Matrix multiplication is probably one of the most basic and important operations having a wide range of applications in studying Linear algebra. We will be primarily focused on reviewing different algorithms for the multiplication of two n×n matrices. The most Naive method has a complexity of O(n3) but interestingly, algorithms with better complexity have been achieved. We will get to see how the complexity can be improved further, occasionally giving a survey.

We formulate a more straightforward, symmetry-based techniques in manually computing the determinant of any n×n matrix by revisiting Dodgson’s condensation method and strategically applying elementary row operations and the definition and properties of determinants. The result yields a more simplified mnemonical algorithm than the usual reduction by elementary row operations and co-factor expansion and with minimal terminal divisions performed compared to Dodgson’s internal divisions.

Peatlands in Indonesia are one of the world's largest carbon sinks, helping to regulate greenhouse gas emissions and global climate change. Lophostemon suaveolens is a plant found in Papua's endemic peat ecosystem that grows well in wet areas with low fertility. It is geographically dispersed and has the potential for peatland re-habilitation. We present an efficient and reproducible protocol for in vitro multiplication of plantlets using nodal segments and shoot apices collected from plantlets. The Murashige and Skoog (MS) medium was sup-plemented with single and double kinds of PGRs (BAP and NAA). The concentration used in this study is based on the best findings of earlier preliminary research on single PGR treatment (0.5 ml/l BAP and 0.1 ml/l NAA) for the initiation stage. After 3 months of in vitro axillary shoot initiation, elongated shoots from plant-lets were separated from the clumps and multiplied. Until 6 months incubation, the potential multiplication of axillary shoots was determined by counting the number of nodule multiplication coefficients (NMC), shoot length, root length, and number of leaves (6 successive subculture). The highest average per plantlet of NMC (8.4 0.13) was observed in MS medium with a combination of 0.5 ml/l BAP and 0.1 ml/l NAA (double PGRs), with the longest shoots (5.91 0.17 cm), the longest root length (8.83 0.69 cm), and the most leaves (32 0.69). They were cultured in a pH 5 environment with a light photoperiod (40 mol/m2/s intensity) for 16 hours. When BAP and NAA were used together after acclimatization, seedling survival was highest. This is the first report of L. suaveolens in vitro multiplication, and the protocol could be used to propagate this peatland species on a large scale.

Sechium compositum (Cucurbitaceae) is a wild species that is distributed in the Soconusco region, Chiapas, Mexico, and the border with Guatemala. This species has an intangible biochemical value resulting from the pharmacological relevance of its secondary metabolites. However, as a consequence of the lack of knowledge about its importance, it is being displaced from its habitat at an accelerated rate, incurring the risk of genetic loss. Therefore, an in vitro culture protocol with two experimental phases was evaluated to propagate, conserve, and regenerate this species. The first phases considered the multiplication of seedlings, adding seven concentrations (0.1, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2 mg mL-1) of 6-benzylaminopurine (BA) and thidiazuron (TDZ) and evaluating the number of buds, shoots, and seedling height. The best multiplication response was recorded with 0.1, 0.2, 0.4, and 1.0 mg L-1 of BA and 0.1 mg L-1 of TDZ, as well as the MS base culture medium. With 0.1 mg L-1 of BA, 52-mm height, 1.36 shoots, and 9.22 buds were obtained, surpassing the MS control (MS culture medium alone); likewise, twice more roots (80%) and a 14% reduction in the bud structure were packed in 50 repetitions. The second phase consisted of inducing callus formation from stem and leaf explants through the addition of 0.5, 1.0, and 2.0 mg L-1 of TDZ and 2,4-Dichlorophenoxyacetic acid (2,4-D) to the medium. The results demonstrated that the induction of callogenesis in S. compositum has a better start from the stem in a medium with 2.0 mg L-1 of 2,4-D, whose value was 4.89 —equivalent to 97.8% callus formation around the explant. The addition of 500 mg L-1 of polyvinylpyrrolidone (PVP) is also suggested to reduce oxidation. This protocol represents a significant advance in the conservation, multiplication, and callogenesis of S. compositum and contributes to its rescue and revaluation in the face of the danger of extinction.

Thermal safety of lithium-ion traction batteries is a highly concerned issue in the field of electric transportation. The large amount of gas emission during the thermal runaway process of batteries has highly safety hazards such as fire and explosion. Quantitative analysis of emission is one of the important challenges for testing and evaluating battery safety. Focus on quantifying the gas emission through large-scale thermal propagation in battery modules and packs, based on the idea of cell result multiplication, this article conducts thermal runaway emission analysis on single cell and modules, and compares the behavior of thermal runaway and gas emission of the cell and module from the aspects of temperature, pressure, gas composition and battery morphology. The feasibility of the cell result multiplication method was verified from the perspective of experimental data.

o estudo visa explicar os efeitos diferentes entre as equações de multiplicação, soma, divisão e subtração. Nesse sentido, as equações de multiplicação e soma desenvolvem um novo elemento ou fato, enquanto isso, as equações de divisão expressar informações dentro do elemento ou fato, a subtração é a seleção de elemento ou fato Alem disso, para uma equação de multiplicação e soma resulta no novo elemento ou fato é necessario dependência de conceitos físicos específicos, sem alteração extrema, pois resultaria na alteração do elemento ou fato. vale ressaltar, a importância de valores fixos na equação para descreve a natureza específica do elemento ou fato. O estudo possibilita o entendimento das equações em aberta(sem solução) na matematica e fisica, além de compreender equações existentes. O estudo faz referência à teoria da necessidade obrigatória e teoria da diferenças entre elementos.

An optical frequency comb (OFC) generator based on an electro-optic Talbot laser and cross-phase modulation (XPM) in a high nonlinear fiber (HNLF) is designed and demonstrated. The Talbot laser is an electro-optic frequency shifting loop to produce repetition rate multiplied pulses, and these pulses working as a pump signal induce XPM process in the HNLF to modulate the phase of a probe signal. At the output of the HNLF, OFCs with a multiplied line-spacing can be generated. The effects of the pump power and the HNLF length on the performance of the generated OFCs are theoretically analyzed. In the experiments, the line-spacing of the generated OFCs is respectively multiplied to be 10 GHz, 15 GHz and 20 GHz with a factor of 2, 3 and 4. The center of the OFCs is tuned in a 4-nm range by adjusting the wavelength of the probe signal.

In this paper, we introduce as a pedagogical strategy an internal division-free, straightforward, and symmetrically progressing algorithm in manually computing matrix inverse and solving systems of linear equations by revisiting the application of elementary row operations in the Gauss-Jordan reduction method and connecting it to the determinant method. The proposed cross-multiplication-flip (CMF) algorithm employs cross-multiplication similar to the butterfly movement in computing determinants as a strategic application of elementary row operations to efficiently reduce the rows and then applies flipping of rows and entries to put an upper triangular matrix into lower triangular form to continue the reduction process.

Live attenuated vaccines have through history proved themselves as safe and efficient. Pasteur has developed a vaccine against rabies, through a long process of passage of the virus wild type through rabbits. The result was one of the most efficient attenuated live-virus vaccines. This paper suggests a method based on calculations of biothermodynamic properties of potential tissues for vaccine application, predilected host tissue and virus wild type. Gibbs energy of biosynthesis represents the thermodynamic driving force for virus multiplication. The attenuated strain of the virus should possess Gibbs energy of biosynthesis, which is less negative than the predilected target tissue, but more negative than the vaccine portal tissue. In that way, the attenuated virus strain should be able to multiply in the vaccine portal of entry tissue and cause an immune response, making it efficient. On the other hand, the attenuated strain cannot multiply in the predilected host tissue, making it safe. The attenuation is achieved by adding a gene to the virus, which encodes a ballast protein. Production of the ballast protein would make virus multiplication less favorable, since it would require energy, but be useless to the virus.

Many authors proposed analytical methods for solving fully fuzzy Sylvester matrix equation (FFSME) based on Vec-operator and Kronecker product. However, these methods are restricted to nonnegative fuzzy numbers and cannot be extended to FFSME with near-zero fuzzy numbers. The main intention of this paper is to develop a new numerical method for solving FFSME with near-zero trapezoidal fuzzy numbers that provides a wider scope of trapezoidal fully fuzzy Sylvester matrix equation (TrFFSME) in scientific applications. This numerical method can solve the trapezoidal fully fuzzy Sylvester matrix equation with arbitrary coefficients and find all possible finite arbitrary solutions for the system. In order to obtain all possible fuzzy solutions, the TrFFSME is transferred to a system of non-linear equations based on newly developed arithmetic fuzzy multiplication between trapezoidal fuzzy numbers. The fuzzy solutions to the TrFFSME are obtained by developing a new two-stage algorithm. To illustrate the proposed method numerical example is solved.

In this paper, we present a high-speed, unified elliptic curve cryptography (ECC) processor for arbitrary Weierstrass curves over GF(p), which to the best of our knowledge, outperforms other similar works in terms of execution time. Our approach employs the combination of the schoolbook long and Karatsuba multiplication algorithm for the elliptic curve point multiplication (ECPM) to achieve better parallelization while retaining low complexity. In the hardware implementation, the substantial gain in speed is also contributed by our n-bit pipelined Montgomery Modular Multiplier (pMMM), which is constructed from our n-bit pipelined multiplier-accumulators that utilizes DSP primitives as digit multipliers. Additionally, we also introduce our unified, pipelined modular adder/subtractor (pMAS) for the underlying field arithmetic, and leverage a more efficient yet compact scheduling of the Montgomery ladder algorithm. The implementation on the 7-series FPGA: Virtex-7, Kintex-7, and XC7Z020, yields 0.139, 0.138, and 0.206 ms of execution time, respectively. Furthermore, since our pMMM module is generic for any curve in Weierstrass form, we support multi-curve parameters, resulting in a unified ECC architecture. Lastly, our method also works in constant time, making it suitable for applications requiring high speed and SCA-resistant characteristics.

Elliptic curve cryptography (ECC) over prime fields relies on scalar point multiplication realized by point addition and point doubling. Point addition and point doubling operations consist of many modular multiplications of large operands (256 bits for example), especially in projective and Jacobian coordinates which eliminate the modular inversion required in affine coordinates for every point addition or point doubling operation. Accelerating modular multiplication is therefore important for high-performance ECC. This paper presents the hardware implementations of modular multiplication algorithms, including 1) Interleaved modular multiplication (IMM), 2) Montgomery modular multiplication (MMM), 3) Shift-sub modular multiplication (SSMM), 4) SSMM with advance preparation (SSMMPRE), and 5) SSMM with CSAs and sign detection (SSMMCSA) algorithms, and evaluates their execution time (the number of clock cycles and clock frequency) and required hardware resources (ALMs and registers). Experimental results show that SSMM is 1.76 times faster than IMM, and SSMMCSA is 3.21 times faster than IMM. We also present the ECC hardware implementations based on the Secp256k1 protocol in affine, projective, and Jacobian coordinates using the IMM, SSMM, SSMMPRE, and SSMMCSA algorithms, and investigate their cost and performance. Our ECC implementations can be applied to the design of hardware security module systems.

SARS-CoV-2 belongs to the group of RNA viruses with a pronounced tendency to mutate. Omicron BA.2.75 is a subvariant believed to be able to suppress the currently dominant BA.5. Omicron BA.2.75 is characterized by a greater infectivity compared to earlier Omicron variants. However, Gibbs energy of biosynthesis of virus particles is slightly less negative compared to those of other variants. Thus, the multiplication rate of Omicron BA.2.75 is lower than that of other variants. This leads to slower accumulation of newly formed virions and less damage to host cells, indicating evolution of SARS-CoV-2 towards decreasing pathogenicity.