Areas of Trapezoids

Pentagon ABCDE is split into three triangles. To find the area of ABCDE, you could addthe area so fthetri angles.This is one reason you learned how to find the area of atriangle.

On the beginning of the Japanese Upper Paleolithic:A review of recent archaeological and anthropological evidence

The beginning of the Japanese Upper Paleolithic has mainly been examined using two major models:the Middle Paleolithic evolutionary model within the archipelago and the continental Upper Paleolithic diffusion/migration model.However,recent archeological data from Japan and nearby countries are challenging such simple models.This paper critically reviews previous chronology of the Japanese Paleolithic,including possible Lower and Middle Paleolithic(LP/MP),and attempts to show an alternative model of the beginning of the Japanese Upper Paleolithic.This paper suggests several possible specimens of LP/MP and recommends further geoarchaeological investigation to understand the reliability and cultural relationship between possible LP/MP specimens and the Early Upper Paleolithic(EUP).The start of the Japanese EUP is presently characterized by a flake industry with trapezoids and denticulates around 39-37 kaBP cal on Paleo-Honshu Island,which has partial resemblance with contemporary assemblages in China and the Korean Peninsula,although trapezoids are endemic only to the Japanese EUP and may have derived from the ancestral lithic tradition.Blade technology appeared earliest on Central Paleo-Honshu Island,about 1000 years later than the earliest flake technology.Although blade technology may have originated from the elongated flake technology of the previous period,the sudden simultaneous emergence implies that it diffused from the Korean Peninsula.This paper proposes that blade technology from the Korean Peninsula arrived on the northeastern Paleo-Honshu Island,including the Japan Sea coastal region of western Honshu,rather than the southwest,where flake technology long prospered,due to differences in ecological settings and adaptation strategies between the two regions.

A Generalized Array Factor for Time-Modulated Hexagonal Based Antenna Array Geometry With Novel Trapezoidal Switching

The concept of the time-modulated array has been emerging as an alternative to the complex phase shifters,which lowers the cost of the array feeding network due to the utilization of radio frequency(RF)switches.The various forms of hexagonal antenna array geometries can be used for applications like surveillance tracking in phased array radar and wireless communication systems.This work proposes the generalized array factor(AF)for the hexagonal antenna array geometry based on time modulation.The time modulation in generalized hexagonal geometry can maintain the fixed static amplitude excitation,giving more flexibility over time.Furthermore,a novel trapezoidal switching function is also proposed and applied to the generalized array factor to enable future researchers to use this array factor in the field of advancement to observe how switching schemes like trapezoidal and rectangular affect the array pattern's side lobe level(SLL).The generalized equation can be utilized for the analysis and synthesis of radiation characteristics of the time-modulated hexagonal array(TMHA),time-modulated concentric hexagonal array(TMCHA),time-modulated hexagonal cylindrical array(TMHCA),and time-modulated hexagonal concentric cylindrical array(TMHCCA).The numerical result illustrates the generation of AF of time-modulated hexagonal structures and also shows that the trapezoidal switching sequence outperforms the rectangular switch using the cat swarm optimization(CSO)approach.

Study of a Hydraulic Jump in an Asymmetric Trapezoidal Channel with Different Sluice Gates

In this study,the main properties of the hydraulic jump in an asymmetric trapezoidal flume are analyzed experimentally,including the so-called sequent depths,characteristic lengths,and efficiency.In particular,an asymmetric trapezoidal flume with a length of 7 m and a width of 0.304 m is considered,with the bottom of the flume transversely inclined at an angle of m=0.296 and vertical lateral sides.The corresponding inflow Froude number is allowed to range in the interval(1.40<F1<6.11).The properties of this jump are compared to those of hydraulic jumps in channels with other types of cross-sections.A relationship for calculating hydraulic jump efficiency is proposed for the considered flume.For F1>5,the hydraulic jump is found to be more effective than that occurring in triangular and symmetric trapezoidal channels.Also,when■mes>8 and■>5,the hydraulic jump in the asymmetrical trapezoidal channel downstream of a parallelogram sluice gate is completely formed as opposed to the situation where a triangular sluice is considered.

Sliding behaviors of the trapezoidal roof rock block under a lateral dynamic disturbance

The surrounding rock of underground space is always affected by external dynamic disturbance from the side position,such as blasting vibration from a stope at the same level or seismic waves from adjacent strata.A series of laboratory tests,numerical simulations and theoretical analyses were carried out in this study to disclose the sliding mechanism of roof rock blocks under lateral disturbance.Firstly,the experiments on trapezoidal key block under various clamping loads and disturbance were conducted,followed by numerical simulations using the fast Lagrangian analysis of continua(FLAC3D).Then,based on the conventional wave propagation model and the classical shear-slip constitutive model,a theoretical model was proposed to capture the relative displacement between blocks and the sliding displacement of the key block.The results indicate that the sliding displacement of the key block increased linearly with the disturbance energy and decreased exponentially with the clamping load when the key block was disturbed to slide(without instability).Meanwhile,when the key block was disturbed to fall,two types of instability process may appear as immediate type or delayed type.In addition,the propagation of stress waves in the block system exhibited obvious low-velocity and lowfrequency characteristics,resulting in the friction reduction effect appearing at the contact interface,which is the essential reason for the sliding of rock blocks.The results can be applied to practical underground engineering and provide valuable guidance for the early detection and prevention of rockfalling disasters.

Composite Fractional Trapezoidal Rule with Romberg Integration

The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator.This would be achieved by first establishing a fractional-order version of the 2-point Trapezoidal rule and then by proposing another fractional-order version of the(n+1)-composite Trapezoidal rule.In particular,the so-called divided-difference formula is typically employed to derive the 2-point Trapezoidal rule,which has accordingly been used to derive a more accurate fractional-order formula called the(n+1)-composite Trapezoidal rule.Additionally,in order to increase the accuracy of the proposed approximations by reducing the true errors,we incorporate the so-called Romberg integration,which is an extrapolation formula of the Trapezoidal rule for integration,into our proposed approaches.Several numerical examples are provided and compared with a modern definition of the Riemann-Liouville fractional integral operator to illustrate the efficacy of our scheme.

Multiple attribute decision making method based on trapezoid fuzzy linguistic variables

The problem of multiple attribute decision making under fuzzy linguistic environments, in which decision makers can only provide their preferences （attribute values）in the form of trapezoid fuzzy linguistic variables（TFLV）, is studied. The formula of the degree of possibility between two TFLVs is defined, and some of its characteristics are studied. Based on the degree of possibility of fuzzy linguistic variables, an approach to ranking the decision alternatives in multiple attribute decision making with TFLV is developed. The trapezoid fuzzy linguistic weighted averaging （TFLWA） operator method is utilized to aggregate the decision information, and then all the alternatives are ranked by comparing the degree of possibility of TFLV. The method can carry out linguistic computation processes easily without loss of linguistic information, and thus makes the decision results reasonable and effective. Finally, the implementation process of the proposed method is illustrated and analyzed by a practical example.

Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision making problems

Intuitionistic trapezoidal fuzzy numbers and their operational laws are defined. Based on these operational laws, some aggregation operators, including intuitionistic trapezoidal fuzzy weighted arithmetic averaging operator and weighted geometric averaging operator are proposed. Expected values, score function, and accuracy function of intuitionitsic trapezoidal fuzzy numbers are defined. Based on these, a kind of intuitionistic trapezoidal fuzzy multi-criteria decision making method is proposed. By using these aggregation operators, criteria values are aggregated and integrated intuitionistic trapezoidal fuzzy numbers of alternatives are attained. By comparing score function and accuracy function values of integrated fuzzy numbers, a ranking of the whole alternative set can be attained. An example is given to show the feasibility and availability of the method.

Assessment of gas and dust explosion in coal mines by means of fuzzy fault tree analysis

During the past decade, coal dust and gas explosions have been the most two serious types of disasters in China, threatening the lives of miners and causing significant losses in terms of national property. In this paper, an evaluation model of coal dust and gas explosions was constructed based on a fuzzy fault tree by taking the Xingli Coal Mine as a research site to identify the risk factors of coal dust and gas explosions.Furthermore, the hazards associated with such explosions were evaluated for this particular coal mine.After completing an on-site investigation, the fuzzy probabilities of basic events were obtained through expert scoring, and these expert opinions were then aggregated as trapezoidal fuzzy numbers to calculate the degrees of importance of all basic events. Finally, these degrees of importance were sorted. According to the resulting order, the basic events with higher probabilities were determined to identify key hazards in the daily safety management of this particular coal mine. Moreover, effective measures for preventing gas and coal dust explosions were derived. The fuzzy fault tree analysis method is of high significance in the analysis of accidental coal mine explosions and provides theoretical guidance for improving the efficiency of coal mine safety management in a scientific and feasible manner.

Meshless Method for Analysis of Permeable Breakwaters in the Proximity of A Vertical Wall

In the present work, the improved version of the meshless singular boundary method(ISBM) is developed for analyzing the performance of bottom standing submerged permeable breakwaters in regular normally incident waves and in the proximity of a vertical wall. Both single and dual prismatic breakwaters of rectangular and trapezoidal forms are examined. The physical problem is cast in terms of the Laplace equation governing an irrotational flow and incompressible fluid motion with appropriate mixed type boundary conditions, and solved numerically using the ISBM. To model the permeability of the breakwaters fully absorbing boundary conditions are assumed. Numerical results are presented in terms of hydrodynamic quantities of the reflection coefficients. These are firstly validated against the results of a multi-domain boundary element method(BEM) developed independently for a previous study. The agreement between the results of the two methods is excellent. The coefficients of reflection are then computed and discussed for a variety of structural conditions including the breakwaters height, width, spacing, and absorbing permeability. Effects of the proximity of the vertical plane wall are also investigated. The breakwater's width is found to have only marginal effects compared with its height. Permeability tends to decrease the minimum reflections. These coefficients show periodic variations with the spacing relative to the wavelength. Trapezoidal breakwaters are found to be more cost-effective than the rectangular breakwaters. Dual breakwater systems are confirmed to perform much better than single structures.

New insights in the treatment of acromioclavicular separation

A direct force on the superior aspect of the shoulder may cause acromioclavicular(AC) dislocation or separation. Severe dislocations can lead to chronic impairment, especially in the athlete and high-demand manual laborer. The dislocation is classified according to Rockwood. Types Ⅰ?and Ⅱ are treated nonoperatively, while types Ⅳ, Ⅴ and Ⅵ are generally treated operatively. Controversy exists regarding the optimal treatment of type Ⅲ dislocations in the high-demand patient. Recent evidence suggests that these should be treated nonoperatively initially. Classic surgical techniques were associated with high complication rates, including recurrent dislocations and hardware breakage. In recent years, many new techniques have been introduced in order to improve the outcomes. Arthroscopic reconstruction or repair techniques have promising short-term results. This article aims to provide a current concepts review on the treatment of AC dislocations with emphasis on recent developments.

Predictive model for stage-discharge curve in compound channels with vegetated floodplains

The governing equation of the discharge per unit width, derived from the flow continuity equation and the momentum equation in the vegetated compound chan- nel, is established. The analytical solution to the discharge per unit width is presented, including the effects of bed friction, lateral momentum transfer, drag force, and secondary flows. A simple＇ but available numerical integral method, i.e., the compound trapezoidM formula, is used to calculate the approximate solutions of the sub-area discharge and the total discharge. A comparison with the published experimental data from the U. K. Flood Channel Facility （UK-FCF） demonstrates that this model is capable of predicting not only the stage-discharge curve but also the sub-area discharge in the vegetated com- pound channel. The effects of the two crucial parameters, i.e., the divided number of the integral interval and the secondary flow coefficient, on the total discharge are discussed and analyzed.

Application of pole-zero cancellation circuit in nuclear signal filtering and shaping algorithm

In radiation measurement and digital nuclear spectrum systems,traditional nuclear signal processing circuits in nuclear electronics have been gradually replaced by digital algorithm modules with the application of highperformance programmable hardware logic devices(such as FPGA or DSP).Referring to the digital realization method of inverse RC integral circuit systems,the function of the pole-zero cancellation(PZC)circuit was analyzed,a new modified cascade equivalent model of PZC was established,and the time-domain digital PZC(DPZC)recursive algorithm was derived in detail in this study.Two parameters kIand k_(D)are included in the new algorithm,where kIshould match the exponential decay time constant of the input signal to realize the pole-zero compensation,while the decay time constant of the output signal can be changed with the adjustable parameter k_(D)(which is larger than the decay time constant of the input signal).Based on the new DPZC algorithm module,two trapezoidal(triangular)shaping filters were designed and implemented.The amplitude–frequency characteristics of the output signal of the proposed trapezoidal shaping algorithm and the convolution trapezoidal shaping algorithm were compared,with fixed peaking time.The results show that the trapezoidal shaping algorithm based on DPZC can better suppress high-frequency noise.Finally,based on the Na I(Tl)scintillator(u75 mm×75 mm)detector and^(137)Cs source,the effect of the k_(D)value on the energy resolution of the DPZC trapezoidal(triangular)shaping algorithm was studied.The experimental results show that,with an increase in k_(D),the energy resolution of the system improved and reached the maximum when k_(D)was greater than 10,and the optimal energy resolution of the system was 7.72%.

Characteristics from a hydrodynamic model of a trapezoidal artificial reef

Flume experiments and numerical simulation were conducted to characterize the hydrodynamics of a trapezoid artificial reef.Measurements in particle image velocimetry were conducted to observe the formation of upwelling and vortices;and forces for the reef model were measured by load cell.The results of flume experiments agree well with the numerical data.In addition,the flow structure around a reef combining trapezoidal and cubic blocks was simulated numerically under two deployment schemes,showing a more complicated flow structure than that of a stand-alone reef.Relationship between drag coefficient and Reynolds number suggest that the degree of turbulence can be assessed from the value of drag coefficient downstream from the reef.The role of the reef in water flow is to reduce flow velocity and generate turbulence.

Experimental Study of Wave Attenuation in Trapezoidal Floating Breakwaters

A comprehensive experimental study was carried out on the regular wave attenuation with a trapezoidal pontoontype floating breakwater(FB) in deep water. The functionalities of two simple FB geometries consist of a rectangle and a trapezoid with the slope of 60° were investigated under the wave attack. A two-dimensional wave flume was used in the experiment; the incident, transmitted waves, mooring line forces and motion responses of the floating breakwaters were measured. Also the influence of the sea state conditions(incident wave height and wave period)and structural parameters(draught of the structure) were investigated using the trapezoidal FB. Our experimental results indicated that the trapezoidal FB significantly reduced the wave transmission and mooring line force when compared with rectangular FBs. A new formula was developed in order to predict the value of the transmission coefficient in trapezoidal FBs with the slope of 60°. Experimental data showed to be consistent with the results of the formula.

Effect of loading rates on crack propagating speed,fracture toughness and energy release rate using single-cleavage trapezoidal open specimen under impact loads

The former studies indicate that loading rates significantly affect dynamic behavior of brittle materials,for instance,the dynamic compressive and tensile strength increase with loading rates.However,there still are many unknown or partially unknown aspects.For example,whether loading rates have effect on crack dynamic propagating behavior(propagation toughness,velocity and arrest,etc).To further explore the effect of loading rates on crack dynamic responses,a large-size single-cleavage trapezoidal open(SCTO)specimen was proposed,and impacting tests using the SCTO specimen under drop plate impact were conducted.Crack propagation gauges(CPGs)were employed in measuring impact loads,crack propagation time and velocities.In order to verify the testing result,the corresponding numerical model was established using explicit dynamic software AUTODYN,and the simulation result is basically consistent with the experimental results.The ABAQUS software was used to calculate the dynamic SIFs.The universal function was calculated by fractal method.The experimental-numerical method was employed in determining initiation toughness and propagation toughness.The results indicate that crack propagating velocities,dynamic fracture toughness and energy release rates increase with loading rates;crack delayed initiation time decreases with loading rates.

Average temperature calculation for straight single-row-piped frozen soil wall

The average temperature of frozen soil wall is an essential parameter in the process of design, construction, and safety manage- ment of artificial ground freezing engineering. It is the basis of calculating frozen soil＇s mechanical parameters, fiarther prediction of bearing capacity and, ultimately, safety evaluation of the frozen soil wall. Regarding the average temperature of sin- gle-row-piped frozen soil wall, this paper summarizes several current calculation methods and their shortcomings. Furthermore, on the basis of Bakholdin＇s analytical solution for the temperature field under straight single-row-piped freezing, two new calcula- tion models, namely, the equivalent trapezoid model and the equivalent triangle model, are proposed. These two approaches are used to calculate the average temperature of a certain cross section which indicates the condition of the whole frozen soil wall. Considering the possible parameter range according to the freezing pipe layout that might be applied in actual construction, this paper compares the average temperatures of frozen soil walls obtained by the equivalent trapezoid method and the equivalent tri- angle method with that obtained by numerical integration of Bakholdin＇s analytical solution. The results show that the discrepancies are extremely small and these two new approaches are better than currently prevailing methods. However, the equivalent triangle method boasts higher accuracy and a simpler formula compared with the equivalent trapezoid method.

Dynamic Characteristics of Double-Helical Planetary Gear Sets Under Time-Varying Mesh Stiffness

Internal and external meshes are two of primary excitation sources which induce vibration while double-helical planetary gear sets are in transmission. Based on the analysis of tooth movement principle,three cases of mesh stiffness are derived via investigating the length of action lines,and catalogued in terms of β < β0,β = β0and β > β_0. The simulation demonstrates mesh stiffness between gear pairs performs as a trapezoid waveform( TW) and changes along with the line of action simultaneously,total mesh stiffness comes from the superposition of each engaged gear. While governing equations of motion contained 16 DOFs( degree of freedom) are constructed and effectively solved through the combination of numerical approaches. Comparing with sinusoidal waveform mesh stiffness( SW),the results show that dynamical factors and perturbation under the excitation of TW( β < β_0) are greater and remarkable than that from SW,with respect to the mean dynamic factors about 1. 51 and 1. 28,respectively. The fluctuation response between ring- planet( R- P) is stronger than sun-planet( S-P) which is also validated by both approach studies,frequency spectra analyses identifies larger distinct rotational resonance and more frequencies under TW excitation.

Effect of trapezoidal collars as a scour countermeasure around wing-wall abutments

Local scour around bridge abutments is a widespread problem that can result in structural failure. Collars can be used as a countermeasure to reduce the scour depth. In this study, the temporal scour development around a wing-wall abutment was investigated with and without collars. The tests were carried out under clear-water conditions for different abutment lengths, with collars of different sizes placed at the bed level. When no collar was used in the experiments, 70% of the maximum scour depth occurred in less than 2 h. However, when a collar with a width greater than the length of the abutment was used, no scour was observed for up to 200 min from the beginning of the experiments. The results show that an increase in the collar width not only led to a lag time for the onset of scouring but also reduced the maximum scour depth. Moreover, an increased collar width led to a better performance in mitigating scouring around smaller abutments. Generally, the scour depth decreased by 9%-37% with different collar widths.

A STUDY OF TEARING PROPERTIES OF MEMBRANE COATED FABRICS

In this paper,an equation expressing relationship between trapezoid tear strength and fabrictensile properties,weave texture and geomitry of specimen was studied.It could be used to predicttrapezoid tear strength for uncoated fabrics well,and also for coated fabrics if multipling a stressconcentration coefficient.The four tear test standard methods(single-rip,double-rip,wing-ripand trapezoid tear)have been adopted to measure the tear strength of fifteen coated fabrics whichwere used for architecture and gencral industrial purposes and the nine corresponding uncoatedfabrics respectively.Analysis and comparison of experimental results showed that there are an ex-cellent correlation between each two methods with the four tear parameters respectively and a goodconsistency of the order of the tear strength for different samples.In addition,the experimental results showed that a good correlation exists between the tenslieproperties and tear strength of coated and uncoated fabrics.