals with one parameter, related to the isotropic square Ising model, and corresponding, respectively, to the n-particle contributions of the magnetic susceptibility, to the (lattice) form factors, to the two-point correlation functions and to their λ-extensions. The univariate analytic functions defined by these integrals are holonomic and even G-functions: they satisfy Fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties. We recall the explicit forms, found in previous work, of these Fuchsian equations, as well as their Russian-doll and direct sum structures. These differential operators are selected Fuchsian linear differential operators, and their remarkable properties have a deep geometrical origin: they are all globally nilpotent, or, sometimes, even have zero p-curvature. We also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known. Focusing on the factorized parts of all these operators, we find out that the global nilpotence of the factors (resp. p-curvature nullity) corresponds to a set of selected structures of algebraic geometry: elliptic curves, modular curves, curves of genus five, six,…, and even a remarkable weight-1 modular form emerging in the three-particle contribution χ(3) of the magnetic susceptibility of the square Ising model. Noticeably, this associated weight-1 modular form is also seen in the factors of the differential operator for another n-fold integral of the Ising class, Φ(3)H, for the staircase polygons counting, and in Apéry's study of ζ(3). G-functions naturally occur as solutions of globally nilpotent operators. In the case where we do not have G-functions, but Hamburger functions (one irregular singularity at 0 or ∞) that correspond to the confluence of singularities in the scaling limit, the p-curvature is also found to verify new structures associated with simple deformations of the nilpotent property.

On the rigidity of the cotangent complex at the prime 2

In [D. Quillen, On the (co)homology of commutative rings, Proc. Symp. Pure Math. 17 (1970) 65–87; L. Avramov, Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology, Annals of Math. 2 (150) (1999) 455–487] a conjecture was posed to the effect that if R → A is a homomorphism of Noetherian commutative rings then the flat dimension, as defined in the derived category of A -modules, of the associated cotangent complex L A / R satisfies: fd A L A / R < ∞ ⟹ fd A L A / R ≤ 2 . The aim of this paper is to initiate an approach for solving this conjecture when R has characteristic 2 using simplicial algebra techniques. To that end, we obtain two results. First, we prove that the conjecture can be reframed in terms of certain nilpotence properties for the divided square γ 2 and the André operation ϑ as it acts on Tor R ( A , ℓ ) , ℓ any residue field of A . Second, we prove the conjecture is valid in two cases: when fd R A < ∞ and when R is a Cohen–Macaulay ring.

Time-optimal control of disturbance-rejection tracking systems for discrete-time time-delayed systems by state feedback

In this paper, we solve the disturbance-rejection and tracking problem for linear multivariable discrete-time systems with time-delayed controlled inputs. A set of necessary and sufficient conditions under which the proposed problem is controllable is defined. Also, the nilpotency properties of such systems is established and used as the basis of a comprehensive design procedure. This general procedure is illustrated by designing a time-optimal disturbance rejection tracking system for a stirred-tank with time-delayed control inputs.

The Nilpotency Properties of the Leibniz Algebra Mn(C)D

We study the nilpotency properties of the Leibniz algebras constructed by means of D-mappings on the algebra of complex square matrices Mn(C). In particular, we obtain a criterion for nilpotency of these algebras in terms of the properties of a D-mapping. We prove also that the Leibniz algebras under consideration cannot be simple.

Nilpotency Property in Table Algebras

Table algebras form an important class of C-algebras. The dual of a table algebra may not be a table algebra, but just a C-algebra. It is not known under what conditions the dual of a table algebra is also a table algebra. In this paper we prove that if a table algebra has nilpotency property then its dual also is a table algebra. Finally we conclude that if (A,B) is a nilpotent table algebra then its dual also is a nilpotent table algebra.

THREE-POINT FUNCTIONS ON THE SPHERE OF CALABI-YAU d-FOLDS

Using mirror symmetry in Calabi-Yau manifolds M, we study three-point functions of A(M) model operators on the genus 0 Riemann surface in cases of one-parameter families of d-folds realized as Fermat type hypersurfaces embedded in weighted projective spaces and a two-parameter family of d-folds embedded in a weighted projective space Pd+1 [2,2,2,...,2,2,1,1] (2 (d + 1)). These three-point functions [Formula: see text] are expanded by indeterminates [Formula: see text] associated with a set of Kähler coordinates {tl}, and their expansion coefficients count the number of maps with a definite degree which map each of the three-points 0, 1 and ∞ on the world sheet on some homology cycle of M associated with a cohomology element. From these analyses, we can read the fusion structure of Calabi-Yau A(M) model operators. In our cases they constitute a subring of a total quantum cohomology ring of the A(M) model operators. In fact we switch off all perturbation operators on the topological theories except for marginal ones associated with Kähler forms of M. For that reason, the charge conservation of operators turns out to be a classical one. Furthermore, because their first Chern classes c1 vanish, their topological selection rules do not depend on the degree of maps (in particular, a nilpotent property of operators [Formula: see text] is satisfied). Then these fusion couplings {κl} are represented as some series adding up all degrees of maps.

Affine Toda solitons and vertex operators

Affine Toda theories with imaginary couplings associate with any simple Lie Algebra g generalisations of sine-Gordon theory which are likewise integrable and posses soliton solutions. The solitons are “created” by exponentials of quantities F ̂ i(z) which lie in the untwisted affine Kac-Moody algebra ĝ and ad-diagonalise the principal Heisenberg subalgebra. When g is simply laced and highest-weight irreducible representations at level one are considered, F ̂ i(z) can be expressed as a vertex operator whose square vanishes. This nilpotency property is extended to all highest-weight representations of all affine untwisted Kac-Moody algebras in the sense that the highest non-vanishing power becomes proportional to the level. As a consequence, the exponential series mentioned terminates and the soliton solutions have a relatively simple algebraic expression whose properties can be studied in a general way. This means that various physical properties of the soliton solutions can be directly related to the algebraic structure. For example, a classical version of Dorey's fusing rule follows from the operator product expansion of two F ̂ ' s , at least when g is simply laced. This adds to the list of resemblances of the solitons with respect to the particles which are the quantum excitations of the fields.

Nilpotence in finitary skew linear groups

We study nilpotence properties (upper central series, Engel elements, central heights, etc.) of groups of finitary linear transformations of a vector space over a division ring that is locally finite-dimensional over its centre. Not surprisingly properties of skew linear groups over these divisions rings do not usually extend to this much more general setting. However, not all is lost. Although the group itself may not have the property, frequently it has a local system of normal subgroups with the property. In many ways this is a surprisingly strong conclusion. Most of the difficulties involve unipotent elements. This paper really concerns the development of bounds for unipotent skew linear groups of various kinds that are independent of the degree and hence are potentially meaningful in the finitary, infinite-dimensional situation. Some of these give new insights even in the ordinary linear case. Once this is done the nilpotence results hinted at above are then easy corollaries.

Properties that characterize classes of nilpotent groups

Obviously we have not covered aU possible ascendant and descendant nilpotent properties. Using (1.1), (1.2), or other methods, and Lemma 1.2 it is possible to provide many more such properties. For exampleCq∼(Cpm×Cpm) is a metabelian group having infinitely many non-isomorphic Sylow-p-subgroups [6]. Thus, having one isomorphism class of Sylow-p-subgroups is a descendant nilpotent property.

An upper radical property and an answer to a problem of L.C.A. van Leeuwen and G.A.P. Heyman

1. It is well known that the Baer's lower radical property B, coincides with the lower radical property determined by the class of all nilpotent rings, and also with the upper radical property determined by the class of all semiprime rings, that is, rings without nonzero nilpotent ideals (N. J. Dwp~srcY [5]). L. C. A. VAN LF.EUW~N and G.A.P. HEYMAN [10] in an almost recent paper generalizing the notion of nilpotent rings introduced the class of almost nilpotent rings, that is, of rings every proper homomorphic image of which, is nilpotent ring,* and proved analogously, that the lower radical property L~ determined by the class of all almost nilpotent rings coincides with the upper radical property determined by the class of all rings without nonzero almost nilpotent ideals and it holds B~L2~Bq, where B~0 is the Andrunakiev 6's antisimple radical property (V. A. ANDRUNAKIEVI~ [1], [2]). Evidently, for every almost nilpotent ring R it holds R~ where ~o is the first limit ordinal number, and R~ ('1 R". n(N Generalizing further the notion of nilpotent rings, the ring R is said to be trausfinitely nilpotent if there exists an ordinal number r such that Rr=0. We recall that the power R' is defined as follows: If r is a nonlimit ordinal number then Rr=R "-I.R while if r is a limit ordinal number then Rr= ORbb<r The transfinitely nilpotent property was first discussed by R. BAER [3] and thirty years later F. A. Sz2{sz [15] asked whether the lower radical property determined by the class of all transfinitely nilpotent rings is special, and attributed an upper bound to this class. B. J. GARDNER [6] proved that this lower radical property is trivial because it coincides with the class of all rings, rejecting so the existence of an upper bound and restricting its interest. It is now natural to ask: Which is the upper rddical property ~ determined by the class M of all rings without nonzero transfinitely nilpotent ideals? In the present paper this radical property is studied. It is proved that the class M is weakly special and that B~ T<=B~,. Further, T~J and J~ T, where J is the Jacobson's radical class. Finally, this paper answers a problem raised by L.C.A. van Leeuwen and G.A.P. Heyman, proving B<L~ ~:<T<--B~, by an example which is the Jacobson radical of a Noetherian principal left ideal domain

Design of time-optimal disturbance-rejection tracking systems for linear multivariable discrete-time time-delayed plants

The design of time-optimal disturbance-rejection tracking systems for linear multivariable discrete-time plants with time-delayed control inputs is considered. The controllability and nilpotency properties of such systems are established and used as the basis of a comprehensive design procedure. This general procedure is illustrated by designing a timeoptimal disturbance-rejection tracking system for a stirred tank with time-delayed control inputs under computer control.

Nilpotency properties of linear multivariable discrete-time disturbance-rejection systems

In this paper the nilpotency properties of deadbeat linear multivariable discrete-time disturbance-rejection systems incorporating full-order or reduced-order observers for plants with accessible or inaccessible statea are investigated. The theory is illustrated by the presentation of the results of computer simulation studies which indicate that the presence of an observer affects only the transient behaviour of deadbeat discrete-time disturbance-rejection systems and that the use of a reduced-order observer results in markedly superior transient behaviour than the use of a full-order observer.

Nilpotency properties of linear multivariable discrete-time tracking systems

In this paper the nilpotency properties of deadbeat linear multivariable discrete-time tracking systems incorporating full-order or reduced-order observers for plants with accessible or inaccessible states are investigated. The theory is illustrated by the presentation of the results of computer simulation studies which indicate that the presence of an observer affects only the transient behaviour of deadbeat discrete-time tracking systems and that, the use of a reduced-order observer results in markedly superior transient behaviour than the use of a full-order observer.

Nilpotency and related properties of group extensions

Gilbert Baums lag has given sufficient conditions for a group G to be nilpotent in terms of properties of a normal subgroup H and the quotient G/H, and has shown these conditions to be necessary when G is a wreath product and H its base group. His results are extended here to give corresponding conditions for weaker properties ranging from “bounded local nilpotency” to “being an Engel group”. Again we ask whether the sufficient conditions found are necessary when G is non-trivially a wreath product, or is of the form the answer is “yes” for the stronger properties considered, but for the weaker ones the question remains open.

Nilpotency property of the discrete regulator

The optimal regulator control law for linear discrete systems possesses many interesting algebraic and geometric properties. One such property, that of the nilpotency of the resultant closed-loop matrix, is developed in this correspondence. The nilpotency property is then used to obtain many well-known results which have been previously developed by more classical approaches.

An addition to Ado’s theorem

For the suggestion that this nilpotency property of p might be secured I am indebted to Leonard Ross who used the characteristic 0 case of Theorem 1 in his proof of Ado's Theorem for graded Lie algebras (Thesis, Cohomology of graded lie algebras, University of California, Berkeley, 1964). In the case of characteristic 0, it is known that there exists a faithful finite-dimensional representation of L whose restriction to the maximum nilpotent ideal of L is nilpotent [1, pp. 202-203]. Hence, in order to establish Theorem 1 in the case of characteristic 0, it suffices to make the following observation: Let L be a finite-dimensional Lie algebra over a field of characteristic 0, and let M be a finite-dimensional L-module on which the maximum nilpotent ideal N of L is nilpotent. Let x be an element of L whose adjoint image a(x) is nilpotent. Then x is nilpotent on M. PROOF. Write L = S+R, where R is the radical of L and S is a semisimple subalgebra of L. Accordingly, write x=s+r, with s in S and r in R. Since a(x) is nilpotent, it is clear that the adjoint representation of S sends s onto a nilpotent derivation of S. Since S