Order P4 Research Articles

On Linear Codes over Local Rings of Order p4

Suppose R is a local ring with invariants p,n,r,m,k and mr=4, that is R of order p4. Then, R=R0+uR0+vR0+wR0 has maximal ideal J=uR0+vR0+wR0 of order p(m−1)r and a residue field F of order pr, where R0=GR(pn,r) is the coefficient subring of R. In this article, the goal is to improve the understanding of linear codes over small-order non-chain rings. In particular, we produce the MacWilliams formulas and generator matrices for linear codes of length N over R. In order to accomplish that, we first list all such rings up to isomorphism for different values of p,n,r,m,k. Furthermore, we fully describe the lattice of ideals in R and their orders. Next, for linear codes C over R, we compute the generator matrices and MacWilliams identities, as shown by numerical examples. Given that non-chain rings are not principal ideals rings, it is crucial to acknowledge the difficulties that arise while studying linear codes over them.

Non-Abelian extensions of degree p3 and p4 in characteristic p > 2

This paper describes in terms of Artin-Schreier equations field extensions whose Galois group is isomorphic to any of the four non-cyclic groups of order p3 or the ten non-Abelian groups of order p4, p an odd prime, over a field of characteristic p.

Groups of nilpotency class 2 of order p4 as additive groups of local nearrings

Groups of nilpotency class 2 of order p4 as additive groups of local nearrings

Electrical Impedance Spectroscopic Analysis on Whole Blood Cells to Correlate Severity Level of Ischemic Stroke Patients

Measuring the impedance value of biological materials in the form of blood samples has been widely used to determine a person's health status. Ischemic stroke patients in terms of impedance values and correlates with morphological tests on changes in red blood cells. The Electrical Impedance Spectroscopy (EIS) method is applied by providing a blood sample which is detected by electrodes and read using a BIA tool set so that the results are obtained on a laptop. The results obtained from the EIS test are in the form of Bode graphs, Bode phase graphs, Nyquist graphs, and impedance values. The EIS method uses a frequency of 100 Hz to 100 kHz by injecting a current of 10 μA. In this study, the impedance value obtained for normal people was 1058 Ω to 709 Ω, while for ischemic stroke patients with various levels of condition, it was from 517 Ω to 761 Ω. When counting the number of morphological cells in the patient's blood, changes were found ranging from the most cell changes to the least, namely with a value of 44% to 19%. The severity level of ischemic stroke patients obtained based on impedance tests and morphological tests includes the order P4, P5, P3, P2, P10, P21, P15, P19, P1, P6, P13, P18, P14, P17, P16, P20, P12, P11, P8, P7, P9.

Validation of a pre-established triage protocol for critically ill patients in a COVID-19 outbreak under resource scarcity: A retrospective multicenter cohort study.

In case of COVID-19 related scarcity of critical care resources, an early French triage algorithm categorized critically ill patients by probability of survival based on medical history and severity, with four priority levels for initiation or continuation of critical care: P1 -high priority, P2 -intermediate priority, P3 -not needed, P4 -not appropriate. This retrospective multi-center study aimed to assess its classification performance and its ability to help saving lives under capacity saturation. ICU patients admitted for severe COVID-19 without triage in spring 2020 were retrospectively included from three hospitals. Demographic data, medical history and severity items were collected. Priority levels were retrospectively allocated at ICU admission and on ICU day 7-10. Mortality rate, cumulative incidence of death and of alive ICU discharge, length of ICU stay and of mechanical ventilation were compared between priority levels. Calculated mortality and survival were compared between full simulated triage and no triage. 225 patients were included, aged 63.1±11.9 years. Median SAPS2 was 40 (IQR 29-49). At the end of follow-up, 61 (27%) had died, 26 were still in ICU, and 138 had been discharged. Following retrospective initial priority allocation, mortality rate was 53% among P4 patients (95CI 34-72%) versus 23% among all P1 to P3 patients (95CI 17-30%, chi-squared p = 5.2e-4). The cumulative incidence of death consistently increased in the order P3, P1, P2 and P4 both at admission (Gray's test p = 3.1e-5) and at reassessment (p = 8e-5), and conversely for that of alive ICU discharge. Reassessment strengthened consistency. Simulation under saturation showed that this two-step triage protocol could have saved 28 to 40 more lives than no triage. Although it cannot eliminate potentially avoidable deaths, this triage protocol proved able to adequately prioritize critical care for patients with highest probability of survival, hence to save more lives if applied.

Twisted group ring isomorphism problem and infinite cohomology groups

We continue our investigation of a variation of the group ring isomorphism problem for twisted group algebras. Contrary to previous work, we include cohomology classes which do not contain any cocycle of finite order. This allows us to study the problem in particular over any field of characteristic 0. We prove that there are finite groups G and H which can not be distinguished by their rational twisted group algebras, while G and H can be identified by their semi-simple twisted group algebras over other fields. This is in contrast with the fact that the structural information on G which can be obtained from all the semi-simple group algebras of G is already encoded in its rational group algebra.We further show that for an odd prime p there are groups of order p4 which can not be distinguished by their twisted group algebras over F for any field F of characteristic different from p. On the other hand we prove that the groups constructed by E. Dade, which have isomorphic group algebras over any field, can be distinguished by their rational twisted group algebras. We also answer a question about sufficient conditions for the twisted group ring isomorphism problem to hold over the complex numbers.

Commutators and commutator subgroups of finite p-groups

We present a characterisation, up to isoclinism, of finite p-groups G with γ2(G), the commutator subgroup of G, of order p4 and exponent p such that not all elements of γ2(G) are commutators.

Semifield Planes Admitting the Quaternion Group Q8

We discuss a well-known conjecture that the full automorphism group of a finite projective plane coordinatized by a semifield is solvable. For a semifield plane of order pN (p > 2 is a prime, 4|p − 1) admitting an autotopism subgroup H isomorphic to the quaternion group Q8, we construct a matrix representation of H and a regular set of the plane. All nonisomorphic semifield planes of orders 54 and 134 admitting Q8 in the autotopism group are pointed out. It is proved that a semifield plane of order p4, 4|p−1, does not admit SL(2, 5) in the autotopism group.

High order symmetric direct discontinuous Galerkin method for elliptic interface problems with fitted mesh

In this article, we aim to develop a high order symmetric direct discontinuous Galerkin method solving elliptic interface problems with zero or non-zero solution jump and flux jump interface conditions. We focus on the case, in which the mesh is partitioned along the curved interface. The two interface jump conditions are naturally and simultaneously built into the numerical flux definition on the curved triangular elements' edges that overlap with the interface. We obtain a stable and high order method, regardless of the combination of the two interface jump conditions. The two interface jump conditions are both weakly enforced in the scheme formulation. Optimal (k+1)th order L2 norm error estimate is proved for polygonal interfaces. A sequence of numerical examples is carried out to verify the optimal convergence of the symmetric direct discontinuous Galerkin method with high order P2, P3 and P4 approximations. Uniform optimal convergence order, which is independent of the diffusion coefficient ratio inside and outside of the interface, is obtained. The symmetric direct discontinuous Galerkin method is shown to be capable of handling interface problems with complicated geometries.

Minimal Proper Quasifields with Additional Conditions

We investigate the finite semifields which are distributive quasifields, and finite near-fields which are associative quasifields. A quasifield Q is said to be a minimal proper quasifield if any of its sub-quasifield H ̸= Q is a subfield. It turns out that there exists a minimal proper near-field such that its multiplicative group is a Miller–Moreno group. We obtain an algorithm for constructing a minimal proper near-field with the number of maximal subfields greater than fixed natural number. Thus, we find the answer to the question: Does there exist an integer N such that the number of maximal subfields in arbitrary finite near-field is less than N? We prove that any semifield of order p4 (p be prime) is a minimal proper semifield

The non-abelian tensor square of p-groups of order p4

In this paper, in the class of [Formula: see text]-groups of order [Formula: see text], we obtain the non-abelian exterior square, the exterior center, the non-abelian tensor square, the tensor center and the third homotopy group of suspension of an Eilenberg–MacLane space [Formula: see text] of such groups.

Tetravalent half-arc-transitive graphs of order p5

A graph is half-arc-transitive if its full automorphism group acts transitively on vertices and edges, but not on arcs. Let p be a prime. It is known that there exists no tetravalent half-arc-transitive graph of order p or p2. All the tetravalent half-arc-transitive graphs of order p3 or p4 have been classified in two previous papers [9,23]. As a continuation, in this paper, a classification is given of all tetravalent half-arc-transitive graphs of order p5.

Finite dimensional Nichols algebras over certain p-groups

ABSTRACTWe investigate pointed Hopf algebras over finite nilpotent groups of odd order, with nilpotency class 2. For such a group G, we show that if its commutator subgroup coincides with its center, then there exists no non-trivial finite-dimensional pointed Hopf algebra with kG as its coradical. We apply these results to non-abelian groups of order p3, p4 and p5, and list all the pointed Hopf algebras of order p6, whose group of grouplikes is non-abelian.

Simple Twisted Group Algebras of Dimension p4 and their Semi-Centers

For a simple twisted group algebra over a group G, if G∣ is Hall subgroup of G, then the semi-center is simple. Simple twisted group algebras correspond to groups of central type. We classify all groups of central type of order p4 where p is prime and use this to show that for odd primes p there exists a unique group G of order p4, such that there exists simple twisted group algebra over G with a commutative semi-center. Moreover, if 1 < |G| <64, then the semi-center of simple twisted group algebras over G is noncommutative and this bounds are strict.

On Blocks with Abelian Defect Groups of Small Rank

Let B be a p-block of a finite group with abelian defect group D. Suppose that D has no elementary abelian direct summand of order p4. Then, we show that B satisfies Brauer’s k(B)-Conjecture (i.e. \({k(B)\le|D|}\)). Together with former results, it follows that Brauer’s k(B)-Conjecture holds for all blocks of defect at most 3. We also obtain some related results.

THE SCHUR MULTIPLIER OF PAIRS OF NONABELIAN GROUPS OF ORDER P4

Let (G,N) be a pair of groups where G is any group and N is a normal subgroup of G, then the Schur multiplier of pairs of groups is a functorial abelian group. The notion of the Schur multiplier of pairs of groups is an extension from the Schur multiplier of a group G. In this research, the Schur multiplier of pairs of finite nonabelian groups of order p4, where p is an odd prime, is determined.

Elementary proof that [formula omitted] is a DCI-group

A finite group R is a DCI-group if, whenever S and T are subsets of R with the Cayley graphs Cay(R,S) and Cay(R,T) isomorphic, there exists an automorphism φ of R with Sφ=T.Elementary abelian groups of order p4 or smaller are known to be DCI-groups, while those of sufficiently large rank are known not to be DCI-groups. The only published proof that elementary abelian groups of order p4 are DCI-groups uses Schur rings and does not work for p=2 (which has been separately proven using computers). This paper provides a simpler proof that works for all primes. Some of the results in this paper also apply to elementary abelian groups of higher rank, so may be useful for completing our determination of which elementary abelian groups are DCI-groups.

Nilpotent Rings of Order p 4 and Nilpotency Index 3

We give a complete description of irreducible rings of order p 4 and nilpotency index 3 provided that the identity x 2 = 0 and partially describe nilpotent such rings if the identity x 2 = 0. fails.

Interaction between CO2 and ionic liquids confined in the nanopores of SAPO-11

TPD-CO2 profiles of P4,4,4,6Triz confined in the nanopores of SAPO-11 (30 wt%).

Fibonacci lengths of all finite p-groups of exponent p2

The Fibonacci lengths of finite p-groups were studied by Dikici and coauthors since 1992. All considered groups are of exponent p and the lengths depend on the Wall number k(p). The p-groups of nilpotency class 3 and exponent p were studied in 2004 also by Dikici. In the paper, we study all p-groups of nilpotency class 3 and exponent p 2. Thus, we complete the study of Fibonacci lengths of all p-groups of order p 4 by proving that the Fibonacci length is k(p 2).