Defining Relations of Minimal Degree of the Trace Algebra of 3×33 \times 3 Matrices

The trace algebra C(n,d) over a field of characteristic 0 is generated by all traces of products of d generic nxn matrices, n,d>1. Minimal sets of generators of C(n,d) are known for n=2 and n=3 for any d as well as for n=4 and n=5 and d=2. The defining relations between the generators are found for n=2 and any d and for n=3, d=2 only. Starting with the generating set of C(3,d) given by Abeasis and Pittaluga in 1989, we have shown that the minimal degree of the set of defining relations of C(3,d) is equal to 7 for any d>2. We have determined all relations of minimal degree. For d=3 we have also found the defining relations of degree 8. The proofs are based on methods of representation theory of the general linear group and easy computer calculations with standard functions of Maple

Computing with rational symmetric functions and applications to invariant theory and PI-algebras

Let the formal power series f in d variables with coefficients in an arbitrary field be a symmetric function decomposed as a series of Schur functions, and let f be a rational function whose denominator is a product of binomials of the form (1 - monomial). We use a classical combinatorial method of Elliott of 1903 further developed in the Partition Analysis of MacMahon in 1916 to compute the generating function of the multiplicities (i.e., the coefficients) of the Schur functions in the expression of f. It is a rational function with denominator of a similar form as f. We apply the method to several problems on symmetric algebras, as well as problems in classical invariant theory, algebras with polynomial identities, and noncommutative invariant theory.Comment: 37 page

Polynomial identities on superalgebras and exponential growth

n/

*-Subvarieties of the Variety Generated by

AbstractLet be a field of characteristic zero, and * = t the transpose involution for the matrix algebra M2(). Let be a proper subvariety of the variety of algebras with involution generated by . We define two sequences of algebras with involution Rp, Sq, where p, q ∊ . Then we show that T*() and T*(Rp ⊕ Sq) are *-asymptotically equivalent for suitable p, q

Defining Relations of Noncommutative Trace Algebra of Two 3×33 \times 3 Matrices

The noncommutative (or mixed) trace algebra $T_{nd}$ is generated by $d$ generic $n\times n$ matrices and by the algebra $C_{nd}$ generated by all traces of products of generic matrices, $n,d\geq 2$. It is known that over a field of characteristic 0 this algebra is a finitely generated free module over a polynomial subalgebra $S$ of the center $C_{nd}$. For $n=3$ and $d=2$ we have found explicitly such a subalgebra $S$ and a set of free generators of the $S$-module $T_{32}$. We give also a set of defining relations of $T_{32}$ as an algebra and a Groebner basis of the corresponding ideal. The proofs are based on easy computer calculations with standard functions of Maple, the explicit presentation of $C_{32}$ in terms of generators and relations, and methods of representation theory of the general linear group.Comment: 19 page

Electromechanical and robotic devices for gait and balance rehabilitation of children with neurological disability: a systematic review

In the last two decades, a growing interest has been focused on gait and balance robot-assisted rehabilitation in children with neurological disabilities. Robotic devices allow the implementation of intensive, task-specific training fostering functional recovery and neuroplasticity phenomena. However, limited attention has been paid to the protocols used in this research framework. This systematic review aims to provide an overview of the existing literature on robotic systems for the rehabilitation of gait and balance in children with neurological disabilities and their rehabilitation applications. The literature search was carried out independently and synchronously by three authors on the following databases: MEDLINE, Cochrane Library, PeDro, Institute of Electrical and Electronics Engineers, ScienceDirect, and Google Scholar. The data collected included three subsections referring to clinical, technical, and regulatory aspects. Thirty-one articles out of 81 found on the primary literature search were included in the systematic review. Most studies involved children with cerebral palsy. Only one-third of the studies were randomized controlled trials. Overall, 17 devices (nine end-effector systems and eight exoskeletons) were investigated, among which only 4 (24%) were bore the CE mark. Studies differ on rehabilitation protocols duration, intensity, and outcome measures. Future research should improve both rehabilitation protocols\u2019 and devices\u2019 descriptions

Robot-assisted upper limb training for patients with multiple sclerosis: an evidence-based review of clinical applications and effectiveness

Upper extremities limitation is a common functional impairment in patients with Multiple Sclerosis (PwMS). Novel technological devices are increasingly used in neurorehabilitation to support motor function improvement and the quantitative assessment of motor performance during training in patients with neurological diseases. In this review, we systematically report the evidence on clinical applications and robotic-assisted arm training (RAT) in functional recovery in PwMS. PubMed/MEDLINE, the Cochrane Library, and the Physiotherapy Evidence Database (PEDro) databases were systematically searched from inception to March 2021. The 10-item PEDro scale assessed the study quality for the RCT, and the AMSTAR-2 was used to assess the quality of the systematic review. The 5-item Oxford CEBM scale was used to rate the level of evidence. A total of 10 studies (161 subjects) were included. The selected studies included one systematic review, four RCTs, one randomized crossover, and four case series. The RCTs were scored as high-quality studies, while the systematic review was determined to be of low quality. Shoulder range of motion, handgrip strength, and proximal arm impairment improved after RAT. Manual dexterity, arm function, and use in daily life also ameliorated arm function. The high clinical heterogeneity of treatment programs and the variety of robot devices affects the generalizability of the study results; therefore, we emphasize the need to standardize the intervention type in future studies that evaluate the role of robotic-assisted training in PwMS. Robot-assisted treatment seems safe and useful to increase manual dexterity and the quality of movement execution in PwMS with moderate to severe disability. Additional studies with an adequate sample size and methodological rigour are warranted to drive definite conclusion

Effects of robotic upper limb treatment after stroke on cognitive patterns: A systematic review

Background: Robotic therapy (RT) has been internationally recognized for the motor rehabilitation of the upper limb. Although it seems that RT can stimulate and promote neuroplasticity, the effectiveness of robotics in restoring cognitive deficits has been considered only in a few recent studies. Objective: To verify whether, in the current state of the literature, cognitive measures are used as inclusion or exclusion criteria and/or outcomes measures in robotic upper limb rehabilitation in stroke patients. Methods: The systematic review was conducted according to PRISMA guidelines. Studies eligible were identified through PubMed/MEDLINE and Web of Science from inception to March 2021. Results: Eighty-one studies were considered in this systematic review. Seventy-three studies have at least a cognitive inclusion or exclusion criteria, while only seven studies assessed cognitive outcomes. Conclusion: Despite the high presence of cognitive instruments used for inclusion/exclusion criteria their heterogeneity did not allow the identification of a guideline for the evaluation of patients in different stroke stages. Therefore, although the heterogeneity and the low percentage of studies that included cognitive outcomes, seemed that the latter were positively influenced by RT in post-stroke rehabilitation. Future larger RCTs are needed to outline which cognitive scales are most suitable and their cut-off, as well as what cognitive outcome measures to use in the various stages of post-stroke rehabilitation

Upper limb robotic rehabilitation for patients with cervical spinal cord injury: a comprehensive review

The upper extremities limitation represents one of the essential functional impairments in patients with cervical spinal cord injury. Electromechanics assisted devices and robots are increasingly used in neurorehabilitation to help functional improvement in patients with neurological diseases. This review aimed to systematically report the evidence-based, state-of-art on clinical applications and robotic-assisted arm training (RAT) in motor and functional recovery in subjects affected by cervical spinal cord injury. The present study has been carried out within the framework of the Italian Consensus Conference on "Rehabilitation assisted by robotic and electromechanical devices for persons with disability of neurological origin" (CICERONE). PubMed/MEDLINE, Cochrane Library, and Physiotherapy Evidence Database (PEDro) databases were systematically searched from inception to September 2021. The 10-item PEDro scale assessed the study quality for the RCT and the AMSTAR-2 for the systematic review. Two different authors rated the studies included in this review. If consensus was not achieved after discussion, a third reviewer was interrogated. The five-item Oxford CEBM scale was used to rate the level of evidence. A total of 11 studies were included. The selected studies were: two systematic reviews, two RCTs, one parallel-group controlled trial, one longitudinal intervention study and five case series. One RCT was scored as a high-quality study, while the systematic review was of low quality. RAT was reported as feasible and safe. Initial positive effects of RAT were found for arm function and quality of movement in addition to conventional therapy. The high clinical heterogeneity of treatment programs and the variety of robot devices could severely affect the generalizability of the study results. Therefore, future studies are warranted to standardize the type of intervention and evaluate the role of robotic-assisted training in subjects affected by cervical spinal cord injury