On Hadamard Product of Hypercomplex Numbers

Certain product rules take various forms in the set of hypercomplex numbers. In this paper, we introduce a new multiplication form of the hypercomplex numbers that will be called «the Hadamard product», inspired by the analogous product in the real matrix space, and investigate some algebraic properties of that, including the norm of inequality. In particular, we extend our new definition and its applications to the complex matrix theory.

New Approach for Secant Update Generalized Version of PSB

Working with Quasi-Newton methods in optimization leads to one important challenge, being to find an estimate of the Hessian matrix as close as possible to the real matrix. While multisecant methods are regularly used to solve root finding problems, they have been little explored in optimization because the symmetry property of the Hessian matrix estimation is generally not compatible with the multisecant property. In this paper, we propose a solution to apply multisecant methods to optimization problems. Starting from the Powell-Symmetric-Broyden (PSB) update formula and adding pieces of information from the previous steps of the optimization path, we want to develop a new update formula for the estimate of the Hessian. A multisecant version of PSB is, however, generally mathematically impossible to build. For that reason, we provide a formula that satisfies the symmetry and is as close as possible to satisfy the multisecant condition and vice versa for a second formula. Subsequently, we add enforcement of the last secant equation to the symmetric formula and present a comparison between the different methods.

Ferrate (VI) as efficient oxidant for elimination of sulfamethazine in aqueous wastes: Real matrix implications

The presence of antibiotics in aquatic environments has become a serious concern since they develop the antibiotic/multi-drug-resistant bacteria which further affect to living beings. The study intended to assess the freshly synthesized ferrate (VI) in the degradation of an important emerging micro-pollutant i.e., sulfamethazine (SMZ). Moreover, the real matrix implications are extensively conducted for implication of ferrate (VI) technology as safer and viable options. Batch reactor studies enabled the molar ratio of ferrate (VI) to sulfamethazine is 2:1 with overall rate constant 6,128 mM-2.min-1. Percentage elimination of sulfamethazine was observed Ca. 80% at initial sulfamethazine concentration 0.02 mM and ferrate (VI) dose 0.1 mM. Presence of several co-ions NaCl, Na2HPO4, NaNO3, oxalic acid and NaNO2 showed insignificant effect on elimination of sulfamethazine; whereas the efficiency of ferrate (VI) was lowered due to glycine and EDTA. Mineralization of sulfamethazine is significantly increased at lower pH value (pH 5.0). Further, the removal of sulfamethazine in the real water matrix showed that the elimination efficiency of sulfamethazine is almost unaffected as compared to the distilled water treatment. This implied that ferrate (VI) is a viable and greener option for treatment of emerging water pollutants to enhance the efficiency of existing wastewater treatment plants.

On the Eventual Exponential Positivity of Some Tree Sign Patterns

An n×n matrix A is called eventually exponentially positive (EEP) if etA=∑k=0∞tkAkk!>0 for all t≥t0, where t0≥0. A matrix whose entries belong to the set {+,−,0} is called a sign pattern. An n×n sign pattern A is called potentially eventually exponentially positive (PEEP) if there exists some real matrix realization A of A that is EEP. Characterizing the PEEP sign patterns is a longstanding open problem. In this article, A is called minimally potentially eventually exponentially positive (MPEEP), if A is PEEP and no proper subpattern of A is PEEP. Some preliminary results about MPEEP sign patterns and PEEP sign patterns are established. All MPEEP sign patterns of orders n≤3 are identified. For the n×n tridiagonal sign patterns Tn, we show that there exists exactly one MPEEP tridiagonal sign pattern Tno. Consequently, we classify all PEEP tridiagonal sign patterns as the superpatterns of Tno. We also classify all PEEP star sign patterns Sn and double star sign patterns DS(n,m) by identifying all the MPEEP star sign patterns and the MPEEP double star sign patterns, respectively.

Riesz transforms of a general Ornstein–Uhlenbeck semigroup

We consider Riesz transforms of any order associated to an Ornstein–Uhlenbeck operator with covariance given by a real, symmetric and positive definite matrix, and with drift given by a real matrix whose eigenvalues have negative real parts. In this general Gaussian context, we prove that a Riesz transform is of weak type (1, 1) with respect to the invariant measure if and only if its order is at most 2.

Acceleration of Hessenberg reduction for nonsymmetric matrix.

The worth of finding a general solution for nonsymmetric eigenvalue problems is specified in many areas of engineering and science computations, such as reducing noise to have a quiet ride in automotive industrial engineering or calculating the natural frequency of a bridge in civil engineering. The main objective of this thesis is to design a hybrid algorithm (based on CPU-GPU) in order to reduce general non-symmetric matrices to Hessenberg form. A new blocks method is used to achieve great efficiency in solving eigenvalue problems and to reduce the execution time compared with the most recent related works. The GPU part of proposed algorithm is thread based with asynchrony structure (based on FFT techniques) that is able to maximize the memory usage in GPU. On a system with an Intel Core i5 CPU and NVIDA GeForce GT 635M GPU, this approach achieved 239.74 times speed up over the CPU-only case when computing the Hessenberg form of a 256 * 256 real matrix. Minimum matrix order (n), which the proposed algorithm supports, is sixteen. Therefore, supporting this matrix size is led to have the large matrix order range.

Acceleration of Hessenberg reduction for nonsymmetric matrix.

The worth of finding a general solution for nonsymmetric eigenvalue problems is specified in many areas of engineering and science computations, such as reducing noise to have a quiet ride in automotive industrial engineering or calculating the natural frequency of a bridge in civil engineering. The main objective of this thesis is to design a hybrid algorithm (based on CPU-GPU) in order to reduce general non-symmetric matrices to Hessenberg form. A new blocks method is used to achieve great efficiency in solving eigenvalue problems and to reduce the execution time compared with the most recent related works. The GPU part of proposed algorithm is thread based with asynchrony structure (based on FFT techniques) that is able to maximize the memory usage in GPU. On a system with an Intel Core i5 CPU and NVIDA GeForce GT 635M GPU, this approach achieved 239.74 times speed up over the CPU-only case when computing the Hessenberg form of a 256 * 256 real matrix. Minimum matrix order (n), which the proposed algorithm supports, is sixteen. Therefore, supporting this matrix size is led to have the large matrix order range.