REFINEMENTS OF THE NORM OF TWO ORTHOGONAL PROJECTIONS

In this paper,some refinements of norm equalities and inequalities of combination of two orthogonal projections are established.We use certain norm inequalities for positive contraction operator to establish norm inequalities for combination of orthogonal projections on a Hilbert space.Furthermore,we give necessary and sufficient conditions under which the norm of the above combination of o`rthogonal projections attains its optimal value.

Optimal Design of High-Speed Partial Flow Pumps using Orthogonal Tests and Numerical Simulations

To investigate the influence of structural parameters on the performances and internal flow characteristics of partial flow pumps at a low specific speed of 10000 rpm,special attention was paid to the first and second stage impeller guide vanes.Moreover,the impeller blade outlet width,impeller inlet diameter,blade inclination angle,and number of blades were considered for orthogonal tests.Accordingly,nine groups of design solutions were formed,and then used as a basis for the execution of numerical simulations(CFD)aimed at obtaining the efficiency values and heads for each design solution group.The influence of impeller geometric parameters on the efficiency and head was explored,and the“weight”of each factor was obtained via a range analysis.Optimal structural parameters were finally chosen on the basis of the numerical simulation results,and the performances of the optimized model were verified accordingly(yet by means of CFD).Evidence is provided that the increase in the efficiency and head of the optimized model was 12.11%and 23.5 m,respectively,compared with those of the original model.

Dynamically enhanced Autler–Townes splitting by orthogonal XUV fields

Based on numerical solutions of the time-dependent Schr ¨odinger equation, we theoretically investigate the photoelectron spectrum of hydrogen atoms ionized by a pair of ultrashort, intense, and orthogonally polarized laser pulses with a relative time delay in a pump–probe configuration. The pump pulse resonantly excites electrons from the 1s and 2p levels,inducing Rabi oscillations. The resulting dynamically enhanced Autler–Townes(AT) splitting is observed in the photoelectron energy spectrum upon interaction with the second probe pulse. In contrast to the previous parallel-polarization scheme, the proposed orthogonal-polarization configuration enables the resolution of dynamically enhanced AT splitting over a considerably wider range of probe photon energies.

Manipulating the electron dynamics in the non-sequential double ionization process of Ar atoms by an orthogonal two-color laser field

Electron dynamics during non-sequential double ionization(NSDI) is one of the most attractive areas of research in the field of laser–atom or laser–molecule interaction. Based on the classic two-dimensional model, we study the process of NSDI of argon atoms driven by a few-cycle orthogonal two-color laser field composed of 800 nm and 400 nm laser pulses. By changing the relative phase of the two laser pulses, a localized enhancement of NSDI yield is observed at 0.5πand 1.5π, which could be attributed to a rapid and substantial increase in the number of electrons returning to the parent ion within extremely short time intervals at these specific phases. Through the analysis of the electron–electron momentum correlations within different time windows of NSDI events and the angular distributions of emitted electrons in different channels, we observe a more pronounced electron–electron correlation phenomenon in the recollision-induced ionization(RII) channel. This is attributed to the shorter delay time in the RII channel.

Optimization of Center of Gravity Position and Anti-Wave Plate Angle of Amphibious Unmanned Vehicle Based on Orthogonal Experimental Method

When the amphibious vehicle navigates in water,the angle of the anti-wave plate and the position of the center of gravity greatly influence the navigation characteristics.In the relevant research on reducing the navigation resistance of amphibious vehicles by adjusting the angle of the anti-wave plate,there is a lack of scientific selection of parameters and reasonable research of simulation results by using mathematical methods,and the influence of the center of gravity position on navigation characteristics is not considered at the same time.To study the influence of the combinations of the angle of the anti-wave plate and the position of the center of gravity on the resistance reduction characteristics,a numerical calculation model of the amphibious unmanned vehicle was established by using the theory of computational fluid dynamics,and the experimental data verified the correctness of the numerical model.Based on this numerical model,the navigation characteristics of the amphibious unmanned vehicle were studied when the center of gravity was located at different positions,and the orthogonal experimental design method was used to optimize the parameters of the angle of the anti-wave plate and the position of the center of gravity.The results show that through the parameter optimization analysis based on the orthogonal experimental method,the combination of the optimal angle of the anti-wave plate and the position of the center of gravity is obtained.And the numerical simulation result of resistance is consistent with the predicted optimal solution.Compared with the maximum navigational resistance,the parameter optimization reduces the navigational resistance of the amphibious unmanned vehicle by 24%.

Optimization of short-circuit transition parameters of Q235B steel GMAW based on orthogonal gray relational analysis

In gas metal arc welding(GMAW)process,the short-circuit transition was the most typical transition observed in molten metal droplets.This paper used orthogonal tests to explore the coupling effect law of welding process parameters on the quality of weld forming under short-circuit transition,the design of 3 factors and 3 levels of a total of 9 groups of orthogonal tests,welding current,welding voltage,welding speed as input parameters:effective area ratio,humps,actual linear power density,aspect ratio,Vickers hardness as output paramet-ers(response targets).Using range analysis and trend charts,we can visually depict the relationship between input parameters and a single output parameter,ultimately determining the optimal process parameters that impact the single output index.Then combined with gray the-ory to transform the three response targets into a single gray relational grade(GRG)for analysis,the optimal combination of the weld mor-phology parameters as follows:welding current 100 A,welding voltage 25 V,welding speed 30 cm/min.Finally,validation experiments were conducted,and the results showed that the error between the gray relational grade and the predicted value was 2.74%.It was observed that the effective area ratio of the response target significantly improved,validating the reliability of the orthogonal gray relational method.

Optimization of Extraction Process of Fagopyri Dibotryis Rhizoma by Orthogonal Experiment

[Objectives]To optimize the water extraction process of Fagopyri Dibotryis Rhizoma.[Methods]The entropy weight method was used to determine the weight of epicatechin extraction rate and dry extract rate and calculate the comprehensive score.The water extraction process of Fagopyri Dibotryis Rhizoma was optimized by orthogonal design with the comprehensive score as the indicator and the amount of water,extraction time and extraction times as the factors.[Results]The optimum extraction process of Fagopyri Dibotryis Rhizoma was as follows:adding 10 times of water,extracting 3 times,and extracting for 60 min each time.[Conclusions]The optimized extraction process is stable and feasible,and can be used for the extraction of Fagopyri Dibotryis Rhizoma.

Model Order Reduction Methods for Discrete Systems via Discrete Pulse Orthogonal Functions

This paper explores model order reduction(MOR)methods for discrete linear and discrete bilinear systems via discrete pulse orthogonal functions(DPOFs).Firstly,the discrete linear systems and the discrete bilinear systems are expanded in the space spanned by DPOFs,and two recurrence formulas for the expansion coefficients of the system’s state variables are obtained.Then,a modified Arnoldi process is applied to both recurrence formulas to construct the orthogonal projection matrices,by which the reduced-order systems are obtained.Theoretical analysis shows that the output variables of the reducedorder systems can match a certain number of the expansion coefficients of the original system’s output variables.Finally,two numerical examples illustrate the feasibility and effectiveness of the proposed methods.

Steering the energy sharing of electrons in nonsequential double ionization with orthogonally polarized two-color field

Using the semiclassical ensemble model,the dependence of relative amplitude for the recollision dynamics in nonsequential double ionization(NSDI)of neon atom driven by the orthogonally polarized two-color field(OTC)laser field is theoretically studied.And the dynamics in two typical collision pathways,recollision-impact-ionization(RII)and recollisionexcitation with subsequent ionization(RESI),is systematically explored.Our results reveal that the V-shaped structure in the correlated momentum distribution is mainly caused by the RII mechanism when the relative amplitude of the OTC laser field is zero,and the first ionized electrons will quickly skim through the nucleus and share few energy with the second electron.As the relative amplitude increases,the V-shaped structure gradually disappears and electrons are concentrated on the diagonal in the electron correlation spectrum,indicating that the energy sharing after electrons collision is symmetric for OTC laser fields with large relative amplitudes.Our studies show that changing the relative amplitude of the OTC laser field can efficiently control the electron–electron collisions and energy exchange efficiency in the NSDI process.

Selecting the Technology of Sodium Silicate Modified Poplar with the Highest Performance by Fuzzy Orthogonal Method

Sodium silicate modification can improve the overall performance of wood.The modification process has a great influence on the properties of modified wood.In this study,a new method was introduced to analyze the wood modification process,and the properties of modified wood were studied.Poplar wood was modified with sodium silicate by vacuum-pressure impregnation.After screening using single-factor experiments,an orthogonal experiment was carried out with solution concentration,impregnation time,impregnation pressure,and the cycle times as experimental factors.The modified poplar with the best properties was selected by fuzzy mathematics and characterized by SEM,FT-IR,XRD and TG.The results showed that some lignin and hemicellulose were removed from the wood due to the alkaline action of sodium silicate,and the orderly crystal area of poplar became disorderly,resulting in the reduction of crystallinity of the modified poplar wood.FT-IR analysis showed that sodium silicate was hydrolyzed to form polysilicic acid in wood,and structural analysis revealed the formation of Si-O-Si and Si-O-C,indicating that sodium silicate reacted with fibers on the wood cell wall.TG-DTG curves showed that the final residual mass of modified poplar wood increased from 25%to 67%,and the temperature of the maximum loss rate decreased from 343℃ to 276℃.The heat release and smoke release of modified poplar wood decreased obviously.This kind of material with high strength and fire resistance can be used in the outdoor building and indoor furniture.

The investigation of multiple factors on elastic anisotropy of artificial shale based on the orthogonal experiment

Elastic anisotropy of shales is critical to accurate constraints for rock physical models,quantitative interpretation and hydraulic fracturing.However,the causes of elastic anisotropy of shales are very complicated,and the understanding of how multiple influence factors affect the elastic anisotropy of shales is still not clear.Hence,the orthogonal experiment,as an effective multiple factors experimental method,is adopted in this study to analyze the effect of multiple factors for shale elastic anisotropy.Three factors,clay content,organic matter(OM)content and compaction stress are selected as independent variables,the orthogonal test table L_(16)(4^(3))with four levels for each factor is adopted.According to the designed orthogonal table,sixteen artificial shales are constructed based on the cold-pressing method,and all the dry artificial shales are measured by the ultrasonic measurements.The influence of each factor on the elastic anisotropy and the sensitivity orders of three factors are obtained using the range analysis.The orders of sensitivity for selected factors follow the sequence clay content>compaction stress>OM content for velocity anisotropy parameters.The compaction mechanism of artificial shales is also discussed by the compaction factor,which are positively correlated with the velocity anisotropy parameters.The clay platelets orientation distribution function(ODF)of samples is evaluated by a theoretical model,the ODF coefficients are significantly affected by the clay content and compaction stress,and W200 are much more sensitive to these factors than W400.The results can provide a critical rock physics basis for quantitative interpretation and reservoir prediction of the low-maturity or maturity shale reservoir.

A Spider Monkey Optimization Algorithm Combining Opposition-Based Learning and Orthogonal Experimental Design

As a new bionic algorithm,Spider Monkey Optimization(SMO)has been widely used in various complex optimization problems in recent years.However,the new space exploration power of SMO is limited and the diversity of the population in SMO is not abundant.Thus,this paper focuses on how to reconstruct SMO to improve its performance,and a novel spider monkey optimization algorithm with opposition-based learning and orthogonal experimental design(SMO^(3))is developed.A position updatingmethod based on the historical optimal domain and particle swarmfor Local Leader Phase(LLP)andGlobal Leader Phase(GLP)is presented to improve the diversity of the population of SMO.Moreover,an opposition-based learning strategy based on self-extremum is proposed to avoid suffering from premature convergence and getting stuck at locally optimal values.Also,a local worst individual elimination method based on orthogonal experimental design is used for helping the SMO algorithm eliminate the poor individuals in time.Furthermore,an extended SMO^(3)named CSMO^(3)is investigated to deal with constrained optimization problems.The proposed algorithm is applied to both unconstrained and constrained functions which include the CEC2006 benchmark set and three engineering problems.Experimental results show that the performance of the proposed algorithm is better than three well-known SMO algorithms and other evolutionary algorithms in unconstrained and constrained problems.

Generation of quasi-chirp-free isolated attosecond pulses from atoms under the action of orthogonal two-color combined pulse of fundamental frequency and higher intensity second harmonic fields

The intrinsic chirp of high-order harmonic generation is an important factor limiting the production of ultrashort attosecond pulses.Based on numerically solving the time-dependent Schrodinger equation,the generation process of highorder harmonic from the He atom under the action of orthogonal two-color combined pulse of fundamental frequency and higher intensity second harmonic fields is studied.In this paper,we propose to achieve quasi-chirp-free isolated attosecond pulses by superimposing a higher second-harmonic field on the orthogonal direction of the fundamental frequency field.It is found that the high-energy part of its harmonic emission exhibits small chirp characteristics,which can be used to synthesize isolated attosecond pulses.Through the analysis of the wave packets evolution and the classical motion trajectories of the electron,it is demonstrated that the quasi-chirp-free harmonic can be attributed to the simultaneous return of electrons ionized at different times to the parent particle.The influence of the relative phase of the two pulses on the harmonics is further analyzed,and it is observed that this phenomenon is sensitive to the relative phase,but it can still generate isolated attosecond pulses within a certain phase.

The formation of orthogonal joint systems and cuboidal blocks:New insights gained from flat-lying limestone beds in the region of Havre-Saint-Pierre(Quebec,Canada)

Vertical orthogonal joints are a common feature in shallow crustal rocks.There are several competing theories for their formation despite the ubiquity.We examined the exceptional exposures of orthogonal joints in flat-lying Ordovician limestone beds from the Havre-Saint-Pierre Region in Quebec,Canada(north shore of Saint-Lawrence River)to test conceptual models of joint formation in a natural setting.In the region,the spacing of cross-joints is consistently larger than the spacing of systematic joints by a factor of 1.5 approximately.The joint-spacing-to-bed-thickness ratios(s/t)are much larger in these beds(s/t=4.3 for systematic joints,and 6.4 for cross-joints)than those in higher strained strata along the south shore of the Saint-Lawrence River(s/t=1),highlighting the effect of tectonic strain in decreasing fracture spacing and block size.The high values of s/t indicate that cross-joint formation was unlikely caused by a switch from compression to tension once a critical s/t ratio for systematic joints was reached(as hypothesized in previous studies).We proposed a new model for the formation of orthogonal joint systems where the principal stress axes locally switch during the formation of systematic fractures.The presence of ladder-shaped orthogonal joints suggests a state of effective stress withσ_(1)^(∗)≫0>σ_(2)^(∗)>σ_(3)^(∗)and whereσ_(2)^(∗)-σ_(3)^(∗)is within the range of fracture strength variability at the time of fracture.This research provides a new mechanical model for the formation of orthogonal joint systems and cuboidal blocks.

Proper orthogonal decomposition based seismic source wavefield reconstruction for finite element reverse time migration

The large storage requirement is a critical issue in cross-correlation imaging-condition based reverse time migration(RTM),because it requires the operation of the source and receiver wavefields at the same time.The boundary value method(BVM),based on the finite difference method(FDM),can be used to reconstruct the source wavefield in the reverse time propagation in the same way as the receiver wavefield,which can reduce the storage burden of the RTM data.Considering that the FDM cannot well handle models with discontinuous material properties and rough interfaces,we develop a source wavefield reconstruction strategy based on the finite element method(FEM),using proper orthogonal decomposition(POD)to enhance computational efficiency.In this method,we divide the whole time period into several segments,and construct the POD basis functions to get a reduced order model(ROM)for the source wavefield reconstruction in each segment.We show the corresponding quantitative analysis of the storage requirement of the POD-FEM.Numerical tests on the homogeneous model show the effectiveness of the proposed method,while the layered model and part of the Marmousi model tests indicate that the POD-FEM can keep an excellent balance between computational efficiency and memory usage compared with the full-stored method(FSM)and the BVM,and can be effectively applied in imaging.

Finite Element Orthogonal Collocation Approach for Time Fractional Telegraph Equation with Mamadu-Njoseh Polynomials

Finite element method (FEM) is an efficient numerical tool for the solution of partial differential equations (PDEs). It is one of the most general methods when compared to other numerical techniques. PDEs posed in a variational form over a given space, say a Hilbert space, are better numerically handled with the FEM. The FEM algorithm is used in various applications which includes fluid flow, heat transfer, acoustics, structural mechanics and dynamics, electric and magnetic field, etc. Thus, in this paper, the Finite Element Orthogonal Collocation Approach (FEOCA) is established for the approximate solution of Time Fractional Telegraph Equation (TFTE) with Mamadu-Njoseh polynomials as grid points corresponding to new basis functions constructed in the finite element space. The FEOCA is an elegant mixture of the Finite Element Method (FEM) and the Orthogonal Collocation Method (OCM). Two numerical examples are experimented on to verify the accuracy and rate of convergence of the method as compared with the theoretical results, and other methods in literature.

Effects of Orthogonal Heat Treatment on Microstructure and Mechanical Properties of GN9 Ferritic/Martensitic Steel

Microstructure and mechanical properties of GN9 Ferritic/Martensitic steel for sodium-cooled fast reactors have been investigated through orthogonal design and analysis.Scanning electron microscopy(SEM),transmission electron microscopy(TEM),differential scanning calorimeter(DSC),tensile and impact tests were used to evaluate the heat treatment parameters on yield strength,elongation and ductile-to-brittle transition temperature(DBTT).The results indicate that the microstructures of GN9 steel after orthogonal heat treatments consist of tempered martensite,M23C6,MX carbides and MX carbonitrides.The average prior austenite grains increase and the lath width decreases with the austenitizing temperature increasing from 1000°C to 1080°C.Tempering temperature is the most important factor that influences the dislocation evolution,yield strength and elongation compared with austenitizing tempera-ture and cooling methods.Austenitizing temperature,tempering temperature and cooling methods show interactive effects on DBTT.Carbide morphology and distribution,which is influenced by austenitizing and tempering tempera-tures,is the critical microstructural factor that influences the Charpy impact energy and DBTT.Based on the orthogo-nal design and microstructural analysis,the optimal heat treatment of GN9 steel is austenitizing at 1000°C for 0.5 h followed by air cooling and tempering at 760°C for 1.5 h.

Optimization of Compound Coptis Ointment Extraction through the Integration of Pharmacodynamic Indices and Orthogonal Test

[Objectives]To determine the optimal reflux extraction process conditions for Compound Coptis Ointment.[Methods]The study employed the orthogonal experimental design method and drug-sensitive disc agar diffusion method to evaluate the extraction rate,berberine hydrochloride content,and bacteriostatic ring diameter of the extract as comprehensive indices,and optimized the extraction process conditions of Compound Coptis Ointment using ethanol reflux.[Results]Based on the results of comprehensive indices,the optimal reflux extraction conditions for Compound Coptis Ointment were determined to be soaking in 1.5 times the amount of 70%ethanol for 2 h,followed by two extractions with 6 times the amount of 70%ethanol for 1 h each time.Three verification tests were conducted under the optimal process conditions.The yield of the extract was 28.32%±0.53%,the content of berberine hydrochloride was 4.68%±0.45%,and the diameter of the bacteriostatic ring was(2.5±0.2)cm.[Conclusions]The extract had higher drug content and exhibited better antibacterial effects.The optimized extraction process is simple,stable,and reliable,and can be effectively used to optimize the extraction process of Compound Coptis Ointment.

Coherence Based Sufficient Condition for Support Recovery Using Generalized Orthogonal Matching Pursuit

In an underdetermined system,compressive sensing can be used to recover the support vector.Greedy algorithms will recover the support vector indices in an iterative manner.Generalized Orthogonal Matching Pursuit(GOMP)is the generalized form of the Orthogonal Matching Pursuit(OMP)algorithm where a number of indices selected per iteration will be greater than or equal to 1.To recover the support vector of unknown signal‘x’from the compressed measurements,the restricted isometric property should be satisfied as a sufficient condition.Finding the restricted isometric constant is a non-deterministic polynomial-time hardness problem due to that the coherence of the sensing matrix can be used to derive the sufficient condition for support recovery.In this paper a sufficient condition based on the coherence parameter to recover the support vector indices of an unknown sparse signal‘x’using GOMP has been derived.The derived sufficient condition will recover support vectors of P-sparse signal within‘P’iterations.The recovery guarantee for GOMP is less restrictive,and applies to OMP when the number of selection elements equals one.Simulation shows the superior performance of the GOMP algorithm compared with other greedy algorithms.

Optimization for Microbial Degumming of Ramie with Bacillus subtilis DZ_(5) in Submerged Fermentation by Orthogonal Array Design and Response Surface Methodology

As a kind of natural fiber,ramie fiber has distinctive advantages in textile application,but the application is limited due to the traditional degumming mode.Compared with the traditional degumming process,the microbial degumming process has many advantages.To obtain the optimal conditions for degumming ramie with Bacillus subtilis DZ_(5)(BS DZ_(5)),a combined statistical approach of orthogonal array design(OAD)and response surface methodology(RSM)was used.The influences of initial pH of the bacteria medium,culture temperature,shaking speed,degumming time and inoculum size on submerged fermentation degumming were evaluated by using fractional factorial design.The main factors in the analysis were culture temperature,shaking speed and initial pH.The residual gum mass fraction was used as the optimization index,and the optimal conditions for degumming were determined by central composite design and RSM.Thus with only a limited number of experiments,an optimal ramie microbial degumming condition was found as the culture temperature of 40℃,the initial pH in the culture medium of 8.5,the shaking speed of 205 r/min,the degumming time of 96 h and the inoculum size of 5%.After microbial degumming of ramie under the optimal conditions,there was only 10.6%residual gum by mass in the fiber.In addition,the effective degumming of BS DZ_(5)was also confirmed by a scanning electron microscope(SEM).