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It is known that the so-called monadic decomposition, applied to the adjunction connecting the category of bialgebras to the category of vector spaces via the tensor and the primitive functors, returns the usual adjunction between bialgebras and (restricted) Lie algebras. Moreover, in this framework, the notions of augmented monad and combinatorial rank play a central role. In order to set these results into a wider context, we are led to substitute the monadic decomposition by what we call the adjoint decomposition. This construction has the advantage of reducing the computational complexity when compared to the first one. We connect the two decompositions by means of an embedding and we investigate its properties by using a relative version of Grothendieck fibration. As an application, in this wider setting, by using the notion of augmented monad, we introduce a notion of combinatorial rank that, among other things, is expected to give some hints on the length of the monadic decomposition.

The purpose of this paper is to study Rota-Baxter operators for BiHom-associative algebras. Moreover, we introduce and discuss the properties of the notions of BiHom-(tri)dendriform algebra, BiHom-Zinbiel algebra and BiHom-quadri-algebra. We construct the free Rota-Baxter BiHom-associative algebra and present some observations about categories and functors related to Rota-Baxter structures.

Motivated by an example related to the tensor algebra, a stronger version of the notion of separable functor, called heavily separable (h-separable for short), was introduced and investigated in [1] . Here we study h-coseparable coalgebras in monoidal categories with special concern with the monoidal category T A ♯ of right transfer morphisms through an algebra A in a monoidal category. We characterize the h-separability of the forgetful functor from the category of entwined modules associated to a cowreath to the base category using suitable Casimir morphisms. Even if there are non trivial examples of h-coseparable coalgebras over a field [1, Theorem 4.4] , here we provide non trivial examples of h-coseparable coalgebras in the monoidal category T A ⊗ H o p # where H = H 4 is the Sweedler 4-dimensional Hopf algebra over a field k and A = C l ( α , β , γ ) the Clifford algebra.

We contribute to the study of Rota–Baxter operators on types of algebras other than associative and Lie algebras. If A is an algebra of a certain type and R is a Rota–Baxter operator on A, one can define a new multiplication on A by means of R and the previous multiplication and ask under what circumstances the new algebra is of the same type as A. Our first main result deals with such a situation in the case of BiHom-Lie algebras. Our second main result is a BiHom analogue of Aguiar’s theorem that shows how to obtain a pre-Lie algebra from a Rota–Baxter operator of weight zero on a Lie algebra. The BiHom analogue does not work for BiHom-Lie algebras, but for a new concept we introduce here, called left BiHom-Lie algebra, at which we arrived by defining first the BiHom version of Leibniz algebras.

The main objective of this paper is to implement a high-gain “practical” observer with the interval arithmetic in the case of sampled measurements. In order to pursue this objective, two closed-form expressions are proposed to compute certified bounds on the solution of a single-input linear system, with distinct, real, and negative eigenvalues, when both the initial condition and the control input lie in given intervals. Furthermore, it is shown how such formulas can be extended in the case of sampled input measurements. These results are used to design an interval high-gain observer for non-linear systems in observer normal form. In particular, it is shown how to implement a high-gain “practical” observer with the interval arithmetic, both in continuous-time and in discrete-time, so to compute guaranteed interval estimates of the state of the system.

We study BiHom-Novikov-Poisson algebras, which are twisted generalizations of Novikov-Poisson algebras and Hom-Novikov-Poisson algebras, and find that BiHom-Novikov-Poisson algebras are closed under tensor products and several kinds of perturbations. Necessary and sufficient conditions are given under which BiHom-Novikov-Poisson algebras give rise to BiHom-Poisson algebras., 21 pages

In this paper, mechanical systems subject to impacts and contacts, that would be not controllable if the impacts were absent (usually called jugglers), are considered. On the basis of an algorithm taken from the literature and of a new procedure to determine a reference trajectory for such a class of systems, a fully algorithmic procedure, able to compute a control input that achieves dead–beat regulation of the “uncontrollable” subsystem just by using impacts, is given. Such a procedure exploits some tools borrowed from algebraic geometry that allow to solve parametric systems of equalities and inequalities. A practical example of application of the given procedure is reported.

Purpose To evaluate the impact of a novel postprocessing denoising technique on accuracy and precision in myocardial T1 mapping. Materials and Methods This study introduces a fast and robust denoising method developed for magnetic resonance T1 mapping. The technique imposes edge-preserving regularity and exploits the co-occurence of spatial gradients in the acquired T1-weighted images. The proposed approach was assessed in simulations, ex vivo data and in vivo imaging on a cohort of 16 healthy volunteers (12 males, average age 39 ± 8 years, 62 ± 9 bpm) both in pre- and postcontrast injection. The method was evaluated in myocardial T1 mapping at 3T with a saturation-recovery technique that is accurate but sensitive to noise. ROIs in the myocardium and left-ventricle blood pool were analyzed by an experienced reader. Mean T1 values and standard deviation were extracted and compared in all studies. Results Simulations on synthetic phantom showed signal-to-noise ratio and sharpness improvement with the proposed method in comparison with conventional denoising. In vivo results demonstrated that our method preserves accuracy, as no difference in mean T1 values was observed in the myocardium (precontrast: 1433/1426 msec, 95%CI: [−40.7, 55.9], p = 0.75, postcontrast: 766/759 msec, 95%CI: [−60.7, 77.2], p = 0.8). Meanwhile, precision was improved with standard deviations of T1 values being significantly decreased (precontrast: 223/151 msec, 95%CI: [27.3, 116.5], p = 0.003, postcontrast: 176/135 msec, 95%CI: [5.5, 77.1], p = 0.03). Conclusion The proposed denoising method preserves accuracy and improves precision in myocardial T1 mapping, with the potential to offer better map visualization and analysis. Level of Evidence: 3 Technical Efficacy: Stage 1 J. Magn. Reson. Imaging 2017;46:1377–1388.

International audience; Purpose: To correct with post-processing effects of incomplete recovery of the longitudinal magnetization before a new inversion pulse in the Modified Look-Locker Inversion recovery sequence (MOLLI) sequence.Theory and methods: We model such effects as a temporal shift ([Formula: see text]) of the signal of the Look-Locker block following next inversion pulses. After using the following equation [Formula: see text], a temporal registration of [Formula: see text] is applied to the signal of the affected block to adjust the sampling time of the recovery signal and correct the underlying effect on quantitative T1. To test our approach, simulations, phantoms, and five volunteers' data were used while applying different MOLLI sampling schemes at different heart rates and compared to the reference three-parameter fit.Results: The temporal registration of the affected signals allows to reach higher accuracy on long T1 when compared to the reference three parameters fit (10.15 vs 22.12% for T1 = 1785 ms; 8.22 vs 14.65% for T1 = 1278 ms), and lower average variation in case of rest-period deletion (62 vs 231 ms).Conclusion: The proposed approach leads to more accurate T1 in case of incomplete recovery. It is less sensitive to parameters affecting the recovery such as the rest period or the sampling scheme; and, therefore, supports multi-center studies with different MOLLI protocols.

In this paper, by exploiting the concept of polynomial greatest common divisor, some algebraic tests are proposed to certify the structural properties of both discrete-time and continuous-time linear systems. Furthermore, by exploiting the concept of parametric greatest common divisor, such results are extended to certify the structural properties of systems whose dynamical matrices depend polynomially on some parameters.

By using techniques borrowed from algebraic geometry, some tests are proposed to certify structural properties (such as reachability, observability, controllability, constructibility, stabilisability, and detectability) of discrete-time and continuous-time linear time-invariant systems. These results are then generalised for linear time-invariant systems depending polynomially on some real parameters, by exploiting the notion of Gröbner cover. The main innovation of the proposed results with respect to existing techniques is that they provide exact certificates for the structural properties of linear time-invariant systems. Examples of application are given all throughout the study to illustrate and validate the theoretical results.

In this paper, a dynamical interval Newton’s method is proposed to solve systems of nonlinear equations. Differently from other interval techniques available in the literature, the proposed method is capable of determining the roots of vector-valued time-varying functions. The design is carried out by firstly proposing an interval Newton’s method capable of determining the roots of a scalar time-varying function and then extending these results to the vector-valued case. If the function to be zeroed is polynomial, then it is shown how to couple the proposed scalar and vector Newton’s methods to determine improved estimates of its roots. Examples of application of the proposed procedure to the inverse kinematics of a robotic manipulator and to the problem of designing an observer for a nonlinear system are reported.

In this paper, some computational tools are proposed to determine the largest invariant set, with respect to either a continuous-time or a discrete-time system, that is contained in an algebraic set. In particular, it is shown that if the vector field governing the dynamics of the system is polynomial and the considered analytic set is a variety, then algorithms from algebraic geometry can be used to solve the considered problem. Examples of applications of the method &#x0028 spanning from the characterization of the stability to the computation of the zero dynamics &#x0029 are given all throughout the paper.

In this paper, algebraic tests are proposed to establish asymptotic stability of parametric linear systems assuming that the parameters lie in a given variety. In particular, by using tools borrowed from algebraic geometry, necessary and sufficient condition are proposed to test whether a continuous-time or a discrete-time linear parametric system is asymptotically stable for all the specializations belonging to a given variety. Both the cases of zero dimensional and non-zero dimensional varieties are considered.

In this paper, the single-gene regulatory system for the lac operon is considered and a method to build a Boolean network able to represent its dynamics is given. A procedure, based on techniques borrowed from algebraic geometry, to compute an optimal control policy minimising a cost function is given for such a Boolean network. A tool to apply such an optimal control policy to the continuous-time system is reported. The proposed control method is tested in two simulative examples.

In this paper monoidal Hom-Lie algebras, Lie color algebras, Lie superalgebras and other type of generalized Lie algebras are recovered by means of an iterated construction, known as monadic decomposition of functors, which is based on Eilenberg–Moore categories. To this aim we introduce the notion of Milnor–Moore category as a monoidal category for which a Milnor–Moore type Theorem holds. We also show how to lift the property of being a Milnor–Moore category whenever a suitable monoidal functor is given and we apply this technique to provide examples.

We introduce the concept of a $\{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\}$ -Rota–Baxter operator, as a twisted version of a Rota–Baxter operator of weight zero. We show how to obtain a certain $\{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\}$ -Rota–Baxter operator from a solution of the associative (Bi)Hom-Yang–Baxter equation, and, in a compatible way, a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.

In this paper, an algorithm based on linear algebra tools is proposed to compute a weighted sum of squares decomposition of a given form whose length is lower than the number of variables. Such an objective is pursued by using linear algebra techniques to perform tasks that are usually carried out through computational algebraic geometry tools. Several examples are reported to show that the use of linear algebra rather than algebraic geometry leads to a reduction of the execution times, without affecting the effectiveness of the algorithm. Applications of the given procedure to system analysis and to control design problems are reported as well as a detailed complexity analysis.

The main objective of this paper is the characterisation of the notion of uniform algebraic observability for multi-switching linear systems. In particular, it is shown that a multi-switching linear system is 'generically' uniformly algebraically observable if and only if the number of available measurements is greater than the number of switching signals acting on the systems. This result is established without requiring any knowledge about the (possibly, stochastic) process generating the switching pattern. Moreover, since such systems can be used to model sensor networks, the results given in this paper can 'generically' be used to design simple observers for sensor networks able to cope with sensor failures and data losses.

We introduce and study infinitesimal BiHom-bialgebras, BiHom-Novikov algebras, BiHom-Novikov-Poisson algebras, and find some relations among these concepts. Our main result is to show how to obtain a left BiHom-pre-Lie algebra from an infinitesimal BiHom-bialgebra, Comment: 22 pages. v2, final version, we modified the definition of a BiHom-Novikov-Poisson algebra in section 3

Among the various phenomena that can be modeled by Boolean networks, i.e., discrete-time dynamical systems with binary state variables, gene regulatory interactions are especially well known. Therefore, the analysis of Boolean networks is critical, e.g., to identify genetic pathways and to predict the effects of mutations on the cell functionality. Two methodologies (i.e., the semi-tensor product and the Grobner bases over finite fields) have recently been proposed to tackle the problem of determining cycles and attractors (with the corresponding basin of attraction) for such systems. Here, it is shown that, by suitably coupling methodologies taken from these two fields (i.e., linear algebra and algebraic geometry), it is not only possible to determine cycles and attractors, but also to find closed-form solutions of the Boolean network. Such a goal is pursued by finding an immersion that recasts the Boolean dynamics in a linear form and by computing the closed-form solution of the latter system. The effectiveness of this technique is demonstrated by fully computing the solutions of the Boolean network modeling the differentiation of the Th-lymphocyte, a type of white blood cells involved in the human adaptive immune system.

In this paper, a symbolic, algorithmic procedure to compute an immersion that recasts a polynomial system into a linear one up to an output injection is proposed. Such a technique is based on computing, through algebraic geometry methods, the set of all the embeddings of the system and on matching the coefficients of these polynomials with the ones of the embeddings of a linear system up to an output injection. The given algorithm is then relaxed to compute an immersion that recasts a polynomial system into a form that is linear up to a finite order and an output injection and to compute an approximation of the immersion.

This paper deals with the problem of inverting a nonlinear map. The proposed solution consists in a nonlinear state observer, which mimics a Newton-like algorithm, that allows to determine the inverse of a given diffeomorphism in finite time. The results are illustrated by application to the inverse kinematics of a three DOF planar manipulator.

Prompted by an example related to the tensor algebra, we introduce and investigate a stronger version of the notion of separable functor that we call heavily separable. We test this notion on several functors traditionally connected to the study of separability.

An algorithmic procedure is given for obtaining a certificate of global asymptotic stability for planar polynomial systems having the origin as equilibrium point. The procedure is not Lyapunov based and uses methods from Algebraic Geometry to study the sign of polynomial functions.

Sum of squares (SOS) decompositions for positive semidefinite polynomials are usually computed numerically, using convex optimization solvers. The precision of the decompositions can be improved by increasing the number of digits used in the computations, but, when the number of variables is greater than the length (i.e., the minimum number of squares needed for the decomposition) of the polynomial, it is difficult to obtain an exact SOS decomposition with the existing methods. A new algorithm, which works well in “almost all” such cases, is proposed here. The results of randomly generated experiments are reported to compare the proposed algorithm with those based on convex optimization.

A complete procedure for the design of a robust output feedback regulator is proposed for a class of uncertain linear hybrid systems with periodic jumps, using a hybrid extension of the classical internal model principle. Simple conditions, testable on the plant nominal data, for the problem solvability are given. The plant is not restricted to be minimum phase, square, or single input single output. The proposed regulator has a key feature of containing an internal model composed by two main units, a flow internal model , in charge of providing the correct input to achieve regulation during flows, and a jump internal model , in charge of suitably resetting the state of the regulator at each period. The proposed procedure is illustrated by its application to a physically motivated example, for which the output regulation problem is not solvable by methods appeared thus far in the literature.

High-gain observers can be easily designed for nonlinear systems and they present very good convergence properties, when the observability mapping of the system is injective in the whole. If the observability mapping is injective in the whole, the system can be transformed into a normal form, from which the estimation of the successive output derivatives can be easily carried out. If the observability mapping is not injective in the whole and the state trajectory of the system passes close to some set where the property of observability is lost (about which the observability mapping is not injective), the resulting high-gain observers may be subject to significant numerical errors. The objective of the paper is to propose an improvement of the classical procedure for the high-gain observer design, so that the nonlinear system is transformed into a modified normal form, through a transformation that is obtained from the observability mapping, by removing singularities belonging to a certain class. Hence, a new high-gain structure, containing two sets of small parameters, is proposed for designing a high-gain observer for this modified normal form. The new observer is applied to a Bullard dynamo, to show that the new observer obtains good state estimates even for trajectories very close to unobservability, for which the classical high-gain observer is affected by significant numerical errors.

In this paper the output regulation problem for a class of hybrid linear systems in the presence of uncertain time domain is considered. Uncertainty is modeled by assuming that the time domain is not known to the controller and by allowing for arbitrarily close, even simultaneous, jumps. Considering the full information setting, the geometric characterization of the relevant regulation manifolds is given. Finally, the theory is illustrated and validated by means of numerical examples.

In this paper, it is shown that if a discrete-time nonlinear system admits a Lie symmetry that can be linearized by a state diffeomorphism, then, under some technical conditions, such a discrete-time nonlinear system can be immersed into a linear one, thus generalizing a similar result valid for continuous-time nonlinear systems.

In this paper, using the concept of semi-invariant, an algebraic version of the internal model principle is proposed for a class of nonlinear systems, thus extending a similar characterization valid for linear systems.

A simple necessary and sufficient condition for asymptotic stability of the origin is derived for a large class of planar nonlinear systems that cannot be studied by means of their linearization, which is nilpotent and not zero. The condition is based on the use of Darboux polynomials, to derive a Lyapunov function for the first approximation of the system with respect to a vector function representing a dilation. Results by Andreev (1958), on the qualitative behaviour of equilibrium points, and the Belitskii normal form are used in the proofs. The proposed condition is applied to the asymptotic stabilization of the origin, for a class of planar systems whose linearization cannot be stabilized.

In this paper, it is shown that if a nonlinear system admits a Lie symmetry that can be transformed into its Poincare-Dulac normal form by a state diffeomorphism, then, under some technical conditions, such a nonlinear system can be immersed into a linear one. This allows us to compute in closed-form the flow, all algebraic invariant curves (through semi-invariants) of the nonlinear system, and Lyapunov functions to study stability properties.

In our paper we heavily used the result that two constituent bialgebroids in a Hopf algebroid possess isomorphic comodule categories. This statement was based on [T. Brzezinski, A note on coring extensions, Ann. Univ. Ferrara Sez. VII Sci. Mat. LI (2005) 15–27. A corrected version is available at http://arxiv.org/abs/math/0410020v3 , Theorem 2.6], whose proof turned out to contain an unjustified step. Here we prove the main results in our paper without using [T. Brzezinski, A note on coring extensions, Ann. Univ. Ferrara Sez. VII Sci. Mat. LI (2005) 15–27. A corrected version is available at http://arxiv.org/abs/math/0410020v3 , Theorem 2.6] and the derived isomorphism of comodule categories.

The paper is devoted to prove a version of Milnor–Moore Theorem for connected braided bialgebras that are infinitesimally cocommutative. Namely in characteristic different from 2, we prove that, for a given connected braided bialgebra ( A , c A ) which is infinitesimally λ-cocommutative for some element λ ≠ 0 that is not a root of one in the base field, then the infinitesimal braiding of A is of Hecke-type of mark λ and A is isomorphic as a braided bialgebra to the symmetric algebra of the braided subspace of its primitive elements.

We study comodule functors for comonads arising from mixed distributive laws. Their Galois property is reformulated in terms of a (so-called) regular arrow in Street's bicategory of comonads. Between categories possessing equalizers, we introduce the notion of a regular adjunction. An equivalence is proven between the category of pre-torsors over two regular adjunctions $(N_A,R_A)$ and $(N_B,R_B)$ on one hand, and the category of regular comonad arrows $(R_A,\xi)$ from some equalizer preserving comonad ${\mathbb C}$ to $N_BR_B$ on the other. This generalizes a known relationship between pre-torsors over equal commutative rings and Galois objects of coalgebras.Developing a bi-Galois theory of comonads, we show that a pre-torsor over regular adjunctions determines also a second (equalizer preserving) comonad ${\mathbb D}$ and a co-regular comonad arrow from ${\mathbb D}$ to $N_A R_A$, such that the comodule categories of ${\mathbb C}$ and ${\mathbb D}$ are equivalent., Comment: 34 pages LaTeX file. v2: a few typos corrected

Braided bialgebras of type one in abelian braided monoidal categories are characterized as braided graded bialgebras which are strongly $\mathbb{N}$-graded both as an algebra and as a coalgebra.