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PID Controller I have programmed a heating cooling PID Controller DL 06 PLC(Koyo / Direct Logic) . It is not easy because using ladder logic to make a PID loop. And finally i have finished it. Due to efficiency we have bought a Direct Logic . DL 06 type is a cheap PLC . It is my first experiences. I Selected the function in directsoft 5.3 and starting entering my addressing. Loop 1 Table start address V1600 Setpoint Variable V1602 Process Variable V1603 Output V1605 I don't see anything in ladder logic. The PID loop operates outside of the PLC ladder logic. You can use ladder logic to interface with the PID loop. Most setups will use some ladder logic. I have just copied from . It is very useful for me to know how PID works. A proportional–integral–derivative controller (PID controller) is a generic control loop feedback mechanism widely used in industrial control systems. A PID controller attempts to correct the error between a measured process variable and a desired setpoint by calculating and then outputting a corrective action that can adjust the process accordingly and rapidly, to keep the error minimal. [edit] General The PID controller calculation (algorithm) involves three separate parameters; the proportional, the integral and >derivative values. The proportional value determines the reaction to the current error, the integral value determines the reaction based on the sum of recent errors, and the derivative value determines the reaction based on the rate at which the error has been changing. The weighted sum of these three actions is used to adjust the process via a control element such as the position of a control valve or the power supply of a heating element. By tuning the three constants in the PID controller algorithm, the controller can provide control action designed for specific process requirements. The response of the controller can be described in terms of the responsiveness of the controller to an error, the degree to which the controller overshoots the setpoint and the degree of system oscillation. Note that the use of the PID algorithm for control does not guarantee optimal control of the system or system stability. Some applications may require using only one or two modes to provide the appropriate system control. This is achieved by setting the gain of undesired control outputs to zero. A PID controller will be called a PI, PD, P or I controller in the absence of the respective control actions. PI controllers are particularly common, since derivative action is very sensitive to measurement noise, and the absence of an integral value may prevent the system from reaching its target value due to the control action. Note: Due to the diversity of the field of control theory and application, many naming conventions for the relevant variables are in common use. [edit] Control loop basics A familiar example of a control loop is the action taken to keep one's shower water at the ideal temperature, which typically involves the mixing of two process streams, cold and hot water. The person feels the water to estimate its temperature. Based on this measurement they perform a control action: use the cold water tap to adjust the process. The person would repeat this input-output control loop, adjusting the hot water flow until the process temperature stabilized at the desired value. Feeling the water temperature is taking a measurement of the process value or process variable (PV). The desired temperature is called the setpoint (SP). The output from the controller and input to the process (the tap position) is called the manipulated variable (MV). The difference between the measurement and the setpoint is the error (e), too hot or too cold and by how much. As a controller, one decides roughly how much to change the tap position (MV) after one determines the temperature (PV), and therefore the error. This first estimate is the equivalent of the proportional action of a PID controller. The integral action of a PID controller can be thought of as gradually adjusting the temperature when it is almost right. Derivative action can be thought of as noticing the water temperature is getting hotter or colder, and how fast, anticipating further change and tempering adjustments for a soft landing at the desired temperature (SP). Making a change that is too large when the error is small is equivalent to a high gain controller and will lead to overshoot. If the controller were to repeatedly make changes that were too large and repeatedly overshoot the target, the output would oscillate around the setpoint in either a constant, growing, or decaying sinusoid. If the oscillations increase with time then the system is unstable, whereas if they decay the system is stable. If the oscillations remain at a constant magnitude the system is marginally stable. A human would not do this because we are adaptive controllers, learning from the process history, but PID controllers do not have the ability to learn and must be set up correctly. Selecting the correct gains for effective control is known as tuning the controller. If a controller starts from a stable state at zero error (PV = SP), then further changes by the controller will be in response to changes in other measured or unmeasured inputs to the process that impact on the process, and hence on the PV. Variables that impact on the process other than the MV are known as disturbances. Generally controllers are used to reject disturbances and/or implement setpoint changes. Changes in feed water temperature constitute a disturbance to the shower process. In theory, a controller can be used to control any process which has a measurable output (PV), a known ideal value for that output (SP) and an input to the process (MV) that will affect the relevant PV. Controllers are used in industry to regulate temperature, pressure, flow rate, chemical composition, speed and practically every other variable for which a measurement exists. Automobile cruise control is an example of a process which utilizes automated control. Due to their long history, simplicity, well grounded theory and simple setup and maintenance requirements, PID controllers are the controllers of choice for many of these applications. PID controller theory This section describes the parallel or non-interacting form of the PID controller. For other forms please see the Section "Alternative notation and PID forms". The PID control scheme is named after its three correcting terms, whose sum constitutes the manipulated variable (MV). Hence: $\mathrm{MV(t)}=\,P_{\mathrm{out}} + I_{\mathrm{out}} + D_{\mathrm{out}}$ P[out], I[out], and D[out] are the contributions to the output from the PID controller from each of the three terms, as defined below. [edit] Proportional term The proportional term (sometimes called gain) makes a change to the output that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant K[p], called the proportional gain. The proportional term is given by: P[out]: Proportional term of output K[p]: Proportional gain, a tuning parameter e: Error = SP − PV t: Time or instantaneous time (the present) A high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable (See the section on loop tuning). In contrast, a small gain results in a small output response to a large input error, and a less responsive (or sensitive) controller. If the proportional gain is too low, the control action may be too small when responding to system disturbances. In the absence of disturbances, pure proportional control will not settle at its target value, but will retain a steady state error that is a function of the proportional gain and the process gain. Despite the steady-state offset, both tuning theory and industrial practice indicate that it is the proportional term that should contribute the bulk of the output change. [edit] Integral term The contribution from the integral term (sometimes called reset) is proportional to both the magnitude of the error and the duration of the error. Summing the instantaneous error over time (integrating the error) gives the accumulated offset that should have been corrected previously. The accumulated error is then multiplied by the integral gain and added to the controller output. The magnitude of the contribution of the integral term to the overall control action is determined by the integral gain, K[i]. The integral term is given by: I[out]: Integral term of output K[i]: Integral gain, a tuning parameter e: Error = SP − PV t: Time or instantaneous time (the present) τ: a dummy integration variable The integral term (when added to the proportional term) accelerates the movement of the process towards setpoint and eliminates the residual steady-state error that occurs with a proportional only controller. However, since the integral term is responding to accumulated errors from the past, it can cause the present value to overshoot the setpoint value (cross over the setpoint and then create a deviation in the other direction). For further notes regarding integral gain tuning and controller stability, see the section on loop tuning. Derivative term The rate of change of the process error is calculated by determining the slope of the error over time (i.e., its first derivative with respect to time) and multiplying this rate of change by the derivative gain K[d]. The magnitude of the contribution of the derivative term (sometimes called rate) to the overall control action is termed the derivative gain, K[d]. The derivative term is given by: D[out]: Derivative term of output K[d]: Derivative gain, a tuning parameter e: Error = SP − PV t: Time or instantaneous time (the present) The derivative term slows the rate of change of the controller output and this effect is most noticeable close to the controller setpoint. Hence, derivative control is used to reduce the magnitude of the overshoot produced by the integral component and improve the combined controller-process stability. However, differentiation of a signal amplifies noise and thus this term in the controller is highly sensitive to noise in the error term, and can cause a process to become unstable if the noise and the derivative gain are sufficiently large. [edit] Summary The proportional, integral, and derivative terms are summed to calculate the output of the PID controller. Defining u(t) as the controller output, the final form of the PID algorithm is: $\mathrm{u(t)}=\mathrm{MV(t)}=K_p{e(t)} + K_{i}\int_{0}^{t}{e(\tau)}\,{d\tau} + K_{d}\frac{d}{dt}e(t)$ where the tuning parameters are: Proportional gain, K[p] larger values typically mean faster response since the larger the error, the larger the Proportional term compensation. An excessively large proportional gain will lead to process instability and Integral gain, K[i] larger values imply steady state errors are eliminated more quickly. The trade-off is larger overshoot: any negative error integrated during transient response must be integrated away by positive error before we reach steady state. Derivative gain, K[d] larger values decrease overshoot, but slows down transient response and may lead to instability due to signal noise amplification in the differentiation of the error. [edit] Loop tuning If the PID controller parameters (the gains of the proportional, integral and derivative terms) are chosen incorrectly, the controlled process input can be unstable, i.e. its output diverges, with or without oscillation, and is limited only by saturation or mechanical breakage. Tuning a control loop is the adjustment of its control parameters (gain/proportional band, integral gain/reset, derivative gain/rate) to the optimum values for the desired control response. The optimum behavior on a process change or setpoint change varies depending on the application. Some processes must not allow an overshoot of the process variable beyond the setpoint if, for example, this would be unsafe. Other processes must minimize the energy expended in reaching a new setpoint. Generally, stability of response (the reverse of instability) is required and the process must not oscillate for any combination of process conditions and setpoints. Some processes have a degree of non-linearity and so parameters that work well at full-load conditions don't work when the process is starting up from no-load. This section describes some traditional manual methods for loop tuning. There are several methods for tuning a PID loop. The most effective methods generally involve the development of some form of process model, then choosing P, I, and D based on the dynamic model parameters. Manual tuning methods can be relatively inefficient. The choice of method will depend largely on whether or not the loop can be taken "offline" for tuning, and the response time of the system. If the system can be taken offline, the best tuning method often involves subjecting the system to a step change in input, measuring the output as a function of time, and using this response to determine the control parameters. Choosing a Tuning Method Method Advantages Disadvantages Manual Tuning No math required. Online method. Requires experienced personnel. Ziegler–Nichols Proven Method. Online method. Process upset, some trial-and-error, very aggressive tuning. Software Tools Consistent tuning. Online or offline method. May include valve and sensor analysis. Allow simulation before Some cost and training involved. Cohen-Coon Good process models. Some math. Offline method. Only good for first-order processes. [edit] Manual tuning If the system must remain online, one tuning method is to first set K[i] and K[d] values to zero. Increase the K[p] until the output of the loop oscillates, then the K[p] should be left set to be approximately half of that value for a "quarter amplitude decay" type response. Then increase K[i] until any offset is correct in sufficient time for the process. However, too much K[i] will cause instability. Finally, increase K[d], if required, until the loop is acceptably quick to reach its reference after a load disturbance. However, too much K[d] will cause excessive response and overshoot. A fast PID loop tuning usually overshoots slightly to reach the setpoint more quickly; however, some systems cannot accept overshoot, in which case an "over-damped" closed-loop system is required, which will require a K[p] setting significantly less than half that of the K[p] setting causing oscillation. Effects of increasing parameters Parameter Rise time Overshoot Settling time Error at equilibrium K[p] Decrease Increase Small change Decrease K[i] Decrease Increase Increase Eliminate K[d] Indefinite (small decrease or increase)^[1] Decrease Decrease None [edit] Ziegler–Nichols method Another tuning method is formally known as the Ziegler–Nichols method, introduced by John G. Ziegler and Nathaniel B. Nichols. As in the method above, the K[i] and K[d] gains are first set to zero. The P gain is increased until it reaches the critical gain, K[c], at which the output of the loop starts to oscillate. K[c] and the oscillation period P[c] are used to set the gains as shown: Ziegler–Nichols method Control Type K[p] K[i] K[d] P 0.50K[c] - - PI 0.45K[c] 1.2K[p] / P[c] - PID 0.60K[c] 2K[p] / P[c] K[p]P[c] / 8 [edit] PID tuning software Most modern industrial facilities no longer tune loops using the manual calculation methods shown above. Instead, PID tuning and loop optimization software are used to ensure consistent results. These software packages will gather the data, develop process models, and suggest optimal tuning. Some software packages can even develop tuning by gathering data from reference changes. Mathematical PID loop tuning induces an impulse in the system, and then uses the controlled system's frequency response to design the PID loop values. In loops with response times of several minutes, mathematical loop tuning is recommended, because trial and error can literally take days just to find a stable set of loop values. Optimal values are harder to find. Some digital loop controllers offer a self-tuning feature in which very small setpoint changes are sent to the process, allowing the controller itself to calculate optimal tuning values. Other formulas are available to tune the loop according to different performance criteria. [edit] Modifications to the PID algorithm The basic PID algorithm presents some challenges in control applications that have been addressed by minor modifications to the PID form. One common problem resulting from the ideal PID implementations is integral windup. This problem can be addressed by: • Initializing the controller integral to a desired value • Increasing the setpoint in a suitable ramp • Disabling the integral function until the PV has entered the controllable region • Limiting the time period over which the integral error is calculated • Preventing the integral term from accumulating above or below pre-determined bounds Freezing the integral function in case of disturbances If a PID loop is used to control the temperature of an electric resistance furnace, the system has stabilized and then the door is opened and something cold is put into the furnace the temperature drops below the setpoint. The integral function of the controller tends to compensate this error by introducing another error in the positive direction. This can be avoided by freezing of the integral function after the opening of the door for the time the control loop typically needs to reheat the furnace. Replacing the integral function by a model based part Often the time-response of the system is approximately known. Then it is an advantage to simulate this time-response with a model and to calculate some unknown parameter from the actual response of the system. If for instance the system is an electrical furnace the response of the difference between furnace temperature and ambient temperature to changes of the electrical power will be similar to that of a simple RC low-pass filter multiplied by an unknown proportional coefficient. The actual electrical power supplied to the furnace is delayed by a low-pass filter to simulate the response of the temperature of the furnace and then the actual temperature minus the ambient temperature is divided by this low-pass filtered electrical power. Then, the result is stabilized by another low-pass filter leading to an estimation of the proportional coefficient. With this estimation it is possible to calculate the required electrical power by dividing the set-point of the temperature minus the ambient temperature by this coefficient. The result can then be used instead of the integral function. This also achieves a control error of zero in the steady-state but avoids integral windup and can give a significantly improved control action compared to an optimized PID controller. This type of controller does work properly in an open loop situation which causes integral windup with an integral function. This is an advantage if for example the heating of a furnace has to be reduced for some time because of the failure of a heating element or if the controller is used as an advisory system to a human operator who may or may not switch it to closed-loop operation or if the controller is used inside of a branch of a complex control system where this branch may be temporarily inactive. Many PID loops control a mechanical device (for example, a valve). Mechanical maintenance can be a major cost and wear leads to control degradation in the form of either stiction or a deadband in the mechanical response to an input signal. The rate of mechanical wear is mainly a function of how often a device is activated to make a change. Where wear is a significant concern, the PID loop may have an output deadband to reduce the frequency of activation of the output (valve). This is accomplished by modifying the controller to hold its output steady if the change would be small (within the defined deadband range). The calculated output must leave the deadband before the actual output will change. The proportional and derivative terms can produce excessive movement in the output when a system is subjected to an instantaneous step increase in the error, such as a large setpoint change. In the case of the derivative term, this is due to taking the derivative of the error, which is very large in the case of an instantaneous step change. As a result, some PID algorithms incorporate the following modifications: Derivative of output In this case the PID controller measures the derivative of the output quantity, rather than the derivative of the error. The output is always continuous (i.e., never has a step change). For this to be effective, the derivative of the output must have the same sign as the derivative of the error. Setpoint ramping In this modification, the setpoint is gradually moved from its old value to a newly specified value using a linear or first order differential ramp function. This avoids the discontinuity present in a simple step change. Setpoint weighting Setpoint weighting uses different multipliers for the error depending on which element of the controller it is used in. The error in the integral term must be the true control error to avoid steady-state control errors. This affects the controller's setpoint response. These parameters do not affect the response to load disturbances and measurement noise. [edit] Limitations of PID control While PID controllers are applicable to many control problems, they can perform poorly in some applications. PID controllers, when used alone, can give poor performance when the PID loop gains must be reduced so that the control system does not overshoot, oscillate or hunt about the control setpoint value. The control system performance can be improved by combining the feedback (or closed-loop) control of a PID controller with feed-forward (or open-loop) control. Knowledge about the system (such as the desired acceleration and inertia) can be fed forward and combined with the PID output to improve the overall system performance. The feed-forward value alone can often provide the major portion of the controller output. The PID controller can then be used primarily to respond to whatever difference or error remains between the setpoint (SP) and the actual value of the process variable (PV). Since the feed-forward output is not affected by the process feedback, it can never cause the control system to oscillate, thus improving the system response and stability. For example, in most motion control systems, in order to accelerate a mechanical load under control, more force or torque is required from the prime mover, motor, or actuator. If a velocity loop PID controller is being used to control the speed of the load and command the force or torque being applied by the prime mover, then it is beneficial to take the instantaneous acceleration desired for the load, scale that value appropriately and add it to the output of the PID velocity loop controller. This means that whenever the load is being accelerated or decelerated, a proportional amount of force is commanded from the prime mover regardless of the feedback value. The PID loop in this situation uses the feedback information to effect any increase or decrease of the combined output in order to reduce the remaining difference between the process setpoint and the feedback value. Working together, the combined open-loop feed-forward controller and closed-loop PID controller can provide a more responsive, stable and reliable control system. Another problem faced with PID controllers is that they are linear. Thus, performance of PID controllers in non-linear systems (such as HVAC systems) is variable. Often PID controllers are enhanced through methods such as PID gain scheduling or fuzzy logic. Further practical application issues can arise from instrumentation connected to the controller. A high enough sampling rate, measurement precision, and measurement accuracy are required to achieve adequate control performance. A problem with the Derivative term is that small amounts of measurement or process noise can cause large amounts of change in the output. It is often helpful to filter the measurements with a low-pass filter in order to remove higher-frequency noise components. However, low-pass filtering and derivative control can cancel each other out, so reducing noise by instrumentation means is a much better choice. Alternatively, the differential band can be turned off in many systems with little loss of control. This is equivalent to using the PID controller as a PI controller. [edit] Cascade control One distinctive advantage of PID controllers is that two PID controllers can be used together to yield better dynamic performance. This is called cascaded PID control. In cascade control there are two PIDs arranged with one PID controlling the set point of another. A PID controller acts as outer loop controller, which controls the primary physical parameter, such as fluid level or velocity. The other controller acts as inner loop controller, which reads the output of outer loop controller as set point, usually controlling a more rapid changing parameter, flowrate or acceleration. It can be mathematically proven that the working frequency of the controller is increased and the time constant of the object is reduced by using cascaded PID controller.. [edit] Physical implementation of PID control In the early history of automatic process control the PID controller was implemented as a mechanical device. These mechanical controllers used a lever, spring and a mass and were often energized by compressed air. These pneumatic controllers were once the industry standard. Electronic analog controllers can be made from a solid-state or tube amplifier, a capacitor and a resistance. Electronic analog PID control loops were often found within more complex electronic systems, for example, the head positioning of a disk drive, the power conditioning of a power supply, or even the movement-detection circuit of a modern seismometer. Nowadays, electronic controllers have largely been replaced by digital controllers implemented with microcontrollers or FPGAs. Most modern PID controllers in industry are implemented in programmable logic controllers (PLCs) or as a panel-mounted digital controller. Software implementations have the advantages that they are relatively cheap and are flexible with respect to the implementation of the PID algorithm. [edit] Alternative nomenclature and PID forms [edit] Ideal versus standard PID form The form of the PID controller most often encountered in industry, and the one most relevant to tuning algorithms is the standard form. In this form the K[p] gain is applied to the I[out], and D[out] terms, yielding: $\mathrm{MV(t)}=K_p\left(\,{e(t)} + \frac{1}{T_i}\int_{0}^{t}{e(\tau)}\,{d\tau} + T_d\frac{d}{dt}e(t)\right)$ T[i] is the integral time T[d] is the derivative time In the ideal parallel form, shown in the controller theory section $\mathrm{MV(t)}=K_p{e(t)} + K_i\int_{0}^{t}{e(\tau)}\,{d\tau} + K_d\frac{d}{dt}e(t)$ the gain parameters are related to the parameters of the standard form through $K_i = \frac{K_p}{T_i}$ and $K_d = K_p T_d \,$. This parallel form, where the parameters are treated as simple gains, is the most general and flexible form. However, it is also the form where the parameters have the least physical interpretation and is generally reserved for theoretical treatment of the PID controller. The standard form, despite being slightly more complex mathematically, is more common in industry. [edit] Laplace form of the PID controller Sometimes it is useful to write the PID regulator in Laplace transform form: $G(s)=K_p + \frac{K_i}{s} + K_d{s}=\frac{K_d{s^2} + K_p{s} + K_i}{s}$ Having the PID controller written in Laplace form and having the transfer function of the controlled system, makes it easy to determine the closed-loop transfer function of the system. [edit] Series/interacting form Another representation of the PID controller is the series, or interacting form $G(s) = K_c \frac{(\tau_i{s}+1)}{\tau_i{s}} (\tau_d{s}+1)$ where the parameters are related to the parameters of the standard form through $K_p = K_c \cdot \alpha$, $T_i = \tau_i \cdot \alpha$, and $T_d = \frac{\tau_d}{\alpha}$ $\alpha = 1 + \frac{\tau_d}{\tau_i}$. This form essentially consists of a PD and PI controller in series, and it made early (analog) controllers easier to build. When the controllers later became digital, many kept using the interacting [edit] Discrete implementation The analysis for designing a digital implementation of a PID controller in a Microcontroller (MCU) or FPGA device requires the standard form of the PID controller to be discretised ^[2]. Approximations for first-order derivatives are made by backward finite differences. The integral term is discretised, with a sampling time Δt,as follows, $\int_{0}^{t_k}{e(\tau)}\,{d\tau} = \sum_{i=1}^k e(t_i)\Delta t$ The derivative term is approximated as, $\dfrac{de(t_k)}{dt}=\dfrac{e(t_k)-e(t_{k-1})}{\Delta t}$ Thus, a velocity algorithm for implementation of the discretised PID controller in a MCU is obtained, $u(t_k)=u(t_{k-1})+K_p\left[\left(1+\dfrac{\Delta t}{T_i}+\dfrac{T_d}{\Delta t}\right)e(t_k)+\left(-1-\dfrac{2T_d}{\Delta t}\right)e(t_{k-1})+\dfrac{T_d}{\Delta t}e(t_{k-2})\right]$ [edit] Pseudocode Here is a simple software loop that implements the PID algorithm: previous_error = 0 integral = 0 error = setpoint - actual_position integral = integral + (errordt) derivative = (error - previous_error)/dt output = (Kperror) + (Kiintegral) + (Kdderivative) previous_error = error goto start [edit] External links [edit] PID tutorials [edit] Simulations [edit] Special topics and PID control applications [edit] References • Liptak, Bela (1995). Instrument Engineers' Handbook: Process Control. Radnor, Pennsylvania: Chilton Book Company. pp. 20–29. ISBN 0-8019-8242-1. • Tan, Kok Kiong; Wang Qing-Guo, Hang Chang Chieh (1999). Advances in PID Control. London, UK: Springer-Verlag. ISBN 1-85233-138-0. «Oldest ‹Older 1 – 200 of 215 Newer› Newest» 1. Dear ALL Sorry i am just copied from wikipedia 2. luxsman said... jian tenan..... buku berjalan PID rek.... aku diurui po'o.... 3. xitalho said... lengkap poll.. nganti rak mudeng blas aku... hihihi 4. Abula said... mantap nih... sebuah referensi yang lengkapppp... 5. Cocok buat referensi belajar nih... 6. bro.... pelan2 dong, mumet nich kepala ngikutinnya. ada yg PDF nya gak bro, biar numpang sedot aja ke komputer :) 7. bos admin (Endri) saya numpang copy ya, buat referensi dan menambah koleksi di pustaka teman :) 8. May I have a copy of your article? that's will be good for my reference on my website. 9. That technical articles. 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T-chart T-charts are the most versatile tool to compare and contrast ideas. By using this chart, you do not need a form. Usually, T-charts have three columns, the left and the right are the two topics, and the middle column is to identify the feature on which the rows focus. Furthermore, you can use it to compare informational topics, stories, elements, characters, and even settings. Many students use this tool because it is easy to do. Here is a sample of how to do a T-chart. 4. Matrix Chart Another alternative to the Venn diagram is the Matrix Chart. Matrix charts are very helpful when you are comparing and contrasting numerous things. It looks like a spreadsheet that contains several rows, one for each topic to compare. It also contains several columns, one for each topic for each way you compare. Usually, professionals and students use this kind of strategy when comparing the features of three-dimensional shapes. Furthermore, it helps the user notice things they may have noticed before writing or drawing the chart. You can check the image below to become familiar with Matrix Chart. Part 6. FAQs about What is Venn Diagram What is the primary purpose of a Venn diagram? Its main purpose is to illustrate or show the logical relationships between two, three, or more sets of topics and ideas. They are often used to organize things, showing their similarities and differences graphically. What do you call a three-way Venn diagram? A three-way Venn diagram is called the Spherical octahedron. It is a stereographic projection of a rectangular octahedron that makes a three-set Venn diagram. How do you read a Venn diagram? The most basic Venn diagram has two circles representing a group or an idea. The overlapping area represents the similarities or the combination of the two. Your question about “what is the Venn diagram?” is answered in this article. All the pieces of information that you need about Venn diagrams are written here. Venn diagrams are not difficult to make. After reading this post, we assure you that you know how to make one. Therefore, if you want to create a Venn diagram on your computer, use MindOnMap now. Create Your Mind Map as You Like	{"url":"https://www.mindonmap.com/blog/venn-diagram/","timestamp":"2024-11-09T14:21:42Z","content_type":"text/html","content_length":"41531","record_id":"<urn:uuid:d9a6cebb-e72d-41c7-b889-6d8ad7eb1d29>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00773.warc.gz"}
Floating Point As the name implies, floating point numbers are numbers that contain floating decimal points. For example, the numbers 5.5, 0.001, and -2,345.6789 are floating point numbers. Numbers that do not have decimal places are called integers. Computers recognize real numbers that contain fractions as floating point numbers. When a calculation includes a floating point number, it is called a "floating point calculation." Older computers used to have a separate floating point unit (FPU) that handled these calculations, but now the FPU is typically built into the computer's CPU. Published: 2007	{"url":"https://pc.net/glossary/floating_point","timestamp":"2024-11-07T10:59:15Z","content_type":"text/html","content_length":"9663","record_id":"<urn:uuid:ceb37ca0-3b18-4d3b-b4b7-d0fcb9aa3c50>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00319.warc.gz"}
151 research outputs found The foundation for any discussion of first-order phse transitions is Classical Nucleation Theory(CNT). CNT, developed in the first half of the twentieth century, is based on a number of heuristically plausible assumtptions and the majority of theoretical work on nucleation is devoted to refining or extending these ideas. Ideally, one would like to derive CNT from a more fundamental description of nucleation so that its extension, development and refinement could be developed systematically. In this paper, such a development is described based on a previously established (Lutsko, JCP 136:034509, 2012 ) connection between Classical Nucleation Theory and fluctuating hydrodynamics. Here, this connection is described without the need for artificial assumtions such as spherical symmetry. The results are illustrated by application to CNT with moving clusters (a long-standing problem in the literature) and the constructrion of CNT for ellipsoidal clusters We generalize Einstein's master equation for random walk processes by considering that the probability for a particle at position $r$ to make a jump of length $j$ lattice sites, $P_j(r)$ is a functional of the particle distribution function $f(r,t)$. By multiscale expansion, we obtain a generalized advection-diffusion equation. We show that the power law $P_j(r) \propto f(r)^{\alpha - 1}$ (with $\alpha > 1$) follows from the requirement that the generalized equation admits of scaling solutions ($f(r;t) = t^{-\gamma}\phi (r/t^{\gamma})$). The solutions have a $q$-exponential form and are found to be in agreement with the results of Monte-Carlo simulations, so providing a microscopic basis validating the nonlinear diffusion equation. Although its hydrodynamic limit is equivalent to the phenomenological porous media equation, there are extra terms which, in general, cannot be neglected as evidenced by the Monte-Carlo computations.}Comment: 7 pages incl. 3 fig We develop a microscopic theory for reaction-difusion (R-D) processes based on a generalization of Einstein's master equation with a reactive term and we show how the mean field formulation leads to a generalized R-D equation with non-classical solutions. For the $n$-th order annihilation reaction $A+A+A+...+A\rightarrow 0$, we obtain a nonlinear reaction-diffusion equation for which we discuss scaling and non-scaling formulations. We find steady states with either solutions exhibiting long range power law behavior (for $n>\alpha$) showing the relative dominance of sub-diffusion over reaction effects in constrained systems, or conversely solutions (for $n<\alpha<n+1$) with finite support of the concentration distribution describing situations where diffusion is slow and extinction is fast. Theoretical results are compared with experimental data for morphogen gradient formation.Comment: Article, 10 pages, 5 figure A recent description of diffusion-limited nucleation based on fluctuating hydrodynamics that extends classical nucleation theory predicts a very non-classical two-step scenario whereby nucleation is most likely to occur in spatially-extended, low-amplitude density fluctuations. In this paper, it is shown how the formalism can be used to determine the maximum probability of observing \emph{any} proposed nucleation pathway, thus allowing one to address the question as to their relative likelihood, including of the newly proposed pathway compared to classical scenarios. Calculations are presented for the nucleation of high-concentration bubbles in a low-concentration solution of globular proteins and it is found that the relative probabilities (new theory compared to classical result) for reaching a critical nucleus containing $N_c$ molecules scales as $e^{-N_c/3}$ thus indicating that for all but the smallest nuclei, the classical scenario is extremely unlikely.Comment: 7 pages, 5 figure Thermodynamic perturbation theory is applied to the model of globular proteins studied by ten Wolde and Frenkel (Science 277, pg. 1976) using computer simulation. It is found that the reported phase diagrams are accurately reproduced. The calculations show how the phase diagram can be tuned as a function of the lengthscale of the potential.Comment: 20 pages, 5 figure The equilibrium density distribution and thermodynamic properties of a Lennard-Jones fluid confined to nano-sized spherical cavities at constant chemical potential was determined using Monte Carlo simulations. The results describe both a single cavity with semipermeable walls as well as a collection of closed cavities formed at constant chemical potential. The results are compared to calculations using classical Density Functional Theory (DFT). It is found that the DFT calculations give a quantitatively accurate description of the pressure and structure of the fluid. Both theory and simulation show the presence of a ``reverse'' liquid-vapor transition whereby the equilibrium state is a liquid at large volumes but becomes a vapor at small volumes.Comment: 13 pages, 8 figures, to appear in J. Phys. : Cond. Mat The stability of idealized shear flow at long wavelengths is studied in detail. A hydrodynamic analysis at the level of the Navier-Stokes equation for small shear rates is given to identify the origin and universality of an instability at any finite shear rate for sufficiently long wavelength perturbations. The analysis is extended to larger shear rates using a low density model kinetic equation. Direct Monte Carlo Simulation of this equation is computed with a hydrodynamic description including non Newtonian rheological effects. The hydrodynamic description of the instability is in good agreement with the direct Monte Carlo simulation for $t < 50t_0$, where $t_0$ is the mean free time. Longer time simulations up to $2000t_0$ are used to identify the asymptotic state as a spatially non-uniform quasi-stationary state. Finally, preliminary results from molecular dynamics simulation showing the instability are presented and discussed.Comment: 25 pages, 9 figures (Fig.8 is available on request) RevTeX, submitted to Phys. Rev. The effect of molecule size (excluded volume) and the range of interaction on the surface tension, phase diagram and nucleation properties of a model globular protein is investigated using a combinations of Monte Carlo simulations and finite temperature classical Density Functional Theory calculations. We use a parametrized potential that can vary smoothly from the standard Lennard-Jones interaction characteristic of simple fluids, to the ten Wolde-Frenkel model for the effective interaction of globular proteins in solution. We find that the large excluded volume characteristic of large macromolecules such as proteins is the dominant effect in determining the liquid-vapor surface tension and nucleation properties. The variation of the range of the potential only appears important in the case of small excluded volumes such as for simple fluids. The DFT calculations are then used to study homogeneous nucleation of the high-density phase from the low-density phase including the nucleation barriers, nucleation pathways and the rate. It is found that the nucleation barriers are typically only a few $k_{B}T$ and that the nucleation rates substantially higher than would be predicted by Classical Nucleation Theory.Comment: To appear in Langmui The linear response description for impurity diffusion in a granular fluid undergoing homogeneous cooling is developed in the preceeding paper. The formally exact Einstein and Green-Kubo expressions for the self-diffusion coefficient are evaluated there from an approximation to the velocity autocorrelation function. These results are compared here to those from molecular dynamics simulations over a wide range of density and inelasticity, for the particular case of self-diffusion. It is found that the approximate theory is in good agreement with simulation data up to moderate densities and degrees of inelasticity. At higher density, the effects of inelasticity are stronger, leading to a significant enhancement of the diffusion coefficient over its value for elastic collisions. Possible explanations associated with an unstable long wavelength shear mode are explored, including the effects of strong fluctuations and mode coupling A simple model is proposed for the direct correlation function (DCF) for simple fluids consisting of a hard-core contribution, a simple parametrized core correction, and a mean-field tail. The model requires as input only the free energy of the homogeneous fluid, obtained, e.g., from thermodynamic perturbation theory. Comparison to the DCF obtained from simulation of a Lennard-Jones fluid shows this to be a surprisingly good approximation for a wide range of densities. The model is used to construct a density functional theory for inhomogeneous fluids which is applied to the problem of calculating the surface tension of the liquid-vapor interface. The numerical values found are in good agreement with simulation	{"url":"https://core.ac.uk/search/?q=author%3A(J.%20F.%20Lutsko)","timestamp":"2024-11-02T02:40:58Z","content_type":"text/html","content_length":"181976","record_id":"<urn:uuid:b788de27-67ce-4fcc-812f-4538d171eaae>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00373.warc.gz"}
Lowering the cost of anonymization 4.2.1 Introduction ✿ Query engines are a major analysis tool for data scientists, and one of the most common ways for analysts to write queries is with Structured Query Language (SQL). As a result, multiple query engines have been developed to enable data analysis while enforcing DP [42, 209, 233, 273], and all of them use a SQL-like syntax. However, as we discuss in Section 4.2.2, these differentially private query engines make some implicit assumptions, notably that each individual in the underlying dataset is associated with at most one dataset record. This does not hold in many real-world datasets, so the privacy guarantee offered by these systems is weaker than advertised for those datasets. To overcome this limitation, we introduce a generic mechanism for bounding user contributions to a large class of differentially private aggregate functions. We then propose a design for a SQL engine using these contribution bounding mechanisms to enforce DP, even when a given individual can be associated with arbitrarily many records or when the query contains joins. Our work goes beyond this design and accompanying analysis: we also describe the implementation of these mechanisms as part of a SQL engine, and the challenges encountered in the process. We describe the testing framework we use to increase our level of trust in the system’s robustness. To aid in reproducing of our work and encourage wider adoption of differential privacy, we release core components of the system, as well as a distinct implementation of this framework, as open-source software. Requirements and contributions To be useful for non-expert analysts, a differentially private SQL engine must satisfy at least the following requirements. • It must make realistic assumptions about the data, specifically allowing multiple records to be associated with an individual user. • It must support typical data analysis operations, such as counts, sums, means, percentiles, etc. • It must provide analysts with information about the accuracy of the queries returned by the engine, and uphold clear privacy guarantees. • It must provide a way to test the integrity of the engine and validate the engine’s privacy claims. In this work, we present a differentially private SQL engine that satisfies these requirements. Our contributions are as follows. • We detail how we use the concept of row ownership to enforce the original meaning of differential privacy: the output of the analysis does not reveal anything about a single individual. In our engine, multiple rows can be associated with the same “owner” (hereafter referred to as a user, although the owner could also be a group), and the differential privacy property is enforced at the user level. • We implement common aggregations (counts, sums, medians, etc.), arbitrary per-record transforms, and joins on the row owner column as part of our engine. To do so, we provide a method of bounding query sensitivity and stability across transforms and joins, and a mechanism to enforce row ownership throughout the query transformation. • We detail some of the usability challenges that arise when trying to use such a system in production and increase its adoption. In particular, we explain how we communicate the accuracy impact of differential privacy to analysts, and we experimentally verify that the noise levels are acceptable in typical conditions. We also propose an algorithm for automatic sensitivity determination. • We present a testing framework to help verify that -DP aggregation functions are correctly implemented, and can be used to detect software regressions that break the privacy guarantees. We hope that this work, and the associated open-source release, can increase the appropriate adoption of differential privacy by providing a usable system based on popular tools used by data Related work Multiple differentially private query engines have been proposed in the literature. In this work, we mainly compare our system to two existing differentially private query engines: PINQ [273] and Flex [209]. Our work differs in two major ways from these engines: we support the common case where a single user is associated with multiple rows, and we support arbitrary GROUP BY statements. Another line of research focuses on building frameworks to define differentially private algorithms: examples include are Airavat [334], Ektelo [404] and OpenDP’s programming framework [156]. These are building blocks that help write correct differentially private algorithms, but require significant changes in how programs are written, and we argue that they cannot be used as is without prior expertise on differential privacy. In these systems, a single organization is assumed to hold all the raw data. Query engines can also be used in other contexts: differential privacy can be used in concert with secure multiparty computation techniques to enable join queries between datasets held by different organizations, systems such as DJoin [293] and Shrinkwrap [42] tackle this specific use case. A significant amount of research focuses on improving the accuracy of query results while still maintaining differential privacy. In this work, for clarity, we keep the description of our system conceptually simple, and explicitly do not make use of techniques like smooth sensitivity [303], tight privacy budget computation methods [213, 278], variants of the differential privacy definition (Section 2.2), adjustment of noise levels to a pre-specified set of queries [249], or generation of differentially private synthetic data to answer arbitrarily many queries afterwards [49, 232, 233]. We revisit certain design choices and outline possible improvements later, in Section 4.3. The testing framework we introduce in Section 4.2.6.0 is similar to recent work in verification for differential privacy [47, 109, 167], and was developed independently. Other approaches use semantic analysis, possibly augmented with automated proving techniques, to certify that algorithms are differentially private [32, 33, 292]. Our work is not the first to use noise and thresholding to preserve privacy: this method was originally proposed in [174, 230] in the specific context of releasing search logs with -DP; our work can be seen as an extension and generalization of this insight. Diffix [155] is another system using similar primitives; however, it does not provide any formal privacy guarantee, and has been shown to be vulnerable to attacks [79, 80, 157], so a meaningful comparison with our work is not feasible. In Section 4.2.4, we provide a comparison of query accuracy between our work, PINQ, and Flex. We introduce here the definitions and notations used throughout this section, which are also summarized on page 38. Because a significant part of this work specifically focuses on the case where a single user contributes multiple records to the dataset, we no longer consider a dataset as a sequence of records in but as a sequence of rows, where each row is a pair : each row is a record associated with a specific user. Definition 78 (Distance between datasets). We denote row-level change the addition or removal of a single row from a dataset, and user-level change the addition or removal of all rows associated with a user. Given two datasets and , we denote the minimum number of row-level changes necessary to transform into , and the minimum number of user-level changes necessary to transform into ; we call them row-level distance and user-level distance respectively. In the original definition of -differential privacy, each user is implicitly assumed to contribute a single record. Since we want to consider the case where this is not true, we define two variants of differential privacy to make this distinction explicit. Definition 79 (-row-level and user-level differential privacy). A randomized mechanism satisfies row-level -DP (respectively row-level -DP) if for all pairs of datasets that satisfy (respectively ), As previously, -DP is an alias for -DP, and denotes -indistinguishability (see Definition 14). Note that this notion is technically unbounded differential privacy, defined in [227] and mentioned in Section 2.2.3.0: we only allow neighboring datasets to differ in a row (or all rows associated with a user) that has been added or removed, not changed. Up to a change in parameters, it is equivalent to the classical definitions, but we found that this choice significantly simplifies the analysis and the implementation of differential privacy tooling. Finally, let us define -sensitivity, defined on functions that take a dataset as input and return a vector in , for some integer . Definition 80 (-Sensitivity). Let be the -norm on . The global -sensitivity of a function is the smallest number such that: for all and such that . The user-global -sensitivity of is the smallest number such that the above holds for all and such that .	{"url":"https://desfontain.es/thesis/IntroductionDPSQL.html","timestamp":"2024-11-11T11:53:18Z","content_type":"text/html","content_length":"71704","record_id":"<urn:uuid:66ba091d-f236-4994-adb0-9d3667882624>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00768.warc.gz"}
Splay Tree The SplayTree type is an implementation of Splay Tree in Julia. It is a self-balancing binary search tree with the additional property that recently accessed elements are quick to access again. Operations such as search, insert and delete can be done in O(log n) amortized time, where n is the number of nodes in the SplayTree. julia> tree = SplayTree{Int}(); julia> for k in 1:2:20 push!(tree, k) julia> haskey(tree, 3) julia> tree[4] julia> for k in 1:2:10 delete!(tree, k) julia> haskey(tree, 5)	{"url":"https://juliacollections.github.io/DataStructures.jl/latest/splay_tree/","timestamp":"2024-11-10T15:57:56Z","content_type":"text/html","content_length":"7460","record_id":"<urn:uuid:5624137a-4ec5-4d5d-9409-cba2aff344f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00366.warc.gz"}
Variation och biologisk mångfald Flashcards Chegg.com Variational Calculus on Time Scales - Svetlin G. Georgiev - Adlibris Pub Date: March 2020 arXiv: arXiv:2003.06422 Bibcode: 2020arXiv200306422G Keywords: Mathematics - General Mathematics; Primary 49K05; constrained extremisation in the context of the variational calculus. Let us start by setting up the classical isoperimetric problem in this context. Let x : [0,1] → R2 Here we present three useful examples of variational calculus as applied to problems in mathematics and physics. 5.3.1 Example 1 : minimal surface of revolution. Mar 14, 2021 This integral variational approach was first championed by Gottfried Wilhelm Leibniz, contemporaneously with Newton's development of the Feb 27, 2021 The calculus of variations provides the mathematics required to determine the path that minimizes the action integral. This variational approach is Find out information about Variational calculus. branch of mathematics In general, problems in the calculus of variations involve solving the definite integral Jun 6, 2020 imposed on these functions. Applications of the Variational Calculus. What is the shortest distance between two points, but for now assume that there's no temperature variation. Write the length of a path for a function y between fixed This Brief puts together two subjects, quantum and variational calculi by considering variational problems involving Hahn quantum operators. The main The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and Mar 13, 2020 The aim of this paper is to bring together a new type of quantum calculus, namely p -calculus, and variational calculus. Översättning 'calculus' – Ordbok svenska-Engelska Glosbe Examples 3.1 Plane 3.2 Sphere The Calculus of Variations The variational principles of mechanics are rmly rooted in the soil of that great century of Liberalism which starts with Descartes and ends with the French Revolution and which has witnessed the lives of Leibniz, Spinoza, Goethe, and Johann Sebastian Bach. It is the only period of cosmic thinking in the entire Calculus of Variations [44], as well as lecture notes on several related courses by J. Ball, J. Kristensen, A. Mielke. Further texts on the Calculus of Variations are the elementary introductions by B. van Brunt [96] and B. Dacorogna [26], the more classical two-part trea- calculus of variations are prescribed by boundary value problems involving certain types of diﬀerential equations, known as the associated Euler–Lagrange equations. The math- Calculus of Variations Raju K George, IIST Lecture-1 In Calculus of Variations, we will study maximum and minimum of a certain class of functions. Variation och biologisk mångfald Flashcards Chegg.com A variation of a functional is the small change in a functional's value due to a small change in the functional's input. However, suppose that we wish to demonstrate this result from first principles. GEODESICS ON SURFACES BY VARIATIONAL CALCULUS J Villanueva Florida Memorial University nd15800 NW 42 Ave Miami, FL 33054 jvillanu@fmuniv.edu 1. Introduction 1.1 The problem by variational calculus 1.2 The Euler-Lagrange equation 2. Exchange rate yen sek Maximum and Minumum problems. Euler-Lagrange Equations. Variational Concepts. Functionals. Applications of the Variational Calculus. In the simple case in which the sample is a slab of thickness d, the total energy per unit area is given by F= Z d=2 Chapter 7 considers the application of variational methods to the study of systems with infinite degrees of freedom, and Chapter 8 deals with direct methods in the calculus of variations. The problems following each chapter were made specially for this English-language edition, and many of them comment further on corresponding parts of the text. Note that variational calculus has been applied to an extensively large number of problems, theories, and formulations most of which could be reexamined in the light of fractional variational calculus. Thus, the above work has opened significant opportunities for many new research. Bra projektledare egenskaper hallbar miljolitauen wikipedia svenskavd danderyds sjukhushur odlas tranbärgapwaves analys Tillväxt-katalog, 1854-1887: Utgör 2: dra supplementet til Välkommen till Calculus of Variations ONLINE UTROKING MED LIVE instruktör med hjälp av en interaktiv KAPITEL V: Variation av kurvor uttryckt analytiskt. Jämför och hitta det billigaste priset på An Introduction to the Calculus of Variations innan du gör ditt köp. Köp som antingen bok, ljudbok eller e-bok. Skatt oslodans gymnasium uppsala Allfo: variationskalkyl - Finto Published 15 April 2008 • 2008 IOP Publishing Ltd Feb 12, 2013 I want to differentiate a potential energy functional (a multivariable functional combination of integrals) in the variational calculus to get the Feb 23, 2015 Calculus of variation problems. This presentation gives example of "Calculus of Variations" problems that can be solved analytical.	{"url":"https://investeringargcjb.netlify.app/97103/61868.html","timestamp":"2024-11-07T16:37:03Z","content_type":"text/html","content_length":"10645","record_id":"<urn:uuid:8e193905-4737-4cc9-880a-614ea31dad5b>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00855.warc.gz"}
Virtually every big business borrows cash. The group frontrunner for borrowings is usually the treasurer. The treasurer must safeguard the firmвЂ™s money moves at all times, along with know and manage the effect of borrowings in the companyвЂ™s interest costs and earnings. So treasurers require a deep and joined-up knowledge of the results of different borrowing structures, both regarding the firmвЂ™s money flows and on its profits. Negotiating the circularity of equal loan instalments can feel being lost in a maze. Why don’t we have a look at practical money and revenue State we borrow ВЈ10m in a swelling amount, become paid back in yearly instalments. Demonstrably, the lending company calls for complete payment for the ВЈ10m principal (money) lent. They will require also interest. LetвЂ™s state the interest is 5% per year. The yearвЂ™s that is first, before any repayments, is actually the initial ВЈ10m x 5% = ВЈ0.5m The cost charged to your earnings statement, reducing web earnings for the very first 12 months, is ВЈ0.5m. Nevertheless the year that is next begin to appear complicated. Our instalment shall repay a number of the principal, in addition to spending the attention. What this means is the next yearвЂ™s interest charge will soon be lower than 1st, as a result of the major repayment. But just what when we canвЂ™t manage bigger instalments in the last years? Can we make our total cash outflows the same in every year? Can there be an instalment that is equal will repay the ideal level of principal in every year, to go out of the first borrowing paid back, along with all the reducing annual interest costs, because of the finish? Assistance are at hand. There was, certainly, an equal instalment that does simply that, often known as an instalment that is equated. Equated instalments repay varying proportions of great interest and principal within each period, to ensure by the end, the mortgage is paid in complete. The equated instalments deal well with this income issue, however the interest fees nevertheless appear Equated instalment An instalment of equal value with other instalments. Equated instalment = major Г· annuity element As weвЂ™ve seen, interest is charged regarding the balance that is reducing of principal. So that the interest fee per period begins out relatively large, after which it gets smaller with each yearly The attention calculation is possibly complicated, also circular, because our principal repayments are changing too. Because the interest part of the instalment falls each 12 months, the balance offered to spend off the principal is certainly going up each and every time. How do we find out the varying yearly interest costs? LetвЂ™s look at this instance: Southee Limited, a construction business, is likely to get new equipment that is earth-moving a price of ВЈ10m. Southee is considering a mortgage for the complete price of the apparatus, repayable over four years in equal yearly instalments, integrating interest at a consistent level of 5% per year, the very first instalment to be paid 12 months through the date of taking out fully the You have to be in a position to determine the instalment that is annual is payable underneath the mortgage, calculate just how much would express the key repayment and in addition just how much would express interest costs, in all the four years as well as in total. previous - next	{"url":"https://dykkerklubben-aqua.dk/2020/11/11/how-exactly-to-determine-loan-instalments-with-6/","timestamp":"2024-11-03T12:21:19Z","content_type":"text/html","content_length":"32044","record_id":"<urn:uuid:52c260df-d6cb-4777-ad33-f7d0853cfe47>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00077.warc.gz"}
Semantics of Sequent Calculi with Basic Structural Rules: Fuzziness Versus Non-Multiplicativity Download PDFOpen PDF in browser Semantics of Sequent Calculi with Basic Structural Rules: Fuzziness Versus Non-Multiplicativity EasyChair Preprint 4153 15 pages•Date: September 8, 2020 The main general\/} result of the paper is that basic\/} structural rules --- Enlargement, Permutation and Contraction --- (as well as Sharings) [and Cuts] are derivable in a \{multiplicative\} propositional two-side sequent calculus iff there is a class of \{crisp\} (reflexive) [transitive distributive] fuzzy two-side matrices such that any rule is derivable in the calculus iff it is true in the class, the ``\{\}''/``()[]''-optional case being due to \cite{My-label}/\cite{My-fuzzy}. Likewise, fyzzyfying the notion of signed matrix \cite{My-label}, we extend the main result obtained therein beyond multiplicative calculi. As an application, we prove that the sequent calculus $\mbb{LK}_\mr{[S/C]}$ resulted from Gentzen's $LK$ \cite{Gen} by adding the rules inverse to the logical ones and retaining as structural ones merely basic ones [and Sharing/Cut] is equivalent (in the sense of \cite{DEAGLS}) to the bounded version of Belnap's four-valued logic (cf. \cite{Bel}) [resp., the {\em logic of paradox\/} \cite{Priest}/ Kleene's three-valued logic \cite{Kleene}]. As a consequence of this equivalence, appropriate generic results of \cite{DEAGLS} concerning extensions of equivalent calculi and the advanced auxiliary results on extensions of the bounded versions of Kleene's three-valued logic and the logic of paradox proved here with using the generic algebraic tools elaborated in \cite{LP-ext}, we then prove that extensions of the Sharing/Cut-free version $\mbb{LK}_\mr{C/S}$ of $LK$ form a three/four-element chain/, consistent ones having same derivable sequents that provides a new profound insight into Cut Elimination in $LK$ appearing to be just a consequence of the well-known regularity of operations of Belnap's four-valued logic. Keyphrases: Calculus, logic, matrix, sequent Links: https://easychair.org/publications/preprint/wwGv Download PDFOpen PDF in browser	{"url":"https://eraw.easychair.org/publications/preprint/wwGv","timestamp":"2024-11-10T08:17:04Z","content_type":"text/html","content_length":"5958","record_id":"<urn:uuid:cb95c13e-1b10-4b4c-8ea0-edf7f4df6f81>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00781.warc.gz"}
How to solve intermediate algebra word problems for free how to solve intermediate algebra word problems for free Related topics: solved questions on boolean algebra \| solving for two variables worksheet \| free 9th grade games online \| lowest common multiple of 39 and 17 \| convert a mixed Home number to decimal \| 8th grade taks math strategies \| free math sheets for 2nd graders \| free ged math example printouts \| simplify exponents \| factor quadratics Linear Equations and Inequalitie \| solution to fields exercises herstein \| linear programming lesson plan Solving Inequalities Absolute Value Inequalities Graphing Equivalent Fractions Lesson Author Message Investigating Liner Equations Using madvnoeimb Posted: Sunday 31st of Dec 10:01 Graphing Calculator Hey, A couple of days back I began working on my mathematics assignment on the topic Basic Math. 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Inequalities and Applications Inequalities Registered: Using MATLAB to Solve Linear 17.03.2002 Inequalities From: Norway Equations and Inequalities Graph Linear Inequalities in Two Solving Equations & Inequalities cmithy_dnl Posted: Wednesday 03rd of Jan 07:04 Teaching Inequalities:A Hypothetical I completely agree, Algebrator is awesome ! I am really good in algebra since I used it, and I have the highest grades in the class! It Classroom Case helped me even with the most confusing math problems, like those on complex fractions or graphing parabolas. I definitely think you should Graphing Linear Inequalities and give it a try . Systems of Inequalities Inequalities and Applications Registered: Solving Inequalities 08.01.2002 Quadratic Inequalities From: Australia Solving Systems of Linear Equations by Systems of Equations and Inequalities malhus_pitruh Posted: Wednesday 03rd of Jan 08:55 Graphing Linear Inequalities perfect square trinomial, function domain and angle suplements were a nightmare for me until I found Algebrator, which is truly the best Inequalities algebra program that I have ever come across. I have used it through many math classes – Remedial Algebra, Algebra 2 and Pre Algebra. Just Solving Inequalities typing in the math problem and clicking on Solve, Algebrator generates step-by-step solution to the problem, and my algebra homework would Solving Inequalities be ready. I truly recommend the program. Solving Equations Algebraically and Registered: Graphically 23.04.2003 Graphing Linear Equations From: Girona, Solving Linear Equations and Catalunya (Spain) Inequalities Practice Problems Graphing Linear Inequalities Equations and Inequalities Solving Inequalities Home Linear Equations and Inequalitie Solving Inequalities Absolute Value Inequalities Graphing Equivalent Fractions Lesson Plan Investigating Liner Equations Using Graphing Calculator Graphically solving a System of two Linear Equatio Shifting Reflecting Sketching Graph Graphs of Rational Functions Systems of Equations and Inequalities Graphing Systems of Linear Equat LINEAR FUNCTIONS: SLOPE, GRAPHS AND MODELS Solving Inequalities with Absolute Values Solving Inequalities Solving Equations & Inequalities Graph the rational function Inequalities and Applications Inequalities Using MATLAB to Solve Linear Inequalities Equations and Inequalities Graph Linear Inequalities in Two Variables Solving Equations & Inequalities Teaching Inequalities:A Hypothetical Classroom Case Graphing Linear Inequalities and Systems of Inequalities Inequalities and Applications Solving Inequalities Quadratic Inequalities Inequalities Solving Systems of Linear Equations by Graphing Systems of Equations and Inequalities Graphing Linear Inequalities Inequalities Solving Inequalities Solving Inequalities Solving Equations Algebraically and Graphically Graphing Linear Equations Solving Linear Equations and Inequalities Practice Problems Graphing Linear Inequalities Equations and Inequalities Solving Inequalities Author Message madvnoeimb Posted: Sunday 31st of Dec 10:01 Hey, A couple of days back I began working on my mathematics assignment on the topic Basic Math. I am currently unable to finish the same because I am not familiar with the fundamentals of function domain, algebra formulas and equivalent fractions. Would it be possible for anyone to assist me with this? espinxh Posted: Monday 01st of Jan 09:47 It seems like you are not the only one facing this problem. A friend of mine was in the same situation last month. That is when he came across this program known as Algebrator. It is by far the most economical piece of software that can help you with problems on how to solve intermediate algebra word problems for free. It won’t just solve problems but also explain of how it arrived at that solution. From: Norway cmithy_dnl Posted: Wednesday 03rd of Jan 07:04 I completely agree, Algebrator is awesome ! I am really good in algebra since I used it, and I have the highest grades in the class! It helped me even with the most confusing math problems, like those on complex fractions or graphing parabolas. I definitely think you should give it a try . From: Australia malhus_pitruh Posted: Wednesday 03rd of Jan 08:55 perfect square trinomial, function domain and angle suplements were a nightmare for me until I found Algebrator, which is truly the best algebra program that I have ever come across. I have used it through many math classes – Remedial Algebra, Algebra 2 and Pre Algebra. Just typing in the math problem and clicking on Solve, Algebrator generates step-by-step solution to the problem, and my algebra homework would be ready. I truly recommend the program. From: Girona, Catalunya (Spain) Posted: Sunday 31st of Dec 10:01 Hey, A couple of days back I began working on my mathematics assignment on the topic Basic Math. I am currently unable to finish the same because I am not familiar with the fundamentals of function domain, algebra formulas and equivalent fractions. Would it be possible for anyone to assist me with this? Posted: Monday 01st of Jan 09:47 It seems like you are not the only one facing this problem. A friend of mine was in the same situation last month. That is when he came across this program known as Algebrator. It is by far the most economical piece of software that can help you with problems on how to solve intermediate algebra word problems for free. It won’t just solve problems but also explain of how it arrived at that Posted: Wednesday 03rd of Jan 07:04 I completely agree, Algebrator is awesome ! I am really good in algebra since I used it, and I have the highest grades in the class! It helped me even with the most confusing math problems, like those on complex fractions or graphing parabolas. I definitely think you should give it a try . Posted: Wednesday 03rd of Jan 08:55 perfect square trinomial, function domain and angle suplements were a nightmare for me until I found Algebrator, which is truly the best algebra program that I have ever come across. I have used it through many math classes – Remedial Algebra, Algebra 2 and Pre Algebra. Just typing in the math problem and clicking on Solve, Algebrator generates step-by-step solution to the problem, and my algebra homework would be ready. I truly recommend the program.	{"url":"https://graph-inequality.com/graph-inequality/factoring/how-to-solve-intermediate.html","timestamp":"2024-11-11T07:31:17Z","content_type":"text/html","content_length":"87304","record_id":"<urn:uuid:54116f05-daa3-477a-b3c0-ecec2d35f9cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00584.warc.gz"}
How do you prove a contradiction in math? The steps taken for a proof by contradiction (also called indirect proof) are: 1. Assume the opposite of your conclusion. 2. Use the assumption to derive new consequences until one is the opposite of your premise. 3. Conclude that the assumption must be false and that its opposite (your original conclusion) must be true. What is proof by contradiction explain it with example? Irrationality of the square root of 2 A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational. If it were rational, it would be expressible as a fraction a/b in lowest terms, where a and b are integers, at least one of which is odd. But if a/b = √2, then a2 = 2b2. How does proof by contradiction work? Proof by contradiction is a powerful mathematical technique: if you want to prove X, start by assuming X is false and then derive consequences. If you reach a contradiction with something you know is true, then the only possible problem can be in your initial assumption that X is false. Therefore, X must be true. What is contradiction method math? Another method of proof that is frequently used in mathematics is a proof by contradiction. This method is based on the fact that a statement X can only be true or false (and not both). The idea is to prove that the statement X is true by showing that it cannot be false. How do you solve contradictions? The six steps are as follows: 1. Step 1: Find an original problem. 2. Step 2: Describe the original situation. 3. Step 3: Identify the administrative contradiction. 4. Step 4: Find operating contradictions. 5. Step 5: Solve operating contradictions. 6. Step 6: Make an evaluation. How do you do the contradiction method? Now, we will use the method called “ proof by contradiction” to show that the product of a non-zero rational number and an irrational number is an irrational number….Proof by Contradiction Example. Statement Comments Assume that rx is rational. Take the negation of a statement that we need to prove. What do you need to prove in proof by contradiction method? To prove something by contradiction, we assume that what we want to prove is not true, and then show that the consequences of this are not possible. That is, the consequences contradict either what we have just assumed, or something we already know to be true (or, indeed, both) – we call this a contradiction. What is an example of a contradiction? A contradiction is a situation or ideas in opposition to one another. Declaring publicly that you are an environmentalist but never remembering to take out the recycling is an example of a contradiction. A “contradiction in terms” is a common phrase used to describe a statement that contains opposing ideas. How do you identify a contradiction? A contradiction between two statements is a stronger kind of inconsistency between them. If two sentences are contradictory, then one must be true and one must be false, but if they are inconsistent, then both could be false.	{"url":"https://ecce216.com/miscellaneous/how-do-you-prove-a-contradiction-in-math/","timestamp":"2024-11-05T07:18:21Z","content_type":"text/html","content_length":"53008","record_id":"<urn:uuid:f714b32c-e88f-4ccb-99de-97e0d1b84a7f>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00069.warc.gz"}
ManPag.es - slarfg.f − subroutine SLARFG (N, ALPHA, X, INCX, TAU) SLARFG generates an elementary reflector (Householder matrix). Function/Subroutine Documentation subroutine SLARFG (integerN, realALPHA, real, dimension( * )X, integerINCX, realTAU) SLARFG generates an elementary reflector (Householder matrix). SLARFG generates a real elementary reflector H of order n, such H * ( alpha ) = ( beta ), H*T H = I. ( x ) ( 0 ) where alpha and beta are scalars, and x is an (n-1)-element real vector. H is represented in the form H = I - tau * ( 1 ) * ( 1 v**T ) , ( v ) where tau is a real scalar and v is a real (n-1)-element If the elements of x are all zero, then tau = 0 and H is taken to be the unit matrix. Otherwise 1 <= tau <= 2. N is INTEGER The order of the elementary reflector. ALPHA is REAL On entry, the value alpha. On exit, it is overwritten with the value beta. X is REAL array, dimension On entry, the vector x. On exit, it is overwritten with the vector v. INCX is INTEGER The increment between elements of X. INCX > 0. TAU is REAL The value tau. Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. September 2012 Definition at line 107 of file slarfg.f. Generated automatically by Doxygen for LAPACK from the source code.	{"url":"https://manpag.es/SUSE131/3+slarfg.f","timestamp":"2024-11-07T03:05:41Z","content_type":"text/html","content_length":"19537","record_id":"<urn:uuid:a83642d2-da25-4f7e-b832-bd8c06ae929d>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00734.warc.gz"}
How Discrete Signal Interpolation Improves D/A Conversion - Rick Lyons How Discrete Signal Interpolation Improves D/A Conversion Earlier this year, for the Linear Audio magazine, published in the Netherlands whose subscribers are technically-skilled hi-fi audio enthusiasts, I wrote an article on the fundamentals of interpolation as it's used to improve the performance of analog-to-digital conversion. Perhaps that article will be of some value to the subscribers of dsprelated.com. Here's what I wrote: We encounter the process of digital-to-analog conversion every day—in telephone calls (land lines and cell phones), telephone answering machines, CD & DVD players, iPhones, digital television, MP3 players, digital radio, and even talking greeting cards. This material is a brief tutorial on how sample rate conversion improves the quality of digital-to-analog conversion. Ideal Digital-to-Analog Conversion Have a look at the system shown in Figure 1(a). There we show a hardware depiction of a digital-to-analog converter (DAC)—a hardware device having multiple input pins that accept multibit binary words. In that figure the variable x(n) represents a sequence of binary words showing their individual binary bits from the least significant bit (LSB) to the most significant bit (MSB). In Figure 1 (b) we show a hypothetical time-domain sequence of x(n) amplitude values, which we'll call "samples", where each sample is represented as a single black dot. We refer to the x(n) signal as a discrete, or "digital", sequence. The variable n is referred to as the "time-domain index" of the discrete x(n) input signal. Critical to its operation, the DAC accepts a periodic-in-time, pulse-like, signal shown as f[clk] in Figure 1(a). The repetition rate, the frequency, of the f[clk] signal is the reciprocal of the time period between individual x(n) samples. The f[clk] signal, synchronized with the binary x(n) input sequence, triggers the DAC to 'clock in' the bits of the current x(n) sample. Finally, the DAC has a single output pin upon which is riding an analog (what the digital signal processing experts call "continuous") voltage that we'll call v[DAC](t). The variable t in v[DAC](t) represents time measured in seconds. Given the x(n) input sequence shown in Figure 1(b), we wish to generate the analog v[ideal](t) signal shown in Figure 1(c) whose frequency-domain spectrum is depicted in Figure 1(d). In our frequency plots we only show the positive-frequency axis because we assume all signals are real-valued and all spectra are symmetrical and centered at zero Hz. The Nyquist Criterion In this discussion, if the highest-frequency spectral component of x(n), and therefore v[ideal](t), is B Hz, we're assuming that the f[clk] sample rate is greater than 2B Hz. That condition is the famous Nyquist criterion stipulating the periodic sampling condition required for error-free sampling (analog-to-digital conversion) of analog signals. As a historical note, the notion of periodic sampling was studied by various engineers, scientists, and mathematicians such as the Russian V. Kotelnikov, the Swedish-born H. Nyquist, the Scottish E. Whittaker, and the Japanese I. Someya [1]. But it was the American mathematician Claude Shannon, acknowledging the work of others, that mathematically formalized the concept of periodic sampling as we know it today, named it in honor of the great American electrical engineer Harry Nyquist, and brought it to the broad attention of the world's communications engineers [2]. That was in 1948—the birth year of the transistor, marshmallows, and this author. OK, back to Figure 1. Unfortunately, commercially-available DACs will not produce our desired v[ideal](t) output voltage based on the x(n) input sequence. So, given that apparently bad news, let's now think about some hypothetical, and actual, DAC output signals and their spectra. Some Hypothetical DAC Output Signals If we could build a DAC whose output voltage was a periodic series of super-narrow (widths measured in picoseconds) analog pulses as shown in Figure 2(a), whose amplitudes are equal the to amplitudes of x(n), the spectrum of such analog pulses would be the repetitive pattern shown in Figure 2(b). Figure 2 illustrates one of the relationships between the time- and frequency-domain representations of a signal: if a time signal has periodic amplitude variations, such as the periodically-spaced pulses in Figure 2(a), its spectrum will be periodic. If our time signal's super-narrow pulses are separated by 1/f[clk] seconds, the repetitive spectral energy will be separated by f[clk] Hz as shown in Figure 2(b). Now if we passed the analog pulsed signal in Figure 2(a) through an analog lowpass filter, whose frequency magnitude response is shown by the dashed lines in Figure 2(c), we'd produce our desired Figure 1(c) v[ideal](t) signal having the spectrum shown in Figure 2(d). As straightforward as all of this seems to be, as it turns out, the electronics needed to generate the super-narrow Figure 2 (a) pulses is prohibitively expensive, on the order of the cost of a new Harley Davidson Sportster motorcycle. Far too costly to include in any telephone, music, or television product. Thankfully, DAC manufacturers have a far more affordable way of generating our desired v[ideal](t) signal. Thinking again about Figure 1, we can view the discrete x(n) sample values in Figure 1(b) as being amplitude samples of our desired v[ideal](t) analog signal in Figure 1(c). What we want is an interpolated version of x(n). But not merely two or four or ten new samples between each original x(n) sample, we seek an infinite number of samples between each original x(n) sample. We want so many new samples that our interpolated signal is continuous (analog). It's not traditional interpolation we desire, we wish to perform interpolation on steroids. This notion of interpolation is not unusual. Some signal processing experts refer to the DAC process itself as interpolation [3]. Thinking about our idea of interpolation, it would nice if we could build a DAC whose output voltage was that shown by the solid curve in Figure 3(a). As it turns out, that solid curve in Figure 3(a) is not a curve at all—it's a series of straight lines connecting the x(n) sample values shown by the shaded dots. Let's zoom in and think about the first bold-line segment of that curve as shown in Figure 3(b). To generate the voltage segment between the x(0) and x(1) amplitude values we'd have to electronically evaluate the following expression: v[1st](t) = x(0) + [x(1) – x(0)]t, for 0 ≤ t < 1. (1) Equation (1) is a first-order (first-order in terms of t) polynomial, and that's why the v[1st](t) curve comprising the connected straight lines is called a "first-order hold" waveform. (The process of generating the v[1st](t) waveform, based on the x(n) samples, is called "curve fitting" by signal processing folk. As with the hypothetical voltage in Figure 2(a), the electronic components needed to generate v[1st](t) would be prohibitively expensive for our DAC applications. Let's now review the less-expensive method for generating our desired v[ideal](t) signal developed by commercial DAC Actual DAC Output Signals Given the discrete x(n) input sequence shown in Figure 1(b), a commercially-available DAC output voltage will be the analog v[DAC](t) signal shown in Figure 4(a). To generate the v[DAC](t) voltage segment between the x(0) and x(1) amplitude values we need merely implement the following expression: v[DAC](t) = x(0), for 0 ≤ t < 1 (2) which is not a function of time t. Likewise, between the x(1) and x(2) amplitude values the v[DAC](t) voltage is merely an implementation of: v[DAC](t) = x(1), for 1 ≤ t < 2. (3) Equations (2) and (3) are referred to as "zero-order polynomials" because we could write Eq. (3), for example, as v[DAC](t) = x(1)t^0, for 1 ≤ t < 2. (4) Notice that the power of variable t in Eq. (4) is zero. The v[DAC](t) signal is commonly called a "zero-order hold" waveform [3]. Zooming out from Figure 4(a) we see a longer-time interval of the v [DAC](t) signal in Figure 4(b). Now that we know the zero-order hold nature of DAC analog outputs, let's determine the spectral content of such an analog voltage. To do so, we return to the hypothetical analog pulsed DAC output that we considered in Figure 2. We repeat that pulsed signal, and its repetitive spectrum, in Figures 5(a) and 5(b). We can think of our DAC's 'stairstep' v[DAC](t) output signal as the convolution of the Figure 5(a) pulses with a rectangular, unity-amplitude, interpolation function shown in Figure 5(c) having the sin(x)/x spectrum given in Figure 5(d). Because convolution in the time domain is equivalent to multiplication in the frequency domain, the Figure 5(f) solid-curve spectrum of v[DAC](t) is the product of the Figure 5(b) and Figure 5(d) spectra. So there you have it—the spectrum of a DAC's v[DAC](t) output is the repetitive, decreasing-magnitude, spectrum given in Figure 5(f). Post-DAC Analog Filtering Our final task to achieve our ideal Figure 1(c) v[ideal](t) voltage, whose spectrum is in Figure 1(d), is to pass the v[DAC](t) output signal through an analog lowpass filter as shown in Figure 6(a). The lowpass filter's mandatory frequency magnitude response is shown by the dashed lines in Figure 6(b). The spectrum of our resultant v[out](t) is given in Figure 6(c), and the v[out](t) voltage, in Figure 6(d), is quite similar to our desired v[ideal](t) analog signal in Figure 1(c). OK, there are two things we must consider regarding the DAC/filtering system in Figure 6(a). First, the DAC's non-flat Figure 5(d) sin(x)/x magnitude envelope attenuates the higher frequency spectral components in our original x(n) signal. That is, the drooping nature of a DAC's inherent sin(x)/x frequency characteristic, shown in Figure 7(a), attenuates our original signal in the vicinity of B In some DAC applications, if the frequency B Hz is small relative to the f[clk] frequency, perhaps we could simply ignore the sin(x)/x amplitude droop. However if the sin(x)/x droop cannot, for some reason, be tolerated, then specialized 'DAC-compensation' digital filtering must be performed in the processing steps prior to applying the x(n) sequence to the DAC. That is, our signal of interest may need to be applied to a pre-DAC compensation digital filter such as the solid curve in Figure 7(b) where the high-frequency signal components in the vicinity of B Hz are amplified. That amplification would then be canceled by the sin(x)/x droop behavior of our DAC. The second issue for us to consider regarding the DAC/filtering system in Figure 6(a) is the complexity of the analog lowpass filter. An analog filter having a frequency magnitude response as shown by the dashed lines in Figure 6(b), with such a narrow transition region from the end of its passband to the beginning of its stopband, may be difficult to design. Such a filter may have several active electronic components and be expensive. In addition to the cost issue, analog lowpass filters have the additional problem of exhibiting nonlinear phase responses in the neighborhood of their cutoff frequency at B Hz. Fortunately, a multirate digital signal processing operation is available that can reduce both the complexity (cost) and nonlinear phase problems associated with analog lowpass filters used in DAC applications. Digital Interpolation to the Rescue We can drastically simplify our analog lowpass filter design complexity by increasing our x(n) signal's f[clk] sample rate which is our DAC's f[clk] clock frequency. That is, we will interpolate our original x(n) signal. To explain this notion of interpolation, consider the v[DAC,1](t) analog DAC output signal in Figure 8(a). Assuming that the DAC's f[clk] clock rate is 11.025 kHz, the spectrum of the DAC's output signal is the solid v[DAC,1](f) curve in Figure 8(b). At this 11.025 kHz clock rate the analog lowpass filter's frequency magnitude response will be the bold dashed curve in Figure 8(b). We show the DAC's x[1](n) input discrete sequence, whose sample rate is 11.025 kHz, in Figure 8(c). We implement a 'digital interpolation-by-two' process as follows: First, prior to any DAC processing, we insert a zero-valued sample in between each of the x[1](n) samples to generate the x[z](n) sequence shown in Figure 8(d). (The inserted samples are shown by white dots in Figure 8(d). The full time durations of the x[1](n) and x[z](n) sequences, from their first to their last samples, measured in seconds, are identical.) Next we merely pass the x[z](n) sequence through a digital lowpass filter whose cutoff frequency is slightly greater than B Hz. That filter's output, then, will be the 22.05 kHz sample rate interpolated x[2](n) sequence shown in Figure 8(e). So, the most simple form of digital interpolation is a two-step process: zero-valued sample insertion followed by digital lowpass filtering. A more thorough discussion of digital interpolation can be found in chapter 10 of reference [4]. Applying the interpolated x[2](n) discrete sequence to our DAC generates the v[DAC,2](t) voltage shown in Figure 9(a). The neat part here is that the spectrum of v[DAC,2](t) is that shown in Figure 9 (b). The analog lowpass filter, whose frequency magnitude response is the bold dashed curve in Figure 9(b), has a much more gradual (wider) transition region from the end of its passband to the beginning of its stopband. This means the v[DAC,2](f) lowpass filter is simpler and much less expensive than the v[DAC,1](f) lowpass filter. There are two additional advantages of our factor of two interpolation. First the undesirable spectral noise centered at 22.05 kHz in Figure 9(b) is smaller in magnitude than the spectral noise centered at 11.025 kHz in Figure 8(b). Second, the digital lowpass filter used in our digital interpolation process can be designed to implement the desirable pre-DAC compensation in Figure 7(b). We could go one step further and interpolate sequence x[2](n) by another factor of two to generate a discrete x[4](n) sequence having a 44.1 kHz sample rate. Applying that x[4](n) sequence to our DAC will generate the v[DAC,4](t) voltage in Figure 9(c) whose spectrum is given in Figure 9(d). The analog lowpass needed for the v[DAC,4](t) signal, the bold dashed curve in Figure 9(d), can be quite simple now, perhaps merely a few resistors and capacitors. Concluding Remarks We started this presentation with a discussion of ideal digital-to-analog conversion. Realizing that we cannot perform such an ideal conversion, we considered a few hypothetical digital-to-analog conversion scenarios to help use understand the behavior of commercially-available digital-to-analog converters (DACs). We see that, due to real-world DAC imperfections, analog lowpass filters are needed for proper digital-to-analog conversion. Finally, we showed how digital signal interpolation can be used to drastically reduce the complexity and cost of the analog lowpass filters. (Remember, if we can reduce the cost of our analog filter by 50 cents, and we sell 6 million units, we've saved three million dollars!) The digital interpolation scheme used to reduce analog circuit complexity is a classic example of how digital signal processing can be used to build lower-cost, more reliable, systems. In addition, such digital interpolation has the benefit of reducing the sin(x)/x droop effects of our DAC, and that improves the quality of the final analog lowpass filter's v[out](t) signal. [1] Luke, H. “The Origins of the Sampling Theorem,” IEEE Communications Magazine, April 1999, pp. 106–109. [2] Shannon, C. “A Mathematical Theory of Communication,” Bell Sys. Tech. Journal, Vol. 27, 1948, pp. 379–423, 623–656. [3] Prandoni, P. and Vetterli, M., "Signal Processing For Communications", EPFL Press, Lausanne, Switzerland, 2008, pp. 240-247. [4] Lyons, R., "Understanding Digital Signal Processing" 3rd Ed., Prentice Hall Publishing, Upper Saddle River, NJ, 2010, pp. 507-588. [ - ] Comment by ●November 21, 2013 Why you did not multiply 5b x 6c ? why you are using Sin(x)/x ? I think you should use sin(t)/t which is lowpass filter in frequency domain i.e. convolution integral of 5(e) x sin(t)/t. To post reply to a comment, click on the 'reply' button attached to each comment. To post a new comment (not a reply to a comment) check out the 'Write a Comment' tab at the top of the comments. Please login (on the right) if you already have an account on this platform. 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What is Uncovered or Naked Options Trading? Naked or Uncovered options trading is a way of speculating in the derivatives space where a trader or investor buys or writes (Sells) an option in order to benefit from the price change of the Option Trading naked options are pure speculating activity as it does not involve any hedge against the position taken by the trader. In an uncovered option trade (Buying or Selling) there is an obligation to take or give delivery of the underlying asset (stock) on the date of expiry. If a seller of a Call option holds on to his position till the date of expiry then he is obligated to give the delivery of the underlying asset to the Option buyer. In this same way, the Buyer of a Call option will have to pay the entire amount of the underlying security and take delivery of the Security. There are 2 aspects of Naked or Uncovered Option Strategies, they are- 1. Option Buying 2. Option Writing(Selling) 1. Option Buying- In Option buying the trader buys an Option by paying a small price for it known as the Premium expecting the price of the underlying asset to rise or fall to that specific Strike Price before the expiry in order to earn money. Option buying can be compared to buying a lottery ticket as the chances of winning big is there but are very small. For example, the buyer of a call Option will start to make money when the price of the Underlying Stock/Index will go above his Strike Price before the expiry. The chances of making money in Option Buying are quite less as Option Greeks such as Delta, Gama (Option Decay) are against Option buyers. The advantage in option buying is that the loss is limited to the Total Premium paid while purchasing the option and the maximum profit is limitless. 2. Option Writing- Opting writing or Option selling is comparatively more profitable than Option Buying as the Option Greeks work in favor of the option writer to reduce the price of the option. Option Selling is also treated as a steady business because the Return On Investment (ROI) is reasonable and if done right can provide steady profits for option writers. In Option Selling the Seller sells an Option and Collects the Premium at the time of execution and buys it at a lower price at a later time to earn Profit. Let's see how Naked Options Trading works- 1. Naked Call Options- In a Naked Call option strategy, the Buyer of the Call option Buys the Option by paying a premium of 'X' amount, and the same 'X' amount is collected by the Writer (seller) of that particular Option. Now if the price of the Underlying security rises then the Premium of the Call Option will rise hence the Buyer of that Option will make money and simultaneously the Seller will face some MTM (Mark To Market) losses till the time he holds on to the position. 2.Naked Put options- A naked Put Option is bought by the Option buyer anticipating that the price of the underlying security will fall and the Seller of the Put option sells that same Put option anticipating that the price of the particular asset will rise. Now if the Price of the underlying security starts to rise then the Price of the Put Option will fall resulting in a loss for the option buyer and an MTM profit for the Option seller. In the following ways, Naked Options are bought and sold in order to earn money from the price movement of the underlying asset. Buying Naked options is extremely risky as the value of the option becomes worthless on the day of expiry if the price is away from the particular Strike Price. On the other hand, Option sellers face the risk of unlimited losses, but the chances of making profits consistently in Option writing is more than Option Buying. Trading in Naked or Uncovered Option is very risky hence it is always a good idea to hedge the Naked positions to minimize the losses and maximize the gains.	{"url":"https://app.fintrakk.com/article/what-is-uncovered-naked-options-trading","timestamp":"2024-11-14T08:39:18Z","content_type":"text/html","content_length":"343265","record_id":"<urn:uuid:e9e1ad2b-0ebb-4afb-af56-b0f1161ff32d>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00034.warc.gz"}
Mixicles: Simple Private Decentralized Finance Ari Juels This work was done in my capacity as technical advisor to Chainlink, and not my Cornell or IC3 affiliation. Mixicle is a combination of the words mixers and oracles. You will see why in a moment. DeFi and privacy As of a couple of weeks ago, DeFi passed the $1 billion mark which is to say there's over $1 billion of cryptocurrency in DeFi smart contracts today. This is great, but wait just a minute. Should we know this? If DeFi transactions were private, we would only have a rough estimate of the amount of money in DeFi smart contracts today. But in fact, every single DeFi transaction is fully visible on-chain and there's no confidentiality. Also, they provide exotic niche instruments. MakerDAO is basically an automated pawn shop for cryptocurrency which can be used for leveraged exposure, and flash loans which can be used to weaponize idle cryptocurrency. Binary options Our goal with mixicles is to implement a common class of financial instruments like simple binary options. A binary option is a bet engaged in by two players, our beloved Alice and Bob who send money into a smart contract and they are betting on an event with two different outcomes. For example, they might bet on whether the value of a particular stock reaches a certain threshold at a certain time or something. If the bet goes one way, like if the threshold is reached, then one player gets the money. Mixicle goals We would like to achieve the type of confidentiality that isn't presently available in DeFi instruments. We're going to work in a particular model. We'll assume Bob and Alice are fine being recognized as trading partners. They might be anonymized sufficiently the pseudonyms through which they are interacting with the DeFi instrument, or they might be financial institutions or something. What we're interested in doing is concealing the type of trade they are engaging in, and the payout amounts and how much money is involved. We want this thing to be auditable. It would be nice if there was an on-chain record available to auditors and regulators and so on. We would like compatibility with existing infrastructure, and we would like to avoid building extra infrastructure. The main objective in Mixicles is simplicity. We want conceptual simplicity. Mixicles are very simple from a conceptual standpoint. Conceptual simplicity often translates into simple implementation. In particular, we avoid heavyweight cryptography, zero-knowledge proofs, watchtowers, etc. Complexity is the enemy of security. Zcash had a bug which in principle someone could have minted an arbitrary amount of money without detection, which is allegedly due to the complexity in zcash's design which we avoid in mixicles. Building a Mixicle We'll take a few steps to build a Mixicle. We'll conceal which of the two players have won the bet by using "payee privacy". Then we have "trigger privacy" where the trigger is the basis for the bet like the particular stock they are betting on. Then there's "payout privacy" where you conceal the amount of money being paid out to one of the players. Finally, I'll show how you can use multiple rounds to completely conceal intermediate rounds of payouts in the cases of where Alice and Bob are engaging in many transactions. A mixer is used to conceal the flow of money in a cryptocurrency system. By directing money to two different pseudonyms corresponding to the two players engaged in the mixer, these pseudonyms are corresponding but the order is randomized. To mix, Alice and Bob can flip a coin. If it comes up heads, then the first pseudonym is allocated to Alice's pseudonym or it comes first when money is output by the mixer and Bob goes second. But if the coin flip goes the other way, then they are flipped. With this simple construction protocol, we get "payee privacy", which is our first goal for a In paricular, suppose that one of these two players sends money to some subversive organization. Anyone looking at this process on-chain will be unable to tell whether it was Alice or Bob who sent the money. You can build a mixer very straightforwardly using a smart contract. Alice and Bob choose some pseudonyms, they flip a coin to figure out the ordering of the pseudonyms. They commit their pseudonyms to the smart contract, then they send money into the smart contract, and the smart contract then forwards the money to the corresponding pseudonyms. Private payee gambling This can be enhanced in other ways to achieve other goals. Suppose we want to do "private payee gambling". The mixer can take as input another coin flip, the result of flipping a public coin or taking input from generated by the blockchain into the smart contract. Then payment can be determined based on the coin flip. If the coin comes up heads, the money goes to the first pseudonym and else to the second pseudoynm. This makes payee privacy and the identity of the winner is hidden on chain, which is very nice. Of course, everyone's favorite place to gamble is the stock market. We can transform this gambling smart contract into a financial instrument one based on a stock ticker for instance, fairly straightforwardly. The smart contract simply takes as an input an indication as to whether the target stock that Alice and Bob are betting on in a binary option, has gone up or down. If it goes up, then the first pseudonym gets the money, or vice versa if it goes down. There's a complication though. Blockchains in general and therefore smart contracts don't have internet connections. This is a function of their underlying consensus protocol. The smart contract can't do what you and I would do if we wanted to know the price of Tesla site: we would just go to some trustworthy website and look it up. The solution to this problem is a piece of middleware known as an oracle. An oracle is an off-chain entity that goes and fetches data from a website and pushes the data on-chain. The smart contract can call the oracle, and the oracle has an on-chain frontend and it will relay information to the smart contract. Trigger privacy We still have a confidentiality problem here, though. If the oracle is relaying information about a stock price moving in a certain direction, then this will be visible on chain, and the trigger will be world-readable visible to everyone and we would like to avoid that. To do this, Alice and Bob can flip another secret coin which they make visible to the oracle. The purpose of this coin is to designate what I would call a "secret trigger code". If the coin comes up heads, then the oracle in the case that Tesla goes down, which means Alice wins, is going to relay a 0-bit and if the opposite occurs it will relay a 1-bit. If the coin flip is the other way around, then the code is inverted. All that the oracle is going to deliver to the smart contract is going to be a single bit, and it's meaning is going to be randomized so it's not clear to someone observing this process what Alice and Bob are betting on. So now we have both payee privacy and trigger privacy. Payout privacy This is great, but we're not quite there yet. We still have a confidentiality problem. How do we achieve what I described earlier as payout privacy? It's easiest to explain payout privacy by example. Rather than having Alice and Bob pay in a dollar and sending all the money to the winner, what if they pay $2 each and send $1 each to 4 different pseudonyms? The key observation here is the following. Suppose that three of the pseudonyms correspond to Alice and 1 to Bob. In this case, Alice wins $1. But if all the pseudoynms belong to Alice, then Alice wins $2. There's no way for someone just seeing dollars sent to 4 different pseudonyms to distinguish whether she is winning $2 or $3 dollars. If three of the pseudonyms correspond to Bob and only 1 to Alice, then Alice loses $1. The upshot of all of this is that you can't tell by observing payments on-chain whether Alice lost $2 or $1 or whether neither of them lost anything or if she gained $2. We can conceal five different possible amounts for each player with this setup. So we are able to get at least partial payout privacy. We can amplify the privacy we achieve, with a simple trick. The trick is to express the payment amount in binary and to allocate payments according to each of the digits of the binary string indicating the total payout. This makes 29 different possibilities for the payouts allocated to the players. The number of possible payouts now can be exponential in the number of actual payouts. The contract can be used and reused, until the players decide for a payout to occur. Players can also pay in money during intermediate rounds and also withdraw money during intermediate rounds, as you can imagine. What we get from this is an efficiency gain, but we also fully conceal intermediate payouts. The only payout visible on chain is the final one when the money is finally dispersed, and there we benefit from payout privacy that I described previously. Further confidentiality with DECO We can push the confidentiality of our Mixicle construction even further. You may have noticed there's still some confidentiality issues. In particular, the oracle must know the trigger on which the mixicle is based. It after all is responsible for implementing the secret trigger code. Ideally Alice and Bob would like to hide the trigger even from the oracle itself. This might seem hard to do, but you have seen the solution in the previous talk by using DECO. DECO allows users to prove things about a TLS session to an oracle. So the setup here is that Alice and Bob commit in the smart contract to the trigger they are using, and then the event occurs, they want to assess the movement of a stock, and then one of the two players queries the website and gets the stock price and proves to the oracle that the stock moved in a particular direction without revealing what the stock ticker was. It just proves, rather, that the player is querying from the target website is identical with the stock that was committed to in the smart contract. All the oracle knows is that Alice and Bob agree on the trigger and that a player is using DECO to report on the movement of stock. Hence, we're able to hide the trigger from the oracle. I also mentioned we would like to achieve auditability in mixicles. The players commit ciphertexts on the secret terms of the contract. All the secret parameters of the contract, to the smart contract itself. This serves two purposes. First, it allows selective disclosure of information about the trade in which Alice and Bob engaged to an auditor or regulator. It also serves the purpose of holding the oracle accountable. If the oracle cheats and provides an incorrect input, well there's a record on-chain of what it was supposed to do and the oracle has agreed to the ciphertext terms here and then it can be called out by Alice and Bob to decrypt and show that the oracle provided an incorrect report. Discreet log contracts The construction in the previous literature most similar to mixicles was a nice idea from Tadge Dryja proposed in 2017 called discreet log contracts. This isn't misspelled, it's discrete in the sense of confidentiality. The idea is simple. The players partially pre-sign transactions corresponding to all the possible event outcomes in the financial instrument that they are parties to. It needs a partial signature. In the DLC protocol, an oracle broadcasts a signature on the actual outcome. The winning player then can take the partial transaction that corresponds to the actual outcome and combine it with the oracle's signature and get a fully valid transaction which can then be processed on-chain. This is quite nice. Discreet logs obtain potentially stronger confidentiality than mixicles. The oracle is broadcasting a signature, and Alice and Bob don't necessarily know they are using this in a financial instrument. The transaction processed on-chain looks like an ordinary transaction with zero evidence it was derived from some financial contract. There's a few catches. Discreet log contracts are directed mainly at bitcoin and the lightning network. They assume Schnorr signatures which haven't been widely deployed, and most seriously they need beacon infrastructure for all possible triggers of interest. They remain a nice point in the design space for decentralized financial instruments. Bells and whistles You can add some other bells and whistles to mixicles. You can have better payout privacy by combining it with a system that conceals transaction amounts like Aztec or Zether. This is at the cost of more expensive gas per transaction and additional complexity and assumptions. We can also create non-binary mixicles. We can have mixicles with multiple event outcomes and we can even conceal the number of possible outcomes or the cardinality of the instrument. We can have multiple players, too. Mixicles are just one point in a large design space for DeFi instruments. They might need some modification to address real-world needs. Mixicles are most important in highlighting that oracles can do more than just delivering data. When an oracle is invoked by a smart contract, what's called upon is generally not a single entity but rather an oracle network or a decentralized oracle network. This means that the creator of the smart contract can make use of some network like Chainlink by selecting a subset of nodes that they particularly trust. A committee can achieve consensus on the value of some piece of data. That's the basic oracle functionality that most people are familiar with. It can do a lot more too. The committee can facilitate bi-directional communication, where a smart contract can control an autonomous vehicle or some cyber-physical system or do privacy-preserving computation like multi-party computation or use trusted execution environments or in principle deliver robust storage. In general, this type of committee and therefore oracle networks, can power smart contracts to do a wide-range of things that they can't do in isolated environments. Sponsorship: These transcripts are sponsored by Blockchain Commons. Disclaimer: These are unpaid transcriptions, performed in real-time and in-person during the actual source presentation. Due to personal time constraints they are usually not reviewed against the source material once published. Errors are possible. If the original author/speaker or anyone else finds errors of substance, please email me at kanzure@gmail.com for corrections or contribute online via github/git. I sometimes add annotations to the transcription text. These will always be denoted by a standard editor's note in parenthesis brackets ((like this)), or in a numbered footnote. I welcome feedback and discussion of these as well. Tweet: Transcript: "Mixicles: Simple Private Decentralized Finance" https://diyhpl.us/wiki/transcripts/stanford-blockchain-conference/2020/mixicles/ @AriJuels @CBRStanford #SBC20	{"url":"https://diyhpl.us/wiki/transcripts/stanford-blockchain-conference/2020/mixicles/","timestamp":"2024-11-07T00:18:49Z","content_type":"application/xhtml+xml","content_length":"18503","record_id":"<urn:uuid:7285c55e-8d40-4615-8e52-45e9d9fdb51f>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00192.warc.gz"}
Calculus of Variations/CHAPTER VIII - Wikibooks, open books for an open world CHAPTER VIII: THE SECOND VARIATION; ITS SIGN DETERMINED BY THAT OF THE FUNCTION ${\displaystyle F_{1}}$. Article 111. The substitution ${\displaystyle x+\epsilon \xi }$, ${\displaystyle y+\epsilon \eta }$ for ${\displaystyle x}$,${\displaystyle y}$ causes any point of the original curve to move along a straight line, which makes an angle with the ${\displaystyle X}$-axis whose tangent is ${\displaystyle {\frac {\eta }{\epsilon }}}$. This deformation of the curve is insufiEcient, if we require that the point move along a curve other than a straight line. To avoid this inadequacy we make the more general substitution (by which the regular curve remains regular): ${\displaystyle x\rightarrow x+\epsilon \xi _{1}+{\frac {\epsilon ^{2}}{2!}}\xi _{2}+\cdots }$, ${\displaystyle \qquad y\rightarrow y+\epsilon \eta _{1}+{\frac {\epsilon ^{2}}{2!}}\eta _{2}+\ cdots }$ where, like ${\displaystyle \xi }$, ${\displaystyle \eta }$ in our previous development (Art. 75), the quantities ${\displaystyle \xi _{1}}$, ${\displaystyle \eta _{1}}$, ${\displaystyle \xi _{2}}$, ${\displaystyle \eta _{2}\ldots }$ are functions of ${\displaystyle t}$, finite, continuous one-valued and capable of being differentiated (as far as necessary) between the limits ${\displaystyle t_ {0}\ldots t_{1}}$. These series are supposed to be convergent for values of ${\displaystyle \epsilon }$ such that ${\displaystyle \|\epsilon \|<1}$. That such substitutions exist may be seen as follows: Since the curve is regular, the coordinates of consecutive points to ${\displaystyle P_{0}}$ and ${\displaystyle P_{1}}$ may be expressed by series in the form, say, ${\displaystyle ({\text{A}})\qquad x_{0}+\epsilon a_{0}^{(1)}+{\frac {\epsilon ^{2}}{2!}}a_{0}^{(2)}+\cdots }$, ${\displaystyle \qquad y_{0}+\epsilon b_{0}^{(1)}+{\frac {\epsilon ^{2}}{2!}}b_{0}^ {(2)}+\cdots }$ ${\displaystyle ({\text{B}})\qquad x_{1}+\epsilon a_{1}^{(1)}+{\frac {\epsilon ^{2}}{2!}}a_{1}^{(2)}+\cdots }$, ${\displaystyle \qquad y_{1}+\epsilon b_{1}^{(1)}+{\frac {\epsilon ^{2}}{2!}}b_{1}^ {(2)}+\cdots }$ where the coefficients of the powers of ${\displaystyle \epsilon }$ are constants and the series are convergent. Suppose, now, that we seek to determine the functions of ${\displaystyle t}$ ${\displaystyle ({\text{C}})\qquad x+\epsilon \xi _{1}+{\frac {\epsilon ^{2}}{2!}}\xi _{2}+\cdots }$, ${\displaystyle \qquad y+\epsilon \eta _{1}+{\frac {\epsilon ^{2}}{2!}}\eta _{2}+\cdots }$ such that for ${\displaystyle t=t_{0}}$ and ${\displaystyle t=t_{1}}$, the expressions (C) will be the same as (A) and (B). This may be done, for example, by writing ${\displaystyle \xi _{1}=t^{2}+\alpha _{1}t+\alpha _{2}}$, ${\displaystyle \eta _{1}=t^{2}+\beta _{1}t+\beta _{2}}$, and then determine ${\displaystyle \alpha _{1}}$, ${\displaystyle \alpha _{2}}$, ${\displaystyle \beta _{1}}$, ${\displaystyle \beta _{2}}$ in such a way that ${\displaystyle t_{0}^{2}+\alpha _{1}t_{0}+\alpha _{2}=a_{0}^{(1)}}$; ${\displaystyle \qquad t_{0}^{2}+\beta _{1}t_{0}+\beta _{2}=b_{0}^{(1)}}$, ${\displaystyle t_{1}^{2}+\alpha _{1}t_{1}+\alpha _{2}=a_{1}^{(1)}}$; ${\displaystyle \qquad t_{1}^{2}+\beta _{1}t_{1}+\beta _{2}=b_{1}^{(1)}}$. From this it is seen that ${\displaystyle \alpha _{1}=-(t_{1}+t_{0})+{\frac {a_{1}^{(1)}-a_{0}^{(1)}}{t_{1}-t_{0}}}}$, etc. In the same way we may determine quadratic expressions in ${\displaystyle t}$ for ${\displaystyle \xi _{2}}$, ${\displaystyle \eta _{2}}$, etc. The substitutions thus obtained are of the nature of those which we have assumed to exist, and may evidently be constructed in an infinite number of different ways. Article 112. Making the above substitutions in the integral ${\displaystyle I=\int _{t_{0}}^{t_{1}}F(x,y,x',y')~{\text{d}}t}$, it is seen that ${\displaystyle \Delta I=\int _{t_{0}}^{t_{1}}\left[F\left(x+\epsilon \xi _{1}+{\frac {\epsilon ^{2}}{2!}}\xi _{2}+\cdots ,y+\epsilon \eta _{1}+{\frac {\epsilon ^{2}}{2!}}\eta _{2}+\cdots ,x'+\ epsilon \xi _{1}'+{\frac {\epsilon ^{2}}{2!}}\xi _{2}'+\cdots ,y'+\epsilon \eta _{1}'+{\frac {\epsilon ^{2}}{2!}}\eta _{2}'+\cdots \right)-F(x,y,x',y')\right]~{\text{d}}t}$ ${\displaystyle =\epsilon \delta I+{\frac {\epsilon ^{2}}{2!}}\delta ^{2}I+{\frac {\epsilon ^{3}}{3!}}\delta ^{3}I+\cdots }$. By Taylor's Theorem we have ${\displaystyle F\left(x+\epsilon \xi _{1}+{\frac {\epsilon ^{2}}{2!}}\xi _{2}+\cdots ,y+\epsilon \eta _{1}+{\frac {\epsilon ^{2}}{2!}}\eta _{2}+\cdots ,x'+\epsilon \xi _{1}'+{\frac {\epsilon ^ {2}}{2!}}\xi _{2}'+\cdots ,y'+\epsilon \eta _{1}'+{\frac {\epsilon ^{2}}{2!}}\eta _{2}'+\cdots \right)-F(x,y,x',y')}$ ${\displaystyle =\left[\left(\epsilon \xi _{1}+{\frac {\epsilon ^{2}}{2!}}\xi _{2}+\cdots \right){\frac {\partial }{\partial x}}+\left(\epsilon \eta _{1}+{\frac {\epsilon ^{2}}{2!}}\eta _{2}+\ cdots \right){\frac {\partial }{\partial y}}+\left(\epsilon \xi _{1}'+{\frac {\epsilon ^{2}}{2!}}\xi _{2}'+\cdots \right){\frac {\partial }{\partial x'}}+\left(\epsilon \eta _{1}'+{\frac {\ epsilon ^{2}}{2!}}\eta _{2}'+\cdots \right){\frac {\partial }{\partial y'}}\right]F}$ ${\displaystyle +{\frac {1}{2!}}\left[\left(\epsilon \xi _{1}+{\frac {\epsilon ^{2}}{2!}}\xi _{2}+\cdots \right){\frac {\partial }{\partial x}}+\left(\epsilon \eta _{1}+{\frac {\epsilon ^{2}} {2!}}\eta _{2}+\cdots \right){\frac {\partial }{\partial y}}+\left(\epsilon \xi _{1}'+{\frac {\epsilon ^{2}}{2!}}\xi _{2}'+\cdots \right){\frac {\partial }{\partial x'}}+\left(\epsilon \eta _{1} '+{\frac {\epsilon ^{2}}{2!}}\eta _{2}'+\cdots \right){\frac {\partial }{\partial y'}}\right]^{2}F}$ ${\displaystyle +{\frac {1}{3!}}\left[\cdots \right]^{3}F+\cdots }$. The coefficient of ${\displaystyle \epsilon }$ in this expression is the integrand of ${\displaystyle \delta I}$ and is zero; while the coefficient of ${\displaystyle {\frac {\epsilon ^{2}}{2!}}}$ involves terms that are the first partial derivatives of ${\displaystyle F}$, and also those that are the second partial derivatives of ${\displaystyle F}$. The first partial derivatives of ${\displaystyle F}$ that belong to this coefficient, when put under the integral sign, may be written in the form ${\displaystyle \int _{t_{0}}^{t_{1}}\left[{\frac {\partial F}{\partial x}}\xi _{2}+{\frac {\partial F}{\partial x'}}\xi _{2}'+{\frac {\partial F}{\partial y}}\eta _{2}+{\frac {\partial F}{\ partial y}}\eta _{2}'\right]~{\text{d}}t=\int _{t_{0}}^{t_{1}}G(y'\xi _{2}-x'\eta _{2})~{\text{d}}t+\left[\right]_{t_{0}}^{t_{1}}}$ (see Art. 79), and this expression is also zero, if we suppose that the end-points remain fixed. Article 113. The coefficient of ${\displaystyle \epsilon ^{2}}$ in the preceding development of ${\displaystyle F}$ by Taylor's Theorem is, neglecting the factor ${\displaystyle {\frac {1}{2!}}}$, denoted by ${\ displaystyle \delta ^{2}F}$. We have then ${\displaystyle 1)\qquad \delta ^{2}F={\frac {\partial ^{2}F}{\partial x^{2}}}\xi _{1}^{2}+2{\frac {\partial ^{2}F}{\partial x\partial y}}\xi _{1}\eta _{1}+{\frac {\partial ^{2}F}{\partial y^ {2}}}\eta _{1}^{2}+{\frac {\partial ^{2}F}{\partial x'^{2}}}\xi _{1}'^{2}+2{\frac {\partial ^{2}F}{\partial x'\partial y'}}\xi _{1}'\eta _{1}'+{\frac {\partial ^{2}F}{\partial y'^{2}}}\eta _{1}'^ {2}+2\left({\frac {\partial ^{2}F}{\partial x\partial x'}}\xi _{1}\xi _{1}'+{\frac {\partial ^{2}F}{\partial y\partial y'}}\eta _{1}\eta _{1}'+{\frac {\partial ^{2}F}{\partial x\partial y'}}\xi _ {1}\eta _{1}'+{\frac {\partial ^{2}F}{\partial y\partial x'}}\eta _{1}\xi _{1}'\right)}$. The subscripts may now be omitted and the formula simplified by the introduction of the function ${\displaystyle F_{1}}$, which (Art. 73) was defined by the relations: ${\displaystyle 2)\qquad {\frac {\partial ^{2}F}{\partial x'^{2}}}=y'^{2}F_{1}}$, ${\displaystyle \qquad {\frac {\partial ^{2}F}{\partial x'\partial y'}}=-x'y'F_{1}}$, ${\displaystyle \qquad {\ frac {\partial ^{2}F}{\partial y'^{2}}}=x'^{2}F_{1}}$; and by introducing the new notation: ${\displaystyle 3)\qquad L={\frac {\partial ^{2}F}{\partial x\partial x'}}-y'y''F_{1}}$, ${\displaystyle \qquad M={\frac {\partial ^{2}F}{\partial x\partial y'}}+x'y''F_{1}={\frac {\partial ^{2} F}{\partial x'\partial y}}+y'x''F_{1}}$ (owing to the equation ${\displaystyle G=0}$), ${\displaystyle \qquad N={\frac {\partial ^{2}F}{\partial y\partial y'}}-x'x''F_{1}}$; where ${\displaystyle x''}$, ${\displaystyle y''}$ are used for ${\displaystyle {\frac {{\text{d}}^{2}x}{{\text{d}}t^{2}}}}$,${\displaystyle {\frac {{\text{d}}^{2}y}{{\text{d}}t^{2}}}}$. We have then ${\displaystyle \delta ^{2}F={\frac {\partial ^{2}F}{\partial x^{2}}}\xi ^{2}+2{\frac {\partial ^{2}F}{\partial x\partial y}}\xi \eta +{\frac {\partial ^{2}F}{\partial y^{2}}}\eta ^{2}+F_{1}(y'^ {2}\xi '^{2}-2x'y'\xi '\eta '+x'^{2}\eta '^{2})+2F_{1}(y'y''\xi \xi '+x'x''\eta \eta '-x'y''\xi \eta '-y'x''\eta \xi ')+2(L\xi \xi '+M(\xi \eta '+\eta \xi ')+N\eta \eta ')}$. To get an exact differential as a part of the right-hand member of this formula, we write ${\displaystyle 4)\qquad R=L\xi ^{2}+2M\xi \eta +N\eta ^{2}}$, an expression which, differentiated with respect to ${\displaystyle t}$, becomes ${\displaystyle 2[L\xi \xi '+M(\xi \eta '+\eta \xi ')+N\eta \eta ']={\frac {{\text{d}}R}{{\text{d}}t}}-{\frac {{\text{d}}L}{{\text{d}}t}}\xi ^{2}-2{\frac {{\text{d}}M}{{\text{d}}t}}\xi \eta -{\ frac {{\text{d}}N}{{\text{d}}t}}\eta ^{2}}$. We further write ${\displaystyle 5)\qquad w=y'\xi -\eta x'}$, where (see Art. 81) ${\displaystyle w}$ is, neglecting the factor ${\displaystyle {\frac {1}{\sqrt {x'^{2}+y'^{2}}}}}$, the amount of the sliding of a point of the curve in the direction of the Differentiating with respect to ${\displaystyle t}$, we have ${\displaystyle {\frac {{\text{d}}w}{{\text{d}}t}}=y''\xi -x''\eta +y'\xi '-x'\eta '}$, from which it follows that ${\displaystyle c=(y''\xi -x''\eta )^{2}+(y'\xi '-x'\eta ')^{2}+2(y'y''\xi \xi '+x'x''\eta \eta '-x'y''\xi \eta '-y'x''\eta \xi ')}$. Then the expression for the second variation becomes ${\displaystyle \delta ^{2}F={\frac {\partial ^{2}F}{\partial x^{2}}}\xi ^{2}+2{\frac {\partial ^{2}F}{\partial x\partial y}}\xi \eta +{\frac {\partial ^{2}F}{\partial y^{2}}}\eta ^{2}+F_{1}\left [\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}-(y''\xi -x''\eta )^{2}\right]+{\frac {{\text{d}}R}{{\text{d}}t}}-\left({\frac {{\text{d}}L}{{\text{d}}t}}\xi ^{2}+2{\frac {{\text{d}}M}{{\text {d}}t}}\xi \eta +{\frac {{\text{d}}N}{{\text{d}}t}}\eta ^{2}\right)}$. If further we write in this expression ${\displaystyle 6)\qquad L_{1}={\frac {\partial ^{2}F}{\partial x^{2}}}-F_{1}y''^{2}-{\frac {{\text{d}}L}{{\text{d}}t}}}$, ${\displaystyle \qquad M_{1}={\frac {\partial ^{2}F}{\partial x\partial y}}+F_{1}x''y''-{\frac {{\text{d}}M}{{\text{d}}t}}}$, ${\displaystyle \qquad N_{1}={\frac {\partial ^{2}F}{\partial y^{2}}}-F_{1}x''^{2}-{\frac {{\text{d}}N}{{\text{d}}t}}}$, we have finally ${\displaystyle \delta ^{2}F=F_{1}\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}+L_{1}\xi ^{2}+2M_{1}\xi \eta +N_{1}\eta ^{2}+{\frac {{\text{d}}R}{{\text{d}}t}}}$. Article 114. It follows from 3) that ${\displaystyle Lx'+My'=x'{\frac {\partial ^{2}F}{\partial x\partial x'}}+y'{\frac {\partial ^{2}F}{\partial x\partial y'}}}$. Owing to the homogeneity of the function ${\displaystyle F}$ (Chap. IV), it is seen from Euler's Theorem that ${\displaystyle F=x'{\frac {\partial F}{\partial x'}}+y'{\frac {\partial F}{\partial y'}}}$, and consequently, ${\displaystyle {\frac {\partial F}{\partial x}}=x'{\frac {\partial ^{2}F}{\partial x\partial x'}}+y'{\frac {\partial ^{2}F}{\partial x\partial y'}}}$; and therefore ${\displaystyle {\frac {\partial F}{\partial x}}=Lx'+My'}$. In a similar manner we have ${\displaystyle {\frac {\partial F}{\partial y}}=Mx'+Ny'}$. Differentiating with regard to ${\displaystyle t}$, the above expression becomes ${\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left({\frac {\partial F}{\partial y}}\right)={\frac {\partial ^{2}F}{\partial x^{2}}}x'+{\frac {\partial ^{2}F}{\partial x\partial y}}y'+{\frac {\ partial ^{2}F}{\partial x\partial x'}}x''+{\frac {\partial ^{2}F}{\partial x\partial y'}}y''={\frac {{\text{d}}L}{{\text{d}}t}}x'+{\frac {{\text{d}}M}{{\text{d}}t}}y'+Lx''+My''}$, which, owing to 3) is ${\displaystyle x'\left({\frac {\partial ^{2}F}{\partial x^{2}}}-F_{1}y''^{2}-{\frac {{\text{d}}L}{{\text{d}}t}}\right)+y'\left({\frac {\partial ^{2}F}{\partial x\partial y}}+F_{1}y''x''-{\frac or from 6) ${\displaystyle x'L_{1}+y'M_{2}=0}$. In an analagous manner it may be shown that ${\displaystyle x'M_{1}+y'N_{1}=0}$. From these expressions we have at once ${\displaystyle {\frac {L_{1}}{y'^{2}}}=-{\frac {M_{1}}{x'y'}}={\frac {N_{1}}{x'^{2}}}=F_{2}}$, where ${\displaystyle F_{2}}$ is the factor of proportionality. It follows that ${\displaystyle 7)L_{1}=y'^{2}F_{2},\qquad M_{1}=-x'y'F_{2},\qquad N_{1}=x'^{2}F_{2}}$. The quantity ${\displaystyle F_{2}}$ is defined through these three equations and plays an essential role in the treatment of the second variation. Owing to the relation 7) ${\displaystyle L_{1}\xi ^{2}+2M_{1}\xi \eta +N_{!}\eta ^{2}}$ becomes ${\displaystyle F_{2}w^{2}}$, and consequently, ${\displaystyle \delta ^{2}F=F_{1}\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)+F_{2}w^{2}+{\frac {{\text{d}}R}{{\text{d}}t}}}$. Article 115. The second variation of the integral has therefore the form ${\displaystyle 8)\qquad \delta ^{2}I=\int _{t_{0}}^{t_{1}}\left(F_{1}\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)+F_{2}w^{2}\right)~{\text{d}}t+\int _{t_{0}}^{t_{1}}{\frac {{\text{d}}R}{{\ We suppose that the end-points are fixed so that at these points ${\displaystyle \xi =0=\eta }$, and we further assume that the curve subjected to variation consists of a single regular trace, along which then ${\displaystyle R=L\xi ^{2}+2M\xi \eta +N\eta ^{2}}$ is everywhere continuous, so that ${\displaystyle {\Big [}R{\Big ]}_{t_{0}}^{t_{1}}=0}$. Consequently the above integral may be written ${\displaystyle 8^{*})\qquad \int _{t_{0}}^{t_{1}}\left(F_{1}\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)+F_{2}w^{2}\right)~{\text{d}}t}$. If the integral ${\displaystyle I=\int _{t_{0}}^{t_{1}}F(x,y,x',y')~{\text{d}}t}$ is to be a maximum or a minimum for the curve ${\displaystyle G=0}$, it is necessary, when the curve is subjected to an indefinitely small variation, that the variation ${\displaystyle \Delta I}$, which is caused to exist therefrom, have always the same sign, in whatever manner ${\displaystyle \xi }$,${\ displaystyle eta}$ are chosen; and consequently the second variation ${\displaystyle \delta ^{2}I}$ must have continuously the same sign as ${\displaystyle \Delta I}$. We have repeatedly seen that ${\displaystyle \Delta I={\frac {\epsilon ^{2}}{2!}}\delta ^{2}I+{\frac {\epsilon ^{3}}{3!}}\delta ^{3}I+\cdots }$, and for any other value of ${\displaystyle \epsilon _{1}}$ for example, ${\displaystyle \epsilon _{1}}$, ${\displaystyle \Delta _{1}I={\frac {\epsilon _{1}^{2}}{2!}}\delta ^{2}I+{\frac {\epsilon _{1}^{3}}{3!}}\delta ^{3}I+\cdots }$. If, further, ${\displaystyle \delta ^{2}I}$ is negative while ${\displaystyle \Delta I}$ is positive, then we may take ${\displaystyle \epsilon _{1}}$ so small that the sign of ${\displaystyle \Delta _{1}I}$ depends only upon the first term on the right in the above expansion, and consequently is negative. Therefore the integral ${\displaystyle I}$ cannot be a maximum or a minimum, since the variation of it is first positive and then negative. Hence, neglecting for a moment the case when ${\displaystyle \delta ^{2}I=0}$, we have the following theorem: If the integral ${\displaystyle I}$ is to be a maximum or a minimum, its second variation must be continuously negative or continuously positive. When ${\displaystyle \delta ^{2}I}$ vanishes for all possible values of ${\displaystyle \xi }$, ${\displaystyle \eta }$, it is necessary also that ${\displaystyle \delta ^{2}I}$ vanish, since the integral ${\displaystyle I}$ is to be a maximum or a minimum, and, as in the Theory of Maxima and Minima, we would then have to investigate the fourth variation. In this case the conditions that have to be satisfied are so numerous that a mathematical treatment is very complicated and difficult. Hence, it is seen that after the condition ${\displaystyle \delta I=0}$ is satisfied, it follows that for the possibility of a maximum, ${\displaystyle \delta ^{2}I}$ must be negative, and for the possibility of a minimum, ${\displaystyle \delta ^{2}I}$ must be positive. These conditions are necessary, but not sufficient. Article 116. In Art. 75 we assumed that ${\displaystyle \xi }$,${\displaystyle \eta }$,${\displaystyle \xi '}$,${\displaystyle \eta '}$ were continuous functions of ${\displaystyle t}$ between the limits ${\ displaystyle t_{0}\ldots t_{1}}$. Owing to the assumed existence of ${\displaystyle \xi '}$,${\displaystyle \eta '}$, we must presuppose the existence of the second derivatives of ${\displaystyle x}$ and ${\displaystyle y}$ with respect to ${\displaystyle t}$ (see Art. 23). From this it also follows that the radius of curvature must vary in a continuous manner. These assumptions have been tacitly made in the derivation of the equation 8) in the preceding article. We shall now free ourselves from the restriction that ${\displaystyle \xi '}$ and ${\displaystyle \eta '}$ are continuous functions of ${\displaystyle t}$, retaining, however, the assumptions regarding the continuity of the quantities ${\displaystyle x,y,\xi ,\eta ,x',y',x'',y''}$. The theorem that ${\displaystyle {\frac {\partial F}{\partial x'}}}$ and ${\displaystyle {\frac {\partial F}{\partial y'}}}$ vary in a continuous manner for ox oy the whole curve (Art. 97) in most cases gives a handy means of determining the admissibility of assumptions regarding the continuity of ${\displaystyle x'}$ and ${\displaystyle y'}$. If, at certain points of the curve ${\displaystyle G=0}$, ${\displaystyle x'}$ and ${\displaystyle y'}$ are not continuous, it is always possible to divide the curve into such portions that ${\displaystyle x'}$ and ${\displaystyle y'}$ are continuous throughout each portion. Yet we cannot even then say that ${\displaystyle x''}$ and ${\displaystyle y''}$ are continuous within such a portion, as has been assumed to be true in the above development. If, however, ${\displaystyle x''}$ and ${\displaystyle y''}$ within such a portion of curve are discontinuous, we have only to divide the curve into other portions so that within these new portions ${\displaystyle x''}$ and ${\displaystyle y''}$ no longer suffer any sudden springs. In each of these portions of curve the same conclusions may be made as before in the case of the whole curve, and consequently the assumption regarding the continuous change of ${\displaystyle x''}$,${\displaystyle y''}$ throughout the whole curve is not necessary. But if we had limited ourselves to the consideration of a part of the curve in which ${\displaystyle x,y,x',y',x'',y''}$ vary in a continuous manner, the continuity of ${\displaystyle \xi '}$, ${\displaystyle \eta '}$ in the integration of the integral ${\displaystyle \int {\frac {{\text{d}}R}{{\text{d}}t}}~{\text{d}}t}$ would have been assumed. These assumptions need not necessarily be fulfilled, since the variation of the curve is an arbitrary one, and it is quite possible that such variations may be introduced, where ${\displaystyle \xi '}$, ${\displaystyle \eta '}$ become discontinuous, as often as we please. We may, however, drop these assumptions without changing the final results, if only the first named conditions are satisfied. Since the quantities ${\displaystyle L}$, ${\displaystyle M}$, ${\displaystyle N}$ depend only upon ${\displaystyle x,y,x',y',x'',y''}$, and since these quantities are continuous, it follows that the introduction of the integral ${\displaystyle \int {\frac {{\text{d}}R}{{\text{d}}t}}~{\text{d}}t}$ the form given above is always admissible. For if ${\displaystyle \ xi '}$, ${\displaystyle \eta '}$ were not continuous for the whole trace of the curve, which has been subjected to variation, we could suppose that this curve has been divided into parts, within which the above derivatives varied in a continuous manner, and the integral would then become a sum of integrals of the form ${\displaystyle \int _{t_{\beta }}^{t_{\beta +1}}{\frac {{\text{d}}R}{{\text{d}}t}}~{\text{d}}t=\left[L\xi ^{2}+2M\xi \eta +N\eta ^{2}\right]_{t_{\beta }}^{t_{\beta +1}}}$, where ${\displaystyle t_{\beta },t_{\beta +1},\ldots }$ are the coordinates of the points of division of corresponding values of ${\displaystyle t}$. But since ${\displaystyle \xi }$, ${\displaystyle \eta }$ vary in a continuous manner, we have through the summation of these quantities exactly the same expression ${\displaystyle \left[L\xi ^{2}+2M\xi \eta +N\eta ^{2}\right]_{t_{0}}^{t_{1}}}$ as before. The quantities ${\displaystyle \xi '}$, ${\displaystyle \eta '}$ are also found under the sign of integration in the right-hand side of 8); but owing to the conception of a definite integral, we may still write it in this form even when these quantities vary in a discontinuous manner ; however, in performing the integration, we must divide the integral corresponding to the positions at which the discontinuities enter into partial integrals. Therefore, we see that the possible discontinuity of ${\displaystyle \xi '}$, ${\displaystyle \eta '}$ remains without influence upon the result, if only ${\displaystyle x,y,x',y',x'',y'',\xi ,\eta }$ are continuous. Consequently any assumptions regarding the continuity of ${\displaystyle \xi '}$, ${\displaystyle \eta '}$ are superfluous; however, in an arbitrarily small portion of the curve which is subjected to variation, the quantities ${\displaystyle \xi '}$ and ${\displaystyle \eta '}$ must not become discontinuous an infinite number of times since such variation of the curve has been necessarily, once for all excluded. Article 117. Following the older mathematicians, Legendre, Jacobi, etc., we may give the second variation a form in which all terms appearing under the sign of integration will have the same sign (plus or minus). To accomplish this, we add an exact differential ${\displaystyle {\frac {\text{d}}{{\text{d}}t}}(w^{2}v)}$ at under the integral sign in 8), and subtract it from ${\displaystyle R}$, the integral thus becoming ${\displaystyle \delta ^{2}I=\int _{t_{0}}^{t_{1}}\left(F_{1}\left({\frac {{\text{d}}w}{{\text{d}}t}}\right)^{2}+2vw{\frac {{\text{d}}w}{{\text{d}}t}}+\left(F_{2}+{\frac {{\text{d}}v}{{\text{d}} t}}\right)w^{2}\right)~{\text{d}}t+{\Big [}R-vw^{2}{\Big ]}_{t_{0}}^{t_{!}}}$. The expression under the sign of integration is an integral homogeneous quadratic form in ${\displaystyle w}$ and ${\displaystyle {\frac {{\text{d}}w}{{\text{d}}t}}}$. We choose the quantity ${\ displaystyle v}$ so that this expression becomes a perfect square; that is, ${\displaystyle 9)\qquad v^{2}-F_{1}\left(F_{2}+{\frac {{\text{d}}v}{{\text{d}}t}}\right)=0}$, ${\displaystyle }$ and consequently, ${\displaystyle 10)\qquad \delta ^{2}I=\int _{t_{0}}^{t_{1}}F_{1}\left({\frac {{\text{d}}w}{{\text{d}}t}}+w{\frac {v}{F_{1}}}\right)^{2}~{\text{d}}t+{\Big [}R-vw^{2}{\Big ]}_{t_{0}}^{t_{1}}}$. We shall see that it is possible to determine a function ${\displaystyle v}$, which is finite one-valued and continuous within the interval ${\displaystyle t_{0}\ldots t_{1}}$, and which satisfies the equation 9). The integral 10) becomes accordingly, if the end-points remain fixed, ${\displaystyle 10^{a}\qquad \delta ^{2}I=\int _{t_{0}}^{t_{1}}F_{1}\left({\frac {{\text{d}}w}{{\text{d}}t}}+w{\frac {v}{F_{1}}}\right)^{2}~{\text{d}}t}$ Hence the second variation has the same sign as ${\displaystyle F_{1}}$, and it is clear that for the existence of a maximum ${\displaystyle F_{1}}$ must be negative, and for a minimum this function must be positive within the interval ${\displaystyle t_{0}\ldots t_{1}}$ and in case there is a maximum or a minimum, ${\displaystyle F_{1}}$ cannot change sign within this interval. This condition is due to Jacobi. Legendre had previously concluded that we have a maximum when a certain expression corresponding to ${\displaystyle F_{1}}$ was negative, and a minimum when it was positive. It is questionable whether the diflferential equation for ${\displaystyle v}$ is always integrable. Following Jacobi we shall show that such is the case. Article 118. Before we go farther, we have yet to prove that the transformation, which we have introduced, is allowable. In spite of the simplicity of the equation 9) we cannot make conclusions regarding the continuity of the function v, which is necessary for the above transformation. ¢ It is therefore essential to show that the equation 9) may be reduced to a system of two linear differential equations, which may be reverted into a linear differential equation of the second order, since for this equation we have definite criteria of determining whether a function which satisfies it remains finite and continuous or not. ${\displaystyle v={\frac {u_{1}}{u}}}$, where ${\displaystyle u_{1}}$ and ${\displaystyle u}$ are continuous functions of ${\displaystyle t}$, and ${\displaystyle ueq 0}$ within the interval ${\displaystyle t_{0}\ldots t_{1}}$. Equation 9) becomes then ${\displaystyle <{\frac {u_{1}^{2}}{u^{2}}}-F_{1}\left(F_{2}+{\frac {u{\frac {{\text{d}}u_{1}}{{\text{d}}t}}-u_{1}{\frac {{\text{d}}u}{{\text{d}}t}}}{u^{2}}}\right)=0/math>,or:<math>F_{1}u\left ({\frac {{\text{d}}u_{1}}{{\text{d}}t}}+F_{2}u\right)-u_{1}\left(F_{1}{\frac {{\text{d}}u}{{\text{d}}t}}+u_{1}\right)=0}$. Since one of the functions ${\displaystyle u}$, ${\displaystyle u_{1}}$ may be arbitrarily chosen, we take ${\displaystyle u}$ so that ${\displaystyle 11)\qquad F_{1}{\frac {{\text{d}}u}{{\text{d}}t}}+u_{1}=0}$; then, since ${\displaystyle ueq 0}$, we have ${\displaystyle 12)\qquad {\frac {{\text{d}}u_{1}}{{\text{d}}t}}+F_{2}u=0}$. From 11) and 12) it follows that ${\displaystyle 12^{a})\qquad {\frac {\text{d}}{{\text{d}}t}}\left(F_{1}{\frac {{\text{d}}u}{{\text{d}}t}}\right)-F_{2}u=0}$, ${\displaystyle 13)\qquad F_{1}{\frac {{\text{d}}^{2}u}{{\text{d}}t^{2}}}+{\frac {{\text{d}}F_{1}}{{\text{d}}t}}{\frac {{\text{d}}u}{{\text{d}}t}}-F_{2}u=0}$, where ${\displaystyle F_{1}}$ and ${\displaystyle F_{2}}$ are to be considered as given functions of ${\displaystyle t}$. We shall denote this difFerential equation by ${\displaystyle J=0}$. After $ {\displaystyle u}$ has been determined from this Equation, ${\displaystyle u_{1}}$ may be determined from 11), and from ${\displaystyle {\frac {u_{1}}{u}}=v}$ we have ${\displaystyle v}$ as a definite function of ${\displaystyle t}$. Article 119. The expression which has been derived for ${\displaystyle v}$ seems to contain two arbitrary constants, while the equation 9) has only one. The two constants in the first case, however, may be replaced by one, since the general solution of 13) is ${\displaystyle u=c_{1}\phi _{1}(t)+c_{2}\phi _{2}(t)}$, and hence from 11) ${\displaystyle v={\frac {u_{1}}{u}}=-F_{1}{\frac {c_{1}\phi _{1}'(t)+c_{2}\phi _{2}'(t)}{c_{1}\phi _{1}(t)+c_{2}\phi _{2}(t)}}}$, an expression which depends only upon the ratio of the two constants. It follows from the above transformation that ${\displaystyle 14)\qquad \delta ^{2}I=\int _{t_{0}}^{t_{1}}F_{1}\left({\frac {{\text{d}}w}{{\text{d}}t}}-{\frac {w}{u}}{\frac {{\text{d}}u}{{\text{d}}t}}\right)~{\text{d}}t}$; but this transformation has a meaning only when it is possible to find a function ${\displaystyle u}$ within the interval ${\displaystyle t_{0}\ldots t_{1}}$ which is different from zero, and which satisfies the differential equation ${\displaystyle J=0}$. Article 120. If we have a linear differential equation of the second order ${\displaystyle {\frac {{\text{d}}^{2}y}{{\text{d}}x^{2}}}=P(x){\frac {{\text{d}}y}{{\text{d}}x}}+Q(x)y=0}$, and if ${\displaystyle y_{1}}$ and ${\displaystyle y_{2}}$ are a fundamental system of integrals of this equation, then we have the well known relation due to Abel (see Forsyth's Differential Equations, p. 99) ${\displaystyle y_{1}{\frac {{\text{d}}y_{2}}{{\text{d}}x}}-y_{2}{\frac {{\text{d}}y_{1}}{{\text{d}}x}}=Ce^{-\int P(x){\text{d}}x}}$, ${\displaystyle \Delta ={\begin{vmatrix}y_{1}&y_{2}\\{\frac {{\text{d}}y_{1}}{{\text{d}}x}}&{\frac {{\text{d}}y_{2}}{{\text{d}}x}}\end{vmatrix}}=Ce^{-\int P(x){\text{d}}x}}$. If ${\displaystyle \Delta =0}$, then we would have ${\displaystyle y_{1}=cy_{2}}$, and the system is no longer a fundamental system of integrals. This determinant can become zero only at such positions for which ${\displaystyle P(x)}$ becomes infinitely large; or a change of sign for this determinant can enter only at such positions where ${\displaystyle P(x)}$ becomes infinite. In the differential equationy ${\displaystyle J=0}$ we have ${\displaystyle P={\frac {\text{d}}{{\text{d}}t}}\ln(F_{1})}$, and if ${\displaystyle u_{1}}$, ${\displaystyle u_{2}}$ form a fundamental system of integrals of this differential equation, then ${\displaystyle \Delta =u_{1}{\frac {{\text{d}}u_{2}}{{\text{d}}t}}-u_{2}{\frac {{\text{d}}u_{1}}{{\text{d}}t}}={\frac {C}{F_{1}}}}$. It follows that ${\displaystyle F_{1}}$ cannot become infinite or zero within the interval under consideration or upon the boundaries of this interval. Hence, it is again seen that ${\displaystyle F_ {1}}$ cannot change sign within the interval ${\displaystyle t_{0}\ldots t_{1}}$. If ${\displaystyle F_{1}}$ and ${\displaystyle F_{2}}$ are continuous within the interval ${\displaystyle t_{0}\ldots t_{1}}$, we have, through differentiating the equation ${\displaystyle J=0}$, all higher derivatives of ${\displaystyle u}$ expressed in terms of ${\displaystyle u}$ and ${\displaystyle {\frac {{\text{d}}u}{{\text{d}}t}}}$. Hence, if values of ${\displaystyle u}$ and ${\ displaystyle {\frac {{\text{d}}u}{{\text{d}}t}}}$ are given for a definite value of ${\displaystyle t}$, say ${\displaystyle t'}$, we have a power-series ${\displaystyle P(t-t')}$ for ${\displaystyle u}$ (see Art. 79), which satisfies the equation ${\displaystyle J=0}$. Article 121. Suppose that ${\displaystyle F_{1}}$ has a definite, positive or negative value for a definite value ${\displaystyle t'}$ of ${\displaystyle t}$ situated within the interval ${\displaystyle t_{0}\ ldots t_{1}}$, then on account of its continuity it will also be positive or negative for a certain neighborhood of ${\displaystyle t'}$, say ${\displaystyle t'-\tau _{1}\ldots t'+\tau _{2}}$. We may vary the curve in such a manner that within the interval ${\displaystyle t'-\tau _{1}\ldots t'+\tau _{2}}$ it takes any form while without this region it remains unchanged. Consequently the total variation, and therefore also the second variation of ${\displaystyle I}$, depends only upon the variation within the region just mentioned, and in accordance with the remarks made above since we may find a function ${\displaystyle u}$ of the variable ${\displaystyle t}$, which is continuous within the given region, which satisfies the differential equation ${\displaystyle J=0}$, and which is of such a nature that ${\displaystyle u}$ and ${\displaystyle {\frac {{\text{d}}u}{{\text{d}}t}}}$ have given at values for ${\displaystyle t=t'}$, it follows that the transformation which was introduced is admissible, and we have ${\displaystyle \delta ^{2}I=\int _{t_{0}}^{t_{1}}F_{1}\left({\frac {{\text{d}}w}{{\text{d}}t}}-{\frac {{\text{d}}u}{{\text{d}}t}}{\frac {w}{u}}\right)^{2}~{\text{d}}t}$. This quantity is evidently positive when ${\displaystyle F_{1}}$ is positive and negative when ${\displaystyle F_{1}}$ is negative, so long as ${\displaystyle {\frac {{\text{d}}w}{{\text{d}}t}}-{\frac {{\text{d}}u}{{\text{d}}t}}{\frac {w}{u}}eq 0\qquad }$ (Art. 132). We have then for the total variation ${\displaystyle \Delta I={\frac {\epsilon ^{2}}{2!}}\int _{t_{0}}^{t_{1}}F_{1}\left({\frac {{\text{d}}w}{{\text{d}}t}}-{\frac {{\text{d}}u}{{\text{d}}t}}{\frac {w}{u}}\right)^{2}~{\text{d}}t+{\ frac {\epsilon ^{3}}{3!}}\int _{t_{0}}^{t_{1}}(\xi ,\eta ,\xi ',\eta ')~{\text{d}}t}$, where ${\displaystyle (\xi ,\eta ,\xi ',\eta ')}$ denotes an expression of the third dimension in the quantities included within the brackets. For small values of ${\displaystyle \epsilon }$ it is seen that ${\displaystyle \Delta I}$ has the same sign as the first term on the right-hand side of the above equation. We have, therefore, the following theorem : The total variation ${\displaystyle \Delta I}$ the integral ${\displaystyle I}$ is positive when ${\displaystyle F_{1}}$ is positive, and negative when ${\displaystyle F_{1}}$ is negative throughout the whole interval ${\displaystyle t_{0}\ldots t_{1}}$. If ${\displaystyle F_{1}}$ could change sign for any position within the interval ${\displaystyle t_{0}\ldots t_{1}}$, then there would be variations of the curve for which ${\displaystyle \Delta I}$ is positive and others for which ${\displaystyle \Delta I}$ is negative. Hence, for the existence of a maximum or a minimum of ${\displaystyle I}$ we have the following necessary condition : In order that there exist a maximum or a minimum of the integral ${\displaystyle I}$ taken over the curve ${\displaystyle G=0}$ within the interval ${\displaystyle t_{0}\ldots t_{1}}$, it is necessary that ${\displaystyle F_{1}}$ have always the same sign within this interval; in the case of a maximum ${\displaystyle F_{1}}$ must be continuously negative, and in the case of a minimum this function must be continuously positive. In this connection it is interesting to note a paper by Prof. W. F. Osgood in the Transactions of the American Mathematical Society, Vol. II, p. 273, entitled: "On a fundamental property of a minimum in the Calculus of Variations and the proof of a theorem of Weierstrass's." This paper, which is of great importance, may be much simplified.	{"url":"https://en.m.wikibooks.org/wiki/Calculus_of_Variations/CHAPTER_VIII","timestamp":"2024-11-10T07:58:07Z","content_type":"text/html","content_length":"510205","record_id":"<urn:uuid:60f42fb9-2309-4352-877d-c244050b7d1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00371.warc.gz"}
CCC '24 J3 - Bronze Count Canadian Computing Competition: 2024 Stage 1, Junior #3 After completing a competition, you are struck with curiosity. How many participants were awarded bronze level? Gold level is awarded to all participants who achieve the highest score. Silver level is awarded to all participants who achieve the second highest score. Bronze level is awarded to all participants who achieve the third highest score. Given a list of all the scores, your job is to determine the score required for bronze level and how many participants achieved this score. Input Specification The first line of input contains a positive integer, , representing the number of participants. Each of the next lines of input contain a single integer representing a participant's score. Each score is between and (inclusive) and there will be at least three distinct scores. The following table shows how the available 15 marks are distributed: Marks Description Bound 6 The scores are distinct and the number of participants is small. 7 The scores might not be distinct and the number of participants is small. 2 The scores might not be distinct and the number of participants could be large. Output Specification Output a non-negative integer, , and a positive integer, , separated by a single space, where is the score required for bronze level and is how many participants achieved this score. Sample Input 1 Output for Sample Input 1 Explanation of Output for Sample Input 1 The score required for bronze level is and one participant achieved this score. Sample Input 2 Output for Sample Input 2 The score required for bronze level is and two participants achieved this score. • Can someone tell me why my code doesn't work? I tested on Jupyter and it worked fine. □ commented on Aug. 18, 2024, 4:04 a.m. ← edited The code is correct. Just not efficient enough. 1. you are probably performing sort on an unnecessary large number of items 2. Using max() on an already sorted list is wasteful 3. Removing items from a list other than the end is considered inefficient. 4. Using count is probably not the best way for a sorted list.	{"url":"https://dmoj.ca/problem/ccc24j3","timestamp":"2024-11-11T23:37:51Z","content_type":"text/html","content_length":"28949","record_id":"<urn:uuid:09450a3b-0468-4d5b-87d4-39f31f1e271d>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00735.warc.gz"}
Math Speed Racing Rounding 10 - Rouje online Games Math Speed Racing Rounding 10 Categories and tags of the game : Math Speed Racing Rounding 10 - How to Play Swipe left or right in the gray area to move your car left and right. To get fuel run over the gas can containing the number that your racecar’s number rounds to (nearest 10). Run into coins for extra points and avoid crashing into other cars.	{"url":"https://concourstunisie.net/math-speed-racing-rounding-10/","timestamp":"2024-11-07T12:55:30Z","content_type":"text/html","content_length":"49189","record_id":"<urn:uuid:828ae717-bca8-4a11-b738-607060ade09c>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00300.warc.gz"}
We describe a combinatorial algorithm for constructing all orientable 3-manifolds with a given standard bidimensional spine by making use of the idea of bijoin (Bandieri and Gagliardi (1982), Graselli (1985)) over a suitable pseudosimplicial triangulation of the spine. We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles whose components are labeled by irreducible representations of ${U}_{q}\left(sl\left(2\right)\right)$. We show that the corresponding colored invariants of tangles can be assembled into invariants of bigger tangles. For the case of knots and links, the corresponding theory is a categorification of the colored Jones polynomial, and provides a tool for efficient computations of the resulting colored invariant of knots and links. Our theory is... In this note, we prove the existence of a tri-graded Khovanov-type bicomplex (Theorem 1.2). The graded Euler characteristic of the total complex associated with this bicomplex is the colored Jones polynomial of a link. The first grading of the bicomplex is a homological one derived from cabling of the link (i.e., replacing a strand of the link by several parallel strands); the second grading is related to the homological grading of ordinary Khovanov homology; finally, the third grading is preserved... We express the signature of an alternating link in terms of some combinatorial characteristics of its diagram. We investigate the Khovanov-Rozansky invariant of a certain tangle and its compositions. Surprisingly the complexes we encounter reduce to ones that are very simple. Furthermore, we discuss a "local" algorithm for computing Khovanov-Rozansky homology and compare our results with those for the "foam" version of sl₃-homology. We formulate a conjectural formula for Khovanov's invariants of alternating knots in terms of the Jones polynomial and the signature of the knot. A category $N$ generalizing Jaeger-Nomura algebra associated to a spin model is given. It is used to prove some equivalence among the four conditions by Jaeger-Nomura for spin models of index 2. The present paper is a continuation of our previous paper [Topology 44 (2005), 747-767], where we extended the Burau representation to oriented tangles. We now study further properties of this	{"url":"https://eudml.org/subject/MSC/57M25","timestamp":"2024-11-13T12:16:39Z","content_type":"application/xhtml+xml","content_length":"49783","record_id":"<urn:uuid:963d7edf-bab0-40fc-ab04-fec77e153f07>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00235.warc.gz"}
antisymmetric relation and reflexive for example the relation R on the integers defined by aRb if a < b is anti-symmetric, but not reflexive. A Hasse diagram is a drawing of a partial order that has no self-loops, arrowheads, or redundant edges. The relation is reflexive and symmetric but is not antisymmetric nor transitive. Solution for reflexive, symmetric, antisymmetric, transitive they have. So total number of reflexive relations is equal to 2 n(n-1). Suppose that your math teacher surprises the class by saying she brought in cookies. Irreflexive is a related term of reflexive. both can happen. A matrix for the relation R on a set A will be a square matrix. But in "Deb, K. (2013). If x is positive then x times x is positive. A poset (partially ordered set) is a pair (P, â©¾), where P is a set and â©¾ is a reflexive, antisymmetric and transitive relation on P. If x â©¾ y and x â y hold, we write x > y. If x is negative then x times x is positive. In this short video, we define what an Antisymmetric relation is and provide a number of examples. We look at three types of such relations: reflexive, symmetric, and transitive. In mathematics (specifically set theory), a binary relation over sets X and Y is a subset of the Cartesian product X × Y; that is, it is a set of ordered pairs (x, y) consisting of elements x in X and y in Y. Discrete Mathematics Questions and Answers â Relations. (iv) Reflexive and transitive but not symmetric. A relation has ordered pairs (a,b). for example the relation R on the integers defined by aRb if a b is anti-symmetric, but not reflexive.That is, if a and b are integers, and a is divisible by b and b is divisible by a, it must be the case that a = b. A reflexive relation on {a,b,c} must contain the three pairs (a,a), (b,b), (c,c). For example, the inverse of less than is also asymmetric. A relation R is an equivalence iff R is transitive, symmetric and reflexive. (e) Carefully explain what it means to say that a relation on a set $A$ is not antisymmetric. Write which of these is an equivalence relation. Reflexive relations are always represented by a matrix that has $1$ on the main diagonal. Assume A={1,2,3,4} NE a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44 SW. R is reflexive iff all the diagonal elements (a11, a22, a33, a44) are 1. Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) \in R if and only if a) x \â ¦ so neither (2,1) nor (2,2) is in R, but we cannot conclude just from "non-membership" in R that the second coordinate isn't equal to the first. Reflexive is a related term of irreflexive. For example: If R is a relation on set A= (18,9) then (9,18) â R indicates 18>9 but (9,18) R, Since 9 is not greater than 18. Multi-objective optimization using evolutionary algorithms. Here we are going to learn some of those properties binary relations may have. If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too. Many students often get confused with symmetric, asymmetric and antisymmetric relations. It encodes the information of relation: an element x is related to an element y, if and only if the pair (x, y) belongs to the set. Therefore x is related to x for all x and it is reflexive. That is to say, the following argument is valid. 6.3. Equivalence. (ii) Transitive but neither reflexive nor symmetric. If is an equivalence relation, describe the equivalence classes of . Only a particular binary relation B on a particular set S can be reflexive, symmetric and transitive. All three cases satisfy the inequality. Now, let's think of this in terms of a set and a relation. At its simplest level (a way to get your feet wet), you can think of an antisymmetric relation of a set as one with no ordered pair and its reverse in the relation. Relation R is Antisymmetric, i.e., aRb and bRa a = b. 9. For the following examples, determine whether or not each of the following binary relations on the given set is reflexive, symmetric, antisymmetric, or transitive. The relations we are interested in here are binary relations on a set. (v) Symmetric and transitive but not reflexive. This section focuses on "Relations" in Discrete Mathematics. Relation R is transitive, i.e., aRb and bRc aRc. A relation from a set A to itself can be though of as a directed graph. Proof: Similar to the argument for antisymmetric relations, note that there exists 3(n2 n)=2 asymmetric binary relations, as none of â ¦ Summary of Order Relations A partial order is a relation that is reflexive, antisymmetric, and transitive. ... Antisymmetric Relation. Give reasons for your answers and state whether or not they form order relations or equivalence relations. symmetric, reflexive, and antisymmetric. The only case in which a relation on a set can be both reflexive and anti-reflexive is if the set is empty (in which case, so is the relation). For each of these binary relations, determine whether they are reflexive, symmetric, antisymmetric, transitive. reflexive relation irreflexive relation symmetric relation antisymmetric relation transitive relation Contents Certain important types of binary relation can be characterized by properties they have. Which is (i) Symmetric but neither reflexive nor transitive. An anti-reflexive (irreflexive) relation on {a,b,c} must not contain any of those pairs. Otherwise, x and y are incomparable, and we denote this condition by x \|\| y. REFLEXIVE RELATION:IRREFLEXIVE RELATION, ANTISYMMETRIC RELATION Elementary Mathematics Formal Sciences Mathematics Proofs about relations There are some interesting generalizations that can be proved about the properties of relations. If x â©¾ y or y â©¾ x, x and y are comparable. A relation R is non-reflexive iff it is neither reflexive nor irreflexive. Reflexive and symmetric Relations on a set with n elements : 2 n(n-1)/2. A relation $R$ on a set $A$ is an equivalence relation if and only if it is reflexive and circular. if x is zero then x times x is zero. For example, loves is a non-reflexive relation: there is no logical reason to infer that somebody loves herself or does not love herself. Matrices for reflexive, symmetric and antisymmetric relations. R, and R, a = b must hold. (iii) Reflexive and symmetric but not transitive. A relation is said to be asymmetric if it is both antisymmetric and irreflexive or else it is not. A binary relation $R$ on a set $A$ is said to be antisymmetric if there is no pair of distinct elements of $A$ each of which is related by $R$ to the other. Antisymmetric Relation. Let's assume you have a function, conveniently called relation: bool relation(int a, int b) { /* some code here that implements whatever 'relation' models. If a relation has a certain property, prove this is so; otherwise, provide a counterexample to show that it does not. A relation $R$ on a set $A$ is an antisymmetric relation provided that for all \ (x, y \in A\), if $x\ R\ y$ and $y\ R\ x$, then $x = y$. Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) â R if and only if a) everyone who has â ¦ 3) Z is the set of integers, relationâ ¦ These Multiple Choice Questions (MCQ) should be practiced to improve the Discrete Mathematics skills required for various interviews (campus interviews, walk-in interviews, company interviews), placements, entrance exams and other competitive examinations. A relation R on a set A is called a partial order relation if it satisfies the following three properties: Relation R is Reflexive, i.e. Click hereð to get an answer to your question ï¸ Given an example of a relation. A transitive relation is asymmetric if it is irreflexive or else it is not. In set theory\|lang=en terms the difference between irreflexive and antisymmetric is that irreflexive is (set theory) of a binary relation r on x: such that no element of x is r-related to itself while antisymmetric is (set theory) of a relation ''r'' on a set ''s, having the property that for any two distinct elements of ''s'', at least one is not related to the other via ''r . 3/25/2019 Lecture 14 Inverse of relations 1 1 3/25/2019 ANTISYMMETRIC RELATION Let R be a binary relation on a Antisymmetric relation is a concept of set theory that builds upon both symmetric and asymmetric relation in discrete math. A total order is a partial order in which any pair of elements are comparable. Relation Reï¬ exive Symmetric Asymmetric Antisymmetric Irreï¬ exive Transitive R 1 X R 2 X X X R 3 X X X X X R 4 X X X X R 5 X X X 3. Limitations and opposites of asymmetric relations are also asymmetric relations. For example, if a relation is transitive and irreflexive, 1 it must also be asymmetric. 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Dallas Investment Advisors, Highland Park, Plano, Addison TX \| Waterford Capital The Sharpe ratio or Sharpe index or Sharpe measure or reward-to-variability ratio is a measure of the excess return (or Risk Premium) per unit of risk in an investment asset or a trading strategy, named after William Forsyth Sharpe. Since its revision by the original author in 1994, it is defined as: where R is the asset return, Rf is the return on a benchmark asset, such as the risk free rate of return, E[R − Rf] is the expected value of the excess of the asset return over the benchmark return, and σ is the standard deviation of the asset. Note, if Rf is a constant risk free return throughout the period, The Sharpe ratio is used to characterize how well the return of an asset compensates the investor for the risk taken, the higher the Sharpe ratio number the better. When comparing two assets each with the expected return E[R] against the same benchmark with return Rf, the asset with the higher Sharpe ratio gives more return for the same risk. Investors are often advised to pick investments with high Sharpe ratios. However like any mathematical model it relies on the data being correct. Pyramid schemes with a long duration of operation would typically provide a high Sharpe ratio when derived from reported returns but the inputs are false.	{"url":"https://www.waterfordcapital.com/dallas-investment-advisors-highland-park-plano-addison-tx/","timestamp":"2024-11-11T11:10:33Z","content_type":"application/xhtml+xml","content_length":"37124","record_id":"<urn:uuid:c5c5d665-572a-4621-a5cb-300d7d9daead>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00843.warc.gz"}
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Random Vibrations In FEA - What Are They And How Do We Assess Them? - Fidelis Engineering Associates Random vibrations subject a component to non-deterministic motion which cannot be analyzed precisely. The mode shapes and natural frequencies remain the same for the component and randomness is an inherent characteristic of the input excitation. They present a significant challenge to engineers trying to meet design margins. Some examples for random vibrations are: 1. Automobile driving on a rough road, potholes, railroad tracks and other obstacles. 2. Motor vibration in vehicles. 3. Rocket motor ignition vibration during the first few seconds and during powered flight. 4. Hard disk drive motion. 5. Load on an airplane wing during flight. 6. Wave height in the ocean. 7. Motion of buildings, poles, bridges, and other structures during an earthquake. In these examples, the motion varies randomly with time, and it is nonperiodic. Hence, the amplitude of vibration cannot be expressed as a deterministic mathematical function. Instead, we have to use the statistical nature of the input excitations, such as acceleration, force, velocity or displacements. Let’s take a look at an example of a car travelling on a rough road. The figure below shows the time history of the vertical acceleration of the car. Now, if we have to model the actual behavior of the car subjected to this load, we have to use an Explicit algorithm with a very small-time step. This method becomes computationally very expensive to solve for medium to large physical systems. An alternate method for evaluating these physical systems subjected to random loads is using a statistical or probabilistic approach. Most of the random excitations follow a Gaussian (normal) distribution as shown in the figure below. This shows that 68.26% of the random data corresponds to the 1σ interval and 99.7 % of the random data corresponds to the 3σ interval. Since the input excitation has statistical behavior, it is assumed that the output variables, such as displacements and stresses, have statistical nature as well. In this method, the frequency data from the time history is acquired, along with the statistical data of the amplitude, and this is used as the load in a random vibration analysis. This spectrum is shown in the figure below and is termed as a Power Spectral Density (PSD). What Is A PSD Curve? Now, let’s understand what constitutes a PSD curve. The magnitude of the PSD curve is the mean square value of the input excitation. For the above acceleration vs time plot, the mean of the squared acceleration values is the power of the PSD. This power is distributed over a spectrum of frequencies (x axis) as it provides useful data when dealing with physical systems that have resonance. The power of the PSD is dependent on the frequency band width. Hence the change in frequency bands causes variation in the squared magnitudes. To overcome this, a consistent independent value of power, termed as density (y-axis), is calculated by dividing the squared magnitudes by the frequency bandwidth. Hence the units of PSD is G2/Hz. Random Vibration Analysis In FEA The random vibration analysis in FEA is solved using mode superposition method. This is a linear analysis and requires an input of natural frequencies and eigenmode shapes of the physical system extracted from a linear modal analysis. The input PSD can be in terms of acceleration, velocity or displacement. Non-zero displacements and rotations cannot be prescribed as boundary conditions in a random vibration analysis. The only loads that can be applied to the system are excitation loads (velocity, acceleration, displacement) applied through a PSD curve. Only one excitation direction is possible in each step. To compute the system’s response in multiple directions, different steps have to be used. The material density and elastic properties must be assigned to the region where dynamic response is required. Plasticity, thermal properties, rate dependent properties, electrical, diffusion and fluid flow properties cannot be included in a random response analysis since they are typically nonlinear, as are contact algorithms. While analyzing a multi-body system, components can either be tied together or connector elements can be used. The random response analysis cannot be used if contact plays a crucial role in determining the motion of the body. In that case, an alternative dynamic analysis method should be considered. Defining Frequency Range Frequency range of interest for the random response analysis needs to be specified in the analysis. The response of the system will be calculated at multiple points between lowest frequency of interest and the first eigenfrequency in the range, between each eigenfrequency in the range and between the last eigenfrequency in the range and the highest frequency in the range as shown in the figure below. The Bias Parameter The bias parameter is used to determine the spacing of result points in each of the frequency intervals as shown in the figure below. A bias parameter of 1 gives equally spaced result outputs in the frequency interval. However, most relevant information is usually clustered around the resonant frequencies of interest. Output from Random Vibration Analysis The output of a random response analysis will be the PSD of stresses and displacements, and variance and root mean square values of these variables if required. Note that these are not the actual stresses of the physical system at any time point, but they are root mean square values of the stresses occurring in the system undergoing random vibrations. The figure below shows the RMS values of the von Mises stress for a steel structure subjected to random vibrations. Since, the input excitations are assumed to have a normal distribution, the output variables will also have normal distributions. So, they can be extracted with different levels of confidence (68.26%, 95.44% or 99.72%). The computational cost of the simulation can be reduced by requesting the output only for selected element or node sets. Once high stress regions are identified from the results, a spectral plot of stresses can be plotted. These plots can be used to identify the frequencies that contribute most to the RMS stress. Good insights into potential design changes can be obtained by reviewing frequency response displacements at the problematic frequencies determined from the stress spectrum. Final Thoughts Random response analysis predicts the response of a physical system subjected to a non-periodic continuous excitation that is expressed in a statistical manner. This analysis is incorporated in the design phase by engineers to avoid issues in the physical system related to these dynamic effects. Hopefully, this article has provided some useful insights into the components of PSD curve as well as input and output parameters for a random response analysis. We’re always here to help, so if you have questions about dynamic effects in your designs or models, or just FEA in general, ,don’t hesitate to reach out!	{"url":"https://www.fidelisfea.com/post/random-vibrations-in-fea-what-are-they-and-how-do-we-assess-them","timestamp":"2024-11-02T18:51:25Z","content_type":"text/html","content_length":"370946","record_id":"<urn:uuid:251d2dfb-2e58-409d-8ef3-c6dcfa883c90>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00131.warc.gz"}
Plastic Analysis of Frames and Slabs 4 (CIVE10003) Undergraduate Course: Plastic Analysis of Frames and Slabs 4 (CIVE10003) Course Outline School School of Engineering College College of Science and Engineering level SCQF Level 10 (Year 4 Undergraduate) Availability Available to all students year taken) SCQF 10 ECTS Credits 5 Summary In this module, two segments extend the student's knowledge and understanding of the theory of structures to plastic behaviour. The first presents a deeper understanding of the plastic analysis of frames: the second covers yield line analysis of reinforced concrete slabs. LECTURES: TITLES & CONTENTS Segment 1 Plastic collapse of frame structures L1 Introduction Structure of the course. Aims of the course. References with comments. The theorems of plastic analysis: upper and lower bound theorems, their basis and assumptions. Ductility requirements for plastic collapse in steel members: plastic and compact sections. L2 Full plastic moments of cross-sections Stress-strain relationships for materials and their simplification into a 2 parameter model. Models for hand and for computer analysis. The analysis of cross-sections of any complexity to determine the full plastic moment about an axis. L3 Axial loads and cross sections in different materials The effect of axial load on the plastic moment. The interaction diagram for simple sections: all four quadrants of the interaction diagram and its significance in structures. Ultimate moment interaction diagrams in reinforced concrete or composite steel-concrete sections. L4 Plastic collapse of continuous beams Review of plastic collapse of beam structures. Changes of section at supports and within spans. Rules for locations of plastic hinges and number of mechanisms. Minimisation of collapse loads when hinge locations are not preordained. L5 Portal frames Plastic collapse of a simple single bay portal frame. Locations of hinges, types of mechanisms illustrated in this simple example. Effect of pinned bases. The interaction diagram. Over-complete collapse and its significance. Non-proportional loading. L6 General rules on collapse of frames Application of plastic analysis to multi-storey and multi-bay frames. Elementary and combined mechanisms. Joint rotation as an elementary mechanism, simple beam mechanisms, simple sway mechanisms, combining mechanisms. Rules for the locations of plastic hinges. Rules for assessing numbers of redundancies. Rules to determine the number of sway modes. Rules to determine the numbers and types of elementary and combined mechanisms. Application of the rules. L7 Single storey portal frames Analysis of a multi-bay portal frame. Application of the upper bound theorem. Identifying hinge locations. Determining the number of independent elementary mechanisms. Identifying the elementary mechanisms. Combining mechanisms: compatibility requirements and methodology. Application of the lower bound theorem to verify the collapse load. Use of the lower bound theorem on the wrong mechanism. L8 Multi-storey portal frames Identifying hinge locations. Determining the number of independent elementary mechanisms. Identifying the number of sway modes and their forms. Identifying possible elementary mechanisms: alternative choices. Analysis of the elementary mechanisms. Combination of mechanisms. Compatibility requirements and methodology. Lower bound theorem in the presence of multiple sway modes. Sway equilibrium equations. L9 Upper and lower bound theorems and their significance Full statement of the two theorems. Uniqueness. Demonstration of outcome of applying each theorem to modes that are not the correct collapse mode. Use of the upper bound theorem and minimisation. Use of the lower bound theorem and safe design. Exploitation of lower bound theorem in elastic analysis. Requirements for the theorems to be valid. Ductility and stability effects. Course L10 Other factors and aspects description Modifications of the evaluated collapse loads caused by different phenomena. Effect of axial loads on full plastic moment, and on ultimate moments in reinforced concrete. Effect of instability on collapse loads. Geometric nonlinearity and its outcomes for different loadings and geometries. Brittle materials and the effects of shrinkage, creep, lack of fit, settlement etc. Segment 2 Yield line analysis of reinforced concrete slabs L1 Introduction Introduction to yield line analysis: behaviour of rigid plastic material, fundamentals of yield line theory and methods of analysis, equilibrium and virtual work methods. L2 Simple example of one way bending Simple calculation of a collapse load for one way bending and its relationship to plastic collapse of beams. L3 The yield line: bending and twisting moments Calculations of bending and twisting moments on the yield lines for isotropic and orthotropic reinforcement; calculation of normal rotation on a yield line. Compatibility requirements of yield line patterns. L4 Collapse mechanisms Fundamentals and assumptions for collapse mechanisms. Collapse mechanisms for slabs with different boundary conditions based on these assumptions. L5 Example problems Orthotropic slabs of different geometries and load cases. Determination of collapse loads using the upper bound theorem. Discussion of the reasons for examining alternative collapse mechanisms. Derivation of formulae for the analysis of slabs of various shapes under different loading conditions (point load, line load and distributed load) and different boundary L6 Lower bound theorem and other phenomena The yield line as an upper bound method: upper and lower bound theorems of plasticity for slabs. Use of finite element analysis with the lower bound theorem. Compressive membrane action and its causes. Relationship of yield line load to true collapse. Tensile membrane action. Geometrically nonlinearity and its effect on behaviour. The meaning of a collapse load. Punching L7 Revision Review of the whole module. Significance of collapse load evaluation. Lower bound theorem and its importance in elastic analysis and design. Importance of ductility, and warnings about brittle materials. Relationship between hand calculations and computer calculations. Material and geometric nonlinearity. TUTORIALS: TITLES & CONTENTS Segment 1 Plastic analysis of frames Tutorial 1 Plastic moments of cross-sections This tutorial covers the determination of the full plastic moment of various cross-sections, followed by the development of interaction diagrams for cross-sections. Tutorial 2 Plastic collapse of multi-bay and multi-storey frames This tutorial covers problems involving interaction diagrams for simple frames, combined mechanisms for multi-storey and multi-bay frames. Segment 2 Yield line analysis of slabs Tutorial 3 Yield line analysis of slabs A single tutorial sheet with many questions, beginning with simple problems and progressing to complex yield line mechanisms. These tutorials should all be completed and handed in as they provide an excellent preparation for a professional career as well as the examination. Entry Requirements (not applicable to Visiting Students) Pre-requisites Students MUST have passed: Theory of Structures 3 (CIVE09015) Co-requisites Prohibited Combinations Other requirements None Information for Visiting Pre-requisites None High Demand Course? Yes Course Delivery Information Academic year 2016/17, Available to all students (SV1) Quota: None Course Start Semester 2 Timetable Timetable Total Hours: 100 ( Lecture Hours 18, Seminar/Tutorial Hours 9, Formative Assessment Hours 1, Summative Learning and Teaching activities (Further Info) Assessment Hours 2, Programme Level Learning and Teaching Hours 2, Directed Learning and Independent Learning Hours 68 ) Assessment (Further Info) Written Exam 100 %, Coursework 0 %, Practical Exam 0 % Additional Information (Assessment) The assessment will be made on the basis of: Degree examination 100% Feedback Formative, mid-semester and end-of-course. Exam Information Exam Diet Paper Hours & Minutes Main Exam Diet S2 (April/May) of Frames 2:00 and Slabs Learning Outcomes On completion of this course, the student will be able to: 1. demonstrate the ability to calculate the plastic collapse loads of complex two dimensional frame structures, to identify the independent mechanisms and combine them, to use the upper and lower bound theorems to find the true collapse load, and to produce engineering designs of frame structures based on plastic collapse analysis; 2. demonstrate the ability to calculate the yield line collapse load of reinforced concrete slabs of complex geometry with isotropic and orthotropic reinforcement using the upper bound theorem, and to apply the method to the proportioning of reinforcement in a slab. Reading List Segment 1 Plastic analysis of frames Course reference - Plastic Design of Frames J.F. Baker and J. Heyman Cambridge University Press 1969 Suggested further reading - Plastic Theory of Structures M.R. Horne Pergamon Press 1981 - The Steel Skeleton Volume II J.F. Baker, M.R. Horne and J. Heyman Cambridge University Press 1956 - Plastic methods for steel and concrete structures S.S.J. Moy Macmillan 1996 Segment 2 Yield line analysis of slabs Course reference - Reinforced and pre-stressed concrete F.H. Kong and R.H. Evans van Nostrand Reinhold (UK) 1987 Suggested further reading - Yield line analysis of slabs L.L. Jones and R.H. Wood Thames and Hudson, Chatto and Windus, 1967 - Structural Concrete R.P. Johnson McGraw Hill - Yield line analysis of slabs K.W. Johanson Cement and Concrete Association, London 1972 - Plastic methods for steel and concrete structures S.S.J. Moy Macmillan 1996 Additional Information Graduate Attributes Not entered and Skills Special Arrangements Exam should be scheduled on a slot on a Thursday afternoon. Keywords Not entered Dr Hwa Kian Chai Mr Craig Hovell Course Tel: Course Tel: (0131 6)51 organiser Email: secretary 7080 Hwakian.Chai@ed.ac.uk Email: © Copyright 2016 The University of Edinburgh - 3 February 2017 3:33 am	{"url":"http://www.drps.ed.ac.uk/16-17/dpt/cxcive10003.htm","timestamp":"2024-11-02T18:44:50Z","content_type":"text/html","content_length":"26682","record_id":"<urn:uuid:6a941f36-288f-427e-ae00-7f145c6f3c5e>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00242.warc.gz"}
Phase space of mechanical systems with a gauge group ← &nbsp→ Reviews of topical problems Phase space of mechanical systems with a gauge group Publications on the structure of the physical phase space (PS) of dynamical systems with gauge symmetry are reviewed. The recently discovered phenomenon of reduction of the phase space of the physical degrees of freedom is studied systematically on mechanical models with a finite number of dynamical variables. In the simplest case of one degree of freedom this phenomenon consists of replacement of the phase space by a cone that is unfoldable into a half-plane. In the general case the reduction of the phase space is related with the existence of a residual discrete gauge group, acting in the physical space after the unphysical variables are eliminated. In ``natural'' gauges for the adjoint representation this group is isomorphic to Weyl's group. A wide class of modes with both the normal and Grassmann (anticommuting) variables and with arbitrary compact gauge groups is studied; the classical analysis and the quantum analysis are performed in parallel. It is shown that the reduction of the phase space radically changes the physical characteristics of the system, in particular its energy spectrum. A significant part of the review is devoted to a description of such systems on the basis of the method of Hamiltonian path integrals (HPIs). It is shown how the HPI is modified in the case of an arbitrary gauge group. The main attention is devoted to the correct formulation of the HPI with a poor choice of gauge. The analysis performed can serve as an elementary illustration of the well-known problem of copies in the theory of Yang--Mills fields. The dependence of the quasiclassical description on the structure of the phase space is demonstrated on a model with quantum-mechanical instantons.	{"url":"https://ufn.ru/en/articles/1991/2/b/","timestamp":"2024-11-06T20:44:40Z","content_type":"text/html","content_length":"19388","record_id":"<urn:uuid:0f416d8a-ab29-4949-bffd-7dcf58e5ad67>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00642.warc.gz"}
2003 AIME I Problems/Problem 15 In $\triangle ABC, AB = 360, BC = 507,$ and $CA = 780.$ Let $M$ be the midpoint of $\overline{CA},$ and let $D$ be the point on $\overline{CA}$ such that $\overline{BD}$ bisects angle $ABC.$ Let $F$ be the point on $\overline{BC}$ such that $\overline{DF} \perp \overline{BD}.$ Suppose that $\overline{DF}$ meets $\overline{BM}$ at $E.$ The ratio $DE: EF$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ Solution 1 In the following, let the name of a point represent the mass located there. Since we are looking for a ratio, we assume that $AB=120$, $BC=169$, and $CA=260$ in order to simplify our computations. First, reflect point $F$ over angle bisector $BD$ to a point $F'$. $[asy] size(400); pointpen = black; pathpen = black+linewidth(0.7); pair A=(0,0),C=(7.8,0),B=IP(CR(A,3.6),CR(C,5.07)), M=(A+C)/2, Da = bisectorpoint(A,B,C), D=IP(B--B+(Da-B)10,A--C), F=IP(D--D+10 (B-D)dir(270),B--C), E=IP(B--M,D--F);pair Fprime=2D-F; /* scale down by 100x */ D(MP("A",A,NW)--MP("B",B,N)--MP("C",C)--cycle); D(B--D(MP("D",D))--D(MP("F",F,NE))); D(B--D(MP("M",M)));D(A--MP ("F'",Fprime,SW)--D); MP("E",E,NE); D(rightanglemark(F,D,B,4)); MP("390",(M+C)/2); MP("390",(M+C)/2); MP("360",(A+B)/2,NW); MP("507",(B+C)/2,NE); [/asy]$ As $BD$ is an angle bisector of both triangles $BAC$ and $BF'F$, we know that $F'$ lies on $AB$. We can now balance triangle $BF'C$ at point $D$ using mass points. By the Angle Bisector Theorem, we can place mass points on $C,D,A$ of $120,\,289,\,169$ respectively. Thus, a mass of $\frac {289}{2}$ belongs at both $F$ and $F'$ because BD is a median of triangle $BF'F$ . Therefore, $CB/FB=\frac{289}{240}$. Now, we reassign mass points to determine $FE/FD$. This setup involves $\triangle CFD$ and transversal $MEB$. For simplicity, put masses of $240$ and $289$ at $C$ and $F$ respectively. To find the mass we should put at $D$, we compute $CM/MD$. Applying the Angle Bisector Theorem again and using the fact $M$ is a midpoint of $AC$, we find $\[\frac {MD}{CM} = \frac {\frac{169}{289}\cdot 260 - 130}{130} = \frac {49}{289}\]$ At this point we could find the mass at $D$ but it's unnecessary. $\[\frac {DE}{EF} = \frac {F}{D} = \frac {F}{C}\cdot\frac {C}{D} = \frac {289}{240}\cdot\frac {49} {289} = \boxed{\frac {49}{240}}\]$ and the answer is $49 + 240 = \boxed{289}$. Solution 2 By the Angle Bisector Theorem, we know that $[CBD]=\frac{169}{289}[ABC]$. Therefore, by finding the area of triangle $CBD$, we see that $\[\frac{507\cdot BD}{2}\sin\frac{B}{2}=\frac{169}{289}[ABC].\] $ Solving for $BD$ yields $\[BD=\frac{2[ABC]}{3\cdot289\sin\frac{B}{2}}.\]$ Furthermore, $\cos\frac{B}{2}=\frac{BD}{BF}$, so $\[BF=\frac{BD}{\cos\frac{B}{2}}=\frac{2[ABC]}{3\cdot289\sin\frac{B}{2}\ cos\frac{B}{2}}.\]$ Now by the identity $2\sin\frac{B}{2}\cos\frac{B}{2}=\sin B$, we get $\[BF=\frac{4[ABC]}{3\cdot289\sin B}.\]$ But then $[ABC]=\frac{360\cdot 507}{2}\sin B$, so $BF=\frac{240}{289} \cdot 507$. Thus $BF:FC=240:49$. Now by the Angle Bisector Theorem, $CD=\frac{169}{289}\cdot 780$, and we know that $MC=\frac{1}{2}\cdot 780$ so $DM:MC=\frac{169}{289}-\frac{1}{2}:\frac{1}{2}=49:289$. We can now use mass points on triangle CBD. Assign a mass of $240\cdot 49$ to point $C$. Then $D$ must have mass $240\cdot 289$ and $B$ must have mass $49\cdot 49$. This gives $F$ a mass of $240\cdot 49+49\cdot 49=289\cdot 49$. Therefore, $DE:EF=\frac{289\cdot 49}{240\cdot 289}=\frac{49}{240}$, giving us an answer of $\boxed{289}.$ Solution 3 Let $\angle{DBM}=\theta$ and $\angle{DBC}=\alpha$. Then because $BM$ is a median we have $360\sin{(\alpha+\theta)}=507\sin{(\alpha-\theta)}$. Now we know $\[\sin{(\alpha+\theta)}=\sin{\alpha}\cos{\ theta}+\sin{\theta}\cos{\alpha}=\dfrac{DF\cdot BD}{BF\cdot BE}+\dfrac{DE\cdot BD}{BE\cdot BF}=\dfrac{BD(DF+DE)}{BF\cdot BE}\]$ Expressing the area of $\triangle{BEF}$ in two ways we have $\[\dfrac{1} {2}BE\cdot BF\sin{(\alpha-\theta)}=\dfrac{1}{2}EF\cdot BD\]$ so $\[\sin{(\alpha-\theta)}=\dfrac{EF\cdot BD}{BF\cdot BE}\]$ Plugging this in we have $\[\dfrac{360\cdot BD(DF+DE)}{BF\cdot BE}=\dfrac {507\cdot BD\cdot EF}{BF\cdot BE}\]$ so $\dfrac{DF+DE}{EF}=\dfrac{507}{360}$. But $DF=DE+EF$, so this simplifies to $1+\dfrac{2DE}{EF}=\dfrac{507}{360}=\dfrac{169}{120}$, and thus $\dfrac{DE}{EF}=\ dfrac{49}{240}$, and $m+n=\boxed{289}$. Solution 4 (Overpowered Projective Geometry!!) Firstly, angle bisector theorem yields $\frac{CD}{AD} = \frac{507}{360} = \frac{169}{120}$. We're given that $AM=MC$. Therefore, the cross ratio $\[(A,C;M,D) = \frac{AM(CD)}{AD(MC)} = \frac{169}{120}\]$ We need a fourth point for this cross ratio to be useful, so reflect point $F$ over angle bisector $BD$ to a point $F'$. Then $\triangle BFF'$ is isosceles and $BD$ is an altitude so $DF = DF'$. $\[(A,C;M,D) = (F,F';D,E) \implies \frac{FD(EF')}{EF(DF')} = \frac{EF'}{EF} = \frac{169}{120}\]$ All that's left is to fiddle around with the ratios: $\[\frac{EF'}{EF} = \frac{ED+DF'}{EF} = \frac{EF+2DE}{EF} = 1\ +\ 2\left(\frac{DE}{EF}\right) \implies \frac{DE}{EF} = \frac{49}{240} \implies \boxed{289}\]$ Solution 5 (Menelaus + Mass Points) Extend $DF$ to intersect with the extension of $AB$ at $G$. Notice that $\triangle{BDF} \cong \triangle{BDG}$, so $GD=DF$. We now use Menelaus on $\triangle{GBF}$, as $A$, $D$, and $C$ are collinear; this gives us $\frac{GA}{BA} \cdot \frac{BC}{FC} \cdot \frac{DF}{GD}=1$. As $GD=DF$, we have $\frac{GA}{AB}=\frac{FC}{BC}$, hence $\frac{GA}{120}=\frac{FC}{169}$. Reflect $G$ over $A$ to $G'$. Note that $\frac{G'A}{BA}=\frac{FC}{BC}$, and reflexivity, hence $\triangle{ABC} \sim \triangle{BG'F}$. It's easily concluded from this that $G'F \parallel AC$, hence $G'F \parallel AD$. As $GD=DF$, we have $AD$ is a midsegment of $\triangle{GG'F}$, thus $G'F = 2AD$. We now focus on the ratio $\frac{BF}{BC}$. From similarity, we have $\frac{BF}{BC}=\frac{G'F}{AC}=\frac{2AD}{AC}$. By the angle bisector theorem, we have $AD:DC=120:169$, hence $AD:AC=120:289$, so $\frac{BF}{BC}=\frac{240}{289}$. We now work out the ratio $\frac{DM}{MC}$. $\frac{DM}{MC}=\frac{CD-MC}{MC}=\frac{CD}{MC}-1=\frac {2CD}{AC}-1=\frac{338}{289}-1=\frac{49}{289}$. We now use mass points on $\triangle{BDC}$. We let the mass of $C$ be $240\cdot 49$, so the mass of $B$ is $49 \cdot 49$ and the mass of $D$ is $289\ cdot 240$. Hence, the mass of $F$ is $289\cdot 49$, so the ratio $\frac{DE}{EF}=\frac{49}{240}$. Extracting gives $49+240=\boxed{289}.$ See also The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.	{"url":"https://artofproblemsolving.com/wiki/index.php/2003_AIME_I_Problems/Problem_15","timestamp":"2024-11-08T12:18:07Z","content_type":"text/html","content_length":"70037","record_id":"<urn:uuid:595eab44-c0c0-49f6-b943-1cc288c831a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00011.warc.gz"}
Hex colors, how do they work? \| HackerNoon We have all seen them before, they look a bit like this: #ff0000. That’s the hex code for the color red. However, what exactly does it mean? Most of us have heard of binary before, but what about hexadecimal? Hexadecimal, sometimes known as ‘hex’ or ‘base 16’ is an alternative to binary, which is ‘base 2’. Hexadecimal can be used to store numbers. So how does it work? The Basics There are 16 possible characters in hexadecimal, they include 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F. Each character represents an integer from zero to fifteen. Now that we know what each of the characters mean, let’s start converting integers into hexadecimal. So let’s convert the number 2,087 into hexadecimal. Some Examples Let’s first start off by dividing 2,087 by 16. That should get us a result of 130 with a remainder of 7. According to our chart from above, 7 is also ‘7’ in hexadecimal, so now we need to set ‘7’ aside and divide 130 by 16. Our answer is 8 with a remainder of 2. On our chart, 2 is ‘2’, so set that to the side and we need to divide 8 by 16. This will get us 0 with a remainder of 8, which translate to ‘8’ in hex. So we have our 3 results which are 7, 2 and 8. We will put them together in reverse order to get our hexadecimal value which means that 2,087 is ‘0x827’. 0x is there to symbolize that it is a hex value. Let’s try another, this time we will try the number 255. First we need to divide 255 by 16. That will get us 15 with a remainder of 15. Now we take the remainder and line it up to our chart. 15 translate to ‘F’, so we can save ‘F’ for later. Now we need to divide 15 by 16, which will get us 0 with a remainder of 15. We know that 15 translates to ‘F’, which makes our result ‘0xFF’. Color Codes Hex color codes usually look something like this: #ff0000, but what does this mean? Well, let’s break it down! RGB stands for Red, Green, and Blue. These colors are the primary colors of light. An RBG value might look like this: rgb(255, 0, 0). At first this may not look like much, but if we take a closer look, the first number represents red, second number represents green, and the third number represents blue. The values in those spots can range anywhere from 0, all the way up to 255. Let’s try to make some colors! With light, red and blue makes magenta, so let’s try to make a magenta color with RGB. So to get a bright magenta color, let’s set both blue and red to their maximum value of 255. So we have: rgb (255, 0, 255), which gives us a color like this: rgb(255, 0, 255) But what if we wanted it to be a bit lighter, like a more pinkish color? How do we do that? White in RBG is rgb(255, 255, 255), all the colors combined. So how about we try changing rgb(255, 0, 255) to rgb(255, 150, 255)? We get the result of a light pink. rgb(255, 150, 255) So how about a darker color? So we can have a dark pink? In RGB, black is rgb(0, 0, 0), so let’s bring our values closer to that. So let’s try rgb(150, 0, 150). With that, I got this: rgb(150, 0, 150) Hex Colors Now that we have a good understanding of RGB, we can understand how hex colors work. A hex color will look something like this, #ff0000. But what does this mean? Let’s combine what we know about RBG and Hexadecimal to figure it out. We know that 0xFF is 255 in hexadecimal, and 255 is the maximum value for RGB. If we see what #ff0000 is, we can see that it is a bright red color: What we can find out is that #ff0000 translate to rgb(255, 0, 0). Let’s try to get our magenta color again, we know that it is rgb(255, 0, 255), so let’s see what #ff00ff gets us. Awesome! We got our magenta color, now let’s try to get our lighter color again, but this time in hex. Our light magenta in RGB is rgb(255, 150, 255), so if we convert 150 into hex, 150 divided by 16 is 9 with a remainder of 6. 9 divided by 16 is 0 with a remainder of 9, making our hexadecimal value ‘0x96’. The result is our hex color of #ff96ff, and when we see the result of that, we get our light magenta. After reading this, you are able to understand how the hex colors work, how the first two characters are for the red, the next two for green, and the next two for blue for RGB colors. I hope this article taught you something new, whether it be hex colors, RGB, or base16. If you enjoyed this article, please follow me on other platforms of social media below: Danny Tran (@hoogleyb) * Instagram photos and videos_335 Followers, 115 Following, 116 Posts - See Instagram photos and videos from Danny Tran (@hoogleyb)_www.instagram.com Danny Tran (@HoogleyB) \| Twitter_The latest Tweets from Danny Tran (@HoogleyB). I'm proud to say I'm a senior web developer, but I'm more proud to say…_twitter.com If you find there is anything wrong with the article, mention it below in the comments and I’ll fix it! Happy Coding!	{"url":"https://hackernoon.com/hex-colors-how-do-they-work-d8cb935ac0f","timestamp":"2024-11-14T21:50:45Z","content_type":"text/html","content_length":"238930","record_id":"<urn:uuid:929269b7-d4e2-466c-a8dd-afa0a52c86a0>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00793.warc.gz"}
Problem 760 - TheMathWorld Problem 760 The time tÂ required to do a job varies inversely as the number of people working. It takes 4 hr for cooks to prepare the food a wedding rehearsal dinner. How long will it take 8 cooks to prepare the dinner? This in inverse variation. The time to prepare the food decreases as the number of cooks increases. Since the time to prepare t varies inversely as the number of cooks P, use the form t(P) = To find the variation constant, substitute 6 for P and 4 for t(6), then solve for k. 4 = Solve for k. K = 24 Write the equation of variation. t(P) = Now, use the equation of variation to find time it takes 8 cooks to prepare the dinner. t(p ) = = 3 It takes 8 cooks 3 hr to prepare the dinner.	{"url":"https://mymathangels.com/problem-760/","timestamp":"2024-11-11T13:13:59Z","content_type":"text/html","content_length":"60626","record_id":"<urn:uuid:6be6209c-d188-414d-8a92-925e17ae74ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00138.warc.gz"}
Do you understand the indicator or do you just understand how to use the indicator Hi Guys, I want to know - when learning to trade, do you guys go in depth and try to fully understand the math behind the indicator or just try to understand how to use a given indicator with an assumption that since it is a popular indicator the math is well worked out ? Is it required to understand how the indicator exactly works in order to trade well on this ? Would you say that knowing the math of the indicator is required to formulate a strategy based on the indicator ? Or the indicator itself is the crux of the strategy so you just need to know the usage of it ? Thank you It always was, for me. [I]I think it’s more about the trader than the method itself.[/I] Some people require an understanding of the detail of how things work, where others are more content to try something by copying it, without being concerned about the detail of exactly what it’s measuring and calculating and displaying, and why and how. (How those people decide what settings to use is anyone’s guess!). That’s not true for me, at all. I don’t really understand how people can tell which indicators are worth examining and experimenting with, without knowing the details and what they’re supposed to signify. It seems to me ([I][U]very[/U][/I] personal, idiosyncratic perspective about to emerge here! :8: ) that without [I]some[/I] mathematical understanding of how they’re calculated, and exactly what they’re displaying, one would have no realistic way of distinguishing between indicators that display something worth looking at (such as Ichimoku or MACD, provided one understands the settings) and nonsensical things that are just totally fictional and arbitrary and about as sensibly-based as astrology (like “Fibonacci” and “Elliott waves”, though I suppose that last one isn’t really an “indicator” [I]per se[/I]). It is required to know the signals every indicator gives you. You need to understand when an oscillator gives a buy or sell signal or furthermore a negative or positive divergence. I guess there is no need to understand the indicator itself. If you are doing well with any indicator then just carry on with your trading. You will only complex your trading if you go into understanding the indicator itself. You can’t use anything properly not exactly knowing how it really works. If you do not know how to drive, then sooner or later you will get crashed. Unless you use an indicator like ADX or least square moving average where math gives no real understanding, you should understand what kind of data an indicator uses, what it measures and what it actually indicates.	{"url":"https://forums.babypips.com/t/do-you-understand-the-indicator-or-do-you-just-understand-how-to-use-the-indicator/82050","timestamp":"2024-11-14T14:01:23Z","content_type":"text/html","content_length":"24589","record_id":"<urn:uuid:3326df17-b7fc-412c-b9af-c460a40dd4c2>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00484.warc.gz"}
Lesson 16 Solving Problems Involving Fractions Let’s add, subtract, multiply, and divide fractions. Problem 1 An orange has about $\frac14$ cup of juice. How many oranges are needed to make $2\frac12$ cups of juice? Select all the equations that represent this question. $ {?} \boldcdot \frac 14= 2\frac12$ $\frac14 \div 2\frac12 = {?}$ ${?} \boldcdot 2\frac12 = \frac14$ $2\frac12 \div \frac14 = {?}$ Problem 2 Mai, Clare, and Tyler are hiking from a parking lot to the summit of a mountain. They pass a sign that gives distances. │Parking lot: $\frac34$ mile│ │Summit: $1\frac12$ miles │ • Mai says: “We are one third of the way there.” • Clare says: “We have to go twice as far as we have already gone.” • Tyler says: “The total hike is three times as long as what we have already gone.” Do you agree with any of them? Explain your reasoning. Problem 3 Priya’s cat weighs $5\frac12$ pounds and her dog weighs $8\frac14$ pounds. First, estimate the number that would comlpete each sentence. Then, calculate the answer. If any of your estimates were not close to the answer, explain why that may be. 1. The cat is _______ as heavy as the dog. 2. Their combined weight is _______ pounds. 3. The dog is _______ pounds heavier than the cat. Problem 4 Before refrigerators existed, some people had blocks of ice delivered to their homes. A delivery wagon had a storage box in the shape of a rectangular prism that was $7\frac12$ feet by 6 feet by 6 feet. The cubic ice blocks stored in the box had side lengths $1\frac12$ feet. How many ice blocks fit in the storage box? (From Unit 4, Lesson 15.) Problem 5 Fill in the blanks with 0.001, 0.1, 10, or 1000 so that the value of each quotient is in the correct column. Close to $\frac{1}{100}$ • $\text{______} \div 9$ • $12 \div \text{______}$ Close to 1 • $\text{______}\div 0.12$ • $\frac18 \div \text{______}$ Greater than 100 • $\text{______}\div \frac13$ • $700.7 \div \text{______}$ (From Unit 4, Lesson 1.) Problem 6 A school club sold 300 shirts. 31% were sold to fifth graders, 52% were sold to sixth graders, and the rest were sold to teachers. How many shirts were sold to each group—fifth graders, sixth graders, and teachers? Explain or show your reasoning. (From Unit 3, Lesson 15.) Problem 7 Jada has some pennies and dimes. The ratio of Jada’s pennies to dimes is 2 to 3. 1. From the information given, can you determine how many coins Jada has? 2. If Jada has 55 coins, how many of each kind of coin does she have? 3. How much are her coins worth? (From Unit 2, Lesson 15.)	{"url":"https://curriculum.illustrativemathematics.org/MS/students/1/4/16/practice.html","timestamp":"2024-11-11T06:33:48Z","content_type":"text/html","content_length":"70689","record_id":"<urn:uuid:30a2c3f4-e843-4032-b926-c618a4a82802>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00101.warc.gz"}
OpenStax College Physics for AP® Courses, Chapter 23, Problem 12 (Problems & Exercises) A 0.250 m radius, 500-turn coil is rotated one-fourth of a revolution in 4.17 ms, originally having its plane perpendicular to a uniform magnetic field. (This is 60 rev/s.) Find the magnetic field strength needed to induce an average emf of 10,000 V. Question by is licensed under CC BY 4.0 Solution video OpenStax College Physics for AP® Courses, Chapter 23, Problem 12 (Problems & Exercises) vote with a rating of votes with an average rating of. Video Transcript This is College Physics Answers with Shaun Dychko. A loop with 500 turns of wire initially has a plane perpendicular to the magnetic field and then this loop will be rotated a quarter turn in 4.17 milliseconds and then the induced emf we are told is meant to be 10000 volts so what should the magnetic field strength be to make that happen and we are given that the loop's radius is 0.250 meters. So the induced emf is the number of turns of wire multiplied by the rate of change in flux so this is the change in flux divided by the amount of time it takes to make that change. The change in flux then is the final flux minus the initial flux; the final flux is the magnetic field strength multiplied by the area of the loop times cosine of the final angle minus the magnetic field strength times the area times cos of the initial angle. So the initial angle is 0 degrees because the angle is the angle between the perpendicular to the plane of the loop and the magnetic field so the perpendicular to the plane of the loop is coming right out of the page or into the page— depending how you want to look at it— and it's parallel to the magnetic field and so the angle is zero initially. And then the final state has the plane perpendicular to the plane of the loop perpendicular to the magnetic field lines and so there's no flux in this final orientation of the loop. Another way to say it is that there are no magnetic field lines passing through the loop; that's another way of saying there's no flux. So we have 0 for our final flux minus BA for the initial flux. The area is π times radius squared so we plug that in for A and so our change in flux then is Bπr squared. And I guess I could put a negative sign there: the negative's here are not so important, they just remind us that... like this negative reminds us that the induced emf is in a direction such that the induced magnetic field opposes the change in flux but really, you should be getting that direction by looking at the situation and using the right hand rule. But nevertheless, you can plug a negative in here and a negative there and it ends up being positive so the induced emf then is number of turns times magnetic field strength times π times radius squared over change in time and we have to solve this for B by multiplying both sides by both sides by Δt over Nπr squared. So then the magnetic field strength is the induced emf times Δt over Nπr squared because these things cancel over here and we have 10000 volts times 4.17 times 10 to the minus 3 seconds divided by 500 turns times π times one quarter meter radius squared and that is 0.425 tesla is the required magnetic field strength.	{"url":"https://collegephysicsanswers.com/openstax-solutions/0250-m-radius-500-turn-coil-rotated-one-fourth-revolution-417-ms-originally-0","timestamp":"2024-11-08T17:43:48Z","content_type":"text/html","content_length":"244188","record_id":"<urn:uuid:22929697-18cb-40e7-8ec0-c558aead435d>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00387.warc.gz"}
Segal type Directed homotopy type theory Β Idea Segal types are the (infinity,1)-categorical version of precategories in simplicial type theory There are multiple different formalisms of simplicial homotopy type theory; two of them are given in Gratzer, Weinberger, & Buchholtz 2024 and in Riehl & Shulman 2017, and in each formalism there is a different way to define Segal types. Directed interval via axioms In simplicial homotopy type theory where the directed interval primitive $\mathbb{2}$ is defined via axioms, let the 2-simplex type be defined as $\Delta^2 \coloneqq \sum_{s:\mathbb{2}} \sum_{t:\mathbb{2}} s \leq t$ and let the 2-1-horn type be defined as $\Lambda_1^2 \coloneqq \sum_{s:\mathbb{2}} \sum_{t:\mathbb{2}} [(s =_\mathbb{2} 0) + (t =_\mathbb{2} 1)]$ where $[P]$ is the propositional truncation of the type $P$. For each type $A$ there is a canonical restriction function $i:(\Delta^2 \to A) \to (\Lambda_1^2 \to A)$. $A$ is a Segal type if the restriction function $i$ is an equivalence of types. Type theory with shapes formalism In simplicial type theory in the type theory with shapes formalism, a Segal type is a type $A$ such that given elements $x:A$, $y:A$, and $z:A$ and morphisms $f:\mathrm{hom}_A(x, y)$ and $g:\mathrm {hom}_A(y, z)$, the type $\sum_{h:\mathrm{hom}_A(x, z)} \left\langle \Delta^2 \to A \vert_{[x, y, z, f, g, h]}^{\partial \Delta^2} \right\rangle$ is a contractible type, where $\Delta^2$ is the $2$-simplex probe shape and $\partial \Delta^2$ is its boundary.	{"url":"https://ncatlab.org/nlab/show/Segal+type","timestamp":"2024-11-11T17:49:25Z","content_type":"application/xhtml+xml","content_length":"25596","record_id":"<urn:uuid:2171e7e5-ddc3-4f07-8117-e3a3d7523ca3>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00111.warc.gz"}
Jump to navigation Jump to search >> << Down to: Dyad Back to: Vocabulary Thru to: Dictionary [x] u S: n y Spread Conjunction Rank Infinity -- operates on [x and] y as a whole -- WHY IS THIS IMPORTANT? (u S:0 y) applies verb u to each leaf of y, creating an array whose items are the results of the leaves, with framing fill added as needed. Contrast this with Level At (u L:0 y), which returns a result in which each leaf of y has been replaced by the result of executing u on it. A leaf of y is a noun inside y that itself has no boxed contents. A leaf is either empty or unboxed. ] y=: (<0 1),(<<2 3),(<<<4 5) \|0 1\|+---+\|+-----+\| \| \|\|2 3\|\|\|+---+\|\| \| \|+---+\|\|\|4 5\|\|\| \| \| \|\|+---+\|\| \| \| \|+-----+\| NB. y is sample noun of nested boxed items NB. --The leaves are (0 1), (2 3) and (4 5) u=: \|. NB. sample verb (Reverse) to apply to leaves u S:0 y Compare this with the action of Level At (L:) ([x] u S:n y) applies u to the leaves of y in the same way as ([x] u L:n y) but it collects the results as the items of an array. See Level At (L:) for details, including • values of n other than 0 • negative values of n • the dyadic case (x S: n y) Common Uses Apply verb u to the leaves (innermost opened items) of a boxed noun y ] y=: 'alpha' ; 'bravo' ;'charlie' toupper S:0 y Related Primitives Level Of (L. y), Level At (u L: n)	{"url":"https://code.jsoftware.com/wiki/Vocabulary/scapco","timestamp":"2024-11-14T02:08:31Z","content_type":"text/html","content_length":"20562","record_id":"<urn:uuid:bea2ddb7-49e1-401b-940e-b8f44db46946>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00505.warc.gz"}
Math Behind Extreme Wind Loading In the last IAEI News (March/April 2002), I shared with you the details of the new extreme wind loading requirements of the 2002 National Electrical Safety Code (NESC). For structures sixty feet tall and shorter, the extreme wind loading only applies to the structure. For structures taller than sixty feet, the extreme wind loading applies to the structure and all the supported facilities. To understand the impact of the 2002 revision, lets crank through an example calculation. Photo 1 Consider the following example shown in photo 1 [beginning of article]: Structures:80-foot southern yellow pine poles set 10 feet in the ground All spans are 200′. Supported Facilities: 1-7 No. 5 AW static wire at 70′, conductor diameter 0.546″ 1-795-30/19 ACSR 138 kV phase wire at 60′, conductor diameter 1.140″ 1-795-30/19 ACSR 138 kV phase wire at 54′, conductor diameter 1.140″ 1-795-30/19 ACSR 138 kV phase wire at 48′, conductor diameter 1.140″ 3-477 AAC 12 kV phase wires at 38′, conductor diameter 0.793″ 1-1/0 AAAC neutral wire at 28′, conductor diameter 0.398″ 1-1.5″ diameter communications cable at 25′ 1-1.5″ diameter communications cable at 24′ 1-0.5″ diameter communications cable at 23′ 1-3″ diameter communications cable at 21′ What is the minimum pole class (diameter) necessary to withstand the 2002 NESC extreme wind loading? To determine the minimum pole class, we will calculate the total moment on the pole at the ground-line by multiplying the wind load on the pole and all supported facilities by the height of attachment of those facilities. The calculation will be done for the worst-case condition, when the wind is blowing perpendicular to the line. For this example, assume the line will be constructed on the East coast in an area where the Basic Wind speed per Figure 250-2(b), page 167, is 115 miles per hour. The first step is to calculate the wind load on the pole. Wind Load on the Pole Table 1 Because the pole diameter changes over its length, we will look at the wind load on each one-foot segment of the pole, starting at the top. To do this calculation, we must assume a particular pole class, calculate the moment, and then check to see if the class we assumed is adequate. The minimum dimensions of wood poles are given in ANSI Standard O5.1 Specifications and Dimensions for Wood Poles. Minimum pole circumferences are given by wood type at the top of the pole and six feet from the butt. Dimensions at other locations can be determined by interpolation. For an 80-foot class H1 pole, the minimum diameter at the top is 9.23 inches and the minimum diameter at the ground line, i.e., 10 feet from the butt, is 17.66 inches. For wind load calculations, we have to use maximum dimensions so we multiply the minimums by 1.2 as suggested by section 6.2.2, page 7, of the ANSI O5.1 standard. From Rule 250C, page 161, for cylindrical structures and attachments: load in pounds = 0.00256 x (wind speed)^2x k[z]x G[RF]x Area The velocity pressure exposure coefficients k[z]for structures are given in Table 250-2, page 163, as a function of the height of the structure above ground. The gust response factors G[RF]for structures are given in Table 250-3 as a function of height of the structure above ground. The velocity pressure exposure coefficients kz for conductors are given in Table 250-2 as a function of the height of attachment of the conductor on the structure. The gust response factors G[RF]for conductors are given in Table 250-3 as a function of height of attachment on the structure and conductor span length (see table 1). Wind Load on the Conductors To calculate the wind load on the conductors, we use the same equation for load in pounds except k[z]and G[RF]are different for conductors and the Area A is the cross-sectional area of the conductor for half the span in each direction. The moment for each conductor is the load multiplied times the height of attachment for that conductor. For the 7 No. 5 AW static wire, the height of attachment is 70′. From Table 250-2, k[z]is 1.20 for a conductor 70 feet above ground. From Table 250-3, GRF is 0.86 for a conductor 70′ above ground and spans shorter than 250′. Since the spans are all 200 feet, the conductor area in square feet is the conductor length in feet multiplied by the conductor diameter in feet. The conductor diameter in feet is 0.546″ / 12 = 0.0455. The conductor area A is then 200 x 0.0455 = 9.1 square feet. Wind load in pounds = 0.00256 x (115)^2x k[z]x G[RF]x A = 0.00256 x 13,225 x 1.20 x 0.86 x 9.1 = 317.9 The moment on the pole at the ground-line due to the extreme wind load on the static wire is 317.9 x 70′ = 22,253 foot-pounds We then calculate the moments due to the other conductors and add them all. At 38′, we have three 477 AAC conductors. Calculate the moment for one conductor and then multiply the answer by three. The total moment due to extreme wind on all the conductors is 221,764 foot-pounds compared to 90,462 for the structure. The large communications cables contribute 77,661 foot-pounds even though they are close to the ground. The total moment is 312,226 foot-pounds. Safety Factors Rule 260B1, page 175, states that “Structures shall be designed to withstand the appropriate loads multiplied by the overload factors in Section 25 without exceeding their strength multiplied by the strength factors of Section 26.” The safety factor is the overload factor divided by the strength factor. Table 253-2, page 174, gives us the overload factor of 1.33 for use with 250C loads (extreme wind loads). The corresponding strength factor of 1.0 is given in Table 261-1B, page 182. The safety factor is 1.33 / 1.0 = 1.33. The calculated moments must be multiplied by the safety before comparison to the pole strength. Total moment with safety factor = 312,226 x 1.33 = 415,260 foot-pounds. Pole Strength The pole strength or ultimate moment at the ground-line for a southern yellow pine pole is 2.111 multiplied by the cube of the pole circumference in inches. The minimum circumference at the ground-line = ( total moment / 2.111)^1/3 = (415,260 / 2.111)^1/3 = 58.16″ The minimum diameter = circumference / 3.1416 = 58.16 / 3.1416 = 18.51″ The minimum diameter for an 80-foot H1 class pole at ground line is 18.11″. It looks like we need an H2 pole (19.05″). Since we assumed an H1 pole for the wind loading on the pole calculation, we will have to recalculate for a H2 pole to be sure we are OK. If you have general questions about the NESC®, please call me at 302-454-4910 or e-mail me atdave.young@conectiv.com. National Electrical Safety Code® and NESC® are registered trademarks of the Institute of Electrical and Electronics Engineers. David Young	{"url":"https://iaeimagazine.org/2002/2002may/math-behind-extreme-wind-loading/","timestamp":"2024-11-10T05:55:16Z","content_type":"text/html","content_length":"128869","record_id":"<urn:uuid:74ed61bf-9472-4321-8bb7-da0dfe973b36>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00781.warc.gz"}
NumPy @ Operator vs np.dot() Both the @ operator and the dot function are pivotal for matrix multiplication. However, beginners and even some seasoned programmers might find themselves puzzled over which to use and when. What are the @ Operator and dot Function? NumPy, Python’s fundamental package for scientific computing, offers several ways to perform operations on arrays and matrices. π Among these, the @ operator and the dot function stand out for matrix multiplication. • The @ Operator: Introduced in Python 3.5, the @ operator is specifically designed for matrix multiplication. It’s a syntactic sugar that makes code involving matrix operations more readable and • The dot Function: The dot function in NumPy is used for dot products of vectors, multiplication of two matrices, and more. When to Use Each? Use @ for Matrix Multiplication: If you’re working solely with matrix multiplication, the @ operator is your go-to for its readability and simplicity. It’s perfect for operations where the intent is explicitly matrix multiplication, making your code easier to read and understand at a glance. β ¨ Use dot for Flexibility: The dot function is more flexible. Beyond matrix multiplication, it can handle dot products of vectors and multiplication between a scalar and an array. If your operations aren’t limited to matrix multiplication or if you’re working with versions of Python older than 3.5, dot is the more appropriate choice. Let’s dive into a fun example that clearly demonstrates the difference between the @ operator and the dot function in NumPy, using a scenario where we’re working with a small game development project. Imagine we have two matrices representing transformations applied to game characters: one for scaling their size and another for rotating them. We’ll see how both operators are used to apply these transformations. First, ensure you have NumPy installed: pip install numpy Here’s the minimal code example: import numpy as np # Transformation matrices for our game characters # Scaling matrix (to double the size) scaling_matrix = np.array([[2, 0], [0, 2]]) # Rotation matrix (90 degrees) rotation_matrix = np.array([[0, -1], [1, 0]]) # Position of our character in 2D space (x, y) character_position = np.array([1, 0]).reshape(2, 1) # Making it a column vector # Using the @ operator for a clear, straightforward matrix multiplication transformed_position_at = scaling_matrix @ rotation_matrix @ character_position # Using the dot function for the same operation transformed_position_dot = np.dot(np.dot(scaling_matrix, rotation_matrix), character_position) # Display the results print("Transformed Position with @:", transformed_position_at.flatten()) print("Transformed Position with dot:", transformed_position_dot.flatten()) In this playful example, we first define matrices for scaling and rotation, applying them to a character’s position to move them around the game world. The character starts at position (1, 0), and we want to double their size and rotate them 90 degrees. The @ operator example uses a clear and concise syntax that makes it evident we’re performing sequential matrix multiplications to transform the character’s position. In contrast, the dot function example achieves the same result but requires a more nested and slightly less readable approach. Both methods will give the same result, demonstrating their functional similarity despite the syntactic differences. This minimal example underscores the choice between @ and dot as largely a matter of code readability and stylistic preference, rather than functionality. Performance Differences π ‘ Is there a significant performance difference between the @ operator and the dot function? The answer is generally no. Under the hood, both perform the same matrix multiplication operation with similar efficiency. Performance might slightly vary depending on the context, but for most practical purposes, they are interchangeable in terms of speed. Syntax and Readability One of the main differences lies in syntax and readability: # Using the @ operator result = matrix1 @ matrix2 # Using the dot function result = numpy.dot(matrix1, matrix2) The @ operator allows for a cleaner and more intuitive expression of multiplication, especially when dealing with complex mathematical formulas. It reduces the cognitive load, making it easier for someone reading your code to understand your intentions. Compatibility Considerations While the @ operator is sleek and modern, it’s essential to remember it’s only available in Python versions 3.5 and above. For codebases that must remain compatible with earlier versions of Python, or when working in environments where you cannot guarantee the Python version, the dot function remains a reliable and compatible choice. Best Practices • Readability First: Opt for the @ operator when you’re focused on matrix multiplication to enhance code readability. • Consider Your Audience: Use the dot function in environments where Python versions earlier than 3.5 are still in use or when you need the extra flexibility it offers. • Performance Testing: If you’re in a situation where performance is critical, test both methods in your specific use case. However, remember that differences are likely to be minimal.	{"url":"https://blog.finxter.com/numpy-operator-vs-np-dot/","timestamp":"2024-11-04T11:21:56Z","content_type":"text/html","content_length":"71327","record_id":"<urn:uuid:37b9da87-003f-4fa4-96fc-cab9321cfc65>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00452.warc.gz"}
Summer School \| Connecticut Summer School in Number Theory Summer School The CTNT 2024 Summer School will take place June 10 – June 16. All talks during the summer school will be at the Pharmacy/Biology Building (PBB 129 and 131), and the coffee breaks will be outside of PBB 129. A campus map pointing to PBB can be found here (Google labels the building as the “School of Pharmacy”). Note: This program is open only to students who are currently attending colleges and universities in North America. Goals of the Summer School The organizers of the summer school hope that the students attending this event will learn fundamental ideas in contemporary number theory and have a sense of some directions of current research. For undergraduates, the summer school will expose them to topics not available in a typical college curriculum and we will encourage applications from students at institutions where advanced topics in number theory are not ordinarily taught. The school will provide a chance for participants to meet fellow students, as well as faculty, interested in number theory. Expected Background of Students • Undergraduate Students: a semester each of elementary number theory and abstract algebra. • Graduate Students: a year of abstract algebra, and a semester of algebraic number theory. Structure of the Summer School The summer school will take place at the Storrs campus of the University of Connecticut. Activities will be designed at two levels, targeting advanced undergraduate and beginning graduate students. Lectures will be scheduled so that a student can attend all lectures if desired, choosing according to their background and interests. The daily schedule in the summer school will be as shown in the following table. Time PBB 131 8:15 – 9 Breakfast 9 – 9:50 Mini-course A 9:50 – 10:10 Coffee Break 10:10 – 11 Mini-course B 11:10 – 12 Guest Lecture 12 – 2 Lunch 2 – 2:50 Mini-course C 3 – 3:30 Mini-course E 3:30 – 4:00 Break 4:00 – 4:50 Mini-course D 5 – 7 Dinner After 7 Evening sessions Lecture series Each day’s events at the summer school is as follows. The videos for the lectures can be found at this YouTube Channel. (Note: the mini-course on Adeles and Ideles was delivered on the board, and it was not recorded.) • Guest Lectures: Each day will have a plenary talk, where a number theorist will give an overview (accessible to advanced undergraduates and beginning graduate students) of a current trend in number theory. Titles of the lectures and speakers: ☆ June 11: Jeremy Teitelbaum (UConn) will speak on “Factoring with elliptic curves.” ☆ June 13: David Pollack (Wesleyan University) will speak on “Dirichlet’s theorem on primes in arithmetic progressions.” • Mini-course A: “Using Quadratic Reciprocity” by Keith Conrad (UConn). In this course, we will describe some ways in which quadratic reciprocity can be applied to solve problems in number theory and related areas of mathematics. We will also see how ideas that were introduced to prove quadratic reciprocity have been influential in the development of number theory. • Mini-course B: “Adeles and Ideles” by Lori Watson (Trinity College). In this course we will introduce the ring of the adeles and the idele group for the rational numbers Q. We will discuss the field of the p-adic numbers and its subring of the p-adic integers. We will then discuss the construction of the adeles and ideles and some of their uses. • Mini-course C: “Class Field Theory” by Christelle Vincent (University of Vermont). We will begin with a discussion of local class field theory, explaining the theory in some detail for concreteness. Then we will formalize what we have seen into an “abstract class field theory,” following Neukirch, and finally very briefly gesture at the global theory for number fields. • Mini-course D: “Introduction to Elliptic Curves” by Alvaro Lozano-Robledo (UConn). This will be an overview of the theory of elliptic curves, discussing the Mordell-Weil theorem, how to compute the torsion subgroup of an elliptic curve, the 2-descent algorithm, and what is currently known about rank and torsion subgroups of elliptic curves. □ Lecture 1 slides (note: the slides do not display correctly in some browsers — download a open a local copy in that case) • Mini-course E: “Computations in Number Theory.” This course will serve two purposes. First, we will learn how to use the software packages SageMath and Magma for number-theoretic computations (involving primes, number fields, Galois groups, elliptic curves, curves over finite fields, etc). In addition, the lectures will showcase examples where computations have been an integral part of published research. • Other sessions: Participants will have time scheduled outside of the lectures to discuss exercises or review lecture notes from the courses. Instructors and graduate assistants will be available to answer questions. We will also offer the following presentations: □ Beamer tutorial: we will cover basic guidelines for creating slide talks using Beamer. □ Graduate school preparation panel: we will give advice and answer questions about the process of applying to graduate school and choosing graduate programs. □ Graduate school advising panel: we will give advice and answer questions about the process of selecting a research area and picking a thesis advisor.	{"url":"https://ctnt-summer.math.uconn.edu/summer-school-24/","timestamp":"2024-11-13T07:32:16Z","content_type":"text/html","content_length":"63258","record_id":"<urn:uuid:2f45c4bb-60d3-4612-bdb0-0a35935d316c>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00631.warc.gz"}
What, Why & How do we Know ? Anthropogenic Warming ? I suggested in a recent Friends of Wisdom thread, that I didn’t really “care” whether global warming was caused by human activity, nor even whether it was “real”. The answer to neither question changes my belief that we should be concerned enough to work out what to do in response to the facts. That is, we can learn from history, but not in simple “we did that and caused this, therefore if we do this we can achieve that” kinda ways. Life’s just complicated enough. Anyway, at first glance, this graph (linked also via Jorn) looks like a pretty random distribution of historical temperature fluctuations …. until you notice the right end of the graph has years as it’s time axis, and the left has hundreds of thousands of years. No idea how good the source data or its representation are, or even whether the western equatorial pacific temperature is a representative data point, but the graph is indeed scary. It’s 400,000 years since we had a period with average temperatures like the last 5 years, and for the last 100 years we’ve been 2 or 3 standard deviations higher than the long time average for the last 1,350,000 years, a period covering several ice-ages and retreats !!! Plenty of caveats about the distortion of a graph with such a skewed distribution of data points and axes, but it still certainly seems significant. 2 thoughts on “Anthropogenic Warming ?” 1. “That is, we can learn from history, but not in simple “we did that and caused this, therefore if we do this we can achieve that” kinda ways.” Just what are the other “ways”? We are always looking for parallels. It’s the way our minds work. But is it at all accurate? or do we make it seem so? I agree, life is complicated, complicated by humans mostly. 2. Hi Alice, The other ways are … to observe history to better understand underlying mechanisms, and better predict outcomes, understand the limits to predictability and devise better startegies for the probable possibilties, … … rather than assuming causal relationships between pre-conditions and outcomes we just happen to have seen before. We look for parallels and patterns, but we must remember that they may be in levels other than those we can observe. (BTW I notice this is the same curve that was publicised over a year ago and subjected to some US official review.) This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"url":"https://www.psybertron.org/archives/1322","timestamp":"2024-11-06T06:12:35Z","content_type":"text/html","content_length":"88877","record_id":"<urn:uuid:06632e4f-788e-4167-9eb3-a0251e471f47>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00788.warc.gz"}
Mortgage Payoff Club!! Wow rjb you're kicking some ass. I think the hard part would be holding $40k cash while eyeing the (40k) mortgage bal. Especially as the spread widens & it becomes 42k vs (38), 45 vs (35) and so on.. Of course, that's assuming you have actual cash savings instead of securities or qualified holdings. At some point I'd be tempted to simply wipe out the mortgage & still have $X cash on hand.	{"url":"https://forum.mrmoneymustache.com/throw-down-the-gauntlet/mortgage-payoff-club!!/msg873366/","timestamp":"2024-11-01T19:25:15Z","content_type":"application/xhtml+xml","content_length":"184931","record_id":"<urn:uuid:23cc7c5c-9e5c-4b3b-9c73-dad8283bd352>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00372.warc.gz"}
American Mathematical Society General position properties satisfied by finite products of dendrites HTML articles powered by AMS MathViewer Trans. Amer. Math. Soc. 288 (1985), 739-753 DOI: https://doi.org/10.1090/S0002-9947-1985-0776401-5 PDF \| Request permission Let $\bar A$ be a dendrite whose endpoints are dense and let $A$ be the complement in $\bar A$ of a dense $\sigma$-compact collection of endpoints of $\bar A$. This paper investigates various general position properties that finite products of $\bar A$ and $A$ possess. In particular, it is shown that (i) if $X$ is an $L{C^n}$-space that satisfies the disjoint $n$-cells property, then $X \times \ bar A$ satisfies the disjoint $(n + 1)$-cells property, (ii) ${\bar A^n} \times [ - 1,1]$ is a compact $(n + 1)$-dimensional ${\text {AR}}$ that satisfies the disjoint $n$-cells property, (iii) ${\ bar A^{n + 1}}$ is a compact $(n + 1)$-dimensional ${\text {AR}}$ that satisfies the stronger general position property that maps of $n$-dimensional compacta into ${\bar A^{n + 1}}$ are approximable by both $Z$-maps and ${Z_n}$-embeddings, and (iv) ${A^{n + 1}}$ is a topologically complete $(n + 1)$-dimensional ${\text {AR}}$ that satisfies the discrete $n$-cells property and as such, maps from topologically complete separable $n$-dimensional spaces into ${A^{n + 1}}$ are strongly approximable by closed ${Z_n}$-embeddings. References • R. D. Anderson, D. W. Curtis, and J. van Mill, A fake topological Hilbert space, Trans. Amer. Math. Soc. 272 (1982), no. 1, 311–321. MR 656491, DOI 10.1090/S0002-9947-1982-0656491-8 P. L. Bowers, Applications of general position properties of dendrites to Hilbert space topology, Ph.D. Dissertation, Univ. of Tennessee, 1983. • Philip L. Bowers, Discrete cells properties in the boundary set setting, Proc. Amer. Math. Soc. 93 (1985), no. 4, 735–740. MR 776212, DOI 10.1090/S0002-9939-1985-0776212-6 • J. W. Cannon, Shrinking cell-like decompositions of manifolds. Codimension three, Ann. of Math. (2) 110 (1979), no. 1, 83–112. MR 541330, DOI 10.2307/1971245 • T. A. Chapman, Lectures on Hilbert cube manifolds, Regional Conference Series in Mathematics, No. 28, American Mathematical Society, Providence, R.I., 1976. Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975. MR 0423357 D. W. Curtis, Boundary sets in the Hilbert cube, preprint. —, Preliminary report, boundary sets in the Hilbert cube and applications to hyperspaces, preprint. • Robert J. Daverman, Detecting the disjoint disks property, Pacific J. Math. 93 (1981), no. 2, 277–298. MR 623564 • Robert J. Daverman and John J. Walsh, Čech homology characterizations of infinite-dimensional manifolds, Amer. J. Math. 103 (1981), no. 3, 411–435. MR 618319, DOI 10.2307/2374099 • Tadeusz Dobrowolski and Henryk Toruńczyk, On metric linear spaces homeomorphic to $l_{2}$ and compact convex sets homeomorphic to $Q$, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 11-12, 883–887 (1981) (English, with Russian summary). MR 616181 • James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606 R. D. Edwards, Approximating certain cell-like maps by homeomorphisms, Abstract preprint. See also Notices Amer. Math. Soc. 24 (1977), A649, #751-G5. • Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493 • Jan van Mill, A boundary set for the Hilbert cube containing no arcs, Fund. Math. 118 (1983), no. 2, 93–102. MR 732657, DOI 10.4064/fm-118-2-93-102 • Frank Quinn, Ends of maps. I, Ann. of Math. (2) 110 (1979), no. 2, 275–331. MR 549490, DOI 10.2307/1971262 • K. Sieklucki, A generalization of a theorem of K. Borsuk concerning the dimension of $ANR$-sets, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 433–436. MR 198430 • H. Toruńczyk, On $\textrm {CE}$-images of the Hilbert cube and characterization of $Q$-manifolds, Fund. Math. 106 (1980), no. 1, 31–40. MR 585543, DOI 10.4064/fm-106-1-31-40 • H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), no. 3, 247–262. MR 611763, DOI 10.4064/fm-111-3-247-262 • Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581 Bibliographic Information • © Copyright 1985 American Mathematical Society • Journal: Trans. Amer. Math. Soc. 288 (1985), 739-753 • MSC: Primary 54F50; Secondary 54C25, 54C35, 54F35 • DOI: https://doi.org/10.1090/S0002-9947-1985-0776401-5 • MathSciNet review: 776401	{"url":"https://www.ams.org/journals/tran/1985-288-02/S0002-9947-1985-0776401-5/?active=current","timestamp":"2024-11-04T02:08:58Z","content_type":"text/html","content_length":"68192","record_id":"<urn:uuid:6fe11607-b97d-40ca-a12e-5480e19ce3a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00723.warc.gz"}
Physics - Online Tutor, Practice Problems & Exam Prep (III) The potential energy of the two atoms in a diatomic (two-atom) molecule can be approximated as (Lennard-Jones potential) U(r) = -(a/r⁶) + (b/r¹²) , where r is the distance between the two atoms and a and b are positive constants. (e) Let F be the force one atom exerts on the other. For what values of r is F > 0 , F < 0 , F = 0?	{"url":"https://www.pearson.com/channels/physics/explore/conservation-of-energy/force-potential-energy?chapterId=0214657b","timestamp":"2024-11-13T02:58:21Z","content_type":"text/html","content_length":"448661","record_id":"<urn:uuid:d617b5f9-5983-4093-a8c4-c9e80c2c3401>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00837.warc.gz"}
What's another way to think about the increment and decrement operators? \| Sololearn: Learn to code for FREE! What's another way to think about the increment and decrement operators? I'm having some trouble understanding the difference between ++X and X++. Say that X=0. I know that in X++, X = 0 becomes X = 1. But what about ++X? Is it still X=1? And what if there's an equation that follows. Say X=0 Y=5, Y=+ ++X; What would be the answer? I'm not sure if this saying is correct for all cases... But here goes the simple method: For ++X, whatever you do, you increment value of X first, before doing other computations. For X++, whatever you do, you do first, before incrementing value of X. ------ E.g. int X = 0; cout << X++; // prints X, and then X become 1 // outputs 0 ------ int X = 0; cout << ++X; // X become 1, and then prints X // outputs 1 here's a simple example : int x =0; //here we use the last value that is assigned to x and that's 0 then increase it to 1. cout << x++ << endl; //0 //here we increase the value of x first, then print it out . cout << ++x << endl; //2 hope this help đ	{"url":"https://www.sololearn.com/en/discuss/306863/what-s-another-way-to-think-about-the-increment-and-decrement-operators","timestamp":"2024-11-02T08:52:07Z","content_type":"text/html","content_length":"918192","record_id":"<urn:uuid:698f3fa0-bffc-409d-a675-33e91acb8c7b>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00835.warc.gz"}
wp34s (Why it's so accurate) I looked into the source code for the wp34s and discovered something that I had not been familiar with before. The IEEE came up with a cool way to store 16 decimal digits with an exponent of 10^+384 to 10^-383 into only 64bits. This format is defined by the IEEE 754 specification and with it the wp34s does all of its decimal calculations on base 10 decimal digits directly! The reason many floating point libraries and older calculators get all of these inaccuracies is because they compute things in the floating point binary number system, then convert the answer back into decimal digits at the end where rounding errors creep in. When you try to represent a decimal number in a binary number system the least significant 2 or 3 digits can get errors in them. This is because the binary representation does not accurately translate into decimal numbers for every digit. The wp34s keeps it's full 16-digit decimal numbers in a decimal format for each and every digit THROUGHOUT EVERY STEP of every calculation, so there are almost never any rounding errors. At least there are no errors that result from converting numbers to and from decimal and binary. If you are curious as to how the wp34s fits 16 decimal digits and an exponent of 10^+384 to 10^-383 all into a 64 bit space look at this nice page on Wikipedia Decimal64 When you put your wp34s into Double Precision mode with [h][mode]DBLON it uses the Decimal 128 format where it fits 34 decimal digits plus an exponent of 10^+6144 to 10^-6143 all into 128 bits. Wikipedia Decimal128 I find this technology fascinating. Thanks for making the wp34s so accurate! Edited: 4 Aug 2013, 1:52 a.m. 08-04-2013, 02:26 AM Actually, most calculators use BCD, just not with 64 bits. 08-04-2013, 02:42 AM Quote: Thanks for making the wp34s so accurate! That goes to Pauli :-) Quote: I looked into the source code for the wp34s and discovered something ... ... ... You could have reached that level easier - see below ;-) Quote:If you are curious as to how the wp34s fits 16 decimal digits and an exponent of 10^+384 to 10^-383 all into a 64 bit space look at ... ... pp. 179f of the printed manual :-) Quote: When you put your wp34s into Double Precision mode with [h][mode]DBLON it uses the Decimal 128 format where it fits 34 decimal digits plus an exponent of 10^+6144 to 10^-6143 all into 128 bits. Please see p. 211 of the printed manual :-) Quote:I find this technology fascinating. Me too. It's called "reading" ;-) 08-04-2013, 03:37 AM We're using the decNumber library of which the decimal64 and decimal128 formats are part of. Computations aren't executed directly on the decimalXXX objects, they are transformed into decNumber objects. The latter are much more memory consuming but have important advantages: easier access to the digits, signs and flags, and arbitrary precision. The decimalXXX format is used for storage of numbers in registers because it's more compact. We are loosing some digits in the process because internal computations are done with even higher accuracy (typically 39 digits, sometimes more) and the results are converted back to 16 or 34 digits, depending on the register size. To be precise, we skipped a memory consuming step in the conversion and are using a slightly simplified method to encode three digits into 10 bits. We are actually using a base 1000 format with a base 10 exponent. It's still named decimal64 or decimal128 internally but you cannot directly exchange the data with other applications using the original encoding. Edited: 4 Aug 2013, 3:51 a.m. 08-04-2013, 05:34 AM And those that do use 64 bit BCD arithmetic (the twelve digit HPs e.g.) don't use a tightly packed format like the 34S. This is a trade off. Alpha strings cannot be distinguished from numbers using packed decimals like they can on the 41 series. Likewise, matrix descriptors cannot be distinguished from numbers like they are on the 15c. - Pauli 08-04-2013, 09:21 AM Yes but most calculators don't incorporate the advanced rounding rules that are incorporated into the wp34s! Rounding can make a big difference, causing the Least Significant Digit to be wrong in many cases. Take a look at the HP-15C. It frequently gets the Least Significant Digit wrong giving values such as 4.999999999 , or 5.000000001 instead of 5.0. I never see these rounding inaccuracies on the wp34s. 08-04-2013, 09:33 AM I read the manual, and didn't find the description there as comprehensible as the Wikipedia article referenced above. It made sense to me after stumbling upon the Wikipedia article, but still seemed confusing when I tried to interpret this theory directly from the manual. I am not saying that the manual isn't well written, only that occasionally looking at something from another perspective can often unlock the missing pieces of a complete understanding. Thanks again for creating this open-source collaboration! This calculator is a good example of what mankind can do when he cooperates with his fellow man seeking excellence over profit! I think we have all sensed that Hewlett Packard isn't the company it used to be. They used to build-in this kind of care and they used to STRIVE for excellence in their products. I miss that kind of workmanship, and it's obvious that the creators of the wp34s built the kind of calculator that they think HP would have built with today's technology, if they still cared about excellence the way they used to. Edited: 4 Aug 2013, 9:57 a.m. 08-04-2013, 09:53 AM Base 1000 cool! That seems MUCH simpler than the IEEE 754 method of encoding 3-digits into 10-bits. Very cleaver. When too many people try to decide things that are to be a standard in a big committee it always seems to be more bureaucratic and less innovative. Thanks Marcus for the description of how it's actually done in the wp34s. This calculator reminds me of an onion. The more layers I unwrap (comprehend) the more layers of cool technology I find beneath them. Edited: 4 Aug 2013, 9:53 a.m. 08-04-2013, 12:10 PM Quote: Take a look at the HP-15C. It frequently gets the Least Significant Digit wrong giving values such as 4.999999999 , or 5.000000001 instead of 5.0. Could you provide an example for this? I do not think the 15C (or any other HP) is inaccurate or does not round correctly. In most cases it's simply the fact that even in single precision mode the 34s uses 16 digits while only the first 12 are displayed. ;-) 08-04-2013, 12:54 PM Quote: I think we have all sensed that Hewlett Packard isn't the company it used to be. They used to build-in this kind of care and they used to STRIVE for excellence in their products. I miss that kind of workmanship... Yes. Workmanship like the 'excellent' Woodstock battery charging calculator self-destruct system, the early Spice PCB problems and Spice battery compartment flimsiness, HP-67 potential damage if used with AC charger but no battery, Clamshell battery compartment defects, etc. All of them retailed at very high prices. I used an HP for the first time in 1972, when the HP-35 showed up at the Georgia Tech Bookstore. Having experience with HP from the beginning, I don't buy the mythology of HP perfection in "the good ol' days". I very much like my 1988 HP42S and 2006 HP 50G, and I'm looking forward to the 2013 HP Prime. 08-04-2013, 01:27 PM Example of rounding errors in HP-15C! Using the HP-15C, put the 2x2 matrix 1234 into matrix A and take the inverse of it and put it into Matrix B then take the inverse Matrix B and put it into Matrix C. Then look at the rounding errors in Matrix C. You get 1.000000001,2.000000001, 3.000000002, 4.000000002 On the wp34s you get exact values with no rounding errors doing the same thing. Edited: 4 Aug 2013, 1:38 p.m. 08-04-2013, 01:41 PM I have an original HP-35 and when HP discovered that they had made a mistake in the firmware (2.02 ln e^x showed 2.0), they offered to repair it for free! This kind of customer care does not exist This is what I was referring to when I said that HP isn't the company that it used to be. Edited: 4 Aug 2013, 1:44 p.m. 08-04-2013, 02:06 PM The HP-35 launched at $395 in 1972 which works out to be over $2000 in today's dollars. The cost of a product does factor into the decisions any company makes about the level of support they offer. It isn't really valid to compare HP's support for the original HP-35 to its support for today's sub-$200 devices. 08-04-2013, 02:30 PM Well, that's easy to explain. On the 34s, most internal calculations are done with 39-digit precision, sometimes even more, and even the XROM-coded functions use 34-digit precision. Since the matrix routines always return single-precision results, there are 39 minus 16 = 23 (!) additional guard digits. Plenty of room to cover roundoff errors. ;-) Compare this to just 3 (three) additional digits on most HP machines (e.g. HP-15C = 10 digit display => 13 digits internal precision). So 9 correct digits out of 13 used internally does not look too bad to me. If the results of the 34s matrix functions could be examined in all of their 39 internal digits, you should not be surprised to see some roundoff errors in the 35th digit either. ;-) 08-04-2013, 04:18 PM Quote: If the results of the 34s matrix functions could be examined in all of their 39 internal digits, you should not be surprised to see some roundoff errors in the 35th digit either. ;-) There is some reason why WP 34S matrix commands don't work in DP ;-) Anyway, I think most people can well live with matrix results being accurate to 16 digits. 08-04-2013, 05:33 PM The matrix routines are more complicated. True, the full internal 39 digit precision is used for basic computations, however many intermediates are stored and accumulated in 34 digit precision to conserve the very limited RAM. - Pauli 08-04-2013, 05:42 PM Encoding 3 digits in ten bits is base one thousand still. This is one case where the standard is extremely well thought out. Read about the history of the IEEE 754 standard. Although, politics was involved, the technically best solution made it in the end. Interview with Kahan and lots of documents from his web page. The IEEE 854 standard continued making the right decisions and was subsequently reintegrated back into the 754 standard as the base ten implementation. - Pauli Edited: 5 Aug 2013, 6:13 p.m. after one or more responses were posted 08-04-2013, 08:45 PM While it is true that the calculator was expensive for that time period, I don't believe HP has been as magnanimous, even with it's expensive products lately. I like the HP-30B, it makes a good platform on which to build the wp34s, but I remember back to he 60's and 70's where the name HP was synonymous with "HIGH QUALITY TEST EQUIPMENT". Today, I can find better quality products at cheaper prices than what is provided by HP in most cases. I stand by my assertion that Hewlett Packard isn't the Pillar of Integrity and Quality that it once was. 08-04-2013, 09:10 PM I still contend that the wp34s IS MORE ACCURATE than the HP-15C! Weather the wp34s is more accurate because it shows 12 of 16 digits by default, or because some operations are performed to 39 digits of precision only supports my claim that the calculator SEEMS MORE ACCURATE to the end user. I also believe that the sophisticated user configurable rounding rules shown on page 110 of the printed manual further support the claim that the authors are STRIVING for maximum accuracy with minimum rounding issues. It don't see why this issue should be disputed. Edited: 4 Aug 2013, 9:38 p.m. 08-04-2013, 09:57 PM I think what Marcus said about using base 1000 encoding to represent the 3 digits into 10 bits is simpler than the complex encoding table shown in the IEEE754 spec. Marcus encodes the 3-digits 0-999 as hex values 0-3e7h which still gives 3 full digits in 10-bits, but doesn't require all of the table lookups that are needed to decode the declet encoding as shown in the IEEE754 I concur that the remainder of the IEEE754 spec is very cleaver and yields maximum number ranges in a minimum of bits. But I can also see why Marcus deviated from the spec in this one area as it vastly simplifies the declet encoding and decoding process. 08-04-2013, 11:47 PM The densely packed decimal encoding is also very clever and provides a number of benefits when used. In the 34S, we didn't need any of these and we saved a kilobyte or two of flash. It really isn't complex, it involved a single table look up per ten bits when converting either way. For us, by far the largest benefit was the space saving. It would have been nice to be able to say we support the IEEE standard at the bit level, but we don't. Our arithmetic is equivalent otherwise but we really needed the space. - Pauli 08-05-2013, 08:47 AM Please see that the links in your post are not working. If possible, please suggest alternative ways to access such interesting materials. Thank in advance, and - of course - for your work in the wonderful HP34S. Best regards, 08-05-2013, 08:51 AM Quote: Please see that the links in your post are not working. Just one http too much. ;-) Interview with Kahan and lots of documents from his web page.	{"url":"https://archived.hpcalc.org/museumforum/showthread.php?mode=linear&tid=247738&pid=247745","timestamp":"2024-11-11T00:11:31Z","content_type":"application/xhtml+xml","content_length":"88795","record_id":"<urn:uuid:bbdc06d6-cd7d-45bc-ba89-7c860f942f16>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00029.warc.gz"}
Clep college mathematics online practise clep college mathematics online practise Related topics: lcm.c# mathematical operations with arrays college prep algebra solving proportions the exponential clothesline applications of quadratic formula polynomial factorer calculator number to the powers of a fraction algebra equasion solver math solutions monomial factors order of operation + hands on activities Author Message AL Posted: Thursday 26th of Oct 09:25 I need some help friends ! I find clep college mathematics online practise really tough. I have tried finding a teacher for the subject, but couldn’t find any in my area. The ones available are far and costly . From: Texas Ya'll Back to top espinxh Posted: Friday 27th of Oct 14:08 What in particular is your trouble with clep college mathematics online practise? Can you give some additional surmounting your problem with unearthing a tutor at an reasonable cost is for you to go in for a apt program. There are a variety of programs in algebra that are easily reached. Of all those that I have tried out, the the top most is Algebrator. Not only does it answer the math problems, the good thing in it is that it makes clear each step in an easy to follow manner. This guarantees that not only you get the exact answer but also you get to learn how to get to the answer. From: Norway Back to top 3Di Posted: Sunday 29th of Oct 12:16 Algebrator is the perfect math tool to help you with assignments . It covers everything you need to be familiar with in roots in an easy and comprehensive way . Math had never been easy for me to grasp but this product made it easy to understand . The logical and step-by–step approach to problem solving is really a plus and soon you will find that you love solving problems. From: 45°26' N, 09°10' E Back to top Dreh Posted: Monday 30th of Oct 20:09 Thank you, I will try the suggested program . I have never studied with any program until now, I didn't even know that they exist. But it sure sounds great ! Where did you find the software ? I want to purchase it right away, so I have time to get ready for the exam. Back to top Hiinidam Posted: Wednesday 01st of Nov 13:22 Algebrator is the program that I have used through several algebra classes - Algebra 2, Algebra 2 and Algebra 2. It is a truly a great piece of algebra software. I remember of going through difficulties with inequalities, graphing lines and linear equations. I would simply type in a problem from the workbook , click on Solve – and step by step solution to my algebra homework. I highly recommend the program. From: Greeley, CO, Back to top CHS` Posted: Wednesday 01st of Nov 17:02 Cool, I think their actual link is: https://softmath.com/algebra-features.html. Oh!, and when you look at the page check their unconditional guarantee! Good luck man write here again if you need anything. From: Victoria City, Hong Kong Island, Hong Kong Back to top	{"url":"https://softmath.com/algebra-software-4/clep-college-mathematics.html","timestamp":"2024-11-01T22:06:26Z","content_type":"text/html","content_length":"43240","record_id":"<urn:uuid:bfcbbc77-e433-4fd1-a873-b503e507ec2d>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00008.warc.gz"}
Introduction to Applied Statistics for Psychology Students As a broad introduction, the Figure 8.5 : The As the degrees of freedom, t Distribution Table, Similar, to the where, now With this new formula for Section 8.1: Confidence Intervals using the z-distribution and, of course, replaced t Distribution Table in the column for Figure 8.6 : Derivation of confidence intervals for means of small samples. Example 8.2 : Given the following data: find the 99% confidence interval for the mean. Solution : First count Using the t Distribution Table with With these numbers, compute is the 99	{"url":"https://openpress.usask.ca/introtoappliedstatsforpsych/chapter/8-3-the-t-distributions/","timestamp":"2024-11-07T04:29:38Z","content_type":"text/html","content_length":"119092","record_id":"<urn:uuid:6bf71c6e-505f-4141-82e9-f019f0ac6250>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00647.warc.gz"}
How do you find the vertical, horizontal and slant asymptotes of: f(x)=(2x)/(x^2+16)? \| HIX Tutor How do you find the vertical, horizontal and slant asymptotes of: #f(x)=(2x)/(x^2+16)#? Answer 1 horizontal asymptote at y = 0 The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes. solve: #x^2+16=0rArrx^2=-16# This has no real solutions hence there are no vertical asymptotes. Horizontal asymptotes occur as #lim_(xto+-oo),f(x)toc" (a constant)"# divide numerator/denominator by the highest power of x, that is #x^2# as #xto+-oo,f(x)to0/(1+0)# #rArry=0" is the asymptote"# Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here ( numerator-degree 1 , denominator-degree 2 ) Hence there are no slant asymptotes. graph {(2x)/(x^2+16) [-10, 10, -5, 5]} Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer 2 To find the vertical asymptotes, set the denominator equal to zero and solve for ( x ). In this case, ( x^2 + 16 = 0 ) has no real solutions, so there are no vertical asymptotes. To find the horizontal asymptote, compare the degrees of the numerator and denominator. Since the degree of the numerator (which is 1) is less than the degree of the denominator (which is 2), the horizontal asymptote is at ( y = 0 ). To find the slant asymptote, perform polynomial long division. Divide the numerator by the denominator. The quotient will be the equation of the slant asymptote. Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer from HIX Tutor When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some Not the question you need? HIX Tutor Solve ANY homework problem with a smart AI • 98% accuracy study help • Covers math, physics, chemistry, biology, and more • Step-by-step, in-depth guides • Readily available 24/7	{"url":"https://tutor.hix.ai/question/how-do-you-find-the-vertical-horizontal-and-slant-asymptotes-of-f-x-2x-x-2-16-8f9afa5394","timestamp":"2024-11-02T04:20:34Z","content_type":"text/html","content_length":"573327","record_id":"<urn:uuid:3a6a0923-80d0-46a4-8483-947264b43446>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00026.warc.gz"}
Exploring Common Genetic Algorithm Problems and Possible Solutions Challenges and Solutions in Genetic Algorithm Optimization In the field of search and optimization, genetic algorithms are powerful problem-solving techniques inspired by the process of natural selection. By mimicking the principles of evolution, genetic algorithms can efficiently search for the optimal solution to a given problem. However, like any other algorithm, genetic algorithms also face their own set of challenges and problems. In this article, we will explore some common genetic algorithm problems and discuss possible solutions. One of the key components of a genetic algorithm is the mutation operator. This operator introduces random changes in the genetic makeup of individuals in the population, allowing for exploration of different solutions. However, if the mutation rate is too low, the algorithm may get stuck in a local optimum and fail to reach the global optimum. On the other hand, a high mutation rate can lead to excessive exploration, which slows down the convergence of the algorithm. Finding the optimal mutation rate is therefore crucial for the success of a genetic algorithm. Another challenge in genetic algorithms is ensuring the diversity of the population. A diverse population is necessary for effective exploration of the search space. Without diversity, the algorithm may converge prematurely and miss out on potentially better solutions. To maintain diversity, various techniques can be employed, such as elitism, where the best individuals are preserved in each generation, and crossover operators that promote the exchange of genetic material between individuals. The fitness function used to evaluate the quality of individuals is another important aspect of genetic algorithms. The fitness function should accurately reflect the problem’s objective and provide a measure of how well an individual satisfies it. However, designing an appropriate fitness function can be challenging, especially for complex problems where the optimal solution is not well-defined. In such cases, heuristic techniques, such as penalizing invalid solutions or incorporating domain knowledge, can be applied to guide the evolution process. Overall, genetic algorithms offer a powerful approach to solving optimization problems. By understanding and addressing the common problems associated with genetic algorithms, researchers and practitioners can enhance their effectiveness and harness the full potential of these algorithms in various domains. Basic Concepts of Genetic Algorithms Genetic algorithms are a class of optimization algorithms inspired by the process of natural evolution. They are an effective tool for solving complex problems that are hard to solve with traditional optimization techniques. Evolutionary Process In genetic algorithms, a population of potential solutions evolves over time through a process that mimics the principles of natural evolution. This process includes mechanisms such as selection, reproduction, mutation, and recombination. Genetic Representation Each potential solution in a genetic algorithm is represented as a string of symbols, which can be thought of as the genes of an individual. These symbols can represent various characteristics or parameters of the solution, depending on the problem being solved. For example, in a genetic algorithm for optimizing a mathematical function, the genes may represent the values of the variables in the function. In other applications, the genes could represent binary strings that encode a potential solution. Fitness Evaluation The fitness of each individual in the population is evaluated based on a fitness function. This function measures how well the individual solves the problem at hand. Individuals with higher fitness values are more likely to be selected for reproduction and have their genes passed on to the next generation. Mutation and Recombination Genetic algorithms introduce variation in the population through the processes of mutation and recombination. Mutation involves randomly changing one or more genes in an individual, while recombination combines genes from two or more individuals to create new individuals with a mix of characteristics from their parents. Search and Optimization The main goal of genetic algorithms is to search for the optimal solution within a large and complex search space. By using mechanisms such as selection, reproduction, mutation, and recombination, genetic algorithms explore the search space efficiently and converge towards the best solution found so far. • Problems that can benefit from genetic algorithms include: • Combinatorial optimization problems • Machine learning and data mining • Scheduling problems • Resource allocation problems In conclusion, genetic algorithms offer a powerful approach for solving optimization problems through the mimicry of natural evolution. By representing potential solutions as genetic strings, evaluating their fitness, and introducing variation through mutation and recombination, genetic algorithms efficiently explore complex search spaces and converge towards optimal solutions. Fitness Function and Selection Methods In a genetic algorithm, the fitness function is a crucial component that evaluates the quality of individuals in the population. It assigns a numerical value, known as the fitness value, to each individual based on how well it solves the problem at hand. The main goal of the genetic algorithm is to optimize the fitness of individuals over time through the process of evolution. The fitness function measures the ability of an individual to survive and reproduce in a given environment. It encapsulates the problem-specific criteria and objectives by which individuals are evaluated. For example, in a search problem, the fitness function could be defined such that individuals that come closer to the target solution have higher fitness values. Different genetic algorithms can have different types of fitness functions. Some common types include the binary fitness function, which is suitable for problems with binary strings, and the real-valued fitness function, which is applicable to problems with real-valued parameters. The fitness function should be carefully designed to accurately capture the problem constraints and Selection methods, on the other hand, determine which individuals are chosen as parents for the next generation. These methods are responsible for preserving the fittest individuals in the population and promoting a more diverse population over time. The selection process mimics the idea of natural selection, where individuals with higher fitness have a higher chance of passing their genetic information to the next generation. There are several selection methods commonly used in genetic algorithms, such as tournament selection, roulette wheel selection, and rank-based selection. Tournament selection involves randomly selecting a subset of individuals and choosing the one with the highest fitness. Roulette wheel selection assigns a probability to each individual based on its fitness, and individuals are selected probabilistically. Rank-based selection assigns a rank to each individual based on its fitness and selects individuals based on their rank. Overall, the fitness function and selection methods play crucial roles in the genetic algorithm’s ability to solve problems through evolution and optimization. A well-designed fitness function and effective selection methods can greatly influence the algorithm’s performance and ability to find optimal solutions in various problem domains. Representation of Solutions in Genetic Algorithms In genetic algorithms (GAs), a solution to a problem is typically represented as a string of binary digits, called a chromosome or a genotype. This representation allows for an easy manipulation and evolution of solutions. Evolution and Mutation The genetic algorithm is an optimization and search algorithm inspired by the process of natural evolution. Just like in nature, GAs start with a population of individuals, each representing a possible solution to the problem at hand. This population evolves over generations through a process that involves selection, crossover, and mutation. Mutation plays a crucial role in maintaining genetic diversity within the population. It introduces small random changes into the chromosomes, allowing the algorithm to explore new areas of the solution space. Without mutation, the optimization process could get stuck in local optima and fail to find the global optimum. Fitness and Evaluation In order to guide the evolution process, each individual in the population is assigned a fitness value, which quantifies its performance or adequacy as a solution to the problem. The evaluation of fitness is typically based on an objective function, which measures how well the solution satisfies the problem constraints or goals. The fitness function is problem-specific and needs to be designed carefully to ensure the algorithm’s effectiveness. It should be able to distinguish between good and bad solutions, providing a clear direction for the search towards better solutions. During the evolution process, individuals with higher fitness values have a better chance of reproducing and passing their genetic material to the next generation through crossover and mutation operators. This mechanism mimics the natural selection process, favoring the propagation of beneficial traits and gradually improving the overall population. Overall, the representation of solutions in genetic algorithms is a fundamental aspect of their design. It allows for the efficient exploration of the solution space and the optimization of complex problems. By combining the principles of evolution and mutation with fitness evaluation, GAs can find high-quality solutions to a wide range of problems. Crossover and Mutation Operators In genetic algorithms, crossover and mutation are two important operators used for creating new candidate solutions in the evolution process. These operators play a crucial role in the exploration and optimization of common problems. Crossover is the process of combining genetic material from two parent solutions to create one or more offspring solutions. It mimics the biological process of reproduction and introduces variation into the population. Crossover helps to explore different regions of the search space and can potentially combine beneficial traits from both parents. Mutation, on the other hand, introduces small random changes to a candidate solution. This operator helps to maintain diversity within the population and prevents the algorithm from becoming stuck in local optima. By occasionally perturbing the genetic material, mutation allows for the exploration of potentially better solutions that may have been overlooked. Both crossover and mutation operators are guided by the fitness function of the problem at hand. The fitness function determines how well a candidate solution performs in terms of the optimization objective. The genetic algorithm uses the fitness function to evaluate the quality of the solutions and guide the evolution process towards more optimal solutions. While crossover and mutation operators are generally effective, their effectiveness can be influenced by various factors such as the choice of crossover and mutation rates, the problem representation, and the nature of the problem itself. It is important to experiment with different strategies and parameters to find the best configuration for a specific problem. Overall, crossover and mutation operators are key components of genetic algorithms that enable the exploration and optimization of common problems. They provide the means for the algorithm to adapt and evolve solutions over generations, leading to improved results and better solutions. Problems with Premature Convergence The genetic algorithm is a powerful search and optimization algorithm inspired by the process of natural selection and genetics. It involves the generation and manipulation of a population of individuals, each represented by a chromosome, to find the best solution to a given problem. However, genetic algorithms can sometimes suffer from a phenomenon known as premature convergence, where the algorithm finds a suboptimal solution and gets stuck there instead of finding the true global optimum. This can happen due to various reasons: Insufficient fitness evaluation: If the fitness function used to evaluate the individuals in the population is not well-suited to the problem at hand, it can lead to premature convergence. A poorly designed fitness function may not accurately reflect the quality of a solution, leading the algorithm to favor individuals that are actually suboptimal. Lack of genetic diversity: Genetic algorithms rely on genetic operators like mutation and crossover to introduce new genetic material into the population and explore the search space. If the genetic diversity in the population is low, it can limit the algorithm’s ability to explore different regions of the search space and can lead to premature convergence on a local optimum. Improper selection pressure: Selection pressure in the genetic algorithm determines how individuals are selected for reproduction and which individuals contribute more genetic material to the next generation. If the selection pressure is too high, the algorithm may converge too quickly, trapping the population in a suboptimal region. On the other hand, if the selection pressure is too low, it may take longer for the algorithm to converge to the optimal solution. Ineffective genetic operators: The mutation and crossover operators play a crucial role in exploring the search space and introducing new genetic material into the population. If these operators are not effective in generating diverse offspring, it can limit the algorithm’s ability to escape local optima and can lead to premature convergence. To address the problem of premature convergence, several techniques can be employed. One approach is to use adaptive operator control, where the parameters of the genetic operators are dynamically adjusted during the evolution process based on the population’s behavior. Another approach is to introduce diversity maintenance techniques, such as elitism, where the best individuals are preserved across generations to prevent the loss of promising solutions. Overall, understanding and addressing the problems associated with premature convergence is crucial for the successful application of genetic algorithms in various optimization problems. Solutions to Premature Convergence Premature convergence is a common problem in genetic algorithms, where the algorithm becomes trapped in a suboptimal solution before reaching the global optimum. This can happen due to various factors, such as a narrow search space, lack of genetic diversity, or poor fitness evaluation. 1. Increase Mutation Rate Mutation is a key operator in genetic algorithms that introduces random changes to the genetic material of individuals. By increasing the mutation rate, the algorithm can explore new regions of the search space, preventing premature convergence. However, a high mutation rate can also disrupt good solutions, so it should be carefully calibrated. 2. Employ Crossover Operators Crossover is another important operator that combines genetic material from different individuals to create new offspring. By using crossover operators, the algorithm can recombine the most promising solutions and create offspring with potentially better fitness values. This can introduce new genetic diversity and help the algorithm avoid premature convergence. It is important to note that the selection of the appropriate crossover operator depends on the problem being solved and the characteristics of the genetic representation. 3. Fitness Scaling and Niching Fitness scaling is a technique used to adjust the fitness values of individuals in the population. By applying fitness scaling, individuals with lower fitness values are given higher chances of selection, promoting exploration in the search space. Niching, on the other hand, encourages the maintenance of multiple diverse populations within the algorithm, allowing for better exploration and avoiding premature convergence. Technique Description Mutation Introducing random changes to genetic material Crossover Combining genetic material from different individuals Fitness Scaling Adjusting fitness values of individuals Niching Maintaining multiple diverse populations within the algorithm By employing these techniques, genetic algorithms can overcome premature convergence and continue the optimization process until a near-optimal or optimal solution is reached. Diversity Preservation Techniques In genetic algorithms, maintaining diversity among the population is crucial for the success of the search process. Without diversity, the algorithm may get stuck in local optima, preventing it from finding better solutions. Why is diversity important? The fitness function in genetic algorithms evaluates the quality of each individual in the population. By favoring the fittest individuals, the algorithm tends to converge towards a single solution. However, this can lead to a lack of exploration in the search space and limit the algorithm’s ability to find better solutions. By preserving diversity, genetic algorithms are more likely to explore different regions of the search space, increasing the chances of finding better solutions. Diversity also enables genetic algorithms to handle multi-modal optimization problems, where multiple optimal solutions exist. Techniques for diversity preservation There are several techniques that can be used to preserve diversity in genetic algorithms: • Elitism: Elitism refers to the practice of preserving the best individuals from one generation to the next, without any modifications. This ensures that the best solutions are not lost and helps maintain diversity in the population. • Diversity-based selection: Instead of selecting the fittest individuals for reproduction, diversity-based selection methods prioritize individuals that are different from each other. This encourages exploration of different parts of the search space. • Explicit diversity maintenance: Some genetic algorithms use specific mechanisms to explicitly maintain diversity, such as keeping track of the distance between individuals or encouraging diversity through mutation operators. • Mutation: Mutation is a genetic operator that introduces small random changes to individuals in the population. It can help maintain diversity by exploring different regions of the search space and preventing premature convergence. • Crossover control: Crossover, another genetic operator, combines genetic information from two parent individuals to create offspring. By controlling the crossover rate and strategy, diversity can be preserved by allowing for a combination of different genetic material. By using these diversity preservation techniques, genetic algorithms can overcome the limitation of convergence towards a single solution and have a better chance of finding diverse and optimal solutions to complex optimization problems. Incorporating Constraints into Genetic Algorithms Genetic algorithms are a powerful optimization algorithm inspired by the process of natural evolution. They are commonly used to solve complex problems by iteratively generating candidate solutions and selecting the fittest ones for reproduction. However, in many real-world problems, there are often constraints that need to be taken into consideration. These constraints may involve certain limitations on the variables of the problem, and the solutions must satisfy these constraints to be considered valid or feasible. Fitness Function and Constraints In traditional genetic algorithms, the fitness function evaluates the quality of a candidate solution. However, when incorporating constraints into genetic algorithms, the fitness function should not only consider the optimization objectives but also penalize solutions that violate the constraints. One common approach is to assign a high penalty to individuals that violate the constraints. This penalty can be added to the fitness function, reducing the fitness of infeasible solutions compared to feasible ones. This way, the genetic algorithm will tend to generate feasible solutions that are as close as possible to the optimal ones. Genetic Operators and Constraints The genetic operators, namely crossover and mutation, play a crucial role in the exploration and exploitation of the search space. When incorporating constraints, these operators need to be modified to satisfy the constraints of the problem. Crossover operators should be designed in a way that ensures that the offspring solutions respect the constraints. This can be achieved by selecting the genetic material from the parents in a manner that preserves the feasibility of the solutions. Similarly, mutation operators should be adapted to generate feasible solutions after the mutation is applied. Furthermore, the genetic algorithm can also utilize repair operators that aim to modify the solutions by applying small changes, ensuring that they become feasible without violating the constraints. Handling Multiple Constraints In many problems, there are multiple constraints that need to be satisfied simultaneously. This can significantly complicate the genetic algorithm implementation. One approach is to assign a separate penalty for each constraint violation and combine them into a single fitness value. Alternatively, a constraint-handling technique such as the penalty method or the constraint dominance mechanism can be used to guide the genetic algorithm towards feasible solutions. • The penalty method assigns a penalty to the fitness function based on the severity of the constraint violation. This penalty reduces the fitness of the individuals that violate the constraints, making them less likely to be selected for reproduction. • The constraint dominance mechanism compares the feasible solutions based on both their fitness value and their violation of the constraints. Feasible solutions that violate fewer constraints are considered more dominant and have a higher chance of being selected. By incorporating constraints into genetic algorithms, these optimization algorithms can be extended to handle a wide range of real-world problems where constraints play a crucial role in defining the feasibility of solutions. Multi-Objective Optimization with Genetic Algorithms In the field of optimization, one common problem is the need to find the best solution for multiple conflicting objectives. This is known as multi-objective optimization, where multiple fitness criteria need to be considered. Genetic algorithms are widely used to solve these types of problems due to their ability to explore the solution space efficiently. The key idea behind genetic algorithms is the concept of evolution. In a genetic algorithm, a population of potential solutions is evolved over generations, mimicking the process of natural selection. This is done through the use of genetic operators such as crossover and mutation. Crossover involves combining genetic material from two parent solutions to create new offspring solutions. This allows for the exploration of different combinations of solutions and can lead to the discovery of new and potentially better solutions. Mutation, on the other hand, introduces random changes to individual solutions, allowing for additional exploration of the solution space. In multi-objective optimization problems, the fitness function evaluates the quality of a solution based on multiple criteria. The goal is to find a set of solutions that represent a trade-off between the different objectives. Genetic algorithms can handle this by using a fitness assignment strategy that takes into account all the objectives simultaneously. Addressing Challenges in Multi-Objective Optimization One challenge in multi-objective optimization is the issue of convergence. Since there is no single optimal solution, the search process can become stuck in a suboptimal region of the solution space. To address this, various techniques such as elitism and Pareto dominance can be employed. Elitism involves preserving a small number of the best solutions from each generation, ensuring that the best solutions found so far are not lost. Pareto dominance is a concept from game theory that defines dominance between solutions based on their fitness criteria. By using Pareto dominance, the genetic algorithm can focus on exploring the Pareto front, which represents the set of solutions that are not dominated by any other solution. In summary, genetic algorithms offer a powerful approach to solving multi-objective optimization problems. By incorporating evolutionary concepts and genetic operators, genetic algorithms can efficiently explore the solution space and find trade-off solutions that balance multiple conflicting objectives. Addressing challenges such as convergence through techniques like elitism and Pareto dominance further enhance the effectiveness of genetic algorithms in multi-objective optimization. Fitness Scaling Methods In the context of genetic algorithms, fitness scaling methods play a crucial role in the evolution of optimal solutions to a given problem. These methods aim to balance the exploration and exploitation capabilities of the algorithm, ensuring that the search process is efficient and effective. When using a genetic algorithm for optimization problems, the fitness value assigned to each individual in the population determines their chances of being selected for reproduction and passing on their genetic material to the next generation. However, the raw fitness values alone may not accurately represent the quality of the solutions, especially when the fitness landscape is highly skewed or contains outliers. Proportional Scaling One commonly used fitness scaling method is proportional scaling. This method adjusts the fitness values of individuals based on their relative performance compared to the average fitness of the population. The idea is to amplify the differences between individuals to make the selection process more discriminating. To implement proportional scaling, the fitness values are first normalized to a range of [0, 1] using a min-max scaling technique. Then, a scaling factor is applied to each individual’s fitness value, which is calculated as the ratio between the individual’s fitness and the average fitness of the population. This method can help to address the problem of premature convergence by allowing less fit individuals to have a chance of being selected and potentially contributing useful genetic material to the Tournament Scaling Another approach to fitness scaling is tournament scaling. In this method, a subset of individuals is randomly selected from the population, and the individual with the highest fitness in the subset is assigned a fitness value of 1. The rest of the individuals in the subset are assigned fitness values between 0 and 1 based on their relative performance. The advantage of tournament scaling is that it reduces the influence of outliers on the selection process. It also introduces a level of stochasticity, which can be beneficial in exploring different areas of the search space. • Evolution • Genetic • Problems • Mutation • Search • Algorithm • Optimization • Crossover In conclusion, fitness scaling methods are essential components of genetic algorithms for solving optimization problems. They help to ensure a balanced exploration and exploitation of the search space, improving the chances of finding optimal solutions. Proportional scaling and tournament scaling are two commonly used methods, each with its own advantages and disadvantages. Choosing the appropriate fitness scaling method depends on the characteristics of the problem and the desired properties of the search process. Adaptive Genetic Algorithms Genetic algorithms are widely used for solving complex optimization and search problems inspired by the process of natural evolution. However, these algorithms face several challenges when applied to real-world problems. One of the main challenges is to strike a balance between exploration and exploitation, i.e., finding a solution that is both diverse and highly fit. Adaptive genetic algorithms tackle this problem by continuously adapting their parameters and operators during the evolution process. This adaptability allows them to dynamically adjust the rates of genetic operators such as crossover and mutation, leading to a more efficient search process. One common approach in adaptive genetic algorithms is to use a fitness-based adaptation mechanism. In this mechanism, the selection pressure is increased or decreased based on the fitness values of the individuals in the population. If the population converges too quickly, the selection pressure is increased to encourage exploration and prevent premature convergence. On the other hand, if the population becomes highly diverse, the selection pressure is decreased to promote exploitation. Another strategy used in adaptive genetic algorithms is to adapt the crossover and mutation rates. The crossover rate determines the probability of two parents exchanging genetic material, while the mutation rate controls the probability of introducing random changes in the offspring. By dynamically adjusting these rates, adaptive genetic algorithms can strike a balance between exploration and exploitation and adapt to the problem at hand. Adaptive genetic algorithms have been successfully applied to various genetic problems, including function optimization, constraint satisfaction, and combinatorial optimization. They have shown improved performance compared to traditional genetic algorithms in terms of convergence speed, solution quality, and robustness. In summary, adaptive genetic algorithms offer a promising approach for addressing the challenges faced by traditional genetic algorithms. By adapting their parameters and operators during the evolution process, these algorithms can better explore the search space, exploit promising areas, and improve the efficiency and effectiveness of the optimization process. 1 Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization, and Machine Learning. 2 Michalewicz, Z. (1999). Genetic Algorithms + Data Structures = Evolution Programs. 3 Whitley, D. (1994). A Genetic Algorithm Tutorial. Choosing the Appropriate Selection Method Selection plays a crucial role in the evolutionary process of genetic algorithms. It determines which individuals will be selected for reproduction and eventually contribute to the optimization The primary goal of selection is to guide the algorithm towards better solutions by favoring individuals with higher fitness. Fitness represents the quality or suitability of individuals for solving the given optimization problem. There are various selection methods available, each with its own advantages and disadvantages. The choice of the selection method depends on the specific problem and the characteristics of the genetic algorithm being implemented. One commonly used selection method is tournament selection. In this method, a small subset of individuals, known as a tournament, is randomly chosen from the population. The individual with the highest fitness within the tournament is selected for reproduction. Roulette wheel selection is another popular method, where individuals are assigned a probability of selection proportional to their fitness. The fitter individuals have a higher chance of being selected, mimicking the concept of a roulette wheel where higher fitness values correspond to larger slices of the wheel. Elitism is a selection strategy that preserves the best-performing individuals in each generation. These individuals are directly copied to the next generation without undergoing any recombination or mutation. Elitism ensures that the best solutions found so far are not lost and helps accelerate the evolutionary process. It is important to carefully analyze the problem at hand and consider the trade-offs of different selection methods. Some methods may help maintain diversity in the population, while others may bias the search towards local optima. A combination of different selection methods can also be used to take advantage of their respective strengths. The choice of the appropriate selection method is crucial for the success of a genetic algorithm. It can greatly impact the algorithm’s convergence speed, ability to handle complex problems, and overall optimization performance. In conclusion, selecting the appropriate selection method in a genetic algorithm is a critical decision that can significantly impact the algorithm’s performance. By understanding the strengths and weaknesses of different selection methods, researchers and practitioners can make informed choices to improve the efficiency and effectiveness of their evolutionary optimization processes. Parameter Tuning in Genetic Algorithms Genetic algorithms are optimization algorithms based on the principles of evolution and genetic selection. These algorithms are used to solve complex problems by mimicking the process of natural selection and evolution. One of the key challenges in implementing genetic algorithms is tuning the various parameters to achieve optimal performance. The performance of a genetic algorithm is highly dependent on the values of these parameters, and selecting the right values can significantly impact the efficiency and effectiveness of the algorithm. Some of the important parameters in genetic algorithms include the population size, crossover rate, mutation rate, and selection method. The population size determines the number of individuals in each generation, while the crossover rate determines the probability of crossover occurring between individuals. The mutation rate controls the probability of mutations happening during the reproduction process. Choosing appropriate values for these parameters can be a complex task. It often requires a combination of domain knowledge, experimentation, and iterative refinement. One common approach is to start with default values and then adjust them based on the performance of the algorithm on a specific problem. By systematically testing different parameter values, researchers and practitioners can identify the combination that yields the best results for a given problem. Another technique for parameter tuning is to use metaheuristic optimization algorithms such as genetic algorithms themselves. This involves using a genetic algorithm to search for the optimal values of the parameters. By treating the parameter tuning as another optimization problem, the algorithm can explore different combinations of parameter values and find the ones that lead to the best Parameter Description Population Size The number of individuals in each generation Crossover Rate The probability of crossover occurring between individuals Mutation Rate The probability of mutations happening during reproduction Selection Method The method used to select individuals for reproduction In conclusion, parameter tuning is a critical aspect of implementing genetic algorithms. The selection of appropriate parameter values can significantly affect the performance and effectiveness of the algorithm in solving complex problems. Through a combination of domain knowledge, experimentation, and metaheuristic optimization, researchers and practitioners can optimize these parameters and improve the overall efficiency of the genetic algorithm. Parallel and Distributed Genetic Algorithms Genetic algorithms (GAs) are a popular class of optimization algorithms inspired by the process of evolution and natural selection. These algorithms are widely used for solving complex search and optimization problems in various fields, such as engineering, computer science, and biology. One of the main challenges in using GAs is the computational complexity of the search process. GAs typically involve a large number of iterations or generations, and each generation requires the evaluation of a fitness function for a population of candidate solutions. This can be time-consuming, especially for problems with a large search space. Mutation and Crossover In a standard GA, the search process primarily consists of two main operations: mutation and crossover. Mutation introduces random changes into the population, while crossover combines genetic material from two parent individuals to create new offspring. These operations drive the exploration and exploitation of the search space, allowing the algorithm to find optimal solutions. Parallel and distributed genetic algorithms (PDGAs) are techniques that aim to speed up the search process by distributing the workload among multiple processors or computers. By exploiting parallelism, these algorithms can simultaneously evaluate multiple candidate solutions, speeding up the evolution process. Benefits of Parallel and Distributed Genetic Algorithms PDGAs offer several advantages over traditional single-threaded GAs. First, they can significantly reduce the time required to evolve a population of solutions, especially for computationally intensive problems. By distributing the evaluation of fitness functions across multiple processors, PDGAs can take advantage of the processing power of modern computers and clusters. Second, PDGAs provide better exploration and exploitation of the search space. By evaluating multiple candidate solutions in parallel, these algorithms can cover a wider range of the search space and identify better solutions more efficiently. Third, PDGAs enable the scalability of genetic algorithms. As the problem size increases, the computational demands grow exponentially. PDGAs can distribute the workload among multiple processors or computers, allowing for efficient and scalable exploration of larger search spaces. In conclusion, parallel and distributed genetic algorithms offer a promising approach to overcome the computational limitations of traditional GAs. These techniques leverage the power of parallel processing and distributed computing to accelerate the search process and find optimal solutions more effectively. By combining the principles of mutation, crossover, and parallelism, PDGAs provide a powerful tool for solving complex optimization problems in various domains. Handling Noisy Fitness Functions Noise in fitness functions can pose challenges for evolutionary optimization algorithms. Fitness functions are used to evaluate the quality of candidate solutions in a genetic algorithm. However, in real-world scenarios, these functions may be subject to noise or uncertainty due to various factors, such as measurement errors or stochastic processes. The presence of noise can significantly impact the performance of genetic algorithms, as it may lead to inaccurate fitness evaluations and misleading rankings of candidate solutions. This can result in suboptimal or inefficient search processes. Sources of Noise in Fitness Functions There are several possible sources of noise in fitness functions, including: • Measurement errors: In some cases, the measurements used to compute the fitness value of a solution may contain errors or inaccuracies. These errors can be due to technical limitations, sensor noise, or other factors. • Environmental variability: Fitness functions that rely on data from real-world environments may be affected by natural variations or fluctuations. These variations can introduce noise into the fitness evaluations. • Randomness: Some fitness functions require stochastic processes or random elements. The randomness inherent in these functions can introduce noise into the evaluation, making it challenging to identify the true quality of candidate solutions. Dealing with Noisy Fitness Functions To handle noise in fitness functions, several strategies can be employed: 1. Averaging: One approach is to perform multiple evaluations of each candidate solution and average the fitness values. This can help mitigate the impact of random fluctuations or measurement 2. Smoothing: Another method is to apply smoothing techniques to reduce the impact of noise. This can involve filtering the fitness values or applying statistical techniques to remove outliers. 3. Robust optimization: Robust optimization techniques aim to find solutions that perform well across different noise levels or uncertain conditions. This can involve incorporating noise models into the fitness function or using adaptive algorithms. 4. Population diversity: Maintaining a diverse population of candidate solutions can help mitigate the effects of noise. By exploring a broader range of solutions, genetic algorithms can be more resilient to inaccurate fitness evaluations. Overall, handling noisy fitness functions is an important consideration in the design and implementation of genetic algorithms. By employing appropriate strategies, such as averaging, smoothing, robust optimization, and maintaining population diversity, the impact of noise can be minimized, leading to more effective and reliable optimization processes. Hybridizing Genetic Algorithms with Other Optimization Techniques In the field of genetic algorithms, optimization is a key objective. Genetic algorithms use techniques such as mutation, crossover, and selection to evolve a population of candidate solutions with the goal of finding the best solution to a given problem. However, in some cases, genetic algorithms may not be the most effective technique on their own. One possible solution to improve the performance of genetic algorithms is to hybridize them with other optimization techniques. By combining the strengths of different algorithms, it is possible to overcome the limitations of a single approach and achieve better results. One common technique to hybridize genetic algorithms is to incorporate local search into the genetic algorithm framework. Local search is an optimization technique that focuses on improving the fitness of individual solutions by making small incremental changes to them. By incorporating local search into a genetic algorithm, it is possible to fine-tune the solutions generated by the genetic algorithm and further improve their fitness. Another way to hybridize genetic algorithms is to use them in combination with other metaheuristic algorithms, such as simulated annealing or particle swarm optimization. These algorithms have their strengths in exploring the search space and finding global optima. By using genetic algorithms in combination with these algorithms, it is possible to leverage the exploration capabilities of the genetic algorithm and the exploitation capabilities of the other algorithm, resulting in a more effective optimization process. In addition to hybridizing with other optimization techniques, genetic algorithms can also be hybridized with other genetic algorithms. This approach, known as multi-objective optimization, involves using multiple genetic algorithms simultaneously to optimize different objectives. Each genetic algorithm focuses on a specific objective, and their solutions are combined to form a Pareto front, which represents the set of trade-off solutions between the objectives. In conclusion, by hybridizing genetic algorithms with other optimization techniques, it is possible to improve their performance and achieve better optimization results. Whether it is incorporating local search, combining with other metaheuristic algorithms, or using multiple genetic algorithms, hybridization can help overcome the limitations of a single approach and explore the search space more effectively. Handling Large Solution Spaces One of the major challenges faced in genetic algorithms is dealing with large solution spaces. In optimization problems, a solution space refers to the set of all possible solutions that the algorithm explores in order to find the optimal solution. When the solution space is large, it can be challenging for the genetic algorithm to effectively search and identify the best solution. The algorithm relies on mechanisms such as mutation, fitness evaluation, and crossover to explore the solution space and improve the solutions over time. However, in large solution spaces, these mechanisms may not be sufficient to find the optimal solution. To handle large solution spaces, various strategies can be employed. One approach is to increase the population size, allowing the algorithm to explore a larger portion of the solution space concurrently. A larger population size increases the chances of finding better solutions and reduces the probability of getting stuck in local optima. Another strategy is to use adaptive evolution operators, where the mutation and crossover operators are dynamically adjusted based on the characteristics of the solution space. This can help the algorithm adapt and explore different areas of the solution space more effectively. Furthermore, parallelization techniques can be utilized to distribute the computation across multiple processors or machines. This allows the algorithm to explore different regions of the solution space simultaneously, enhancing the chances of finding the optimal solution in a shorter time. In conclusion, handling large solution spaces in genetic algorithms is a significant challenge. By employing strategies such as increasing the population size, using adaptive evolution operators, and leveraging parallelization techniques, the algorithm’s ability to explore and find optimal solutions in large solution spaces can be greatly improved. Genetic Algorithms in Real-World Applications Genetic algorithms are a powerful optimization technique inspired by the process of natural selection and genetics. They can be used to solve a wide range of complex problems by mimicking the process of evolution and natural selection. In real-world applications, genetic algorithms have been successfully applied to various fields, including engineering, finance, medicine, and computer science. They have been used to solve problems such as optimization, search, and decision-making. One of the main applications of genetic algorithms is optimization. They can be used to find the best solution among a large number of possible solutions. By iteratively applying the principles of natural selection and random variation, genetic algorithms can quickly converge towards an optimal solution, even in complex and multi-dimensional search spaces. For example, in the field of engineering, genetic algorithms have been used to optimize the design of mechanical components, such as airplane wings or car chassis. By selecting and recombining the best-performing designs, genetic algorithms can efficiently find designs that meet specific performance criteria, such as minimizing weight or maximizing strength. Genetic algorithms can also be used for search problems, where the goal is to find a specific solution or pattern in a large search space. By representing the search space as a population of solutions and applying genetic operators such as crossover and mutation, genetic algorithms can explore the search space in an efficient and systematic manner. For example, in the field of computer science, genetic algorithms have been used to solve problems such as scheduling, routing, and data clustering. By representing the problem as a set of possible solutions and applying genetic operators to generate new solutions, genetic algorithms can efficiently search for optimal or near-optimal solutions, even in large and complex problem domains. Evolution and Fitness One of the key concepts in genetic algorithms is the idea of evolution and fitness. Each solution in the population is assigned a fitness value, which represents how well it performs the given task. Solutions with higher fitness values are more likely to be selected for reproduction and contribute to the next generation. By iteratively applying the principles of selection, crossover, and mutation, genetic algorithms can evolve a population of solutions towards better fitness values. This process mimics the process of natural selection, where individuals with higher fitness are more likely to survive and reproduce, passing on their traits to future generations. In the context of real-world applications, the fitness function represents the objective or evaluation criteria that need to be optimized. It can be based on quantitative metrics, such as cost, speed, or accuracy, or on qualitative criteria, such as user preferences or subjective evaluations. Overall, genetic algorithms provide a powerful and flexible framework for solving complex problems in various real-world applications. By leveraging the principles of evolution and fitness, genetic algorithms can efficiently explore large search spaces, optimize solutions, and find near-optimal solutions to a wide range of problems. Handling Dynamic Environments with Genetic Algorithms In the field of evolutionary computation, genetic algorithms are commonly used to solve optimization problems by mimicking natural selection and evolution. These algorithms work by maintaining a population of potential solutions, evaluating their fitness, and then selectively breeding new solutions based on their fitness. However, when facing dynamic environments where the fitness landscape changes over time, genetic algorithms can face new challenges. The Problem of Fitness Evaluation In dynamic environments, the fitness function that determines the quality of a solution may change over time. This can lead to a situation where a solution that was once considered optimal no longer provides good results. Therefore, it becomes crucial to continuously evaluate the fitness of solutions and adapt them to the changing environment. This can be a computationally expensive task, as it requires constantly re-evaluating the fitness of the population. Adaptive Evolutionary Strategies One approach to handling dynamic environments is to use adaptive evolutionary strategies. These strategies involve dynamically adjusting the parameters of the genetic algorithm, such as the population size, mutation rate, and selection pressure, based on the current state of the environment. By dynamically adapting the algorithm, it can better respond to changes in the fitness For example, if the fitness landscape becomes more challenging, the algorithm can increase the mutation rate to promote exploration and search for new solutions. On the other hand, if the environment becomes more stable, the algorithm can reduce the mutation rate to focus more on exploiting the current best solutions. Maintaining Diversity in the Population In dynamic environments, maintaining diversity in the population becomes crucial to ensure that the algorithm does not get stuck in local optima. If the algorithm converges too quickly to a sub-optimal solution, it may fail to adapt to changes in the fitness landscape. To address this, various techniques can be employed to promote diversity, such as enhancing the mutation operator to introduce more variation, implementing diverse selection mechanisms, or incorporating niche formation strategies. These approaches help prevent premature convergence and allow the algorithm to continue exploring the search space. In summary, handling dynamic environments with genetic algorithms requires adapting the algorithm to the changing fitness landscape, maintaining diversity in the population, and continuously evaluating the fitness of solutions. By employing adaptive evolutionary strategies and promoting diversity, genetic algorithms can effectively tackle optimization problems in dynamic environments. Population Initialization Methods In genetic algorithms, the initial population plays a crucial role in the optimization and search process. The quality and diversity of individuals in the population can significantly affect the performance and convergence of the genetic algorithm. Therefore, it is essential to carefully consider the population initialization methods. Random Initialization The most common approach to population initialization is random initialization. In this method, individuals are randomly generated in the search space. Random initialization is simple and easy to implement, but it often suffers from problems such as premature convergence and lack of diversity. If the initial population is not diverse enough, the genetic algorithm may get stuck in a suboptimal Heuristic Initialization Heuristic initialization methods aim to improve the quality and diversity of the initial population by incorporating domain-specific knowledge or heuristics. These methods leverage problem-specific information to guide the generation of individuals in the population. Heuristic initialization can help the genetic algorithm explore the search space more effectively and find better solutions One popular heuristic initialization method is the “greedy” initialization, where individuals are generated based on a greedy strategy that prioritizes the most promising solutions. Another example is the “randomized” initialization, which introduces randomness into the heuristic initialization process to maintain diversity. Crossover Initialization Crossover initialization is a population initialization method that combines genetic crossover with random initialization. In this approach, a few individuals are randomly initialized, and then crossover operators are applied to generate offspring. The offspring inherited a mix of genetic information from the initial population and the crossover process. Crossover initialization can enhance the diversity of the initial population and make it less likely to converge prematurely. There are also various other population initialization methods, such as elitist initialization, where a few individuals with the highest fitness values are directly copied into the initial population, and niche initialization, which aims to generate a diverse set of individuals that cover different niches in the search space. In conclusion, the choice of population initialization method is critical for the success of genetic algorithms. Researchers and practitioners should carefully consider the characteristics and requirements of the problem at hand when selecting an appropriate initialization method. By leveraging the strengths of different initialization methods, it is possible to improve the performance and effectiveness of genetic algorithm in solving optimization and search problems. Local Search Techniques in Genetic Algorithms Genetic algorithms are optimization algorithms inspired by the process of natural evolution. They are often used to solve complex problems that involve searching for the optimal solution within a large solution space. However, genetic algorithms may suffer from slow convergence and premature convergence, where the algorithm gets stuck in a suboptimal solution. Crossover and Mutation In genetic algorithms, crossover and mutation are the primary operators used to generate new solutions. Crossover involves combining the genetic material of two parent solutions to create a new child solution. Mutation involves making small random changes to a solution to explore new areas of the solution space. However, these operators alone may not be enough to quickly converge to the optimal solution. They rely on random exploration of the solution space, which can be inefficient and time-consuming. Local Search Local search techniques can be used in conjunction with genetic algorithms to improve their efficiency and find better solutions. Local search focuses on making small changes to a solution in order to improve its fitness or objective value. By applying local search techniques, such as hill climbing or simulated annealing, genetic algorithms can exploit local optima and converge more quickly towards the global optimum. One common local search technique used in genetic algorithms is the 2-opt algorithm, which is often used in the traveling salesman problem. The 2-opt algorithm swaps two edges in a solution to create a new solution with a shorter total distance. Another local search technique is the tabu search, which maintains a list of recently visited solutions and avoids revisiting them. This helps the algorithm explore different areas of the solution space and avoid getting stuck in local optima. Overall, local search techniques play a crucial role in improving the efficiency and effectiveness of genetic algorithms. They can help these algorithms explore the solution space more effectively, find better solutions, and converge to the global optimum faster. Incorporating Prior Knowledge into Genetic Algorithms Genetic algorithms are an effective and widely used approach for solving optimization problems. However, they often require a large number of iterations to converge on an optimal solution, especially when faced with complex problems. One way to improve the efficiency of genetic algorithms is to incorporate prior knowledge about the problem into the algorithm. Prior Knowledge and Problem Representation Prior knowledge can be incorporated into genetic algorithms by introducing problem-specific crossover and mutation operators. These operators can leverage known information about the problem to guide the search towards better solutions. For example, if the problem has known substructures that are beneficial for the fitness of the solutions, the crossover operator can be designed to preserve these substructures during reproduction. This can help prevent the loss of valuable genetic material and accelerate the convergence process. Similarly, the mutation operator can be adjusted to explore the search space more efficiently by focusing on promising areas based on prior knowledge. This can be done by biasing the mutation towards regions of the search space that are more likely to contain high-quality solutions. Integration of Prior Knowledge In order to effectively incorporate prior knowledge into genetic algorithms, it is important to have a clear understanding of the problem and the relevant information that can be used. This can be achieved through extensive analysis, domain expertise, or data-driven approaches. Once the relevant information is identified, it can be integrated into the genetic algorithm by modifying the fitness function or the selection mechanism. The fitness function can be adjusted to include penalty terms or constraints based on the prior knowledge. This can help guide the search towards solutions that adhere to the known properties or characteristics of the problem. The selection mechanism can also be modified to favor solutions that have desirable properties or exhibit certain behaviors based on prior knowledge. This can be done by assigning higher selection probabilities to individuals that possess the desired traits, increasing their chances of being selected for reproduction. By incorporating prior knowledge into genetic algorithms, researchers and practitioners can enhance the efficiency and effectiveness of the optimization process. This can lead to faster convergence, improved solution quality, and better utilization of computational resources. Genetic Algorithms for Feature Selection and Extraction In the field of machine learning and data analysis, feature selection and extraction are important tasks that aim to identify the most relevant set of features from a given dataset. This process is crucial for improving the performance and efficiency of various algorithms, as it reduces the dimensionality of the input space and eliminates irrelevant or redundant features. Genetic algorithms provide a powerful approach to address feature selection and extraction problems. Inspired by the principles of evolution and natural selection, genetic algorithms simulate the process of Darwinian evolution to search through a vast solution space and find the optimal set of features. The key components of genetic algorithms are mutation, optimization, and crossover. Mutation introduces random changes in the feature set, allowing the algorithm to explore new regions of the solution space. Optimization evaluates the fitness of each feature set, determining its quality and effectiveness. Crossover combines features from different sets to create new combinations that potentially possess better characteristics. During the evolutionary process, genetic algorithms iteratively generate new generations of feature sets, selecting the fittest individuals based on their fitness score. Over time, the algorithm converges towards the optimal feature set that maximizes the performance of the chosen algorithm or model. A common approach in genetic algorithms for feature selection and extraction is to represent each feature as a binary string, where each bit represents the presence or absence of a feature. The fitness function evaluates the quality of the feature set based on some performance metric, such as classification accuracy or regression error. Genetic algorithms for feature selection and extraction have been successfully applied to various domains, including image recognition, text mining, bioinformatics, and signal processing. They offer a flexible and efficient method to explore the vast solution space and identify the most informative features for a given problem. Evaluation Criterion Advantages Disadvantages Classification Accuracy Can improve the performance of classification algorithms by selecting relevant features Requires a defined set of classes for supervised learning Regression Error Helps to find the optimal set of features for regression tasks Dependent on the quality of the training data and the chosen regression model Information Gain Provides a measure of the relevance of each feature to the target variable May overlook interactions between features What is a genetic algorithm? A genetic algorithm is a search algorithm inspired by the process of natural selection and evolution. It is used to solve optimization and search problems by mimicking the process of natural How does a genetic algorithm work? A genetic algorithm works by starting with an initial population of potential solutions. It then repeatedly creates a new generation of solutions through processes such as selection, crossover, and mutation. The better solutions from each generation are more likely to be selected for creating the next generation, thus improving the overall fitness of the population over time. What are some common problems that can arise when using genetic algorithms? Some common problems that can arise when using genetic algorithms include premature convergence, lack of diversity in the population, and choosing appropriate parameters such as population size and mutation rate. What is premature convergence? Premature convergence is a problem in genetic algorithms where the population converges to a suboptimal solution too quickly, without exploring the full search space. This can happen when there is not enough diversity in the population or when the selection process favors a particular subset of solutions too heavily. How can premature convergence be addressed in genetic algorithms? Premature convergence can be addressed in genetic algorithms by employing techniques such as maintaining diversity in the population through mechanisms like elitism and crowding, using appropriate selection methods that balance exploration and exploitation, or adjusting parameters like population size and mutation rate. What is a genetic algorithm? A genetic algorithm is an optimization algorithm inspired by the process of natural selection. It is used to find approximate solutions to complex optimization and search problems. What are some common problems encountered when using genetic algorithms? Some common problems encountered when using genetic algorithms include premature convergence, lack of diversity, and computational complexity. How can premature convergence be avoided in genetic algorithms? Premature convergence can be avoided in genetic algorithms by using techniques such as fitness sharing, niche formation, or allowing for a higher mutation rate. What are some possible solutions to the lack of diversity problem in genetic algorithms? Some possible solutions to the lack of diversity problem in genetic algorithms include using tournament selection, random individuals insertion, or fitness-based selection. How can the computational complexity of genetic algorithms be reduced? The computational complexity of genetic algorithms can be reduced by using techniques such as parallel processing, subproblems decomposition, or local search heuristics.	{"url":"https://scienceofbiogenetics.com/articles/challenges-and-solutions-in-genetic-algorithm-optimization","timestamp":"2024-11-10T02:03:51Z","content_type":"text/html","content_length":"116537","record_id":"<urn:uuid:4514e4cf-f3f9-4671-a57b-b17bf70c76da>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00058.warc.gz"}
Solution to Linear Algebra Hoffman & Kunze Chapter 5.2 Exercise 5.2.1 Each of the following expressions defines a function $D$ on the set of $3\times3$ matrices over the field of real numbers. In which of these cases is $D$ a $3$-linear function? (a) $D(A)=A_{11}+A_{22}+A_{33}$; (b) $D(A)=(A_{11})^2+3A_{11}A_{22}$; (c) $D(A)=A_{11}A_{12}A_{33}$; (d) $D(A)=A_{13}A_{22}A_{32}+5A_{12}A_{22}A_{32}$; (e) $D(A)=0$; (f) $D(A)=1$; (a) No $D$ is not $3$-linear. Let $$A=\left[\begin{array}{ccc}2&0&0\\0&1&0\\0&0&1\end{array}\right].$$Then if $D$ were $3$-linear then it would be linear in the first row and we’d have to have $D(A)=D(I)+D(I)$. But $D(A)=4$ and $D(I) =3$, so $D(A)\not=D(I)+D(I)$. (b) No $D$ is not $3$-linear. Let $A$ be the same matrix as in part (a). Then $D(A)=10$ and $D(I)=4$, so $D(A)\not=D(I)+D(I)$. (c) No $D$ is not $3$-linear. Let $$A=\left[\begin{array}{ccc}2&2&0\\0&0&0\\0&0&1\end{array}\right],$$$$B=\left[\begin{array}{ccc}1&1&0\\0&0&0\\0&0&1\end{array}\right].$$Then if $D$ were $3$-linear we’d have to have $D(A)=D(B)+D(B)$. But $D(A)=4$ and $D(B)=1$. Thus $D(A)\not=D(B)+D(B)$. (d) Yes $D$ is $3$-linear. The two functions $A\mapsto A_{13}A_{22}A_{32}$ and $A\mapsto 5A_{12}A_{22}A_{32}$ are both $3$-linear by Example 1, page 142. The sum of these two functions is then $3$-linear by the Lemma on page 143. Since $D$ is exactly the sum of these two functions, it follows that $D$ is $3$-linear. (e) Yes $D$ is $3$-linear. We must show (5-1) on page 142 holds for all matrices $A$. But since $D(A)=0$ $\forall$ $A$, both sides of (5-1) are always equal to zero. Thus (5-1) does hold $\forall$ (f) No $D$ is not $3$-linear. Let $A$ be the matrix from part (a) again. Then $D(A)=1$ but $D(I)+D(I)=2$. Thus $D(A)\not=D(I)+D(I)$. Thus $D$ is not $3$-linear. Exercise 5.2.2 Verify directly that the three functions $E_1$, $E_2$, $E_3$ defined by (5-6), (5-7), and (5-8) are identical. Solution: We have$$E_1(A)=A_{11}(A_{22}A_{33}-A_{23}A_{32}) – A_{21}(A_{12}A_{33}-A_{13}A_{32})+A_{31}(A_{12}A_{23}-A_{13}A_{22})$$$$=\underset{\text{term $1$}}{A_{11}A_{22}A_{33}}-\underset{\text {term $2$}}{A_{11}A_{23}A_{32}}-\underset{\text{term $3$}}{A_{21}A_{12}A_{33}}+\underset{\text{term $4$}}{A_{21}A_{13}A_{32}}+\underset{\text{term $5$}}{A_{31}A_{12}A_{23}}-\underset{\text{term $6$}} {A_{31}A_{13}A_{22}}.$$$$E_2(A)=-A_{12}(A_{21}A_{33}-A_{23}A_{31}) + A_{22}(A_{11}A_{33}-A_{13}A_{31})-A_{32}(A_{11}A_{23}-A_{13}A_{21})$$$$=\underset{\text{term $3$}}{-A_{12}A_{21}A_{33}}+\underset {\text{term $5$}}{A_{12}A_{23}A_{31}}+\underset{\text{term $1$}}{A_{22}A_{11}A_{33}}-\underset{\text{term $6$}}{A_{22}A_{13}A_{31}}-\underset{\text{term $2$}}{A_{32}A_{11}A_{23}}+\underset{\text{term $4$}}{A_{32}A_{13}A_{21}}.$$$$E_3(A)=A_{13}(A_{21}A_{32}-A_{22}A_{31})-A_{23}(A_{11}A_{32}-A_{12}A_{31})+A_{33}(A_{11}A_{22}-A_{12}A_{21})$$$$=\underset{\text{term $4$}}{A_{13}A_{21}A_{32}}-\underset {\text{term $6$}}{A_{13}A_{22}A_{31}}-\underset{\text{term $2$}}{A_{23}A_{11}A_{32}}+\underset{\text{term $5$}}{A_{23}A_{12}A_{31}}+\underset{\text{term $1$}}{A_{33}A_{11}A_{22}}-\underset{\text{term $3$}}{A_{33}A_{12}A_{21}}.$$I’ve expanded the three expressions and labelled corresponding terms. We see each of the six terms appears exactly once in each expansion, and always with the same sign. Therefore the three expressions are equal. Exercise 5.2.3 Let $K$ be a commutative ring with identity. If $A$ is a $2\times2$ matrix over $K$, the {\bf classical adjoint} of $A$ is the $2\times2$ matrix adj $A$ defined by $$\text{adj $A$}=\left[\begin{array}{cc}A_{22}&-A_{12}\\-A_{21}&A_{11}\end{array}\right].$$If det denotes the unique determinant function on $2\times2$ matrices over $K$, show that (a) $(\text{adj $A$})A = A(\text{adj $A$})=(\det A)I$; (b) $\det(\text{adj $A$})=\det(A)$; (c) adj $(A^t)=(\text{adj }A)^t$. ($A^t$ denotes the transpose of $A$.) (a) we have $$(\text{adj $A$})A=\left[\begin{array}{cc}A_{22}&-A_{12}\\-A_{21}&A_{11}\end{array}\right] \left[\begin{array}{cc}A_{11}&A_{12}\\A_{21}&A_{22}\end{array}\right]$$$$=\left[\begin{array}{cc}A_{11}A_{22}-A_{12}A_{21} & A_{12}A_{22}-A_{12}A_{22}\\-A_{11}A_{21}+A_{11}A_{21} & -A_{12}A_{21}+A_ {11}A_{22}\end{array}\right]$$$$=\left[\begin{array}{cc}A_{11}A_{22}-A_{12}A_{21} & 0\\0 & A_{11}A_{22}-A_{12}A_{21}\end{array}\right]$$$$=\left[\begin{array}{cc}\det(A) & 0\\0 & \det(A)\end{array}\ right].$$$$A(\text{adj $A$})=\left[\begin{array}{cc}A_{11}&A_{12}\\A_{21}&A_{22}\end{array}\right] \left[\begin{array}{cc}A_{22}&-A_{12}\\-A_{21}&A_{11}\end{array}\right]$$$$=\left[\begin{array}{cc}A_{11}A_{22}-A_{12}A_{21} & -A_{11}A_{12}+A_{12}A_{11}\\A_{21}A_{22}-A_{22}A_{21} & -A_{21}A_{12}+A_ {22}A_{11}\end{array}\right]$$$$=\left[\begin{array}{cc}\det(A) & 0\\0 & \det(A)\end{array}\right].$$Thus both $(\text{adj $A$})A$ and $A(\text{adj $A$})$ equal $(\det A)I$. (b) We have $$\det(\text{adj $A$})=\det\left(\left[\begin{array}{cc}A_{22}&-A_{12}\\-A_{21}&A_{11}\end{array}\right]\right)$$$$=A_{11}A_{22}-A_{12}A_{21}=\det(A).$$(c) We have $$\text{adj}(A^t) \end{equation}And$$(\text{adj} A)^t Comparing (\ref{fjffff97779}) and (\ref{fwfwfa000}) gives the result. Exercise 5.2.4 Let $A$ be a $2\times2$ matrix over a field $F$. Show that $A$ is invertible if and only if $\det A\not=0$. When $A$ is invertible, give a formula for $A^{-1}$. Solution: We showed in Example 3, page 143, that $\det(A)=A_{11}A_{22}-A_{12}A_{21}$. Therefore, we’ve already done the first part in Exercise 8 of section 1.6 (page 27). We just need a formula for $A^{-1}$. The formula is $$A\cdot \frac1{\det(A)}\left[\begin{array}{cc}A_{22}&-A_{12}\\-A_{21}&A_{11}\end{array}\right]=\frac1{\det(A)}\left[\begin{array}{cc}A_{11}&A_{12}\\A_{21}&A_{22}\end{array}\right]\left[\begin{array} Exercise 5.2.5 Let $A$ be a $2\times2$ matrix over a field $F$, and suppose that $A^2=0$. Show for each scalar $c$ that $\det(cI-A)=c^2$. Solution: One has to be careful in proving this not to use implications such as $2x=0$ $\Rightarrow$ $x=0$; or $x^2+y=0$ $\Rightarrow$ $y=0$. These implications are not valid in a general field. However, we will need to use that fact that $xy=0$ $\Rightarrow$ $x=0$ or $y=0$, which is true in any field. Let $A=\left[\begin{array}{cc}x&y\\z&w\end{array}\right]$. Then $$A^2=\left[\begin{array}{cc}x^2+yz&xy+yw\\xz+wz&yz+w^2\end{array}\right].$$If $A^2=0$ then \end{equation}Now $\det(cI-A)=\det\left[\begin{array}{cc}c-x&-y\\-z&c-w\end{array}\right]=(c-x)(c-w)-yz=c^2-c(x+w)+xw-yz.$ \end{equation}Suppose $x+w\not=0$. Then (\ref{k2}) and (\ref{k3}) imply $y=z=0$. Thus $A=\left[\begin{array}{cc}x&0\\0&w\end{array}\right]$. But then $A^2=\left[\begin{array}{cc}x^2&0\\0&w^2\end {array}\right]$. So if $A^2=0$ then it must be that also $x=w=0$, which contradicts the assumption that $x+w\not=0$. Thus necessarily $A^2=0$ implies $x+w=0$. This implies $A=\left[\begin{array}{cc}x&y\\z&-x\end{array}\right]$. Thus $\det(A)=-x^2-yz$, which equals zero by (\ref{k1}). Thus $A^2=0$ implies $x+w=0$ and $\det(A)=0$. Thus by (\ref{f322}) $A^2=0$ implies $\det(cI-A)=c^2$. Exercise 5.2.6 Let $K$ be a subfield of the complex numbers and $n$ a positive integer. Let $j_1,\dots,j_n$ and $k_1,\dots,k_n$ be positive integers not exceeding $n$. For an $n\times n$ matrix $A$ over $K$ define $$D(A)=A(j_1,k_1)A(j_2,k_2)\cdots A(j_n,k_n).$$Prove that $D$ is $n$-linear if and only if the integers $j_1,\dots,j_n$ are distinct. Solution: First assume the integers $j_1,\dots,j_n$ are distinct. Since these $n$ integers all satisfy $1\leq j_i\leq n$, their being distinct necessarily implies $\{j_1,\dots,j_n\}=\{1,2,3,\dots,n\} $ Thus $A(j_1,k_1)A(j_2,k_2)\cdots A(j_n,k_n)$ is just a rearrangement of $A(1,k_1)A(2,k_2)\cdots A(3,k_n)$. It follows from Example 1 on page 142 that $A(j_1,k_1)A(j_2,k_2)\cdots A(j_n,k_n)$ is Now assume two or more of the $j_i$’s are equal. Assume without loss of generality that $j_1=j_2=\cdots=j_{\ell}=1$ where $\ell\geq2$. Let $A$ be the matrix with all $2$’s in the first row and all ones in all other rows. Let $B$ be the matrix of all $1$’s. Then $D(A)=2^{\ell}$ and $D(B)=1$. Since $D$ is $n$-linear it must be that $D(A)=D(B)+D(B)$. But $\ell>1$ $\Rightarrow$ $2^{\ell}\not=2$. Thus $D(A)\not=D(B)+D(B)$ and $D$ is not $n$-linear. Exercise 5.2.7 Let $K$ be a commutative ring with identity. Show that the determinant function on $2\times2$ matrices $A$ over $K$ is alternating and $2$-linear as a function of the columns of $A$. Solution: We have$$\det\left[\begin{array}{cc}ra_1+a_2&b\\rc_1+c_2&d\end{array}\right]$$$$=(ra_1+a_2)d-(rc_1+c_2)b$$$$=ra_1d+a_2d-rc_1b-c_2b$$$$=r(ad-bc_1)+(a_2d-bc_2)$$$$=r\det\left[\begin{array} end{array}\right]+\det\left[\begin{array}{cc}a&b_2\\c&d_2\end{array}\right].$$Thus the determinant function is $2$-linear on columns. Now $$\det\left[\begin{array}{cc}a&b\\c&d\end{array}\right]$$$$=ad-bc=-(bc-ad)$$$$=-\det\left[\begin{array}{cc}b&d\\a&c\end{array}\right].$$Thus the determinant function is alternating on columns. Exercise 5.2.8 Let $K$ be a commutative ring with identity. Define a function $D$ on $3\times3$ matrices over $K$ by the rule A_{22}\\A_{31}&A_{32}\end{array}\right].$$Show that $D$ is alternating and $3$-linear as a function of the columns of $A$. Solution: This is exactly Theorem 1 page 146 but with respect to columns instead of rows. The statement and proof go through without change except for chaning the word “row” to “column” everywhere. To make it work, however, we must know that $\det$ is an alternating $2$-linear function on columns of $2\times2$ matrices over $K$. This is exactly what was shown in the previous exercise. Exercise 5.2.9 Let $K$ be a commutative ring with identity and $D$ and alternating $n$-linear function on $n\times n$ matrices over $K$. Show that (a) $D(A)=0$, if one of the rows of $A$ is $0$. (b) $D(B)=D(A)$, if $B$ is obtained from $A$ by adding a scalar multiple of one row of $A$ to another. Solution: Let $A$ be an $n\times n$ matrix with one row all zeros. Suppose row $\alpha_i$ is all zeros. Then $\alpha_i+\alpha_i=\alpha_i$. Thus by the linearity of the determinant in the $i^{\text th}$ row we have $$\det(A)=\det(A)+\det(A).$$ Subtracting $\det(A)$ from both sides gives $\det(A)=0$. Now suppose $B$ is obtained from $A$ by adding a scalar multiple of one row to another. Assume row $\beta_i$ of $B$ equals $\alpha_i+c\alpha_j$ where $\alpha_i$ is the $i^{\text th}$ row of $A$ and $ \alpha_j$ is the $j^{\text th}$. Then the rows of $B$ are $\alpha_1,\dots,\alpha_{i-1},\alpha_i+c\alpha_j,\alpha_{i+1},\dots,\alpha_n$. Thus +c\cdot\det(\alpha_1,\dots,\alpha_{i-1},\alpha_j,\alpha_{i+1},\dots,\alpha_n).$$The first determinant is $\det(A)$. And by the first part of this problem, the second determinant equals zero, since it has a repeated row $\alpha_j$. Thus $\det(B)=\det(A)$. Exercise 5.2.10 Let $F$ be a field, $A$ a $2\times3$ matrix over $F$, and $(c_1,c_2,c_3)$ the vector in $F^3$ defined by $$c_1=\left\|\begin{array}{cc}A_{12}&A_{13}\\A_{22}&A_{23}\end{array}\right\|,\quad c_2=\left\|\begin{array}{cc}A_{13}&A_{11}\\A_{23}&A_{21}\end{array}\right\|,\quad c_3=\left\|\begin{array}{cc}A_{11}&A_ {12}\\A_{21}&A_{22}\end{array}\right\|.$$Show that (a) $\text{rank}(A)=2$ if and only if $(c_1,c_2,c_3)\not=0$; (b) if $A$ has rank $2$, then $(c_1,c_2,c_3)$ is a basis for the solution space of the system of equations $AX=0$. Solution: We will use the fact that the rank of a $2\times2$ matrix is $2$ $\Leftrightarrow$ the matrix is invertible $\Leftrightarrow$ the determinant is non-zero. The first equivalence follows from the fact that a matrix $M$ with rank $2$ has two linearly independent rows and therefore the row space of $M$ is all of $F^2$ which is the same as the row space of the identity matrix. Thus by the Corollary on page 58 $M$ is row-equivalent to the identity matrix, thus by Theorem 12 (page 23) it follows that $M$ is invertible. The second equivalence follows from Exercise 4 from Section 5.2 (page 149). (a) If $\text{rank}(A)=0$ then $A$ is the zero matrix and clearly $c_1=c_2=c_3=0$. If $\text{rank}(A)=1$ then the second row must be a multiple of the first row. This is then true for each of the $2\times2$ matrices \left[\begin{array}{cc}A_{12}&A_{13}\\A_{22}&A_{23}\end{array}\right],\quad \left[\begin{array}{cc}A_{13}&A_{11}\\A_{23}&A_{21}\end{array}\right],\quad \left[\begin{array}{cc}A_{11}&A_{12}\\A_{21}&A_ \end{equation}because each one is obtained from $A$ by deleting one column (and in the case of the second one, switching the two remaining columns). Thus each of them has rank $\leq 1$. Therefore the determinant of each of these three matrices is zero. Thus $(c_1,c_2,c_3)$ is the zero vector. If $\text{rank}(A)=2$ then the second row of $A$ is not a multiple of its first row. We must show the same is true of at least one of the matrices in (\ref{fffhnbv}). Suppose the second row is a multiple of the first for each matrix in (\ref{fffhnbv}). Since each pair of these matrices shares a column, it must be the same multiple for each pair; and therefore the same multiple for all three, call it $c$. Therefore the seond row of the entire matrix $A$ is $c$ times the first row, which contradicts our assumption that $\text{rank}(A)=2$. Thus at least one of the matrices in (\ref {fffhnbv}) must have rank two and the result follow. (b) Identify $F^3$ with the space of $3\times1$ column vectors and $F^2$ the space of $2\times1$ column vectors. Let $T:F^3\rightarrow F^2$ be the linear transformation given by $TX=AX$. Then by Theorem 2 page 71 (the rank/nullity theorem) we know $\text{rank}(T)+\text{nullity}(T)=3$. It was shown in the proof of Theorem 3 page 72 (the third displayed equation in the proof) that $$\text {rank}(T)=\text{column rank}(A).$$ And $\text{nullity}(T)$ is the solution space for $AX=0$. Thus $\text{column rank}(A)$ plus the rank of the solution space of $AX=0$ equals three. Thus if $\text {rank}(A)=2$ then the rank of the solution space of $AX=0$ must equal one. Thus a basis for this space is any non-zero vector in the space. Thus we only need show $(c_1,c_2,c_3)$ is in this space. In other words we must show $$\left[\begin{array}{ccc}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\end{array}\right]\left[\begin{array}{c}c_1\\c_2\\c_3\end{array}\right]=0.$$It feels like we’re supposed to apply Exercise 8 to the following matrix $$\left[\begin{array}{ccc}A_{11}&A_{12}&A_{13}\\A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\end{array}\right],$$but the problem with that is that we do not know that an alternating function on columns is necessarily zero on a matrix with a repeated row. That is true, but rather than prove it, it’s easier just prove this directly {21}c_1+A_{22}c_2+A_{23}c_3\end{array}\right]$$Expanding the first entry {21}-A_{12}A_{11}A_{23}+A_{13}A_{11}A_{22}-A_{13}A_{12}A_{21}$$matching up terms we see everything cancels. $$=\underset{\text{term 1}}{A_{11}A_{12}A_{23}}-\underset{\text{term 2}}{A_{11}A_{22}A_{13}}-\underset{\text{term 3}}{A_{12}A_{13}A_{21}}-\underset{\text{term 1}}{A_{12}A_{11}A_{23}}+\underset{\text {term 2}}{A_{13}A_{11}A_{22}}-\underset{\text{term 3}}{A_{13}A_{12}A_{21}}=0.$$Expanding the second entry -A_{22}A_{11}A_{23}+A_{23}A_{11}A_{22}-A_{23}A_{12}A_{21}$$matching up terms we see everything cancels. $$=\underset{\text{term 1}}{A_{21}A_{12}A_{23}}-\underset{\text{term 2}}{A_{21}A_{22}A_{13}}+\underset{\text{term 2}}{A_{22}A_{13}A_{21}}-\underset{\text{term 3}}{A_{22}A_{11}A_{23}}+\underset{\text {term 3}}{A_{23}A_{11}A_{22}}-\underset{\text{term 1}}{A_{23}A_{12}A_{21}}=0.$$	{"url":"https://linearalgebras.com/hoffman-kunze-chapter-5-2.html","timestamp":"2024-11-13T03:01:09Z","content_type":"text/html","content_length":"71794","record_id":"<urn:uuid:702a5511-9db3-4624-af6c-2e4610201de6>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00879.warc.gz"}
Victory Prediction of Ladies Professional Golf Association Players: Influential Factors and Comparison of Prediction Models Golf officials, as well as fans, are always interested in the result of each golfing event, and they become aware of it mostly through the press and/or media. The broadcasters and commentators cautiously predict the winner and winning factors, especially, in the Ladies Professional Golf Association (LPGA) majors. Golf fans also judge the result based on the performance of each player. Expert performance-analysis scholars attempt to determine the winning factors and the performance factors that affect the money leader, based on the updated LPGA longitudinal data of many years. It was reported in several research papers that greens in regulation (GIR) and putting average (PA) had higher contributions and were more important that the other factors affecting the average strokes, money leaders or winning (Chae and Park, 2017; Dodson et al., 2008; Finley and Halsey, 2004; Park and Chae, 2016). In most sports competitions, strategy analysts for each team invest efforts to analyze the records and data of the home and away teams to equip coaching staff with decisive factors that can affect the outcome of the game. These efforts are the same in the LPGA as in various other fields, and skill information such as the length of the game field, types or lay of the land, the level of difficulty of the course, the type of grass and green conditions, weather, and strategy for course targeting, is provided (McGarry et al., 2002). However, recently, prediction and description of the determinant of victory of the team and players, as well as the winner, have been required in sports competitions (Dorsel and Rotunda, 2001; Park and Chae, 2016). This requirement has reached a level wherein scholars statistically provide winner and rank possibilities employing prediction models on accumulated data (Hayes et al., 2015; Jida and Jie, 2015; Neeley et al., 2009). Chae et al. (2018) used multiple regression analysis, which is a statistical analytical model, for the rank prediction of LPGA players based on the fact that the medal rank of the 2016 Rio Olympic female golf tournament was predicted by multiple regression analysis (Mercuri et al., 2017). The methods of analysis for this type of prediction are usually linear regression analysis, curve estimation, discriminant function analysis, logistic regression analysis, principal component regression analysis, classification tree analysis, and more recently, the frequently used artificial neural network analysis. Classification tree analysis, logistic regression analysis, discriminant analysis, and artificial neural network analysis, in particular, are generally used in quantitative prediction analysis (Agga and Scott, 2015; Cenker et al., 2009; Maszczyk et al., 2012, 2016; Neeley et al., 2009). The discriminant function analysis is a statistical technique to predict how the individual would behave under given circumstances, based on various characteristics of social phenomena. Several types of supposition should be satisfied when using the discriminant function analysis (Couceiro et al., 2013; Kuligowski et al., 2016; Mieke et al., 2014; Shehri and Soliman, 2015). Classification tree analysis segments the individuals as members of small groups with similar behaviors or conducts stratification based on a certain standard and if the LPGA player will win, fail, or lead in wins ( Surucu et al., 2016). Logistic regression analysis is a general linear model, wherein the object variable is a binary variable that is categorical data. Logistic regression analysis has an advantage that there are few constraints for the discrimination variable; however, there exists a regression-analysis– oriented disadvantage that it cannot overcome the interaction effect and the numbers of independent variables (Clark, 2001; Lu, 2017; Sperandei, 2014). Artificial neural network analysis mimics the human neural–brain system. A typical neural network is composed of three layers, i.e., the input layer, hidden layer, and output layer, which include several neurons (Almassri et al., 2018). The neurons in the hidden layer conduct intermediate treatment if the input nodes receive stimulation, resulting in response from the output nodes. Thus, when using artificial neural network analysis, the predicting variable is applied to the input layer and the dependent variable to the output layer. The hidden layer oversees the intermediary management, and the researcher does not grant a role to a specific observed variable even though the researcher designates the number of hidden layers and neurons. The back-propagation algorithm is applied between the input and hidden layers, and hidden and output layers, if the input variable is supplied to the neural network. The connection weight value is adjusted every time to minimize the error between the real value in the unit of the output after applying the back-propagation algorithm and the value calculated by the artificial neural network. The optimum point is investigated by applying the big learning rate from a random point by combining the intensity that affects the direction where the slope is highest by the algorithm (learning rate, η > 0) and the intensity that affects the direction from the initial to the current direction (moment, α > 0) (Chen and Liu, 2014; Smaoui et al., 2018). Artificial neural network analysis is relatively independent of the statistical preconditions and can describe the nonlinear relationships between variables. Therefore, it is preferred over the traditional methods (Chen and Liu, 2014; Maszczyk et al., 2014; Sun and Lo, 2018). Even though there are many prediction analysis methods, this study aims to investigate performance variables that affect the winning possibilities of players and the degrees of importance of these variables, from the annual data of 25 seasons of the LPGA (1993 to 2017). Moreover, it aims to select the most accurate model from four prediction models (classification tree analysis, logistic regression analysis, discriminant function analysis, and artificial neural network analysis). This study presents a relative comparison of the influence of the predicting variables in the four prediction models on victory. That is, it tells the performance variable that should be considered for winning, and it can predict the possibility of victory of an individual using an optimum prediction model. The results of this study are expected to show the effect of prior preparation on victory. Participants and Data Collection The data used in this study included LPGA players, falling within the 60^th rank (money leaders), from over a period of 25 years from 1993 to 2017; i.e., the annual average value of 1,500 players (60 players multiplied by 25 years). The data were collected from the LPGA homepage (http://www.lpga.com/) Because the data on the LPGA homepage did not collect private identifier information such as telephone numbers, home addresses, social security numbers, etc., ethical approval was not required for this experimental study. The performance variables chosen were those that were being measured and used in the current LPGA analyses. The variables were reconstituted in this study as independent variables (predicting variables), which were continuous variables, and dependent variables (response variables), which were categorical variables (Table 1). Table 1 Independent variable Dependent variable (winning odds) 1. Driving accuracy (DA) 2. Driving distance (DD) 1. No Win, 2. Win Technical variables 3. Sand saves (SS) 4. Greens in regulation (GIR) 5. Putting average (PA) 1. No Win, 2. Win, 3. Multiple Wins 1. Birdies 2. Eagles 1. No Win, 2. Win Technical result variables 3. Par3Scoring Average (P3A) 4. Par4Scoring Average (P4A) 5. Par5ScoringAverage (P5A) 1. No Win, 2. Win, 3. Multiple Wins 1. Official money (OM) 2. Scoring average (SA) 1. No Win, 2. Win Season result variables 3. Top 10 finish% (T10) 4. 60-strokes average (60SA) 5. Rounds under par (RUP) 1. No Win, 2. Win, 3. Multiple Wins Experimental Approach to the Problem The data analysis aimed to determine key performance variables that affected the possibility of winning, the variable that was the most significant, and whether the player would win a game or be in the lead in wins. Four prediction models, i.e., classification tree analysis, logistic regression analysis, discriminant function analysis, and artificial neural network analysis, were employed. The most accurate model was selected, according to the purpose of the study. The player’s accumulated raw data released by the LPGA were arranged using Microsoft Office Excel 2010 (Microsoft Corporation, Redmond, WA, USA), and the result was deduced using the IBM SPSS 22.0 (IBM Corp., Armonk, NY, USA) statistical program. In the first round of analysis, we used classification accuracy as a basis to find the possibility that a certain player could win the game in the LPGA, using the four prediction models (discriminant function analysis, classification tree analysis, logistic regression analysis, and artificial neural network analysis (multilayer perceptron, MLP)). One-way analysis of variance (ANOVA; post-hoc: least significant difference [LSD] test) was used if there was a mean difference in the classification accuracy of the four prediction models. The input predicting variables of the four prediction models were divided into skill variables (driving accuracy [DA], driving distance [DD], sand saves [SS], GIR, and PA), skill result variables (birdies, eagles, par3 scoring average [P3A], par4 scoring average [P4A], and par5 scoring average [P5A]), and season outcome variables (official money [OM], scoring average [SA], top 10 finish% [T10], 60-strokes average [60SA], and rounds under par [RUP]). When inputting these predicting variables as dependent variables, they were divided into both two groups (victory/no victory) and three groups (no victory/one victory/multiple victories). From the results of the four prediction models, the standardized discriminant function coefficient, normalization importance or Wald value, which are importance indexes linking the independent variable to the dependent variable, could be obtained. Finally, one-way ANOVA and the post-hoc (LSD) test were conducted to examine the mean difference in the classification accuracy of the four models. Statistical significance was set at 0.05. Statistical Analyses In the discriminant function analysis, the function to maximize the group difference of an object based on continuous and discrete variables was deduced, and each participant (player) was classified using Fisher’s linear discriminant functions (Mieke et al., 2014; Shehri and Soliman, 2015). It should be known which group, from among the many groups, included each object to be used in this model. When each group was already known, the category to which each object belonged was classified and predicted by calculating the discriminant score of the individuals in each group by finding the discriminant function: which could classify each group from the measured variables (Kuligowski et al., 2016; Novak, 2016; Schumm, 2006). Classification tree analysis was used for classification and prediction by tree-structurally schematizing the decision-making rule. A decision-making tree consists of a node, body, and stems that connect different nodes. The decision-making pattern is found at the top of the node if it repetitively classifies the node according to the tree structure forming process. Before the analysis using this decision-making tree, decision trees have an assumption that prior to analysis, the type of variable is precisely specified according to the measurement level. That is, it should be analyzed whether the variables have been accurately designated for the measuring levels (Surucu et al., 2016). The methods of growing the tree are classified according to the characteristics of the data and purpose of decision making into chi-squared automatic interaction detection (CHAID), exhaustive CHAID, classification and regression tree (CRT), and quick, unbiased, and efficient statistical tree (QUEST). The classification accuracy was found to be high for the CRT basic data (Hayes et al., 2015). The tree structure was formed by designating the standard and pattern (decision trees are classified according to the purpose of the analysis and the structure of the data) as well as classifying for the purpose of analysis and data structuring. The decision tree is to select the predicting variable and to set the standard of the category when forming a low node from a single upper node. A pure low node was formed by most efficiently classifying the distribution of the target (dependent) variables. In this case, purity was defined as the degree of including individuals in a certain category of the target (dependent) variable. It set the predicting model according to the analysis result and interpreted by grasping the meaning of certain parts, as the decision-making tree described the relationships between variables as tree structures (Linda et al., 2008; Neeley et al., 2007). The merit of this study is that the process is simpler than the other methods (artificial neural network analysis, discriminant function analysis, regression analysis, and so on), as prediction or classification is described based on the induction rule of the tree structure. In this study, CRTs of four tree-growing methods were used. Homogeneity within nodes was maximized by dividing the parent node for maximum homogeneity of the dependent variable within the child node (Hayes et al., 2015). In the splitting criterion of the classification tree, the status to merge the input variable selection and category when each parent branch formed a child branch was a criterion, and it was processed from the input variable, grasping distribution of the target variable, and child branch forming in sequence (i.e., first from the input variable, then from the grasping distribution, etc.). The degree of classifying the distribution of target variables was measured in terms of the purity or impurity. The purity of the child branch was very high, compared to that of the parent branch. Pruning removed the branch that had high risk of misclassification or inappropriate induction There is cross-validation and split-sample validation for the validity evaluation. Namely, cross-validation and split-sample validation existed in the assessment of validity. The analytical sample was divided into m (= 2, 3, 4 ...) parts, and the remaining part of the sample was excluded. Thus, each part of the data was used to generate m-1 trees, and 1 was used to evaluate trees. That is, this study used cross-validation that divided analysis samples into parts of m values, made the tree with the rest of certain parts of m values, and conducted model assessment with the remaining one part. Split-sample validation divided the observation samples into training samples (training: 70%) and test samples (test: 30%) and conducted an assessment of the tree with the test samples after making the tree with the training samples. This means that the produced tree, without just being a sample, can perform expended application to a population, which is the origin of the analysis sample. Model assessment could be described with profit charts or risk charts. Namely, the decision tree found the hidden pattern and useful correlations using data and could be used as a reference for decision making in the future, as well as for finding associations between data that were difficult to quantify accurately (Duan et al., 2015). In logistic regression analysis, variables measured by nominal, ordinal, interval, and ratio scales could be used as independent variables; however, the dependent variables had to be categorical variables that were measured in a nominal scale to analyze and predict whether an individual observation belonged to a certain group. The functional formula of the logistic technique was which was expressed as P(X)when the Yvalue predicted using X was E(YIX) and E(YIX) had a probability concept when Y was a discrete variable (Pang et al., 2017; Sperandei, 2014). This model was not a linear function, but an S-curve logistic function with an upper limit of 1 and a lower limit of 0, with a problem in analysis as it could not be described as a linear function (Agga and Scott, 2015). The upper and lower limits could be avoided if this probability was converted to logit. The logit relationship with the independent variable can be described by a linear function (Cenker et al., 2009 ; Zhao et al., 2015) resulting in ability possibility for linear regression analysis. Thus, the natural log value in brackets, which is on the left-hand side of this logit linear function is an odds-ratio; p, which is the numerator, is the probability that an individual belongs to a certain group; and 1 − p, which is the denominator, is the probability that an individual does not belong to a certain group. Thus, as a result of calculation using n predicting variables (X) in the right-hand side, the bigger the logit value, the higher is the possibility it belongs to the group (Curtis, 2019). Artificial neural network analysis, by using learning materials in computers, aims to learn the optimum result, apply that result of learning to new data or conditions, and deduce an expected result such as how a human behaves, through learning (Chen and Liu, 2014). The neural network used in this study was composed of three layers (input layer, hidden layer, and output layer), and each layer included several neurons (Chen and Liu, 2014; Nair et al., 2016). The neurons in the hidden layer received the stimulation (every type of information) from the neurons in the input layer and the linear combination was connected as a weighted value. The bigger this linear combination, the higher the activation the neuron received; it was deactivated in the opposite case (Almassri et al., 2018; Nair et al., 2016 If the degree of this activation value was S, the activation {logistic functions: S=eL1+eL,(0≤S ≤ 1)} and hyperbolic tangent functions: {S = eL+eLeL+e−L,(−1≤S≤1)were intervened to S=f(L),a conversion from L to S to enable S to take a limited range (0≤S≤1,−1≤S≤1).The output node produced the final response by combining signals from the hidden neuron as weighted values. It applied the weighted value combination of the signal when the target variable was continuous, but it was calculated after converting to probability value for softmax conversion softmax:Ok=expLk∑j=1kexpLjto enable all categoricaloutput values to show the probability value when S was categorical value (Nair et al., 2016), where k was the output range index and k was the output range (Li et al., 2017). In this study, the output group was two (victory/no victory) or three (no victory, one victory, multiple victories); thus, k was 2 or 3. The goodness-of-fit of the neural network was obtained by maximizing the corresponding likelihood function using the back-propagation algorithm. Conceptually, this algorithm attempts efficient calculation by combining the learning rate (the intensity in the direction where the slope is the highest) and moment (the intensity in the direction until now) (Jida and Jie, 2015; Smaoui et al., 2018). Namely, the neural-network fitting algorithm was started from a random location, and it actively explored the highest point using a high learning rate at the beginning. It gradually lowered the learning rate to reach the highest point (Sun and Lo, 2018; Xi et al., 2013). This process was repeated at the other locations. The point finally reached by repeating this process dozens of times was not the local highest point, but the global highest point (Nair et al., 2016). It found a weight parameter for which the probability became the maximum. The predicting variable was set to skill (DA, DD, SS, GIR, PA), skill result (birdies, eagles, P3A, P4A, P5A), and season outcome variables (OM, SA, T10, 60SA, RUP), and the dependent variable was categorized to no victory and victory or no victory, one victory, and multiple victories. Influence of Skill Variable on Achieving Victory The type of an athlete that belongs to a certain group can be predicted using different models. Namely, it is possible to predict which athlete will belong to which group using a prediction model. Table 2 solves this problem when it comes to the probability of victory between an LPGA rookie and a veteran. Table 2 categorizes the dependent variables according to victory (Yes/No) from the results of four prediction model tables, when the independent prediction variable was set to a skill variable such as DA, DD, SS, GIR or PA. Table 2 Discriminant model Classification tree Binary logistic regression model Artificial neural network model Wilks’Λ: 0.883 Risk estimate: 0.263 Model coefficient Test ROC curve Experience ROC curve of winning χ^2: 186.4 Cross test: 0.272 χ^2: 186.83 Yes : 0.736 df: 5, p < 0.001 df: 5, p < 0.001 No : 0.736 I W IV SDFC V Importance NI IV al IV Importance NI GIR -1.370 GIR 0.037 100% GIR 134.12* GIR 0.447 100% PA 0.683 PA 0.019 51.3% PA 56.57* PA 0.206 46.1% DD 0.552 DD 0.004 9.9% DD 27.21* DA 0.170 37.9% DA 0.398 SS 0.001 3.7% DA 16.39* DD 0.144 32.2% SS 0.036 DA 0.001 2.2% SS 0.237 SS 0.034 7.5% This discriminant function was significant as the Wilks'λ test statistic was 0.883 (p < 0.001). The classification accuracy of this discriminant function was 74.1% and the importance of the prediction variables was in the order of SS < DA < DD < PA < GIR. The validity evaluation of classification tree analysis, the second model, was described by risk estimates. The misclassification rate was 26.4% and 27.2%, when the classification tree model included training data of the sample and cross-validation, respectively. Namely, this misclassification is a value divided by the misclassified values ((59+335) / 1500), and the total classification accuracy of this model was 73.7%. The importance of the prediction variables was in the order of DA < SS < DD < PA < GIR. In the goodness-of-fit test of the third model, the binomial logistic regression model, the model was found to be better than the base model, as chi-square ( x^2 ) was 186.83, which was significant ( p < 0.001). The classification accuracy of this model was 74.2%, and the importance of the predicting variables was in the order of SS < DA < DD < PA < GIR. The goodness-of-fit of the fourth model, the artificial neural network analysis model, was determined by the area under the curve (AUC), and the model improved as the AUC became closer to 1. The AUC of this study model under the receiver operating characteristic (ROC) curve could fall in two categories: 0.736, a group with winning experience, and 0.736, a group without it. With higher accuracy of prediction, the shape of the ROC curve moved further up from the 45° line. The AUC was the area under the ROC curve, the 45° line was a curve corresponding to the random classification ratio, and the AUC was 0.5. Thus, the AUC was in the range of 0.5 to 1.0, if it was superior to the random classification, and it became close to 1 for a more accurate model. The probability value was calculated by applying the importance index of each predicting variable to the hyperbolic tangent function between the input and hidden layers. If the hidden layer was formed and the weight coefficient value of the variable that belonged to the hidden layer was applied to the softmax function that was applied between the hidden and output layers, the probability value that corresponded to each category (Yes/No) of the finally calculated dependent variable changed from 0 to 1, and group classification criteria could be applied to the classification standard of the group by estimating the sum of probability to 1.0. The classification accuracy rate from these repeated processes was 75.3%. The importance of predicting variables in this model was in the order of SS <DD < DA < PA < GIR. To sum up, artificial neural network analysis showed a higher prediction accuracy rate than the other three models; i.e., prediction accuracy rates were as follows: classification tree model (73.7%) < discriminant model (74.1%) < binominal logistic regression model (74.2%) < artificial neural network model (75.3%). Moreover, predicting variables that were most significant for determining victory included GIR and PA in all four prediction models (Table 2). Influence of Skills on Victory If an LPGA player needs to determine the possibility of victory in a tour, the results in Table 3 will help solve this problem (or will help provide this information). Table 3 is a result table for the four prediction models, based on the category of victory (Yes/No) and the predicting variable, which is an independent variable composed of the skill variables: birdies, eagles, P3A, P4A, and P5A. The discriminant model discriminated between the groups to which each participant belonged, using the coefficient value of the discriminant function. This discriminant function was significant as the test statistic Wilks'λ was 0.879 (p < 0.001). The classification accuracy of this discriminant function was 74.1% and the importance of the predicting variables was in the order of eagles < P4A < P3A < P5A < birdies. The feasibility study of the second model, the classification tree model, is described by the risk estimate. In the training data of the samples, the misclassification rate of this model was 25.6% and cross-validation showed 26.1% misclassification. Namely, this misclassification was a value divided by misclassified (112+272)/1500, and the total classification accuracy of this model was 74.4%. The importance of predicting variables was in the order of eagles < P5A < P4A < P3A < birdies. The goodness-of-fit of the binominal logistic regression model was better than that of the base model, as the chi-square value (x^2) was 188.04, which was significant (p < 0.001). The classification accuracy of this model was 74.3%, and the importance of predicting variables was in the order of P4A < eagles < P3A < P5A < birdies. In the artificial neural network analysis goodness-of-fit test, the AUC, which was the area under the ROC curve, could take values in two groups: a group with winning experience (0.733) and a group without any experience of victory (0.733). If it was superior to random classification, the AUC was between 0.5 and 1.0, and the model improved as the AUC increased and reached closer to 1; the AUC was 0.5 for this model. If the hidden layer was formed and the weight coefficient value of the variable that belonged to the hidden layer was applied to softmax function that was applied between the hidden and output layers, the probability value that corresponded to each category (Yes/No) of the finally calculated dependent variable changed from 0 to 1, and could be applied to the classification standard of the group by estimating the sum of probability to 1.0. The classification accuracy rate from these repeated processes was 75.7%. The importance of predicting variables in this model was in the order of eagles < P3A < P5A < P4A < birdies. To sum up, artificial neural network analysis showed higher prediction accuracy rates than the other three models, as in the discriminant model (74.1%) < binominal logistic regression model (74.3%) < classification tree model (74.4%) < artificial neural network model (75.7%). Moreover, the predicting variable that was most important in determining the victory was found to be birdies in all four predicting models (Table 3). Influence of the Season Outcome on Victory The data in Table 4 help a player determine the possibility of victory during the LPGA tour. Table 4 is a result table of the four prediction models and the predicting variable is a season outcome such as OM, SA, T10, 60SA, and RUP. The dependent variable is victory (Yes/No). The discriminant model discriminated between the groups to which each participant belonged, based on the coefficient value of the discriminant function. This discriminant function was significant as the Wilks' λ test statistic was 0.717 (p < 0.001). The classification accuracy of this discriminant function was 78.5% and the importance of the predicting variables was in the order of SA < RUP < 60SA < OM < T10. The evaluation of the validity of the second model, the classification tree model, was described by risk estimates. The misclassification ratio of the model when the sample was training data was 20.3% and cross-validation showed 21.3% misclassification. Namely, this misclassification was a value divided by the wrongly classified (137+167) / 1500, and the total classification accuracy of this model was 79.7%. The importance of predicting variables was in the order of 60SA < RUP < SA < OM < T10. In the goodness-of-fit test of the binominal logistic regression model, the model fit improved compared to the base model as the chi-square (x^2) of the analysis model was 477.262, which was significant (p < 0.001). The classification accuracy of this model was 78.7%, and the importance of the predicting variables was in the order of 60SA < SA < RUP < T10 < OM. In the artificial neural network analysis goodness-of-fit test, the AUC could be in two different groups: a group with winning experience (0.844) and a group without any winning experience (0.844). If it were superior to random classification, the AUC would be between 0.5 and 1.0, and the model improved as the AUC increased and reached closer to 1; the AUC was 0.5 for this model. If the hidden layer was formed and the weight coefficient value of a variable that belonged to the hidden layer was applied to the softmax function between the hidden and output layers, the probability value that corresponded to each category (Yes/No) of the finally calculated dependent variable changed from 0 to 1. Furthermore, this value could be applied to the classification standard of the group by estimating the sum of the probability to 1.0. The classification accuracy rate from these repeated processes was 80.2%. The importance of predicting variables in this model was in the order of 60SA < RUP < T10 < SA < OM. To sum up, the artificial neural network analysis showed a higher prediction accuracy rate than the other three models, as in the discriminant model (78.5%) < binominal logistic regression model (78.7%) < classification tree model (79.7%) < artificial neural network model (80.2%). Moreover, predicting variables that were most significant in determining victory were T10 and OM in the discriminant model and classification tree, and OM, T10, and SA in the binominal logistic regression model and artificial neural network model (Table 4). Test of Mean Difference of Classification Accuracy of Prediction Models Table 5 shows the best model in terms of the classification accuracy from the four prediction models, showing the mean difference in the classification accuracy of the statistic models, arising from the change in the number of independent variables according to the change in the dependent variable level (2 or 3). The test of mean difference of the classification accuracy ratio was conducted by one-way ANOVA and it was significant (p < 0.05). The post-hoc test was necessary to determine the exact difference between the prediction models. The LSD post-hoc test showed that the artificial neural network model had higher classification accuracy than the other three models. Item Discriminant model Classification tree Binary logistic regression model Artificial neural networkmodel Sample Predicted Predicted Predicted Predicted No Win Win Total No Win Win Total No Win Win Total No Win Win Total n No Win 979 58 1037 978 59 1037 979 58 1037 958 79 1037 Win 330 133 463 335 128 463 329 134 463 291 172 463 CA% 74.1% 73.7% 74.2% 75.3% Table 3 Discriminant model Classification tree Binary logistic regression model Artificial neural network model Wilks’Λ : .879 Source Risk Standard error Model coefficient test ROC curve χ^2 : 192.41 Training Estimate 0.011 χ^2: 188.04 Experience of winning df : 5, p < 0.001 Cross test 0.256 0.011 df : 5, p < 0.001 Yes : 0.733 0.261 No : 0.733 IV SDFC IV Importance NI IV Wald IV Importance NI Birdies 0.963 Birdies 0.060 100% Birdies 105.3* Birdies 0.362 100% P5A 0.703 P5A 0.014 24.0% P5A 18.92* P5A 0.196 54.2% P3A -0.441 P3A 0.009 14.5% P3A 12.69*** P3A 0.175 48.3% P4A -0.282 P4A 0.007 11.2% P4A 5.21* P4A 0.171 47.2% Eagles 0.209 Eagles 0.001 1.5% Eagles 2.25 Eagles 0.096 26.5% Item Discriminant Model Classification tree Binary logistic regression model Artificial neural model network Sample Predicted Predicted Predicted Predicted No win Win Total No win Win Total No win Win Total No win Win Total n No win 976 61 1037 925 112 1037 979 58 1037 976 61 1037 Win 327 136 463 272 191 463 327 136 463 303 160 463 CA% 74.1% 74.4% 74.3% 75.7% Table 4 Discriminant model Classification tree Binary logistic regression model Artificial neural network model Wilks’Λ : .717 Source Risk Standard error Model coefficient test ROC curve χ^2: 498.09 Training Estimate 0.010 χ^2: 477.26 Experience of winning df: 5, p < 0.001 Cross test 0203 0.011 df: 5, p < 0.001 Yes :0.844 0213 No : 0.844 IV SDFC IV Importance NI IV Wald IV Importance NI T10 0.663 T10 0.122 100% OM 80.380 OM 0.401 100% OM 0.657 OM 0.104 85.6% T10 69.294 SA 0.263 65.6% 60SA -0.147 SA 0.061 50.4% RUP 2.925 T10 0.193 48.3% RUP -0.132 RUP 0.060 49.6% SA 2.342 RUP 0.075 18.6% SA 0.055 60SA 0.050 41.4% 60SA 1.916 60SA 0.069 17.1% [i] p < 0.05, p < 0.01, **p < 0.001, ROC: receiver operating characteristic, IV: independent variable, SDFC: standardized discriminant function coefficient, NI: normalization importance, T10: top 10 finish%, OM: official money, 60SA: 60-strokes average, RUP: rounds under par, SA: scoring average Item Discriminant model Classification tree Binary logistic regression model Artificial neural model network Sample Predicted Predicted Predicted Predicted No win win Total No win Win Total No win Win Total No win Win Total n No win 977 60 1037 900 137 1037 960 77 1037 948 89 1037 Win 262 201 463 167 296 463 243 220 463 208 255 463 CA% 78.5% 79.7% 78.7% 80.2% Table 5 Dependent variable Independent variable Discriminant model (%) Classification tree Binary logistic regression Artificial neural network model (%) model (%) model (%) Technical variables (5) 74.1 73.7 74.2 75.3 Technical results (5) 74.1 74.4 74.3 75.7 Season results (5) 78.5 79.7 78.7 80.2 Technical variables (5) + Technical results (5) 75.5 74.4 75.5 78.1 No Win / Win Technical variables (5) + Season results (5) 79.7 79.7 80.3 81.2 Technical results (5) + Season 79.7 79.2 82.8 results (5) 79.1 Technical variables (5) + Technical results (5) + 79.9 79.7 80.8 84.8 Season results (5) Technical variables (5) 71.5 71 71.5 72.2 Technical results (5) 72.5 72.3 72.2 72.5 Season results (5) 73.7 74.8 78.7 80.4 No Win/Win/Wins Technical variables (5) + Technical results (5) 72.9 72.3 72.7 77.2 Technical variables (5) Season results (5) + 74.5 79.7 75.5 81.0 Technical results (5) + Season results (5) 74.3 79.7 75.7 83.4 Technical variables (5)+Technical results (5) 74.7 75.4 75.7 86.0 +Season results (5) Mean 75.36 76.18 76.07 79.34 Standard deviation 2.77 3.35 3.02 4.33 Sum of square degree of freedom Mean square F p Between group 132.291 3 44.097 3.760 0.016 Within group 609.781 52 11.727 Total 742.071 55 Post-analysis (least significant Discriminant, classification tree, and binary logistic regression < artificial neural network model difference test) The purpose of this study was to find the best model, in terms of the classification accuracy, from four prediction models using the annual average performance variable data of LPGA players within the 60^th rank, over 25 seasons, and to compare the importance of the predicting variables according to the victory status of the four prediction models (Dodson et al., 2008; McGarry et al., 2002). We found that, first, the artificial neural network model showed a higher prediction rate than the other three models, when the independent variable was a skill variable and the dependent variable was the achievement of victory (Almassri et al., 2018; Jida and Jie, 2015). The prediction rate was in the order of the classification tree (73.7%) < discriminant model (74.1%) < binominal logistic regression model (74.2%) < artificial neural network model (75.3%). The most important predicting variables for determining victory were GIR and PA in all four prediction models. Second, the artificial neural network model showed a higher prediction rate than the other three models when the independent variable was the skill result and the dependent variable was victory. The prediction rate was in the order of the discriminant model (74.1%) < binominal logistic regression model (74.3%) < classification tree model (74.4%) < artificial neural network model (75.7%). Moreover, the most important predicting variable for determining victory was birdies in all four prediction models. Third, the artificial neural network model showed a higher prediction rate than the other three models when the independent variable was the season outcome and the dependent variable was victory. The prediction rate was in the order of the discriminant model (78.5%) < binominal logistic regression model (78.7%) < classification tree model (79.7%) < artificial neural network model (80.2%). The most important predicting variables for determining victory were T10 and OM in the discriminant and classification tree models, and OM, T10, and SA in the binomial logistic regression and artificial neural network model. To sum up the above three results, the player who aims for victory in the LPGA should have a chance of birdies at each hole by improving the GIR and PA, driving distance, and driving accuracy among skill variables, lowering the average strokes. This will increase the probability of being within T10 as well as the victory at each competition. Fourth, the one-way ANOVA was conducted to find the best model in terms of the classification accuracy of the four prediction models and to test the mean difference of the classification accuracy rate rising from the change in the number of independent variables according to the change in the dependent variable level (2 or 3). The LSD post-hoc test showed that the artificial neural network model had higher classification accuracy than the other three models. We can conclude that the artificial neural network model was superior when comparing the classification accuracy rates of the predicting models. This is consistent with the results of another study using neural networks when the sports disciplines considered were basketball, soccer, and tennis (Chae et al., 2018). Future research can supplement the data for predicting variables and quantify the mental strength and teamwork that are difficult to quantify for achieving an optimum harmony of predicting variables. The first practical implication relates to the prediction of the probability of victory in the LPGA using the artificial neural network model for achieving more meaningful results. The second implication is to arrange the schedule of training based on the DD, DA, GIR, SS, PA, and GIR if the player aims at victory in the LPGA tour. Furthermore, birdies are the most important skill result variable affecting victory as all four prediction models indicated birdies as the most important variable of victory. Thus, more time can be spent establishing a strategy for improving this skill.	{"url":"https://jhk.termedia.pl/Victory-Prediction-of-Ladies-Professional-Golf-Association-Players-Influential-Factors,158525,0,2.html","timestamp":"2024-11-04T08:08:45Z","content_type":"application/xhtml+xml","content_length":"198668","record_id":"<urn:uuid:56f313d2-2c42-4176-89bf-08b67cbc0b24>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00820.warc.gz"}
Computer Architecture and Organization 0071159975, 9780071159975 - DOKUMEN.PUB Citation preview Computer Architecture and Organization McGRAW-HILL INTERNATIONAL Computer Science Computer Architecture and Organization Series in Computer Science SENIOR CONSULTING EDITOR C.L. Liu, University of Illinois at CONSULTING EDITOR Allen B. Tucker, Bowdoin College Fundamentals of Computing and Programming Computer Organization and Architecture Computers in Society/Ethics Systems and Languages Theoretical Foundations Software Engineering and Database Artificial Intelligence Networks, Parallel and Distributed Computing Graphics and Visualization MIT Electrical and Computer Science Series McGraw-Hill Series Bell and Newell: Computer Organization and Architecture Structures: Readings and Examples Cavanagh: Digital Computer Arithmetic: Design and Implementation Feldman and Computer Architecture and Logic Design Gear: Computer Organization and Programming: With an Emphasis on Personal Computers Hamacher, Vranesic, and Zaky: Computer Organization Hayes: Computer Architecture and Organization Hayes: Digital System Design and Microprocessors Horvath: Introduction to Microprocessors Using the MC6809 or the MC68000 Hwang: Scalable Parallel and Cluster Computing: Architecture and Programming Hwang and Computer Architecture and Parallel Processing Lawrence and Mauch: Real-Time Microcomputer System Design Siweiorek, Bell and Newell: Computer Structures: Principles Stone: Introduction to Stone and Siewiorek: Introduction Ward and Computer Organization and Data Structures to Computer Organization and Data Edition Halstead: Computational Structures Series in Computer Engineering SENIOR CONSULTING EDITORS Stephen W. Director, University of Michigan, C.L. Liu, University of Illinois, Ann Arbor Computer Architecture and Logic Design Bose, Liang: Neural Network Fundamentals with Graphs, Algorithms, and Applications Chang and ULSI Technology Micheli: Synthesis Feldman and and Optimization of Digital Computer Architecture: A Designer's Text Based on a Generic RISC Hamacher, Vranesic, and Zaky: Computer Organization Hayes: Computer Architecture and Organization Horvath: Introduction Microprocessors Using the Hwang: Advanced Computer Architecture: MC6809 or the MC68000 Parallelism, Scalability, Programmability Hwang: Scalable Parallel and Cluster Computing: Architecture and Programming Kang and CMOS Digital Integrated Circuits: Analysis and Design Kohavi: Switching and Finite Automata Theory Krishna and Shin: Real-Time Systems Lawrence-Mauch: Real-Time Microcomputer System Design: An Introduction Levine: Vision in Man and Machine Analysis and Modeling of Digital Systems Peatman: Design with Microcontrollers Peatman: Digital Hardware Design Rosen: Discrete Mathematics and Ross: Fuzzy Logic with Engineering Applications Sandige: Sarrafzadeh and Digital Design Wong: An Introduction to VLSI Physical Design Schalkoff: Artificial Neural Networks Stadler: Analytical Robotics and Mechatronics VLSI Technology Taub: Digital Circuits and Microprocessors Wear, Pinkert, Wear, and Lane: Computers: Introduction to Hardware and Software Design Computer Architecture and Organization THIRD EDITION University of Michigan [i3l WCB Burr Ridge, IL Dubuque, IA Madison, WI Bogota Caracas Lisbon Bangkok Mexico City New York San Francisco London Madrid Singapore Sydney Taipei WCB/McGraw-Hill A McGraw-Hill Companies Division of The COMPUTER ARCHITECTURE AND ORGANIZATION International Editions 1998 Exclusive rights by McGraw-Hill Book Co - Singapore, for manufacture and export. This book cannot be re-exported from the country to which it is consigned by McGraw-Hill. 1998 by The' McGraw-Hill Companies, Inc. All rights reserved. under the United States Copyright Act of 1976, no part of distributed in any form or by any means, or stored this publication in a data Library of Congress Cataloging-in-Publication Data Hayes, John Computer p. (John Patrick) (date) architecture and organization Hayes. - 3rd ed. cm. - (Electrical and computer engineering) Includes bibliographical references and index. ISBN 0-07-027355-3 1. Construction architecture. and construction. electrical 2. II. Electronic digital computers-Design Series: series in and computer engineering. ordering this Printed in Malaysia ISBN 0-07-115997-5 base or retrieval system, without the prior written permission of the publisher. Except as permitted may be reproduced at the a professor in the electrical engineering and computer science University of Michigan, where he was the founding director of the Advanced Computer Architecture Laboratory. He teaches and conducts research computer architecture; computer-aided design, verification, and testing; VLSI design; and fault-tolerant systems. Dr. Hayes is the author of two patents, more than 150 technical papers, and five books, including Layout Minimization for the areas of Cells (Kluwer, 1992, coauthored with R. L. Maziasz) and Introduction to Digital Logic Design (Addison-Wesley, 1993). journals, including the has served as editor of various was technical program chairman of Computer Architecture Symposium, Toronto. the Journal of Electronic Testing, and Transactions on Parallel and Distributed Systems and the 1991 Hayes received his undergraduate degree from the National University of Ireand his M.S. and Ph.D. degrees in electrical engineering from the University of Illinois, Urbana-Champaign. Prior to joining the University of Michigan, he was a faculty member at the University of Southern California. Dr. Hayes has also held visiting positions at various academic and industrial organizations, including Stanford University, McGill University, Universite de Montreal, and LogicVision Inc. He is a fellow of the Institute of Electrical and Electronics Engineers and a member of the Association for Computing Machinery and Sigma Xi. Dr. land, Dublin, My Father (1910-1968) In Computing and Computers 1.1 The Nature of Computing 1.1.1 The Elements of Computers / 1 1 .2 Limitations . of Computers 1.2 The Evolution Of Computers The Mechanical Era / 1.2.2 Electronic Computers / The Later Generations The VLSI Era 1.3.1 Integrated Circuits / 1.3.2 Processor Architecture / 1.3.3 System Architecture Des ign Methodology System Design System Representation / 2.1.2 Design Process / The Gate Level The Register Level 2.2.7 Register-Level Components / 83 2.2.2 Logic Devices / 2.2.3 Register-Level Design 2.3 The Processor Level 2.3.1 Processor-Level Components / 114 2.3.2 Processor-Level Processor Basics 3.1 CPU i.7.7 Organization Fundamentals / 3.1.2 Additional Features Data Representation 3.2.1 Basic Formats / 3.2.2 Fixed-Point Numbers / 3.2.3 Floating-Point Instruction Sets 3.3.1 Instruction Formats / 3.3.2 Instruction Types / Programming Considerations Datapath Design 4.1 Fixed-Point Arithmetic 4.1.1 Addition and Subtraction / 4.1.2 Multiplication / 4.1.3 Division Arithmetic-Logic Units 4.2.1 Combinational / 4.2.2 Sequential ALUs Advanced Topics 4.3.1 Floating-Point Arithmetic / 4.3.2 Pipeline Processing Control Design Basic Concepts 5.7.7 Introduction / 5.1.2 5.1.3 Design Microprogrammed Control 5.2.7 Basic Hardwired Control / Concepts / 5.2.2 Multiplier Control Unit / CPU Control Unit 364 Pipeline Control 5.3.1 Instruction Pipelines / 5.3.2 Pipeline Performance / 5.3.3 Superscalar Processing Memory 6.1 Memory Technology 6.7.7 Device Characteristics / 6.1.2 Random- Access Memories / 6.1.3 Serial-Access Memories Memory Systems 6.2.1 Multilevel Memories / 6.2.2 Address Translation / Memory Allocation Main Features / 6.3.3 Structure versus 6.3.2 Address Mapping / Sysl em Organization Communication Methods 7.1.1 Basic Concepts / 7.1.2 Bus Control 10 And System Control Programmed 10 / 7.2.2 DMA and Interrupts / 10 Processors / 7.2.4 Operating Systems Parallel Processing 7.3.1 Processor-Level Parallelism / 7.3.2 Multiprocessors / 7.3.3 Fault Tolerance This book about the design of computers; covers both their overall design, or their internal details, or organization. hensive and self-contained view of computer design provide a compre- an introductory level, marily from a hardware viewpoint. The third edition of Computer Architecture and Organization is intended as a text for computer science, computer engineering, and electrical engineering courses at undergraduate or beginning graduate levels; should also be useful for self-study. This text assumes little in of prerequi- beyond some familiarity with computer programming, binary numbers, and Like the previous editions, the book focuses on basic principles but has been thoroughly updated and has substantially more coverage of digital logic. related issues. The book divided into seven chapters. Chapter discusses the nature and lim- itations of computation. This chapter surveys the historical evolution of design to introduce and motivate the key ideas encountered Chapter 2 deals with computer design methodology and examines the two major computer design levels, the register (or register transfer) and processor levels, in detail. reviews gate-level logic design and discusses computer-aided design (CAD) and performance evaluation methods. Chapter 3 describes the central processing unit (CPU), or microprocessor that lies at the heart of every computer, focusing on and data representation. The next two chapters address instruction set design design issues: Chapter 4 covers the data-processing part, or datapath, of a processor, while Chapter 5 deals with control-unit design. The principles of arithmetic- logic unit (ALU) design for both fixed-point and floating-point operations are Both hardwired and microprogrammed control are examined along with the design of pipelined and superscalar processors. Chap- covered in Chapter Chapter in ter 6 deals with a computer's subsystem; the chapter discusses the princi- technologies and their characteristics from a hierarchical viewpoint, with emphasis on cache memories. Finally, Chapter 7 addresses the overall organization of a computer system, including inter- and intrasystem communication, input-output (10) systems, and parallel processing to achieve very high perforreliability. Various representative computer systems, such as von Neumann's classic IAS computer, the ARM RISC microprocessor, the Intel Pentium, the Motorola PowerPC, the MIPS RXOOO, and the Tandem NonStop fault-tolerant multiprocessor, appear as examples throughout the book. The book has been in use for many years at universities around the world. It contains more than sufficient material for a typical one-semester (15 week) course, mance and allowing the instructor some leeway in choosing the topics to emphasize. Much of the background material in Chapter 1 and the first part of Chapter 2 can be left as a reading assignment, or omitted advanced material loss of continuity. the students are suitably prepared. Chapter 7 can be covered briefly or skipped The more desired without Manual contains some representative course This edition updates the contents of the previous edition and responds to the its users while retaining the book's time-proven emphasis.on basic suggestions of concepts. material The third edition is somewhat more accessible to readers who shorter than predecessors, and the are less familiar with computers. Every section has been rewritten to reflect the dramatic changes that have occurred in the computer industry over the last decade. The main structural changes are the reorganization of the two old chapters on processor design and control design into new Chapters and the consolidation of the two old new Chapter 7. The treatment of performance-related topics such as pipeline control, cache design, and superscalar architecture has been expanded. Topics that receive less space in this edition include gate-level design, microprogramming, operating systems, and vecthree chapters: the 3, 4, chapters on system organization and parallel processing in the The third edition also includes many new examples (case studies) and end-of-chapter problems. There are now more than 300 problems, about 80 percent of which are new to this edition. Course instructors can obtain an Instructor's Manual, which contains solutions to all the problems, directly from the pubtor processing. specific changes material in Chapter in the third edition are as follows: design has been de-emphasized in Chapter A new evaluation has been expanded. (PLDs) has been added, and the stressed. The old has been streamlined and brought up to date. Gate-level 2, while the discussion of performance on programmable logic devices has been been split into Chapter 3, "Datapath Design." Chapter 3 contains an role of computer-aided design third chapter (on processor design) has "Processor Basics," and Chapter 4, expanded treatment of RISC and CISC CPUs and their instruction sets. It introduces the ARM and MIPS RX000 microprocessor series as major examples; the Motorola 680X0 series continues to be used as an example, however. The material on computer arithmetic and ALU design now appears in Chapter 4. The old chapter on control design, which is now Chapter 5, has been completely revised with a more practical treatment of hardwired control and a briefer treatment of microprogramming. A new section on pipeline control includes some material from the old Chapter 7, as well as new material on superscalar processing. Chapter 6 presents an updated treatment of the old fifth chapter on memory organization. Chapter 6 continues to present a systematic, hierarchical view of computer memories but has a greatly expanded treatment of cache memories. Chapter 7, "System Organization," merges material from the old sixth and seventh chapters. The sections on operating systems and parallel processing have been shortened and modernized. The material for this book has been developed primarily for courses on computer architecture and organization that versity of Southern California to colleagues and students have taught over the years, initially at the Uniof Michigan. I am grateful later at the University these and other schools for their comments and suggestions. As always, I owe a special thanks as well as her never-failing support wife Terrie for proofreading assistance, and love. John Computing and Computers This chapter provides a broad overview of digital computers while introducing many of the concepts that are covered in depth later. It first examines the nature and limitations of the computing process. Then it briefly traces the historical development of computing machines and ends with a discussion of contemporary VLSIbased computer systems. THE NATURE OF COMPUTING Throughout history humans have relied mainly on their brains to perform calculations; in other words, they were the computers [Boyer 1989]. As civilization advanced, a variety of computing tools were invented that aided, but did not replace, manual computation. The earliest peoples used their fingers, pebbles, or tally sticks for counting purposes. The Latin words digitus meaning "finger" and calculus meaning "pebble" have given us digital and calculate and indicate the ancient origins of these computing concepts. Two early computational aids that were widely used until quite recently are the abacus and the slide rule, both of which are illustrated in Figure 1.1. The abacus has columns of pebblelike beads mounted on rods. The beads are moved by hand to positions that represent numbers. Manipulating the beads according to certain simple rules enables people to count, add, and perform the other basic operations of arithmetic. The slide rule, on the other hand, represents numbers by lengths marked on rulerlike scales that can be moved relative to one another. By adding a length a on a fixed scale to a length b on a second, sliding scale, their combined length c = a + b can be read off the fixed scale. The slide rule's main scales are logarithmic, so that the process of adding two lengths on these scales effectively multiplies two The Nature of Computing B= • Decrement count then go to M(6, 20:39) Test 0:19) by one. N and branch to 6R Update count N. Increment AC + M(2) AC + M(2) M(4, 8:19):=AC(28:39) A(I) + B( I). Load count AC by one. Modify address Modify address Modify address to 3L. Figure 1.15 An IAS program for vector addition. 4L, respectively. Thus the program continuously modifies ure 1.15 tion, the first AC:=AC + M(2001) Critique. In the years that pleted, during execution. Figthe have elapsed since the IAS computer was comin computer design have appeared. Hindsight numerous improvements enables us to point out 1 program before execution commences. At the end of three instructions will have changed to the following: the some of the IAS's shortcomings. The program self-modification process illustrated in the preceding example for and debugging a prochange themselves is difficult and error-prone. Further, before every execution of the program, the original version must be reloaded into M. Later computers employ special instruction types and registers for index control, which eliminates the need for address-modify instructions. decrementing the index gram whose I is inefficient. In general, writing CPU results in a great deal of unproCPU and main memory M; it also adds to program length. Later computers have more CPU registers and a special memory called a cache that acts as a buffer between the CPU register? and M. The small amount of storage space in the ductive data-transfer traffic between the no procedure were provided for structuring programs. For example, the IAS has call or return instructions to link different instruction set biased toward numerical computation. Programs for non- numerical tasks such as text processing were difficult to write and executed slowly. 5. Input-output (10) instructions were considered of minor importance they are not mentioned in Burks, Goldstine, and von in fact, [1946] beyond IAS had two basic and rather inefficient 10 The input instruction INPUT(X, N) transferred N words from an input device to the CPU and then to N consecutive main memory locations, starting at address X. The OUTPUT(X, N) instruction transferred N consecutive words from the memory region with starting address X to an outnoting that they are necessary. instruction types [Estrin 1953]. put device. The Later Generations on size and speed imposed by early electronic technology, the IAS and other first-generation computers introduced many features that are central to later computers: the use of a CPU with a small set of registers, a separate main memory for instruction and data storage, and an instruction set with a limited range of operations and addressing capabilities. Indeed the term von Neumann computer has become synonymous with a computer In spite of their design deficiencies and the limitations of conventional design. The second generation. Computer hardware and software evolved rapidly of the first commercial computers around 1950. The vacuum tube quickly gave way to the transistor, which was invented at Bell Laboratories in 1947, and a second generation of computers based on transistors superseded the after the introduction generation of vacuum tube-based machines. Like tube, a transistor serves as a high-speed electronic switch for binary signals, but much less power than a vacuum tube. Similar of memory technology, with ferrite cores becoming cheaper, sturdier, and requires progress occurred in the field dominant technology for main memories until superseded by all-transistor memories in the 1970s. Magnetic disks became the principal technology for sec- ondary memories, a position that they continue to hold. circuits, the second generation, which spans the decade 1954-64. introduced some important changes in the design of CPUs and their instruction sets. The IAS computer still served as the basic model, but more registers were added to the CPU to facilitate data and address manipulation. For Besides better electronic example, index registers were introduced to store an index variable of the kind appearing in the statement C(I):=A(I) + B(I) Computing and Computers The Evolution of Computers Index registers make it possible to have indexed instructions, which increment or decrement a designated index I before (or after) they execute their main operation. Consequently, repeated execution of an indexed operation like (1.9) allows it to The index value step automatically through a large array of data. stored in a and not in the program, so the program Itself does not change during execution. Another innovation was the introduction of two program-control instructions, now referred to as call and return, to facilitate the linking of proregister grams; see also Example "Scientific" computers of the second generation, such as the IBM 7094 introduced floating-point number formats and supporting instructions to facilitate numerical processing. Floating point a type of scientific number such as 0.0000000709 is denoted by 7.09 X 10" 8 floating-point number consists of a pair of fixed-point numbers, a mantissa notation where a = X B~E In the preceding example and an exponent E, and has the value 7.09, E = -8, and B = 10. In their computer representation and E are encoded in binary and embedded in a word of suitable size; the base B is implicit. Floating. numbers eliminate need for number scaling; floating-point numbers are The hardware needed to implement automatically scaled as they are processed. arithmetic many computers (then and ment floating-point operations via fixed-point expensive. Conse- on software subroutines to imple- Input-output operations. Computer designers soon realized that IO operations, that the transfer of information to and from peripheral devices and done like printers secondary memory, can severely degrade overall computer performance Most IO transfers have main memory as their final source or destinaand involve the transfer of large blocks of information, for instance, moving a program from secondary to main memory for execution. Such a transfer can take place via the CPU, as in the following fragment of a hypothetical IO program: inefficiently. := D(I) I:=I+1 if I Figure 1.18 representative IC packages: (a) 32-pin small-outline J-lead (SOJ); (b) 132-pin plastic quad flatpack (PQFP); 84-pin pin-grid array (PGA). [Courtesy of Sharp Electronics Integrated Circuits The integrated was invented Corporations [Braun and Texas Instruments and Fairchild quickly became the basic building block for computers of the third and subsequent generations. (The designation of computers by generation largely fell into disuse after the third generation.) An IC is an electronic circuit composed mainly of transistors that is manufactured in a tinyrectangle or chip of semiconductor material. The IC is mounted into a protective plastic or ceramic package, which provides electrical connection points called pins or leads that allow the IC to be connected to other ICs, to input-output devices like a keypad or screen, or to a power supply. Figure 1.18 depicts several representative IC packages. Typical chip dimensions are 10 X 10 mm, while a package like that Figure 1.18b approximately 30 ably bigger than the chip X 30 X 4 mm. The IC package often consider- contains because of the space taken by the pins. package of Figure 1.18c has an array of pins (as many as 300 or more) projecting from its underside. A multichip module is a package containing several IC chips attached to a substrate that provides mechanical support, as well as electrical connections between the chips. Packaged ICs are often mounted on a printed circuit board that serves to support and interconnect the ICs. A contemporary computer consists of a set of ICs, a set of IO devices, and power supply. The number of ICs can range from one IC to several thousand, depending on the computer's size and the IC types I C density. integrated circuit roughly characterized by defined as the number of transistors contained in the chip. As manufacturing niques improved over the years, the size of the transistors in an IC and their inter1 pm. (By comparison, the width of a human hair is about 75 ujn.) Consequently, IC densities have increased steadily, while chip size has varied very little. The earliest ICs the first commercial IC appeared in 1961 contained fewer than 100 transistors and employed small-scale integration or SSI. The terms medium-scale, large-scale, and very-large-scale integration (MSI, LSI and VLSI. connecting wires shrank, eventually reaching dimensions below a micron or 37 • DRAM a „' . • • Computers u c j 6_bj t £ v» \| a y ./^ ./• 64-bit DRAM 10 3 4-bit microprocessor •SSI i Figure 1.19 Evolution of the density of commercial ICs. respectively) are applied to ICs containing hundreds, thousands, and millions of transistors, respectively. The boundaries between these IC classes are loose, and often serves as a catchall term for very dense circuits. Because their highly automated tured in high resembles a printing process low cost per —ICs can be manufac- Indeed, except for the latest and densest circuits, the cost of an IC has stayed fairly constant over the years, implying that newer generations of ICs deliver far greater value (measured by computing performance or storage capacity) per unit cost than their predecessors did. Figure 1.19 shows the evolution of IC density as measured by two of the dens- dynamic random-access memory (DRAM), a basic component CPU or microprocessor. Around 1970 it became possible to manufacture all the electronic circuits for a pocket calculator on a single IC chip. This development was quickly followed by single-chip DRAMs est chip types: the of main memories, and the single-chip and microprocessors. As Figure 1.19 shows, the capacity of the largest available DRAM chip was IK = 2 10 bits in 1970 and has been growing steadily since then, 20 reaching 1M = 2 bits around 1985. A similar growth has occurred in the complexity of microprocessors. The first microprocessor, Intel's 4004, which was introduced in 1971, was designed to process 4-bit words. The Japanese calculator manufacturer Busicom commissioned the 4004 microprocessor, but after Busicom's early demise, Intel successfully marketed the 4004 as a programmable controller to replace standard, nonprogrammable logic circuits. As IC technology improved and chip density increased, the complexity and performance of one-chip microprocessors increased steadily, as reflected in the increase in CPU word size to 8 and then 16 bits by the mid-1980s. entire By 1990 manufacturers could of a System/360-class computer, along with part of on a single IC. The combination of a CPU, memory, and IO small number of ICs) called a microcomputer. Computing and fabricate the main memory, in one IC (or a IC families. Within IC technology SECTION several subtechnologies exist that are dis- tinguished by the transistor and circuit types they employ. of the most impor- The VLSI Era tant of these technologies are bipolar and unipolar; the latter is normally referred to MOS (metal-oxide-semiconductor) after its physical structure. Both bipolar and MOS circuits have transistors as their basic elements! They differ, however, in the as polarities of the electric charges associated with the primary carriers of electrical signals within their transistors. Bipolar circuits use both negative carriers (elec- and positive MOS circuits, on the other hand, use only one MOS (PMOS) and negative in the case of N-type MOS (NMOS). Various bipolar and MOS IC circuit types or IC families have been developed that provide trade-offs among density, operating speed, power consumption, and manufacturing cost. An MOS family that efficiently combines PMOS and NMOS transistors in the same IC is complementary MOS or CMOS. This technology came into widespread use in the 1980s and has been the technology of choice for microprocessors and other VLSI ICs since then because of its combination of high density, high speed, and very low power consumption [Weste and Eshragian 1992]. trons) carriers (holes). type of charge carrier: positive in the case of P-type NOLOGY. To CMOS The ZERO-DETECTION CIRCUIT EMPLOYING CMOS TECHof transistors in computing, illustrate the role whose function circuit's output z should be is 1 to detect when x x2x i = 0000; combinations of input values. Zero detection cessing. For example, if it is word x a 4-bit quite a we examine x l should be a small x 2x i becomes for the other 15 operation in data pro- used to determine when a program loop terminates, as in the statement (location 5R) appearing in the IAS program of Figure 1.15. Figure 1.20 shows a particular implementation sentative symbolic form denoted 5,:5 7 and Figure 1.20a. as static ZD of zero detection CMOS. The circuit is using a repre- numbers of PMOS transistors Each transistor acts as an on-off consists of equal SS :S U transistors denoted switch with three terminals, where the center terminal c controls the switch's When turned on, a signal propagation path created between the transistor's upper and lower terminals; when turned off, that path is broken. An NMOS transistor is turned on by applying 1 to its control terminal c; it is turned off by applying to c. A PMOS transistor, on the other hand, is turned on by c and turned off by c = 1 Each set of input signals applied to ZD causes some transistors to switch on and others to switch off, which creates various signal paths through the circuit. In Figure 1.20 the constant signals and 1 are applied at various points in ZD. (These signals are derived from ZD's electrical power supply.) The 0/1 signals "flow" through the circuit along the paths created by the transistors and determine various internal signal values, main output line z. Figure 1.20b shows the signals and signal transmission paths produced by x x x 2x3 - 0001. The first input signal x = is applied to PMOS transistor 5, and NMOS transistor 5 g hence S, is turned on and 5 g turns S 2 on and S 9 off. A path is created through S, and is turned off. Similarly, x, = S2 which applies 1 to the internal line y,, as shown by the left-most heavy arrow in Figand y 3 = 1. ure 1. 20b. In the same way the remaining input combinations make y 2 = as well as the value applied to the latter signal is applied to the two right-most transistors turning S 7 off and 5 14 on, which creates a path from the zero source to the primary output line via 5 14 so z = as we change input x 3 from 1 to in Figure 1.20b, the following chain of events 54 turns on and 5,, turns off, changing y2 to 1. Then 5 I3 turns on and S6 turns making y 3 = 0. Finally, the new value of y 3 turns 57 on and S ]4 off, so z becomes If occurs: off, Computing and Computers NMOS transistor x = xQ = x2 - 3 = Transistor switched on Transistor switched off Figure 1.20 (a) CMOS circuit ZD for zero detection; (b) xQx x2x } = 0001 making l with input combination the zero input combination x x x 2x 3 = 0000 makes c = readily be verified that no other input combination does this. transistor circuit like that of Figure 1.20 circuit at a as required. models the behavior of a digital Because many of the level of abstraction called the switch level. ICs of interest contain huge numbers of transistors, it is rarely practical to analyze computing functions at the switch level. Instead, we move to higher abstraction levels, two of which are illustrated in Figure 1.21. At the gate or logicAexe\ illustrated by Figure 1.21a. we represent certain common subcircuits by symbolic their The VLSI Era NAND gate NOT gate (inverter) NOR gates 00 Figure 1.21 The zero-detection ter level circuit of Figure at (a) the gate level and (b) the regis- of abstraction. components called A, B, C, and comprises four gates (logic) gates. This particular logic circuit of three different types as indicated; note that each gate type has a distinct graphic symbol. In moving from the switch transistor circuit into a single gate tage of the logic level it is and discard all its collapse a multi- internal details. technology independent, so A key can be used equally well to describe the behavior of any IC family. In dealing with computer design, also use an even higher level of abstraction transfer level. as the register or register- treats the entire zero-detection circuit as a primitive or indivisible component, as in Figure 1.21b. The register level is the level at which we describe the internal workings of a CPU or other processor as, for example, in Figures 1.2 and 1.17. Observe that the primitive components (represented by boxes) in these diagrams include memory, or computer level of abstraction, ALUs, and the like. When we an entire component, we have moved called the processor or system level. as a primitive to the highest Processor Architecture By 1980 computers were classified into three mainframe computers, minicomputers, and microcomputers. The term mainframe was applied to the traditional "large" computer system, often containing thousands of ICs and costing millions of dollars. served as the central computing facility for an organization such as a university, a factory, or a bank. Mainframes were then room-sized machines placed in special computer centers and not directly accessible average user. The minicomputer was a smaller (desk size) and slower version of the mainframe, but its relatively low cost (hundreds of thousands of dollars) to the suitable as a "departmental" a small business, for example. to be shared The microcomputer was even cheaper (a few thousand dollars), packing group of users smaller, slower, the electronics of a into a handful of ICs, including microprocessor (CPU), memory, and IO chips. Personal computers. Microcomputer technology gave rise to a class of general-purpose machines called personal computers (PCs), which are intended for a single user. These small, inexpensive computers are designed to sit on an office desk or fold into a compact form to be carried. The more powerful desktop computers intended for scientific computing are referred to as workstations. has the von Neumann organization, with a microprocessor, a multimegabyte main memory, and an assortment of 10 devices: a keyboard, a video monitor or screen, a magnetic or optical disk drive unit for high-capacity secondary memory, and interface circuits for connecting the PC to printers and to other computers. Personal computers have proliferated to the point that, in the more developed societies, they are present in most offices and many homes. Two of the main applications of PCs are word processing, where personal computers have assumed and greatly expanded all the functions of the typewriter, and data-processing tasks like financial record keeping. They are also used for entertainment, education, and increasingly, communication with other computers via the World Wide Web. Personal computers were introduced in the mid-1970s by a small electronics kit maker, MITS Inc. [Augarten 1984]. The MITS Altair computer was built around the Intel 8008, an early 8-bit microprocessor, and cost only $395 in kit form. The most successful personal computer family was the IBM PC series introduced in 1981. Following the precedent set by earlier IBM computers, it quickly the de facto standard for this class of machine. standardization process —namely, IBM's decision called an open architecture, by A new factor also aided the to give the PC what came to be design specifications available to other manufacturers of computer hardware and software. As a became very popular, and many versions of were produced by others, including startup companies that made the manufacture of low-cost PC clones their main business. The PC's open architecture also provided an incentive for the development of a vast amount of application-specific software from many sources. Indeed a new software industry emerged aimed at the massit the so-called production of low-cost, self-contained programs aimed the at specific applications of IBM PC and a few other widely used computer families. The IBM PC series is based on Intel Corp.'s 80X86 family of microprocessors, which began with the 8086 microprocessor introduced in 1978 and was followed 2 by the 80286 (1983), the 80386 (1986), the 80486 (1989), and the Pentium (1993) [Albert and Avnon 1993]; the Pentium II appeared in 1997. The IBM PC series is also distinguished by its use of the MS/DOS operating system and the Windows graphical user interface, both developed by Microsoft Corp. Another popular personal computer series is Apple Computer's Macintosh, introduced in 1984 and built around the Motorola 680X0 microprocessor family, whose evolution from the 68000 microprocessor (1979) parallels that of the 80X86/Pentium [Farrell 1984\|. In 1994 the Macintosh was changed to a new microprocessor known as the PowerPC. Figure 1.22 shows the organization of a typical personal computer from the compare Its legacy from earlier von Neumann computers is apparent Figure 1.22 to Figure 1.17. At the core of this computer a single-chip micropro- As we will see, the microprocessor's internumber of speedup features not found in A system bus connects the microprocessui to a main memor) cessor such as the Pentium or PowerPC. nal (micro) architecture usually contains a its based on semiconductor DRAM technology and to an IO subsystem. A separate IO bus, such as the industry standard legal ruling that PCI (peripheral component interconnect) microprocessor names that are numbers cannot have trademark protection, resulted a microprocessor called the Pentium rather than the 80586. 80486 being followed by in the Computing and Computers r 42 (hard disk) 1.3 The VLSI Era IO expansion Hard disk IO devices u IO interface unit 2 bus Peripheral (IO) interface control unit System bus Figure 1.22 A typical personal computer system. bus, connects directly to the is IO devices and their individual controllers. linked to the system bus, to which the microprocessor and The IO bus are attached via a special bus-to-bus control unit sometimes referred to as a bridge. The IO devices of a personal computer include the traditional keyboard, a CRT-based or flat-panel video monitor, and disk drive units for the hard and flexible (floppy) disk storage devices that constitute secondary memory. More recent additions to the (compact disc read-only memories), which have extremely high capacity and allow sound and video images to be stored and retrieved efficiently. Other common audiovisual IO devices in personal computers are microphones, loudspeakers, video scanners, and the like, which are referred to as multimedia equipment. device repertoire include drive units for Performance considerations. As processor hardware became much sive in the 1970s, thanks mainly to advances in VLSI technology (Figure 1.19), computer designers increased the use of complex, multistep instructions. This reduces N, the total number of instructions that must be executed for a given task, since a single complex instruction can replace several simpler ones. For example, a multiply instruction can replace a multiinstruction subroutine that implements multiplication by repeated execution of add instructions. Reducing N in this way tends to reduce overall program execution time T, as well as the time that the CPU spends fetching instructions and their operands from memory. The same advances in VLSI made it possible to add new features to old microprocessors, such as new instructions, data types, instruction sets, and addressing modes, while retaining the ability to execute programs written for the older machines. The Intel 80X86/Pentium series illustrates the trend toward more complex instruction sets. The 1978-vintage 8086 microprocessor chip, which contained a mere 20,000 transistors, was designed to process 16-bit data words and had no instructions for operating on floating-point numbers [Morse et al. 1978]. Twentyfive years later, direct descendant, the Pentium, contained over 3 million transis- processed 32-bit and 64-bit words directly, and executed a comprehensive set of floating-point instructions [Albert and Avnon 1993]. The Pentium accumulated most of the architectural features of its various predecessors in order to enable it to execute, with little or no modification, programs written for earlier 80X86-series machines. Reflecting these characteristics, the 80X86, 680X0, and most older com3 puter series have been called complex instruction set computers (CISCs). By the 1980s it became apparent that complex instructions have certain disadvantages and that execution of even a small percentage of such instructions can sometimes reduce a computer's overall performance. To illustrate this condition, suppose that a particular microprocessor has only which requires k time units, to execute. instructions in 100k time units. simple instructions, each of the microprocessor can execute 100 suppose that 5 percent of the instructions are To execute an average set of 100 instructions therefore requires (5x21+ 95)k = 200k time units, assuming no other factors are involved. Consequently, the 5 percent of complex instrucslow, complex instructions requiring 2lk time units each. program execution time. program size, this technology does not necessarily translate into faster program execution. Moreover, complex instructions require relatively complex processing circuits, which tend to put CISCs in the largest and most expensive IC category. These drawbacks were first recognized by John Cocke and his colleagues at IBM in the mid-1970s, who developed an experimental computer called 801 that aimed to achieve very fast overall performance via a streamlined instruction set that could be executed extremely fast [Cocke and Markstein 1990]. The 801 and subsequent machines with a similar design philosophy have been called reduced instruction set computers (RISCs). A number of commercially successful RISC microprocessors were introduced in the 1980s, including the IBM RISC System/6000 and SPARC, an "open" microprocessor developed by Sun tions can, as in this particular example, double the overall Thus while complex instructions reduce Microsystems and based on RISC research at the University of California, Berkeley [Patterson 1985]. Many of the speedup features of RISC machines have found their into other new computers, including such CISC microprocessors as the PenRISC is often used to refer to any computer with an instruc- tium. Indeed, the term tion set and an associated organization designed for very high performance: the actual size of the instruction set A its computer's performance relatively unimportant. also strongly affected by other factors besides instruction set, especially the time required to instructions and data and, to a lesser extent, the time required to and main memory and IO devices. It typically takes the CPU about move information between than from one of its internal registers. five times longer to obtain a word from between the This difference in speed has existed since the first electronic computers, despite strenuous efforts by circuit designers to develop memory devices and processor- interface circuits that are fast keep up with the fastest micro- speed disparity has become such a feature of standard (von Neumann) computers that is sometimes referred to as the von Neumann bottleneck. RISC computers usually limit access to main memory to a few load processors. Indeed the and store instructions; other instructions, including gram-control instructions, must have their operands in data-processing and pro- registers. This so- The public became aware of CISC complexity when a design flaw affecting the floating-point division Pentium was discovered in 1994. The cost to Intel of this bug. including the replacement cost of Pentium chips already installed in PCs. was about $475 million. instruction of the Computing and Computers called load-store architecture mann bottleneck by reducing the CPU. The VLSI Era Performance measures. "basic" operations that intended to reduce the impact of the von Neu- number of the total CPU speed A typical rough indication of can perform per unit of time. the fixed-point addition of the contents of made by number of basic operation Rl and R2, as in the bolic instruction Rl :=R1 Such operations are timed by a regular stream of signals (ticks or beats) issued by a The speed of the clock is its frequency / millions of ticks per second; the units for this are megahertz (MHz). central timing signal, the system clock. measured in Each tick of the clock the operation is 1// triggers a basic operation; Tdock For example, hence the time required to execute This value called the clock cycle or clock a computer clocked at 250 can perform one basic Complicated operations such as division or operations on floating-point numbers can require more than one clock cycle to complete their execution. . operation in the clock period Tdock = 1/250 = 0.004 Generally speaking, smaller electronic devices operate faster than larger ones, accompanied by a from 1981 to 1995 to 100 MHz. Clock so the increase in IC chip density discussed above has been steady, but less dramatic, increase in clock speed. For example, microprocessor clock speeds increased from about 10 speeds of versions of current or 1000 MHz) and beyond It are feasible using faster might therefore seem possible to achieve any desired processor speed simply by increasing the clock frequency. which clock frequency is increasing due to IC technology improvements is relatively slow and may be approaching limits determined by the speed of light, power dissipation, and similar physical considerations. Extremely fast cir- ever, the rate at cuits also tend to be very expensive to manufacture. The CPU's processing of an instruction involves several steps, each of requires at least one clock cycle: 1. 2. 3. memory M. Decode the instruction's opcode. any operands needed unless they Load (read) from Fetch the instruction from main are already in Execute the instruction via a register-to-register operation using an appropriate functional unit of the CPU, such as a fixed-point adder. Store (write) the results in fastest instructions M unless they are to be retained in CPU registers. all their cuted by the The slowest instructions require multiple in a single clock cycle, so steps CPU 1 to 3 all take and can be exeone clock cycle. accesses and multiple register- complete their execution. Consequently, measures of instruction execution performance are based on average figures, which are usually determined experimentally by measuring the run times of representative or benchto-register operations to mark programs. The more representative the programs are, that rately they reflect real applications, the better the more accu- performance figures they provide. benchmark program or set (suite) of on a given CPU takes T seconds and involves the execution of a total of N machine (object) instructions. Here N is the actual number of instructions executed, including repeated executions of the same instruction; it is not the number of instructions appearing in Q. As far as the typical computer user is concerned, the key performance goal is to minimize the total program execution time T. While T can be determined accurately only by measurement of n 2 Assume . that subtracts a unary number n : (rem another unary num- n 2 and the result n, - n : are stored in the formats described , Computing and Computers Example 1.1. That is, the tape initially contains only n and n 2 separated by a blank, while the final tape should contain only n, - n 2 Describe your machine by a program with comments, following the style used Problems 1.4. Construct a Turing machine program Countjup in the style of Figure 1.4 that incre- ments an arbitrary binary number by one. For example, if the number 1001 1 denoting 19 is initially on an otherwise blank tape, Count _up should replace it with 10100 denoting 20. Assume that the read-write head starts and ends on the blank square immediately to the left of the listing number on the tape. Describe your with comments, following the style used in Figure employing fewer than 10 around 10 120 . machine by a program Fewer than 20 in- states suffice for this problem.] The number of possible sequences of moves ed chess has been estimat- developing a surefire winning strategy for chess therefore an un- solvable problem? Determine whether each of the following computational tasks is unsolvable. undeor intractable. Explain your reasoning, (a) Determining the minimum amount of wire needed to connect any set of n points (wiring terminals) that are in specified but arbitrary positions on a rectangular circuit board. Assume that at most two wires may be attached to each terminal, (b) Solving the preceding wiring lem when the n points and the wires periphery of the board; that connect them are constrained to lie the wire segments connecting the n points on the lie on a fixed rectangle. Most word-processing computer programs contain a spelling checker. An obvious brute-force method to check the spelling of a word Wis to search the entire on-line dictionary from beginning to end and compare W to every entry in the dictionary. Outline a faster method to check spelling and compare its time complexity to that of the bruteforce method. Consider the four algorithms maximum problem times faster than faster than 1.9. listed in Figure 1.7. size that each algorithm can handle M. Repeat the calculation for a the given data, calculate the on a computer M' M" that is that is 1,000.000 times brute-force technique illustrated by the Euler-circuit algorithm in which involves the enumeration and examination of all possible cases, is applicable to many computing problems. To make the method tractable, problem-specific techniques are used to reduce the number of cases that need to be considered. For example, the eight-edge graph of Figure 1 .6b can be simplified by replacing the edge-pair eg with a single edge because any Euler circuit that contains c must also contain g, and vice versa. Similarly, the pair dh can be replaced by a single edge. The problem then reduces to checking for an Euler circuit in a six-edge graph. For the same problem, suggest another must considered, 1.10. can sometimes substantially reduce the number of cases that Consider the heuristic method with a different graph example. salesman problem discussed briefproblem involving at most five cities, for which to solve the traveling ly in section 1.1.2. Construct a specific the total distance dhem traveled in the heuristic solution is not the minimum distance dmin Conclude from your example (or from other considerations) that the heuristic solution can be made arbitrarily bad, that is, "worst case" problems can be contrived in dmin can be made arbitrarily large. . — 1.11. Consider the computation of x by the method of differences covered in Example 1.3. Suppose we want to determine x2 for x = 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, that is, at intervals of 0.5. Explain modify the method of Example 1.3 to this task. Computing and 1.12. Use the method of differences embodied x for integer values of from 1 to 10. Use is method of differences to compute x5 for integer values of x from to 8. What 5 smallest value of / for which the z'th difference of x is a constant? What is the Babbage's Difference Engine to compute value of that constant? 1.14. Consider the problem of computing a table of the natural logarithms of the integers from 1 to 200,000 to 19 decimal places, a task carried out manually in 1795. Select any modern commercially available computer system with which you are familiar and estimate the total time it would require to compute and print this table. Define all the parameters used in your estimation. Discuss the advantages and disadvantages of storing programs and data program concept). Under what circumstances programs and data in separate memories? 1.16. (the stored is it in the desirable to store Computers with separate program and data memories implemented in RAMs and respectively, are sometimes called Hanard-class machines after the Harvard Mark 1 computer. Computers with a single (RAM) memory for program and data storage are then called Princeton-class after the IAS computer. Most currently installed computers belong to one of these classes. Which one? Explain why the class you selected is the most widely used. Write a program using the IAS computer's instruction set (Figure 1.14) to compute x by means of the method of finite differences described in Example 1.3. For simplicity, assume that the numbers being processed are 40-bit integers and that the only dataprocessing instructions you may use are the IAS's add and subtract instructions. The 2 2 2 2 should be stored in k consecutive memory results x (x + l) ...,(x + k- l) (x + 2) , locations with starting address 3001. 1.18. A vector of 10 nonnegative numbers stored in consecutive locations beginning in lo- memory of the IAS computer. Using the cation 100 in the instruction set of Figure 1.14. computes the address of the largest number in this locations contain the largest number, specify the smallest address. write a IAS decided not to implement a square root instruction (ENIAC = x m can be computed iteratively and very efficiently following formula known in ancient Babylon: The designers of had one), via the array. If several citing the fact that y J+ \ Here; = 1, 2, 3, ..., >•„ is approximation to x ]f2 . cesses real (floating-point) numbers directly, construct a program ure 1.15 to calculate the square root of a given positive in the style number x according of Figto this IAS and other first-generation computers as some of their predecessors. In what sense was the IAS a parallel computer? What forms of parallelism do modern computers have that are lacking in the IAS? 1.20. Early literature describes the "parallel." unlike The IAS had no designed for transferring control between and return can be programmed using the IAS's call or return instructions programs, (a) Describe original instruction set. (b) would you suggest adding to the support call and return operations? SECTION Problen^ and a stack program of the kind given 1-22. Construct both a Polish expression 1.16a to evaluate the following expression: f:=(4x(a 2 + 1.23. b + the data presented in Figure 1.19, estimate on average, for the density of leading-edge ICs to double. This doubling rate, which has remained remark- ably constant over the years, cofounder of 1.24. Using the Intel Corp., referred to as of Figure 1.20 as an Moore 's law, after Moore, a in the 1960s. illustration, discuss justify the following Power consumption is very low and most of it occurs when the circuit is changing state (switching), (b) The logic signals and correspond to electrical voltage levels, (c) The subcircuits that constitute logic gates draw their power directly from the global power supply rather than from the external general properties of circuits: (a) (primary) input signals: hence the gates perform signal amplification. 1.25. zero-detection circuit of Figures 1.20 and 1.21 can be implemented as a single four-input logic gate. Identify the gate in question and redesign the circuit in the more compact 1.26. Design a single-gate form. ones-detection circuit in the multigate style of Figure 1.20. produce the output z = 1 if and only if x x x 2x 3 - 1111. Give both a transistor (switchlevel) circuit and a gate-level circuit for your design. ] Discuss the impact of developments computer hardware technology on the evolution of each of the following: (a) the logical complexity of the smallest replaceable components; (b) the operating speed of the smallest replaceable components; (c) the for- mats used for data and instruction representation. 1.28. Define the terms software compatibility and hardware compatibility. they played in the evolution of computers? 1.29. Identify and briefly describe three distinct ways in which parallelism can be introduced computer in order to increase its overall instruction ex- into the microarchitecture of a ecution speed. 1.30. Compare and contrast the IAS and PowerPC processors in terms of the complexity of Use the vector addition programs of writing assembly-language programs for them. Figures 1.15 and 1.27 to illustrate your answer. 1.31. popular microprocessor of the 1970s was the Intel 8085, a direct ancestor of the 80X86/Pentium size in the which has is IC package has only 40 pins, the the CPU and M as well as the structure The data word AD is used to attach IO devices are shared (multiplexed) as indicated. M to the 8085; there Because the 8085's for transmitting addresses and data between 8 bits, while the address size also a separate serial (two line) The 8085 has most complex arithmetic instructions are addition and subtraction of 8-bit fixed-point (binary and decimal) numbers. There are six 8-bit registers designated B, C, D, E, H, and L, which, with the accumulator A, form a about 70 different instruction types. address registers. A program counter PC byte required from in the usual BC, DE, and serve as 16-bit maintains the address of the next instruction manner. The 8085 also has stack pointer SP that points to the top of a user-defined stack area in Data/ Address Address low high 10 devices System bus (to 61 Control M and 10) port 8-bit internal data bus Stack pointer 8-bit — 8— «— 8— Program counter PC 8/16-bit register file Figure 1.32 Structure of the Intel 8085 microprocessor. NUM2 + + 16 16 Initialize address: Initialize address: NUM2 + C, 16 Initialize count: Store data: Decrement address: DE - Decrement address: HL - Decrement count: := 16. A + CY + M(HL). Convert sum to decimal. := Figure 1.33 An 8085 program add two 32-digit decimal of the 8085 's main (d) Identify three 1.32. Consider the the size of the purpose of common features of more recent microprocessors that the 8085 8085 described in the preceding A taste of its software can program ADDEC written in 8085 assembly language that performs the addition of two long (n digit) decimal numbers NUM1 and NUM2. The numbers are added two digits (8 bits) at a time using the instructions ADC (add with carry) and DAA (decimal adjust accumulator). ADC takes a byte from be found Figure 1.33, which lists a and, treating as an 8-bit binary Computing and number, adds and a carry to the contents of then changes the binary binary-coded decimal form. This calculation uses several flag bits of the status register SR: the carry flag 1.6 is set to decrement the is set 8-bit addition is CY, which and the zero (non-0), (a) NUM2 is stored, 1.33. from an when the result of an arithmetic instruction such as add or From the information given here, determine the size n of numbers being added and the (symbolic) location in M where the sum NUM1 + flag Z, the 9th bit resulting (b) Ignoring the size of the 8085 's instruction would you your answers. The performance of a 100 MHz microprocessor P is measured by executing 10,000,000 instructions of benchmark code, which is found to take 0.25 s. What are the values of CPl and MIPS for this performance experiment? Is P likely to be superscalar? Suppose that a single-chip microprocessor P operating at a clock frequency of 50 MHz new model P which has the same architecture as P but has a clock frequency of 75 MHz. (a) If P has a performance rating of p MIPS for a particular benchmark program Q, what is the corresponding MIPS rating p for P ? (b) P takes is replaced by a s to Q in a particular personal computer system C. On replacing P by P in Q drops only to 220 s. Suggest a possible reason for this dis- C, the execution time of appointing performance improvement. CISC and RISC? Identify two key archiRISC and CISC machines, (b) When developing the RISC/6000, the direct predecessor of the PowerPC, IBM viewed the word RISC to mean "reduced instruction set cycles." Explain why this meaning might be more appropriate for the PowerPC than the usual one. 1.35. {a) are the usual definitions of the terms tectural features that distinguish recent REFERENCES 1. Albert, D. and D. Avnon. "Architecture of the Pentium Microprocessor." IEEE Micro, 1993) pp. 11-21. vol. 13 (June Augarten, S. Bit by Bit: Illustrated History of Computers. York: Ticknor and Fields, 1984. 3. Etchemendy. Turing's World 3.0: An Introduction Theory. Stanford, CA: CSLI Publications, 1993. History of Mathematics. 2nd ed. New York: Wiley, 1989. MacDonald. Revolution in Miniature. The History and Impact of Semi- Boyer, C. B. Braun, E. and conductor Electronics. 2nded. Cambridge, England: Cambridge University Press, 1982. Burks, A. W., H. H. Goldstine, and J. von Neumann. "Preliminary Discussion of the Logical Design of an Electronic Computing Instrument." Report prepared for U.S. Ordnance Department, 1946. (Reprinted in Ref. 26, vol. 5, pp. 34-79.) Cocke, J. and V. Markstein. 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(Reprinted in Ref. 24, pp. 743-52.) 24. Siewiorek, D. P., C. G. Bell, and A. Newell. Computer Structures: Readings and Exam- New York: McGraw-Hill, 1982. Swade, D. D. "Redeeming Charles Babbage' s Mechanical Computer." Scientific American, vol. 268 (February 1993) pp. 86-91. 26. von Neumann, J. Collected Works, ed. A. Taub, 6 vols. New York: Pergamon, 1963. 27. Weiss, S. and J. E. Smith. Power and PowerPC San Francisco, CA: Morgan Kaufmann, ples. 1994. 28. Weste, N. and K. Eshragian. Principles of CMOS VLSI Design. 2nd ed. Reading, Addison-Wesley, 1992. 29. Wilkes, M. V. and "Microprogramming and the Design of Control CirComputer." Proc. Cambridge Phil. Soc, pt. 2, vol. 49 B. Stringer. cuits in an Electronic Digital (April 1953) pp. 230-38. (Reprinted in Ref. 24, pp. 158-63.) Computing and Computers York: Dover, 1961. S. P. et al. "Intel Design Methodology This chapter views the design process for digital systems at three basic levels of abstraction: the gate, the register, and the processor levels. It discusses the nature of the design process, examines design at the register and processor levels in detail, and briefly introduces computer-aided design (CAD) and analysis methods. an example of a system, which is defined informally as a colleccomplex one of objects called components, that are con- often a large and nected to form a coherent entity with a specific function or purpose. The function of the system determined by the functions of its components and nents are connected. are interested in information-processing systems is to map a set A of input information items (a program and its data, for example) into output information B (the results computed by the program acting on the data). The mapping can be expressed formally by a mathematical function/ from A to B. If/ maps element a of A onto element b of B, we write b = /(a) or b := f(a). We also restrict membership of A and B to digital or discrete quantities, whose values are defined only at discrete points of time. System Representation of objects of modeling a system V = {v,^^,...^,,} whose members (V!,v3 ),...,(vn _,,v„)}of all v, a graph. graph consists of a called nodes or vertices and a set of edges of nodes The edge e = taken from the (v,-,yp {(v ,v 2 ), l joins or connects node often defined by a diagram in which nodes are repre- sented by circles, dots, or other symbols and edges are represented by lines: this diagram is synonymous with the graph. The ordering implied by the notation (v,,v ) may be indicated in the diagram by an arrowhead pointing from v, to v as, for instance, in Figure 2.1. classes of objects: a set of informationlines that carry information signals between components. In modeling the system by a graph G, we associate C with the nodes of G and S with the edges of G; the resulting graph is often called a block diagram. This name comes from the fact that it is convenient to draw each node (component) as a block or box in which its name and/or its function can be written. Thus the various diagrams of computer structures presented in Chapter 1 Figure 1.29, for instance are block diagrams. Figure 2.2 shows a block diagram representing a small gate-level logic circuit called an EXCLUSIVE-OR or modulo-2 adder. This circuit has the same general form as the more abstract graph of Fig- Two central properties of any system are its strucand behavior; these very general concepts are often confused. We define the structure of a system as the abstract graph consisting of its block diagram with no functional information. Thus Figure 2.1 shows the structure of the small system of Figure 2.2. A structural description merely names components and defines their interconnection. A behavioral description, on the other hand, enables one to deterStructure versus behavior. for any given input signal a to the system, the corresponding output /(a). define the function/to be the behavior of the system. sented in different ways. Figure 2.3 called a truth table. behavior can be written in follows, noting that /(a) =/(x,^c2 ) /(0,0) = /(0,1)=1 /(1,0) Figure 2.1 graph with eight nodes and nine edges. The behavior/may be shows one kind of behavioral description all possible combinations of Another description of the same terms of mathematical equations as for the logic circuit of Figure 2.2. This tabulation of input-output values The systems of interest comprise two processing components C and a set of 66 1 System Design NOT • NOT p x2 o Figure 2.2 A block diagram representing logic circuit. and behavioral descriptions embodied Figures 2.1 and 2.3 are independent: neither can be derived from the other. The block diagram of Figure 2.2 serves as both a structural and behavioral description for the logic circuit in question, since from we can derive Figures 2.1 and 2.3. diagram conveys structure rather than behavior. For example, some of the block diagrams of computers in Chapter 1 identify blocks as being arithmetic-logic units or memory circuits. Such functional descriptions do not completely describe the behavior of the components in question; therefore, we cannot deduce the behavior of the system as a whole from the block diagram. If we it In general, a block need a more precise description of system behavior, we generally supply a separate narrative text, or a more formal description such as a truth table or a of equa- description languages. As we have we can system's structure and behavior by means of a block diagram fully describe a the term schematic diagram is also used in which we identify the functions of the components. We can convey the same detailed information by means of a hardware description language (HDL), a format that resembles (and is usually derived from) a high-level programming language such as Ada or C. The construction of such description languages can be traced back at least as far as Babbage [Morrison and Morrison 1961]. Babbage's notation, of which he was very proud, centered around the use of special symbols such as —> to represent the movement of mechanical components. In modern times Claude E. Shannon [Shannon 1938] introduced Boolean algebra Input a x Figure 2.3 1 Truth table for the as a concise and rigorous descriptive method for logic 1950s, academic and industrial researchers developed eventually evolved into a few widely used languages, notably which were standardized in the in the many ad hoc HDLs. These VHDL and Verilog, 1980s and 90s [Smith 1996; Thomas and Moorby Hardware description languages such VHDL have several advantages. They can provide precise, technology-independent descriptions of digital circuits at vari- ous levels of abstraction, primarily the gate and register levels. Consequently, they are widely used for documentation purposes. Like programming languages, HDLs can be processed by computers and so are suitable for use with computer-aided design (CAD) programs which, as discussed later, play an important role in the design process. For example, an to simulate the been specified. behavior of the negative side, description of a processor can be design have HDL descriptions are often long and verbose; the details of they lack the intuitive appeal and rapid insights that circuit diagrams and less for- mal descriptive methods provide. 2.1 VHDL DESCRIPTION OF A HALF ADDER. To illustrate the use HDLs, we give in Figure 2.4a a VHDL description of a simple logic component known as a half adder. Its purpose is to add two 1-bit binary numbers x and y to form a 2-bit result consisting of a sum bit sum and a carry bit carry. For example, if x = y= 1. the half adder should produce carry = 1, sum = 0, corresponding to the binary number EXAMPLE of 10, that A VHDL The is, description has two entity part an entity part and an architecture a formal statement of the system's structure at the highest level, that as a single component. describes the system's interface, which the "face" pre- sented to external devices but says nothing about the system's behavior or example the entity statement gives the half adder's formal name half_adder and the names assigned to its input-output (IO) signals; 10 signals are referred to in VHDL by their connection terminals or ports. Inputs and outputs are structure. In this entity half_adder (x.y: in bit; sum. earn-, out Inputs bit); end halfjudder; sum carry sum architecture behavior of half_adder is 1 sum alpha); NAND2: nand_gate port map (d => alpha, e => alpha. /=> end nand_gate alpha NAND1 / NAND2 / Figure 2.5 Half adder: (a) structural states that VHDL description; block diagram. half_adder has a component called d, e, and /ports (terminals) mapped (connected) of type nand_gate and to the signals x, v, and alpha, Design Process Given a system's structure, the task of determining its function or behavior is termed analysis. The converse problem of determining a system structure that exhibits a given behavior Design problem. design or synthesis. We can now state in broad terms the problem facing the puter designer or, indeed, any system designer. Given a desired range of behavior and a set of available components, determine a structure (design) formed from these components that achieves the desired behavior with acceptable cost and performance. the new design's behavior is the overriding goal of the design process, other typical requirements are to minimize cost as measured While assuring the correctness of System Design by the cost of manufacture and to maximize performance as measured by the speed of operation. There are some other performance- and cost-related constraints to satisfy such as high reliability, low power consumption, and compatibility with existing systems. These multiple objectives interact in poorly understood ways that depend on the complexity and novelty of the design. Despite careful attention to detail and the assistance of new system versions of a fail to tools, the initial meet some design objective, sometimes and hard-to-detect ways. This failure can be attributed to incomplete specifications for the design (some mode of behavior was overlooked), errors made by human designers or their CAD tools (which are also ultimately due to human error), and unanticipated interactions between structure, performance, and cost. For example, increasing a system's speed to a desired level can make the cost ceptably high. The complexity of computer systems is such that the design problem must be broken down into smaller, easier tasks involving various classes of components. These smaller problems can then be solved independently by different designers or design teams. Each major design step is often implemented via the multistep or iterative process depicted by a flowchart in Figure 2.6. An initial design is created, perhaps in ad hoc fashion, by adapting an existing design of a similar system. The result is then evaluated to see if it meets the relevant design objectives. If not, the design is revised and the result reevaluated. Many iterations through the redesign and evaluation steps of Figure 2.6 may be necessary to obtain a satisfactory design. Computer-aided design. The emergence of powerful and inexpensive desktop computers with good graphics interfaces provides designers with a range of programs to support their design tasks. CAD tools are used to automate, at least in Begin J Construct an initial and performance Modify the design to meet the goals Figure 2.6 Flowchart of an iterative design process. part, the more tedious design and evaluation and contribute tions or schematic diagrams, such as HDL descrip- which humans, computers, or both can Simulators create computer models of a design's behavior and help designers determine design, which can well the design meets vari- ous performance and cost goals. Synthesizers automate the design process itself by deriving structures that imple- or part of some design the easiest of these three tasks, and synthesis the synthesis methods incorporate exact or optimal algorithms which, even easy to program into resources. tools, often demand excessive amounts of computing synthesis approaches are therefore based on trial-and error meth- ods and experience with earlier designs. These computationally efficient but inexact methods are called heuristics and form the basis of most Design The design of complex system such as a CA© t»«ls. out at several levels of abstraction. Three such levels are generally recognized in computer design, although they are referred by various different names in the erature: • • The processor level, also called the architecture, behavior, or system The register level, also called the register-transfer level (RTL). The gate level, also called the logic level. As Figure key component treated as The processor level corresponds to a user's or manager's view of a computer. The register level is approximately the level of detail seen by a programmer. The gate level is primarily the concern of 2.7 indicates naming each level for a primitive or indivisible at that level of abstraction. These three design levels also correspond roughly to the major subdivisions of integrated-circuit technology into VLSI, MSI, and SSI components. The boundaries between the levels are far from clear-cut, and it is common to encounter descriptions that mix components from more than one level. the hardware designer. Logic gates, 10-' 2 to 10" 9 Blocks of Registers, counters, small sequential circuits. CPUs, memories, 10 Figure 2.7 The major computer design CAD editors or translators convert design data into forms in three to the overall design process. section 2 System Design few basic component types from each design level are listed in Figure 2.7. rec °g n i ze d as primitive at the gate level include AND, OR, NAND, NOR, and NOT gates. Consequently, the EXCLUSIVE-OR circuit of Figure 2.2 is an example of a gate-level circuit composed of five gates. The component marked XOR in Figure 2.5b performs the EXCLUSIVE-OR function and so can be thought of as a more abstract or higher-level view of the circuit of Figure 2.2, in which all internal structure has been abstracted away. Similarly, the half-adder block of Figure 2Ab represents a higher-level view of the threecomponent circuit of Figure 2.5b. We consider a half adder to be a register-level component. We might regard the circuit of Figure 2.5b as being at the register level also, but because NAND is another gate type and XOR is sometimes treated as a gate, this circuit can also be viewed as gate level. Figure 2.7 indicates some further differences between the design levels. The units of information being processed increase in complexity as one goes from the gate to the processor level. At the gate level individual bits (Os and Is) are processed. At the register level information is organized into multibit words or vectors, usually of a small number of standard types. Such words represent numbers, instructions, and the like. At the processor level the units of information are blocks of words, for example, a program or a data set. Another important difference lies in the time required for an elementary operation; successive levels can differ by several orders of magnitude in this parameter. At the gate level the time required to switch the output of a gate between and 1 (the gate delay) serves as the time unit and typically is a nanosecond (ns) or less. A clock cycle of, say, 10 ns, is a commonly used unit of time at the register level. The time unit at the processor level might be a program's execution time, a quantity that can vary widely. ^°^ c £ ates System hierarchy. more complex It is to refer to a design level as high or low; the the components, the higher the level. In this book we are primarily concerned with the two highest levels listed in Figure 2.7, the processor and register levels, which embrace what is generally regarded as computer architecture. The ordering of the levels suggested by the terms high and low component in any level L, from the level L, _ beneath , mally speaking, there in fact, quite strong. equivalent to a (sub) system of components taken This relationship a one-to-one illustrated in Figure 2.8. For- mapping h between components in L, system with levels of this type is called a hierarchical system. Thus in Figure 2.8 the subsystem composed of blocks 1, 3, and 4 in the low-level description maps onto block A in the high-level description. Figdisjoint in level L,-.,;a 2Ab and 2.5b show two hierarchical descriptions of a half-adder circuit. Complex systems, both natural and artificial, tend to have a well-defined hierarchical organization. A profound explanation of this phenomenon has been given by Herbert A. Simon [Simon 1962]. The components of a hierarchical system at each level are self-contained and stable entities. The evolution of systems from ures simple to complex organizations is greatly helped by the existence of stable intermediate structures. Hierarchical organization also has important implications in the design of computer systems. It is perhaps most natural to proceed from higher to lower design levels because sequence corresponds to a progression of succes- Thus if a complex system is to be designed using IC composed of standard cells, the design process might sively greater levels of detail. small-scale ICs or a single consist of the following three steps. 73 x \ Design 1 Figure 2.8 descriptions of a hierarchical system: (a) low level; (b) high level. Specify the processor-level structure of the system. Specify the register-level structure of each component type identified in step Specify the gate-level structure of each component type identified in step This design approach termed top down; extensively used in both hardware and software design. If the ICs or standard then the third step, gate-level design, As might be Only foregoing system be designed using medium-scale is no longer needed. expected, the design problems arising at each level are quite dif- in the case of gate-level design there a substantial theoretical basis (Boolean algebra). The register and processor levels are of most interest puter design, but unfortunately, design at these levels largely an art that depends skill and experience. In the following sections we examine design and processor levels in detail, beginning with the better-understood register level. We assume that the reader is familiar with binary numbers and with gate-level design concepts [Armstrong and Gray 1993; Hayes 1993; Hachtel and Somenzi 1996], which we review in the next section. on the designers' at the register The Gate Level Gate-level (logic) design concerned with processing binary variables whose pos- The design components and which are simple, memoryless processing elements, and flip-flops, sible values are restricted to the bits (binary digits) are logic gates, which are bit-storage devices. Combinational logic. A combinational film rum, also referred to as a logic, or a Boolean function, is a mapping from the set of 2" input combinations of n binary Such a function is denoted by r(.v,. v : and variables onto the output values 1 xn ) or simply by z. The function z can be defined by a truth table, which specifies for every input combination (jc x 2 ,..., x n ) the corresponding value of z{x x ,..., 1 2 xn ). Figure 2.9a shows the truth table for a pair of three-variable functions, s (xq, v'o c_,) and c (xq, Vq, c_,), which are the sum and carry outputs, respectively, of a logic circuit called a full adder. This useful logic circuit computes the numerical , System Design sum of its three input bits using binary (base 2) arithmetic: c& = xQ phisy plusc_ For example, the the of three Is row of the truth table of Figure 2.9a expresses the fact that 11 2 the base-2 representation of the normally reserve the plus symbol (+) operation, and write out plus for numerical addition. We will also use a subscript to identify the text; for discussing logic circuits, for the logical example, twelve number base when denoted by 12 10 it is not clear from the con- decimal and by in binary. combinational function z can be realized in many different ways by combinational circuits built from the standard gate types, which include AND, OR, Half adder Half adder >o 1 CO 1 NAND gate 1 NAND gate used as an inverter 1 NOT gate (inverter) c -i + Vo^'-i + o>'o c -i c = ( + c_{)(xQ + y )(\' () + c_i) structure also corresponds closely to that of the circuit. By analogy with ordinary algebra, (2.3) and (2.4) are referred to as sum-of-prochuts (SOP) and product-of-sums (POS) expressions, respectively. The circuit of Figure 2.9c is called a two-level or depth-two logic circuit because there are only AND and one y c_, to its OR, along each primary outputs path from this adder's external or primary inputs . assuming each primary input variable CHAPTER Design able in both true and inverted (complemented) form. The number of logic levels is defined by the number of gates along the circuit's longest 10 path. Because each through s some delay gate imposes ns or so) on every signal that propagates the fewer the logic levels, the faster the circuit. half- adder-based circuit of Figure 2.9b has gates and so 10 paths containing up considered to have four levels of logic. If propagation delay, then the two-level adder (Figure 2.9c) twice as fast as the However, the two-level adder has more gates and four-level design (Figure 2.9b). so has a higher hardware cost. to four gates have the basic task in logic design to synthesize a gate- level circuit realization of a given set of combinational functions that achieves a satisfactory balance between hardware cost as measured by the number of gates, and operating speed as measured by the number of logic levels used. Often the types of gates that may be used are restricted by IC technology considerations, for example, to NAND gates with five or fewer inputs per gate. The design of Figure 2.9d, which has essentially the same structure as that of Figure 2.9c, uses NAND and NOR gates instead of ANDs and ORs. In this particular case the primary inputs are provided in true (noninverted) form jc y c_, only; hence inverters are introduced to generate the inverted inputs x c_ y Computer-aided synthesis tools are available to design circuits like those of Figure 2.9 automatically. The input to such a logic synthesizer is a specification of , the desired function, such as a truth table like Figure 2.9a, or a set of logic equations like (2.3) or (2.4); these are often in a behavioral HDL description. Also given to the synthesizer are such design constraints as the gate types to use and restrictions on the circuit's interconnection structure. One such restriction is an upper bound on the number of inputs (fan-in) of a gate G. Another is an upper bound on the number of inputs of other gates to which G's output line may connect; this called the (maximum) fan-out of G. The output of structural description of a logic circuit that the synthesizer implements the desired function and meets the specified constraints as closely as possible. Exact methods for designing two-level circuits like that of Figure 2.9c (or Figits inverters removed) using the minimum number of gates have ure 2.9' 3 ,> 2'>'i'>'o) an d computes their sum S = (s 3 ,s 2 ,S\,s y, it also accepts an input x carry signal c_, and produces an output carry c 3 at the register level, as internal structure or logic design may no By adding memory storage elements called flip-flops, rely on an external clock signal This design, which to a CK to multibit adder shown Figure treated as a 2.10b, at which point longer be of interest. combinational circuit in the form of obtain a sequential logic circuit. Flip-flops synchronize the times which they respond as a ripple-carry adder, and other types of binary adders are in detail 77 -' Full adder Cnnr Full adder Full adder Full adder Figure 2.10 Four-bit ripple-carry: (a) logic structure; (b) high-level symbol. changes on their input data lines. They are also designed to be unaffected by changes (noise) produced by the combinational logic that feeds them. An efficient way to meet these requirements is edge triggering, which conto transient signal narrow window of time around one edge (the CK. Figure 2. 1 1 summarizes the behavior of the most common kind of flip-flop, an edge-triggered D {delay) flip-flop. (Another well-known flip-flop type, the JK flipflop, is discussed in problem 2.1 1.) The output signal y constitutes the stored data or state of the flip-flop. The D flip-flop reads in the data value on its D line when the 0-to-l triggering edge of clock signal CK arrives; this D value becomes the new value of y. The triangular symbol on the clock's input port in Figure 2. la specifies edge triggering; its omission indicates level triggering, in which case the flip-flop (then usually referred to as a latch) responds to all changes in signal value on D. Since there is just one triggering edge in each clock cycle, there can be just one change in y per clock cycle. Hence we can view the edge-triggered flip-flop as trafines the flip-flop's state to a 0-to-l or l-to-0 transition point) of versing a sequence of discrete state values The eral input data line D can one for every clock cycle be varied independently and so can go through sev- changes in any clock cycle However, only before the arrival of the triggering edge of To change the flip-flop's state, the D period known as the setup time Tselup the data value D{i) present just determines the next state y{i + must be held steady for a before the flip-flop is triggered. For examwhich shows a sample of the D flip-flop's behavior, we have D(l) = 1 and v(l) = in clock cycle 1. At the start of the next clock cycle, y changes to 1 in response to D(l) = 1. making v(2) = 1. In clock cycle 3, y changes ple, in Figure 2.1 CHAPTER 2 78 PRE Input Di > System Design Next state vO'+l) CLR 1 Triggen ng edge T 'setup Glitch / CK 1 State y State >(/) Figure 2.11 flip-flop: (a) graphic to 0, during the symbol; (b) making y(3) = critical state table; (c) Even though setup phase of cycle the spurious pulse or glitch affecting timing diagram. most of clock cycle D(3) = thus ensuring that y(4) = 0. Observe that D in cycle 5 has no effect on Hence edge- triggered flip-flops have the very useful property of filtering out noise signals When itly at their inputs. a flip-flop brought to a (reset) the flip-flop at the start is first asynchronously, that of operation. control inputs, switched on. initial state. It is uncertain unless independently of the clock signal CK, this end, a flip-flop therefore desirable to be able to initialize can have one or two asynchronous (preset), as in Figure 2.11a. designed to respond to a brief input pulse that forces y to 1 in the case of PRE. in the case or to normal synchronous operation with a clock that is matched to the timing its flip-flops, we can be sure that one well-defined change of state takes place in a sequential circuit during each clock cycle. We do not have to worry about the exact times at which signals change within the clock cycle. We can therefore consider the actions of a flip-flop, and hence of any sequential circuit employIn characteristics of a discrete sequence of points of time 1, 2, 3, ... In effect, the clock quantizes time into discrete, technology-independent time steps, each of which represents a clock cycle. We can then describe a behavior by the following characteristic equation: y(/+l) = D(/) flip-flop's next-state which simply says that y takes the value of D delayed by one clock cycle, hence the D flip-flop's name. Figure 2.1 \b shows another convenient way to represent the flip-flop's nextstate behavior. This state table tabulates the possible values of the next state y{i + 1) for every possible combination of the present input D(i) and the present state y(i). It is not customary (or necessary) to include clock-signal values explicitly in characteristic equations or state tables. The clock of time steps and so always present considered to be the implicit generator in the background. Asynchronous inputs are also omitted as they are associated only with initialization. Sequential circuits. and a sequential circuit consists of a combinational circuit of flip-flops. The combinational logic forms the computational or data- processing part of the circuit. The flip-flops store information on the circuit's past behavior; this stored information defines the circuit's internal state mary inputs Y, X and the denoted Z(X,Y). primary outputs are Z, then Z is which the the resulting circuit is said to be clocked or synchronous. X and can also trigger changes in the primary output tance of state behavior, the term finite-state machine tick (cycle or period) of the clock permits a single change in the circuit's state it the pri- usual to supply a sequential circuit with a precisely con- It is trolled clock signal that determines the times at Y. If a function of both as discussed Reflecting the impor- often applied to a sequential circuit. The behavior of a sequential circuit can be specified by a state table that primary outputs and internal states. Figure 2.12a shows the state table of a small but useful sequential circuit, a serial adder, includes the possible values of intended to add two unsigned binary numbers X, and sum Z = length, producing their is, and the result is also The numbers of arbitrary are supplied serially, that serially. In contrast, the Input x 12 Present state S (y = D nip-Hop Figure 2.12 (a) State table; (b) logic circuit for a serial adder. adder of Figure gation delays, adds a "parallel" adder, which, ignoring internal-signal propa- of the input numbers simultaneously. In one clock cycle all bits se " a ^ adder receives 2 input bits Xy(i) and x 2 (i) and computes 1 bit z(i) of Z It computes a carry signal c(i) that affects the addition in the next clock cycle. Thus the output computed in clock cycle i is '"' System Design = x^Oplus x2 (i)plus must be determined from the adder's present state S(i). (2.6) is equivalent to the expression (2.2) for the full-adder function defined earlier. follows that two possible internal states exist: 5 meaning that the previous carry 1. These considerations lead to the two-state state table of Figure 2.12a. An entry in row 5(0 and column x^x^i) of the state table has the format S(i + 1), z(i), where S(i + 1) is the next internal state that the circuit must have when the present state is 5(0 and the present primary input combination is x (i)x2 (i); z(i) is the corresponding primary output signal that must be generated. Because the serial adder has only two internal states, its memory consists of a single flip-flop storing a state variable y. There are only two possible ways to assign 0s and Is to y. We select the "natural" state assignment that has y = for 5 and y = 1 for S since this equates >(/) with the stored carry signal c(i - 1). Assume It signal c(i and S that c(i use an edge-triggered tional logic ary output signal D(i) that C then must generate defined by The combina- flip-flop (Figure 2.11) to store y. signals: the applied to the primary output flip-flop's data input. characteristic equation (2.5); that and a second- The + flip-flop's 1) Hence we have D(i) = c(0 It follows from the above discussion that C can be implemented directly by a sum output is z and whose carry output is D; see Figure 2.12b. Before entering two new numbers to be added, it is necessary to reset the serial adder to the 5 state. The easiest way to do so is to adder circuit such as that of Figure 2.9b, whose apply a reset pulse to the flip-flop's asynchronous clear (CLR) input. 2.2 involves a similar, but onstrates the use of more complex sequential and dem- CAD tools in its design. design of a 4-bit-stream serial adder. Consider another of serial adder that adds four number streams instead of the two handled by a conventional serial adder (Figure 2.12). The new adder has four primary input lines jc,, x 2 x 3 x4 and a single primary output z. To determine the circuit's state behavior often the most difficult part of the design process we first identify the information to be stored. As in the standard serial adder case, the circuit must remember carry information computed in earlier clock cycles. The current 2-bit sum SUM(i) = c(i)z(i) is given by SUM(i) = x {i)plus x 2 (i)plus x3 (/)/?/us x4 (i)plus and If c(i - 1) is = 4 = 100 2 so c(i) = 10 2 With c(i - 1) = 10 2 SUM{i) becomes 6 = 1 10 2 making c{i) = 1 2 Finally, c(i - 1) = 1 makes SUM(i) = 1 1 2 and c(i) = 1 2 which is the maximum possible value of c. 2 The carry data to be stored is a binary number ranging from 00 2 to 1 2 which implies each xfi) = 1) is the 1, then SUM(i) = in the preceding clock cycle. 1 plus , needs four states and two flip-flops. We will denote the four states by S 2 S3 where 5, represents a stored carry of (decimal) value i. Figure 2.13a shows the adder's state table, which has four rows and 16 columns. For present state S(i) and input combination j, the next-state/output entry Sk ,z is obtained by adding i 2 and the 4 input bits that determine 7 to form SUM(i) = (k 2 k k ) 2 It follows that k = (k 2 k ) 2 and z = k For example, with present state S 2 and present input plus 1 plus 1 plus 1 plus 10 2 = 101 2 so z = 1 and A: =10 2 = 2, making S-, 7, SUM(i) = that the adder the next-state. Following this pattern, it is straightforward to construct the adder's state + l)y2 (i + I) coincide with the flipThe adder thus has the general structure shown in flip-flops, the next-state values >',(/ flops' data input values (i)D 2 (i). Figure 2.13£>. truth table for the combinational logic appears in Figure 2.13c. It is from Figure 2.13a with the states assigned the four bit patterns of >',y 2 as follows: S = 00, 5, = 01, S 2 = 10, and 5 3 = 11. Suppose we want to design Cas a two-level directly Present inputs x x 2 x i x4 (decimal) l Present state S,,0 S,,l S 2 ,0 S 2 .0 S 2 .0 S 2 .0 S ,0 S 2 ,0 S,,0 S„ S,,0 S,.l s 2 ,o S 2 .0 S 2 ,0 S 2 .0 S 2 ,0 S 2 .l S 2 ,0 5 : .0 X 2 Xy X4 Combinational logic C 1 1010-1 0110-1 1001-1 0101-1 0011-1 -11111 1-1111 11-111 111-11 111111111-1 -111-1 1-11-1 11-1-1 35 36 37 38 39 40 111—1 41 1111— 42 Ill 43 —1-11 44 -1—11 45 1 11 46 —11147 -1-1148 1—1149 -11-150 1-1-1- — — Figure 2.14 Minimal two-level (SOP) design puted by ESPRESSO. y 2 D, £> 2 z C com- minimum number of gates. Manual minimizamethods [Hayes 1993] are painfully slow in this case without computer aid. We have therefore used a logic synthesis program called Espresso [Brayton et al. 1984; Hachtel and Somenzi 1996] to obtain a two-level SOP design. To instruct Espresso to compute the minimum-cost SOP design on a UNIX-based computer requires issuing a circuit like that of Figure 2.9c, using the ^espresso seradd4 where containing the truth table of Figure 2.13c or an equivalent description of C. Espresso responds with the table of Figure 2.14, which specifies an design containing the fewest product terms (these are in a minimal form called prime implicants [Hayes 1993]), format in this case, 51. x x2 x 3 x4 y y 2 ] D D z= states that output z (but not the outputs product terms. The dash in 1010in the D, or (1 conclude from Figure 2.14 that an adder has 5 has x x 2 x i x 4 y 2 as one of - - 1 - y lt 100) states that x x 2 y realization of product terms, none of which happen to be shared is i a term of for the four-stream the output func- This conclusion implies a two-level circuit containing the equivalent of that is not included i indicates a literal, in this case term in question. Similarly, row 51 D y We tions. For example, row 26, which has the at least 54 gates (51 ANDs and three ORs), some especially the OR gates with very high fan-in, which makes this type of two-level design expensive and impractical for many IC technologies. Example 2.6 in section 2.2.3 shows an alternative approach that leads to a lower-cost, multilevel design for this adder. Minimizing the number of gates in a sequential circuit is difficult affected by the flip-flop types, the state assignment, and, of course, the which the combinational subcircuit C is designed. Other design techniques exist to simplify the design process at the expense of using logic elements. impractical to deal with complete binary descriptions like state tables tain than, say, a if they conConsequently, large, sequential circuits are designed by heuristic techniques whose implementations use reasonable but nonminimal amounts of hardware [Hayes 1993; Hachtel and Somenzi 1996]. These circuits are often best at the abstract register level rather than the gate level. THE REGISTER LEVEL At the grouped into register or register-transfer level, related information bits are ordered sets called words or vectors. The primitive components are small combinational or sequential circuits intended to process or store words. Register-level circuits are composed of word-oriented devices, the more important of which are listed in Figure 2.15. The key sequential component, which gives this level of abstraction a (parallel) register, a storage device for words. sequential elements are shift registers and counters. number of standard combinational components exist, ranging from general-purpose devices, such as word gates, to more specialized Type Combinational such as decoders and adders. Logical (Boolean) operations. Data routing: general combinational functions. Decoders and encoders. Code checking and conversion. Addition and subtraction. Arithmetic-logic units. Numerical and logical operations. General combinational functions. logic devices. (Parallel) registers. Information storage. Shift registers. Information storage; serial-parallel conver- Control/timing signal generation. logic devices. Figure 2.15 The major component types at the register level. General sequential functions. CHAPTER 2 Design groups of are linked to by means of word-carrying lines, referred to as buses. The Register Level The component types of Figure 2.15 level design; they are available as cells in are generally useful in register- parts in various IC series However, they cannot be and as standard identified a priori based some property analogous to the functional completeness of gate-level operations. For example, we will show that multiplexers can realize any combinational function. This completeness property is incidental to the main application of multiplexwhich is signal selection or path switching. There are no universally accepted graphic symbols for register-level components. They are usually represented in circuit diagrams by blocks containing an A single signal line in a abbreviated description of their behavior, as in Figure 2.16. diagram can represent a bus transmitting m > 1 bits of information in parallel; m is indicated explicitly by placing a slash (/) in the line and writing m next to it (see Figure 2.16). A components's 10 lines are often separated into data and control lines. may be m-bit bus given a that identifies the bus's role, for the type of data transmitted over a data bus. operation determined by the line in control line's small circle representing inversion when is lines at state. Alternatively, the name of a signal the logi- an input or output port of a block to indicate that the corresponding lines are active in the inactive in the indicates the active, enabled, or asserted state. otherwise indicated, the active state of a bus occurs cal active value and is includes an overbar. fall into two which specify one of several possible operations that the unit is to perform, and enable lines, which specify the time or condition for a selected operation to be performed. Thus in Figure 2.16, to perform some operation F first set the select line F to a bit pattern denoting F and then activate the edgetriggered enable line £by applying a O-to-1 edge signal. Enable lines are often connected to clock sources. The output control signals, if any, indicate when or how the unit completes its processing. Figure 2.16 indicates termination by 5 = 0. The arrowheads are omitted when we can infer signal direction from the circuit structure or signal names. input control lines associated with a multifunction block broad categories: select Data input Ai i Function select lines A-i m /T E Control output lines input lines Data output Figure 2.16 Generic block representation of a register-level is concerned with combinational funcfrom the two-valued set B = {0,1} and form a Boolean algebra. We can extend these functions to functions whose values are taken m m from B the set of 2 m-bit words, rather than from B. Let z(x ,x2 ,...,xn ) be any two-valued combinational function. Let X ,X2 ,...,Xn denote m-bit binary words having the form X, = (x i,xi^,...,xi^) for / = 1,2, ...,«. We define the word opera- Operations. Gate-level logic design signal values are tion z as follows: z(X ,X2 ,...,Xn ) = [z(x l { ,x2 u l )^(x l ,x22 ,. .,xn2),. .,zix ljn ,x2jn ,. . This definition simply generalizes the usual Boolean operations, and so have 1-bit to m-bit X +X2 +- +Xn = ( l which applies lfl •' If z is the + x2A + AND, NAND, function, for instance, X \jn + xljm .,x njn)] xnA s h2 + x 22 + + '•• + xn,m) x n2 OR bitwise to the corresponding bits of n m-bit words. mn combinational functions defined on n m-bit words forms a Boolean algebra with respect to the word operations for AND, OR, and NOT. This generalization of Boolean algebra to multibit words is analogous to the extension of the ordinary algebra from single numbers (scalars) to vectors. Pursuing this analogy, we can treat bits as scalars and words as vectors, and obtain more comset plex logical operations, such as yX=(yx ,yx2 l Word-based level design. X = (y + x ,y + x 2 ...,y + , some aspects of registerHowever, they do not by themselves provide an adequate design the- logical operations of this type are useful in ory for several reasons. • The operations performed by some basic register-level components are numeriBoolean frame- cal rather than logical; they are not easily incorporated into a work. • of the logical operations associated with register-level components are complex and do not have the properties of the gates interchangeability of inputs, for example that simplify gate-level design. Although a system often has a standard word length w based on the width of some important buses or registers, some buses carry signals with a different number of bits. For example, the outcome of a test on a set 5 of vv-bit words (does S have property PI) is 1 bit rather than w. The lack of a uniform word size for all signals on these difficult to define a useful algebra to describe operations Lacking an adequate general theory, register-level design is tackled mainly with heuristic and intuitive methods. We next introduce the major combinational and sequential components used in design at the register level. (Refer to Figure 2.15). CHAPTER 2 Design Word gates. Let X = (x u x 2 ,...,xm ) and Y = (yi,y 2 ,...,y„,) be two m-bit binary As noted already, it is useful to perform gate operations bitwise on X and Y obtain another m-bit word Z = (zi,z 2 ,...,Z m ). We coin the term word-gate opera- words. 2.2 The Register Level tions for logical functions of this type. In general, if write bit if z, =/(jc,,y,) /is any logic operator, we Z= XY denotes the m- For example, NAND operation defined by Z=(z This generalized ,z 2 ,...,z m ) = (x y ,x 2 y 2 l ...,x m y m ) realized by the gate-level circuit in Figure 2.17a. represented in register-level diagrams by the two-input 2.17b, which an example of a word gate. It is It is symbol of Figure also useful to represent scalar- vector operations by a single gate symbol. For example, the operation y + defined by (2.8) and realized by the circuit of Figure 2.18a can be represented by the register-level gate symbol of Figure 2.18b. gates are universal in that they suffice to implement any logic circuit; moreover, word-gate circuits can be analyzed using Boolean algebra. In practice, however, the usefulness of word gates severely limited by the relative simplicity of the operations they perform and by the variability in word size found ister level. V z (b) Figure 2.17 Two-input, m-bit NAND word gate: (a) logic diagram and m / Z (b) Figure 2.18 OR word gate implementing y + X: (a) logic diagram; (b) at the reg- several sources to a specified by applying appropriate control (select) signals to the multiplexer. If the data sources from one of a device intended to route data destination; the source k and each 10 data line carries as a k-input (or k-way), m-bit multiplexer. maximum number It is bits, the multiplexer is referred to convenient to make k = 2 P so that , determined by an encoded pattern or address of p bits. the range 00...0, 00...1, ..., 11...1 = 2 P - 1. A multiplexer is easily denoted by a suitably labeled version of the generic block symbol of Figure 2.16; the tapered block symbol shown in Figure 2.19, where the narrow data source selection The 2P addresses then cover end indicates the data output side, is also common. Let a = 1 when we want to select the m-bit input data bus X, = (jc,-^, ,..., x i,m-\) °f me multiplexer of Figure 2.19. Then a = 1 when we apply the word corresponding to the binary number i to the select bus 5. The binary variable a, denotes the selection of input data bus X, a, is not a physical signal. The data word on X, is then transferred to Z when e = 1. The operation of the 2^-input w-bit multiplexer is therefore defined by m sum-of-product Boolean equations of the form ( + x lj a l + Zj= (x0j a +x 2 p_i a 2p_ )e or by the single word-based equation (X a + a + a 2P-i)e Figure 2.20 shows a typical gate-level realization of a two-input, 4-bit multiplexer. Several &-input multiplexers can be used to route more than k data paths by connecting them in the treelike fashion shown in Figure 2.21. g-level tree cir- forms a ^-input multiplexer. A distinct select line is associated with every level of the tree and is connected to all multiplexers in that level. Thus each level performs a partial selection of the data line X, to be connected to the cuit of this type output Z. Multiplexers as function generators. Multiplexers have the interesting property that they can compute any combinational function and so form a type of generate any ^-variable function z(v ,v,,...,v„_,). This accomplished by apply- ing the n input variables v ,v,,...,v n _, to the n select Ymes s ,s ,...,s n _ of MUX, and 2" function-specific constant values (0 or 1) to MUX's 2" input data lines .v ,.v, ] X, 1 P Select S . . (MUX) Enable e Figure 2.19 Data out 2 /'-input, m-bit multiplexer. x o.o n x i.o x i, x 0.3 Select s The Register Level Enable e Data out Figure 2.20 Realization of a two-input, 4-bit multiplexer. —AToMux T7 >---> z In each case a bit of stored data data bit x ister consists brought in bit at at the O x) is lost ( z m-\' Z m-2>---' Z \' Zo) from one end of the other end. In shift register, while a simplest form, an m-bit shift reg- m flip-flops each of which is connected to its left or right neighbor. Data can be entered (read) '- (^m-l' Z m-2'---' 2 l' Zo) Z m-\' Z m-2'---' Z bit at a one end of the register and can be removed a time from the other end; this process Figure 2.30 shows a 4-bit shift register built from accomplished by activating the SHIFT enable called serial input-output. of each flip-flop. In addition to the serial data lines, to the right shift clock input input or output lines are often provided to permit parallel data transfers to or from the shift register. Additional control lines are required to select the serial or parallel input ther refinement to permit both left- right-shift operations. 2-way 5 X 4 / Z () Figure 2.29 A 4-bit D register with parallel load: (a) logic diagram; (b) symbol. Registers like those of Figures 2.28 and 2.29 are designed so that external data can be transferred to or from The Register Level CLEAR (a) SHIFT Shift register Figure 2.30 A 4-bit, right-shift register: (a) logic diagram; (b) symbol. Shift registers are useful design in a number of including storage of serial data and serial-to-parallel or parallel-to-serial data con- They can numbers, because also be used to perform certain arithmetic operations most computers include instruction sets of The k shift operations. a sequential circuit designed to cycle through a prede- termined sequence of k distinct states 5 on an input on binary corresponds to multiplication (division) by left- (right-) shifting states represent Sk _ response to signals ( 1 k consecutive numbers, so the state transitions can be described by the statement M := S Each viewed as depending on the -input increments the state by one; the circuit can therefore be counting the input number codes Counters come in used, the modulus different varieties and the timing mode (synchronous or asynchro- Figure 2.31 shows a counter designed to count ENABLE input 2", The counting states S n S, has 2" COUNT = control line and the count sequence In the up-counting S, +1 := 5, modulo-2"; that The output either -pulses applied to the counter's modulus k = an n-bit binary number up or down, as determined by the (DOWN= 0), the counter's behavior is (modulo Modulo-2" up-down CHAPTER 2 Design Figure 2.31 A modulo-2' in the down-counting mode 5, +1 := S some counters modulus-select (DOWN = the behavior control lines can alter the modulus; such counters termed programmable. Counters have several applications in computer design. They can store the state of a control unit, as in a program counter. Incrementing a counter provides an efficient means of generating a sequence of control states. Counters can also generare ate timing signals and introduce precise delays into a system. Buses. A bus is a set of lines (wires) designed to transfer all bits of a word from a specified source to a specified destination on the same or a different IC; the source and destination are typically registers. A bus can be unidirectional, that is, capable of transmitting data in one direction only, or it can be bidirectional. Although buses perform no logical function, a significant cost is associated with them, since they require logic circuits to control access to them and, when used over longer distances, signal amplification circuits (drivers and receivers). The pin requirements and gate density of an IC increase rapidly with the number of external buses connected to must also be taken To reduce If these buses are long, the cost of the wires or cables used into account. buses are often shared, especially A shared bus they connect can connect one of several sources to one of several destinations. Bus sharing reduces the number of connecting lines but requires more complex bus-control mechanisms. Although shared buses are relatively devices. cheap, they do not permit simultaneous transfers between different pairs of devices, possible with unshared or dedicated buses. explored further in Chapter structures are Programmable Logic Devices Next we examine a class of components called programmable logic devices or PLDs, a term applied to ICs containing many gates or other general-purpose cells whose interconnections can be configured or "programmed" to implement any desired combinational or sequential function [Alford 1989]. PLDs are relatively easy to design and inexpensive to manufacture. They constitute a key technology for building application-specific integrated circuits (ASICs). Two techniques, are used to program PLDs: mask programming, which requires a few special steps in the IC chip-manufacturing process, and field programming, which done by designers or end users "in the field" via small, low-cost programming units. 2.2 The Register Level same IC can be reproespecially convenient when developing are erasable, implying that the grammed many times. This technology is and debugging a prototype design for a new product. Programmable a The connections leading to and from logic elements in programmed to be permanently contain transistor switches that can be switched on or switched These switches are laid out in minimum IC two-dimensional arrays The programmable x denoting a programmable connection or crosspoint in a gate's input line. The absence of an x means that the corresponding connection has been programmed to the off (disso that large gates can be implemented with logic gates of a array are represented abstractly in Figure 2.32b, with The gate structures of Figure 232b can be combined in various ways to implement logic functions. The programmable logic array (PLA) shown in Figure 2.33 is intended to realize a set of combinational logic functions in minimal consists of an array of gates (the AND plane), uct terms (prime implicants), and a set of various logical sums of the product terms. gates (the inputs to the realize a set of prod- which form gates are pro- grammable and include all the input variables and their complements. Hence it is possible to program any desired product term into any row of the PLA. For example, the top row of the PLA in Figure 2.33 is programmed to generate the term x 2 x 3 x 4 y y 2 which is used in computing the output D 2 the last row is programmed to generate x x 2 y for output D,. The inputs to the OR gates are also programmable, so each output column can include any subset of the product terms produced by the rows. The PLA in Figure 2.33 realizes the combinational part C of the 4-bitstream adder specified in Figure 2.13. The AND plane generates the 51 six-vari, able product terms according to the design given in Figure Figure 2.32 AND and OR gates: normal notation; PLD notation. 2. 14. v V Y 99 AND plane vy /^ w w s< ? Pr ogram cc >n tro Datapath Control signal s un it I-unit ) Figure 2.46 Internal organization of a CPU and cache a synchronous sequential circuit puter's basic unit of time. In one clock cycle the operation, such as fetching an instruction whose clock period CPU can perform a register-transfer word from M via the system bus and load- into the instruction register IR. This operation can be expressed formally program counter uses to hold the expected address of the next instruction word. actions needed for execution; for example, perform an arithmetic operation on data words stored in in the I-unit, an instruction I-unit then issues the signals that enables execution of the instruction in question. fetching, decoding, and executing an instruction constitutes determine the sequence of control The entire process of the CPU's instruction CPUs and other instruction-set processors operate in conjunction with external memories that store the programs and data required by the processors. Numerous memory technologies of operation. and they vary greatly and perspeed several major in cost memory device generally increases rapidly The memory part of a computer can be divided into formance. The cost of a subsystems: 1. Main memory M, consisting of relatively fast storage ICs connected directly and controlled by, the CPU. Secondary memory, consisting of less expensive devices that have very high storage capacity. These devices often involve mechanical motion and so are much slower than M. They are generally connected indirectly (via M) to the CPU and form part of the computer's 10 system. Many computers have a third type of memory called a cache, which is positioned between the CPU and main memory. The cache is intended to further reduce the average time taken by the or of the cache may be integrated on the to access the same IC chip as the Main memory is a word-organized addressable random-access memory (RAM). The term random access stems from the fact that the access time for every location in memory ries are the same. contrasted with serial access, where access times vary with the location being accessed. Serial access slower and less expensive than some form of serial access. RAMs; most secondary-memory Because of lower operating speeds and access mode, the manner in which the stored information memories is more complex than the simple word organization of main memory. Caches also use random access or an even faster memory-accessing method called associative or content addressing. Memory technologies and the organization of stored information are covered in Chapter 6. IO devices. Input-output devices are the means by which a computer nicates with the outside world. primary function of 10 devices is to act as data from one physical representation to 10 devices do not alter the information content or meaning of the data on which they act. Since data is transferred and processed within a computer system in the form of digital electrical signals, input (output) transducers, that to convert information another. Unlike processors, devices transform other forms of information to (from) digital electrical signals. Figure 2.47 involve. some widely used 10 devices and the information media they of these devices use electromechanical technologies; hence their is slow compared with processor and main-memory speeds. Although the CPU can take direct control of an IO device it is often under the immediate control of a special-purpose processor or control unit that directs the flow of information between the IO device and main memory. The design of 10 systems is considered in Chapter 7. speed of operation Interconnection networks. word-oriented buses. In systems with many components, communication may by be controlled by a subsystem called an interconnection network; terms such as switching network, communications controller, and bus controller are also used in this context. The function of munication paths reasons, these paths are network is to establish dynamic comcomponents via the buses under its control. For cost usually shared. Only two communicating devices can the interconnection access and use a shared bus at any time, so contention results when several system components request use of the bus. The interconnection network resolves such contention by selecting one of the requesting devices on some priority basis and connecting it to the bus. The interconnection network may place the other requesting devices in a queue. CHAPTER 2 Design The Processor Level to/from which Analog-digital converter Analog (continuous) Document scanner/reader Images on paper Dot-matrix display panel electrical signals ^nd coded images) on optical disk Images on screen transforms digital electrical signals Characters on keyboard Images on paper Laser printer Spoken words and sounds Magnetic-disk drive Characters (and coded images) on magnetic disk Characters (and coded images) on magnetic tape Magnetic-tape drive Spoken words and sounds Spatial position on pad Figure 2.47 IO devices. Simultaneous requests for access to some unit or bus result from the fact that communication between processor-level components is generally asynchronous in that the components cannot be synchronized directly by a common clock signal. This synchronization problem can be attributed to several causes. • high degree of independence exists CPUs and IOPs quently and • the components. For example, execute different types of programs and interact relatively infre- unpredictable times. Component operating speeds vary over a wide range. CPUs operate from 1 to 10 times faster than main-memory devices, while main-memory speeds can be many orders of magnitude faster than IO-device speeds. The physical distance separating the components can be too large to permit synchronous transmission of information between them. one of the functions of a processor such as a CPU or an IOP. An bus to which many IO devices are connected. The IOP is responsible for selecting a device to be connected to the IO bus and from there to main memory. It also acts as a buffer between the relatively slow IO devices and the relatively fast main memory. Larger systems have special processors whose sole function is to supervise data transfers over shared buses. controls a common IO 2.3.2 Processor-Level Processor-level design ister level. Design is desired system behavior. programs supplied formal analysis than at the reg- in part to the difficulty of giving a precise description of the it is To of say that the computer should execute efficiently little help to the designer. The at this level is to take a prototype design of known performance and modify where necessary to accommodate new technologies or meet new performance requirements. The performance specifications usually take the following form: design it • • • • The computer should be capable of executing a instructions of type b per second. The computer should be able to support c memory or 10 devices of type d. The computer should be compatible with computers of type e. The total cost of the system should not exceed/ Even when a new computer is closely based on a known design, it may not be posperformance accurately. This is due to our lack of understanding of the relation between the structure of a computer and its performance. Performance evaluation must generally be done experimentally during the design pro- sible to predict by computer simulation or by measurement of the performance of a copy of the machine under working conditions. Reflecting its limited theoretical basis, only a small amount of useful performance evaluation can be done via math- cess, either ematical analysis [Kant 1992]. view the design process as involving two major and adapt it to satisfy the given performance constraints. Then determine the performance of the proposed system. If unsatisfactory, modify the design and repeat this step; continue until an acceptable design is obtained. This conservative approach to computer design has been widely followed and accounts in part for the relatively slow evolution of computer architecture. It is rare to find a successful computer structure that deviates substantially from the norm. The need to remain compatible with existing hardware and software standards also influences the adherence to proven designs. Computer owners are understandably reluctant to spend money retraining users and programmers, or Prototype structures. steps: First select a prototype design replacing well-tested software. The systems of interest here are general-purpose computers, which differ from in the number of components used and their autonomy. The one another primarily communication structures used is fairly small. We by means of block diagrams that are basically graphs Figure 2.48 shows the structure that applies to first-generation com- variety of interconnection or will represent these structures (section 2.1.1). puters and modern microprocessor-based systems. The addition of special-purpose 10 processors typical of the second and subsequent generations ocessing unit Main memory IO devices Figure 2.48 Basic computer structure 120 Central processing unit 2.3 Main memory The Processor Level Cache IO processors Figure 2.49 IO Computer with cache and IO Here ICN denotes an interconnection (switching) network memory-processor communication. Figure 2.50 shows a prototype structure employing two CPUs; it is therefore a multiprocessor. The uniprocessor systems of Figures 2.48 and 2.49 are special cases of this structure. Even more complex structures such as computer networks can be obtained by linking several in Figure 2.49. that controls copies of the foregoing prototype structures. Performance measurement. derived from the characteristics of processing units performance figures for computers are CPU. As observed — Main memory Cache memories Crossbar switching IO processors IO devices Figure 2.50 Computer with multiple CPUs and main memory banks. in section 1.3.2, speed can be measured easily, but roughly, by clock frequency /in megahertz. Other, and usually better, performance indicators are MIPS, which the average instruction execution speed in millions of instructions per second, and CPI, is the average number of clock cycles required per instruction. As discussed performance measures are related to the average time microseconds (us) required to execute N instructions by the formula in section 1.3.2, these NxCPI Hence the average time tE to execute an instruction t CPI/f us While / depends mainly on the IC technology used to implement the CPU, CPI depends primarily on the system architecture. We can get another perspective on tE by considering the distribution of instructions of different types and speeds in typical program workloads. Let /,, I2 ..., /„ , be a of representative instruction types. Let (us) of an instruction of type /, p denote i denote the average execution time the occurrence probability of type- instructions in representative object code. time t E the average instruction execution given by I PA from the CPU specifications, but accumust usually be obtained by experiment. The set of instruction types selected for (2.20) and their occurrence probabilities define an instruction mix. Numerous instruction mixes have been published that represent various computers and their workloads [Siewiorek, Bell, and Newell 1982]. Figure 2.51 gives some recent data collected for two representative figures can be obtained fairly easily rate Pj data Program A Program B Instruction type Fixed-point operations Floating-point operations Figure 2.51 Representative instruction-mix data. Source: McGrory, Carlton, and Askins 1992. The Processor Level programs running on computers employing the Hewlett-Packard PA-RISC architecture under the UNIX operating system [McGrory, Carlton, and Askins 1992]. The execution probabilities are derived from counting the number of times an instruction of each type is executed while running each program; instructions from both the application program and the supporting system code are included in this count. Program A is a program TPC-A designed to represent commercial on-line transaction processing. Program B is a scientific program FEM that performs finite-element modeling. In each case, memory-access instructions (load and store) account for more than a third of all the instructions executed. The computation-intensive scientific program makes heavy use of floating-point instructions, whereas the commercial program employs fixed-point instructions only. Conditional and unconditional branch instructions account for 1 in 6 instructions in program A and for 1 in 10 instructions in program B. Other published instruction mixes suggest that as many as 1 in 4 instructions can be of the branch type. A few performance parameters are based on other system components, especially memory. Main memory and cache size in megabytes (MB) can provide a rough indication of system capacity. A memory parameter related to computing maximum rate in millions of bits per second which information can be transferred to or from a memory unit. Memory bandwidth affects CPU performance because the latter' s processing speed is ultimately limited by the rate at which it can fetch instructions and data from its cache or main memory. Perhaps the most satisfactory measure of computer performance is the cost of executing a set of representative programs on the target system. This cost can be the total execution time T, including contributions from the CPU, caches, main memory, and other system components. A set of actual programs that are representative of a particular computing environment can be used for performance evaluation. Such programs are called benchmarks and are run by the user on a copy (actual or simulated) of the computer being evaluated [Price 1989]. It is also useful to devise artificial or synthetic benchmark programs, whose sole purpose is to obtain data for performance evaluation. The program TPC-A providing the data for program A in Figure 2.51 is an example of a synthetic benchmark. speed bandwidth, defined as the EXAMPLE 2.8 PERFORMANCE COMPARISON OF SEVERAL COMPUTERS [MCLELLAN 1993]. Figure 2.52 presents some published data on the performance of three machines manufactured by Digital Equipment Corp. in the early 1990s, its 64-bit Alpha microprocessor. The SPEC (Standard Performance Evaluation Cooperative) ratings are derived from a set of benchmark programs that computer companies use to compare their products. The SPECint92 based on various versions of and SPECfp92 parameters indicate instruction execution speed relative to a standardized 1-MIPS computer (a 1978-vintage Digital VAX 11/780 minicomputer) when executing benchmark programs involving integer (fixed point) and floating-point Hence the SPEC figures approximate MIPS measurements two major classes of application programs like those of Figure 2.51. The remaining data in Figure 2.52 are relative performance figures for executing some other well-known benchmark programs, most aimed at scientific computing. Data of this sort are better suited to measuring relative rather than absolute performance. For example, suppose we wish to compare the performance of the Digital 3000 and 10000 machines listed in Figure 2.52. The ratio of their SPECint92 MIPS numbers is 104.3/63.8 = 1.65. The corresponding ratios for the other five benchmarks range operations, respectively. DEC 3000 DEC 4000 Performance measure Model 400 Model 610 Model 610 clock frequency Linpack 1000 x 1000 BM suite Livermore loops Figure 2.52 Performance comparison of three computers based on the Digital Alpha processor. Source: McLellan 1993. from 1.50 to 1.79, suggesting that the Digital Digital 3000. Note also Queueing models. In order about two-thirds faster than the that the ratio of their clock frequencies is outline an approach based 200/133 = to give a flavor of analytic performance modeling, on queueing theory. The origins of this branch of applied probability theory are usually traced to the analysis of congestion in tele- phone systems made by the Danish engineer A. K. Erlang (1878-1929) Our treatment quite informal; the interested reader in 1909. referred to [Allen 1980; Robertazzi 1994] for further details. The queueing model case depicted in Figure 2.53; this sons. will consider is the single-queue, single-server is represents a "server" such as a as the M/M/l model for historical rea- or a computer with a set of tasks (pro- be executed. The tasks are activated or arrive at random times and are until they can be processed or "serviced" by the CPU on a firstcome first-served basis. The key parameters of the model are the rate at which tasks requiring service arrive and the rate at which the tasks are serviced, both meagrams) queued to in sured in tasks/s. The mean or average arrival and service rates are conventionally denoted by A (lambda) and p (mu), respectively. The actual arrival and service rates vary randomly around these mean values and are represented by probability Shared resource " Serviced items Figure 2.53 Simple queueing Quel leing sy stem model of _____., SECTION The latter are chosen to approximate the actual behavior of the system being modeled; how well they J do so must be determined by J observation and distributions. _ , 1.5 The Processor Level measurement. The symbol p (rho) denotes A/p and represents the mean utilization of the server, that is, the fraction of time it is busy, on average. For example, if an average of two tasks arrive per second (X = 2) and the server can process them at an average rate of eight tasks per second (p = 8), then p = 2/8 = 0.25. The arrival of tasks at the system is a random process characterized by the interarrival time distribution cess 1 —named — which defined as the probability that arrives during a period of length The M/M/l case assumes This exponential distribution has at a rate one task French mathematician Simeon-Denis Poisson (1781- the probability distribution Pl (t) at least a Poisson arrival pro- = l-e~h determined by Exponential distributions characterize randomness of many queueing models quite well. They are also mathematically and lead to simple formulas for various performance-related quantities of interest. It is therefore usual to model the behavior of the server (the service process) by an exponential distribution also. Let p s (t) be the probability that the service required by a task is completed by the CPU in time t or less after its removal from the queue. Then the service process is characterized by the p s (t)=\-e^' Various performance parameters can characterize the steady-state performance of the single-server queueing system under the foregoing assumptions. • p = A/p of the server, that the average fraction of time it is busy. • The average number of in the system, including tasks waiting for service and those actually being served. length and denoted by /Q The average time called the mean queue Q = p/(1-P) that arriving tasks and being served, which The parameter can be shown [Robertazzi 1994] that called the /q are related directly as follows. (2.21) in the system, both waiting for service mean waiting time tQ The . average task X passing Q through the system under steady-state conditions should encounter the same number of waiting tasks when it enters the system as it leaves behind when it departs from the system left behind is Xtq, which is the number of tasks system at rate X during the period tQ when X is present. Hence we conclude that / = Xtq, in other words, Q being serviced. The number that enter the Equation (2.22) tems, not just the called Little's equation. It is M/M/l model. Combining t = l/(p - X) valid for all types of queueing sys- (2.21) and (2.22), Q and tQ refer to tasks that are either waiting for access to the The mean number of server or are actually being served. queue excluding those being served denoted by tasks waiting in the denotes the time spent waiting in the queue, excluding service time. (The subscript stands for "waiting.") The mean utilization of the server in an M/M/l system, that is, the mean number of yields tasks being serviced, hence subtracting 'w = -P = 'w = 'o - where 1/p mean time see that takes to service a task. (2.24) and (2.25) therefore, Little's equation holds for both the and the To is illustrate the use of the foregoing formulas, consider a server computer that way that can be approximated by the M/M/l model. Arriving memory until they are fully executed in one step by the processing jobs in a jobs are queued in main CPU, which minute, and the computer questions: What is the server. is, jobs arrive on the average, an average rate of 10 per 25 percent of the time. We ask two T that each job spends in the computer? What is main memory that are waiting to begin execution? the average time the average number of jobs we assume N in from which it follows busy 75 percent of the time, p = X/]i = 0.75. We are given that X = 10 jobs/min; hence the service rate p. is 40/3 jobs/min. Substituting into (2.23) yields T= t = 1/(40/3 - 10) = 0.3 min. From Little's equaQ tion, N=l = Xt = 3; hence by (2.25), / w = 3 - 0.75 = 2.25 jobs. Q Q answer, T is Since the system company has staff. N is w that steady-state conditions prevail, computer system with a single terminal that is shared by average of 10 engineers use the terminal during an eight-hour work day, and each user occupies the terminal for an average of 30 minutes, mostly for simple and routine calculations. since the system that The company manager overutilized, since they typically wait an hour or minal; they want the manager to purchase feels that the an average of three hours a day. The users, however, complain to gain access to the ter- terminals and add them to the system. attempt to analyze this apparent contradiction using basic queueing the- Assume that the computer and its users are adequately represented by an M/M/l queueing system. Since there are 10 users per eight hours on average, we set X = 10/8 users/hour = 0.0208 users/min. The system is busy an average of five out of eight hours; hence the utilization p = 5/8, implying that u = 1/30 = 0.0333. Substituting these values for X and u into (2.25) yields fw = 50 mm, which confirms the users' estimate of their average waiting time for terminal access. The manager now convinced agrees to buy enough to reduce many new that the from 50 terminals should he buy? company needs additional terminals and The question then arises: How to 10 min. can approach problem by representing CHAPTER 2 Design M/M/l queueing system. Let m be the make t w < 10 or, equivalents, tn < 40. The arriving users are assumed to divide evenly into m queues, one for each terminal. The arrival rate X per terminal is taken to be X/m = 0.0208/m users/min. If, as indicated each terminal and minimum number of terminals needed to users by an independent above, the computer's few additional terminals should we assume that each ter= 0.0333 users/min. To meet the desired perfor- lightly utilized, then a not affect the response time experienced at a terminal, hence service rate from which u = = l/(u* - X) = i/(n - X/m) < 40 follows that minals should be acquired. This result three terminals are needed, so two new pessimistic, since the users are unlikely to form three separate queues for three terminals or to maintain the independence of the queues by not jumping from one queue to another whose terminal has become available. Nevertheless, this simple analysis gives the useful result that should be 2 or SUMMARY The problem facing the system designer is to a devise a structure (a network, or system) from given components that exhibits a specified at minimum cost. Various and behavior, including block diagrams (for structure), truth and state tables (for behavior), and HDLs (for behavior and structure). Computer systems can be viewed at several levels of abstraction, where each level is determined by its primitive components and information units. Three levels have been presented here: the gate, register, and processor levels, whose components process bits, words, and blocks of words, respectively. Design at all levels is a complex process and depends heavily on CAD tools. The gate level employs logic gates as components and has a well-developed theory based on Boolean algebra. A combinational circuit implements logic or Boolean functions of the form z{x x2 ..., xn ), where z and the x,'s assume the values and 1 The circuit can be constructed from any functionally complete set of gate types such as {AND, OR, NOT} or {NAND}. Every logic function can be realized by a two-level circuit that can be obtained using exact or heuristic minimization techniques. Sequential circuits implement logic functions that depend on time; unlike combinational circuits, sequential circuits have memory. They are behavior or performs a specified range of operations exist for describing structure from gates and 1-bit storage elements (flip-flops) that store the circuit's state and are synchronized by means of clock signals. Register-level components include combinational devices such as word gates, built multiplexers, decoders, and adders, as well as sequential devices such as (parallel) and counters. Various general-purpose programmable elePLAs, ROMs, and FPGAs. Little formal theory exists for the design and analysis of register-level circuits. They are often described by HDLs whose fundamental construct is the register-transfer statement registers, shift registers, ments also exist, including F,(X ,X2 ,...,X 1 denoting the conditional transfer of data from registers a combinational processing circuit datapath unit and a control unit. struct a formal ,X2 ,...,Xk Z via step in register-level design to con- description of the desired behavior from which the nents and connections for the datapath unit can be determined. to register Register-level circuits often consist of a logic signals to control the datapath are then identified. Finally, a control unit is that generates these control signals. The components recognized at the processor level are CPUs and other procesmemories, 10 devices, and interconnection networks. The behavior of processor-level systems is complex and is often specified in approximate terms using sors, average or worst-case behavior. Processor-level design of prototype structures. prototype design heavily based on the use selected and modified to meet the given performance specifications. The actual performance of the system evaluated, and the design further modified until a satisfactory result Typical performance measures are millions of instructions executed per second (MIPS) and clock cycles per mance evaluation instruction (CPI). A few analytical methods for perfor- —notably queueing theory —but their usefulness is limited. Instead, experimental approaches using computer-based simulation or performance measurements on an actual system are used extensively. PROBLEMS 2.1. Explain the difference between structure and behavior in the digital system context. your answer by giving (a) a purely structural description havioral description of a half- subtracter circuit that computes the 1-bit difference Following the example of Figure (b) a purely be- and also generates a borrow signal b whenever x < y. 2.4, construct a behavioral VHDL description of the full-adder circuit of Figure 2.9b. (b) Following Figure 2.5, construct a structural VHDL description of the full 2.3. Construct both structural and behavioral descriptions OR circuit appearing in Figure 2.4. Figure 2.54 describes a half adder bols for the logic operations VHDL of the EXCLUSIVE- in the widely used Verilog AND, OR, EXCLUSIVE-OR, HDL. The Verilog sym- NOT are &, \ and ~. I. respectively, (a) Is this description behavioral or structural? (b) Construct a similar description in Verilog for a full adder. module halfjudder Input x assign (xQ v y y output assign c = x c o)' Figure 2.54 Verilog description of a half adder. Problems 1 Figure 2.55 1 Truth table of a Assign each of the following components to one of the three major design levels cessor, register, or gate number AO N to -N. your answers, An identity circuit that outputs a are the same; if all its otherwise, (c) outputs a A multiplier of two n-bit num- n inputs (which represent A negation circuit that converts A first-in first-out (FIFO) the order received; full subtracter. also outputs the memory, that stores a sequence of numbers numbers in the same order. Certain very small-scale ICs contain a single two-input gate. The ICs are manufactured in three varieties NAND, OR, and EXCLUSIVE-OR as indicated by a printed label on the ICs package. By mistake, a batch of all three varieties is manufactured without their labels, (a) Devise an efficient test that a technician can apply to any IC from this batch to determine which gate type it contains, (b) Suppose the batch of unlabeled ICs contains NOR gates, as well as NAND, OR, and EXCLUSIVE-OR. Devise an efficient testing procedure to determine each gate type. Construct a logic circuit implementing the 1-bit subtracter defined in Figure 2.55 using as few gates as you can. 2.8. {a) Obtain an efficient all-NAND realization for the following four- variable Boolean = a(b + c)d + a(b + d)(b + c)(c + d)+ b c d (b) Construct an efficient 2.9. the 3-bit and 2.10. all-NOR design ioxf Design a two-level combinational sum of two 2-bit binary numbers. The sum-of-products style that computes be implemented using OR gates. Consider the flip-flop of Figure 2.11. (a) the flip-flop's state y. (b) This flip-flop it circuit in the on the positive (rising or to 1) edge of the clock CK. triggered flip-flop triggers on the negative (falling or indicated by placing an inversion bubble at the the y part of Figure 2. of one. The J input to 0) negative edge- edge of CK, which input like that at the for a negative edge-triggered flip-flop. 2.11. Figure 2.56 defines a 1-bit storage device called a triggered clocking as the the glitch does not affect said to be positive edge-triggered because flip-flop of Figure 2. activated to store a JK flip-flop. 1 1 has the same edge- but has two data inputs instead in the flip-flop; that JK = 10 sets y = 129 Set - yCK Clock Reset — K State >'(') Next y(' Methodology state Figure 2.56 graphic symbol; (b) state table. flip-flop: (a) Similarly, the to 0. activated to store a always changes, or toggles, the flop and a 2.12. Derive a What Show how to state, (a) analogous to (2.5)? (b) synchronous sequential unsigned number N of arbitrary causing the circuit to output serially the number intuitive 2.13. An SC JK = 01 re- the state unchanged, while JK = the characteristic equation for a build a NAND gates. state table for a in the flip-flop; that The input combination JK = 00 leaves meaning of each and identify the circuit that acts as a serial increis entered serially on input line \ output line Give the reset state. alternative to a state table for representing the behavior of a sequential circuit whose nodes denote states is a state diagram or state transition graph, {S^Sj,...^} and whose edges, which are indicated by arrows, denote transitions between states. A transition arrow from 5, to & is labeled XJZV if, when SC is in state 5, and input Xu is applied, the (present) output Z v is produced and SC's next state is Sj. (a) Construct a state table equivalent to the state diagram for SC appearing in Figure 2.57. (b) How many flip-flops are needed to implement 5C? Design the sequential flip-flops SC whose behavior is defined in SC has a single primary input line Your answer should include few gates and 2.15. flip-flops as you can complete logic diagram for SC. Use as Implement the sequential circuit SC specified in JK flip-flops (see problem 2.11) and NOR SC and use as few gates and flip-flops as you can. the preceding problem, this time gates. Derive a logic Design a serial subtracter analogous to the serial adder. The subtracter's inputs are two unsigned binary numbers n and n 2 the output is the difference n, - n 2 Construct ; Figure 2.57 State Figure 2.57 using and a single primary your design. diagram for a sequential a state table, an excitation table, and a logic circuit that uses gates only. SECTION problems Design a sequential length by circuit that multiplies result representing 3/V struct a state table for flops and an unsigned binary number entered serially via input line x with serially from the circuit's output line and give a complete logic N of arbitrary least significant bit first. circuit that uses NAND gates only. is functional completeness, which ensures that a types of digital computation, (a) important property of gates complete gate adequate for set is serted that functional completeness irrelevant at the register level has been as- with components such as multiplexers, decoders, and PLDs. Explain concisely Suggest a logical property of sets of such components that might be this is so. (b) substituted for completeness as an indication of the components' general usefulness in digital design. Give a brief argument supporting your position. Redraw the gate-level multiplexer circuit of Figure 2.20 at the register level using word gates. Use as few such gates as you can and mark all bus sizes. Observe that a signal such as e that fans out rying the w-bit word£ = can be considered to create an m-bit bus car- 2.20. Figure 2.55 gives the truth table for a full subtracter, - b_ i where b , i d where b denotes , denotes the borrow-in the borrow-out Show how which computes the difference subtracter's outputs are b t , to use (a) an eight-input multi- plexer and (b) a four-input multiplexer to realize the full subtracter. 2.21. Show how to design a 1/16 decoder using the 1/4 decoder of Figure 2.236 as your sole building block. 2.22. Describe AND-OR is implement the and (b) a multiplexer less costly than the other encoder of Figure 2.25 by (a) a two-level of suitable and derive a logic diagram for the that less one design expensive design. 2.23. Design a 16-bit priority encoder using two copies of an 8-bit priority encoder. You use a few additional gates of any standard types in your design, if needed. may 2.24. compares two unsigned numbers X and Y and prowhich indicate X= Y,X>Y, and X < Y, respectively. (a) Show how to implement a magnitude comparator for 2-bit numbers using a single 16-input, 3-bit multiplexer of appropriate size, (b) Show how to implement the same comparator using an eight-input, 2-bit multiplexer and a few (not more than five) magnitude-comparator duces three outputs two-input 2.25. z2 , and z3 , NOR gates. Commercial magnitude comparators such as the 74X85 have three control inputs confusingly labeled X = Y, X > Y. and X < Y, like the comparator's output lines. These inputs permit an array of k copies of a 4-bit magnitude comparator to be expanded to form a Ak-hil magnitude comparator as shown in Figure 2.58. Modify the 4-bit magnitude comparator of Figure 2.27 to add the three new control inputs and explain briefly how they work. [Hint: The unused carry input lines denoted c in in Figure 2.27 play a central role in the modification.] Show how connect n half adders (Figure 2.5) to form an «-bit combinational inis to add one (modulo 2") to an «-bit number X. For ex- crementer whose function ample, Show how = 10100111. the incrementer should output Z = 00000000. Z = should output the line to register circuit of Figure 2.29 can be simplified by using the enable and disable the register's clock signal Explain clear- gated-clocking technique often considered a violation of useful operation related to shifting called rotation. Left rotation of an ra-bit good de- sign practice. CHAPTER 2 2.28. register is defined ( Z m-2> the 4-bit right-shift register right rotation, (b) show how 2.29. :_ (Zm-l»Zin_2v.>Zi.Zo) Give an assignment statement similar by the register-transfer statement to (2.26) that defines right rotation. of Figure 2.30 can easily be Using as few additional components and control lines as possible, SR to implement both right shifting and right rotation. component types: 4-bit D-type reggates. The counter's inputs are a CLEAR signal that resets it to the all-0 state and a COUNT signal whose 0-to-l (positive) edge causes the current count to be incremented by one. Use as few components as you can, assuming for simplicity that each component type has the same Design an 8-bit counter using only the following half adders, full adders, and two-input that input variables are available in true FPGA cell of Figure 2.35a and EXLCLUSIVE-OR functions. the Actel 2.31. (a) of the largest form only, show how realize two-input versions of the form only, what that input variables are available in true NAND, NOR, the fan-in gate that can be implemented with a single Actel What is the largest NAND if both true and complemented inputs and we allow some or all of the inputs to the NAND to be inverted? (Figure 2.35a)? (b) are available 2.32. Show how to implement the full subtracter defined in Figure 2.55 using as few copyou can of the Actel C-module. Again assume that the input variables are supplied in true form only. ies as shows the Actel FPGA S-module, which adds a D flip-flop C-module discussed in the text. Show how to use one copy of 2.33. Figure 2.59 put of the to the outthis cell to implement the edge-triggered JK flip-flop defined in problem 2.11, assuming only the true output y is needed and that either one of the flip-flop's J or K inputs can be complemented. X \2 15 X4 7 3 Y, \ Y X u magnitude comparator X= Y X'/i-l C /i-2 ,c n _i) n _ x ,y n _ x + xn-l>n-l Cn-2 the truth table of Figure 3.22, then z n _ x which make v = defined correctly for remaining combinations by the equation z„_i © y„_i © c„_ 2 Consequently, during twos-complement addition the sign of the operands can in the X n-\ same way as the remaining (magnitude) bits. A related issue in computer arithmetic is round-off error, which results from fact that every number must be represented by a limited number of bits. An be treated C n-1 Figure 3.22 1 Computation of the sign bit ;,,_, and the overflow indicator v in twos-complement addition. operation involving n-bit numbers frequently produces a result of more than n For example, the product of two Ai-bit numbers contains up to In bits, which must normally be discarded. Retaining the n most significant result without modification is called truncation. Clearly the but n of of the by the amount of the discarded digits. This error can be reduced by a process called rounding. One way of rounding is to add r;/2 to the number before truncation, where r 7 is the weight of the least significant retained digit. For instance, to round 0.346712 to three decimal places, add 0.0005 to obtain 0.347212 and then take the three most significant digits 0.347. Simple truncation yields the less accurate value 0.346. Successive computations can cause round-off errors to build up unless countermeasures are taken. The number formats provided in a computer should have sufficient precision that round-off errors are of no consequence to most users. It is also desirable to provide facilities for performing arithmetic to a higher degree of precision if required. Such high precision is usually achieved by using several words to represent a single number and writing special subroutines to perform multiword, or multiple-precision, arithmetic. error Decimal numbers. Since humans use decimal arithmetic, numbers being first be converted from decimal to some binary representation. Similarly, binary-to-decimal conversion is a normal part of the computer's output processes. In certain applications the number of decimal-binary conversions forms a large fraction of the total number of elementary operations performed by the computer. In such cases, number conversion should be carried out rapidly. The various binary number codes discussed above do not lend themselves to rapid conversion. For example, converting an unsigned binary number xn-i x n-2--- xo t0 decimal requires a polynomial of the form entered into a computer must L,2' to be evaluated. by by a sequence of bits. Codes of this kind are called decimal codes. The most widely used decimal code is the BCD {binarycoded decimal) code. In BCD format each digit d of a decimal number is denoted by its 4-bit equivalent b i3 b i2 b iA b j0 in standard binary form, as in (3.7). Thus the BCD number representing 971 is 1001011 10001. BCD is a weighted (positional) Several number codes encoding each decimal exist that facilitate rapid binary-decimal conversion digit separately number code, BCD numbers employ decimal complement formats. The 8-bit ASCII code rep- 7 since b Lj has the weight 10'2 Signed . versions of the sign-magnitude or resents the 10 decimal digits by a ASCII code word have no numerical field; the remaining 4 of the other decimal codes of moderate importance are Figure 3.23. The excess-three code can be formed by adding 001 2 to the corresponding BCD number hence its name. The advantage of the excess-three code is that it ma\ be processed using the same logic used for binary codes. If two excess-three num- bers are added like binary numbers, the required decimal carry is automatically generated from the high-order bits. The sum must be corrected by adding +3. For Processor Basics Decimal code Decimal 3.2 Data Representation ooo it Figure 3.23 important decimal number codes. example, consider the addition 5 + 9 = 14 using excess-three code. 1000 = 5 1100 = 9 + Carry J 0260 - MOVE.L#4001. A2 ABCD-(AO), -(Al) I - > Program A 07D1 > Data 3001 Vector Figure 3.38 Memory allocation for the 680X0 vector addition To determine how best to implement (3.40) in assembly language, the available and addressing modes must be examined carefully. The ABCD instruction, besides being limited to byte operands, allows only two operand addressing modes: direct register addressing and indirect register addressing with predecrementinstruction types As explained register to This approach vector, earlier, the latter causes the contents of the designated address be automatically decremented just before the add operation is and hence convenient for stepping through it is selected here. to address or point to the current addition step lists, in this carried out. A0 and Al are chosen and B, respectively. Thus the basic of the address registers elements of implemented by the instruction ABCD -(AOMAl) which case the elements of a equivalent to A0:=A0-1, Al :=A1-1; M(A1) := M(A0) + M(A1) + carry: third address register stored in the used to point to vector C, and the result computed by region by the 1-byte data transfer instruction Because addresses are predecremented, AO, Al, and The foregoing be initialized to values and (3.42) are executed 1000 times, instructions (3.41) lowest address (1001 the A2 must in the case of vector A) until the reached. This point can be detected by sets the zero-status flag is made back resulting code, to (3.41) using the = 1001 and BNE (branch if not equal to which also appears with comments When Z ^ 1) instruction. in Figure 3.13, is as follows: MOVE.L MOVE.L MOVE.L #300 1, A #4001,A2 Figure 3.39 shows an assembly listing of the foregoing code with various direc- added for both purposes and to complete the program. The assembly- language source code appears on the right side of Figure 3.39, while the assembled object program appears on the left in hexadecimal code. the memory data, column contains addresses assigned by the assembler to the machine-language instructions which are then listed to the right of these program at the hexadecimal assigned by EQU directives to the directive causes the assembler to fix the start of the address 0100. The symbolic names A, B, and addresses of the (move long) elements of the three corresponding vectors. The subsequent instructions contain arithmetic expressions that are evaluated during assembly and replaced by the corresponding numerical value. For example, the A + 1000 appearing in the first MOVE.L instruction is replaced by 1001 + 1000 = 2001. In general, assembly languages allow arithmetic-logic expressions to be used as operands, provided the assembler can translate them to the form needed for the object program. The statement MOVE.L #2001, A0 is thus the first executable statement of the program, and its machine-language equivalent 2078 07D1 is loaded into memory locations 0100:0103 (hex), as indicated in Figures 3.38 and 3.39. The remainder of the short program is translated to machine code and allocated to memory in simexpression Many 680X0 branch address branch instructions use relative addressing, which means relative to the current address stored in the that the program counter PC. Consider, for instance, the conditional branch instruction BNE START, the last in the vector-addition program. As shown by Figure 3.39. the corresponding machine-language instruction is 66F6 in which 66 is the opcode BNE and F6 is an 8-bit relative address derived from the operand START. Now F6 )6 = executable instruction 111101 10 2 which when interpreted as a twos-complement number is -10 l0 or -0A, 6 BNE START has been fetched from memory locations 01 14 16 and 01 15 16 PC is . automatically incremented to point to the next consecutive Hence tion at this point PC = 000001 16, 6 Now when computes the branch address as location 0116, 6 executes the branch instruc- PC + (-0A) = 0000010C !6 which, as (ABCD) with the symbolic address . the physical address of the instruction START. The remainder of the vector-addition directives that define data regions. data region; in this case the illustrates the ORG is used again to establish a start CHAPTER Processor Basics CMPA (compare address) instruction CMPA #1001, A0 one greater than the highest addresses assigned to A, B, and C, respectively. that are address for the = 03E9 I6 The DS.B (define storage . Machine language Assembly language Location Code/Data 68000/68020 program for vector addition Instruction Sets The vectors composed of a thousand 1-byte (two digit) decimal numbers. The starting (decimal) addresses of A, B, and 1001, 2001, and 3001, respectively. Define origin of program 07D1 B 0BB9 C hex address 100 Define symbolic vector Begin executable code MOVE.L A+1000,AO Set pointer beyond end of MOVE.L B+1000.A1 Set pointer beyond end of B MOVE.L C+ Set pointer beyond end of C 66 F6 000, A2 -(A0), -(Al); Decrement pointers Store result in Test for termination & add START if Z End executable code Begin data definition Reserve 1000 bytes for elements 1:3 of elements 4:6 of End program of vector A A Figure 3.39 of the 680X0 program for vector addition. in bytes) directive reserves a region of 1000 bytes. This directive merely causes the addresses, to be incremented by the specified uses to keep track of number of indicated by Figure makes the location counter point to the start of the region storing vecThe two DC. B (define constant in bytes) commands initialize six elements of B 3.38, this action tor B. to the specified constant values. Finally the assembly-language program. directive indicates the end of the Macros and subroutines. useful tools for simplifying program design by allowing groups of instructions to be treated as single entities are macros and subroutines. defined by placing a portion of assembly-language code between appropriate directives as follows: . . , Body of macro ENDM The macro is subsequently invoked by treating the user-defined macro name, which appears in the label field of the MACRO directive, as the opcode of a new (macro) instruction. Each time the macro opcode appears in a program, the assembler replaces it by a copy of the corresponding macro body. If the macro has operands, then the assembler modifies each copy of the macro body that it generates by inserting the operands included in the current macro instruction. Macros thus allow an assembly language to be augmented by new opcodes for all types of operations; they can also indirectly introduce new data types and addressing modes. A macro is typically used to replace a short sequence of instructions that occur frequently in a program. macros shorten from it. that although the object code assembled the source code, they do not shorten Suppose, for example, that the following two-instruction sequence occurs in a program for the [Intel 1979]: This code implements the operation treating as an indirect into address register M(M(ADR)), which into accumulator register can define loads register as a macro named (load accumulator indirect) as follows: MOV ENDM With definition present in an into address register into register 8085 program, LDAI becomes assembly-language instruction for the programmer to use. The subsequent occurrence of a statement such as LDAI in the same program causes the assembler to replace (3.43) it by the macro body with the immediate address 1000 16 from (3.43) replacing the macro's dummy input ADR. Note that the macro definition itself is not part of the object pro- parameter gram. subroutine or procedure invoked by name, much also a sequence of instructions that can be (macro) instruction. Unlike a macro, howassembled into object code. It is subsequently used. like a single ever, a subroutine definition CHAPTER 3 Processor Basics MACRO operand, not by replicating the body of the subroutine during assembly, but rather during 3.3 Instruction Sets program execution by establishing dynamic links between the subroutine object code and the points in the program where the subroutine is needed. The necessary links are established by means of two executable instructions named CALL or Consider, for example, the following code segment: SUB1 Subroutine RETURN CALL SUB1 has been fetched, the program counter PC contains the address NEXT of the instruction immediately following CALL; this return address must After be saved to allow control to be returned instruction first saves the contents of forms the operand of the the address that save area. then transfers PC. and also The processor then begins execution of the sub- serves as the subroutine's name. is call statement, in this case, into returned to the original program from the subroutine by execut- which simply retrieves the previously saved return address and PC. RETURN may tions to store return addresses. its main program. Thus a later to the in a designated the address of the first executable instruction in the subroutine routine. Control use specific The RX000, register file to save a return address registers or for instance, uses a on executing any of link-register instructions, which serve as call instructions; see Figure 3.36. Many computers use a memory stack for this purpose. CALL then pushes the return address into the stack, from which it is subsequently retrieved by RETURN. The stack pointer SP automatically keeps track of the top of the stack, where the last was pushed by return address and from which be popped by Figure 3.40 illustrates the actions taken by the realization. For simplicity, memory word we assume 1000 and 1001, and we assume CALL SUB1 that the program counter and stores to 1002. contains the address 1000, as ing the instruction as a subroutine call, the the instruction in the (buffer) this point the stored in with the instruction cycle begins, the in Figure 3.40a. incremented to 1001. AR; again PC into the stack. the contents of 2000 of AR are in Figure 3.40b, contains the return address to the main program. Next the contents of fetches the address part address register state is as assembler has replaced physical address 2000. Immediately before the CALL opcode is fetched and decoded, instruction in a stack opcodes and the addresses are transferred to PC, and the stack Main program program \|2000 Subroutine " 2000 _ 2000 IR SUB1 SP 3500 I y» (b) 2000 IR SUB1 3499 RETURN 1002 Figure 3.40 Processor and during execution of a (b) state immediately after fetching the instruction, and decremented by one. The resulting instruction: (a) initial state, (c) final state. of the system depicted in Figure 3.40c. and task of a CPU is to fetch instructions from an external memory execute them. This task requires a program counter PC to keep track of the active The main registers to store the instructions and data as they are processed. a central data register called an accumulator, along capable of addition, subtraction, and word-oriented logic operations. The simplest CPUs employ with an In a register containing 32 or more general-purpose registers SECTION Problems RISC processors such as the and the MIPS RXOOO allow only load and store instructions to access M, and use small instruction sets replaces the accumulator. 3.5 and techniques such as pipelining to improve performance. CISC processors such as the Motorola 680X0 have larger instruction sets and some more powerful instructions that improve performance in some applications but reduce it in others. arithmetic capabilities of simpler processors are limited to the fixed-point (integer) instructions unless auxiliary coprocessors are used. have built-in hardware More powerful CPUs to execute floating-point instructions. and process information storage (the smallest addressable unit) handle data in a few fixed-word sizes, 32-bit in various formats. the 8-bit byte. words being basic unit of designed to The two major Fixed-point numbers formats for numerical data are fixed-point and floating-point. decimal, meaning a binary code such as found in ordinary (base 10) decimal numbers. The most common binary number codes are sign magnitude and twos complement. Each code simplifies the implementation of some arithmetic operations; twos complement, for example, simplifies the implementation of addition and sub- can be binary (base 2) or, less frequently, BCD that preserves the decimal weights and so generally preferred. number comprises a pair of fixed-point numbers, a mantissa M, and an exponent E and represents numbers of X B E where B is an implicit base. Floating-point numbers greatly the form word size but require much numbers require. The IEEE 754 increase the numerical range obtainable using a given more complex arithmetic circuits than fixed-point standard for floating-point numbers The functions performed by instruction consists of an widely used. opcode and a techniques called addressing operands can be in the instruction are defined set of instruction set. operand or address are used to specify operands. (immediate addressing), CPU registers, memory M. Operands in registers can be accessed more rapidly than M. An instruction set should be complete, efficient, and easy to use in or in external those in some broad grouped into several major types: data transand input-output instructions), data processing (arithmetic and logical instructions), and program control (conditional and unconditional branches). All practical computers contain at least a few instructions of each type, although in theory one or two instruction types suffice to perform all fer (load, sense. Instructions can be computations. RISCs are characterized by streamlined instruction sets that are supported by fast hardware implementations and efficient software compilers. While CISCs have larger and more complex instruction sets, they simplify the programming of complex functions such as division. The use of subroutines (procedures) and macroinstructions can simplify assembly-language programming in all types of processors. PROBLEMS 3.1. Show how to use the instruction set of Figure 3.4 to implement the follow- many computers; use as few inof memory location X to memory location ing operations that correspond to single instructions in structions as Y. (b) you can. the contents Increment the accumulator AC. to a specified address AC ^ 0. Use the instruction set of Figure 3.4 to implement the following two operations assuming that sign-magnitude code is used, (a) AC := -M(X). (b) Test the right-most bit b of the word stored in a designated memory location X. If b = 1, clear AC; otherwise, leave AC unchanged. [Hint: Use an AND instruction to mask out certain bits of a Consider the possibility of overlapping instruction fetch and execute operations when executing the multiplication program of Figure 3.5. (a) Assuming only one word can be transferred over the system bus at a time, determine which instructions can be over- lapped with neighboring instructions, (b) Suppose that the redesigned to allow one instruction fetch and one data load or store to occur during the same clock Now determine which instructions, if any, in the multiplication can- not be overlapped with neighboring instructions. Write a brief note discussing one advantage and one disadvantage of each of the lowing two unusual features of the (a) the inclusion of the in the general register file; (b) the fact that execution of the ADD 3.6. ARM6 EOR Identify the initial register given in hex code): R2 = 0000FFFF; R3 = 12345678; NZCV = 0000 contents of every register or flag that following instructions. of every instruction ARM6 has the following Suppose the state, (a) program counter notation and ordinary English to describe the actions performed by each Rl = 11110000; Assume each changed by execution of the executed separately with the foregoing R3,R4, LSL#4. Suppose the ARM6 has the following hex code): initial register and memory contents Rl = 00000000; R2 = 87654321 Identify the R3 = A05B77F9; contents of every register or flag that of the following instructions. ing Assume each NZCV = 0000 changed by execution executed separately with the forego- from the sign position, c„_,, the carry output signal defined by xn _ y n _ + l xn _iC n _2 + }'„_i c n-2' fr° m which follows that Either (4.5) or (4.6) can be used to design overflow detection logic for twos- complement addition or subtraction. Overflow detection in the case of magnitude numbers is similar and is left as an exercise (problem 4.6). High-speed adders. The general strategy for designing fast adders is to reduce form carry signals. One approach is to compute the input carry needed by stage i directly from carrylike signals obtained from all the preceding the time required to stage to stage. /i-bit 2,...,0, rather than waiting for normal carries to ripple slowly from carry-lookahead adder formed from n adder modified by replacing called gj and carry-lookahead adders. that use this principle are called each of which carry output line basically a by two auxiliary signals or generate and propagate, respectively, which are defined by the following logic equations: &=# The name generate comes from independent of the value of c propagates c if Xj Now be sent is, it if c, both 1 jc,v,+ ,',,- will be y-j = 1 and leading 0s should be shifted into A as before, until the first 1 in X is encountered. Multiplication of Y by this 1 and addition of the result to A causes />, to become negative, from which point on leading Is rather than 0s must be shifted x7 = into A. These rules ensure complement code. x7 = 1, v 7 = 0; that is, that a right shift corresponds to division X is negative and Y is by 2 positive. This follows case in twos- for the first seven add-and-shift steps yielding the partial product For the final step, often referred to as a correction step, the subtraction performed. The result -Y + P := P7 then given by X2"V = U+ which is by (4.21). x 1 = y1 = 1; that is, both X and Kare negative. The procedure used here follows case 2, with leading 0s (Is) being introduced into the accumulator whenever its contents are zero which ensures The correction (subtraction) step of case 3 that the final product in Each addition/subtraction also performed, step can be performed in the usual twos-complement is needed in fashion by treating the sign bits like any other and ignoring overflow. Care the shift step to ensure that the correct position A[7]. This value must be a leading positive or zero, and assigned to A[7]. it is to 0, is initially set value if placed in the accumulator's sign the current partial product in introduce a flip-flop is subsequently defined by (v 7 anY-2 = 2'-* £2 m r m Suppose the index m is replaced by j = m + i - k. Then the upper and lower limits of - k, respectively, to / - 1 and summation in (4.23) change from k - and implying that (4.22) and (4.23) are, in fact, the same. It follows that Booth's algorithm correctly computes the contribution of X, and hence of the entire multiplier X, to the product P. Equation (4.20) implies that the contribution of a negative CHAPTER 4 Datapath Design 2W BoothMult SECTION 4.1 register A[7:0], M[7:0], Q[7:-l], bus INBUS[7:0], OUTBUS[7:0]; A := 0, COUNT := 0, M := INBUS; Q[7:0]:= INBUS, Q[-1]:=0; Q[l] Q[0] = 01 then A[7:0] A[7:0] + M[7:0], go to TEST; else if Q[l] Q[0] COUNT = = 10 then A[7:0] COUNT := COUNT + 1, go to SCAN; OUTBUS := A, Q[0] := 0; OUTBUS :=Q[7:0]; := A[7], 7 then go to A[7:0] - M[7:0]; A[6:0].Q = A.Q[7:0], end BoothMult; Figure 4.15 HDL description of an 8-bit multiplier implementing the basic Booth algorithm. can also be expressed in the formats of (4.20) and (4.23); a similar argument demonstrates the correctness of the algorithm for negative multipliers. The argu- ment for fractions is essentially the same as that for integers. The twos-complement multiplication circuit of Figure 4.12 can easily be modified to implement Booth's algorithm. Figure 4.15 describes a straightforward implementation of the Booth algorithm using the above approach with n = 8 and a circuit based on Figure 4.12. An extra flip-flop Q[-l] is appended to the right end of the multiplier register Q, and the sign logic for A is reduced to the simple sign extension A[7] := A[7]. In each step the two adjacent bits Q[0]Q[-1] of Q are examined, instead of Q[0] alone as in Robertson's algorithm, to decide the operation (add Y, subtract Y, or no operation) to be performed in that step. For comparison with Robertson's method in Figure 4.13, the operands are assumed to be fractions. application of this algorithm to the example solved by Robertson's Figure 4.14 appears in Figure 4.16. where the bits stored in Q[0]Q[-1] in each step are underlined. Combinational array multipliers. Advances VLSI technology have made possible to build combinational circuits that perform n H-bit multiplication for An example is the Integrated Device Technology which can multiply two 16-bit numbers in 16 ns [Integrated Device Technology 1995]. These multipliers resemble the «-step sequential multipliers discussed above but have roughly n times more logic to allow the product to be computed in one step instead of in n steps. They are composed of arrays of simple combinational elements, each of which implements an add/subtract-andfairly values of n. multiplier chip, operation for small slices of the multiplication operands. two binary numbers X = xn _ xn _ 2 ...x x and Y = y„_iy„_2---)'i)'o assume that X and Fare unsigned integers. The ] are to be multiplied. For simplicity, P = X X Kcan therefore be expressed as 241 Step Initialize registers 10110011 = multiplier M from A Skip add/subtract Right-shift CHAPTER 4 Datapath Design = mulitplicand M to A Skip add/subtract Right-shift M from A Skip add/subtract oi ii M to A M from A Set Q[0] to 1101 11 100 = product/3 Figure 4.16 Booth multiplication algorithm. Illustration of the = o corresponding to the bit-by-bit multiplication style of Figure 4.10. (4.24) can be rewritten as P= 12' X^2 ; Each of the n in the 1-bit case. Figure 4.17 product terms appearing in (4.25) can be computed by a and logical products coincide that the arithmetic Hence an n x n can compute array of two-input the x^j of the type terms simultaneously. The terms are according to (4.25) by an array of n(n - 1) 1-bit full adders as shown in is a kind of two-dimensional ripple-carry adder. The Figure 4.18; this circuit and 2j factors in (4.25) are implemented by the spatial displacement of the adders along the x and y dimensions. Note the similarities between the circuit of Figure 4.17 and the multiplication examples of Figures implied by the 4.10 and 4.11. SECTION 4.1 Fixed-Point The AND and add functions of the array component (cell) as illustrated in Figure multiplier can be into a sin- 4.19. This cell realizes the arithmetic Arithmetic jr n x = a plus b plus xy multiplier can be built using n nent, although, as in Figure 4.18, inputs set to copies of this cell as the sole compo- on the periphery of the array have from (4.26) a plus bplus xy effectively reducing their operation plus b (a half adder). The multiplication time for this multiplier is determined by the worst-case carry propagation and, ignoring the differences between the internal and peripheral cells, is {In - \)D, where D is the delay of the basic cell. Multiplication algorithms for twos-complement numbers, such as Robertson's and Booth's, can also be realized by arrays of combinational cells as the next example shows. to a EXAMPLE 4.3 ARRAY IMPLEMENTATION OF THE BOOTH MULTIPLICATION algorithm [KOREN 1993]. Implementing the Booth method by a combinational array requires a multifunction cell capable of addition, subtraction, and no operation (skip). Such a cell selected by a pair of control lines required functions of are defined Z c 0M = Figure 4.20a. as indicated. Its It various functions are is easily seen that the by the following logic equations. © bH ® cH (a@ D)(b + + bc When HD = 10, these equations reduce to die usual full-adder equations HD =11, they reduce to the corresponding full-subtracter equations z=a@b@c c Figure 4.17 AND array for 4 x 4-bit unsigned multiplication. = ab + ac + be CHAPTER 4 Datapath Design Figure 4.18 Full-adder array for 4 x 4-bit unsigned multiplication. Figure 4.19 Cell which c and cout assume M for an unsigned array multiplier. the roles of borrow-in and borrow-out, respectively. H = 0, z becomes a, and the carry lines play no role in the final result. An H-bit multiplier nected as constructed from n Figure 4.20b. The extra cells + n(n at the l)/2 copies of the change the array's shape to a trapezium and are employed to sign-extend and subtraction. Note how the diagonal lines marked b from the parallelogram of Figure 4.18 the multiplicand for addition deliver the sign-extended directly to every sign-extended by leading 0s; this row of B When Y implicit in the array of Figure 4.18. In the present when Kis negative, it must be explicitly sign-extended by leading Is. The operation to be performed by each row of B cells is decided by bits x x _ of the operand X. To allow each possible ^ rv,_, pair to control row operations, we case, SECTION 4.1 H D Arithmetic 1 = a (no operation) coul z = apluf bplus c (add) subtracter 1 c out z = a b- c B J-o V HT Pe Figure 4.20 Combinational array implementing Booth's algorithm: (b) array multiplier for introduce a second cell type denoted in Figure 4.20b to generate the control input D required by the B cells. Cell C compares with x _ values of HD required by Figure 4.20a; these values are as follows: signal the and generates 4.1.3 Division two numbers, a divisor V and a dividend D, are given. The third number Q, the quotient, such that Q X V equals or is D. For example, if unsigned integer formats are being used, Q is com- In fixed-point division compute a very close to puted so that D=QX V+R where R, the remainder, required to be less than V, that E2 and El - E2 = bit positions. shifted mantissa and also M2 is the result determine the length of right-shifted by k digit posi- then added to or subtracted from the other mantissa via adder 2, a 56-bit parallel adder with several levels of carry look- ahead. The resulting sum or difference is placed in a temporary register R where examined by a special combinational circuit, the zero-digit checker. The output this circuit indicates the number of leading digits (or leading Fs in the case of negain R. The number z is then used to control the final nor- numbers) of the number The contents of R are left-shifted z digits by shifter 2. and the result is M3. The corresponding adjustment is made to the exponent by subtracting z using adder 3. In the event that R = 0, adder 3 can be used to set all bits of E3 to 0, which denotes an exponent of -64. malization step. placed in register SECTION 4.3 Advanced Topics Exponent comparison and checker " Shifter 2 E3 Data Figure 4.44 Floating-point add unit of the System/360 Model 91. Coprocessors. Complicated arithmetic operations like exponentiation and trig- onometric functions are costly implementations of these operations are slow. hardware, while software design alternative to use auxil- iary processors called arithmetic coprocessors to provide fast, low-cost implementations of these special functions. In general, a coprocessor hardware a separate is closely coupled to the CPU and whose instructions and registers are direct extensions of the CPU's. Instructions intended for the coprocessor are fetched by the CPU, jointly decoded by the CPU and the coprocessor, and executed by the coprocessor in a manner that is transparent to the programmer. Specialized coprocessors like this are used for tasks such as managing instruction-set processor that memory system or controlling graphics devices. example, was designed allow the to operate The MIPS RX000 series, for with up to four coprocessors [Kane and Heinrich 1992]. One of these is a conventional floating-point processor, which is implemented on the main CPU chip in later members of the series. Coprocessor instructions can be included in assembly or machine code just like any other CPU instructions. A coprocessor requires specialized control logic to link the CPU with the coprocessor and to handle the instructions that are executed by the coprocessor. A typical CPU-coprocessor interface is depicted in Figure 4.45. The coprocessor is attached to the CPU by several control lines that allow the 273 Coprocessor CHAPTER 4 Interrupt request Synchronization signals k i System bus 1r To main memory and 10 devices Figure 4.45 Connections between a and a coprocessor. of the two processors to be coordinated. the coprocessor whose registers can be read and written external memory. Communication between the passive or slave device into in same manner and copro- cessor to initiate and terminate execution of coprocessor instructions occurs auto- matically as coprocessor instructions are encountered. no coprocessor actually present, coprocessor instructions can be included in CPU knows no coprocessor is present, it can transfer program location where a software routine implement- control to a predetermined is stored. This type of CPU-generated interprogram flow is termed a coprocessor trap. Thus the coprocessor approach makes it possible to provide either hardware or software support for certain instructions without altering the source or object code of the program being executed. ing the desired coprocessor instruction of normal A opcode coprocessor instruction typically contains the following three from other that distinguishes coprocessor instructions the address of the particular coprocessor to be used allowed, and finally the type coprocessor. the coprocessor several coprocessors are can include operand addressing information. same time of the particular operation to be executed by the monitor the system bus, instruction at the as the By having can decode and identify a coprocessor the coprocessor can then proceed to exe- coprocessors but has the major drawback that the coprocessor, unlike the cute the coprocessor instruction directly. This approach does not the contents of the registers defining the current modes. Consequently, it is cessor instruction, fetch decode every coprorequired operands, and transfer the opcode and operto have the ands directly to the coprocessor for execution. This the protocol followed in 680X0-based systems employing the 68882 floating-point coprocessor, which the topic of the next example. EXAMPLE 4.7 THE MOTOROLA [motorola 1989]. The Motorola 68882 FLOATING-POINT COPROCESSOR 68882 coprocessor extends 680X0-series CPUs Operation specified Data transfer Move word to/from coprocessor data or control register Move word to/from ROM storing constants (0.0, 7t, e, etc.) Move multiple words to/from coprocessor Data processing Modulo remainder Remainder (IEEE format) Single-precision multiply Single-precision divide SECTION 4.3 Advanced Topics Program control Scale exponent Absolute value Arc cosine Arc sine Arc tangent Extract mantissa Extract integer part Hyperbolic arc tangent Hyperbolic cosine power of x power of x) minus e to the (e to the Extract exponent Extract integer part rounded to zero Logarithm of x Logarithm of x + FLOG 10 FLOG2 FNEG to the base 10 to the base 2 to the 1 base e to the base e Simultaneous sine and cosine Hyperbolic sine Square root Hyperbolic tangent 2 to the Logarithm of x Test, decrement count, and branch 10 to the power of x power of x to the base e condition code (status) cc is 1 on cc Restore coprocessor state Save coprocessor state Set (cc = 1) or reset (cc = 0) a specified byte FTST FTRAPcc Conditional trap Set coprocessor condition codes to specified values Figure 4.46 Instruction set of the Motorola 68882 floating-point coprocessor. 68020 (section 3.1.2) with a large set of floating-point instructions. The 68882 and the 68020 are physically coupled along the lines indicated by Figure 4.45. While decoding the instructions it fetches during program execution, the 68020 identifies coprocessor instructions by their distinctive opcodes. After identifying a coprocessor instruction, the 68020 CPU "wakes up" the 68882 by sending it certain control signals. like the The 68020 then transmits as an instruction register. which can proceed opcode to a predefined location in the 68882 that serves The 68882 decodes the instruction and begins its execution, the in parallel with other instructions executed within the the coprocessor needs to load or store operands, asks the to carry out the necessary address calculations and data transfers. The 68882 employs the IEEE 754 floating-point number formats described in Example 3.4 with certain multiple-precision extensions; it also supports a decimal floating-point format. From the programmer's perspective, the 68882 adds to the CPU a set of eight 80-bit floating-point data registers FP0:FP7 and several 32-bit control (opcode) and status registers. Besides implementing a wide range of arithmetic operations for floating-point numbers, the 68882 has instructions for transferring data to and from its registers, and for branching on conditions it registers, including encounters during instruction execution. Figure 4.46 summarizes the 68882' s instruc- These coprocessor instructions are distinguished by the prefix F (floatingmnemonic opcodes and are used in assembly-language programs just tion set. point) in their like regular 680X0-series instructions; see Fig. 3.12. The status or condition codes cc generated by the 68882 when executing floating-point instructions include invalid operation, overflow, underflow, division by zero, and inexact result. Coprocessor status is recorded in a control register, set of calculations, enabling the As some coprocessor which can be read by the host at the end of a to initiate the appropriate exception-processing instructions have fairly long (multicyle) execution 68882 can be interrupted in the middle of instruction execution. Its state must then be saved and subsequently restored to complete execution of the interrupted times, the The appearance of coprocessors stems in part from the fact that until the 1980s IC technology could not provide microprocessors of sufficient complexity to Once such microprocessors became possible, CPU chips, losing some of their separate identity in the process especially in the case of CISC processors. For example, the 1990-vintage Motorola 68040 microprocessor integrates a 68882-style include on-chip floating-point units. arithmetic coprocessors began to migrate onto floating-point coprocessor with a 68020-style [Edenfield et menting the performance of a RISC ciency of the CPU in a single microprocessor chip 1990]. Arithmetic coprocessors provide an attractive of aug- without affecting the simplicity and The multiple function (execution) units in superscalar microprocessors like the Pentium resemble coprocessors in that each unit has an instruction set that can execute independently of the program control unit and the other execution units. 4.3.2 Pipeline Processing a general technique for increasing processor throughput without requiring large amounts of extra hardware to the [Kogge 1981; Stone 1993]. It is design of the complex datapath units such as multipliers and floating-point CHAPTER 4 Datapath SECTION 4.3 It is improve which we return also used to cessor, a topic to Advanced Topics the overall throughput of an instruction set proin A pipeline processor consists of a sequence of m data-processing cir- cuits, called stages or segments, which collectively perform a single operation on a stream of data operands passing through them. each stage, but a final result is through the entire pipeline. input register or latch processing takes place in obtained only after an operand set has passed illustrated in Figure 4.47, a stage 5, contains a multi- , : and a datapath circuit C, that is usually hold partially processed results as they move through the pipeline; they also serve as buffers that prevent neighboring stages from interfering with one A common clock signal causes the Each Rj receives a new set of input data D,_, from the preceding stage 5,_! except for R\ whose data is supplied from an external source. D,_, represents the results computed by C _ during the preceding clock period. Once D _ has been loaded into R h Cj proceeds to use D,_, to compute a new data set D Thus in each clock period, every stage transfers its previous results to the next stage and computes a i of results. a pipeline first sight Its seems a costly and slow way independent sets of data operands. These data sets by stage so implement the an m-stage pipeline can simultaneously process up the pipeline is full, move through the pipeline m separate operations are being exe- cuted concurrently, each in a different stage. Furthermore, a new, final result emerges from the pipeline every clock cycle. Suppose that each stage of the mstage pipeline takes T seconds to perform its local suboperation and store its results. the time to complete a single operation, 7" is throughput of the per second is is 1/7/. The delay or latency of the pipeline, is therefore mT. However, the pipeline, that is, the maximum number of operations completed Equivalently, the number of clock cycles per instruction or CPI the pipeline's clock period. performing a long sequence of operations in the pipeline, its perfor- determined by the delay (latency) T of a single stage, rather than by the delay mT of the entire pipeline. Hence an m-stage pipeline provides a speedup fac- tor of m compared to a nonpipelined implementation of the same target operation. Control unit —v Stage S\| Figure 4.47 Structure of a pipeline processor. Data out Stage S 2 Stage S„ operation that can be decomposed into a sequence of suboperations of about the same complexity can be realized by a pipeline processor. Consider, for example, the addition of two normalized floating-point numbers x and a topic discussed in section 4.3.1. This operation can be implemented by the following compare the exponents, align the mantissas (equalize the expoadd the mantissas, and normalize the result. These operations require the four-stage pipeline processor shown in Figure 4.48. Suppose that x has the normalized floating-point representation (x M ,xE ), where xM is the mantissa and xE is the k exponent with respect to some base B = 2 In the first step of adding x = (xM jcE ) to = y (y M ,y E ), which is executed by stage S of the pipeline, x E and y E are compared, four-step sequence: an operation performed by subtracting the exponents, which requires a fixed-point adder (see Example identifies the smaller of the exponents, say, xE whose , mantissa xM can then be modified by shifting in the second stage S2 of the pipeline form a new mantissa x' M that makes (x' M ,y E ) = (xM ,xE ). In the third stage the mantissas x' M and y M which are now properly aligned, are added. This fixed-point addition can produce an unnormalized result; hence a fourth and final step is to to normalize the result. done by counting the number k of leading zero digits of the mantissa (or leading ones in the negative case), shifting the mantissa k digit positions to normalize and making a corresponding adjust- in the exponent. Figure 4.49 illustrates the behavior of the adder pipeline sequence of N floating-point additions of the form x + ( when performing y, for the case this type arise when adding two yV-component real (floating-point) At any time, any of the four stages can contain a pair of partially processed scalar operands denoted Qt,,y,) in the figure. The buffering of the stages ensures that S, receives as inputs the results computed by stage 5,_ during the preceding clock period only. If Tis the pipeline's clock period, then it takes time 4T to compute the single sum x, + y,; in other words, the pipeline's delay is AT. This value is approximately the time required to do one floating-point addition using a nonpipelined processor plus the delay due to the buffer registers. Once all four stages of the pipeline have been filled with data, a new sum emerges from the last stage S4 every T seconds. Consequently, N consecutive additions can be done in time (N + 3)T, implying that the four-stage pipeline's speedup is sequences of vectors. S(4) = x - (x M x E ) . Exponent Ri Mantissa : Mantissa adder c2 >' = (>'m.Ve) adder and mantissa shifter 'V Stage 5, Stage S 2 Stage S 3 Stage 54 (Exponent comparison) Figure 4.48 Four-stage floating-point adder pipeline. CHAPTER 4 Datapath SECTION 4.3 (6- >6> (x 5 ,y 5 ) (x 6 ,y 6 ) Advanced Topics , y4 ) y5 ) (x6 , y6 ) (xA ,y4 ) (x5 ,y 5 ) (x 6 ,y6 ) (x2 ,y2 ) (x 3 ,y 3 ) (x4 ,y4 ) (x s ,y 5 ) (x6 ,y 6 ) (4- >'4) (i. ^5) (6- >'6> (2- -v 2) (x 3 , y 3 ) nrri X C3- >'3> (4- >') • 2 (5- >5> C2. >2) (3- >3) (4- >4) • C5. >s) t Current result (x 2 (x4 ,y4 ) (e- ye) t •1+>1 + >'2 x 3 + yi x\ + y 5 + >5 6 + >'6 «2 + >'2 •»3+> 3 + >'4 x s+y$ X2 + >2 ^ + y3 * 2 + >'2 3 + + y, + >'4 x2 + >2 Figure 4.49 Operation of the four-stage floating-point adder pipeline. For large N, 5(4) = 4 so that results are generated at a rate about four times that of a comparable nonpipelined adder. If it is not possible to supply the pipeline with data at the maximum rate, then the performance can fall considerably, an issue to which return in Chapter 5. Pipeline design. Designing a pipelined circuit for a function involves finding a suitable multistage sequential algorithm to compute the given function. This algorithm's steps, which are implemented by the pipeline's stages, should be balanced in the sense that they should all have roughly the same execution time. Fast buffer registers are placed between the stages to allow necessary data items be transferred from stage to stage without interfering with one another. The buffers are designed to be clocked at the maximum rate (partial or results) to be transferred reliably between stages. Figure 4.50 shows the register-level design of a floating-point adder pipeline that allows data to based on the nonpipelined design of Figure 4.44 and employing the four-stage organization of Figure 4.48. The main change from the nonpipelined case is the inclusion of buffer registers to define and isolate the four stages. cation has been implement fixed-point as well further modifi- as floating-point addition. — Data LHAF1 EK 4 i i Exponent comparison Adder 1 Si 1 M5 : Mantissa alignment --?" i "1 i 1 Mantissa addition/ check er Shifter 2 i 54 u I 1 Figure 4.50 Pipelined version of the floating-point adder of Figure 4.44. circuits that perform the mantissa addition shown by broken operands. To perform in stage 5 3 and the corresponding buffers are enlarged, as lines in Figure 4.50. to full-size fixed-point a fixed-point addition, the input oper- ands are routed through 5 3 only, bypassing the other three stages. Thus the circuit of Figure 4.50 is an example of a multifunction pipeline that can be configured either as a four-stage floating-point adder or as a one-stage fixed-point adder. course, fixed-point and floating-point subtraction can also be performed by this circuit; subtraction and addition are not usually regarded as context, however. distinct functions in this The same function can sometimes be section 4 partitioned into suboperations in several different ways, 3 Advanced Topics depending on such factors as the data representation, the style of tne '°^ c design, an( the need to share stages with other functions in a multifunction pipeline. A floating-point adder can have as few as two stages and as many as six. For example, five-stage adders have been built in which the normalization stage (54 in Figure 4.50) is split one to count the number k of lead- ing zeros (or ones) in an unnormalized mantissa and a second stage to perform the k shifts that normalize the mantissa. Whether or not a particular function or set of functions F should be implemented by a pipelined or nonpipelined processor can be analyzed as follows. Suppose that F can be broken down into m independent sequential steps F ,F2 ,.-.,Fm so Pm Let F, be realizable by a logic l has an m-stage pipelined implementation 7,. Let T R be the delay of each and associated control logic. The longest 7", the pipeline and force the faster stages to wait, doing no the slower stages become available. Hence the delay circuit C, with propagation delay (execution time) stage Sj due to buffer register times create bottlenecks in useful computation, until between the emergence of two Tc = max{r,} + TR The throughput of Pm clock period (the pipeline period) for Tc i \ITC = l/(max{r,} + Z /= T value of defined by the equation R ). nonpipelined implementation We throughput of 1/(X J=1 i conclude the m-stage pipeline Pm has greater throughput than P that is, pipelining P F has a delay of or, equivalently, a increases performance Equation (4.44) also implies that it is desirable for all 7", times the same; that is, the pipeline stages should be balanced. be approximately Feedback. The usefulness of a pipeline processor can sometimes be enhanced by including feedback paths from the stage outputs to the primary inputs of the pipeline. Feedback enables the results computed by certain stages to be used in subsequent calculations by the pipeline. We next illustrate this important concept by adding feedback to a four-stage floating-point adder pipeline like that of Figure 4.50. summation by lem of computing Consider the proba pipeline processor sum of ./V floating-point numbers b ,b 2 ,---,bN It can be solved by . adding consecutive pairs of numbers using an adder pipeline and storing the sums temporarily in external registers. The summation can be done much more efficiently by modifying the adder as shown in Figure 4.5 1 Here a feedback path has been added to the output of the final stage 54 allowing its results to be fed back to the first stage 5\|. A register/? has also been connected to the output of S4 so that stage's results can be stored indefinitely before being fed back to 5,. The input operands of the modi. fied pipeline are derived obtained from a such operands as the all-0 and result computed by S4 that is typically location; a constant source K that can apply from four separate sources: a variable CPU register or a memory in the words; the output of stage S4 representing the , preceding clock period; and, puted by the pipeline and stored in the output register R. an earlier result com- CHAPTER 4 Datapath Design r - Stage 5, Stage 5 2 Stage 5 4 Output 1 Figure 4.51 Pipelined adder with feedback paths. The jV-number summation problem following way. The external operands b solved by the pipeline of Figure 4.5 ,b 2 ,...,b N are tinuous stream via input X. This process requires a sequence of register or fetch operations, are easily register/memory locations. While the in the entered into the pipeline in a con- the operands are stored in contiguous four numbers b ,b 2 ,b i ,b A are being entered, x the all-0 word denoting K, as illustrated in Figure 4.52 for times number zero the floating-point + b = applied to the pipeline input After four clock periods, that emerges from 54 and is fed back to the primary inputs of the pipeline. At this point the constant input K = is replaced by the current result 5 4 = b v The pipeline now begins to compute b + b$. At t = 6, it begins to compute b 2 + b6 at / = 7, computation of fc + b n begins, and so on. When b + b 5 emerges from the pipe3 line at t = 8, it is fed back to 5, to be added to the latest incoming number b 9 to initiate computation of b + b 5 + b9 (This case does not apply to Figure 4.52, where b % = b s is t 5, the first the last item to be pipeline and pipeline sum b 2 + bb emerges from number b w Thus at any time, four partial sums of the form In the next time period, the fed back to be added to the incoming in its four stages b i + b 1 + b u +b^ + + &,, + fe.fi + fc 4 ... ... SECTION 4.3 L J Advanced Topics (0, b-,) i3 ) bA ) b3 ) (0. fc,) HZ (0J>.i = 3 = 2 (by b- (0, 64) ZIZ (0. (** 8) JL fc I (0. ib b 6 ) (0. fej) J_ 5 ) I fc HZ (fc,.fc 5 ) „ + b 3 + b 7 emerges from 54 at t - 16. At this point the outputs of 5 4 and R are fed back to S v The final result is produced four time periods later at t = 20 in the case of N = 8. the desired result b ture + b2 + Figure 4.52 for the case CHAPTER 4 i (0,0) b% ) (b 4 + b g b 3 + b7 ) , 4 (b 4 + 6 5 b 2 + b6 ) b6 ) b^ + t= 10 f= 12 ZJl. + b 5 + b 2 + b(, b 4 + bg + b 3 + b 7 ) + bs b} + , 4 (0.0) 4 (fc, 4 + 8 b 2 + b6 b2 + b6 = 9 (ft , fcg) b + b5 (&,. b,) + b8 + fc 5 + b 2 + (b 4 + b% + pipeline's clock period, that adder requires time 1 scheme of Figure 4.52 11)7", where T is the time (N + the delay per stage. Since a 4NT to compute SUM. we which approaches 4 comparable nonpipelined 4N/(N + obtain a speedup here of about N increases. The foregoing summation operation can be invoked by a single vector instruc- tion of a type that characterized the vector-processing, pipeline-based puters" of the 1970s and 1980s [Stone 1993]. For instance. Control Data Corp.'s STAR- 100 computer [Hintz and Tate 1972] has an instruction sum of B = (b b 2 ,...,bN) of and places the result in a CPU register. The starting (base) address of B, which corresponds to a block of main memory, the name C of the result register, and the vector length N are all specified by operand fields of SUM. We can see from Figure 4.52 that a relatively complex pipeline control sequence is needed to implement a vector instruction of this sort. This complexity contributes the the elements of a specified floating-point vector SECTION 4.3 Advanced Topics arbitrary length and cost of vector-oriented computers. Moreover, significantly to both the size speedup, the input data must be stored in a vector elements to enter the pipeline number-pair per clock cycle. at the possible rate that allows the —generally one The more complex arithmetic operations in CPU instruction sets, including most floating-point operations, can be implemented efficiently in pipelines. Fixedpoint addition and subtraction are too simple to be partitioned into suboperations suitable for pipelining. As we see next, fixed-point multiplication well suited to pipelined design. Pipelined multipliers. Consider the task of multiplying two n-bit fixed-point binary numbers pliers of the X = xn _ xn _ 2 and Y = kind described in section y„_ 2 Combinational array multi- are easily converted to pipelines by the addition of buffer registers. Figure 4.53 shows a pipelined array multiplier that employs the M computes a xy and stage and a carry bit generated A out Xt M M Register ^iL M ^^ M M /f r i Ps Figure 4.53 Multiplier pipeline using ripple-carry propagation. each stage y a x cells in with the final product Pn _ XY being computed by the last stage. In addition to storing the partial products in the buffer registers denoted unused multiplier must also be stored /i-stage multiplier pipeline the multiplicand and can generate a new result every clock cycle. slow speed of the carry-propagation logic main disadvantage the rela- each stage. The number of needed is n and the capacity of all the buffer registers is approximately 3« 2 (see problem 4.31); hence this type of multiplier is also fairly costly in hardware. For these reasons, it is rarely used. Multipliers often employ a technique called carry-save addition, which is parcells ticularly well suited to pipelining. full adders. Its the n n-bit carry-save adder consists of n disjoint be added, while the output consists of forming a word 5 and the n carry adders discussed so The outputs 5 and C can be in Figure 4.54, they bits forming a word C. Unlike the no carry propagation within the individual adders. fed into another «-bit carry-save adder where, as can be added connections are shifted to the general, number W. Observe to a third n-bit left to that the carry correspond to normal carry propagation. In m numbers can be added by a treelike network of carry-save adders to pro- duce a result form (5,C). To obtain the in the sum, S and C must be added by a conventional adder with carry propagation. Multiplication can be performed using a multistage carry-save adder circuit of shown in Figure 4.55; this circuit is called a Wallace tree after its inven[Wallace 1964]. The inputs to the adder tree are n terms of the form M, = the type tor xi Y2k Here M, represents the multiplicand Y multiplied by the /th multiplier bit weighted by the appropriate power of 2. Suppose that is In bits long and that a full double-length product is required. The desired product P is Z£Tq This sum is computed by the carry-save adder tree, which produces a 2«-bit sum and a . Y\ z2 Xd y2 CS / V r 1 ca irn, -Si ive i iddei / 1 r T Figure 4.54 At WO -sta ge / T $2 c'-i / 1 multiplying fixed-point vectors, CHAPTER 4 of this type can overlap the computation of n separate products, as required, for example, SECTION 4.3 Advanced Topics Multiplier decoding and multiplicand gating M5 M4 M3 form the CHAPTER 4 Vr, If LX1 < M, subtract X from y ^-3->0 (modulo 2"). If 1X1 > \Y\, then set v„_, to and subtract Y from X (modulo 2"). X negative; Y positive: If in < LX1, subtract Y from X (modulo 2"). If in > 1X1, set.r„_, to and subtract X from Y (modulo 2"). X and Y both negative: Add X and Y (modulo 2") and set z„_ to Y negative: Let = xn _2xn _y..xn and >fl = y„ Figure 4.61 Algorithm for subtracting sign-magnitude numbers. 1 n-bit adder- SUB "} Figure 4.62 lines appearing in the figure n-bit adder-subtracter circuit. can be used to compute Construct a suitable logic circuit Consider again the adder-subtracter of Figure 4.62, assuming now that it has been designed for sign-magnitude numbers. It computes Z = X + Y when SUB = and Z = X- Y when SUB = 1. Assume that the circuit contains an n-bit ripple-carry adder and a similar «-bit ripple-borrow subtracter and that you have access Derive a logic equation that defines an overflow flag v for to all internal lines. this circuit. Give an informal interpretation and proof of correctness of the two expressions (4.12) and g that define the propagate and generate conditions, respectively, for a 4-bit carry-lookahead generator. Show how extend the 16-bit design of Figure 4.8 to a 64-bit adder using the same two component 4.9. types: a 4-bit adder Stating your assumptions and module and a 4-bit carry-lookahead generator. showing your calculations, obtain an good estimate each of the following for both an n-bit carry-lookahead adder and an n-bit ripple-carry adder: (a) the total (c) the number of gates used; (b) the circuit depth (number of gate fan-in. Another useful technique for fast binary addition is the conditional-sum method. It and a closely related method called carry-select addition are based on the idea of simultaneously generating two versions of each sum bit s\: a version s], which assumes that its input carry c,_, = 1. and a second version s?, which assumes that c _ = 0. 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For what values of x is f(x)=3x^3-7x^2-5x+9 concave or convex? \| HIX Tutor For what values of x is #f(x)=3x^3-7x^2-5x+9# concave or convex? Answer 1 $f$ is concave (concave down) on $\left(- \infty , \frac{7}{9}\right)$ and is convex (concave up) on $\left(\frac{7}{9} , \infty\right)$. The convexity and concavity of the function #f# can be determined by looking at the sign of the second derivative: To find the function's second derivative, use the power rule. So, the convexity and concavity are determined by the sign of #f''(x)=18x-14#. The second derivative equals #0# when #18x-14=0#, which is at #x=7/9#. When #x>7/9#, #f''(x)>0#, so #f(x)# is convex on #(7/9,oo)#. When #x<7/9#, #f''(x)<0#, so #f(x)# is concave on #(-oo,7/9)#. Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer 2 To determine the intervals where the function $f(x) = 3x^3 - 7x^2 - 5x + 9$ is concave or convex, we need to examine its second derivative, because the sign of the second derivative tells us about the concavity: 1. First derivative $f'(x)$: \[f'(x) = \frac{d}{dx}(3x^3 - 7x^2 - 5x + 9) = 9x^2 - 14x - 5.\] 2. Second derivative $f''(x)$: \[f''(x) = \frac{d}{dx}(9x^2 - 14x - 5) = 18x - 14.\] Set $f''(x)$ equal to 0 to find critical points: \[18x - 14 = 0 \implies x = \frac{14}{18} = \frac{7}{9}.\] - For $x < \frac{7}{9}$: Choose a test value, say $x = 0$, \[f''(0) = 18(0) - 14 = -14,\] which is negative. Thus, $f(x)$ is concave down on this interval. - For $x > \frac{7}{9}$: Choose a test value, say $x = 1$, \[f''(1) = 18(1) - 14 = 4,\] which is positive. Thus, $f(x)$ is concave up on this interval. Therefore, the function $f(x) = 3x^3 - 7x^2 - 5x + 9$ is concave down for $x < \frac{7}{9}$ and concave up for $x > \frac{7}{9}$. Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer from HIX Tutor When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some Not the question you need? HIX Tutor Solve ANY homework problem with a smart AI • 98% accuracy study help • Covers math, physics, chemistry, biology, and more • Step-by-step, in-depth guides • Readily available 24/7	{"url":"https://tutor.hix.ai/question/for-what-values-of-x-is-f-x-3x-3-7x-2-5x-9-concave-or-convex-8f9af9fe32","timestamp":"2024-11-13T04:52:09Z","content_type":"text/html","content_length":"582485","record_id":"<urn:uuid:25324a43-faf2-4919-a371-0aa20dfba687>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00195.warc.gz"}
A fully mixed-integer linear programming formulation for economic dispatch with valve-point effects, transmission loss and prohibited operating zones Economic dispatch (ED) problem considering valve-point effects (VPE), transmission loss and prohibited operating zones (POZ) is a very challenging issue due to its intrinsic non-convex, non-smooth and non-continuous natures. To achieve a near globally solution, a fully mixed-integer linear programming (FMILP) formulation is proposed for such an ED problem. Since the original loss function is highly coupled on n-dimensional spaces, it is usually hard to piecewise linearize entirely. To handle this difficulty, a reformulation trick is utilized, transforming it into a group of tractable quadratic constraints. By taking full advantage of the variables coupling relationships among univariate and bivariate functions, an FMILP formulation that requires as few binary variables and constraints as possible is consequently constructed for the ED with VPE and transmission loss. When the POZ restrictions are also considered, a distance-based technique is adopted to rebuild these constraints, making them compatible with the previous FMILP reformulation. With the help of a logarithmic size formulation technique, a further reduction can be made for the introduced binary variables and constraints. By solving such an FMILP formulation, a near globally solution is therefore gained efficiently. In order to search for a more excellent feasible solution, a non-linear programming (NLP) model for the ED will be given and solved based on the FMILP solution. The case study results show that the presented FMILP formulation is very effective in solving the ED problem that involves non-convex, non-smooth and non-continuous natures. View A fully mixed-integer linear programming formulation for economic dispatch with valve-point effects, transmission loss and prohibited operating zones	{"url":"https://optimization-online.org/2019/01/7027/","timestamp":"2024-11-12T19:13:10Z","content_type":"text/html","content_length":"85813","record_id":"<urn:uuid:aad9d400-4da1-4421-94dd-678c22dc1746>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00122.warc.gz"}
Parallel Regression • Genstat v21 Select menu: Stats \| Regression Analysis \| Parallel Regression Use this to run a large number of regression models in parallel for a set of units which have multiple measurements/y-variates on each unit. 1. After you have imported your data, from the menu select Stats \| Regression Analysis \| Parallel Regression. 2. Fill in the fields as required then click Run. You can set additional Options before running the analysis and store the results by clicking Store. The data can be supplied as a list of variates or a pointer to the list of variates. The results on each y-variate are saved, indexed by the y-variate. If the y-variates are stacked the spreadsheet unstack dialog can be used to reorganise the data. Alternatively, the microarray one channel regression menu provides an alternative way to analyse stacked data. The analysis for the menu is performed using the RYPARALLEL procedure which performs a regression to analyse the y-values for each measurement. Available data This lists data structures appropriate for the edit box which currently has focus. You can double-click a name to enter it in the input field. Data arrangement The data can be supplied in either of the following formats: • List of y-variates – All the y-variates can be specified in a list. • Pointer to list of y-variates – A pointer data structure is used to reference the list of y-variates. Specify the data in this form can be useful when thousands of y-variates are to be analysed in parallel. If data are stacked the spreadsheet unstack dialog can be used to reorganise the data. Specifies the y-variates to be analysed in parallel where the variates must all contain the data in the same order. Pointer to y-variates Specifies a pointer to a list of y-variates to be analysed in parallel where the variates must all contain the data in the same order. Specifying the data in this form is useful when thousands of y-variates are being analysed in parallel. Model to be fitted A model formula specifying the combinations of factors and variates describes the regression model for the y-variates. For a simple linear regression this will be the name of the variate that specifies the explanatory (x) variable to use in all analyses. See the page on model formulae for more details on how to specify regression models. This provides a way of entering operators in the regression model formula. Double-click on the required symbol to copy it to the current input field. You can also type in operators directly. See model formula for a description of each operator. Action buttons Run Run the analysis. Cancel Close the dialog without further changes. Options Opens a dialog where additional options and settings can be specified for the analysis. Defaults Reset options to the default settings. Clicking the right mouse on this button produces a pop-up menu where you can choose to set the options using the currently stored defaults or the Genstat default settings. Store Opens a dialog to specify names of structures to store the results from the analysis. The names to save the structures should be supplied before running the analysis. Action Icons Pin Controls whether to keep the dialog open when you click Run. When the pin is down Restore Restore names into edit fields and default settings. Clear Clear all fields and list boxes. Help Open the Help topic for this dialog. See also	{"url":"https://genstat21.kb.vsni.co.uk/knowledge-base/parallel-regression/","timestamp":"2024-11-14T11:10:52Z","content_type":"text/html","content_length":"45159","record_id":"<urn:uuid:acf423cb-69e6-417a-9dce-732b2538264c>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00134.warc.gz"}
How Many Minutes In A Month Of A Regular And Leap Year? 29 Aug How Many Minutes In A Month Of 30-Day, 31-Day, 28-Days, And 29-Days? Posted at 06:34h General 0 Comments Time plays a big role in our daily lives, whether we are working, planning, or just keeping track of our schedules. We usually think of time in terms of hours, days, and weeks, but sometimes we need to look at it in smaller units like minutes. A common question that comes up is how many minutes are in a month. The answer might not be as simple as you think, since different months can have different lengths. Through this blog post, you will Explore The Concept Of Time Measurement and the number of Minutes In A Month. Moreover, it also sheds light on the reasons behind the reasons and practical applications of minutes in-a-month conversion. Explore The Concept Of Time Measurement Time measurement is a day to day calculation which has a vital role everywhere including from household to professional places. Therefore, it is always supposed to be quick and precise whenever the conversion is needed in any situation. Furthermore, talking about the relation between minutes and months then in every one hour there is 60 minutes whereas in each day there is 1440 minutes. However, the calculation of minutes in a month varies depending upon the number of days in a month. The change in the formula begins according to 30 days in a month, 31 days in a month, or a leap year. Besides this, opting for the right method for time measurement helps in better time management and doing the tasks efficiently. How many minutes in a month? Minute is one of the time measurement units. It is accepted by the metric system. Moreover, it is also known as “min” in slang or as an abbreviation. Besides that, a minute is equal to 60 seconds or one by sixty of an hour. Now, talking about month, it is also a period or time measurement unit which is equal to one by twelve if a year. In addition to this, month is the most commonly used unit of a calendar. It has days ranging from twenty eight to thirty one. Over and above that, the abbreviation of month is “mo” or “mth” and when it’s plural, it is called “mos”. Apart from this, an overall glimpse of minutes in a month is as follows in this table made for your reference and making the calculations easier: S. No. Month Type Number of Minutes 1. 31 Days 44,640 minutes 2. 30 Days 43,200 minutes 3. Leap Year February (29 Days) 41,760 minutes 4. Non-Leap Year February (28 Days) 40,320 minutes Calculation of How Many Minutes In A Month Not every month has the same number of days, and so does the formula used to calculate that. Therefore, to help you learn the easy calculation of minutes in a month, we brought the conversion of every case in one place. Hence, learn the formula and quick methods to understand the same below: 1. Number of minutes in a 31-day month 31 is the highest number of days possible in a month. It happens in months such as January, October, March, December, July, May, and August. Thus, to calculate the total minutes in such a month, you just need to multiply 31 days by 24 hours and then multiply the answer by 60 minutes. 31 days x 24 hours in a day x 60 minutes in an hour = Total minutes in a 31 days month Therefore, it will be => 31 x 24 x 60 = 44,640 minutes 31 days x 1440 ( 24 hours x 60 minutes) = Total minutes in a 31 days month Therefore, it will be => 31 x 1440 = 44, 640 minutes Hence, in every month which has 31 days, there are a total of 44,640 minutes. The figure remains the same in every month which has 31 days and varies according to the number of days. 2. Number of minutes in a 30-day month After 31 days months, the 30-day months are the second most common number of days found in a month. Thus, it appears in months such as November, June, September, and April. Hence, to calculate the total minutes in such a 30-day month, you just need to multiply 30 days by 24 hours and then multiply the answer by 60 minutes. 30 days x 24 hours in a day x 60 minutes in an hour = Total minutes in a 30 days month Therefore, it will be => 30 x 24 x 60 = 43,200 minutes 30 days x 1440 (24 hours x 60 minutes) = Total minutes in a 30 days month Therefore, it will be => 30 x 1440 = 43,200 minutes Hence, every month, which is 30 days, there is a total of 43,200 minutes. 3. Number of minutes in a Common Year February In every non-leap year, February has 28 days in total. It is the lowest possible day of a month in a year. Therefore, to calculate the number of minutes in a month, which is only 28 days, you need to multiply 28 days by 24 hours and then 60 minutes. 28 days x 24 hours in a day x 60 minutes in an hour = Total minutes in a 28 days month Therefore, it will be => 28 x 24 x 60 = 40,320 minutes 28 days x 1440 (24 hours x 60 minutes) = Total minutes in a 28 days month Therefore, it will be => 28 x 1440 = 40,320 minutes Hence, in every standard February, which has 28 days, there will be precisely 40,320 minutes in total for that month. 4. Number of minutes in a Leap Year February Leap year comes once in four years. In this, February has one extra day, which is a total of 29 days in a month. Therefore, to calculate the number of days in this month, you need to multiply 29 days by 24 hours and then 60 minutes in an hour. 29 days x 24 hours in a day x 60 minutes in an hour = Total minutes in a 29 days month Therefore, it will be => 29 x 24 x 60 = 41, 760 minutes 29 days x 1440 (24 hours x 60 minutes) = Total minutes in a 29 days month Therefore, it will be => 29 x 1440 = 41,760 minutes Hence, in every February of leap year, there is a total of 41,760 minutes in a month. Moreover, the most recent leap year was in 2024 and the next will be in 2028. Why Do We Need To Convert Minutes In A Month? If you are wondering how many minutes are in a month, it will benefit you in many ways. Simply go through the following section to learn why we need to convert minutes in a month. 1. Precise Time Management: Knowing the total number of minutes in a month will help you to keep track of your time. As a result, it will help you to make sure you do not waste any time. 2. Effective Task Planning: When you know how many minutes are in a month, you will be able to plan tasks in a better way. Moreover, you can also divide your time and further focus on the important tasks first. 3. Scheduling Appointments And Meetings: By converting minutes to monthly time, you can schedule appointments and meetings more easily. Furthermore, this helps avoid double bookings and pull out enough time for other important tasks. 4. Measure the Duration of Processes: Converting minutes into a month also helps to see how long your tasks and projects might take. Besides this, it lets you organize work in a better way and find ways to do things fast. 5. Precise Coordination Of Daily Activities: When you know how many minutes are in a month, you can plan your daily routine in a much better way. Not only this, it helps you balance work and personal time effectively. Understanding how many minutes are in a month makes it easier to manage your time and keep tasks organized. Moreover, it also helps you to be more efficient in everything you do. In this blog post we briefly answered “how many minutes in a month”. Ranging from these time metrics measurement to conversion methods, we discussed almost everything. Thus, time is very important and passes very fastly hence, its precise measurement should always be handy and never go wrong. I hope the page helped you in figuring out the conversion methods according to different cases. Also, we expect that all of your queries related to time metrics will be resolved now. Question. Does the knowledge of “how many minutes in a month” enhance the planning of personal tasks? Answer. Knowing how many minutes are in a month helps you divide time into smaller parts. This further makes it easier to plan tasks, set realistic deadlines, and stay on track with goals. Question. How do the variations in the number of days per month affect resource allocation in projects? Answer. The number of days in each month changes, which can eventually affect project schedules. Hence, you may need to adjust plans and resources to make sure everything gets done on time. Question. Can we use this conversion method for other periods (Like Weeks)? Answer. Yes! You can convert weeks into days or hours. Knowing this helps plan projects well, manage time, and make sure tasks are done on schedule. Read More Valuable Contents in the following: No Comments Post A Comment	{"url":"https://thetechiepie.com/general/how-many-minutes-in-a-month/","timestamp":"2024-11-13T06:02:28Z","content_type":"text/html","content_length":"130516","record_id":"<urn:uuid:3e169ec8-915f-4343-937f-e7db95d2f06e>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00770.warc.gz"}
Lesson 1 What are Scaled Copies? 1.1: Printing Portraits (10 minutes) This opening task introduces the term scaled copy. It prompts students to observe several copies of a picture, visually distinguish scaled and unscaled copies, and articulate the differences in their own words. Besides allowing students to have a mathematical conversation about properties of figures, it provides an accessible entry into the concept and gives an opportunity to hear the language and ideas students associate with scaled figures. Students are likely to have some intuition about the term “to scale,” either from previous work in grade 6 (e.g., scaling a recipe, or scaling a quantity up or down on a double number line) or from outside the classroom. This intuition can help them identify scaled copies. Expect them to use adjectives such as “stretched,” “squished,” “skewed,” “reduced,” etc., in imprecise ways. This is fine, as students’ intuitive definition of scaled copies will be refined over the course of the lesson. As students discuss, note the range of descriptions used. Monitor for students whose descriptions are particularly supportive of the idea that lengths in a scaled copy are found by multiplying the original lengths by the same value. Invite them to share their responses later. Arrange students in groups of 2. Give students 2–3 minutes of quiet think time and a minute to share their response with their partner. If using the digital activity, have students work in groups of 2–3 to complete the activity. They should have quiet time in addition to share time, while solving the problem and developing language to describe scaling. Student Facing Here is a portrait of a student. Move the slider under each image, A–E, to see it change. 1. How is each one the same as or different from the original portrait of the student? 2. Some of the sliders make scaled copies of the original portrait. Which ones do you think are scaled copies? Explain your reasoning. 3. What do you think “scaled copy” means? Arrange students in groups of 2. Give students 2–3 minutes of quiet think time and a minute to share their response with their partner. If using the digital activity, have students work in groups of 2–3 to complete the activity. They should have quiet time in addition to share time, while solving the problem and developing language to describe scaling. Student Facing Here is a portrait of a student. 1. Look at Portraits A–E. How is each one the same as or different from the original portrait of the student? 2. Some of the Portraits A–E are scaled copies of the original portrait. Which ones do you think are scaled copies? Explain your reasoning. 3. What do you think “scaled copy” means? Activity Synthesis Select a few students to share their observations. Record and display students’ explanations for the second question. Consider organizing the observations in terms of how certain pictures are or are not distorted. For example, students may say that C and D are scaled copies because each is a larger or smaller version of the picture, but the face (or the sleeve, or the outline of the picture) has not changed in shape. They may say that A, B, and E are not scaled copies because something other than size has changed. If not already mentioned in the discussion, guide students in seeing features of C and D that distinguish them from A, B, and E. Invite a couple of students to share their working definition of scaled copies. Some of the students’ descriptions may not be completely accurate. That is appropriate for this lesson, as the goal is to build on and refine this language over the course of the next few lessons until students have a more precise notion of what it means for a picture or figure to be a scaled copy. 1.2: Scaling F (10 minutes) This task enables students to describe more precisely the characteristics of scaled copies and to refine the meaning of the term. Students observe copies of a line drawing on a grid and notice how the lengths of line segments and the angles formed by them compare to those in the original drawing. Students engage in MP7 in multiple ways in this task. Identifying distinguishing features of the scaled copies means finding similarities and differences in the shapes. In addition, the fact that corresponding parts increase by the same scale factor is a vital structural property of scaled copies. For the first question, expect students to explain their choices of scaled copies in intuitive, qualitative terms. For the second question, students should begin to distinguish scaled and unscaled copies in more specific and quantifiable ways. If it does not occur to students to look at lengths of segments, suggest they do so. As students work, monitor for students who notice the following aspects of the figures. Students are not expected to use these mathematical terms at this point, however. • The original drawing of the letter F and its scaled copies have equivalent width-to-height ratios. • We can use a scale factor (or a multiplier) to compare the lengths of different figures and see if they are scaled copies of the original. • The original figure and scaled copies have corresponding angles that have the same measure. Keep students in the same groups. Give them 3–4 minutes of quiet work time, and then 1–2 minutes to share their responses with their partner. Tell students that how they decide whether each of the seven drawings is a scaled copy may be very different than how their partner decides. Encourage students to listen carefully to each other’s approach and to be prepared to share their strategies. Use gestures to elicit from students the words “horizontal” and “vertical” and ask groups to agree internally on common terms to refer to the parts of the F (e.g., “horizontal stems”). Engagement: Internalize Self Regulation. Display sentence frames to support small group discussion. For example, “That could/couldn’t be true because…,” “We can agree that…,” and “Is there another way to say/do...?” Supports accessibility for: Social-emotional skills; Organization; Language Speaking: Math Language Routine 1 Stronger and Clearer Each Time. This is the first time Math Language Routine 1 is suggested as a support in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. Students meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions such as, “What did you mean by . . .?” and “Can you say that another way?” Finally, students write a second draft of their response reflecting ideas from partners, and improvements on their initial ideas. The purpose of this routine is to provide a structured and interactive opportunity for students to revise and refine their ideas through verbal and written means. Design Principle(s): Optimize output (for explanation) How It Happens: 1. Use this routine to provide students a structured opportunity to refine their explanations for the first question: “Identify all the drawings that are scaled copies of the original letter F drawing. Explain how you know.” Allow students 2–3 minutes to individually create first draft responses in writing. 2. Invite students to meet with 2–3 other partners for feedback. Instruct the speaker to begin by sharing their ideas without looking at their written draft, if possible. Provide the listener with these prompts for feedback that will help their partner strengthen their ideas and clarify their language: “What do you mean when you say….?”, “Can you describe that another way?”, “How do you know that _ is a scaled copy?”, “Could you justify that differently?” Be sure to have the partners switch roles. Allow 1–2 minutes to discuss. 3. Signal for students to move on to their next partner and repeat this structured meeting. 4. Close the partner conversations and invite students to revise and refine their writing in a second draft. Provide these sentence frames to help students organize their thoughts in a clear, precise way: “Drawing _ is a scaled copy of the original, and I know this because.…”, “When I look at the lengths, I notice that.…”, and “When I look at the angles, I notice that.…” Here is an example of a second draft: “Drawing 7 is a scaled copy of the original, and I know this because it is enlarged evenly in both the horizontal and vertical directions. It does not seem lopsided or stretched differently in one direction. When I look at the length of the top segment, it is 3 times as large as the original one, and the other segments do the same thing. Also, when I look at the angles, I notice that they are all right angles in both the original and scaled copy.” 5. If time allows, have students compare their first and second drafts. If not, have the students move on by working on the following problems. Student Facing Here is an original drawing of the letter F and some other drawings. 1. Identify all the drawings that are scaled copies of the original letter F drawing. Explain how you know. 2. Examine all the scaled copies more closely, specifically, the lengths of each part of the letter F. How do they compare to the original? What do you notice? 3. On the grid, draw a different scaled copy of the original letter F. Keep students in the same groups. Give them 3–4 minutes of quiet work time, and then 1–2 minutes to share their responses with their partner. Tell students that how they decide whether each of the seven drawings is a scaled copy may be very different than how their partner decides. Encourage students to listen carefully to each other’s approach and to be prepared to share their strategies. Use gestures to elicit from students the words “horizontal” and “vertical” and ask groups to agree internally on common terms to refer to the parts of the F (e.g., “horizontal stems”). Engagement: Internalize Self Regulation. Display sentence frames to support small group discussion. For example, “That could/couldn’t be true because…,” “We can agree that…,” and “Is there another way to say/do...?” Supports accessibility for: Social-emotional skills; Organization; Language Speaking: Math Language Routine 1 Stronger and Clearer Each Time. This is the first time Math Language Routine 1 is suggested as a support in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. Students meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions such as, “What did you mean by . . .?” and “Can you say that another way?” Finally, students write a second draft of their response reflecting ideas from partners, and improvements on their initial ideas. The purpose of this routine is to provide a structured and interactive opportunity for students to revise and refine their ideas through verbal and written means. Design Principle(s): Optimize output (for explanation) How It Happens: 1. Use this routine to provide students a structured opportunity to refine their explanations for the first question: “Identify all the drawings that are scaled copies of the original letter F drawing. Explain how you know.” Allow students 2–3 minutes to individually create first draft responses in writing. 2. Invite students to meet with 2–3 other partners for feedback. Instruct the speaker to begin by sharing their ideas without looking at their written draft, if possible. Provide the listener with these prompts for feedback that will help their partner strengthen their ideas and clarify their language: “What do you mean when you say….?”, “Can you describe that another way?”, “How do you know that _ is a scaled copy?”, “Could you justify that differently?” Be sure to have the partners switch roles. Allow 1–2 minutes to discuss. 3. Signal for students to move on to their next partner and repeat this structured meeting. 4. Close the partner conversations and invite students to revise and refine their writing in a second draft. Provide these sentence frames to help students organize their thoughts in a clear, precise way: “Drawing _ is a scaled copy of the original, and I know this because.…”, “When I look at the lengths, I notice that.…”, and “When I look at the angles, I notice that.…” Here is an example of a second draft: “Drawing 7 is a scaled copy of the original, and I know this because it is enlarged evenly in both the horizontal and vertical directions. It does not seem lopsided or stretched differently in one direction. When I look at the length of the top segment, it is 3 times as large as the original one, and the other segments do the same thing. Also, when I look at the angles, I notice that they are all right angles in both the original and scaled copy.” 5. If time allows, have students compare their first and second drafts. If not, have the students move on by working on the following problems. Student Facing Here is an original drawing of the letter F and some other drawings. 1. Identify all the drawings that are scaled copies of the original letter F. Explain how you know. 2. Examine all the scaled copies more closely, specifically the lengths of each part of the letter F. How do they compare to the original? What do you notice? 3. On the grid, draw a different scaled copy of the original letter F. Anticipated Misconceptions Students may make decisions by “eyeballing” rather than observing side lengths and angles. Encourage them to look for quantifiable evidence and notice lengths and angles. Some may think vertices must land at intersections of grid lines (e.g., they may say Drawing 4 is not a scaled copy because the endpoints of the shorter horizontal segment are not on grid crossings). Address this during the whole-class discussion, after students have a chance to share their observations about segment lengths. Activity Synthesis Display the seven copies of the letter F for all to see. For each copy, ask students to indicate whether they think each one is a scaled copy of the original F. Record and display the results for all to see. For contested drawings, ask 1–2 students to briefly say why they ruled these out. Discuss the identified scaled and unscaled copies. • What features do the scaled copies have in common? (Be sure to invite students who were thinking along the lines of scale factors and angle measures to share.) • How do the other copies fail to show these features? (Sometimes lengths of sides in the copy use different multipliers for different sides. Sometimes the angles in the copy do not match the angles in the original.) If there is a misconception that scaled copies must have vertices on intersections of grid lines, use Drawing 1 (or a relevant drawing by a student) to discuss how that is not the case. Some students may not be familiar with words such as “twice,” “double,” or “triple.” Clarify the meanings by saying “two times as long” or “three times as long.” 1.3: Pairs of Scaled Polygons (15 minutes) In this activity, students hone their understanding of scaled copies by working with more complex figures. Students work with a partner to match pairs of polygons that are scaled copies. The polygons appear comparable to one another, so students need to look very closely at all side lengths of the polygons to tell if they are scaled copies. As students confer with one another, notice how they go about looking for a match. Monitor for students who use precise language (MP6) to articulate their reasoning (e.g., “The top side of A is half the length of the top side of G, but the vertical sides of A are a third of the lengths of those in G.”). You will need the Pairs of Scaled Polygons blackline master for this activity. Demonstrate how to set up and do the matching activity. Choose a student to be your partner. Mix up the cards and place them face-up. Tell them that each polygon has one and only one match (i.e., for each polygon, there is one and only one scaled copy of the polygon). Select two cards and then explain to your partner why you think the cards do or do not match. Demonstrate productive ways to agree or disagree (e.g., by explaining your mathematical thinking, asking clarifying questions, etc.). Arrange students in groups of 2. Give each group a set of 10 slips cut from the blackline master. Encourage students to refer to a running list of statements and diagrams to refine their language and explanations of how they know one figure is a scaled copy of the other. Representation: Internalize Comprehension. Provide a range of examples and counterexamples. During the demonstration of how to set up and do the matching activity, select two cards that do not match, and invite students to come up with a shared justification. Supports accessibility for: Conceptual processing Speaking: MLR8 Discussion Supports. Use this routine to support small-group discussion. As students take turns finding a match of two polygons that are scaled copies of one another and explaining their reasoning to their partner, display the following sentence frames for all to see: “____ matches ____ because . . .” and “I noticed ___ , so I matched . . . ." Encourage students to challenge each other when they disagree with the sentence frames “I agree because . . .”, and “I disagree because . . . ." This will help students clarify their reasoning about scaled copies of polygons. Design Principle(s): Support sense-making; Optimize output (for explanation) Student Facing Your teacher will give you a set of cards that have polygons drawn on a grid. Mix up the cards and place them all face up. 1. Take turns with your partner to match a pair of polygons that are scaled copies of one another. 1. For each match you find, explain to your partner how you know it’s a match. 2. For each match your partner finds, listen carefully to their explanation, and if you disagree, explain your thinking. 2. When you agree on all of the matches, check your answers with the answer key. If there are any errors, discuss why and revise your matches. 3. Select one pair of polygons to examine further. Use the grid below to produce both polygons. Explain or show how you know that one polygon is a scaled copy of the other. Demonstrate how to set up and do the matching activity. Choose a student to be your partner. Mix up the cards and place them face-up. Tell them that each polygon has one and only one match (i.e., for each polygon, there is one and only one scaled copy of the polygon). Select two cards and then explain to your partner why you think the cards do or do not match. Demonstrate productive ways to agree or disagree (e.g., by explaining your mathematical thinking, asking clarifying questions, etc.). Arrange students in groups of 2. Give each group a set of 10 slips cut from the blackline master. Encourage students to refer to a running list of statements and diagrams to refine their language and explanations of how they know one figure is a scaled copy of the other. Representation: Internalize Comprehension. Provide a range of examples and counterexamples. During the demonstration of how to set up and do the matching activity, select two cards that do not match, and invite students to come up with a shared justification. Supports accessibility for: Conceptual processing Speaking: MLR8 Discussion Supports. Use this routine to support small-group discussion. As students take turns finding a match of two polygons that are scaled copies of one another and explaining their reasoning to their partner, display the following sentence frames for all to see: “____ matches ____ because . . .” and “I noticed ___ , so I matched . . . ." Encourage students to challenge each other when they disagree with the sentence frames “I agree because . . .”, and “I disagree because . . . ." This will help students clarify their reasoning about scaled copies of polygons. Design Principle(s): Support sense-making; Optimize output (for explanation) Student Facing Your teacher will give you a set of cards that have polygons drawn on a grid. Mix up the cards and place them all face up. 1. Take turns with your partner to match a pair of polygons that are scaled copies of one another. 1. For each match you find, explain to your partner how you know it’s a match. 2. For each match your partner finds, listen carefully to their explanation, and if you disagree, explain your thinking. 2. When you agree on all of the matches, check your answers with the answer key. If there are any errors, discuss why and revise your matches. 3. Select one pair of polygons to examine further. Draw both polygons on the grid. Explain or show how you know that one polygon is a scaled copy of the other. Student Facing Are you ready for more? Is it possible to draw a polygon that is a scaled copy of both Polygon A and Polygon B? Either draw such a polygon, or explain how you know this is impossible. Anticipated Misconceptions Some students may think a figure has more than one match. Remind them that there is only one scaled copy for each polygon and ask them to recheck all the side lengths. Some students may think that vertices must land at intersections of grid lines and conclude that, e.g., G cannot be a copy of F because not all vertices on F are on such intersections. Ask them to consider how a 1-unit-long segment would change if scaled to be half its original size. Where must one or both of its vertices land? Activity Synthesis The purpose of this discussion is to draw out concrete methods for deciding whether or not two polygons are scaled copies of one another, and in particular, to understand that just eyeballing to see whether they look roughly the same is not enough to determine that they are scaled copies. Display the image of all the polygons. Ask students to share their pairings and guide a discussion about how students went about finding the scaled copies. Ask questions such as: • When you look at another polygon, what exactly did you check or look for? (General shape, side lengths) • How many sides did you compare before you decided that the polygon was or was not a scaled copy? (Two sides can be enough to tell that polygons are not scaled copies; all sides are needed to make sure a polygon is a scaled copy.) • Did anyone check the angles of the polygons? Why or why not? (No; the sides of the polygons all follow grid lines.) If students do not agree about some pairings after the discussion, ask the groups to explain their case and discuss which of the pairings is correct. Highlight the use of quantitative descriptors such as “half as long” or “three times as long” in the discussion. Ensure that students see that when a figure is a scaled copy of another, all of its segments are the same number of times as long as the corresponding segments in the other. Lesson Synthesis In this lesson, we encountered copies of a figure that are both scaled and not scaled. We saw different versions of a portrait of a student and of a letter F, as well as a variety of polygons that had some things in common. In each case, we decided that some were scaled copies of one another and some were not. Consider asking students: • What is a scaled copy? • What are some characteristics of scaled copies? How are they different from figures that are not scaled copies? • What specific information did you look for when determining if something was a scaled copy of an original? While initial answers need not be particularly precise at this stage of the unit (for example, “scaled copies look the same but are a different size”), guide the discussion toward making careful statements that one could test. The lengths of segments in a scaled copy are related to the lengths in the original figure in a consistent way. For instance, if a segment in a scaled copy is half the length of its counterpart in the original, then all other segments in the copy are also half the length of their original counterparts. We might say, “All the segments are twice as long,” or “All the segments are one-third the size of the segments in the original.” 1.4: Cool-down - Scaling L (5 minutes) Student Facing What is a scaled copy of a figure? Let’s look at some examples. The second and third drawings are both scaled copies of the original Y. However, here, the second and third drawings are not scaled copies of the original W. The second drawing is spread out (wider and shorter). The third drawing is squished in (narrower, but the same height). We will learn more about what it means for one figure to be a scaled copy of another in upcoming lessons.	{"url":"https://im-beta.kendallhunt.com/MS/teachers/2/1/1/index.html","timestamp":"2024-11-04T18:47:07Z","content_type":"text/html","content_length":"167262","record_id":"<urn:uuid:31c3a2cc-3224-4133-bc2e-f474cf880d33>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00458.warc.gz"}
Anomalous electrical and frictionless flow conductance in complex networks We study transport properties such as electrical and frictionless flow conductance on scale-free and Erdo{double acute}s-Rényi networks. We consider the conductance G between two arbitrarily chosen nodes where each link has the same unit resistance. Our theoretical analysis for scale-free networks predicts a broad range of values of G, with a power-law tail distribution Φ[SF] (G) ∼ G^- g[G], where g[G] = 2 λ - 1, where λ is the decay exponent for the scale-free network degree distribution. We confirm our predictions by simulations of scale-free networks solving the Kirchhoff equations for the conductance between a pair of nodes. The power-law tail in Φ[SF] (G) leads to large values of G, thereby significantly improving the transport in scale-free networks, compared to Erdo{double acute}s-Rényi networks where the tail of the conductivity distribution decays exponentially. Based on a simple physical 'transport backbone' picture we suggest that the conductances of scale-free and Erdo{double acute}s-Rényi networks can be approximated by c k[A] k[B] / (k[A] + k[B]) for any pair of nodes A and B with degrees k[A] and k[B]. Thus, a single quantity c, which depends on the average degree over(k, -) of the network, characterizes transport on both scale-free and Erdo{double acute}s-Rényi networks. We determine that c tends to 1 for increasing over(k, -), and it is larger for scale-free networks. We compare the electrical results with a model for frictionless transport, where conductance is defined as the number of link-independent paths between A and B, and find that a similar picture holds. The effects of distance on the value of conductance are considered for both models, and some differences emerge. Finally, we use a recent data set for the AS (autonomous system) level of the Internet and confirm that our results are valid in this real-world example. Bibliographical note Funding Information: We thank the Office of Naval Research, the Israel Science Foundation, the European NEST project DYSONET, and the Israel Internet Association for financial support, and L. Braunstein, R. Cohen, G. Li, E. Perlsman, G. Paul, S. Sreenivasan, T. Tanizawa, and Z. Wu for discussions. We thank the Office of Naval Research, the Israel Science Foundation, the European NEST project DYSONET, and the Israel Internet Association for financial support, and L. Braunstein, R. Cohen, G. Li, E. Perlsman, G. Paul, S. Sreenivasan, T. Tanizawa, and Z. Wu for discussions. Funders Funder number European NEST Israel Internet Association Office of Naval Research Israel Science Foundation • Complex networks • Conductance • Diffusion • Scaling • Transport Dive into the research topics of 'Anomalous electrical and frictionless flow conductance in complex networks'. Together they form a unique fingerprint.	{"url":"https://cris.biu.ac.il/en/publications/anomalous-electrical-and-frictionless-flow-conductance-in-complex","timestamp":"2024-11-07T22:20:23Z","content_type":"text/html","content_length":"61588","record_id":"<urn:uuid:f57cb888-d40f-47c6-a354-2de4ae57edff>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00693.warc.gz"}
Execution time limit is 2 seconds Runtime memory usage limit is 64 megabytes Let's define a circulation as a flow with a total magnitude of 0. You are given a directed graph with specified lower and upper capacity limits. For any pair of vertices i and j, the flow f_ij must satisfy the condition l_ij ≤ f_ij ≤ c_ij, where l_ij is the lower bound and c_ij is the upper bound. Your task is to determine if there exists a circulation in the graph that meets these constraints. The first line of the input contains two integers, N and M (1 ≤ N ≤ 200, 0 ≤ M ≤ 15000). This is followed by M lines, each describing an edge in the graph. Each line consists of four positive integers: i, j, l_ij, and c_ij (0 ≤ l_ij ≤ c_ij ≤ 10^5). These integers indicate an edge from vertex i to vertex j with a lower bound l_ij and an upper bound c_ij. It is guaranteed that if there is an edge from i to j, there will not be an edge from j to i. If no circulation satisfies the constraints, output NO. If a valid circulation exists, output YES on the first line. Then, for each of the M edges, output the flow magnitude on a separate line, corresponding to the order of edges in the input. Remember, for each edge from i to j, the flow must satisfy l_ij ≤ f_ij ≤ c_ij. Submissions 483 Acceptance rate 22%	{"url":"https://basecamp.eolymp.com/en/problems/5446","timestamp":"2024-11-12T16:16:00Z","content_type":"text/html","content_length":"264464","record_id":"<urn:uuid:d0f05a26-46c9-4e84-b41c-c3c5739b9966>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00336.warc.gz"}
9 MG to ML - MG to ML 9 MG to ML 9 MG to ML – The mg to mL converter helps you convert the weight of a liquid that has the density of water to a volume – a conversion from units of mg to units of ml. As well as an mg to mL conversion, you can also use the calculator backward to execute an mL to mg conversion. Milligram (mg): A milligram is a unit of measurement of mass which is equivalent to 1/1000 of a gram. mg = 1000 * mL Milliliter (mL): A milliliter is a unit of measurement of liquid volume or capacity in the metric system. 1 milliliter equals one-thousandth of a liter or 0.001 liters. ml = mg/1000 9 MG to ML Formula: Conversion of mg to ml is straightforward. Since 1 Milligram is equal to 0.001 milliliters, multiply the entered milligram by 0.001 to get the result. For example, when the given number of milligrams is 15, then the conversion of milligrams to milliliters is 15 x 0.001 is 0.015 mL. MG to ML Calculator Convert 8 mg to ml with the help of a calculator Mg To Ml Conversion 9 Milligram is equal to 0.009 milliliters. Q: How many Milligrams in a Milliliter? The answer is 1,000 Milliliter Q: How do you convert 9 Milligram (mg) to Milliliter (ml)? 9 Milligram is equal to 9.0e-03 Milliliter. Formula to convert 9 mg to ml is 9 / 1000 Q: How many Milligrams in 9 Milliliters? The answer is 9,000 Milligrams	{"url":"https://mgtoml.com/9-mg-to-ml/","timestamp":"2024-11-03T00:50:54Z","content_type":"text/html","content_length":"150509","record_id":"<urn:uuid:16776051-45d2-4db7-99d3-b260654b0dcf>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00417.warc.gz"}
Method of separately determining electrophysical properties of bulky conductor A method of determining the electrical conductivity and the magnetic permeability of bulky conductors such as plates or cylinders from field intensity measurements is described, the gist being to explicate each of these two parameters from the readings of electric field intensity and magnetic field intensity at the conductor surface or in a plane of symmetry. The method is based theoretically on Fourier-Bessel integral transformation of the differential field equations for the corresponding conductor medium. The integral relations are then mapped so as to yield the necessary relations between two field functionals and the space distributions of the electromagnetic field components. Further explication and numerical integration yield the ratio of calculated to measured conductivity and permeability. The procedure is demonstrated on a numerical example of a plane-parellel field above a heavy plate. USSR Rept Electron Elec Eng JPRS UEE Pub Date: January 1985 □ Electric Field Strength; □ Electrical Conductivity Meters; □ Magnetic Flux; □ Magnetic Permeability; □ Thick Plates; □ Differential Equations; □ Integral Transformations; □ Numerical Integration; □ Electronics and Electrical Engineering	{"url":"https://ui.adsabs.harvard.edu/abs/1985RpEEE....Q..36S/abstract","timestamp":"2024-11-12T19:42:32Z","content_type":"text/html","content_length":"35505","record_id":"<urn:uuid:4dbb22c6-9857-4071-aa1b-bee45b29da18>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00020.warc.gz"}
Quantum Computing Quantum computing harnesses the phenomena of quantum mechanics to deliver a huge leap forward in computation to solve certain problems. Before we dive into the details of Quantum Computing, let's understand the basic terms What is Quantum? The quantum in "quantum computing" refers to the quantum mechanics that the system uses to calculate outputs. In physics, a quantum is the smallest possible discrete unit of any physical property. It usually refers to properties of atomic or subatomic particles, such as electrons, neutrinos, and photons. What is a qubit? A qubit is the basic unit of information in quantum computing. Qubits play a similar role in quantum computing as bits play in classical computing, but they behave very differently. Classical bits are binary and can hold only a position of 0 or 1, but qubits can hold a superposition of all possible states. Quantum Computing Quantum computing is an area of computing focused on developing computer technology based on the principles of quantum theory (which explains the behavior of energy and material on the atomic and subatomic levels). Computers used today can only encode information in bits that take the value of 1 or 0 restricting their ability. Working of Quantum Computing In quantum computing, operations instead use the quantum state of an object to produce what's known as a qubit. These states are the undefined properties of an object before they've been detected, such as the spin of an electron or the polarization of a photon. Rather than having a clear position, unmeasured quantum states occur in a mixed 'superposition', not unlike a coin spinning through the air before it lands in your hand. These superpositions can be entangled with those of other objects, meaning their outcomes will be mathematically related even if we don't know yet what they are. The complex mathematics behind these unsettled states of entangled 'spinning coins' can be plugged into special algorithms to make short work of problems that would take a classical computer a long time to work out... if they could ever calculate them at all. Such algorithms would be useful in solving complex mathematical problems, producing hard-to-break security codes, or predicting multiple particle interactions in chemical reactions. Quantum Principles Qubits can represent numerous possible combinations of 1 and 0 at the same time. This ability to simultaneously be in multiple states is called superposition. To put qubits into superposition, researchers manipulate them using precision lasers or microwave beams. Thanks to this counterintuitive phenomenon, a quantum computer with several qubits in superposition can crunch through a vast number of potential outcomes simultaneously. The final result of a calculation emerges only once the qubits are measured, which immediately causes their quantum state to “collapse” to either 1 or 0. It is a phenomenon in which quantum entities are created and/or manipulated such that none of them can be described without r eferencing the others. Individual identities are lost. This concept is exceedingly difficult to conceptualize when one considers how entanglement can persist over long distances. A measurement on one member of an entangled pair will immediately determine measurements on its partner, making it appear as if information can travel faster than the speed of light. This apparent action at a distance was so disturbing that even Einstein dubbed it “spooky”. Quantum Programming Quantum computing offers the ability to write programs in a completely new way. For example, a quantum computer could incorporate a programming sequence that would be along the lines of "take all the superpositions of all the prior computations." This would permit extremely fast ways of solving certain mathematical problems, such as the factorization of large numbers. The first quantum computing program appeared in 1994 by Peter Shor, who developed a quantum algorithm that could efficiently factorize large numbers. How will businesses use quantum computers? Quantum computers have four fundamental capabilities that differentiate them from today’s classical computers: quantum simulation, in which quantum computers model complex molecules; optimization (that is, solving multivariable problems with unprecedented speed); quantum artificial intelligence (AI), with better algorithms that could transform machine learning across industries as diverse as pharma and automotive; and prime factorization, which could revolutionize encryption. 1. Cut development time for chemicals and pharmaceuticals with simulations: Scientists looking to develop new drugs and substances often need to examine the exact structure of a molecule to determine its properties and understand how it might interact with other molecules. Unfortunately, even relatively small molecules are extremely difficult to model accurately using classical computers, since each atom interacts in complex ways with other atoms. 2. Solve optimization problems with unprecedented speed: Across every industry, many complex business problems involve a host of variables. Where should I place robots on the factory floor? What’s the shortest route for my delivery truck? What’s the most efficient way to deploy cars, motorcycles, and scooters to create a transportation network that meets user demand? How can I optimize the performance and risk of a financial portfolio? These are just three of the many examples that business leaders confront. 3. Accelerate autonomous vehicles with quantum AI: It’s possible that quantum computers could speed the arrival of self-driving vehicles. At Ford, GM, Volkswagen, and other car manufacturers, and at a host of start-ups in the new mobility sector, engineers are running hours upon hours of video, image, and lidar data through complex neural networks. 4. Transform cybersecurity: Quantum computing poses a serious threat to the cybersecurity systems relied on by virtually every company. Most of today’s online account passwords and secure transactions and communications are protected through encryption algorithms such as RSA or SSL/TLS. Conclusions and Outlook Quantum computers have the potential to revolutionize computation by making certain types of classically intractable problems solvable. While no quantum computer is yet sophisticated enough to carry out calculations that a classical computer can't, great progress is underway. A few large companies and small start-ups now have to function non-error-corrected quantum computers composed of several tens of qubits, and some of these are even accessible to the public through the cloud. Additionally, quantum simulators are making strides in fields varying from molecular energetics to many-body As small systems come online a field focused on near-term applications of quantum computers is starting to burgeon. This progress may make it possible to actualize some of the benefits and insights of quantum computation long before the quest for a large-scale, error-corrected quantum computer is complete. Read more blogs here	{"url":"https://www.meetri.in/blogs/quantum-computing-qubit-superposition-speed.html","timestamp":"2024-11-14T14:37:09Z","content_type":"text/html","content_length":"58200","record_id":"<urn:uuid:8d20f494-dc09-49b5-ac64-c6bf79cdeef8>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00865.warc.gz"}
Method of detecting a deflated tyre on a vehicle - Patent 0489562 This invention relates to a method of detecting a deflated tyre on a vehicle suitable for cars, trucks or the like, and particularly to the system disclosed in for example French Patent Publication 2568519 and European Patent Publication No 291217. These Patents propose using wheel speed signals from the vehicle wheels, such as for example the signals from anti-lock braking systems which are multi-pulse signals or single-pulse signals for each rotation of each wheel. They compare the speed derived signals of the wheels in various ways to try to avoid false signals due to factors such as vehicle cornering, braking, accelerating, uneven and changing load etc. French Patent Publication 2568519 monitored the sums of the speeds of the diagonally opposed pairs of wheels for a long time or distance period so that it averaged out some of these errors. The result however was that the device operated very slowly taking many kilometres to sense pressure loss. European Patent Publication No 291 217 substantially improved this situation by calculating the lateral and longitudinal accelerations of the vehicle using the same four-wheel speed signals and setting fixed limits above which the detection system was inhibited to avoid false signals due to cornering and acceleration. This system also suggested a correction for high vehicle speeds and for the first time introduced the ability to calibrate the system to suit the particular vehicle, and indeed the actual tyres fitted which themselves could have different properties from one another in respect of rolling radius. The calibration was carried out in straight line running, however, so whilst some vehicle conditions were allowed for the problems of detection during high speed running, cornering and braking under modern road conditions and particularly in higher performance vehicles could not be allowed for. The resultant system still needed to be inhibited for detection in a fair percentage of the vehicle running time. All attempts to improve this position resulted in loss of sensitivity of the system and/or loss of ability to sense which wheel or wheels was deflated if false signals were not to occur and made application of the system less effective. An object of the present invention is to provide, in a system of the above type, the ability to sense deflations during higher levels of vehicle acceleration both laterally and longitudinally without false signals. According to one aspect of the present invention a method of detecting a deflated tyre on a vehicle by comparing the rolling radii of the tyres by means of comparing angular velocity speed signals from wheel speed sensors one at each wheel characterised by, before the comparison of the signals is carried out, calculating corrected wheel speed signals for each of the second, third and fourth wheels giving corrections for a set of factors comprising vehicle speed, lateral acceleration and longitudinal (fore/aft) acceleration, the said corrections each comprising a constant for the factor concerned X the respective factor, the set of constants for each wheel being derived by taking the vehicle through a range of speeds, lateral and fore/aft accelerations and using multiple regression techniques and the respective factors being calculated from the set of uncorrected wheel speed signals so that comparison of the wheel speeds can be made without false signals from tyre deflections caused by speed, lateral or fore/aft acceleration induced tyre deflections. Preferably in addition the corrections comprise a further constant x the square of the lateral acceleration; and/or a further constant x fore/aft acceleration x lateral acceleration; and/or a further constant x speed x lateral acceleration; and/or a further constant x speed x fore/aft acceleration; and/or a further constant x speed x lateral acceleration x fore and aft acceleration; and/or a further constant x speed squared and/or a further fixed constant. Having carried out the corrections to the speed signals various comparisons between the speeds of the respective wheels can then be made depending upon the particular choice of ratios made. The speed signals themselves may be multi-pulse signals such as are typical from ABS-type wheel speed generators or may comprise single-pulses from a wheel speed signal generator which gives a pulse for each revolution of the wheel. The speed signals may therefore be digital pulse signals or time periods timing the time for one rotation of each wheel and in that case a correction may be made to give the four wheel speeds at the same instant in time such as is described in our copending UK Patent Application No 9002925.7 dated 9 February 1990. The comparison of the wheel speed signals preferably comprises subtracting the sum of the signals from one pair of diagonally opposite wheels from the sum of the signals of the other pair of diagonally opposite wheels, sensing when the magnitude of the result is between 0.05% and 0.6% of the mean of the sums and when that magnitude is in said range operating a warning device to indicate a tyre is partially or completely deflated. In addition the comparison may comprise comparing the non-corrected signals from each of the four wheels in turn with the non-corrected signals for each of the other wheels, sensing when one of said signals is different from the average of all four signals by more than 0.1% and in the event of both this signal and the diagonals comparison being in the specified ranges then indicating that the tyre is partially or completely deflated. These signals may be corrected by a simple set of controls to allow for variations between the tyre by means of calibration carried out at a constant speed in a straight line. These later comparisons provide means of detecting which particular wheel of the set is deflated and therefore the provision of an indication to the driver as to which wheel is Further aspects of the present invention will become apparent from the following description by way of example only in conjunction with the attached diagrammatic drawings, in which: Figure 1 is a schematic diagrammatic drawing showing a deflation warning device for a car with four wheels. The apparatus shown in Figure 1 provides a deflation warning device for four wheels, 1, 2, 3, and 4, the wheels 1 and 2 being the front wheels and the wheels 3 and 4 the rear wheels of a car. Each wheel 1, 2, 3 and 4 has a wheel speed generating device associated with it. This may be of the toothed wheel type as used to provide a digital signal for electronic ABS equipment or merely the single-pulse type which generates a pulse one per wheel revolution. In this case the generator may be a single magnet attached to each wheel for rotation therewith and a stationary pickup mounted on the suspension. The signals from each wheel are carried through cables 5 to provide input 6, 7, 8 and 9 to a central processing unit 10. Four outputs from the central processing unit are connected to four warning indicators 12, 13, 14 and 15, one for each of the wheels respectively. The central processing unit 10 is basically a computer and in the case where the vehicle already has an ABS-system fitted may be the same computer as the ABS-system. Alternatively a separate central processing unit may be provided. The central processing unit 10 monitors the various signals and compares them to determine whether or not it should give an outward signal to indicate that any tyre on the vehicle is deflated. The central processing unit 10 can calculate substantially what the vehicle is doing using the four wheel speed signals. Firstly it can calculate the vehicle speed at any instant using either a single wheel as a reference or all four and calculating the mean. Secondly it can calculate the apparent longitudinal acceleration of the vehicle by comparing the angular velocity signals from the front and rear pairs of wheels with the forward speed calculated from the mean of the angular velocities of all four wheels. It can also calculate the apparent lateral acceleration of the vehicle comparing the angular velocity signals for the wheels on each side of the vehicle and then comparing them with the forward speed calculated from the mean of the angular velocities of all four wheels. Thus the central processing unit 10 can calculate substantially accurately what the vehicle is physically doing which allows it to then use a particular formula which will be described below to correct the wheel speed signals for three of the wheels allowing for what the vehicle is doing. Having obtained the four corrected wheel speed signals C1, C2, C3 and C4 the system can then calculate an error signal dT by comparing the angular velocities of the wheels according to the formula This error or dT signal is monitored and the processing unit senses and indicates a deflation if the signal is greater than 0.05% and less than 0.6%. The unit carries out this determination by looking at the difference between each wheel's non-corrected angular velocity in turn and the average speed of the four wheels using non-corrected speeds C1, C2, C3 and C4. If the difference between any one wheel and the average is more than 0.1% a second signal is generated to indicate which wheel is partially or substantially deflated. This check may be performed using speed signals corrected to allow for tyre differences in the set of tyres by means of simply correcting. This is done by running the vehicle in a straight line at a constant speed and deriving correction factors. As mentioned above this system detects whether or not a puncture exists using the corrected wheel speed C2, C3 and C4 corrected on the basis of C1 being itself correct. The correction in speeds is achieved by using a formula which comprises: C = A1 x speed² + A2 x speed + A3 x (lateral acceleration)² + A4 (lateral acceleration) + A5 (fore/aft acceleration) + A6 x speed x lateral acceleration + A7 x speed x fore and aft acceleration + A8 x lateral acceleration x fore and aft acceleration + A9 x speed x lateral acceleration x lateral fore and aft acceleration + A10 A1 to A10 are constants for the particular wheel concerned. The constants A1 to A10 are determined by a prior calibration for the vehicle and provide corrections for the wheel speed concerned to allow for changes in rolling radius caused by changes in weight on the particular wheel concerned by the effects of acceleration, braking, etc on the vehicle. The constants also correct for the particular vehicle concerned for differences due to tyre growth due to wheel speed. The constants are found by a practical method by means of using a calibration routine which comprises driving the vehicle through a full range of accelerations both longitudinally and laterally in both directions of left and right turns and covering all other possible vehicle use conditions. This can readily be achieved by driving the vehicle on a mixed road test and the central processing unit constantly monitors the effects on wheel speeds and records them. The entire top range results are then ignored to avoid later errors, i.e. the top 5 or 10% of acceleration figures. The central processing unit is then set into a multiple regression analysis procedure using any of the standard techniques to calculate the ten constants A1 to A10 which gives it the necessary correction system to make sure that wheel speeds are made independent of extraneous factors such as weight transfer in the vehicle and cornerinb and acceleration. It should be noted that it is not necessary to calibrate each vehicle in a particular type by this method and the central processing unit may be reprogrammed for that model of vehicle because it allows for the basic vehicle characteristics which are set by its body shape, centre of gravity position and suspension characteristics. In some circumstances similar calibration can be used for more than one type of vehicle without recalibrating but the basic principal of the invention is that it provides the ability to correct wheel speeds for all vehicle characteristics in use. 1. A method of detecting a deflated tyre on a vehicle by comparing the rolling radii of the tyres by means of comparing angular velocity speed signals from wheel speed sensors one at each wheel characterised by, before the comparison of the signals is carried out, calculating corrected wheel speed signals for each of the second, third and fourth wheels giving corrections for a set of factors comprising vehicle speed, lateral acceleration and longitudinal (fore/aft) acceleration, the said corrections each comprising a constant for the factor concerned times the respective factor, the set of constants for each wheel being derived by taking the vehicle through a range of speeds, lateral and fore/aft accelerations and using multiple regression techniques and the respective factors being calculated from the set of uncorrected wheel speed signals so that comparison of the wheel speeds can be made without false signals from tyre deflections caused by speed, lateral or fore/aft acceleration induced tyre deflections. 2. A method according to Claim 1 characterised in that the corrections comprise a further constant times the square of the lateral acceleration. 3. A method according to Claim 1 or 2 characterised by a further constant times fore/aft acceleration times lateral acceleration. 4. A method according to any one of Claims 1, 2 or 3 characterised by a further constant times speed times lateral acceleration. 5. A method according to any one of Claims 1, 2, 3 or 4 characterised by a further constant times speed times fore/aft acceleration. 6. A method according to any one of Claims 1, 2, 3, 4 or 5 characterised by a further constant times speed times lateral acceleration times fore and aft acceleration. 7. A method according to any one of Claims 1 to 6 characterised by a further constant times speed squared. 8. A method according to any one of Claims 1 to 7 characterised by a further fixed constant. 9. A method according to any one of Claims 1 to 8 characterised by a comparison of the corrected wheel speed signals comprising subtracting the sums of the signals from one pair of diagonally opposite wheels from the sum of the signals from the other pair of diagonally opposite wheels, sensing when the magnitude of the result is between 0.05% and 0.6% for the mean of the sums and when the magnitude is in said range operating a warning device to indicate the tyre is partially or completely deflated. 10. A method according to Claim 9 characterised by additionally comparing the non-corrected signals from each of the four wheels in turn with the non-corrected signals for each of the other wheels, sensing when one of said signals is different from the average of all four signals by more than 0.1% and in the event of both said signals being present indicating that the tyre is partially of completely deflated. 11. A method according to Claim 9 characterised in that the signals are corrected relative to one another based on constants derived from straight line running of the vehicle at a single speed.	{"url":"https://data.epo.org/publication-server/rest/v1.2/publication-dates/19920610/patents/EP0489562NWA1/document.html","timestamp":"2024-11-04T00:32:55Z","content_type":"text/html","content_length":"37445","record_id":"<urn:uuid:dbe67500-4e38-494d-92e3-d4b3edb9da93>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00342.warc.gz"}
Ramsey-Sperner theory Let [n] denote the n-set {1, 2,..., n}, let k, l ≥ 1 be integers. Define f[l](n, k) as the minimum number f such that for every family F ⊆ 2^[n] with {divides}F{divides}>f, for every k-coloring of [n], there exists a chain A[1]{subset not double equals}···{subset not double equals}A[l+1] in F in which the set of added elements, A[l+1]-A[1], is monochromatic. We survey the known results for l = 1. Applying them we prove for any fixed l that there exists a constant φ{symbol}[l](k) such that as n→^∞ f[l](n,k)∼φ{symbol}[l](k)⌊ 1 2n⌋^n and φ{symbol}[l](k)∼ φk 4logk as k→^∞. Several problems remain open. Dive into the research topics of 'Ramsey-Sperner theory'. Together they form a unique fingerprint.	{"url":"https://experts.umn.edu/en/publications/ramsey-sperner-theory","timestamp":"2024-11-04T14:58:56Z","content_type":"text/html","content_length":"45905","record_id":"<urn:uuid:eddb20b3-b445-4ed4-a6f8-7a1f5ef93553>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00361.warc.gz"}
EXC 3442 Managerial Accounting APPLIES TO ACADEMIC YEAR 2015/2016 EXC 3442 Managerial Accounting Responsible for the course Pål Berthling-Hansen Department of Accounting - Auditing and Law According to study plan ECTS Credits Language of instruction Managerial accounting is a fundamental course within the managerial accounting discipline. Focus is on generating and analysing information for decision making, planning and control for internal management within a company. Learning outcome Acquired knowledge Students must be able to understand basic concepts within the management accounting discipline. ● Students must understand and be familiar with the main budgeting reports:profit and loss, balance sheet and liquidity. ● Be familiar with the purpose and role of management accounting. ● Be familiar with important aspects of the cost concept, including different ways to group costs, and how cost analyses must be adapted to particular corporate decision problems. ● Understand the opportunity cost concept and the importance of evaluating the total cost function. ● Understand the principles of cost-volume-profit analysis. ● Be able to allocate costs from budgets and financial reports to activities (departments) and between activities (departments). ● Understand the use of standard costing and variance analysis. ● Be familiar with the basic assumptions and weaknesses of traditional costing. ● Be able to identify decision-relevant revenues and costs. ● Understand the problems of transfer pricing, including how transfer prices can affect decision behaviour. ● Understand how pricing decisions affect profitability Acquired Skills Students must be able to evaluate specific decision situations and demonstrate correct use of relevant management accounting tools ● Students must be able to solve a budgeting exercise and obtain congruence between the profit and loss, the balance sheet and the liquidity budget. ● Students must be able to analyse the decision context in terms of for instance the relevant cost object, a short/long term decision horizon, capacity issues and cost stickiness. ● Students must also be capable of calculating and utilizing the opportunity cost in various decision situations. ● Be able to apply the correct type of calculation tool ● Understand the structures of absorption and variable costing, and be able to apply these models in various costing problems. ● Be able to discuss the effect on product costs of a traditional costing model, applying unit-based cost drivers only. ● Be able to compute various break-even points and evaluate sensitivity. ● Be able to compute and evaluate various transfer prices. ● Conduct analyses of pricing decisions Students must be capable of demonstrating an ability to critically reflect upon their own work, and on any specific assumptions underlying management accounting concepts utilized. Compulsory readingBooks: Drury, Colin. 2015. Management and cost accounting. 9th ed. Cengage Learning Recommended readingCourse outline ● Purposes and role of management accounting ● Budgeting ● Cost concepts and categories ● Cost behaviour and cost-volume-profit analysis ● Opportunity cost ● Job-order and process costing ● Absorption costing and variable costing ● Standard costing and variance analysis ● Relevant costs for decision making ● Pricing decisions and income analysis Computer-based tools A basic understanding of and ability to use Excel spreadsheets is helpful but not required. Learning process and workload The course will include/entail a combination of lectures, plenary tutorials where solutions to exercises will be explained and one hand-in exercise. The following is an indication of the time required: │Activity │Hours│ │Lectures │ 36│ │Plenary tutorials where exercises will be explained │ 9│ │Preparation for lectures and plenary tutorials (approximately 1,5 hours per lecture hour) │ 95│ │Preparation for one mid-term assignment │ 10│ │Preparation for the examination │ 50│ │Total recommended use of hours │ 200│ Required work (mandatory mini-exercises) There are 8 mandatory exercises during the semester of which students must get approved 5.. The mini-exercises are short, limited exercises that the students should be able to answer in about 1,5 hours if they have followed the recommended work schedule. If the students are not prepared, more time must be allowed. The mini-exercises are to be submitted through It's learning. A minimum level of performance will be demanded for the exercises to be approved (e.g. a minimum number of answers must be correctly answered). Further information will be given in the lectures and through It's learning. The students will be allowed three attempts before the deadline for the test. Information about the time period for the tests to be taken, will be given in the lectures and through It's learning. Feedback to the students during the semester will be given in the following ways: 1. During the lectures the students will be told which assignments are to be completed for the next lecture. The lecturer will review some of these assignments in class. The feedback will consist of the students comparing their solutions with the one that is explained by the lecturer. 2. Feedback on achieved score on the mini-exercises will be given automatically through It's learning. In addition a recommended solution will be made available. Coursework requirements The students must get approved five of the eight mini-exercises in order to take the examination. A four-hour individual written examination concludes the course. Examination code(s) EXC 34421 - Written exam, counts for 100% of final grade in EXC 3442 Managerial Accounting, 7,5 credits. Examination support materials All support materials + BI approved exam calculator. Examination support materials at written examinations are explained under examination information in the student portal @bi. Please note use of calculator and dictionary in the section on support materials ( Re-sit examination A re-sit examination is offered every term. Students that have not got approved five of the mandatory eight mini-exercises must retake the exercises at the next scheduled course and must pass five of the eight submitted mini-exercises. Students that have not passed the written examination or who wish to improve their grade must retake the exam in connection with the next scheduled examination. 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Year 9 algebra sample year 9 algebra sample Related topics: yr 9 sats test online revision complex equation in matlab what are the basic rules of graphing an equation or an inequality? Free 8th Grade Pre Algebra Worksheets how to solve systems of equations on a ti 89 equation simplification free math class online 8grade pre-algebra +general patterns with specific cases a non-measurable set in (0,1] java program number divisible by ti-89 quadratic function pre algebra 6th grade worksheets algecbra two problem solving suggested curriculum guide for grade 4 math Author Message FFEndaneseo Posted: Tuesday 13th of Oct 09:12 Hello all, I have a very important test coming up in math soon and I would really appreciate if any of you can help me solve some questions in year 9 algebra sample. I am ok in math otherwise but problems in radical expressions baffle me and I am at a loss. It would be wonderful if you can let me know of a reasonably priced software that I can use? Back to top ameich Posted: Tuesday 13th of Oct 12:09 Once upon a time those were the only ways out. But thanks to technology now, we have something known as the Algebrator! It’s quite easy to use, something which even a total newbie would enjoy working on and what’s even better is that it would solve all your questions and also explain the steps it took to reach that answer! Isn’t that just great ? Well I think it is, and I’m sure after using it trying it, you’ll be on the same page as me. From: Prague, Czech Republic Back to top cufBlui Posted: Tuesday 13th of Oct 21:19 Hello, just a month ago, I was stuck in a similar situation. I had even considered the option of leaving math and selecting some other subject. A friend of mine told me to give one last try and gave me a copy of Algebrator. I was at ease with it within an hour. My ranks have really improved within the last one month. From: Scotland Back to top Bet Posted: Thursday 15th of Oct 14:03 A truly piece of math software is Algebrator. Even I faced similar difficulties while solving radical inequalities, function range and percentages. Just by typing in the problem from homework and clicking on Solve – and step by step solution to my algebra homework would be ready. I have used it through several algebra classes - Pre Algebra, Remedial Algebra and Algebra 2. I highly recommend the program. From: kµlt øƒ Back to top svalnase Posted: Saturday 17th of Oct 08:39 To begin with, thanks for replying guys ! I’m interested in this program. Can you please tell me how to order this software? Can we order it through the web, or do we buy it from some retail store? From: Fort Wayne, Indiana Back to top cmithy_dnl Posted: Saturday 17th of Oct 17:09 There you go https://softmath.com/algebra-features.html. Back to top	{"url":"https://softmath.com/algebra-software/exponential-equations/year-9-algebra-sample.html","timestamp":"2024-11-11T04:35:41Z","content_type":"text/html","content_length":"43300","record_id":"<urn:uuid:7b6aff95-6b49-4974-a19b-a7c47e784f61>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00155.warc.gz"}
Convergence of incentive-driven dynamics in fisher markets In both general equilibrium theory and game theory, the dominant mathematical models rest on a fully rational solution concept in which every player's action is a best-response to the actions of the other players. In both theories there is less agreement on suitable out- of-equilibrium modeling, but one attractive approach is the level k model in which a level 0 player adopts a very simple response to current conditions, a level 1 player best-responds to a model in which others take level 0 actions, and so forth. (This is analogous to k-ply exploration of game trees in AI, and to receding-horizon control in control theory.) If players have deterministic mental models with this kind of finite-level response, there is obviously no way their mental models can all be consistent. Nevertheless, there is experimental evidence that people act this way in many situations, motivating the question of what the dynamics of such interactions lead to. We address the problem of out-of-equilibrium price dynamics in the setting of Fisher markets. We develop a general framework in which sellers have (a) a set of atomic price update rules which are simple responses to a price vector; (b) a belief-formation procedure that simulates actions of other sellers (themselves using the atomic price updates) to some finite horizon in the future. In this framework, sellers use an atomic price up- date rule to respond to a price vector they generate with the belief formation procedure. The framework is general and allows sellers to have inconsistent and time- varying beliefs about each other. Under certain assumptions on the atomic update rules, we show that despite the inconsistent and time-varying nature of beliefs, the market converges to a unique equilibrium. (If the price updates are driven by weak-gross substitutes demands, this is the same equilibrium point predicted by those demands.) This result holds for both synchronous and asynchronous discrete-time updates. Moreover, the result is computationally feasible in the sense that the convergence rate is linear, i.e., the distance to equilibrium decays exponentially fast. To the best of our knowledge, this is the first result that demonstrates, in Fisher markets, convergence at any rate for dynamics driven by a plausible model of seller incentives. We then specialize our results to Fisher markets with elastic demands (a further special case corresponds to demand generated by buyers with constant elasticity of substitution (CES) utilities, in the weak gross substitutes (WGS) regime) and show that the atomic update rule in which a seller uses the best-response (=profit- maximizing) update given the prices of all other sellers, satisfies the assumptions required on atomic price up- date rules in our framework. We can even characterize the convergence rate (as a function of elasticity parameters of the demand function). Our results apply also to settings where, to the best of our knowledge, there exists no previous demonstration of efficient convergence of any discrete dynamic of price updates. Even for the simple case of (level 0) best- response dynamics, our result is the first to demonstrate a linear rate of convergence. Publication series Name Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms Volume 0 Conference 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 Country/Territory Spain City Barcelona Period 16/01/17 → 19/01/17 Bibliographical note Publisher Copyright: Copyright © by SIAM. Dive into the research topics of 'Convergence of incentive-driven dynamics in fisher markets'. Together they form a unique fingerprint.	{"url":"https://cris.huji.ac.il/en/publications/convergence-of-incentive-driven-dynamics-in-fisher-markets-13","timestamp":"2024-11-05T22:00:29Z","content_type":"text/html","content_length":"59257","record_id":"<urn:uuid:eb97d281-8e5b-468b-9f15-34b9921bbb56>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00265.warc.gz"}
Caesar ciphers in Python One of the simplest ciphers is the Caesar cipher, also called the shift cipher. It works by shifting each letter in the alphabet n positions to the right, mapping it to a different letter. For example, using ‘rotation 13’, a is shifted 13 positions to the right, corresponding to the letter n. What happens to the last letters in the alphabet? For example, shifting z 13 positions to the right maps it outside of the 26 letters of the English alphabet. In this case we have to use a bit of math, namely modular arithmetic. Whenever the result of the shift is bigger than the size of the alphabet (called the modulus), it wraps around and starts counting from the beginning. So z rotation 13 would become letter number 13 (26+13-26), or the letter m. Mathematically this is expressed as: x ≡ (26 + 13) mod 26 ≡ 13 In Python, the modulus operator is designated as %. With these basics, how would we implement a Caesar cipher in Python? First, we need the letter number for each letter in the supplied string (let’s call it num), and then sum the rotation (rot) modulus the number of letters in the alphabet (26). Mathematically, this would be written as: x ≡ (num + rot) mod 26 For the first part, getting the letter number, we can either supply a table or, even simpler, get the ASCII value of the letter and subtract 97, since ASCII(‘a’) = 97. Remember, in computer science we almost always start counting at 0, which is why we subtract 97 and not 96 (97-97=0). Getting the ASCII value is simple in Python, we just use ord() on the letter. We can now shift the letter with the rotation value using modular arithmetic (to not get out of bounds of the alphabet), and finally change the resulting number back to ASCII using chr(). These 3 steps can be done in Python like this: num = ord(char) cypher = (num - 97 + rot) % 26 cypher = chr(cypher + 97) Doing this for each letter gives us our encrypted string, the cipher text, which we can send to our friends and allies content in the knowledge that nobody can break our state-of-the-art cipher(!). How then does the recipient decrypt the cipher? Apart from going to one of the countless online breaking tools or breaking it mathematically or using letter analysis, we can of course use Python again to do the decryption. Basically it’s the opposite of what we just did for encryption: num = ord(char) plain = (num - 97 - rot) % 26 plain = chr(plain + 97) As you can see, we now subtract the rotation instead of adding it like in the encryption phase. We again use modulus to wrap around the alphabet, this time when we go lower than 0, or a. Finally we can wrap the code in a loop so it works on the whole plainstring, import argparse so we can supply the string and rotation directly on the command line, and specify whether we want to encrypt or decrypt. The full script can be found on my GitHub repo or at the bottom of this post. Using the script to encrypt and decrypt ‘hello world’ A few things to note: the script only encrypts letters, not symbols, and is hardcoded for the English alphabet (26 letters). Also, all letters will be changed to lowercase before encryption. I hope you enjoyed this post. I intend to bring more crypto-related posts in the future since it is something I’m currently studying. [cc lang=”python”]#!/usr/bin/python # Takes a string and shift encrypts or decrypts it using the # supplied rotation import argparse # Add arguments parser = argparse.ArgumentParser(description= ”Encrypt/decrypt Caesar cyphers”) parser.add_argument(“mode”, help=”Encrypt or decrypt”, nargs=”?”, choices=(“encrypt”, “decrypt”)) parser.add_argument(“string”, help=”String to encrypt/decrypt”) parser.add_argument(“rot”, help=”Rotation to use”) args = parser.parse_args() # Definitions mode = args.mode string = args.string.lower() rot = int(args.rot) def encrypt(string, rot): “””Caesar encryption function””” cypherstr = “” for char in string: if not char.isalpha(): cypher = char elif char.isalpha(): num = ord(char) cypher = (num – 97 + rot) % 26 cypher = chr(cypher + 97) cypherstr += cypher return cypherstr def decrypt(string, rot): “””Caesar decryption function””” plainstr = “” for char in string: if not char.isalpha(): plain = char elif char.isalpha(): num = ord(char) plain = (num – 97 – rot) % 26 plain = chr(plain + 97) plainstr += plain return plainstr # Either encrypt or decrypt if mode == “encrypt”: print(encrypt(string, rot)) elif mode == “decrypt”: print(decrypt (string, rot)) [/cc] 3 thoughts on “Caesar ciphers in Python” 1. ChollyMo Yikes… doesn’t work on uppercase or numbers !! True, though it’s relatively easy to make it work with numbers by changing the modulus from 26 to 36 (26 letters + 10 numbers). Keeping capital letters would take a little more code, but is certainly doable. change the 97 for a 65	{"url":"http://almadj.us/index.php/2019/01/28/caesar-ciphers-in-python/","timestamp":"2024-11-02T22:13:21Z","content_type":"text/html","content_length":"119732","record_id":"<urn:uuid:2c1cf43a-f7da-471d-956f-27a123487f84>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00880.warc.gz"}
Universal parallel search set - Implementation Based on QuickUnion (Java language) For the implementation of parallel query set based on basic data type, please refer to QuickUnion. The underlying implementation of user-defined object and query set: linked list + Map mapping [the linked list here is represented as a tree] Logically: multiple trees from bottom to top. The array can no longer meet the requirements of user-defined objects and query sets. You can define an internal class Node to form a Node node to encapsulate each user-defined object. The rank attribute of tree height is provided here. The relative balance of the tree is controlled through rank to improve the efficiency of query set. private static class Node<V>{ V value; Node<V> parent = this; //During initialization, its parent node is itself int rank = 1; //During initialization, the tree height is 1 Provide initSet method to initialize and query the set: public void initSet(V v) { if(nodes.containsKey(v)) return; nodes.put(v, new Node<>(v)); //One v corresponds to one node After the instantiated object calls the initSet method, each user-defined object forms its own set: Define the private findRoot method: continuously probe from the given node until the root node is found. The probe process uses the path halving optimization strategy to improve efficiency. private Node<V> findRoot(V v){ Node<V> node = nodes.get(v); //Get the Node corresponding to v if(node == null) return null; //Probe from node to root node while(!Objects.equals(node.value, node.parent.value)) { node = node.parent.parent; //Path halving optimization is used return node; Define the find method: find the set to which the user-defined object belongs, and the root node of the node encapsulating the user-defined object is its set. public V find(V v) { Node<V> node = findRoot(v); //Get the root node where v is located return node == null ? null : node.value; Define union method: for a given two custom objects, merge and encapsulate the two nodes of the custom object. First get the root node of the two nodes and graft the root node. Node<V> rt1 = findRoot(v1); Node<V> rt2 = findRoot(v2) There are three situations: (1) The height of the left tree is lower than that of the right tree. At this time, the root node of the left tree is directly grafted to the root node of the right tree, and the overall height of the tree remains unchanged. (2) The height of the left tree is higher than that of the right tree. At this time, the root node of the right tree is directly grafted to the root node of the left tree, and the overall height of the tree remains unchanged. (3) The height of the tree on the left is equal to the height of the tree on the right. By default, the root node of the tree on the right is grafted to the root node of the tree on the left, and the overall height of the tree is increased by 1. Code implementation in three cases: if(rt1.rank < rt2.rank) { //Tree height with rt1 as root node < tree height with rt2 as root node rt1.parent = rt2; //Grafting from rt1 to rt2 }else if(rt1.rank > rt2.rank) { //Tree height with rt1 as root node > tree height with rt2 as root node rt2.parent = rt1; //Graft rt2 to rt1 }else { //Tree height with rt1 as root node = tree height with rt2 as root node rt2.parent = rt1; //Default rt2 grafted to rt1 rt1.rank += 1; //Update rt2 the tree height Define isTogether method: judge whether a given two objects belong to the same set. If the root node of the node encapsulating the user-defined object is the same, it belongs to the same set, otherwise it does not belong to different sets. public boolean isTogether(V v1, V v2) { return Objects.equals(find(v1), find(v2)); Since then, the general concurrent query set has been completed, and the basic data types and user-defined objects can be used. Provide complete code blocks for reference: import java.util.HashMap; import java.util.Map; import java.util.Objects; * Underlying implementation: linked list + Map mapping [the linked list here is represented as a simple tree] logically: multiple trees from bottom to top * @author Asus * @param <V> Incoming custom object public class GenericUnionFind<V> { private Map<V, Node<V>> nodes = new HashMap<>(); //Initialize and query set public void initSet(V v) { if(nodes.containsKey(v)) return; nodes.put(v, new Node<>(v)); //One v corresponds to one node //Returns the root node of v public V find(V v) { Node<V> node = findRoot(v); //Get the root node where v is located return node == null ? null : node.value; public void union(V v1, V v2) { //Get the root node of v1 and v2 Node<V> rt1 = findRoot(v1); Node<V> rt2 = findRoot(v2); if(rt1 == null \|\| rt2 == null) return; if(rt1.equals(rt2)) return; //In the same set, there is no need to merge if(rt1.rank < rt2.rank) { //Tree height with rt1 as root node < tree height with rt2 as root node rt1.parent = rt2; //Grafting from rt1 to rt2 }else if(rt1.rank > rt2.rank) { //Tree height with rt1 as root node > tree height with rt2 as root node rt2.parent = rt1; //Graft rt2 to rt1 }else { //Tree height with rt1 as root node = tree height with rt2 as root node rt2.parent = rt1; //Default rt2 grafted to rt1 rt1.rank += 1; //Update rt2 the tree height //Judge whether V1 and V2 belong to the same set public boolean isTogether(V v1, V v2) { return Objects.equals(find(v1), find(v2)); //Find root node private Node<V> findRoot(V v){ Node<V> node = nodes.get(v); //Get the Node corresponding to v if(node == null) return null; //Probe from node to root node while(!Objects.equals(node.value, node.parent.value)) { node = node.parent.parent; //Path halving optimization is used return node; //Define node private static class Node<V>{ V value; int rank = 1; //The default tree height with itself as the root node is 1 Node<V> parent = this; //The default parent node is itself Node(V v){ this.value = v;	{"url":"https://programmer.ink/think/universal-parallel-search-set-implementation-based-on-quickunion-java-language.html","timestamp":"2024-11-05T18:38:46Z","content_type":"text/html","content_length":"13269","record_id":"<urn:uuid:221c26ed-fef1-49f3-b4fc-5a2dbeff802c>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00570.warc.gz"}
Pre-exponential factor - (Geochemistry) - Vocab, Definition, Explanations \| Fiveable Pre-exponential factor from class: The pre-exponential factor is a constant that appears in the Arrhenius equation, representing the frequency of collisions and the likelihood that those collisions will lead to a reaction. It is crucial in understanding reaction rates as it is a key component that, along with activation energy, determines how temperature influences the speed of a chemical reaction. congrats on reading the definition of pre-exponential factor. now let's actually learn it. 5 Must Know Facts For Your Next Test 1. The pre-exponential factor is often denoted as 'A' in the Arrhenius equation and is measured in units that depend on the order of the reaction. 2. It reflects the total number of collisions between reactant molecules that are effective in leading to a reaction, not just any collision. 3. In reactions involving gases, the pre-exponential factor can be significantly affected by changes in pressure and temperature. 4. For complex reactions, determining an accurate value for the pre-exponential factor may require experimental data or sophisticated modeling techniques. 5. The value of the pre-exponential factor generally increases with increasing temperature, as higher temperatures lead to more frequent molecular collisions. Review Questions • How does the pre-exponential factor influence reaction rates in relation to temperature? □ The pre-exponential factor plays a crucial role in determining how temperature affects reaction rates. It represents the frequency of effective collisions between reactants. As temperature increases, molecular motion becomes more vigorous, leading to more frequent collisions and thus a higher value for the pre-exponential factor. This directly contributes to an increase in the overall reaction rate, as described by the Arrhenius equation. • Discuss the relationship between activation energy and the pre-exponential factor within the context of the Arrhenius equation. □ In the Arrhenius equation, both activation energy and the pre-exponential factor are key components that determine the rate constant 'k'. While activation energy represents the barrier that must be overcome for a reaction to occur, the pre-exponential factor quantifies how often collisions occur that could potentially lead to a reaction. Thus, even with a high activation energy, if the pre-exponential factor is large due to frequent collisions, the overall rate constant can still be significant. • Evaluate how changes in reaction conditions might affect both the pre-exponential factor and reaction rates. □ Changes in reaction conditions such as temperature, pressure, or concentration can significantly impact both the pre-exponential factor and reaction rates. For example, increasing temperature typically raises both factors by enhancing molecular motion and collision frequency. However, specific reactions may show different responses; for instance, increasing pressure can affect gaseous reactions differently than liquid-phase reactions. Therefore, understanding these relationships is crucial for predicting how reactions behave under varying conditions. "Pre-exponential factor" also found in: Â© 2024 Fiveable Inc. All rights reserved. APÂ® and SATÂ® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.	{"url":"https://library.fiveable.me/key-terms/geochemistry/pre-exponential-factor","timestamp":"2024-11-11T00:40:29Z","content_type":"text/html","content_length":"146942","record_id":"<urn:uuid:7c6e25ac-d326-440f-9212-9ae5577f109e>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00575.warc.gz"}
Simplify graphs You can use the Simplify Graph tool to remove unnecessary nodes from straight sections and curves of the input streets. It operates on the current street selection or on all the streets when nothing is selected. Segments are combined to longer straight or curved segments based on their angle in-between, except in the following cases: • The node in-between is an intersection (valency > 2). • Their street width is not equal. • One of the segments to combine exceeds the set thresholds. • Their combined length would be larger than the set maximum. You can click Graph > Simplify Graph in the main menu to simplify streets as shown in the following images: Streets with unnecessary nodes Streets simplified with nodes removed Simplify settings Simplify straight sections Straight sections are simplified. Straight threshold angle Angle in degrees. Streets with a lower angle than this are combined to straight segments. Meaningful for both simplify straight sections and simplify curves. Simplify curves Curves are simplified. Curve threshold angle Angle in degrees. Streets with a higher angle than this form boundaries between fitted curves. Only meaningful if the Simplify curves option is checked. Curve threshold segment length Maximum length of a single curve input segment. Only meaningful if the Simplify curves option is checked. Maximum simplified segment length Maximum length of the output straight or curve segment. Straight sections and curves Straight sections only Curves only Curves without limits Valid when they are shorter than the set threshold. Combined curves Combined segments with length limited	{"url":"https://doc.arcgis.com/en/cityengine/latest/help/help-simplify-graph.htm","timestamp":"2024-11-08T09:17:19Z","content_type":"text/html","content_length":"20487","record_id":"<urn:uuid:835fee86-df2f-49e1-bd77-f9b7382ddfd5>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00572.warc.gz"}
Uppbyggnad och validering av emissionsdatabas avseende Rita polära grafer As the degree of the polynomial equation (n) becomes higher, the polynomial equation becomes more complicated and there is a possibility of the model tending to overfit which will be discussed in the later part. Free Polynomial Standard Form Calculator - Reorder the polynomial function in standard form step-by-step This website uses cookies to ensure you get the best experience. By using this website, you agree to our Cookie Policy. 2 dagar sedan · Polynomial Regression Calculator is a tool to define a function for your data that is copied from excel, text, csv or enter manually. From Adam Douglass Sierra Vista, AZ January 09, 2015 at 11:25 AM. Thank you, this website helped my group with my engineering project. We got a Online Calculator Curve Fit Regression Calculator. Online calculator for curve fitting with least square methode for linear, polynomial, power, gaussian, exponential and fourier curves. Adaptation of the functions to any measurements. This online calculator uses several regression models for approximation of an unknown function given by a set of data points. See the webpage Confidence Intervals for Multiple Regression 2020-05-31 Step 3: Polynomial Regression Model. Gratis nedladdning Root Solver and Calculator För Windows Me Online Integral Calculator ». Detailed instructions in: Instrucciones detalladas en: https://carreteras-laser- escaner.blogspot.com/2018/12/regresion-no-lineal-en-android-non.html. Perform This function fits a polynomial regression model to powers of a single predictor by the method of linear least squares. STOR BABYLYCKA I DET BRITTISKA KUNGAHUSET - actforwork.net By using this website, you agree to our Cookie Policy. With polynomial regression, the data is approximated using a polynomial function. A polynomial is a function that takes the form f ( x ) = c0 + c1 x + c2 x2 ⋯ cn xn where n is the degree of the polynomial and c is a set of coefficients. empirisk regressionslinje; linje som erh The thick green line represents a 5-degree polynomial fit, with 95 Trading Calculator Beräkna din marginal, vinst eller förlust jämföra empirical biophysics briefing calculator aversion splendor climaxes ascription Aflace comely polynomial avoider axon bawling? drawn Saukville adjured Casin In http://www.tophar.com/31827.html Barth regression accidents.trapezoidal Kenneth Danielsson, Libors Market Models and the Calculation of Sensiti- Petter Gustafsson, Polynomial Approximation and Analytic Capacity. 2002 Frida Johansson, Anpassning av regressionsmodeller utifrån tidskorrelerat data. scientific calculator ingenjörsräknedosa. calculus cubic polynomial tredjegradspolynom. curl rotation (VA) edge of regression kuspidalkurva,. Using quotations for emphasis This poses some limitations to the used regression model, namely, only linear regression models can be used. That's why, unlike the above-mentioned calculator, this one does not include power and exponential regressions. However, it includes 4th and 5th order polynomial regressions. The method was published in 1805 by Legendre and 1809 by Gauss. The first Polynomial regression model came into being in1815 when Gergonne presented it in one of his papers. It is a very common method in scientific study and research. Online Integral Calculator ». Detailed instructions in: Instrucciones detalladas en: https://carreteras-laser- escaner.blogspot.com/2018/12/regresion-no-lineal-en-android-non.html. Perform This function fits a polynomial regression model to powers of a single predictor by the method of linear least squares. Interpolation and calculation of areas How to calculate power calculation for polynomial regression analysis? I am having trouble finding information on how to calculate (a priori and post hoc) power Windows 3D Scientific Calculator Equation Solver, Regression, linear fit, curve fit, polynomial fit. The equation solver linear / curve fitting tool gives you 9 Calculation instructions for many commercial assay kits recommend the use of a cubic regression curve-fit (also known as 3rd order polynomial regression). Polymath can fit a polynomial of degree n with the general form: P(x) = a0 + a1x + a2x^2 + . Library phone interview questions Logaritmisk: För data som stiger eller faller i snabb takt och sedan jämnas ut. Nu väntar man bara på et Source: polyurethane-volume-calculator.torresdeandalucia.com/, polynomial-regression-python-from-scratch.2hg13.com/, Serie 2: Vilka lägen har fx-991MS Calculator av Sara Learning Home ??? "excel polynomial regression" är people.stfx.ca/bliengme/ExcelTips/ Polynomial.htm A new estimator for the multicollinear Poisson regression Systems of linear equations and online linear algebra calculators are included. Solving systems of polynomial equations using combinations of algebraic geometry Perform a Polynomial Regression with Inference and Scatter Plot with our Free, Easy-To-Use, Online Statistical Software. Polynomial Regression Calculator In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable y is modelled as an nth degree polynomial in x. If you enter 1 for degree value so the regression would be linear. Polynomial Regression Calculator More about this Polynomial Regression Calculator so you can have a deeper perspective of the results that will be provided by this calculator. More about this Polynomial Regression Calculator so you can have a deeper perspective of the results that will be provided by this calculator. Privata vårdcentraler göteborg slogans tipsvad händer om man inte har tpms sensorervad betyder ordet empatisjukskriven student csnlockarps brodnytt lager norrköpingmunken portalen Note that this is a file from 1994 converted to Word 99 The plugin, at the current version, uses the regression npm package as its calculation engine. Important. Only bar, line, and scatter chart types are supported. This poses some limitations to the used regression model, namely, only linear regression models can be used. That's why, unlike the above-mentioned calculator, this one does not include power and exponential regressions. Matematikprogram - Topp 10 gratis programvara 2021 Similar Algebra Calculator Adding Complex Number Calculator. Subtracting Complex Number Calculator. Polynomial Equation Calculator. Multiplying Complex Number Calculator. Carlo-sampling, Polynomial Chaos-konstruktion, transportkartor och många such as regression analyses, based on observations venient for use on a table-calculator as they involve best fit, a polynomial of at least fourth degree is a. Which polynomials p(x) satisfy the following equality: (x-16)p(2x)=16(x-1)p(x)? Klöjm. Svar: En Casio Scientific Calculator fx-180p, duger gott och väl.	{"url":"https://hurmanblirrikcwxonm.netlify.app/65875/9024.html","timestamp":"2024-11-08T10:27:22Z","content_type":"text/html","content_length":"18015","record_id":"<urn:uuid:ab61d7bf-d28a-4a02-a3a2-c03f51f1182d>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00029.warc.gz"}
How do you predict using XGBoost? How do you predict using XGBoost? This tutorial is broken down into the following 6 sections:Install XGBoost for use with Python.Problem definition and download dataset.Load and prepare data.Train XGBoost model.Make predictions and evaluate model.Tie it all together and run the example. How do I interpret XGBoost results? You can interpret xgboost model by interpreting individual trees. Each of xgboost trees looks like this: As long as decision tree doesn’t have too many layers, it can be interpreted. So you can try to build an interpretable XGBoost model by setting maximum tree depth parameter (max_depth) to a low value (less than 4). What does XGBoost CV return? XGBoost has a very useful function called as cv which performs cross-validation at each boosting iteration and thus returns the optimum number of trees required. Tune tree-specific parameters ( max_depth, min_child_weight, gamma, subsample, colsample_bytree) for decided learning rate and number of trees. What is DMatrix in XGBoost? DMatrix is an internal data structure that is used by XGBoost, which is optimized for both memory efficiency and training speed. You can construct DMatrix from multiple different sources of data. Is XGBoost better than random forest? Ensemble methods like Random Forest, Decision Tree, XGboost algorithms have shown very good results when we talk about classification. Both the two algorithms Random Forest and XGboost are majorly used in Kaggle competition to achieve higher accuracy that simple to use. Is XGBoost a classifier? 2. XGBoost Model Performance. XGBoost dominates structured or tabular datasets on classification and regression predictive modeling problems. The evidence is that it is the go-to algorithm for competition winners on the Kaggle competitive data science platform. Why is XGBoost faster than GBM? Both xgboost and gbm follows the principle of gradient boosting. There are however, the difference in modeling details. Specifically, xgboost used a more regularized model formalization to control over-fitting, which gives it better performance. Is XGBoost deep learning? 1. XGBoost, commonly used by data scientists, is a scalable machine learning system for tree boosting which avoids overfitting. It performs well on its own and have been shown to be successful in many machine learning competitions. However, we observe that this model is still unclear for feature learning. Why does XGBoost work so well? It is a highly flexible and versatile tool that can work through most regression, classification and ranking problems as well as user-built objective functions. As an open-source software, it is easily accessible and it may be used through different platforms and interfaces. Why is XGBoost better than logistic regression? OTOH: XGBoost wins tons of Kaggle contests and beats out logistic regression, and boosted decision trees (some years ago) frequently won bake-offs in ML literature. So, in most scenarios, unless your don’t have the time to tune parameters AND perform n training folds on the whole process, XGBoost. Is XGBoost a random forest? XGBoost is normally used to train gradient-boosted decision trees and other gradient boosted models. One can use XGBoost to train a standalone random forest or use random forest as a base model for gradient boosting. … Can XGBoost handle outliers? yes. It is tree based and thus sensitive to order of values but not actual values. Outliers in target variable are another matter. With many loss functions (such as RMSE/L2) you are necessarily sensitive to outliers. How do you identify outliers? A commonly used rule says that a data point is an outlier if it is more than 1.5 ⋅ IQR 1.5\cdot \text{IQR} 1. 5⋅IQR1, point, 5, dot, start text, I, Q, R, end text above the third quartile or below the first quartile. Said differently, low outliers are below Q 1 − 1.5 ⋅ IQR \text{Q}_1-1.5\cdot\text{IQR} Q1−1. Is AdaBoost sensitive to outliers? AdaBoost is known to be sensitive to outliers & noise. Is random forest affected by outliers? Robust to Outliers and Non-linear Data Random forest handles outliers by essentially binning them. It is also indifferent to non-linear features. Is Random Forest an ensemble method? Random forests or random decision forests are an ensemble learning method for classification, regression and other tasks that operate by constructing a multitude of decision trees at training time and outputting the class that is the mode of the classes (classification) or mean/average prediction (regression) of the … Is random forest deep learning? Both the Random Forest and Neural Networks are different techniques that learn differently but can be used in similar domains. Random Forest is a technique of Machine Learning while Neural Networks are exclusive to Deep Learning. What is the difference between outliers and anomalies? Outlier = legitimate data point that’s far away from the mean or median in a distribution. Anomaly detection refers to the problem of ending anomalies in data. While anomaly is a generally accepted term, other synonyms, such as outliers are often used in different application domains. What is considered an outlier? An outlier is an observation that lies outside the overall pattern of a distribution (Moore and McCabe 1999). A convenient definition of an outlier is a point which falls more than 1.5 times the interquartile range above the third quartile or below the first quartile. What is another word for outlier? SYNONYMS FOR outlier ON THESAURUS.COM 2 nonconformist, maverick; original, eccentric, bohemian; dissident, dissenter, iconoclast, heretic; outsider. How do you predict using XGBoost? This tutorial is broken down into the following 6 sections:Install XGBoost for use with Python.Problem definition and download dataset.Load and prepare data.Train XGBoost model.Make predictions and evaluate model.Tie it all together and run the example. How do I interpret XGBoost results? You can interpret xgboost model by interpreting	{"url":"https://bridgitmendlermusic.com/how-do-you-predict-using-xgboost/","timestamp":"2024-11-04T12:38:46Z","content_type":"text/html","content_length":"44171","record_id":"<urn:uuid:60a29ce7-cc97-4e4b-a0de-1ef91a7372d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00884.warc.gz"}
(a) Graph the fourth-degree polynomial $p(x)=a x^{4}-6 x^{2}$ for $a=-3,-2,-1,0,1,2$, and $3 .$ For what values of the constant $a$ does $p$ have a relative minimum or relative maximum? (b) Show that $p$ has a relative maximum for all values of the constant $a$. (c) Determine analytically the values of $a$ for which $p$ has a relative minimum. (d) Let $(x, y)=(x, p(x))$ be a relative extremum of $p$. Show that $(x, y)$ lies on the graph of $y=-3 x^{2}$. Verify this result graphically by graphing $y=-3 x^{2}$ together with the seven curves from part (a).	{"url":"https://www.vaia.com/en-us/textbooks/math/calculus-8-edition/chapter-3/","timestamp":"2024-11-15T02:47:24Z","content_type":"text/html","content_length":"228477","record_id":"<urn:uuid:c16d4e9d-812c-4ca5-9afd-bd3af2f60da1>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00091.warc.gz"}
melanie c independence day Archives - All News Stories Independence Day Lessons, Excercise and Activities Activity 1: The teacher will enter the class and eventually students will greet her/him as usual with good morning/good afternoon etc. The teacher will respond to the class with warm gestures and body movements. Then the teacher will show some pictures of great freedom fighters of India. Then he … Read more	{"url":"https://dadiammakikahani.com/tag/melanie-c-independence-day/","timestamp":"2024-11-12T18:28:55Z","content_type":"text/html","content_length":"91562","record_id":"<urn:uuid:1317ad16-f4cb-4845-9dcd-74ac16a8a145>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00867.warc.gz"}
Chapter 3 – Solving Linear Equations Download Chapter 3 – Solving Linear Equations * Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project Document related concepts Series (mathematics) wikipedia , lookup Lp space wikipedia , lookup Fundamental theorem of calculus wikipedia , lookup Sobolev space wikipedia , lookup Function of several real variables wikipedia , lookup Differential equation wikipedia , lookup Derivation of the Navier–Stokes equations wikipedia , lookup Partial differential equation wikipedia , lookup Chapter 3 – Solving Linear Equations 3.7 – Formulas and Functions 3.7 – Formulas and Functions Today we will learn how to: Solve a formula for one of its variables Rewrite an equation in function form 3.7 – Formulas and Functions So far, we have been working with equations that have one variable. Today, we are going to work with equations that have more than one 3.7 – Formulas and Functions Formula – an algebraic equation that relates two or more real-life quantities 3.7 – Formulas and Functions Example 1 Solve A = lw for w. Find the width of a rectangle that has an area of 42 ft.2 and a length of 6 ft. 3.7 – Formulas and Functions Example 2 Solve K F 32 273 for F. 3.7 – Formulas and Functions Example 3 Solve I = Prt for t. Find the number of years t that $2800 was invested to earn $504 at 4.5% 3.7 – Formulas and Functions When you have an equation with x and y, you can re-write the equation two ways. “y is a function of x” means you rearrange the equation so you have y = “x is a function of y” means you rearrange the equation so you have x = 3.7 – Formulas and Functions Example 4 Rewrite the equation 2x – y = 9 so that y is a function of x. 3.7 – Formulas and Functions Example 5 Write the equation 2x – y = 9 so that x is a function of y. Use the result to find x when y = -2, -1, 0, and 1. 3.7 – Formulas and Functions #11 – 14, 16 – 34 even	{"url":"https://studyres.com/doc/8474121/chapter-3-%E2%80%93-solving-linear-equations","timestamp":"2024-11-10T02:35:58Z","content_type":"text/html","content_length":"63592","record_id":"<urn:uuid:250f5173-716a-4d09-b1b7-5a23638f97c4>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00807.warc.gz"}
Lesson 20 More Practice to Represent and Solve Warm-up: Number Talk: Two Steps (10 minutes) The purpose of this Number Talk is to elicit strategies students have for multiplying single-digit factors and adding two-digit numbers. The expressions involve two operations. They encourage students to look for and make use of structure as they use their understanding of equal-size groups and properties of operations to find products and sums (MP7). The reasoning here will be helpful later when students solve two-step word problems. • Display one expression. • “Give me a signal when you have an answer and can explain how you got it.” • 1 minute: quiet think time • Record answers and strategy. • Keep expressions and work displayed. • Repeat with each expression. Student Facing Find the value of each expression mentally. • $20 + (2 \times 3)$ • $30 + (4 \times 3)$ • $50 + (8 \times 3)$ • $99 + (8 \times 3)$ Activity Synthesis • “How did the first two expressions help you find the value of the last two expressions?” • Consider asking: □ “Who can restate ___ 's reasoning in a different way?” □ “Did anyone have the same strategy but would explain it differently?” □ “Did anyone approach the problem in a different way?” □ “Does anyone want to add on to _____’s strategy?” Activity 1: Info Gap: Introduction (15 minutes) The purpose of this activity is to introduce students to the structure of the MLR4 Information Gap routine. This routine facilitates meaningful interactions by positioning some students as holders of information that is needed by other students. Tell students that first, a demonstration will be conducted with the whole class, in which they are playing the role of the person with the problem card. Explain to students that it is the job of the person with the problem card (in this case, the whole class) to think about what information they need to answer the question. For each question that is asked, students are expected to explain what they will do with the information, by responding to the question, “Why do you need to know (that piece of information)?” If the problem card person asks for information that is not on the data card (including the answer!), then the data card person must respond with, “I don’t have that information.” In explaining their answers, students need to be precise in their word choice and use of language (MP6). Once the students have enough information to solve the problem, they solve the problem independently. The info gap routine requires students to make sense of problems by determining what information is necessary and then ask for information they need to solve them. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). • Groups of 2 • “The problems in this lesson are about setting up for a special event at a school. What’s your favorite special event at school or in your community?” • Share responses. MLR4 Information Gap • Display the Sample Problem Card. • Read the problem aloud. • Listen for and clarify any questions about the context. • “Some of the information you need to solve this problem is missing, and I have it here. What specific information do you need?” • 1–2 minutes: quiet think time • “With your partner, decide what information you need to solve the problem, and create a list of questions you can ask to find out.” • 2–3 minutes: partner discussion • Invite students to share 1 question at a time. • Record each question on a display, and respond with: “Why do you need to know (restate the information requested)?” Students should provide a justification for how they will use the information before the information is revealed. • Answer questions using only information stated on the Sample Data Card (do not reveal). • Record information that is shared on the display. Give students time to decide whether they have enough information to solve the problem. • Repeat until students decide they have enough information to solve, then ask students to solve. • 2–4 minutes: independent work time Activity Synthesis • Invite 1–2 students to share how they solved the problem. • “Which questions helped you find out how many chairs were in the room?” (How many chairs were in the corner? How many chairs were in each row? How many rows were there?) • If there were any questions that the data card cannot answer, discuss them here. Activity 2: Info Gap: Bake Sale (20 minutes) This Info Gap activity gives students an opportunity to determine and request information needed to solve a two-step problem that involves multiplication. The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6). Here is an image of the cards for reference: Representation: Access for Perception. Begin by giving a physical demonstration of the activity’s procedure to support understanding of the activity and understanding of the context. Supports accessibility for: Social-Emotional Functioning, Memory Required Preparation • Create a set of cards from the blackline master for each group of 2. • Keep set 1 separate from set 2. MLR4 Information Gap • Display the task statement, which shows a diagram of the Info Gap structure. • 1–2 minutes: quiet think time • Read the steps of the routine aloud. • “I will give you either a problem card or a data card. Silently read your card. Do not read or show your card to your partner.” • Distribute the cards. • 1–2 minutes: quiet think time • Remind students that after the person with the problem card asks for a piece of information the person with the data card should respond with “Why do you need to know (restate the information • 3–5 minutes: partner work time • After students solve the first problem, distribute the next set of cards. Students switch roles and repeat the process with Problem Card 2 and Data Card 2. Student Facing Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner. Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner. Activity Synthesis • “What kinds of questions were the most useful to ask?” • Select 1–2 students to share different strategies used to solve one of the problems. Try to feature a student-drawn tape diagram. • Consider asking: □ “Did anyone solve the problem in a different way?” □ “Did anyone use a tape diagram?” □ “How did you know if your answer made sense?” □ “How could we represent the second problem with an equation with a letter for the unknown quantity?” ($230 - (7 \times 10) = {d}$) Lesson Synthesis “Today we learned the Information Gap routine. How did this routine help you make sense of the problems you solved?” (The routine gave me a chance to focus on what was important in the problem. I had to think about what I needed to know to solve the problem. I had to think about why some information was needed to solve the problem. It helped me make sense of what was happening in the problem.) Cool-down: Reflection (5 minutes) Student Facing In this section, we used rounding to estimate answers to problems. This helped us decide if our answers to problems made sense based on the situation and the numbers in the situation. We also wrote equations with an unknown and used diagrams to solve for the exact answer in problems. Mai had 104 beads. She bought 2 more packs of beads and each pack has 10 beads in it. How many beads does she have now? Equation with an unknown:	{"url":"https://im.kendallhunt.com/K5/teachers/grade-3/unit-3/lesson-20/lesson.html","timestamp":"2024-11-07T06:09:39Z","content_type":"text/html","content_length":"97363","record_id":"<urn:uuid:34d0d89f-909b-4b40-830a-1731c86b3e09>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00029.warc.gz"}
How to do Binning in R? » finnstats How to do Binning in R? Binning in R, you will learn about data binning in this tutorial. Binning develops distinct categories from numerical data that are frequently continuous. It’s very handy for comparing different sets of data. Binning is a pre-processing procedure for numerical numbers that can be used to group them. Why do we need binning? Binning can sometimes increase the predictive model’s accuracy. To have a better grasp of the data distribution, you can use data binning to group a set of numerical values into a smaller number of bins. For example, the variable “ArrDelay” has 2855 unique values and a range of -73 to 682 and can categorize “ArrDelay” variable as [0 to 5], [6 to 10], [11 to 15], and so on. Binning in R In this tutorial, arrival delays can be divided into four bins by quartiles using binning. The borders that divide observations into four distinct intervals are referred to as quartiles. They’re frequently calculated using data point values and how they compare to the rest of the dataset. Binning is simple to implement in tidyverse. Assume you want four bins with the same number of observations, in which case you’ll need three numbers as dividers: The 1st, 2nd, and 3rd quartiles are the first, 2nd, and 3rd quartiles, respectively. The dataset is divided into two half by the median. The median of the lower half of the dataset is the 1st quartile or lower quartile. This quartile is referred to as Q1. The median of the entire dataset is in the second quartile, Q2. The median of the upper half of the dataset is the upper quartile, or 3rd quartile, Q3. Plotting a histogram before binning can give you an idea of how the data looks. Based on the above plot, most of the flights experience no delays which are roughly bell-shaped and right-skewed. Let’s get binning now. To begin, divide “ArrDelay” into four buckets, each with an equal amount of observations of flight arrival delays, using the dplyr ntile() function. Then, make a list called “rank” with four bins named “1”, “2”, “3”, and “4”, accordingly. This categorizes the data into different bins based on the number of minutes the planes were delayed. The longer the flight was delayed, the larger the bin label. You can execute the same based on a one-liner code. binning<-data %>% mutate(rank=ntile(data$ArrDelay,4)) Binning is a data pre-processing technique that groups a series of numerical values into a set of bins, as you learned in this tutorial. Binning can help you better understand the distribution of your data and increase the accuracy of predictive models. You also learned how to improve data analysis by using a binning method that separates numerical values into quartiles. 2 Responses 1. Base R has everything needed, and is simpler and more direct (in my opinion, that is): > x bins table(bins) (-2.77,-0.687] (-0.687,0.0878] (0.0878,0.757] (0.757,3.27] ## an alternative would be > ints <- findInterval(x, quantile(x)) ## or using the example data bins <- with(data, cut(ArrDelay , quantile(ArrDelay) ) ) For a small but interesting side trip, investigate the type argument of quantile(), for variations in how to interpolate the quantiles from the data. □ Great. Thanks	{"url":"https://finnstats.com/2021/10/18/how-to-do-binning-in-r/","timestamp":"2024-11-05T06:36:15Z","content_type":"text/html","content_length":"290712","record_id":"<urn:uuid:73271e37-36a3-4963-8de6-9edbc1dde8c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00564.warc.gz"}
Jianzhi Offer Explanation \| # Translating Numbers into Strings # Dynamic Programming Title Address: Translating Numbers into Strings. [Dynamic Planning] ① Analysis question: The number that can be decoded will not exceed 26, which means the effective decoding range is [1,26]. ② Determine the state: Taking the array "12345" as an example, two points need to be clarified first: According to the question, all numbers in the array must participate in decoding. For example, if one decoding method of the array "1234" is "1,2,3,4"", then when another element "5" is added to the end of the array, the previous decoding method will become "1,2,3,4,5", that is, "1,2,3,4" and "1,2,3,4,5" are the same decoding method. Since all numbers need to participate in decoding, any single digit "0" in the decoding method is invalid. When adding a new number" 5" In addition to being used as a separate number for decoding, it can also be combined with the number on its left to form a number and then participate in decoding, known as "45". At this point, the number of decoding methods depends on the remaining number, which is "123". This is the same principle as before, but at this point, "4" and "5" are considered as a whole, It can be understood that the original decoding method for array "123" was "1,2,3", but now adding "45" changes the decoding method to "1,2,3,45". Due to the effective decoding range being [1,26], the newly added digit "5" only needs to be considered as a 2-digit combination with its left digit, without the need to consider forming a larger combination, such as a 3-digit "345", etc. The combination of numbers must be ten digits, for example, "02" composed of "0" and "2" does not meet the decoding requirements. After understanding the above two points, now let's start from the substring of array "12345"; 1" Start the analysis, then gradually add elements backwards, and let the array x have f (x) effective decoding methods: When the array is "1", there are the following decoding methods: ① 1. F ("1")= one. Next, add numbers" 2", When the array is "12", there are the following decoding methods: ① 1,2. ② 12. F ("12")= two. The first type evolved from the decoding method of array "1", which is to add a single number "2" on top of it. Next, add numbers" 3", When the array is "123", there are the following decoding methods: ① 1,2,3. ② 12,3. ③ 1,23. F ("123")= three. The first and second methods evolved from the decoding method of array "12", which is to add a single number "3" on top of it; The third method evolved from the decoding method of array "1", which is to add the combination of numbers "23" on top of it. Next, add numbers" 4", When the array is "1234", there are the following decoding methods: ① 1,2,3,4. ② 12,3,4. ③ 1,23,4. ④ 1,2,34. ⑤ 12,34. The first, second, and third decoding methods evolved from the decoding method of array "123", which is to add a single number "4" on top of it; The fourth and fifth methods have evolved from the decoding method of the array "12", which is to add the combination of numbers "34" on top of it; Due to "34" not being within the range of effective decoding, these two decoding methods will be discarded. F ("1234")= three. As can be seen, the decoding method of array "1234" is determined by the array" 123" The decoding method and array; 12" Composed of decoding methods. According to the above analysis, when the array continues to expand to "12345", its decoding method is: When the newly added number "5" participates in decoding as a single digit, the decoding method is equivalent to the decoding method of the array "1234". This set of decoding methods is effective because the single digit "5" is within the effective decoding range; When "5" and its preceding digit form a tens digit "45", the decoding method at this time is equivalent to the decoding method of the array "123", and whether this decoding set is effective depends on whether "45" is within the valid decoding range. I.e. f ("12345")= F ("12345" - "5")+ F ("12345" - "45")= F ("1234")+ F ("123")= 3+ 3= However, since "45" is not within the valid decoding range, the result of f ("123") cannot be included, so the final result should be 3. It can be seen that when enumerating to a certain number, how many decoding methods there are at this time can be obtained by adding up the previous decoding methods, that is, the current state can utilize the previous state, which is a typical dynamic programming. ③ State transition equation: f (x)= F (x-1)+ F (x-2) (where x represents the length of the array, f (x) represents how many effective decoding methods there are; Whether to add f (x-1) depends on whether nums [x] is within [1,9], that is, nums [x] needs to satisfy non zero; Whether to add f (x-2) depends on whether the numerical combination of nums [i-1] and nums [i] is within [10,26]. Time complexity: O (N), requires traversing the array once. Space complexity: O (N), requires declaring a state array record f (x). C# Code: class Solution { public int solve (string nums) { int[] codesNum=new int[nums.Length]; //state array, recording f(x) //initial condition : if(nums[0].ToString() != "0"){ //enumerating from the second bit of the array to the last bit : for(int i=1;i<nums.Length;++i){ //judge nums[i] is it not 0 if(nums[i].ToString() != "0"){ //judge nums[i-1] related to nums[i] is the combination of numbers in the [10,26] within int temp=int.Parse(nums[i-1].ToString() + nums[i].ToString()); if(temp >= 10 && temp <= 26){ if(i==1){ //namely f (2) , because it does not exist f (0), so special handling is required : return codesNum[nums.Length-1];	{"url":"http://iopenv.com/V4AQIF7FR/Jianzhi-Offer-Explanation-Translating-Numbers-into-Strings-Dynamic-Programming","timestamp":"2024-11-14T23:36:44Z","content_type":"text/html","content_length":"19940","record_id":"<urn:uuid:e69b6f62-e348-4d96-b3d5-b2941584368e>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00614.warc.gz"}
Ants and Chips Imagine a bunch of wood chips randomly distributed on a surface. Now add an ant, randomly walking around amongst the chips. Whenever it bumps into a chip, the ant picks up the chip; if it bumps into another chip, it drops the one it is carrying and keeps walking. How will such a system evolve over time? We can speculate: since the ant drop its loads when bumping into another chip, piles of chips should grow. But as these piles grow, it is more likely that the ant will pick up a chip from one pile and then drop it on another pile: in other words, we should not expect all chips eventually to end up on a single pile; instead, we should reach a steady state, where piles (on average) neither grow nor shrink. The problem is that it might take a while to reach the steady state! I first became aware of this system through the book “The Computational Beauty of Nature” by Gary William Flake, quite a few years ago. At that point, waiting long enough to reach the steady state took a lot of patience, now it is a matter of seconds, for realistic system sizes. Here are two movies for 1000 chips, distributed on a 100x100 field, with periodic boundary conditions. The simulation was carried through to 35 million steps taken by the ant. The movies differ in the initial configuration: on the left, chips are initially randomly distributed, on the right, they start out in 5 equally sized heaps. It takes a while, but eventually, both systems reach “equivalent” (or: equivalent looking) steady states. To make the notion of “equivalence” a bit more rigorous, we can evaluate the correlation function $g(r)$: this is essentially the probability to find a chip at a distance $r$ from another. By construction, correlation functions obey: $g(0) = 1$ and $g(r \to \infty) = \rho$, where $\rho$ is the overall density of chips (for 1000 chips on a 100x100 field: $\rho = 0.1$). Correlation functions typically decay exponentially: $g(r) = \exp(-x/b)$, where the correlation length $b$ is a measure for the distance over which chips are correlated — we can think of it as a measure of the “radius” of the piles. A rough estimate for $b$ can be obtained by examining the initial decay of the correlation function near the origin: by equating it to the derivative $g^\prime(r) = -\exp(-r/b)/b \to -1/b$, we obtain a simple estimate for $b$. (This rough estimate is entirely deterministic and does not require iteration. With more effort and attention, it is of course also possible to fit a functional form of the correlation function over more data points.) The results are shown below for the systems in the movies. It is evident that the correlation function (and that means: the typical cluster size) tend towards the same limit, regardless of initial configuration. (Notice that the correlation length grows when starting from the random configuration, indicating the growing of clusters; but for the other configuration it shrinks as the initial set up disappears). More movies, for different chip densities can be found here. The simulation code can be found below. The program uses a goroutine to calculate the correlation function concurrently to the main simulation loop; thus leading to improved CPU utilization. The code includes a feature to give the ant a “sense of direction”. In this case, the ant does not decide at each step which direction to choose next, but instead has a (adjustable) higher probability to continue in the direction of the last step. I would expect that this leads to the system reaching its steady state faster, but I have not experimented with this option.	{"url":"https://janert.me/blog/2024/ants-and-chips/","timestamp":"2024-11-11T18:11:03Z","content_type":"text/html","content_length":"19835","record_id":"<urn:uuid:3e7b059f-e391-4c7e-949b-8c41ff746262>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00276.warc.gz"}
A satellite control problem A numerical approach is described for calculating the optimal policy in the stochastic control problem of keeping a satellite close to a fixed point in space when it is subject to random forces. The random forces are modelled by Brownian Motion. A policy is evaluated in terms of its long run expected average cost. The running costs consist of a charge for fuel used plus a charge of x sub 1 squared per unit of time when the satellite is x sub 1 units away from the target. The space is one-dimensional. The method used is to apply backward induction to a bounded discrete space, discrete time version of the problem. Incidentally a solution is presented for the deterministic version of the problem where there are no random forces. NASA STI/Recon Technical Report N Pub Date: December 1977 □ Brownian Movements; □ Satellite Attitude Control; □ Stochastic Processes; □ Cost Analysis; □ Fuel Consumption; □ Mathematical Models; □ Probability Distribution Functions; □ Launch Vehicles and Space Vehicles	{"url":"https://ui.adsabs.harvard.edu/abs/1977STIN...7821194C/abstract","timestamp":"2024-11-07T17:41:58Z","content_type":"text/html","content_length":"34617","record_id":"<urn:uuid:b2dc7adb-8102-4e8e-8b8d-aad9f62aa5f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00366.warc.gz"}
tatistical hypothes What is it and who needs it? Verification (test) statistical hypotheses are a way of mathematically determining the validity of some statement based on distribution law. Having mastered this method, you will be able to make mathematically sound conclusions, for example: Example #1 You make dice for the dice game and to make sure that the dice is perfectly balanced, you conduct a test - roll the dice 600 times and decide that if each number falls 100±10 times, then the dice is In production, 5% of products are rejected, you have developed a new technology and want to check whether it will decrease the amount of marriage. Basic terms, definitions and formulas Null and alternative statistical hypotheses Mathematically, the condition of the statistical test is written in the form of the main (null) hypothesis H[0] and the alternative (competing) hypothesis H[1]. The main hypothesis implies a certain parameter value. An alternative hypothesis is used to indicate an area that we may also be interested in. Now in the examples: In the first example, we want to find out if the number of each thrown number is equal to 100±10, while for us both more than 110 and less than 90 will be unsuccessful H[0]: μ = 100±10 H[1]: μ ≠ 100±10 the scientific record looks like this: H[0]: μ = 100 H[1]: μ ≠ 100 α = 0.1 In the second example, we want to find out if the new technology is better than the old one? At the same time, we are not interested in whether it has become worse, but only if there are improvements. Suppose that if the number of defects remained at the level of 5% 0.25%, then the process did not get better, if the number of defects is less than 4.75%, then there are H[0]: p = 5±0.25% H[1]: p < 4.75% the scientific record looks like this: H[0]: p = 0.05 H[1]: p < 0.05 α = 0.05 Critical area and two errors The area of values in which the main hypothesis is incorrect is the critical area, the size of this area is set as the significance level: We have values from 100 to 200 and want to check, we assume that the main hypothesis is incorrect in the critical area, if our assumption is incorrect, then we have made a mistake, such an error is called error of the first kind. For an alternative hypothesis, we can also make an error, such an error will be called a error of the second kind We formulate the hypothesis in such a way that the incorrect rejection of the main hypothesis is more significant for our solution, than incorrect acceptance of an alternative, here is an example: A study is being conducted on whether there is a link between smoking and cancer, the main hypothesis is put forward as follows: smoking causes cancer. If we reject this statement, and it turns out to be true - we are endangering human lives (a mistake of the first kind). At the same time, if smoking does not cause cancer, and during the experiment we confirmed what causes it, then it will not cause any special consequences (error of the second kind). In terms of making a decision, we want to control the level of errors of the first kind, i.e. if we need to make a decision about a certain statement, we must set some significance level α and subsequent calculations will depend on this parameter. It is necessary to check, Significance level, statistical power The significance level α is the probability of making a mistake of the first kind. The significance level and the error of the first kind are the same thing. Statistical power is associated with a second kind of error (β), statistical power is the probability of rejecting the main hypothesis when true alternative. The probability of error of the second kind and statistical power in total give 100%, respectively, the greater the statistical power, the less likely it is to make a mistake of the second kind. So we have: Statistical hypothesis testing is a mathematical representation of a certain statement Null hypothesis (H[0]) - an assumption about a certain parameter θ, H[0]: θ = θ[0] Alternative hypothesis (H[1]) is an assumption about a certain parameter θ, H[1]: θ = θ[0] Critical region is the area in which the main hypothesis H[0] is incorrect Error of the first kind - the probability of rejecting the main hypothesis when it is true Type II error - the probability of accepting the main hypothesis when it is incorrect Mathematical record of the hypothesis that the average value of the general population is 2 H[0] : μ = 2 H[1] : μ ≠ 2 Another example Mathematical record of the hypothesis that the average value of sample A and the average value of sample B are equal H[0] : μ[A] = μ[B] H[1] : μ[A] ≠ μ[B] That would certainly Mathematical record of the hypothesis that the average value of sample A is less than the average value of sample B H[0] : μ[A] < μ[B] H[1] : μ[A] ≥ μ[B] Significance level α The significance level (it could also be called the "Degree of confidence") is a parameter that means what is the probability, that the correct hypothesis will not be accepted. This parameter can be obtained, or it can be pre-set by a condition, I give two examples: • Can we be 90% sure (the significance level is 10%) that the car will not need to be repaired within a year? After testing the hypothesis, we will get the result "yes" or "no" • How much can we be sure that the car will not need to be repaired during the year? After testing the hypothesis, we will get the result as a percentage Hypothesis errors When we make a statement about a certain hypothesis, we can make two mistakes: Error of the first kind α For example, we conducted a test of a certain sample and based on the results decided that the parameter X does not correspond to the general population. If the selection was if it is done incorrectly and the parameter X describes the general population, then we made a mistake of the first kind - we abandoned the main hypothesis when it is true. α = P(error of the first kind) = P(rejection of H[0] \|H[0] is true) The error of the first kind and the level of significance are absolutely the same. We weighed 10 rabbits, their average weight is 5.1±0.5 kg. Suppose that the rabbit's weight obeys normal law, then: σ = 0.5/√(10) = 0.16 μ = 5.1 Hypothesis condition: α = P(H[0] is incorrect \| H[0] is true) = P(x< ) Error of the second kind β The reverse case of the error of the first kind is when we accepted the main hypothesis, but it turned out to be incorrect β = P(error of the second kind) = P(acceptance of H[0] \|H[0] is erroneous) Statistical hypothesis testing Checking a statistical hypothesis means performing the following steps: 1. Building a random sample 2. Calculating the parameter X of the sample 3. Testing the hypothesis using the obtained value of X	{"url":"https://en.k-tree.ru/articles/statistics/data_analysis/statistical_hypotesis","timestamp":"2024-11-04T12:28:06Z","content_type":"application/xhtml+xml","content_length":"388944","record_id":"<urn:uuid:0a1b13ce-5da2-4134-b58f-1f2a21e7074f>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00729.warc.gz"}
The Poisson distribution - Dave Tang's blog The Poisson distribution A Poisson distribution is the probability distribution that results from a Poisson experiment. A probability distribution assigns a probability to possible outcomes of a random experiment. A Poisson experiment has the following properties: 1. The outcomes of the experiment can be classified as either successes or failures. 2. The average number of successes that occurs in a specified region is known. 3. The probability that a success will occur is proportional to the size of the region. 4. The probability that a success will occur in an extremely small region is virtually zero. A Poisson random variable is the number of successes that result from a Poisson experiment. Given the mean number of successes that occur in a specified region, we can compute the Poisson probability based on the following formula: $P(x; \mu) = \frac{(e^{-\mu})(\mu^x)}{x!}$ which is also written as: $Pr(X = k) = e^{-\lambda} \frac{\lambda^k}{k!} \ \ k = 0, 1, 2, \dotsc$ The average number of homes sold is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow? $P(3; 2) = \frac{(e^{-2}) (2^3)}{3!}$ Calculating this manually in R: e <- exp(1) [1] 0.180447 Using dpois(): dpois(x = 3, lambda = 2) [1] 0.180447 The Poisson distribution can be used to estimate the technical variance in high-throughput sequencing experiments. My basic understanding is that the variance between technical replicates can be modelled using the Poisson distribution. Check out Why Does Rna-Seq Read Count Fit Poisson Distribution? on Biostars. Calculating confidence intervals Calculate the confidence intervals using R. Create data with 1,000,000 values that follow a Poisson distribution with lambda = 20. n <- 1000000 data <- rpois(n, 20) Functions for calculating the lower and upper tails. poisson_lower_tail <- function(n) { qchisq(0.025, 2n)/2 poisson_upper_tail <- function(n) { qchisq(0.975, 2(n+1))/2 Lower limit for lambda = 20. [1] 12.21652 Upper limit for lambda = 20. [1] 30.88838 How many values in data are lower than the lower limit? FALSE TRUE How many values in data are higher than the upper limit? FALSE TRUE What percentage of values were outside of the 95% CI? (sum(data<poisson_lower_tail(20)) + sum(data>poisson_upper_tail(20))) * 100 / n [1] 5.2548 Using the Poisson Confidence Interval Calculator and lambda = 20 returns: • 99% confidence interval: 10.35327 - 34.66800 • 95% confidence interval: 12.21652 - 30.88838 • 90% confidence interval: 13.25465 - 29.06202 which matches our 95% CI values. This work is licensed under a Creative Commons Attribution 4.0 International License. This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"url":"https://davetang.org/muse/2012/02/14/the-poisson-distribution/","timestamp":"2024-11-03T22:25:52Z","content_type":"text/html","content_length":"114329","record_id":"<urn:uuid:33855dcc-e391-4b58-8639-9b1497019c34>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00866.warc.gz"}
Stability of Unsteady MHD Flow in a Rectangular Duct In this study, the time dependent and coupled magnetohydrodynamic (MHD) flow equations are solved in the cross-section of a rectangular pipe (duct) by using the radial basis function approximation (RBF). The MHD studies electrically conducting fluids in the presence of a magnetic field. It has wide range of industrial applications such as MHD generators, MHD pumps, plasma physics and nuclear fusion [1]. The velocity and the induced magnetic field are obtained by approximating the inhomogeneities using thin plate splines (r2 lnr) [2]. Then, particular solution is found satisfying both the MHD equations and the boundary conditions which are the no-slip and insulated wall conditions. The Euler time integration scheme is used for advancing the solution to steady-state together with a relaxation parameter for achieving stable solution. It is shown that, as Hartman number (M) increases the flow develops boundary layers of order M-1 and M-1/2 on the Hartmann walls (perpendicular to the applied magnetic field) and side walls (parallel to the magnetic field), respectively. The induced magnetic field also exhibits boundary layers at the Hartmann walls, and the flow flattens and becomes stagnant at the center of the duct with an increase in the Hartmann number. These are the well-known characteristics of the MHD flow. The stability analysis is carried in terms of spectral radius of the coefficient matrix in the final discretized system, requiring the boundedness of spectral radius by one. The implemented scheme “Euler in time - radial basis function approximation in space” gives stable solution by using quite large time increment and relaxation parameter although the Euler scheme is an explicit method. International Conference on Applied Mathematics in Engineering (ICAME) (27-29 Haziran 2018) M. Tezer, “Stability of Unsteady MHD Flow in a Rectangular Duct,” presented at the International Conference on Applied Mathematics in Engineering (ICAME) (27-29 Haziran 2018), Balıkesir, Türkiye, 2018, Accessed: 00, 2021. [Online]. Available: https://hdl.handle.net/11511/84805.	{"url":"https://open.metu.edu.tr/handle/11511/84805","timestamp":"2024-11-11T03:16:23Z","content_type":"application/xhtml+xml","content_length":"55441","record_id":"<urn:uuid:4772b278-0ca5-452a-9251-e3159cf0eefa>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00172.warc.gz"}
Single-arc Variational Equations Propagation Single-arc Variational Equations Propagation For propagation of the variational equations alongside the system state, a different sort of simulator object - a VariationalSimulator - has to be used. VariationalSimulator objects contain a Simulator object, which means that they can do anything that a Simulator can plus the added functionality of propagating variational equations. To propagate the variational equations alongside the single-arc system state, the SingleArcVariationalSimulator derivative of the VariationalSimulator base class should be used. With the basic simulation setup (system of bodies, integrator settings, propagator settings) and the parameter settings for the variational equations, a variational equations solver can be set up. The setup works similarly to the normal dynamics simulator: variational_equations_solver = estimation_setup.SingleArcVariationalSimulator( bodies, integrator_settings, propagator_settings, estimation_setup.create_parameters_to_estimate(parameter_settings, bodies) The state history, state transition matrices, and sensitivity matrices can then be extracted: states = variational_equations_solver.state_history state_transition_matrices = variational_equations_solver.state_transition_matrix_history sensitivity_matrices = variational_equations_solver.sensitivity_matrix_history For a complete example of propagation and usage of the variational equations, please see the tutorial Linear sensitivity analysis of perturbed orbit.	{"url":"https://docs.tudat.space/en/latest/_src_user_guide/state_propagation/propagating_variational_equations/single_arc_variational_propagation.html","timestamp":"2024-11-07T02:56:07Z","content_type":"text/html","content_length":"27787","record_id":"<urn:uuid:fae42c21-a371-4d9c-aad8-6ccf4d1c9461>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00236.warc.gz"}
Quantum Instruction Set - Computerphile \| Video Summary and Q&A \| Glasp Quantum Instruction Set - Computerphile \| Summary and Q&A Quantum Instruction Set - Computerphile Writing software for quantum computers is similar to writing software for traditional computers, but the basic building block is a qubit instead of a bit. Key Insights • 🖱️ Writing software for quantum computers is similar to writing software for traditional computers but with the use of qubits as the building blocks. • ❓ Qubits are resource-like and can represent various physical systems with two states and probabilities. • 🔬 Quantum programming involves manipulating probabilities through instructions like Hadamard and CNOT gates. • 👻 Quantum computers can interact with multiple qubits, allowing for powerful computations. • ✋ Quantum programming languages and libraries, like PyQuil, provide higher-level abstractions for writing quantum programs. • 💻 Quantum computers can solve specific problems efficiently, such as the Fourier Transform. • 🏃 Running programs on a quantum computer requires handling noise and collecting statistics to obtain accurate answers. We talked a bit about the hardware, you know, people are working on the hardware of quantum computing, yep What about software? where do you start thinking about that? Writing software for a quantum computer In my opinion is actually not very very different from how we write software for just a normal computer and we think about software in terms o... Read More Questions & Answers Q: What is the fundamental difference between writing software for quantum computers and traditional computers? While the concept of writing instructions is similar, the fundamental building block in quantum computers is a qubit, which can exist in two states with probabilities. Q: How are qubits represented in quantum computers? Qubits can be represented by various physical systems, such as photons' polarization or superconducting charge qubits. These systems have two states and probabilities associated with them. Q: How do qubits interact in a quantum computer? Qubits in a quantum computer can interact with each other, allowing for powerful computations. Each additional qubit exponentially increases the number of probabilities that can be manipulated. Q: How are probabilities changed in a quantum computer? Instructions, such as Hadamard and CNOT gates, are used to change the probabilities of qubits. These instructions act on the probabilities and can be represented by matrices. Summary & Key Takeaways • Writing software for quantum computers is not very different from writing software for traditional computers. • Qubits are the fundamental building blocks of quantum computers and have two states with probabilities. • Quantum computers can interact with multiple qubits, allowing for powerful computations. • Quantum programming involves changing probabilities through instructions like Hadamard and CNOT gates. Explore More Summaries from Computerphile 📚	{"url":"https://glasp.co/youtube/p/quantum-instruction-set-computerphile","timestamp":"2024-11-12T08:20:08Z","content_type":"text/html","content_length":"358767","record_id":"<urn:uuid:7a1893ca-8b76-438c-95fe-88dd0ce8d374>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00074.warc.gz"}
How to Sum If Cell Contains a Text in Excel (6 Suitable Examples) - ExcelDemy We will be using a sample product price list as a dataset to demonstrate all the methods throughout the article. Let’s have a sneak peek of it: Method 1 – Using SUMIF Function to Sum If Cell Contains a Text in Excel In the spreadsheet, we have a product price list with categories. So, in this section, we will try to calculate the total price of the products under the Wafer category. • Select cell C15. • Put the following formula inside the cell: =SUMIF(B5:B12,"Wafer", E5:E12) ␥ Formula Breakdown: Syntax: SUMIF(range, criteria, [sum_range]) • B5:B12 is the range where the SUMIF function will look for the word “Wafer”. • “Wafer” the search keyword. • E5: E12 is the sum range. • =SUMIF(B5:B12,”Wafer”, E5:E12) returns the total price of the products under the “Wafer” category. Method 2 – Applying Excel SUMIFS Function to Add Up Data If Cell Contains a Specific Text • Select cell C15. • Input the following formula: ␥ Formula Breakdown: Syntax: SUMIFS(sum_range, criteria_range1, criteria1, [criteria_range2, criteria2], …) • E5: E12 is the sum range. • B5:B12 is the range where the SUMIFS function will look for the word “Wafer”. • “Wafer” is the search keyword. • =SUMIFS(E5:E12, B5:B12,”Wafer”) returns the total price of the products under the “Wafer” category. Method 3 – Applying SUMIF Function to Sum If Cell Contains Text in Another Cell in Excel • Create new cells to store the search terms and result. • Select cell C15. • Input the following formula. ␥ Formula Breakdown: Syntax: SUMIF(range, criteria, [sum_range]) • B5:B12 is the range where the SUMIF function will look for the word “Wafer”. • “”&C14&”” refers to the address of the cell that contains the search keyword “Wafer”. • E5: E12 is the sum range. • =SUMIF(B5:B12,””&C14&””,E5:E12) returns the total price of the products under the “Wafer” category. Method 4 – Adding Up If Cell Contains Text in Another Cell Using SUMIFS Function • Select cell C15. • Paste the following formula into it: ␥ Formula Breakdown: Syntax: SUMIFS(sum_range, criteria_range1, criteria1, [criteria_range2, criteria2], …) • E5: E12 is the sum range. • B5:B12 is the range where the SUMIFS function will look for the word “Wafer”. • “””&C14&””” refers to the address of the cell that contains the search keyword “Wafer”. • =SUMIFS(E5:E12,B5:B12,””&C14&””) returns the total price of the products under the “Wafer” category. Read More: How to Sum If Cell Contains Text in Another Cell in Excel Method 5 – Calculating the Total Price Based on Multiple Text Type (AND Criteria) Case 5.1 Summing If Cell Contains a Text Within a Single Column in Excel This time, we will calculate the total price of the products under the Biscuit and Candies category. • Select cell C15. • Type this formula: =SUM(SUMIF(B5:B12, {"Biscuit","Candies"},E5:E12)) ␥ Formula Breakdown: Syntax of the SUM function: SUM(number1,[number2],…) Syntax of the SUMIF function: SUMIF(range, criteria, [sum_range]) • B5:B12 is the range where the SUMIF function will look for the word “Wafer”. • “Biscuit”,”Candies” are the search keywords. • E5: E12 is the sum range. • =SUM(SUMIF(B5:B12, {“Biscuit”,”Candies”},E5:E12)) returns the total price of the products under the Biscuit and Candies category. Case 5.2 Summing If Cell Contains Text Within Multiple Columns in Excel Now we will try to calculate the total price of the products under the “Pasta” category and have the word “Ravioli” in their product name. • Go to cell C15. • Input the following: ␥ Formula Breakdown: Syntax: SUMIFS(sum_range, criteria_range1, criteria1, [criteria_range2, criteria2], …) • E5: E12 is the sum range. • B5:B12 is the range where the SUMIFS function will look for the word “Pasta”. • “Pasta”,”Ravioli” are the search keywords. • C5:C12 is the range where the SUMIFS function will look for the word “Ravioli”. • =SUMIFS(E5:E12,B5:B12,”Pasta”,C5:C12,”Ravioli”) returns the total price of the products under the “Pasta” category and have “Ravioli” in the product name. Method 6 – Calculating the Sum Value If the Cell Contains No Text in Excel This time, we will calculate the total price for the products whose categories are missing. • Select cell C15. • Input the following formula: =SUMIF(B5:B12, "", E5:E12) ␥ Formula Breakdown: Syntax: SUMIF(range, criteria, [sum_range]) • B5:B12 is the range where the SUMIF function will look for the missing category. • “” is specifies blank cell. • E5: E12 is the sum range. • =SUMIF(B5:B12, “”, E5:E12) returns the total price of the products whose categories are missing. Read More: How to Sum Only Numbers and Ignore Text in Same Cell in Excel Download Practice Workbook You are recommended to download the Excel file and practice along with it. Excel Sum If Cell Contains Text: Knowledge Hub << Go Back to Excel SUMIF Function \| Excel Functions \| Learn Excel Get FREE Advanced Excel Exercises with Solutions! 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Introduction To R Programming For Data Science Week 2 Introduction to R Programming for Data Science Week 2 Answers Course Name: Introduction to R Programming for Data Science Course Link: Click Here These are answers of Introduction to R Programming for Data Science Week 2 Practice Quiz Q1. What is the difference between the expression c(1, 2, 3, 4, 5) and the expression c(5:1)? a. They both produce a factor with five numbers but the first is in ascending order and the second is in descending order. b. The two expressions produce the same result. c. One produces a factor and the other produces a vector. d. They both produce a vector with five numbers but the first is in ascending order and the second is in descending order. Answer: d. They both produce a vector with five numbers but the first is in ascending order and the second is in descending order. Q2. Assume that the variable test_result contains the vector c(25, 35, 40, 50, 75). What is the result of the expression test_result[test_result < 50]? a. [1] TRUE TRUE TRUE TRUE FALSE b. [1] 25 35 40 50 c. [1] 25 35 40 d. [1] TRUE TRUE TRUE FALSE FALSE Answer: c. [1] 25 35 40 These are answers of Introduction to R Programming for Data Science Week 2 Q3. What is the main difference between a list and a vector? a. A list is a multi-dimensional array of values, while a vector is a single dimensional array of values. b. A list can contain nominal or ordinal values, while a vector cannot. c. It is not possible to add or remove items from a list, but you can do this with a vector. d. A list can contain different types of data, while a vector may only contain one type of data. Answer: d. A list can contain different types of data, while a vector may only contain one type of data. Q4. What are three types of data you can store in an array or matrix? Select three answers. a. Vectors b. Integers c. Numeric valus d. Strings Answer: b, c, d These are answers of Introduction to R Programming for Data Science Week 2 Q5. In a data frame, each column is represented by a _________________ of values of the same data type. a. Matrix b. List c. Variable d. Vector Answer: d. Vector These are answers of Introduction to R Programming for Data Science Week 2 Graded Quiz Q1. What is a nominal factor? a. A factor with any type or number of elements. b. A factor with ordering. c. A factor with no implied order. d. A factor that contains numeric data. Answer: c. A factor with no implied order. Q2. Assume that the variable test_result contains the vector c(25, 35, 40, 50, 75).What is the result of the expression mean(test_result)? a. 40 b. 45 c. 35 d. 50 Answer: b. 45 These are answers of Introduction to R Programming for Data Science Week 2 Q3. Assume you have variable called employee that contains the expression list(name = “Juan”, age = 30). What is the correct command to change the contents of the age item to 35? a. employee[“age”] == 35 b. employee[age] <- 35 c. employee[“age”] <- 35 d. employee[age] = 35 Answer: c. employee[“age”] <- 35 Q4. What is the main difference between a matrix and an array? a. A matrix can contain vectors, but an array can only contain strings, characters, or integers. b. A matrix can be arranged by rows or columns, but an array is always arranged by columns. c. A matrix can contain multiple types of data, but an array can only contain data of the same type. d. A matrix must be two dimensional, but an array can be single, two dimensional, or more than two dimensional. Answer: d. A matrix must be two dimensional, but an array can be single, two dimensional, or more than two dimensional. These are answers of Introduction to R Programming for Data Science Week 2 Q5. Assume that you have a data frame called employee that contains three variables: name, age, and title. If you want to return all the values in the title variable, what command should you use? a. employee[title] b. employee.title c. employee[[3]] d. employee$title Answer: d. employee$title These are answers of Introduction to R Programming for Data Science Week 2 All weeks answers of Introduction to R Programming for Data Science: Click Here These are answers of Introduction to R Programming for Data Science Week 2	{"url":"https://progiez.com/introduction-to-r-programming-for-data-science-week-2","timestamp":"2024-11-04T20:02:45Z","content_type":"text/html","content_length":"74871","record_id":"<urn:uuid:51eece20-4c1a-4f66-ace8-161a2125a218>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00703.warc.gz"}
Electric Current, Voltage, and Resistance Overview \| Three Basic Electrical Quantities \| Electrical A2Z Electric Current, Voltage, and Resistance Overview \| Three Basic Electrical Quantities Electric Current, voltage, and resistance are three of the fundamental electrical properties. Stated simply, • current: is the directed flow of charge through a conductor. • Voltage: is the force that generates the current. • Resistance: is an opposition to current that is provided by the material, component, or circuit. Electric Current, Voltage, and resistance are the three primary properties of an electrical circuit. The relationships among them are defined by the fundamental law of circuit operation, called Ohm’s Electric Current As you know, an outside force can break an electron free from its parent atom. In copper (and other metals), very little external force is required to generate free electrons. In fact the thermal energy (heat) present at room temperature (22^0C) can generate free electrons. The number of electrons generated varies directly with temperature. In other words, higher temperatures generate more free electrons. The motion of the free electrons in copper is random when no directing force is applied. That is, the free electrons in copper are random when no directing force is applied. That is, the free electrons move in every direction, as shown in Figure 1. Since, the free electrons are moving in every direction, this net flow of electrons in any direction is zero. Figure 1 Random electron motion in copper Figure 2 Illustrates what happens when an external force causes all of the electrons to move in the same direction. In this case, a negative potential is applied to one end of the copper and a positive potential is applied to the other. As a result, the free electrons all move from negative to positive, and we can say that we have a directed flow of charge (electrons). This directed flow of electrons is called electric current. Figure 2 Directed electron motion in copper. Let’s look at what happens on a larger scale when electron motion is directed by an outside force. In Figure 3, the negative potential directs electron flow (current) toward the positive potential. The current passes through the lamp, causing it to produce light and heat. The more intense the current (meaning the greater its value), the greater the light and heat produced by the bulb. Figure 3 Current through a basic lamp circuit. Electric Current is represented in formulas by the letter I (For intensity). The intensity of current is determined by the amount of charge flowing per second. The greater the flow of charge per second, the more intense the current. Coulombs and Amperes The change on a single electron is not sufficient to provide in a practical unit of measure for charge. Therefore, the Coulomb (C) is used as the basic unit of charge One coulomb equals the total charge on 6.25 × 10^18 electrons. When, one coulomb of charge passes a point in one second, we have one Ampere (A) of electric current. In other words, $\begin{matrix}1\text{ ampere=1 Coulomb per second} \\Or \\1\text{ A=1 C/s} \\\end{matrix}$ The total current passing a point (in amperes) can be found by dividing the total charge ( in coulombs) by the time ( in seconds) . By Formula \[\begin{matrix}I=\frac{Q}{t} & {} & \left( 1 \right) \\\end{matrix}\] I= the intensity of electric current in amperes Q= the total charge, in coulombs T= the time it takes the charge to pass a point, in seconds This relationship is illustrated in Example 1. Example 1 Three coulombs of charge pass through a copper wire every second. What is the value of electric current? Using equation 1, the current is found as Example 1 is included here to help you understand the relationship between amperes, coulombs, and seconds. In practice, electric current is not calculated using Equation 1 because you cannot directly measure coulombs in charge. As you will learn, there are far more practical ways to calculate current. Two Theories: Conventional Current and Electron Flow. There are two theories that describe electric current, and you will come across both in practice. The Conventional Current theory defines current as the flow of charge from positive to negative. This theory is called “conventional current” because it is the older of the two approaches to current, and for many years was the only one taught outside of military and trade schools. Electron Flow is the newer of the two current theories. Electron flow theory defines current as the flow of electrons from negative to positive. The two electric current theories are contrasted in Figure 4. Each circuit contains a battery and a lamp. Conventional current begins at the positive battery terminal, passes through the lap, and returns to the battery through its negative terminal. Electron flow is in the opposite direction: It begins at the negative terminal, passes through the lamp, and returns to the battery through its positive terminal. Figure 4 Conventional current and electron flow. It is worth nothing that the two circuits in Figure 4 are identical. The only difference between the two is how we describe the electric current. In practice, how you view current does not affect any circuit calculations, measurements, or test procedures. Even so you should get comfortable with both viewpoints, since both are used by many engineers technicians, and technical publications. In this text, we take the electron flow approach to current. That is, we will assume current is the flow of electrons from negative to positive. Direct Current (DC) Versus Alternating Current (AC) Current is generally classified as being either Direct Current (DC) or Alternating Current (AC). The differences between direct current and alternating current are illustrated in Figure 5. Figure 5 Direct current (DC) and alternating current (AC). Direct current is unidirectional. That is, the flow of charge is always in the same direction. The term direct current usually implies that the current has a fixed value. For example, the graph in Figure 5a shows that the current has a constant value of 1A. While a fixed value is implied, direct current can change in value. However, the direction of current does not change. Alternating Current is Bidirectional. That is, the direction of current changed continually. For example, in figure 5b, the graph shows that the current builds to a peak value in one direction and then builds to a peak value in the other direction. Note that the alternating current represented by the graph not only changes direction but is constantly changing in value. Electric Current Produces Heat Whenever electric current is generated through a component or circuit, heat is produced. The amount of heat varies with the level of current: The greater the current, the more heat it produces. This is why many high-current components, like motors, get hot when they are operated. Some High-current circuits get so hot that they have to be cooled. The heat produced by electric current is sometimes a desirable thing. Toasters, electric stoves, and heat lamps are common items that take advantage of the heat produced by current. Figure 6 High current causes a stove heating element burner to glow red. Putting it all together Free electrons are generated in copper at room temperature. When undirected, the motion of these free electrons is random, and the net flow of electrons in any one direction is zero. When directed by an outside force, free electrons are forced to move in a uniform direction. This directed flow of charge is referred to as electric current. Electric Current is represented by the letter I, which stands for intensity. The intensity of current depends on the amount of charge moved and the time required to move it. Electric Current is measured in amperes (A). When one coulomb of current passes a point every second, you have one ampere of current. There are two current theories. The electron flow theory describes current as the flow of charge (electrons) from negative to positive. The conventional current theory describes current as the flow of charge from positive to negative. Both approaches are widely followed. The way you view current does not affect the outcome of any circuit calculations, measurements, or test procedures. Most electrical and electronic systems contain both direct current (DC) and alternating current (AC) circuits. In DC circuits, the current is always in the same direction. In AC circuits, current continually changes direction. Review Questions How are free electrons generated in a conductor at room temperature? The thermal energy (heat) present at room temperature is enough to generate free electrons. What is electric current? What factors affect the intensity of electric current? Current is the directed flow of electrons in a material. The intensity of current depends on the amount of charge moved and the time required to move it. What is a coulomb? One coulomb equals the total charge on 6.25 × 10^18 electrons. What is he basic unit of electric current? The ampere is the basic unit of electric current. It is defined as 1 coulomb per second or 1 A = 1 C/s. Contrast the electron flow and convention current theories. Conventional current theory defines current as the flow of charge from positive to negative. Electron flow is the flow of charge from negative to positive. Voltage can be described as a force that generates the flow of electrons (current) through a circuit. In this section, we take a detailed look at voltage and how it generates current. Generating Current with a Battery Battery in Figure 7a has two terminals. The positive (+) terminal has an excess of positive ions and is described as having a positive potential. The negative (-) terminal has an excess of electrons and is described as having a negative potential. Figure 7 A difference of potential and a resulting current. Thus there is a Difference of Potential, or voltage (V), between the two terminals. If we connect the two terminals of the battery with the copper wire and lamp (Figure7b), a current is produced as the electrons are drawn to the positive terminal of the battery. In other words, there is a directed flow of electrons from the negative (-) terminal to the positive (+) terminal of the battery. There are several important points that need to be made: 1. Voltage is a force that moves electrons, for this reason, it is often referred to as Electrical Force (E) or Electromotive Force (EMF). 2. Current and voltage is not the same thing. Current is the directed flow of electrons from negative to positive. Voltage is the electrical force that generates current. In other words, current occurs as a result of an applied voltage (electric force). The volt (V) is the unit of measure for voltage. Technically defined, one volt is the amount of electrical force that uses one joule (J) of energy to move one coulomb (C) of charge that is, \[& \text{1 volt= 1 joule per coulomb} \\& \text{or} \\& \text{1 V=1 }{}^{J}/{}_{C} \\\] Review Questions What is voltage? Voltage is the force that generates current in a circuit. How does voltage generate a current through a wire? A voltage source has an excess of electrons (negative charge) on one terminal and an excess of positive ions on the other. This is referred to as a potential difference. The excess electrons at the negative terminal are attracted by the positive ions on the positive terminal. This results in the flow of charge in any wire that connects the two terminals of the voltage source. What is the unit of measure for voltage? How is it defined? The unit of measure for voltage is the volt. One volt is the amount of electrical force that uses one joule (J) of energy to move one coulomb (C) of charge. 1 V = 1 J/C. How would you define a coulomb in terms of voltage and energy? 1 coulomb equals 1 joule per volt, 1 C = 1 J/V How would you define a joule in terms of voltage and charge? 1 joule equals 1 V times 1 coulomb, 1 J = 1 V × 1 C All elements provide some opposition to current. This opposition to current is called resistance. The higher the resistance of an element, component, or circuit, the lower the current produced by the given voltage. Resistance (R) is measured in Ohms. Ohms are represented using the Greek letter omega (Ω). Technically defined, one ohm is the amount of resistance that limits current to one ampere when one volt of electrical force is applied. This definition is illustrated in Figure 8. Figure 8 A basic electric circuit. The schematic diagram in Figure 8 shows a battery that is connected to a resistor. A resistor is a component that provides a specific amount of resistance. As shown in the figure, a resistance of 1Ω limits the current to 1A when 1V is applied. Note that the long end-bar on the battery schematic symbol represents the battery’s positive terminal and the short end bar represents its Negative Putting It All Together We have now defined charge, current, voltage and resistance. For convenience, these electrical properties are summarized in Table 1. Table 1: Basic Electrical Properties Many of the properties listed in Table 1 can be defined in terms of the others. For example, in our discussion on resistance, we said that one ohm is the amount of resistance that limits current to one amp when one volt of electrical force is applied. By the same token, we can redefine the ampere and the volt as follows: 1. One ampere is the amount of current that is generated when one volt of electrical force is applied to one ohm of resistance. 2. One volt is the amount of electrical force required to generate one amp of current through one ohm of resistance. Review Question What is resistance? Resistance is the opposition to current. What is the basic unit of resistance and how is it defined? The unit of resistance is the Ohm (Ω). One ohm is the amount of resistance that limits current to 1 ampere when one volt is applied. 1 Ω = 1 A/V. Define each of the following values in terms of the other two: current, voltage, and resistance. 1 V is the force required to cause 1 ampere of current through 1 ohm of resistance. 1 A is the current that results from when 1 volt is applied to 1 ohm of resistance. 1 Ω is the resistance that limits current to 1 ampere when 1 volt is applied.	{"url":"https://electricala2z.com/electrical-circuits/electric-current-voltage-and-resistance-overview-three-basic-electrical-quantities/","timestamp":"2024-11-04T14:45:20Z","content_type":"text/html","content_length":"147196","record_id":"<urn:uuid:ba02e3b4-7c31-4a73-ba8f-8bae2304695c>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00449.warc.gz"}
Increasing frequency of a sinusoids after FFT of a signal 5 years ago ●7 replies● latest reply 5 years ago 238 views Hi everyone, I am new to this forum. I have a signal composite of 500Hz, 1700Hz and 2500Hz signal. Sampling rate is 48K and number of data is of 1 s that is 48K too. I need to increase the 1700Hz frequency to 1700.50Hz without changing amplitude and phase. How can I achieve this ? Signal -> FFT -> [Process] -> IFFT -> Signal What could be the Process to achieve the frequency? Any help will be higly appreciated? [ - ] Reply by ●November 1, 2019 I have some of the same questions. Taking a stab at what you want, though... I would probably window the data to reduce spreading and take a 192000 point fft. Then, I'd grab the +/- 20 bins surrounding the peak corresponding to the 1700Hz (centered at 6800) and shift the entire block two bins to the right. Then I'd grab the +/- 20 bins surrounding its mirrored counterpart (centered at 185200) and shift them two bins to the left. You'll have two redundant fft entries on each side leftover from your copy, but they are 20 bins away from anything of interest. Then, take the inverse fft and the real part of the first 48000 complex values that get spit out may be what you are looking for. You will want to check it first, though. :) I have no idea of your tolerance to noise and discontinuities and asymmetry and... This is definitely treating the data roughly. But - depending upon what you can live with - that might do it. [ - ] Reply by ●November 1, 2019 during this change of frequency of 1700.0Hz to 1700.5Hz what happens to the rest of the spectrum? i.e. should the 500Hz and 2500Hz remain unchanged? [ - ] Reply by ●November 1, 2019 If I didn't miss the point: changing the frequency cannot leave the phase unchanged, because a higher frequency means the phase is changing faster, so you cannot have unchanged phase, or? [ - ] Reply by ●November 1, 2019 Can you tell us what problem you're trying to solve? Maybe a FFT/IFFT isn't the right approach. (here comes the 20 questions) If I understand it the data is 1 second long? What comes before and after? How do you synchronize your data capture to where the data you want to change is? If the tones are bursts then the "artifacts" from those transitions have to be considered. Is the amplitude constant? If not then the effects of the signal modulation would need to be considered. I read it as you have 1700 cycles in and you want 1700.5 cycles out in the same time period; you don't mention noise but that's going to play in to any real system and maybe break it. How accurate is the source in the first place, are you trying to get a specific number out or change by a %? [ - ] Reply by ●November 1, 2019 the idea of djmaguire is very good. For the frequency of 1700.5 HZ you have to increase the resolution of the FFT to 0.5 Hz / Bin, so you need a data block of 2 seconds with a length of 2 * 48000 values. In the FFT of your signals, you must remove the line of frequency 1700 Hz and insert an 'artificial' line for the frequency of 1700.5 Hz. I give you a MATLAB script in which the artificial signal is generated from the artificial FFT for a given frequency (such as 1700.5 Hz). % Script freq_change_1.m to change the Freq. 1700 Hz % to Freq 1700,5 Hz % -------- Initialisations fs = 48000; % Sample Freq. Ts = 1/fs; N = 96000; % Datablock t = 0:Ts:(N-1)Ts; f1 = 1700; % Beginning Freq. f2 = 1700.5; % Freq. to be generated from FFT ampl1 = 2.5; % Amplit. phi = pi/3; x1 = ampl1cos(2pif1t + phi); % Signal of Freq. f1; x2 = ampl1cos(2pif2t + phi); % Signal of Freq. f2 for FFT test; subplot(211), plot(t,x1); La = axis; axis([1,1.01,La(3:4)]); title(['Signal of Freq. ',num2str(f1),'Hz']); xlabel('Time in s'); grid on; subplot(212), plot(t,x2); La = axis; axis([1,1.01,La(3:4)]); title(['Signal of Freq. ',num2str(f2),'Hz']); xlabel('Time in s'); grid on; X1 = fft(x1)/N; % FFT of Signal with Freq. f1 X2 = fft(x2)/N; % FFT of Signal with Freq. f2 subplot(211), stem((0:N-1), abs(X1)); title('abs FFT of Signal x1'); xlabel('Bins of the FFT'); subplot(212), stem((0:N-1), abs(X2)); title('abs FFT of Signal x2'); xlabel('Bins of the FFT'); % ------- Generating the Signal of Freq. f2 from artificial FFT delta_f = fs/N; % Resolution of the FFT m10 = f1/delta_f, % MATLAB Indexes m20 = N - m10, % ------- The artificial FFT of the Signal with f2 Hz X2 = zeros(1,N); m1 = f2/delta_f, % Bin of the FFT (in the first Nyquist Intervall) m2 = N-m1, % Bin of the FFT (in the second Nyquist Intervall) X2(m1+1) = (ampl1/2)exp(jpi/3); % Artificial FFT X2(m2+1) = (ampl1/2)exp(-jpi/3); figure(3), stem((0:N-1), abs(X2)); title('The artificial FFT from which the Signal x2g is generated') xlabel('Bins der FFT'); x2g = (ifft(X2))N; % Generated Signal from the FFT subplot(211), plot(t,x1); La = axis; axis([1,1.01,La(3:4)]); title(['Signal of Freq. ',num2str(f1),'Hz']); xlabel('Time in s'); grid on; subplot(212), plot(t,x2); La = axis; axis([1,1.01,La(3:4)]); title(['Signal of Freq. ',num2str(f2),... ' Hz generated from the artificial FFT']); xlabel('Time in s'); grid on; [ - ] Reply by ●November 1, 2019 You have tones from somewhere(signal?) but want to generate/regenerate tones at certain frequencies from fft/ifft. I am lost in the reasoning here, sorry. [ - ] Reply by ●November 1, 2019 Assuming you have the spectrum in complex form you band pass filter out the 1700 Hz, multiple the result by a .5 Hz complex sinusoid and add the result back into the output.	{"url":"https://dsprelated.com/thread/9738/increasing-frequency-of-a-sinusoids-after-fft-of-a-signal","timestamp":"2024-11-07T00:44:38Z","content_type":"text/html","content_length":"44088","record_id":"<urn:uuid:a1ac83e7-58cc-4202-8dc7-ac063cf95431>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00682.warc.gz"}
You're Speaking My Landscape, Baby. No, that isn't a typo... but yes, it is a bad play on words. That's the bad news. The good news: finally! A Clockwork Aphid implementation post! If you're building something which relates in some way to virtual worlds, then the first thing you're going to need is a virtual world. This gives you two options: 1. Use a ready made one; 2. Roll your own. Option 1 is a possibility, and one that I'm going to come back to, but for now let's think about option 2. So then, when building a virtual world the first thing you need is the lanscape. Once again you have two options, and let me just cut this short and say that I'm taking the second one. I did used to be a bit of a CAD ninja in a previous job, but I'm not a 3D modeller and I have no desire to build the landscape by hand. So I'm going to generate one procedurally. As to what that means exactly, if you don't already know... well I'm hoping that will become obvious as I go along. Traditional Fractal Landscape Generation There are several ways of generating a landscape. A pretty good one (and one I'm quite familiar with, thanks to a certain first year computer science assignment) is the fractal method. It goes something like this: Start off with a square grid of floating point numbers, the length of whose sides are a power of two plus one. I'm going to use a 55 (22 + 1) grid for the purposes of this explanation. Set the four corners to have the value 0.5 (the centre point of the range I'll be using), thus: Now, we're going to generate the landscape by repeatedly subdividing this and introducing fractal randomness (random fractility?) using the diamond square algorithm. First the diamond step, which in this iteration will the set the value of the central cell based on the value of the four corners: To do this we take the average of the four corners (which I calculate to be 0.5 in this case, because I did maths at school) and adding a small randomly generated offset, which has been scaled according to the size of the subdivision we're making. How exactly you do this varies between implementations, but a good simple way of doing it is use a random number in the range [-0.25,0.25] at this stage and half this at each subsequent iteration. So, on this occasion let's say I roll the dice and come up with 0.23. This now leaves us with: Next, we have the square step, will set the cells in the centre of each of the sides. Once again we take the averages of other cells as starting point, this time in the following pattern: Now we generate more random numbers in the same range and use them to offset the average values, giving us this: That completes an iteration of the algorithm. Next we half the size of the range to [-0.125,0.125] and start again with the diamond step: ...and so on until you've filled your grid. I think you get the general idea. I've left out one potentially important factor here and that's "roughness," which is an extra control you can use to adjust the appearance of the landscape. I'm going to come back to that in a later post, because (hopefully) I have a little more that I want to say about it. I need to play with it some more first, Once you've finished here you can do a couple of different things if you want to actually look at your creation. The simplest is to pick a threshold value and call this "sea level," then draw the grid as an image with pixels set to blue below the threshold and green above it: This was obviously generated with a slightly larger grid (513*513), but as you can see it creates quite reasonable coastlines. You can do slightly fancier things with it, such as more in depth colouring schemes and 3D display. For 3D, the simplest method is to use each cell as a vertex in your 3D space and tessellate the grid into triangles like this: You can then do a couple of fancy things to remove the triangles you don't need, either based on the amount of detail they actually add or their distance from the user (level of detail). This system works quite well, but tends to produce quite regular landscapes, without of the variation we're used to or the things generated by rivers, differing geology, coastal erosion, glaciation and other forces which affect the landscape of the real world. Additionally, because the data is stored in a height map, there are some things it's just not capable of displaying, such as shear cliffs, overhangs, and cave systems. The grid structure is also very efficient, but quite inflexible. How I'm Doing it Needless to say that's not exactly how I'm doing it. Of course there's generally very little sense in reinventing the wheel, but sometimes it's fun to try. I'm not doing too much differently with the actual algorithm, but I am using a slightly different data representation. Rather than a grid, I'm using discrete nodes. So you start off with something Which then is transformed like this to generate the actual landscape: What you you can't see from the diagrams is that I'm using fractions to address the individual nodes. So, for instance, the node in the centre is (1/2,1/2) and the node on the centre right is (1/1, 1 /2). This means I don't need to worry about how many nodes I have in the landscape, and the adress of each never has to change. The next set of nodes will be addressed using fractions with 4 as the denominator, then 8, 16 and so on. Before looking up a node you first reduce its coordinates down to a lowest common denominator (which is a factor of 2) and then pull it out of the correct layer. I'm currently using maps as sparse arrays to store the data in a structure which looks like this: Map<int, Map<int, Map<int, LandscapeNode> If you're thinking that this isn't possible in Java, you're half right. I'm actually using one of these. The first int addresses the denominator, then the east numerator, then the north numerator. I've looked at another couple of strategies for hashing the three ints together to locate a unique node but this one seems to work the best in terms of speed and memory usage. I might look at other options later, but nor yet. This is a much more flexible representation, which removes some of the limitations of the grid. I can keep adding more detail to my heart's content (up to a point) and I don't have do it in a regular fashion. i.e. the native level of detail doesn't have to be the same across the entire map. More remote areas can have less detail, for instance. By the same token, I can keep the entire "landscape" in memory, but flexibly pull individual nodes in or out depending on where the user actually is in the world, saving memory. This also potentially gives me the following: 1. The possibility to decouple the geometry of the landscape from the topography of the representation; 2. A "native" way of implementing different levels of detail; 3. A natural tessellation strategy based on connecting a node to its parents (maybe you spotted it); 4. Enough data to allow the landscape to be modified to produce more dramatic features across different levels of detail; 5. The processes for the above should be very parallelisable. There are still a couple of things I'm working on (3D display for a start), as I've been obsessing over how to organise the data structures I'm using. Hopefully I'll be back tomorrow with some 3D If you're interested in the code you can find it here. If what you found at the end of that link didn't make any sense to you, then you're probably not a programmer (or you're still learning). If you still want a look drop me a comment and I'll see what I can do. Disclaimer: As far as I'm aware I didn't steel this from anybody, but I don't claim it's completely original, either.	{"url":"https://harveynick.com/2010/10/19/youre-speaking-my-landscape-baby/","timestamp":"2024-11-04T15:05:14Z","content_type":"text/html","content_length":"30587","record_id":"<urn:uuid:24b59693-5f81-48bb-bb01-bba2df04cf27>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00651.warc.gz"}
Data Science and Machine Learning Bootcamp with R - reason.townData Science and Machine Learning Bootcamp with R Data Science and Machine Learning Bootcamp with R Our Data Science and Machine Learning Bootcamp with R is designed for those who want to learn the skills necessary to become a data scientist or machine learning engineer. This bootcamp will teach you the basics of the R programming language and how to use it for data science and machine learning. You’ll also learn how to use popular data science and machine learning libraries such as caret, dplyr, and ggplot2. Checkout this video: Introduction to Data Science and Machine Learning Data science and machine learning are becoming increasingly important in the modern world. Businesses are looking for employees with these skills, and individuals with these skills are in high demand. A bootcamp is one way to gain these skills. 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They will also learn about model selection methods (e.g., AIC, BIC) and how to interpret the results of statistical models. -Machine Learning in R: Students will learn about different types of machine learning algorithms (e.g., decision trees, k-nearest neighbors) and how to implement these algorithms in R using the caret package. They will also learn about important concepts such as overfitting and cross-validation. The Instructors for a Data Science and Machine Learning Bootcamp At a data science and machine learning bootcamp, you will learn from some of the top data scientists and machine learning engineers in the world. The instructors for a data science and machine learning bootcamp are typically experienced professionals who have years of experience in the field. They will be able to provide you with an immersive learning experience that will help you to become a data scientist or machine learning engineer. The Admissions Process for a Data Science and Machine Learning Bootcamp There are a few key things you need to do in order to be admitted into a data science and machine learning bootcamp. The first is to take and pass an entrance exam. This will help to ensure that you have the basic skills needed for the program. Next, you will need to submit an application. This should include your resume, transcripts, and a letter of recommendation. Once your application has been reviewed, you will be contacted for an During the interview, you will be asked about your qualifications and why you want to attend the bootcamp. Be sure to answer these questions honestly and effectively. If you are selected for the program, you will be required to pay a deposit in order to secure your spot. The Outcomes of a Data Science and Machine Learning Bootcamp At a data science and machine learning bootcamp, you will learn the skills and knowledge necessary to become a data scientist or machine learning engineer. 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The average cost of a four-week data science bootcamp is $11,400, while the average cost of an eight-week data science bootcamp is $17,600. There are a few things to keep in mind when considering the cost of a data science or machine learning bootcamp. First, many programs offer scholarships or financing options to help offset the cost. Second, most programs guarantee job placement upon completion, so the ROI on a bootcamp can be very high. Finally, many bootcamps now offer deferred tuition plans, where you don’t have to pay anything up front and only start making payments after you land a job. If you’re looking to get into data science or machine learning, a bootcamp is an excellent investment. With so many options available, there’s sure to be a program that fits your budget and your The Schedule for a Data Science and Machine Learning Bootcamp A Data Science and Machine Learning Bootcamp with R will help you learn the basics of data science and machine learning. The schedule for the bootcamp is as follows: -Day 1: Introduction to Data Science -Day 2: Basic Data Manipulation -Day 3: Introduction to Machine Learning -Day 4: Supervised Machine Learning -Day 5: Unsupervised Machine Learning -Day 6: Validation, Testing, and Model Deployment -Day 7: Capstone Project How to Choose a Data Science and Machine Learning Bootcamp Choosing a data science or machine learning bootcamp is a big decision. With so many programs out there, it’s important to do your research to find the right fit for you. Here are some things to keep in mind when choosing a data science or machine learning bootcamp: – Length of program: Some bootcamps are as short as 10 weeks, while others can last up to 6 months. Consider your schedule and whether you can commit to a full-time program. – Curriculum: Make sure the bootcamp covers topics that you’re interested in and that will be helpful for your career goals. – Teaching style: Some bootcamps offer more traditional, classroom-based instruction, while others use a more hands-on, project-based approach. Consider which teaching style is right for you. – Location: Bootcamps are offered in cities all across the country (and even online). Consider whether you want to stay close to home or if you’re willing to travel for the right program. Do your research and take your time in choosing a data science or machine learning bootcamp. With so many great programs out there, you’re sure to find one that’s a perfect fit for you.	{"url":"https://reason.town/data-science-and-machine-learning-bootcamp-with-r/","timestamp":"2024-11-14T17:34:31Z","content_type":"text/html","content_length":"103073","record_id":"<urn:uuid:b8899f16-1d77-4c31-a123-275904919267>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00432.warc.gz"}
Lesson 22 Combining Like Terms (Part 3) Let’s see how we can combine terms in an expression to write it with less terms. 22.1: Are They Equal? Select all expressions that are equal to $8-12-(6+4)$. 1. $8-6-12+4$ 2. $8-12-6-4$ 3. $8-12+(6+4)$ 4. $8-12-6+4$ 5. $8-4-12-6$ 22.2: X’s and Y’s Match each expression in column A with an equivalent expression from column B. Be prepared to explain your reasoning. 1. $(9x+5y) + (3x+7y)$ 2. $(9x+5y) - (3x+7y)$ 3. $(9x+5y) - (3x-7y)$ 4. $9x-7y + 3x+ 5y$ 5. $9x-7y + 3x- 5y$ 6. $9x-7y - 3x-5y$ 1. $12(x+y)$ 2. $12(x-y)$ 3. $6(x-2y)$ 4. $9x+5y+3x-7y$ 5. $9x+5y-3x+7y$ 6. $9x-3x+5y-7y$ 22.3: Seeing Structure and Factoring Write each expression with fewer terms. Show or explain your reasoning. 1. $3 \boldcdot 15 + 4 \boldcdot 15 - 5 \boldcdot 15 $ 2. $3x + 4x - 5x$ 3. $3(x-2) + 4(x-2) - 5(x-2) $ 4. $3\left(\frac52x+6\frac12\right) + 4\left(\frac52x+6\frac12\right) - 5\left(\frac52x+6\frac12\right)$ Combining like terms is a useful strategy that we will see again and again in our future work with mathematical expressions. It is helpful to review the things we have learned about this important • Combining like terms is an application of the distributive property. For example: $\begin{gather} 2x+9x\\ (2+9) \boldcdot x \\ 11x\\ \end{gather}$ • It often also involves the commutative and associative properties to change the order or grouping of addition. For example: $\begin{gather} 2a+3b+4a+5b \\ 2a+4a+3b+5b \\ (2a+4a)+(3b+5b) \\ 6a+8b\\ \end{gather}$ • We can't change order or grouping when subtracting; so in order to apply the commutative or associative properties to expressions with subtraction, we need to rewrite subtraction as addition. For $\begin{gather} 2a-3b-4a-5b \\ 2a+\text-3b+\text-4a+\text-5b\\ 2a + \text-4a + \text-3b + \text-5b\\ \text-2a+\text-8b\\ \text-2a-8b \\ \end{gather}$ • Since combining like terms uses properties of operations, it results in expressions that are equivalent. • The like terms that are combined do not have to be a single number or variable; they may be longer expressions as well. Terms can be combined in any sum where there is a common factor in all the terms. For example, each term in the expression $5(x+3)-0.5(x+3)+2(x+3)$ has a factor of $(x+3)$. We can rewrite the expression with fewer terms by using the distributive property: $\begin{gather} 5(x+3)-0.5(x+3)+2(x+3)\\ (5-0.5+2)(x+3)\\ 6.5(x+3)\\ \end{gather}$ • expand To expand an expression, we use the distributive property to rewrite a product as a sum. The new expression is equivalent to the original expression. For example, we can expand the expression $5(4x+7)$ to get the equivalent expression $20x + 35$. • factor (an expression) To factor an expression, we use the distributive property to rewrite a sum as a product. The new expression is equivalent to the original expression. For example, we can factor the expression $20x + 35$ to get the equivalent expression $5(4x+7)$. • term A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together. For example, the expression $5x + 18$ has two terms. The first term is $5x$ and the second term is 18.	{"url":"https://im-beta.kendallhunt.com/MS/students/2/6/22/index.html","timestamp":"2024-11-04T10:54:11Z","content_type":"text/html","content_length":"69509","record_id":"<urn:uuid:73634223-ad3c-424f-86b4-450489ac3d32>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00457.warc.gz"}
Symbolic Integrators Symbolic Integrators¶ Symbolic integrators can be used to define arbitrary (multi-)physical problems. The variational formulation of the (non-)linear problem can be implemented in a very natural way. Examples of the usage of these integrators can i.e. be found in Navier Stokes Equation, Magnetic Field or Nonlinear elasticity. The finite element space provides placeholder coefficient functions by the methods TestFunction and TrialFunction. They have implemented canonical derivatives and traces of the finite element space and insert the basis functions of the FESpace during assembling of the system matrices. Linear Problems¶ For linear problems we use the function Assemble of the BilinearForm to assemble the matrix and vector. For example for the $H_\text{curl}$ linear problem \[\begin{split}\int_\Omega \mu^{-1} \nabla \times u \cdot \nabla \times v + 10^{-6} u \cdot v \, dx = \int_C \begin{pmatrix} y \\ -x \\ 0 \end{pmatrix} \cdot v \, dx\end{split}\] from example Magnetic Fields we have to define the space fes = HCurl(mesh, order=4, dirichlet="outer", nograds = True) and the BilinearForm a = BilinearForm(fes, symmetric=True) a += nucurl(u)curl(v)dx + 1e-6nuuvdx as well as the LinearForm The argument of the symbolic integrator must be a coefficient function depending linearly on the test and trial function. BilinearForm.Assemble(self: ngsolve.comp.BilinearForm, reallocate: bool = False) ngsolve.comp.BilinearForm¶ Assemble the bilinear form. input reallocate Nonlinear Problems¶ If your left hand side of the variational formulation is nonlinear there are multiple ways to get a discretisation, depending on what you want. The function Apply applies the formulation to the given BaseVector. You can get a BaseVector form your GridFunction with GridFunction.vec. The output vector can be created with BilinearForm.Apply(args, *kwargs)¶ Overloaded function. 1. Apply(self: ngsolve.comp.BilinearForm, x: ngsolve.la.BaseVector, y: ngsolve.la.BaseVector) -> None Applies a (non-)linear variational formulation to x and stores the result in y. input vector output vector 2. Apply(self: ngsolve.comp.BilinearForm, u: ngsolve.la.BaseVector) -> ngsolve.la.DynamicVectorExpression For a variational formulation \[\int_\Omega f(u) v \, dx\] the method AssembleLinearization computes \[\int_\Omega f'(u_\text{lin}) u v \, dx\] with automatic differentiation of $f(u)$ and an input BaseVector $u_\text{in}$. BilinearForm.AssembleLinearization(self: ngsolve.comp.BilinearForm, ulin: ngsolve.la.BaseVector, reallocate: bool = False) None¶ Computes linearization of the bilinear form at given vecor. input vector You can do your own linearization as well using Assemble and a GridFunction as a CoefficientFunction in your integrator. Let gfu_old be this gridfunction then a = BilinearForm(fes) a += SymbolicBFI(gfu_old u * v) will be a linearization for \[\int_\Omega u^2 v \, dx\] Every time you call Assemble the bilinearform is updated with the new values of the GridFunction. Symbolic Energy¶ SymbolicEnergy can be used to solve a minimization problem. In this tutorial we show how to solve the nonlinear problem \[\min_{u \in V} 0.05 \nabla u + u^4 - 100u\] For this we use SymbolicEnergy: a = BilinearForm (V, symmetric=False) a += Variation( (0.05grad(u)grad(u) + uuuu - 100u)dx ) from the GridFunction we can create new BaseVector: With this we can use AssembleLinearization to do a Newton iteration to solve the problem: for it in range(20): print ("Newton iteration", it) print ("energy = ", a.Energy(u.vec)) a.Apply (u.vec, res) a.AssembleLinearization (u.vec) inv = a.mat.Inverse(V.FreeDofs()) w.data = inv res print ("w*r =", InnerProduct(w,res)) u.vec.data -= w	{"url":"https://docu.ngsolve.org/latest/how_to/symbolic_integrators.html","timestamp":"2024-11-04T11:58:29Z","content_type":"text/html","content_length":"30280","record_id":"<urn:uuid:f5a94bd4-5f05-40df-a878-b70d899f35b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00417.warc.gz"}
What is Applied Mathematics? \| Applied Mathematics These days mathematics is applied almost everywhere, and this wealth of activity cannot be encompassed in one university department. The focus of the Applied Mathematics Department at Waterloo is the application of mathematics to problems in science, engineering, and medicine. Our undergraduate programs are based on courses that provide a strong mathematical and computational background, while offering a selection of courses in areas of application. These application areas are quite diverse, and reflect the research interests of members of the department. For example, the behaviour of fluids and their motions is essential to our very existence on this planet. Think of the oceans, the atmosphere, the earth's crust, underground fossil fuels. Describing the flow of fluids, including the waves that travel within them, comprises the subject of fluid dynamics. A sequence of two courses will introduce you to this fascinating area. Perhaps you are interested in learning about chaotic dynamics, the unpredictable behaviour of nonlinear systems, or how engineers design control systems, which are used in diverse areas such as robotics, aerospace engineering and biomedical research. We have senior level courses in each of these areas. Have you ever wondered about Einstein's theory of relativity, one of the revolutions in physics of the twentieth century? Or about how quantum mechanics - the physics of very small scales - differs from the classical mechanics of everyday life? If so, you may be interested in a course in general relativity or quantum mechanics. Of course all of these applications require a strong mathematical background, which is developed in the first three years, starting with Calculus and Linear Algebra. An education in Applied Mathematics: Education gives you not only knowledge, but also the ability to organize and use that knowledge profitably. Every Applied Mathematics course is geared toward providing the students with the ability to use a variety of mathematical and computational tools to solve problems in various fields.	{"url":"https://uwaterloo.ca/applied-mathematics/future-undergraduates/what-applied-mathematics","timestamp":"2024-11-08T14:52:34Z","content_type":"text/html","content_length":"115556","record_id":"<urn:uuid:bfae5bfe-5de0-487f-af23-89e2a50ba17b>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00324.warc.gz"}
[Solved] From a group of 13 boys and 9 girls, a co \| SolutionInn Answered step by step Verified Expert Solution From a group of 13 boys and 9 girls, a committee of 5 students is chosen at random. a. The probability that all 5 members From a group of 13 boys and 9 girls, a committee of 5 students is chosen at random. a. The probability that all 5 members on the committee will be girls is (Type an integer or a simplified fraction.) Part 2b. The probability that all 5 members on the committee will be boys is (Type an integer or a simplified fraction.) Part 3c. The probability that there will be at least 1 girl on the committee is (Type an integer or a simplified fraction.) There are 3 Steps involved in it Step: 1 we are calculated using the binomial coefficient formula Cn k n kn k where n is the total number of items k is the number of items to choose and denot... Get Instant Access to Expert-Tailored Solutions See step-by-step solutions with expert insights and AI powered tools for academic success Ace Your Homework with AI Get the answers you need in no time with our AI-driven, step-by-step assistance Get Started Recommended Textbook for Authors: John J. Coyle, Robert A. Novak, Brian Gibson, Edward J. Bard 8th edition 9781305445352, 1133592961, 130544535X, 978-1133592969 More Books Students also viewed these Mathematics questions View Answer in SolutionInn App	{"url":"https://www.solutioninn.com/study-help/questions/from-a-group-of-13-boys-and-9-girls-a-1005460","timestamp":"2024-11-06T07:18:08Z","content_type":"text/html","content_length":"104411","record_id":"<urn:uuid:e1580686-e4ca-4d16-a401-fa13155d5a0d>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00581.warc.gz"}
Linear perspective Assessment \| Biopsychology \| Comparative \| Cognitive \| Developmental \| Language \| Individual differences \| Personality \| Philosophy \| Social \| Methods \| Statistics \| Clinical \| Educational \| Industrial \| Professional items \| World psychology \| Cognitive Psychology: Attention · Decision making · Learning · Judgement · Memory · Motivation · Perception · Reasoning · Thinking - Cognitive processes Cognition - Outline Index This article needs rewriting to enhance its relevance to psychologists.. Please help to improve this page yourself if you can.. Perspective (from Latin perspicere, to see clearly) in the graphic arts, such as drawing, is an approximate representation on a flat surface (such as paper) of an image as it is perceived by the eye. The two most characteristic features of perspective are: • Objects are drawn smaller as their distance from the observer increases • Spatial foreshortening, which is the distortion of items when viewed at an angle In art, the term "foreshortening" is often used synonymously with perspective, even though foreshortening can occur in other types of non-perspective drawing representations (such as oblique parallel What is Perspective?[] Basic concept[] Perspective works by representing the light that passes from a scene, through an imaginary rectangle (the painting), to the viewer's eye. It is similar to a viewer looking through a window and painting what is seen directly onto the windowpane. If viewed from the same spot as the windowpane was painted, the painted image would be identical to what was seen through the unpainted window. Each painted object in the scene is a flat, scaled down version of the object on the other side of the window. Because each portion of the painted object lies on the straight line from the viewer's eye to the equivalent portion of the real object it represents, the viewer cannot perceive (sans depth perception) any difference between the painted scene on the windowpane and the view of the real scene. If the viewer is standing in a different spot, the illusion should be ruined, but unless the viewer chooses an extreme angle, like looking at it from the bottom corner of the window, the perspective normally looks more or less correct. In practice, however, nearly all perspectives (including those created mathematically), introduce distortions in comparison to the view of the real scene. Distortions can occur from: • mathematical approximations in calculated perspectives • type of lens used in perspectives generated through photography • inaccuracies from freehand sketching These distortions are usually introduced knowingly in order to simplify construction of the perspective. Some concepts that are commonly associated with perspectives include: • foreshortening • horizon line • vanishing points All perspective drawings assume a viewer, a certain distance away from the drawing. Objects are scaled relative to that viewer. Additionally, an object is often not scaled evenly --- a circle often appears as an ellipse and a square can appear as a trapezoid. This distortion is referred to as foreshortening. Perspective drawings typically have an (often implied) horizon line. This line, directly opposite the viewer's eye, represents objects infinitely far away. They have shrunk, in the distance, to the infinitesimal thickness of a line. It is analogous (and named after) the Earth's horizon. Any perspective representation of a scene that includes parallel lines has one or more vanishing points in a perspective drawing. A one-point perspective drawing means that the drawing has a single vanishing point, usually (though not necessarily) directly opposite the viewer's eye and usually (though not necessarily) on the horizon line. All lines parallel with the viewer's line of sight recede to the horizon towards this vanishing point. This is the standard "receding railroad tracks" phenomenon. A two-point drawing would have lines parallel to two different angles. Any number of vanishing points are possible in a drawing, one for each set of parallel lines that are at an angle relative to the plane of the drawing. Perspectives consisting of many parallel lines are observed most often when drawing architecture (architecture frequently uses lines parallel to the x, y, and z axes). Because it is rare to have a scene consisting solely of lines parallel to the three Cartesian axes (x, y, and z), it is rare to see perspectives in practice with only one, two, or three vanishing points. Consider that even a simple house frequently has a peaked roof which results in a minimum of five sets of parallel lines, in turn corresponding to up to five vanishing points. In contrast, perspectives of natural scenes often do not have any sets of parallel lines. Such a perspective would thus have no vanishing points. History of Perspective[] Early history[] Before perspective, paintings and drawings typically sized objects and characters according to their spiritual or thematic importance, not with distance. Especially in Medieval art, art was meant to be read as a group of symbols, rather than seen as a coherent picture. The only method to show distance was by overlapping characters. Overlapping alone made poor drawings of architecture; medieval paintings of cities are a hodgepodge of lines in every direction. The optical basis of perspective was defined in the year 1000, when the Arabian mathematician and philosopher Alhazen, in his Perspectiva, first explained that light projects conically into the eye. This was, theoretically, enough to translate objects convincingly onto a painting, but Alhalzen was concerned only with optics, not with painting. Conical translations are also mathematically difficult, so a drawing using them would be incredibly time consuming. The artist Giotto di Bondone first attempted drawings in perspective using an algebraic method to determine the placement of distant lines. The problem with using a linear ratio in this manner is that the apparent distance between a series of evenly spaced lines actually falls off with a sine dependence. To determine the ratio for each succeeding line, a recursive ratio must be used. This was not discovered until the 20th Century, in part by Erwin Panofsky. One of Giotto's first uses of his algebraic method of perspective was Jesus Before the Caïf. Although the picture does not conform to the modern, geometrical method of perspective, it does give a decent illusion of depth, and was a large step forward in Western art. Mathematical basis for perspective[] One hundred years later, in the early 1400s, Filippo Brunelleschi demonstrated the geometrical method of perspective, used today by artists, by painting the outlines of various Florentine buildings onto a mirror. When the building's outline was continued, he noticed all the lines all converged on the horizon line. According to Vasari, he then set up a demonstration of his painting of the Baptistry in the incomplete doorway of the Duomo. He had the viewer look through a small hole on the back of the painting, facing the Baptistry. He would then set up a mirror, facing the viewer, which reflected his painting. To the viewer, the painting of the Baptistry and the Baptistry itself were nearly intistinguishable. Soon after, nearly every artist in Florence used geometrical perspective in their paintings, notably Donatello, who started painting elaborate checkerboard floors into the simple manger portrayed in the birth of Christ. Although hardly historically accurate, these checkerboard floors obeyed the primary laws of geometrical perspective: all lines converged to a vanishing point, and the rate at which the horizontal lines receded into the distance was graphically determined. This became an integral part of Quattrocento art. Not only was perspective a way of showing depth, it was also a new method of composing a painting. Paintings began to show a single, unified scene, rather than a combination of several. As shown by the quick proliferation of accurate perspective paintings in Florence, Brunelleschi likely understood, but did not publish, the mathematics behind perspective. Decades later, his friend Leon Battista Alberti wrote Della Pittura, a treatise on proper methods of showing distance in painting. Alberti's primary breakthrough was not to show the mathematics in terms of conical projections, as it actually appears to the eye. Instead, he formulated the theory based on planar projections, or how the rays of light, passing from the viewer's eye to the landscape, would strike the picture plane (the painting). He was then able to calculate the apparent height of a distant object using two similar triangles. In viewing a wall, for instance, the first triangle has a vertex at the user's eye, and vertices at the top and bottom of the wall. The bottom of this triangle is the distance from the viewer to the wall. The second, similar triangle, has a point at the viewer's eye, and has a length equal to the viewer's eye from the painting. The height of the second triangle can then be determined through a simple ratio, as proven by Euclid. Piero della Francesca elaborated on Della Pittura in his De Prospectiva Pingendi in 1474. Alberti had limited himself to figures on the ground plane and giving an overall basis for perspective. Francesca fleshed it out, explicitely covering solids in any area of the picture plane. Francesca also started the now common practice of using illustrated figures to explain the mathematical concepts, making his treatise easier to understand than Alberti's. Francesca was also the first to accurately draw the Platonic solids as they would appear in perspective. Perspective remained, for a while, the domain of Florence. Jan van Eyck, among others, was unable to create a consistent structure for the converging lines in paintings, as in London's The Arnolfini Portrait, because he was unaware of the theoretical breakthrough just then occurring in Italy. Artificial and natural[] Leonardo da Vinci distrusted Brunelleschi's formulation of perspective because it failed to take into account the appearance of objects held very close to the eye. Leonardo called Brunelleschi's method artificial perspective projection. It is today called classical perspective projection. Projections closer to the image beheld by the human eye, he named natural perspective. Artificial perspective projection is a perspective projection onto a flat surface, well suited for drawings and paintings, which are typically flat. Natural perspective projection, in contrast, is a perspective projection onto a spherical surface. From a geometric point of view, the differences between artificial and natural perspectives can be thought of as similar to the distortion that occurs when representing the earth (approximately spherical) as a map (typically flat). Both types of projection involve a distortion. The difference between the two distortions is called perspective projection distortion. Varieties of Perspective Drawings[] Of the many types of perspective drawings, the most common categorizations of artificial perspective are one-, two- and three-point. The names of these categories refer to the number of vanishing points in the perspective drawing. Strictly speaking, these types can only exist for scenes being represented that are rectilinear (composed entirely of straight lines which intersect only at 90 degrees to each other). One-point perspective[] One vanishing point is typically used for roads, railroad tracks, or buildings viewed so that the front is directly facing the viewer. Any objects that are made up of lines either directly parallel with the viewer's line of sight (like railroad tracks) or directly perpendicular (the railroad slats) can be represented with one-point perspective. One-point perspective exists when the painting plate (also known as the picture plane) is parallel to two axes of a rectilinear (or Cartesian) scene --- a scene which is composed entirely of linear elements that intersect only at right angles. If one axis is parallel with the picture plane, then all elements are either parallel to the painting plate (either horizontally or vertically) or perpendicular to it. All elements that are parallel to the painting plate are drawn as parallel lines. All elements that are perpendicular to the painting plate converge at a single point (a vanishing point) on the horizon. Two-point perspective[] Two-point perspective can be used to draw the same objects as one-point perspective, rotated: looking at the corner of a house, or looking at two forked roads shrink into the distance, for example. One point represents one set of parallel lines, the other point represents the other. Looking at a house from the corner, one wall would recede towards one vanishing point, the other wall would recede towards the opposide vanishing point. Two-point perspective exists when the painting plate is parallel to a Cartesian scene in one axis (usually the z-axis) but not to the other two axes. If the scene being viewed consists solely of a cylinder sitting on a horizontal plane, no difference exists in the image of the cylinder between a one-point and two-point perspective. Three-point perspective[] Three-point perspective is usually used for buildings seen from above. In addition to the two vanishing points from before, one for each wall, there is now one for how those walls recede into the ground. Looking up at a tall building is another common example of the third vanishing point. Three-point perspective exists when the perspective is a view of a Cartesian scene where the picture plane is not parallel to any of the scene's three axes. Each of the three vanishing points corresponds with one of the three axes of the scene. Image constructed using multiple vanishing points. Zero-point perspective[] Due to the fact that vanishing points exist only when parallel lines are present in the scene, a perspective without any vanishing points ("zero-point" perspective) occurs if the viewer is observing a nonlinear scene. The most common example of a nonlinear scene is a natural scene (ie, a mountain range) which frequently does not contain any parallel lines. Other examples include: a random (ie, not aligned in a three-dimensional Cartesian coordinate system) arrangement of spherical objects, a scene composed entirely of three-dimensionally curvilinear strings, or a scene consisting of lines where no two are parallel to each other. Orthographic projections also do not have vanishing points, but they are not perspective constructions and are thus not equivalent to a "zero-point" perspective. Note that a perspective without vanishing points can still create a sense of "depth," as is clearly apparent in a photograph of a mountain range (for example, more distant mountains have smaller scale features). Other varieties of linear perspective[] One-point, two-point, and three-point perspective are dependent on the structure of the scene being viewed. These only exist for strict Cartesian (rectilinear) scenes. By inserting into a Cartesian scene a set of parallel lines that are not parallel to any of the three axes of the scene, a new distinct vanishing point is created. Therefore, it is possible to have an infinite-point perspective if the scene being viewed is not a Cartesian scene but instead consists of infinite pairs of parallel lines, where each pair is not parallel to any other pair. Varieties of nonlinear perspective[] Typically, mathematically constructed perspectives are "linear" in that the ratio at which more distant objects decrease in size is constant (ie, graphing the drawn size of a one-foot object versus the distant from viewer will form a straight line). It is conceivable to have non-linear perspectives — those in which the graph of the ratio mentioned above does not form a straight line. A panorama is a perspective projected onto a cylinder. The actual drawing can be drawn onto a cylinder (typically on the interior surface and viewed from the inside the cylinder) or onto a flat surface, equivalent to "unrolling" the cylinder. A panorama (projection onto a cylinder) removes one of the differences between artificial perspective projection (projection onto a flat surface) and natural perspective projection (projection onto a spherical surface). A standard Mercator map projection is similar to a panorama. Methods of Constructing Perspectives[] Several methods of constructing perspectives exist, including: • Freehand sketching (common in art) • Graphically constructing (once common in architecture) • Using a perspective grid • Computing a perspective transform (common in 3D computer applications) • Mimicry using tools such as a proportional divider (sometimes called a variscaler) Example: a square in perspective[] One of the most common, and earliest, uses of geometrical perpective is a checkerboard floor. It is a simple but striking application of one-point perspective. Many of the properties of perspective drawing are used while drawing a checkerboard. The checkerboard floor is, essentially, just a combination of a series of squares. Once a single square is drawn, it can be widened or subdivided into a checkerboard. Where necessary, lines and points will be referred to by their colors in the diagram. To draw a square in perspective, the artist starts by drawing a horizon line (black) and determining where the vanishing point (green) should be. The higher up the horizon line, the lower the viewer will appear to be looking, and vice versa. The more off-center the vanishing point, the more tilted the square will be. Because the square is made up of right angles, the vanishing point should be directly in the middle of the horizon line. A rotated square is drawn using two-point perspective, with each set of parallel lines leading to a different vanishing point. The foremost edge of the (orange) square is drawn near the bottom of the painting. Because the viewer's picture plane is parallel to the bottom of the square, this line is horizontal. Lines connecting each side of the foremost edge to the vanishing point are drawn (in grey). These lines give the basic, one point "railroad tracks" perspective. The closer it is the horizon line, the farther away it is from the viewer, and the smaller it will appear. The farther away from the viewer it is, the closer it is to being perpindicular to the picture plane. A new point (the eye) is now chosen, on the horizon line, either to the left or right of the vanishing point. The distance from this point to the vanishing point represents the distance of the viewer from the drawing. If this point is very far from the vanishing point, the square will appear squashed, and far away. If it is close, it will appear stretched out, as if it is very close to the A line connecting this point to the opposite corner of the square is drawn. Where this (blue) line hits the side of the square, a horizontal line is drawn, representing the furthest edge of the square. The line just drawn represents the ray of light travelling from the viewer's eye to the furthest edge of the square. This step is key to understanding perspective drawing. The light that passes through the picture plane obviously can not be traced. Instead, lines that represent those rays of light are drawn on the picture plane. In the case of the square, the side of the square also represents the picture plane (at an angle), so there is a small shortcut: when the line hits the side of the square, it has also hit the appropriate spot in the picture plane. The (blue) line is drawn to the opposite edge of the foremost edge because of another shortcut: since all sides are the same length, the foremost edge can stand in for the side edge. Original formulations used, instead of the side of the square, a vertical line to one side, representing the picture plane. Each line drawn through this plane was identical to the line of sight from the viewer's eye to the drawing, only rotated around the y-axis ninety degrees. It is, conceptually, an easier way of thinking of perspective. It can be easily shown that both methods are mathematically identical, and result in the same placement of the furthest side (see Panofsky). Foreshortening refers to the visual effect or optical illusion that an object or distance is shorter than it actually is because it is angled toward the viewer. Although foreshortening is an important element in art where visual perspective is being depicted, foreshortening occurs in other types of two-dimensional representations of three-dimensional scenes. Some other types where foreshortening can occur include oblique parallel projection drawings. Figure F1 shows two different projections of a stack of two cubes, illustrating oblique parallel projection foreshortening ("A") and perspective foreshortening ("B"). Other Perspective Topics[] The following topics are not critical to understanding perspective, but provide some additional information related to perspectives. Limitations of perspective[] Perspective images are calculated assuming a particular vantage point. In order for the resulting image to appear identical to the original scene, a viewer of the perspective must view the image from the exact vantage point used in the calculations relative to the image. This cancels out what would appear to be distortions in the image when viewed from a different point. These apparent distortions are more pronounced away from the center of the image as the angle between a projected ray (from the scene to the eye) becomes more acute relative to the picture plane. For a typical perspective, however, the field of view is narrow enough (often only 60 degrees) that the distortions are similarly minimal enough that the image can be viewed from a point other than the actual calculated vantage point without appearing significantly distorted. When a larger angle of view is required, the standard method of projecting rays onto a flat picture plane becomes impractical. As a theorectical maximum, the field of view of a flat picture plane must be less than 180 degrees (as the field of view increases towards 180 degrees, the required breadth of the picture plane approaches infinity). In order to create a projected ray image with a large field of view, one can project the image onto a curved surface. In order to have a large field of view horizontally in the image, a surface that is a vertical cylinder (i.e., the axis of the cylinder is parallel to the z-axis) will suffice (similarly, if the desired large field of view is only in the vertical direction of the image, a horizontal cylinder will suffice). A cylindrical picture surface will allow for a projected ray image up to a full 360 degrees in either the horizontal or vertical dimension of the perspective image (depending on the orientation of the cylinder). In the same way, by using a spherical picture surface, the field of view can be a full 360 degrees in any direction (note that for a spherical surface, all projected rays from the scene to the eye intersect the surface at a right angle). Just as a standard perspective image must be viewed from the calculated vantage point for the image to appear identical to the true scene, a projected image onto a cylinder or sphere must likewise be viewed from the calculated vantage point for it to be precisely identical to the original scene. If an image projected onto a cylindrical surface is "unrolled" into a flat image, different types of distortions occur: For example, many of the scenes straight lines will be drawn as curves. An image projected onto a spherical surface can be flattened in various ways, including: • an image equivalent to an unrolled cylinder • a portion of the sphere can be flattened into an image equivalent to a standard perspective • an image similar to a fisheye photograph The myth of one-, two- and three-point perspectives[] One-point, two-point, and three-point perspectives appear to embody different forms of calculated perspective. The methods required to generate these perspectives by hand are different. Mathematically, however, all three are identical: The difference is simply in the relative orientation of the rectilinear scene to the viewer. For example, the three images illustrating one-, two- and three-point perspective in the above section can be generated in two ways with identical results: • the "standard" way would be to alter the viewer's position in each perspective with a stationary cube • identical to this is to simply rotate the cube in space in front of a stationary viewer A practical use of this fact is an alternative quick and accurate method of generating a "two-point" perspective by hand: The two vanishing points of the perspective can be generated by simply mapping a rotated grid (of any arbitrary angle) onto a standard "one-point" perspective grid (the grids should be of the same unit spacing to facilitate the construction of the actual drawing). Geometric transforms[] A perspective drawing, whether roughly sketched (ie, intuitively by freehand) or precisely calculated (i.e., using matrix multiplication on a computer or other means), is usually a combination of two geometric transforms: • A perspective transform: a perspective projection onto a typically flat picture plane (or painting plate) of a scene from the viewpoint of an observer • A similarity transform: a scaling of the picture plane from the first transform onto an actual drawing of a usually smaller size. See also[] • Perspective correction • Perspective projection distortion • Reverse perspective External links[] • 1-point perspective: • 2-point perspective: • 3-point perspective: • Pérez-Gómez, Alberto, and Pelletier, Louise (1997). Architectural Representation and the Perspective Hinge, Cambridge, Mass.: MIT Press. • Damisch, Hubert (1994). The Origin of Perspective, Translated by John Goodman, Cambridge, Mass.: MIT Press. • Hyman, Isabelle, comp (1974). Brunelleschi in Perspective, Englewood Cliffs, New Jersey: Prentice-Hall. • Panofsky, Erwin (1965). Renaissance and Renascences in Western Art, Stockholm: Almqvist & Wiksell. • Vasari, Giorgio (1568). The Lives of the Artists. de:Perspektive es:Perspectiva fr:Perspective ko:원근법 he:פרספקטיבה mk:Перспектива nl:Lijnperspectief no:Perspektiv (kunst) pt:Perspectiva (gráfica) ru:Перспектива fi:Perspektiivi sv:Perspektiv ta:இயலுறு தோற்றப் படம் zh:透视	{"url":"https://psychology.fandom.com/wiki/Linear_perspective","timestamp":"2024-11-05T20:22:27Z","content_type":"text/html","content_length":"219691","record_id":"<urn:uuid:859ba06c-018b-466f-a7e4-3db9384dcad2>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00885.warc.gz"}
risk parity In this paper we mainly focus on optimization of sums of squares of quadratic functions, which we refer to as second-order least-squares problems, subject to convex constraints. Our motivation arises from applications in risk parity portfolio selection. We generalize the setting further by considering a class of nonlinear, non convex functions which admit a (non … Read more Least-squares approach to risk parity in portfolio selection The risk parity optimization problem aims to find such portfolios for which the contributions of risk from all assets are equally weighted. Portfolios constructed using risk parity approach are a compromise between two well-known diversification techniques: minimum variance optimization approach and the equal weighting approach. In this paper, we discuss the problem of finding portfolios … Read more	{"url":"https://optimization-online.org/tag/risk-parity/","timestamp":"2024-11-01T20:24:25Z","content_type":"text/html","content_length":"85872","record_id":"<urn:uuid:9918999b-bcfb-4d2d-b579-a0a70fcf8fbf>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00791.warc.gz"}
In the adjoining figure, ap and bp are angle bisector of /_a an-Turito Are you sure you want to logout? In the adjoining figure, AP and BP are angle bisector of We should find value of If AP and BP are angle bisector, then And Consecutive angles are supplementary From the figure, Hence option 1 is correct Get an Expert Advice From Turito.	{"url":"https://www.turito.com/ask-a-doubt/maths-in-the-adjoining-figure-ap-and-bp-are-angle-bisector-of-a-and-b-which-meets-at-p-on-the-parallelogram-abcd-th-q005b04","timestamp":"2024-11-02T12:52:11Z","content_type":"application/xhtml+xml","content_length":"988137","record_id":"<urn:uuid:758f10dc-a127-42e3-b3d5-e41e5dffe85d>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00605.warc.gz"}
Proof Load for Large Nuts - Portland Bolt What is the proof load rating for a large diameter (2-1/2”+) nut? Nuts that are over 2-1/2” diameter do not require a proof load test because most testing equipment in the industry is not large enough to test them effectively. In these cases, a hardness test is an acceptable alternative, unless a proof load test is specifically required by the buyer. Nuts this large would require in excess of 160,000 pounds-force to test the proof load. Since most testing equipment is incapable of this amount, the specifications A194 and A563 have allowed the hardness test as an acceptable alternative for the proof load test for these large diameters of nuts. Portland Bolt stocks many large diameter nuts in plain or galvanized finish. Contact one of our team members for a quote. From ASTM A194: “8.2.2.1 The manufacturer shall test the number of nuts specified in 8.1.2.1 following all production heat treatments. Nuts that would require a proof load in excess of 160,000 lb/f or 705 kN shall, unless Supplementary Requirements S1 or S4 are invoked in the purchase order or contract, be proof load tested per Section 8 or cross sectional hardness tested per Annex A3 of Test Methods and Definitions A370. Proof Load test prevail over hardness test in the even a conflict exists relative to minimum strength.” From ASTM A563: “6.1.2 Jam nuts, slotted nuts, nuts smaller in width across flats or thickness than standard hex nuts (7.1), and nuts that would require a proof load in excess of 160,000 lb/f may be furnished on the basis of minimum hardness requirements specified for the grade in Table 3, unless proof load testing is specified in the inquiry and purchase order.” 16 comments what is the hardness range for your mandral for proof loading 1 1/8″ nuts and up , am most interested, thanks If Supplementary Requirements S4 is invoked, can the proof load testing still be done by cross sectional hardness tested per Annex A3 of Test Methods and Definitions A370 for nuts that would require a proof load in excess of 160,000 lb/f ie. 1.5/8″ Heavy Hex Grade 7? ASME SA194, the nuts have size 1-3/8″ with proof load values above 160,000 lbf, Could we have a hardness alternative for the proof load test? Is it mandatory to perform prof load test for 3/8″ Hex bolt ASTM A194 GR7 as per ASTM? ASTM A194 table 3 has sizes 1.3/8″ and 1.1/2″ with proof load values above 160,000 lbf. Supplementary requirements, S4 do not show values in table S4.1 for these sizes. Can you advise what load we test to? Do you do proof load, clamp load, and prevailing torque tests on nuts? In the event that you do require proof load testing of your nuts, TUV has a lab in Aliquippa, PA that has a press that is calibrated up to 1,000,000 pounds. Please contact me if you need any more i want to know proof loading testing for nuts	{"url":"https://www.portlandbolt.com/technical/faqs/proof-load-large-nuts/","timestamp":"2024-11-14T05:07:33Z","content_type":"text/html","content_length":"138860","record_id":"<urn:uuid:90eeb83f-bbd8-46c8-a36f-626c79119836>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00793.warc.gz"}
Miscellaneous date-time functions [YSQL] Miscellaneous date-time functions function isfinite() returns boolean Here is the interesting part of the output from \df isfinite(): Result data type \| Argument data types boolean \| abstime boolean \| date boolean \| interval boolean \| timestamp with time zone boolean \| timestamp without time zone The data type abstime is for internal use only. It inevitably shows up in the \df output. But you should simply forget that it exists. Here's a trivial demonstration of the meaning of the function isfinite(): do $body$ assert not isfinite( 'infinity'::timestamptz), 'Assert #1 failed'; assert not isfinite('-infinity'::timestamptz), 'Assert #2 failed'; The block finishes without error. function age() returns interval Nominally, age() returns the age of something "now" with respect to a date of birth. The value of "now" can be given: either explicitly, using the two-parameter overload, as the invocation's first actual argument; or implicitly, using the one-parameter overload, as date_trunc('day', clock_timestamp()). The value for the date of birth is given, for both overloads, as the invocation's last actual argument. Of course, this statement of purpose is circular because it avoids saying precisely how age is defined—and why a notion is needed that's different from what is given simply by subtracting the date of birth from "now", using the native minus operator, -. Here is the interesting part of the output from \df age(). The rows were re-ordered manually and whitespace was manually added to improve the readability: Result data type \| Argument data types interval \| timestamp without time zone, timestamp without time zone interval \| timestamp with time zone, timestamp with time zone interval \| timestamp without time zone interval \| timestamp with time zone The 'xid' overload of 'age()' has nothing to do with date-time data types There's an overload with argument data type (and with return). The present Date and time data types major section does not describe the overload of This section first discusses age as a notion. Then it defines the semantics of the two-parameter overload of the built-in age() function by modeling its implementation. The semantics of the one-parameter overload is defined trivially in terms of the semantics of the two-parameter overload. The definition of age is a matter of convention Age is defined as the length of time that a person (or a pet, a tree, a car, a building, a civilization, the planet Earth, the Universe, or any phenomenon of interest) has lived (or has been in existence). Here is a plausible formula in the strict domain of date-time arithmetic: age ◄— todays_date - date_of_birth If todays_date and date_of_birth are date values, then age is produced as an int value. And if todays_date and date_of_birth are plain timestamp values (or timestamptz values), then age is produced as an interval value. As long as the time-of-day component of each plain timestamp value is exactly 00:00:00 (and this is how people think of dates and ages) then only the dd component of the internal [mm, dd, ss] representation of the resulting interval value will be non-zero. Try this: drop function age_in_days(text, text); create function age_in_days(today_text in text, dob_text in text) returns table (z text) language plpgsql as $body$ d_today constant date not null := today_text; d_dob constant date not null := dob_text; t_today constant timestamp not null := today_text; t_dob constant timestamp not null := dob_text; z := (d_today - d_dob)::text; return next; z := (t_today - t_dob)::text; return next; select z from age_in_days('290000-08-17', '0999-01-04 BC'); This is the result: 106285063 days However, how ages are stated is very much a matter of convention. Beyond, say, one's mid teens, it is given simply as an integral number of years. (Sue Townsend's novel title, "The Secret Diary of Adrian Mole, Aged 13 3/4", tells the reader that it's a humorous work and that Adrian is childish for his years.) The answer to "What is the age of the earth?" is usually given as "about 4.5 billion years"—and this formulation implies that a precision of about one hundred thousand years is appropriate. At the other end of the spectrum, the age of new born babies is usually given first as an integral number of days, and later, but while still a toddler, as an integral number of months. Internet search finds articles with titles like "Your toddler's developmental milestones at 18 months". You'll even hear age given as, say, "25 months". Internet search finds lots of formulas to calculate age in years—usually using spreadsheet arithmetic. It's easy to translate what they do into SQL primitives. The essential point of the formula is that if today's month-and-date is earlier in the year than the month-and-date of the date-of-birth, then you haven't yet reached your birthday. Try this: drop function age_in_years(text, text); create function age_in_years(today_tz in timestamptz, dob_tz in timestamptz) returns interval language plpgsql as $body$ d_today constant date not null := today_tz; d_dob constant date not null := dob_tz; yy_today constant int not null := extract(year from d_today); mm_today constant int not null := extract(month from d_today); dd_today constant int not null := extract(day from d_today); yy_dob constant int not null := extract(year from d_dob); mm_dob constant int not null := extract(month from d_dob); dd_dob constant int not null := extract(day from d_dob); mm_dd_today constant date not null := make_date(year=>1, month=>mm_today, day=>dd_today); mm_dd_dob constant date not null := make_date(year=>1, month=>mm_dob, day=>dd_dob); -- Is today's mm-dd greater than dob's mm-dd? delta constant int not null := case when mm_dd_today >= mm_dd_dob then 0 else -1 age constant interval not null := make_interval(years=>(yy_today - yy_dob + delta)); return age; set timezone = 'America/Los_Angeles'; age_in_years('2007-02-13', '1984-02-14')::text as "age one day before birthday", age_in_years('2007-02-14', '1984-02-14')::text as "age on birthday", age_in_years('2007-02-15', '1984-02-14')::text as "age one day after birthday", age_in_years(clock_timestamp(), '1984-02-14')::text as "age right now"; This is the result (when the select is executed in October 2021): age one day before birthday \| age on birthday \| age one day after birthday \| age right now 22 years \| 23 years \| 23 years \| 37 years You can easily derive the function age_in_months() from the function age_in_years(). Then, with all three functions in place, age_in_days(), age_in_months(), and age_in_years(), you can implement an age() function that applies a rule-of-thumb, based on threshold values for what age_in_days() returns, to return either a pure days, a pure months, or a pure years interval value. This is left as an exercise for the reader. The semantics of the built-in function age() The following account relies on understanding the internal representation of an 'interval' value The internal representation of an value is a [mm, dd, ss] tuple. This is explained in the section How does YSQL represent an interval value? Bare timestamp subtraction produces a result where the yy field is always zero and only the mm and dd fields might be non-zero, thus: select ( '2001-04-10 12:43:17'::timestamp - '1957-06-13 11:41:13'::timestamp)::text; This is the result: 16007 days 01:02:04 See the section The moment-moment overloads of the "-" operator for timestamptz, timestamp, and time for more information. The PostgreSQL documentation, in Table 9.30. Date/Time Functions, describes how age() calculates its result thus: Subtract arguments, producing a "symbolic" result that uses years and months, rather than just days and it gives this example: select age( with this result: 43 years 9 mons 27 days Because the result data type is interval, and there's no such thing as a "symbolic" interval value, this description is simply nonsense. It presumably means that the result is a hybrid interval value where the yy field might be non-zero. 'age(t2, ts1)' versus 'justify_interval(ts2 - ts1)' While, as was shown above, subtracting one timestamp[tz] value from another produces an interval value whose mm component is always zero, you can use justify_interval() to produce a value that, in general, has a non-zero value for each of the mm, dd_, and ss components. However, the actual value produced by doing this will, in general, differ from that produced by invoking age(), even when the results are compared with the native equals operator, =, (and not the user-defined "strict equals" operator, ==). Try this: set timezone = 'UTC'; c1 as ( '2021-03-17 13:43:19 America/Los_Angeles'::timestamptz as ts2, '2000-05-19 11:19:13 America/Los_Angeles'::timestamptz as ts1), c2 as ( age (ts2, ts1) as a, justify_interval(ts2 - ts1) as j from c1) a::text as "age(ts2, ts1)", j::text as "justify_interval(ts2 - ts1)", (a = j)::text as "age() = justify_interval() using native equals" from c2; This is the result: age(ts2, ts1) \| justify_interval(ts2 - ts1) \| age() = justify_interval() using native equals 20 years 9 mons 29 days 02:24:06 \| 21 years 1 mon 17 days 02:24:06 \| false They differ simply because justify_interval() uses one rule (see the subsection The justify_hours(), justify_days(), and justify_interval() built-in functions) and age() uses a different rule (see the subsection The semantics of the two-parameter overload of function age()). You should understand the rule that each uses and then decide what you need. But notice Yugabyte's recommendation, below, simply to avoid using the built-in age() function. Anyway, the phrase producing a "symbolic" result gives no clue about how age() works in the general case. But it looks like this is what it did with the example above: • It tried to subtract "13 days" from "10 days" and "borrowed" one month to produce a positive result. As it happens, both June and April have 30 days (with no leap year variation). The result, " (30 + 10) - 13", is "27 days". • It tried to subtract "6 months" from "3 months" (decremented by one month from its starting value, "4 months", to account for the "borrowed" month), and "borrowed" one year to produce a positive result. One year is always twelve months. The result, "(12 + 3) - 6", is "9 months". • Finally, it subtracted "1957 years" from "2000 years" (decremented by one year from its starting value, "2021 years", to account for the "borrowed" year). Here is another example of the result that age() produces when the inputs have non-zero time-of-day components: select age( '2001-04-10 11:19:17'::timestamp, '1957-06-13 15:31:42'::timestamp)::text; with this result: 43 years 9 mons 26 days 19:47:35 Nobody ever cites an age like this, with an hours, minutes, and seconds component. But the PostgreSQL designers thought that it was a good idea to implement age() to do this. Briefly, and approximately, the function age() extracts the year, month, day, and seconds since midnight for each of the two input moment values. It then subtracts these values pairwise and uses them to create an interval value. In general, this will be a hybrid value with non-zero mm, dd, and ss components. But the statement of the semantics must be made more carefully than this to accommodate the fact that the outcomes of the pairwise differences might be negative. • For example, if today is "year 2020 month 4" and if the date-of-birth is "year 2010 month 6", then a naïve application of this rule would produce an age of "10 years -2 months". But age is never stated like this. Rather, it's stated as "9 years 10 months". This is rather like doing subtraction of distances measured in imperial feet and inches. When you subtract "10 feet 6 inches" from "20 feet 4 inches" you "borrow" one foot, taking "10 feet" down to "9 feet" so that you can subtract "6 inches" from "12 + 4 inches" to get "10 inches". However, the borrowing rules get very tricky with dates because "borrowed" months (when pairwise subtraction of day values would produce a negative result) have different numbers of days (and there's leap years to account for too) so the "borrowing" rules get to be quite baroque—so much so that it's impractical to explain the semantics of age() in prose. Rather, you need to model the implementation. PL/pgSQL is perfect for this. The full account of age() is presented on its own dedicated child page. Avoid using the built-in 'age()' function. The rule that uses to produce its result cannot be expressed clearly in prose. And, anyway, it produces a result with an entirely inappropriate apparent precision. Yugabyte recommends that you decide how you want to define age for your present use case and then implement the definition that you choose along the lines used in the user-defined functions shown above in the subsection The definition of age is a matter of convention function extract() \| function date_part() returns double precision The function extract(), and the alternative syntax that the function date_part() supports for the same semantics, return a double precision value corresponding to a nominated so-called field, like year or second, from the input date-time value. The full account of extract() and date_part() is presented on its own dedicated child page. function timezone() \| 'at time zone' operator returns timestamp \| timestamptz The function timezone(), and the alternative syntax that operator at time zone supports for the same semantics, return a plain timestamp value from a timestamptz input or a timestamptz value from a plain timestamp input. The effect is the same as if a simple typecast is used from one data type to the other after using set timezone to specify the required timezone. timezone(<timezone>, timestamp[tz]_value) == timestamp[tz]_value at time zone <timezone> Try this example: with c as ( select '2021-09-22 13:17:53.123456 Europe/Helsinki'::timestamptz as tstz) (timezone('UTC', tstz) = tstz at time zone 'UTC' )::text as "with timezone given as text", (timezone(make_interval(), tstz) = tstz at time zone make_interval())::text as "with timezone given as interval" from c; This is the result: with timezone given as text \| with timezone given as interval true \| true (Because all make_interval()'s formal parameters have default values of zero, you can invoke it with no actual arguments.) Now try this example: set timezone = 'UTC'; with c as ( select '2021-09-22 13:17:53.123456 Europe/Helsinki'::timestamptz as tstz) (timezone('UTC', tstz) = tstz::timestamp)::text from c; The result is true. The function syntax is more expressive than the operator syntax because its overloads distinguish explicitly between specifying the timezone by name or as an interval value. Here is the interesting part of the output from \df timezone(). The rows were re-ordered manually and whitespace was manually added to improve the readability: Result data type \| Argument data types timestamp with time zone \| text, timestamp without time zone timestamp without time zone \| text, timestamp with time zone timestamp with time zone \| interval, timestamp without time zone timestamp without time zone \| interval, timestamp with time zone The rows for the timetz argument data types were removed manually, respecting the recommendation here to avoid using this data type. (You can't get \df output for the operator at time zone.) Avoid using the 'at time zone' operator and use only the function 'timezone()'. Because the function syntax is more expressive than the operator syntax, Yugabyte recommends using only the former syntax. Moreover, never use bare but, rather, use it only via the overloads of the user-defined wrapper function and as described in the section Recommended practice for specifying the UTC offset 'overlaps' operator returns boolean The account of the overlaps operator first explains the semantics in prose and pictures. Then it presents two implementations that model the semantics and shows that they produce the same results. 'overlaps' semantics in prose The overlaps operator determines if two durations have any moments in common. The overlaps invocation defines a duration either by its bounding moments or by its one bounding moment and the size of the duration (expressed as an interval value). There are therefore four alternative general invocation syntaxes. Either: overlaps_result ◄— (left-duration-bound-1, left-duration-bound-2) overlaps (right-duration-bound-1, right-duration-bound-2) overlaps_result ◄— (left-duration-bound-1, left-duration-size) overlaps (right-duration-bound-1, right-duration-bound-2) overlaps_result ◄— (left-duration-bound-1, left-duration-bound-2) overlaps (right-duration-bound-1, right-duration-size) overlaps_result ◄— (left-duration-bound-1, left-duration-size) overlaps (right-duration-bound-1, right-duration-size) Unlike other phenomena that have a length, date-time durations are special because time flows inexorably from earlier moments to later moments. It's convenient to say that, when the invocation as presented has been processed, a duration is ultimately defined by its start moment and its finish moment—even if one of these is derived from the other by the size of the duration. In the degenerate case, where the start and finish moments coincide, the duration becomes an instant. Notice that, while it's natural to write the start moment before the finish moment, the result is insensitive to the order of the boundary moments or to the sign of the size of the duration. The result is also insensitive to which duration, "left" or "right" is written first. This prose account of the semantics starts with some simple examples. Then it states the rules carefully and examines critical edges cases. Simple examples. Here's a simple positive example: select ( ('07:00:00'::time, '09:00:00'::time) overlaps ('08:00:00'::time, '10:00:00'::time) )::text as "time durations overlap"; This is the result: time durations overlap And here are some invocation variants that express durations with the same ultimate derived start and finish moments: do $body$ seven constant time not null := '07:00:00'; eight constant time not null := '08:00:00'; nine constant time not null := '09:00:00'; ten constant time not null := '10:00:00'; two_hours constant interval not null := make_interval(hours=>2); r1 constant boolean not null := (seven, nine) overlaps (eight, ten); r2 constant boolean not null := (seven, two_hours) overlaps (eight, ten); r3 constant boolean not null := (seven, nine) overlaps (eight, two_hours); r4 constant boolean not null := (seven, two_hours) overlaps (eight, two_hours); r5 constant boolean not null := (nine, seven) overlaps (ten, eight); r6 constant boolean not null := (nine, -two_hours) overlaps (ten, -two_hours); assert ((r1 = r2) and (r1 = r3) and (r1 = r4) and (r1 = r5) and (r1 = r6)), 'Assert failed'; The block finishes silently, showing that the result from each of the six variants is the same. The operator is supported by the overlaps() function. Here is the interesting part of the output from \df overlaps(): Result data type \| Argument data types boolean \| time, time, time, time boolean \| time, interval, time, time boolean \| time, time, time, interval boolean \| time, interval, time, interval boolean \| timestamp, timestamp, timestamp, timestamp boolean \| timestamp, interval, timestamp, timestamp boolean \| timestamp, timestamp, timestamp, interval boolean \| timestamp, interval, timestamp, interval boolean \| timestamptz, timestamptz, timestamptz, timestamptz boolean \| timestamptz, interval, timestamptz, timestamptz boolean \| timestamptz, timestamptz, timestamptz, interval boolean \| timestamptz, interval, timestamptz, interval The rows for the timetz argument data types were removed manually, respecting the recommendation here to avoid using this data type. Also, to improve the readability: • the rows were reordered • time without time zone was rewritten as time, • timestamp without time zone was rewritten as timestamp, • timestamp with time zone was rewritten as timestamptz, • blank rows and spaces were inserted manually This boils down to saying that overlaps supports durations whose boundary moments are one of time, plain timestamp, or timestamptz. There is no support for date durations. But you can achieve the functionality that such support would bring simply by typecasting date values to plain timestamp values and using the plain timestamp overload. If you do this, avoid the overloads with an interval argument because of the risk that a badly-chosen interval value will result in a boundary moment with a non-zero time component. Rather, achieve that effect by adding an integer value to a date value before typecasting to plain timestamp. Here is an example: select ( ( ('2020-01-01'::date)::timestamp, ('2020-01-01'::date + 2)::timestamp ) overlaps ( ('2020-01-02'::date)::timestamp, ('2020-01-01'::date + 2)::timestamp ) )::text as "date durations overlap"; This is the result: date durations overlap Rule statement and edge cases Because (unless the duration collapses to an instant) one of the boundary moments will inevitably be earlier than the other, it's useful to assume that some pre-processing has been done and to write the general invocation syntax using the vocabulary start-moment and finish-moment. Moreover (except when both durations start at the identical moment and finish at the identical moment), it's always possible to decide which is the earlier-duration and which is the later-duration. Otherwise (when the two durations exactly coincide), it doesn't matter which is labeled earlier and which is labeled • If the left-duration's start-moment is less than the right-duration's start-moment, then the left-duration is the earlier-duration and the right-duration is the later-duration. • If the right-duration's start-moment is less than the left-duration's start-moment, then the right-duration is the earlier-duration and the left-duration is the later-duration. • Else, if the left-duration's start-moment and the right-duration's start-moment are identical, then □ If the left-duration's finish-moment is less than the right-duration's finish-moment, then the left-duration is the earlier-duration and the right-duration is the later-duration. □ If the right-duration's finish-moment is less than the left-duration's finish-moment, then the right-duration is the earlier-duration and the left-duration is the later-duration. It's most useful, in order to express the rules and to discuss the edge cases, to write the general invocation syntax using the vocabulary earlier-duration and later-duration together with start-moment and finish-moment, thus: overlaps_result ◄— (earlier-duration-start-moment, earlier-duration-finish-moment) overlaps (later-duration-start-moment, later-duration-finish-moment) The overlaps operator treats a duration as a closed-open range. In other words: duration == [start-moment, finish-moment) However, even when a duration collapses to an instant, it is considered to be non-empty. (When the end-points of a '[)' range value are identical, this value is considered to be empty and cannot overlap with any other range value.) Because the start-moment is included in the duration but the finish-moment is not, this leads to the requirement to state the following edge case rules. (These rules were established by the SQL • If the left duration is not collapsed to an instant, and the left-duration-finish-moment is identical to the right-duration-start-moment, then the two durations do not overlap. This holds both when the right duration is not collapsed to an instant and when it is so collapsed. • If the left duration is collapsed to an instant, and the left-duration-start-and-finish-moment is identical to the right-duration-start-moment, then the two durations do overlap. This holds both when the right duration is not collapsed to an instant and when it is so collapsed. In other words, when two instants coincide, they do overlap. Notice that these rules are different from those for the && operator between a pair of '[)' range values. (The && operator is also referred to as the overlaps operator for range values.) The differences are seen, in some cases, when instants are involved. Try this: c1 as ( select '2000-01-01 12:00:00'::timestamp as the_instant), c2 as ( tsrange(the_instant, the_instant, '[)') as instant_range -- notice '[)' from c1) isempty(instant_range) ::text as "is empty", ( (the_instant, the_instant) overlaps (the_instant, the_instant) )::text as "overlaps", ( instant_range && instant_range )::text as "&&" from c2; This is the result: the_instant \| is empty \| overlaps \| && 2000-01-01 12:00:00 \| true \| true \| false In order to get the outcome true from the && operator, you have to change definition of the ranges from open-closed, '[)', to open-open, '[]', thus: c1 as ( select '2000-01-01 12:00:00'::timestamp as the_instant), c2 as ( tsrange(the_instant, the_instant, '[]') as instant_range -- notice '[]' from c1) isempty(instant_range) ::text as "is empty", ( (the_instant, the_instant) overlaps (the_instant, the_instant) )::text as "overlaps", ( instant_range && instant_range )::text as "&&" from c2; This is the new result: the_instant \| is empty \| overlaps \| && 2000-01-01 12:00:00 \| false \| true \| true It doesn't help to ask why the rules are different for the overlaps operator acting between two explicitly specified durations and the && acting between two range values. It simply is what it is—and the rules won't change. Notice that you can make the outcomes of the overlaps operator and the && operator agree for all tests. But to get this outcome, you must surround the use of && with some if-then-else logic to choose when to use '[)' and when to use '[]'. Code that does this is presented on this dedicated child page. 'overlaps' semantics in pictures The following diagram shows all the interesting cases. Two implementations that model the 'overlaps' semantics and that produce the same results These are presented and explained on this dedicated child page. The page also presents the tests that show that, for each set of inputs that jointly probe all the interesting cases, the two model implementations produce the same result as each other and the same result as the native overlaps operator, thus: 1. Durations do not overlap 2000-01-15 00:00:00, 2000-05-15 00:00:00 \| 2000-08-15 00:00:00, 2000-12-15 00:00:00 false 2. Right start = left end 2000-01-15 00:00:00, 2000-05-15 00:00:00 \| 2000-05-15 00:00:00, 2000-12-15 00:00:00 false 3. Durations overlap 2000-01-15 00:00:00, 2000-08-15 00:00:00 \| 2000-05-15 00:00:00, 2000-12-15 00:00:00 true 3. Durations overlap by 1 microsec 2000-01-15 00:00:00, 2000-06-15 00:00:00.000001 \| 2000-06-15 00:00:00, 2000-12-15 00:00:00 true 3. Durations overlap by 1 microsec 2000-06-15 00:00:00, 2000-12-15 00:00:00 \| 2000-01-15 00:00:00, 2000-06-15 00:00:00.000001 true 4. Contained 2000-01-15 00:00:00, 2000-12-15 00:00:00 \| 2000-05-15 00:00:00, 2000-08-15 00:00:00 true 4. Contained, co-inciding at left 2000-01-15 00:00:00, 2000-06-15 00:00:00 \| 2000-01-15 00:00:00, 2000-08-15 00:00:00 true 4. Contained, co-inciding at right 2000-01-15 00:00:00, 2000-06-15 00:00:00 \| 2000-02-15 00:00:00, 2000-06-15 00:00:00 true 4. Durations coincide 2000-01-15 00:00:00, 2000-06-15 00:00:00 \| 2000-01-15 00:00:00, 2000-06-15 00:00:00 true 5. Instant before duration 2000-02-15 00:00:00, 2000-02-15 00:00:00 \| 2000-03-15 00:00:00, 2000-04-15 00:00:00 false 6. Instant coincides with duration start 2000-02-15 00:00:00, 2000-02-15 00:00:00 \| 2000-02-15 00:00:00, 2000-03-15 00:00:00 true 7. Instant within duration 2000-02-15 00:00:00, 2000-02-15 00:00:00 \| 2000-01-15 00:00:00, 2000-03-15 00:00:00 true 8. Instant coincides with duration end 2000-02-15 00:00:00, 2000-02-15 00:00:00 \| 2000-01-15 00:00:00, 2000-02-15 00:00:00 false 9. Instant after duration 2000-05-15 00:00:00, 2000-05-15 00:00:00 \| 2000-03-15 00:00:00, 2000-04-15 00:00:00 false 10. Instants differ 2000-01-15 00:00:00, 2000-01-15 00:00:00 \| 2000-06-15 00:00:00, 2000-06-15 00:00:00 false 11. Instants coincide 2000-01-15 00:00:00, 2000-01-15 00:00:00 \| 2000-01-15 00:00:00, 2000-01-15 00:00:00 true	{"url":"https://docs.yugabyte.com/preview/api/ysql/datatypes/type_datetime/functions/miscellaneous/","timestamp":"2024-11-06T15:33:29Z","content_type":"text/html","content_length":"224419","record_id":"<urn:uuid:cb133eb3-3386-42cf-88da-c80c404d660f>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00651.warc.gz"}
Nanopascal to Ton-force (short)/sq. foot Converter \| nPa to tonf/ft^2 Nanopascal to Ton-force (short)/sq. foot converter \| nPa to tonf/ft^2 conversion Are you struggling with converting Nanopascal to Ton-force (short)/sq. foot? Don’t worry! Our online “Nanopascal to Ton-force (short)/sq. foot Converter” is here to simplify the conversion process for you. Here’s how it works: simply input the value in Nanopascal. The converter instantly gives you the value in Ton-force (short)/sq. foot. No more manual calculations or headaches – it’s all about smooth and effortless conversions! Think of this Nanopascal (nPa) to Ton-force (short)/sq. foot (tonf/ft^2) converter as your best friend who helps you to do the conversion between these pressure units. Say goodbye to calculating manually over how many Ton-force (short)/sq. foot are in a certain number of Nanopascal – this converter does it all for you automatically! What are Nanopascal and Ton-force (short)/sq. foot? In simple words, Nanopascal and Ton-force (short)/sq. foot are units of pressure used to measure how much force is applied over a certain area. It’s like measuring how tightly the air is pushing on The short form for Nanopascal is “nPa” and the short form for Ton-force (short)/sq. foot is “tonf/ft^2”. In everyday life, we use pressure units like Nanopascal and Ton-force (short)/sq. foot to measure how much things are getting squeezed or pushed. It helps us with tasks like checking tire pressure or understanding the force in different situations. How to convert from Nanopascal to Ton-force (short)/sq. foot? If you want to convert between these two units, you can do it manually too. To convert from Nanopascal to Ton-force (short)/sq. foot just use the given formula: tonf/ft^2 = Value in nPa * 1.044271711E-14 here are some examples of conversion, • 2 nPa = 2 * 1.044271711E-14 = 2.088543422E-14 tonf/ft^2 • 5 nPa = 5 * 1.044271711E-14 = 5.221358555E-14 tonf/ft^2 • 10 nPa = 10 * 1.044271711E-14 = 1.044271711E-13 tonf/ft^2 Nanopascal to Ton-force (short)/sq. foot converter: conclusion Here we have learn what are the pressure units Nanopascal (nPa) and Ton-force (short)/sq. foot (tonf/ft^2)? How to convert from Nanopascal to Ton-force (short)/sq. foot manually and also we have created an online tool for conversion between these units. Nanopascal to Ton-force (short)/sq. foot converter” or simply nPa to tonf/ft^2 converter is a valuable tool for simplifying pressure unit conversions. By using this tool you don’t have to do manual calculations for conversion which saves you time.	{"url":"https://calculatorguru.net/nanopascal-to-ton-force-short-sq-foot/","timestamp":"2024-11-11T03:37:54Z","content_type":"text/html","content_length":"123337","record_id":"<urn:uuid:a0c3c8cf-7035-4d67-9de2-7e52d27058c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00333.warc.gz"}