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Describe the water-jug problem. Also, give a suitable state space representation for this problem. Consider a scenario where you have a 3-litre jug and a 5-litre jug, and you need to measure precisely 4 litres of water. Visualize the scenario by imagining the two jugs and an infinite water source to fill them. The challenge here is to determine a sequence of actions that will allow you to reach the desired measurement of 4 litres, taking into account the constraints and capacities of the The Water Jug Problem in AI involves constraints and objectives that make it a puzzle: • Constraint 1: The jugs have limited capacities. • Constraint 2: You can only fill or pour water between the jugs or from the source. • Objective: The goal is to measure a specific quantity of water accurately, typically by combining and transferring water between the jugs. In AI problem-solving, we work with a state space (all possible states) and an action space (all possible actions). In the Water Jug Problem, the state space comprises all possible configurations of water levels in the jugs. The action space includes the actions you can take, such as filling a jug, emptying a jug, or pouring water from one jug to another. • Initial State: The initial state is where you start. In the classic scenario, this typically means both jugs are empty. • Goal State: The goal state is where you want to reach, representing the desired water level, e.g., 4 liters. • Actions: Actions are the operations you can perform on the jugs, such as filling, emptying, or pouring water between them. Now, let us formally state the problem. Suppose you have two jugs: a small one with a capacity of 𝑀 litres, and a large one with a capacity of 𝑁 liters (𝑀<𝑁). You are also given a target volume of water 𝑉 liters that you need to measure out using these jugs. The objective is to determine a sequence of pour operations that will result in obtaining exactly 𝑉 liters of water in one of the jugs or both. Here's an example of a suitable state space representation for this problem: A suitable state representation could be a tuple (𝑥,𝑦), where 𝑥 represents the amount of water in the small jug and 𝑦 represents the amount of water in the large jug. The initial state would be (0,0) (both jugs empty), and the goal state would be any state where 𝑥=𝑉 or 𝑦=𝑉.	{"url":"https://www.crookshanksacademy.com/post/describe-the-water-jug-problem-also-give-a-suitable-state-space-representation-for-this-problem","timestamp":"2024-11-11T19:24:22Z","content_type":"text/html","content_length":"1050086","record_id":"<urn:uuid:d590ad4e-75f8-4065-812e-5980b8130fc0>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00784.warc.gz"}
Area Conversion Calculator The area conversion calculator allows you to convert between 9 different units for area. Enter a value in the box for the known unit, press TAB and the other boxes will chance to their equivalent value. You can convert metric units, such as hectares, to other metric units such as square metres or square centimetres. You can also convert metric units, such has hectares, to imperial units such as acres, square mile or square feet. The area conversion calculator also works the other way. You can convert imperial units to metric units. The area conversion calculator will help you convert square metres to square feet, convert acres to hectares or convert acres to square mile. In fact when you enter one measure and press the TAB button you will see the conversion to the other 8 units of measure.	{"url":"https://calculatenow.biz/conversions/area.html","timestamp":"2024-11-04T12:25:10Z","content_type":"text/html","content_length":"14191","record_id":"<urn:uuid:74a65d3c-5bce-461d-b368-313df2f4b829>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00668.warc.gz"}
Thermal Analysis of a Flux-Switching Permanent Magnet Machine for Hybrid Electric Vehicles School of Electrical Engineering, Southeast University, Nanjing 210096, China Author to whom correspondence should be addressed. Submission received: 24 April 2023 / Revised: 12 May 2023 / Accepted: 17 May 2023 / Published: 19 May 2023 This paper investigates the loss and thermal characteristics of a three-phase 10 kW flux-switching permanent magnet (FSPM) machine, which is used as an integrated starter generator (ISG) for hybrid electric vehicles (HEVs). In this paper, an improved method considering both DC-bias component and minor hysteresis loops in iron flux-density distribution is proposed to calculate core loss more precisely. Then, a lumped parameter thermal network (LPTN) model is constructed to predict transient thermal behavior of the FSPM machine, which takes into consideration various losses as heat sources determined from predictions and experiments. Meanwhile, a simplified one-dimensional (1D) steady heat conduction (1D-SHC) model with two heat sources in cylindrical coordinates is also proposed to predict the thermal behavior. To verify the two methods above, transient and steady thermal analyses of the FSPM machine were performed by computational fluid dynamics (CFD) based on the losses mentioned above. Finally, the predicted results from both LPTN and 1D-SHC were verified by the experiments on a prototyped FSPM machine. 1. Introduction With the energy dilemma and environmental pollution becoming worse, electric-powered vehicles have attracted considerable attention due to their lower green gas emission and oil consumption [ ]. Recent hybrid electric vehicles (HEVs) provide long-distance operation due to the optimal match between an engine powered by oil and electric motors powered by electricity stored in batteries. HEVs are widely researched by academics and have become dominant commercial products in EV markets [ ]. For micro-hybrid EVs, an integrated starter generator (ISG) is the key component of both driving and generating systems, since it plays two major roles, namely, as a starter and a generator. Due to space and weight limitations, the rotor of an ISG is normally directly coupled to the flywheel of the engine in HEVs, where high torque (power) density, large overload torque capability, and high efficiency are expected. Hence, the flux-switching permanent magnet (FSPM) machine is considered as a promising candidate to be applied in electric vehicles, and aerospace and ship propulsion due to its high torque (power) density, high efficiency, and compact structure [ With the ever-increasing demand for power (torque) density, research on machine loss and temperature has become a hot topic. An accurate thermal model is an essential tool not only at the machine design stage but also for online prediction of temperature distribution [ ]. While finite element method (FEM)-based and computational fluid dynamics (CFD)-based thermal models can achieve high accuracy, a lumped parameter thermal network (LPTN)-based model is often preferred thanks to its lower computational requirement and good accuracy [ In [ ], the thermal influence of vehicle integration on the thermal load of an ISG was discussed by a FEM-based thermal model. In [ ], an axially segmented FEM model of a FSPM machine was proposed to analyze the coupled electromagnetic–thermal performances. A thermal resistance network was established based on a nine-node model for an interior PM (IPM) machine and the transient temperature characteristics were obtained [ ]. In [ ], a numerical approach for estimation of convective heat transfer coefficient in the end region of an ISG was proposed, and both the local and averaged heat transfer coefficients were estimated. A systematic procedure to study the impact of each thermal phenomenon in IPM machines used for ISG was presented in [ ]. In [ ], a reduced model in a multi-physical electric machine optimization procedure was proposed. The contribution of this paper is to propose two temperature prediction models for a 10 kW FSPM machine as an ISG for micro-hybrid vehicles; namely, a LPTN thermal model, and a one-dimensional (1D) steady heat conduction (SHC) (1D-SHC) model. The two methods can both quickly predict the internal temperature distribution of the FSPM machine. The results were verified by CFD and experiments to prove their accuracy. Section 2 will propose an improved core loss model considering both the DC-bias component and minor hysteresis loops in iron flux-density, and the core loss of the FSPM machine is calculated and verified by experiments. Then, in Section 3 a LPTN model is proposed firstly to predict transient thermal behavior. After that, a simplified 1D-SHC thermal model is proposed to reveal the relationship between design parameters of a cooling jacket and thermal distribution of stator, and verified by experiments under different cooling conditions. In Section 4 , both steady and transient thermal predictions are compared with those from ANSYS fluent-based CFD. Experiments with rising temperatures were conducted on a prototyped FSPM machine and are detailed Section 5 , followed by conclusions in Section 6 2. Loss Prediction Model The key dimensions of the studied FSPM machine are listed in Table 1 . In addition to electromagnetic parameters, the thermal conductivities, the specific heats, and densities of materials used for transient thermal analysis are presented in Table 2 . The employed PM material was N35 and the silicon steel sheet was 35WW310. The loss of the FSPM machine includes winding joule loss, core loss, eddy current loss in PMs, housing and frictional loss, and excess loss. According to the Bertotti G. model [ ], the core loss of PM machines consists of hysteresis loss, eddy current loss and excess loss, and the core loss yields: $P F e = P h + P c + P e = k h f B m α + k c f 1.5 B m 1.5 + k e f 2 B m 2$ is hysteresis loss in W, is the classical eddy current loss in W, is the excess loss in W, , and are the corresponding coefficient of the above losses, respectively, is the fundamental frequency of a magnetizing flux in Hz, and is the maximum flux density in core in T. However, Equation (1) only works given a purely sinusoidal magnetizing flux. To exactly obtain the magnetizing flux-density characteristics in the FSPM machine core, eight key points located in stator and rotor respectively are selected, as shown in Figure 1 a. Correspondingly, the resultant loci of the flux-density radial and tangential components ( ) are predicted, as shown in Figure 1 b. Clearly, for the stator points 1 and 2, the surrounded areas by the loci are to be almost zero, which means the averaged values are nearly zero. However, for points 3 and 4, the corresponding areas (the blue one and the pink one) are not centrosymmetric, which means a DC-biased component exists. For the points 5~8 in the rotor, the loci are all centrosymmetric and the averaged values are close to zero. A typically DC-biased component and a minor hysteresis loop are shown in Figure 1 c,d, respectively. Unfortunately, the influence of magnetized DC-biased components and minor hysteresis loops are not well recognized in the commercial FEM software packages [ ]. Hence, to predict the core loss of FSPM machines more precisely, an improved model considering both DC-biased component and minor hysteresis loops is proposed as follows. Assuming that the minor loop is similar to the major loop, the core loss can be divided into radial and tangential components. The total core loss, including the hysteresis loss [ ], eddy current loss [ ], and excess loss [ ], can be obtained as follows: $P h = k h f L a ∑ i = 1 N e l e m Δ A i [ ( ∑ j = 1 N p r i B r m i j 2 ) ε ( Δ B r ) + ( ∑ j = 1 N p t i B t m i j 2 ) ε ( Δ B t ) ]$ $P c = K c 2 π 2 L a N s e t p ⋅ ∑ k = 1 N s t e p ∑ i = 1 N e l e m Δ A i [ ( B r m i k + 1 − B r m i k Δ t ) 2 + ( B t m i k + 1 − B t m i k Δ t ) 2 ]$ $P e = K e L a N s e t p ∑ k = 1 N s t e p ∑ i = 1 N e l e m Δ A i [ ( B r m i k + 1 − B r m i k Δ t ) 1.5 + ( B t m i k + 1 − B t m i k Δ t ) 1.5 ]$ $P F e = P h + P c + P e$ is the finite elements number, Δ is the th finite element area in m are the radial and tangential minor loops numbers of the th element during one period, respectively, is the calculation steps number, are the maximum radial and tangential flux-densities of the th hysteresis loop in the th element in T, respectively, are the maximum radial and tangential flux-density of the th element in the th calculation step in T, and Δ is the time step in s. According to Equations (2)–(5), the core loss can be obtained by a combination platform of ANSYS and MATLAB, where based on ANSYS the detailed results of each meshed iron element can be acquired, and based on MATLAB the core loss versus rotating speeds under different conditions can be assessed by a series of data processing calculations according to Equations (2)–(5). The no-load core loss density distribution of the stator and rotor is shown in Figure 2 . Consequently, the predicted core losses versus rotor speed are compared with those obtained by commercial software, e.g., by JMAG and ANSYS EM as shown in Figure 3 . It can be seen that the core losses obtained by the improved method are slightly higher than those from software, which validates the influence of the DC-biased component and minor hysteresis loop, and also validates the feasibility of the improved core loss prediction method. In addition, the no-load eddy current density distribution derived by 3D-FEM is shown in Figure 4 . The resulting no-load eddy loss in PMs and housing versus rotating speeds are shown in Figure 5 . It can be found that with the increase of the speed, the eddy current losses in PMs and housing increase gradually, which is caused by the air-gap harmonic fields and can be calculated by Equation (6) [ $P e d d y = 1 T ∫ t c ∑ i = 1 k J e 2 Δ A i σ r − 1 L a d t$ is the eddy current loss in PM and housing in W, is the current density in each element in A/m , Δ is the th element area in m is the conductivity of the eddy current zone in S/m, and is the time corresponding to a period in each element in s. The frictional loss of the FSPM machine yields [ $P f r i = 16 N r ( v r 40 ) 3 L a 19 × 10 3$ is the rotor peripheral speed in m/s. It is found that = 10.7 W under the rated speed of 1000 r/min. Finally, a no-load test under different rotation speeds is conducted on a prototyped FSPM machine to verify the predicted results, where the machine is controlled by a DSP-based controller and the input power is obtained by a power analyzer. The frictional loss is so small that it can be neglected. Therefore, the input power is equal to the total loss. The total losses versus rotor speeds by different methods are compared in Figure 6 . Compared with the results obtained by commercial software, the predicted core losses derived by the improved method agree with the measurements with the smallest deviations. 3. Two Thermal Models For the prototyped FSPM machine, a circumferential water jacket with one cooling duct is introduced in the stator housing as shown in Figure 7 . The coolant channel of the casing adopts a single-layer water jacket cooling structure. Figure 7 a shows the schematic diagram of the FSPM machine structure. Figure 7 b–d show the housing, coolant flow path, and machine assembly, respectively. Figure 7 b shows the cross-sectional diagram of the machine, and Figure 7 c shows the cross-sectional diagram of the cooling duct. Arrows are used in Figure 7 c to indicate the flow path of the coolant. The blue arrow represents the low temperature coolant near the inlet, while the red arrow represents the coolant that has been heated through heat exchange. The fluid running inside the cooling duct can be modeled as the movement of fluid in a rectangular channel using dimensionless numbers. Consequently, the convection coefficient can be obtained in the following stages. Firstly, with the cooling jacket cross-section in Figure 7 c, the Prandtl number of the fluid is the specific heat capacity of fluid in J/(kg·°C), is fluid dynamic viscosity in N·s/m , and is the fluid thermal conductivity in W/(m·°C). Secondly, the Reynolds number of fluid is the velocity of fluid in m/s, is the fluid kinetic viscosity in m /s, and is the hydraulic radius in m by Equation (10). $d e = 4 A c s s = 4 b h 2 ( b + h )$ is a cross-section area of a single cooling duct in m , and are the wetted perimeter, width, and height of cooling duct in m, respectively. Approximately, in a circumferential cooling duct, the velocity of fluid can be figured out as is the fluid quantity in kg/s. According to the value of , the fluid flow can be divided into turbulence flow and laminar flow. The Nusselt number of the laminar flow yields [ $N u f l = 0.644 ( R e 0.5 ) P r f 1 3$ For turbulence flow, the Nusselt number $N u f t = 0.023 ( R e 0.8 ) P r f 0.4 ( η f η w ) 0.14$ is the dynamic viscosity of housing in N·s/m Based on the similarity criterion of fluid [ ], the convection heat transfer coefficient Considering that turbulence flow shows a better heat dissipation than laminar flow, the former is employed in the cooling jacket, where the convection heat transfer coefficient of the cooling jacket is affected by the geometric parameters and the velocity of the fluid, and can be given by $h f = λ f b + h 2 b h ( 2 Q v f ( b + h ) ) 0.8 P r f 0.4 ( η f η w ) 0.14$ From Equation (15), when the fluid quantity keeps constant, as the cross-sectional area of the cooling ducts decreases, the convection coefficient of the cooling jacket increases. However, a small cross-section cooling duct is not only difficult to manufacture, but also may lead to high inlet velocity and high hydraulic pressure, which causes the corrosion of cooling duct and deteriorates the operation stability. In the following, two thermal models are proposed, one a LPTN model and the other a 1D-SHC model. 3.1. Lumped Parameter Thermal Network Model A LPTN model enables the heat flow and the temperature distribution inside the machine by means of an equivalent thermal circuit, which is composed of heat sources, thermal resistances, and thermal capacitances. For the convenience of calculation and improved accuracy, three assumptions are made as follows [ • Symmetrical temperature distribution and the same cooling conditions along the circumference; • Uniformly distributed thermal capacity and heat generation; • Independent heat flow in radial and axial directions To simplify the calculation load, only 1/24 of the FSPM machine is modeled as shown in Figure 8 due to symmetry, where the heat sources including stator/rotor core losses, PM/housing eddy current losses, and windings joule loss are considered. The thermal resistances and capacitances can be determined according to the machine geometry and the physical properties of materials. Table 3 Table 4 list the corresponding resistances and capacitances of the LPTN model. A preliminary selection of the resistances and capacitances can be determined according to the machine geometry and physical properties of the materials used [ With the convection heat transfer coefficient, the thermal resistance = 1, 2, 3) representing the heat dissipation by cooling medium convection between the housing external surface and ambient can be calculated by Equation (16) [ $R c o n v i = 1 h c o n v i A c o n v i$ is the thermal resistance due to convection heat transfer in °C/W, is the convection heat transfer coefficient in W/(m ·°C), and is the convective area in m . Here, the area of the end-part winding is considered. Since the heat exchange between stator and rotor through the air-gap is assumed to be only by convection, Equation (16) is also used for the calculation of the thermal resistances R[airi]. In addition to heat convection, heat conduction is also an important way for heat dissipation. The resistance representing the heat flow in the radial direction is modeled by using Equation (17), $R r a d i a l = l n ( r o / r i ) 2 π λ L$ are the outer and inner diameter of the cylinder in m, is the thermal conductivity of the material in W/m/°C, and is the cylinder length in m. In the tangential direction, the thermal resistance due to conduction heat transfer is given by, $R t a n g e n t i a l = l λ A c o n d$ is a portion length of the path considered in m, and is the area for the conduction in m Figure 8 a,b show the 3D module structure and modular stator element of the FSPM machine. Based on the thermal resistances above, a thermal resistance network of the 1/24 machine is constructed as shown in Figure 8 c, where , and represent the thermal resistances of the housing, stator yoke, air-gap, stator tooth, stator winding coils, and PMs. represent the thermal resistances of rotor tooth, rotor yoke, and shaft, respectively. , and represent the thermal capacitances of the housing, stator yoke, stator tooth, winding coils, PMs, and rotor tooth. Here, since the heat dissipated by forced convection is much larger than that by radiation, the radiation heat dissipation is ignored. Under two typical operation conditions, i.e., the speed of 1000 r/min and the phase current of 30.7 A (RMS) and 60 A (RMS), the transient temperature rises of different components under forced water cooling are obtained by the LPTN model, and the results are shown in Figure 9 . It takes around 70 min for the machine under water cooling to reach a thermal steady state where the armature windings have the highest temperatures (45.7 ° [email protected] A and 82.8 °C@60 A under water cooling). In addition, since the PMs are mounted on the stator, the temperature of the PMs is very close to that of the stator core, exhibiting the advantage of FSPM 3.2. One-Dimensional Steady Heat Conduction Model Generally, LPTN and FEM have been widely employed in thermal analysis of electrical machines. However, these two methods are normally time-consuming and require complicated modeling. For water-cooling machines, in order to select a reasonable flow rate of coolant, a 1D-SHC approach is proposed based on heat transfer and fluid mechanics as shown in Figure 10 , and the relationship between the internal temperature of the stator and the coolant flow rate and coolant temperature is obtained. It should be noted that the 1D-SHC model is based on the following assumptions. (1) The loss is uniformly distributed in each component of the machine. (2) The heat generated by joule loss and stator core loss is only dissipated by the housing. (3) The stator laminations, windings, and PMs are simplified to a homogeneous heating unit and the equivalent averaged thermal conductivity yields [ $λ a v e = A s + A w i n d + A p m A s λ s + A w i n d λ c u + A p m λ p m$ , and are the cross-section area of the stator, windings, and PMs in m , respectively, , and are the thermal conductivity of stator, windings, and PMs in W/(m·°C). The winding and insulation layering is used to calculate the thermal conductivity of the stator windings. Figure 11 shows the equivalent diagram of the winding structure, where the insulation and windings are arranged with intervals. The slot filling factor is set as 0.35 according to the prototyped machine. The equivalent winding thermal conductivity yields: $λ w i n d = ∑ i = 1 n δ i / ∑ i = 1 n δ i λ i$ is the thickness of the th layer in m, is the thermal conductivity of the th layer in W/(m·°C). Then, the equivalent volumetric heat generation of stator and rotor can be obtained by ${ P e a v e = P s V s + P c u V c u + P p m V p m V s + V c u + V p m q V s = P e a v e V e q u q V r = P r V r$ is equivalent average loss in W, , and are the stator core loss, winding joule loss, eddy current loss in PMs, and rotor core loss in W, respectively, , and are the volume of stator, winding, PM, equivalent stator, and rotor in m , respectively. Since the thermal model is simplified into a 1D-SHC model, the heat flux density of stator/rotor ( ) yields ${ q s = q V s / S s q r = q V r / S r$ is the cross-section area of stator/rotor lamination in m The inlet and outlet temperature of the cooling fluid can be detected by a hand-held infrared thermometer. Thus, the temperature of the fluid is the inlet/outlet temperature of the fluid in °C. According to the 1D thermal circuit in Figure 10 b, the temperature of housing can be derived by $T s h = T f + q r π h f ( R s h o − R s h a o ) + q s π h f ( R s h o − R s i )$ is the fluid convection coefficient in W/(m is the housing outer radius in m, is the shaft outer radius in m, and is the stator inner radius in m. The temperature of stator yoke $T s y = T s h + ( q r + q s ) l n ( R s h o / R s o ) 2 π h s h$ is the stator outer radius in m, and is the housing thermal conductivity in W/(m·°C). The differential equations of the heat conduction and the boundary conditions for a cylinder with uniform heat generation are as follows [ ${ 1 r ⋅ d d r ( r ⋅ d t d r ) + q r λ r + q r + q s λ a v g = 0 T = T s y , r = R s o ; d t d r = 0 , r = 0$ is the rotor thermal conductivity in W/(m·°C). The thermal distribution of the machine can be given by ${ h f = λ f b + h 2 b h ( 2 v v f ) 0.8 P r f 0.4 ( η f η w ) 0.14 T ( r ) = R s o 2 4 ( q r λ r + q r + q s λ a v g )$ Hence, the stator teeth temperature can be obtained by $T s t = T i + T o 2 + ( q r + q s ) ( l n ( R s h o / R s o ) 2 π h s h + R s o 2 − R s i 2 4 λ a v g ) + 1 π h f ( q r R s h o − R s h a o + q s R s h o − R s i ) + ( R s o 2 − R s i 2 ) q r 4 λ r$ According to Equations (11), (15) and (28), as the average velocity of fluid increases, the convection coefficient of the cooling jacket increases and the stator temperature decreases. According to the prototype dimensions, when the no-load machine is running at the speed of 1000 r/min, the relationship between the temperature of the equivalent stator core (marked in Figure 10 ) and the inlet velocity can be obtained as shown in Figure 12 . Obviously, as the inlet velocity of water increases to 0.6 m/s, the equivalent stator core temperature decreases almost linearly. When the inlet velocity increases to 1 m/s, the stator core temperature varies nonlinearly and slowly, so 1 m/s is set as the rated cooling inlet velocity of the FSPM machine, corresponding to a pump flow of 1800 L/h. 4. CFD-Based 3D Temperature Field Verification In order to verify the proposed LPTN and 1D-SHC models, based on the loss calculated by FEM, a 3D-CFD thermal model is built as shown in Figure 13 Figure 13 b corresponds to water cooling. When the cooling jacket is injected with an total inlet flow of 1800 L/h, a phase current of 30.7 A and 60 A, as well as a speed of 1000 r/min, the transient temperature rises of different components under forced water cooling are obtained as shown in Figure 14 . It can be seen that the armature windings achieve the highest temperature under forced water-cooling conditions, whereas the armature windings temperature difference is 31 °C when the armature current is 30.7 A and 60 A, respectively. In addition, compared with the results obtained by the LPTN model shown in Figure 9 , both the steady-state and transient results agree well. On the other hand, to verify the 1D-SHC model, the predicted equivalent stator core temperatures vs. inlet velocity by 1D-SHC and CFD are compared in Figure 15 , where the operation status and cooling conditions are consistent with the 1D-SHC model. In addition to predicted temperatures, the time consumed by the three methods is compared in Table 5 . Obviously, both the LTPN and 1D-SHC methods can save considerable time, which is favorable for the optimal design of machines. 5. Experiment Verification To validate the proposed thermal prediction models, a prototyped FSPM machine was manufactured and tested as shown in Figure 16 . The prototyped FSPM machine was driven by an inverter supplied by a DC power source and the output shaft was directly connected with a dynamometer machine, in which a torque transducer and a resolver were equipped to measure torque and (rotor position) speed, respectively. Figure 17 compares the FEA-predicted and experimental results of torque versus phase currents. It can be seen that good agreements can be achieved with a deviation below 8%. To verify the LPTN model, experiments on transient temperature rise were performed. Under the phase current of 30.7 A and rated speed of 1000 r/min, the transient temperature rises of different components under forced water cooling were obtained as shown in Figure 18 , where the temperature was detected with a hand-held infrared thermometer. Under forced water-cooling conditions, the measured highest temperature was 47.1 °C. Compared with the results obtained by LPTN ( Figure 9 ) and CFD ( Figure 14 ), it was found that the steady-state and transient results of the three methods were very close. Figure 19 shows the experimental steady-state temperatures under forced water cooling. Obviously, agreement between the experiments and LPTN was achieved, validating the effectiveness of the LPTN model. To verify the 1D-SHC model, experiments with temperature rising and various fluid inlet velocities of the no-load machine at the rated speed of 1000 r/min were conducted, as shown in Figure 20 . It can be seen that when the inlet velocity was bigger than 1.1 m/s, the temperature of the equivalent stator core decreased slowly, which agrees with the simulations giving both 1D-SHC and CFD Overall, satisfactory agreement was achieved between the calculation and measured results, considering manufacturing and testing tolerances. 6. Conclusions In this paper, a LPTN model is constructed to predict transient thermal behavior of the FSPM machine. Meanwhile, a simplified 1D-SHC model is also proposed to obtain the relationship between the internal temperature of the stator and the coolant flow rate and coolant temperature. The time consumption of the LPTN and 1D-SHC models was significantly less than that of the CFD model, which has advantages in machine design and optimization with large amounts of data. Based on the housing water jacket cooling FSPM machine studied in this manuscript, the LPTN and 1D-SHC methods have accelerated the steady-state temperature calculation speed by 330 and 1100 times, respectively, compared to the CFD method, and have accelerated the transient calculation speed by 857 and 4285 times, respectively. The static and transient temperatures under different conditions were verified by the CFD calculations and experiments. The predicted results from the models agree well with experimental results. This work will be useful in further investigation of thermal analysis of FSPM machines. Author Contributions Conceptualization, W.H.; methodology, W.Y. and Z.W.; software, W.Y.; investigation, W.Y. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Science Fund for Distinguished Young Scholars of China under Grant 51825701 and the Major Program of National Natural Science Foundation of China under Grant Data Availability Statement The data presented in this study are available on request from the corresponding author. The data are not publicly available due to the project being still in progress. Conflicts of Interest The authors declare no conflict of interest. Figure 1. The flux-density loci of key points in the FSPM machine. (a) The key stator and/rotor core points in the FSPM machine. (b) The B[gr]/B[gt] loci of key stator and rotor core points. (c) The DC-biased components of B[gr]/B[gt]. (d) The local minor hysteresis loop. Figure 7. The cooling system of the FSPM machine. (a) FSPM machine structure, (b) housing, (c) coolant flow path, and (d) machine assembly. (e) Cross-section of the cooling jacket, and (f) schematic diagram of cooling duct. Figure 8. The LPTN model of the FSPM machine: (a) 3D module structure; (b) stator module; (c) the LPTN model of the 1/24 FSPM machine. Figure 9. The predicted temperature rises by the LPTN model under water-cooling conditions @ nN = 1000 r/min and Iph = 30.7 A and 60 A (RMS). Figure 10. One-dimensional steady heat conduction model of the FSPM machine. (a) Equivalent heat conduction model; (b) equivalent heat flow path. Figure 13. Three-dimensional CFD thermal model of the FSPM machine and steady-state temperatures @30.7 A and 60 A; (a) 3D-CFD thermal model; (b) water cooling. Figure 19. Steady-state temperature distribution of the FSPM machine under forced water-cooling conditions. Parameter Symbol Value Unit DC-link voltage U[DC] 144 V Phase number m 3 - Stator slots N[s] 12 - Rotor pole pairs N[r] 10 - PM pole pairs N[PM] 6 - Rated power P[N] 10 kW Rated speed n[N] 1000 r/min Rated torque T[N] 95.5 Nm Stator outer diameter D[so] 260 mm Rotor inner diameter D[ri] 50 mm Air-gap length g[0] 0.9 mm Stack length L[a] 55 mm Materials Thermal Conductivity (W/m/°C) Specific Heat Capacity (J/kg/°C) Density (kg/m^3) Steel silicon 23 460 7650 Copper 380 385 8978 PM 9 504 7500 Aluminum 237 833 2688 Air 0.02624 1005 1.205 Thermal Resistances Value (°C/W) Thermal Resistances Value (°C/W) R[sh][1], R[sh][2], R[sh][3] 0.0004307 R[st][1], R[st][2] 0.1479 R[sh][4] 0.1094 R[st][3] 0.01449 R[sh][5] 0.07664 R[coil][1] 0.6297 R[sh][6], R[sh][7], R[sh][8] 0.0004469 R[coil][2] 4.895 R[sy][1], R[sy][2] 0.00359 R[coil][3] 0.7244 R[sy][3] 0.2107 R[pm][1] 0.01197 R[sy][4] 0.1128 R[pm][2] 0.04699 R[sy][5] 0.003729 R[pm][3] 0.05441 R[sy][6] 0.01632 R[pm][4] 0.2054 R[air][1], R[air][2], R[air][3] 1.096 R[pm][5] 0.04829 R[air][4], R[air][5] 596 R[rt][1] 0.02435 R[air][6], R[air][7] 27.84 R[ry][1] 0.03246 R[air][8] 30.76 R[shaft][1] 0.2613 R[air][9] 56.14 - - Thermal Capacitance Value (J/°C) Thermal Capacitance Value (J/°C) C[sh][1] 30.82 C[st][1] 71.37 C[sh][2] 21.86 C[coil][1] 0.0036 C[sh][3] 10.75 C[pm][1] 10.67 C[sy][1] 31.63 C[pm][2] 33.14 C[sy][2] 22.98 C[r] 497.2 Temperature Prediction Methods Steady State Transient LPTN method 2 s 7 s 1D-SHC method 0.6 s 1.4 s CFD method 11 min 100 min Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https: Share and Cite MDPI and ACS Style Yu, W.; Wu, Z.; Hua, W. Thermal Analysis of a Flux-Switching Permanent Magnet Machine for Hybrid Electric Vehicles. World Electr. Veh. J. 2023, 14, 130. https://doi.org/10.3390/wevj14050130 AMA Style Yu W, Wu Z, Hua W. Thermal Analysis of a Flux-Switching Permanent Magnet Machine for Hybrid Electric Vehicles. World Electric Vehicle Journal. 2023; 14(5):130. https://doi.org/10.3390/wevj14050130 Chicago/Turabian Style Yu, Wenfei, Zhongze Wu, and Wei Hua. 2023. "Thermal Analysis of a Flux-Switching Permanent Magnet Machine for Hybrid Electric Vehicles" World Electric Vehicle Journal 14, no. 5: 130. https://doi.org/ Article Metrics	{"url":"https://www.mdpi.com/2032-6653/14/5/130","timestamp":"2024-11-06T05:45:30Z","content_type":"text/html","content_length":"507184","record_id":"<urn:uuid:505bbb1b-d954-4851-a45d-88727d13afff>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00638.warc.gz"}
Readings \| Double Affine Hecke Algebras in Representation Theory, Combinatorics, Geometry, and Mathematical Physics \| Mathematics \| MIT OpenCourseWare Required Readings In order to prepare for class, students are required to read selections from the course notes. These readings can be found on the lecture notes page. WEEK # TOPICS READINGS 1 Classical and quantum Olshanetsky-Perelomov systems for finite Coxeter groups Chapter 2 2 The rational Cherednik algebra I Chapter 3, sections 3.1-3.13 The rational Cherednik algebra II Chapter 3, sections 3.14-3.17 Finite Coxeter groups and the Macdonald-Mehta integral Chapter 4, section 4.1 4 The Macdonald-Mehta integral Chapter 4, sections 4.2-4.4 5 Parabolic induction and restriction functors for rational Cherednik algebras Chapter 5 The Knizknik-Zamolodchikov functor Chapter 6 Rational Cherednik algebras for varieties with group actions Chapter 7, sections 7.1-7.5 7 Hecke algebras for varieties with group actions Chapter 7, sections 7.6-7.15 8 Symplectic reflection algebras I Chapter 8, sections 8.1-8.7 9 Symplectic reflection algebras II Chapter 8, sections 8.8-8.13 10 Calogero-Moser spaces Chapter 9 11 Quantization of Calogero-Moser spaces Chapter 10 Supplemental Readings Bezrukavnikov, R., and P. Etingof. “Parabolic Induction and Restriction Functors for Rational Cherednik Algebras.” Selecta Math 14, nos. 3-5 (2009): 397-425. Etingof, P., and V. Ginzburg. “Symplectic Reflection Algebras, Calogero-Moser Space, and Deformed Harish-Chandra Homomorphism.” arXiv:math/0011114. Rouquier, R. “Representations of Rational Cherednik Algebras.” arXiv:math/0504600. Etingof, P. Lectures on Calogero-Moser Systems. arXiv:math/0606233. ———. “Cherednik and Hecke Algebras of Varieties With a Finite Group Action.” arXiv: math.QA/0406499. ———. “A Uniform Proof of the Macdonald-Mehta-Opdam Identity for Finite Coxeter Groups.” arXiv:0903.5084. ———. “Supports of Irreducible Spherical Representations of Rational Cherednik Algebras of Finite Coxeter Groups.” arXiv:0911.3208.	{"url":"https://ocw.mit.edu/courses/18-735-double-affine-hecke-algebras-in-representation-theory-combinatorics-geometry-and-mathematical-physics-fall-2009/pages/readings/","timestamp":"2024-11-05T10:51:07Z","content_type":"text/html","content_length":"43674","record_id":"<urn:uuid:3fb90b25-5193-4ed0-8055-34e980ebbfa8>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00747.warc.gz"}
Let’s break the encryption Published by: Amit Nikhade July 7 . 2021 Shor’s Algorithm exposed/simplified/clarified with a quick overview. Prof.Peter Williston Shor Once you read this article, you’ll get a clear overview of how exactly Shor’s algorithm works. I tried explaining it, without going into a much deeper section, because it will definitely confuse at the beginner stage. RSA breaching isn’t only the application of Shor’s algorithm but there are several use-cases like quantum simulation, spin-off technology, Quantum cryptography, etc. Shor’s algorithm was developed by Peter Shor (American mathematician) in the year 1994. It performs integer factorization in polynomial time. The requirement of the algorithm is the quantum computer. The algorithm says that the quantum computer is capable of performing factorization on a very large number in polynomial time. There are many algorithms made for integer multiplication, but there weren’t any of them for factorizing an integer Let us take a deep overview of this useful stuff. According to Quantitative Church’s thesis, Any physical computing device can be simulated by a Turing machine in a number of steps polynomial in the resources used by the computing device. Shor’s algorithm Contents Eagle eye view The algorithm is composed of two parts as follows: 1. Turning the factoring problem into the problem of finding the period of a function 2. Finding the period using the quantum Fourier transform. Run time on the classical computer is O[exp (L1/3(log L)2/3)] and that on the quantum computer is O(L3). The algorithm requires both classical as well as quantum computation. The first part is performed classically and the second one utilizes quantum power and responsible for quantum speedup. Shor’s algorithm is probabilistic in nature, it addresses results with high probabilities, and the one with failure can be reduced by looping the algorithm until we get the desired output. Salient terms 1. Prime number: The number that can only be divided by itself like, 3, 7, 11, etc. 2. Polynomial Time: An algorithm is said to be solvable in polynomial time if the number of steps required to complete the algorithm for a given input is O(n^k), where n is the number of steps to solve the algorithm and k is constant i.e the non-negative number. The polynomial-time algorithm is much faster as compared to the exponential time algorithms. Problems that can be solved by a polynomial-time algorithm are called tractable problems. 3. Euclid algorithm: The Euclidien algorithm is used to find the GCD of two numbers i.e the greatest common divisor. Where a and b are two numbers, the GCD(a, b) = GCD(b, a%b), if a=0 then GCD(a, b) =b and if b=0 then GCD (a, b) = b. The algorithm finds the greatest common divisor in polynomial time. 4. Quantum Fourier transforms: The quantum Fourier transform is a unitary transformation. In quantum Fourier transform the (DFT) Discrete Fourier transform is applied over the amplitudes of the quantum states. The Discrete Fourier transform can be defined as: The Fourier transform is mainly used to map a signal or a function into an alternate representation, characterized by sine and cosines. It’s just a fancy term used to change a form of representation. Discrete Fourier transform has various applications, it is used to digitize the signals, like in Fourier analysis where we describe the internal frequencies of a function. However, the quantum Fourier transform is exponentially faster than the DFT as well as its faster version FFT. Quantum Fourier transformation circuit is composed of Hadamard gates and conditional phase shift gates. Quantum Fourier transform can be defined as shown in the below figure, where x is the basis state. Quantum Fourier transform The Quantum Fourier Transform is a generalization of the Hadamard transform by the Hadamard gate. 6. Period finding: Period finding is the basic building block of Shor’s algorithm, similar to the simons algorithm. If finding the period of a function is done efficiently, we would be able to factor integers quickly. To begin with, the states are initialized to state∣0⟩ and apply Hadamard gate to put them into superposition, Another approach would be the use of Quantum Fourier transform. Finding the period heavily depends on the ability of the Quantum computer to achieve superposition. The quantum Fourier transform helps us finding the superior result returning them with their probabilities. The period finding can be also done using the classical approach but can be done in a better way with quantum computers. You may hear something like “Factoring reduces to order-finding” (Can be done classically) which means that if there is an algorithm to solve order-finding efficiently, we can efficiently solve the factoring problem as well by a polynomial-time reduction from factoring to order-finding. The order finding is similar to the Period finding. Shor’s algorithm consists of the following steps: 1. Choose a random positive integer m. Compute the greatest common divisor GCD using the euclidean method (m, N) where N is the set of natural numbers N={1, 2, 3, 4, 5…….}. If the greatest common divisor gcd(m, N) != 1, then the value obtained is the non-trivial factor of N. But if gcd(m, N) = 1, then we need to determine the period. 2. This step involves finding the period r of the function f(x) = a^x mod N. Such that f(x)= f(x+r), we need to find the smallest value of r. (This step can be brought into use only with a Quantum 3. If r is an odd number, then we need to again start from the beginning else we’ll proceed to the next step. 4. If m^r/2 + 1 = 0 mod N, then we again need to go to step 1. else we need to compute the greatest common divisor of (m^r/2- 1, N) using the Euclidean algorithm, And finally we obtain non-trivial factors of N Calculating the GCD Suppose we wist to factor 91 i.e N=91. And we know that square of 64 = 4096 i.e 4591+1, so that x=64 is a solution of square of x = 1 (mod 91) . The above explanation implies that 4591=square of 64–1 = 63*65 Summary of Classical computation 1. Choose a random integer. 2. Calculate the GCD. If GCD is equal to one, proceed to the next step i.e the period finding. else if GCD is not one, it’s done. 3. If the period is odd, start from step 1. Else proceed to the next step. 4. If m^r/2 + 1 = 0 mod N, then we again need to go to step 1, Else it’s done with just calculating the GCD. Quantum Period finding subroutine 1. Firstly, we need to initialize the registers state, like 2. Next, we need to apply the Hadamard gate to register one, for superpositioning them. where ⊗ denotes the tensor product. Where the \|0⟩ represents the empty second register 3. furthermost, we need to construct the f(x) as a quantum function and Apply the linear transformation Uf to the two registers to obtain Here, We have entangled both registers where Uf is the unitary transformation that maps\|x⟩ \|0⟩ to \|x⟩ \|f(x)⟩. 4. This step, we apply the Quantum Fourier transform to the first register, which concludes with 5. At the almost end we measure the register one, which yields y. We also find the period r via the continued fraction expansion method by y/2^L. Continued fraction Expansion 6. The Continued fraction method helped us obtaining the period r. Now we simply have to perform some classical post-processing. Here if the resulting period value is an odd integer, the process has to be restarted and if it is even we can proceed to further classical computing steps. The Probabilities can be given by this spectrum as shown above The probabilities of y, something similar to this Summary of Period Finding Algorithm 1. Step 1: Initialize Registers with their states 2. Step 2: Apply Hadamard gates to qubit in register 1. 3. Step 3: Apply Unitary transform to the registers 4. Step 4: Apply quantum Fourier transform to register 1. 5. Step 5: Perform Measurement of register 1. Implementing the code We’ll try creating a code for Shor’s algorithm in python. Let’s see how it works. We need to install Qiskit firstly. Below are the installation steps. Qiskit is tested and supported on the following 64-bit systems: • Ubuntu 16.04 or later • macOS 10.12.6 or later • Windows 7 or later Qiskit supports Python 3.6 or later. Create a conda environment. conda create -n qenv python=3 Activate your new environment. conda activate qenv Next, install the Qiskit package. pip install qiskit And you are done. Next, you have to set up the IBM Quantum Experience, Just create an account on it, and get your API token. Now you are fully set for coding with Qiskit. Let dive into the code now from qiskit import IBMQ from qiskit.aqua import QuantumInstance from qiskit.aqua.algorithms import Shor IBMQ.enable_account(‘API’) # Enter your API token here provider = IBMQ.get_provider(hub=‘ibm-q’) backend = provider.get_backend(‘ibmq_qasm_simulator’) # Specifies the quantum device factors = Shor(21) #Function to run Shor’s algorithm where 21 is the integer to be factored result_dict = factors.run(QuantumInstance(backend, shots=1, skip_qobj_validation=False)) Factors = result_dict[‘factors’] # Get factors from results The factors resulted are 3 and 7. Quantum computing is a flagship technology that can yield advancements beyond imagination. Shor’s algorithm has been an immense achievement in quantum computing, but still, we are not able to implement it practically due to the Quantum computers haven’t yet reached a significant level that can perform factorization on a very big number. Hope we’ll soon able to see them. Originally Published on amitnikhade.com I hope you found it insightful. Do hit those claps and follow me for more such amazing stuff.	{"url":"https://havric.com/2021/07/07/lets-break-the-encryption-shors-algorithm/","timestamp":"2024-11-04T11:43:26Z","content_type":"text/html","content_length":"135584","record_id":"<urn:uuid:47bf5a69-c748-4444-8e7e-034f0313794a>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00758.warc.gz"}
Loan to Value (LTV) Ratio Calculator 2024 \| Casaplorer® Loan to Value (LTV) Ratio Calculator 2024 This Page Was Last Updated: August 17, 2022 CASAPLORER^®Trusted & Transparent How much is the home you’re looking to buy? How much downpayment do you have? Your Loan-to-Value (LTV) is The loan-to-value calculator measures the relationship between the mortgage amount and the value of the home and is used by lenders to determine the risk associated with a secured loan. What is a Loan-to-Value (LTV) Ratio? The LTV ratio is an important financial indicator used by mortgage lenders as it shows the amount of initial equity the borrower has in the home. The ratio is a crucial component of the mortgage underwriting process and is used as a requirement for a majority of mortgage programs. The loan-to-value ratio is used for all home transactions such as buying a home, second mortgage, refinancing a mortgage, home equity loans, and even for a Home Equity Line of Credit (HELOC). The LTV ratio is used as a requirement for all types of mortgages, fixed and adjustable-rate mortgages (ARM). Adjustable-rate mortgages are mortgage rates that are linked to a benchmark index such as the Prime Rate which is linked to the FED Funds Rate. How do I Calculate the Loan-to-Value Ratio? The LTV ratio can be determined by having two inputs, the total mortgage amount and the value of the home. The mortgage amount is divided by the home value to determine the LTV ratio. The following formula is used: LTV Ratio Mortgage Amount Home Value For example, if the home selling price is $300,000 and the mortgage amount is $250,000 then the LTV ratio is 83% ($250,000/$300,000). The LTV ratio is impacted by three main factors, home selling price, down payment, and the appraised value of the home. In most cases, the appraised value of your home is not known till the mortgage process begins, hence, instead of the appraised value, the home selling price is used in the calculation. What should my Loan-to-Value Ratio be? A good LTV ratio to have is any value less than 80%. What this means is that the total mortgage loan taken from the lender is 80% of the home value. The other 20% comes in the form of a down payment and is your ownership stake in the home from the start of the mortgage. The reason for the 80% benchmark is because if the down payment is less than 20%, such that the LTV ratio is greater than 80%, then Private Mortgage Insurance (PMI) is required. Mortgage insurance can make the total mortgage more expensive and even result in higher monthly mortgage payments. Should I have a High or a Low Loan-to-Value Ratio? It is preferred to have a lower LTV ratio as it demonstrates that the borrower has a larger stake in the home which means the loan amount is a smaller proportion of the total value of the home. A higher LTV ratio suggests a larger mortgage amount and more risk for the lender. Lenders use the debt-to-income (DTI) ratio and the LTV ratio as risk-measuring tools during the underwriting process. If the mortgage amount is closer to the value of the home, that is a higher LTV ratio, it presents a larger risk for lenders if the borrower defaults on their mortgage payments as there is a smaller equity cushion. In the event of foreclosure, the lender will have to cover a larger portion of the cost as the home equity owned by the borrower is smaller. Therefore, a higher LTV ratio often results in a higher mortgage interest rate to compensate the lender for the additional risk. An LTV ratio below 80% is preferred and this can be achieved by making a larger down payment. What is the relationship between the Loan-to-Value Ratio and Down Payment? The loan-to-value ratio and down payment are two sides of the same coin, such that a higher down payment results in a lower LTV ratio, and a lower down payment results in a higher LTV ratio. A minimum down payment requirement for different mortgages is equivalent to having a maximum LTV ratio requirement. The sum of the LTV ratio and the percentage of a down payment is equal to the home value as seen in the table: Home Value Mortgage Amount Down Payment (%) LTV Ratio $285,000 $15,000 (5%) 95% $270,000 $30,000 (10%) 90% $300,000 $255,000 $45,000 (15%) 85% $240,000 $60,000 (20%) 80% $225,000 $75,000 (25%) 75% The table shows the various values of the mortgage amount, down payment, and LTV ratio for a home value of $300,000. As the down payment rises from 5% to 25%, the LTV ratio falls from 95% to 75%. In each case the sum of the down payment, that is the home equity of the borrower, and the mortgage loan taken is the value of the home. How do Lenders View the Loan-to-Value Ratio? The LTV ratio is an important requirement for different mortgage lenders. Here is a general summary: LTV Ratio Down Payment Private Mortgage Insurance (PMI) Lender Perception LTV > 97% Down Payment < 3% Yes Poor LTV 90% < LTV < 97% 3% < Down Payment < 10% Yes Acceptable LTV 80% < LTV < 90% 10% < Down Payment < 20% Yes Average LTV LTV < 80% Down Payment > 20% No Good LTV If an LTV ratio is greater than 97%, such as when the downpayment is less than 3% of the home’s value, you will not be able to get a mortgage excluding special programs such as Conventional 97, VA Loan, and USDA Loan. LTV greater than 97% is a very poor LTV ratio and will result in the requirement of mortgage insurance and higher mortgage rates . An LTV ratio greater than 90% but less than 97% is acceptable and requires PMI. Apart from the above loan options, FHA Loan is also available as this program requires an LTV of 96.5%. An LTV greater than 80% but less than 90% is an average LTV ratio and accepted by most mortgage programs but still requires mortgage insurance. An LTV below 80% is good as mortgage insurance is not required. Larger mortgages such as Jumbo Loans usually require an LTV ratio of less than 80%. What is the Loan-to-Value Ratio Requirement for different Home Financing Options? The loan-to-value ratio is essential in determining if you are eligible for various home financing transactions: 1. Buying a Home: An LTV ratio below 80% is good as mortgage insurance will not be required. However, an LTV ratio of up to 97% is also accepted by certain programs. If you are planning to buy a house with bad credit, then it is unlikely that you will be able to get a loan with an LTV ratio of 97%. 2. Refinance: Similar to first time home purchase, an LTV ratio of less than 80% is good. However, greater than 80% is accepted with mortgage insurance (PMI) . 3. Home Equity Line of Credit (HELOC): An LTV ratio below 80% is good. Most lenders will underwrite a HELOC with this ratio. An LTV greater than 80% is too high and more of the mortgage should be paid off before considering a HELOC. 4. Second Mortgage & Home Equity Loan: An LTV ratio below 65% is good and will be accepted by lenders. An LTV greater than 85% is very high and more of the mortgage should be paid. What is the Loan-to-Value Ratio Requirement for different Mortgage Programs? LTV ratio is used as a requirement for various mortgage programs: Conventional mortgages can allow for an LTV ratio of up to 97% or a downpayment as low as 3%. On a $500,000 home, only $15,000 ($500,000*3%) is required as a down payment. Jumbo mortgages require an LTV ratio of 80% or lower, which is a down payment of at least 20%. These mortgages are larger than the conforming loan limits set by the Federal Housing Finance Association (FHFA) and require a lower LTV ratio because of the additional risk involved. FHA loans are mortgages that are insured by the Federal Housing Administration and require the LTV ratio to be below 96.5% or at least a 3.5% down payment. The FHA loan program’s goal is to make housing affordable for low to moderate-income earners. The VA home loan is insured by the Department of Veterans Affairs and the USDA loan is backed by the Department of Agriculture and neither of them has an LTV ratio requirement. If you meet the other requirements of USDA loans and VA loans, these loans allow for as low as zero downpayment. Adding closing costs to your mortgage will increase your LTV ratio. This might be done if a borrower wants to finance their closing costs rather than pay them upfront, such as with a no-closing-cost mortgage. Can I Lower my Loan-to-Value Ratio? Yes, you can try to reduce your LTV ratio which can help get you a better mortgage rate for your home. Two strategies can be used to reduce the loan-to-value ratio: 1. Save up for a larger Down Payment: One of the key determinants of the LTV ratio is the amount of money that you put upfront for the home. If your down payment is larger, then the mortgage amount will be a smaller percentage of the home value. Therefore, if you want to get a better mortgage rate with a lower LTV ratio, you can try to save for a larger down payment. 2. Buy a Cheaper Home: If you can’t increase your down payment then the next best alternative is to try to find a home with a lower price tag. By buying a home with a lower price, the same down payment will result in a lower LTV ratio. For example, say you have saved $20,000 for a down payment. If the home price was $400,000, that results in an LTV ratio of 95% ($380,000/$400,000), whereas if the home price was $200,000 the LTV ratio would reduce to 90% ($180,000/$200,000). Check how much house you can afford using our home affordability calculator. It is in your best interest to try to reduce your LTV ratio as much as possible, before applying for a mortgage. If you can get the LTV ratio below 80% you can avoid private mortgage insurance (PMI). • Any analysis or commentary reflects the opinions of Casaplorer.com (a part of Wowa Leads Inc.) analysts and should not be considered financial advice. Please consult a licensed professional before making any decisions. • The calculators and content on this page are for general information only. Casaplorer does not guarantee the accuracy and is not responsible for any consequences of using the calculator. • Interest rates are sourced from financial institutions' websites.	{"url":"https://casaplorer.com/ltv-calculator","timestamp":"2024-11-11T04:31:05Z","content_type":"text/html","content_length":"117810","record_id":"<urn:uuid:67578379-1969-4c89-8aa1-8d49a90a5c7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00557.warc.gz"}
The evaluation of entropy-based algorithm towards the production of closed-loop edge Crysdian, Cahyo ORCID: https://orcid.org/0000-0002-7488-6217 (2023) The evaluation of entropy-based algorithm towards the production of closed-loop edge. Joiv: International Journal on Informatics Visualization, 7 (4). pp. 2481-2488. ISSN 25499904 18705.pdf - Published Version Available under License Creative Commons Attribution Share Alike. Download (4MB) This research concerns with the common problem of edge detection that suffers from producing a disjointed and incomplete edge leading to misdetection of visual object. The entropy-based algorithm has a potential to solve this problem by classifying the pixel belonging to which objects in an image. Hence, the paper aims to evaluate the performance of entropy-based algorithm to produce the closed-loop edge representing the formation of object boundary. The research utilizes the concept of entropy to sense the uncertainty of pixel membership to the existing objects in order to classify pixel as the edge or object. Six entropy-based algorithms are evaluated, i.e. the optimum entropy based on Shannon formula, the optimum of relative-entropy based on Kullback-Leibler divergence, the maximum of optimum entropy neighbour, the minimum of optimum relative-entropy neighbour, the thinning of optimum entropy neighbour, and the thinning of optimum relative-entropy neighbour. The experiment is held to compare the developed algorithms against Canny as a benchmark by employing five performance parameters, i.e., the average number of detected objects, the average number of detected edge pixels, the average size of detected objects, the ratio of the number of edge pixel per object, and the average of ten biggest size. The experiment shows that the entropy-based algorithms significantly improve the production of closed-loop edge, and the optimum of relative-entropy neighbour based on Kullback-Leibler divergence becomes the most desired approach among others due to the production of bigger closed-loop edge in the average. This finding suggests that the entropy-based algorithm becomes the best choice to support edge-based segmentation. The effectiveness of entropy in segmentation task is addressed for further research. Downloads per month over past year Origin of downloads Actions (login required)	{"url":"http://repository.uin-malang.ac.id/18705/","timestamp":"2024-11-07T07:50:44Z","content_type":"application/xhtml+xml","content_length":"30886","record_id":"<urn:uuid:768f7a4f-0dfe-4ce4-b48a-637a601575f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00200.warc.gz"}
Type I Hypothesis Testing Error A Type I Hypothesis Testing Error is a hypothesis rejection decision that is a false positive prediction. • Context: • Example(s): • Counter-Example(s): • (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/type_I_and_type_II_errors Retrieved:2020-10-5. □ In statistical hypothesis testing, a type I error is the rejection of a true null hypothesis (also known as a "false positive" finding or conclusion; example: "an innocent person is convicted"), while a type II error is the non-rejection of a false null hypothesis (also known as a "false negative" finding or conclusion; example: "a guilty person is not convicted"). Much of statistical theory revolves around the minimization of one or both of these errors, though the complete elimination of either is a statistical impossibility for non-deterministic algorithms. By selecting a low threshold (cut-off) value and modifying the alpha (p) level, the quality of the hypothesis test can be increased. The knowledge of Type I errors and Type II errors is widely used in medical science, biometrics and computer science. Intuitively, type I errors can be thought of as errors of commission, and type II errors as errors of omission. For example, in the context of binary classification, when trying to decide whether an input image X is an image of a dog: an error of commission (type I) is classifying X as a dog when it isn't, whereas an error of omission (type II) is classifying X as not a dog when it is.	{"url":"https://www.gabormelli.com/RKB/type_I_error","timestamp":"2024-11-03T22:06:14Z","content_type":"text/html","content_length":"40412","record_id":"<urn:uuid:a659cf59-bf14-4024-ae72-822a8e43b7e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00738.warc.gz"}
Math in Focus Grade 6 Chapter 12 Lesson 12.3 Answer Key Volume of Prisms This handy Math in Focus Grade 6 Workbook Answer Key Chapter 12 Lesson 12.3 Volume of Prisms detailed solutions for the textbook questions. Math in Focus Grade 6 Course 1 B Chapter 12 Lesson 12.3 Answer Key Volume of Prisms Math in Focus Grade 6 Chapter 12 Lesson 12.3 Guided Practice Answer Key Find the volume of each rectangular prism. Question 1. Length = 5$\frac{1}{4}$ in. Width = 6 in. Height = 12 in. V = lwh = ^3 V = lwh Length = 5$\frac{1}{4}$ in. (5×$\frac{4}{4}$+$\frac{1}{4}$) = $\frac{21}{4}$ V = $\frac{21}{4}$ × 6 × 12 = 378 cu.in Question 2. Length = 8 cm Width = 7.2 cm Height = 3 cm V = lwh = ^3 V = 8 × 7.2 × 3 172.8 cu.cm Question 3. Length = 4 ft Width = 3 ft Height = 8 $\frac{1}{3}$ ft Volume = = ^3 V = lwh Height = 8 $\frac{1}{3}$ ft (8 × $\frac{3}{3}$)+$\frac{1}{3}$ = $\frac{25}{3}$ Volume = 4×3×$\frac{25}{3}$ = 100 cu.ft Tell whether slices parallel to each given slice will form uniform cross sections. If not, explain why not. Question 4. No, in the above given figure, the slice parallel to each given slice will not form uniform cross-sections. Because, the area of the cross-section is larger than the area’s of the cube’s face. Question 5. In the above given figure, the slice parallel to each given slice will form uniform cross-sections. Question 6. No, in the above given figure, the slice parallel to each given slice will not form uniform cross-sections. Because, the area of the cross-section is smaller than the area’s of the rectangular’s face. Find the volume of each prism. Question 7. Length = 6 cm Width = 5.5 cm Height = 9 cm Area of base = = ^2 Volume of prism = = ^2 The volume of the prism is Length = 6 cm Width = 5.5 cm Height = 9 cm Area of base = 6×5.5 = 33 sq.cm Volume of prism = Base area of prism × height Volume of prism = 33×9 = 297 cu.cm The volume of the prism is 297 cubic centimeters. Question 8. Base of triangle = 10 in. Height of triangle = 3$\frac{1}{2}$ in. Height of prism = 14 in. Area of base = $\frac{1}{2}$ • = ^2 Volume of prism = = ^3 The volume of the prism is Base of triangle = 10 in. Height of triangle = 3$\frac{1}{2}$ in. (3×$\frac{2}{2}$)+$\frac{1}{2}$ = $\frac{7}{2}$ Height of prism = 14 in. Area of triangular base = $\frac{1}{2}$ × base × height Area of base = $\frac{1}{2}$ × 10 × $\frac{7}{2}$ = $\frac{35}{2}$ sq.in Volume of prism = Base area of prism × height Volume of prism = $\frac{35}{2}$ × 14 = 245 cu.in Question 9. Length of shorter base of trapezoid = 4 ft Length of longer base of trapezoid = 10 ft Height of trapezoid = 2 ft Height of prism = 12 ft Area of base = $\frac{1}{2}$ • = $\frac{1}{2}$ • = ^2 Volume of prism = = ^3 The volume of the prism is The formula to find the base area = $\frac{1}{2}$ × height × (sum of parallel sides) Area of base = $\frac{1}{2}$ × 2 × (4+10) = $\frac{1}{2}$ × 2 × 14 = 14 sq.ft Volume of prism = Base area of prism × height Volume of prism = 14×12 = 168 cu.ft Hands-On Activity Work in pairs. Step 1: Build the cube and the rectangular prism using unit cubes. Step 2: Find the volume of the cube. Find the volume of the rectangular prism. What can you say about the volumes of the cube and the rectangular prism? Volume of cube = lwh Volume of the cube = 2×2×2 = 8 cu.cm Volume of the rectangular prism = 2×4×1 = 8 cu.cm The volumes of the cube and the rectangular prism are same. Step 3: Find the surface area of the cube. Draw its net if it helps you. Find the surface area of the rectangular prism. Draw its net if it helps you. What can you say about the surface areas of the cube and the rectangular prism? Area of one square will be 2×2 = 4 sq.cm Volume of cuboid = 6 × Area of base There are 6 faces for a cube, therefore the surface area will be 6×4 = 24 sq.cm The surface areas of the rectangular prism will be 2(2×4 + 2×1 + 1×4) = 28 sq.cm The surface areas of the cube and the rectangular prism are different. Step 4: Now build these rectangular prisms using unit cubes. Step 5: Find the volume of the cube. Find the volume of the rectangular prism. What can you say about their volumes? Volume of the cube = 3×3×3 = 27 cu.cm Volume of the rectangular prism = 3×9×1 = 27 cu.cm The volumes of the cube and the rectangular prism are same. Step 6: Find the surface area of the cube. Find the surface area of the . rectangular prism. Draw their nets if it helps you. What can you say about their surface areas? Area of one square will be 3×3 = 9 sq.cm There are 6 faces for a cube, therefore the surface area will be 6×9 = 54 sq.cm The surface areas of the rectangular prism will be 2(3×9 + 9×1 + 1×3) = 78 sq.cm The surface areas of the cube and the rectangular prism are different. Math Journal Based on the activity, what can you conclude about prisms with the same volume? Discuss with your partner and explain your thinking. Two prisms of different measurements might have the same volume, they might not have the same surface area. For example: A rectangular prism with side lengths of 1 cm, 2 cm, and 2 cm has a volume of 4 cu cm and a surface area of 16 sq cm. A rectangular prism with side lengths of 1 cm, 1 cm, and 4 cm has the same volume but a surface area of 18 sq cm Math in Focus Course 1B Practice 12.3 Answer Key Question 1. A cube has edges measuring 9 inches each. Find the volume of the cube. Given that: A cube has edges measuring 9 inches each. The base area will be equal to 9×9=81 sq.in Volume of cube = Area of base × height Volume of the cube will be 81×9=729 cu.in Question 2. A cube has edges measuring 6.5 centimeters each. Find the volume of the cube. Given that: A cube has edges measuring 6.5 cm each. The base area will be equal to 6.5×6.5=42.25 sq.cm Volume of cube = Area of base × height Volume of the cube will be 81×9=274.625 cu.cm Question 3. A storage container is shaped like a rectangular prism. The container is 20 feet long, 10 feet wide, and 5$\frac{1}{2}$ feet high. Find the volume of the storage container. Given that: A rectangular prism is 20 feet long, 10 feet wide, and 5$\frac{1}{2}$ feet high. Area of base will be 20×10=200 sq.ft Volume of prism = Area of base × height Volume of the storage container will be 200×5$\frac{1}{2}$ = 200×(5×$\frac{2}{2}$ + $\frac{1}{2}$) = 200×($\frac{10}{2}$ + $\frac{1}{2}$) = 200×$\frac{11}{2}$ = 100×11 = 1100 cu.ft Question 4. Find the volume of the peppermint tea box on the right. Given that: The peppermint tea box is 12.6 cm long, 6.7 cm wide, and 7.8 cm high. Area of base will be 12.6×6.7 = 84.42 sq.cm Volume of prism = Area of base × height Volume of the box will be 7.8×84.42 = 658.476 cu.cm Question 5. The solid below is made of idential cubes. Each cube has an edge length of 2 inches. Find the volume of the solid. Given that: Each cube has an edge length of 2 inches. Let us first find the volume of a single cube Area of the base of a single cube will be 2×2=4 sq.in Volume of cube = Area of base × height Volume of a single cube will be 4×2=8 cu.in The given solid is made of idential cubes. There are 9 such cubes. Therefore, the volume of all the 9 identical cubes or the given solid figure will be 9×8=72 cu.in Find the volume of the triangular prism. Question 6. Given that: A triangular prism is 15ft long, 10ft wide and 6ft high. Area of the triangular prism = $\frac{1}{2}$ × base × height Area of the triangular prism will be $\frac{1}{2}$ × 10 × 6 = 10 × 3 = 30 sq.ft Volume of prism = Area of base × height Volume of the given triangular prism will be 30×15=450 cu.ft Question 7. Given that: A triangular prism is 12cm long, 6.7cm wide and 3cm high. Area of the triangular prism = $\frac{1}{2}$ × base × height Area of the triangular prism will be $\frac{1}{2}$ × 6.7 × 3 = $\frac{1}{2}$ × 20.1 = 10.05 sq.cm Volume of prism = Area of base × height Volume of the given triangular prism will be 10.05×12=120.6 cu.cm Tell whether slices parallel to each given slice will form uniform cross-sections. If not, explain why not. Question 8. In the above given figure, the slice parallel to each given slice will form uniform cross-sections. Question 9. No, in the above given figure, the slice parallel to each given slice will not form uniform cross-sections. Because, the area of the cross-section is smaller than the area’s of the circular face. Question 10. No, in the above given figure, the slice parallel to each given slice will not form uniform cross-sections. Because, the area of the cross-section is smaller than the area’s of the rectangular’s face. Copy the solid. Draw a slice that has the same cross section as the bases In each prism. Question 11. The bases for the above given figure is square. Therefore, a slice that has the same cross section is drawn for the given prism. Question 12. The bases for the above given figure is hexagon. Therefore, a slice that has the same cross section is drawn for the given prism. Question 13. The bases for the above given figure is triangle. Therefore, a slice that has the same cross section is drawn for the given prism. Question 14. The bases of the prism shown are trapezoids. Find the volume of the prism. Length of shorter base of trapezoid = 2 m Length of longer base of trapezoid = 6 m Height of trapezoid = 2 m Height of prism = 10 m The area of the given figure = $\frac{1}{2}$ × height × (sum of parallel sides) Area of base = $\frac{1}{2}$ × 2 × (2+6) = $\frac{1}{2}$ × 2 × 8 = 8 sq.m Volume of prism = base area × height The volume of prism = 10×8 = 80 cu.m Question 15. A cube has a volume of 125 cubic inches. Find the length of its edge. Given that: A cube has a volume of 125 cu.in Let us assume the length of a cube be ‘a’ unit. As cube has all equal sides, the base area will be a×a = a² sq.units Now, the volume will be a²×a = a³ cubic units. Therefore we can say that the volume of a cube is the length of an edge taken to the third power. a³ = 125 and 125 in cube can be written as 5³ a³ = 5³ a = 5 Therefore, the length of its edge will be 5 inches. Question 16. The volume of a triangular prism is 400 cubic centimeters. Two of its dimensions are given in the diagram. Find the height of a triangular base. Given that: The volume of a triangular prism is 400 cubic centimeters. The three rectangular faces measures 10cm long and 8cm wide. The area of three rectangular faces will be 3×10×8 = 240 sq.cm. Inorder to get the areas of the triangular bases, we need to subtract the area of the rectangular faces from the total volume. Area of two triangular bases will be 400-240 = 160 sq.cm Area of one triangular base will be 160÷2 = 80 sq.cm From the given figure, the base length of the triangle is 10 cm and the height is ‘h’ cm. Area of triangular base = $\frac{1}{2}$ × base × height Area of the triangle is $\frac{1}{2}$×10×h = 80 5×h = 80 h = 80÷5 h = 16 cm. Question 17. A cross-section of the triangular prism shown below is parallel to a base. The area of the cross-section is 24 square feet. The ratio of DM to MA is 3 : 5 and the length of $\overline{F O}$ is 6 feet. Find the volume of the triangular prism. Given that the area of cross-section is 24 sq.ft and ratio of DM to MA is 3 : 5. The length of FO is 6 ft. By observing the FO length and the ratio, we can say that the FO length is twice of the ratio of DM. Therefore, the length of OC will be twice of the ratio of MA. Length of OC = 10 ft. Total length = FO + OC = 6+10 = 16 ft. Volume of the triangular prism = Area of triangular base × length Volume of the triangular prism = 24×16 Volume of the triangular prism = 384 cu.ft Question 18. The volume of the rectangular prism shown below is 2,880 cubic inches. The cross-section shown is parallel to a base. The area of the cross-section is 180 square inches. The length of $\overline{A B}$ is x inches, and the length of $\overline{B C}$ is 4x inches. a) Find the length of $\overline{A C}$. Given that: The volume of the rectangular prism is 2,880 cubic inches and the area of the cross-section is 180 square inches. The length of AB is x inches and BC is 4x inches. Length of AC will be AB+BC = x+4x = 5x b) Find the value of x. Volume of the rectangular prism = 2,880 cubic inches Volume of the rectangular prism = Area of rectangular base × length 2,880 = 180 × 5x 5x = 2880 ÷ 180 5x = 16 x = $\frac{16}{5}$ Length of AB will be $\frac{16}{5}$ in. Length of BC will be 4×$\frac{16}{5}$ = $\frac{64}{5}$ in. Length of AC will be $\frac{80}{5}$ = 16 in. Question 19. In the diagram of a cube shown below, points A, B, C, and D are vertices. Each of the other points on the cube is a midpoint of one of its sides. Describe a cross-section of the cube that will form each of the following figures. a) a rectangle b) an isosceles triangle c) an equilateral triangle d) a parallelogram a) a rectangle By joining the CDLJ points, a rectangular cross section can be formed. b) an isosceles triangle By joining the MHL points, an isosceles triangle cross section can be formed. c) an equilateral triangle By joining the CDM points,an equilateral triangle cross section can be formed. d) a parallelogram By joining the MHGL points, a parallelogram cross section can be formed. Solve. Use graph paper. Question 20. Points A, B, C, and D form a square. The area of the square is 9 square units. a) Find the side length of square ABCD. Given that the area of the square is 9 square units. Area of square = length × length 9 = length × length 3 × 3 = length × length Length of the square will be 3 units. b) The coordinates of point A are (2, 6). Points B and C are below $\overline{A D}$. Point B is below point A, and point D is to the right of point A. Plot the points in a coordinate plane. Connect the points in order to draw square ABCD. The coordinates of point A are (2, 6). Point B and C are below point A, so they will be (0,2) and C will be (0,8). D will be (8,6) c) The points E, F, G, and H also form a square that is the same size as square ABCD. Point E is 4 units to the right of point A, and 3 units up. Points F and G are below $\overline{E H}$. Point F is below point E, and point H is to the right of point E. Plot the points in the coordinate plane. Draw $\overline{E H}$ and $\overline{G H}$ with solid lines, and $\overline{E F}$ and $\ overline{F G}$ with dashed lines. Point E is 4 units to the right of point A, and 3 units up. So, it will be (6,9). Points F and G are below, so they will be (6,3) and (12,3). Point H will be (12,9). d) Draw $\overline{A E}$, $\overline{D H}$, and $\overline{C G}$ with solid lines, and $\overline{B F}$ with a dashed line, Use the solid and dashed lines to see the figure as a solid. Name the type of prism formed. After joining AE, DH, CG and BF.Below prism will be formed. Since all the sides are of same length, it will form a cube. e) If the height of the prism is 7 units, find the volume of the prism. Area of the base 7×7 = 49 sq.units Volume of the prism = 49×7 = 343 cu.units Leave a Comment	{"url":"https://mathinfocusanswerkey.com/math-in-focus-grade-6-chapter-12-lesson-12-3-answer-key/","timestamp":"2024-11-05T13:42:17Z","content_type":"text/html","content_length":"175209","record_id":"<urn:uuid:ffbada98-cb92-4861-834e-9d4179cba5b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00677.warc.gz"}
Potentiometer Math I have an application where a potentiometer may be useful. In fact, it would be useful if the potentiometer had a logarithmic resistance characteristic, which is also called an audio taper for reasons that I will cover later. I have never used a potentiometer with a logarithmic characteristic before and I thought it would be worth documenting what I learned during this effort. Logarithmic Misnomer What is normally referred to as a logarithmic taper is really an exponential characteristic. A typical logarithmic taper potentiometer characteristic is shown in Figure 1 (Source). Each vendor will have a different "series" label for the logarithmic potentiometers, which often have names like "series A" or "series W." The series designation indicates a different set of resistance curves. Potentiometer Specifications Not all vendors include a graph of the resistance characteristics of their logarithmic potentiometers. Many of the vendors include a specification that says something similar to the following quote ( The “W” taper attains 20% resistance value at 50% of clockwise rotation (left-hand). This specification means that the potentiometer has • 0% of it full scale resistance value with the wiper at 0% of its full scale position • 20% of it full scale resistance value with the wiper at 50% of its full scale position • 100% of the total resistance value at 100% of its full scale position. This type of specification gives you sufficient information to create an exponential curve fit, which I illustrate in Figure 2. The math associated with this curve fitting is shown in Figure 3. Equation 1 illustrates the basic form of the logarithmic potentiometer's resistance characteristic R(x), where x is the wiper position as a percentage of full scale. Eq. 1 $R(x)={{R}_{0}}\cdot \left( {{e}^{{{R}_{1}}\cdot x}}-1 \right)$ • R[0] is a curve fitting parameter • R[1] is a curve fitting parameter • x is the wiper position as a percentage of full scale range Human Hearing The logarithmic taper is commonly called an audio taper because it is often used audio applications for loudness control. Understanding why involves knowing a little bit about human hearing. Figure 4 (Source) illustrates a human's perception of loudness relative to the Sound Pressure Level (SPL). Sound pressure level is proportional to an audio amplifier's output power. However, the ear is sensitive to the log of the sound pressure level. This is why the "logarithmic" taper is useful. When people are adjusting the loudness of their audio gear, they prefer that the loudness increase by an amount proportional to amount of dial or slide movement. If a linear potentiometer is used to control output power (and therefore loudness), you will need to use larger and larger amounts of wiper movement to get the same loudness change. To get the same amount of loudness change for a the same amount of wiper movement, the potentiometer resistance needs to increase exponentially. Compensating for Hearing Figure 5 shows what how the loudness is perceived by a person as the potentiometer's wiper is moved. You can see that the perceived loudness increases approximately linearly for wiper positions above After all this research, I ended up not using the logarithmic potentiometer because it was not logarithmic. I ended up using another approach which I will discuss in a later post. 14 Responses to Potentiometer Math 1. how do you work out that the tapper position is at 20% Vout at 50% position mathenatically □ 20% of full-scale resistance at 50% wiper full-scale travel is actually the manufacturer's specification. In Figure 1, I referred to vOUT as defined in the manufacturer's test circuit. The manufacturer applied a voltage to the non-wiper contacts on the pot and measured the percentage of that applied voltage at the wiper. 2. I made an interactive pot curve using above on Drag the slider or the orange point to change values. Turn on and off curves using the circled square buttons on the left side. Open folders using the triangles to see equations and comments. □ This is pretty cool! What tool did you use to program it? ☆ That site desmos.com actually provides all (for free), you just have to type the equations and limits and get plots (it's all done in JavaScript, I think). I used wolframalpha.com though and some guess work to figure out how to get the curve parameter at 50% from your numerical solution. You might need to click into the equations to see them fully. See also https:// ○ Thanks for the reference information. I am very impressed. □ Those graphs look sick! If you could add some notes to explain the meaning of each variable & constant, it would be sweet. I didn't realise Desmos was that powerful, I'm so glad I stumble upon your comment 🙂 3. This is nice work, there is just one snag. Most commercial 'Audio' pots are not exponential, but piece-wise linear. I suspect the intermediate value quoted in the specs is the 'knee point' and that the behaviour is linear between 0% and the knee and again between the knee and 100%. A resistor in parallel with a linear pot provides a non-linear response. □ No disagreement. Most commercial logarithmic potentiometers are actually piecewise linear. Here is a good visual on the difference in characteristics (source). You can clearly see the two I did open up an audio pot years ago. The resistive track had two different widths, which caused the change in the slope of the resistance curve. Unfortunately, I did not take a picture. For this discussion, I was just treating the pots as if they were ideal. Thanks for writing. 4. Thanks, was useful for me. 5. Those graphs look sick! If you could add some notes to explain the meaning of each variable & constant, it would be sweet. I didn't realise Desmos was this powerful, I'm so glad I stumbled upon your reply 🙂 This entry was posted in Electronics and tagged circuits, electronics. Bookmark the permalink.	{"url":"https://www.mathscinotes.com/2011/12/potentiometer-math/","timestamp":"2024-11-09T13:57:33Z","content_type":"text/html","content_length":"59569","record_id":"<urn:uuid:b98114ff-71a7-4489-9e95-c186062558c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00513.warc.gz"}
Time Converter Use this time converter to convert instantly between milliseconds, seconds, minutes, hours, days, weeks, months, years and other metric and imperial time units. Disclaimer: Whilst every effort has been made in building our calculator tools, we are not to be held liable for any damages or monetary losses arising out of or in connection with their use. Full To learn about weeks and years conversions, see our article how many weeks are in a year? Alternatively, you may wish to calculate the number of days between two dates. Some modern and vintage watches and clocks represent the hours of the day using Roman numerals. If you need help translating these you can use our roman numerals translation tool. Conversion units for the Time Converter Attoseconds (as), Days (d), Femtoseconds (fs), Hours (h), Microseconds (µs), Milliseconds (ms), Minutes (min), Months, Nanoseconds (ns), Picoseconds (ps), Seconds (s), Shakes, Weeks, Years (a), Popular individual converters: Hours and seconds, minutes and hours, minutes and days To help with productivity, we now set a cookie to store the last units you have converted from and to. This means that when you re-visit this time converter, the units will automatically be selected for you.	{"url":"https://www.thecalculatorsite.com/conversions/time.php","timestamp":"2024-11-08T21:48:19Z","content_type":"text/html","content_length":"308029","record_id":"<urn:uuid:214ba356-a628-40d4-a9b9-77969b4336d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00190.warc.gz"}
NIPS 2016 Mon Dec 5th through Sun the 11th, 2016 at Centre Convencions Internacional Barcelona Reviewer 1 This paper proposes a new interpretation of parameter-free algorithms by showing the characteristic of these parameter-free algorithms to be coin-betting with maximal reward. It starts from using a betting algorithm based on the Krichevsky-Trofimov estimator with coin-betting potential function, and then generalizes it to online learning optimization over Hibert Space and Learning with expert advices. Real-world datasets also demonstrate the effectiveness of proposed algorithm. Qualitative Assessment This paper provides novel understanding of parameter-free algorithms as coin-betting with maximal reward. Based on their understanding, several new algorithms for online convex optimization and learning with experts. The whole paper is well organized and readers can easily follow this paper. Overall, this is a complete work with novel interpretations. Confidence in this Review 2-Confident (read it all; understood it all reasonably well) Reviewer 2 The present paper addresses the online convex optimization setting with a decision set equal to either (i) a Hilbert space or (ii) the probability simplex. The authors present a novel reduction from this problem to a coin betting problem for which optimal and simple algorithms are well known. In particular they show that the Krichevsky-Trofimov strategy (or any generalization of it), which is optimal for coin betting, leads to parameter-free algorithms in (i) and (ii). "Parameter-free" means that the learning algorithm is not tuned as a function of the unknown norm of the competitive vector (for (i)) or the unknown KL divergence of the a good weight vector w.r.t. the the prior at hand (for (ii)). This general framework helps generalize previous works on parameter-free algorithms in either (i) or (ii). Qualitative Assessment My opinion about the paper is very good. It is technically solid, very well written, and it helps generalize existing algorithms for Online Linear Optimization (OCO) or Learning with Expert Advice (LEA) through a novel reduction to a coin betting problem. It should be of great interest to people in the online learning community. I must admit that I do not find the coin betting reduction intuitive (especially for LEA, cf. Section 4.2). However, the generality that this approach provides and the simplicity of the resulting algorithms makes it quite attractive. "Parameter-free": you assume that all gains are bounded by 1 (in Euclidean norm for OLO or in sup norm for LEA). Imagine for a second that all gains were bounded by b instead, with an unknown range b. Can you please answer the following? 1. Of course we could rescale all gains by b and apply your machinery to the rescaled gains to get regret bounds that would be proportional to b. However I have the impression that the resulting outputs w_t would depend on the unknown value of b. Is it correct? 2. Fortunately, for the particular case of the Krichevsky-Trofimov potential, the w_t are proportional to the gains. Therefore, in the LEA framework, the resulting weight vectors p_t defined by (12) would not depend on the value of b. Am I correct? 3. Unfortunately it seems that this invariance property does not hold in the OLO framework. Is it true? 4. What happens if we play with losses instead of gains? Again, if all losses are bounded by b, I have the impression that the transformation gain <- 1-loss /b works for LEA (i.e., the algorithm does not depend on b), but we may have a problem for OLO. Is it indeed the case? Please add a discussion of all the four points above in the paper. In particular if I am not wrong for 3. or 4., then the authors should be careful when using the word "parameter-free". Other (minor) comments: - l.16: the Hilbert space --> a Hilbert space - Lemma 1: is equivalent to: implies? I do not understand the equivalence. - Definition 2: can you give examples of potentials at this point? (As such, the reader has to wait a little bit.) - l.193, Gamma function: t^{x-1} instead of t^{-x} - end of l.394: the cases u_i=0 or \pi_i=0 should be treated separately (but the result is ok). Confidence in this Review 3-Expert (read the paper in detail, know the area, quite certain of my opinion) Reviewer 3 The paper presents a new intuitive framework to design parameter-free algorithms based on a reduction to betting on outcomes of an adversarial coin. The method shows a new interpretation of parameter-free algorithms as coin-betting algorithms, which reveals the common hidden structure of previous parameter-free algorithms and also allows the design of new algorithms. The paper has also run an empirical evaluation to demonstrate the theory. Qualitative Assessment strengths of the paper: 1. The paper has proposed a novel framework to design parameter-free algorithms. The proposed method is simple with no parameters to be tuned. 2. They have both demonstrated in theory and experiments that the proposed algorithm improve or match previous results in terms of regret guarantee and per-round complexity. Confidence in this Review 1-Less confident (might not have understood significant parts) Reviewer 4 In this paper, the authors provide a new approach for solving the classical problem of learning with expert advice (LEA) as well as the problem of online linear optimization (OLO) over Hilbert space. The idea is to reduce these two problem to another classical problem, the coin-betting problem. Then based on an existing coin-betting protocol, the authors are able to obtain parameter-free algorithms for these two problems. The new algorithm for the LEA problem achieves a smaller regret than previous ones, while the new algorithm for the OLO problem matches the regret bound of an existing one. Qualitative Assessment There seems to be a mistake in the proof of Lemma 14, so we are not sure about the correctness of Corollaries 5&6. More precisely, the second inequality below line 430 does not hold: it should be \ leq instead \geq because \ln(1+x) \leq x-(1-\ln(2))x^2. Our guess is that Lemma 14 is still correct, but the proof needs to be rewritten. Assuming that the bug can indeed be fixed, we find the paper interesting since it provides a new approach for designing online algorithms. This is achieved by establishing a somewhat unexpected connection to another classical problem, the coin-betting problem, which has a long history of works itself. With the help of such a connection, the authors are able to convert an existing coin-betting protocol into new algorithms for the LEA problem and the OLO problem. The resulting algorithms are parameter free, which means that no parameter tuning is needed, and this adds to the strength of this paper. In addition, the paper is well-written and easy to follow in general. Two possible weakness we find are the following. First, the potential function used by this paper seems rather complex and unintuitive, which may limit its impact. It is not clear if it is easy to design different potential functions to yield different online algorithms, just as what the mirror-descent algorithm can provide. Next, the OLO problem for which the new algorithm works is the unconstrained version, instead of the more popular constrained one. It is not clear if the ideas developed in this paper can be used to solve the more general OLO problem over any convex feasible set. Confidence in this Review 2-Confident (read it all; understood it all reasonably well) Reviewer 5 Many online learning algorithms are based on a learning parameter which controls how far the algorithm changes its answer after receiving new data. While many of these algorithms fix this rate in advance, others change it as they progress. The main problem discussed in this paper is trying to construct an algorithm that is comparable to the case where we know in advance all the data and not receive it part by part. In this paper, the authors construct a strategy for a coin betting game such that in each turn the gambler needs to decide how much to bet and on which outcome. This strategy is based on suitably chosen potential functions. Each such family of functions produce a betting strategy which also gives an upper bound on the regret for the game, i.e. how much money the gambler could have won if he knew all the coin tosses in advance compared to how much he won using this strategy. The authors then use this coin betting strategy to solve other online learning problems such as the Online Learning Optimization and Learning with Expert Advice problems. This strategy seems to generalize other parameter-free algorithms which already exist. Qualitative Assessment The coin betting game and both the Online learning optimization and Learning with expert advice are interesting problems in online learning, and the choice of the learning rate is fundamental to many learning algorithms. Thus any framework that provides a way to chose this rate in a smart way is highly interesting and can have many applications. The main issue with the paper is the definition of Coin betting potential which lead to the choice of the learning rate. While the formula below line 137 explains why the condition in line 130 is useful, and lines 138-140 explain from where the expression for beta_t comes from (though it is a bit confusing explanation), it is not at all clear how to construct such a function. Later on it is shown that the known online learning algorithms have corresponding potential functions. It would be really interesting if the authors can give some intuition on how to produce more such functions, thus saying that not only their algorithm generalizes the known results, but actually can produce new ones (and not just other similar potentials). some other minor comments: line 16 - "the Hilbert space" should be "a Hilbert space" unless this space is defined before. line 24-29 - It might be better to add some explanation what is a learning rate (and not just how you use it in the examples of OGD). If the reader didn't know what learning rate is before reading this paper, he would just be more confused after this introduction. Also, it might be a good idea to write that "parameter free" means that we do not set the learning rate in advance. It can also be as simple as writing "the learning parameter" or the "learning rate or leaning parameter" instead of just the "learning rate". line 49 - what do you mean by dom(f) not empty? Is the domain not all of V? if this is possible, then you should write it explicitly. line 109 - The inequality in (5) doesn't seem trivial. If it is just some calculations, maybe add another step or say that this is the case. line 114 - why are the names of Krichevsky and Trofimov in Blue? line 141 - The coin betting seems to be one dimensional always. What is the meaning of infinite dimensional in this line? line 196 - is this "peculiar property" has any significance ? 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Find the constrained minimum of a real function. Attention: Valid only with Altair Advanced Optimization Extension x = arsm(@func,x0) x = arsm(@func,x0,A,b) x = arsm(@func,x0,A,b,Aeq,beq) x = arsm(@func,x0,A,b,Aeq,beq,lb,ub) x = arsm(@func,x0,A,b,Aeq,beq,lb,ub,nonlcon) x = arsm(@func,x0,A,b,Aeq,beq,lb,ub,nonlcon,options) [x,fval,info,output] = arsm(...) The function to minimize. An estimate of the location of the minimum. A matrix used to compute Ax for inequality contraints. Use [ ] if unneeded. The upper bound of the inequality constraints Ax<=b. Use [ ] if unneeded. A matrix used to compute Aeqx for equality contraints. Use [ ] if unneeded. The upper bound of the equality constraints Aeqx=beq. Use [ ] if unneeded. The design variable lower bounds. Use [ ] if unbounded. Support for this option is limited. See Comments. The design variable upper bounds. Use [ ] if unbounded. Support for this option is limited. See Comments. The nonlinear constraints function. The function signature is as follows: function [c, ceq] = ConFunc(x) , where c and ceq contain inequality and equality contraints, respectively. The ceq output can be omitted if it is empty. Inequality constraints are assumed to have upper bounds of 0, and equality constraints are equal to 0. For constraints of the form g(x) < b, where b ~= 0, the recommended best practice is to write the constraint as (g(x) - b) / abs(b) < 0. Use [ ] if unused and the options argument is needed. A struct containing options settings. See arsmoptimset for details. The location of the function minimum. The minimum of the function. The convergence status flag. □ info = 3: Converged with a constraint violation within TolCon. □ info = 1: Function value converged to within TolX, TolFunAbs, or TolFunRel. □ info = 0: Reached maximum number of iterations, or the algorithm aborted because it was not converging. □ info = -2: The function did not converge. A struct containing iteration details. The members are as follows. The number of iterations. The candidate solution at each iteration. The objective function value at each iteration. The constraint values at each iteration. The columns will contain the constraint function values in the following order: 1. linear inequality contraints 2. linear equality constraints 3. nonlinear inequality contraints 4. nonlinear equality constraints Minimize the function ObjFunc, subject to the linear inequality constraint: x1 + 4x2 > 27. The constraint must be expressed with an upper bound: -x1 - 4x2 < -27 function obj = ObjFunc(x) obj = 2(x(1)-3)^2 - 5(x(1)-3)(x(2)-2) + 4(x(2)-2)^2 + 6; init = [8, 6]; % initial estimate A = [-1, -4]; % inequality contraint matrix b = [-27]; % inequality contraint bound lb = [-10, -10]; % lower variable bounds ub = [10, 10]; % upper variable bounds [x,fval] = arsm(@ObjFunc,init,A,b,[],[],lb,ub) x = [Matrix] 1 x 2 7.00001 5.00000 fval = 14 Modify the previous example to pass an extra parameter to the objective function using a function handle. function obj = ObjFunc(x,offset) obj = 2(x(1)-3)^2 - 5(x(1)-3)(x(2)-2) + 4(x(2)-2)^2 + offset; handle = @(x) ObjFunc(x,7); [x,fval] = arsm(handle,init,A,b,[],[],lb,ub) x = [Matrix] 1 x 2 7.00001 5.00000 fval = 15 arsm uses an Adaptive Response Surface Method. See the fmincon optimization tutorial, Compose-4030: Optimization Algorithms in OML, for an example with nonlinear constraints. Options are specified with . The defaults are as follows: • MaxIter: 25 • MaxFail: 20000 • TolX: 0.001 • TolCon: 0.5 (%) • TolFunAbs: 0.001 • TolFunRel: 1.0 (%) • ConRet: 50.0 (%) • MoveLim: 0.15 • PertM: 'initial' • PertV: 1.1 • Display: 'off' Unbounded limits design variable limits are not fully supported and are set to -1000 and 1000. Use of large limits is discouraged due to the size of the search area. To pass additional parameters to a function argument, use an anonymous function.	{"url":"https://help.altair.com/compose/help/en_us/topics/reference/oml_language/Optimization/arsm.htm","timestamp":"2024-11-03T03:24:31Z","content_type":"application/xhtml+xml","content_length":"76848","record_id":"<urn:uuid:dead7f03-1517-4ea5-b47e-9ee6c7fe8e0d>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00175.warc.gz"}
Modelica by Example Modelica Standard Library The power of packages in Modelica is the ability to take commonly needed types, models, functions, etc. and organize them into packages for reuse later by simply referencing them (rather than repeating them). But there is still a repetition problem if every user is expected to make their own packages of commonly needed definitions. For this reason, the Modelica Association maintain a package called the Modelica Standard Library. This library includes definitions that are generally useful to scientists and engineers. In this section, we will provide an overview of the Modelica Standard Library so readers are aware of what they can expect to find and utilize from this library. This is not an exhaustive tour and because the Modelica Standard Library is constantly being updated and improved, it may not reflect the latest version of the library. But it covers the basics and hopefully provides readers with an understanding of how to locate useful definitions. The Modelica Standard Library contains definitions of some common physical and machine constants. The library is small enough that we can include the source code for this package directly. The following represents the Modelica.Constants package from version 3.2.2 of the Modelica Standard Library (with a few cosmetic changes): within Modelica; package Constants "Library of mathematical constants and constants of nature (e.g., pi, eps, R, sigma)" import SI = Modelica.SIunits; import NonSI = Modelica.SIunits.Conversions.NonSIunits; // Mathematical constants final constant Real e=Modelica.Math.exp(1.0); final constant Real pi=2Modelica.Math.asin(1.0); // 3.14159265358979; final constant Real D2R=pi/180 "Degree to Radian"; final constant Real R2D=180/pi "Radian to Degree"; final constant Real gamma=0.57721566490153286060 "see http://en.wikipedia.org/wiki/Euler_constant"; // Machine dependent constants final constant Real eps=ModelicaServices.Machine.eps "Biggest number such that 1.0 + eps = 1.0"; final constant Real small=ModelicaServices.Machine.small "Smallest number such that small and -small are representable on the machine"; final constant Real inf=ModelicaServices.Machine.inf "Biggest Real number such that inf and -inf are representable on the machine"; final constant Integer Integer_inf=ModelicaServices.Machine.Integer_inf "Biggest Integer number such that Integer_inf and -Integer_inf are representable on the machine"; // Constants of nature // (name, value, description from http://physics.nist.gov/cuu/Constants/) final constant SI.Velocity c=299792458 "Speed of light in vacuum"; final constant SI.Acceleration g_n=9.80665 "Standard acceleration of gravity on earth"; final constant Real G(final unit="m3/(kg.s2)") = 6.67408e-11 "Newtonian constant of gravitation"; final constant SI.FaradayConstant F = 9.648533289e4 "Faraday constant, C/mol"; final constant Real h(final unit="J.s") = 6.626070040e-34 "Planck constant"; final constant Real k(final unit="J/K") = 1.38064852e-23 "Boltzmann constant"; final constant Real R(final unit="J/(mol.K)") = 8.3144598 "Molar gas constant"; final constant Real sigma(final unit="W/(m2.K4)") = 5.670367e-8 "Stefan-Boltzmann constant"; final constant Real N_A(final unit="1/mol") = 6.022140857e23 "Avogadro constant"; final constant Real mue_0(final unit="N/A2") = 4pi1.e-7 "Magnetic constant"; final constant Real epsilon_0(final unit="F/m") = 1/(mue_0c*c) "Electric constant"; final constant NonSI.Temperature_degC T_zero=-273.15 "Absolute zero temperature"; Noteworthy definitions are those for pi, e, g_n and eps. The first two, pi and e, are mathematical constants representing and , respectively. Having these constants available not only avoids having to provide your own numerical value for these (irrational) constants, but by using the version defined in the Modelica Standard Library, you get a value that has the highest possible precision. The next constant, g_n, is a physical constant representing the gravitational constant on earth (for computing things like potential energy, i.e., ). Finally, eps is a machine constant that represents a “small number” for whatever computing platform is being used. SI Units As we discussed previously, the use of units not only makes your code easier to understand by associating concrete units with parameters and variables, it also allows unit consistency checking to be performed by the Modelica compiler. For this reason it is very useful to associate physical types with parameters and variables whenever possible. The Modelica.SIunits package is very large and full of physical units that are rarely used. They are included for completeness in adhering to the ISO 31-1992 specification. The following are examples of how common physical units are defined in the SIunits package: type Length = Real (final quantity="Length", final unit="m"); type Radius = Length(min=0); type Velocity = Real (final quantity="Velocity", final unit="m/s"); type AngularVelocity = Real(final quantity="AngularVelocity", final unit="rad/s"); type Mass = Real(quantity="Mass", final unit="kg", min=0); type Density = Real(final quantity="Density", final unit="kg/m3", displayUnit="g/cm3", min=0.0); type MomentOfInertia = Real(final quantity="MomentOfInertia", final unit="kg.m2"); type Pressure = Real(final quantity="Pressure", final unit="Pa", type ThermodynamicTemperature = Real( final quantity="ThermodynamicTemperature", final unit="K", min = 0.0, start = 288.15, nominal = 300, "Absolute temperature (use type TemperatureDifference for relative temperatures)"; type Temperature = ThermodynamicTemperature; type TemperatureDifference = Real(final quantity="ThermodynamicTemperature", final unit="K"); The Modelica Standard Library includes many different domain specific libraries inside of it. This section provides an overview of each of these domains and discusses how models in each domain are The Modelica Standard Library includes a collection of models for building causal, block-diagram models. The definitions for these models can be found in the Modelica.Blocks package. Examples of components that can be found in this library include: □ Input connectors (Real, Integer and Boolean) □ Output connectors (Real, Integer and Boolean) □ Gain block, summation blocks, product blocks □ Integration and differentiation blocks □ Deadband and hysteresis blocks □ Logic and relational operation blocks □ Mux and demux blocks The Blocks package contains a wide variety of blocks for performing operations on signals. Such blocks are typically used for describing the function of control systems and strategies. The Modelica.Electrical package contains sub-packages specifically related to analog, digital and multi-phase electrical systems. It also includes a library of basic electrical machines as well. In this library you will find components like: □ Resistors, capacitors, inductors □ Voltage and current actuators □ Voltage and current sensors □ Transistor and other semiconductor related models □ Diodes and switches □ Logic gates □ Star and Delta connections (multi-phase) □ Synchronous and Asynchronous machines □ Motor models (DC, permanent magnet, etc.) □ Spice3 models The Modelica.Mechanics library contains three main libraries. The translational library contains component models used for modeling one-dimensional translational motion. This library contains components like: □ Springs, dampers and backlashes □ Masses □ Sensors and actuators □ Friction The rotational library contains component models used for modeling one-dimensional rotational motion. This library contains components like: □ Springs, dampers and backlashes □ Inertias □ Clutches and Brakes □ Gears □ Sensors and Actuators The multibody library contains component models used for modeling three-dimensional mechanical systems. This library contains components like: □ Bodies (including associated inertia tensors and 3D CAD geometry) □ Joints (e.g., prismatic, revolute, universal) □ Sensors and Actuators The Modelica.Magnetic library contains two sub-packages. The first is the FluxTubes package which is used to construct models of lumped networks of magnetic components. This includes components to represent the magnetic characteristics of basic cylindrical and prismatic geometries as well as sensors and actuators. The other is the FundamentalWave library which is used to model electrical fields in rotating electrical machinery. The Modelica.Thermal package has two sub-packages: The HeatTransfer is for modeling heat transfer in lumped solids. Models in this library can be used to build lumped thermal network models using components like: □ Lumped thermal capacitances □ Conduction □ Convection □ Radiation □ Ambient conditions □ Sensors Normally, the Modelica.Fluid and Modelica.Media libraries should be used to model thermo-fluid systems because they are capable of handling a wide range of problems involving complex media and multiple phases. However, for a certain class of simpler problems, the FluidHeatFlow library can be used to build simple flow networks of thermo-fluid systems. The Utilities library provides support functionality to other libraries and model developers. It includes several sub-packages for dealing with non-mathematical aspects of modeling. The Modelica.Utilities.Files library provides functions for accessing and manipulating a computers file system. The following functions are provided by the Files package: □ list - List contents of a file or directory □ copy - Copy a file or directory □ move - Move a file or directory □ remove - Remove a file or directory □ createDirectory - Create a directory □ exist - Test whether a given file or directory already exists □ fullPathName - Determine the full path to a named file or directory □ splitPathName - Split a file name by path □ temporaryFileName - Return the name of a temporary file that does not already exist. □ loadResource - Convert a Modelica URL into an absolute file system path (for use with functions that don’t accept Modelica URLs). The Streams package is used reading and writing data to and from the terminal or files. It includes the following functions: □ print - Writes data to either the terminal or a file. □ readFile - Reads data from a file and return a vector of strings representing the lines in the file. □ readLine - Reads a single line of text from a file. □ countLines - Returns the number of lines in a file. □ error - Used to print an error message. □ close - Closes a file. The Strings package contains functions that operate on strings. The general capabilities of this library include: □ Find the length of a string □ Constructing and extracting strings □ Comparing strings □ Parsing and searching strings The System package is used to interact with the underlying operating system. It includes the following functions: □ getWorkingDirectory - Get the current working directory. □ setWorkingDirectory - Set the current working directory. □ getEnvironmentVariable - Get the value of an environment variable. □ setEnvironmentVariable - Set the value of an environment variable. □ command - Pass a command to the operating system to execute. □ exit - Terminate execution.	{"url":"https://mbe.modelica.university/components/packages/msl/","timestamp":"2024-11-09T20:23:41Z","content_type":"text/html","content_length":"231231","record_id":"<urn:uuid:de79ffa4-c744-4d1b-ac04-8042b99ab709>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00325.warc.gz"}
Find Out Who's Concerned About What Is Discrete Math and Why You Should Pay Attention - lancasterartificialgrasscompany.com Details of What Is Discrete Math Students also learn to utilize Euler and Hamilton circuits to discover the best solutions in a number of real-world scenarios, including determining the most effective www.papernow.org approach to schedule airline travel. The date an article was initially made available online is going to be carried over. If an excessive amount of time has passed, then grading changes might not be made. MATH 1126Q might be taken concurrently. Otherwise it is known as a disconnected graph. In the event the graphs are infinite, that’s usually specifically stated. You ought to know the fundamental formulas. Each math topic has many unique forms of math worksheets to cover various sorts of problems you may decide to work on. Despite the fact that you’re have a 50 marks theory paper, which is quite easy to prepare. 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The history of discrete mathematics has involved quite a few challenging difficulties that have focused attention within regions of the area. In reality, there are two or three courses provided by Princeton’s COS department that are really discrete mathematics courses in disguise. The fundamental math might not be difficult but it’s always preferable to overprepare and do some type of review besides simply a couple practice tests. Our online interactive classroom has all of the tools you require to receive your math questions answered. Here you will have a lot of math help and a lot of fun whilst learning and teaching math step-by-step. When you have obtained your account, you’ll need to register with our grading computer software. Mathematics majors have to report individually on a minumum of one topic of a moderate amount of difficulty for a demonstration of their resourcefulness, ability, and achievement in the sphere of mathematics. Varsity Tutors is happy to provide completely free practice tests for all degrees of math education. They have the opportunity to participate in industry internships and competitive research programs in the United States and abroad. Learning combinatorics enables us to answer questions like that. Basic concepts are explained through attractive examples concerning our day-to-day life experience. They are produced by a number of national weather services around the globe as well as the private sector. It isn’t difficult to modify the P-1 method such that it is going to get the factor q whenever k is extremely composite. Squares of numbers that aren’t prime numbers will probably have more than 3 If you own a question, your very best choice is to post a message to the newsgroup. Also, notice there are many human circumstances which would be independent in a perfect just ideal Earth, but regrettably aren’t independent in a true world full of inequities. There was a lovely approach to help with that problem, however. http://www.lancasterartificialgrasscompany.com/wp-content/uploads/2018/02/Logo-Lancaster-Artificial-Grass.gif 0 0 wp_admin_project http://www.lancasterartificialgrasscompany.com/wp-content/uploads/ 2018/02/Logo-Lancaster-Artificial-Grass.gif wp_admin_project2019-03-27 13:27:122019-05-14 08:34:20Find Out Who's Concerned About What Is Discrete Math and Why You Should Pay Attention	{"url":"https://www.lancasterartificialgrasscompany.com/find-whos-concerned-discrete-math-pay-attention/","timestamp":"2024-11-08T14:09:41Z","content_type":"text/html","content_length":"43899","record_id":"<urn:uuid:a0e37c66-4cd1-4bc6-90d1-90fd57ec956b>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00166.warc.gz"}
Scale times and scale lengths associated with charged particle fluctuations in the lower ionosphere The concept of chemical fluctuations associated with electron attachment and detachment processes in the terrestrial D-region implies the existence of scale times and scale lengths over which these phenomena can be observed. An extension of Chapman's definition of scale times and scale lengths to fluctuating quantities leads to an eigenvalue problem which can be solved numerically, and for which simple analytical approximations are obtained. Two scale times associated with negative ion chemistry and with neutralization, respectively, differ by several orders of magnitude. One scale length indicates how the classical Debye length is modified by the presence of negative ions. Two other scale lengths are the spatial correspondence of the scale times. Whereas the scale length associated with negative ion chemistry ranges from 10 to 80 cm, the scale length corresponding to recombination processes ranges from 5 to 10 m. Planetary and Space Science Pub Date: October 1986 □ Charged Particles; □ D Region; □ Ionospheric Electron Density; □ Ionospheric Ion Density; □ Debye Length; □ Poisson Equation; □ Vertical Distribution; □ Geophysics	{"url":"https://ui.adsabs.harvard.edu/abs/1986P%26SS...34..979K/abstract","timestamp":"2024-11-08T21:52:36Z","content_type":"text/html","content_length":"37663","record_id":"<urn:uuid:a0c28698-0e03-4544-9871-f07ba58f54bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00005.warc.gz"}
New Papers 6 - 50 questions,1 hour,min CGPA 5.8, Question paper in two sets A & B 3 round of interview In a party 48 person r there,2/3 r men, remaining are women, half of women r doctor,no of doctor in party From a cuboids of 18158 size how many 666 cube can be cut The age of father is thrice then his son, after 15 yr the the father age is twice as the son, Find the age of father.ans-45 yr Which is greater 30% of 40,5 more than the square of 3,aur are hi 2 the yad nahi hai. A man purchase 3 tv @11500 rs per piece,one sold at 20% gain,other two 7% loss,find net gain/loss. One ball drops fron height 150 m,and bounces back to 5/6 of previous height,till it becomes stationary,calculate total height trevelled. Two question related to the position of chair(11 person(5 men.6 women),kucch sharat hai),find the location. Apple:fruit then wood:tree and 3 more One question related to ration and proption values r not adjact---one material @1.60 rs per kg is mixed with second material @1.48 rs per kg,in what propotin they r added so the mixed material costs 1.54 rs per kg. In a polygon,if line is drawn towards outside,then sum of angle at any vertex is----not remembered exactly. Find the min value of 6Xsquare-12X One question from set theory--in a survey of college of 2000 student 54% like coffie,68% like tea, some %(48%)like smoke from total and 30% like coffie and tea,32 % like tea and smoke,some percentage likea coffie and smoke,6 % not like anything, Find the % of Coffie only.Valus r not exact. The sum of two no is 72,then max value if their multiple of those no is----? A function is given,you have to find the valve of X at which the finctin in min--Y is function of (arithmetic If the income of A is 25 % less than the income of B.then the income of B is greater than -----%? Three coin r tossed,The probabaility of getting atleast one head is----? In a Glass a mix of milk and water(4 times milk and 1 times water).Glass se kitana mix nakal lo ki usake place me water dalane per mix ka ratio 50% water and 50% milk ho jaye. ans 3/8 In the expansion of(X2+1/X2)12---X square plus one upon X square ka whole power twelve----the coefficient of term which is free from X If she is y year older from x year,then how many year older from Z year. Three ques depding on relations---3 condition---and you have to give ans----it is too easy---u can do in 1 min. Paragraph complete karana hota hai. Logical sequence of sentence like in Engg services	{"url":"https://www.yuvajobs.com/download-aditi-placement-papers/new-papers-6-aditi-whole-testpaper-508.html","timestamp":"2024-11-10T14:33:18Z","content_type":"text/html","content_length":"28017","record_id":"<urn:uuid:6522ace1-3bc2-4033-8b4b-1a3e6d39e36b>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00006.warc.gz"}
Word Problems on Addition Worksheet \| 2nd Grade Addition Word Problems Word Problems on Addition Worksheet Let us practice the given questions of 2-digit word problems on addition (without carrying or carry if needed). 1. There were 28 students in one school bus and 35 students in another school bus. How many students were there in the two buses altogether? 2. There were 26 marbles in a bag. A boy puts 38 more marbles in the bag. How many marbles are there in the bag now? 3. Lena's father bought 23 apples, 18 mangoes and 21 bananas. How many fruits did she buy altogether? 4. In a school, Class 2 has three sections. If there are 36 students in section A, 34 students in section B and 23 students in section C,what is the total number of students in Class 2? 5. Sonia had 34 books. She bought 9 more books. How many books does she have altogether? 6. Alisha sold 56 orange candies and 25 mango candies. How many candies did she sell in total? 7. Maria colored 87 pages of the colouring book. Navya coloured 10 more pages. How many pages did they colour in all? 8. Rita read 96 pages of a story book. She had to read 40 more pages to finish the book. How many pages did the book have? 9. Mary and Ron were selling tickets for the school fair. Mary sold 69 tickets and Ron sold 28 tickets. How many tickets did they sell in all? 10. A vegetable vendor sold 34 potatoes and 62 tomatoes and 25 onions. How many vegetables did he sell in all? From Word Problems on Addition Worksheet to HOME PAGE Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need. New! Comments Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.	{"url":"https://www.math-only-math.com/word-problems-on-addition-worksheet.html","timestamp":"2024-11-07T05:34:48Z","content_type":"text/html","content_length":"30925","record_id":"<urn:uuid:2d8bef02-91db-433b-a377-a3427826c726>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00737.warc.gz"}
Pawel Sobocinski I am Professor of Trustworthy Software Technologies at TalTech. I lead the Laboratory for Compositional Systems and Methods. I am currently PI of the Estonian Research Council grant PRG1210, Automata in Learning, Interaction and Concurrency (ALICE). I take part in CHESS, the Cyber-security Excellence Hub in Estonia and South Moravia, funded by the European Commission. I am also a TalTech PI of EXAI, the Estonian Centre of Excellence in Artificial Intelligence, financed by the Estonian Ministry of Education and Research. My research can be understood as the study of how to connect open systems (of various kinds: programs, networks, computing devices, circuits, ...) in a way that the description of the connections--i.e. the language that we describe subsystems and how to compose them---is compatible with the behaviour of the system. Therefore, what can be observed of the global system is entirely derivable from the observations made of the component subsystems. This is because each composition operation in the language we use for describing systems gives rise to an analogous operation on the behaviours. This property is known as compositionality. I focus on compositional modelling of systems, developing the underlying mathematics (usually category theory), and applying it to real-life problems such as verification. I work on graph transformation, Petri nets, process algebras, dynamical and cyberphysical systems, as well as mainstream concurrent programming. Some time ago I wrote the Graphical Linear Algebra blog about rediscovering linear algebra in a compositional way, with string diagrams.	{"url":"https://cs.ioc.ee/~pawel/","timestamp":"2024-11-08T17:29:27Z","content_type":"application/xhtml+xml","content_length":"25627","record_id":"<urn:uuid:b4ede751-9335-4ea3-85a5-55a9efde8120>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00244.warc.gz"}
Histograms (2 of 4) Learning Objectives • Describe the distribution of quantitative data using a histogram. We have discussed two types of graphs that summarize a distribution of a quantitative variable: dotplots and histograms. From a dotplot, we also described the pattern in the data with statements about shape, center, and spread. We have to be more cautious making similar statements using a histogram because our perception of shape, center, and spread can be affected by how the bins are defined. We investigate this important point in the next example. We used the same set of data to construct these three histograms of student scores. Are you surprised by how different the distribution looks in each histogram? The histogram on the left has a bin width of 20. The first bin starts at 40. To create the middle histogram, we changed the bin width to 10 but kept the first bin starting at 40. To create the last histogram, we kept the bin width at 10 but started the first bin at 45. These changes affect our description of the shape, center, and spread of this set of data. For example, in the histogram on the left, the distribution looks symmetric with a central peak. In the histogram on the right, the distribution looks slightly skewed to the right. Based on the middle histogram, we might estimate that most students scored between 70 and 80. But the histogram on the right suggests that typical students scored between 65 and 75. Why does changing the bin size and the starting point of the first bin change the histogram so drastically? When we change the bins, the data gets grouped differently. The different grouping affects the appearance of the histogram. To illustrate this point, we highlighted the five students who scored in the 70s in each histogram. • In the histogram on the left, these five students are grouped in the middle bin with other students who scored between 60 and 80. • In the histogram in the middle, these five students form a bin of their own, since no other students scored between 70 and 80. • In the histogram on the right, these five students are in separate bins. Which histogram gives the most helpful summary of the distribution? For this situation, the middle histogram is probably the most useful summary because the intervals correspond to letter grades. Our general advice is as follows: • Avoid histograms with large bin widths that group data into only a few bins. A histogram constructed with large bin widths will show the distribution as a “skyscraper.” This does not give good information about variability in the distribution. • Avoid histograms with small bin widths that group data into lots of bins. A histogram constructed with small bin widths will show the distribution as a “pancake.” This does not help us see the pattern in the data. Use the simulation below to answer the questions in the next Learn By Doing. Click here to open this simulation in its own window. These next exercises focus on recognizing the shape of a distribution using a histogram. We know that changes in the bin width can change the appearance of the distribution. But a histogram with an appropriate bin width can give good information about the shape of the distribution.	{"url":"https://courses.lumenlearning.com/atd-herkimer-statisticssocsci/chapter/histograms-2-of-4/","timestamp":"2024-11-03T09:04:29Z","content_type":"text/html","content_length":"53786","record_id":"<urn:uuid:d94a8b79-1724-4abf-9c58-3964a676659d>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00041.warc.gz"}
CCNA 1 Module 1 Exam Solutions - ITPRC CCNA 1 Module 1 Exam Solutions August 16, 2018 Exam Preps 9 Comments Cisco’s exams can be a lot of hard work- we know. But the worst thing you can do to yourself is to simply look up the answers to exam questions. Not only do you not learn anything- but you’re hurting your future in the process. Instead, shouldn’t you get a helpful hint and get pointed in the right direction of where to look for an answer? That way, you can keep your dignity, hopes for the future, and of course your grade. The questions below are from the CCNA 1 module 1 exam- but instead of straight answers, we reason out why the answers are correct. If any type of math or decision making is involved, we leave that up to you (But we will of course give you the resources you need to complete the question.) 1. Several computers in the company require new NICs. A technician has located a good price on the Internet for the purchase of these NICs. Before these NICs are purchased and installed, what details must be verified? (Choose three.) A) The MAC address on the NIC B) The size of the RAM on the NIC C) The bandwidth supported by the NIC D) The type of media supported by the NIC E) The type of network architecture supported by the NIC More info: What is a Network Interface Card? Explanation: After reading the material “What is a Network Interface Card,” we find out that a MAC address is unique to every NIC. Therefore, we don’t really have a need to be selective, and thus A is wrong. And NICs do have RAM, but it isn’t as vital to a NIC as it is for something like your computer- so B is wrong. The remaining three answers are therefore correct. 2. What is the hexadecimal equivalent for the binary number 00100101? A) 15 B) 20 C) 25 D) 30 E) 37 F) 40 More Info: A Guide to Network math Explanation: The above problem involves converting a binary number to a hexadecimal number. The above article explains that to do so, you need to split the binary number into two nibbles and then find their decimal value. After you find the decimal value, you may finally find the hex number. Consult the article above more a more in-depth explanation. 3. Which phrases describe a byte? (Choose two.) A) a single binary digit B) +5 volts of electricity C) the value range 0 to 127 D) a grouping of eight binary digits E) a single addressable data storage location More Info: A Guide to Network math Explanation: From reading the above article, we know that one binary digit is a bit, not a byte- A is wrong. +5 volts of electricity may in some instances indicate a bit, but certainly not a byte- B is wrong. C is wrong because the value ranges from 0 to 255, not 127. Finally we get D and E as our answers- since we know that there are 8 bits in a byte, and each byte is considered to be a single addressable data storage location. 4. Which specialized equipment is used to make a physical connection from a PC to a network? A) router B) RAM C) CD ROM D) network interface card More Info: What is a Network Interface Card? Explanation: You don’t necessarily need to read the above article to know this answer. RAM and CD-ROM components do not take a part in a network- and a router, like the name implies, routes data rather than connects a PC to a network. Obviously, the answer should be D. 5. What is the binary equivalent for the decimal number 248? A) 11100000 B) 11110000 C) 11110100 D) 11111000 More Info: A Guide to Network math Explanation: The above question wants you to convert the number 248 to binary. By reading the above article, we know that to do so we need to see which bits fit into 248. This is commonly done by the “Will this number go into 248?” method. If you need more of a descript explanation (complete with diagrams), consult the above article. 6. Convert the binary number 01011011 into its hexadecimal equivalent. Select the correct answer from the list below. A) 5A B) 5B C) 5C D) 5D E) 6B F) 7A More Info: A Guide to Network math Explanation: The above problem involves converting a binary number to a hexadecimal number. The above article explains that to do so, you need to split the binary number into two nibbles and then find their decimal value. After you find the decimal value, you may finally find the hex number. Consult the article above more a more indepth explanation. 7. What is the binary equivalent for decimal number 149? A) 10010111 B) 10010101 C) 10011001 D) 10010111 E) 10101011 F) 10101101 More Info: A Guide to Network math Explanation: The above question wants you to convert the number 149 to binary. By reading the above article, we know that to do so we need to see which bits fit into 149. This is commonly done by the “Will this number go into 149?” method. If you need more of a descript explanation (complete with diagrams), consult the above article. 8. In an 8 bit binary number, what is the total number of combinations of the eight bits? A) 128 B) 254 C) 255 D) 256 E) 512 F) 1024 More Info: A Guide to Network math Explanation: This question can be tricky. By reading the above article, we know that it is either 255 or 256. We know that we can go up to 255 with each byte of data. But don’t be so sure it’s C- it is, in fact, D. This is because we count 0 as a number, and thus, there are 256 combinations. Be on the lookout for more sly questions Cisco throws at us to make sure we’re still awake. 9. Which device connects a computer with a telephone line by providing modulation and demodulation of incoming and outgoing data? A) NIC B) CSU/DSU C) router D) modem E) telco switch More Info: What is a Modem? Explanation: This question should be fairly easy. If you know the definition of a modem, you should get this question right even without our help. Nonetheless, the article above will provide extra information on modems if needed. 10. What is the binary equivalent for the decimal number 162? A) 10101010 B) 10100010 C) 10100100 D) 10101101 E) 10110000 F) 10101100 More Info: A Guide to Network math Explanation: The above question wants you to convert the number 162 to binary. By reading the above article, we know that to do so we need to see which bits fit into 162. This is commonly done by the “Will this number go into 162?” method. If you need more of a descript explanation (complete with diagrams), consult the above article. 11. Which of the following are popular web browsers? (Choose two.) A) Acrobat B) Internet Explorer C) Macromedia Flash D) Netscape Navigator E) Quicktime F) World Wide Web More Info: (none) Explanation: This is another question that you should know from basic computer knowledge. Although we ourselves hesitated and asked, “Where’s FireFox, and why is Netscape Navigator on the list?” (It seems Cisco is a little behind on the times.) 12. Convert the Base 10 number 116 into its eight bit binary equivalent. Choose the correct answer from the following list: A) 01111010 B) 01110010 C) 01110100 D) 01110110 E) 01110111 F) 01010110 More Info: A Guide to Network math Explanation: The above question wants you to convert the number 116 to binary. By reading the above article, we know that to do so we need to see which bits fit into 116. This is commonly done by the “Will this number go into 116?” method. If you need more of a descript explanation (complete with diagrams), consult the above article. 13. What is the hexadecimal equivalent for the binary number 10001110? A) 22 B) 67 C) 142 D) AE E) 8E More Info: A Guide to Network math Explanation: The above problem involves converting a binary number to a hexadecimal number. The above article explains that to do so, you need to split the binary number into two nibbles and then find their decimal value. After you find the decimal value, you may finally find the hex number. Consult the article above more a more indepth explanation. 14. Represented as a decimal number, what is the result of the logical ANDing of binary numbers 00100011 and 11111100? A) 3 B) 32 C) 35 D) 220 E) 255 More Info: A Guide to Network math Explanation: This question involves the AND logical operator. Essentially, you compare the two binary numbers. If there is a binary 1 in the same location of both numbers, then you count that bit. Think of it as a test: you need 1 AND 1 for any addition to take place. A binary 1 and binary 0, or binary 0 and binary 0, will both not work. B is the correct answer since the 5th bit from the right is the only bit that can be ANDed- and 2^5 = 32. (Note that it is not the 6th bit, since we start counting at 0, not 1.) 15. Convert the decimal number 231 into its binary equivalent. Select the correct answer from the list below. A) 11110010 B) 11011011 C) 11110110 D) 11100111 E) 11100101 F) 11101110 More Info: A Guide to Network math Explanation: The above question wants you to convert the number 231 to binary. By reading the above article, we know that to do so we need to see which bits fit into 231. This is commonly done by the “Will this number go into 231?” method. If you need more of a descript explanation (complete with diagrams), consult the above article. 16. What are three conditions that would require a network administrator to install a new NIC? (Choose three.) A) whenever a NIC is damaged B) when there is a need to provide a secondary or backup NIC C) when there is a change from copper media to wireless D) whenever operating system security patches are applied E) whenever the PC has been moved to a different location More Info: What is a Network Interface Card? Explanation: A is correct since when something is broke, we should fix it (or in this case replace it). Obviously if we have the need for a backup NIC, we will need to install a new one- B is correct too. C is correct since some NICs are specific to a certain type of media, so changing that media warrants a new NIC. D is not true since the operating system does not have an effect on the NIC. E is not true because the physical location of a computer will not interfere with your NIC. 9 Comments	{"url":"https://www.itprc.com/ccna-1-module-1-exam-solutions/","timestamp":"2024-11-05T10:50:08Z","content_type":"text/html","content_length":"284877","record_id":"<urn:uuid:7a352cc5-8c29-4888-81bd-a424ad255d99>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00651.warc.gz"}
A geometrical approximation for π You're reading: Irregulars A geometrical approximation for π If you were paying very close attention last week, you’ll have noticed my attempt to come up with an estimate of π, geometrically, as part of The Aperiodical’s π Day challenge (even if it’s not really π Day): Oo, my second effort at estimating π came to 3.14151, correct to 0.003%! cc @aperiodical pic.twitter.com/2vuvys0mka — Colin Beveridge (@icecolbeveridge) March 10, 2015 Viewed on its own, that’s probably a bit mysterious, so I thought I’d write a little article to explain what was going on, and explore some of the maths behind it. The approximation The first thing I did, after watching the video, was think about (read: Google) what other strategies one might use to approximate π. I thought about measuring a cylinder (too much work); I thought about something to do with the Catalan numbers (would involve research); and I finally settled on a geometric method that relies on a cool almost-identity: \[x \approx \frac{3 \sin(x)}{2 + \cos(x)} \] There are two immediate questions: • Why does that work? • How does it help? Why it works is fairly simple: a good approximation for $\sin(x)$ is $x – \frac{1}{6}x^3 + \frac{1}{120}x^5 – …$, and a good approximation for $\cos(x)$ is $1 – \frac{1}{2}x^2+ \frac{1}{24}x^4 – …$. That means the fraction becomes: \[ \frac{x\left(3 – \frac{1}{2}x^2 + \frac{1}{40}x^4 – …\right)}{3 – \frac{1}{2}x^2 + \frac{1}{24}x^4 – …} \] Ignoring the $x$, the rest of the numerator and denominator only differ in the $x^4$ term (and onward), which, for a small angle, makes for a very small error. How does it help? Well, sine and cosine are circular functions, which means they can be measured off of a circle. Given a pair of perpendicular axes and a circle centred on them, any point on the circumference is $R \sin(\theta)$ above the horizontal axis and $R \cos(\theta)$ to the right of the vertical axis, assuming the circle has radius $R$ and the point forms an angle $\theta$ with the horizontal axis. That means, if you construct an angle of, say, $\frac{\pi}{6}$, you should be able to construct and measure $\frac{3R \sin(\theta)}{2R + R\cos(\theta)}$, which is approximately $\theta$. The picture So, I drew that kind of circle. I constructed perpendiculars. I extended lines. I measured them. And I did some simplification. If I wanted an estimate for π, I’d need to multiply everything by 6; meanwhile, I’d measured double $R\sin(\theta)$, the chord of the circle (AB in the diagram), so I’d need to multiply that number — 74mm — by 9 to get 666. Oooo! On the bottom, I’d extended my horizontal chord (CA) by two radii to get $2R + R\cos(\theta)$, which I measured to be 212mm (CG). Then I did some tedious long division to get 3.14151, which isn’t bad for something knocked up on the sofa with a borrowed geometry kit. It’s almost too good to be true. Well, yes, of course I cheated Estimates of π that are good to nearly five significant figures don’t pop off of such pages, at least, not without a great deal of preparation. For example, an unscrupulous geometer might fire up Desmos, draw the line $y = \pi x$, and see where it falls unusually close to a lattice point. (Better mathematicians than me would use a continued fraction to get the best estimate possible — although I did consider $\frac{355}{113}$, I couldn’t make it look natural. That was my first attempt.) I found that $\frac{333}{106}$ was pretty much bang on the money — certainly, good enough for this. However, I needed the top to be a multiple of 18; it’s already a multiple of 9, so doubling it would work. I also know that $\sin\left(\frac {\pi}{6}\right) = \frac{1}{2}$, giving me a radius of $\frac{666}{18} = 74\text{mm}$. From there, all that remained to do was fudge the horizontal chord marks ever so slightly so they coincided with the 212mm I’d pre-measured, and boom! A natural-looking, but ever-so-good estimate of	{"url":"https://aperiodical.com/2015/03/a-geometrical-approximation-for-pi/","timestamp":"2024-11-06T02:00:08Z","content_type":"text/html","content_length":"40821","record_id":"<urn:uuid:06b80958-76f0-468d-b7ec-e20c57ed57f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00467.warc.gz"}
Minimum Message Length Encoding, Evolutionary Trees and Multiple-Alignment. Appears in Hawaii Int. Conf. Sys. Sci. 25, Vol.1, pp.663-674, January 1992 L. Allison, C. S. Wallace and C. N. Yee, Department of Computer Science, Monash University, Australia 3168. Abstract: A method of Bayesian inference known as minimum message length encoding is applied to inference of an evolutionary-tree and to multiple-alignment for k>=2 strings. It allows the posterior odds-ratio of two competing hypotheses, for example two trees, to be calculated. A tree that is a good hypothesis forms the basis of a short message describing the strings. The mutation process is modelled by a finite-state machine. It is seen that tree inference and multiple-alignment are intimately connected. Q: What is the difference between a hypothesis and a theory? A: Think of a hypothesis as a card. A theory is a house made of hypotheses. (From rec.humor.funny, attributed to Marilyn vos Savant.) Keywords: inductive inference, minimum message/ description length, multiple-alignment, evolutionary-tree, phylogenetic tree. 1. Introduction. In previous papers[1, 2], minimum message length (MML) encoding was used to infer the relation, if any, between 2 strings and to compare models of mutation or relation. Here we apply MML encoding to the problems of multiple-alignment and inference of an evolutionary-tree given k>=2 strings of average length n. MML encoding is a method of Bayesian inference. We consider the message length of an explanation of given data D. An explanation consists of a message which first states a hypothesis H, then states the data using a code which would be optimal were H true. It is a basic fact of coding theory that the message length (ML), under optimal encoding, of an event of probability p is -log[2](p) bits. From Bayes' theorem in the discrete case: P(H&D) = P(H).P(D\|H) = P(D).P(H\|D) ML(H&D)= ML(H)+ML(D\|H)= ML(D)+ML(H\|D) P(D\|H) is called the likelihood of the hypothesis. It is a function of the data given the hypothesis; it is not the probability of the hypothesis. P(H) is the prior probability of the hypothesis and P(H\|D) is its posterior probability. If ML(H) and ML(D\|H) can be calculated for given H and D then the posterior odds-ratio of 2 hypotheses can be calculated. Analogous results hold in the continuous case and imply an optimum accuracy for estimating real-valued parameters, although the calculations can be difficult. MML encoding can be used to compare models, test hypotheses and estimate parameters. Wallace and Freeman[25] give the statistical foundations. Maximum-likelihood methods and MML encoding are related. A significant difference is that the former do not cost the model or hypothesis complexity, ML(H), and so tend to favour overly complex or detailed models. In sequence analysis the issue covers: the estimation of parameters to optimal accuracy, the complexity of models of mutation and of evolution and the fact that a single alignment is too precise a hypothesis. In cases where model complexity is constant, it can be ignored and then maximum-likelihood and MML encoding give the same results. We next link inference over 2 strings to problems over k strings. Given strings A and B, the following questions can be raised: Q1: Are they related? Q2: Are they related by a particular model of relation and parameter values (~ time since divergence)? Q3: Are they related by a particular alignment? Q4: Are they related by descent from a particular common ancestor? An answer to one of these questions can "only" be a hypothesis. An explanation of the strings can be based on such an answer. Explanations based on successive questions carry increasing amounts of information and must have increasing message lengths. A and B have infinitely many possible common ancestors. It is possible to dismiss most as wildly improbable but impossible to identify a common ancestor with certainty without extra information. An alignment implies a subset of possible ancestors in a natural way. Conventional alignment programs return a single alignment which is optimal under some explicit or implicit assumptions - a model. A given model can relate A and B by many different alignments and we call the answer to Q2 the r-theory (for relation). It states the inferred parameters for the evolution of A and B followed by A and B in a code based on all possible alignments. A by-product is a probability density plot of all alignments. The answer to Q1 would require the sum over all models of the integral over all parameter values. This calculation is impractical and the result would be of little use; Q2 is the question usually asked. There is a natural null-theory in sequence analysis. If the strings are unrelated, it is not possible to do better than transmit them one after the other, at about 2 bits per character for DNA. The lengths of the strings must also be transmitted. Any explanation that has a message length greater than that of the null-theory is not acceptable. The approach to problems over 2 strings generalises to problems over k>=2 strings. Questions analogous to Q1-Q4 arise when given k strings which might be related by an evolutionary-tree: R1 (a): Are they related or, slightly more detailed, R1 (b): are they related by a particular tree-topology? Consideration is usually restricted to binary trees and we follow this convention here. It is not hard to allow arbitrary branching factors. Forests should be allowed so as to permit independent families; the message length of a forest includes the sum of those of its trees and the generalisation is obvious. R2: Are they related by a particular tree (topology + parameter values for each edge). Each edge of the tree represents a simple 2-string relation and the inferred parameters for each edge (~ time) are required. This is the generalised r-theory. R3: Are they related by a particular multiple-alignment? R4: Are they related by descent from particular ancestors? The chain Q1:Q2:Q3:Q4 matches the chain R1:R2:R3:R4 and in particular Q2:Q3 matches R2:R3, although this requires some discussion. Many misunderstandings come from confusing answers to R3 with those to R2. A tree (topology & edge parameters) implies a probability distribution on character-tuples and thus on multiple-alignments. The probability of a tree and the strings (R2) is the product of the prior probability of the tree and the sum of the probabilities of all such multiple-alignments given the tree. When k>2, the number of alignments is very large and heuristics are needed to make the computations feasible. R3 can be taken in 2 ways. We take the word `related' to mean `related by an evolutionary-tree'. A tree has 2k-3 edges and thus has O(k) parameters but there are 5^k-1 possible tuples in a multiple-alignment. There is a smaller but still exponential number of `patterns'. Many of the tuples' probabilities are locked together under an evolutionary-tree hypothesis. If `related' were taken to mean any arbitrary relation, there would be more degrees of freedom in the form of hypotheses which might enable the data to be fitted better but at the higher overhead of stating more parameters. Reichert et al[21] were the first to apply compact encoding to the alignment of 2 strings. They used one particular model of relation and sought a single optimal alignment. This was difficult to do efficiently for the model that they chose. Bishop and Thompson[5] defined maximum-likelihood alignment for what is a 1-state model although they recognised the possibility of other models. Allison et al[2] objectively compared 1, 3 and 5-state models of mutation using MML encoding. Felsenstein[11,12] developed a maximum-likelihood method for inferring evolutionary-trees. To date, it is the method with the soundest statistical credentials. His program does not do alignment itself and requires a multiple-alignment as input. Friday[13] discussed the problem of assigning figures of merit to inferred trees. Wallace and Boulton developed MML methods for classification[24,7], and for hierarchical classification[8], of arbitrary "things". In classification, a class usually contains many things. Cheeseman and Kanefsky[9] applied MML encoding to evolutionary-trees over macro-molecules. Milosavljevic[20] examined the MML classification of macro-molecules. The methods described in this paper relate to other methods as follows. They generalise Felsenstein's approach by allowing inserts and deletes to be inferred and by including the cost of the model. In contrast to Cheeseman and Kanefsky, ancestral strings are not inferred, for to do so involves arbitrary choices. However probabilities can be assigned to possible ancestors and to possible alignments given the parameters that are inferred. string A= TAATACTCGGC string B= TATAACTGCCG mutation instructions: change(ch) NB. ch differs from the corr' char in string A. a mutation sequence: copy; copy; delete; copy; copy; ins(A); copy; copy; del; copy; change(C); copy; generation instructions: mismatch(ch[A],ch[B]) NB. ch[A]<>ch[B] a generation sequence: match(T); match(A); ins[A](A); match(T); match(A); ins[B](A); match(C); match(T); ins[A](C); match(G); mismatch(G,C); match(C); ins[B]G equivalent alignment: \|\| \|\| \|\| \| \| Figure 1: Basic Models. 2. Mutation-Models. We consider 2 broad models to relate 2 strings A and B (Figure 1). In the mutation-model a simple machine reads string A and a sequence of mutation instructions and outputs string B. In the generation-model a simple machine reads a sequence of generation instructions and outputs strings A and B on a single tape 2 characters wide. There is an obvious correspondence between a mutation sequence, a generation sequence and an alignment. Results based on these notions are equivalent and they can be used interchangeably as convenient. The machines that we consider are probabilistic finite-state machines as described in [2]; they are equivalent to hidden Markov models. We usually consider symmetric machines where p(ins) = p(del) = p(ins[A]) = p(ins[B]). The message length of an instruction is -log[2] of its probability plus either 2 bits for 1 character or log[2](12) bits for 2 differing characters; for simplicity only, we assume that all characters are equally probable. In highly conserved sequences, the probabilities of those instructions that do not conserve characters are low and their message lengths are high. The probability of an instruction also depends on the current state, S, of the machine: ML(match(c)\|S) = -log[2](P(match\|S)) + 2 ML(mismatch(c,c')\|S) = -log[2](P(mismatch\|S)) + log[2](12) ML(ins[A](c)\|S) = ML(ins[B](c)\|S) = -log[2](P(ins[A]\|S)) + 2 A 1-state machine corresponds to Sellers' edit distance[23] with adjustable weights. A 3-state machine can correspond to linear indel costs or weights. Machines with more than 3 states can correspond to piece-wise linear indel costs. Multiple states allow an instruction to have different probabilities in different contexts. eg. For a 3-state machine a run of indels can have a higher probability of continuing than of beginning, giving a linear message length or cost as a function of run length. A message based on one generation sequence (or alignment) states: (i) the parameter values of the machine which give its instruction probabilities and (ii) the sequence of instructions. The parameters are estimated by an iterative process: Initial values are chosen and an optimal sequence found. An optimal code for this sequence should be based on probability estimates derived from the observed frequencies of instructions. This change in the parameter values cannot increase the message length but can make another sequence optimal, so iteration is necessary and is guaranteed to converge. A few iterations, typically 4 to 8, are sufficient for convergence. The process could find only a local minimum in theory but this is not a problem in practice. The parameters must be stated to an optimum accuracy; the instructions are chosen from a multi-state distribution and Boulton and Wallace[6] give the necessary calculations. The r-theory (Q2) is based on all instruction sequences. The algorithms[2] that compute the r-theory for the various machines are dynamic programming algorithms [DPA] that perform <+,logplus> on message lengths in the inner step or <,+> on probabilities, instead of the usual <+,min> on edit distances or <+,max> on lengths of common subsequences. (logplus(log(A), log(B)) = log(A+B)) The iterative process is used to estimate the parameter values based on weighted frequencies. TATAA TA-TAA T-ATAA \|\| \| \|\| \|\| \| \|\| \| TAATA 2 TAATA- 2 TAAT-A 2 etc. -CGGC CGGC- CGGC \| \| \| \| GCCG- 3 -GCCG 3 GCCG 4 --CGGC CGGC-- CG-GC- \|\| \|\| \| \| GCCG-- 4 --GCCG 4 -GC-CG 4 \| \| GC--CG 4 etc Figure 2: Possible Sub-alignments. Results reported in [2] support the widely held view that the 3 and 5-state machines are better models for DNA than the 1-state machine. The 3 and the 5-state machines have increasing numbers of parameters to be stated. This tends to increase the message length, but is more than made up for by a saving on the strings. There are practical difficulties in going beyond about 5-states. We do not yet have enough data to reliably judge between the 3 and 5-state machines. Traditionally, alignment programs have been based on Sellers' metric[23], or on linear and piece-wise linear cost functions for indels[14,15], or on arbitrary concave cost functions[19]. Some of these programs have several parameters requiring ad-hoc adjustments. MML encoding provides an objective way to compare models of differing complexities and to infer their parameter values. We now examine the questions Q1-Q4 informally on the example strings A = TAATACTCGGC and B = TATAACTGCCG. Note, results on short strings must be taken with a pinch of salt and we use the 1-state model only because it is the simplest to discuss. Q1: Are strings A and B related? They probably are related but it is hard to see exactly how. They have at the least diverged considerably. Q2: How far and in what ways have A and B diverged? The front sections, TATAA:TAATA can be related in several ways involving either 2 mismatches or 2 indels. The central CTs probably match. The end sections, GCCG:CGGC, can be related with 2 indels and a mismatch, or 4 indels or 4 mismatches. On average we expect about 7 matches, 1 or 2 mismatches and a few indels. Q3: What is the best alignment? It is only possible to say that it probably fits the scheme T[A:AA]T[AA:A]CT[GCCG:CGGC] (Figure 2). As noted, the 1-state model was used only for simplicity. The 3 and 5-state models would penalise isolated indels more heavily. Q4: What was the most recent common ancestor of A and B? It is only possible to say that it probably fits the pattern TA[A]TA[A]CTalpha where alpha involved 2 to 4 Cs and/or Gs. T A T A A C T G C C G - - . T: - * - - . . A: - # * + - . . A: . - * + + - - . . T: . - * * - - - . . A: . . - # * - - - . C: . . - + * + - - . T: . - + + * # + . . C: . . . + # + # + . G: . . - + # # # + G: . . + # + # C: . . + # * 1-state machine key: * # + - . MML plus: 0..1..2..4..8..16 bits 1 avg algn = 64.9 bits null-theory = 56.0 bits r-theory = 53.8 bits = : <> : indel = 7 : 1 : 5.9 probability related = 0.8 Figure 3: Probability Density Plot. When the r-theory algorithm is run on strings A and B the results (Figure 3) quantify the informal arguments made above. The optimal alignments stand out in the density plot but there are several of them and many other near-optimal alignments. Any 1 alignment has a message length greater than the null-theory and is not acceptable but taking them all together the r-theory is acceptable. 3. Trees over Two Strings. Given 2 related strings A and B, we think of them as having evolved from a most recent common ancestor P. . . M[A]. .M[B] . . A B Figure 4: Common Ancestor. This involves 2 unknown steps or mutation machines, M[A] and M[B] (Figure 4). The models are reversible and mutating P into A by M[A] is equivalent to mutating A into P by M[A]'. The composition of 2 s-state machines M[A]' and M[B] is not in general equivalent to an s-state machine for s>1. Fortunately, for typical overall indel rates a single s-state machine is a good approximation to the composition of 2 or more s-state machines. It is thus acceptable to relate A and B directly by a single s-state machine. For example, Figure 5 plots -log[2] probability of indel length for the composition of 1, 2 and 512 machines, for machines that can cause short (s) or long (L) indels and that give asymptotically piece-wise linear costs. The calculations are based on machines that can copy or insert only. The similarity of form of the curves is important, not their absolute values. This also suggests that when considering any 2 out of k strings, and the path of machines (edges) between them in a tree, in order to get partial information on a multiple-alignment, it is acceptable to combine 2 or more adjacent s-state machines into one. Note that there are 2 forests over A and B. The r-theory tree (Figure 4) forms 1 forest. The forest of 2 singleton trees corresponds to the null-theory. If the ancestor P and the mutation instruction sequences that M[A] and M[B] executed were known, it would be possible to form an equivalent generation sequence (Figure 6) for a single generation machine G. However neither P nor the mutation sequences are known. The idea to circumvent this problem is to synchronise M[A] and M[B] as they read P and to remove dependency on the unknown P by summing over all possibilities. M[A] and M[B] are forced to execute copy, change and delete instructions in parallel; each of these instructions reads 1 character from P. Equivalently, G executes a match, mismatch, ins[A] or ins[B] instruction as appropriate except that if both M[A] and M[B] execute a delete, G does nothing. Note that an insert reads no character from P. If M[A] is about to execute an insert instruction and M[B] is about to execute some other kind of instruction, M[B] is delayed and G executes an ins[A], and v.v. If both M[A] and M[B] are about to execute 1 or more inserts, they and G can be made to do one of several things; double-counting can be avoided if this is done carefully. Instruction. Explanations and Rates. x x x<---x--->x x<---y--->x pA'(c).pB'(c) pA'(ch).pB'(ch)/3 x y x<---x--->y x<---y--->y pA'(c).pB'(ch) pA'(ch).pB'(c) x _ x<---x--->_ x<---y--->_ pA'(c).pB'(d) pA'(ch).pB'(d) _ x _<---x--->x _<---y--->x pA'(d).pB'(c) pA'(d).pB'(ch) _ _ _<---x--->_ "invisible", pA'(d).pB'(d) not seen c-copy, ch-change, i-insert, d-delete define pA'(instn) = pA(instn)/(1-pA(i)) pB'(instn) = pB(instn)/(1-pB(i)) rates per character of P Note pA'(c)+pA'(ch)+pA'(d) = pB'(c)+pB'(ch)+pB'(d) = 1 sum rates = (1+pA'(i)+pB'(i)) = (1-pA(i).pB(i))/((1-pA(i)).(1-pB(i))) Figure 6: Explanations for Generation Instructions. The inserts can be interleaved in different ways, eg. G can execute an ins[A] and then an ins[B] or v.v. Each of these options corresponds to a part of an alignment. Some, such as an ins[A] immediately followed by an ins[B] are of low, but non-zero, probability and are not usually seen in alignments. For example, a single optimal alignment always explains a pair of characters <A[i],B [j]> as `mismatch(A[i],B[j])' and never as `ins[A](A[i]);ins[B](B[j])' and thus gives biased estimates. We expect P, A and B to have the same number of characters on average if pA(i)=pA(d) and pB(i)=pB(d). However we expect the alignment to be longer than P, for a tuple is "gained" whenever M[A] or M [B] does an insert but a character is only "lost" when both M[A] and M[B] delete it. For this reason, rates of instructions per character of P, ie. pA' and pB' in figure 6, have a denominator of (1-pA(i)) or (1-pB(i)). The expected number of (visible and invisible pairs) in the alignment per character of P is 1+pA'(i)+pB'(i). The above points are generalised to k strings in the next section. Note that null is a non-character, not a proper character. Similarly an edge with a null at each end represents a non-operation for a machine, introduced only to synchronise M[A] and M[B]. The parameters of G could be calculated from those of M[A] and M[B] but neither P, M[A] nor M[B] are known. In this case parameter values for G can still be inferred from A and B under the r-theory. The values embody the weighted average over all possible P and instruction sequences for M[A] and M[B]. If a constant rate of mutation is assumed, M[A]=M[B] and it is possible to infer the parameters of M[A]. For 3 or more strings it is possible to get a 3-way fix on all hypothetical strings and estimate all machines without assuming a constant rate of mutation. 4. Trees and Multiple-Alignment. Of the questions, R1-R4, over k>=2 strings, we consider R2 to be the most important with R3 a close second, to recap: R2: Are the strings related by a particular tree? R3: Are they related [under an evolutionary-tree] by a particular multiple-alignment? The topology of the tree is important but of little use without the amount and form of mutation on each edge. Strictly, R3 is too detailed to judge a tree on but an optimal alignment is often useful. As for the 2-string case, an optimal multiple-alignment is not unique in general and even sub-optimal alignments may be plausible. An MML encoding of the strings based on an answer to R2 or R3 consists of (i) the value of k, (ii) the tree topology, (iii) the parameter values for each edge and either (iv) the strings, based on an all alignments (R2) or (iv') the strings based on 1 optimal alignment (R3). The tree that best explains the strings gives the shortest message length. The difference in message lengths of 2 trees gives -log[2] of their posterior odds-ratio. The most colourless assumption is that all topologies are equally likely so part (ii) is of constant length, given k, and is only needed for comparison with the null-theory. A rooted tree can only be inferred if a (near) constant rate of mutation is assumed. If this assumption is not made, an unrooted tree is inferred. R2 and R3 can be broken down into sub-problems: S1: What is the best tree-topology? S2: What are the parameters for each edge? S3: What is the message length of a tuple? S4: What is the message length of an alignment? S5: What is the message length of the strings given the tree, considering all alignments taken together? (R2) S5': What is the message length of the strings given the tree and based on 1 optimal alignment? (R3) The answers to these sub-problems are not independent. The topology and parameter values define the tuple probabilities and thus the alignment probabilities and the plausibility of each tree. Two competing sets of answers to the sub-problems can be judged easily and objectively but finding the best answers is hard. On the other hand, algorithms for the sub-problems are largely independent and can be studied as such. A prototype program has been written for R3 based on one set of approaches to these problems. Some other possibilities have also been investigated. These are discussed below. 4.1. S1: Tree-Topology. There are U[k] unrooted tree topologies over k strings: U[2] = U[3] = 1 U[k] = 1.3.5. ... .(2k-5), k>=4 Each is considered equally probable apriori and log[2](U[k]) bits are required to state its identity: The prototype program requires the user to specify one or more trees to be evaluated. Standard distance-matrix methods could be used to propose plausible topologies. The ideal distance is proportional to time. It is thought that the parameters derived from the 2-string r-theory give more accurate pair-wise distances than a simple edit-distance. Felsenstein[12] describes how local rearrangements can be used to find at least a locally optimal tree. There is no great difficulty in allowing n-ary trees, except that there are more of them. Note that a multiple-alignment method that does not explicitly consider a tree nevertheless embodies an implicit model of relation of some kind, possibly a fixed tree. 4.2. S2: Edge Parameters. Each edge of the tree relates 2, possibly hypothetical, strings and is represented by a machine. Any realistic machine requires at least 1 parameter (~time) to specify instruction probabilities. The parameter(s) of each of the 2k-3 machines must be included in the message. The instructions are chosen from a multi-state distribution and Boulton and Wallace[6] give the calculations for the precision with which to state their probabilities. The prototype program is based on 1-state machines, each having 3 parameters - P(copy), P(change), P(indel) - and 2 degrees of freedom. No correlation between P(change) and P(indel) is assumed. MML encoding is invariant under smooth monotonic transformations of parameters so the probabilities are stated rather than being translated into a measure of time. If a constant rate of mutation were assumed, it would be sufficient to describe k-1 machines. This would permit an objective test of this much discussed hypothesis: There is a fall of approximately 50% in the cost of stating parameters but presumably a rise in the cost of stating the strings. An intermediate theory, based on a near constant rate of mutation is also possible, with different trade-offs. An iterative scheme, similar to that used for 2 strings, is used to estimate the parameter values of the machines (edges). This is described later. Actual machine parameters: copy change indel m1: 0.9 0.05 0.05 m2: 0.9 0.08 0.02 m3: 0.7 0.2 0.1 m4: 0.8 0.1 0.1 m5: 0.75 0.1 0.15 A A . . m1. .m3 . A:0.43 A:0.40 . .C:0.56 C:0.59. . m5 . . . m2. .m4 . . C C Probable hypothetical characters. Prior tuple probability = 0.00027 Estimated operations carried out: copy change indel m1: 0.43 0.57 m2: 0.57 0.43 m3: 0.40 0.60 m4: 0.59 0.41 m5: 0.94 0.06 (Probabilities less than 0.01 omitted.) Figure 7: Example, explanations for tuple ACAC. 4.3. S3: Tuple Costs. Here we consider the tree topology and parameters to be fixed. The probability of an alignment depends on the probabilities of the k-tuples of characters forming it. Each tuple can be thought of as an instruction of a k-string generation machine that generates all of the k strings in parallel. This machine is implied by the 2-string machines on the edges of the tree. The probability of a tuple can be found by assigning its characters, possibly null, to the leaves of the tree and considering all explanations of the tuple. An explanation assigns values to the internal, hypothetical characters of the tree. This defines an operation on each edge and a probability can be calculated for the explanation. The probability of the tuple is the sum of the probabilities of all explanations and its message length is -log[2] of its probability. We first consider the case where the mutation-model on 2 strings is a 1-state model. Felsenstein[12] gave the necessary algorithm to calculate the probability of a tuple of characters given the probability of a change on each edge of the tree. It takes O(k . \|alphabet\|^2) time per tuple. It can be used directly by allowing the possibility of indels. Care must be taken that only 1 insert is allowed per explanation of a tuple, as an insert begins a new tuple. The tuple's probability is computed in a post-order traversal of the tree. The probabilities of hypothetical characters and of machine operations, given the tuple, (Figure 7) are computed in a subsequent pre-order traversal. The program computes the probabilities of tuples as they are encountered and stores them in a hash-table for reuse. There is a considerable saving when n>>\|alphabet\|. Sankoff and Cedergren[22] give an algorithm that finds the single most probable explanation for a tuple; this is too detailed a hypothesis in general and is another case where a max over probabilities should be replaced with a `+'. It gives a good approximation when all mutation probabilities are small and all edges are similar. In that case the complexity of their algorithm is only O(k) per tuple. string A: - A - A - A string B: B - - B B - string C: C C C - - - Figure 8: Indel Patterns, k=3. The situation is complex when the model relating 2 strings has more than 1 state. The 3-state model, giving linear indel costs, is of particular interest. There are 2^k-2 patterns of indels possible in a tuple (Figure 8). The k-string generation machine therefore has 2^k-1 states. (2^k-1 is also the number of neighbours that each cell depends on in a k-dimensional DPA matrix.) Each tuple can therefore occur in 2^k-1 contexts with a different probability in each one. This is a serious problem unless a simplification can be found. There is some hope because the probabilities of instructions in these contexts are not independent as there are only O(k) free parameters in the tree. 4.4. S4: Alignment Message Length. The multiple-alignment part of the message states the length of the alignment and the tuples in the alignment. The former is encoded under the log^* prior. The message length of a tuple was given in the previous section. It is therefore easy to compare 2 alignments on the basis of message length; the hard part is to find a good one. 4.5. S5': Optimal Multiple-Alignment. The dynamic-programming algorithm [DPA] can be naively extended to k strings to find an optimal multiple-alignment. The k-dimensional DPA matrix has O(n^k) entries. Each entry depends on 2^k-1 neighbours. If the mutation-model has 1 state, the extension is straightforward and the time complexity is O(n^k.2^k). "Only" O(n^k-1) space is needed to calculate the value of the alignment. The n^k factor becomes painful, in both time and space, for n>=200 and k as small as 3. It is necessary to reduce the volume of the DPA matrix that is searched to make the algorithms practical and there are a number of possible approaches. Altschul and Lipman[4] devised a safe method to restrict the volume investigated for an edit-distance based on fixed tree-costs and also for other costs. (The tree-cost of a tuple is the minimum number of mutations required to give the tuple.) Pair-wise alignments are done first and used to delimit a volume that is guaranteed to contain an optimal multiple-alignment. However, no results are available to indicate how practical the method is for tree-costs. The method relies on the triangle inequality and might be applicable to MML calculations. A related heuristic is to use alignment-regions as fuzzy strings. Pair-wise alignments are performed on sibling leaf strings. A threshold (in edit-distance or in message length) is applied to limit the area of each alignment. Fuzzy strings are aligned in the obvious way until a complete multiple-alignment is obtained. The method is fast if the threshold is tight but is not guaranteed to find an optimal alignment. We have done some experiments with such an algorithm using fixed tree-costs. With a threshold of +1, up to 8 strings of 200 characters can be aligned on a DEC-5000. The model of computer makes only a small difference to the practical limits for this kind of complexity. Hein[16] gave a multiple-alignment algorithm which uses directed acyclic graphs (DAGs) to represent fuzzy ancestral strings. The algorithm is fast but an optimal alignment is not guaranteed. Fixed tree-costs are used but this is by no means necessary. The value of DAGs is that in sections where there is consensus between the strings, the effective number of neighbours of a cell in the DPA "matrix" is much reduced. Long exact matches between strings are fast to find using hash-tables or suffix trees. Heuristics for multiple-alignment based on splitting the strings on a long exact match, ideally near their centres, are clearly possible. The prototype program operates in 2 stages. During an initial post-order traversal, strings and then alignments are aligned recursively until a complete multiple-alignment is obtained: procedure RA(var Result:Alignment; T:Tree); var Left, Right :Alignment; begin if leaf(T) then StringToAlignment(T^.s, Result) else RA(Left, T^.left); RA(Right, T^.right); Align(Left, Right, Result) end if The machine parameters are held constant, at some reasonable but otherwise arbitrary values, during the initial stage. An improvement stage follows; the complete alignment is projected onto subalignments which are realigned during another post-order traversal: procedure Improve(var Algn :Alignment); var A, B :Alignment; ss :StringSet; procedure Im(T:Tree; var ss:StringSet); var ss1, ss2 :StringSet; begin if not leaf(T) then Im(T^.left, ss1); Im(T^.right, ss2); ss := ss1+ss2; Project(Algn, ss1, A, B); Align(A, B, Algn); if ss <> AllStrings then Project(Algn, ss2, A, B); Align(A, B, Algn) end if else {leaf(T)} ss := [T^.s] end if begin Im(T, ss) The machine parameters are continually reestimated during this stage. The overall process takes O(kn^2) time and global optimality is not guaranteed. The improvement stage can be iterated. More research needs to be done on alignment strategies. Note that if the mutation-model on 2 strings has 3 states, to model linear indel costs, the k-string generation machine has 2^k-1 states and each entry in the DPA matrix has 2^k-1 entries. This also adds a multiplicative factor of 2^k-1 to the time complexity. 4.6. S5: Sum over all Alignments. The part of the message that describes the strings proper is ideally based on all alignments. The number of bits required to state the strings is -log[2] of the sum of the probabilities of all alignments. This can be computed by a <+,logplus> dynamic programming algorithm. It suffers from the same combinatorial explosion as finding an optimal alignment. Fortunately most of the probability is concentrated in a narrow region which raises hopes for a fast approximate algorithm. This has been used to implement a fast, approximate r-theory algorithm for 2 strings[2]. Feng[3] has performed experiments on 3 strings which give good results. First, probability density plots are calculated for pair-wise comparisons. A "threshold" is applied to give a small alignment region. The regions are projected into the DPA cube. The r-theory is calculated over this volume. A further approach that we intend to implement is to to sum over all alignments in a region within a distance `w' of a multiple-alignment found as described in the previous section. Pairs of states must be considered at each transition if the generation machine has `s' states, So if linear indel costs are modelled, a multiplicative factor of (2^k-1)^2, or O(2^2k), is included. This is a severe problem unless a simplification or an approximation can be found. 4.7. Estimating Edge Parameters. The probability of a tuple can be calculated, as described previously, if all edge parameters are known. An iterative scheme is used to estimate unknown parameters. It is a generalisation of the scheme used for 2 strings: Initial values are assumed and used to find an optimal alignment (say) and to compute its message length and hence that of the tree hypothesis. For each tuple, the frequencies of operations for each machine are calculated. These frequencies are accumulated and used to calculate the instruction probabilities for the next iteration. The new values cannot increase the message length of the alignment but they may cause another alignment to become optimal. This must reduce the message length further so the process converges. A test program has been written to investigate the estimation process alone for the 1-state model. It assumes that a long and "perfect" alignment is known, tuples occurring with their predicted frequencies. It then estimates the true machine parameters from arbitrary starting values. Convergence is rapid. The message length usually converges to within 1% of the optimum value within 3 iterations. All parameters are reestimated simultaneously. Note that the perfect alignment would be sub-optimal! Its statistical properties could only derive from a weighted average over all This test program is also used to calculate limits for an acceptable hypothesis based on a single alignment for asymptotically long strings where the overhead of stating parameter costs and string lengths can be ignored. The critical point is where the null-theory and the alignment have equal message lengths. For example, when the machines in a tree are all identical and P(change) = P(indel) then the following results are obtained: k: 2 3 4 5 critical P(copy): 0.68 0.79 0.82 0.83 The critical values of P(copy) would be higher for short strings. They would be lower if the probabilities of all alignments were summed; recall that a single alignment is an answer to R3 not to R2. Empirical results for 2 strings are given in [1]. The estimation scheme described above has been built into the alignment program. It will also be used in any r-theory algorithm in which case weighted frequencies will be used for estimation. The programs do not correct for the "invisible" tuple which has a probability of approximately (k-2).p(del)^3. This results in an underestimate of indels by a similar amount but this is considerably less than the accuracy with which parameters can be estimated. 4 - k acgtacgtacagt - string 1 actgtacgtacgt - 2 acgtactagct - 3 accgtactgagct - 4 ((1 2)(3 4)) - tree 1 ((1 3)(2 4)) - 2 ((1 4)(2 3)) - 3 Tree 1: M.L.=108 bits * estimate of parameters: mc1: 0.88 0.02 0.09 mc2: 0.88 0.02 0.09 mc3: 0.94 0.03 0.03 mc4: 0.81 0.02 0.17 mc5: 0.71 0.18 0.11 s1 s4 . . .m1 .m4 . m5 . . . .m2 .m3 . . s2 s3 s1[1..]: ac--gtacgt-acagt s2[1..]: ac-tgtacgt-ac-gt s3[1..]: ac--gtac-t-ag-ct s4[1..]: acc-gtac-tgag-ct Tree 2: M.L.=113 bits Tree 3: M.L.=122 bits Null-Theory= 124.5 bits NB. all trees are unrooted. CPU time = 7 secs/tree (Pyramid) Figure 9: Example Run. 4.8. Null-Theory. The null-theory is that the k strings are not related. Its statement consists of (i) the value of k, (ii) the lengths of the strings and (iii) the characters in the strings. Part (iii) requires 2 bits per character for DNA if all characters are equi-probable. Part (ii) is potentially a large penalty over a tree hypothesis because the latter states only a single length - of an instruction sequence. A reasonable approach, which mitigates this penalty, is to transmit the total length of the strings, encoded under the log^* prior, followed by the set of lengths under a k-nomial prior. The second part is short if the lengths are similar. This generalises the binomial distribution used for 2 strings in [2]. The total message length of the null-theory is therefore: ML[null] = -log[2](P(k)) where total=length[1]+...+length[k] Any hypothesis that has a message length longer than that of the null-theory is not acceptable. The previous section included predictions of critical values of P(copy) for a single alignment, asymptotically long strings and some special trees. Extensive tests have not been done but when the alignment program is run on artificially generated data the results of the significance test are consistent with the predictions. In particular, random strings are always detected as being unrelated. 4.9. Results. Figure 9 shows the results of the prototype program on a small example. All 3 trees are acceptable hypotheses but tree 1 is the most plausible by a significant margin. All trees are un-rooted. Note that little benefit is gained from the hash-table for such short strings. s1: HUMPALA Human prealbumin (transthyretin) mRNA, complete cds s2: RATPALTA Rat prealbumin (transthyretin) mRNA, complete s3: MUSPALB Mouse mRNA for prealbumin s4: sheep Transthyretin cDNA s5: chicken TTR cDNA transthyretin Tree ((1 2)(3 4)): M.L.=3693 bits Tree ((1 3)(2 4)): M.L.=3661 bits Tree ((1 4)(2 3)): M.L.=3569 bits * Null-Theory = 4781 bits Tree ((1 4) (2(5 3))): M.L.=4846 bits Tree ((1 4) ((2 5)3)): M.L.=4844 bits Tree (((1 5)4) (2 3)): M.L.=4807 bits Tree ((1(4 5)) (2 3)): M.L.=4758 bits Tree (((1 4)5) (2 3)): M.L.=4750 bits* incl: 16.6 length, 67.1 params, 4655.9 alignment Null-Theory = 6134 bits s1 \|m5 s2 . \| . .m1 \| .m2 . m6 \| m7 . . . .m4 .m3 . . s4 s3 estimate of parameters: copy change indel mc1: 0.89 0.09 0.02 mc2: 0.84 0.04 0.12 mc3: 0.95 0.05 0.01 mc4: 0.88 0.08 0.05 mc5: 0.74 0.18 0.08 mc6: 0.96 0.04 0.00 mc7: 0.88 0.10 0.03 CPU time approx' 5 min/tree (sparc station). Figure 10: Transthyretin DNA/RNA sequences. Figure 10 gives results on transthyretin DNA and RNA sequences[10]. The grouping ((human sheep) (mouse rat)) is favoured on those 4 sequences. Joining chicken to the edge between the 2 pairs is narrowly the best option. There is still debate over the branching pattern of human, chimpanzee and gorilla in the evolutionary tree of the primates. Figure 11 shows results on some globin pseudo-gene sequences. They favour the grouping (gorilla (human chimp)) and are consistent with the conclusions of Holmes[18, Ch 13] on these sequences. First, the 3 trees over gorilla, human, chimpanzee and owl-monkey sequences were evaluated. Owl-monkey was a consistent outsider. The orangutan sequence was then added and 3 trees were evaluated. 5. Conclusions. We have shown how to use MML encoding to infer evolutionary-trees and multiple-alignments. We have argued that these problems are intimately connected. The posterior probability of a tree depends on the sum of the implied probabilities of all multiple-alignments. In this we agree with the sentiment, but not the detailed methods, of Hogeweg and Hesper[17] that "the alignment of sets of sequences and the construction of phyletic trees cannot be treated separately." The use of a single multiple-alignment leads to an underestimate of the posterior probability of the tree hypothesis and a bias in the parameter estimates. Nevertheless it is a reasonable approach when the strings are similar and a prototype program of this kind has been written for the 1-state model. The MML method can compute the posterior odds-ratios of competing hypotheses such as different trees. The evolutionary-tree hypothesis is explicitly built into the message. The existence of a natural null-theory leads to an in-built significance test. These remarks are qualified with `subject to the validity of the underlying model'. The MML method requires the model to be defined explicitly. In principle, 2 competing models can be compared objectively; this was done for 2 strings in [2]. 1: GGBGLOP:Gorilla beta-type globin pseudogene 2: HSBGLOP:Human beta-type globin pseudogene standard; DNA; PRI; 2153 BP 3: PTBGLOP:Chimpanzee beta-type globin pseudogene standard; DNA; PRI; 2149 BP 4: ATRB1GLOP:Owl-monkey psi beta 1 globin pseudogene. 2124 bp ds-DNA 5: ORAHBBPSE:Orangutan bases 1593 to 3704 Tree ((1 2)(3 4)): M.L.=7241 bits Tree ((1 3)(2 4)): M.L.=7236 bits Tree ((1 4)(2 3)): M.L.=7197 bits Null-Theory = 17193 bits Elapsed time ~50 min's (Sparc station) Tree (((1 2) 3)(4 5)): M.L.=8023 bits Tree (((1 3) 2)(4 5)): M.L.=8031 bits Tree ((1 (2 3))(4 5)): M.L.=7980 bits* *incl: 18.7 length, 98.8 params, 7852 alignment Null-Theory = 21426 bits Elapsed time ~50 min's (Sparc station) estimate of parameters: copy change indel mc1: 0.989 0.011 0.000 mc2: 0.994 0.004 0.003 mc3: 0.988 0.012 0.001 mc4: 0.868 0.087 0.045 mc5: 0.969 0.016 0.015 mc6: 0.996 0.001 0.003 mc7: 0.985 0.011 0.005 m2 m6 \| m7 m4 \| \| \|m3 \|m5 \| \| s3:PTB \| Figure 11: Primate Globin Pseudo-genes. The biggest doubts about the model used are as follows. We have not, as yet, modelled different transition and transversion rates but this would be straightforward. It seems to be difficult to bring a multi-state model of mutation properly into multiple-alignment. Large mutation events, such as block transpositions, duplications and reversals, could be modelled easily but finding an optimal "alignment" would become even harder. Finally, what is completely lacking from our method and from most if not all sequence analysis programs is any model of the pressure of selection. The program discussed above is written in Pascal and is available, free for non-profit, non-classified teaching and research purposes, by email (only) from: lloyd at bruce.cs.monash.edu.au. The alignment programs for 2 strings, based on 1, 3 and 5-state models of relation and discussed in [2] are written in C and are also available. A good work-station with floating point hardware is the minimum requirement for their use. Acknowledgment: This work was partially supported by Australian Research Council grant A49030439. 6. References. • [1] Allison L. & C. N. Yee. Minimum message length encoding and the comparison of macro-molecules. Bull. Math. Biol. 52(3) 431-453 1990. • [2] Allison L., C. S. Wallace & C. N. Yee. Inductive inference over macro-molecules. AAAI Spring Symposium on A.I. & Mol. Biol. Stanford March 1990 (extended abstract), and TR 90/148 Dept. Comp. Sci. Monash University 1990 [HTML] (in full). • [3] Allison L. & Du Xiaofeng. Relating three strings by minimum message length encoding (abstract). Genes, Proteins and Computers, Chester, April 1990. • [4] Altschul S. F. & D. J. Lipman. Trees, stars and multiple biological sequence alignment. SIAM J. Appl. Math. 49(1) 197-209 1989. • [5] Bishop M. J. & E. A. Thompson. Maximum likelihood alignment of DNA sequences. J. Mol. Biol. 90 159-165 1986. • [6] Boulton D. M. & C. S. Wallace. The information content of a multistate distribution. J. Theor. Biol. 23 269-278 1969. • [7] Boulton D. M. & C. S. Wallace. A program for numerical classification. Comp. J. 13 63-69 1970. • [8] Boulton D. M. & C. S. Wallace. An information measure for hierarchic classification. Comp. J. 16 254-261 1973. • [9] Cheeseman P. & B. Kanefsky. Evolutionary tree reconstruction (extended abstract). AAAI Spring Symposium on the Theory and Application of Minimal-Length Encoding Stanford 35-39 March 1990. • [10] Duan W., M. G. Achen, S. J. Richardson, M. C. Lawrence, R. E. H. Wettenhall, A. Jaworowski & G. Schreiber. Isolation, characterization, cDNA cloning and gene expression of an avian transthyretin: implications for the evolution of the structure and function of transthyretin in vertebrates. Euro. J. Biochem. to appear. • [11] Felsenstein J. Evolutionary trees from DNA sequences: a maximum likelihood approach. J. Mol. Evol. 17 368-376 1981. • [12] Felsenstein J. Inferring evolutionary trees from DNA sequences. Statistical Analysis of DNA Sequence Data. B. S. Weir (ed) Marcel Dekker 133-150 1983. • [13] Friday A. Quantitative aspects of the estimation of evolutionary trees. Folia Primatol. 53 221-234 1989. • [14] Gotoh O. An improved algorithm for matching biological sequences. J. Mol. Biol. 162 705-708 1982. • [15] Gotoh O. Optimal sequence alignment allowing for long gaps. Bull. Math. Biol. 52(3) 359-373 1990. • [16] Hein J. A new method that simultaneously reconstructs ancestral sequences for any number of homologous sequences when the phylogeny is given. Mol. Biol. Evol. 6 649-668 1989. • [17] Hogeweg P. & B. Hesper. The alignment of sets of sequences and the construction of phyletic trees: an integrated method. J. Mol. Evol. 20 175-186 1984. • [18] Holmes E. C. Pattern and Process in the Molecular Evolution of the Order Primates. PhD Thesis, Cambridge University, 1989. • [19] Miller W. & E. W. Myers. Sequence comparison with concave weighting functions. Bull. Math. Biol. 50(2) 97-120 1988. • [20] Milosavljevic A. D. Categorization of macro-molecular sequences by minimum length encoding. University of California at Santa Cruz, UCSC CRL 90-41 1990. • [21] Reichert T.A., D. N. Cohen & K. C. Wong. An application of information theory to genetic mutations and the matching of polypeptide sequences. J. Theor. Biol. 42 245-261 1973. • [22] Sankoff D. & R. J. Cedergren. Simultaneous comparison of three or more sequences related by a tree. Time Warps, String Edits and Macromolecules: The Theory and Practice of Sequence Comparison. D. Sankoff & J. B. Kruskal (eds) Addison Wesley 253-263 1983. • [23] Sellers P. H. On the theory and computation of evolutionary distances. SIAM J. Appl. Math. 26(4) 787-793 1974. • [24] Wallace C. S. & D. M. Boulton. An information measure for classification. Comp. J. 11 185-195 1968. [HTML] • [25] Wallace C. S. & P. R. Freeman. Estimation and inference by compact coding. J. Royal Stat. Soc. B 49(3) 240-265 1987. [paper] Also see [Bioinformatics] and [MML].	{"url":"https://allisons.org/ll/Publications/1992.HICSS_25/","timestamp":"2024-11-12T06:43:57Z","content_type":"text/html","content_length":"57452","record_id":"<urn:uuid:973f2ac8-eedf-4ff4-88c6-94c039247656>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00411.warc.gz"}
Haskell for all Rewrite rules are a powerful tool that you can use to optimize Haskell code without breaking backwards compatibility. This post will illustrate how I use rewrite rules to implement a form of shortcut fusion for my pipes stream programming library. I compare pipes performance before and after adding shortcut fusion and I also compare the peformance of pipes-4.0.2 vs. conduit-1.0.10 and This post also includes a small introduction to Haskell's rewrite rule system for those who are interested in optimizing their own libraries. Edit: This post originally used the term "stream fusion", but Duncan Coutts informed me that the more appropriate term for this is probably "short-cut fusion". Rule syntax The following rewrite rule from pipes demonstrates the syntax for defining optimization rules. {-# RULES "for p yield" forall p . for p yield = p #-} -- ^^^^^^^^^^^^^ ^ ^^^^^^^^^^^^^^^ -- Label \| Substitution rule -- \| -- `p` can be anything -+ All rewrite rules are substitution rules, meaning that they instruct the compiler to replace anything that matches the left-hand side of the rule with the right-hand side. The above rewrite rule says to always replace for p yield with p, no matter what p is, as long as everything type-checks before and after substitution. Rewrite rules are typically used to substitute code with equivalent code of greater efficiency. In the above example, for p yield is a for loop that re-yields every element of p. The re-yielded stream is behaviorally indistinguishable from the original stream, p, because all that for p yield does is replace every yield in p with yet another yield. However, while both sides of the equation behave identically their efficiency is not the same; the left-hand side is less efficient because for loops are not free. Rewrite rules are not checked for correctness. The only thing the compiler does is verify that the left-hand side and right-hand side of the equation both type-check. The programmer who creates the rewrite rule is responsible for proving that the substitution preserves the original behavior of the program. In fact, rewrite rules can be used to rewrite terms from other Haskell libraries without limitation. For this reason, modules with rewrite rules are automatically marked unsafe by Safe Haskell and they must be explicitly marked TrustWorthy to be used by code marked Safe. You can verify a rewrite rule is safe using equational reasoning. If you can reach the right-hand side of the equation from the left-hand side using valid code substitutions, then you can prove (with some caveats) that the two sides of the equation are functionally identical. pipes includes a complete set of proofs for its rewrite rules for this purpose. For example, the above rewrite rule is proven here, where (//>) is an infix synonym for for and respond is a synonym for yield: for = (//>) yield = respond This means that equational reasoning is useful for more than just proving program correctness. You can also use derived equations to optimize your program , assuming that you already know which side of the equation is more efficient. Rewrite rules have a significant limitation: we cannot possibly anticipate every possible expression that downstream users might build using our libraries. So how can we optimize as much code as possible without an excessive proliferation of rewrite rules? I anecdotally observe that equations inspired from category theory prove to be highly versatile optimizations that fit within a small number of rules. These equations include: • Category laws • Functor laws • Natural transformation laws (i.e. free theorems) The first half of the shortcut fusion optimization consists of three monad laws, which are a special case of category laws. For those new to Haskell, the three monad laws are: -- Left Identity return x >>= f = f x -- Right Identity m >>= return = m -- Associativity (m >>= f) >>= g = m >>= (\x -> f x >>= g) If you take these three laws and replace (>>=)/return with for/yield (and rename m to p, for 'p'ipe), you get the following "for loop laws": -- Looping over a yield simplifies to function application for (yield x) f = f x -- Re-yielding every element returns the original stream for p yield = p -- You can transform two passes over a stream into a single pass for (for p f) g = for p (\x -> for (f x) g) This analogy to the monad laws is precise because for and yield are actually (>>=) and return for the ListT monad when you newtype them appropriately, and they really form a Monad in the Haskell sense of the word. What's amazing is that these monad laws also double as shortcut fusion optimizations when we convert them to rewrite rules. We already encountered the second law as our first rewrite rule, but the other two laws are useful rewrite rules, too: {-# RULES "for (yield x) f" forall x f . for (yield x) f = f x ; "for (for p f) g" forall p f g . for (for p f) g = for p (\x -> for (f x) g) Note that the RULES pragma lets you group multiple rules together, as long as you separate them by semicolons. Also, there is no requirement that the rule label must match the left-hand side of the equation, but I use this convention since I'm bad at naming rewrite rules. This labeling convention also helps when diagnosing which rules fired (see below) without having to consult the original rule definitions. Free theorems These three rewrite rules alone do not suffice to optimize most pipes code. The reason why is that most idiomatic pipes code is not written in terms of for loops. For example, consider the map function from Pipes.Prelude: map :: Monad m => (a -> b) -> Pipe a b m r map f = for cat (\x -> yield (f x)) The idiomatic way to transform a pipe's output is to compose the map pipe downstream: p >-> map f We can't optimize this using our shortcut fusion rewrite rules unless we rewrite the above code to the equivalent for loop: for p (\y -> yield (f y)) In other words, we require the following theorem: p >-> map f = for p (\y -> yield (f y)) This is actually a special case of the following "free theorem": -- Exercise: Derive the previous equation from this one p1 >-> for p2 (\y -> yield (f y)) = for (p1 >-> p2) (\y -> yield (f y)) A free theorem is an equation that you can prove solely from studying the types of all terms involved. I will omit the proof of this free theorem for now, but I will discuss how to derive free theorems in detail in a follow-up post. For now, just assume that the above equations are correct, as codified by the following rewrite rule: {-# RULES "p >-> map f" . forall p f . p >-> map f = for p (\y -> yield (f y)) With this rewrite rule the compiler can begin to implement simple map fusion. To see why, we'll compose two map pipes and then pretend that we are the compiler, applying rewrite rules at every opportunity. Every time we apply a rewrite rule we will refer to the rule by its corresponding string label: map f >-> map g -- "p >-> map f" rule fired = for (map f) (\y -> yield (g y)) -- Definition of `map` = for (for cat (\x -> yield (f x))) (\y -> yield (g y)) -- "for (for p f) g" rule fired = for cat (\x -> for (yield (f x)) (\y -> yield (g y))) -- "for (yield x) f" rule fired = for cat (\x -> yield (g (f x))) This is identical to a single map pass, which we can prove by equational reasoning: for cat (\x -> yield (g (f x))) -- Definition of `(.)`, in reverse = for cat (\x -> yield ((g . f) x)) -- Definition of `map`, in reverse = map (g . f) So those rewrite rules sufficed to fuse the two map passes into a single pass. You don't have to take my word for it, though. For example, let's say that we want to prove that these rewrite rules fire for the following sample program, which increments, doubles, and then discards every number from 1 to 100000000: -- map-fusion.hs import Pipes import qualified Pipes.Prelude as P main = runEffect $ for (each [1..10^8] >-> P.map (+1) >-> P.map (*2)) discard The -ddump-rule-firings flag will output every rewrite rule that fires during compilation, identifying each rule with the string label accompanying the rule: $ ghc -O2 -ddump-rule-firings map-fusion.hs [1 of 1] Compiling Main ( test.hs, test.o ) Rule fired: p >-> map f Rule fired: for (for p f) g Rule fired: for (yield x) f I've highlighted the rule firings that correspond to map fusion, although there are many other rewrite rules that fire (including more shortcut fusion rule firings). Shortcut fusion We don't have to limit ourselves to just fusing maps. Many pipes in Pipes.Prelude has an associated free theorem that rewrites pipe composition into an equivalent for loop. After these rewrites, the "for loop laws" go to town on the pipeline and fuse it into a single pass. For example, the filter pipe has a rewrite rule similar to map: {-# RULES "p >-> filter pred" forall p pred . p >-> filter pred = for p (\y -> when (pred y) (yield y)) So if we combine map and filter in a pipeline, they will also fuse into a single pass: p >-> map f >-> filter pred -- "p >-> map f" rule fires for p (\x -> yield (f x)) >-> filter pred -- "p >-> filter pred" rule fires for (for p (\x -> yield (f x))) (\y -> when (pred y) (yield y)) -- "for (for p f) g" rule fires for p (\x -> for (yield (f x)) (\y -> when (pred y) (yield y))) -- for (yield x) f" rule fires for p (\x -> let y = f x in when (pred y) (yield y)) This is the kind of single pass loop we might have written by hand if we were pipes experts, but thanks to rewrite rules we can write high-level, composable code and let the library automatically rewrite it into efficient and tight loops. Note that not all pipes are fusible in this way. For example the take pipe cannot be fused in this way because it cannot be rewritten in terms of a for loop. These rewrite rules make fusible pipe stages essentially free. To illustrate this I've set up a criterion benchmark testing running time as a function of the number of map stages in a pipeline: import Criterion.Main import Data.Functor.Identity (runIdentity) import Pipes import qualified Pipes.Prelude as P n :: Int n = 10^6 main = defaultMain [ bench' "1 stage " $ \n -> each [1..n] >-> P.map (+1) , bench' "2 stages" $ \n -> each [1..n] >-> P.map (+1) >-> P.map (+1) , bench' "3 stages" $ \n -> each [1..n] >-> P.map (+1) >-> P.map (+1) >-> P.map (+1) , bench' "4 stages" $ \n -> each [1..n] >-> P.map (+1) >-> P.map (+1) >-> P.map (+1) >-> P.map (+1) bench' label f = bench label $ whnf (\n -> runIdentity $ runEffect $ for (f n) discard) (10^5 :: Int) Before shortcut fusion (i.e. pipes-4.0.0), the running time scales linearly with the number of map stages: warming up estimating clock resolution... mean is 24.53411 ns (20480001 iterations) found 80923 outliers among 20479999 samples (0.4%) 32461 (0.2%) high severe estimating cost of a clock call... mean is 23.89897 ns (1 iterations) benchmarking 1 stage mean: 4.480548 ms, lb 4.477734 ms, ub 4.485978 ms, ci 0.950 std dev: 19.42991 us, lb 12.11399 us, ub 35.90046 us, ci 0.950 benchmarking 2 stages mean: 6.304547 ms, lb 6.301067 ms, ub 6.310991 ms, ci 0.950 std dev: 23.60979 us, lb 14.01610 us, ub 37.63093 us, ci 0.950 benchmarking 3 stages mean: 10.60818 ms, lb 10.59948 ms, ub 10.62583 ms, ci 0.950 std dev: 61.05200 us, lb 34.79662 us, ub 102.5613 us, ci 0.950 benchmarking 4 stages mean: 13.74065 ms, lb 13.73252 ms, ub 13.76065 ms, ci 0.950 std dev: 61.13291 us, lb 29.60977 us, ub 123.3071 us, ci 0.950 Stream fusion (added in pipes-4.0.1) makes additional map stages essentially free: warming up estimating clock resolution... mean is 24.99854 ns (20480001 iterations) found 1864216 outliers among 20479999 samples (9.1%) 515889 (2.5%) high mild 1348320 (6.6%) high severe estimating cost of a clock call... mean is 23.54777 ns (1 iterations) benchmarking 1 stage mean: 2.427082 ms, lb 2.425264 ms, ub 2.430500 ms, ci 0.950 std dev: 12.43505 us, lb 7.564554 us, ub 20.11641 us, ci 0.950 benchmarking 2 stages mean: 2.374217 ms, lb 2.373302 ms, ub 2.375435 ms, ci 0.950 std dev: 5.394149 us, lb 4.270983 us, ub 8.407879 us, ci 0.950 benchmarking 3 stages mean: 2.438948 ms, lb 2.436673 ms, ub 2.443006 ms, ci 0.950 std dev: 15.11984 us, lb 9.602960 us, ub 23.05668 us, ci 0.950 benchmarking 4 stages mean: 2.372556 ms, lb 2.371644 ms, ub 2.373949 ms, ci 0.950 std dev: 5.684231 us, lb 3.955916 us, ub 9.040744 us, ci 0.950 In fact, once you have just two stages in your pipeline, pipes greatly outperforms conduit and breaks roughly even with io-streams. To show this I've written up a benchmark comparing pipes performance against these libraries for both pure loops and loops that are slightly IO-bound (by writing to /dev/null): import Criterion.Main import Data.Functor.Identity (runIdentity) import qualified System.IO as IO import Data.Conduit import qualified Data.Conduit.List as C import Pipes import qualified Pipes.Prelude as P import qualified System.IO.Streams as S criterion :: Int -> IO () criterion n = IO.withFile "/dev/null" IO.WriteMode $ \h -> [ bgroup "pure" [ bench "pipes" $ whnf (runIdentity . pipes) n , bench "conduit" $ whnf (runIdentity . conduit) n , bench "io-streams" $ nfIO (iostreams n) , bgroup "io" [ bench "pipes" $ nfIO (pipesIO h n) , bench "conduit" $ nfIO (conduitIO h n) , bench "iostreams" $ nfIO (iostreamsIO h n) pipes :: Monad m => Int -> m () pipes n = runEffect $ for (each [1..n] >-> P.map (+1) >-> P.filter even) discard conduit :: Monad m => Int -> m () conduit n = C.enumFromTo 1 n $= C.map (+1) $= C.filter even $$ C.sinkNull iostreams :: Int -> IO () iostreams n = do is0 <- S.fromList [1..n] is1 <- S.map (+1) is0 is2 <- S.filter even is1 S.skipToEof is2 pipesIO :: IO.Handle -> Int -> IO () pipesIO h n = runEffect $ each [1..n] >-> P.map (+1) >-> P.filter even >-> P.map show >-> P.toHandle h conduitIO :: IO.Handle -> Int -> IO () conduitIO h n = C.enumFromTo 1 n $= C.map (+1) $= C.filter even $= C.map show $$ C.mapM_ (IO.hPutStrLn h) iostreamsIO :: IO.Handle -> Int -> IO () iostreamsIO h n = do is0 <- S.fromList [1..n] is1 <- S.map (+1) is0 is2 <- S.filter even is1 is3 <- S.map show is2 os <- S.makeOutputStream $ \ma -> case ma of Just str -> IO.hPutStrLn h str _ -> return () S.connect is3 os main = criterion (10^6) The benchmarks place pipes neck-and-neck with io-streams on pure loops and 10% slower on slightly IO-bound code. Both libraries perform faster than conduit: warming up estimating clock resolution... mean is 24.50726 ns (20480001 iterations) found 117040 outliers among 20479999 samples (0.6%) 45158 (0.2%) high severe estimating cost of a clock call... mean is 23.89208 ns (1 iterations) benchmarking pure/pipes mean: 24.04860 ms, lb 24.02136 ms, ub 24.10872 ms, ci 0.950 std dev: 197.3707 us, lb 91.05894 us, ub 335.2267 us, ci 0.950 benchmarking pure/conduit mean: 172.8454 ms, lb 172.6317 ms, ub 173.1824 ms, ci 0.950 std dev: 1.361239 ms, lb 952.1500 us, ub 1.976641 ms, ci 0.950 benchmarking pure/io-streams mean: 24.16426 ms, lb 24.12789 ms, ub 24.22919 ms, ci 0.950 std dev: 242.5173 us, lb 153.9087 us, ub 362.4092 us, ci 0.950 benchmarking io/pipes mean: 267.7021 ms, lb 267.1789 ms, ub 268.4542 ms, ci 0.950 std dev: 3.189998 ms, lb 2.370387 ms, ub 4.392541 ms, ci 0.950 benchmarking io/conduit mean: 310.3034 ms, lb 309.8225 ms, ub 310.9444 ms, ci 0.950 std dev: 2.827841 ms, lb 2.194127 ms, ub 3.655390 ms, ci 0.950 benchmarking io/iostreams mean: 239.6211 ms, lb 239.2072 ms, ub 240.2354 ms, ci 0.950 std dev: 2.564995 ms, lb 1.879984 ms, ub 3.442018 ms, ci 0.950 I hypothesize that pipes performs slightly slower on IO compared to io-streams because of the cost of calling lift, whereas iostreams operates directly within the IO monad at all times. These benchmarks should be taken with a grain of salt. All three libraries are most frequently used in strongly IO-bound scenarios, where the overhead of each library is pretty much negligible. However, this still illustrates how big of an impact shortcut fusion can have on pure code paths. pipes is a stream programming library with a strong emphasis on theory and the library's contract with the user is a set of laws inspired by category theory. My original motivation behind proving these laws was to fulfill the contract, but I only later realized that the for loop laws doubled as fortuitous shortcut fusion optimizations. This is a recurring motif in Haskell: thinking mathematically pays large dividends. For this reason I like to think of Haskell as applied category theory: I find that many topics I learn from category theory directly improve my Haskell code. This post shows one example of this phenomenon, where shortcut fusion naturally falls out of the monad laws for ListT.	{"url":"https://www.haskellforall.com/2014/01/?m=0","timestamp":"2024-11-03T07:02:02Z","content_type":"application/xhtml+xml","content_length":"89510","record_id":"<urn:uuid:5c1d8269-fd09-49d6-8652-f967bea81864>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00680.warc.gz"}
PDL::DSP::Windows - Window functions for signal processing use PDL; use PDL::DSP::Windows 'window'; # Get a piddle with a window's samples with the helper my $samples = window( 10, tukey => { params => .5 }); # Or construct a window object with the same parameters my $window = PDL::DSP::Windows->new( 10, tukey => { params => .5 }); # These two are equivalent $samples = $window->samples; # The window object gives access to additional methods print $window->coherent_gain, "\n"; $window->plot; # Requires PDL::Graphics::Gnuplot This module provides symmetric and periodic (DFT-symmetric) window functions for use in filtering and spectral analysis. It provides a high-level access subroutine "window". This functional interface is sufficient for getting the window samples. For analysis and plotting, etc. an object oriented interface is provided. The functional subroutines must be either explicitly exported, or fully qualified. In this document, the word function refers only to the mathematical window functions, while the word subroutine is used to describe code. Window functions are also known as apodization functions or tapering functions. In this module, each of these functions maps a sequence of $N integers to values called a samples. (To confuse matters, the word sample also has other meanings when describing window functions.) The functions are often named for authors of journal articles. Be aware that across the literature and software, some functions referred to by several different names, and some names refer to several different functions. As a result, the choice of window names is somewhat arbitrary. The "kaiser($N,$beta)" window function requires PDL::GSLSF::BESSEL. The "dpss($N,$beta)" window function requires PDL::LinearAlgebra. But the remaining window functions may be used if these modules are not installed. The most common and easiest usage of this module is indirect, via some higher-level filtering interface, such as PDL::DSP::Fir::Simple. The next easiest usage is to return a pdl of real-space samples with the subroutine "window". Finally, for analyzing window functions, object methods, such as "new", "plot", "plot_freq" are provided. In the following, first the functional interface (non-object oriented) is described in "'FUNCTIONAL INTERFACE'". Next, the object methods are described in "METHODS". Next the low-level subroutines returning samples for each named window are described in "'WINDOW FUNCTIONS'". Finally, some support routines that may be of interest are described in "'AUXILIARY SUBROUTINES'". $win = window({ OPTIONS }); $win = window( $N, { OPTIONS }); $win = window( $N, $name, { OPTIONS }); $win = window( $N, $name, $params, { OPTIONS }); $win = window( $N, $name, $params, $periodic ); Returns an $N point window of type $name. The arguments may be passed positionally in the order $N, $name, $params, $periodic, or they may be passed by name in the hash OPTIONS. # Each of the following return a 100 point symmetric hamming window. $win = window(100); $win = window( 100, 'hamming' ); $win = window( 100, { name => 'hamming' }); $win = window({ N => 100, name => 'hamming' }); # Each of the following returns a 100 point symmetric hann window. $win = window( 100, 'hann' ); $win = window( 100, { name => 'hann' }); # Returns a 100 point periodic hann window. $win = window( 100, 'hann', { periodic => 1 }); # Returns a 100 point symmetric Kaiser window with alpha = 2. $win = window( 100, 'kaiser', { params => 2 }); The options follow default PDL::Options rules. They may be abbreviated, and are case-insensitive. (string) name of window function. Default: hamming. This selects one of the window functions listed below. Note that the suffix '_per', for periodic, may be ommitted. It is specified with the option periodic => 1 ref to array of parameter or parameters for the window-function subroutine. Only some window-function subroutines take parameters. If the subroutine takes a single parameter, it may be given either as a number, or a list of one number. For example 3 or [3]. number of points in window function (the same as the order of the filter). As of 0.102, throws exception if the value for N is undefined or zero. If value is true, return a periodic rather than a symmetric window function. Defaults to false, meaning "symmetric". list_windows STR list_windows prints the names all of the available windows. list_windows STR prints only the names of windows matching the string STR. my $win = PDL::DSP::Windows->new(ARGS); Create an instance of a window object. If ARGS are given, the instance is initialized. ARGS are interpreted in exactly the same way as arguments for the "window" subroutine. For example: my $win1 = PDL::DSP::Windows->new( 8, 'hann' ); my $win2 = PDL::DSP::Windows->new({ N => 8, name => 'hann' }); Initialize (or reinitialize) a window object. ARGS are interpreted in exactly the same way as arguments for the "window" subroutine. As of 0.102, throws exception if the value for N is undefined or For example: my $win = PDL::DSP::Windows->new( 8, 'hann' ); $win->init( 10, 'hamming' ); Generate and return a reference to the piddle of $N samples for the window $win. This is the real-space representation of the window. The samples are stored in the object $win, but are regenerated every time samples is invoked. See the method "get_samples" below. For example: my $win = PDL::DSP::Windows->new( 8, 'hann' ); print $win->samples, "\n"; Generate and return a reference to the piddle of the modulus of the fourier transform of the samples for the window $win. These values are stored in the object $win, but are regenerated every time modfreqs is invoked. See the method "get_modfreqs" below. This sets the minimum number of frequency bins. Defaults to 1000. If necessary, the piddle of window samples are padded with zeroes before the fourier transform is performed. my $windata = $win->get('samples'); Get an attribute (or list of attributes) of the window $win. If attribute samples is requested, then the samples are created with the method "samples" if they don't exist. For example: my $win = PDL::DSP::Windows->new( 8, 'hann' ); print $win->get('samples'), "\n"; my $windata = $win->get_samples; Return a reference to the pdl of samples for the Window instance $win. The samples will be generated with the method "samples" if and only if they have not yet been generated. my $winfreqs = $win->get_modfreqs; my $winfreqs = $win->get_modfreqs(OPTS); Return a reference to the pdl of the frequency response (modulus of the DFT) for the Window instance $win. Options passed as a hash reference will be passed to the "modfreqs". The data are created with "modfreqs" if they don't exist. The data are also created even if they already exist if options are supplied. Otherwise the cached data are returned. This sets the minimum number of frequency bins. See "modfreqs". Defaults to 1000. my $params = $win->get_params; Create a new array containing the parameter values for the instance $win and return a reference to the array. Note that not all window types take parameters. print $win->get_name, "\n"; Return a name suitable for printing associated with the window $win. This is something like the name used in the documentation for the particular window function. This is static data and does not depend on the instance. Plot the samples. Currently, only PDL::Graphics::Gnuplot is supported. The default display type is used. Can be called like this Or this $win->plot_freq({ ordinate => ORDINATE }); Plot the frequency response (magnitude of the DFT of the window samples). The response is plotted in dB, and the frequency (by default) as a fraction of the Nyquist frequency. Currently, only PDL::Graphics::Gnuplot is supported. The default display type is used. This sets the units of frequency of the co-ordinate axis. COORD must be one of nyquist, for fraction of the nyquist frequency (range -1, 1); sample, for fraction of the sampling frequncy (range -0.5, 0.5); or bin for frequency bin number (range 0, $N - 1). The default value is nyquist. This sets the minimum number of frequency bins. See "get_modfreqs". Defaults to 1000. Compute and return the equivalent noise bandwidth of the window. Compute and return the coherent gain (the dc gain) of the window. This is just the average of the samples. Compute and return the processing gain (the dc gain) of the window. This is just the multiplicative inverse of the enbw. Compute and return the scalloping loss of the window. These window-function subroutines return a pdl of $N samples. For most windows, there are a symmetric and a periodic version. The symmetric versions are functions of $N points, uniformly spaced, and taking values from x_lo through x_hi. Here, a periodic function of $N points is equivalent to its symmetric counterpart of $N + 1 points, with the final point omitted. The name of a periodic window-function subroutine is the same as that for the corresponding symmetric function, except it has the suffix _per. The descriptions below describe the symmetric version of each window. The term 'Blackman-Harris family' is meant to include the Hamming family and the Blackman family. These are functions of sums of cosines. Unless otherwise noted, the arguments in the cosines of all symmetric window functions are multiples of $N numbers uniformly spaced from 0 through 2π. Symmetric window functions The Bartlett window. (Ref 1). Another name for this window is the fejer window. This window is defined by 1 - abs(arr) where the points in arr range from -1 through 1. See also triangular. The Bartlett-Hann window. Another name for this window is the Modified Bartlett-Hann window. This window is defined by 0.62 - 0.48 * abs(arr) + 0.38 * arr1 where the points in arr range from -1/2 through 1/2, and arr1 are the cos of points ranging from -PI through PI. The 'classic' Blackman window. (Ref 1). One of the Blackman-Harris family, with coefficients a0 = 0.42 a1 = 0.5 a2 = 0.08 The Blackman-Harris (bnh) window. An improved version of the 3-term Blackman-Harris window given by Nuttall (Ref 2, p. 89). One of the Blackman-Harris family, with coefficients a0 = 0.4243801 a1 = 0.4973406 a2 = 0.0782793 The 'exact' Blackman window. (Ref 1). One of the Blackman-Harris family, with coefficients a0 = 0.426590713671539 a1 = 0.496560619088564 a2 = 0.0768486672398968 The General classic Blackman window. A single parameter family of the 3-term Blackman window. This window is defined by my $cx = arr; .5 - $alpha + ($cx * (-.5 + $cx * $alpha)); where the points in arr are the cos of points ranging from 0 through 2PI. The general form of the Blackman family. One of the Blackman-Harris family, with coefficients a0 = $a0 a1 = $a1 a2 = $a2 The general 4-term Blackman-Harris window. One of the Blackman-Harris family, with coefficients a0 = $a0 a1 = $a1 a2 = $a2 a3 = $a3 The general 5-term Blackman-Harris window. One of the Blackman-Harris family, with coefficients a0 = $a0 a1 = $a1 a2 = $a2 a3 = $a3 a4 = $a4 The Blackman-Harris window. (Ref 1). One of the Blackman-Harris family, with coefficients a0 = 0.422323 a1 = 0.49755 a2 = 0.07922 Another name for this window is the Minimum three term (sample) Blackman-Harris window. The minimum (sidelobe) four term Blackman-Harris window. (Ref 1). One of the Blackman-Harris family, with coefficients a0 = 0.35875 a1 = 0.48829 a2 = 0.14128a3 = 0.01168 Another name for this window is the Blackman-Harris window. The Blackman-Nuttall window. One of the Blackman-Harris family, with coefficients a0 = 0.3635819 a1 = 0.4891775 a2 = 0.1365995 a3 = 0.0106411 The Bohman window. (Ref 1). This window is defined by my $x = abs(arr); (1 - $x) * cos(PI * $x) + (1 / PI) * sin(PI * $x) where the points in arr range from -1 through 1. The Cauchy window. (Ref 1). Other names for this window are: Abel, Poisson. This window is defined by 1 / (1 + (arr * $alpha) 2) where the points in arr range from -1 through 1. The Chebyshev window. The frequency response of this window has $at dB of attenuation in the stop-band. Another name for this window is the Dolph-Chebyshev window. No periodic version of this window is defined. This routine gives the same result as the routine chebwin in Octave 3.6.2. The Cos_alpha window. (Ref 1). Another name for this window is the Power-of-cosine window. This window is defined by arr $alpha where the points in arr are the sin of points ranging from 0 through PI. The Cosine window. Another name for this window is the sine window. This window is defined by where the points in arr are the sin of points ranging from 0 through PI. The Digital Prolate Spheroidal Sequence (DPSS) window. The parameter $beta is the half-width of the mainlobe, measured in frequency bins. This window maximizes the power in the mainlobe for given $N and $beta. Another name for this window is the sleppian window. The Exponential window. This window is defined by 2 ** (1 - abs arr) - 1 where the points in arr range from -1 through 1. The flat top window. One of the Blackman-Harris family, with coefficients a0 = 0.21557895 a1 = 0.41663158 a2 = 0.277263158 a3 = 0.083578947 a4 = 0.006947368 The Gaussian window. (Ref 1). Another name for this window is the Weierstrass window. This window is defined by exp (-0.5 * ($beta * arr )*2), where the points in arr range from -1 through 1. The Hamming window. (Ref 1). One of the Blackman-Harris family, with coefficients a0 = 0.54 a1 = 0.46 The 'exact' Hamming window. (Ref 1). One of the Blackman-Harris family, with coefficients a0 = 0.53836 a1 = 0.46164 The general Hamming window. (Ref 1). One of the Blackman-Harris family, with coefficients a0 = $a a1 = (1-$a) The Hann window. (Ref 1). One of the Blackman-Harris family, with coefficients a0 = 0.5 a1 = 0.5 Another name for this window is the hanning window. See also hann_matlab. The Hann (matlab) window. Equivalent to the Hann window of N+2 points, with the endpoints (which are both zero) removed. No periodic version of this window is defined. This window is defined by 0.5 - 0.5 arr where the points in arr are the cosine of points ranging from 2PI/($N+1) through 2PI$N/($N+1). This routine gives the same result as the routine hanning in Matlab. See also hann. The Hann-Poisson window. (Ref 1). This window is defined by 0.5 (1 + arr1) * exp (-$alpha * abs arr) where the points in arr range from -1 through 1, and arr1 are the cos of points ranging from -PI through PI. The Kaiser window. (Ref 1). The parameter $beta is the approximate half-width of the mainlobe, measured in frequency bins. Another name for this window is the Kaiser-Bessel window. This window is defined by $beta = PI; my @n = gsl_sf_bessel_In($beta sqrt(1 - arr ** 2), 0); my @d = gsl_sf_bessel_In($beta, 0); $n[0] / $d[0]; where the points in arr range from -1 through 1. The Lanczos window. Another name for this window is the sinc window. This window is defined by my $x = PI * arr; my $res = sin($x) / $x; my $mid; $mid = int($N / 2), $res->slice($mid) .= 1 if $N % 2; where the points in arr range from -1 through 1. The Nuttall window. One of the Blackman-Harris family, with coefficients a0 = 0.3635819 a1 = 0.4891775 a2 = 0.1365995 a3 = 0.0106411 See also nuttall1. The Nuttall (v1) window. A window referred to as the Nuttall window. One of the Blackman-Harris family, with coefficients a0 = 0.355768 a1 = 0.487396 a2 = 0.144232 a3 = 0.012604 This routine gives the same result as the routine nuttallwin in Octave 3.6.2. See also nuttall. The Parzen window. (Ref 1). Other names for this window are: Jackson, Valle-Poussin. This function disagrees with the Octave subroutine parzenwin, but agrees with Ref. 1. See also parzen_octave. The Parzen window. No periodic version of this window is defined. This routine gives the same result as the routine parzenwin in Octave 3.6.2. See also parzen. The Poisson window. (Ref 1). This window is defined by exp(-$alpha * abs(arr)) where the points in arr range from -1 through 1. The Rectangular window. (Ref 1). Other names for this window are: dirichlet, boxcar. The Triangular window. This window is defined by 1 - abs(arr) where the points in arr range from -$N/($N-1) through $N/($N-1). See also bartlett. The Tukey window. (Ref 1). Another name for this window is the tapered cosine window. The Welch window. (Ref 1). Other names for this window are: Riez, Bochner, Parzen, parabolic. This window is defined by 1 - arr*2 where the points in arr range from -1 through 1. These subroutines are used internally, but are also available for export. Convert Blackman-Harris coefficients. The BH windows are usually defined via coefficients for cosines of integer multiples of an argument. The same windows may be written instead as terms of powers of cosines of the same argument. These may be computed faster as they replace evaluation of cosines with multiplications. This subroutine is used internally to implement the Blackman-Harris family of windows more efficiently. This subroutine takes between 1 and 7 numeric arguments a0, a1, ... It converts the coefficients of this a0 - a1 cos(arg) + a2 cos(2 arg) - a3 cos(3 * arg) + ... To the cofficients of this c0 + c1 cos(arg) + c2 cos(arg)2 + c3 cos(arg)3 + ... This function is the inverse of "cos_mult_to_pow". This subroutine takes between 1 and 7 numeric arguments c0, c1, ... It converts the coefficients of this c0 + c1 cos(arg) + c2 cos(arg)2 + c3 cos(arg)3 + ... To the cofficients of this a0 - a1 cos(arg) + a2 cos( 2 * arg) - a3 cos( 3 * arg) + ... Returns the value of the $n-th order Chebyshev polynomial at point $x. $n and $x may be scalar numbers, pdl's, or array refs. However, at least one of $n and $x must be a scalar number. All mixtures of pdls and scalars could be handled much more easily as a PP routine. But, at this point PDL::DSP::Windows is pure perl/pdl, requiring no C/Fortran compiler. 1. Harris, F.J. On the use of windows for harmonic analysis with the discrete Fourier transform, Proceedings of the IEEE, 1978, vol 66, pp 51-83. 2. Nuttall, A.H. Some windows with very good sidelobe behavior, IEEE Transactions on Acoustics, Speech, Signal Processing, 1981, vol. ASSP-29, pp. 84-91. John Lapeyre, <jlapeyre at cpan.org> José Joaquín Atria, <jjatria at cpan.org> For study and comparison, the author used documents or output from: Thomas Cokelaer's spectral analysis software; Julius O Smith III's Spectral Audio Signal Processing web pages; André Carezia's chebwin.m Octave code; Other code in the Octave signal package. Copyright 2012-2021 John Lapeyre. This program is free software; you can redistribute it and/or modify it under the terms of either: the GNU General Public License as published by the Free Software Foundation; or the Artistic See http://dev.perl.org/licenses/ for more information. This software is neither licensed nor distributed by The MathWorks, Inc., maker and liscensor of MATLAB.	{"url":"https://metacpan.org/release/ETJ/PDL-DSP-Windows-0.101/view/lib/PDL/DSP/Windows.pm","timestamp":"2024-11-06T04:59:40Z","content_type":"text/html","content_length":"69788","record_id":"<urn:uuid:b8430d6e-891e-4e00-a60c-c33e0ae93488>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00777.warc.gz"}
BMAT Prep - Physics Paper - Q117 & Q209 BMAT Prep Question Analysis - Q117 & Q209 Q117 A 2,000 kg car travelling at 30 m/s crashes into a 5,000 kg truck that is originally at rest. What is the speed of the truck after the collision, if the car comes to rest at the point of impact? (Ignore friction) Q209 The average height of 3 members of a five-a-side football team is 175cm. What does the average height in centimetres of the other 2 players have to be if the average height of the entire team equals 178cm? The sum of the momentums before the collision is equal to the sum of the momentum after the collision (since friction is ignored) and we need to use the conservation of momentum equation to determine the speed of the truck after the impact: M = mass and U = velocity. Mcar U(car)initial + Mtruck U(truck)initial = Mcar U(car)final + Mtruck U(truck)final (2000 x 30) + (5000 x 0) = (2000 × 0) + (5000 x U(truck)final) U(truck)final = 60,000/5,000 = 12 m/s The sum of the heights of the 3 members is 3 x 175 = 525cm. If the average across the whole team of 5 members is 178cm then the sum of their heights is 5 x 178 = 890. So, the sum of the heights of the remaining 2 players is: 890-525=365cm and, consequently, their average height is 365/2 = 182.5cm. Drafted by Quincy (BMAT Prep)	{"url":"https://www.tuttee.co/blog/bmat-prep-physics-paper-q117-q209","timestamp":"2024-11-02T15:46:00Z","content_type":"text/html","content_length":"850448","record_id":"<urn:uuid:6ffb080c-18a9-4773-a1f0-e19bde1f464d>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00853.warc.gz"}
Speculationes circa quasdam insignes proprietates numerorum English Title Speculations about certain outstanding properties of numbers Content Summary Euler writes that "without a doubt, the number of all the different fractions between the endpoints 0 and 1 is infinite; and since the number of all the integers is also infinite, it is manfiest that the multitude of all the ordinary fractions up to infinity is greater; and at the same time, there must be innumerably many different fractions between any two numbers that differ by one." He goes on to define πD as the number of integers less than D and relatively prime to D. He provides a table of πD up to D=100, then makes a table listing the number of fractions with denominators less than or equal to n, for n = 10, 20, …, 100, and poses the problem of finding this number of fractions for any given number N. The solution, of course, involves πN. Published as Journal article Original Source Citation Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 1780: II, pp. 18-30. Opera Omnia Citation Series 1, Volume 4, pp.105-115. Record Created	{"url":"https://scholarlycommons.pacific.edu/euler-works/564/","timestamp":"2024-11-11T19:38:29Z","content_type":"text/html","content_length":"31308","record_id":"<urn:uuid:5c735841-2e90-4f36-b673-0206aceb1f3f>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00188.warc.gz"}
Papa's Freezeria Papa's Freezeria is a game in the Papa Louie series. It's the fourth cooking simulator game (aka "Gameria") in the series, and is the sequel to Papa's Taco Mia. It involves making soft-serve sundaes for the various customers that visit the establishment. Plot[ ] Alberto/Penny (depending on who you chose to play as) has just taken up a job at Papa's Freezeria, an ice cream shop located on Calypso Island. He/she is reading the employee manual, which advertises "stress-free work" and the ability of employees to decorate the store as they please. Excited to start working, Alberto/Penny arrives at the Freezeria in uniform and is given a tour from Papa Louie. Afterwards, Papa Louie leaves the store, leaving Alberto/Penny to start working. Later, Alberto/Penny notices a ship on the horizon; upon closer inspection, he/she sees that it's the S.S. Louie, a cruise ship that Papa Louie uses to bring people back and forth between Calypso Island and the mainland. The S.S. Louie arrives at the dock and a huge horde of customers exits the ship, all of them making their way towards the Freezeria while Alberto/Penny looks on in dismay at the approaching crowd. Gameplay[ ] In this game you play as either Alberto or Penny (you can choose which one you play as when creating a new save file) as they spend their days working at the Freezeria. Each level constitutes one day of business at the Freezeria. During a day, a number of customers come in and place orders for sundaes, and your objective is to create the sundaes to their specifications. This is done through four different stations, one for taking orders and three for assembling the sundaes. When a sundae is completed and sent to a customer, the customer will judge the food based on how long they had to wait for it and how accurately the food matches their order, and give a corresponding score. If a customer gives a high score, they will also give you a tip. The points you get from customer scores give you XP that goes towards achieving a higher rank, allowing you to unlock more of the game's content, and the tips can be used to buy decorations for your lobby and upgrades to make it easier to make sundaes. It's worth noting that unlike in Papa's Taco Mia, lobby decorations can be interchanged and rearranged, and you can also buy flooring and wallpaper to alter the design of the store itself - this is also true of every other Gameria succeeding this one. The ultimate goal of the game is to reach a high enough rank to unlock everything. Aside from customers, there are two other ways to earn tips: weekly paydays and badges. Paydays give you a large amount of tips once every seven days, and badges are in-game achievements that give you a small tip once completed. In addition to earning tips and points, doing well on a customer's order also gives the customer a star. When a customer gets five stars, they get a Customer Medal that causes them to give you higher tips when you do well. The medal gets upgraded every five stars; the first medal customers obtain is the Bronze Medal, then the Silver Medal, and finally the Gold Medal. Aside from normal customers, you also have to contend with Closers; the Closers are a group of seven customers that are more picky about their food than normal customers, and will judge you more harshly when critiquing the food you give them. One Closer always appears as the last customer of each day (except for Day 1), and unlike normal customers, they don't appear randomly; each Closer is tied to one day of the week. On Sundays, the most judgemental Closer of all, Jojo the Food Critic, will visit the Freezeria. His scoring is even harsher than the other six Closers', and unlike any other customer in the game, he orders something completely different every time he visits. If you serve him well, Jojo will reward you with a Blue Ribbon award; the Blue Ribbon stays on your counter for the next three days, causing customers to give you higher tips during that time. Stations[ ] • Order Station: Check on customers and take orders. • Build Station: Assemble the sundaes by selecting a cup size, pouring the ice cream, and adding the mixables and syrups. • Mix Station: Mix the sundaes. • Topping Station: Add toppings. Ingredients[ ] Cup Size[ ] • Medium Cup (Available from the start of the game) • Large Cup (Unlocked at Rank 4) • Small Cup (Unlocked at Rank 10) Mixables[ ] • Nutty Butter Cups (Available from the start of the game) • Strawberries (Available from the start of the game) • Blueberries (Available from the start of the game) • Cookie Dough (Unlocked on Day 2) • Creameo Bits (Unlocked at Rank 5) • Marshmallows (Unlocked at Rank 7) • Pineapple (Unlocked at Rank 10) • Yum n' Ms (Unlocked at Rank 12) Syrups[ ] • Chocolate Syrup (Available from the start of the game) • Vanilla Syrup (Available from the start of the game) • Strawberry Syrup (Unlocked at Rank 2) • Mint Syrup (Unlocked at Rank 6) • Banana Syrup (Unlocked at Rank 13) • Rainbow Sherbet Syrup (Unlocked at Rank 20) Blends[ ] Toppings[ ] • Strawberry Topping (Available from the start of the game) • Chocolate Topping (Available from the start of the game) • Whipped Cream (Available from the start of the game) • Creameos (Available from the start of the game) • Nuts (Available from the start of the game) • Cherries (Available from the start of the game) • Rainbow Sprinkles (Available from the start of the game) • Chocolate Chips (Unlocked at Rank 3) • Bananas (Unlocked at Rank 8) • Butterscotch Topping (Unlocked at Rank 9) • Shaved Mints (Unlocked at Rank 11) • Chocolate Whipped Cream (Unlocked at Rank 14) • Cookies (Unlocked at Rank 16) • Blueberry Toppings (Unlocked at Rank 17) • Gummy Onions (Unlocked at Rank 18) • Tropical Charms (Unlocked at Rank 19) Customers[ ] • Mandi (Tutorial) • Tony (Tutorial) • Alberto/Penny • Wally • Matt • Lisa • Prudence • Marty • Ivy • Claire • Hugo • Utah (Unlocked on Day 2) • Kingsley (Unlocked at Rank 4) • Doan (Unlocked at Rank 5) • Edna (Unlocked at Rank 7) • Chuck (Unlocked at Rank 12) • Sasha (Unlocked at Rank 15) • Sarge Fan (Unlocked at Rank 18) • Connor (Unlocked at Rank 19) • Mindy (Unlocked at Rank 20) • Big Pauly (Unlocked at Rank 21) • Peggy (Unlocked at Rank 22) • Allan (Unlocked at Rank 23) • Cecilia (Unlocked at Rank 24) • Clover (Unlocked at Rank 25) • Rita (Unlocked at Rank 26) • Georgito (Unlocked at Rank 27) • Zoe (Unlocked at Rank 28) • Gino Romano (Unlocked at Rank 29) • Tohru (Unlocked at Rank 30) • Sue (Unlocked at Rank 31) • Mitch (Unlocked at Rank 32) • Carlo Romano (Unlocked at Rank 33) • Kayla (Unlocked at Rank 34) • Rico (Unlocked at Rank 35) • Bruna Romano (Unlocked at Rank 36) • Roy (Unlocked at Rank 37) • Akari (Unlocked at Rank 38) • Cletus (Unlocked at Rank 39) • Vicky (Unlocked at Rank 40) • Franco (Unlocked at Rank 41) • Maggie (Unlocked at Rank 42) • Little Eduardo (Unlocked at Rank 43) • Olga (Unlocked at Rank 44) • Taylor (Unlocked at Rank 45) • Ninjoy (Unlocked at Rank 46) • Kahuna (Monday Closer) • Captain Cori (Tuesday Closer) • Gremmie (Wednesday Closer) • Quinn (Thursday Closer) • Robby (Friday Closer) • Xandra (Saturday Closer) • Jojo (Sunday Closer) • Papa Louie (Unlocked after all the other customers have a Gold Medal) Customers First Appearing in This Game[ ] • Utah • Ivy • Ninjoy • Kahuna • Captain Cori • Gremmie Ranks[ ] 1. Newbie - Starting Rank, Earns $100 on Payday 2. Trainee - Achieved at 300 XP, Earns $105 on Payday 3. Tray Cleaner - Achieved at 750 XP, Earns $110 on Payday 4. Cashier - Achieved at 1,350 XP, Earns $115 on Payday 5. Part-Time Cook - Achieved at 2,100 XP, Earns $120 on Payday 6. Ticket Handler - Achieved at 3,000 XP, Earns $125 on Payday 7. Order Attendant - Achieved at 4,050 XP, Earns $130 on Payday 8. Blender Apprentice - Achieved at 5,250 XP, Earns $135 on Payday 9. Chocolate Champ - Achieved at 6,600 XP, Earns $140 on Payday 10. Strawberry Server - Achieved at 8,100 XP, Earns $145 on Payday 11. Vanilla Fan - Achieved at 9,750 XP, Earns $150 on Payday 12. Banana Pro - Achieved at 11,550 XP, Earns $155 on Payday 13. Mint Master - Achieved at 13,500 XP, Earns $160 on Payday 14. Creameo Pro - Achieved at 15,600 XP, Earns $165 on Payday 15. Peanut Buddy - Achieved at 17,850 XP, Earns $170 on Payday 16. Cherry Champ - Achieved at 20,250 XP, Earns $175 on Payday 17. Marshmallow Master - Achieved at 22,800 XP, Earns $180 on Payday 18. Candy Fan - Achieved at 25,500 XP, Earns $185 on Payday 19. Cookie Dough Pro - Achieved at 28,350 XP, Earns $190 on Payday 20. Gummy Master - Achieved at 31,350 XP, Earns $195 on Payday 21. Pineapple Fan - Achieved at 34,500 XP, Earns $200 on Payday 22. Sprinkle Server - Achieved at 37,800 XP, Earns $205 on Payday 23. Mint Shaver - Achieved at 41,250 XP, Earns $210 on Payday 24. Banana Slicer - Achieved at 44,850 XP, Earns $215 on Payday 25. Butterscotch Lover- Achieved at 48,500 XP, Earns $220 on Payday 26. Cream Whipper - Achieved at 52,500 XP, Earns $225 on Payday 27. Blueberry Expert - Achieved at 56,550 XP, Earns $230 on Payday 28. Sherbert Swirler - Achieved at 60,750 XP, Earns $235 on Payday 29. Blender Buddy - Achieved at 65,100 XP, Earns $240 on Payday 30. Syrup Specialist - Achieved at 69,600 XP, Earns $245 on Payday 31. Sundae Blender - Achieved at 74,250 XP, Earns $250 on Payday 32. Freezer Fan - Achieved at 79,050 XP, Earns $255 on Payday 33. Master of Mixing - Achieved at 84,000 XP, Earns $260 on Payday 34. Blender Expert - Achieved at 89,100 XP, Earns $265 on Payday 35. Sundae Expert - Achieved at 94,350 XP, Earns $270 on Payday 36. Mixable Master - Achieved at 99,750 XP, Earns $275 on Payday 37. Ice Cream Machine - Achieved at 105,300 XP, Earns $280 on Payday 38. Blending Boss - Achieved at 111,000 XP, Earns $285 on Payday 39. Top Topper - Achieved at 116,800 XP, Earns $290 on Payday 40. Part-Time Manager - Achieved at 122,850 XP, Earns $295 on Payday 41. Master of Sundaes - Achieved at 129,000 XP, Earns $300 on Payday 42. Best Blender - Achieved at 135,250 XP, Earns $305 on Payday 43. Ice Cream King - Achieved at 141,750 XP, Earns $310 on Payday 44. Sundae Shop Master - Achieved at 148,350 XP, Earns $315 on Payday 45. Dessert Legend - Achieved at 155,100 XP, Earns $320 on Payday 46. Freezeria Master - Achieved at 162,000 XP, Earns $325 on Payday 47. Better Than Papa! - Achieved at 169,050 XP, Earns $330 on Payday (this goes up by $5 every time you earn an additional 6,300 XP) Upgrades[ ] These items enhance your stations to make it easier to serve the customers. • Doorbell: Alerts you when a new customer enters. Costs $30.00. • Blender Booster 1: Lets you blend sundaes faster on Blender #1 by holding down the booster button. Costs $80.00. • Blender Booster 2: Lets you blend sundaes faster on Blender #2 by holding down the booster button. Costs $80.00. • Blender Booster 3: Lets you blend sundaes faster on Blender #3 by holding down the booster button. Costs $80.00. • Blender Booster 4: Lets you blend sundaes faster on Blender #4 by holding down the booster button. Costs $80.00. • Chunky Alarm: Alerts you when a sundae is blended to Chunky. Costs $90.00. • Regular Alarm: Alerts you when a sundae is blended to Regular. Costs $90.00. • Smooth Alarm: Alerts you when a sundae is blended to Smooth. Costs $90.00. • Auto Ice Cream: Automatically pours the ice cream for you in the Build Station, giving you the best score. Costs $250.00. • Sundae Hat: A wearable item. Improves Waiting score. Costs $60.00. • Straw Hat: A wearable item. Improves Waiting score. Costs $120.00. • Captain Hat: A wearable item. Improves Waiting score. Costs $180.00. • Chef Hat: A wearable item. Improves Waiting score. Costs $180.00. • Pirate Hat: A wearable item. Improves Waiting score. Costs $180.00. • Viking Hat: A wearable item. Improves Waiting score. Costs $300.00. • Royal Crown: A wearable item. Improves Waiting score. Costs $1,000.00. Badges[ ] • Work Up the Ranks: Reach Rank 5 (Gives $30 when earned) • Regular Worker: Reach Rank 10 (Gives $50 when earned) • Freezer Carrier: Reach Rank 20 (Gives $70 when earned) • Long Haul: Reach Rank 30 (Gives $90 when earned) • Better Than Papa!: Reach Rank 47 (Gives $200 when earned) • Spending Spree: Buy any 50 lobby items from the shop (Gives $50 when earned) • Poster Motivation: Buy any 8 items from the shop (Gives $15 when earned) • Interior Decorator: Buy any 8 floor decorations from the shop (Gives $15 when earned) • Mixing Gear: Buy all the Mixer upgrades (Gives $50 when earned) • Serving in Style: Buy all the hats (Gives $100 when earned) • Bronze Beginnings: Earn 5 Bronze Customer Medals (Gives $30 when earned) • Customer Service: Earn 15 Bronze Customer Medals (Gives $50 when earned) • Silver Medal: Earn 10 Silver Customer Medals (Gives $70 when earned) • Coming Back for More: Earn 15 Gold Customer Medals (Gives $100 when earned) • Go for the Gold: Earn Gold Customer Medals for all the customers! (Gives $500 when earned) • Watch the Meter: Get 20 Meter Bonus rewards in the Build Station (Gives $10 when earned) • Ice Cream Rewards: Get 30 GREAT or AWESOME Bonuses on the Ice Cream Machine (Gives $20 when earned) • Mixable Meter: Get 40 GREAT or AWESOME Bonuses on the Mixable Machine (Gives $25 when earned) • Super Syrup Bonus: Get 50 GREAT or AWESOME Bonuses on the Syrup Machine (Gives $30 when earned) • Jackpot!: Get 250 Meter Bonus rewards in the Build Station (Gives $100 when earned) • Order Expert: Get a 100% Waiting score on 20 orders (Gives $70 when earned) • Build Expert: Get a 100% Building score on 20 orders (Gives $70 when earned) • Blender Expert: Get a 100% Mixing score on 20 orders (Gives $70 when earned) • Topping Expert: Get a 100% Topping score on 20 orders (Gives $70 when earned) • Perfect!: Get a perfect score on 30 orders (Gives $100 when earned) • Critically Acclaimed: Earn a Blue Ribbon from Jojo the Food Critic (Gives $40 when earned) • Best Restaurant: Earn five Blue Ribbons from Jojo the Food Critic (Gives $100 when earned) • Quality Assurance: Get 90% Service Quality or higher on five different days (Gives $50 when earned) • High Quality: Get 95% Service Quality or higher on 20 different days (Gives $100 when earned) • Romano Family: Serve everyone in the Romano Quartet - i.e. Bruna, Carlo, Gino, and Little Eduardo (Gives $150 when earned) • Franchise Friends: Serve the employees of Papa's Pizzeria, Burgeria, and Taco Mia - i.e. Roy, Marty, Rita, Mitch, and Maggie (Gives $150 when earned) • Closer Collection: Serve all the Closers (Gives $50 when earned) • The Gang's All Here: Serve all the customers (Gives $300 when earned) • First Paycheck: Get your first paycheck on Payday (Gives $10 when earned) • Month's Pay: Get your paycheck on four different Paydays (Gives $50 when earned) • Syrup Sampler: Unlock all the Mixable Syrups (Gives $25 when earned) • Cup Selection: Unlock all the Cup Sizes (Gives $20 when earned) • Mixable Master: Unlock all the Mixables (Gives $25 when earned) • Topping Pro: Unlock every ingredient in the Topping Station (Gives $25 when earned) • Sundae Cap: Serve 30 sundaes while wearing the Sundae Hat (Gives $50 when earned) • Beach Bum: Serve 40 sundaes while wearing the Straw Hat (Gives $55 when earned) • Anchors Aweigh: Serve 50 sundaes while wearing the Captain Hat (Gives $60 when earned) • Culinary Cap: Serve 60 sundaes while wearing the Chef Hat (Gives $65 when earned) • Seven Seas: Serve 70 sundaes while wearing the Pirate Hat (Gives $75 when earned) • Sundaes from the North: Serve 75 sundaes while wearing the Viking Helmet (Gives $90 when earned) • Freezer Royale: Serve 100 sundaes while wearing the Crown (Gives $150 when earned) • Cherry on Top: Serve 10 sundaes with Cherries (Gives $10 when earned) • Sandwich Cookie: Serve 30 sundaes with Creameo Cookies (Gives $10 when earned) • Cookies Ahoy: Serve 30 sundaes with Cookies (Gives $15 when earned) • Banana Split: Serve 30 sundaes with Bananas (Gives $15 when earned) • Gum'yuns: Serve 20 sundaes with Gummy Onions (Gives $15 when earned) • Peanut Buttery: Serve 20 sundaes with Nutty Butter Cups (Gives $15 when earned) • Fruity Sundae: Serve 30 sundaes with Strawberries (Gives $25 when earned) • Candy Shop: Serve 30 sundaes with Yum n' Ms (Gives $25 when earned) • Tropical Treat: Serve 30 sundaes with Pineapple (Gives $25 when earned) • Do the Dough: Serve 30 sundaes with Cookie Dough (Gives $25 when earned) • Cookies n' Cream: Serve 30 sundaes with Creameo Bits (Gives $25 when earned) • Berry Blast: Serve 30 sundaes with Blueberries (Gives $25 when earned) • S'mores: Serve 30 sundaes with Marshmallows (Gives $25 when earned) • Chocolatey: Serve 30 sundaes mixed with Chocolate Syrup (Gives $25 when earned) • Plain Vanilla: Serve 30 sundaes mixed with Vanilla Syrup (Gives $25 when earned) • Berry Mixer: Serve 30 sundaes mixed with Strawberry Syrup (Gives $25 when earned) • Cool Mint: Serve 30 sundaes mixed with Mint Syrup (Gives $25 when earned) • Bananarama: Serve 30 sundaes mixed with Banana Syrup (Gives $25 when earned) • Follow the Rainbow: Serve 30 sundaes mixed with Rainbow Sherbet Syrup (Gives $25 when earned) • Choc on Top: Serve 30 sundaes with Chocolate Topping (Gives $15 when earned) • Buttery: Serve 30 sundaes with Butterscotch Topping (Gives $25 when earned) • Light and Fluffy: Serve 30 sundaes with Whipped Cream (Gives $25 when earned) • Rich and Creamy: Serve 30 sundaes with Chocolate Whipped Cream (Gives $25 when earned) • Berrylicious: Serve 30 sundaes with Strawberry Topping (Gives $25 when earned) • Do the Blue: Serve 30 sundaes with Blueberry Topping (Gives $25 when earned) • Semi-Sweet: Serve 30 sundaes with Chocolate Chips(Gives $10 when earned) • Rainbow Sprinkles: Serve 30 sundaes with Sprinkles (Gives $10 when earned) • Nuts for Sundaes: Serve 30 sundaes with Nuts (Gives $15 when earned) • After Dinner: Serve 30 sundaes with Shaved Mints (Gives $15 when earned) • Breakfast for Dessert: Serve 20 sundaes with Tropical Charms (Gives $15 when earned) • Getting Started: Serve 4 Sundaes in Medium Cups (Gives $5 when earned) • Medium Master: Serve 30 Sundaes in Medium Cups (Gives $25 when earned) • Super Size: Serve 30 Sundaes in Large Cups (Gives $25 when earned) • Light Dessert: Serve 30 Sundaes in Small Cups (Gives $25 when earned) Trivia[ ] • It's been confirmed through Penny's Flipdeck (Flipdecks are online trading cards that contain short bios of various Flipline Studios characters, viewable on Flipline's website) that she and Alberto began dating during the events of this game. □ This is also referenced in Papa Louie 3, where the two of them are seen holding hands in the game's opening cutscene. 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Multiplication Tables Printable 4 \| Multiplication Chart Printable Multiplication Tables Printable 4 4 Times Table Worksheets Printable Multiplication Facts Worksheets Multiplication Tables Printable 4 Multiplication Tables Printable 4 – A Multiplication Chart is a practical tool for kids to discover exactly how to increase, separate, as well as discover the tiniest number. There are lots of uses for a Multiplication Chart. What is Multiplication Chart Printable? A multiplication chart can be used to help children learn their multiplication facts. Multiplication charts come in several types, from complete page times tables to single page ones. While specific tables are useful for providing portions of info, a complete web page chart makes it less complicated to review truths that have already been understood. The multiplication chart will normally feature a leading row and a left column. When you want to discover the product of two numbers, choose the initial number from the left column and the 2nd number from the top row. Multiplication charts are helpful knowing tools for both youngsters and grownups. Multiplication Tables Printable 4 are available on the Internet as well as can be printed out as well as laminated flooring for toughness. Why Do We Use a Multiplication Chart? A multiplication chart is a layout that reveals how to multiply two numbers. You choose the initial number in the left column, move it down the column, and also then choose the 2nd number from the top row. Multiplication charts are valuable for several reasons, consisting of assisting children find out exactly how to separate and streamline fractions. They can additionally aid children learn exactly how to choose an effective common denominator. Since they offer as a consistent reminder of the trainee’s progress, multiplication charts can likewise be valuable as desk resources. These tools help us develop independent learners that comprehend the fundamental concepts of multiplication. Multiplication charts are additionally useful for aiding pupils remember their times tables. As with any kind of ability, memorizing multiplication tables takes time and also technique. Multiplication Tables Printable 4 4 Times Table 4 Times Table Multiplication Chart Exercise On 4 Times Table Table Of 4 4 Times Table Worksheet The Multiplication Table Multiplication Tables Printable 4 You’ve come to the appropriate place if you’re looking for Multiplication Tables Printable 4. Multiplication charts are readily available in various formats, consisting of full size, half size, and also a range of cute designs. Some are vertical, while others include a horizontal style. You can also find worksheet printables that include multiplication equations and math realities. Multiplication charts and also tables are important tools for youngsters’s education and learning. These charts are terrific for use in homeschool math binders or as class posters. A Multiplication Tables Printable 4 is a valuable tool to enhance math truths as well as can assist a child discover multiplication promptly. It’s also a great tool for avoid checking and learning the moments tables. Related For Multiplication Tables Printable 4	{"url":"https://multiplicationchart-printable.com/multiplication-tables-printable-4/","timestamp":"2024-11-11T17:10:45Z","content_type":"text/html","content_length":"42579","record_id":"<urn:uuid:e26fcea9-4fa5-48c3-8600-f688d480dd0b>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00140.warc.gz"}
JETP Letters: issues online Emergent physics on Mach's principle and the rotating vacuum G. Jannes^+, G. E. Volovik^× ^+Modelling & Numerical Simulation Group, Universidad Carlos III de Madrid, 28911 Leganés, Spain ^Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland ^×Landau Institute for Theoretical Physics of the RAS, 119334 Moscow, Russia Mach's principle applied to rotation can be correct if one takes into account the rotation of the quantum vacuum together with the Universe. Whether one can detect the rotation of the vacuum or not depends on its properties. If the vacuum is fully relativistic at all scales, Mach's principle should work and one cannot distinguish the rotation: in the rotating Universe+vacuum, the co-rotating bucket will have a flat surface (not concave). However, if there are "quantum gravity" effects which violate Lorentz invariance at high energy, then the rotation will become observable. This is demonstrated by analogy in condensed-matter systems, which consist of two subsystems: superfluid background (analog of vacuum) and "relativistic" excitations (analog of matter). For the low-energy (long-wavelength) observer the rotation of the vacuum is not observable. In the rotating frame, the "relativistic" quasiparticles feel the background as a Minkowski vacuum, i.e. they do not feel the rotation. Mach's idea of the relativity of rotational motion does indeed work for them. But rotation becomes observable by high-energy observers, who can see the quantum gravity effects.	{"url":"http://www.jetpletters.ru/ps/2086/article_31372.shtml","timestamp":"2024-11-08T01:43:07Z","content_type":"text/html","content_length":"10952","record_id":"<urn:uuid:7e4831a7-391e-4f93-9e2b-ea4b865212f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00392.warc.gz"}
CS6220 โ CSE6220 Programming Assignment 2 Report Solved - Ideal coders Boqin Zhang (bzhang376), Siying Cen (scen9), Yi Jiao (yjiao47) 1 Algorithm Description The parallel algorithm for calculating the value of polynomial is composed of four steps. Firstly, process 0, as the process having the number of terms of the polynomial and the array of constants, computes the number of terms every process needs to handle and scatter the constants to their corresponding processes. Secondly, let process 0 broadcast the value variable x to every process by hypercubic communication. Thirdly, all processes perform prefix-sum algorithm with variable x as elements and multiplication as operator. Now every process has its xi array. Fourthly, all processes compute the terms aixi. Then all processes perform prefix-sum algorithm with terms aixi as elements and addition as operator. Now the last value of the prefix sum on the last process is the value of the polynomial. At last, let the last process broadcast the value of polynomial to all processes. It will send the value to process 0 at first, then process 0 broadcast it to all other processes by hypercubic communication. Now every process has the final result. 2 Information of Tests To verify the accuracy of the parallel algorithm, a serial function for computing the value of polynomial is developed. A polynomial whose length is 1637809 is generated to be the one to compute. 7 values of variable x is prepared for testing the average computation time. The time of computing the first polynomial value is abandoned, so the time to be analyzed is the average of time of computing 6 values. To let the result of computing time meaningful, the amount of processors used for testing is 2, 4, 6, 8, 12. The submission file is in the folder. 3 Test Results Figure 1: broadcast+PolyEvaluator time 4 Result Analysis 1. The running time of the algorithm is decided by the time of polyEvaluator. The time ofbroadcast is a small part of total running time. 2. With the increasing of amount of processors, the running time is decreasing like an inversefunction. It shows that the algorithm is efficient in the test interval of number of processors. 3. However, it is not a real inverse relationship. If we compute the product of total time and number of processes, we can find that the product becomes larger when more processors are participated in. It means that more resources are used if more processors are used. The test polynomial file is sample-constants2.txt; the test variable file is sample-values2.txt; the job submission file is pa2.pbs. The test results are in testNonp 1637809 final.txt, testnp2 1637809 final.txt, testnp4 1637809 final.txt, testnp6 1637809 final.txt, testnp8 1637809 final.txt, testnp12 1637809 final.txt. There are no reviews yet. Be the first to review “CS6220 – CSE6220 Programming Assignment 2 Report Solved”	{"url":"https://idealcoders.com/product/cs6220-cse6220-programming-assignment-2-report-solved/","timestamp":"2024-11-09T03:12:47Z","content_type":"text/html","content_length":"175565","record_id":"<urn:uuid:27acadc8-acd5-4b95-a925-53e5cf0d156a>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00280.warc.gz"}
Lorentz symmetry group, retardation and energy transformations in a relativistic engine In a previous paper, we have shown that Newton’s third law cannot strictly hold in a distributed system of which the different parts are at a finite distance from each other. This is due to the finite speed of signal propagation which cannot exceed the speed of light in vacuum, which in turn means that when summing the total force in the system the force does not add up to zero. This was demonstrated in a specific example of two current loops with time dependent currents, the above analysis led to suggestion of a relativistic engine. Since the system is effected by a total force for a finite period of time this means that the system acquires mechanical momentum and energy, the question then arises how can we accommodate the law of momentum and energy conservation. The subject of momentum conservation was discussed in a pervious paper, while preliminary results regarding energy conservation where discussed in some additional papers. Here we give a complete analysis of the exchange of energy between the mechanical part of the relativistic engine and the field part, the energy radiated from the relativistic engine is also discussed. We show that the relativistic engine effect on the energy is 4th-order in^1c and no lower order relativistic engine effect on the energy exists. • Electromagnetism • Newton’s third law • Relativity All Science Journal Classification (ASJC) codes • Computer Science (miscellaneous) • Chemistry (miscellaneous) • General Mathematics • Physics and Astronomy (miscellaneous) Dive into the research topics of 'Lorentz symmetry group, retardation and energy transformations in a relativistic engine'. Together they form a unique fingerprint.	{"url":"https://cris.iucc.ac.il/en/publications/lorentz-symmetry-group-retardation-and-energy-transformations-in-","timestamp":"2024-11-04T07:36:26Z","content_type":"text/html","content_length":"50889","record_id":"<urn:uuid:b42d4089-8a07-4b9f-a1b3-988a9f63f5d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00589.warc.gz"}
2024 CMS Summer Meeting Mini-Courses: Applied Topology: DNA Topology Applied Topology: DNA Topology DNA topology is the study of the geometry (supercoiling) and topology (knotting and linking) of DNA. Knots and links in DNA are known to obstruct normal cellular processes and geometric factors such as the amount of writhe or supercoiling are known to affect the formation of entanglements in DNA. Enzymes that act on DNA to cut and reconnect the DNA are natureâ€™s solution for removing these topological obstructions. Polymer models have proved useful for geometric modeling of DNA conformations and methods from knot theory have enabled the characterization of entanglements and the results of enzyme action. Topological methods including tangle analysis, braid theory, classical and quantum knot invariants, and Floer homology can be successfully applied to analyze reactions and characterize entangled structures. The mini-course on DNA topology will be structured as follows: 1. Discussion of motivational problems from DNA experiments [enzyme action on DNA, DNA packing] 2. Overview of knot theoretic foundations and methodology [skein relations, knot polynomials, tangle and braid techniques, homology-theoretic strategies] 3. Hands-on computational tools crash-course demonstrating several packages, programs, and techniques. [Participants will learn about tools such as the Knot Theory package in Sage, KnotPlot, lattice knot simulations, and other computational tools and databases.] Additional Information The CMS is organizing mini-courses to add more value to meetings and make them attractive for students and researchers to attend. The mini-courses will be held on Friday, May 31 and include topics suitable for graduate students, postdocs, and other interested parties. Partial funding may be available for students, postdocs, and early-career researchers to attend the Applied Topology minicourses. To receive more information and apply for this opportunity, please fill out this google form. For full consideration, please apply by May 1 using this form: https://forms.gle/v5FEVmcYC4w2nyXw5 For registration, please complete the Google form for funding above and then enrol through the CMS website. Event Type Scientific, Workshop	{"url":"https://pims.math.ca/events/240531-2csmmcatdt","timestamp":"2024-11-11T15:25:36Z","content_type":"text/html","content_length":"421829","record_id":"<urn:uuid:b249dd0b-bcb6-4ec8-b0d9-3d0ea2e7eccb>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00476.warc.gz"}
Mathematical notes Kermitutor: Math Tutoring List of math newsletters Definitions of Frequency Distribution A possible psuedo math web to be evaluated Advice for solving word problems The web page of Dan Curtin Get help on word problems. Biography of Leonardo Fibonacci Fsu math dept Some email addresses of a few people who are very interested in math An article discussing calculating the nth decimal digit of PI. A discussion of the identity equation of constants e^( i pi) + 1 = 0. Chi Square Tutorial The formula for the sum of powers of consecutive positive integers A discussion of the axiom of determinacy The fortran program source code to find the date for easter in any given year. A note of advice for teaching fractions. A history of Leibniz invention of the calculus downloaded from the internet. A discussion of the generalized integration by parts formula A false hypothesis about how the number e was discovered. A more accurate explanation of how the number e was discovered. A discussion of how to solve the rubix cube. A discussion of how to calculate log base 2. A discussion of the 16 two-variable logic functions of symbolic logic. A six way logic puzzle The guide to mathematical correctness Home page for the math book publishers Saxon Fortran source code for calculating normal distribution School News is a educational resource newsletter. The names of the formal set theory axioms	{"url":"http://kermitrose.com/math.html","timestamp":"2024-11-14T20:08:12Z","content_type":"text/html","content_length":"3931","record_id":"<urn:uuid:ae9cff21-d8fa-43b6-a00f-29dbe2235294>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00109.warc.gz"}
A-Level Maths - Pure/Core 1 Many sites and online services, like Google Maps, suggest HTML code (usually iframes) to embed widgets and content on your page. You can add such HTML widgets in your quizzes. Please note: • In case pasted code is not just iframe tag it will be wrapped into HTTPS iframe. • Only HTTPS sources for iframes, styles and javascript links are supported. • Non-HTTPS sources will be blocked by the browser and won't behave properly.	{"url":"https://mcooke230774.netboard.me/alevelmaths/?tab=670433","timestamp":"2024-11-02T17:07:03Z","content_type":"text/html","content_length":"82341","record_id":"<urn:uuid:074bbace-9224-42ca-8b2f-75744adeb1a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00734.warc.gz"}
§ 25.7520-3 Limitation on the application of section 7520. Federal Code of Regulations § 25.7520-3 Limitation on the application of section 7520. (a) Internal Revenue Code sections to which section 7520 does not apply. Section 7520 of the Internal Revenue Code does not apply for purposes of— (1) Part I, subchapter D of subtitle A (section 401 et. seq.), relating to the income tax treatment of certain qualified plans. (However, section 7520 does apply to the estate and gift tax treatment of certain qualified plans and for purposes of determining excess accumulations under section 4980A); (2) Sections 72 and 101(b), relating to the income taxation of life insurance, endowment, and annuity contracts, unless otherwise provided for in the regulations under sections 72, 101, and 1011 (see, particularly, §§ 1.101-2(e)(1)(iii)(b)(2), and 1.1011-2(c), Example 8); (4) Section 457, relating to the valuation of deferred compensation, unless otherwise provided for in the regulations under section 457; (5) Sections 3121(v) and 3306(r), relating to the valuation of deferred amounts, unless otherwise provided for in the regulations under those sections; (6) Section 6058, relating to valuation statements evidencing compliance with qualified plan requirements, unless otherwise provided for in the regulations under section 6058; (7) Section 7872, relating to income and gift taxation of interest-free loans and loans with below-market interest rates, unless otherwise provided for in the regulations under section 7872; or (8) Section 2702(a)(2)(A), relating to the value of a nonqualified retained interest upon a transfer of an interest in trust to or for the benefit of a member of the transferor's family; and (9) Any other section of the Internal Revenue Code to the extent provided by the Internal Revenue Service in revenue rulings or revenue procedures. (See §§ 601.201 and 601.601 of this chapter). (b) Other limitations on the application of section 7520 — (1) In general — (i) Ordinary beneficial interests. For purposes of this section: (A) An ordinary annuity interest is the right to receive a fixed dollar amount at the end of each year during one or more measuring lives or for some other defined period. A standard section 7520 annuity factor for an ordinary annuity interest represents the present worth of the right to receive $1.00 per year for a defined period, using the interest rate prescribed under section 7520 for the appropriate month. If an annuity interest is payable more often than annually or is payable at the beginning of each period, a special adjustment must be made in any computation with a standard section 7520 annuity factor. (B) An ordinary income interest is the right to receive the income from or the use of property during one or more measuring lives or for some other defined period. A standard section 7520 income factor for an ordinary income interest represents the present worth of the right to receive the use of $1.00 for a defined period, using the interest rate prescribed under section 7520 for the appropriate month. However, in the case of certain gifts made after October 8, 1990, if the donor does not retain a qualified annuity, unitrust, or reversionary interest, the value of any interest retained by the donor is considered to be zero if the remainder beneficiary is a member of the donor's family. See § 25.2702-2. (C) An ordinary remainder or reversionary interest is the right to receive an interest in property at the end of one or more measuring lives or some other defined period. A standard section 7520 remainder factor for an ordinary remainder or reversionary interest represents the present worth of the right to receive $1.00 at the end of a defined period, using the interest rate prescribed under section 7520 for the appropriate month. (ii) Certain restricted beneficial interests. A restricted beneficial interest is an annuity, income, remainder, or reversionary interest that is subject to any contingency, power, or other restriction, whether the restriction is provided for by the terms of the trust, will, or other governing instrument or is caused by other circumstances. In general, a standard section 7520 annuity, income, or remainder factor may not be used to value a restricted beneficial interest. However, a special section 7520 annuity, income, or remainder factor may be used to value a restricted beneficial interest under some circumstances. See paragraphs (b)(2)(v) Example 5 and (b)(4) of this section, which illustrate situations in which special section 7520 actuarial factors are needed to take into account limitations on beneficial interests. See § 25.7520-1(c) for requesting a special factor from the Internal Revenue Service. (iii) Other beneficial interests. If, under the provisions of this paragraph (b), the interest rate and mortality components prescribed under section 7520 are not applicable in determining the value of any annuity, income, remainder, or reversionary interest, the actual fair market value of the interest (determined without regard to section 7520) is based on all of the facts and circumstances if and to the extent permitted by the Internal Revenue Code provision applicable to the property interest. (2) Provisions of governing instrument and other limitations on source of payment — (i) Annuities. A standard section 7520 annuity factor may not be used to determine the present value of an annuity for a specified term of years or the life of one or more individuals unless the effect of the trust, will, or other governing instrument is to ensure that the annuity will be paid for the entire defined period. In the case of an annuity payable from a trust or other limited fund, the annuity is not considered payable for the entire defined period if, considering the applicable section 7520 interest rate on the valuation date of the transfer, the annuity is expected to exhaust the fund before the last possible annuity payment is made in full. For this purpose, it must be assumed that it is possible for each measuring life to survive until age 110. For example, for a fixed annuity payable annually at the end of each year, if the amount of the annuity payment (expressed as a percentage of the initial corpus) is less than or equal to the applicable section 7520 interest rate at the date of the transfer, the corpus is assumed to be sufficient to make all payments. If the percentage exceeds the applicable section 7520 interest rate and the annuity is for a definite term of years, multiply the annual annuity amount by the Table B term certain annuity factor, as described in § 25.7520-1(c)(1), for the number of years of the defined period. If the percentage exceeds the applicable section 7520 interest rate and the annuity is payable for the life of one or more individuals, multiply the annual annuity amount by the Table B annuity factor for 110 years minus the age of the youngest individual. If the result exceeds the limited fund, the annuity may exhaust the fund, and it will be necessary to calculate a special section 7520 annuity factor that takes into account the exhaustion of the trust or fund. This computation would be modified, if appropriate, to take into account annuities with different payment terms. (ii) Income and similar interests — (A) Beneficial enjoyment. A standard section 7520 income factor for an ordinary income interest is not to be used to determine the present value of an income or similar interest in trust for a term of years or for the life of one or more individuals unless the effect of the trust, will, or other governing instrument is to provide the income beneficiary with that degree of beneficial enjoyment of the property during the term of the income interest that the principles of the law of trusts accord to a person who is unqualifiedly designated as the income beneficiary of a trust for a similar period of time. This degree of beneficial enjoyment is provided only if it was the transferor's intent, as manifested by the provisions of the governing instrument and the surrounding circumstances, that the trust provide an income interest for the income beneficiary during the specified period of time that is consistent with the value of the trust corpus and with its preservation. In determining whether a trust arrangement evidences that intention, the treatment required or permitted with respect to individual items must be considered in relation to the entire system provided for in the administration of the subject trust. Similarly, in determining the present value of the right to use tangible property (whether or not in trust) for one or more measuring lives or for some other specified period of time, the interest rate component prescribed under section 7520 and § 1.7520-1 of this chapter may not be used unless, during the specified period, the effect of the trust, will or other governing instrument is to provide the beneficiary with that degree of use, possession, and enjoyment of the property during the term of interest that applicable state law accords to a person who is unqualifiedly designated as a life tenant or term holder for a similar period of time. (B) Diversions of income and corpus. A standard section 7520 income factor for an ordinary income interest may not be used to value an income interest or similar interest in property for a term of years, or for one or more measuring lives, if— (1) The trust, will, or other governing instrument requires or permits the beneficiary's income or other enjoyment to be withheld, diverted, or accumulated for another person's benefit without the consent of the income beneficiary; or (iii) Remainder and reversionary interests. A standard section 7520 remainder interest factor for an ordinary remainder or reversionary interest may not be used to determine the present value of a remainder or reversionary interest (whether in trust or otherwise) unless, consistent with the preservation and protection that the law of trusts would provide for a person who is unqualifiedly designated as the remainder beneficiary of a trust for a similar duration, the effect of the administrative and dispositive provisions for the interest or interests that precede the remainder or reversionary interest is to assure that the property will be adequately preserved and protected (e.g., from erosion, invasion, depletion, or damage) until the remainder or reversionary interest takes effect in possession and enjoyment. This degree of preservation and protection is provided only if it was the transferor's intent, as manifested by the provisions of the arrangement and the surrounding circumstances, that the entire disposition provide the remainder or reversionary beneficiary with an undiminished interest in the property transferred at the time of the termination of the prior interest. (iv) Pooled income fund interests. In general, pooled income funds are created and administered to achieve a special rate of return. A beneficial interest in a pooled income fund is not ordinarily valued using a standard section 7520 income or remainder interest factor. The present value of a beneficial interest in a pooled income fund is determined according to rules and special remainder factors prescribed in § 1.642(c)-6 of this chapter and, when applicable, the rules set forth under paragraph (b)(3) of this section if the individual who is the measuring life is terminally ill at the time of the transfer. (v) Annuity payable from a trust or other limited fund. The present value of an annuity interest (the subject annuity) payable from a trust or other limited fund (the fund) must be determined by taking into account the possibility of exhaustion of the fund. Thus, the present value of any such annuity that will exhaust the fund (and of any other interest dependent on the present value of such an annuity) is determined by using actuarial factors, under the applicable section 7520 mortality and interest rate assumptions, reflecting the term certain period to the exhaustion of the fund. Because it is assumed under the prescribed mortality component, Table 2010CM, that any measuring life may survive past age 109 and until just before age 110, any life annuity could require payments in the person's 110th year. To determine whether a subject annuity that is payable as a level amount paid annually at the end of each year for a life, will exhaust the fund, the annuitant's age must be subtracted from 110 to determine the longest possible duration of the annuity (maximum number of years) under the prescribed mortality table. If the present value of an annuity for the same level payments, payable for a term certain equal to that maximum number of years, is greater than the value of the fund, the annuity is determined to exhaust the fund before the end of that maximum number of years (under the prescribed assumptions), and the present value of the subject annuity is determined on the basis of life annuity factors limited by the period to exhaustion. The period to exhaustion is the shortest period for which the present value of that same annuity payable for a term certain is greater than or equal to the value of the fund, under the prescribed section 7520 interest rate. If the subject annuity is for life (or for a period depending in part on life) and the period to exhaustion is shorter than the longest possible life period, the present value of the subject annuity is determined as the present value of an annuity for the shorter of life or a term of years limited by the period to exhaustion, and the actuarial commutation factors may be used in determining the present value. The actuarial commutation factors can be computed directly by using the formulas in § 25.2512-5(d)(2)(v)(A)(1), the section 7520 rate, and Table 2010CM as set forth in § 20.2031-7(d)(7)(ii) of this chapter. For the convenience of taxpayers, actuarial commutation factors have been computed by the IRS and appear in Table H. The appropriate annuity factors for an annuity payable for a term of years certain is computed by subtracting from 1.000000 the factor for an ordinary remainder interest following the same term certain that is determined under the formula in § 20.2031-7(d)(2)(ii)(A) of this chapter and then dividing the result by the applicable section 7520 interest rate expressed to at least four decimal places. For the convenience of taxpayers, actuarial factors have been computed by the IRS and appear in the “Annuity” column of Table B. Tables B and H can be found on the IRS website at https://www.irs.gov/retirement-plans/actuarial-tables (or a corresponding URL as may be updated from time to time). For an annuity payable for the longer of a life (or lives) or a term of years, the year of the last possible annuity payment is determined based on the later of the end of the term period or the year the youngest measuring life would reach age 110. For an annuity payable for the shorter of a life (or lives) or a term of years, the year of the last possible payment is determined based on the earlier of the end of the term period or the year the youngest measuring life would reach age 110. After determining the point of exhaustion of funds, the approximation method for determining the present value of annuity payments for a life or lives so limited by exhaustion illustrated in the example in paragraph (b)(2)(vi) (E) of this section is to be used if a more exact method (for example, computing the year-by-year present value of each payment until the fund is exhausted) is not used. The selected method must be applied consistently in valuing all interests in the same property. (vi) Examples. The provisions of this paragraph (b)(2) are illustrated by the following examples: (A) Example 1. Unproductive property. The donor transfers corporation stock to a trust under the terms of which all of the trust income is payable to A for life. Considering the applicable federal rate under section 7520 and the appropriate life estate factor for a person A's age, the value of A's income interest, if valued under this section, would be $10,000. After A's death, the trust is to terminate and the trust property is to be distributed to B. The trust specifically authorizes, but does not require, the trustee to retain the shares of stock. The corporation has paid no dividends on this stock during the past 5 years, and there is no indication that this policy will change in the near future. Under applicable state law, the corporation is considered to be a sound investment that satisfies fiduciary standards. The facts and circumstances, including applicable state law, indicate that the income beneficiary would not have the legal right to compel the trustee to make the trust corpus productive in conformity with the requirements for a lifetime trust income interest under applicable local law. Therefore, the life income interest in this case is considered nonproductive. Consequently, A's income interest may not be valued actuarially under this section. (B) Example 2. Beneficiary's right to make trust productive. The facts are the same as in paragraph (b)(2)(vi)(A) of this section (Example 1), except that the trustee is not specifically authorized to retain the shares of corporation stock. Further, the terms of the trust specifically provide that the life income beneficiary may require the trustee to make the trust corpus productive consistent with income yield standards for trusts under applicable state law. Under that law, the minimum rate of income that a productive trust may produce is substantially below the section 7520 interest rate on the valuation date. In this case, because A, the income beneficiary, has the right to compel the trustee to make the trust productive for purposes of applicable local law during A's lifetime, the income interest is considered an ordinary income interest for purposes of this paragraph (b)(2)(vi)(B), and the standard section 7520 life income factor may be used to determine the value of A's income interest. However, in the case of gifts made after October 8, 1990, if the donor was the life income beneficiary, the value of the income interest would be considered to be zero in this situation. See § 25.2702-2. (C) Example 3. Annuity trust funded with unproductive property. The donor, who is age 60, transfers corporation stock worth $1,000,000 to a trust. The trust will pay a 6 percent ($60,000 per year) annuity in cash or other property to the donor for 10 years or until the donor's prior death. Upon the termination of the trust, the trust property is to be distributed to the donor's child. The section 7520 rate for the month of the transfer is 8.2 percent. The corporation has paid no dividends on the stock during the past 5 years, and there is no indication that this policy will change in the near future. Under applicable state law, the corporation is considered to be a sound investment that satisfies fiduciary standards. Therefore, the trust's sole investment in this corporation is not expected to adversely affect the interest of either the annuity beneficiary or the remainder beneficiary. Considering the 6 percent annuity payout rate and the 8.2 percent section 7520 interest rate, the trust corpus is considered sufficient to pay this annuity for the entire 10-year term of the trust, or even indefinitely. The trust specifically authorizes, but does not require, the trustee to retain the shares of stock. Although it appears that neither beneficiary would be able to compel the trustee to make the trust corpus produce investment income, the annuity interest in this case is considered to be an ordinary annuity interest, and a section 7520 annuity factor may be used to determine the present value of the annuity. In this case, the section 7520 annuity factor would represent the right to receive $1.00 per year for a term of 10 years or the prior death of a person age 60. (D) Example 4. Unitrust funded with unproductive property. The facts are the same as in paragraph (b)(2)(vi)(C) of this section (Example 3), except that the donor has retained a unitrust interest equal to 7 percent of the value of the trust property, valued as of the beginning of each year. Although the trust corpus is nonincome-producing, the present value of the donor's retained unitrust interest may be determined by using the section 7520 unitrust factor for a term of years or a prior death. (E) Example 5. Annuity exhausting a trust or other limited fund. (1) The donor, who is age 60 and in normal health, transfers property worth $1,000,000 to a trust created for this purpose on or after June 1, 2023. The trust will pay a 10 percent ($100,000 per year) annuity to a charitable organization for the life of the donor, payable annually at the end of each year, and the remainder then will be distributed to the donor's child. The trust has no other beneficial interests payable before the end of the annuity. The section 7520 rate for the month of the transfer is 4.4 percent. Under section 7520(a)(2) of the Code, the donor has elected to use the section 7520 rate for the month of transfer. For purposes of this example, the relevant factors from Tables B and H(4.4) are: Table 1 to Paragraph (b)(2)(vi)(E)(1) │ Years │ Annuity │ Income interest │ Remainder │ │Factors From Table B │ │Annuity, Income, and Remainder Interests for a Term Certain │ │Interest at 4.4 Percent │ │13 │9.7423 │0.428661 │0.571339 │ │14 │10.2896 │0.452741 │0.547259 │ │50 │20.0878 │0.883862 │0.116138 │ │Factors From Table H(4.4) │ │Commutation Factors—Based on Table 2010CM │ │Interest Rate of 4.4 Percent │ │Age (x) │D[x] │N[x]-factor │M[x]-factor │ │60 │6,694.636 │90,259.34 │2,723.225 │ │73 │3,151.228 │29,432.25 │1,856.209 │ │74 │2,941.075 │26,452.50 │1,777.165 │ (2) First, it is necessary to determine whether the annuity may exhaust the corpus before all planned annuity payments are made. This determination is made by using values from Table B as illustrated in Figure 1 to this paragraph (b)(2)(vi)(E)(2). Figure 1 to Paragraph (b)(2)(vi)(E)(2)—Illustration of Determining Present Value of Term Certain Annuity (3) Because the present value of an annuity for a term of 50 years exceeds the corpus, the annuity may exhaust the trust before all payments are made. Consequently, the annuity must be valued as an annuity payable for a term of years or until the prior death of the annuitant, with the term of years determined by the number of years to exhaustion of the fund, assuming earnings at the section 7520 rate of 4.4 percent. (4) If an annuity of $100,000 payable at the end of each year for a period had an annuity factor of 10.0, it would have a present value exactly equal to the principal available to pay the annuity over the term. The annuity factor for 13 years at 4.4 percent in Table B is 9.7423, so the present value of an annuity of $100,000 at 4.4 percent, payable at the end of each year for 13 years certain, is $100,000 times 9.7423 or $974,230. The annuity factor for 14 years at 4.4 percent is 10.2896, so the present value of an annual annuity of $100,000 per year at 4.4 percent for 14 years certain is $100,000 times 10.2896, or $1,028,960. Therefore, 14 years is the shortest term for which a term certain annuity of $100,000 per year is greater than the fund of $1,000,000. Thus, it is determined, under the prescribed assumptions, that the $1,000,000 initial transfer will be sufficient to make 13 annual payments of $100,000, but not to make the entire 14th payment. Subtracting the present value of the 13-year term certain annuity, $974,230, from the fund of $1,000,000 leaves a remainder of $25,770. Of the initial transfer amount, $25,770 is not needed to make payments for 13 years, so this amount, as accumulated for 14 years, will be available for the final payment. The 14-year accumulation factor at 4.4 percent is 1.827288 ((1 + 0.044)^14 = 1.827288), so the amount available in 14 years is $25,770 times 1.827288 or $47,089.21. Therefore, for purposes of this present value determination, the subject annuity is treated as being composed of two distinct annuity components. The two annuity components taken together must equal the total annual amount of $100,000. The annual amount of the first annuity component is the exact amount that the trust will have available for the final payment, $47,089.21. The annual amount of the second annuity component then must be $100,000 minus $47,089.21, or $52,910.79. Under the section 7520 assumptions, the initial corpus will be able to make payments of $52,910.79 per year for 13 years, as well as payments of $47,089.21 per year for 14 years. The present value of the subject annuity is computed by adding together the present values of two separate component annuities payable for the shorter of a life or a term. (5) The actuarial factor for determining the value of the annuity of $52,910.79 per year payable for 13 years or until the prior death of a person aged 60 is derived by the use of factors involving one life and a term of years, derived from Table H. The factor is determined as illustrated in Figure 2 to this paragraph (b)(2)(vi)(E)(5). Figure 2 to Paragraph (b)(2)(vi)(E)(5)—Illustration of Determining Annuity Factor for Shorter of Life or 13 Years (6) The actuarial factor for determining the value of the annuity $47,089.21 per year payable for 14 years or until the prior death of a person aged 60 is derived by the use of factors involving one life and a term of years, derived from Table H. The factor is determined as illustrated in Figure 3 to this paragraph (b)(2)(vi)(E)(6). Figure 3 to Paragraph (b)(2)(vi)(E)(6)—Illustration of Determining Annuity Factor for Shorter of Life or 14 Years (7) Based on the calculations of paragraph (b)(2)(vi)(E)(5) of this section, the present value of an annuity of $52,910.79 per year payable for 13 years or until the prior death of a person aged 60 is $480,742.15 ($52,910.79 × 9.0859). Based on the calculations of paragraph (b)(2)(vi)(E)(6) of this section, the present value of an annuity of $47,089.21 per year payable for 14 years or until the prior death of a person aged 60 is $448,807.26 ($47,089.21 × 9.5310). Thus, the present value of the charitable annuity interest is the sum of the two component annuities, $929,549.41 ($480,742.15 + (3) Mortality component. The mortality component prescribed under section 7520 may not be used to determine the present value of an annuity, income interest, remainder interest, or reversionary interest if an individual who is a measuring life dies or is terminally ill at the time the gift is completed. For purposes of this paragraph (b)(3), an individual who is known to have an incurable illness or other deteriorating physical condition is considered terminally ill if there is at least a 50 percent probability that the individual will die within 1 year. However, if the individual survives for eighteen months or longer after the date the gift is completed, that individual shall be presumed to have not been terminally ill at the date the gift was completed unless the contrary is established by clear and convincing evidence. (4) Example—terminal illness — (i) Sample factors from actuarial Table S. The provisions of paragraph (b)(3) of this section are illustrated by the example in paragraph (b)(4)(ii) of this section. The appropriate annuity factor for an annuity payable for the life of one individual is computed by subtracting from 1.00000 the factor for an ordinary remainder interest following the life of the same individual that is determined under the formula in § 20.2031-7(d)(2)(ii)(B) of this chapter and then dividing the result by the applicable section 7520 interest rate expressed to at least four decimal places. For the convenience of taxpayers, actuarial factors have been computed by the IRS and appear in the “Annuity” column of Table S. Table S can be found on the IRS website at https://www.irs.gov/ retirement-plans/actuarial-tables. For purposes of the example in paragraph (b)(4)(ii) of this section, the relevant factor from Table S is: Table 2 to Paragraph (b)(4)(i) Factors From Table S—Based on Table │Age│Annuity│Life estate │Remainder │ │Interest at 4.4 Percent │ │75 │8.6473 │0.38048 │0.61952 │ (ii) Example of donor with terminal illness. The donor transfers property worth $1,000,000 to a child on or after June 1, 2023, in exchange for the child's promise to pay the donor $80,000 per year for the donor's life, payable annually at the end of each period. The section 7520 interest rate for the month of the transfer is 4.4 percent. The donor is age 75 but has been diagnosed with an incurable illness and has at least a 50 percent probability of dying within 1 year. Under Table S, the annuity factor at 4.4 percent for a person age 75 in normal health is 8.6473. Thus, if the donor were not terminally ill, the present value of the annuity would be $691,784 ($80,000 × 8.6473). Assuming the presumption provided in paragraph (b)(3) of this section does not apply, because there is at least a 50 percent probability that the donor will die within 1 year, the standard section 7520 annuity factor may not be used to determine the present value of the donor's annuity interest. Instead, a special section 7520 annuity factor must be computed that takes into account the projection of the donor's actual life expectancy. (c) Applicability dates. Paragraph (a) of this section is applicable as of May 1, 1989. The provisions of paragraph (b) of this section, except paragraphs (b)(2)(v), (b)(2)(vi)(E), and (b)(4) of this section, are applicable to gifts made after December 13, 1995. Paragraphs (b)(2)(v), (b)(2)(vi)(E), and (b)(4) of this section are applicable to gifts made on or after June 1, 2023. [T.D. 8540, 59 FR 30177, June 10, 1994, as amended by T.D. 8630, 60 FR 63919, Dec. 13, 1995; T.D. 8819, 64 FR 23228, Apr. 30, 1999; T.D. 8886, 65 FR 36943, June 12, 2000; T.D. 9448, 74 FR 21517, May 7, 2009; T.D. 9540, 76 FR 49642, Aug. 10, 2011; T.D. 9974, 88 FR 37463, June 7, 2023]	{"url":"https://charitableplanning.com/library/documents/471359","timestamp":"2024-11-09T13:02:20Z","content_type":"text/html","content_length":"68389","record_id":"<urn:uuid:fd2f86fb-e27b-4da6-a055-330989a63a51>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00752.warc.gz"}
A continuous-time diffusion limit theorem for dynamical decoupling and intrinsic decoherence We discuss a few mathematical aspects of random dynamical decoupling, a key tool procedure in quantum information theory. In particular, we place it in the context of discrete stochastic processes, limit theorems and completely positive trace-preserving semigroups on matrix algebras. We obtain precise analytical expressions for expectation and variance of the density matrix and fidelity over time in the continuum-time limit depending on the system Lindbladian, which then lead to rough short-time estimates depending only on certain coupling strengths. We prove that dynamical decoupling does not work in the case of intrinsic (i.e., not environment-induced) decoherence, and together with the above-mentioned estimates this yields a novel method of partially identifying intrinsic • central limit theorem • CPT semigroups • dynamical decoupling • intrinsic decoherence Dive into the research topics of 'A continuous-time diffusion limit theorem for dynamical decoupling and intrinsic decoherence'. Together they form a unique fingerprint.	{"url":"https://research.aber.ac.uk/en/publications/a-continuous-time-diffusion-limit-theorem-for-dynamical-decouplin","timestamp":"2024-11-03T08:56:05Z","content_type":"text/html","content_length":"59022","record_id":"<urn:uuid:9c0623dc-f855-4f7f-9c5f-091af4c3886c>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00197.warc.gz"}
Poll: What do you use to analyze and/or visualize data? I asked this same question a couple of years back. I wonder: has the software that people use for visualization and data graphics changed at all? Punch your answer in the poll below. If you select ‘other’ let us know your tool of choice in the comments. P.S. I know many of you use a combination of these. Pick your favorite if that’s the case. 73 Comments • SPSS • Constellation Roamer & Framework from Asterisq • I have used SPSS in the past, but I clicked excel. To be frank, I mostly use you! • Some of us use more than one of the languages/ programs listed. It doesn’t let us vote for more than one. □ I use all of the above. That’s why I said to pick your favorite :) ☆ How can you use “all of the above” being “nothing” one of them? ☆ I have magic powers. ☆ Everything I do starts with the data in an Excel spreadsheet and some initial cleaning/sorting/filtering/calculations there, so I felt obliged to click that. But Tableau Public is my favorite for easy online visualization and R for statistics and for power and flexibility in producing graphics. The following website is also great for quick and easy harnessing of R’s ggplot2 package for common statistical graphics: http://www.yeroon.net/ggplot2/. You can cut and paste the code for later customization in R itself. • agree completely Naomi… I use SPSS/SAS/R and then Excel/Illustrator/Processing depending on volumes and what I’m doing… • IDV Solution’s Visual Fusion • What about adding Stata as an option? □ Stata doesn’t count. And why is Excel even on there? • No MATLABers? □ MATLABer for life, although I’m under the impression I need to familiarize myself with R. • QlikView. Great poll Nathan. Cheers. • Python obviously. □ Indeed! with Matplotlib • STATA mostly • Another vote for python, usually accompanied by JavaScript and canvas or svg. • Stata • ArcGIS • JavaScript + SVG / Canvas, bien sûr! • Matlab • Spotfire. Their new free Silver offering publishes to the cloud: • Stata/R here. Excel is not an analytical tool, poor reproducibility, poor statistical routines and poor graphics (adding a third dimension is totally unnecessary unless you have three variables you are wishing to graph). Tons of reasons why, some links to discussions on mailing lists and peer-reviewed articles at http://slack.ser.man.ac.uk/progs/stata/avoid_excel.html □ But you can make graphs with it, and I think that’s what most people do (even though there are better solutions). • Matlab • For me actually a mixture of Excel, Processing and in some rare cases R. But in most cases Excel is only for pre-analyzing and a first impression of the dataset, and for some exporting/importing stuff. Processing for the actual visual analysis and final visualization. • I use JMP, which is made by SAS. The high school site licence makes it very affordable, and the learners get to work with a program that is professional level. • Another vote for MATLAB • That Excel is “winning” currently somehow saddens me. □ Here’s a positive spin: Excel only has 27% right now, so people are making use of other tools much more. • I jumped into this data world almost a year ago and have been doing most of my work in excel for a couple of reasons: 1. It’s what is on my computer 2. Most of the reports I pull from for my work come in two flavors: pdf or excel 3. Excel, since its on most computers, makes it easy to send a file or let others review my work 4. It’s what I know and have been working with since college 5. Graphs are simple and with features like conditional formatting heatmaps are practically built-in. That being said I, I started reading this blog because I do want to get deeper into this field. So what would you recommend for someone who is comfortable with excel, has limited resources but wants to do more? □ This list of tools and resources might be helpful: • Matlab. • Photoshop, primarily for satellite images and maps. Otherwise I often use Excel or Aabel for graphing, then export a postscript file to Illustrator for cleanup. • Origin. • QlikView @Rob Jensen: there is a free, fully functional personal edition available for download • I must say that Gephi.org is a must have for all graph visualizations and manipulation that happened to be my very first step when trying to explore a new dataset. • All, Of interest to me is how the audience is organized in terms of profession and time on task doing data analysis and visual analytics vs or alongside data/info visualization. Lots of hammers here and lots of nails. What are folks actually building with the tools? SAS neck and neck with pen/paper. Tableau whooping SAS? • Working on my own canvas-based lib ‘Unveil.js’ http://github.com/michael/unveil (work-in-progress). Would enjoy some feedback/ :) — Michael • Matlab!!! • I do a lot of data analysis with Ruby and Python. Otherwise it’s usually Excel, R and sometimes Processing. • I use Microsoft’s SSAS mainly these days, with either C#.net for weird stuff or Excel if it is light lifting. Most of my data sets are of the millions+ range of fact table rows. • Mathematica … • SAS is still my tool of choice for analytics. • @Rob Jensen, We use a number of the tools in this list all day long, but more and more are finding that for commercial purposes, the Tableau (http://www.tableausoftware.com) suite of products cannot be beat. There is also a free version (Tableau Public – http://www.tableaupublic.com) which requires that you store your work in Tableau’s public cloud. Speed, flexibility (http://www.tableausoftware.com/ Full disclosure: We are Tableau evangelists of the first order, so take that for what it is worth. Check out the #Tableau and #TCC2010 stream on Twitter for some feedback from a wider audience of Also, I am happy to have any offline conversation on the matter of tools for the tasks at hand. MANY BLESSINGS! Peace and All Good! Michael W Cristiani • Analyze in SPSS. Play with in Excel. Visualize with actionscript and mxml in Adobe Flex □ I also analyze in SPSS and (sometimes) play with in Excel. • ArcGIS • Mathematica. • python/ numpy/ scipy/ matplotlib xlswt occasionally to export to Excel • I use Matlab. I notice that SPSS is not on the list and only has one vote in the comments. Do folks not use SPSS? I ask because I teach an undergrad stats class for psychology majors and I like to give them some exposure to data analysis software (other than excel). • Hello, Nathan! Bime Desktop – 1 hours, Esankey – 2 hours, Tableau – 14 hours, Excel – 68 hours. Also Parallel Sets, HCE and Google Docs Spreadsheet. • I use Gephi for exploratory network analysis. • I mostly use Excel, but I really like TinkerPlots for disaggregating and looking for patterns. And I also really like the latest visualizations that Google Spreadsheets has, especially the Gapminder-like motion graphs. Oh, and ManyEyes. Love ManyEyes. • Igor Pro for graphing and analysis – awesome program. Excel for some analysis and for “storing” data. Matlab for when nothing else works well. • Igor Pro • SPSS (now PASW) I’m happy to see ArcGIS up there though it’s not my primary. • I’ve used Spotfire for years and now it looks like they’re offering a free version with publishing for 1 year. • gnuplot • ArcGIS and MapInfo • Now Survo, a while ago it was SAS. • Protovis (Javascript) after cleaning up data with Excel. • I like NodeXL + Excel for social network analysis: • Gnuplot – quick and fast for just looking at stuff, but also can churn out publication-quality graphs if necessary (with some tweaking). Also sometimes R and ggplot2, depending on what I’m doing. • 1010data (www.1010data.com) • Chalk me up as another who still defaults to Matlab. Heck, I’ve even been known to write what are essentially shell scripts in Matlab because it’s just that easy. Inefficient, of course. But it beats figuring out the idiosyncrasies between tsch and bash, or, heaven forbid, learning Perl. I keep meaning to migrate to Processing, Protovis, and/or Matplotlib. Just need the right excuse… Anyone for Google Earth, MapServer, Octave, GMT or NCL? • Sure, it would be hard to imagine a world without excel but when it comes to grasping spatial variation in a dataset InstantAtlas is superb. • Python for everything.	{"url":"https://flowingdata.com/2010/09/07/poll-what-do-you-use-to-analyze-andor-visualize-data/","timestamp":"2024-11-12T15:23:47Z","content_type":"text/html","content_length":"129053","record_id":"<urn:uuid:9d8f6942-3380-44b6-be5b-5350377df730>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00179.warc.gz"}
Calculating Accumuldated Depreciation using Mixed References in Excel [Financial Modeling in Excel] Last time we had discussed the use of SumProduct() to ease your life for calculation of consolidated revenues and depreciation. This time we would be using the sum function! Yes you heard it right – The Sum function. But we would use the Sum function with a small trick! We would use it to calculate running cumulative sum! And believe me, you would need this function so many times – to calculate accumulated depreciation, cumulative debt, Profits to Retained Earnings and almost all the accounts that would consolidate into the balance sheet. Using Sum function to calculate cumulative values Most of the accounts in the Balance sheet are cumulative accounts (For example, cumulative debt, accumulated depreciation, etc.). In models, we need to have these as running numbers You would encounter this problem of calculating running cumulative values often and Sum formula (Used with slight intelligence) does come in very handy for the purpose. Using Sum() function for calculating running cumulative values Sum function uses an array as an input. If used intelligently – starting element of the array with fixed reference and the second part of the array as moving, it can help you get a running cumulative value. [read more: Relative & Absolute References in Excel Formulas] The basic trick is very simple. In the first cell in the sum function, I fix the first array argument using “$” and sum till the same cell. Now when I copy this function to the right, the first reference remains constant, whereas the second one keeps moving (as it is relative). This results in a growing array and hence a cumulative sum for accumulated depreciation. Where else can this function be useful in Finance? As I had pointed our earlier, most of the balance sheet numbers are accumulated numbers (Balance sheet is like a bucket, which accumulates values) and hence you can find the application of this running sum in almost all such accounts. I have used it (Can’t remember how many times!!) for converting Profits to Retained Earnings (when there are no dividends paid out), Debt Raised to accumulated debt (when there are no repayments), Debt for Interest During Construction, etc. How do you accumulate numbers? There are obviously multiple ways of doing the same thing in excel. I have shown you one way to get accumulated values. Conceptually accumulation is very simple – New Accumulated value = Old Accumulated Value + New Value. You can use this concept (maybe I will demonstrate in my next tutorial) to get the accumulation. How do you accumulate numbers in excel. I would encourage you to share your experience! Templates to download I have created a template for you, where the subheadings are given and you have use the functions to get the right values for you! You can download the same from here. You can go through the case and fill in the yellow boxes. I also recommend that you try to create this structure on your own (so that you get a hang of what information is to be recorded). Also you can download this filled template and check, if the information you recorded, matches mine or not! Join our Financial Modeling Classes We are glad to inform that our new financial modeling & project finance modeling online class is ready for your consideration. Please click here to learn more about the program & sign-up. The article is written by Paramdeep from Pristine. Chandoo.org has partnered with Pristine to launch a Financial Modeling Course. For details click here. 10 Responses to “Accumulated Depreciation using Mixed References” 1. This is a useful skill to know. It all revolves around the ability to "freeze ranges". I think the true benefits of freezing ranges (using the F4 key) need fully explaining, but this is a good example of where it is useful. 2. well in the example above you cna always use "F15 = E15 + F14" using all relatove referencing 🙂 3. @Stephen: Right! It is a useful skill. In fact you can use this technique in multiple instances. For example, if you want to calculate NPV of the project till various years, it can be used. @Nilay: Yes, that is also a mechanism and a nice one too. I use both the techniques.... 4. Hey; Good to know, it helps to add another additional skill 5. This reminded me of a recent post on the Fast Excel blog: Worth reading when you're doing these (and other) calculations on large data sets. 6. Dear M-B, This is interesting! I would say that if the calculations are very large in number (in the sum formula atleast), then it would make sense to take care in terms of excel operations! Usually in financial models (in equity side) the number of calculations is not so large. You work with data that would be in the range of 5-10 years. So I never thought about optimizing my code! Your post reminds me of coding and optimizing code at the assembly level. It is quite interesting! 🙂 7. Using the F4 key to effective anchor the base point of a calculation, such as producting moving cumulatives is acutally a fairliy basic thing that everyone doing any kind of financial modelling needs to be aware of. 8. I am still trying to find the cumulative sum as this one applied to a table and could grow, but still could not find a correct formula. 9. Hello, Sir, Actually, I have applied to the table and could grow, but still facing the same problem please help me out. 10. Great info and straight to the point. I am not sure if this is in fact the best place to ask but do you guys have any thoughts on where to hire some professional writers? Thank you 🙂	{"url":"https://chandoo.org/wp/accumulated-depreciation-using-mixed-references/","timestamp":"2024-11-11T18:01:19Z","content_type":"text/html","content_length":"453686","record_id":"<urn:uuid:baca3ae3-42b3-41d3-b486-0e118954ca92>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00666.warc.gz"}
Math Colloquia - Quasi-homomorphisms into non-commutative groups A function from a group G to integers Z is called a quasi-morphism if there is a constant C such that for all g and h in G, \|f(gh)-f(g)-f(h)\| < C. Surprisingly, this idea has been useful. I will overview the theory of quasi-morphisms including applications. Then we discuss a recent work with M.Kapovich when we replace the target group from Z to a non-commutative group, for example, a free group.	{"url":"http://my.math.snu.ac.kr/board/index.php?mid=colloquia&sort_index=room&order_type=asc&l=en&page=8&document_srl=766279","timestamp":"2024-11-03T08:54:41Z","content_type":"text/html","content_length":"43830","record_id":"<urn:uuid:b31efb6f-edd2-43bb-b925-4192cdf52819>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00606.warc.gz"}
Excel Line Chart Multiple Series 2024 - Multiplication Chart Printable Excel Line Chart Multiple Series Excel Line Chart Multiple Series – You can create a multiplication graph or chart in Shine by using a web template. You will find several samples of layouts and discover ways to format your multiplication graph using them. Here are several tips and tricks to generate a multiplication graph or chart. When you have a design, all you want do is copy the method and paste it within a new cellular. You can then make use of this method to increase a number of figures by yet another set up. Excel Line Chart Multiple Series. Multiplication kitchen table design You may want to learn how to write a simple formula if you are in the need to create a multiplication table. First, you have to secure row one of many header line, then flourish the amount on row A by cellular B. An additional way to build a multiplication kitchen table is by using blended personal references. In cases like this, you would probably key in $A2 into column A and B$1 into row B. The end result is really a multiplication dinner table with a method that works for both rows and columns. You can use the multiplication table template to create your table if you are using an Excel program. Just available the spreadsheet along with your multiplication dinner table change and template the label for the student’s title. You can even alter the page to fit your individual needs. There is an method to change the color of the tissue to alter the appearance of the multiplication table, also. Then, it is possible to change all the different multiples to meet your requirements. Developing a multiplication graph or chart in Excel When you’re using multiplication dinner table application, it is simple to build a basic multiplication desk in Shine. Just produce a page with rows and columns numbered from one to thirty. Where the columns and rows intersect is definitely the answer. For example, if a row has a digit of three, and a column has a digit of five, then the answer is three times five. The same goes for the opposite. Initial, you may enter into the amounts you need to increase. If you need to multiply two digits by three, you can type a formula for each number in cell A1, for example. To produce the numbers bigger, select the tissues at A1 and A8, and after that click on the proper arrow to pick a selection of tissues. After that you can sort the multiplication formula within the cellular material inside the other columns and rows. Gallery of Excel Line Chart Multiple Series Line Charts With Multiple Series Real Statistics Using Excel Line Charts With Multiple Series Real Statistics Using Excel Excel 2010 Tutorial For Beginners 13 Charts Pt 4 Multi Series Line Leave a Comment	{"url":"https://www.multiplicationchartprintable.com/excel-line-chart-multiple-series/","timestamp":"2024-11-06T10:21:59Z","content_type":"text/html","content_length":"53879","record_id":"<urn:uuid:44712688-4171-4c61-b21b-1237574e83e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00647.warc.gz"}
Research on aero-engine fault diagnosis based on integrated neural network In this paper, the fault diagnosis method of Integrated Neural Network based on oil and Vibration information fusion is studied and applied to Aero-engine Fault Diagnosis. Then, taking an CFM56-3 Aero-engine as an example, the application of Integrated Neural Network Fault Diagnosis method in bearing fault diagnosis of Aero-engine is studied by using the idea of Vibration information and oil information Fusion Diagnosis, and the diagnosis method is validated with specific data. The diagnosis results show that compared with the traditional single information source Fault Diagnosis method, the integrated neural network Fault Diagnosis method is more efficient, can detect more fault modes, and has lower misdiagnosis rate. • The fault modes of integrated neural network diagnosis results are more than those of single diagnosis results. • When the single diagnosis is contradictory, the diagnosis result of integrated neural network can solve the problem of diagnosis conflict well. • Integrated neural network diagnosis can effectively reduce the misdiagnosis rate. 1. Introduction Aero-engine, is a highly complex and sophisticated thermal machine, can provide the power required for flight engines for aircraft. As the heart of the aircraft, it is known as the “flower of industry”, and directly affects the performance, reliability and economy of the aircraft. It is an important manifestation of the strength of a country’s science, technology, industry and national defense. Because of the complex structure and various parts of Aero-engine, the fault diagnosis of Aero-engine has always been a difficult problem in the field of aero-intelligent maintenance. At present, for a certain type of Aero-engine in the field of civil aviation, the main fault diagnosis methods are based on the lubricating oil information and vibration information [1]. Literature [2] puts forward the process neural network method, realizes the prediction of Aero-engine vibration trend. However, many faults of Aero-engine cannot be represented by a single information. It is necessary to integrate the information of various information sources in order to detect these faults. In order to solve these problems effectively, the integrated neural network diagnosis technology emerged as the times require. This paper studies the application of this method to Aero-engine fault diagnosis. 2. Integrated neural network diagnosis technology The diagnostic technology of integrated neural network is to process the information from different information sources. The feature vectors of the processed information are transmitted to the diagnostic sub-networks by the feature information allocation unit. After the diagnosis results of the sub-networks are obtained, the diagnostic results of the sub-networks are transmitted to the decision fusion network, and the final fusion diagnosis results are obtained [3]. This diagnostic technology can combine the advantages of oil information fault diagnosis method and vibration information fault diagnosis method, improve the accuracy of fault diagnosis, and discover the reason of using a single information source [4]. Barrier diagnosis is difficult to detect faults. The structure of the integrated neural network diagnosis system is shown in Fig. 1. Fig. 1Structural diagram of integrated neural network diagnostic system 2.1. Data processing module This module is based on the original data to specific filtering, feature selection and selection of a series of operations, the diagnostic network provides the feature vector for each for ${X}_{i}\ left(i=1,2,...,m\right)$. By using this module, the filtering link can be started directly in the interface, and the data noise can be removed by filtering, so that the necessary data can be prepared automatically to the subsystem to ensure the operation of the diagnosis work. In addition to filtering, sample pretreatment can also be started through the interface to select the features of the selected parameters and prepare data for the training of the neural network [5]. 2.2. Characteristic information allocation unit The feature information distribution unit performs the task of assigning feature information to each diagnostic subnetwork. The distribution of feature information is determined by the structure of each diagnostic sub-network. The feature information allocated by each sub-network can be based on either a single parameter or multiple parameters. In the process of diagnosis, in order to achieve the optimal diagnosis results, feature information can be flexibly allocated according to training requirements, structure settings and expected results [6]. 2.3. Diagnostic subnetwork As the core of the whole diagnostic system, the diagnostic network consists of $M$ sub-networks responsible for diagnostic reasoning. Like rule-based expert system, diagnostic network includes two design parts: reasoning engine and knowledge base. It is only for diagnostic network. The reasoning mechanism adopted by inference engine has replaced the original reasoning based on symbolic rules by numerical calculation based on network structure, and the formation of knowledge base is also replaced by the learning and training of neural network [7]. If the diagnostic sub-network is allocated feature information based on multiple parameters, then assuming that the diagnostic confidence of n diagnostic parameters in the diagnostic sub-network for the same fault mode is ${t}_{i}\left(i=1,2,3,...,n\right)$, the diagnostic confidence of the diagnostic sub-network for the same fault mode is $T$: At the same time, the diagnostic input of the diagnostic sub-network based on multiple parameters can connect the eigenvectors ${e}_{i}\left(i=1,2,...,n\right)$ of all diagnostic parameters and form a new eigenvector $E$, $E=\left({e}_{1},{e}_{2},...,{e}_{n}\right)$. Usually, the three-layer network is used as the structure of each diagnosis sub-network, that is, the dimension of diagnostic input eigenvector is used as the number of input nodes, and the number of fault modes is used as the number of output nodes. In addition, the number of middle layer (hidden layer) nodes should be selected according to the training error requirement and the size of training sample set. 2.4. Decision fusion network The principle of decision fusion network is to fuse the diagnostic conclusions of each diagnostic sub-network and make decisions. Assuming that the integrated neural network diagnosis system diagnoses $n$-type faults, each sub-network diagnoses $n$-type faults based on its own characteristic information at the same time. The output ${R}_{i}$ of subnetwork $N{N}_{i}$ is called fault mode vector: The diagnostic results of each subnetwork for n-type faults are as follows [8]: In the formula, ${X}_{i}\in {R}^{Ni}$ is the diagnostic input eigenvector of function subnetwork $N{N}_{i}$; ${f}_{i}\left(\right)$ is the mapping of the $i$th subnetwork for the design of decision fusion network, which integrates the same nodes of each diagnostic subnetwork, that is, the diagnosis results of the same fault mode are fused by each diagnostic subnetwork, and finally a node is formed as the output of decision fusion network the final diagnosis result. If the diagnostic confidence of the $i$th diagnostic subnetwork for $n$-type faults is ${C}_{i}=\left({C}_{i1},{C}_ {i2},...,{C}_{in}\right)$, then the fault mode matrix $\mathbf{R}$ and the confidence weight matrix $\mathbf{C}$ formed by the parallel combination of the m diagnostic subnetworks are respectively: $\mathbf{R}=\left\{\begin{array}{cccc}{r}_{11}& {r}_{12}& \cdots & {r}_{1m}\\ {r}_{21}& {r}_{21}& \cdots & {r}_{2m}\\ ⋮& ⋮& ⋮& ⋮\\ {r}_{n1}& {r}_{n2}& \cdots & {r}_{nm}\end{array}\right\},$ $\mathbf{C}=\left\{\begin{array}{cccc}{c}_{11}& {c}_{12}& \cdots & {c}_{1n}\\ {c}_{21}& {c}_{22}& \cdots & {c}_{2n}\\ ⋮& ⋮& ⋮& ⋮\\ {c}_{m1}& {c}_{m1}& \cdots & {c}_{mn}\end{array}\right\}.$ The output of decision fusion network is: In the formula, $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}$ represents the diagonal element. The diagnostic probability of the $i$th fault is: In the formula, ${\sum }_{i=1}^{m}{c}_{ij}=$ 1, which is normalized, represents the contribution of each diagnostic subnetwork to fault diagnosis. It can be seen that the algebraic sum method can well reflect the role and influence of various factors in this synthesis. 3. Fault diagnosis of Aero-engine integrated neural network 3.1. Information preprocessing The CFM56-3 Aero-engine is one of the world’s most classic engines, accounting for 55 % of the B737 aircraft. B737 aircraft has been widely used because it meets the needs of the market more closely in terms of range and efficiency. According to the actual application of CFM56-3 Aero-engine, this paper studies the integrated neural network diagnosis technology of Aero-engine Based on the fusion of lubricating oil information and vibration information. It is difficult to analyze and process the diagnostic data because of the difference in both numerical value and dimension. Therefore, before the fusion diagnosis, the original symptoms need to be preprocessed. The specific method is to compare the original data with the standard threshold values of various diagnostic methods. The definition in the normal range is 0, and vice versa is 1, so that the original symptoms data can be converted to 0 and 1. According to the actual situation, the oil analysis mainly relies on spectral information because of the fewer ferrographic data of an Aero-engine. The concentrations of Fe, Cu, Mg, Al, Cr, Ni, Mo, Ti and Ag were selected as the original data for spectral diagnosis. After pretreatment, the diagnostic rules of spectral data are as follows: 1) Fe element concentration exceeding the standard (${S} _{S1}$); 2) Cu element concentration exceeding the standard (${S}_{S2}$); 3) Mg element concentration exceeding the standard (${S}_{S3}$); 4) Al element concentration exceeding the standard (${S}_ {S4}$); 5) Cr element concentration exceeding the standard (${S}_{S5}$); 6) Ni element concentration exceeding the standard (${S}_{S6}$); 7) Mo element concentration exceeding the standard (${S}_{S7} $); 8) Ti element concentration exceeding the standard (${S}_{S8}$); Element concentration exceeded the standard (${S}_{S9}$). At present, there are many vibration parameters applied in engine condition monitoring, and the object and occasion of application will have a greater impact on the diagnostic effect of various parameters. In order to obtain the time domain characteristic parameters of engine vibration signal, this paper extracts the time domain signal based on the noise reduction of the vibration signal, and carries on the time domain statistical analysis. The data of excessive peak value (${S}_{C1}$) and excessive root mean square value (${S}_{C2}$) are selected as vibration data. 3.2. Integrated neural network diagnosis The diagnostic system based on integrated neural network includes spectral information sub-network and vibration information sub-network. After pretreatment of various original symptoms, the Boolean value is obtained and used as the input of the sub-network. The output of each sub-network is the final failure mode. Taking engine rotor failure as an example, the failure modes are determined as follows: 1) normal system (${F}_{1}$); 2) bearing fatigue failure (${F}_{2}$); 3) bearing wear failure (${F}_{3}$); 4) gear gluing or scratch (${F}_{4}$); 5) gear overload fatigue (${F}_{5}$). Fuzzy comprehensive decision-making method is used to make a comprehensive decision on the diagnostic results of each sub-network, so as to consult each sub-diagnostic network. Through the above diagnosis results of each sub-network, matrix $\mathbf{R}$ can be formed, and the final comprehensive decision results can be obtained by multiplying matrix $\mathbf{R}$ and weight matrix $\mathbf{C} A Fuzzy Comprehensive Decision Model for: $\mathbf{R}=\left\{\begin{array}{l}{F}_{S}\\ {F}_{C}\end{array}\right\}=\left\{\begin{array}{lllll}{F}_{S1}& {F}_{S2}& {F}_{S3}& {F}_{S4}& {F}_{S5}\\ {F}_{C1}& {F}_{C2}& {F}_{C3}& {F}_{C4}& {F}_{C5}\ $\mathbf{C}=\left\{\begin{array}{ll}{C}_{11}& {C}_{12}\\ {C}_{21}& {C}_{22}\\ {C}_{31}& {C}_{32}\\ {C}_{41}& {C}_{42}\\ {C}_{51}& {C}_{52}\end{array}\right\}.$ In the matrix $\mathbf{R}$, ${F}_{S1}$, ${F}_{S2}$, ${F}_{S3}$, ${F}_{S4}$ and ${F}_{S5}$ are the diagnostic results of spectral information subnetwork, while ${F}_{C1}$, ${F}_{C2}$, ${F}_{C3}$, ${F} _{C4}$ and ${F}_{C5}$ are the diagnostic results of vibration information subnetwork. Weight matrix $\mathbf{C}$ mainly measures the contribution of various diagnostic methods to different fault According to the actual use of a certain type of engine and expert experience, the weights are determined as follows. 1) For “normal system (${F}_{1}$)”, the contribution of the two diagnostic methods is the same, that is, ${C}_{11}$ and ${C}_{12}$ are 0.5. 2) For “Bearing Fatigue Failure (${F}_{2}$)”, because vibration information has high reliability for the diagnosis of fatigue failure, it mainly relies on vibration information for diagnosis, so the weights of the two diagnostic methods are as follows: the weight ${C}_{21}$ of spectral information is 0.3, and the weight ${C}_{22}$ of vibration information is 0.7. 3) For bearing wear failure (${F}_{3}$), because the metal debris worn on the bearing surface is usually non-ferromagnetic particles and needs to be detected by spectroscopy, the weights of the two diagnostic methods are as follows: the weights of spectral information ${C}_{31}$ are 0.7, and the weights of vibration information ${C}_{32}$ are 0.3. 4) For “gear gluing or scratching (${F}_{4}$)”, because the spectrum can analyze the metal composition and content of gear wear. Vibration information contributes little to such faults, so the weights of the two diagnostic methods are: the weight ${C}_{41}$ of spectral information is 0.9, and the weight ${C}_{42}$ of vibration information is 0.1. 5) For “gear overload fatigue (${F}_{5}$)”, with fault 3, vibration information accounts for the largest proportion in this kind of fault diagnosis, so the weights of the two diagnostic methods are: spectral information weight ${C}_{51}$ is 0.3, vibration information weight ${C}_{52}$ is 0.7. In summary, the weight matrix $\mathbf{C}$ is: $\mathbf{C}=\left\{\begin{array}{ll}0.5& 0.5\\ 0.3& 0.7\\ 0.7& 0.3\\ 0.9& 0.1\\ 0.3& 0.7\end{array}\right\}.$ Vector $\mathbf{F}=\left({F}_{1},{F}_{2},{F}_{3},{F}_{4},{F}_{5}\right)$ comprehensive decision results, is the final diagnosis results. 4. Examples According to the flight parameters and lubricating oil data of an Aero-engine, the vibration value and lubricating oil data of a flight sortie are selected. The spectral data are shown in Table 1. Table 1Spectral data Element name Concentration value / ppm Element name Concentration value / ppm Fe 2.75 Ni 0.83 Cu 3.60 Mo 1.01 Mg 0.42 Ti 1.28 Al 3.22 Ag 1.32 Cr 2.75 – – Because there are too many points in the vibration data, only part of the vibration data are shown in this paper. The vibration data are shown in Table 2. The vibration peak value is 132.47 mm/s and the root mean square value is 20.463 mm/s. Table 2Partial vibration data Serial number 1 2 3 4 5 6 7 8 9 Vibration value / mm.s^-1 132.46 113.73 47.22 23.680 20.317 19.802 19.692 19.693 19.692 According to the vibration threshold value, abrasive concentration value and metal element concentration standard in the “Parameters and Processes of an Engine”, the comparative data analysis is as 1) Spectral raw data. If chromium exceeds the standard and other elements are normal, the sign vector is (${S}_{S1}$, ${S}_{S2}$, ${S}_{S3}$, ${S}_{S4}$, ${S}_{S5}$, ${S}_{S6}$, ${S}_{S7}$, ${S}_{S8} $, ${S}_{S9}$) = (0, 1, 0, 0, 0, 0, 0, 0, 0). 2) Vibration raw data. If the peak value is too large and the root mean square value is normal, the symptom vector is (${S}_{C1}$, ${S}_{C2}$) = (1, 0). The diagnostic confidence of spectral subnetworks, vibration subnetworks and integrated neural networks is shown in Table 3. Table 3Confidence in diagnosis of 3 diagnosis methods Failure mode Diagnostic method Normal system Bearing fatigue Bearing wear Gear scuffing and abrasion Gear overload fatigue Spectroscopic diagnosis 0.0036 0.0025 0.3555 0.9125 0.5646 Vibration diagnosis 0.346 0.9686 0.4540 0.0624 0.0080 Integrated neural network diagnosis 0.0006 0.7650 0.6395 0.6655 0.0366 From Table 3 we can draw the following conclusions. 1) The fault modes of integrated neural network diagnosis results are more than those of single diagnosis results. For example, in the single spectrum diagnosis, only the “gear gluing and scratching” fault is diagnosed, in the single vibration diagnosis, only the “bearing fatigue” fault mode is diagnosed, while in the integrated neural network diagnosis, the fault mode is “bearing wear”, “bearing fatigue” and “gear gluing and scratching”. Obviously, integrated neural network diagnosis can find more faults than single diagnosis. 2) When the single diagnosis is contradictory, the diagnosis result of integrated neural network can solve the problem of diagnosis conflict well. For example, in the single spectrum diagnosis, the “gear gluing and scratching” fault is diagnosed with a confidence of 0.9123, while in the single vibration diagnosis, the “bearing fatigue” fault is diagnosed with a confidence of 0.9685. The diagnosis is contradictory. However, in the conclusion of the integrated neural network diagnosis, it can be seen that the problem of diagnosis conflict has been solved, and the “gear gluing and scratching” fault has been diagnosed at the same time. Injury fault (confidence 0.6654) and bearing fatigue fault (confidence 0.7652). Obviously, the integrated neural network diagnosis results are more reliable and accurate. 3) Integrated neural network diagnosis can effectively reduce the misdiagnosis rate. For example, the spectrum sub-network diagnoses the “gear overload fatigue” fault mode, but the fault is not detected in the conclusion of the integrated neural network diagnosis. Obviously, the misdiagnosis rate of integrated neural network diagnosis is lower. 5. Conclusions In this paper, an integrated neural network diagnosis method based on oil information and vibration information fusion is studied and applied to bearing wear fault diagnosis of civil aviation CFM56-3 engine. The problem of low diagnostic efficiency and accuracy of aero-engine using single information source fault diagnosis method is solved. This method can also be applied to the diagnosis of other faults of aero-engine, which provides a new method for the fault diagnosis of aero-engine. • Wan Peng Research and Implementation of Flight Control Ground Fault Diagnosis System Based on Symptom Analysis. Ph.D. Thesis, University of Electronic Science and Technology, Chengdu, 2018. • Shi Xiangyang Research on fault diagnosis of B737 aircraft electric starting system based on BP neural network. Journal of Binzhou University, Vol. 12, Issue 6, 2018, p. 5-8. • Ling Yiqin Research on Fault Diagnosis Technology of Aircraft System Based on Fault Propagation Mechanism and Petri Net. Ph.D. Thesis, Nanjing University of Aeronautics and Astronautics, Nanjing, • Hao Yichuan Research on Fault Diagnosis Method of Aircraft Electronic System Based on Rough Set and Neural Network. Ph.D. Thesis, China Civil Aviation University, Tianjin, 2015. • Wang Hongwei Research on Key Technologies of Fault Diagnosis and Prediction for Rolling Bearing of Aero-engine. Ph.D. Thesis, Nanjing University of Aeronautics and Astronautics, Nanjing, 2015. • Deng Zheng Research on Aircraft Burst Fault Diagnosis Based on T-S Fuzzy Neural Network. Ph.D. Thesis, China Civil Aviation University, Tianjin, 2014. • Wu Wenjie Fault Diagnosis Method of Aero-Engine Based on Information Fusion. Ph.D. Thesis, University of Electronic Science and Technology of China, Chengdu, 2011. • Jin Xiangyang, Lin Lin, Zhong Shisheng Process neural network method for prediction of aero-engine vibration trend. Vibration, Testing and Diagnosis, Vol. 31, Issue 3, 2011, p. 331-334. About this article integrated neural network fault diagnosis Project supported by National Natural Science Foundation of China, No. 51605037; Binzhou University Scientific Research Fund Project, No. BZXYG1705; Binzhou Soft Science Research Project, No. Copyright © 2019 Shi Xiangyang. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.	{"url":"https://www.extrica.com/article/20636","timestamp":"2024-11-05T12:27:43Z","content_type":"text/html","content_length":"138529","record_id":"<urn:uuid:74cf4d86-6aba-4175-8f7d-d4d20727781c>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00597.warc.gz"}
How to extend the life of electric motors in winches and air compressors A little understanding will help you to extend the life of electric motors in equipment such as winches and air compressors. Here’s how… Warning! The following Unsealed 4X4 story contains mathematical equations… but it’s not too hard to understand, and it will help you extend the life of the electric motors in 4×4 equipment such as winches and air compressors. Save your 12-volt motor: A guide to amp draw. Chances are you’ve got a 12-volt motor somewhere in your four-wheel drive, whether it be in the winch hanging off your bull bar or your compressor for airing back up after a day on the tracks… nearly all of us have one. But when was the last time you gave a thought to its lifespan, or what you can do to extend it? And, interestingly, I’m not talking about stripping it down and rebuilding it. I’m talking about giving it good, quality power. In this yarn, I’m going to walk through the maths around 12-volt systems, amp draw and how this affects your 12-volt gear. The maths Let’s get the maths out of the way first. The calculation, or equation, you need to do is very simple, and can be easily knocked over with the calculator on your phone (even if you still own a 3310!). You need two pieces of information to arrive at the third. The equation looks like this: I = P/V Makes bugger-all sense, right? What if I write it like this: Amps = Watts ÷ Volts It’s basically the same thing; where I is current in amps, P is power in Watts and V is voltage in volts. Pretty simple, right? So that first one, to get amps, you just divide the Watts by the number of volts. What about if you want to find the Watts? You just need to move the equation around: P = V x I Or, more simply (if you weren’t paying attention above): Watts = volts x amps. If you want to get carried away and confirm the specified voltage of a bit of gear, you can work out volts too by: V = P/I But we know we’re working with a 12-volt system, so we don’t need to be too concerned with this equation. So why would you want to know this? We’re glad you asked! Let’s just say you’re trying to work out what amp rating you need for your wiring for something. And let’s also assume you know its power rating in Watts. As an example, car stereo amplifiers are often rated in Watts, and we’re talking RMS Watts here, not PMPO (if you’ve got a 3000W PMPO stereo, it’s actually probably closer to 300W RMS). Anyway, before I get carried away on a ‘fully-sick stezza’ tangent, we know that the amplifier we have will draw 300 Watts, and we also know our four-wheel drive is 12-volts (-ish). So, to work out what size wiring and fuse we need, use this equation: I? = 300w/12v 300 Watts divided by 12-volts is 25 amps (when it’s at absolute max power draw or max volume). So, we need wire that will happily carry 25 amps, and a fuse below our wire rating. So if it were me, I’d be going for some 8AWG wire that’s rated at 50 amps and putting a 30 amp fuse next to the battery. This means that the wire will carry up to 50 amps without melting, and the fuse should be able to hold the amp draw without an issue. But we’ve got a 30 amp fuse because if the wiring does happen to short out, or get crushed, the fuse will blow before the wire (and in turn your four-wheel drive) starts to burn. Mind you, I’m not getting into wire run lengths and voltage drop at this stage; I’m trying to keep it simple. That’s a yarn for another day, I reckon. Ok, so what’s that got to do with compressors and winches? It’s the other side of the equation we want to think about now; amps. The single biggest killer of electric motors is heat. The term ‘burnt-out motor’ isn’t just a tricky name. First off, you should know with an increase in amp draw comes an increase in heat generation, thus dead or burnt-out electric motors. There are two things that will increase amp draw in anything electric; first off, load. An electric motor will run at just about any voltage, quite happily, with no load on it (keeping it simple). Add a load to the motor, whether it be turning a winch drum, pulling a four-wheel drive out of the mud or pumping a piston up and down to generate air pressure, you’re loading it up. In turn, you’re increasing the amps drawn, and thus increasing the heat generated. The second thing that will increase amp draw is the voltage (number of volts) you’re supplying. “But Wes, my four-wheel drive has a 12-volt system,” you might say. You’d be right… and wrong. Hear me When your four-wheel drive is running, your alternator is charging the battery and supplying anywhere between 13.8 volts and 14.4 volts. Turn it off and the battery will still sit around the 12.6 and 12.8 volts. Still more than 12, right? Ok, now run your winch or your compressor with your fourby turned off and all of a sudden you’re down to 11.4 volts to 10-ish volts. When you stop winching or turn off the compressor, the battery will generally recover back up to around 12.6 volts, but while you’re drawing the current from the battery, the voltage drops considerably. Lets go back to the maths (groan!) Let’s say we have a compressor (like the ARB twin compressor I used for the video in this yarn) and it’s got a stated amp draw of 40 amps while under load. This amp draw rating is expressed at 12 volts (unless otherwise stated… and it sometimes is at 13.8 volts). So from our maths above, we know the following: @ 12 volts, we’re drawing 40 amps which we can calculate as 480 Watts (Watts = 12v x 40a). The motor will run at this generating a bit of heat, but within spec. @ 14 volts, we’re drawing 34.2 amps (amps = 480w ÷ 14v), so there’s less amp draw, less heat and a happy compressor. @10 volts, we’re drawing 48 amps (amps = 480w ÷ 10v) , so there’s greater amp draw, greater heat and an angry compressor. Now, an extra 8 amps of power draw doesn’t sound like much, but when you think about that, over the space of 10 minutes pumping up four 33-inch tyres, and then your mates tyres, or the camper, it gets up there. Especially so when you’re pumping up to 40psi. There’s bugger-all load when your compressor is venting to atmosphere but when it’s trying to build pressure, it gets right up there (mind you, apparently those little ARB twin compressors are good for up to 140psi – I’ve gotta find a proper truck tyre and test that!) Where this gets rather interesting is when you exchange an air compressor for a winch. The average winch will draw upwards of 400 amps at 12 volts. So add an extra 80 amp draw to that and watch how quick it gets hot and melts the motor! So you said something about saving money? If you take nothing else from this yarn (and maths aren’t fun… so I understand if you glossed over that bit), make sure you run your bloody four-wheel drive engine when you’re running any electric motor. Even if it’s for 10 seconds, but especially if you’re going to pump a tyre up or try to recover a bogged vehicle. Despite shortening the lifespan of your electric motor’s brushes and commutator, you can have a massive failure of the motor, melt the windings up, and your motor won’t work anymore. If you happen to have bought said compressor or winch recently and it’s still within warranty, I’m sorry to tell you, chances are, wherever you bought it from won’t warrant it. That’s because you’ll have done the wrong thing, which caused the failure. By knowing all this, hopefully you’ll run your engine when you’re airing up or winching, not just put the ‘reds’ on. And chances are this will save you the cost have having to buy a replacement compressor or winch! Anything else I should know? This same principle affects all electrics but motors are one of the more impacted bits of kit you’ll have that will suffer from low voltage/high amperage. And it applies to other electrical items, not just those in your vehicle, such as the fridge in your house (though that’s at 240-ish volts) to the phone in your hand. This is one of the greatest reasons lithium batteries have advanced so much, and have been such an improvement for cordless drills, for example, because they give a constant voltage right up until they’re flat. Old Ni-Cad batteries didn’t, and many a motor and set of brushes were burnt out sooner than they do these days. Another pay to maximise the life of electric motors is to have a bit of mechanical (and electrical) sympathy. Unless you’re in a mad rush, take your time with vehicle recoveries and give your compressor a minute or two to rest between tyres. As well as your electrical components thanking you for years to come, you’ll also get more chin-wagging time with your mates. One last thing… If you happen to be an electrical engineer, and need to point out how I’ve skimmed over the details and over-simplified stuff in this story, or you feel the need to educate me on Ohm’s Law, drop us a line… but yes, I know. Story too long? Check out Wes’ video: Leave a comment Leave a comment	{"url":"https://unsealed4x4.com.au/how-to-extend-the-life-of-electric-motors-in-winches-and-air-compressors/","timestamp":"2024-11-14T04:21:24Z","content_type":"text/html","content_length":"154834","record_id":"<urn:uuid:eadba90a-483e-44b4-af94-708055229337>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00663.warc.gz"}
st: Re: better match-management algorithm [Date Prev][Date Next][Thread Prev][Thread Next][Date index][Thread index] st: Re: better match-management algorithm From Kit Baum <[email protected]> To [email protected] Subject st: Re: better match-management algorithm Date Sat, 4 Aug 2007 09:28:47 -0400 As much as I enjoy Mata programming, I don't think this case obviously calls for Mata. I think it should be feasible to do this without explicit loops at all, using something like the third stanza of the enclosed (the first two just make up some transactions-time data for stock and options quotes: set obs 100 egen transtime = fill(1 2 4) g stockprice = 10uniform()+50 sort transtime save stox, replace set obs 75 egen transtime = fill(1.25 3.75 5.2) g optprice = 10uniform()+45 sort transtime save opts, replace use stox merge transtime using opts sort transtime g prevstock=cond(stockprice[_n-1]<.,stockprice[_n-1], /// cond(stockprice[_n-2]<.,stockprice[_n-2],stockprice[_n-3])) /// if optprice<. l transtime optprice prevstock if optprice<. I am not usually a fan of nested cond() calls, but in this case it seems to work well. If you scale it up to 500,000 stock quotes and 375,000 options quotes, it does the job (without the list) in 4.88 seconds (Stata 10/MP2; it is using parallel computation heavily). Whenever an explicit loop appears in the logic, one should think twice or thrice about whether there is a better way. In Stata there often is. Kit Baum, Boston College Economics and DIW Berlin An Introduction to Modern Econometrics Using Stata: On Aug 4, 2007, at 2:33 AM, Tobias wrote: I encountered the following problem in a finance research project: two tables, one with option prices, the other with (underlying) stock prices. The task is to match the appropriate stock price to each option price observation. My current solution works, but seems to be inefficient due to tremendous processing time (> 4h). My current solution is the following (I refer to the following numbers in the code below): 1) Fetch number of observations from underlying table. 2) Fetch number of obs. from option table 3) Merge the underlying prices to the option prices (one-to-one merge) 4) Using two nested forvalues loops, I iterate over the option observations to find an appropriate underlying price again iterating over the underlying prices in the second forvalues loop. [The matching criteria are an identical ISIN number, identical trading_date, and that the trading time of the subsequent underlying is bigger than the option trading time, i.e. looking for the most recent underlying price.] Before writing down my code, I would have the following questions: A) IS THERE A MORE EFFICIENT WAY TO CARRY OUT THE CONDITIONAL MATCHING WITHOUT HAVING TO ITERATE OVER EACH AND EVERY OBSERVATION ? B) IF NOT, IS IT POSSIBLE TO 'OUTSOURCE' THE TASK TO A MATA PROGRAM, SUCH THAT THE COMPILATION OF THE LOOP-CODE IS DONE ONCE INSTEAD OF A MILLION TIMES ? I thought about the Mata possibility when I read in a presentation by Kit "Your ado-files may perform some complicated tasks which involve many invocations of the same commands. Stata's ado-file language is easy to read and write, but it is interpreted: Stata must evaluate each statement and translate it into machine code. The new Mata programming language (help mata) creates compiled code which can run much faster than ado-file code." * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/	{"url":"https://www.stata.com/statalist/archive/2007-08/msg00174.html","timestamp":"2024-11-12T05:45:23Z","content_type":"text/html","content_length":"10770","record_id":"<urn:uuid:4b9a72fc-c517-437c-84f9-a7b1f1dcf6e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00342.warc.gz"}
Linear Equations Worksheet Answers - Function Worksheets Representing Linear Equations Worksheet – A well-made Capabilities Worksheet with Responses will give you college students with solutions to a variety of crucial queries about … Read more Linear Equations From Tables Worksheet Linear Equations From Tables Worksheet – A highly-developed Features Worksheet with Replies will give you college students with techniques to various essential queries about characteristics. … Read	{"url":"https://www.functionworksheets.com/tag/linear-equations-worksheet-answers/","timestamp":"2024-11-09T16:12:00Z","content_type":"text/html","content_length":"58441","record_id":"<urn:uuid:a0dbcf5b-6dd2-4ded-b9fc-e3bf392edca2>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00082.warc.gz"}
Comparing Fractions with Unlike Numerators Using Journaling This is an introduction to comparing fractions with like denominators and unlike numerators, for students with a basic understanding of fractions as part of a whole, numerators, and denominators. After completing the lesson, students will be able to name and write fractions represented by drawings or models and represent a fraction using models and symbols. Key Understandings/Vocabulary • Fraction: A number used to name a part of a group or a whole. The number below the bar is the denominator, and the number above the bar is the numerator. • Numerator: The top part of a fraction. The numerator represents how many pieces of the whole that are discussed. • Denominator: The bottom part of a fraction. The denominator represents the total number of equal parts in the whole or the set. • Like Denominator: Fractions that have the same number as the denominator. 1. Demonstration a. Journaling. Have students write a short definition of a numerator in their journals. This is meant to access their prior knowledge of fractions, specifically numerators. Example journal entry: Numerators are the smaller number in a fraction. They are on the top of the fraction. b. Review parts of a fraction using models. Draw a square divided into 4 equal parts with 3 parts shaded. Write the fraction 3/4 and review how it represents three of four equal parts shaded. The 3, or numerator, tells how many parts are shaded, while the 4, or denominator, shows how many equal parts the whole is divided into. Ask students to volunteer their definitions of numerators and discuss the correct definition, which could be something like: The number above the line in a fraction. The numerator represents how many pieces of the whole that are discussed. Draw a model of the fraction 2/4 and compare the two fractions. Ask, "Which fraction is greater?" Discuss the idea that the amount of space that is shaded shows which fraction is c. Journaling. Have students answer: How do you know if a fraction is greater than another fraction? This is meant to help students access their thinking and problem solving in comparing fractions. Example journal entry: I know when a fraction is bigger than another fraction by comparing how big they both are. d. Draw a rectangle with 4 parts. Shade one of the equal parts, and write the fraction. Explain what the numerator represents. Draw a similar rectangle directly below the first rectangle, and shade another equal part, so that 2 parts are shaded, and write the fraction, explaining the numerator. Draw another rectangle, directly below the second rectangle, and shade another part, and write the fraction. Point out that the number of shaded parts increases, and discuss how the numerator changes, depending on how many parts are shaded. e. Have students use this handout and square pieces of paper cut to use as fraction squares. Ask them to cover one more part, and write the fraction. Continue until all parts are covered. The fractions they wrote should be: 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, 8/8. f. Journaling. Write what happens when the numerator of a fraction increases. Pair up students, have them exchange notebooks and discuss what they found. As a class, have students share what they wrote, and discuss their findings. Example journal entry: The numerator gets bigger when more of the parts are shaded. The number of equal parts doesn't change when the numerator increases. As the numerator increases, the covered parts increase. g. Compare fractions. Introduce or review the inequality symbols of greater than, > and less than, <. have="" students="" use="" the="" fraction="" pieces="" to="" show="" and="" then="" explain ="" how="" write="" an="" inequality="" from="" left="" right="" read="" it="" as="" a="" sentence.="" for="" example="" is="" less="" than="" three-eighths.="" if="" need="" help="" in="" remembering="" which="" symbol="" you="" might="" want="" them="" think="" of="" animal="" such="" alligator="" who="" wants="" eat="" greatest="" fraction.="" should="" be="" opened=""> 2. Guided Practice To provide guided practice comparing fractions with like denominators, have each student in a pair cover as many parts as they choose, then compare the two fractions that are made. Have pairs determine which is greater, by writing the inequality, and stating the fractions aloud. For example students may write 1/8 > 2/8, and say, "1/8 is greater than 2/8." 3. Sharing Ideas Have students return to their previous writing, and revise their answers to the question: How do you know if a fraction is greater than another fraction? Discuss answers with the class. Journaling. Have students reflect on the lesson and journaling. Have them write about what they liked about journaling and if they learned more about comparing fractions. 4. Independent Practice Have students draw several pictures of two fractions with like denominators, and then determine which is greater. Students can write an inequality comparing the fractions. For example: 2/3 > 1/3 5. Assessment Check student understanding by observing their answers during guided practice, along with checking their independent practice examples. Use a lesson that is an introduction to comparing fractions with like denominators and unlike numerators, for students with a basic understanding of fractions as part of a whole, numerators, and denominators. Students use math journals to complete the lesson.	{"url":"https://www.teachervision.com/lesson/comparing-fractions-unlike-numerators-using-journaling","timestamp":"2024-11-03T20:19:11Z","content_type":"application/xhtml+xml","content_length":"252866","record_id":"<urn:uuid:dbe6780d-cc25-42e3-9b52-786faddb7598>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00635.warc.gz"}
Sonlu farklar yöntemi yardımıyla titmetre ölçümlerinden yapısal deformasyonların belirlenmesi Fen Bilimleri Enstitüsü Similarly, it was considered the polynomial spline function as approximating function. In spite of the fact that cubic piecewise polynomial spline is rather easy for calculating and using, it is the most commonly used piecewise polynomial spline function. Assuming that y(x) be a continuous, bounded function defined in the closed interval a < x ^ b and subdivided the interval by a set of mesh points. a = Xo < Xi < X,,.! < x" = b S(x) cubic spline is a continuous polynomial function for a < x < b such that, (i) S(x) is a different cubic polynomial function in every interval (xj, Xj+i) (İİ) S(x) function and its first, second derivatives S'(x), S"(x) functions, are continuous for while x values on the closed interval a <, x < b. New displacement values were obtained from measurements slope values by cubic piecewise polynomial and optimization techniques be inherent in Matlab software. They are compared with specified displacement values from slope measurement values via finite difference and cubic spline techniques. Statical analysis of the highway bridge is made by computer program, which it is written to be used finite difference method in the Fortran computer languge. The program consider multispans in which the end supports may be either fixed or pinned in torsion and bending, and the girders are equally spaced and have constant radius of curvature, and cross-sectional properties may vary along the span. Cross-sectional properties of the structure (Ix, Iy, Iw Kr that are respectively bending moment of inertia, warping constant, pure tortional constant of beams) are calculated for all different cross-sections. Data file which is consisted of these values, loading cases and physical properties of the structure are prepared, and by mean of result file which is existed by computer program. Displacement values are obtained by means of the result file which is existed by computer program at the measuring points. The last values are compared with the other.values which is to be obtained by cubic piecewise polynomial and finite difference methods, and at last, these results are compared with LVDT measurements and Theoretical solutions that are sent by Wroclaw Technical University, it was discussed whether the results are compared with each other. For further topic, surface impressions may be specified using tiltmeter apparatus by means of finite difference or cubic piecewise spline methods at two dimensional constructional members (esp, slap, plate).Assuming that a function z(x, y) had the partial derivatives Dx, Dy with respect to x, y axes. If h is the constant spacing of the pivotal points in the x axis direction, first central difference of z(x, y) at i point. 2h Dx z{ = Zr - z, + s (h2) Similarly, calling k the constant spacing of the pivotal points in the y axis direction. 2 k Dy z; = za - zb + s (h2) and the second mixed derivative of z(x, y) with respect to x, y is obtained by the product Dx. Dy (Figure 1.1) 4 h k Dxy Zi = 2W - Zai - Zbr + ZW + S (h2) Figure 1.1 Two dimensional pivotal points Its horizontal deformations may be also specified using tiltmeter apparatus at vertical constructional members (esp, pillars of bridges). XIV Similarly, it was considered the polynomial spline function as approximating function. In spite of the fact that cubic piecewise polynomial spline is rather easy for calculating and using, it is the most commonly used piecewise polynomial spline function. Assuming that y(x) be a continuous, bounded function defined in the closed interval a < x ^ b and subdivided the interval by a set of mesh points. a = Xo < Xi < X,,.! < x" = b S(x) cubic spline is a continuous polynomial function for a < x < b such that, (i) S(x) is a different cubic polynomial function in every interval (xj, Xj+i) (İİ) S(x) function and its first, second derivatives S'(x), S"(x) functions, are continuous for while x values on the closed interval a <, x < b. New displacement values were obtained from measurements slope values by cubic piecewise polynomial and optimization techniques be inherent in Matlab software. They are compared with specified displacement values from slope measurement values via finite difference and cubic spline techniques. Statical analysis of the highway bridge is made by computer program, which it is written to be used finite difference method in the Fortran computer languge. The program consider multispans in which the end supports may be either fixed or pinned in torsion and bending, and the girders are equally spaced and have constant radius of curvature, and cross-sectional properties may vary along the span. Cross-sectional properties of the structure (Ix, Iy, Iw Kr that are respectively bending moment of inertia, warping constant, pure tortional constant of beams) are calculated for all different cross-sections. Data file which is consisted of these values, loading cases and physical properties of the structure are prepared, and by mean of result file which is existed by computer program. Displacement values are obtained by means of the result file which is existed by computer program at the measuring points. The last values are compared with the other.values which is to be obtained by cubic piecewise polynomial and finite difference methods, and at last, these results are compared with LVDT measurements and Theoretical solutions that are sent by Wroclaw Technical University, it was discussed whether the results are compared with each other. For further topic, surface impressions may be specified using tiltmeter apparatus by means of finite difference or cubic piecewise spline methods at two dimensional constructional members (esp, slap, plate). XUl Assuming that a function z(x, y) had the partial derivatives Dx, Dy with respect to x, y axes. If h is the constant spacing of the pivotal points in the x axis direction, first central difference of z(x, y) at i point. 2h Dx z{ = Zr - z, + s (h2) Similarly, calling k the constant spacing of the pivotal points in the y axis direction. 2 k Dy z; = za - zb + s (h2) and the second mixed derivative of z(x, y) with respect to x, y is obtained by the product Dx. Dy (Figure 1.1) 4 h k Dxy Zi = 2W - Zai - Zbr + ZW + S (h2) Figure 1.1 Two dimensional pivotal points Its horizontal deformations may be also specified using tiltmeter apparatus at vertical constructional members (esp, pillars of bridges). XIV Similarly, it was considered the polynomial spline function as approximating function. In spite of the fact that cubic piecewise polynomial spline is rather easy for calculating and using, it is the most commonly used piecewise polynomial spline function. Assuming that y(x) be a continuous, bounded function defined in the closed interval a < x ^ b and subdivided the interval by a set of mesh points. a = Xo < Xi < X,,.! < x" = b S(x) cubic spline is a continuous polynomial function for a < x < b such that, (i) S(x) is a different cubic polynomial function in every interval (xj, Xj+i) (İİ) S(x) function and its first, second derivatives S'(x), S"(x) functions, are continuous for while x values on the closed interval a <, x < b. New displacement values were obtained from measurements slope values by cubic piecewise polynomial and optimization techniques be inherent in Matlab software. They are compared with specified displacement values from slope measurement values via finite difference and cubic spline techniques. Statical analysis of the highway bridge is made by computer program, which it is written to be used finite difference method in the Fortran computer languge. The program consider multispans in which the end supports may be either fixed or pinned in torsion and bending, and the girders are equally spaced and have constant radius of curvature, and cross-sectional properties may vary along the span. Cross-sectional properties of the structure (Ix, Iy, Iw Kr that are respectively bending moment of inertia, warping constant, pure tortional constant of beams) are calculated for all different cross-sections. Data file which is consisted of these values, loading cases and physical properties of the structure are prepared, and by mean of result file which is existed by computer program. Displacement values are obtained by means of the result file which is existed by computer program at the measuring points. The last values are compared with the other.values which is to be obtained by cubic piecewise polynomial and finite difference methods, and at last, these results are compared with LVDT measurements and Theoretical solutions that are sent by Wroclaw Technical University, it was discussed whether the results are compared with each other. For further topic, surface impressions may be specified using tiltmeter apparatus by means of finite difference or cubic piecewise spline methods at two dimensional constructional members (esp, slap, plate). XUl Assuming that a function z(x, y) had the partial derivatives Dx, Dy with respect to x, y axes. If h is the constant spacing of the pivotal points in the x axis direction, first central difference of z(x, y) at i point. 2h Dx z{ = Zr - z, + s (h2) Similarly, calling k the constant spacing of the pivotal points in the y axis direction. 2 k Dy z; = za - zb + s (h2) and the second mixed derivative of z(x, y) with respect to x, y is obtained by the product Dx. Dy (Figure 1.1) 4 h k Dxy Zi = 2W - Zai - Zbr + ZW + S (h2) Figure 1.1 Two dimensional pivotal points Its horizontal deformations may be also specified using tiltmeter apparatus at vertical constructional members (esp, pillars of bridges). Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Sosyal Bilimler Enstitüsü, 1997 Anahtar kelimeler Eğim ölçer, Kompozit malzemeler, Sonlu farklar yöntemi, Yük taşımacılığı, Clinometer, Composite materials, Finite differences method, Freight transportation	{"url":"https://polen.itu.edu.tr/items/454699fd-d5a3-46cf-a797-9829c217a59d","timestamp":"2024-11-14T05:26:58Z","content_type":"text/html","content_length":"183794","record_id":"<urn:uuid:65cf544d-80d2-4d84-9ad9-785cf8e8e054>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00321.warc.gz"}
Question #e2e23 \| Socratic Question #e2e23 1 Answer Your tool of choice here will be the conversion factor $\textcolor{b l u e}{\underline{\textcolor{b l a c k}{\text{1 in = 2.54 cm}}}}$ In your case, you want to convert the number of inches to centimeters, so set up the conversion factor as $\text{what you need"/"what you have" = "2.54 cm"/"1 in}$ This means that you have $6.75 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{in"))) * "2.54 cm"/(1color(red)(cancel(color(black)("in")))) = color(darkgreen)(ul(color(black)("17.1 cm}}}}$ The answer is rounded to three sig figs, the number of sig figs you have for the number of inches. Impact of this question 3534 views around the world	{"url":"https://api-project-1022638073839.appspot.com/questions/59a89d25b72cff3490ce2e23","timestamp":"2024-11-12T07:18:52Z","content_type":"text/html","content_length":"33923","record_id":"<urn:uuid:dff483e3-9522-403e-90c2-6790932ea3d7>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00555.warc.gz"}
Tooling for inference and simulation on temporal point processes? Hi all, I’m looking for advice (statistical and computational) to do something like the following simulation/inference for nonhomogeneous point processes (temporal, not considering space): 1. Specify a “prior” for the intensity function as a spline or some other function for which it is easy to enforce constraints (e.g. cyclic over work weeks, monotonic increasing, etc) 2. Simulate from the NHPP under the prior 3. When data arrives, update posterior distribution, do more simulations until more data arrives. I’m not particularly attached to any specific estimation method, in fact having something that could be run relatively fast (MAP, Laplace approximation, etc) would be more valuable than something that took a much longer time. I hope the question isn’t too vague. Thanks for any papers, tutorials/examples, packages, etc that anyone can suggest!	{"url":"https://discourse.julialang.org/t/tooling-for-inference-and-simulation-on-temporal-point-processes/121885","timestamp":"2024-11-14T08:56:58Z","content_type":"text/html","content_length":"20926","record_id":"<urn:uuid:d23de2be-1db2-48ca-a005-1fbaaf6d028d>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00204.warc.gz"}
Concept information 2-methylheptane and 3-methylheptane amount fraction • Amount fraction is used in the construction mole_fraction_of_X_in_Y, where X is a material constituent of Y. 2-methylheptane and 3-methylheptane is a mixture of two compunds and belongs to the group of alkanes. {{#each properties}} {{toUpperCase label}} {{#each values }} {{! loop through ConceptPropertyValue objects }} {{#if prefLabel }} {{#if vocabName }} {{ vocabName }} {{/if}} {{/each}}	{"url":"https://vocabulary.actris.nilu.no/skosmos/actris_vocab/en/page/2-methylheptaneand3-methylheptaneamountfraction","timestamp":"2024-11-08T04:24:29Z","content_type":"text/html","content_length":"20857","record_id":"<urn:uuid:3a9f39b9-5f87-4401-b113-4bf5d57cf608>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00141.warc.gz"}
Tenure Calculator Business Planning Tenure Calculator Understanding the Tenure Calculator The Tenure Calculator is an online tool that helps to determine the future value of an investment or loan over a specified period. This calculator is particularly useful for individuals and businesses planning to engage in financial investments or loan management. By inputting a few key details, users can quickly and accurately predict how much their initial principal amount will grow over time. Applications of the Tenure Calculator This calculator has numerous practical applications. Whether you are a business looking to forecast the future value of a financial venture or an individual planning for retirement savings, the Tenure Calculator provides valuable insights. It helps in making informed decisions about investments, loans, and savings plans. Additionally, financial advisors and planners can use this tool to illustrate potential growth scenarios to their clients. Benefits of the Tenure Calculator The primary benefit of using the Tenure Calculator is the ability to forecast the future value of your investments with precision. By understanding how interest rates and compounding periods affect growth, users can make strategic decisions about where and how to invest their money. It also allows for quick comparisons between different investment options by altering the variables in the Deriving the Answer The result provided by the Tenure Calculator is derived based on the compound interest formula. Here's how it works in simple terms. When you invest a principal amount at a specified annual interest rate, the interest earns on both the initial principal and the accumulated interest over multiple periods. By specifying the number of compounding periods per year, the calculator factors in how often this interest is applied, resulting in the future value of the investment or loan at the end of the given duration. Key Inputs and Their Importance • Principal Amount: This is the initial amount of money that is either borrowed or invested. • Annual Interest Rate: It's the percentage rate at which your investment grows annually. • Compounding Periods per Year: This represents how frequently the interest is applied to the principal amount within a year. Common options include annually, semiannually, quarterly, monthly, and • Duration: The length of time over which the interest is calculated, expressed in years. Armed with this information, users can explore various scenarios by adjusting these inputs in the Tenure Calculator, giving them a clearer picture of potential financial outcomes. What does the Principal Amount represent? The Principal Amount is the initial sum of money that you borrow or invest. It forms the basis on which interest is calculated. How do different compounding periods affect the future value? Compounding periods indicate how often interest is added to the principal. More frequent compounding results in higher future values because the interest is calculated and added to the principal more What is the Annual Interest Rate? The Annual Interest Rate is the percentage at which the investment grows each year. It influences how quickly the principal amount increases over time. What duration should I input? Duration refers to the total time span, in years, over which the principal will earn interest. It determines the number of compounding periods. What formula does the Tenure Calculator use? The calculator uses the compound interest formula: FV = PV * (1 + r/n)^(nt), where FV is the future value, PV is the principal amount, r is the annual interest rate, n is the number of compounding periods per year, and t is the time in years. Can I use the calculator for loan repayment schedules? Yes, you can use this calculator to estimate the total amount repayable on a loan by inputting the loan amount as the principal and the annual interest rate. What does Irregular Compounding mean? Irregular Compounding refers to interest compounding periods that are not standard, such as quarterly or monthly. The Tenure Calculator assumes consistent compounding based on the selected period. How does this tool help financial planning? This tool aids in forecasting the future value of investments, helping you make informed decisions about savings, investment strategies, and loan management. Can I compare different investment scenarios? Yes, by varying the principal amount, interest rate, compounding periods, and duration, you can compare the future values of different investment options. Does the Tenure Calculator account for taxes or fees? No, this calculator assumes a simplistic model where there are no taxes, fees, or other deductions affecting the future value. How accurate are the results? The results are mathematically accurate based on the input values; however, real-world factors like fluctuating interest rates or economic conditions may affect actual outcomes.	{"url":"https://www.onlycalculators.com/finance/business-planning/tenure-calculator/","timestamp":"2024-11-13T05:52:57Z","content_type":"text/html","content_length":"237558","record_id":"<urn:uuid:85f2f6a9-951b-4321-b6f8-c6a4ce50fe33>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00867.warc.gz"}
6th Grade \| MathMVP top of page 1) Jeremy played 9 holes of golf. These are the number of strokes he had on each hole: 5, 4, 6, 3, 5, 7, 3, 6, and 4. How many strokes did Jeremy have on all 9 holes? 2) Chris plays basketball on his school team. Chris scored two jump shots in the first quarter (jump shot = 2 points), one three-point shot in the second quarter, one foul shot in the third quarter (foul shot = 1 point), and two layups in the fourth quarter (layup = 2 points) 3) Jack is the goal-keeper on his soccer team. Jack saved three out of four goals in the first half and four out of six goals in the second half. How many goals did Jack save and how many did the other team score in the whole game? 4) James is the pitcher on his school's baseball team. James pitched the first four innings of their game. James threw two strikeouts in the first inning, no strikeouts in the second inning, three strikeouts in the third inning, and one strikeout in the fourth inning. How many strikeouts did James throw in all four innings? 5) Emma plays field hockey on her school team. Her team played four games this season. These are the number of goals she had in each game: 2, 4, 6, and 1. How many goals did Emma score in the whole bottom of page	{"url":"https://www.mathmvp.org/blank","timestamp":"2024-11-05T15:16:45Z","content_type":"text/html","content_length":"317690","record_id":"<urn:uuid:4eed7820-fef5-4d07-8306-501a8b82c118>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00243.warc.gz"}
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C++ Program to Check Abundant Number \| CodeToFun C++ Basic C++ Interview Programs C++ Program to Check Abundant Number Updated on Oct 06, 2024 By Mari Selvan π οΈ 113 - Views β ³ 4 mins π ¬ 1 Comment Photo Credit to CodeToFun π Introduction In the realm of programming, understanding and identifying special types of numbers is a fascinating exploration. One such category is abundant numbers. An abundant number, also known as an excessive number, is a positive integer that is smaller than the sum of its proper divisors (excluding itself). In this tutorial, we will delve into a C++ program designed to check whether a given number is an abundant number or not. The program will employ logic to calculate the sum of the proper divisors of a number and determine if it exceeds the number itself. π Example Let's explore the C++ code that achieves this functionality. #include <iostream> // Function to check if a number is abundant bool isAbundant(int num) { int sum = 1; // Start with 1 as every number is divisible by 1 // Iterate from 2 to the square root of the number for (int i = 2; i * i <= num; ++i) { if (num % i == 0) { sum += i; if (i != num / i) { sum += num / i; // If the sum of divisors exceeds the number, it is abundant return sum > num; // Driver program int main() { // Replace this value with your desired number int number = 12; // Check if the number is abundant if (isAbundant(number)) { std::cout << number << " is an abundant number." << std::endl; } else { std::cout << number << " is not an abundant number." << std::endl; return 0; π » Testing the Program To test the program with different numbers, simply replace the value of number in the main function. 12 is an abundant number. Compile and run the program to check if the number is an abundant number. π § How the Program Works 1. The program defines a function isAbundant that takes a number as input and returns true if the number is abundant and false otherwise. 2. Inside the function, it iterates through numbers from 2 to the square root of the given number to find its divisors. 3. For each divisor found, it adds it to the sum, including the corresponding divisor on the other side of the square root. 4. Finally, it compares the sum of divisors with the original number to determine if it is abundant. π Between the Given Range Let's dive into the C++ code that checks for abundant numbers in the given range. #include <iostream> // Function to check if a number is abundant bool isAbundant(int num) { int sum = 1; // Sum of proper divisors starts with 1 // Iterate up to the square root of num for (int i = 2; i * i <= num; ++i) { if (num % i == 0) { sum += i; if (i * i != num) { sum += num / i; return sum > num; // Driver program int main() { std::cout << "Abundant numbers between 1 and 50 are: "; // Iterate through the range [1, 50] for (int i = 1; i <= 50; ++i) { if (isAbundant(i)) { std::cout << i << " "; std::cout << std::endl; return 0; π » Testing the Program Abundant numbers between 1 and 50 are: 12 18 20 24 30 36 40 42 48 Run the program to see the abundant numbers within the specified range. π § How the Program Works 1. The program defines a function isAbundant that checks if a given number is abundant. 2. Inside the function, it iterates up to the square root of the number and calculates the sum of proper divisors. 3. The main function iterates through the range of numbers from 1 to 50 and prints the abundant numbers. π § Understanding the Concept of Abundant Number Before diving into the code, let's take a moment to understand abundant numbers. An abundant number is a positive integer for which the sum of its proper divisors (excluding itself) is greater than the number itself. For example, the number 12 is abundant because its divisors (excluding 12) are 1, 2, 3, 4, 6, and the sum of these divisors is 16, which is greater than 12. π ’ Optimizing the Program While the provided program is effective, there are opportunities for optimization, such as caching previously calculated divisor sums to avoid redundant computations. Feel free to incorporate and modify this code as needed for your specific use case. Happy coding! π ¨β π » Join our Community: To get interesting news and instant updates on Front-End, Back-End, CMS and other Frameworks. Please Join the Telegram Channel: π Hey, I'm Mari Selvan For over eight years, I worked as a full-stack web developer. Now, I have chosen my profession as a full-time blogger at codetofun.com. Buy me a coffee to make codetofun.com free for everyone. Buy me a Coffee Share Your Findings to All Search any Post Recent Post (Others) Inline Feedbacks View all comments If you have any doubts regarding this article (C++ Program to Check Abundant Number), please comment here. 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The n-Category Café May 31, 2009 The Mathematics of Music at Chicago Posted by John Baez As a card-carrying Pythagorean, I’m fascinated by the mathematics of music… even though I’ve never studied it very deeply. So, my fascination was piqued when I learned a bit of ‘neo-Riemannian theory ’ from Tom Fiore, a topology postdoc who works on double categories at the University of Chicago. Neo-Riemannian theory is not an updated version of Riemannian geometry… it goes back to the work of the musicologist Hugo Riemann. The basic idea is that it’s fun to consider things like the 24-element group generated by transpositions (music jargon for what mathematicians call translations in $\mathbb{Z}/12$) and inversion (music jargon for negation in $\mathbb{Z}/12$). And then it’s fun to study operations on triads that commute with transposition and inversion. These operations are generated by three musically significant ones called P, L, and R. Even better, these operations form a 24-element group in their own right! I explained why in week234 of This Week’s Finds. For more details try this: Yes, that’s my student Alissa Crans, of Lie 2-algebra fame! Posted at 8:58 PM UTC \| Followups (22) May 28, 2009 Quantum Gravity and Quantum Geometry in Corfu Posted by John Baez This September there will be a physics ‘summer school’ covering loop quantum gravity, spin networks, renormalization and higher gauge theory: I look forward to seeing my quantum gravity friends Abhay Ashtekar, John Barrett and Carlo Rovelli again — it’s been a while. It’s sad how changing one’s research focus can mean you don’t see friends you used to meet automatically at conferences. I’m also eager to meet Vincent Rivasseau, who is a real expert on renormalization and constructive quantum field theory! His book From Perturbative to Constructive Renormalization is very impressive. I had a brief and unsuccessful fling with constructive quantum field theory as a grad student, so it’ll be nice (but a bit scary) to meet someone who’s made real progress in this tough subject. Posted at 4:22 PM UTC \| Followups (5) Metric Coinduction Posted by David Corfield Dexter Kozen and Nicholas Ruozzi have a paper Applications of Metric Coinduction which begins Mathematical induction is firmly entrenched as a fundamental and ubiquitous proof principle for proving properties of inductively defined objects. Mathematics and computer science abound with such objects, and mathematical induction is certainly one of the most important tools, if not the most important, at our disposal. Perhaps less well entrenched is the notion of coinduction. Despite recent interest, coinduction is still not fully established in our collective mathematical consciousness. A contributing factor is that coinduction is often presented in a relatively restricted form. Coinduction is often considered synonymous with bisimulation and is used to establish equality or other relations on infinite data objects such as streams or recursive types. In reality, coinduction is far more general. For example, it has been recently been observed that coinductive reasoning can be used to avoid complicated $\epsilon-\delta$ arguments involving the limiting behavior of a stochastic process, replacing them with simpler algebraic arguments that establish a coinduction hypothesis as an invariant of the process, then automatically deriving the property in the limit by application of a coinduction principle. The notion of bisimulation is a special case of this: establishing that a certain relation is a bisimulation is tantamount to showing that a certain coinduction hypothesis is an invariant of some process. Posted at 9:34 AM UTC \| Followups (9) May 26, 2009 Alm on Quantization as a Kan Extension Posted by Urs Schreiber Recently I was contacted by Johan Alm, a beginning PhD student at Stockholm University, Sweden, with Prof. Merkulov. He wrote that he had thought about formalizing and proving aspects of the idea that appeared as the The $n$-Café Quantum Conjecture about the nature of [[path integral quantization ]]. After a bit of discussion of his work, we thought it would be nice to post some of his notes here: Johan Alm, Quantization as a Kan extension (lab) $n$Café-regulars may be pleased to meet some old friends in there, such as the [[Leinster measure]] starring in its role as a canonical path integral measure. Posted at 7:21 PM UTC \| Followups (45) May 24, 2009 Elsevier Journal Prices Posted by John Baez Do you have data about Elsevier’s journal prices compared to other journals? If so, let me know! Before we launch the revolution, we need to get our facts straight. My friend the physicist Ted Jacobson wants such data. With the help of a librarian, he has compared the prices of Elsevier’s physics journals to other physics journals subscribed to by his Posted at 9:47 PM UTC \| Followups (10) May 22, 2009 Charles Wells’ Blog Posted by John Baez Charles Wells is perhaps most famous for this book on topoi, monads and the category-theoretic formulation of universal algebra using things like ‘algebraic theories’ and ‘sketches’: It’s free online! Snag a copy and learn some cool stuff. But I’ll warn you — it’s a fairly demanding tome. Luckily, Charles Wells now has a blog! And I’d like to draw your attention to two entries: one on sketches, and one on the evil influence of the widespread attitude that ‘the philosophy of math is the philosophy of logic’. Posted at 5:43 PM UTC \| Followups (12) May 19, 2009 Where is the Philosophy of Physics? Posted by David Corfield As the subtitle of this blog says, we run ‘A group blog on math, physics and philosophy’. To what extent, though, do we cover all the interfaces of this triad? Well, we do some philosophy of mathematics here, and we certainly do some mathematical physics. But the question I’ve been wondering about recently is whether we should be doing more philosophy of physics. If we followed the position that physics is the search for more and more adequate mathematical structures to describe the world, perhaps we needn’t take the philosophy of physics to be anything more than a philosophy of mathematics along with an account of how the structures which are most promising for physics are chosen. But this view of physics would be controversial. Posted at 2:01 PM UTC \| Followups (45) TFT at Northwestern Posted by Urs Schreiber Quite unfortunately I couldn’t make it to this event that started yesterday: Topological Field Theories at Northwestern University Workshop: May 18-22, 2009 Conference: May 25-29, 2009 (website, Titles and abstracts) An impressive concentration of extended TFT expertise. But with a little luck $n$Café regulars who are there will provide the regrettable rest of us with reports about the highlights and other lights In fact, Alex Hoffnung already sent me typed notes that he had taken in talks! That’s really nice of him. I am starting to collect this and other material at Posted at 1:57 PM UTC \| Followups (16) May 18, 2009 A Prehistory of n-Categorical Physics Posted by John Baez I’m valiantly struggling to finish this paper: Perhaps blogging about it will help… Posted at 8:58 PM UTC \| Followups (68) Higher Structures in Göttingen III Posted by John Baez Göttingen was famous as a center of mathematics during the days of Gauss, Riemann, Dirichlet, Klein, Minkowksi, Hilbert, Weyl and Courant. One of the founders of category theory, Saunders Mac Lane, studied there! He wrote: In 1931, after graduating from Yale and spending a vaguely disappointing year of graduate study at Chicago, I was searching for a really first-class mathematics department which would also include mathematical logic. I found both in Göttingen. It’s worth reading Mac Lane’s story of how the Nazis eviscerated this noble institution. But now, thanks to the Courant Research Centre on Higher-Order Structures, Göttingen is gaining fame as a center of research on higher structures (like $n$-categories and $n$-stacks) and their applications to geometry, topology and physics! They’re having another workshop soon: Posted at 5:41 PM UTC \| Followups (2) Journal Club – Geometric Infinity-Function Theory – Week 4 Posted by Urs Schreiber In our journal club on [[geometric $\infty$-function theory]] this week Chris Brav talks about chapter 4 of Integral Transforms: Tensor products and integral transforms. This is about tensoring and pull-pushing $(\infty,1)$-categories of quasi-coherent sheaves on perfect stacks. Luckily, Chris has added his nice discussion right into the wiki entry, so that we could already work a bit on things like further links, etc. together. Please see section 4 here. Discussion on previous weeks can be found here: week 1: Alex Hoffnung on Introduction week 2: myself on Preliminaries week 3: Bruce Bartlett Perfect stacks Posted at 7:08 AM UTC \| Followups (6) May 15, 2009 The Relevance of Predicativity Posted by David Corfield If I get around to writing a second book in philosophy of mathematics, one thing I’ll probably need to retract is the ill-advised claim made in the first book that the notion of predicativity is irrelevant to mainstream mathematics. Here’s a passage which goes directly against such a thought, from Nik Weaver’s Is set theory indispensable? Posted at 4:53 PM UTC \| Followups (12) May 11, 2009 Journal Club – Geometric Infinity-Function Theory – Week 3 Posted by Urs Schreiber This week in our Journal Club on [[geometric $\infty$-function theory]] Bruce Bartlett talks about section 3 of “Integral Transforms”: perfect stacks. So far we had Week 1: Alex Hoffnung on Introduction Week 2, myself on Preliminaries See here for our further schedule. We are still looking for volunteers who’d like to chat about section 5 and 6. Posted at 11:04 PM UTC \| Followups (31) May 9, 2009 Smooth Structures in Ottawa II Posted by John Baez guest post by Alex Hoffnung Hi everyone, I am going to even further neglect my duties to the journal club and take a moment to report on the Fields Workshop on Smooth Structures in Logic, Category Theory and Physics which took place this past weekend at the University of Ottawa. The organizers put together a great series of talks giving an overview of the past and current trends and applications in smooth structures. I should right away try to put the idea of smooth structures in some context. Further, I should warn you that I may do this with some amount of bias. Posted at 7:57 PM UTC \| Followups (50) May 8, 2009 In Search of Terminal Coalgebras Posted by David Corfield Tom Leinster has put up the slides for his joint talk – Terminal coalgebras via modules – with Apostolos Matzaris at PSSL 88. It’s all about establishing the existence of, and constructing, terminal coalgebras in certain situations. I realise though looking through the slides that I never fully got on top of the flatness idea, and nLab is a little reluctant to help at the moment (except for flat module). So perhaps someone could help me understand the scope of the result, maybe via an example. Say I take the polynomial endofunctor $\Phi(X) = 1 + X + X^2.$ Given that terminal coalgebras can be said to have cardinality $i$, in which categories will I find such a thing? Posted at 10:07 AM UTC \| Followups (20) May 7, 2009 Odd Currency Puzzle Posted by John Baez Sorry to be posting so much light, frothy stuff lately — but since it’s an odd day, I can’t resist another puzzle. What’s the oddest currency ever used in America? Of course this is a subjective question, so I’d be interested to hear your opinion… Posted at 7:34 PM UTC \| Followups (49) May 6, 2009 nLab - More General Discussion Posted by David Corfield With the previous thread on nLab reaching 343 comments, it’s probably time for a new one. Let me begin discussions by asking whether it is settled that distributor be the term preferred over profunctor. I ask since it would be good to have an entry on the 2-category of small categories, profunctors and natural transformations. Should it be $Dist$ or $Prof$? Posted at 9:06 AM UTC \| Followups (94) May 5, 2009 Posted by David Corfield If we were to have a page at nLab on things to be categorified should it be titled categorifAcienda, categorifIcienda or something else? My suggestions are based on the gerundives formed from verbs such as agenda and Miranda. Concerning verbs more closely resembling ‘categorify’ we have • Satisfacio (satisfy) - satisfaciendus • Efficio (bring to pass) - efficiendus Unfortunately, categorify is a hybrid word, with Greek stem and Latin suffix. I suppose categorize was out of the question. Posted at 4:26 PM UTC \| Followups (14) Journal Club – Geometric Infinity-Function Theory – Week 2 Posted by Urs Schreiber May 4, 2009 The Foibles of Science Publishing Posted by John Baez The latest news about Elsevier journals and Scientific American. Posted at 1:17 AM UTC \| Followups (15)	{"url":"https://golem.ph.utexas.edu/category/2009/05/index.shtml","timestamp":"2024-11-06T05:38:08Z","content_type":"application/xhtml+xml","content_length":"115675","record_id":"<urn:uuid:63255fd0-b943-4819-b649-1ebff0099b1c>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00086.warc.gz"}
Plotting several time frequencies in one chart Sometimes, you may need to show data of different frequencies in one chart. For example, it may be monthly and daily dynamics of stock price returns, or annual and quarterly sales figures. This article discusses the following questions: How to understand which data frequencies are available? To be able to plot different fequencies of a time series in one chart, these frequencies should be available in the dataset. To understand which frequencies are available for a time seires, query this time-series in the Dataset Viewer and look at the "Frequency" box at the top left corner. There are the following frequencies which supported by the platform: • A - annual • H - semi-annual • Q - quarterly • M - monthly • W - weekly • D - daily It is worth mentioning that there are two types of data frequencies: (a) original, and (b) calculated ones. Original frequencies are those which are originally available in the data source while calculated are those which can be derived from original frequencies using aggregations. So, if a dataset originally has only daily frequency, all other frequencies will be available for visualizations through built-in aggregations. On the other hand, if a dataset originally has quarterlty frequency, then only three frequencies will be available to you: quarterly, semi-annual, and To check which original data frequencies are available in the dataset, refer to "Details" tab of the Dataset Viewer. How to convert one frequency to another? There are three ways data frequencies can be aggregated to one another: 1. Sum 2. Average 3. Last value (if you , for example, are aggregating from daily to monthly frequency, it will show the last day of each month) To change the way frequencies are aggregated, pick the one from the "Transformation" box. How to plot different frequencies in one chart? To plot several different frequencies in one chart, open a visualization in the Dashboard Builder and edit it. In the "Dimension Filter" tab, select "Custom" time period in the top right corner. Then, in the "Time" panel at the left, holding "Ctrl" button ("command" on Mac), click an additional frequency you want to plot on a chart. It will be added as a separate series on a chart. Similarly, you can plot multiple frequencies in one chart in the Dataset Viewer.	{"url":"https://help.knoema.com/hc/en-us/articles/16083817862292-Plotting-several-time-frequencies-in-one-chart","timestamp":"2024-11-02T18:11:43Z","content_type":"text/html","content_length":"30332","record_id":"<urn:uuid:e7bc7606-a26c-4d64-b17f-3e09328391f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00082.warc.gz"}
fishkent: Hypothesis test for von Mises-Fisher distribution over Kent... in Directional: A Collection of Functions for Directional Data Analysis Hypothesis test for von Mises-Fisher distribution over Kent distribution R Documentation The null hypothesis is whether a von Mises-Fisher distribution fits the data well, where the altenrative is that Kent distribution is more suitable. x A numeric matrix containing the data as unit vectors in Euclidean coordinates. B The number of bootstrap re-samples. By default is set to 999. If it is equal to 1, no bootstrap is performed and the p-value is obtained throught the asymptotic distribution. A numeric matrix containing the data as unit vectors in Euclidean coordinates. The number of bootstrap re-samples. By default is set to 999. If it is equal to 1, no bootstrap is performed and the p-value is obtained throught the asymptotic distribution. Essentially it is a test of rotational symmetry, whether Kent's ovalness parameter (beta) is equal to zero. This works for spherical data only. This is an "htest"class object. Thus it returns a list including: statistic The test statistic value. parameter The degrees of freedom of the test. If bootstrap was employed this is "NA". p.value The p-value of the test. alternative A character with the alternative hypothesis. method A character with the test used. data.name A character vector with two elements. The degrees of freedom of the test. If bootstrap was employed this is "NA". Rivest L. P. (1986). Modified Kent's statistics for testing goodness of fit for the Fisher distribution in small concentrated samples. Statistics & Probability Letters, 4(1): 1–4. x <- rvmf(100, rnorm(3), 15) fishkent(x) fishkent(x, B = 1) iagesag(x) For more information on customizing the embed code, read Embedding Snippets.	{"url":"https://rdrr.io/cran/Directional/man/fishkent.html","timestamp":"2024-11-11T21:44:16Z","content_type":"text/html","content_length":"33149","record_id":"<urn:uuid:7b8e6278-a42d-4002-a5a8-cfa1c54d678a>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00017.warc.gz"}
Hemisphere - Formula, Properties, Definition \| Hemisphere Shape A day full of math games & activities. Find one near you. A day full of math games & activities. Find one near you. A day full of math games & activities. Find one near you. A day full of math games & activities. Find one near you. A hemisphere, in general, refers to half of the earth such as the northern hemisphere or the southern hemisphere. But in geometry, a hemisphere is referred to as a 3D figure made from cutting a sphere into two equal halves with one flat side. In real life we come across various objects that are in the shape of a hemisphere, for example, if we cut a cherry into half, we get a hemisphere-shaped cherry or if we cut a grapefruit into half we get a hemisphere. Let us learn more about a hemisphere, its properties and formulas in this article. 1. Definition of a Hemisphere 2. Properties of a Hemisphere 3. Difference between Hemisphere and Sphere 4. Volume of a Hemisphere 5. Surface Area of a Hemisphere 6. Lateral Area of a Hemisphere 7. FAQs on Hemisphere Definition of a Hemisphere The word hemisphere can be split into hemi and sphere, where hemi means half and sphere is the 3D shape in math. Hence, a hemisphere is a 3D geometric shape that is half of a sphere with one side flat and the other side as a circular bowl. It is formed when a sphere is cut at the exact center along its diameter leaving behind two equal hemispheres. The flat side of the hemisphere is known as the base or the face of the hemisphere. Therefore, it is considered as an exact half of a sphere. Properties of a Hemisphere Since a hemisphere is the exact half of a sphere, they share very similar properties as well. They are as follows: • A hemisphere has a curved surface area. • Just like a sphere, there are no edges and no vertices in a hemisphere. • It is not a polyhedron since polyhedrons are made up of polygons, but a hemisphere has one circular base and one curved surface. • The diameter of a hemisphere is a line segment that passes through the center and touches the two opposite points on the base of the hemisphere. • The radius of a hemisphere is a line segment from the center to a point on the curved surface of the hemisphere. Difference Between Hemisphere and Sphere We already know that a hemisphere is obtained from a sphere and the two objects share very similar properties, but there are a few differences as well. Listed below are the few differences between a sphere and a hemisphere. │ Hemisphere │ Sphere │ │It is a 3D figure obtained by cutting a sphere in half. │This is a 3D round figure used in geometry that has no │ │ │edges and no vertices. │ │It has one flat side and one curved side. │This has no flat side and is only curved. │ │The volume of a hemisphere = (2/3)πr^3 cubic units │The volume of a sphere = (4/3)πr^3 cubic units │ │Hemisphere has two surface areas, i.e., total surface area, and lateral surface area. The total surface area of hemisphere = 3πr^2 and the│The surface area of sphere = 4πr^2 │ │lateral surface area of hemisphere = 2πr^2. │ │ Volume of a Hemisphere The volume of a hemisphere is the total capacity of the hemisphere and it is the number of unit cubes covered inside that space. The volume of a hemisphere is measured in cubic units and is expressed as m^3, cm^3, in^3, etc. Therefore, the formula to find the volume is: Volume of Hemisphere = (2πr^3)/3 Where π is the constant which is equal to 3.142 or 22/7, and r is the radius of the hemisphere. For a detailed explanation, you can check out this article on Volume of Hemisphere. Surface Area of a Hemisphere The surface area of a hemisphere can be calculated by the area of its circular base along with its curved surface. The hemisphere can either be hollow or a solid, according to that the surface area can be calculated. It is measured in square units and the formula is: Surface Area of Hemisphere = 3πr^2 Where π is the constant which is taken as 3.142 or 22/7, and r is the radius of the hemisphere. For a detailed study, you can check out this article on Surface Area of a Hemisphere. Curved Surface Area of a Hemisphere The curved surface of a hemisphere is considered the lateral area of a hemisphere. If the radius is given, we can find out the lateral surface area using the formula and it is measured in square units. The formula for finding the lateral surface area or the CSA of a hemisphere is: Curved Surface Area of a Hemisphere = 2πr^2 Where π is the constant taken as 3.142 or 22/7, and r is the radius of the hemisphere. A detailed explanation on this topic can be found in this article on Curved Surface Area of a Hemisphere. ☛ Related Articles Listed below are a few interesting topics that are related to the hemisphere. Hemisphere Examples 1. Example 1: Which of these is a hemisphere: a, b, c, or d? Solution: We know that a hemisphere is a 3D figure that is half of a sphere. In the given figures, figure (c) is a hemisphere because it has one curved side and one flat side. Figure (a) is a triangle, figure (b) is a semicircle, and figure (d) is a cylinder. 2. Example 2: Emily has a bowl which is in the shape of a hemisphere. The radius of the bowl is 4 inches. What is the volume of the bowl? (take π = 22/7) Solution: The volume of a hemisphere is half the volume of the sphere. So, the volume of the hemisphere is given by (2πr^3)/3. Let's calculate the volume of the bowl. Volume of the hemisphere shaped bowl = (2πr^3)/3 Volume = (2 × 22 × 4^3) / (7 × 3) Volume = 2816/21 Volume = 134.09 inches^3 Therefore, the volume of the bowl is 134.09 inches^3. 3. Example 3: State true or false. a.) Hemisphere is a 3D figure obtained by cutting a sphere in half. b.) There is 1 edge and 1 vertex in a hemisphere. a.) True, the hemisphere is a 3D figure obtained by cutting a sphere in half. b.) False, there are no edges and no vertices in a hemisphere. View More > Breakdown tough concepts through simple visuals. Math will no longer be a tough subject, especially when you understand the concepts through visualizations. Practice Questions on Hemisphere Try These! > FAQs on Hemisphere What is a Hemisphere Shape in Math? A hemisphere is a 3D figure which is obtained by cutting a sphere into two equal halves through its diameter. It has one curved side and one flat side called the face of the hemisphere or the great circle of the sphere which helps in forming the hemisphere. Some of the real-life examples of a hemisphere are a bowl, igloo, the top part of a mushroom, and so on. What is the Formula to Find the Volume of a Hemisphere? The volume of a hemisphere is expressed in cubic units and the formula which is used for the volume of a hemisphere is, Volume of Hemisphere = (2πr^3)/3, where π is the constant measuring 3.142 or 22 /7, and r is the radius. What is the Formula to Find the Surface Area of a Hemisphere? The surface area of a hemisphere is the curved part of the hemisphere and we calculate it by using this formula, Surface Area of Hemisphere = 3πr^2, where π which is taken as 3.142 or 22/7, and r is the radius of the hemisphere. It should be noted that this is the total surface area of the hemisphere which includes the area of the base too. What is the Formula to Find the CSA Area of a Hemisphere? If the radius of a hemisphere is given, we can calculate the curved surface area (CSA) of a hemisphere by using this formula, Curved surface Area of a Hemisphere = 2πr^2, where π is taken as 3.142 or 22/7, and r is the radius of the hemisphere. The lateral surface area is the area of the curved part of the hemisphere only. It does not include the area of the base which is in the shape of a Are Sphere and Hemisphere the Same? The sphere and hemisphere are very similar in properties since the hemisphere is made from a sphere. When a sphere is cut into two equal halves, these two halves are called hemispheres. But one of the main differences between a sphere and a hemisphere is that a sphere does not have base but only a curved surface whereas a hemisphere has a base and one curved surface. What is a Hollow Hemisphere? If the space inside a hemisphere is hollow, it is known as a hollow hemisphere. A hollow hemisphere has two radii - an internal radius, for the inner circle (hollow region), and an external radius for the outside circle. What is the Formula for the Base Area of a Hemisphere? In a hemisphere, if its radius (r) is given, then its base area is given as Base Area of a Hemisphere = Area of the base circle = πr^2 square units. What does a Hemisphere Look Like? A hemisphere looks like a cherry when cut into half. It also resembles the shape of an igloo. How many Faces does a Hemisphere Have? A hemisphere has one curved surface and one flat face in the shape of a circle. It is different from a sphere which has just one curved surface. Download FREE Study Materials Math worksheets and visual curriculum	{"url":"https://www.cuemath.com/geometry/hemisphere/","timestamp":"2024-11-04T12:34:17Z","content_type":"text/html","content_length":"243067","record_id":"<urn:uuid:92d55bdc-8902-490b-9fa0-1696ee74947e>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00496.warc.gz"}
Using DynamoDB With Python NoSQL databases are non-tabular databases that store and retrieve data differently from SQL databases. While SQL is table-based, NoSQL databases are either key-value pairs. NoSQL databases are for large, distributed systems due to their fast and highly scalable systems. DynamoDB is an example of a NoSQL database. DynamoDB is fast, capable of handling multiple requests, and highly scalable. Dynamo is a NoSQL database provided by Amazon Web Service (AWS). Python is one of the most widely used programming languages and has good support for DynamoDB using the AWS SDK for Python. In this tutorial, we will be using the Boto3 library to create and query tables, load data, and perform CRUD operations in DynamoDB and Python. The source code for this article can be found on GitHub. • Basic knowledge of DynamoDB • Operating system: Windows, macOS, or Linux Getting Started With DynamoDB Using Python DynamoDB is an AWS service that allows you to create database tables for storing and retrieving data and handles request traffic. AWS offers a set of SDKs for interacting with DynamoDB. These SDKs are available for various programming languages; the AWS SDK for Python is known as Boto3. We will be using Boto3 to interact with DynamnoDB. AWS Boto3 allows you to create, configure, and manage different AWS services. Setting up DynamoDB Locally To get DynamoDB running locally, there are a few steps required. Let’s get started! The first step is to download the DynamoDB zip file. This file should be downloaded based on your region. Click here to download the zip file. Once the file is downloaded, extract all the contents of the file and move your file to your preferred directory on your device. Next, open up the command prompt, navigate to the directory where DynamoDBLocal.jar is located, and run this configuration script: java -Djava.library.path=./DynamoDBLocal_lib -jar DynamoDBLocal.jar -sharedDb If you encounter any issues with this process, it is most likely because you don’t have Java installed on your local machine. Run this command to confirm: After confirmation, if you do not have Java on your device, you can download it using this link. The second phase of this setup is accessing DynamoDB through the AWS CLI or your command prompt configuration. This step requires your AWS credentials. Ensure you have these credentials available in a separate environment. Run this command on your terminal to begin: You likely have your AWS credentials already configured on your machine. If this is the case, press enter. However, if you don’t, provide these details: AWS Access Key ID: "yourAccessKeyId" AWS Secret Access Key: "yourAccessKey" Default region name : "yourRegionName" Nicely done! Connecting to the DynamoDB using Python (Boto3) First, install Boto3 by running the following command on your terminal. Next, in your code editor, create a dynamo-python.py file and import the Boto3 library at the top of the file. Finally, we will create the Boto3 DynamoDB resource. This will connect to the local instance of our DynamoDB server. Do this by adding the following line of code: dynamodb = boto3.resource('dynamodb', endpoint_url="http://localhost:8000") Creating a Table in DynamoDB We will be creating a table using the Dynamo create_table function. Here, we call the table "Books." This table will contain attributes for the partition key and sort key. The “title” of the book will be our sort key, and the “book_id” will be our partition key. A sort key is a field in a database that indicates the order in which data is stored (in sorted order by the sort key value). In DynamoDB, the sort key for each field is unique. The partition key is the attribute that identifies an item in a database. Data with the same partition key are stored together to enable you to query the data. The data of a partition key is sorted using the sort key. Now, add the code snippet below to create a table: import boto3 def create_books_table(dynamodb=None): dynamodb = boto3.resource( table = dynamodb.create_table( 'AttributeName': 'book_id', 'KeyType': 'HASH' # Partition key 'AttributeName': 'title', 'KeyType': 'RANGE' # Sort key 'AttributeName': 'book_id', # AttributeType refers to the data type 'N' for number type and 'S' stands for string type. 'AttributeType': 'N' 'AttributeName': 'title', 'AttributeType': 'S' # ReadCapacityUnits set to 10 strongly consistent reads per second 'ReadCapacityUnits': 10, 'WriteCapacityUnits': 10 # WriteCapacityUnits set to 10 writes per second return table if __name__ == '__main__': book_table = create_books_table() print("Status:", book_table.table_status) In the code above, we created a table named Books; book_id is the partition key, and the title is the sort key. Next, we defined our table by declaring a key schema stored in the KeySchema variable. We also declared the data types of the attributes. Where "N" represents a number and "S" represents a string, we also added the ProvisionedThroughput variable to reduce the number of "read" and "write" operations on the database per second. Finally, in the last section of the code snippet, we created an instance of our class. Adding Sample Data to the DynamoDB Table In this section, we will be adding sample data to the DynamoDB table. This data will be written in JSON format. To begin, create a JSON file, data.json, and add the following data: "book_id": 1000, "title": "Atomic habits", "author": "James Clear", "isbn": "34526767", "year_of_publication": "2019" "book_id": 1001, "title": "Americanah", "author": "Chimamanda Adichie", "isbn": "10202223", "year_of_publication": "2013" "book_id": 1002, "title": "Teller of secrets", "author": "Bisi Adjapon", "isbn": "10201120", "year_of_publication": "2013" "book_id": 1003, "title": "Joys of motherhood", "author": "Buchi Emecheta", "isbn": "10110120", "year_of_publication": "1979" "book_id": 1004, "title": "Purple Hibiscus", "author": "Chimamanda Adichie", "isbn": "10001241", "year_of_publication": "2012" Now, we need to load this data to add it to our database. Create a Python file, store_data.py, and add these lines of code: import json from decimal import Decimal import boto3 def load_data(books, dynamodb=None): dynamodb = boto3.resource( books_table = dynamodb.Table('Books') for book in books: book_id = (book['book_id']) title= book['title'] print("Displaying book data:", book_id, title) if __name__ == '__main__': with open("data.json") as json_file: book_list = json.load(json_file, parse_float=Decimal) In the code above, we created a function that will loop through the fields and load the data contained in our JSON file. Run the following command in your terminal to execute the script above. Once the script is running successfully, the result of your loaded data will be displayed on your terminal. Displaying book data: 1000 Atomic habits Displaying book data: 1001 Americanah Displaying book data: 1002 Teller of secrets Displaying book data: 1003 Joys of motherhood Displaying book data: 1004 Purple Hibiscus Successful! Good job so far! CRUD Operations in DynamoDB Using Python We have successfully created a table that contains data (items), and each of these items makes up a set of attributes. These are the core components of DynamoDB. In this section, we will be working on performing CRUD operations using the items in our DynamnoDB table. Let’s dive in! Create Item We will use the put_item() method to add new items to the Books table. To get started, create a new python file, add_book.py, and add the following code snippet: import boto3 def add_book(books, dynamodb=None): dynamodb = boto3.resource('dynamodb') books_table = dynamodb.Table('Books') response = books_table.put_item( "book_id": 1005, "title": "There Was a Country", "author": "Chinua Achebe", "isbn": "0143124030", "year_of_publication": "2012" return response if __name__ == '__main__': book_resp = add_book(books='Books') In the code snippet above, we defined a function that will add items from our Dynamo table. Next, using the put_item() method, we added sample data for the DynamoDB table. Run the scripts in your terminal to add the data: You should get this output: {'ResponseMetadata': {'RequestId': '', 'HTTPStatusCode': 200, 'HTTPHeaders': {'server': 'Server', 'date': 'Thu, 05 May 2022 15:00:40 GMT', 'content-type': 'application/x-amz-json-1.0', 'content-length': '2', 'connection': 'keep-alive', 'x-amzn-requestid': '', 'x-amz-crc32': '2745614147'}, 'RetryAttempts': 0}} Read Item We can also access the item(s) in the DynamoDB table using the get_item() method. We need the primary to be able to access the database. The primary of this project is a combination of the sort key (’title’) and the partition key (’book_id’). Create a python file, get_book.py, and add the following lines of code: import boto3 from botocore.exceptions import ClientError def get_book(book_id, title, dynamodb=None): dynamodb = boto3.resource('dynamodb') books_table = dynamodb.Table('Books') response = books_table.get_item( Key={'book_id': book_id, 'title': title}) except ClientError as e: print(e.response['No item found']) return response['Item'] if __name__ == '__main__': book = get_book(1000, "Atomic habits") if book: From the code above, we imported ClientError from the botocore.exceptions package, which is set to help us navigate and handle errors and exceptions that may be encountered while interacting with the AWS Boto3 SDK. Run this script on your terminal using this command: You should get this output: {'year_of_publication': '2019', 'isbn': '34526767', 'book_id': Decimal('1000'), 'author': 'James Clear', 'title': 'Atomic habits'} Condition Expressions When working with DynamoDB, you are likely to use ConditionExpressions when altering items in a database table. Condition expressions are optional parameters used to manipulate specified items in a DynamoDB table. Conditions expressions are applied using the PutItem, UpdateItem, and DeleteItem operations. These operations are implemented when updating or deleting items. The operation only succeeds if the condition expression value is set to true; otherwise, the operation fails. In the next section, we will be working with condition expressions for updating and deleting items on our table. Update Item We can also update the existing data in our table. This is done by either updating the values of an existing attribute, adding new attributes, or deleting attributes. In this section, we will update the value of an existing attribute using the update_item() method. Let’s get started with an existing attribute in our table: "book_id": 1000, "title": "Atomic habits", "author": "James Clear", "isbn": "34526767", "year_of_publication": "2019" Create a python file, update_book.py, and insert these lines of code: import boto3 def update_book(book_id, title, dynamodb=None): dynamodb = boto3.resource('dynamodb') books_table = dynamodb.Table('Books') response = books_table.update_item( 'book_id' : 1001, 'title': "Americanah" UpdateExpression="set ISBN=:ISBN", ExpressionAttributeValues={':ISBN': "9780307455925"}, return response if __name__ == '__main__': update_response = update_book(1001, 'Americanah') From the code above, we defined a function that will add items from our Dynamo table. Next, using the update_item() method, we intend to update an attribute in the DynamoDB table. Other parameters used include: UpdateExpression: Defines attribute(s) which are to be updated and their new values. ExpressionAttributeValues: This expression holds the substitutes for the attribute to be updated or the new value. ReturnValues: Use this parameter to get the item attributes before or after they are updated. For UpdateItem(), the valid values are NONE \| ALL_OLD \| UPDATED_OLD \| ALL_NEW \| UPDATED_NEW. Run this script on your terminal using this command: {'Attributes': {'ISBN': '9780307455925'}, 'ResponseMetadata': {'RequestId': '3TJHT8856E3GRKFCPN8S5HMRQ3VV4KQNSO5AEMVJF66Q9ASUAAJG', 'HTTPStatusCode': 200, 'HTTPHeaders': {'server': 'Server', 'date': 'Mon, 09 May 2022 10:26:53 GMT', 'content-type': 'application/x-amz-json-1.0', 'content-length': '45', 'connection': 'keep-alive', 'x-amzn-requestid': '3TJHT8856E3GRKFCPN8S5HMRQ3VV4KQNSO5AEMVJF66Q9ASUAAJG', 'x-amz-crc32': '2958762950'}, 'RetryAttempts': 0}} Delete item For the last CRUD operation, we will be deleting an item from our table using the delete_item() method. You can either use the primary key of the item or a ConditionExpression to delete the item. If you wish to use a ConditionExpression the condition expression value must be set to true. However, in this tutorial, we will be using both the primary key and the condition expression. Create a python file, delete_book.py, and these lines of code: import boto3 def delete_book(book_id, title, dynamodb=None): dynamodb = boto3.resource('dynamodb') books_table = dynamodb.Table('Books') response = books_table.delete_item( 'book_id' : 1001, 'title': "Americanah" ExpressionAttributeValues={':ISBN': "9780307455925"}, return response if __name__ == '__main__': delete_response = delete_book(1001, 'Americanah') Run this script on your terminal using this command: {'ResponseMetadata': {'RequestId': 'LQ2M5SKSNKAASF041VPKE6FQCFVV4KQNSO5AEMVJF66Q9ASUAAJG', 'HTTPStatusCode': 200, 'HTTPHeaders': {'server': 'Server', 'date': 'Mon, 09 May 2022 11:56:08 GMT', 'content-type': 'application/x-amz-json-1.0', 'content-length': '2', 'connection': 'keep-alive', 'x-amzn-requestid': 'LQ2M5SKSNKAASF041VPKE6FQCFVV4KQNSO5AEMVJF66Q9ASUAAJG', 'x-amz-crc32': '2745614147'}, 'RetryAttempts': 0}} Query Tables in DynamoDB Querying our database table returns every item in the table with the same partition key. We will query the table using the value of our partition key using the query() method. The partition key in this project is ‘book_id’. Let’s dive in! Create a python file, query_table.py, and insert these lines of code: import boto3 from boto3.dynamodb.conditions import Key def query_book(book_id, dynamodb=None): dynamodb = boto3.resource('dynamodb') books_table = dynamodb.Table('Books') response = books_table.query( return response['Items'] if __name__ == '__main__': query_id = 10001 print(f"Book ID: {query_id}") books_data = query_book(query_id) for book_data in books_data: print(book_data['book_id'], ":", book_data['title']) Run this script on your terminal using this command: Book ID: 10001 1001: Americanah Delete Table In addition to CRUD operations, you can also delete an entire DynamoDB table using the table.delete() method. All you need to do is specify the name of the table you wish to delete. Create a python file, delete_table.py, and insert these lines of code: import boto3 def delete_table(dynamodb=None): dynamodb = boto3.resource('dynamodb') books_table = dynamodb.Table('Books') if __name__ == '__main__': print("DynamoDB table deleted!") Run this script on your terminal using this command: Several DynamoDB operations can be performed by creating a Python script using AWS Boto3. In this tutorial, we created a DynamoDB table using Boto3 to interact with our database and perform CRUD operations, query, and delete a table. I hope you had fun working on this! Happy Coding!🙂	{"url":"https://www.honeybadger.io/blog/using-dynamodb-with-python/","timestamp":"2024-11-07T16:27:40Z","content_type":"text/html","content_length":"88647","record_id":"<urn:uuid:4c0b481f-49fc-47f9-b5c9-3c34d3ffcda9>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00145.warc.gz"}
Finding Particular Solutions Inhomogeneous ODE's \| JustToThePointFinding Particular Solutions Inhomogeneous ODE's A partial differential equation (PDE) is a type of mathematical equation that involves a function of severable variables and its partial derivatives with respect to those variables, e.g., $\frac{∂f} {∂t} = \frac{∂^2f}{∂x^2}, \frac{∂w}{∂t}-\frac{∂^2w}{∂x^2} = 0, etc.$ Partial differential equations are incredibly important in many fields of science and engineering. The Existence and Uniqueness Theorem provides critical insight into the behavior of the solutions to first-order differential equations. It states that: If f(x, y) (the right-hand side of the ODE) is continuous in a neighborhood around a point (x[0], y[0]) and its partial derivative with respect to y, $\frac{∂f}{∂y}$, is also continuous near (x[0], y [0]), then the differential equation y' = f(x, y) has a unique solution to the initial value problem through the point (x[0], y[0]). A first-order linear differential equation (ODE) has the general form: a(x)y' + b(x)y = c(x) where y′ is the derivative of y with respect to x, and a(x), b(x), and c(x) are functions of x. If c(x) = 0, the equation is called homogeneous, i.e., a(x)y’ + b(x)y = 0. The equation can also be written in the standard linear form as: y’ + p(x)y = q(x) where $p(x)=\frac{b(x)}{a(x)}\text{ and }q(x) = \frac{c(x)}{a(x)}$ Finding Particular Solutions for Inhomogeneous ODE’s Consider a second-order linear differential equation with constant coefficients: y’’ + Ay’ + By = f(x) This equation is inhomogeneous because of the non-zero forcing term f(x). Our goal is to find a particular solution y[p] to this equation. The general solution to the ODE will then be the sum of the particular solution and the complementary solution y[c], which solves the associated homogeneous equation y’’ + Ay’ + By = f(x). Hence, the general solution y = y[p] + c[1]y[1] + c[2]u[2] where c[1]y [1] + c[2]u[2] is the complementary or homogeneous solution. We are particularly interested in finding y[p] when f(x) takes certain common form such as: • e^ax, where a is a constant. • sin(wx) and cos(wx), where w is a constant. • More generally, e^(a+iw)x, which is a complex exponential. For simplicity, we’ll denote this as e^αx, where α is a complex number. To make the process clearer, we rewrite the original ODE using differential operator notation. Let D represent the derivative with respect to x. The original ODE: y’’ + Ay’ + By = f(x) becomes (D^2 + AD + B)y = f(x). We can then introduce the operator p(D) defined as: p(D) = D^2 + AD + B. p(D) can be understood as a linear operator on functions and also a formal polynomial in D. This allows us to rewrite the ODE as: p(D)y = f(x) The substitution Rule. A key result for solving such equations is the Substitution Rule for exponentials. This rule states that: p(D)e^αx = p(α)e^αx. Apply the differential operator to the exponential function e^αx: p(D)e^αx = (D^2 + AD + B)e^αx =[By linearity, calculate each of them] D^2e^αx + ADe^αx + Be^αx = α^2e^αx + Aαe^αx + Be^αx =[Factor out e^ax] (α^2 +Aα + B)e^αx = p(α)e^αx ⇒ p(D)e^αx = p(α)e^αx∎ Exponential input theorem. It states that if f(x) = e^ax, the inhomogeneous ODE: y'' + Ay' + By = e^αx has a particular solution of the form $y_p = \frac{e^{αx}}{p(α)}$ assuming that p(α) ≠ 0. Start with the equation p(D)y[p] = e^αx and assume $y_p = \frac{e^{αx}}{p(α)}$ Substituting this into the equation: $p(D)y_p = p(D)\frac{e^{αx}}{p(α)} =[\text{Use the substitution Rule}] \frac{p(α)e^{αx}}{p(α)} = e^{αx}$ ∎ Exercise. Find the particular solution for the equation y’’ -y’ +2y = 10e^-xsin(x) Step 1: Convert to Complex Form To handle the trigonometric function sin(x), express it as the imaginary part of the complex exponential e^(-1+i)x. Thus, we solve the complex ODE: $(D^2-D+2)\tilde{y} = 10e^{(-1+i)x}$ Step 2: Apply the Exponential Input Theorem The characteristic polynomial is p(α) = α^2 - α + 2. Substituting α = −1+i: $p(-1+i) = (-1+i)^2-(-1+i)+2 = 3 -3i$ Thus, the complex particular solution is: $\tilde{y_p} = \frac{e^{αx}}{p(α)} = \frac{10^{(-1+i)x}}{3 -3i} = \frac{10e^{(-1+i)x}}{3(1 -i)} = \frac{10}{3}\frac{1+i}{(1-i)(1+i)}e^{(-1+i)x} = \frac{10} {3}\frac{1+i}{2}e^{-x}(cos(x)+isin(x)) = \frac{5}{3}(1+i)e^{-x}(cos(x)+isin(x))$ Extract the imaginary part: $y_p = Im(\tilde{y_p}) = \frac{5}{3}e^{-x}(cos(x)+sin(x)) =[\text{Alternatively, we can express this as:}] \frac{5}{3}e^{-x}\sqrt{2}cos(x-\frac{π}{4})$ Recall: Acos(x) + Bsin(x) = Rcos(x -ϕ) where A = B = 1, R is the amplitude, R = $\sqrt{1+1}=\sqrt{2}$ and ϕ is the phase shift, cos(ϕ) = sin(ϕ) = ^1⁄[√2]. Since both the sine and cosine of ϕ are equal, we recognize that: ϕ = ^π⁄[4] Special case: p(α) = 0 When p(α)=0, the simple approach using the Exponential Input Theorem fails. This occurs when α is a root of the characteristic equation. In this case, we must modify the form of the particular Exponential-shift rule. If p(α) = 0, it states that: p(D)e^axu(x) = e^axp(D +a)u(x). Particular case p(D) = D p(D)e^axu(x) =[Particular case p(D) = D] De^axu(x) = [Product rule] e^axu’ + ae^axu =[Factor e^ax] e^ax(u’ + au) = e^ax(D + a)u = e^axp(D + a)u(x) p(D) = D^2 D^2e^axu = D(De^axu) =[Previous result] D(e^ax(D + a)u) = [Previous result, but u is in this case (D + a)u] e^ax(D + a)(D + a)u = e^ax(D + a)^2u. This result is derived from the product rule of differentiation and can be applied iteratively to higher-order terms. Let y’’ + Ay’ + By = e^ax be an inhomogeneous ODE where a is a constant. Then, a particular solution is given by $y_p = \frac{xe^{ax}}{p’(a)}$ assuming that p(a) = 0 and a is a simple root of the characteristic equation p(D) = D^2 + AD + B. If a is a double root, the particular solution is $y_p = \frac{x^2e^{ax}}{p’’(a)}$ Proof (Simple root case) If a is a simple root of p(D), then p(D) = D^2 +AD + B. If a is a root of this characteristic equation, we know that p(a) = a^2 + Aa + B = 0. In the case of a simple root, we differentiate p(D) with respect to D, p’(D) = 2D + A. Substituting D = a, we get, p’(a) = 2a + A. Establish the claim: p(D)y[p] = $p(D)\frac{e^{ax}·x}{p’(a)} = e^{ax}$ $p(D)y_p = p(D)\frac{e^{ax}·x}{p’(a)}$ Now, use the Exponential Shift Rule, which states that: p(D)e^axu(x) = e^axp(D+a)u(x), and apply this rule to e^ax·x where u(x) = x, so $p(D)y_p = p(D)\frac{e^{ax}·x}{p’(a)} = \frac{e^{ax}}{p’(α)}·p(D+a)x$ Next, calculate p(D + a) ↭[p(D) = D^2 + AD + B] (D + a)^2 + A(D + a) + B^2 = D^2 + 2aD + a^2 + AD + Aa + B =[a is a root, a^2 +Aa + B = 0] D^2 +(2a + A)D Now, applying this to x, we calculate: p(D + a)x = (D^2 +(2a + A)D)x = D^2(x) + (2a + A)D(x) =[D(x) = 1, D’’(x) = 0] 0 + (2a + A)·1 = (2a + A) = p’(a) The operator D is shorthand for “taking the derivative with respect to x”, D(x) is 1, D(1) = 0 because the derivative of a constant is zero, D^2(x) = 0. $p(D)y_p = p(D)\frac{e^{ax}·x}{p’(a)} = \frac{e^{ax}}{p’(a)}p(D+a)x = \frac{e^{ax}}{p’(a)}p’(a) = e^{ax}$∎ • Find a particular solution to the inhomogeneous ODE y’’ -3y’ + 2y = e^x. To apply the theorem, we first need to determine the characteristic equation associated with the homogeneous part of the ODE: y’’ -3y’ + 2y = e^x. The characteristic equation is obtained by replacing y′′ with D^2, y′ with D, and y with 1, where D is the differential operator: D^2 -3D +2 = 0. Solve the characteristic equation D^2 -3D +2 =[Factoring the quadratic equation gives:] (D -1)(D -2), so the roots are D = 1 and D = 2. These roots corresponds to the solutions e^x and e^2x for the homogeneous ODE y’’ -3y’ + 2y = 0. Since the right-hand side of the inhomogeneous ODE is e^x, we look at the root D = 1. 1 is a simple root of the characteristic equation because it only appears once in the factorization (D−1)(D−2). By the theorem, when the right-hand side of the ODE is of the form e^ax and a is a simple root of the characteristic equation, the particular solution is given by: $y_p = \frac{xe^{ax}}{p’(a)} = \frac{xe^x}{p’(1)}$ The characteristic polynomial is: p(D) = D^2 -3D +2. Differentiate this with respect to D: p’(D) = 2D -3. Now, substitute D = 1: p’(1) = 2 -3 = -1. y[p] $= \frac{xe^x}{p’(1)} = \frac{xe^x}{-1} = -xe^x$ Resonance in Differential Equations We are exploring the phenomenon of resonance through the following second-order linear inhomogeneous differential equation: y’’ + w[0]^2y = cos(w[1]t). This equation describes a system (such as a mass-spring system or a pendulum) that experiences external forcing, represented by cos(w[1]t) where • w[0]^2 is the natural frequency of the system (how it naturally oscillates without external forces). • w[1] is the input frequency of the external force (the frequency of the external “push”, like pushing the pendulum at regular intervals). We assume w[1] ≠ w[0], meaning the input frequency is not exactly the same as the system’s natural frequency. Convert the Equation to Differential Operator Form: First, we rewrite the differential equation using D, where D = d/dt represents the derivative with respect to t. So: (D^2 + w[0]^2)y = cos(w[1]t). This is a second-order linear inhomogeneous differential equation. Switch to Complex Exponentials (Easier to Solve): Since working with cosine functions can be tricky, we switch to using complex exponentials, as they simplify the process. Using Euler’s formula, we know that: $cos(w_1t)=Re(e^{iw_1t})$ Thus, solving the equation with $e^{iw_1t}$ will give us a complex solution, and we’ll take the real part at the end to get our solution for the cosine function. The equation becomes: $(D^2+w_0^2)\tilde{y} = e^{iw_1t}$ where $\tilde{y}$ is the complex solution, and we are only interested in the real part of $\tilde{y}$ at the end. Find a Particular Solution: To solve for $\tilde{y_p}$, we use the fact that e^iw[1]t is a simple exponential, and we substitute into the equation: [Exponential input theorem. It states that if f(x) = e^ax, the inhomogeneous ODE: y'' + Ay' + By = e^αx has a particular solution of the form $y_p = \frac{e^{αx}}{p(α)}$ assuming that p(α) ≠ 0.] p(D) is the characteristic polynomial of the differential operator D^2 + w[0]^2. p(iw[1]) = $(iw_1)^2+w_0^2 = w_0^2-w_1^2$ Thus, the particular solution becomes: $\tilde{y_p} = \frac{e^{iw_1t}}{(iw_1t)^2+w_0^2} =[\text{Simplifying the denominator:}] \frac{e^{iw_1t}}{w_0^2-w_1^2}$ Now, we take the real part of $y_p$, to find the solution for y[p]: $y_p = Re(\tilde{y_p}) = \frac{cos(w_1t)}{w_0^2-w_1^2}$. This is the particular solution when the input frequency is not equal to the system’s natural frequency, w[1] ≠ w[0] Resonance (When w[1] ≈ w[0]): Resonance occurs when the input frequency w[1] is close to the natural frequency w[0] of the system. In such cases, the response of the system becomes amplified (the denominator $w_0^2-w_1^2$ get closer to zero, which leads to the amplitude of the response growing significantly), as the forcing frequency aligns with the system’s natural oscillation (Refer to Figure A for a visual representation and aid in understanding it) Special Case: Resonance (When w[0] ≈ w[1]) If w[0] ≈ w[1], we have a resonance condition. In this case, the earlier approach doesn’t work because the denominator becomes zero, meaning the solution must be treated differently. We start with (D^2 + w[0]^2)y = cos(w[1]t). Again, we convert to complex exponentials: $(D^2+w_0^2)\tilde{y} = e^{iw_0t}$ but iw[0] is a root of the characteristic polynomial, $(D^2+w_0^2)(iw_0) = (iw_0)^2+w_0^2 = 0$ To find a particular solution [$y_p = \frac{xe^{ax}}{p’(a)}, p’(D) = 2D$], $p’(iw_0) = 2iw_0$ $\tilde{y_p} = \frac{te^{iw_0t}}{2iw_0}$ Finally, we take the real part of this to get the solution for the cosine case: $y_p = Re(\tilde{y_p}) = \frac{tsin(w_0t)}{2w_0}$. When w[1] ≈ w[0], the solution has a steadily increasing amplitude. In particular: • The term tsin(w[0]t) shows that the amplitude grows linearly with time. • This is a characteristic of resonance: the system’s response builds over time, leading to larger and larger oscillations (Refer to Figure C for a visual representation and aid in understanding If there’s no damping or limit to the oscillation, resonance can cause the amplitude to grow indefinitely, which is why resonance can be both powerful and dangerous (e.g., bridges collapsing due to resonance from wind or foot traffic) Resonance in Second-Order Linear Differential Equations We are examining resonance by working with the following second-order linear inhomogeneous differential equation: y’’ + w[0]^2y = cos(w[1]t) This equation describes a system (such as a mass-spring system or a pendulum) that experiences external forcing, represented by cos(w[1]t) The general solution to this inhomogeneous differential equation has two parts: 1. The complementary solution y[c], which solves the associated homogeneous equation y’’ + w[0]^2y = 0. 2. The particular solution y[p], which accounts for the inhomogeneous part cos(w[1]t) The complementary solution corresponds to the general solution of the homogeneous equation: y’’ + w[0]^2y = 0. The characteristic equation for this is r^2 + w[0]^2y = 0. Solving this gives complex roots r = ±iw[0]. Therefore, the complementary solution is: y[c] = c[1]cos(w[0]t) + c[2]sin(w[0]t) where c[1] and c[2] are arbitrary constants determined by initial conditions. This describes the natural oscillatory behavior of the system with frequency w[0]. We have previously found a particular solution: $y_p = \frac{cos(w_1t)}{w_0^2-w_1^2}$. This is the particular solution when the input frequency is not equal to the system’s natural frequency, w[0] ≠ Special Case: Resonance (w[1] ≈ w[0]) When the forcing frequency w[1] approaches the natural frequency w[0], the system experiences resonance. In this case, the particular solution $y_p = \frac{cos(w_1t)}{w_0^2-w_1^2}$ becomes problematic as the denominator approaches zero, leading to a singularity. To deal with this, we find another particular solution that takes into account the resonance condition. We start with the complementary solution: $y_c = -\frac{cos(w_0t)}{w_0^2-w_1^2}$ So another particular solution is $\frac{cos(w_1t)}{w_0^2-w_1^2} -\frac{cos(w_0t)}{w_0^2-w_1^2}$ Taking the limit as w[1] → w[0]: $\lim_{w_1 \to w_0} \frac{cos(w_1t)}{w_0^2-w_1^2} -\frac{cos(w_0t)}{w_0^2-w_1^2} = \lim_{w_1 \to w_0} \frac{cos(w_1t)-cos(w_0t)}{w_0^2-w_1^2} $[L’Hopital Rule, please note the variable is w[1]] $\ lim_{w_1 \to w_0} \frac{-sin(w_1t)·t}{-2·w_1} = \frac{sin(w_0t)}{2w_0}$ This result is the particular solution when w[1] = w[0], which corresponds to the resonance case. This solution grows linearly in time (t), reflecting the amplification of oscillations characteristic of resonance. The amplitude of the oscillation increases over time without bound, which can lead to destructive consequences if no damping is present. Geometric Interpretation of the Resonance Condition (w[1] ≈ w[0]) We can also provide a geometric interpretation of the behavior as w[1] approaches w[0]. Consider the difference: $\frac{cos(w_1t)-cos(w_0t)}{w_0^2-w_1^2}= [\text{Using the identity trigonometry: }cosB -cosA = 2sin(\frac{A-B}{2})sin(\frac{A+B}{2})] = \frac{2sin(\frac{w_0-w_1}{2})t+sin(\frac{w_0+w_1}{2})t}{w_0^ 2-w_1^2}$ This equation consists of two parts: (Refer to Figure B for a visual representation and aid in understanding it) • $\frac{2sin(\frac{w_0-w_1}{2})t}{w_0^2-w_1^2}$. This represents oscillations with very small frequency (since w[1]-w[0] is small), resulting in a slowly varying amplitude. The period of this oscillation is large. Because, the term $\frac{2}{w_0^2-w_1^2}$, it increases at a steadily increasing amplitude as w[1] approaches w[0]. • $\frac{sin(\frac{w_0+w_1}{2})t}{w_0^2-w_1^2} ≈ \frac{sin(w_0)t}{w_0^2-w_1^2}$, this is a pure oscillatory term with frequency close to w[0]. As w[1] approaches w[0], the system exhibits a phenomenon known as beats, where the amplitude of the oscillations grows slowly over time. At exact resonance (w[1] = w[0]), the oscillation amplitude grows linearly over time, leading to increasingly large oscillations. This growth can continue indefinitely in undamped systems, resulting in increasingly large oscillations that may cause structural damage, failure or other dangerous effects. This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License and is based on MIT OpenCourseWare [18.01 Single Variable Calculus, Fall 2007]. 1. NPTEL-NOC IITM, Introduction to Galois Theory. 2. Algebra, Second Edition, by Michael Artin. 3. LibreTexts, Calculus and Calculus 3e (Apex). Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson). 4. Field and Galois Theory, by Patrick Morandi. Springer. 5. Michael Penn, and MathMajor. 6. Contemporary Abstract Algebra, Joseph, A. Gallian. 7. YouTube’s Andrew Misseldine: Calculus. College Algebra and Abstract Algebra. 8. MIT OpenCourseWare [18.03 Differential Equations, Spring 2006], YouTube by MIT OpenCourseWare.	{"url":"https://justtothepoint.com/calculus/inhomogeneous2/","timestamp":"2024-11-14T04:11:16Z","content_type":"text/html","content_length":"35295","record_id":"<urn:uuid:9d4b1539-14b2-4699-bcdd-6bcf3cac739d>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00764.warc.gz"}
Wolfram Cloud A Candidate Geometrical Formalism for the Foundations of Mathematics and Physics Formal Correspondences Between Homotopy Type Theory and the Wolfram Model By Jonathan Gorard (University of Cambridge / Wolfram Research, Inc.)This bulletin is a writeup of work done in collaboration with Xerxes Arsiwalla and Stephen Wolfram, as publicly presented and discussed in https://www.youtube.com/watch?v=x5v3KFFWv2o. The field of “metamathematics” - a term first popularized by David Hilbert in the context of Hilbert’s program to clarify the foundations of mathematics in the early 20th century - may ultimately be regarded as the study of “pre-mathematics”. In other words, while mathematics studies what the properties of particular mathematical structures (such as groups, rings, fields, topological spaces, etc.) may be, metamathematics studies what the properties of mathematics itself, when regarded as a mathematical structure in its own right, are. Metamathematics is thus “pre”-mathematics in the sense that it is the structure from which the structure of mathematics itself develops (much like “pre-geometry” in theoretical cosmology is regarded as the structure from which the structure of the universe itself develops). In much the same way, the Wolfram Physics Project may be thought of as being the study of “pre-physics”. The Wolfram Model is not a model for fundamental physics in and of itself, but rather an infinite family of models (or, if you prefer, a “formalism”), within which concepts and theories of fundamental physics can be systematically investigated and validated. However, one of the most remarkable things that we have discovered over the year-or-so that we’ve been working on this project is that a shockingly high proportion of the known laws of physics (most notably, general relativity and quantum mechanics) appear to be very “generic” features of the formalism, i.e. they are exhibited by a wide class of models, as opposed to being particular features of our specific universe, as one might otherwise expect. This leads to the tantalizing possibility that, just as metamathematics aims to explain why mathematics has the structure that it does, the Wolfram Model may serve to explain why physics has the structure that it does. The correspondence between the Wolfram Model and metamathematics goes far deeper than this, though. It does indeed appear that many of the recent innovations in the foundations of mathematics (including dependent type theory, homotopy type theory, the univalence axiom, synthetic geometry, etc.) have considerable overlaps with our recent insights into the foundations of physics (particularly surrounding concepts such as the topology and geometry of rulial space, foliations and fibrations of the rulial multiway system, completion procedures and the inevitability of quantum mechanics, etc.) Indeed, what I aim to sketch out in this bulletin is a potential scheme for a new, computable foundation for both mathematics and physics, which can be thought of as an abstract unification of homotopy type theory and the Wolfram Physics Project. The Curry-Howard Isomorphism and its Physical Generalization Let me begin by describing the broad philosophical setup of this bulletin. In mathematical logic and theoretical computer science, there is a famous result named after Haskell Curry and William Alvin Howard (and deeply connected to the core ideas of intuitionistic logic and its operational interpretation) known as the “Curry-Howard isomorphism” or “Curry-Howard correspondence”. Loosely speaking, what it says is that all mathematical proofs can be interpreted as computer programs, and (somewhat more surprisingly), all computer programs can be interpreted as mathematical proofs. In other words, the core problems of proof theory in mathematical logic and programming language theory in theoretical computer science are actually identical. At some level, the Curry-Howard isomorphism provides the formal justification for an otherwise obvious intuition. In pure mathematics, one starts from a set of “axioms”, one applies a “proof” (in accordance with a set of “rules of inference”), and one obtains a “theorem”. The abstract notion of computation can thus be thought of as being an ultimately desiccated version of the concept of mathematical proof, in which one starts with some “input”, one applies a “computation” (in accordance with a set of “algorithmic rules”), and one obtains an “output”. In other words, one has the formal correspondence: <\|"Axioms""Input","Proof""Computation","Rules of Inference""Algorithmic Rules","Theorem""Output"\|> Indeed, the symbolic nature of the Wolfram Language allows us to see this correspondence explicitly. If we prove a trivial mathematical theorem (such as the theorem that a==c, starting from the axioms that a==b and b==c, i.e. a proof of the transitivity of equality), then we obtain a symbolic proof object: Axiom 1 ba Axiom 2 bc Hypothesis 1 ac SubstitutionLemma 1 ac Conclusion 1 True Each statement in the proof, such as “a==c”, can be interpreted as a symbolic expression. Each application of an inference rule, such as “a->b” (which is, in turn, derived from the axiom “a==b”) can be interpreted as an application of symbolic rewrite rule. Thus, the entire proof can be implemented as a symbolic piece of Wolfram Language code, in which the key steps of the proof are performed using MapAt[...] and Replace[...] functions: Running this piece of code hence allows us to determine that the corresponding theorem is actually true: As such, we can interpret the Curry-Howard isomorphism as being the statement that every ProofObject in the Wolfram Language has an associated proof function (which we can explicitly demonstrate, as above), and moreover that every piece of symbolic Wolfram Language code has a corresponding interpretation as a ProofObject for some theorem. The latter direction is easy to see once one appreciates that all Wolfram Language programs are ultimately reducible to sequences of transformation operations being applied to symbolic expressions, just like a mathematical proof. To be a bit more rigorous, the Curry-Howard isomorphism involves using the so-called “Propositions as Types” interpretation of metamathematics; in other words, Curry-Howard states that a program that produces an output of a given data type can be interpreted as a proof that the corresponding type is inhabited, and vice versa. In the Wolfram Physics Project, and in the field of physics more generally, one also considers an ultimately desiccated notion of what a “physical system” truly is. More specifically, one imagines the system beginning in some “initial state”, undergoing “motion” (in accordance with some “laws of motion”), and ending up in some final state. This immediately suggests a further refinement of the Curry-Howard isomorphism, in which programs, proofs and physical systems are really the same kind of thing: <\|"Initial State""Input","Motion""Computation","Laws of Motion""Algorithmic Rules","Final State""Output"\|> Viewed in this way, the Church-Turing thesis is neither a definition, a theorem nor a conjecture; rather it is a hypothesis of fundamental physics. Namely, what the Church-Turing thesis truly says in this context is that the set of partial functions that can be computed by a universal Turing machine is exactly the set of partial functions that is instantiated by the laws of physics. Or, more concretely, the set of possible motions of a (physically realized) universal Turing machine is in one-to-one correspondence with the set of possible motions of any physical system, anywhere in the The rest of this bulletin will be dedicated to fleshing out the details and implications of this correspondence, both for the foundations of physics, and for the foundations of math. Homotopy Type Theory, Univalent Foundations and Synthetic Geometry Homotopy type theory is an augmentation of type theory (or, more specifically, of Per Martin-Löf’s intuitionistic type theory) with one key additional axiom, namely Vladimir Voevodsky’s axiom of univalence, whose significance we shall discuss later. The key philosophical idea underpinning homotopy type theory is a slight extension of the usual “propositions as types” interpretation of the Curry-Howard correspondence, in which now “types” correspond to topological spaces (homotopy spaces), “terms” of a given type correspond to points in those spaces, proofs of equivalence between terms of a given type correspond to paths connecting the associated points, proofs of equivalence between proofs correspond to (potentially higher) homotopies between the associated paths, etc. In other words, homotopy type theory is a way of endowing type theory (and the foundation of mathematics more generally) with a kind of in-built topological structure. We know from ordinary topology that, a homotopy is just a continuous deformation between paths. More precisely, given a pair of continuous functions f and g from a topological space X to another topological space Y, one can define a homotopy as being a continuous function h from the product space of X with the (closed) unit interval to Y, where the unit interval can be thought of as “parameterizing” the homotopy. To see this explicitly, suppose that we define function f to just be an ordinary Sin function: In homotopy type theory, we would interpret this overall space as being a type, the endpoints of this path would be terms of that type, and the path defined by the function f would correspond to the proof that those terms are equivalent. In ordinary mathematics, we are used to the idea that a given theorem can have many (apparently totally different) proofs, such as in the case of the fundamental theorem of algebra, in which there is a great variety of possible proofs stemming from topology, complex analysis, algebra, Riemannian geometry, etc. So suppose now that we define g to be a slightly different function connecting the same two points: Now, we can interpret this path as being a different possible proof of the same proposition (namely the proposition that the terms corresponding to the endpoints of the path are equivalent). We can then define a homotopy between these two paths as follows: Although this homotopy lives naturally in this higher-dimensional space, we can easily project it back down onto our original space: Finally, this homotopy can be interpreted as a proof that these two different proofs are actually equivalent (in the sense that they are both valid proofs of the same theorem). As we have now seen explicitly, this homotopy can itself be interpreted as a path in some higher-dimensional space, so we can also proceed to define homotopies between those paths, which would correspond to proofs of equivalence between proofs of equivalence of proofs, etc. Thus, there is really a whole infinite hierarchy of all possible higher-order homotopies, corresponding to the infinite hierarchy of all possible higher-order proofs of equivalence, and this hierarchy will becomes relevant soon, once we start considering its implications for the geometry of rulial space. The effort to formalize mathematics in terms of homotopy type theory is commonly known as the “univalent foundations program”, in reference to the central role played by Voevodsky’s univalence axiom. One of the “big picture” questions driving the advancement of univalent foundations (as well as many related areas, such as Michael Shulman’s formulations of “synthetic topology”, “cohesive geometry”, etc.) is a desire to better understand the role of “space” in the foundations of mathematics. In many ways, it is a great mystery why so many different mathematical structures (such as groups, rings, fields, Boolean algebras, lattices, etc.) also come equipped with a corresponding notion of “space”. For instance, in the context of group theory, a group is usually defined purely algebraically (i.e. as a closed algebraic structure obeying the axioms of associativity, identity and inverse), and yet one nevertheless has Lie groups, topological groups, matrix groups, loop spaces, etc., within all of which there naturally appears some corresponding spatial structure (such as a differentiable manifold in the case of a Lie group, or a topological space in the case of a topological group, etc.), which seems to “come along for the ride”, in the sense that the spatial structure is somehow respected by all of the underlying algebraic operations. We can see this explicitly by considering the case of the Lie group SO(3). First, consider the set of all 3x3 matrices of determinant 1; here, we shall look directly at a finite subset: Then, SO(3) can be defined purely syntactically in terms of a binary operation on this set of elements (in this case, corresponding to matrix multiplication) that obeys the axioms of associativity: However, because of the interpretation of 3x3 matrices of determinant 1 as 3D rotation matrices, SO(3) comes naturally equipped with a spatial structure corresponding to the following differentiable manifold (and, hence, is indeed a Lie group): But why? The ZFC axioms of set theory are purely syntactical and algebraic: there is no inherent notion of “space” in ZFC. So why should objects (like groups) defined within ZFC in a purely syntactical and algebraic way induce a notion of space as if by magic? Homotopy type theory aims to address the mystery of the origin of space, by considering instead a foundation for mathematics that is explicitly spatial and topological at the outset, and in which the syntactical structure of mathematics is somehow emergent. In the context of the Wolfram Physics Project, we found ourselves addressing a very similar question. Our models are defined purely combinatorially, in terms of replacement operations on set systems (or, equivalently, in terms of transformation rules on hypergraphs). And yet, the correspondence between our models and known laws of physics depends upon their being a reasonable interpretation of these combinatorial structure in terms of continuous spaces. Our derivation of the Einstein field equations, for instance, depends upon spatial hypergraphs having a reasonable continuum limit in terms of Riemannian manifolds (and hence being a reasonable candidate for physical space): And, moreover, it depends upon causal graphs having a reasonable continuum limit in terms of Lorentzian manifolds (and hence being a reasonable candidate for spacetime): And, at a more abstract and speculative level, some of our current conjectures regarding the quantum mechanical aspects of our formalism depend upon the continuum limit of branchial graphs in multiway systems as being projective Hilbert spaces: Or multiway causal graphs as being twistor correspondence spaces: So could these two questions be related? Could the explanations offered by homotopy type theory for the origins of spatial structure in mathematics also imply, after appropriate translation, equivalent explanations for the origins of spatial structure in physics? And if so, what would the things we’ve discovered in the course of the Wolfram Physics Project so far imply regarding the foundations of mathematics? Multiway Systems as Models for Homotopy Type Theory As discussed above, the axioms of a mathematical theory can be interpreted as defined transformation rules between symbolic expressions. For instance, in the axioms of group theory: The associativity axiom can be interpreted as the following pair of (delayed) pattern rules: Therefore, a proof of a mathematical proposition can ultimately be represented as a multiway system. Indeed, at some level, this is precisely what a ProofObject is: A ProofObject is formatted a little differently to the way multiway systems are conventionally formatted. The green (downwards-pointing) arrows are the axioms, the turquoise diamond is the hypothesis and the red square is the conclusion (in this case, the conclusion being that the hypothesis is true). Each of the orange circles is a substitution lemma, and can be thought of as an ordinary multiway state node: it’s an intermediate lemma that is deduced along the way by simply applying the axiomatic transformation rules described above, just like in a regular multiway system. However, each of the dark orange triangles is a critical pair lemma, which is an indication of a place where the multiway system bifurcates, and in which an additional lemma is necessary in order to prevent it from doing so (in other words, it corresponds to a place where a completion procedure has been applied). Looking at a specific example: Here, we see that two rules, indicated by “Rule” and “MatchingRule” (derived from Axiom 2 and Axiom 1, respectively) both match a common subpattern, indicated by “Subpattern”, thus producing a bifurcation in the multiway system. However, since both elements of the bifurcation were derived from a common expression by valid application of the rewrite rules, this bifurcation itself constitutes a proof that the elements are equivalent, and hence we can declare them to be equal (which is the critical pair lemma). This declaration of equivalence immediately forces the bifurcation to collapse, so by adding a sufficient number of critical pair lemmas to the rewrite system, we can therefore force the multiway system to have only a single path (up to critical pair equivalence). This, incidentally, is identical to the process of adding critical pair completions so as to force causal invariance, which is the basis of our current model for quantum measurement; the correspondence with quantum mechanics will be explored in more detail later. Therefore, the proof graph can be thought of as being an intermediate between a single evolution thread and a whole multiway system; the substitution lemmas by themselves indicate the path of a single evolution thread, but the critical pair lemmas indicate the places where that evolution thread has been “plumbed into” a large, bifurcating multiway system. We can see this correspondence more directly by considering a simple string multiway system: We can clearly see that the strings “AA” and “ABBBABBB” are connected by a path in the multiway system - a fact that we can visualize explicitly: In effect, we can interpret this path as being a proof of the proposition that “AA”->”ABBBABBB”, given the axiom “A”->”AB”. To make the correspondence manifest, with a small amount of additional code we can also represent this path directly as a ProofObject: Of course, a key idea in homotopy type theory is that we can consider the existence of multiple paths connecting the same pair of points, and hence multiple proofs of the same proposition. For instance, take: Homotopy type theory then tells us that we can represent the proof of equivalence of these proofs as a homotopy between the associated paths. Since an (invertible) homotopy is ultimately just a mapping from points on one path to corresponding points on the other path (and back again), this is easy to represent discretely: However, this homotopy is kind of a “fake”, in the sense that it has been explicitly grafted on to the multiway system after-the-fact. The more “honest” way to enact a homotopy of this kind would be to define explicit multiway rules mapping from states along one path onto states along the other, thus yielding: As Xerxes Arsiwalla pointed out, it is therefore correct to think of the multiway rules (like “A”->”AB”) as being analogous to type constructors in mathematical logic (i.e. rules for building new types), so as to allow the multiway systems themselves to be interpreted as inductive types, just as homotopy type theory requires. Moreover, these completion procedures described above that allow us to define homotopies between paths are a kind of higher-order rule (i.e. they are rules for generating new rules), with the result being that the higher-dimensional multiway system that one obtains after applying such a completion procedure is the analog of a higher inductive type. Since there exists an infinite hierarchy of higher inductive types, it follows that there must exist a corresponding infinite hierarchy of higher-order multiway systems, with the multiway system at level n being a partially “completed” version of the multiway system at level n-1 (i.e. in which certain explicit homotopies have been defined between certain pairs of paths). So a natural question to ask is: what kind of mathematical structure would such a hierarchy represent? Groupoids, Topoi and Grothendieck’s Hypothesis A “category” is just a collection of objects (which can be represented as nodes), along with a collection of “morphisms” between those objects (which can be represented as directed edges), and with the property that all morphisms are both associative and reflexive (i.e. there exists an identity morphism for each object). For instance, starting from a directed graph: We can force every directed edge (i.e. morphism) to be associative by computing the transitive closure: And then we can enforce reflexivity by adding a self-loop (i.e. an identity morphism) for each vertex: Thus, this directed graph has a bona fide interpretation as a small category (small because the corresponding directed graph is finite). On the other hand, a “groupoid” is a special case of a category in which all morphisms are invertible (and hence are isomorphisms), which we can again demonstrate purely graph-theoretically by converting our directed graph into an undirected one: Or, to be even more explicit, we could add two-way directed edges to indicate isomorphisms: Groupoids are so named because they generalize ordinary algebraic groups; a group can essentially be thought of as a special case of a groupoid that contains just a single object, and where the invertible morphism corresponds to the (unary) group inverse operation. Closely related to groupoids are objects known as “topoi”, which can be thought of as being the abstract categorical generalization of point set topology. A “topos” is a category that “behaves like” the category of sets (or, to be more precise, like the category of sheaves of sets on a given topological space) in some suitably defined sense. The crucial feature of a topos that relates it to a groupoid, however, is that topoi necessarily possess a notion of “localization”. In ordinary commutative algebra, rings are not required to possess multiplicative inverses for all elements, so one can “localize” the ring by introducing denominators (such that the localization is like a restricted version of the field of fractions), thereby introducing multiplicative for elements where they did not previously exist. Similarly, the localization of a category introduces some inverse morphisms where they did not previously exist, hence converting some morphisms into isomorphisms, and thus causing the category to behave “locally” like a groupoid. All of our multiway systems are naturally equipped with a notion of localization; for instance, consider the string multiway system we explored earlier: We can now select 10 edges at random and introduce their reverse directed edges, to simulate adding inverse morphisms for a collection of ten random morphisms. Much like in the foliation example given above, this method of localization is kind of a “fake”, since we are artificially grafting these inverted edges onto a pre-existing multiway graph. Instead, we can simply adjoin the reversed edges as new set of multiway rules: Applying this localization procedure to the entire multiway rule, unsurprisingly, yields the following groupoid-like structure: Since, in the category of sets, all objects are sets and all morphisms are functions from sets to sets, because for any function from set A to set B there can exist (at least locally) an inverse function from set B to set A, the category of sets is trivially localizable. It is in this sense that we can say that more general topoi “behave like” the category of sets. One can make this correspondence more mathematically rigorous by realizing (as noted by Xerxes) that, when interpreting multiway systems as inductive types and the associated multiway rules as type constructors, then so long as one includes a “subobject classifier” (where a special object in the category that generalizes the notion of a subset identifier, where all subobjects of a given object in the category correspond to morphisms from the subobject onto the subobject classifier), and assuming that finite limits exist (where a limit is a categorical construction that generalizes the universal constructions of products and pullbacks), the resulting multiway system that one obtains is precisely an elementary topos. We have furthermore made the conjecture (although we do not yet know how to prove it) that this subobject classifier is what endows our multiway systems with a concept of “foliation” that is so crucial for constructing branchial spaces, and for deducing both quantum mechanics and general relativity in the appropriate continuum limits, since at least intuitively each hypersurface in the foliation may be thought of as corresponding to a subobject in the corresponding category. We shall explore this conjecture more closely, and examine precisely what foliations of multiway systems mean from a metamathematical standpoint, within the next section. For the time being, however, it suffices to realize that such an elementary topos comes naturally equipped (thanks to various results in cohesive geometry) with a functor (that is, a map between categories) to the topos of sets, as well as a pair of adjoint functors (where adjunction here refers to a kind of “weak” equivalence relation) admitting the discrete and indiscrete topologies, respectively. Free topoi possessing this particular collection of functors provide a means of formalizing the notion of a topological space. Gauge-Dependence of Mathematical Truth and a “Relativistic” Interpretation of the Incompleteness Theorems Kurt Gödel’s proof of the incompleteness theorems in 1931 established, via the ingenious technique of Gödel numbering, that Peano arithmetic (the standard axiom system for integer arithmetic) was capable of universal computation. Therefore, in particular, he demonstrated that Peano arithmetic could be set up so as to encode the statement “This statement is unprovable.” as a statement about natural numbers. Clearly, if this statement is true, then it is unprovable (so Peano arithmetic is incomplete), and if it is false, then it is provable (so Peano arithmetic is inconsistent). As such, one way to phrase the first incompleteness theorem is that “If Peano arithmetic is consistent, then there will exist propositions that are independent of the Peano axioms.” In other words, there will exist propositions where neither the proposition nor its negation are (syntactically) provable using the axioms of Peano arithmetic in any finite time. Completions (i.e. quantum mechanics) are applications of the univalence axiom. Ability to perform quantum measurement is a statement of multiway topology (not too much “branchiness”). Mystery: Define something set-theoretically, often a space “appears”. Why? Define a causal network, often a spacetime “appears”. ZFC: Axiom of Extensionality - like “completions” in set theory. It’s a StateEquivalenceFunction for set theory. Ultimate nature of mathematics/physics: Things that can be considered equivalent can also be considered identical. “Cayley Graph” of a multiway system is an “infinitely completed” states graph (JG conjecture: this is a groupoid.) Axiom of Extensions/Univalence/etc. are byproducts of us as observers choosing to model mathematics in the same way we model physics. Ability to do mathematics arises in the same way as ability to understand physics (the existence of computational reducibility/rulial black holes/etc.) Xerxes: “Conventional” uncertainty principle is a limit on what can be measured in physics. This formalism is constructivist by nature (because of univalence, so no axiom of choice, no law of excluded middle, etc.), so there are limits on what can be constructed in mathematics. Is this the uncertainty principle? Grothendieck’s Hypothesis: All infinity-groupoids are topological (homotopy) spaces. Groupoids: A category in which all morphisms are isomorphisms. Topos: A category that “behaves like” the category of sets (i.e. “Set”). More precisely, a category that possesses a “localization” notion. (Analogy to commutative algebra: ring localization introduces denominators.) First incompleteness theorem: If PA/ZFC is consistent, there exist propositions that are independent of PA/ZFC, i.e. propositions where neither it nor its negation are syntactically provable (in finite time). (Proof-theoretic statement) CH is independent of ZFC => There exist models of ZFC such that ZFC+CH is consistent <=> ZFC is consistent (Godel), and models such that ZFC+CH is inconsistent (Cohen, forcing). (Model-theoretic implication) First incompleteness theorem (WPP): Gauge-dependence in mathematics. Proof theory: Defines a partial order on mathematics. Model theory: Gauge choice that allows one to construct a total order. Ordinary relativity: There is a causal partial order (conformal structure of spacetime). Timelike ordering is gauge-invariant, spacelike is gauge-dependent. There exist propositions (defined by proof theory) whose truth is gauge-invariant (syntactically provable), and those (defined by model theory) whose truth is gauge-dependent. Non-constructive proofs are like boundaries of light cones (proofs that travel “at the speed of proof”), so cannot be witnessed by any “subliminal” model. Black holes are like “inevitable non-constructivism” in mathematics. => (Axiom of Choice is a black hole?) SR: Model-independence of proof theory (syntactically provable statements). => A model-theoretic speedup theorem (analog of Lorentz transformation). Inertial frames: “Free” models? (Look this up...) Superposition in QM: There exists proof redundancy. Godel’s speed-up theorem: “This statement is unprovable using fewer than n symbols.” (takes n symbols to prove in PA, but is trivial to prove in PA+Con(PA).) For any pair of propositions, determining whether there exists a path between them may take an arbitrarily large amount of computational effort. (Defining truth/falsehood in terms of consistency/inconsistency with the total/partial order.) (Syntactical truth is consistency with partial order, semantic truth is consistency with model-induced total order.) General relativity: what is the name for timelike geodesics that are so long that the endpoints appear to be spacelike-separated? * timelike singularity!! * Connection to Malament-Hogarth spacetimes? Mathematics is partially ordered, because mathematicians are computationally bounded? Propositional extensionality (logic): If propositions are path-connected then they are defined to be “equivalent”. Voevodsky’s Univalence Axiom: (If propositions are equivalent, then they are defined to be “identical”.) <- This implication is itself an equivalence. (Xerxes): Take a multiway system, include a subobject classifier (the categorical analog of a subset identifier) as one of the type constructors, then you get an elementary topos (assuming that finite limits exist). Interpretation of the Schwarzschild radius: There is a limit to the height of the “abstraction tower” that you can build in mathematics whilst still being a bona fide constructivist. Increased “abstraction” (energy-momentum in “proof space”), there is gravitational lensing, creating more inequivalent proofs of the same propositions. i.e. abstraction => (proof-theoretic) redundancy. (Conjecture): Subobject classifiers allow you to define foliations. (Xerxes): Such an elementary topos comes naturally equipped with a functor from the topos to the topos of sets, as well as a pair adjoint functors admitting the discrete and indiscrete topologies. Free (and local) topoi with this collection of functors formalize the notion of a topological space. => Locality comes from the subobject classifier, and hence from the “locality” of foliations. Fibration: Formalization of the idea that you can parameterize one topological space (fibre) with another one (base). Connection: A rule allowing you to “lift” a step in the base space to a step in the fibre. Fibration relaxes ordinary fibre bundle, since fibres don’t have to be in the same topological space (or indeed homeomorphic); you just need homotopy-equivalence (i.e. the “homotopy lifting Categorical Quantum Mechanics (Abramsky and Coecke): Symmetric monoidal category. Monoidal category: Category equipped with a tensor product on objects. Symmetric monoidal category: Monoidal category where the product is “as commutative possible”. (Xerxes): So long as we have a tensor product constructor, we recover categorical QM upon foliating the rulial multiway system.	{"url":"https://www.wolframcloud.com/obj/wolframphysics/WorkingMaterial/2020/Metamathematics-08.nb","timestamp":"2024-11-05T08:47:13Z","content_type":"text/html","content_length":"713239","record_id":"<urn:uuid:3ef3a283-991a-49ed-a5e5-547385fb4f75>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00738.warc.gz"}
Hall effect in the two-dimensional metal Sr2RuO4 We report a detailed study of the Hall effect in Sr2RuO4, the first layered perovskite superconductor which does not contain copper (T-c congruent to 1 K). The Hall coefficient (R(H)) was measured at temperatures between 20 mK and 300 K, and an unusual dependence of R(H) on the applied magnetic field was observed. R(H) has a strong temperature dependence below 25 K, but below 1 K it saturates at a value of -1.15x10(-10) m(3)/C. The Fermi surface of Sr2RuO4 is known from quantum oscillation measurements, and since it is nearly two dimensional, it is possible to derive a simple expression for R(H) using methods developed by Ong [Phys. Rev. B 43, 193 (1991)]. We show that if the mean free path, l, is assumed to be isotropic at low temperatures, it is possible to make an accurate quantitative calculation of R(H) on the basis of the known Fermi surface parameters. Dive into the research topics of 'Hall effect in the two-dimensional metal Sr2RuO4'. Together they form a unique fingerprint.	{"url":"https://research-portal.st-andrews.ac.uk/en/publications/hall-effect-in-the-two-dimensional-metal-sr2ruo4","timestamp":"2024-11-06T17:06:09Z","content_type":"text/html","content_length":"54412","record_id":"<urn:uuid:7cad5c1f-8a19-422a-a72b-6450494253c5>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00811.warc.gz"}
Data Downloads MathArticles.com provides relevant articles from renowned math journals. The articles are coordinated to the topics of Larson Calculus. Visit MathArticles.com to access articles from: Journal Organizations AMATYC Review American Mathematical Association of Two-Year Colleges American Mathematical Monthly Mathematical Association of America College Mathematics Journal Mathematical Association of America Journal of Chemical Education American Chemical Society Math Horizons Mathematical Association of America Mathematical Gazette Mathematical Association of (UK) Mathematics Magazine Mathematical Association of America Mathematics Teacher National Council of Teachers of Mathematics Physics Teacher American Association of Physics Teachers Scientific American Scientific American UMAP Journal Consortium for Mathematics and its Applications	{"url":"https://www.collegeprepalgebra.com/cpa/content/data-downloads/","timestamp":"2024-11-14T20:34:50Z","content_type":"text/html","content_length":"69952","record_id":"<urn:uuid:0ae1b4bc-b9b9-4b99-854b-505862f9ccaa>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00485.warc.gz"}
Lookup Tables Math Operations Signal Routing User-Defined Functions Blocks Supported by BlockImporter The following Simulink(R) blocks are supported by BlockImporter. Derivative computes the time derivative of the input. Integrator computes the integral of the input with respect to time. State-Space simulates a constant-coefficient state-space system. Transfer-Fcn simulates a transfer function. Zero-Pole simulates a zero-pole block. Saturation limits the range of a signal. Lookup Table (1D) approximates a one-dimensional function with a lookup table. Abs computes the absolute value of the input. Gain scales the input. Supports element-wise and matrix multiplication. Product multiplies the inputs. Supports element-wise and matrix multiplication. The sign of the exponent of each input is configurable. Math Function applies a selected math function to the input. Sum sums the inputs. The sign (+/-) of each input is configurable. Trigonometric Function applies a trigonometric function to the input. Bus Creator combines sets of signals into a bus. Bus Selector selects signals from a bus. Mux combines sets of signals into a vector signal. Demux separates the sets of signals from a vector signal. Selector selects specified signals from a bus. From connects a signal from a Goto block. Goto connects a signal to a From block. Display generates a numeric display of input values. Scope displays scope. Terminator terminates output signals. ToWorkspace writes input to an array in the workspace. BandLimited White-Noise acts as a dummy connection. Chirp generates a sinusoidal output whose frequency increases with time. Clock generates an output proportional to the simulation time. Constant generates a constant output. FromWorkspace acts as a dummy connection. Ground generates a constant zero output. Ramp generates a waveform with a constant slope. Signal Generator generates one of three waveforms: a sine wave, a square wave, or a sawtooth waveform. The Simulink signal generator allows a random waveform, however, that is currently not supported. Sine generates a sine waveform. Step generates a step waveform. In1 an input port of a subsystem. Out1 an output port of a subsystem. Subsystem a collection of blocks that form a unit. Fcn applies a C-style expression to the input. MATLABFcn applies a function to the input.	{"url":"https://cn.maplesoft.com/support/help/content/11814/BlockImporter-supportedblocks.mw","timestamp":"2024-11-03T06:30:06Z","content_type":"application/xml","content_length":"19731","record_id":"<urn:uuid:ed61134c-4146-47b9-ad42-41101040d67e>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00258.warc.gz"}
a. Try searching through different options for $k$, starting at $k = 1$ and increasing $k$. b. What happens to the decimal expansion of a number when you multiply it by $10^k$? What happens to the decimal expansion of a number of the form $0.a_1a_2a_3\ldots$ when you add an integer to it? Do the same things hold true in the world of binary expansions? c. From part (b) you have two different binary expansions for the same number. By Fact 2 at the beginning of the list of problems, these two binary expansions must be identical. Compare these expressions to each other. d. Start by finding integers $k$ and $n$ so that $2^k\cdot \frac{3}{11} = \frac{3}{11} + n$. Such integers do exist. Keep searching for them!	{"url":"https://cemc.uwaterloo.ca/sites/default/files/documents/2024/POTM-24-1-oct-H-158.html","timestamp":"2024-11-09T14:01:07Z","content_type":"application/xhtml+xml","content_length":"140564","record_id":"<urn:uuid:cf8fc10e-c91f-47b8-8dea-b699c90d570a>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00521.warc.gz"}
The Stacks project Let $k$ be a field of characteristic $0$. In this section we prove that the functor defines a Weil cohomology theory over $k$ with coefficients in $k$ as defined in Weil Cohomology Theories, Definition 45.11.4. We will proceed by checking the constructions earlier in this chapter provide us with data (D0), (D1), and (D2') satisfying axioms (A1) – (A9) of Weil Cohomology Theories, Section 45.14. Throughout the rest of this section we fix the field $k$ of characteristic $0$ and we set $F = k$. Next, we take the following data For our $1$-dimensional $F$ vector space $F(1)$ we take $F(1) = F = k$. For our functor $H^$ we take the functor sending a smooth projective scheme $X$ over $k$ to $H^_{dR}(X/k)$. Functoriality is discussed in Section 50.3 and the cup product in Section 50.4. We obtain graded commutative $F$-algebras by Lemma 50.4.1. For the maps $c_1^ H : \mathop{\mathrm{Pic}}\nolimits (X) \to H^2(X)(1)$ we use the de Rham first Chern class introduced in Section 50.9. We are going to show axioms (A1) – (A9) hold. In this paragraph, we are going to reduce the checking of the axioms to the case where $k$ is algebraically closed by using Weil Cohomology Theories, Lemma 45.14.18. Denote $k'$ the algebraic closure of $k$. Set $F' = k'$. We obtain data (D0), (D1), (D2') over $k'$ with coefficient field $F'$ in exactly the same way as above. By Lemma 50.3.5 there are functorial isomorphisms for $X$ smooth and projective over $k$. Moreover, the diagrams \[ \xymatrix{ \mathop{\mathrm{Pic}}\nolimits (X) \ar[r]_{c^{dR}_1} \ar[d] & H_{dR}^2(X/k) \ar[d] \\ \mathop{\mathrm{Pic}}\nolimits (X_{k'}) \ar[r]^{c^{dR}_1} & H_{dR}^2(X_{k'}/k') } \] commute by Lemma 50.9.1. This finishes the proof of the reduction. Assume $k$ is algebraically closed field of characteristic zero. We will show axioms (A1) – (A9) for the data (D0), (D1), and (D2') given above. Axiom (A1). Here we have to check that $H^_{dR}(X \coprod Y/k) = H^_{dR}(X/k) \times H^_{dR}(Y/k)$. This is a consequence of the fact that de Rham cohomology is constructed by taking the cohomology of a sheaf of differential graded algebras (in the Zariski topology). Axiom (A2). This is just the statement that taking first Chern classes of invertible modules is compatible with pullbacks. This follows from the more general Lemma 50.9.1. Axiom (A3). This follows from the more general Proposition 50.14.1. Axiom (A4). This follows from the more general Lemma 50.15.6. Already at this point, using Weil Cohomology Theories, Lemmas 45.14.1 and 45.14.2, we obtain a Chern character and cycle class maps \[ \gamma : \mathop{\mathrm{CH}}\nolimits ^(X) \longrightarrow \bigoplus \nolimits _{i \geq 0} H^{2i}_{dR}(X/k) \] for $X$ smooth projective over $k$ which are graded ring homomorphisms compatible with pullbacks between morphisms $f : X \to Y$ of smooth projective schemes over $k$. Axiom (A5). We have $H_{dR}^(\mathop{\mathrm{Spec}}(k)/k) = k = F$ in degree $0$. We have the Künneth formula for the product of two smooth projective $k$-schemes by Lemma 50.8.2 (observe that the derived tensor products in the statement are harmless as we are tensoring over the field $k$). Axiom (A8). Let $X$ be a smooth projective scheme over $k$. By the explanatory text to this axiom in Weil Cohomology Theories, Section 45.14 we see that $k' = H^0(X, \mathcal{O}_ X)$ is a finite separable $k$-algebra. It follows that $H_{dR}^(\mathop{\mathrm{Spec}}(k')/k) = k'$ sitting in degree $0$ because $\Omega _{k'/k} = 0$. By Lemma 50.20.2 we also have $H_{dR}^0(X, \mathcal{O}_ X) = k'$ and we get the axiom. Axiom (A6). Let $X$ be a nonempty smooth projective scheme over $k$ which is equidimensional of dimension $d$. Denote $\Delta : X \to X \times _{\mathop{\mathrm{Spec}}(k)} X$ the diagonal morphism of $X$ over $k$. We have to show that there exists a $k$-linear map such that $(1 \otimes \lambda )\gamma ([\Delta ]) = 1$ in $H^0_{dR}(X/k)$. Let us write \[ \gamma = \gamma ([\Delta ]) = \gamma _0 + \ldots + \gamma _{2d} \] with $\gamma _ i \in H_{dR}^ i(X/k) \otimes _ k H_{dR}^{2d - i}(X/k)$ the Künneth components. Our problem is to show that there is a linear map $\lambda : H_{dR}^{2d}(X/k) \to k$ such that $(1 \ otimes \lambda )\gamma _0 = 1$ in $H^0_{dR}(X/k)$. Let $X = \coprod X_ i$ be the decomposition of $X$ into connected and hence irreducible components. Then we have correspondingly $\Delta = \coprod \Delta _ i$ with $\Delta _ i \subset X_ i \times X_ i$. It follows that \[ \gamma ([\Delta ]) = \sum \gamma ([\Delta _ i]) \] and moreover $\gamma ([\Delta _ i])$ corresponds to the class of $\Delta _ i \subset X_ i \times X_ i$ via the decomposition \[ H^_{dR}(X \times X) = \prod \nolimits _{i, j} H^_{dR}(X_ i \times X_ j) \] We omit the details; one way to show this is to use that in $\mathop{\mathrm{CH}}\nolimits ^0(X \times X)$ we have idempotents $e_{i, j}$ corresponding to the open and closed subschemes $X_ i \times X_ j$ and to use that $\gamma $ is a ring map which sends $e_{i, j}$ to the corresponding idempotent in the displayed product decomposition of cohomology. If we can find $\lambda _ i : H_{dR}^{2d}(X_ i/k) \to k$ with $(1 \otimes \lambda _ i)\gamma ([\Delta _ i]) = 1$ in $H^0_{dR}(X_ i/k)$ then taking $\lambda = \sum \lambda _ i$ will solve the problem for $X$. Thus we may and do assume $X$ is Proof of Axiom (A6) for $X$ irreducible. Since $k$ is algebraically closed we have $H^0_{dR}(X/k) = k$ because $H^0(X, \mathcal{O}_ X) = k$ as $X$ is a projective variety over an algebraically closed field (see Varieties, Lemma 33.9.3 for example). Let $x \in X$ be any closed point. Consider the cartesian diagram \[ \xymatrix{ x \ar[d] \ar[r] & X \ar[d]^\Delta \\ X \ar[r]^-{x \times \text{id}} & X \times _{\mathop{\mathrm{Spec}}(k)} X } \] Compatibility of $\gamma $ with pullbacks implies that $\gamma ([\Delta ])$ maps to $\gamma ([x])$ in $H_{dR}^{2d}(X/k)$, in other words, we have $\gamma _0 = 1 \otimes \gamma ([x])$. We conclude two things from this: (a) the class $\gamma ([x])$ is independent of $x$, (b) it suffices to show the class $\gamma ([x])$ is nonzero, and hence (c) it suffices to find any zero cycle $\alpha $ on $X$ such that $\gamma (\alpha ) \not= 0$. To do this we choose a finite morphism To see such a morphism exist, see Intersection Theory, Section 43.23 and in particular Lemma 43.23.1. Observe that $f$ is finite syntomic (local complete intersection morphism by More on Morphisms, Lemma 37.62.10 and flat by Algebra, Lemma 10.128.1). By Proposition 50.19.3 we have a trace map \[ \Theta _ f : f_\Omega ^\bullet _{X/k} \longrightarrow \Omega ^\bullet _{\mathbf{P}^ d_ k/k} \] \[ \Omega ^\bullet _{\mathbf{P}^ d_ k/k} \longrightarrow f_\Omega ^\bullet _{X/k} \] is multiplication by the degree of $f$. Hence we see that we get a map such that $\Theta \circ f^$ is multiplication by a positive integer. Hence if we can find a zero cycle on $\mathbf{P}^ d_ k$ whose class is nonzero, then we conclude by the compatibility of $\gamma $ with pullbacks. This is true by Lemma 50.11.4 and this finishes the proof of axiom (A6). Below we will use the following without further mention. First, by Weil Cohomology Theories, Remark 45.14.6 the map $\lambda _ X : H^{2d}_{dR}(X/k) \to k$ is unique. Second, in the proof of axiom (A6) we have seen that $\lambda _ X(\gamma ([x])) = 1$ when $X$ is irreducible, i.e., the composition of the cycle class map $\gamma : \mathop{\mathrm{CH}}\nolimits ^ d(X) \to H_{dR}^{2d}(X/k)$ with $\lambda _ X$ is the degree map. Axiom (A9). Let $Y \subset X$ be a nonempty smooth divisor on a nonempty smooth equidimensional projective scheme $X$ over $k$ of dimension $d$. We have to show that the diagram \[ \xymatrix{ H_{dR}^{2d - 2}(X/k) \ar[rrr]_{c^{dR}_1(\mathcal{O}_ X(Y)) \cap -} \ar[d]_{restriction} & & & H_{dR}^{2d}(X) \ar[d]^{\lambda _ X} \\ H_{dR}^{2d - 2}(Y/k) \ar[rrr]^-{\lambda _ Y} & & & k } \] commutes where $\lambda _ X$ and $\lambda _ Y$ are as in axiom (A6). Above we have seen that if we decompose $X = \coprod X_ i$ into connected (equivalently irreducible) components, then we have correspondingly $\lambda _ X = \sum \lambda _{X_ i}$. Similarly, if we decompoese $Y = \coprod Y_ j$ into connected (equivalently irreducible) components, then we have $\lambda _ Y = \sum \lambda _ {Y_ j}$. Moreover, in this case we have $\mathcal{O}_ X(Y) = \otimes _ j \mathcal{O}_ X(Y_ j)$ and hence \[ c_1^{dR}(\mathcal{O}_ X(Y)) = \sum \nolimits _ j c^{dR}_1(\mathcal{O}_ X(Y_ j)) \] in $H_{dR}^2(X/k)$. A straightforward diagram chase shows that it suffices to prove the commutativity of the diagram in case $X$ and $Y$ are both irreducible. Then $H_{dR}^{2d - 2}(Y/k)$ is $1$-dimensional as we have Poincaré duality for $Y$ by Weil Cohomology Theories, Lemma 45.14.5. By axiom (A4) the kernel of restriction (left vertical arrow) is contained in the kernel of cupping with $c^{dR}_1(\mathcal{O}_ X(Y))$. This means it suffices to find one cohomology class $a \in H_{dR}^{2d - 2}(X)$ whose restriction to $Y$ is nonzero such that we have commutativity in the diagram for $a$. Take any ample invertible module $\mathcal{L}$ and set Then we know that $a\|_ Y = c^{dR}_1(\mathcal{L}\|_ Y)^{d - 1}$ and hence \[ \lambda _ Y(a\|_ Y) = \deg (c_1(\mathcal{L}\|_ Y)^{d - 1} \cap [Y]) \] by our description of $\lambda _ Y$ above. This is a positive integer by Chow Homology, Lemma 42.41.4 combined with Varieties, Lemma 33.45.9. Similarly, we find \[ \lambda _ X(c^{dR}_1(\mathcal{O}_ X(Y)) \cap a) = \deg (c_1(\mathcal{O}_ X(Y)) \cap c_1(\mathcal{L})^{d - 1} \cap [X]) \] Since we know that $c_1(\mathcal{O}_ X(Y)) \cap [X] = [Y]$ more or less by definition we have an equality of zero cycles \[ (Y \to X)_\left(c_1(\mathcal{L}\|_ Y)^{d - 1} \cap [Y]\right) = c_1(\mathcal{O}_ X(Y)) \cap c_1(\mathcal{L})^{d - 1} \cap [X] \] on $X$. Thus these cycles have the same degree and the proof is complete. Proposition 50.22.1. Let $k$ be a field of characteristic zero. The functor that sends a smooth projective scheme $X$ over $k$ to $H_{dR}^(X/k)$ is a Weil cohomology theory in the sense of Weil Cohomology Theories, Definition 45.11.4. Proof. In the discussion above we showed that our data (D0), (D1), (D2') satisfies axioms (A1) – (A9) of Weil Cohomology Theories, Section 45.14. Hence we conclude by Weil Cohomology Theories, Proposition 45.14.17. Please don't read what follows. In the proof of the assertions we also used Lemmas 50.3.5, 50.9.1, 50.15.6, 50.8.2, 50.20.2, and 50.11.4, Propositions 50.14.1, 50.17.3, and 50.19.3, Weil Cohomology Theories, Lemmas 45.14.18, 45.14.1, 45.14.2, and 45.14.5, Weil Cohomology Theories, Remark 45.14.6, Varieties, Lemmas 33.9.3 and 33.45.9, Intersection Theory, Section 43.23 and Lemma 43.23.1, More on Morphisms, Lemma 37.62.10, Algebra, Lemma 10.128.1, and Chow Homology, Lemma 42.41.4. $\square$ Remark 50.22.2. In exactly the same manner as above one can show that Hodge cohomology $X \mapsto H_{Hodge}^(X/k)$ equipped with $c_1^{Hodge}$ determines a Weil cohomology theory. If we ever need this, we will precisely formulate and prove this here. This leads to the following amusing consequence: If the betti numbers of a Weil cohomology theory are independent of the chosen Weil cohomology theory (over our field $k$ of characteristic $0$), then the Hodge-to-de Rham spectral sequence degenerates at $E_1$! Of course, the degeneration of the Hodge-to-de Rham spectral sequence is known (see for example [Deligne-Illusie] for a marvelous algebraic proof), but it is by no means an easy result! This suggests that proving the independence of betti numbers is a hard problem as well and as far as we know is still an open problem. See Weil Cohomology Theories, Remark 45.11.5 for a related question. Your email address will not be published. Required fields are marked. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). All contributions are licensed under the GNU Free Documentation License. In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FWC. Beware of the difference between the letter 'O' and the digit '0'. The tag you filled in for the captcha is wrong. You need to write 0FWC, in case you are confused.	{"url":"https://stacks.math.columbia.edu/tag/0FWC","timestamp":"2024-11-08T02:41:49Z","content_type":"text/html","content_length":"29896","record_id":"<urn:uuid:2b6849ab-f1b3-4300-9e2d-891120114466>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00757.warc.gz"}
Integrating R into a SAS shop \| R-bloggersIntegrating R into a SAS shop Integrating R into a SAS shop [This article was first published on Realizations in Biostatistics , and kindly contributed to ]. (You can report issue about the content on this page ) Want to share your content on R-bloggers? if you have a blog, or if you don't. I work in an environment dominated by SAS, and I am looking to integrate R into our environment. Why would I want to do such a thing? First, I do not want to get rid of SAS. That would not only take away most of our investment in SAS training and hiring good quality SAS programmers, but it would also remove the advantages of SAS from our environment. These advantages include the following: • Many years of collective experience in pharmaceutical data management, analysis, and reporting • Workflow that is second to none (with the exception of reproducible research, where R excels) • Reporting tools based on ODS that are second to none • SAS has much better validation tools than R, unless you get a commercial version of R (which makes IT folks happy) • SAS automatically does parallel processing for several common functions So, if SAS is so great, why do I want R? • SAS’s pricing model makes it so that if I get a package that does everything I want, I pay thousands of dollars per year more than the basic package and end up with a system that does way more than I need. For example, if I want to do a CART analysis, I have to buy Enterprise Miner, which does way more than I would need. • R is more agile and flexible than SAS • R more easily integrates with Fortran and C++ than SAS (I’ve tried the SAS integration with DLLs, and it’s doable, but hard) • R is better at custom algorithms than SAS, unless you delve into the world of IML (which is sometimes a good solution). I’m still looking at ways to do it, although the integration with IML/IML studio is promising.	{"url":"https://www.r-bloggers.com/2012/08/integrating-r-into-a-sas-shop/","timestamp":"2024-11-10T05:51:55Z","content_type":"text/html","content_length":"93448","record_id":"<urn:uuid:00a41bd5-95d2-46d6-af90-0d7b4dfbb1d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00070.warc.gz"}
Class Summary Class Description PbBase Abstract class for all basic objects. PbDataMapping Abstract class for data mapping. PbDateFormatMapping Defines the date format and mapping. PbDomain Class to define a domain. PbIsovaluesList Class to define a list of isovalues. PbLinearDataMapping Class to define linear data mapping. PbMiscTextAttr Class to define a numeric display format. PbNonLinearDataMapping Class to define non linear data mapping. PbNonLinearDataMapping2 Class to define non linear data mapping. PbNumericDisplayFormat Class to define a numeric display format. PoAngularAxis Class to build an angular axis. PoArrow Class to build a 2D arrow. PoArrow3 Class to build a 3D arrow. PoAutoCubeAxis Class to build a set of axes on a parallelepiped relating to the view. PoAutoValueLegend Abstract class for automatic value legend. PoAxis Abstract class for axis representations. PoBar Abstract base class for bar representations. PoBase Abstract base class for all Graph Master & 3D Data Master classes. PoBaseAxis Base class for all axis objects. PoBiErrorPointField Builds a bi-error point field. PoCartesianAxis Abstract class for cartesian axes. PoChart Abstract base class for all charting representations. PoCircle Abstract class for 2D circle representation. PoCircle3 Abstract class for 3D circle representation. PoCircle3CenterRadius Class to build a 3D circle. PoCircle3ThreePoints Class to build a 3D circle. PoCircleArc Abstract class for 2D circle arc representation. PoCircleArc3 Abstract class for 3D circle arc representation. PoCircleArc3CtrPtAngle Class to build a 3D circle arc. PoCircleArc3CtrTwoPts Class to build a 3D circle arc. PoCircleArc3ThreePts Class to build a 3D circle arc. PoCircleArcCtrPtAngle Class to build a 2D circle arc. PoCircleArcCtrRadTwoAngle Class to build a 2D circle arc. PoCircleArcCtrTwoPts Class to build a 2D circle arc. PoCircleArcThreePts Class to build a 2D circle arc. PoCircleCenterRadius Class to build a 2D circle. PoCircleThreePoints Class to build a 2D circle. PoConicBar Class to build conic bars. PoCoordinateSystemAxis Class for a 3D axes system. PoCurve Builds a 2D curve. PoCurve3 Builds a 3D curve. PoCurveFilling Class to build 3D filled curve. PoCurveLine Class to build a 2D line curve. PoCylindricalBar Class to build cylindrical bars. PoErrorCurve Class to build an error curve represention. PoErrorPointField Builds points field with X and Y margin error. PoGenAxis Class to build a generalized axis. PoGeneralizedBar Class to build generalized bars. PoGeneralizedScatter Class to build a 2D generalized scatter. PoGraphMaster Abstract base class for all Graph Master classes. PoGroup2Axis Class to build a group of two axes. PoGroup3Axis3 Class to build a group of three axes. PoGroup4Axis Class to build a group of four axes. PoGroup6Axis3 Class to build a group of six axes. PoHighLowClose Class to build a high low close representation. PoHistogram Abstract class for histogram representations. PoItemLegend Class to build an items legend. PoLabel Class to build a label field. PoLabelField Class to build a label field. PoLegend Abstract class for legend representations. PoLinearAxis Class to build a linear axis. PoLinearBar Class to build line bars. PoLinearValueLegend Class to build a linear auto value legend. PoLogAxis Class to build a logarithmic axis. PoMultipleHistogram Class to build a multiple histogram. PoNonLinearValueLegend1 Class to build a non linear legend (first representation). PoNonLinearValueLegend2 Class to build a non linear legend (second representation). PoNonLinearValueLegend3 Class to build a non linear legend (third representation). PoParallelogram Class for a 2D parallelogram. PoParallelogram3 Class for a 3D parallelogram. PoPieChart Abstract class for pie chart representation. PoPieChart2D Class for 2D pie chart representation. PoPieChart3D Class for 3D pie chart representation. PoPieChartRep Class to build a 3D pie chart. PoPointsFieldBars Class to build a points field bars. PoPolarAxis Abstract class for polar axis. PoPolarLinAxis Class to build a polar linear axis. PoPolarLogAxis Class to build a logarithmic polar axis. PoProfileBar Class to build profile bars. PoRectangle Class for a 2D rectangle. PoRibbon Class to build a 2D ribbon curve. PoScatter Class to build a 2D scatter. PoSingleHistogram Class to build a single histogram. PoTimeAxis Class to build a time axis. PoTube Class to build a 2D tube curve. PoValuedMarkerField Class for a valued markers field. PoValueLegend Abstract class for values legend. Class to build a set of axes on a parallelepiped relating to the view. Abstract base class for all Graph Master & 3D Data Master classes. Enum Summary Enum Description PbDomain.BoundingBoxTypes Bounding box interpretation. PbDomain.TransformTypes Transform type. PbNonLinearDataMapping2.Types Type of data mapping. PoAngularAxis.GradFits Enumerations. PoArrow.PatternTypes Type of pattern at the arrow extremities. PoArrow3.PatternTypes Type of pattern at the arrow extremities. PoAutoCubeAxis.AxisTypes Type of axes on the parallelepiped edges. PoAxis.AxisReverses Axis reverse type. PoAxis.GradPositions Graduation position type. PoAxis.MarginTypes Margin type. PoAxis.TextPaths Text path type. PoAxis.TickPositions Tick position type. PoAxis.TickSubDefs Sub-tick type. PoAxis.TitlePositions Title position type. PoAxis.Visibilities Enumerations. PoBar.Orientations Orientation of the bars. PoBase.NodeWriteFormats Type of write format. PoBase.TextTypes Type of Text. PoBase.UpdateMethodTypes Type of update method. PoBiErrorPointField.VariationTypes Type of interpretation of the fields lowX,lowY and highX,highY. PoCartesianAxis.Types Type of axis orientation. PoChart.ColorBindings Color binding. PoCircleArc.ArcTypes Type of the circle arc. PoCircleArc3.ArcTypes Type of the circle arc. PoCurve.CurveReps Curve representation. PoCurve.FilterTypes Filter type. PoCurve3.CurveReps Curve representation. PoCurveFilling.Orientations Orientation of the filled bar. PoCurveLine.ThicknessBindings Thickness binding. PoErrorCurve.ErrorCurveReps Type of error curve representation. PoErrorCurve.VariationTypes Type of interpretation of the fields lowY and highY. PoErrorPointField.ShapeTypes Type of shape used to represent each point. PoErrorPointField.SkeletonTypes Type of skeleton used to represent each point. PoGroup2Axis.AxisTypes Type of axis. PoGroup3Axis3.AxisTypes Type of axis. PoGroup4Axis.AxisTypes Type of axis. PoGroup6Axis3.AxisTypes Type of axis. PoHighLowClose.HorCloseBarPositions Position of the horizontal close bar. PoHistogram.BarSpaceTypes Type of spacing between bars. PoHistogram.Colorings Type of coloration of the bars. PoHistogram.Positions Type of positions relative to a histogram bar. PoHistogram.TextPaths Type of text path. PoHistogram.Types Type of orientation of the histogram's bars. PoHistogram.Visibilities Type of visibility. PoLabel.AxisType values computation. PoLabel.Positions Position of the labels. PoLabel.ValueTypes Type of value displayed by the labels. PoLabelField.ConcatTypes Type of string concatenation. PoLabelField.CoordinateTypes Type of coordinates. PoLegend.IncrementTypes Type of values incrementation. PoLegend.MarginTypes Type of margins. PoLegend.Positions Type of position. PoLegend.TextPaths Text path. PoLegend.Visibilities Type of visibility. PoLinearAxis.GradFits First graduation rounded or not. PoLinearValueLegend.ValueDistributions Type of distribution of the values. PoLogAxis.DecadeListDefs Decade list computed automatically or given by the user. PoLogAxis.DecadeReps Type of presentation of the axis decades. PoLogAxis.MultFactorPositions Type of position of the multiplicative factor. PoLogAxis.TenPowGradReps Type of presentation of the power of ten. PoMultipleHistogram.Representations Type of presentation of multiple histogram. PoPieChart.Alignments Type of annotation alignment. PoPieChart.ExtAnnotPositions Type of external annotation position. PoPieChart.IntAnnotPositions Type of internal annotation position. PoPieChart.PercentStatus Type of threshold for the grouping slice. PoPolarAxis.MultFactorPositions Type of position of the multiplicative factor. PoPolarLinAxis.GradFits First graduation rounded or not. PoPolarLogAxis.DecadeListDefs Decade list computed automatically or given by the user. PoPolarLogAxis.DecadeReps Type of presentation of the axis decades. PoPolarLogAxis.TenPowGradReps Type of presentation of the power of ten. PoTimeAxis.Languages Language used for date. PoTimeAxis.Types Axis orientation.	{"url":"https://developer.openinventor.com/refmans/latest/RefManJava/com/openinventor/meshviz/graph/package-summary.html","timestamp":"2024-11-06T12:10:39Z","content_type":"text/html","content_length":"53622","record_id":"<urn:uuid:ebf7fc70-0af8-4b6f-8d58-085433845930>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00776.warc.gz"}
How to Calculate the IRR of Private Equity \| Bizfluent Finance Your Business How to Calculate the IRR of Private Equity IRR, or an Internal Rate of Return, is typically used by private equity investors to compare the profitability of multiple investment scenarios. IRR is also present in many private equity and joint venture agreements, and is often used to define a minimum level of return for a preferred investor. IRR can be represented by the formula: NPV = c(0) + c(1)/(1+r)^t(1) + c(2)/(1+r)^t(2) + .... + c(n) Convert the date of all cash inflows and outflows to period of years from the date of the start of the IRR calculation. Typically the first cash outflow is the start of the IRR calculation and is labeled time period 0. If a cash inflow occurs at 6 months from the start of the IRR calculation, it is labeled time period 0.5. If another cash inflow of occurs at 1 year from the start of the IRR calculation, it is labeled time period 1.0. Input all cash flows into the formula: NPV = c(0) + c(1)/(1+r)^t(1) + c(2)/(1+r)^t(2) + .... + c(n)/(1+r)n^t(n), where c = the dollar amount of the cashflow, t = the time period determined in Step 1, n = the number of cash inflows or outflows and NPV = Net Present Value. In a simple IRR scenario where a $100 outflow occurs at time 0, a $50 inflow occurs at 1 year (t = 1) and a $100 inflow occurs at 2 years (t =2), the formula is represented as follows: NPV = $100 + $50/(1+r)^1 + $60/(1+r)^2. Set the NPV as equal to zero. By definition, IRR is the discount rate that makes the net present value of the cash inflows and outflows equal to zero. In our example: $0 = $100 + $50/(1+r)^1 + $60/ Solve for r. The value of r is the IRR. The method for solving will depend on the number of periods of inflows and outflows. Most investment professionals will use a scientific calculator or Excel's "IRR" function to solve. In our example above, r = 6.81%.	{"url":"https://bizfluent.com/how-10045140-calculate-irr-private-equity.html","timestamp":"2024-11-03T00:17:12Z","content_type":"text/html","content_length":"140488","record_id":"<urn:uuid:18d83f1f-7633-4e49-af86-3f447b319e1b>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00299.warc.gz"}
Sinking Feeling Two vases are cylindrical in shape. Can you work out the original depth of the water in the larger vase? Two vases are cylindrical in shape. The larger vase has diameter 20 cm. The smaller vase has diameter 10 cm and height 16 cm. The larger vase is partially filled with water. Then the empty smaller vase, with the open end at the top, is slowly pushed down into the water, which flows over its rim. When the smaller vase is pushed right down, it is half full of water. What was the original depth of the water in the larger vase? This problem is taken from the UKMT Mathematical Challenges Student Solutions The water completely fills the space between the cylinders up to the height of the smaller cylinder, and also half of the smaller cylinder. The volume of a cylinder is $V=\pi r^2h$, so can calculate the total volume of water by adding these two volumes together. Between the cylinders the volume is: $$V = \pi[10^2-5^2]\times 16=1200\pi$$ The volume in the smaller cylinder is $$V = \pi (5^2 \times 8)=200\pi$$ So the total volume is: $$V= 1200\pi + 200\pi = 1400\pi$$ In the large cylinder without the smaller cylinder, this volume occupies up to a height $h$ that satisfies: $$100h\pi = 1400\pi \Rightarrow h=14$$	{"url":"https://nrich.maths.org/problems/sinking-feeling","timestamp":"2024-11-07T12:13:30Z","content_type":"text/html","content_length":"37928","record_id":"<urn:uuid:9d3cdcbd-64bc-49a8-9ff2-5ecc370866aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00285.warc.gz"}
Closed curves in R^3: a characterization in terms of curvature and torsion, the Hasimoto map and periodic solutions of the Filament Equation If a curve in R^3 is closed, then the curvature and the torsion are periodic functions satisfying some additional constraints. We show that these constraints can be naturally formulated in terms of the spectral problem for a 2x2 matrix differential operator. This operator arose in the theory of the self-focusing Nonlinear Schrodinger Equation. A simple spectral characterization of Bloch varieties generating periodic solutions of the Filament Equation is obtained. We show that the method of isoperiodic deformations suggested earlier by the authors for constructing periodic solutions of soliton equations can be naturally applied to the Filament Equation. eprint arXiv:dg-ga/9703020 Pub Date: March 1997 □ Mathematics - Differential Geometry; □ Nonlinear Sciences - Exactly Solvable and Integrable Systems LaTeX, 27 pages, macros "amssym.def" used	{"url":"https://ui.adsabs.harvard.edu/abs/1997dg.ga.....3020G","timestamp":"2024-11-05T15:53:30Z","content_type":"text/html","content_length":"36796","record_id":"<urn:uuid:48a03bf7-a070-4291-9a90-bc7adb9cfac2>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00709.warc.gz"}
Q.241st OR question only.reply as soon as possibe please.. Integral Calculus> Q.24 1 st OR question only. reply as soon... 3 Answers Aditya Gupta Last Activity: 4 Years ago it is not clear which ques u r asking. i am assuming it is the following: vectors 2i – 3j + 4k and – 4i + 6j – 8k are.......... note that – 4i + 6j – 8k= – 2(2i – 3j + 4k) hence, they are collinear. but more specifically, they are anti parallel vectors, bcoz they are opposite in direction. KINDLY APPROVE THE ANSWER PLZZ :)) Last Activity: 4 Years ago Dear student In 24 question Take log (1 +x/1-x) = t Then dx/(x² -1 ) = dt Hence it becomes integration of 1 = t + constant = Log (1 +x/1-x) + constant Hope it helps Aditya Gupta Last Activity: 4 Years ago oops sorry u mean to ask about integrate from 0 to pi/4 (x^2+sin^2x)/(1+x^2) dx. separate it as integral from 0 to pi/4 (x^2)/(1+x^2) dx + integral from 0 to pi/4 (sin^2x)/(1+x^2) dx clearly integral from 0 to pi/4 (x^2)/(1+x^2) dx is elementary. but integral from 0 to pi/4 (sin^2x)/(1+x^2) dx is NOT POSSIBLE TO CALCULATE. the ques is wrong. Provide a better Answer & Earn Cool Goodies Enter text here... Ask a Doubt Get your questions answered by the expert for free Enter text here...	{"url":"https://www.askiitians.com/forums/Integral-Calculus/q-24-1-st-or-question-only-reply-as-soon-as-possi_259377.htm","timestamp":"2024-11-06T02:32:35Z","content_type":"text/html","content_length":"187110","record_id":"<urn:uuid:5b3d9892-bc93-4dfd-9fb3-6589b9b311c0>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00407.warc.gz"}
Math, Grade 6, Ratios, Glide Ratio Compare Glide Ratios Work Time Compare Glide Ratios Based on their glide ratios, which of these can glide farther from a given altitude: • The white-backed vulture or the sailplane or the paraglider? • Explain and support your answer using the representations you made. When representing and comparing the glide ratios, pay attention to which numbers you use for the forward distances and which you use for the vertical descent distances.	{"url":"https://goopennc.oercommons.org/courseware/lesson/4996/student/?section=5","timestamp":"2024-11-14T05:04:29Z","content_type":"text/html","content_length":"31679","record_id":"<urn:uuid:1ff53b42-f2a6-4432-893c-b33176e6931a>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00125.warc.gz"}
2238 grams to ounces Convert 2238 Grams to Ounces (gm to oz) with our conversion calculator. 2238 grams to ounces equals 78.94312248 oz. Enter grams to convert to ounces. Formula for Converting Grams to Ounces: ounces = grams ÷ 28.3495 By dividing the number of grams by 28.3495, you can easily obtain the equivalent weight in ounces. Converting grams to ounces is a common task that many people encounter, especially when dealing with recipes, scientific measurements, or everyday activities. Understanding how to perform this conversion can help bridge the gap between the metric and imperial systems, making it easier to communicate measurements across different contexts. The conversion factor between grams and ounces is essential for accurate measurement. One ounce is equivalent to approximately 28.3495 grams. This means that to convert grams to ounces, you need to divide the number of grams by this conversion factor. Knowing this allows you to easily switch between the two units of measurement. To convert 2238 grams to ounces, you can use the following formula: Ounces = Grams ÷ 28.3495 Now, let’s break down the calculation step-by-step: 1. Start with the total grams you want to convert: 2238 grams. 2. Use the conversion factor: 28.3495. 3. Perform the division: 2238 ÷ 28.3495. 4. The result of this calculation is approximately 78.83 ounces when rounded to two decimal places. This conversion is particularly important in various fields. For instance, in cooking, many recipes use ounces, especially in the United States, while ingredients may be measured in grams in other parts of the world. Understanding how to convert between these units ensures that you can follow recipes accurately, regardless of the measurement system used. In scientific measurements, precise conversions are crucial for experiments and data analysis. Whether you are measuring chemicals, biological samples, or other materials, being able to convert grams to ounces can help maintain consistency and accuracy in your work. Everyday use also benefits from this conversion. For example, if you are purchasing food items that list their weight in grams but you are more familiar with ounces, knowing how to convert can help you make informed decisions about portion sizes and nutritional content. In summary, converting 2238 grams to ounces is a straightforward process that can enhance your understanding and application of measurements in various contexts. By mastering this conversion, you can navigate between the metric and imperial systems with ease, making your cooking, scientific work, and daily activities more efficient and accurate. Here are 10 items that weigh close to 2238 grams to ounces – • Standard Laptop Shape: Rectangular Dimensions: 15.6 x 10.0 x 0.8 inches Usage: Used for personal computing, work, and entertainment. Random Fact: The average laptop weighs between 2 to 5 pounds, making it portable for daily use. • Medium-Sized Dog Shape: Quadrupedal Dimensions: Varies by breed, typically around 18-24 inches tall Usage: Companionship, service, and working roles. Random Fact: The average weight of a medium-sized dog ranges from 20 to 50 pounds. • Large Bag of Flour Shape: Rectangular Dimensions: 12 x 8 x 4 inches Usage: Used in baking and cooking. Random Fact: A standard bag of flour typically weighs 5 pounds, so 5 bags would be around 2238 grams. • Small Microwave Oven Shape: Cubic Dimensions: 20 x 15 x 12 inches Usage: Used for heating and cooking food quickly. Random Fact: The average microwave weighs between 20 to 30 pounds. • Weighted Blanket Shape: Rectangular Dimensions: 60 x 80 inches Usage: Used for relaxation and improving sleep quality. Random Fact: Weighted blankets are often recommended for anxiety relief and can weigh anywhere from 5 to 30 pounds. • Portable Air Conditioner Shape: Cylindrical Dimensions: 30 x 15 x 15 inches Usage: Used for cooling small spaces. Random Fact: Portable air conditioners can weigh between 50 to 80 pounds, depending on the model. • Large Backpack Shape: Rectangular Dimensions: 20 x 12 x 8 inches Usage: Used for carrying books, gear, and personal items. Random Fact: A fully loaded large backpack can weigh around 20-30 pounds. • Box of Books Shape: Rectangular Dimensions: 18 x 12 x 12 inches Usage: Used for reading and education. Random Fact: A box containing 10-15 average-sized books can weigh around 20-25 pounds. • Electric Guitar Shape: Contoured Dimensions: 39 x 12 x 2 inches Usage: Used for music performance and recording. Random Fact: The weight of an electric guitar typically ranges from 6 to 10 pounds. • Full-Sized Suitcase Shape: Rectangular Dimensions: 28 x 18 x 10 inches Usage: Used for travel and storage of clothing and personal items. Random Fact: A fully packed suitcase can weigh anywhere from 20 to 50 pounds, depending on its contents. Other Oz <-> Gm Conversions –	{"url":"https://www.gptpromptshub.com/grams-ounce-converter/2238-grams-to-ounces","timestamp":"2024-11-13T02:49:48Z","content_type":"text/html","content_length":"186977","record_id":"<urn:uuid:e1c9317a-9efe-4156-9f83-2249135191ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00135.warc.gz"}
Recent Publications Spin Rotations in a Bose-Einstein Condensate Driven by Counterflow and Spin-Independent Interactions Spin dynamics in cold atomic gases exhibit rich phenomena due to the interplay of particle interactions, quantum coherence, and particle statistics. In a Bose-Einstein condensate (BEC) with only contact interactions, spin dynamics can be induced by the spin dependence of the interaction strengths between particles. If interspin and intraspin interaction strengths are the same, on the other hand, there is only one scattering length, a.	{"url":"https://cqiqc.physics.utoronto.ca/research/recent-publications/","timestamp":"2024-11-11T18:27:09Z","content_type":"text/html","content_length":"52787","record_id":"<urn:uuid:c20f945b-dbd2-47d6-9874-d8b1b39129a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00667.warc.gz"}
Unscramble PEEEARDPAR How Many Words are in PEEEARDPAR Unscramble? By unscrambling letters peeeardpar, our Word Unscrambler aka Scrabble Word Finder easily found 114 playable words in virtually every word scramble game! Letter / Tile Values for PEEEARDPAR Below are the values for each of the letters/tiles in Scrabble. The letters in peeeardpar combine for a total of 23 points (not including bonus squares) • P [3] • E [1] • E [1] • E [1] • A [1] • R [5] • D [2] • P [3] • A [1] • R [5] What do the Letters peeeardpar Unscrambled Mean? The unscrambled words with the most letters from PEEEARDPAR word or letters are below along with the definitions.	{"url":"https://www.scrabblewordfind.com/unscramble-peeeardpar","timestamp":"2024-11-06T00:44:45Z","content_type":"text/html","content_length":"61354","record_id":"<urn:uuid:1985d463-d9fd-4d10-9f17-bf58a52badf1>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00488.warc.gz"}
On testing (EC)DH and (EC)DSA I found on Twitter an interesting blog post on breaking (EC)DSA and was writing a lengthy comment when some weird combination of keystrokes shutdown my browser and ate my comment, so I thought I'd write a blog post instead. The post raises an important point, namely that one should always validate domain parameters, but I'm not sure whether the issues it describes can be found in real world applications. I haven't seen any protocol or application in which (EC)DSA signers take domain parameters from some external source. In practice, signers either generate these parameter themselves (e.g., using openssl dsaparam), or pick one of the well-known, standard parameters (e.g., named curves in ECDSA). Also, if Mallory can choose g and if Alice doesn't validate it, infinite loop is the least of Alice's concern, because Mallory might as well choose g such that Alice's private key has small order, and thus can be easily recovered. A corollary to the Sylow theorems shows that given a finite group G and a prime number p dividing the order of G, then there exists an element (and hence a subgroup) of order p in G. Since the order of a DSA group is always p - 1 = 2 * q there exists a g that generates a subgroup of order 2. That g is p - 1. If q is not a prime (which is not required in FIPS 186 ) there exists many g that generate small subgroups -- any such g can be used to conduct the aforementioned key recovery attack. The same attack should work against ECDSA. The blog post mentioned invalid curve attacks in the context of ECDSA, but the issue is actually key recovery. If Mallory can choose not only the base point, but also the curve, he can choose an arbitrary curve and a base point with small order on that curve. If Mallory can choose only the base point, but not the curve, he's out of luck because the curves used in ECDSA usually have prime order that is also the order of all but the identity point, which usually doesn't have a valid encoding. In summary, I suspect that the issues highlighted in the blog post might never happen in real life, but when they do they are definitely less of a concern than losing private keys. But if one looks at these issues in the context of (EC)DH things get more interesting. There are many real-world scenarios where the (EC)DH params come from a somewhat untrusted source. For example: 1/ In SSL during the handshake the server can choose DH domain parameters. Failure to validate DH parameters has led to the Triple Handshake Attack . It seems that the server can also choose ECDH domain parameters, but I'm not sure whether Openssl or Boringssl supports arbitrary curves -- they might support only named curves. 2/ Since generating DH parameters is expensive, people usually use only one set of parameters, stored in a file. With temporary write access to this file one can install a permanent backdoor by modifying one or two bits. Last year it was found that the hardcoded DH params in socat contained a non-prime p To find and fix these issues, I'm thinking about adding some tests to Project Wycheproof to ensure that key generation utilities behave correctly when fed with malicious domain parameters. For ECDSA/ECDH, the key generation utilities should reject domain parameters if one of these conditions are not met: 1/ p is a prime 2/ base point G is on curve 3/ the order of G is a prime and order * G is the identity 4/ cofactor is small (<= 8?) 5/ order * cofactor is within the Hasse's bound But for ECDSA/ECDH one should always use named curves, and reject everything else. For DSA/DH, the conditions are: 1/ p is a prime 2/ 1 < g < p - 1 3/ the order of the subgroup q is a prime 4/ g^q = 1 (mod p) 5/ q and p have relatively reasonable sizes (e.g., if p has 2048 bits, q should have minimum 112 bits) As Bleichenbacher , the order q is usually missing from the DH parameter spec, thus we cannot really test condition #3, #4 and #5. To fill in the gap of the missing q, Bleichenbacher uses a brilliant trick that you should check out. Unknown said… Thanks you for the follow-up post! I agree that the scenario in which the domain parameters are possibly provided by the attacker is a scenario I am yet to encounter in real life and I must confess I missed the small subgroup attacks possibilities. What is interesting is that FIPS 186 is actually allowing the domain parameter to be provided by a third party, even an untrusted one. IMO, this means that libraries implementing digital signature schemes should also support this somewhat peculiar use case and not dismiss it as "not actually used", since it's not because we have never heard of such a case that it cannot Regarding your concern for non-prime q in the DSA case, the FIPS 186 still require it to be a "a prime divisor of (p – 1)" by definition, hence domain parameters validation would also mitigate that problem... If libraries were to provide their users with a mean to perform such validation! I'd also like to mention that when performing "only" signatures in the frame of a communication with an attacker, leaking the private key generated for the attacker's domain parameters is a big problem, for sure, but if it is an ephemeral key, then it's not that much of a concern, however the DoS ability a simple signature grants to the attacker is still a non-negligible problem, and a pernicious one. I've also already seen implementations falling into an infinite loop on signature verification on attacker provided public keys... (But because of a bug in the arithmetic operations) Imagine how SSH servers could be affected. Regarding the ECDH, it's on my "todo list" since a while, because I'm curious about how many implementations are checking those correctly. What's interesting also with DSA and ECDSA, is that currently, I've only found CryptoPP getting those correctly. Most libraries are completely lacking domain parameters validation. Thai Duong said… Hi Yolan, thanks for the comment :) > I agree that the scenario in which the domain parameters are possibly provided by the attacker is a scenario I am yet to encounter in real life and I must confess I missed the small subgroup attacks possibilities. What is interesting is that FIPS 186 is actually allowing the domain parameter to be provided by a third party, even an untrusted one. IMO, this means that libraries implementing digital signature schemes should also support this somewhat peculiar use case and not dismiss it as "not actually used", since it's not because we have never heard of such a case that it cannot happen. Yeah, I agree that this might happen. > Regarding your concern for non-prime q in the DSA case, the FIPS 186 still require it to be a "a prime divisor of (p – 1)" by definition, hence domain parameters validation would also mitigate that problem... If libraries were to provide their users with a mean to perform such validation! Even if they validate q and check 1 < g < p -- which should prevent both issues you mentioned -- setting g = p - 1 still works. > I'd also like to mention that when performing "only" signatures in the frame of a communication with an attacker, leaking the private key generated for the attacker's domain parameters is a big problem, for sure, but if it is an ephemeral key, then it's not that much of a concern, however the DoS ability a simple signature grants to the attacker is still a non-negligible problem, and a pernicious one. I've also already seen implementations falling into an infinite loop on signature verification on attacker provided public keys... (But because of a bug in the arithmetic operations) Imagine how SSH servers could be affected. I agree that infinite loop is a concern, but I'm not following the logic on ephemeral key. If the signer generates an ephemeral ECDSA key for every session, how does the verifier authenticate the key? Usually the signer has to sign the ephemeral key with some other long-term key. That means the ephemeral key can be reused, unless it's short-lived. I haven't see any protocol that relies on ephemeral ECDSA keys though. > Regarding the ECDH, it's on my "todo list" since a while, because I'm curious about how many implementations are checking those correctly. What's interesting also with DSA and ECDSA, is that currently, I've only found CryptoPP getting those correctly. Most libraries are completely lacking domain parameters validation. If you're into automated crypto testing, I recommend taking a look at Wycheproof. It has a lot of tests and has uncovered many bugs in many libraries. Cheers :) Unknown said… > I agree that infinite loop is a concern, but I'm not following the logic on ephemeral key. If the signer generates an ephemeral ECDSA key for every session, how does the verifier authenticate the key? Usually the signer has to sign the ephemeral key with some other long-term key. That means the ephemeral key can be reused, unless it's short-lived. I haven't see any protocol that relies on ephemeral ECDSA keys though. You are right, I was stuck trying to imagine a scenario in which I could envision attacker-provided domain parameters and it involved ephemeral keys, but yeah it does not make sense in any practically used case I'm aware of. But so does the domain parameters being provided by the client. What I meant was that if there exist some scenario in which such attacker-provided domain parameters may happen, I agree that we should be worried by both problems. > If you're into automated crypto testing, I recommend taking a look at Wycheproof. It has a lot of tests and has uncovered many bugs in many libraries. Yup, I've discovered it in December and added to my todo list to try out your test-vectors when you'll release them in a nicely formatted way, as announced. See you, hopefully at BH :) PS : to avoid data loss on "cat walking on the keyboard causing random browser shutdowns", the Lazarus browser extension used to be excellent, however its development ended.	{"url":"https://vnhacker.blogspot.com/2017/04/testing-ecdh-and-ecdsa.html","timestamp":"2024-11-05T16:36:49Z","content_type":"application/xhtml+xml","content_length":"144095","record_id":"<urn:uuid:d7aa0675-24ff-4f2c-bcc6-9b2f4998439c>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00621.warc.gz"}
A short antenna tuner This article describes a method to match short antennas to standard transmission lines. It's mainly intended for radio amateurs working on 137 kHz and 500 kHz bands, because when dealing with long waves, matching the antenna is always a problem. Traditional matchboxes designed for short waves do not have enough capacitance and inductance to work on low frequencies, and large commercially available variable capacitors and inductors are expensive and hard to find. Furthermore, installing an appropriate antenna, let's say a quarter wavelength tower, is out of reach for the common amateur. So, all long wave ham antennas are always too short and are a nightmare to match to a 50Ω line. A typical short antenna What one usually does is trying to run a wire as long as possible as high as possible. Any short antenna can be matched with this method, such as towers with or without capacitive hat, T antennas, inverted-L antennas, random wires, and so on, as long as they are small compared to the wavelength; shorter than a quarter wavelength in any case. Let's imagine one wire going vertically for 8 meters and than horizontally for 45, and take this antenna as an example, as shown in the figure below. This is already a big antenna for a backyard, but really small compared to the 2200 m (600 m) wavelength of the 137 kHz (500 kHz) band. Determining the antenna impedance The impedance of an antenna can be simulated with a NEC based program by "drawing" and approximating its structure and four our example the results are as follows: Simulation Antenna Equivalent Radiation frequency impedance capacitance resistance 137 kHz (0.7 – j3900) Ω 298 pF 0.01 Ω 500 kHz (1.4 – j960) Ω 330 pF 0.12 Ω The impedance has a very low resistance in series with a very high capacitive reactance which is typical of short antennas and the results look plausible. If the antenna cannot be simulated, one can still build an antenna as high and as long as possible and then measure the static capacitance. Since we are far below the first resonance, the antenna will look like a capacitor, and with some luck we can measure its static capacitance with a capacitance meter. Please remark that unfortunately some capacitance meters use a too high frequency and gives inconsistent results: in order to be accurate, the frequency should be as low as possible, but definitively not higher than the design frequency (the antenna has to be short at the measuring frequency). Connecting a capacitance meter to the antenna taken as example gives 295 pF; not far from the simulated values and the static capacitance is a really good starting point. Another method to measure the static capacitance of the antenna is to connect an inductor of known value in parallel to its feed point and to measure the resonance frequency. Again, this frequency has to be as low as possible to make sure the antenna is still "short" and a large inductor must be used. Once the capacitance has been determined, the capacitive reactance can be easely calculated with the formula X[c] = 1/(2πfC). The two above described measuring methods only determine the capacitive reactance but not the resistance. It's pretty hard to measure values as low as a few ohms while having kiloOhms of reactance in series, but knowing the exact real part of the impedance is not really important, since this will have little effect on the design of the match, as we will see later. Assuming just one or two Ohms is accurate enough. Determining the matching network So, the antenna looks like a big capacitor with a small resistor in series, which we would like to transform in purely resistive impedance Z[c], let's say 50 Ω with a matching network. If we plot the antenna impedance Z[a] on the complex plane, we find a point very close to the vertical axis, much lower than the horizontal axis. Z[c] is of course located on the horizontal axis on the right, as one can see in the figure below: The goal is to reach Z[c] starting from Z[a] as shown by the dashed line. No component can achieve this in one step and we will match the impedance in two steps. We have several choices: if we connect a series capacitor C[s] to the antenna, a parallel inductor L[p] or a step down transformer N:1 (more turns on the transmitter side), Z[a] will be transformed in a higher impedance (we are moving away from the origin) and this situation must be avoided because a high impedance means a high voltage, much higher than the already high voltage typical of a short antenna. Let's imagine to connect a 1 kW transmitter on our antenna at 137 kHz: with an impedance around 3900 Ω the voltage on the feed point will be 2 kV! This is already high enough to make some trouble and we definitively don't want to deal with higher voltages in the matching network. If we connect a parallel capacitor C[p] or a step up transformer 1:N (more turns on the antenna side), Z[a] will be transformed in a lower impedance (we are moving towards the origin) and this situation must be avoided because this reduces both the reactance and the resistance. The resistance of a short antenna is already very low (in the Ohm range), reducing it will dramatically increase the losses in the matching network and this ahs to be avoided as well. So, the only choice is to connect a series inductor L[s] to the antenna in order to reduce ("resonate away") the huge capacitive reactance of the antenna without reducing the resistive part of the impedance. In the figure this means moving vertically in the up direction. It's good to remark that the best choice is also consistent with common sense, as one would most probably think in matching a capacitive load with a series inductor. Now that we determined that the first element of the matching network is a series inductor L[s] connected to the antenna, we have to determine the second element that will finish the match. As it can be seen in the figure above, here we have three choices. The first idea is to resonate away completely the capacitive reactance with L[s] and find the intermediate impedance Z[i2] which lays in the real axis and than use a step down transformer N:1 (more turns on the transmitter side) to reach Z[c]. This solution could work but requires a transformer which takes longer to build, adds some difficulties to the design and requires more copper and has therefore higher losses. The second possibility is to over-compensate the capacitive reactance of the antenna with a slightly larger series inductor L[s] reaching point Z[i3] which is slightly inductive and lays on the constant admittance circle passing through Z[c]. Using a parallel capacitor C[p] we can reach Z[c]. This solution can work as well and good capacitors have very little losses, but the capacitance required is very large. In order to match the antenna of our example at 137 kHz some 200 nF are required: not easy to find and not easy to adjust as variable capacitors are usually in the few 100 pF The third and preferred option is to slightly under-compensate the capacitive reactance of the antenna with a slightly smaller series inductor L[s] reaching point Z[i1] which is slightly capacitive and lays on the constant admittance circle passing through Z[c]. Using a parallel inductor L[p] we can reach Z[c]. This solution has several advantages: first it requires only a small parallel inductor; it has the shortest path between Z[a] and Z[c] (meaning low losses) and is easier to build. The final matching network So, after having analyzer the different options we have determined that the impedance follows the path shown in the plot below: With a series inductor L[s] connected at the antenna port and a parallel inductor L[p] connecter at the transmitter port. Our matching network looks like the following diagram: Now, to determine the value of the two inductors, one can use the two equations above and with our antenna we get: Frequency Antenna Series Parallel Series Parallel impedance reactance reactance inductor inductor 137 kHz (0.7–j3900) Ω 3894 Ω 6.0 Ω 4.5 mH 6.9 μH 500 kHz (1.4–j960) Ω 952 Ω 8.5 Ω 300 μH 2.7 μH In order to simplify this operation the following calculator can be handy: Just enter the working frequency f, the transmitter impedance Z[c] (usually 50 Ω), the antenna impedance Z[a] and hit the button to calculate the required series and parallel inductances L[s] and L Determining the number of turns To calculate the required number of turns, an empirical formula that works well is described in the ARRL Handbook. Transformed for metric units is as follows: Where d is the diameter in mm, l is the length in mm and N is the number of turns. The calculated inductance L is in μH. When designing a large coil, it's more practical to use the turns spacing s in mm than the total length. s is defined as l/N. In order to simplify coil design, the following calculator can be handy: Just enter the core diameter d, turns spacing s, the desired inductance L and hit the "Calculate N" button to computer the number of turns N and the overall coil length l. As verification, one can enter the number of turns N and hit the "Calculate L" button to compute the inductance L and the overall coil length l, based on the same diameter d and spacing s. Variable inductors The values we have just calculated are approximate since are based on simulations, on low precision measurements, on arbitrary assumptions and on a simplified theoretical model. Therefore real inductors require some adjustment and must be variable. Changing the transmit frequency inside the band requires retuning as well. Experience shows that only L[s] needs adjustment, L[p] is less sensitive to frequency changes and once tuned can stay fixed for the whole band. There are several ways to make a variable inductance and the easiest is to change the number of turns. One could wind a coil with more turns than needed and than make connections to intermediate taps, as shown in the figure below. There is always debate about shorting unused turns or not (the dashed line in the figure). Shorting them results in higher losses while leaving them open may result in high induced voltage and arcing. As we will discuss later, in an air core solenoid the coupling factor between the turns is low and doesn't work well as autotransformer; on the other hand if the unused coil section happens to be resonant at the working frequency high voltage may result. This is quite unlikely since the working frequency is low and fairly fixed; a resonance requires significant stray capacitance which is unlikely with a small coil section. It's therefore wiser to leave unused turns open reducing losses unless arcing takes place leaving no other choice than shorting them. If the unused turns have to be shorted it's better not to short them all but to leave a small gap to let the field "escape", as shown in the figure below: Another way to have a variable inductor without involving too much mechanics is to build a variometer as summarized below: The coil is split in half and a smaller coil is inserted inside the first one and connected in series. By turning the small coil inside it's possible to increase the total inductance (when both coils are aligned in phase) or to decrease it (when both coils are aligned with opposite phase). A variometer has higher losses because the wire is longer, but on the other hand it's easy to build and has no sliding contact. The final solution is to use a tapped coil for L[p] and a variometer for L[s]. L[p] is small and is not adjusted frequently, usually on once when the antenna is first tuned, so a tapped coil is acceptable. L[s] needs to be adjusted every time the working frequency is moved, even a few hundred Hz, making the variometer solution more practical because of its continuously adjustable value. Building a variometer A variometer is easy to build with readily available materials. Two plastic pipes of different diameters, insulated copper wire and a plastic rod for the shaft are the main components. The figure below shows the cross section and the way the coils are connected together. Whit the formula (or the calculator) we discussed before, on determines the required number of turns on the external (stator) coil. This coil is split in two to let same space for the shaft to cross the core. A second coil, small enough to turn inside, is built and connected to the shaft and forms the rotor. All inductors are connected in series. Since the rotor only needs to move 180°, there is no need for sliding contact since flexible wire will do nicely the job. It's hard to determine exactly the range of variation of the inductance of a variometer, since this strongly depends on the coupling factor between stator and rotor coil, but as a rule of thumb, the experience shows that the total inductance will be something in-between L[stat]±L[rot] and L[stat]±2L[rot] and is a fair starting point. The best swing is obtained when all the coils are symmetrical. If the variation happens to be too small, one can still add a second layer of turns on the rotor. A single inductor instead of two A question still remains: do we really need two separate inductors? Well, theoretically yes, the two inductors need to be separate and not coupled at all. On the other hand, if we talk about air core solenoids, the answer may be yes under certain conditions. In an air core solenoid, the coupling between turns is fairly low and further decreases as the distance between the turns increases. It's therefore possible to wind on the same core both L[s] and L[p], adjacent to each other. The whole matching network looks than as an autotransformer, but it's not (or it's a very bad one), as explained later. L[p] can be adjusted by selecting a suitable tap on the main coil and L[s] can be adjusted by turning the variometer. This works only if the coupling factor is low: meaning that there is no magnetic core inside the coil, the turns are on only one layer in proximity of L[p] (coupling factor is much higher for two windings one inside the other)the variometer inside L[s] is far from L[p], and L[p] is small compared to L[s]. It's not really an autotransformer The final antenna tuner looks like the above diagram. This really looks like an autotransformer, but it's not really one. If it was an autotransformer it could not cancel the reactive component of the impedance. The antenna could not be matched, since it will just transform down the complex impedance of the antenna into another smaller complex impedance, something like (0.01–j50) Ω, but to match the antenna it's necessary to remove the reactance and to transform the resistance to 50 Ω. Just adjusting one parameter is not enough: the matching network must transform both. On the other hand, if we imagine that part of the coil works like an inductance, cancelling out the reactive part of the antenna impedance, the rest of the coil cannot be an autotransformer either, because once the reactance has been resonated away, the impedance is real and very low. It would require a step-down transformer (more turns on the transmitter side) instead of a step-up transformer, which is how the antenna tuner looks like. In order to check the coupling between the turns and the autotransformer effect, the setup shown in the figure below has been built: The coil has a diameter of 200 mm, a length of 280 mm, 100 turns and a tap on the 4^th turn. The transformer turn ratio is therefore 1:25. A voltage U[p] of 1 V[pp] has been applied on the 4 turns and the voltage across the whole 100 turns U[s] has been measured. Several measurements were taken at different frequencies and show that for this the voltage U[s] is around 4 V[pp] from a few kHz until about 400 kHz. The transformer ratio is therefore 1:4, very far from the expected 1:25, meaning that only the first few turns are coupled and that the rest of the coil is just an inductor. In this case, if the frequency is increased to 463 kHz we have a resonance and U[s] goes up to 55 V[pp], much more than what a 1:25 step-up transformer would do. Further increasing the frequency lowers U[s] down again to 4 V[pp] until other resonances occurs. It's best to think of this antenna tuner as two separate inductors, one in parallel with the transmitter and the other in series with the antenna, but because of the coupling between the two, the required values will be slightly different, especially for the small L[p], but this it's not a big issue, since, it's just a matter of finding the correct tap on the coil. Practical remarks Now that we have determined how our antenna tuner will be and how many turn we'll have to wind, let's spend a few words about the construction. First of all, we should try to design a tuner with minimal losses, but this is not an easy task. The wire used to wind the coils will be responsible for the majority of the losses and choosing a large diameter wire is a good idea. "Litz" wire is definitively the best option, but this is almost impossible to find in large diameters. The second preferred option is enameled copper wire, which is easier to find but expensive. The third option is to use PVC insulated electrical installation wire. This wire is relatively cheap, easy to find and available in large diameters. There is one detail one should take care of: the copper must look "red" and not be tinned. Tinned wire has very high losses at radio frequency and should be avoided. PVC insulation is not the best option because of its poor tg(δ), but the frequency is low and can be accepted; Teflon isolation would be a better option. Concerning the core, the best core is no core at all, but this won't be possible for such a big coil. Ceramic cores are very, but very hard to find in large sizes, so the third option is to use PVC pipes. These pipes are available in many large diameters, are cheap and are easy to work. They are a bit lossy, but again, the frequency is low and can be accepted for amateur use. In short antennas, the resistive part of the impedance is highly dominated by losses. Trying to reduce losses is always a good idea. For example one could use a few (or more) buried radials to increase ground conductivity near the antenna (where the current is higher). Another idea is to use more than one aerial conductor running parallel at some distance apart: this will split the current and reduce the losses. Using the antenna tuner Using this antenna tuner is quite easy: since the topology of the matching network has only two parameters to adjust, there is no risk to find a "wrong" and lossy match. If one happens to have an antenna analyzer that works at the desired frequency, this is by far the best option to adjust the tuner. If not, just connect the antenna and the transmitter (with a suitable SWR meter) and select low power operation. Adjust the variometer first is to find the position where the SWR is the lowest, then move the tap between L[s] and L[p] to find the best possible match (don't forget to switch the transmitter off before moving the tap). There is nothing more to do as this will usually allow match as good as 1.2:1 if there are enough taps. If the match is not good enough, try to repeat these two operations again maybe adding some more intermediate taps to the coil. If frequency is changed, usually only the variometer needs to be adjusted again and the tap can stay fixed for the whole band. Please remark that the majority of SWR meters designed for short-waves operations do not work for long waves. One should check before with a 50 Ω dummy load if the instrument gives reasonable results at the desired frequency. Some real antenna tuners The above described matching networks have been built and tested. The following pictures show a variometer designed for the 137 kHz band. The main coil can be adjusted between 3.75 and 4.63 mH and matches the antenna used as example at 137.3 kHz with an inductance of 4.39 mH. We can therefore calculate the capacitance of the antenna in 306 pF. In order to achieve the required inductance with less wire, some additional turns have been added inside the main coil (on the opposite side of L[p]). The rotor of the variometer is also a double layer coil. The loaded Q of this matching unit (loaded with the antenna) is about 60. The second variometer is designed for the 500 kHz band and can be adjusted from 214 to 598 μH by moving the variometer and selecting one of four taps on the upper side of the coil. It matches the same antenna at 500 kHz with 288 μH and we can calculate the capacitance at this new frequency in 352 pF. The loaded Q of this antenna tuner is about 80. A simple way of matching an electrically short antenna to a standard 50 Ω transmission line has been explained. Both theoretical and practical aspects have been addressed. The goal is to help radio amateurs in tuning their (short) antenna with the hope that more and more hams will be interested in medium and long wave frequency bands.	{"url":"https://www.giangrandi.org/electronics/shortanttuner/shortanttuner.shtml","timestamp":"2024-11-08T02:43:34Z","content_type":"text/html","content_length":"34860","record_id":"<urn:uuid:abc8c533-61f6-4707-9679-61fe6f311e28>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00481.warc.gz"}
Dough Rise Calculator for Pizza & Bread BeanAnimal's Reef > Tool > Dough Rise Calculator for Pizza & Bread Dough Rise Calculator for Pizza & Bread Dough rise calculator for perfect pizza and bread. Adjustable temperature and yeast for multi-step fermentation. Fermentation Calculator Instructions This dough fermentation calculator (yeast calculator or dough rise calculator) is fairly self explanatory. Select a yeast type and concentration and a starting temperature. The dough calculator will indicate how many hours REMAIN for full rise (fermentation). The dough rise calculator can easily accommodate multiple stages. If you select a rise time that does not result in full fermentation (full rise) the calculator will show an additional fermentation stage row. Up to different rise stages can be set. About this Dough Rise Tool First and foremost, this dough rise calculator is based on the hard work and research by TXCraig1 over at the pizzmaking.com forums. Craig’s Baker’s Yeast Quantity Prediction Model was used as both a data source and informational resource that are the foundation of this tool. I took the Baker’s Yeast Quantity Prediction Model chart that Craig published and charted each temperature and concentration relationship (time in hours). I then accurately fitted an equation to each curve. This calculator uses those equations to interpolate any value in the chart. Multi-step fermentation is a bit more complex. Craig’s chart is also useful in this regard. This calculator solves multi-stage fermentation using the same logic that Craig describes in the original While Craig’s data and chart were used as the basis for this dough calculator and are part of the public domain, the equations derived from the data and application of said equations are my work. Please don’t ask, they are not available or for sale. 8 comments • Any plans to do the sourdough version? • Maybe one day – That model is a bit different and somewhat harder to implement. • It should be pretty straight forward to add the sourdough variant. • Thank you for the response. For known yeasts, the calculable activity for a broad range of temperatures and concentrations allow for a wide range of (predictable) fermentation options. For SD, the calculable temperatures and somewhat unpredictable concentrations result in a much smaller dataset where reliable predictions can be made. Therefore, the SD calculator would be of extremely limited use due to the unpredictability. One or more steps on the edge of the predictable range would generate erratic results, even if they appeared to be valid. Please see here for a brief in context conversation: • Would love to have a poolish implementation in this <3 • Any chance you could share how you implemented the calculator? I’ve seen the posts on https://www.pizzamaking.com/ where people are sharing TXCraig1’s screenshots of excel tables. Did you just turn those tables into csvs and linearly interpolate the values in between, or is something more going on here? I’m not looking to throw my hat in the ring and make yet another online calculator, but I would like to write my own personal python notebook and I’m curious what others are doing. • TXCraig1’s hard work on the tables is the basis for this calculator. It is a bit more complicated than that, but interpolation will get you close. Long story short, I built and fitted equations to get this to work easily and quickly in a webform. • If I ever get around to updates, I will keep that in mind.	{"url":"https://beananimal.com/tools/dough-fermentation-calculator/","timestamp":"2024-11-09T20:55:26Z","content_type":"text/html","content_length":"169963","record_id":"<urn:uuid:e91c2bc9-815d-4927-8cb3-84d56b2453c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00438.warc.gz"}
DSA: Heap - questions 1. Basic Heap Operations · Implement a Min Heap · Implement a Max Heap · Insert an Element into a Min Heap · Insert an Element into a Max Heap · Delete the Minimum Element from a Min Heap · Delete the Maximum Element from a Max Heap · Peek the Minimum Element in a Min Heap · Peek the Maximum Element in a Max Heap · Heapify an Array (Build a Heap) · Convert an Array into a Min Heap or Max Heap 2. Heap Construction and Maintenance · Convert a Min Heap to a Max Heap · Convert a Max Heap to a Min Heap · Implement Heap Sort (Ascending and Descending Order) · Decrease Key in a Min Heap · Increase Key in a Max Heap · Find the Kth Largest Element in an Array (Using a Max Heap) · Find the Kth Smallest Element in an Array (Using a Min Heap) · Merge K Sorted Lists (Using a Min Heap) · Merge K Sorted Arrays (Using a Min Heap) · K Closest Points to the Origin (Using a Min Heap) 3. Heap-Based Problem Solving · Top K Frequent Elements (Using a Min Heap) · Find Median of a Stream of Integers (Using Two Heaps) · Sliding Window Maximum (Using a Double-Ended Queue or Heap) · Kth Largest Element in a Stream (Using a Min Heap) · Kth Smallest Element in a Stream (Using a Max Heap) · Shortest Path in a Weighted Graph (Dijkstra’s Algorithm with a Min Heap) · Find the Range of a Given Subarray with Minimum Sum (Using a Min Heap) · Find the Median of a Set of Numbers (Using Two Heaps) · Longest Subarray with Sum Less Than K (Using a Min Heap) · Reconstruct a Heap from a Given Set of Values 4. Advanced Heap Problems · Find All Valid Combinations with Sum Equal to Target (Using a Min Heap) · Implement a Priority Queue Using Heaps · Find the Maximum Sum of a Subarray of Size K (Using a Max Heap) · Find the Maximum Sum of k Non-Overlapping Subarrays (Using a Max Heap) · Heap-Based Approach to Solve Job Scheduling Problems · Implement a Median Maintenance Algorithm (Using Two Heaps) · Find Top K Elements in a Matrix (Using a Min Heap) · Sort K-Sorted Array (Using a Min Heap) · Rearrange Characters in a String so No Two Adjacent Characters are the Same (Using a Max Heap) · Implement a Heap-Based Algorithm to Solve the Traveling Salesman Problem (Approximate Solution) 5. Heap Applications in Graph Algorithms · Implement Prim’s Algorithm for Minimum Spanning Tree (Using a Min Heap) · Implement Kruskal’s Algorithm for Minimum Spanning Tree (Using Union-Find and Min Heap) · Find the Shortest Path in a Graph with Non-Negative Weights (Using Dijkstra’s Algorithm with a Min Heap) · Find the Longest Path in a Graph with Positive Weights (Using a Max Heap) · Compute the Minimum Cost Path in a Weighted Grid (Using a Min Heap) · Find All Pair Shortest Paths (Using Floyd-Warshall with Heap Optimization) · Implement A* Search Algorithm (Using a Min Heap) · Compute the Shortest Path Tree from a Source Node (Using Dijkstra’s Algorithm with a Min Heap) · Find the Most Frequent Path in a Graph (Using a Max Heap) · Compute the Minimum Cost to Connect All Nodes (Using Prim’s Algorithm with a Min Heap) Top comments (0) For further actions, you may consider blocking this person and/or reporting abuse	{"url":"https://practicaldev-herokuapp-com.global.ssl.fastly.net/jayaprasanna_roddam/dsa-heap-interview-preparation-questions-4l4f","timestamp":"2024-11-12T23:09:16Z","content_type":"text/html","content_length":"64745","record_id":"<urn:uuid:d8c72bf4-154d-4b50-a3e5-e90b2bc9a950>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00616.warc.gz"}
How to Use Multiple Input Production Functions in Managerial Economics - dummies Multiple-input production functions allow you to account for more complexity in your firm’s decision-making processes. Although single-input production functions are useful for illustrating many concepts, usually, they’re too simplistic to represent a firm’s production decision. In other words, you’re dealing with two or more variable inputs. Consider the production function q = f(L,K), which indicates the quantity of output produced is a function of the quantities of labor, L, and capital, K, employed. The specific form of this function may be the following Cobb-Douglas function Production isoquants: All input combinations are equal The relationship between labor, capital, and the quantity of output produced in the previous equation is graphically described by using a production isoquant. A production isoquant shows all possible combinations of two inputs that produce a given quantity of output. The curve labeled q[1] represents all combinations of capital and labor that produce 3,200 units of output. Marginal product Marginal product is the change in total product that occurs given one additional unit of an input. With a production isoquant, the amount of output you gain from using one more unit of labor is exactly offset by the amount of output you lose by using less capital. Every input combination on the production isoquant produces the same level of output — output is constant. Capital can change by more or less than one unit. What is critical is that total product remains constant as you increase labor and decrease capital. Marginal rate of technical substitution The marginal rate of technical substitution measures the change in the quantity of the input on the vertical axis of the diagram that’s necessary per one-unit change of the input on the horizontal axis of the diagram in order for total product to remain constant. That’s too technical of a definition, so remember this instead — the marginal rate of technical substitution is simply the production isoquant’s slope. The production isoquant’s slope equals the marginal product of the input on the horizontal axis divided by the marginal product the input on the vertical axis. Isocost curves: All input combinations cost the same An important factor in your production decision is how much the inputs cost. If an additional worker (unit of labor) costs less than an additional unit of capital, but the worker produces the same quantity of output as the capital, it’s a good deal to hire the additional worker. So, you need to add cost to your decision-making process. The isocost curve illustrates all possible combinations of two inputs that result in the same level of total cost. The isocost curve is presented as an equation. For a situation with two inputs — labor and capital — the isocost curve’s equation is In this equation, C is a constant level of cost, p[L] is the price of labor, L is the quantity of labor employed, p[K] is the price of capital, and K is the quantity of capital employed. On a graph, the isocost curve’s slope equals the price of the input on the horizontal axis divided by the price of the vertical axis input. Assume your total cost is $4,000 a day, and labor costs $20 per hour, and capital costs $5 per machine-hour. Given this information, your isocost curve equation is Some possible combinations of labor and capital you can employ for a total cost of $4,000 are 50 hours of labor and 600 machine-hours of capital, 100 hours of labor and 400 machine-hours of capital, and 150 hours of labor and 200 machine hours of capital. Any combination of labor and capital that results in total cost being $4,000 would be on the same — $4,000 — isocost curve. Changes in input prices shift the isocost curve. If the input on the horizontal axis becomes cheaper, the isocost curve rotates out on that axis. If the input on the vertical axis becomes cheaper, the isocost curve rotates. More expensive inputs cause shifts in the opposite direction. How to minimize costs In order to maximize profits, you must produce your output at the lowest possible cost. The costs of producing a given quantity of output are minimized at the point where the production isoquant is just tangent — or, in other words, just touching — the isocost curve. This point is illustrated as point A. The cost-minimizing combination of labor and capital are the quantities L[0] and K[0]. Now comes the easy part. At the point where you minimize costs, the production isoquant and isocost curve are tangent. This means the slopes of these two curves are equal. Therefore Or, if you rearrange that equation Thus, you minimize costs when the marginal product per dollar spent on each input is equal for all inputs. And this holds true no matter how many inputs you use! Economists call the preceding concept the least cost criterion, and it’s an application of the equimarginal principle. To produce goods with the lowest possible production cost, you equate the marginal product per dollar spent. About This Article This article can be found in the category:	{"url":"https://www.dummies.com/article/business-careers-money/business/economics/how-to-use-multiple-input-production-functions-in-managerial-economics-166973/","timestamp":"2024-11-14T10:43:42Z","content_type":"text/html","content_length":"88987","record_id":"<urn:uuid:346798bd-6ca0-474a-b75e-883b391581a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00459.warc.gz"}
Differential Geometry Let $B$ be a Kähler-Einstein Fano manifold, and $L \to B$ be a suitable root of the canonical bundle. We give a construction of complete Calabi-Yau metrics and gradient shrinking, steady, and expanding Kähler-Ricci solitons on the total space $M$, ${\rm dim}_{\mathbb{C}} M = n$ of certain vector bundles $E \to B$, composed of direct sums of powers of $L$. We employ the theory of hamiltonian 2-forms [2, 3] as an Ansatz, thus generalizing recent work of the author and Apostolov on $\mathbb{C}^n$ [5], as well as that of Cao, Koiso, Feldman-Ilmanen-Knopf, Futaki-Wang, and Chi Li [10, 26, 23, 24, 30] when $E$ has Calabi symmetry. As a result, we obtain new examples of asymptotically conical Kähler shrinkers, Calabi-Yau metrics with ALF-like volume growth, and steady solitons with volume growth $R^{\frac{4n-2}{3}}$. I prove a scalar curvature rigidity theorem for spheres. In particular, I prove that geodesic balls of radii strictly less than $\frac{\pi}{2}$ in $n+1~(n\geq 2)$ dimensional unit sphere are rigid under smooth perturbations that increase scalar curvature preserving the intrinsic and extrinsic geometry of the boundary, and such rigidity result fails for the hemisphere. The proof of this assertion requires the notion of a real Killing connection and solution of the boundary value problem associated with its Dirac operator. The result serves as the sharpest refinement of the now-disproven Min-Oo conjecture. We present a proof of the folklore result that any length metric on $\mathbb R^d$ can be approximated by conformally flat Riemannian distance functions in the uniform distance. This result is used to study Liouville quantum gravity in another paper by the same authors. Let (X,I,J,K) be a compact hypercomplex manifold, i.e. a smooth manifold X with an action of the quaternion algebra (Id,I,J,K) on the tangent bundle TX, inducing integrable almost complex structures. For any $(a, b, c) \in S^2$, the linear combination $L := aI + bJ + cK$ defines another complex structure on X. This results in a $C P^1$-family of complex structures called the twistor family. Its total space is called the twistor space. We show that the twistor space of a compact hypercomplex manifold is never Moishezon and, moreover, it is never Fujiki class C (in particular, never Kahler and never projective). Convex geometry and complex geometry have long had fascinating interactions. This survey offers a tour of a few.	{"url":"https://papers.cool/arxiv/math.DG","timestamp":"2024-11-03T10:19:14Z","content_type":"text/html","content_length":"24135","record_id":"<urn:uuid:469770f9-b754-4cfb-bd50-d8a82300b220>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00699.warc.gz"}
Degree Angle Degree Angles Chart - It describes all the negatives and positive angles. Tangent (45 degrees) = 1. Web simply click in the field inside the box and enter a value for the inches and fraction if necessary. The angle in figure 2.1.2 is formed from → ed and → ef. If you want to find the sine or cosine of any arbitrary angle, you first have to look for its reference angle. Web in this topic, we will learn what an angle is and how to label, measure and construct them. The angle types definitions are listed in the glossary below. Web figure 2.1.1 an angle is the union of two rays having a common endpoint. Web based on rotation six types of angles in maths, there are mainly 5 types of angles based on their direction. Web learning / unit circle charts 42 printable unit circle charts & diagrams (sin, cos, tan, cot etc) a unit circle diagram is a platform used to explain trigonometry. Degrees Solved Examples Measure Angles Cuemath It describes all the negatives and positive angles. Web 104 rows slope can be expressed in angles, gradients or grades. We will also explore special types of angles. Or by the three letters on the shape that define the angle, with the middle letter being where the angle. In general, a 52 degrees wedge can cover a distance of about. MEDIAN Don Steward mathematics teaching 45 degree angles Web there are two main ways to label angles: Tangent (45 degrees) = 1. We will also explore special types of angles. Web the unit of measure for an angle in mathematics is called a degree. There are seven types of angles commonly used in mathematics: Angles Chart Printable Printable Word Searches It describes all the negatives and positive angles. These five angle types are the most common ones used in geometry. Tangent (25 degrees) = 0.4663. The angle types definitions are listed in the glossary below. Web half a circle is 180° (called a straight angle) quarter of a circle is 90° (called a right angle) why 360 degrees? Angles Chart Printable Printable Word Searches It describes all the negatives and positive angles. Or by the three letters on the shape that define the angle, with the middle letter being where the angle. The angle in figure 2.1.2 is formed from → ed and → ef. For a quick reference for getting the angle. Think about what your angle signs might mean about your identity,. Pin on Projects to Try Web half a circle is 180° (called a straight angle) quarter of a circle is 90° (called a right angle) why 360 degrees? Tangent (45 degrees) = 1. If you want to find the sine or cosine of any arbitrary angle, you first have to look for its reference angle. Web simply click in the field inside the box and. High School Trigonometry/Radian Measure Wikibooks, open books for an The angle in figure 2.1.2 is formed from → ed and → ef. Web 104 rows slope can be expressed in angles, gradients or grades. Web simply click in the field inside the box and enter a value for the inches and fraction if necessary. For the complementary angle, reverse the rise and run. We will also explore special types. Degrees Degrees Direction The endpoint is called the vertex of the angle, and the two rays are the sides of the angle. Web the manufacturers always give the best and most suitable angle for sharpening the teeth for optimum performance. There are seven types of angles commonly used in mathematics: For a quick reference for getting the angle. We will also explore special. Degree Angle Chart Tangent (31 degrees) = 0.6009. We will also explore special types of angles. Web simply click in the field inside the box and enter a value for the inches and fraction if necessary. Web the manufacturers always give the best and most suitable angle for sharpening the teeth for optimum performance. Zero angle (0° in measure) acute angle (0 to. degree of an angle, measuring temperature, of a polynomial A Maths The angle types definitions are listed in the glossary below. Tangent (31 degrees) = 0.6009. Tangent (25 degrees) = 0.4663. Web the manufacturers always give the best and most suitable angle for sharpening the teeth for optimum performance. The inches box will also accept decimal values making it possible to calculate the degrees for any slope! Measuring angles with a protractor lesson & video Tangent (45 degrees) = 1. It describes all the negatives and positive angles. These five angle types are the most common ones used in geometry. Tangent (31 degrees) = 0.6009. Web any angle can easily be created with a carpenters square and a straight board as indicated in the figure and table above. Acute angles obtuse angles right angles straight angles reflex angles. The angle in figure 2.1.2 is formed from → ed and → ef. These five angle types are the most common ones used in geometry. In general, a 52 degrees wedge can cover a distance of about 85 to 100 yards. So now you can change the angle by dragging a handle or clicking for a random value. Reference angles are useful in trigonometry. If you want to find the sine or cosine of any arbitrary angle, you first have to look for its reference angle. Web figure 2.1.1 an angle is the union of two rays having a common endpoint. Web not many crown charts calculate miter/bevel angles for corners less than 60 degrees (even the bosch angle finder will not currently calculate miter/bevel angles for corners sharper than 60 degrees). Think about what your angle signs might mean about your identity, home, relationships and career. We will also explore special types of angles. Web simply click in the field inside the box and enter a value for the inches and fraction if necessary. So print this chart and carry it in your truck. Web in this topic, we will learn what an angle is and how to label, measure and construct them. Tangent (25 degrees) = 0.4663. Web the unit of measure for an angle in mathematics is called a degree. Web using this strategy and the table above, here is what the tangent of a few selected angles is equal to. Tangent (4 degrees) = 0.0699. It describes all the negatives and positive angles. Tangent (31 degrees) = 0.6009. In General, A 52 Degrees Wedge Can Cover A Distance Of About 85 To 100 Yards. These five angle types are the most common ones used in geometry. Tangent (70 degrees) = 2.7475. Web any angle can easily be created with a carpenters square and a straight board as indicated in the figure and table above. Web there are two main ways to label angles: Web Simply Click In The Field Inside The Box And Enter A Value For The Inches And Fraction If Necessary. We will also explore special types of angles. The angle types definitions are listed in the glossary below. Web click the ctrl to either show or hide the type. Web the manufacturers always give the best and most suitable angle for sharpening the teeth for optimum performance. Web For Help With Descriptions Of Each Sign Download My Astrology Basics Guide Below. Reference angles are useful in trigonometry. Tangent (4 degrees) = 0.0699. Tangent (25 degrees) = 0.4663. The endpoint is called the vertex of the angle, and the two rays are the sides of the angle. Zero Angle (0° In Measure) Acute Angle (0 To 90° In Measure) Right Angle (90° In Measure) Obtuse Angle (90 To 180° In Measure) Straight Angle (180° In Measure) Reflex Angle (180 To 360° In Measure) Complete Angle. Web not many crown charts calculate miter/bevel angles for corners less than 60 degrees (even the bosch angle finder will not currently calculate miter/bevel angles for corners sharper than 60 degrees). Web based on rotation six types of angles in maths, there are mainly 5 types of angles based on their direction. Web half a circle is 180° (called a straight angle) quarter of a circle is 90° (called a right angle) why 360 degrees? The degree of an angle is measured by using a tool called a protractor.a complete circle rotates at 360° and angles can be measured at different angles showcasing different degrees. Related Post:	{"url":"https://time.ocr.org.uk/en/degree-angles-chart.html","timestamp":"2024-11-05T15:49:42Z","content_type":"text/html","content_length":"29674","record_id":"<urn:uuid:32118acd-29ab-4eaa-9304-46bf437aede1>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00007.warc.gz"}
Physics Cup – TalTech 2021 – Pr. 3, Hint 1. The first hint will be a long piece of theory which you can find, in principle, in textbooks, but it might be hard to collect all the required pieces, so let us present these pieces here. Small oscillations of a system of bodies are described by a set of linear differential equations of second order. For instance, if three bodies of masses $m_1$, $m_2$, and $m_3$, respectively, are constrained to move along $x$-axis while connected with springs of stiffness $k$, we have $\ddot x_1=-k(x_1-x_2)$, $\ddot x_2=-k(2x_2-x_1-x_3)$, and $\ddot x_3=-k(x_3-x_2)$, where $x_i$ denotes the displacement of the $i$-th mass. We have no nonlinear terms because, by small amplitudes, we can neglect all quadratic and higher terms in the Taylor expansion. Such equations can be written in matrix form (which isn’t really needed to solve this problem) as $\ddot x_i=a_{ij}x_j$, where we have assumed Einstein’s summation convention and assume summation over repeated indices. For this example of three masses, assuming the masses to be all equal to $m$, we have $a_{11}=a_{33}= - k/m$, $a_{22}= - 2k/m$, $a_{12}=a_{21}= a_{23}=a_{32}= k/m$, and $a_{13}=a_{31}= 0$. Note that the matrix here is symmetric, $a_{ij}=a_{ji}$; this is a property which you may think about as following from Newton’s third law, combined with the fact that all the masses are equal. For the example above, the functions $x_{i}=x_{i}(t)$ are just the Euclidian coordinates of the point masses, but for more complicated situations, these are generalized coordinates the total number of which needs to be equal to the number of degrees of freedom. The number of degrees of freedom is the smallest number of scalar parameters needed to describe fully a dynamical system. For instance, consider three point masses connected with two rigid rods, with a hinged connection at the middle joint, and with everything constrained into a horizontal plane. In order to describe the position of each of the point masses, we would need to use their $x$ and $y$ coordinates, so that three point masses would come with six coordinates in total. However, we have two scalar constraints – the distances between neighboring points, which will reduce the number of required generalized coordinates down by two. As a result, we need four generalized coordinates; for instance, these coordinates can be $x_1=\xi$ and $x_2=\eta$ – the coordinates of the first mass, together with $x_3=\alpha$ and $x_4=\beta$ – the angles of the rods. However, we can always go to a reference frame where the center of mass of the whole system is at rest, at the origin, in which case the number of coordinates is further decreased by two (we won’t need $\xi$ and $\eta$ anymore). The key to an easy solution is a convenient choice of generalized coordinates. In some cases, you may be able to find convenient coordinates, but some scalar constraints remain still unused (i.e. your number of generalized coordinates $N$ is still greater than the number of degrees of freedom $n$). In that case, you may just consider a subspace in the $n$-dimensional space defined by your generalized coordinates. Next, we introduce the concept of natural modes; these are the modes where all the coordinates oscillate with the same frequency. The general theory of coupled oscillators tells us that arbitrary motion of the system is a linear combination (superposition) of the natural modes. For instance, in the case of the example with three equal masses connected with springs, one of the obvious modes is when the central point mass is at rest, and the other two oscillate in opposite phase: $x_1=-x_3=x_0\cos(\omega_1 t+\varphi)$, $x_2=0$. Due to symmetry, it is clear that $x_1$ and $x_3$ remain moving symmetrically while there is no net force exerted onto the mass in the middle. Now we can easily conclude from Newton’s second law for one of the masses that $\omega_1=\sqrt {k/m}$. This motion is described by the eigenvector $\mathbf X=(1,0,-1)$ (i.e. $X_1=-X_3=1$, $X_2=0$); a natural mode can be conveniently expressed in terms of the eigenvector as $\mathbf x=\mathbf Xx_0\cos(\omega_1 t+\ varphi)$. There is also the trivial mode when all the masses move together with constant speed: with eigenvector $\mathbf Y=(1,1,1)$, $\mathbf x=\mathbf Yv_0t$. This can be interpreted as a motion with an infinitely long period, i.e. with $\omega_2=0$. We are missing one more mode which can be easily found here if we know a useful fact: for a symmetric matrix $a_{ij}=a_{ji}$, the eigenvectors corresponding to different natural frequencies are perpendicular to each other, i.e. we need to find such a vector $Z_{i}$ that $\sum_iX_iZ_{i}=0$ and $\sum_iY_iZ_{i}=0$. It is easy to see that these equalities are satisfied with $\mathbf Z=(1,-2,1)$ (keep in mind that our space of eigenvectors is three-dimensional – because we have three degrees of freedom; therefore, all the vectors which are perpendicular to the plane defined by the vectors $\ mathbf X$ and $\mathbf Y$must be parallel to $\mathbf Z$). So, our missing mode is in the form $\mathbf x=\mathbf Zz_0\cos(\omega_3t+\phi)$; let us express the average kinetic and potential energies for such a mode, which must be equal to each other for sinusoidal oscillations, as $\left<E_k\right>=\frac 14m\omega_3^2z_0^2\mathbf Z^2=\frac 32m\omega_3^2z_0^2$; $\left<E_p\right>=\frac 12k[z_0 (1+2)]^2=\frac 92kz_0^2$, hence $\omega_3^2=3k/m$. Please submit the solution of this problem via e-mail to physcs.cup@gmail.com. For full regulations, see the “Participate” tab.	{"url":"https://physicscup.ee/physics-cup-taltech-2021-pr-3-hint-1-2/","timestamp":"2024-11-03T23:35:00Z","content_type":"text/html","content_length":"59056","record_id":"<urn:uuid:f6ae6a59-ca5a-4110-b771-ba0286f253aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00357.warc.gz"}
A Novel Design and Performance Analysis of Piezoelectric Energy Harvester with Application to a Vehicle Suspension System Moving on Uniform Bridges Experimental Methods Design Concept and System Modelling The PEH is proposed to be universally used on a wide range of vehicle suspension types. However, the space available on the suspension section is limited. Also, the vibration level experienced by the suspension is different for every vehicle. It leads to variations of load exerted on the piezoelectric transducer. To harvest the vibration energy optimally, an effective mechanism of the PEH is designed to address those challenges. 2.1. Design Concept Figure 1a shows the proposed PEH design concept (Figure 1b) mounted on a vehicle suspension system. There are two main developed mechanisms: force magnification and piezoelectric protector mechanism (Figure 1c). The former is designed to amplify the excitation force through a linear-spring mounted on the suspension system. It works based on a pressurized liquid cylinder-piston mechanism known as a hydraulic system, consisting of the master cylinder and caliper cylinder components. Due to the vehicle suspension’s limited space, the PEH that can be mounted onto the suspension system is limited in volume. Thus, the master cylinder components with small volumes and easily attached must be chosen. However, some factors such as road surface roughness, traffic conditions, and vehicle speed dynamics may significantly affect excitation force. If the road surface is flat with congested traffic conditions, the vehicle suspension's excitation force may be weak. Thus, the force magnification mechanism must be set to effectively capture the suspension’s vibration while amplifying its magnitude simultaneously. Meanwhile, the piezoelectric protector is an additional mechanism to avoid piezoelectric damage due to excessive unidirectional compressive force as a result of an unpredictable uneven road. It uses springs as the main component for regulating the pressure that deforms the piezoelectric. Figure 1 System: (a) suspension [Adapted from www.pakwheels.com (Ali, 2015)], (b) PEH, (c) diagram block of modeling 2.2. System Modelling The model of the overall system in Figure 1a comprises three main subsystems: the dynamics of the bridge, the vehicle, and the PEH. In this subsection, each will be detailed, and the parameter values of the respective subsystems will be tabulated afterward. 2.2.1. Modelling of Force Magnification Mechanism Figure 2 shows the overall structure and working principle of the proposed force magnification mechanism, which is the main subsystem of the PEH. The spring force F generated by the vehicle suspension's vibration is transmitted to the large piston, which moves the master cylinder and produces force F1. The force is then transferred to the fluid in the closed channel. The fluid will move the small piston and produce F2 in the caliper. The force F2 will deform the piezoelectric bar and generate electrical energy. In short, this mechanism applies Pascal’s Law (Equations 1-2). By taking into account the cross-sectional area of both pistons, A1 and A2, the value of F2 can then be calculated, The ratio of cross-sectional area is denoted by n[p] = A2/A1 where n[p]>1 so the excitation force in the master cylinder into the caliper can be magnified. This magnified force will deform the piezoelectric at the end of the piston in the caliper. By taking the effect of friction force between piston and cylinder and working fluid mass in the pressurized cylinder-piston system are not accounted for, the ratio of the big piston of master cylinder S1 and small piston of caliper S2 to the respective cross-sectional area A1 and A2 is, Since the excitation force exerted by the vehicle suspension system on the PEH is random, impulsive impact force can occur anytime. It then becomes an important consideration for PEH design. A novel design of a piezoelectric protector mechanism is proposed to minimize physical damage. Figure 2 provides the design concept and its components. If the magnitude of force F approaches or exceeds the limit of compressive strength of piezoelectric material, the linear spring will be compressed accordingly. The push rod then sticks to the stopper and discharges to press the piezoelectric. Thus, the piezoelectric will always be within its strength limit and a safe mode. 2.2.2. Modelling of PEH Since the PEH is mounted onto the existing vehicle’s suspension, it can be considered as two main structures: super- and sub-structure. The former is a vehicle, while the latter is a hydraulic system-based PEH. The piezoelectricity cylinder-piston mechanism depicted in Figure 3a can be simplified as a translational lumped mass and is shown in Figure 3b. It consists of an equivalent mass m [eq], damping coefficient c[eq,] and spring constant k[eq]. Two external springs are from master cylinder k[t] and piezoelectric protector k[s]. If the piston thickness and its mass density, t[cp,] and are respectively defined, then the mass of the small piston under pressurized working fluid can be computed using Equation 3, A big piston deforms the piezoelectric transducer in Figure 2c through piston extension where its spring constant is not considered. Spring stiffness k[pb] of the big piston may be computed using Equation 4, Considering the working fluid, the small piston can be modeled as a spring on a mass. The equivalent spring stiffness k[eq][ ]can be calculated using Equation 5. Figure 2 Overall structure and working principle (a) proposed force magnification mechanism (b) Proposed piezoelectric protector mechanism (c) Piezoelectric bar When the work done by the spring force is converted into electricity, there is an equivalent electric resistance. The electrical damping coefficient c[eq] can be derived as follows (Aouali et al., 2021; Pasharavesh, Moheimani, and Dalir, 2020; Wu, Wang, and Xie, 2015; Xie and Wang, 2015), Equation 6 contains three important variables: d[33] is the piezoelectric strain constant in the polling direction, c[a] is the electrical capacity of the piezoelectric bar, and f is the natural vibration frequency of the system. Once the equivalent mass m[eq], spring constant k[eq,] and damping coefficient c[eq] are obtained, the proposed PEH results in a damped single-degree-of-freedom system subjected to the road surface roughness. Considering the equivalent magnified force F[m](t) at the PEH as expressed in Equation 7, the electrical charge that can be stored by the piezoelectric bar is calculated using Equation 8, Figure 3 PEH model (a) Cylinder-piston mechanism (b) Equivalent model of the proposed PEH Equation 7 and Figure 3b contain some important parameters: y[v ]and y[wt] are displacements of the vehicle body and wheel tire, respectively, in the area where the PEH is mounted, y[eq] is the displacement of the PEH, and n is the magnification factor (cross-sectional area ratio). The electrical current resulted by the piezoelectric bar can be calculated by deriving the electrical charge in Equation 8 with respect to time, The electrical voltage resulted by the piezoelectric bar can be found by dividing the electrical charge Q[ep] in Equation 8 over electrical capacity of the piezoelectric material c[a], Generated electrical power is then obtained by multiplying the electrical current in Equation 9 and electrical voltage in Equation 10 with the number of piezoelectric p, Equations 8-11 reflect that the outputs of PEH are electrical parameters such as electrical charge, current, voltage, and power. In particular, electrical charge Q[ep] is seen as a very instrumental parameter and must be produced optimally. The RMS of the electric power within a predefined time duration can be then determined by using Equation 12, 2.2.3. Modelling of Bridge Dynamics Traversed by a Mobile Half-Car Planar Model Quarter-car model is the common model employed. However, such a model type is considered inadequate for many realistic cases, particularly for analyzing the total vehicle dynamics. In this paper, the vehicle consisting of the driver and passenger is modeled as a mobile half-car planar model, as depicted in Figure 4a. The vehicle moves on a bridge above the uneven road surface. The case becomes transverse elastic deformation of the bridge traversed by a mobile half-car planar model carrying the PEHs. They are modeled in Figure 4b. In deriving the governing equations of motion, simplifications and assumptions are defined: 1. The overall system is modeled in linear behavior. 2. A mobile half-car planar model has six DoFs, which consist of a body, two PEHs, two-wheel tires, a driver, and a passenger. The vehicle body is constrained to have the vertical motion (bounce) and the angular motion (pitch), where every wheel-tire bounces in its respective coordinate. Also, the driver and the passenger are considered to have only their vertical oscillations. 3. The PEH is mounted between the vehicle body and the wheel-tire in front or rear positions. 4. Passenger seats, suspension, and wheel-tire systems are modeled as a combination of linear springs and viscous dampers which are connected in parallel arrangements. 5. The resilience and damping of the suspension and wheel-tire systems are expected to be sufficient so as to be more realistic models for simulation and analysis. 6. The wheel-tire system is assumed to be in contact with the surface of the uneven road at all times. To derive the motion equations of the overall system, the energy method is employed. By defining x as the axis along the length of the beam measured from the left to right end support, and t as the travel time, then y(x,t) can be characterized as the vertical deformation of the bridge all the way of the undeformed neural axis of the bridge as depicted by Figure 4b. The kinetic and potential energies of the overall system in Figure 4a using linear strains assumption are given by Equations 13-14, respectively. Both equations belong to the bridge, vehicle body (bounce and pitch), PEH, wheel-tire (front and rear), driver, and passenger. Parameter in Equation 13 is the mass density per unit length of the uniform beam, EI in Equation 14 denotes the flexural rigidity of the beam, while H(x) represents the Heaviside function. In particular, Equation 14 contains variables which point out the locations of the contact points of the front and rear tires with the bridge surface. Rayleigh’s dissipation function and generalized force are expressed in Equations 16-18, By considering the Galerkin approximation, y(x,t) is written in Equation 19, where presents mode shapes and q[i](t) points out the generalized coordinates for the elastic deflection of the beam element. Orthogonality conditions are given by Equation 20, where the term [ ]denotes the Kronecker delta for i,j=1,2,..,n, Figure 4 Vehicle system model (a) the bridge traversed by a mobile half-car planar model (b) transverse elastic deformation of the bridge traversed by a mobile half-car planar model Defining the eight state variables as in Equation 21, Lagrange’s equations for those variables can be expressed in Equations 22-23, Finally, motion equations of the overall system can be derived in general forms, 1. The vertical motion (bounce) for the driver is given by, 2. The vertical motion (bounce) for the passenger is presented by, 3. The vertical motion (bounce) for the vehicle is defined as, 4. The vertical motion for the front PEH is, 5. The angular motion (pitch) for the vehicle can be expressed as, 6. The vertical motion for the rear PEH is, 7. The vertical motion for the front wheel-tire can be written as, 8. The vertical motion for the rear wheel-tire can be written as, 9. The equation motion of the bridge is, Parameter k[tot1] is equivalent spring stiffness between shock-breaker with external spring of master cylinder k[t], k[tot2] is equivalent spring stiffness between wheel-tires with external spring of piezoelectric protector k[s] while two coefficients D[1] and D[2] denote the predefined interval of the motion of the vehicle. The roughness of road surface (Wei and Taghavifar, 2017), G[q](n[0]) indicates the roughness coefficient of the road surface, n[0] refers to a reference spatial frequency with a value of 0.1 m^-1, f[0] is a minimal boundary frequency with a value of 0.0628 Hz, v(t) denotes the vehicle velocity, and w(t) presents a zero-mean white noise. 2.2.4. Numerical Solver Equations 24-32 form a system of nine second-order coupled differential equations. They could be written in the state-space model by converting all equations into first-order differential equation systems. Consequently, state variables become (2+(2·DOF)+n) as the following, From Equation 34, the system of first-order differential equations can be arranged in Equation 35. Considering the computational time and accuracy, the time step is selected as 0.001 s. Consequently, the displacements and velocities of the system at the time t[i+1][ ]can be arranged as in Equation 36. By assuming that the system is in an equilibrium position at i=0 and t[0]=0 s, the initial conditions of all are equal to zero. Results and Discussion The proposed PEH's performance is evaluated using important design factors: the ratio of piston cross-sectional area, vehicle velocity, and road roughness coefficient. Dynamic responses of displacement, velocity, and acceleration in all nodal coordinates of the half-car planar model, particularly at the point of the PEH mounted on the vehicle suspension, are used. For vehicle dynamics, the road roughness coefficient refers to ISO/TC108/SC2N67 (Wei and Taghavifar, 2017), in which only the classes B, E, and H are used, with roughness coefficient (G[q](n[0]), in cm^3) are 64; 4,096; and 65,536 respectively. Meanwhile, Tables 1 and 2 list the beam and vehicle parameters. Dimension and material properties of the PEH are listed in Table 3. The harvested electric power is expressed in RMS value. 3.1. Dynamic Responses Analysis Results are obtained by setting the numerical variables in simulations such as: (1) a random Gaussian white noise with a zero mean shown in Figure 5a, (2) road surface roughness of the car on the road in the class of B for 10 seconds are depicted in Figure 5b, (3) Two types of velocity are used in terms of variable and constant velocities as, where the former is shown in Figure 6a, and the latter is set to be constant of 30 m/s, (4) parameters of the bridge, vehicle, and PEH are based on Tables 1-3. Dynamic displacements are displayed in Figures 6b. Each figure compares the displacement of respective subsystems under both velocities. However, it should be noted that the values are selected only to show the dynamic characteristics. Any arbitrary values can be chosen as long as the time trajectory of the vehicle moving on the bridge is sufficient. Figure 7 reveals that the developed motion equations with the numerical solver can capture the dynamics of the overall system. The dynamic displacement obtained from variable and constant velocities is different. An amplification factor is produced when the maximum value of each displacement is compared. The ratio is averagely found around 1.23. It can be a variation if the trajectory profile is varied accordingly. However, those figures are intended to demonstrate the effect of velocity on the generated electrical power of the proposed PEH model, as justified by Figure 8a. Figure 5 Road data (a) Gaussian white noise with zero mean value (b) road surface roughness Figure 6 Vehicle and bridge system (a) Trajectory profile for vehicle (b) Bridge displacements 3.2. Parametric Study for PEH Three key design parameters are chosen: vehicle speed variations, road roughness coefficient, and ratio of the cross-sectional area of the piston. Each variation is tested with three white noise time series to check the robustness of the generated electric power. The white noise has a similar mean value with different intensities, shown in Figure 8b respectively. The first parametric study involves variations in vehicle velocity and its effect on the harvested electric power, while keeping the road roughness coefficient in the C class and maintaining a piston cross-sectional area ratio of 4. The piezoelectric bar and the cylinder-piston dimensions are based on Tables 2-3. The results show that the highest RMS of the generated power is 135 W. From Figure 9a, it can be seen that the increase in vehicle velocity is followed by the increase of P[rms]. The increment has a linear form. This finding owes to the fact that an increase in velocity leads to an increase in dynamic displacements of the bridge and suspension system, respectively. As a result, the dynamic displacement of the equivalent model of the cylinder-piston mechanism and piezoelectric bar (PEH) in Figure 3, either in the front or rear parts increases correspondingly. Those increments correspond to the increase in charge, voltage, current, and generated electric power of the PEH as indicated by Equations 9-12. Under three white noise time series, the difference slightly deviates within 6% for the generated electrical power. Thus, the proposed PEH model is less sensitive to noise variances. In the next case, we examine variations in the road roughness coefficient while keeping the vehicle velocity at 40 m/s and the cross-sectional area ratio at 4. Road-roughness coefficients are based on ISO/TC108/SC2N67 (Class A-H). Piezoelectric bar dimensions and piston-cylinder dimensions are still based on Table 3. The results are displayed in Figure 9b. Similar to the previous variation, an increase in the road roughness coefficient results in an increase in the RMS of the electric power generated by the proposed PEH. However, this increment follows a nonlinear form. The results show that the highest RMS of the generated power is 230 W. However, this value is only found on a very rough road. Such a road may be found as off-road, which is not common as a public road. City road is commonly found in the range of class A to class D so the road-roughness coefficients are considered in those ranges. In this variation, the proposed PEH model is also less sensitive to noise variations. The difference is within 7.5%. Table 1 Beam parameters │ Parameter │ Value │ │Beam length (m) │ 150 │ │Mass density (kg/m^3) │ 20,000 │ │Modulus of elasticity of the beam (N/m^2) │2.07e^11│ │Moment of inertia (m^4) │ 0.261 │ │Damping Coefficient (N.s/m) │ 2,625 │ Table 2 Vehicle parameters Table 3 Property and dimension of a piezoelectric bar (Piezo, n.d.)] The last case is variations of piston cross-sectional area ratio to the harvested electric power by keeping the vehicle velocity of 30 m/s, and road roughness coefficients in the class of C. The results are displayed in Figure 9c. A nonlinear relation between the RMS of the electric power generated by the proposed PEH and the piston cross-sectional area ratio is found. The results show that the highest RMS of the generated power is 93.5 W under piezoelectric bar and cylinder-piston dimensions in Table 2. This is to be expected since the increase of cross-sectional area ratio n[p] decreases the deflection of the PEH. This can be referred to as magnified force F[m](t) in Equation 7. Since the force increases with the ratio of n, then the P[rms] increases up to a certain limit. When the ratio of n[p] is at higher values, the equivalent damping coefficient c[eq] becomes predominantly, and the P[rms] decrease according to Equation 6. Hence, the optimum value of n[p] is selected to be 4, which can still be accommodated in the design. For this variation, the proposed PEH model also seems less sensitive to the noise variations as in the previous case. The difference is found within 8.7% for the generated electrical power. This result suggests that the proposed PEH is robust due to the deviation being relatively low to the variations of white noise as an environmental factor. Figure 7 Dynamic displacements of (a) vehicle body (b) front tire (c) rear tire (d) PEH Figure 8 Harvested energy (a) Generated electric power (b) White noise time series Figure 9 RMS of generated electric power under variation of (a) vehicle velocity (b) road class (c) piston cross-sectional area ratio Akbar, M., Ramadhani, M.J., Izzuddin, M.A., Gunawan, L., Sasongko, R.A., Kusni, M., Curiel-Sosa, J.L., 2022. Evaluation on Piezoaeroelastic Energy Harvesting Potential of A Jet Transport Aircraft Wing with Multiphase Composite by means of Iterative Finite Element Method. International Journal of Technology, Volume 13(4), pp. 803–815 Alhumaid, S., Hess, D., Guldiken, R., 2022. A Noncontact Magneto–Piezo Harvester-Based Vehicle Regenerative Suspension System: An Experimental Study. Energies, Volume 15, p. 4476 Ali, A., 2015. Comparison Between Pakistani Corolla and Civic’s Suspension: What’s The Difference? 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Physics Procedia, Volume 70, pp. 970–973 Lafarge, B., Grondel, S., Delebarre, C., Cattan, E., 2018. A Validated Simulation of Energy Harvesting with Piezoelectric Cantilever Beams on a Vehicle Suspension Using Bond Graph Approach. Mechatronics, Volume 53, pp. 202–214 Li, Z., Peng, Y., Xu, A., Peng, J., Xin, L., Wang, M., Luo, J., Xie, S., Pu, H., 2021. Harnessing Energy Form Suspension Systems of Oceanic Vehicles with High-Performance Piezoelectric Generators. Energy, Volume 228, pp. 1–14 Morangueira, Y.L.A., Pereira, J.C.C., 2020. Energy Harvesting Assessment with A Coupled Full Car and Piezoelectric Model. Applied Energy, Volume 210, pp. 1–13 Pan, P., Zhang, D., Nie, X., Chen, H., 2017. Development of Piezoelectric Energy-Harvesting Tuned Mass Damper. Science China Technological Sciences, Volume 60, pp. 467–478 Pasharavesh, A., Moheimani, R., Dalir, H., 2020. Nonlinear Energy Harvesting from Vibratory Disc-Shaped Piezoelectric Laminates. 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Engineering Structures, Volume 133, pp. 74–80 Tavares, R., Ruderman, M., 2020. Energy Harvesting Using Piezoelectric Transducers for Suspension Systems. Mechatronics, Volume 65, pp. 1–9 Tian, L., Shen, H., Yang, Q., Song, R., Bian, Y., 2023. A Novel Outer-Inner Magnetic Two Degree-Of-Freedom Piezoelectric Energy Harvester. Energy Conversion and Management, Volume 283, p. 116920 Touairi, S., Mabrouki, M., 2021. Control and Modelling Evaluation of a Piezoelectric Harvester System. International Journal of Dynamics and Control, Volume 9, pp. 1559–1575 Viet, N.V., Al-Qutayri, M., Liew, K.M., Wang, Q., 2017. An Octo-Generator for Energy Harvesting Based on The Piezoelectric Effect. Applied Ocean Research, Volume 64, pp. 128–134 Wang, M., Yin, P., Li, Z., Sun, Y., Ding, J., Luo, J., Xie, S., Peng, Y., Pu, H., 2020. Harnessing Energy from Spring Suspension Systems with A Compressive-Mode High-Power-Density Piezoelectric Transducer. 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Fibonacci Retracement Levels Who was Fibonacci ? What are Fibonacci Retracement Levels ? Fibonacci retracement lines are well known in technical analysis of stocks and other securities. These horizontal lines represent potential support or resistance price levels. Common retracement values are 23.6%, 38.2% and 61.8% of the % price move within a trend. 50% is also used on the charts even though it is not a fibonacci retracement level based on the fibonacci sequence. The fibonacci sequence starts with 0, 1 and continues to infinity. The sequence is written by taking the current number in the sequence and adding it to the previous number in the sequence. That result is the next number in the sequence. So we have... 0, 1 (0+1 = 1) 0, 1, 1 (1+1 = 2) 0, 1, 1, 2 (2+1 = 3) 0, 1, 1, 2, 3 (3+2 = 5) 0, 1, 1, 2, 3, 5 (5+3 = 8) 0, 1, 1, 2, 3, 5, 8 (8+5 = 13) 0, 1, 1, 2, 3, 5, 8, 13 (13+8 = 21) 0, 1, 1, 2, 3, 5, 8, 13, 21 (21+13 = 34) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 (34+21 = 55) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 (55+34 = 89) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.... to infinity ! The fibonacci retracements are based on a unique property of this sequence. Notice when you take any number in the sequence beyond the initial 0, 1 in the sequence and divide it by the next number you get an approximation to 0.618... (a fibonacci retracement level and inverse of the Golden Ratio). And, if you instead divide your number by 2 numbers ahead in the sequence you get an approximation to 0.382... (a fibonacci retracement level). And, if you instead divide your number by 3 numbers ahead in the sequence you get an approximation to 0.236... (a fibonacci retracement level). The accuracy of these approximations improves as the sequence extends to Golden Ratio A fundamental property of the fibonacci sequence is the Golden Ratio (1.6180339887...). The Golden Ratio is an irrational number. Take any number beyond the initial 0, 1, 1 in the sequence and divide it by the previous number and you get an approximation to 1.6180339887... that improves as the sequence continues. The approximation to the Golden Ratio shows up frequently in nature. 1. Dividing the number female bees by the number of male bees in any given hive gives you approximately 1.618. 2. The count of the 2 opposing spiral patterns of seeds in sunflowers has a 1.618 ratio (commonly 89/55 or 55/34 or 34/21). 3. The DNA molecule, the program for all life, is based on the golden section. It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. 34/21 = Golden 4. Anatomy: Measure the length of your arm (from shoulder to tip of middle finger) then divide by the length from your elbow to the tip of your middle finger. The result will approximate 5. Other examples include atoms, pine cones, pineapples, nautilus shells, certain hurricanes and spiral galaxies. You can also use the Golden Ratio to find any number in the fibonacci sequence. Here's how... Let's say you want to find the 19th number in the fibonacci sequence. Let's call that variable y. You get your answer by raising the Golden Ratio to the 18th power and dividing by the square root of 5. Then round to the nearest integer. More generally it is ((Golden Ratio ^ (y-1)) / sqrt (5). Use of Fibonacci Retracements The price levels for fibonacci retracement lines depend on trend identification and determination of the high and low prices within the trend. Trend lines are subjective so caution should be used. Someone may draw a different trend line than the next person which in turn may change the price levels of the fibonacci retracement lines. It is not recommended that you rely solely on Fibonacci retracements or HotCandlestick.com when making trading decisions. Candlestick patterns are nicely complimented by Fibonacci retracement levels. But you still need to do your own research. HotCandlestick.com automatically determines the fibonacci retracement lines using the following method... Identify clear price trends on the chart using the following criteria: A) A trend must run over at least 10 consecutive (daily, weekly, monthly or quarterly) candlesticks. B) A trend is nullified by too much price variation which can include a significant price gap. A solid orange line is drawn on the chart for each of the last 2 up and down trends. For the most recent up trend, a black circle is drawn around the high and low. For the most recent down trend, a red circle is drawn around the high and low. The fibonacci retracement price levels are calculated by removing 23.6%, 38.2%, 50% and 61.8% of the % price move. The 4 fibonacci retracements are plotted as a series of horizontal dashed orange lines with the corresponding price levels shown on the chart. To avoid chart clutter this is done only for the most recent trend line. Fibonacci retracements are automatically included on all daily, weekly, monthly and quarterly charts. Since prices are plotted on a logarithmic scale on the charts this means that any identical vertical distance on the chart is the same % price change. You can easily verify the retracement levels by holding a piece of paper to the chart and marking the distance from the circled high to the 50% retracement. Then measure the distance from the circled low to the 50% retracement. The 2 distances will be equal. Next Page » View the candlestick patterns tracked at HotCandlestick.com Home Disclaimer Privacy Contact Copyright © 2001-2024, All rights reserved. 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How to Plot Accuracy Curve In Tensorflow? To plot the accuracy curve in TensorFlow, you can use the data collected during training and validation of your model. First, store the accuracy values in lists or arrays as the model is trained. Once the training is complete, use a plotting library such as Matplotlib to create a graph of accuracy over training epochs. You can use the epoch number as the x-axis and the accuracy values as the y-axis. This will give you a visual representation of how the accuracy of your model has improved over time. Visualizing the accuracy curve can help you analyze the performance of your model and make any necessary adjustments to improve its accuracy. What is the connection between accuracy and generalization in machine learning? Accuracy and generalization are closely related concepts in machine learning. Accuracy in machine learning refers to the ability of a model to correctly predict the target variable, while generalization refers to the ability of a model to perform well on new, unseen data. A model that achieves high accuracy on the training data but performs poorly on new, unseen data is said to have low generalization. This is often due to overfitting, where the model learns the patterns and noise in the training data rather than the underlying relationships that generalize to new data. On the other hand, a model that achieves high accuracy on both the training and test data is said to have good generalization. This means that the model has successfully learned the underlying relationships in the data and can make accurate predictions on new data. Therefore, the goal in machine learning is to achieve a balance between accuracy and generalization. A model that is both accurate and generalizes well is considered to be a high-quality model that can make reliable predictions on new, unseen data. How to use TensorFlow to plot accuracy curves? To plot accuracy curves using TensorFlow, you can follow these steps: 1. Train your model and keep track of the accuracy values during training. You can use the metrics module in TensorFlow to calculate accuracy. 2. Store the accuracy values in a list or array during each training iteration. 3. After training is complete, use a plotting library such as Matplotlib to create a line plot of the accuracy values over time. 4. Here is an example code snippet to demonstrate how to plot accuracy curves using TensorFlow and Matplotlib: 1 import matplotlib.pyplot as plt 2 from tensorflow import keras 3 from tensorflow.keras import metrics 5 # Define your model and compile it 6 model = keras.Sequential([ 7 keras.layers.Dense(64, activation='relu'), 8 keras.layers.Dense(10, activation='softmax') 9 ]) 10 model.compile(optimizer='adam', loss='sparse_categorical_crossentropy', metrics=['accuracy']) 12 # Train your model and keep track of accuracy values 13 history = model.fit(x_train, y_train, epochs=10, validation_data=(x_val, y_val)) 15 # Extract accuracy values from history object 16 train_acc = history.history['accuracy'] 17 val_acc = history.history['val_accuracy'] 19 # Plot accuracy curves 20 plt.plot(train_acc, label='Training Accuracy') 21 plt.plot(val_acc, label='Validation Accuracy') 22 plt.xlabel('Epoch') 23 plt.ylabel('Accuracy') 24 plt.legend() 25 plt.show() This code snippet assumes you have already trained your model and have access to the training and validation accuracy values. It uses Matplotlib to create a line plot of the accuracy curves over the training epochs. How to incorporate accuracy curves into model selection criteria? Accuracy curves can be a useful tool for model selection criteria by providing a visual representation of how well a model performs across different levels of complexity or hyperparameters. One way to incorporate accuracy curves into model selection criteria is to compare multiple models based on their accuracy curve. This involves plotting the accuracy of each model on a graph, with the x-axis representing the complexity or hyperparameters of the model and the y-axis representing the accuracy of the model. By comparing these curves, you can easily see which model performs better at different levels of complexity. Another way to use accuracy curves for model selection is to look for the point on the curve where the accuracy plateaus or starts to decrease. This can help identify the optimal level of complexity or hyperparameters for the model, as going beyond this point may result in overfitting and decreased accuracy. Overall, incorporating accuracy curves into model selection criteria can provide valuable insights into how well a model performs at different levels of complexity and can help guide the selection of the best model for a given task.	{"url":"https://stlplaces.com/blog/how-to-plot-accuracy-curve-in-tensorflow","timestamp":"2024-11-07T14:29:26Z","content_type":"text/html","content_length":"330784","record_id":"<urn:uuid:a603c5ce-fb40-4b1b-b9cf-dd6230aac55a>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00556.warc.gz"}
1. Introduction Additive manufacturing (AM) technologies have developed intensively over the past decades and are now an essential part of efficient industrial production. 3D-printing no longer serves only as a technology for prototype or small batch production. As a manufacturing technology, the type of 3D-printing not only affects the design of the component along with the material used, but also the development time, functionality, safety, reliability, durability, efficiency, ecological footprint and recyclability [ ]. The working principles of various 3D-printing technologies have enabled the emergence of new methods and procedures for component design. The design of a part is no longer adapted only to traditional manufacturing methods, but also to its application and functionality. It is important that research and development for AM continues to come up with innovative and sophisticated component design methods corresponding to current design advances, due to the increasing number of input requirements for part design and manufacturing [ Manufacturing technology, operating costs, waste management, etc. are limiting factors that influence the component design process, even if they are not directly related to its functionality. Currently, the final design is a compromise that takes into account all specific factors of the product life cycle [ ]. Traditional manufacturing technologies are one of the limiting factors that has a significant impact on product design. Their technological design principles do not always achieve the optimum part properties for a given application. Modern AM (Design for Additive Manufacturing - DfAM) technologies make it possible to implement an innovative component design process. The design of a design is determined only by the application and the specific requirements of the designer for the component [ ]. Based on the new knowledge and understanding of the capabilities of DfAM, the following construction methods were introduced [ • Topology Optimization (TO), • Generative Design (GD), • Multiscale Structure Design) (MSD), • Multimaterial Design (MD), • Parts Consolidation) (PC), • Design for Mass Customization (DfMC), • Lattice Structures (LS), • Thermal Issues in Design (TIiD). Among the above-mentioned design methods, TO is used in the research due to the ever-increasing development of that type of DfAM, especially in the automotive industry but also in other industries. The principles of TO have been known for several decades. The method provides optimal designs of components that are very complex in terms of geometry and are not manufacturable or difficult to manufacture by traditional manufacturing methods (chip machining, casting). It was with the advent of additive manufacturing technologies that it began to be applied to a greater extent in the component design process [ ]. Currently, TO is implemented in the development of components that have the optimum strength, mass and dynamic parameters for a specific function. The resulting design is characterized by organic shapes that are only manufacturable by 3D-printing [ ]. The above method of construction (TO) is a set of mathematical iterations that work on the principle of removing and distributing material in a defined space of the original shape of the part [ ]. The optimization is based on predefined conditions - load, functional elements of the part that must be preserved (contact surfaces, holes, etc.), links, limiting space (original shape of the part). Often TO is confused with the GD method. The main difference is that TO upgrades the design of an existing component and the GD method creates a new component with an optimal design. The optimization itself is preceded by a strength check of the part using the finite element method. The strength check results in stress and strain maps that serve as input to the optimization process ( Figure 1 ) [ Today, there are several innovative CAD software on the market that allow designers to use the full production potential and capabilities of additive manufacturing technologies, many of which include TO tools. One of them is the 3DExperience software, whose licenses are available at the research institute of the Faculty of Mechanical Engineering. The software includes two optimization modules: • Structural Generative Design (SGD) – mass optimization, • Flow Generative Design (FGD) – shape optimization with respect to pressure losses. In the SGD module, the optimization process can be based on different strategies. The TO strategies used are: • maximize the stiffness for a given part weight, • minimize the weight while respecting the constraints, • maximize the value of the lowest natural frequency for a given weight of the part. For strategies 1 and 3, it is necessary to define the desired weight of the optimized part. The disadvantage of these strategies (especially strategy 1) is that the predefined desired weight may not be the lowest possible weight of the component at which the strength requirements are satisfied. Optimization by strategy 2 requires the determination of a limiting factor, such as: • maximum allowable stress, • maximum allowable deformation, • maximum/minimum wall thickness, • manufacturing technology. In this strategy, the material is removed as long as the optimization conditions of the TO are met. A schematic representation of the whole optimization process is shown in Figure 2 The aim of the research activity is to develop, apply and evaluate the effectiveness of an innovative methodology for the design of mobile machine components intended for AM in comparison with traditional design procedures, based on the acquired know-ledge in the field of AM and DfAM methods. The specific aim is to minimize the time required not only to reduce the product development stage but also to manufacture it efficiently, especially for geometrically shape-intensive components, often with the requirement to reduce their weight while maintaining the required strength parameters. To achieve the final result of the research, the following sub-objectives are specified: • to develop a sophisticated design methodology for the construction of mobile machine parts intended for AM based on the current state of the art, • application of the proposed design methodology for shape and weight optimization, • validation of the proposed component manufacturing methodology by evaluating its effectiveness. 2. Materials and Methods 2.1. Traditional method of component designing To evaluate the effectiveness of the new construction methodology, it is necessary to define the traditional construction procedures. The construction procedure used in companies specialized in the development of mobile machines serves as a comparative benchmark. 2.1.1. Design process by traditional design method In companies using CAD software in the development of component construction, whose modules and tools do not use the principles of innovative construction methods, the development of the component construction is an iterative process, consisting of two, maximum three iterations of the design solution. The two-iteration development process of the component construction is composed of the following operations: • application site (available footprint), • function, • load cases, • weight and strength requirements; first iteration of the design solution: • material selection, • design of the structure, • design validation by strength, • analysis of strength check results – failure to meet strength requirements; second iteration of the design solution: • analysis of the original design and strength check, • modification of the design, • validation of the new design by strength check, • analysis of strength check results – meeting strength requirements; • analysis of tolerance chain, • manufacturability analysis, • optimization of the final design for production. The development of the design of a new component, from the given production line, presented in the research consists of two iterations of the design solution, which, based on the data obtained, takes one designer two weeks, i.e. 75 hours. 2.1.2. Applying the traditional method of designing to research From the traditional design approach, the first iteration was implemented on the electric motor mounts. The assembly of the brackets and the electric motor is shown in Figure 3 The weight of the front bracket is 1.9 kg and the weight of the rear bracket is 1.124 kg. However, this design did not meet the strength requirements. The material used was aluminum alloy EN AW-7022 with a yield strength of 370 MPa. In some loading situations, the maximum stress values were 476 MPa and 481 MPa for the front and rear brackets, respectively. The traditional design method no longer proposed a second iteration of the design that would take into account the strength test results of the first iteration. Instead, the decision was taken to apply CAD software to the development of the components, which includes modules and tools that use the principles of innovative design methods. The TO method in the SGD module of the 3DExperience software was used to develop the component design. The whole procedure serves as a template for the development of a sophisticated design methodology for shape and weight optimization of mobile machine components, which is described in detail in the following section of the research. 2.2. TO designing procedure The proposed procedure for the development of the design of the electric motor mounts using the TO method in the SGD module of the 3DExperience software consists of five main stages: • pre-preparation for TO, • TO, • postprocessing, • validation of the final design, • preparation for production. 2.2.1. Pre-preparation for TO Part of the TO pre-preparation is the creation of the shape of the object for optimization, definition of its functional elements, creation of the Body (volume in CAD software) of the optimization object, creation and assignment of the material profile. 2.2.1.1. Object shape creation for optimization The first thing to do is to create an optimization object. The object can be either an existing component or the maximum allowed building volume, the so-called semi-finished product. In this case, the existing components, i.e. the first iterations of the design of the electric motor holders, did not meet the strength requirements. Therefore, the holder blanks were created. The semi-finished volume is the maximum allowable build volume that the future component can occupy. The design of the blank is designed to avoid collision with the frame, interior and other components of the passenger car. The designs of the holder blanks and their comparison with the first iterations of the bracket design are shown in Figure 4 Figure 5 . The individual semi-holders were created as separate 2.2.1.2. Defining the functional elements of an object for optimization Functional elements are those parts of the optimization object volume that are to be preserved after TO. The individual functional elements have been created as separate Bodies. The shape of the Body of the functional elements has a significant impact on the quality of the mesh and thus on the progress of the individual simulations that are part of the TO. Therefore, the shapes of the Bodies of the functional elements were designed to follow the contour of the shape of the semi-finished product with a certain distance. In the case of the front bracket, the following object functional elements were retained for optimization ( Figure 6 • through-holes and recesses for screws (marked in yellow), • the holes for the silent blocks (shown in orange). In the case of the rear bracket, the following functional elements of the optimization object were retained ( Figure 7 • through-holes for screws (marked in yellow), • the threaded holes for the rear arm attachment (shown in orange). 2.2.1.3. Creating the Body object for optimization By linking the Body of the blank with the Body of the functional elements, the Body of the optimization object is created, which was used in the next steps of the optimization process. The Partition Design Space function is used to create the Body object for optimization. 2.2.1.4. Create and assign a material profile to the optimization object In order for a material profile to be usable for a TO, three parameters need to be defined: • density – 𝜌, • Young's modulus – 𝐸, • Poisson's number – 𝜇. Aluminum alloy AlSi10Mg, which is suitable for fabrication by additive manufacturing using DMLS or SLM technologies, was selected for the structural design of electric motor mounts by TO method. Its parameters are: • density: 𝜌=2650 𝑘𝑔.𝑚^-3, • Young's modulus: 𝐸=66 𝐺𝑃𝑎, • Poisson's number: 𝜇=0.325, • yield stress: 𝑅𝑒=260 𝑀𝑃𝑎. 2.2.2. TO The TO process in the SGD module of the 3DExperience CAD software consists of the following sequential steps: • defining the TO strategy, • definition of the optimization object and its functional elements, • definition of force loads, constraints and their assignment to load cases (Load Cases), • mesh creation, • initial strength check of the optimization object, • definition of optimization conditions, • TO, • concept generation, • strength check of the generated concepts. 2.2.2.1. TO strategy In the SGD module, TO can follow three different strategies as mentioned earlier (Chapter 1): • maximize the stiffness with respect to the specified component mass, • minimize the weight while respecting the constraints, • maximize the value of the lowest natural frequency for a given component weight. In the development of the electric motor mounts, the objective was set to achieve the lowest possible weight at which the components meet the strength requirements. Therefore, the strategy TO no. 2 Minimize the weight while respecting the specified constraints was chosen. 2.2.2.2. Object to be optimized and its functional elements The user selects from the list of all available Bodies whose internal structure contains partitions, i.e. the Body of the Optimization Object. In the next step, it is necessary to select which volumes of the optimization object are to be preserved and which are to be optimized. The software also informs the user of the masses of the individual volumes. The total weight of the front bracket blank is 12.776 kg, of which the weight of the functional volumes is 0.188 kg (1.46 %) and the weight of the optimized volume is 12.589 kg (98.54 %). The total weight of the rear bracket blank is 6.668 kg, of which the weight of the functional volumes is 0.198 kg (2.96 %) and the weight of the optimized volume is 6.471 kg (97.04 %). 2.2.2.3. Definition of load situations The loads acting on the components were obtained from measurements performed on a real vehicle during thirty load situations. The individual load situations represent conditions that may occur during vehicle handling, driving or collision. The measured forces were in the silentblocks ( Figure 8 position 1,2 and 3) through which the electric motor with brackets is attached to the vehicle frame. These forces are then applied to the inner circumference of the front bracket hubs and the rear arm hubs, from which the force effects are transferred to the rear bracket. All thirty load situations are represented in the software by load cases. A load case consists of the applied force effects (force, moment, ...) and the constraints. For the front holder, the load case includes ( Figure 9 • two forces (Force type) - reactions from the silentblocks acting on the holder hubs; • six linkages - representing the attachment of the holder to the electric motor by two bolts: two rotational links (Rotate type), four sliding links (Slide type). For the rear holder, the load case includes ( Figure 9 • one force (Remote Force type) - the reaction from the silentblock acting on the rear holder hub; • nine links - representing the attachment of the holder to the electric motor by three bolts: three rotating links (Rotate type), six sliding links (Slide type). 2.2.2.4. Creating a mesh Based on the results obtained from the analysis of the influence on the progression and outcome of the TO, a quadratic configuration with a default number of elements ( Automatically deducated quadratic mesh option ) was used for both holders. The quadratic network with default number of elements of the front holder is shown in Figure 10 a and that of the rear holder in Figure 10 b. The shapes of the function elements were designed to follow the contours of the shape of the blank with some distance to avoid certain qualitative errors of the 2.2.2.5. Initial strength check of the object for optimization The initial strength check is used to demonstrate the optimization potential of the object to be optimized. Its result serves as input to the TO. Based on the stress map, the material is removed or reorganized in the blank volume. The value of the maximum stress from the input strength check must be lower than the value of the allowable stress in the optimization condition. For the purpose of evaluation and analysis, 30 stress situations with strength control were performed. After analyzing the results of the input strength control, it was decided that the parts will be optimized only for the ten most unfavorable-greater load situations. This accelerated the entire development process. The most unfavorable load situations did not develop only on the basis of voltage values, but also on the basis of the sizes of the individual components of the acting forces. 2.2.2.6. Definition of optimization conditions A single optimization condition, i.e. the maximum allowable stress, was defined for the TO of the electric motor holders. In order to be able to run the optimization process at all, it is necessary that the value of the maximum allowable stress is greater than the stress values obtained during the initial strength check of the semi-finished product. On the other hand, it should not be greater than the yield strength or ultimate strength of the material used. On the basis of consultations with companies using additive SLM technology for the production of dynamically stressed mobile machine components made of aluminum alloy AlSi10Mg, it was found that the maximum stress value (reduced Von Mises stress) of the static strength analysis should be in the range of 90 MPa - 110 MPa. Compliance with this condition will achieve the required service life of the components. A maximum allowable stress value of 100 MPa was set as the optimization condition for the TO of both motor 2.2.2.7. TO The TO of the front mount lasted 3 hours and 2 minutes. The optimization converged to a solution that satisfied the optimization conditions after 120 iteration cycles were performed. The results of the TO of the front holder are shown in Figure 11 . The software evaluated the failure to meet the optimization requirement for two load cases . The stress values exceeded the maximum allowable stress value by 0.5%. However, after analyzing the results, it was concluded that the optimization condition was satisfied in all ten load cases. The achieved mass of the front bracket concept was 1.574 kg ( Figure 11 labeled as Mass - Design Space.1: Final Value ), i.e. 12.3% of the weight of the holder. The TO of the rear holder lasted 1 hour and 32 minutes. In this case, the optimization converged to a solution that satisfied the optimization conditions after only 52 iteration cycles had been performed. The results of the TO of the rear holder are shown in Figure 12 . The optimization condition was satisfied in all load cases . The achieved mass of the rear bracket was 0.806 kg ( Figure 12 labeled as Mass - Design Space.1: Final Value ), i.e. 12.1% of the mass of the holder. 2.2.2.8. Concept generation The concept generation process is used to materialize the results of the TO. The Cutting Value (CV) parameter has the greatest influence on the generated shape. It is an informative parameter whose value describes the amount of mass retained after the optimization process. By changing the parameter, material is removed or added with respect to the TO result. Two concepts have been generated for the two holders. One concept was generated with a CV parameter such that its mass is closest to the mass of the concept after the last iteration of the TO ( Figure 11 and 12 labeled as Mass - Design Space.1: Final Value , respectively). The CV value of the front holder and the rear holder was set to 55 and 53, respectively. The second concept was generated with a higher CV parameter value (CV value: front bracket - 90, rear bracket - 80). The generation of two concepts with different CV parameter value allows to analyze the influence of the parameter on the weight, design concept and strength of the generated concepts. From Figure 13 Figure 14 , it can be seen that the weight of the generated concepts with the CV parameter, where the weight is closest to the weight after the last iteration of TO, is larger than that shown in Figure 11 Figure 12 , respectively. This is because the software adds material at certain locations in order to generate a concept with a nice continuous shape without defects. 2.2.2.9. Robustness check of generated concepts The strength check of the generated raw front and rear holder concepts for all ten load cases was used to validate the strength of their design and the accuracy of the TO results. The raw generated front bracket concept with CV 55 did not meet the strength requirements for all load cases (two load cases). However, after analyzing the results, it was concluded that with minor modifications to the design in postprocessing, these relatively high stresses could be avoided. The strength checks proved the accuracy of the TO results. The generated concepts are suitable for use as templates for the final design of components within the postprocessing. 2.2.3. Postprocessing –- creation of the final design In postprocessing, the complicated design of the generated and validated concepts is reconstructed, and functional elements are implemented in the final design. The SGD module has functions and tools for the reconstruction of complex geometry using surfaces [ ]. These are general surfaces whose geometry is controlled by control points. Such surfaces can be connected to each other to form larger units, thus creating the final design of the component. The individual design elements of the concepts are modeled as separate planar entities by the IMA - Tube drawing IMA - Cylinder functions. The final design of the front and rear holder was obtained by trimming the blanks with the final function and implementing the functional elements into the trimmed volume by Boolean operations Figure 15 Figure 16 2.2.4. Validation of the final design To validate the correctness of the final design of the front and rear holder, strength tests were performed for all thirty loading situations. A linear mesh with an element size of 2 mm was used for the front holder and a linear mesh with an element size of 1.5 mm was used for the rear holder. These are the same mesh configurations that were used in the validation of the generated bracket concepts (Chapter 2.2.2.9). The strength tests proved the correctness of the front and rear electric motor bracket design. In neither load case do the stresses (Von Mises type) exceed the maximum allowable stress value of 100 MPa. All the conditions imposed on the design of the bracket construction have been fulfilled. 2.2.5. Final design - preparation for 3D-printing The weight of the final design of the optimized front holder is 1.714 kg. A detailed view of the front holder design of the electric motor is shown in Figure 17 . The weight of the final design of the rear holder is 0.783 kg. A detailed view of the rear electric motor holder design is shown in Figure 18 . The electric motor assembly with optimized front and rear holder is shown in Figure 19 Before the actual additive manufacturing with SLM technology, it is necessary to perform 3D-printing and cooling simulation of the component. This will highlight the behavior of the components throughout the manufacturing process. Based on the results, manufacturing inaccuracies can be predicted, which can be eliminated by changing the design (so called component pre-deforming) or eliminated in the manufacturing finishing operations. Areas that are prone to deformation during 3D-printing or during cooling after the manufacturing process shall be reinforced with supports. These supports are then removed mechanically or by vibration. Additional material shall be applied to functional surfaces which are subject to high demands in terms of precision and roughness. Precision chip machining is used to remove the additive and thus achieve the required precision and roughness of the functional surfaces. 3. Results Table 1 shows the duration times for each stage of the development of the electric motor bracket design by the TO method using the software. These are the times that can be achieved with ideal inputs, ideal operation of the software and ideal outputs from the optimization process. However, real-world use has shown that it is often necessary to rework the inputs (blanks, partitions) to improve the input strength optimization results, and it is often necessary to change the optimization conditions to ensure that the optimization process is working properly and thus obtain usable concepts of the optimized parts. This nature of the software and the TO method itself is accounted for by the reliability coefficient 𝑘𝑠𝜖〈1.3〉. The coefficient interval was defined based on the experience gained by applying the method and software to the development of several components. The comparison of the development time of the electric motor brackets with the TO method and the development time with traditional design methods is shown in Table 2 It is clear that the implementation of the TO method into the process of designing the optimal design of electric motor holders has significantly reduced the development time. A weight saving was also achieved compared to the first iteration of the design of the electric motor holders, which additionally did not undergo a strength check. 4. Discussion The application of the proposed design methodology in practice confirmed its effectiveness at the component development stage. Its implementation in the component design process demonstrated significant savings in development time and component weight. The principles of TO have been known for a long time, but the method has not been used in practice on a large scale. This is due to the relative difficulty of operating optimization software, its reliability and the lack of trained and experienced personnel. In order to ensure the efficiency and reliability of the implementation of optimization software in the development of new components, it is necessary to develop variants of the design methodology that will be tailored for the design process of a specific product line. Author Contributions Conceptualization, P.H. and Martin N.; methodology, L.G. and A.K.; software, A.K. and Miroslav N.; validation, P.H. and V.Ch.; formal analysis, P.H. and Martin N.; investigation, L.G. and V.Ch.; resources, V.Ch.; data curation, Miroslav.N.; writing—original draft preparation, P.H. and Martin N.; writing—review and editing, P.H. and L.G.; visualization, V.Ch.; supervision, L.G.; project administration, L.G.; funding acquisition, L.G. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Institutional Review Board Statement Not applicable. Informed Consent Statement Not applicable. Data Availability Statement The data presented in this study are available on request from the corresponding authors. Conflicts of Interest The authors declare no conflict of interest. Figure 9. Schematic representation of all 30 load cases of the blanks: (a) front holder, (b) rear holder. Figure 10. Network of a quadratic mesh with default number of elements: (a) front holder blank, (b) rear holder blank. Figure 15. Trimming of the front holder blank with the final NURBS-plate and implementation of the functional elements. Figure 16. Trimming of the rear holder blank with the final NURBS-plate and implementation of the functional elements. Development time of electric motor holders by topological optimization method Front Rear Stages Job description holder holder Time (h) creating an object for optimization pre-preparation for topological optimization 3 3 creation of functional elements of the optimization object creation of a material profile and assignment to the optimization object defining the optimisation strategy 0.25 0.25 defining the object for optimisation defining the functional elements of the optimisation object definition of force loads and constraints and their assignment to load cases creating a MESH topological optimization Initial strength check of the object for optimization definition of optimization conditions 3 1,5 topological optimization generation of concepts strength check of generated concepts reconstruction of the complex geometry of the generated and validated concept through NURBS surfaces postprocessing 4 4 implementation of functional elements of the final design validation of the final design 1 1 strength check of the final component design preparation for production 1 1 allowances for finishing after 3D-printing total development time under ideal conditions 15 14 (rounded to whole hours) method reliability coefficientk[s] 3 total development time under realistic conditions 46 41 Comparison of design methods Development time (h) Weight of holders (kg) Method of construction Front Rear Front Rear Front Rear holder holder holder holder holder holder traditional 75 1,9 1,124 topological optimization Ideal conditions Real conditions 1,714 0,783 savings when using the topological optimization method 80% 82% 39% 45% 10% 30% Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/	{"url":"https://www.preprints.org/manuscript/202309.1821/v1","timestamp":"2024-11-13T02:02:30Z","content_type":"text/html","content_length":"713287","record_id":"<urn:uuid:d3bae527-5c4f-4e87-a346-a3dbbd487017>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00804.warc.gz"}
CATBOO – Latex Photography / Models in Catsuits / Hydrasuit It’s time for a new Latex set with one of our favorite Latex Catsuit Model Darina! This time Darina is wearing the Red Latex Catsuit for us, combined with her Black Heel Leather Boots, a pair of Black Latex Gloves and our open face Black Latex Hood! Enjoy this first part of the set Darina modeling for us on her bed – all covered in Shiny Latex! Pics: 52 Latex: Darina Red Latex Catsuit Pt.1/2	{"url":"https://blog.catboo.net/tag/latex-bed","timestamp":"2024-11-05T07:04:52Z","content_type":"text/html","content_length":"43388","record_id":"<urn:uuid:aa18f30e-6da7-44ab-8352-28cbd3112759>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00721.warc.gz"}
Is a TI 84 Plus allowed on the SAT? On the SAT® exam, calculators are allowed on one of the two math sections. Read on for tips to leverage your TI-84 Plus CE for success on test day! Can you use calculator in chemistry exam? No other calculator is allowed, nor may any other device containing a calculator (including those installed on a smartphone or tablet) be brought into the exam room. Is calculator allowed in Chemistry SAT? Yes, all scientific calculators are permitted in SAT Math as long as you’re working in the calculator-permitted section. Does chemistry require a calculator? Chemistry calculations are a bit different from the standard mathematical calculations you come across. One needs to have access to a calculator with all the functions required to carry out these calculations. These functions involve logarithms, roots, trigonometric functions, and more. What calculator is not allowed in GCSE? Please note J560/02 (Foundation tier) and J560/05 (Higher tier) are non-calculator examinations and candidates are not permitted calculators for these. Which type of calculator is not allowed? Calculators with any of the following facilities are prohibited, unless specifically stated otherwise in the syllabus: graphic display. data banks. dictionaries or language translators. Why are calculators not allowed in exams? One major concern is cheating. People can enter information into their calculators that may give them an unfair advantage on the exam, in essence using the calculator as a “cheat sheet”. Having a professor verify that a couple hundred students don’t have any information stored in their calculators isn’t feasible. Is a TI-84 Plus Ce allowed on the act? In other words, all versions of TI-84 Plus and TI-Nspire™ CX (excluding CAS) graphing calculators are allowed on the ACT® exam. Can I bring 2 calculators to the SAT? One last important note on the SAT rules–you are allowed to bring more than one calculator. You can only have one out at a time, but you can switch whenever you feel like it with the spare you have under your desk. It is strongly advised to bring a backup calculator in case your primary has a problem. What percentage of chemistry is math? Overall, at least 20% of the marks in assessments for chemistry will require the use of mathematical skills. How much math is chemistry A-level? In the new AS and A-Level Chemistry exams, the use of maths is required for 20% of the marks — and this brilliant book explains all the maths students will need to learn! Do you need a calculator for biology GCSE? A basic scientific calculator is the minimum you require for exams. However a more advanced model such as the fx-991EX will give you advantages for learning and in the exam, for example with solving Is fx991es allowed in SAT? Is the Casio FX-991ES Plus allowed on the SAT? As mentioned in the “Other good calculators for the SAT” section, a scientific calculator is allowed on both the SAT and ACT. However, it is not a graphing calculator, so you cannot solve any complex equations or plot graphs. What can I bring to SAT? • Face covering. • Your up-to-date admission ticket. (You need to print this out.) • Acceptable photo ID. • Two No. 2 pencils with erasers. • An approved calculator. • Epinephrine auto-injectors (like EpiPens) are permitted without the need for accommodations. What calculators are not allowed on the SAT? • Laptops or other computers, tablets, mobile phones, smartwatches, or wearable technology. • Models that can access the Internet, have wireless, Bluetooth, cellular, audio/video recording and playing, camera, or any other smartphone-type features. What calculator do I need for GCSE? Casio FX-991EX – Recommended for GCSE, IGCSE and A-level This is the calculator I would recommend for GCSE and IGCSE students and is essential for A-level. It can do everything the fx-85 can do, but has lots of extra tools. Can you use calculators for GCSE? exam boards will need to comply with our new rules for all new GCSEs, AS and A levels which allow students to use calculators in exams. exam boards will not need change their approach to calculator use in any of subjects we have already accredited (including GCSE maths), although they can do so if they wish. Are calculators allowed in physics GCSE? 2.1 In legacy GCSEs (graded A* to G) the only restrictions we place on calculators are in GCSE mathematics, where we specify that between 25 and 50 per cent of marks must be awarded for questions that are answered without using a calculator. Which calculator is allowed in exams? Students appearing for the Class 12 or ISC term 2 exams conducted by Council for Indian School Certificate Exam (CISCE) will be allowed to use a scientific calculator during the exam. What type of calculators are allowed in exam? The only models of electronic calculator that students will be permitted to take into the exam room are: CASIO fx 991 (any version) CASIO fx 115 (any version) Which calculator is allowed in colleges? 1. Use of the following calculators in examinations is permitted: Non- Programmable, Silent hand-held, Self-contained, Single-Line Display or Dual-Line Display calculators. Use of the following calculators in examinations is prohibited. a) Programmable calculators are prohibited. Does college let you use a calculator? While calculators might not be allowed on tests and exams, colleges know that tech-savvy students will utilize programs such as Wolfram Alpha, a powerful web-based computational tool, to aid with calculus assignments. Why calculators should not be used in the classroom? If proper examples are used, calculators are almost never needed (except for a few topics in trig and a few others). Calculators make teachers lazy and worse teachers than they should be because they don’t have to make sure the problem has numbers to assure their students learn the skill intended. Is calculator allowed in college exam? In most of the entrance examination for admission into undergraduate program calculator and any other electronic device is strictly prohibited. However in engineering college internal semester examinations calculator are allowed and student takes their own Calculator. Does ACT check your calculator? The ACT Calculator Policy In general, the ACT explains that all test-takers can use “any 4-function, scientific, or graphing calculator, as long as it is not on the prohibited list and it is modified, if needed.” Your best option is to use a calculator you’ve used before.	{"url":"https://scienceoxygen.com/is-a-ti-84-plus-allowed-on-the-sat/","timestamp":"2024-11-04T07:55:55Z","content_type":"text/html","content_length":"305579","record_id":"<urn:uuid:ad9a302a-14d4-451c-b7fc-9c1314032563>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00719.warc.gz"}
Characterization of soliton solutions in 2D nonlinear Schrödinger lattices by using the spatial disorder In this paper, the pattern of the soliton solutions to the discrete nonlinear Schrödinger (DNLS) equations in a 2. D lattice is studied by the construction of horseshoes in l∞-spaces. The spatial disorder of the DNLS equations is the result of the strong amplitudes and stiffness of the nonlinearities. The complexity of this disorder is log(N+1) where N is the number of turning points of the nonlinearities. For the case N=1, there exist disjoint intervals I0 and I1, for which the state um,n at site (m,n) can be either dark (um,n∈I0) or bright (um,n∈I1) that depends on the configuration km,n=0 or 1, respectively. Bright soliton solutions of the DNLS equations with a cubic nonlinearity are also discussed. ASJC Scopus subject areas 深入研究「Characterization of soliton solutions in 2D nonlinear Schrödinger lattices by using the spatial disorder」主題。共同形成了獨特的指紋。	{"url":"https://scholar.lib.ntnu.edu.tw/zh/publications/characterization-of-soliton-solutions-in-2d-nonlinear-schr%C3%B6dinger-2","timestamp":"2024-11-05T07:53:20Z","content_type":"text/html","content_length":"56758","record_id":"<urn:uuid:bdd30c4f-8743-4f6d-ba90-a00c6508ec34>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00805.warc.gz"}
Problem 1938: Destroy The Village Time limit: 0.5s Memory limit: 64MB Input: Output: A village is being raided by a barbarian, who, despite being on his own, shouldn't be underestimated. The village consists of $N$ houses, numbered from $0$ to $N-1$ and arranged in a straight line perpendicular to the shore, with house $i$ being $D_i$ meters away from the shore. The barbarian's strategy is simple: he will raze the house he is in, then move to the closest house that hasn't been razed yet, until all the treasures in the village have been hoarded. If there is more than one house at the minimum distance, the barbarian will move to the one closest to the shore first. The villagers have a plan though, which is to hide in the last house that the barbarian will raid to face him when he's most tired. Trying to predict the barbarian's moves is giving them an headache, moreover, the raid may start from any house. In order to help them, you have to find, for each possible starting house $i$, which house $H_i$ would be destroyed last by the barbarian. Input data The input file consists of: • a line containing integer $N$. • a line containing the $N$ integers $D_{0}, \, \ldots, \, D_{N-1}$. Output data The output file must contain a single line consisting of the $N$ integers $H_{0}, \, \ldots, \, H_{N-1}$. Constraints and clarifications • $2 \le N \le 10^6$. • $0 \le D_i \le 10^{12}$ for each $i=0\ldots N-1$. • $D_i > D_{i-1}$ for each $1 \le i \le N-1$. • For tests worth $12$ points, $N \le 100$, $D_i \le 10^9$, for each $0 \le i \le N-1$. • For tests worth $28$ more points, $N \le 1000$. • For tests worth $35$ more points, $N \le 100\,000$. • When the barbarian starts from house $0$, located $1$ meter away from the shore, he destroys it, then moves to house $1$, at a distance of $3$ meters from house $0$, then to house $2$, then $3$. The last house standing is number $4$. • When the barbarian starts from house $1$, located $4$ meter away from the shore, he destroys it, then moves to house $2$, at a distance of $2$ meters from house $1$, then to house $3$, then $4$. The last house standing is number $0$. • When the barbarian starts from house $2$, located $6$ meter away from the shore, he destroys it, then moves to house $3$, at a distance of $1$ meter from house $2$, then to house $1$. Note that both house $1$ and $4$ are at a distance of $3$ meters from house $3$, but house $1$ is closer to the shore. He then moves to house $0$. The last house standing is number $4$. • When the barbarian starts from house $3$, located $7$ meter away from the shore, he destroys it, then moves to house $2$, at a distance of $1$ meter from house $3$, then to house $1$, then $0$. The last house standing is number $4$. • When the barbarian starts from house $4$, located $10$ meter away from the shore, he destroys it, then moves to house $3$, at a distance of $3$ meters from house $4$, then to house $2$, then $1$. The last house standing is number $0$.	{"url":"https://kilonova.ro/problems/1938","timestamp":"2024-11-02T21:07:49Z","content_type":"text/html","content_length":"62890","record_id":"<urn:uuid:e9bafa18-76ef-4966-9abb-b71e6051717b>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00603.warc.gz"}
Calculator Universe What is Circular Cylinder? A circular cylinder is a three-dimensional geometric shape that consists of two parallel circular bases connected […] Circular Cylinder Calculator Read Post » Circular Permutation Calculator What is Circular Permutation? Circular permutation is a concept in combinatorics and mathematics that deals with the arrangement of objects Circular Permutation Calculator Read Post »	{"url":"https://calculatoruniverse.com/page/7/","timestamp":"2024-11-01T20:43:34Z","content_type":"text/html","content_length":"239380","record_id":"<urn:uuid:52e62f6e-71bf-4665-b6e4-c5c3425b7700>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00835.warc.gz"}
Problems with picking items randomly by chance this is what i got right now: if math.random(0,100) <= weight then return true return false and i have a item thats like 0.0000001% as jokes and alot of the players that played my game got it pretty fast, not sure if it was luck or the system 1 Like Could you provide an example weight value and expected %? 2 Likes Whats the weight? Is this the code for the rare item? If so, then if you are doing something like 0.0000001 as the weight, the odds wont actually be 0.0000001%, it will be 0.99%. This is because math.random(0,100) only picks whole numbers. In other words, the item will be given if 0 is selected out of the 101 possible whole numbers ranging from 0 to 100. What you may be looking for instead local weight = 0.0000000001 -- this is equivalent to 0.0000001%, see below comment if math.random() <= weight then -- math.random() without arguments returns any decimal number between 0 and 1 return true return false 1 Like what do you mean? not much to provide you 1 Like When you declare a range for the math.random function. It returns an integer value from the specified range. If you just use math.random with no parameters it should return a float value. So try this simple change first before anything more advanced can occur. Additionally, you should use the print for checking if the random function is actually returning float values. (As I am writing this, I was not able to completely test this through roblox studio.) local weight = 0.0000000001 if math.random() <= weight then return true return false If you want to do it using integers you could use this structure of code. -- You can use floats to define these variables. local minPercentage = 1 -- Minimum percentage value (e.g., 1%) local maxPercentage = 100 -- Maximum percentage value (e.g., 100%) local randomPercentage = math.random(minPercentage, maxPercentage) print(randomPercentage .. "%") -- Print the random percentage We need the weight being used. Also, check Template_Xx and my answer as well.	{"url":"https://devforum.roblox.com/t/problems-with-picking-items-randomly-by-chance/2501075","timestamp":"2024-11-13T21:30:42Z","content_type":"text/html","content_length":"32801","record_id":"<urn:uuid:2ea975b0-f345-48f6-8ef2-6f7b1260b92b>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00668.warc.gz"}
def boundedDistances(g: m[(t, Int32, t)]): Map[(t, t), Int32] \ Aef[m] with Foldable[m], Order[t] Source Returns the shortest distance between all pairs of nodes in the weighted directed graph g. OBS: No negative cycles must be present. Returns the pairs (a, b) where a can reach b through a number of edges in the directed graph g, including zero. Returns triples (x, cut, y) such that x cannot reach y without using cut (where x, cut and y are all distinct) in the directed graph g. There will at most be one triple for each pair of nodes (which will be the maximum cut of the possible choices). Returns the degree of each node in the directed graph g (the number of times a node exists as an endpoint of an edge). def distance(src: { src = t }, dst: { dst = t }, g: m[(t, Int32, t)]): Option[Int32] \ Aef[m] with Foldable[m], Order[t] Source Returns the shortest distance from src to dst in the weighted directed graph g. def distances(g: m[(t, Int32, t)]): Option[Map[(t, t), Int32]] \ Aef[m] with Foldable[m], Order[t] Source Returns the shortest distance between all pairs of nodes in the weighted directed graph g. Returns None if g contains a negative cycle. def distancesFrom(src: t, g: m[(t, Int32, t)]): Map[t, Int32] \ Aef[m] with Foldable[m], Order[t] Source Returns the shortest distance from src to every other reachable vertex in the weighted directed graph g. Returns the graph where all edges in the directed graph g have their nodes flipped. def frontiersFrom(src: t, g: m[(t, t)]): Map[Int32, Set[t]] \ Aef[m] with Foldable[m], Order[t] Source Returns a mapping from distances to the set of nodes for which the shortest path from src in the directed graph g is of a given length. Returns the in-degree (how many edges end in a given node) of each node in the directed graph g. Returns the inverse graph of the directed graph g. For all nodes in g. The new graph contains exactly those edges that are not in g. OBS: No self-edges are returned no matter the input. Returns true if the directed graph g contains at least one cycle. Returns the out-degree (how many edges start in a given node) of each node in the directed graph g. def reachable(src: { src = t }, dst: { dst = t }, g: m[(t, t)]): Bool \ Aef[m] with Foldable[m], Order[t] Source Returns true if there is a path from src to dst in the directed graph g. Returns the nodes that are reachable from src in the directed graph g. def stronglyConnectedComponents(g: m[(t, t)]): Set[Set[t]] \ Aef[m] with Foldable[m], Order[t] Source Returns the strongly connected components of the directed graph g. Two nodes are in the same component if and only if they can both reach each other. Returns a Graphviz (DOT) string of the directed graph g. The strings of nodes are put in quotes but DOT identifier validity is up to the caller. Returns a Graphviz (DOT) string of the directed graph g. The strings of nodes are put in quotes and existing quotes are escaped. Other than that, DOT identifier validity is up to the caller. Returns a copy of the directed graph g where all flipped edges are added. An undirected graph in directed representation. def toUndirectedLabeled(g: m[(t, Int32, t)]): Set[(t, Int32, t)] \ Aef[m] with Foldable[m], Order[t] Source Returns a copy of the weighted directed graph g where all flipped edges are added. An undirected graph in directed representation. Returns the topologically sorted nodes (all edges go from lower indices to higher indices of the list) in the directed graph g. Unordered nodes are consistently (although not intuitively) ordered. OBS: No cycles must be present. Returns the nodes that are unreachable from src in the directed graph g. def withinDistanceOf(src: t, limit: Int32, g: m[(t, Int32, t)]): Set[t] \ Aef[m] with Foldable[m], Order[t] Source Returns the nodes that are at most limit (inclusive) distance away from src in the weighted directed graph g. OBS: No negative cycles must be present. def withinEdgesOf(src: t, limit: Int32, g: m[(t, t)]): Set[t] \ Aef[m] with Foldable[m], Order[t] Source Returns the nodes that are at most limit (inclusive) edges away from src in the directed graph g.	{"url":"http://api.flix.dev/Graph.html","timestamp":"2024-11-01T19:15:27Z","content_type":"text/html","content_length":"60804","record_id":"<urn:uuid:f240c5fe-4766-4467-b826-a70ac943227a>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00589.warc.gz"}
Centron Tutorials – Expertenwissen für Cloud-Technologien und IT-InfrastrukturQuantile() Function in R - Essential Guide Quantile() Function in R – Essential Guide You can generate the sample quantiles using the quantile() function in R. Hello people, today we will be looking at how to find the quantiles of the values using the quantile() function. Quantile: In laymen terms, a quantile is nothing but a sample that is divided into equal groups or sizes. Due to this nature, the quantiles are also called as Fractiles. In the quantiles, the 25th percentile is called as lower quartile, 50th percentile is called as Median and the 75th Percentile is called as the upper quartile. In the below sections, let’s see how this quantile() function works in R. Quantile() function syntax The syntax of the Quantile() function in R is: quantile(x, probs = , na.rm = FALSE) • X = the input vector or the values • Probs = probabilities of values between 0 and 1. • na.rm = removes the NA values. A Simple Implementation of quantile() function in R Well, hope you are good with the definition and explanations about quantile function. Now, let’s see how quantile function works in R with the help of a simple example which returns the quantiles for the input data. #creates a vector having some values and the quantile function will return the percentiles for the data. 0% 25% 50% 75% 100% In the above sample, you can observe that the quantile function first arranges the input values in the ascending order and then returns the required percentiles of the values. Note: The quantile function divides the data into equal halves, in which the median acts as middle and over that the remaining lower part is lower quartile and upper part is upper quartile. Handle the missing values – ‘NaN’ NaN’s are everywhere. In this data-driven digital world, you may encounter these NaN’s more frequently, which are often called as the missing values. If your data by any means has these missing values, you can end up with getting the NaN’s in the output or the errors in the output. So, in order to handle these missing values, we are going to use na.rm function. This function will remove the NA values from our data and returns the true values. Let’s see how this works. #creates a vector having values along with NaN's Error in quantile.default(df) : missing values and NaN's not allowed if 'na.rm' is FALSE Oh, we got an error. If your guess is regarding the NA values, you are absolutely smart. If NA values are present in our data, the majority of the functions will end up in returning the NA values itself or the error message as mentioned above. Well, let’s remove these missing values using the na.rm function. #creates a vector having values along with NaN's #removes the NA values and returns the percentiles quantile(df,na.rm = TRUE) 0% 25% 50% 75% 100% In the above sample, you can see the na.rm function and its impact on the output. The function will remove the NA’s to avoid the false output. The ‘Probs’ argument in the quantile As you can see the probs argument in the syntax, which is showcased in the very first section of the article, you may wonder what does it mean and how it works?. Well, the probs argument is passed to the quantile function to get the specific or the custom percentiles. Seems to be complicated? Dont worry, I will break it down to simple terms. Well, whenever you use the function quantile, it returns the standard percentiles like 25,50 and 75 percentiles. But what if you want 47th percentile or maybe 88th percentile? There comes the argument ‘probs’, in which you can specify the required percentiles to get those. Before going to the example, you should know few things about the probs argment. Probs: The probs or the probabilities argument should lie between 0 and 1. Here is a sample which illustrates the above statement. #creates the vector of values #returns the quantile of 22 and 77 th percentiles. quantile(df,na.rm = T,probs = c(22,77)) Error in quantile.default(df, na.rm = T, probs = c(22, 77)) : 'probs' outside [0,1] Oh, it’s an error! Did you get the idea, what happened? Well, here comes the Probs statement. Even though we mentioned the right values in the probs argument, it violates the 0-1 condition. The probs argument should include the values which should lie in between 0 and 1. So, we have to convert the probs 22 and 77 to 0.22 and 0.77. Now the input values is in between 0 and 1 right? I hope this makes sense. #creates a vector of values #returns the 22 and 77th percentiles of the input values quantile(df,na.rm = T,probs = c(0.22,0.77)) 22% 77% 10.08 78.00 The ‘Unname’ function and its use Suppose you want your code to only return the percentiles and avoid the cut points. In these situations, you can make use of the ‘unname’ function. The ‘unname’ function will remove the headings or the cut points ( 0%,25% , 50%, 75% , 100 %) and returns only the percentiles. Let’s see how it works! #creates a vector of values quantile(df,na.rm = T,probs = c(0.22,0.77)) #avoids the cut-points and returns only the percentiles. unname(quantile(df,na.rm = T,probs = c(0.22,0.77))) 10.08 78.00 Now, you can observe that the cut-points are disabled or removed by the unname function and returns only the percentiles. The ‘round’ function and its use We have discussed the round function in R in detail in the past article. Now, we are going to use the round function to round off the values. Let’s see how it works! #creates a vector of values quantile(df,na.rm = T,probs = c(0.22,0.77)) #returns the round off values unname(round(quantile(df,na.rm = T,probs = c(0.22,0.77)))) As you can see that our output values are rounded off to zero decimal points. Get the quantiles for the multiple groups/columns in a data set Till now, we have discussed the quantile function, its uses and applications as well as its arguments and how to use them properly. In this section, we are going to get the quantiles for the multiple columns in a data set. Sounds interesting? follow me! I am going to use the ‘mtcars’ data set for this purpose and also using the ‘dplyr’ library for this. #reads the data #returns the top few rows of the data #install required paclages #using tapply, we can apply the function to multiple groups do.call("rbind",tapply(mtcars$mpg, mtcars$gear, quantile)) 0% 25% 50% 75% 100% 3 10.4 14.5 15.5 18.400 21.5 4 17.8 21.0 22.8 28.075 33.9 5 15.0 15.8 19.7 26.000 30.4 In the above process, we have to install the ‘dplyr’ package, and then we will make use of tapply and rbind functions to get the multiple columns of the mtcars datasets. In the above section, we took multiple columns such as ‘mpg’ and the ‘gear’ columns in mtcars data set. Like this, we can compute the quantiles for multiple groups in a data set. Can we visualise the percentiles? My answer is a big YES!. The best plot for this will be a box plot. Let me take the iris dataset and will try to visualize the box plot which will showcase the percentiles as well. Let’s roll! This is the iris data set with top 6 values. Let’s explore the data with the function named – ‘Summary’. In the above image, you can see the mean, median, 25th percentile(1 st quartile), 75 th percentile(3rd percentile) and min and max values as well. Let’s plot this information through a box plot. Let’s do it! #plots a boxplot with labels boxplot(iris$Sepal.Length, main='The boxplot showing the percentiles', col='Orange', ylab='Values', xlab='Sepal Length', border = 'brown', horizontal = T) A box plot can show many aspects of the data. In the below figure I have mentioned the particular values represented by the box plots. This will save some time for you and facilitates your understanding in the best way possible. Quantile() function in R – Wrapping up Well, it’s a longer article I reckon. And I tried my best to explain and explore the quantile() function in R in multiple dimensions through various examples and illustrations as well. The quantile function is the most useful function in data analysis as it efficiently reveals more information about the given data. I hope you got a good understanding of the buzz around the quantile() function in R. That’s all for now. We will be back with more and more beautiful functions and topics in R programming. Till then take care and happy data analyzing!!! Quantile() Function in R – Essential Guide Visualizing Data Trends with Seaborn Line Plots Visualizing data through various graphs is crucial for analyzing trends and relationships within datasets. Among these visualizations, line plots are one of the… Learning Path for Data Analysts: Building Essential Skills for Your Career The role of a data analyst has become increasingly essential in today’s data-driven world. But what does a data… Understanding Covariance and Correlation in R Programming In the field of statistics, analyzing the relationship between variables is crucial, especially when preparing data for machine learning and data science models.…	{"url":"https://www.centron.de/en/tutorial/quantile-function-in-r-essential-guide/","timestamp":"2024-11-10T09:59:15Z","content_type":"text/html","content_length":"312558","record_id":"<urn:uuid:78b22ed6-9968-467c-8240-b1760059ad15>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00353.warc.gz"}
The Outer Product > The outer product and the cross product, are _definitely not_ terms for > one and the same thing > The same thing is true for the inner product, and the dot product, False. (See below.) > the latter being defined as > A.B = Sum[i=1,..,N] A^iB^i (1) No. (See below.) > [snip - cross product] > In 4-space such a vector > can have infinitely many orientations, so "orthogonal" here > is not really defined. In 4-space there are many directions orthogonal to a pair of vectors; orthogonal is perfectly well defined (it means the inner or dot product is zero), but a pair of (non-parallel) vectors does not define a single orthogonal direction. > [snip] > So we > define the inner product between two (1,0) tensors to be > A.B = g_{ij} A^{i} A^{j} (Einstein summation is used) (A^j should be B^j) The tensor g has to satisfy certain properties (so that A.B is positive definite symmetric). > this is a tensor equation, and thus holds true in any basis, > frame, system. Note that the definition of the _dot product_ > as given in (1) is _not_ a tensor equation, and thus only holds > in the space it was defined in. Given an inner product, as you have (correctly) defined it, you can find coordinates such that A.B = sum_i A^i B^i . Here is the right way to think about this. An inner product on the real vector space V is a mapping VxV->R such (A+B).C = A.C + B.C (sA).C = s(A.C) for scalar s A.A>=0, and A.A=0 implies A=0 A.B=B.A . Choose a basis (e_1,...,e_N) for (finite-dimensional) V. Then any vector A in V can be uniquely written as A = A^i e_i . The first two conditions on the inner product imply that there is a collection of numbers g_{ij} so that A.B = g_{ij} A^i B^j . The last condition implies that g_{ij}=g_{ji} for all i,j. (The third condition, positive definiteness, is not as easy to describe in terms of g, so I won't try.) Now, choose another basis (f_1,...,f_N) for V. Then there exist unique matrices L and M such that e_i = L^j_i f_j f_i = M^j_i e_j ; L and M are inverses, L^i_j M^j_k = delta^i_k . Now, remember that we defined our inner product without any reference to a basis; it's just something that satisfies those four properties. It was not until we introduced a basis that we obtained the numbers g_{ij}. We can construct other numbers, h_{ij}, that represent the inner product in terms of the basis (f). That is, if A = a^i f_i , and similarly for B, then A.B = h_{ij} a^i b^j . Note that A = A^i e_i = A^i L^j_i f_j = a^j f_j so (since the expression of a vector with respect to a basis is a^j = L^j_i A^i . Thus (being careful with our indices) A.B = h_{ij} a^i b^j = h_{ij} L^i_k A^k L^j_m B^m = (h_{ij} L^i_k L^j_m) A^k B^m = g_{km} A^k B^m so (since this holds for all A,B) g_{km} = h_{ij} L^i_k L^j_m . What I've just done here is derived the transformation rule for (0,2)-tensors, starting with the definition that a (0,2)-tensor is a bilinear mapping from VxV to R. (I never used the fact that it's an inner product, which is a special kind of (0,2)-tensor.) Now (as I said above) you can always find a basis such that One often refers to the "usual R^N", or the "usual basis for R^N", or the "usual inner product on R^N"; what this means is the basis e_1 = [1;0;0;...;0] e_2 = [0;1;0;...;0] e_N = [0;0;0;...;1] and the inner product defined by e_i.e_j = delta_{ij} . The terms "inner product" and "dot product" are, as far as I know, synonymous, although the term "dot product" is most often used to refer to the usual (or standard) inner product on R^N above. Note, though, that a vector space does not come with an inner product built in; if you want an inner product on a vector space, you have to say which one you want. (There are "standard" inner products on all the R^n, but one can also speak of R^n without referring to this standard [or any other] inner product.) The outer or tensor product, on the other hand, always exists. > The outer product of two tensors of arbitrary rank gives a tensor > of their combined ranks. Suppose we have two (1,0) tensors (A, B) and > get their tensor product to yield a (2,0) tensor T = A(x)B. > The components of T can be put into a matrix. Now we > can obtain the symmetric, and anti-symmetric parts of this matrix > with : > T^{ij}(SYM) = 1/2(T^{ij} + T^{ji}) i,j=0,...,N > T^{ij}(ASYM)= 1/2(T^{ij} - T^{ij}) > The anti-symmetric part of T^{ij} has diagonal elements which are > necessarily 0. Now take N=3, there are now 6 non-zero components of > T^{ij}(ASYM). Now apparently these 6 components can be arranged > into a vector in such a way that we obtain the equation for the > cross product. But _only_ when N=3. There are many ways of looking at this. One is as you described. (But while there are 6 non-zero coefficients, there are only 3 independent non-zero coefficients; it is those three that are arranged into a vector in R^3.) More precisely, one can construct the "exterior product", which is the antisymmetric part of the tensor product. This gives you the exterior algebra, consisting of (0,0)-tensors, (1,0)-tensors, ..., and (N,0)-tensors. (Or (0,0), (0,1), ..., (0,N)-tensors.) An inner product, together with an orientation, gives an isomorphism (called the "Hodge star") between antisymmetric (p,0)-tensors and antisymmetric (N-p,0)-tensors. The inner product also gives an isomorphism between (p,q)-tensors and (p+1,q-1)-tensors (raising and lowering indices). Thus, when N=3, you have isomorphisms: = (1,0)-tensors --[Hodge]--> antisymmetric (2,0)-tensors --[raising/lowering indices]--> antisymmetric (1,1)-tensors = antisymmetric matrices . Let us write the composition of these, vectors->matrices, as M. (That is, if A is a 3-vector, then M(A) is an antisymmetric 3-by-3 matrix.) Then there are a couple of nice things: AxB = M(A)B M(AxB) = [M(A),M(B)] = M(A)M(B)-M(B)M(A) . (The first one gives you an easy way to construct M. The second one is fairly sophisticated, and says that the vector space isomorphism M is an algebra isomorphism from the algebra of cross products to the Lie algebra so(3) of the rotation group, which is a nice fact when you remember how velocity V, position R, and rotational [angular] velocity W are related: V=WxR .) (Note that I snuck the word "orientation" in there. That's an important but easily forgotten element in the construction of the cross product.) The exterior product is often denoted by something that looks sort of like the character "^" (but is wider and is located a bit lower). Perhaps due to the relationship between it and the cross product that I described above, the cross product is sometimes called the exterior product and denoted by the same "^"-like character. The exterior product and the cross product are not, though, the same thing. And the exterior and outer products are different as well, the former being the antisymmetrization of the latter.	{"url":"https://groups.google.com/g/sci.math/c/V-Kh8_u_7qs/m/ZmgaVaO3aoMJ","timestamp":"2024-11-11T04:25:13Z","content_type":"text/html","content_length":"881080","record_id":"<urn:uuid:830d2887-b9ab-4a87-8ac8-5b4405aae6be>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00225.warc.gz"}
Guesser [fx-5800P] - Your Projects Inspired by https://www.youtube....h?v=ud_frfkt1t0 I made a program/game. You think of two positive numbers. No limits in how big they can be as long as you can type them in the calculator. They can be integers, reals, whatever. Just make sure you know which number is the first one, and which one is the second one that you thought of. Eg. the first number can be 190 and the second number can be 42. Then the program will ask you to give it one of this two numbers. Eg it will ask for the second number, so you will type in 42. Then the program will make a prediction, either it will say that this is the smallest of the two numbers, or it will say that it is the biggest. It will ask you whether the prediction was correct so that it can keep score. The "paradox" is that the calculator will have >50% score when you'd expect that there is no way to be more than 50% correct in such a game. Check the video for an explanation on why this is Lbl 0 T⇒(0.5+Ran# )x(A÷T)➔G "THINK OF 2 NUMS"◢ If RanInt#(0,1):Then "WHAT IS THE 2ND"?➔N "WHAT IS THE 1ST"?➔N "IS THIS THE" If N<G:Then "SMALLEST ONE?" "BIGGEST ONE?" "1=YES 0=NO"?➔R "WIN RATIO:" Goto 0 Edited by Tritonio, 11 September 2019 - 10:12 PM.	{"url":"https://community.casiocalc.org/topic/7910-guesser-fx-5800p/","timestamp":"2024-11-01T22:28:20Z","content_type":"text/html","content_length":"161804","record_id":"<urn:uuid:e455c336-2e04-4d16-883a-3ecbb0bcc11b>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00334.warc.gz"}
Geo.8 Conditional Probability In this unit, students extend what they learned about probability in grade 7 by considering events that are combined in various ways including both occurring, at least one occurring, and one event happening under the condition that the other happens as well. This unit introduces the relationship between probabilities of some combinations of events using the Addition Rule, $P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B})$. Conditional probability is discussed. In particular, the Multiplication Rule $P(\text{A and B}) = P(\text{A \| B}) \boldcdot P(\text{B})$ is used to determine conditional probabilities as well as independence of events A and B. Independence is further explored using everyday language as well as through the equation $P(\text{A \| B}) = P(\text{A}) $ when events A and B are independent. Read More	{"url":"https://im-beta.kendallhunt.com/HS/teachers/2/8/index.html","timestamp":"2024-11-07T21:57:48Z","content_type":"text/html","content_length":"81971","record_id":"<urn:uuid:f79f77b0-5958-42e3-8398-c8a7746f902a>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00798.warc.gz"}
aspecd.processing module You're reading an old version of this documentation. For up-to-date information, please have a look at v0.11. aspecd.processing module Data processing functionality. Key to reproducible science is automatic documentation of each processing step applied to the data of a dataset. Such a processing step each is self-contained, meaning it contains every necessary information to perform the processing task on a given dataset. Processing steps, in contrast to analysis steps (see aspecd.analysis for details), not only operate on data of a aspecd.dataset.Dataset, but change its data. The information necessary to reproduce each processing step gets added to the aspecd.dataset.Dataset.history attribute of a dataset. Generally, two types of processing steps can be distinguished: In the first case, the processing is usually handled using the processing() method of the respective aspecd.dataset.Dataset object. Additionally, those processing steps always only operate on the data of a single dataset. Processing steps handling single datasets should always inherit from the aspecd.processing.SingleProcessingStep class. In the second case, the processing step is handled using the processing() method of the aspecd.processing.ProcessingStep object, and the datasets are stored as a list within the processing step. As these processing steps span several datasets. Processing steps handling multiple datasets should always inherit from the aspecd.processing.MultiProcessingStep class. The module contains both, base classes for processing steps (as detailed above) as well as a series of generally applicable processing steps for all kinds of spectroscopic data. The latter are an attempt to relieve the developers of packages derived from the ASpecD framework from the task to reinvent the wheel over and over again. The next section gives an overview of the concrete processing steps implemented within the ASpecD framework. For details of how to implement your own processing steps, see the section below. Concrete processing steps Besides providing the basis for processing steps for the ASpecD framework, ensuring full reproducibility and traceability, hence reproducible science and good scientific practice, this module comes with a (growing) number of general-purpose processing steps useful for basically all kinds of spectroscopic data. Here is a list as a first overview. For details, see the detailed documentation of each of the classes, readily accessible by the link. Processing steps operating on individual datasets The following processing steps operate each on individual datasets independently. Processing steps operating on multiple datasets at once The following processing steps operate each on more than one dataset at the same time, requiring at least two datasets as an input to work. Writing own processing steps Each real processing step should inherit from either aspecd.processing.SingleProcessingStep in case of operating on a single dataset only or from aspecd.processing.MultiProcessingStep in case of operating on several datasets at once. Furthermore, all processing steps should be contained in one module named “processing”. This allows for easy automation and replay of processing steps, particularly in context of recipe-driven data analysis (for details, see the aspecd.tasks module). General advice A few hints on writing own processing step classes: • Always inherit from aspecd.processing.SingleProcessingStep or aspecd.processing.MultiProcessingStep, depending on your needs. • Store all parameters, implicit and explicit, in the dict parameters of the aspecd.processing.ProcessingStep class, not in separate properties of the class. Only this way, you can ensure full reproducibility and compatibility of recipe-driven data analysis (for details of the latter, see the aspecd.tasks module). • Always set the description property to a sensible value. • Always set the undoable property appropriately. In most cases, processing steps can be undone. • Implement the actual processing in the _perform_task method of the processing step. For sanitising parameters and checking general applicability of the processing step to the dataset(s) at hand, continue reading. • Make sure to implement the aspecd.processing.ProcessingStep.applicable() method according to your needs. Typical cases would be to check for the dimensionality of the underlying data, as some processing steps may work only for 1D data (or vice versa). Don’t forget to declare this as a static method, using the @staticmethod decorator. • With the _sanitise_parameters method, the input parameters are automatically checked and an appropriate exception can be thrown in order to describe the error source to the user. Some more special cases are detailed below. For further advice, consult the source code of this module, and have a look at the concrete processing steps whose purpose is described below in more Changing the dimensions of your data If your processing step changes the dimensions of your data, it is your responsibility to ensure the axes values to be consistent with the data. Note that upon changing the dimension of your data, the axes values will be reset to indices along the data dimensions. Hence, you need to first make a (deep) copy of your axes, then change the dimension of your data, and afterwards restore the remaining values from the temporarily stored axes. Changing the length of your data When changing the length of the data, always change the corresponding axes values first, and only afterwards the data, as changing the data will change the axes values and adjust their length to the length of the corresponding dimension of the data. Adding parameters upon processing Sometimes there is the need to persist values that are only obtained during processing the data. A typical example may be averaging 2D data along one dimension and wanting to store both, range of indices and actual axis units. While in this case, typically the axis value of the centre of the averaging window will be stored as new axis value, the other parameters should end up in the aspecd.processing.ProcessingStep.parameters dictionary. Thus, they are added to the dataset history and available for reports and alike.	{"url":"https://docs.aspecd.de/v0.3/api/aspecd.processing.html","timestamp":"2024-11-10T15:49:48Z","content_type":"text/html","content_length":"229047","record_id":"<urn:uuid:5b070ed1-27b2-4d6e-bda8-9dbdb56c2fc8>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00842.warc.gz"}