Every polynomial over a field containing F_16 is a strict sum of four cubes and one expression A^2+A

Luis H. Gallardo 대한수학회 2009 대한수학회보 Vol.46 No.5

Let q be a power of 16. Every polynomial P 2 F_q[t] is a strict sum P = A^2 + A + B^3 + C^3 + D^3 + E^3. The values of A,B,C,D,E are effectively obtained from the coefficients of P. The proof uses the new result that every polynomial Q ∈ F_q[t], satisfying the necessary condition that the constant term Q(0) has zero trace, has a strict and effective representation as: Q = F^2 + F + tG^2. This improves for such q’ and such Q’ a result of Gallardo, Rahavandrainy, and Vaserstein that requires three polynomials F, G,H for the strict representation Q = F^2+F +GH. Observe that the latter representation may be considered as an analogue in characteristic 2 of the strict representation of a polynomial Q by three squares in odd characteristic. Let q be a power of 16. Every polynomial P 2 F_q[t] is a strict sum P = A^2 + A + B^3 + C^3 + D^3 + E^3. The values of A,B,C,D,E are effectively obtained from the coefficients of P. The proof uses the new result that every polynomial Q ∈ F_q[t], satisfying the necessary condition that the constant term Q(0) has zero trace, has a strict and effective representation as: Q = F^2 + F + tG^2. This improves for such q’ and such Q’ a result of Gallardo, Rahavandrainy, and Vaserstein that requires three polynomials F, G,H for the strict representation Q = F^2+F +GH. Observe that the latter representation may be considered as an analogue in characteristic 2 of the strict representation of a polynomial Q by three squares in odd characteristic.

EVERY POLYNOMIAL OVER A FIELD CONTAINING 𝔽₁₆ IS A STRICT SUM OF FOUR CUBES AND ONE EXPRESSION A² + A

Gallardo, Luis H. Korean Mathematical Society 2009 대한수학회보 Vol.46 No.5

Let q be a power of 16. Every polynomial $P\in\mathbb{F}_q$[t] is a strict sum $P=A^2+A+B^3+C^3+D^3+E^3$. The values of A,B,C,D,E are effectively obtained from the coefficients of P. The proof uses the new result that every polynomial $Q\in\mathbb{F}_q$[t], satisfying the necessary condition that the constant term Q(0) has zero trace, has a strict and effective representation as: $Q=F^2+F+tG^2$. This improves for such q's and such Q's a result of Gallardo, Rahavandrainy, and Vaserstein that requires three polynomials F,G,H for the strict representation $Q=F^2$+F+GH. Observe that the latter representation may be considered as an analogue in characteristic 2 of the strict representation of a polynomial Q by three squares in odd characteristic.

Comments on the Article “Results of Simple Conservative Treatment of Midfoot Charcot Arthropathy”: To the Editor

Gallardo-Molina Nicolas 대한정형외과학회 2020 Clinics in Orthopedic Surgery Vol.12 No.2

The article by Kim et al.1) offers important and interesting information on the management of midfoot Charcot arthropathy. This manuscript presents satisfactory results of a conservative treatment without restriction of daily living activities. However, much of the literature manifests other types of treatment for this pathology. The aim of this letter is to provide a short review on the treatment of midfoot Charcot arthropathy with latest evidence. Botek et al.2) stated that offloading is the key to treatment because it gives the time to heal and arrest the progressive tissue damage and deformities. This should remain until the inflammation disappears (3 to 12 months, approximately). However, nowadays surgical correction of the Charcot deformity has good supporting evidence. The different techniques include exostectomies, muscle flaps, arthrodesis with internal or external fixation. The circular external fixation is considered to be biomechanically superior to the others. Indications for surgical intervention are unstable joints, nonhealing or infected ulcers, equinus deformities, and unbraceable deformities. A case report by Higgins et al.3) also presents a 58-year-old diabetic man with an acute Charcot arthropathy, in which offloading was essential for the treatment of the foot. The authors demonstrated that surgery is not useful in acute cases; however, the surgical procedures mentioned previously have demonstrated variable success in the treatment of deformities in chronic Charcot arthropathy. Therefore, the initial treatment of this patient was immobilization with total contact casting, which is the gold standard in acute cases. The offloading therapy is critical in the initial treatment of Charcot arthropathy because it gives the chance of healing properly without weakening deformities and preserves longitudinal arch; however, there are cases that are not successful. For these cases, Rosskopf et al.4) recommended stabilization with the Ilizarov external fixator (or ring fixator) frame as an alternative treatment option for offloading in patients with severe deformity or after removal of osteomyelitic bone fragments. Raspovic et al.5) stated that surgical reconstruction usually consists of a combination of tendon releases/lengthening, fusions, and osteotomies as needed to address the deformity. The goal is to give stability so the patient can achieve free ambulation. One of the surgical techniques is to release the contracted soft tissue to correct deformity. This is done percutaneously or open via three small incisions. On the other hand, the intramedullary fixation of the medial and lateral columns for midfoot Charcot arthropathy reconstruction allows control of the transverse arch of the foot and to this a fusion of the subtalar joint can be added in order to limit frontal and transverse plane torsion and achieve greater stability. However, internal fixation is not recommended in cases of infection. Charcot arthropathy is a severe complication of diabetes mellitus that mainly affects the patient's quality of life. Ambulation is severely limited, so the periodic control of this type of patients is an essential part of reporting early findings of arthropathy, and thus, avoiding invasive interventions that harm rather than benefit the evolution of the disease.

DEVELOPMENT OF BASE METAL OXIDE CATALYST FOR AUTOMOTIVE EMISSION CONTROL

Gallardo, Susan,Aida, Takashi,Niiyama, Hiroo 한국화학공학회 1998 Korean Journal of Chemical Engineering Vol.15 No.5

An application of screw theory to the kinematic analysis of a Delta-type robot

Jaime Gallardo-Alvarado,Albert L. Balmaceda-Santamaría,Eduardo Castillo-Castaneda 대한기계학회 2014 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.28 No.9

This paper reports on the kinematics of a translational parallel manipulator whose topology is close to the architecture of the famousDelta robot. The displacement analysis is presented in closed-form solution by applying a new strategy based on the unknown coordinatesof a point embedded to the moving platform. The input-output equations of velocity and acceleration of the robot are systematicallyobtained by resorting to reciprocal-screw theory. The singularities of the mechanism are explained through the input-output equation ofvelocity. Finally, a numerical example is provided to show the application of the method.

A simple approach to solving the kinematics of the 4-UPS/PS (3R1T) parallel manipulator

Jaime Gallardo-Alvarado,Mario A. García-Murillo,Md. Nazrul Islam,Mohammad H. Abedinnasab 대한기계학회 2016 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.30 No.5

This work reports on the position, velocity and acceleration analyses of a four-degrees-of-freedom parallel manipulator, 4-DoF-PM for brevity, which generates Three-rotation-one-translation (3R1T) motion. Nearly closed-form solutions to solve the forward displacement analysis are easily obtained based on closure equations formulated upon linear combinations of the coordinates of three non-collinear points embedded in the moving platform. Then, the input-output equations of velocity and acceleration of the robot manipulator are systematically established by resorting to the theory of screws. To this end, the Klein form of the Lie algebra se(3) of the Euclidean group SE(3) is systematically applied to the velocity and reduced acceleration state in screw form of the moving platform cancelling the passive joint rates of the parallel manipulator. Numerical examples, which are confirmed by means of commercially available software, are provided to show the application of the method.

Kinematics of the 4-RUU parallel manipulator generator of the Schönflies motion by means of screw theory

Jaime Gallardo-Alvarado,Mario A. García-Murillo,Md. Nazrul Islam,Mohammad H. Abedinnasab 대한기계학회 2017 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.31 No.10

This work deals with the inverse–forward kinematic analysis of a symmetric parallel manipulator equipped with a rotary actuator generator of three independent translations and one rotation motion. The closure equations of the displacement analysis are easily formulated based on the unknown coordinates of two points embedded in the moving platform. The input–output equations of velocity and acceleration of the robot are systematically obtained through the reciprocal-screw theory. The pseudo-kinematic pairs that connect the limbs to the fixed platform and a passive kinematic chain connected to the robot manipulator eliminate the handling of rank-deficient Jacobian matrices, which is an undisputable advantage from the computational point of view. Furthermore, this strategy allows the use of the Lie algebra se(3) without the inherent restrictions associated with the limited mobility of the robot.

A novel six-degrees-of-freedom series-parallel manipulator

J. Gallardo-Alvarado,R. Rodríguez-Castro,C. R. Aguilar-Nájera,L. Pérez-González 대한기계학회 2012 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.26 No.6

This paper addresses the description and kinematic analyses of a new non-redundant series-parallel manipulator. The primary feature of the robot is to have a decoupled topology consisting of a lower parallel manipulator, for controlling the orientation of the coupler platform,assembled in series connection with a upper parallel manipulator, for controlling the position of the output platform, capable to provide arbitrary poses to the output platform with respect to the fixed platform. The forward displacement analysis is carried-out in semi-closed form solutions by resorting to simple closure equations. On the other hand; the velocity, acceleration and singularity analyses of the manipulator are approached by means of the theory of screws. Simple and compact expressions are derived here for solving the infinitesimal kinematics by taking advantage of the concept of reciprocal screws. Furthermore, the analysis of the Jacobians of the robot shows that the lower parallel manipulator is practically free of singularities. In order to illustrate the performance of the manipulator, a numerical example which consists of solving the inverse/forward kinematics of the series-parallel manipulator as well as its singular configurations is provided.

An approach to solving the forward kinematics of the 5-RPUR (3T2R) parallel manipulator

Jaime Gallardo-Alvarado,Mario A. Garcia-Murillo,Luis D. Aguilera-Camacho,Luis A. Alcaraz-Caracheo,X. Yamile Sandoval-Castro 대한기계학회 2023 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.37 No.3

This work is devoted to simplifying the formulation and solution of the closure equations associated with the forward kinematic problem (FKP) of the 5-RPUR parallel manipulator, a limited-DOF robot able to perform 3T2R motion. The analysis yields a set of eighteen nonlinear equations that are solved numerically through a combination of the homotopy continuation method and the usual Newton-Raphson technique. Unlike existing methods, the proposed approach is easy to follow and can be easily translated into computer codes. Numerical examples are provided with the aim to illustrate the potential and correctness of the proposed method.

Long-Term Grey Matter Changes in First Episode Psychosis: A Systematic Review

Ruth Gallardo-Ruiz,Benedicto Crespo-Facorro,Esther Setié,n-Suero,Diana Tordesillas-Gutierrez 대한신경정신의학회 2019 PSYCHIATRY INVESTIGATION Vol.16 No.5

Objective: To determine possible progressive changes of the grey matter at the first stages of the schizophrenia spectrum disorders, and to determine what regions are involved in these changes. Methods: We searched the literature concerning studies on longitudinal changes in grey matter in first-episode psychosis using magnetic resonance imaging, especially studies with an interval between scans of more than a year. Only articles published before 2018 were searched. We selected 19 magnetic resonance imaging longitudinal studies that used different neuroimaging analysis techniques to study changes in cerebral grey matter in a group of patients with a first episode of psychosis. Results: Patients with first episode of psychosis showed a decrease over time in cortical grey matter compared with a group of control subjects in frontal, temporal (specifically in superior regions), parietal, and subcortical regions. In addition to the above, studies indicate that patients showed a grey matter decrease in cerebellum and lateral ventricles volume. Conclusion: The results suggest a decrease in grey matter in the years after the first episode of psychosis. Furthermore, the results of the studies showed consistency, regardless of the methods used in their analyses, as well as the time intervals between image collections.